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diff --git a/.gitattributes b/.gitattributes new file mode 100644 index 0000000..d7b82bc --- /dev/null +++ b/.gitattributes @@ -0,0 +1,4 @@ +*.txt text eol=lf +*.htm text eol=lf +*.html text eol=lf +*.md text eol=lf diff --git a/LICENSE.txt b/LICENSE.txt new file mode 100644 index 0000000..6312041 --- /dev/null +++ b/LICENSE.txt @@ -0,0 +1,11 @@ +This eBook, including all associated images, markup, improvements, +metadata, and any other content or labor, has been confirmed to be +in the PUBLIC DOMAIN IN THE UNITED STATES. + +Procedures for determining public domain status are described in +the "Copyright How-To" at https://www.gutenberg.org. + +No investigation has been made concerning possible copyrights in +jurisdictions other than the United States. Anyone seeking to utilize +this eBook outside of the United States should confirm copyright +status under the laws that apply to them. diff --git a/README.md b/README.md new file mode 100644 index 0000000..252846c --- /dev/null +++ b/README.md @@ -0,0 +1,2 @@ +Project Gutenberg (https://www.gutenberg.org) public repository for +eBook #74864 (https://www.gutenberg.org/ebooks/74864) diff --git a/old/74864-0.txt b/old/74864-0.txt deleted file mode 100644 index 7706e0d..0000000 --- a/old/74864-0.txt +++ /dev/null @@ -1,34220 +0,0 @@ - -*** START OF THE PROJECT GUTENBERG EBOOK 74864 *** - -<b>Transcriber’s notes</b>: - -The text of this book has been preserved as closely as practicable -to its original form. However, the author used some unusual symbols, -and I have taken the liberty of using Unicode characters with similar -appearance (ꖌ ᔕ) as substitutes, disregarding their official meaning -and aware that they might not display on all devices. An archaic symbol -used by the author to indicate the mathematical ‘factorial’ function -has been replaced by the modern equivalent, viz. ! Unusual placements -of various sub- and superscripted symbols remain as in the original text. - -In this plain-text version, italic text is denoted by *asterisks*; -superscripted and subscripted characters are enclosed within {curly} -brackets preceded by a caret (^) or underline (_) respectively. - -Inconsistencies of punctuation have been corrected silently, but -inconsistent spellings such as *Roemer, Römer, Rœmer* have not been -altered. A list of corrected spellings is appended at the end of the -book. - -Footnotes have been renumbered consecutively and positioned below the -relevant paragraphs. A missing footnote marker has been inserted on -p.751 after tracking down the original document. A missing negative -symbol has been added to an exponent in a formula on p.327. - -There is a misleading calculation on p.194 and the table that follows, -regarding progressive powers of two: ((2^{2})^{2})^{2} is equivalent to -(16)^{2} which equals 256 not 65,356 as stated, but 2^{16} does equal -65,356. - -[sic] has been inserted on p.179 alongside a statement that the -alphabet contains 24 letters; however, the statement may well be -correct given that it was written in 1704 by a Flemish author and the -language is not specified. - -A large arithmetic triangle on p.184 cannot be displayed optimally in -this plain-text version for lack of space - - - - -THE PRINCIPLES OF SCIENCE. - -[Illustration] - -[Illustration: THE LOGICAL MACHINE.] - - - - THE PRINCIPLES OF SCIENCE: - - *A TREATISE ON LOGIC* - - *AND* - - *SCIENTIFIC METHOD.* - - - BY - - W. STANLEY JEVONS, - - LL.D. (EDINB.), M.A. (LOND.), F.R.S. - - - London: - - MACMILLAN AND CO. - 1883. - - - *The Right of Translation and Reproduction is Reserved.* - - - - LONDON: - R. Clay, Sons, & Taylor, Printers, - BREAD STREET HILL. - - - Stereotyped Edition. - - - - -PREFACE - -*TO THE FIRST EDITION*. - - -It may be truly asserted that the rapid progress of the physical -sciences during the last three centuries has not been accompanied by -a corresponding advance in the theory of reasoning. Physicists speak -familiarly of Scientific Method, but they could not readily describe -what they mean by that expression. Profoundly engaged in the study -of particular classes of natural phenomena, they are usually too -much engrossed in the immense and ever-accumulating details of their -special sciences to generalise upon the methods of reasoning which they -unconsciously employ. Yet few will deny that these methods of reasoning -ought to be studied, especially by those who endeavour to introduce -scientific order into less successful and methodical branches of -knowledge. - -The application of Scientific Method cannot be restricted to the sphere -of lifeless objects. We must sooner or later have strict sciences of -those mental and social phenomena, which, if comparison be possible, -are of more interest to us than purely material phenomena. But it -is the proper course of reasoning to proceed from the known to the -unknown--from the evident to the obscure--from the material and -palpable to the subtle and refined. The physical sciences may therefore -be properly made the practice-ground of the reasoning powers, -because they furnish us with a great body-of precise and successful -investigations. In these sciences we meet with happy instances of -unquestionable deductive reasoning, of extensive generalisation, of -happy prediction, of satisfactory verification, of nice calculation of -probabilities. We can note how the slightest analogical clue has been -followed up to a glorious discovery, how a rash generalisation has at -length been exposed, or a conclusive *experimentum crucis* has decided -the long-continued strife between two rival theories. - -In following out my design of detecting the general methods of -inductive investigation, I have found that the more elaborate and -interesting processes of quantitative induction have their necessary -foundation in the simpler science of Formal Logic. The earlier, and -probably by far the least attractive part of this work, consists, -therefore, in a statement of the so-called Fundamental Laws of -Thought, and of the all-important Principle of Substitution, of which, -as I think, all reasoning is a development. The whole procedure of -inductive inquiry, in its most complex cases, is foreshadowed in -the combinational view of Logic, which arises directly from these -fundamental principles. Incidentally I have described the mechanical -arrangements by which the use of the important form called the Logical -Alphabet, and the whole working of the combinational system of Formal -Logic, may be rendered evident to the eye, and easy to the mind and -hand. - -The study both of Formal Logic and of the Theory of Probabilities has -led me to adopt the opinion that there is no such thing as a distinct -method of induction as contrasted with deduction, but that induction -is simply an inverse employment of deduction. Within the last century -a reaction has been setting in against the purely empirical procedure -of Francis Bacon, and physicists have learnt to advocate the use of -hypotheses. I take the extreme view of holding that Francis Bacon, -although he correctly insisted upon constant reference to experience, -had no correct notions as to the logical method by which from -particular facts we educe laws of nature. I endeavour to show that -hypothetical anticipation of nature is an essential part of inductive -inquiry, and that it is the Newtonian method of deductive reasoning -combined with elaborate experimental verification, which has led to all -the great triumphs of scientific research. - -In attempting to give an explanation of this view of Scientific Method, -I have first to show that the sciences of number and quantity repose -upon and spring from the simpler and more general science of Logic. -The Theory of Probability, which enables us to estimate and calculate -quantities of knowledge, is then described, and especial attention -is drawn to the Inverse Method of Probabilities, which involves, as -I conceive, the true principle of inductive procedure. No inductive -conclusions are more than probable, and I adopt the opinion that the -theory of probability is an essential part of logical method, so -that the logical value of every inductive result must be determined -consciously or unconsciously, according to the principles of the -inverse method of probability. - -The phenomena of nature are commonly manifested in quantities of -time, space, force, energy, &c., and the observation, measurement, -and analysis of the various quantitative conditions or results -involved, even in a simple experiment, demand much employment of -systematic procedure. I devote a book, therefore, to a simple and -general description of the devices by which exact measurement is -effected, errors eliminated, a probable mean result attained, and -the probable error of that mean ascertained. I then proceed to the -principal, and probably the most interesting, subject of the book, -illustrating successively the conditions and precautions requisite -for accurate observation, for successful experiment, and for the sure -detection of the quantitative laws of nature. As it is impossible to -comprehend aright the value of quantitative laws without constantly -bearing in mind the degree of quantitative approximation to the truth -probably attained, I have devoted a special chapter to the Theory of -Approximation, and however imperfectly I may have treated this subject, -I must look upon it as a very essential part of a work on Scientific -Method. - -It then remains to illustrate the sound use of hypothesis, to -distinguish between the portions of knowledge which we owe to empirical -observation, to accidental discovery, or to scientific prediction. -Interesting questions arise concerning the accordance of quantitative -theories and experiments, and I point out how the successive -verification of an hypothesis by distinct methods of experiment yields -conclusions approximating to but never attaining certainty. Additional -illustrations of the general procedure of inductive investigations are -given in a chapter on the Character of the Experimentalist, in which -I endeavour to show, moreover, that the inverse use of deduction was -really the logical method of such great masters of experimental inquiry -as Newton, Huyghens, and Faraday. - -In treating Generalisation and Analogy, I consider the precautions -requisite in inferring from one case to another, or from one part of -the universe to another part; the validity of all such inferences -resting ultimately upon the inverse method of probabilities. The -treatment of Exceptional Phenomena appeared to afford an interesting -subject for a further chapter illustrating the various modes in which -an outstanding fact may eventually be explained. The formal part of -the book closes with the subject of Classification, which is, however, -very inadequately treated. I have, in fact, almost restricted myself to -showing that all classification is fundamentally carried out upon the -principles of Formal Logic and the Logical Alphabet described at the -outset. - -In certain concluding remarks I have expressed the conviction which -the study of Logic has by degrees forced upon my mind, that serious -misconceptions are entertained by some scientific men as to the -logical value of our knowledge of nature. We have heard much of -what has been aptly called the Reign of Law, and the necessity and -uniformity of natural forces has been not uncommonly interpreted as -involving the non-existence of an intelligent and benevolent Power, -capable of interfering with the course of natural events. Fears have -been expressed that the progress of Scientific Method must therefore -result in dissipating the fondest beliefs of the human heart. Even -the ‘Utility of Religion’ is seriously proposed as a subject of -discussion. It seemed to be not out of place in a work on Scientific -Method to allude to the ultimate results and limits of that method. I -fear that I have very imperfectly succeeded in expressing my strong -conviction that before a rigorous logical scrutiny the Reign of Law -will prove to be an unverified hypothesis, the Uniformity of Nature an -ambiguous expression, the certainty of our scientific inferences to a -great extent a delusion. The value of science is of course very high, -while the conclusions are kept well within the limits of the data on -which they are founded, but it is pointed out that our experience is -of the most limited character compared with what there is to learn, -while our mental powers seem to fall infinitely short of the task of -comprehending and explaining fully the nature of any one object. I -draw the conclusion that we must interpret the results of Scientific -Method in an affirmative sense only. Ours must be a truly positive -philosophy, not that false negative philosophy which, building on a few -material facts, presumes to assert that it has compassed the bounds -of existence, while it nevertheless ignores the most unquestionable -phenomena of the human mind and feelings. - -It is approximately certain that in freely employing illustrations -drawn from many different sciences, I have frequently fallen into -errors of detail. In this respect I must throw myself upon the -indulgence of the reader, who will bear in mind, as I hope, that the -scientific facts are generally mentioned purely for the purpose of -illustration, so that inaccuracies of detail will not in the majority -of cases affect the truth of the general principles illustrated. - - *December 15, 1873.* - - - - -PREFACE - -*TO THE SECOND EDITION*. - - -Few alterations of importance have been made in preparing this second -edition. Nevertheless, advantage has been taken of the opportunity to -revise very carefully both the language and the matter of the book. -Correspondents and critics having pointed out inaccuracies of more -or less importance in the first edition, suitable corrections and -emendations have been made. I am under obligations to Mr. C. J. Monro, -M.A., of Barnet, and to Mr. W. H. Brewer, M.A., one of Her Majesty’s -Inspectors of Schools, for numerous corrections. - -Among several additions which have been made to the text, I may mention -the abstract (p. 143) of Professor Clifford’s remarkable investigation -into the number of types of compound statement involving four classes -of objects. This inquiry carries forward the inverse logical problem -described in the preceding sections. Again, the need of some better -logical method than the old Barbara Celarent, &c., is strikingly shown -by Mr. Venn’s logical problem, described at p. 90. A great number -of candidates in logic and philosophy were tested by Mr. Venn with -this problem, which, though simple in reality, was solved by very few -of those who were ignorant of Boole’s Logic. Other evidence could -be adduced by Mr. Venn of the need for some better means of logical -training. To enable the logical student to test his skill in the -solution of inductive logical problems, I have given (p. 127) a series -of ten problems graduated in difficulty. - -To prevent misapprehension, it should be mentioned that, throughout -this edition, I have substituted the name *Logical Alphabet* for -*Logical Abecedarium*, the name applied in the first edition to the -exhaustive series of logical combinations represented in terms of -*A*, *B*, *C*, *D* (p. 94). It was objected by some readers that -*Abecedarium* is a long and unfamiliar name. - -To the chapter on Units and Standards of Measurement, I have added two -sections, one (p. 325) containing a brief statement of the Theory of -Dimensions, and the other (p. 319) discussing Professor Clerk Maxwell’s -very original suggestion of a Natural System of Standards for the -measurement of space and time, depending upon the length and rapidity -of waves of light. - -In my description of the Logical Machine in the *Philosophical -Transactions* (vol. 160, p. 498), I said--“It is rarely indeed that -any invention is made without some anticipation being sooner or later -discovered; but up to the present time I am totally unaware of even -a single previous attempt to devise or construct a machine which -should perform the operations of logical inference; and it is only, -I believe, in the satirical writings of Swift that an allusion to an -actual reasoning machine is to be found.” Before the paper was printed, -however, I was able to refer (p. 518) to the ingenious designs of the -late Mr. Alfred Smee as attempts to represent thought mechanically. Mr. -Smee’s machines indeed were never constructed, and, if constructed, -would not have performed actual logical inference. It has now just -come to light, however, that the celebrated Lord Stanhope actually did -construct a mechanical device, capable of representing syllogistic -inferences in a concrete form. It appears that logic was one of the -favourite studies of this truly original and ingenious nobleman. There -remain fragments of a logical work, printed by the Earl at his own -press, which show that he had arrived, before the year 1800, at the -principle of the quantified predicate. He puts forward this principle -in the most explicit manner, and proposes to employ it throughout his -syllogistic system. Moreover, he converts negative propositions into -affirmative ones, and represents these by means of the copula “is -identic with.” Thus he anticipated, probably by the force of his own -unaided insight, the main points of the logical method originated in -the works of George Bentham and George Boole, and developed in this -work. Stanhope, indeed, has no claim to priority of discovery, because -he seems never to have published his logical writings, although they -were put into print. There is no trace of them in the British Museum -Library, nor in any other library or logical work, so far as I am -aware. Both the papers and the logical contrivance have been placed -by the present Earl Stanhope in the hands of the Rev. Robert Harley, -F.R.S., who will, I hope, soon publish a description of them.[1] - - [1] Since the above was written Mr. Harley has read an account of - Stanhope’s logical remains at the Dublin Meeting (1878) of the - British Association. The paper will be printed in *Mind*. (Note added - November, 1878.) - -By the kindness of Mr. Harley, I have been able to examine Stanhope’s -logical contrivance, called by him the Demonstrator. It consists of a -square piece of bay-wood with a square depression in the centre, across -which two slides can be pushed, one being a piece of red glass, and -the other consisting of wood coloured gray. The extent to which each -of these slides is pushed in is indicated by scales and figures along -the edges of the aperture, and the simple rule of inference adopted -by Stanhope is: “To the gray add the red and subtract the *holon*,” -meaning by holon (ὅλον) the whole width of the aperture. This rule -of inference is a curious anticipation of De Morgan’s numerically -definite syllogism (see below, p. 168), and of inferences founded on -what Hamilton called “Ultra-total distribution.” Another curious point -about Stanhope’s device is, that one slide can be drawn out and pushed -in again at right angles to the other, and the overlapping part of the -slides then represents the probability of a conclusion, derived from -two premises of which the probabilities are respectively represented -by the projecting parts of the slides. Thus it appears that Stanhope -had studied the logic of probability as well as that of certainty, here -again anticipating, however obscurely, the recent progress of logical -science. It will be seen, however, that between Stanhope’s Demonstrator -and my Logical Machine there is no resemblance beyond the fact that -they both perform logical inference. - -In the first edition I inserted a section (vol. i. p. 25), on -“Anticipations of the Principle of Substitution,” and I have reprinted -that section unchanged in this edition (p. 21). I remark therein that, -“In such a subject as logic it is hardly possible to put forth any -opinions which have not been in some degree previously entertained. The -germ at least of every doctrine will be found in earlier writings, and -novelty must arise chiefly in the mode of harmonising and developing -ideas.” I point out, as Professor T. M. Lindsay had previously done, -that Beneke had employed the name and principle of substitution, and -that doctrines closely approximating to substitution were stated by the -Port Royal Logicians more than 200 years ago. - -I have not been at all surprised to learn, however, that other -logicians have more or less distinctly stated this principle of -substitution during the last two centuries. As my friend and successor -at Owens College, Professor Adamson, has discovered, this principle can -be traced back to no less a philosopher than Leibnitz. - -The remarkable tract of Leibnitz,[2] entitled “Non inelegans Specimen -Demonstrandi in Abstractis,” commences at once with a definition -corresponding to the principle:-- - -“Eadem sunt quorum unum potest substitui alteri salva veritate. Si sint -*A* et *B*, et *A* ingrediatur aliquam propositionem veram, et ibi in -aliquo loco ipsius *A* pro ipso substituendo *B* fiat nova propositio -æque itidem vera, idque semper succedat in quacunque tali propositione, -*A* et *B* dicuntur esse eadem; et contra, si eadem sint *A* et *B*, -procedet substitutio quam dixi.” - - [2] Leibnitii *Opera Philosophica quæ extant*. Erdmann, Pars I. - Berolini, 1840, p. 94. - -Leibnitz, then, explicitly adopts the principle of substitution, but -he puts it in the form of a definition, saying that those things are -the same which can be substituted one for the other, without affecting -the truth of the proposition. It is only after having thus tested -the sameness of things that we can turn round and say that *A* and -*B*, being the same, may be substituted one for the other. It would -seem as if we were here in a vicious circle; for we are not allowed -to substitute *A* for *B*, unless we have ascertained by trial that -the result is a true proposition. The difficulty does not seem to be -removed by Leibnitz’ proviso, “idque semper succedat in quacunque -tali propositione.” How can we learn that because *A* and *B* may -be mutually substituted in some propositions, they may therefore -be substituted in others; and what is the criterion of likeness of -propositions expressed in the word “tali”? Whether the principle -of substitution is to be regarded as a postulate, an axiom, or a -definition, is just one of those fundamental questions which it seems -impossible to settle in the present position of philosophy, but this -uncertainty will not prevent our making a considerable step in logical -science. - -Leibnitz proceeds to establish in the form of a theorem what is -usually taken as an axiom, thus (*Opera*, p. 95): “Theorema I. Quæ -sunt eadem uni tertio, eadem sunt inter se. Si *A* ∝ *B* et *B* ∝ *C*, -erit *A* ∝ *C*. Nam si in propositione *A* ∝ *B* (vera ea hypothesi) -substituitur *C* in locum *B* (quod facere licet per Def. I. quia *B* ∝ -*C* ex hypothesi) fiet *A* ∝ *C*. Q. E. Dem.” Thus Leibnitz precisely -anticipates the mode of treating inference with two simple identities -described at p. 51 of this work. - -Even the mathematical axiom that ‘equals added to equals make equals,’ -is deduced from the principle of substitution. At p. 95 of Erdmann’s -edition, we find: “Si eidem addantur coincidentia fiunt coincidentia. -Si *A* ∝ *B*, erit *A* + *C* ∝ *B* + *C*. Nam si in propositione *A* -+ *C* ∝ *A* + *C* (quæ est vera per se) pro *A* semel substituas *B* -(quod facere licet per Def. I. quia *A* ∝ *B*) fiet *A* + *C* ∝ *B* + -*C* Q. E. Dem.” This is unquestionably the mode of deducing the several -axioms of mathematical reasoning from the higher axiom of substitution, -which is explained in the section on mathematical inference (p. 162) in -this work, and which had been previously stated in my *Substitution of -Similars*, p. 16. - -There are one or two other brief tracts in which Leibnitz anticipates -the modern views of logic. Thus in the eighteenth tract in Erdmann’s -edition (p. 92), called “Fundamenta Calculi Ratiocinatoris”, he says: -“Inter ea quorum unum alteri substitui potest, salvis calculi legibus, -dicetur esse æquipollentiam.” There is evidence, also, that he had -arrived at the quantification of the predicate, and that he fully -understood the reduction of the universal affirmative proposition to -the form of an equation, which is the key to an improved view of logic. -Thus, in the tract entitled “Difficultates Quædam Logicæ,”[3] he says: -“Omne *A* est *B*; id est æquivalent *AB* et *A*, seu *A* non *B* est -non-ens.” - - [3] Erdmann, p. 102. - -It is curious to find, too, that Leibnitz was fully acquainted with the -Laws of Commutativeness and “Simplicity” (as I have called the second -law) attaching to logical symbols. In the “Addenda ad Specimen Calculi -Universalis” we read as follows.[4] “Transpositio literarum in eodem -termino nihil mutat, ut *ab* coincidet cum *ba*, seu animal rationale -et rationale animal.” - -“Repetitio ejusdem literæ in eodem termino est inutilis, ut *b* est -*aa*; vel *bb* est *a*; homo est animal animal, vel homo homo est -animal. Sufficit enim dici *a* est *b*, seu homo est animal.” - - [4] Ibid. p. 98. - -Comparing this with what is stated in Boole’s *Mathematical Analysis of -Logic*, pp. 17–18, in his *Laws of Thought*, p. 29, or in this work, -pp. 32–35, we find that Leibnitz had arrived two centuries ago at a -clear perception of the bases of logical notation. When Boole pointed -out that, in logic, *xx* = *x*, this seemed to mathematicians to be a -paradox, or in any case a wholly new discovery; but here we have it -plainly stated by Leibnitz. - -The reader must not assume, however, that because Leibnitz correctly -apprehended the fundamental principles of logic, he left nothing for -modern logicians to do. On the contrary, Leibnitz obtained no useful -results from his definition of substitution. When he proceeds to -explain the syllogism, as in the paper on “Definitiones Logicæ,”[5] -he gives up substitution altogether, and falls back upon the notion -of inclusion of class in class, saying, “Includens includentis est -includens inclusi, seu si *A* includit *B* et *B* includit *C*, etiam -*A* includet *C*.” He proceeds to make out certain rules of the -syllogism involving the distinction of subject and predicate, and -in no important respect better than the old rules of the syllogism. -Leibnitz’ logical tracts are, in fact, little more than brief memoranda -of investigations which seem never to have been followed out. They -remain as evidence of his wonderful sagacity, but it would be difficult -to show that they have had any influence on the progress of logical -science in recent times. - - [5] Erdmann, p. 100. - -I should like to explain how it happened that these logical writings -of Leibnitz were unknown to me, until within the last twelve months. I -am so slow a reader of Latin books, indeed, that my overlooking a few -pages of Leibnitz’ works would not have been in any case surprising. -But the fact is that the copy of Leibnitz’ works of which I made -occasional use, was one of the edition of Dutens, contained in Owens -College Library. The logical tracts in question were not printed in -that edition, and with one exception, they remained in manuscript in -the Royal Library at Hanover, until edited by Erdmann, in 1839–40. -The tract “Difficultates Quædam Logicæ,” though not known to Dutens, -was published by Raspe in 1765, in his collection called *Œuvres -Philosophiques de feu M^{r.} Leibnitz*; but this work had not come to -my notice, nor does the tract in question seem to contain any explicit -statement of the principle of substitution. - -It is, I presume, the comparatively recent publication of Leibnitz’ -most remarkable logical tracts which explains the apparent ignorance of -logicians as regards their contents and importance. The most learned -logicians, such as Hamilton and Ueberweg, ignore Leibnitz’ principle -of substitution. In the Appendix to the fourth volume of Hamilton’s -*Lectures on Metaphysics and Logic*, is given an elaborate compendium -of the views of logical writers concerning the ultimate basis of -deductive reasoning. Leibnitz is briefly noticed on p. 319, but without -any hint of substitution. He is here quoted as saying, “What are the -same with the same third, are the same with each other; that is, if *A* -be the same with *B*, and *C* be the same with *B*, it is necessary -that *A* and *C* should also be the same with one another. For this -principle flows immediately from the principle of contradiction, and is -the ground and basis of all logic; if that fail, there is no longer any -way of reasoning with certainty.” This view of the matter seems to be -inconsistent with that which he adopted in his posthumous tract. - -Dr. Thomson, indeed, was acquainted with Leibnitz’ tracts, and refers -to them in his *Outline of the Necessary Laws of Thought*. He calls -them valuable; nevertheless, he seems to have missed the really -valuable point; for in making two brief quotations,[6] he omits all -mention of the principle of substitution. - - [6] Fifth Edition, 1860, p. 158. - -Ueberweg is probably considered the best authority concerning the -history of logic, and in his well-known *System of Logic and History -of Logical Doctrines*,[7] he gives some account of the principle of -substitution, especially as it is implicitly stated in the *Port Royal -Logic*. But he omits all reference to Leibnitz in this connection, -nor does he elsewhere, so far as I can find, supply the omission. -His English editor, Professor T. M. Lindsay, in referring to my -*Substitution of Similars*, points out how I was anticipated by Beneke; -but he also ignores Leibnitz. It is thus apparent that the most learned -logicians, even when writing especially on the history of logic, -displayed ignorance of Leibnitz’ most valuable logical writings. - - [7] Section 120. - -It has been recently pointed out to me, however, that the Rev. Robert -Harley did draw attention, at the Nottingham Meeting of the British -Association, in 1866, to Leibnitz’ anticipations of Boole’s laws of -logical notation,[8] and I am informed that Boole, about a year after -the publication of his *Laws of Thought*, was made acquainted with -these anticipations by R. Leslie Ellis. - - [8] See his “Remarks on Boole’s Mathematical Analysis of Logic.” - *Report of the 36th Meeting of the British Association, Transactions - of the Sections*, pp. 3–6. - -There seems to have been at least one other German logician who -discovered, or adopted, the principle of substitution. Reusch, in his -*Systema Logicum*, published in 1734, laboured to give a broader basis -to the *Dictum de Omni et Nullo*. He argues, that “the whole business -of ordinary reasoning is accomplished by the substitution of ideas in -place of the subject or predicate of the fundamental proposition. This -some call the *equation of thoughts*.” But, in the hands of Reusch, -substitution does not seem to lead to simplicity, since it has to be -carried on according to the rules of Equipollence, Reciprocation, -Subordination, and Co-ordination.[9] Reusch is elsewhere spoken of[10] -as the “celebrated Reusch”; nevertheless, I have not been able to find -a copy of his book in London, even in the British Museum Library; it is -not mentioned in the printed catalogue of the Bodleian Library; Messrs. -Asher have failed to obtain it for me by advertisement in Germany; and -Professor Adamson has been equally unsuccessful. From the way in which -the principle of substitution is mentioned by Reusch, it would seem -likely that other logicians of the early part of the eighteenth century -were acquainted with it; but, if so, it is still more curious that -recent historians of logical science have overlooked the doctrine. - - [9] Hamilton’s Lectures, vol. iv. p. 319. - - [10] Ibid. p. 326. - -It is a strange and discouraging fact, that true views of logic should -have been discovered and discussed from one to two centuries ago, and -yet should have remained, like George Bentham’s work in this century, -without influence on the subsequent progress of the science. It may be -regarded as certain that none of the discoverers of the quantification -of the predicate, Bentham, Hamilton, Thomson, De Morgan, and Boole, -were in any way assisted by the hints of the principle contained in -previous writers. As to my own views of logic, they were originally -moulded by a careful study of Boole’s works, as fully stated in my -first logical essay.[11] As to the process of substitution, it was -not learnt from any work on logic, but is simply the process of -substitution perfectly familiar to mathematicians, and with which I -necessarily became familiar in the course of my long-continued study of -mathematics under the late Professor De Morgan. - - [11] *Pure Logic, or the Logic of Quality apart from Quantity; - with Remarks on Boole’s System, and on the Relation of Logic and - Mathematics.* London, 1864, p. 3. - -I find that the Theory of Number, which I explained in the eighth -chapter of this work, is also partially anticipated in a single -scholium of Leibnitz. He first gives as an axiom the now well-known law -of Boole, as follows:-- - -“Axioma I. Si idem secum ipso sumatur, nihil constituitur novum, seu -*A* + *A* ∝ *A*.” Then follows this remarkable scholium: “Equidem in -numeris 4 + 4 facit 8, seu bini nummi binis additi faciunt quatuor -nummos, sed tunc bini additi sunt alii a prioribus; si iidem essent -nihil novi prodiret et perinde esset ac si joco ex tribus ovis facere -vellemus sex numerando, primum 3 ova, deinde uno sublato residua 2, ac -denique uno rursus sublato residuum.” - -Translated this would read as follows:-- - -“Axiom I. If the same thing is taken together with itself, nothing new -arises, or *A* + *A* = *A*. - -“Scholium. In numbers, indeed, 4 + 4 makes 8, or two coins added to two -coins make four coins, but then the two added are different from the -former ones; if they were the same nothing new would be produced, and -it would be just as if we tried in joke to make six eggs out of three, -by counting firstly the three eggs, then, one being removed, counting -the remaining two, and lastly, one being again removed, counting the -remaining egg.” - -Compare the above with pp. 156 to 162 of the present work. - -M. Littré has quite recently pointed out[12] what he thinks is an -analogy between the system of formal logic, stated in the following -pages, and the logical devices of the celebrated Raymond Lully. Lully’s -method of invention was described in a great number of mediæval books, -but is best stated in his *Ars Compendiosa Inveniendi Veritatem, seu -Ars Magna et Major*. This method consisted in placing various names -of things in the sectors of concentric circles, so that when the -circles were turned, every possible combination of the things was -easily produced by mechanical means. It might, perhaps, be possible to -discover in this method a vague and rude anticipation of combinational -logic; but it is well known that the results of Lully’s method were -usually of a fanciful, if not absurd character. - - [12] *La Philosophie Positive*, Mai-Juin, 1877, tom. xviii. p. 456. - -A much closer analogue of the Logical Alphabet is probably to be found -in the Logical Square, invented by John Christian Lange, and described -in a rare and unnoticed work by him which I have recently found in the -British Museum.[13] This square involved the principle of bifurcate -classification, and was an improved form of the Ramean and Porphyrian -tree (see below, p. 702). Lange seems, indeed, to have worked out his -Logical Square into a mechanical form, and he suggests that it might -be employed somewhat in the manner of Napier’s Bones (p. 65). There -is much analogy between his Square and my Abacus, but Lange had not -arrived at a logical system enabling him to use his invention for -logical inference in the manner of the Logical Abacus. Another work -of Lange is said to contain the first publication of the well known -Eulerian diagrams of proposition and syllogism.[14] - - [13] *Inventum Novum Quadrati Logici*, &c., Gissæ Hassorum, 1714, 8vo. - - [14] See *Ueberweg’s System of Logic*, &c., translated by Lindsay, - p. 302. - -Since the first edition was published, an important work by Mr. George -Lewes has appeared, namely, his *Problems of Life and Mind*, which -to a great extent treats of scientific method, and formulates the -rules of philosophising. I should have liked to discuss the bearing -of Mr. Lewes’s views upon those here propounded, but I have felt it -to be impossible in a book already filling nearly 800 pages, to enter -upon the discussion of a yet more extensive book. For the same reason -I have not been able to compare my own treatment of the subject of -probability with the views expressed by Mr. Venn in his *Logic of -Chance*. With Mr. J. J. Murphy’s profound and remarkable works on -*Habit and Intelligence*, and on *The Scientific Basis of Faith*, I -was unfortunately unacquainted when I wrote the following pages. They -cannot safely be overlooked by any one who wishes to comprehend the -tendency of philosophy and scientific method in the present day. - -It seems desirable that I should endeavour to answer some of the -critics who have pointed out what they consider defects in the -doctrines of this book, especially in the first part, which treats -of deduction. Some of the notices of the work were indeed rather -statements of its contents than critiques. Thus, I am much indebted -to M. Louis Liard, Professor of Philosophy at Bordeaux, for the very -careful exposition[15] of the substitutional view of logic which he -gave in the excellent *Revue Philosophique*, edited by M. Ribot. -(Mars, 1877, tom. iii. p. 277.) An equally careful account of the -system was given by M. Riehl, Professor of Philosophy at Graz, in -his article on “Die Englische Logik der Gegenwart,” published in the -*Vierteljahrsschrift für wissenschaftliche Philosophie*. (1 Heft, -Leipzig, 1876.) I should like to acknowledge also the careful and able -manner in which my book was reviewed by the *New York Daily Tribune* -and the *New York Times*. - - [15] Since the above was written M. Liard has republished this - exposition as one chapter of an interesting and admirably lucid - account of the progress of logical science in England. After a brief - but clear introduction, treating of the views of Herschel, Mill, and - others concerning Inductive Logic, M. Liard describes in succession - the logical systems of George Bentham, Hamilton, De Morgan, Boole, - and that contained in the present work. The title of the book is as - follows:--*Les Logiciens Anglais Contemporains*. Par Louis Liard, - Professeur de Philosophie à la Faculté des Lettres de Bordeaux. - Paris: Librairie Germer Baillière. 1878. (Note added November, 1878.) - -The most serious objections which have been brought against my -treatment of logic have regard to my failure to enter into an analysis -of the ultimate nature and origin of the Laws of Thought. The -*Spectator*,[16] for instance, in the course of a careful review, says -of the principle of substitution, “Surely it is a great omission not -to discuss whence we get this great principle itself; whether it is a -pure law of the mind, or only an approximate lesson of experience; and -if a pure product of the mind, whether there are any other products -of the same kind, furnished by our knowing faculty itself.” Professor -Robertson, in his very acute review,[17] likewise objects to the want -of psychological and philosophical analysis. “If the book really -corresponded to its title, Mr. Jevons could hardly have passed so -lightly over the question, which he does not omit to raise, concerning -those undoubted principles of knowledge commonly called the Laws of -Thought.... Everywhere, indeed, he appears least at ease when he -touches on questions properly philosophical; nor is he satisfactory -in his psychological references, as on pp. 4, 5, where he cannot -commit himself to a statement without an accompaniment of ‘probably,’ -‘almost,’ or ‘hardly.’ Reservations are often very much in place, but -there are fundamental questions on which it is proper to make up one’s -mind.” - - [16] *Spectator*, September 19, 1874, p. 1178. A second portion of - the review appeared in the same journal for September 26, 1874, - p. 1204. - - [17] *Mind*: a Quarterly Review of Psychology and Philosophy. No. II. - April 1876. Vol. I. p. 206. - -These remarks appear to me to be well founded, and I must state why -it is that I have ventured to publish an extensive work on logic, -without properly making up my mind as to the fundamental nature of the -reasoning process. The fault after all is one of omission rather than -of commission. It is open to me on a future occasion to supply the -deficiency if I should ever feel able to undertake the task. But I do -not conceive it to be an essential part of any treatise to enter into -an ultimate analysis of its subject matter. Analyses must always end -somewhere. There were good treatises on light which described the laws -of the phenomenon correctly before it was known whether light consisted -of undulations or of projected particles. Now we have treatises on the -Undulatory Theory which are very valuable and satisfactory, although -they leave us in almost complete doubt as to what the vibrating medium -really is. So I think that, in the present day, we need a correct and -scientific exhibition of the formal laws of thought, and of the forms -of reasoning based on them, although we may not be able to enter into -any complete analysis of the nature of those laws. What would the -science of geometry be like now if the Greek geometers had decided that -it was improper to publish any propositions before they had decided -on the nature of an axiom? Where would the science of arithmetic be -now if an analysis of the nature of number itself were a necessary -preliminary to a development of the results of its laws? In recent -times there have been enormous additions to the mathematical sciences, -but very few attempts at psychological analysis. In the Alexandrian and -early mediæval schools of philosophy, much attention was given to the -nature of unity and plurality chiefly called forth by the question of -the Trinity. In the last two centuries whole sciences have been created -out of the notion of plurality, and yet speculation on the nature of -plurality has dwindled away. This present treatise contains, in the -eighth chapter, one of the few recent attempts to analyse the notion of -number itself. - -If further illustration is needed, I may refer to the differential -calculus. Nobody calls in question the formal truth of the results of -that calculus. All the more exact and successful parts of physical -science depend upon its use, and yet the mathematicians who have -created so great a body of exact truths have never decided upon the -basis of the calculus. What is the nature of a limit or the nature of -an infinitesimal? Start the question among a knot of mathematicians, -and it will be found that hardly two agree, unless it is in regarding -the question itself as a trifling one. Some hold that there are no such -things as infinitesimals, and that it is all a question of limits. -Others would argue that the infinitesimal is the necessary outcome of -the limit, but various shades of intermediate opinion spring up. - -Now it is just the same with logic. If the forms of deductive and -inductive reasoning given in the earlier part of this treatise are -correct, they constitute a definite addition to logical science, and -it would have been absurd to decline to publish such results because I -could not at the same time decide in my own mind about the psychology -and philosophy of the subject. It comes in short to this, that my book -is a book on Formal Logic and Scientific Method, and not a book on -psychology and philosophy. - -It may be objected, indeed, as the *Spectator* objects, that Mill’s -System of Logic is particularly strong in the discussion of the -psychological foundations of reasoning, so that Mill would appear to -have successfully treated that which I feel myself to be incapable of -attempting at present. If Mill’s analysis of knowledge is correct, then -I have nothing to say in excuse for my own deficiencies. But it is -well to do one thing at a time, and therefore I have not occupied any -considerable part of this book with controversy and refutation. What I -have to say of Mill’s logic will be said in a separate work, in which -his analysis of knowledge will be somewhat minutely analysed. It will -then be shown, I believe, that Mill’s psychological and philosophical -treatment of logic has not yielded such satisfactory results as some -writers seem to believe.[18] - - [18] Portions of this work have already been published in my - articles, entitled “John Stuart Mill’s Philosophy Tested,” printed - in the *Contemporary Review* for December, 1877, vol. xxxi. p. 167, - and for January and April, 1878, vol. xxxi. p. 256, and vol. xxxii. - p. 88. (Note added in November, 1878.) - -Various minor but still important criticisms were made by Professor -Robertson, a few of which have been noticed in the text (pp. 27, 101). -In other cases his objections hardly admit of any other answer than -such as consists in asking the reader to judge between the work and the -criticism. Thus Mr. Robertson asserts[19] that the most complex logical -problems solved in this book (up to p. 102 of this edition) might be -more easily and shortly dealt with upon the principles and with the -recognised methods of the traditional logic. The burden of proof here -lies upon Mr. Robertson, and his only proof consists in a single case, -where he is able, as it seems to me accidentally, to get a special -conclusion by the old form of dilemma. It would be a long labour to -test the old logic upon every result obtained by my notation, and I -must leave such readers as are well acquainted with the syllogistic -logic to pronounce upon the comparative simplicity and power of the -new and old systems. For other acute objections brought forward by Mr. -Robertson, I must refer the reader to the article in question. - - [19] *Mind*, vol. i. p. 222. - -One point in my last chapter, that on the Results and Limits of -Scientific Method, has been criticised by Professor W. K. Clifford in -his lecture[20] on “The First and the Last Catastrophe.” In vol. ii. -p. 438 of the first edition (p. 744 of this edition) I referred to -certain inferences drawn by eminent physicists as to a limit to -the antiquity of the present order of things. “According to Sir W. -Thomson’s deductions from Fourier’s *theory of heat*, we can trace down -the dissipation of heat by conduction and radiation to an infinitely -distant time when all things will be uniformly cold. But we cannot -similarly trace the Heat-history of the Universe to an infinite -distance in the past. For a certain negative value of the time, the -formulæ give impossible values, indicating that there was some initial -distribution of heat which could not have resulted, according to known -laws of nature, from any previous distribution.” - - [20] *Fortnightly Review*, New Series, April 1875, p. 480. Lecture - reprinted by the Sunday Lecture Society, p. 24. - -Now according to Professor Clifford I have here misstated Thomson’s -results. “It is not according to the known laws of nature, it is -according to the known laws of conduction of heat, that Sir William -Thomson is speaking.... All these physical writers, knowing what they -were writing about, simply drew such conclusions from the facts which -were before them as could be reasonably drawn. They say, here is a -state of things which could not have been produced by the circumstances -we are at present investigating. Then your speculator comes, he reads -a sentence and says, ‘Here is an opportunity for me to have my fling.’ -And he has his fling, and makes a purely baseless theory about the -necessary origin of the present order of nature at some definite point -of time, which might be calculated.” - -Professor Clifford proceeds to explain that Thomson’s formulæ only give -a limit to the heat history of, say, the earth’s crust in the solid -state. We are led back to the time when it became solidified from the -fluid condition. There is discontinuity in the history of the solid -matter, but still discontinuity which is within our comprehension. -Still further back we should come to discontinuity again, when the -liquid was formed by the condensation of heated gaseous matter. Beyond -that event, however, there is no need to suppose further discontinuity -of law, for the gaseous matter might consist of molecules which had -been falling together from different parts of space through infinite -past time. As Professor Clifford says (p. 481) of the bodies of the -universe, “What they have actually done is to fall together and get -solid. If we should reverse the process we should see them separating -and getting cool, and as a limit to that, we should find that all these -bodies would be resolved into molecules, and all these would be flying -away from each other. There would be no limit to that process, and we -could trace it as far back as ever we liked to trace it.” - -Assuming that I have erred, I should like to point out that I have -erred in the best company, or more strictly, being a speculator, I -have been led into error by the best physical writers. Professor Tait, -in his *Sketch of Thermodynamics*, speaking of the laws discovered by -Fourier for the motion of heat in a solid, says, “Their mathematical -expressions point also to the fact that a uniform distribution of heat, -or a distribution tending to become uniform, must have arisen from some -primitive distribution of heat of a kind not capable of being produced -by known laws from any previous distribution.” In the latter words it -will be seen that there is no limitation to the laws of conduction, -and, although I had carefully referred to Sir W. Thomson’s original -paper, it is not unnatural that I should take Professor Tait’s -interpretation of its meaning.[21] - - [21] Sir W. Thomson’s words are as follows (*Cambridge Mathematical - Journal*, Nov. 1842, vol. iii. p. 174). “When *x* is negative, the - state represented cannot be the result of any *possible* distribution - of temperature which has previously existed.” There is no limitation - in the sentence to the laws of conduction, but, as the whole paper - treats of the results of conduction in a solid, it may no doubt be - understood that there is a *tacit* limitation. See also a second - paper on the subject in the same journal for February, 1844, vol. iv. - p. 67, where again there is no expressed limitation. - -In his new work *On some Recent Advances in Physical Science*, -Professor Tait has recurred to the subject as follows:[22] “A profound -lesson may be learned from one of the earliest little papers of Sir W. -Thomson, published while he was an undergraduate at Cambridge, where -he shows that Fourier’s magnificent treatment of the conduction of -heat [in a solid body] leads to formulæ for its distribution which -are intelligible (and of course capable of being fully verified by -experiment) for all time future, but which, except in particular -cases, when extended to time past, remain intelligible for a finite -period only, and *then* indicate a state of things which could -not have resulted under known laws from any conceivable previous -distribution [of heat in the body]. So far as heat is concerned, modern -investigations have shown that a previous distribution of the *matter* -involved may, by its potential energy, be capable of producing such -a state of things at the moment of its aggregation; but the example -is now adduced not for its bearing on heat alone, but as a simple -illustration of the fact that all portions of our Science, especially -that beautiful one, the Dissipation of Energy, point unanimously to a -beginning, to a state of things incapable of being derived by present -laws [of tangible matter and its energy] from any conceivable previous -arrangement.” As this was published nearly a year after Professor -Clifford’s lecture, it may be inferred that Professor Tait adheres to -his original opinion that the theory of heat does give evidence of “a -beginning.” - - [22] Pp. 25–26. The parentheses are in the original, and show - Professor Tait’s corrections in the verbatim reports of his lectures. - The subject is treated again on pp. 168–9. - -I may add that Professor Clerk Maxwell’s words seem to countenance the -same view, for he says,[23] “This is only one of the cases in which a -consideration of the dissipation of energy leads to the determination -of a superior limit to the antiquity of the observed order of things.” -The expression “observed order of things” is open to much ambiguity, -but in the absence of qualification I should take it to include the -aggregate of the laws of nature known to us. I should interpret -Professor Maxwell as meaning that the theory of heat indicates the -occurrence of some event of which our science cannot give any further -explanation. The physical writers thus seem not to be so clear about -the matter as Professor Clifford assumes. - - [23] *Theory of Heat* 1871, p. 245. - -So far as I may venture to form an independent opinion on the subject, -it is to the effect that Professor Clifford is right, and that the -known laws of nature do not enable us to assign a “beginning.” Science -leads us backwards into infinite past duration. But that Professor -Clifford is right on this point, is no reason why we should suppose -him to be right in his other opinions, some of which I am sure are -wrong. Nor is it a reason why other parts of my last chapter should be -wrong. The question only affects the single paragraph on pp. 744–5 of -this book, which might, I believe, be struck out without necessitating -any alteration in the rest of the text. It is always to be remembered -that the failure of an argument in favour of a proposition does not, -generally speaking, add much, if any, probability to the contradictory -proposition. I cannot conclude without expressing my acknowledgments -to Professor Clifford for his kind expressions regarding my work as a -whole. - - 2, The Chestnuts, - West Heath, - Hampstead, N. W. - - *August 15, 1877.* - - - - -CONTENTS. - - - BOOK I. - - FORMAL LOGIC, DEDUCTIVE AND INDUCTIVE. - - - CHAPTER I. - - INTRODUCTION. - - SECTION PAGE - - 1. Introduction 1 - - 2. The Powers of Mind concerned in the Creation of Science 4 - - 3. Laws of Identity and Difference 5 - - 4. The Nature of the Laws of Identity and Difference 6 - - 5. The Process of Inference 9 - - 6. Deduction and Induction 11 - - 7. Symbolic Expression of Logical Inference 13 - - 8. Expression of Identity and Difference 14 - - 9. General Formula of Logical Inference 17 - - 10. The Propagating Power of Similarity 20 - - 11. Anticipations of the Principle of Substitution 21 - - 12. The Logic of Relatives 22 - - - CHAPTER II. - - TERMS. - - 1. Terms 24 - - 2. Twofold meaning of General Names 25 - - 3. Abstract Terms 27 - - 4. Substantial Terms 28 - - 5. Collective Terms 29 - - 6. Synthesis of Terms 30 - - 7. Symbolic Expression of the Law of Contradiction 31 - - 8. Certain Special Conditions of Logical Symbols 32 - - - CHAPTER III. - - PROPOSITIONS. - - 1. Propositions 36 - - 2. Simple Identities 37 - - 3. Partial Identities 40 - - 4. Limited Identities 42 - - 5. Negative Propositions 43 - - 6. Conversion of Propositions 46 - - 7. Twofold Interpretation of Propositions 47 - - - CHAPTER IV. - - DEDUCTIVE REASONING. - - 1. Deductive Reasoning 49 - - 2. Immediate Inference 50 - - 3. Inference with Two Simple Identities 51 - - 4. Inference with a Simple and a Partial Identity 53 - - 5. Inference of a Partial from Two Partial Identities 55 - - 6. On the Ellipsis of Terms in Partial Identities 57 - - 7. Inference of a Simple from Two Partial Identities 58 - - 8. Inference of a Limited from Two Partial Identities 59 - - 9. Miscellaneous Forms of Deductive Inference 60 - - 10. Fallacies 62 - - - CHAPTER V. - - DISJUNCTIVE PROPOSITIONS. - - 1. Disjunctive Propositions 66 - - 2. Expression of the Alternative Relation 67 - - 3. Nature of the Alternative Relation 68 - - 4. Laws of the Disjunctive Relation 71 - - 5. Symbolic Expression of the Law of Duality 73 - - 6. Various Forms of the Disjunctive Proposition 74 - - 7. Inference by Disjunctive Propositions 76 - - - CHAPTER VI. - - THE INDIRECT METHOD OF INFERENCE. - - 1. The Indirect Method of Inference 81 - - 2. Simple Illustrations 83 - - 3. Employment of the Contrapositive Proposition 84 - - 4. Contrapositive of a Simple Identity 86 - - 5. Miscellaneous Examples of the Method 88 - - 6. Mr. Venn’s Problem 90 - - 7. Abbreviation of the Process 91 - - 8. The Logical Alphabet 94 - - 9. The Logical Slate 95 - - 10. Abstraction of Indifferent Circumstances 97 - - 11. Illustrations of the Indirect Method 98 - - 12. Second Example 99 - - 13. Third Example 100 - - 14. Fourth Example 101 - - 15. Fifth Example 101 - - 16. Fallacies Analysed by the Indirect Method 102 - - 17. The Logical Abacus 104 - - 18. The Logical Machine 107 - - 19. The Order of Premises 114 - - 20. The Equivalence of Propositions 115 - - 21. The Nature of Inference 118 - - - CHAPTER VII. - - INDUCTION. - - 1. Induction 121 - - 2. Induction an Inverse Operation 122 - - 3. Inductive Problems for Solution by the Reader 126 - - 4. Induction of Simple Identities 127 - - 5. Induction of Partial Identities 130 - - 6. Solution of the Inverse or Inductive Problem, involving - Two Classes 134 - - 7. The Inverse Logical Problem, involving Three Classes 137 - - 8. Professor Clifford on the Types of Compound Statement - involving Four Classes 143 - - 9. Distinction between Perfect and Imperfect Induction 146 - - 10. Transition from Perfect to Imperfect Induction 149 - - - BOOK II. - - NUMBER, VARIETY, AND PROBABILITY. - - - CHAPTER VIII. - - PRINCIPLES OF NUMBER. - - 1. Principles of Number 153 - - 2. The Nature of Numbe 156 - - 3. Of Numerical Abstraction 158 - - 4. Concrete and Abstract Number 159 - - 5. Analogy of Logical and Numerical Terms 160 - - 6. Principle of Mathematical Inference 162 - - 7. Reasoning by Inequalities 165 - - 8. Arithmetical Reasoning 167 - - 9. Numerically Definite Reasoning 168 - - 10. Numerical meaning of Logical Conditions 171 - - - CHAPTER IX. - - THE VARIETY OF NATURE, OR THE DOCTRINE OF COMBINATIONS - AND PERMUTATIONS. - - 1. The Variety of Nature 173 - - 2. Distinction of Combinations and Permutations 177 - - 3. Calculation of Number of Combinations 180 - - 4. The Arithmetical Triangle 182 - - 5. Connexion between the Arithmetical Triangle and the - Logical Alphabet 189 - - 6. Possible Variety of Nature and Art 190 - - 7. Higher Orders of Variety 192 - - - CHAPTER X. - - THEORY OF PROBABILITY. - - 1. Theory of Probability 197 - - 2. Fundamental Principles of the Theory 200 - - 3. Rules for the Calculation of Probabilities 203 - - 4. The Logical Alphabet in questions of Probability 205 - - 5. Comparison of the Theory with Experience 206 - - 6. Probable Deductive Arguments 209 - - 7. Difficulties of the Theory 213 - - - CHAPTER XI. - - PHILOSOPHY OF INDUCTIVE INFERENCE. - - 1. Philosophy of Inductive Inference 218 - - 2. Various Classes of Inductive Truths 219 - - 3. The Relation of Cause and Effect 220 - - 4. Fallacious Use of the Term Cause 221 - - 5. Confusion of Two Questions 222 - - 6. Definition of the Term Cause 224 - - 7. Distinction of Inductive and Deductive Results 226 - - 8. The Grounds of Inductive Inference 228 - - 9. Illustrations of the Inductive Process 229 - - 10. Geometrical Reasoning 233 - - 11. Discrimination of Certainty and Probability 235 - - - CHAPTER XII. - - THE INDUCTIVE OR INVERSE APPLICATION OF THE THEORY - OF PROBABILITY. - - 1. The Inductive or Inverse Application of the Theory 240 - - 2. Principle of the Inverse Method 242 - - 3. Simple Applications of the Inverse Method 244 - - 4. The Theory of Probability in Astronomy 247 - - 5. The General Inverse Problem 250 - - 6. Simple Illustration of the Inverse Problem 253 - - 7. General Solution of the Inverse Problem 255 - - 8. Rules of the Inverse Method 257 - - 9. Fortuitous Coincidences 261 - - 10. Summary of the Theory of Inductive Inference 265 - - - BOOK III. - - METHODS OF MEASUREMENT. - - - CHAPTER XIII. - - THE EXACT MEASUREMENT OF PHENOMENA. - - 1. The Exact Measurement of Phenomena 270 - - 2. Division of the Subject 274 - - 3. Continuous quantity 274 - - 4. The Fallacious Indications of the Senses 276 - - 5. Complexity of Quantitative Questions 278 - - 6. The Methods of Accurate Measurement 282 - - 7. Conditions of Accurate Measurement 282 - - 8. Measuring Instruments 284 - - 9. The Method of Repetition 288 - - 10. Measurements by Natural Coincidence 292 - - 11. Modes of Indirect Measurement 296 - - 12. Comparative Use of Measuring Instruments 299 - - 13. Systematic Performance of Measurements 300 - - 14. The Pendulum 302 - - 15. Attainable Accuracy of Measurement 303 - - - CHAPTER XIV. - - UNITS AND STANDARDS OF MEASUREMENT. - - 1. Units and Standards of Measurement 305 - - 2. Standard Unit of Time 307 - - 3. The Unit of Space and the Bar Standard 312 - - 4. The Terrestrial Standard 314 - - 5. The Pendulum Standard 315 - - 6. Unit of Density 316 - - 7. Unit of Mass 317 - - 8. Natural System of Standards 319 - - 9. Subsidiary Units 320 - - 10. Derived Units 321 - - 11. Provisional Units 323 - - 12. Theory of Dimensions 325 - - 13. Natural Constants 328 - - 14. Mathematical Constants 330 - - 15. Physical Constants 331 - - 16. Astronomical Constants 332 - - 17. Terrestrial Numbers 333 - - 18. Organic Numbers 333 - - 19. Social Numbers 334 - - - CHAPTER XV. - - ANALYSIS OF QUANTITATIVE PHENOMENA. - - 1. Analysis of Quantitative Phenomena 335 - - 2. Illustrations of the Complication of Effects 336 - - 3. Methods of Eliminating Error 339 - - 4. Method of Avoidance of Error 340 - - 5. Differential Method 344 - - 6. Method of Correction 346 - - 7. Method of Compensation 350 - - 8. Method of Reversal 354 - - - CHAPTER XVI. - - THE METHOD OF MEANS. - - 1. The Method of Means 357 - - 2. Several Uses of the Mean Result 359 - - 3. The Mean and the Average 360 - - 4. On the Average or Fictitious Mean 363 - - 5. The Precise Mean Result 365 - - 6. Determination of the Zero Point 368 - - 7. Determination of Maximum Points 371 - - - CHAPTER XVII. - - THE LAW OF ERROR. - - 1. The Law of Error 374 - - 2. Establishment of the Law of Error 375 - - 3. Herschel’s Geometrical Proof 377 - - 4. Laplace’s and Quetelet’s Proof of the Law 378 - - 5. Logical Origin of the Law of Error 383 - - 6. Verification of the Law of Error 383 - - 7. The Probable Mean Result 385 - - 8. The Probable Error of Results 386 - - 9. Rejection of the Mean Result 389 - - 10. Method of Least Squares 393 - - 11. Works upon the Theory of Probability 394 - - 12. Detection of Constant Errors 396 - - - BOOK IV. - - INDUCTIVE INVESTIGATION. - - - CHAPTER XVIII. - - OBSERVATION. - - 1. Observation 399 - - 2. Distinction of Observation and Experiment 400 - - 3. Mental Conditions of Correct Observation 402 - - 4. Instrumental and Sensual Conditions of Correct Observation 404 - - 5. External Conditions of Correct Observation 407 - - 6. Apparent Sequence of Events 409 - - 7. Negative Arguments from Non-Observation 411 - - - CHAPTER XIX. - - EXPERIMENT. - - 1. Experiment 416 - - 2. Exclusion of Indifferent Circumstances 419 - - 3. Simplification of Experiments 422 - - 4. Failure in the Simplification of Experiments 424 - - 5. Removal of Usual Conditions 426 - - 6. Interference of Unsuspected Conditions 428 - - 7. Blind or Test Experiments 433 - - 8. Negative Results of Experiment 434 - - 9. Limits of Experiment 437 - - - CHAPTER XX. - - METHOD OF VARIATIONS. - - 1. Method of Variations 439 - - 2. The Variable and the Variant 440 - - 3. Measurement of the Variable 441 - - 4. Maintenance of Similar Conditions 443 - - 5. Collective Experiments 445 - - 6. Periodic Variations 447 - - 7. Combined Periodic Changes 450 - - 8. Principle of Forced Vibrations 451 - - 9. Integrated Variations 452 - - - CHAPTER XXI. - - THEORY OF APPROXIMATION. - - 1. Theory of Approximation 456 - - 2. Substitution of Simple Hypotheses 458 - - 3. Approximation to Exact Laws 462 - - 4. Successive Approximations to Natural Conditions 465 - - 5. Discovery of Hypothetically Simple Laws 470 - - 6. Mathematical Principles of Approximation 471 - - 7. Approximate Independence of Small Effects 475 - - 8. Four Meanings of Equality 479 - - 9. Arithmetic of Approximate Quantities 481 - - - CHAPTER XXII. - - QUANTITATIVE INDUCTION. - - 1. Quantitative Induction 483 - - 2. Probable Connexion of Varying Quantities 484 - - 3. Empirical Mathematical Laws 487 - - 4. Discovery of Rational Formulæ 489 - - 5. The Graphical Method 492 - - 6. Interpolation and Extrapolation 495 - - 7. Illustrations of Empirical Quantitative Laws 499 - - 8. Simple Proportional Variation 501 - - - CHAPTER XXIII. - - THE USE OF HYPOTHESIS. - - 1. The Use of Hypothesis 504 - - 2. Requisites of a good Hypothesis 510 - - 3. Possibility of Deductive Reasoning 511 - - 4. Consistency with the Laws of Nature 514 - - 5. Conformity with Facts 516 - - 6. Experimentum Crucis 518 - - 7. Descriptive Hypotheses 522 - - - CHAPTER XXIV. - - EMPIRICAL KNOWLEDGE, EXPLANATION AND PREDICTION. - - 1. Empirical Knowledge, Explanation and Prediction 525 - - 2. Empirical Knowledge 526 - - 3. Accidental Discovery 529 - - 4. Empirical Observations subsequently Explained 532 - - 5. Overlooked Results of Theory 534 - - 6. Predicted Discoveries 536 - - 7. Predictions in the Science of Light 538 - - 8. Predictions from the Theory of Undulations 540 - - 9. Prediction in other Sciences 542 - - 10. Prediction by Inversion of Cause and Effect 545 - - 11. Facts known only by Theory 547 - - - CHAPTER XXV. - - ACCORDANCE OF QUANTITATIVE THEORIES. - - 1. Accordance of Quantitative Theories 551 - - 2. Empirical Measurements 552 - - 3. Quantities indicated by Theory, but Empirically Measured 553 - - 4. Explained Results of Measurement 554 - - 5. Quantities determined by Theory and verified by - Measurement 555 - - 6. Quantities determined by Theory and not verified 556 - - 7. Discordance of Theory and Experiment 558 - - 8. Accordance of Measurements of Astronomical Distances 560 - - 9. Selection of the best Mode of Measurement 563 - - 10. Agreement of Distinct Modes of Measurement 564 - - 11. Residual Phenomena 569 - - - CHAPTER XXVI. - - CHARACTER OF THE EXPERIMENTALIST. - - 1. Character of the Experimentalist 574 - - 2. Error of the Baconian Method 576 - - 3. Freedom of Theorising 577 - - 4. The Newtonian Method, the True Organum 581 - - 5. Candour and Courage of the Philosophic Mind 586 - - 6. The Philosophic Character of Faraday 587 - - 7. Reservation of Judgment 592 - - - BOOK V. - - GENERALISATION, ANALOGY, AND CLASSIFICATION. - - - CHAPTER XXVII. - - GENERALISATION. - - 1. Generalisation 594 - - 2. Distinction of Generalisation and Analogy 596 - - 3. Two Meanings of Generalisation 597 - - 4. Value of Generalisation 599 - - 5. Comparative Generality of Properties 600 - - 6. Uniform Properties of all Matter 603 - - 7. Variable Properties of Matter 606 - - 8. Extreme Instances of Properties 607 - - 9. The Detection of Continuity 610 - - 10. The Law of Continuity 615 - - 11. Failure of the Law of Continuity 619 - - 12. Negative Arguments on the Principle of Continuity 621 - - 13. Tendency to Hasty Generalisation 623 - - - CHAPTER XXVIII. - - ANALOGY. - - 1. Analogy 627 - - 2. Analogy as a Guide in Discovery 629 - - 3. Analogy in the Mathematical Sciences 631 - - 4. Analogy in the Theory of Undulations 635 - - 5. Analogy in Astronomy 638 - - 6. Failures of Analogy 641 - - - CHAPTER XXIX. - - EXCEPTIONAL PHENOMENA. - - 1. Exceptional Phenomena 644 - - 2. Imaginary or False Exceptions 647 - - 3. Apparent but Congruent Exceptions 649 - - 4. Singular Exceptions 652 - - 5. Divergent Exceptions 655 - - 6. Accidental Exceptions 658 - - 7. Novel and Unexplained Exceptions 661 - - 8. Limiting Exceptions 663 - - 9. Real Exceptions to Supposed Laws 666 - - 10. Unclassed Exceptions 668 - - - CHAPTER XXX. - - CLASSIFICATION. - - 1. Classification 673 - - 2. Classification involving Induction 675 - - 3. Multiplicity of Modes of Classification 677 - - 4. Natural and Artificial Systems of Classification 679 - - 5. Correlation of Properties 681 - - 6. Classification in Crystallography 685 - - 7. Classification an Inverse and Tentative Operation 689 - - 8. Symbolic Statement of the Theory of Classification 692 - - 9. Bifurcate Classification 694 - - 10. The Five Predicates 698 - - 11. Summum Genus and Infima Species 701 - - 12. The Tree of Porphyry 702 - - 13. Does Abstraction imply Generalisation? 704 - - 14. Discovery of Marks or Characteristics 708 - - 15. Diagnostic Systems of Classification 710 - - 16. Index Classifications 714 - - 17. Classification in the Biological Sciences 718 - - 18. Classification by Types 722 - - 19. Natural Genera and Species 724 - - 20. Unique or Exceptional Objects 728 - - 21. Limits of Classification 730 - - - BOOK VI. - - CHAPTER XXXI. - - REFLECTIONS ON THE RESULTS AND LIMITS OF SCIENTIFIC METHOD. - - 1. Reflections on the Results and Limits of Scientific Method 735 - - 2. The Meaning of Natural Law 737 - - 3. Infiniteness of the Universe 738 - - 4. The Indeterminate Problem of Creation 740 - - 5. Hierarchy of Natural Laws 742 - - 6. The Ambiguous Expression--“Uniformity of Nature” 745 - - 7. Possible States of the Universe 749 - - 8. Speculations on the Reconcentration of Energy 751 - - 9. The Divergent Scope for New Discovery 752 - - 10. Infinite Incompleteness of the Mathematical Sciences 754 - - 11. The Reign of Law in Mental and Social Phenomena 759 - - 12. The Theory of Evolution 761 - - 13. Possibility of Divine Interference 765 - - 14. Conclusion 766 - - - INDEX 773 - - - - -THE PRINCIPLES OF SCIENCE. - - - - -CHAPTER I. - -INTRODUCTION. - - -Science arises from the discovery of Identity amidst Diversity. The -process may be described in different words, but our language must -always imply the presence of one common and necessary element. In -every act of inference or scientific method we are engaged about a -certain identity, sameness, similarity, likeness, resemblance, analogy, -equivalence or equality apparent between two objects. It is doubtful -whether an entirely isolated phenomenon could present itself to our -notice, since there must always be some points of similarity between -object and object. But in any case an isolated phenomenon could be -studied to no useful purpose. The whole value of science consists -in the power which it confers upon us of applying to one object the -knowledge acquired from like objects; and it is only so far, therefore, -as we can discover and register resemblances that we can turn our -observations to account. - -Nature is a spectacle continually exhibited to our senses, in which -phenomena are mingled in combinations of endless variety and novelty. -Wonder fixes the mind’s attention; memory stores up a record of each -distinct impression; the powers of association bring forth the record -when the like is felt again. By the higher faculties of judgment and -reasoning the mind compares the new with the old, recognises essential -identity, even when disguised by diverse circumstances, and expects to -find again what was before experienced. It must be the ground of all -reasoning and inference that *what is true of one thing will be true -of its equivalent*, and that under carefully ascertained conditions -*Nature repeats herself*. - -Were this indeed a Chaotic Universe, the powers of mind employed in -science would be useless to us. Did Chance wholly take the place of -order, and did all phenomena come out of an *Infinite Lottery*, to use -Condorcet’s expression, there could be no reason to expect the like -result in like circumstances. It is possible to conceive a world in -which no two things should be associated more often, in the long run, -than any other two things. The frequent conjunction of any two events -would then be purely fortuitous, and if we expected conjunctions to -recur continually, we should be disappointed. In such a world we might -recognise the same kind of phenomenon as it appeared from time to time, -just as we might recognise a marked ball as it was occasionally drawn -and re-drawn from a ballot-box; but the approach of any phenomenon -would be in no way indicated by what had gone before, nor would it be a -sign of what was to come after. In such a world knowledge would be no -more than the memory of past coincidences, and the reasoning powers, if -they existed at all, would give no clue to the nature of the present, -and no presage of the future. - -Happily the Universe in which we dwell is not the result of chance, -and where chance seems to work it is our own deficient faculties which -prevent us from recognising the operation of Law and of Design. In -the material framework of this world, substances and forces present -themselves in definite and stable combinations. Things are not in -perpetual flux, as ancient philosophers held. Element remains element; -iron changes not into gold. With suitable precautions we can calculate -upon finding the same thing again endowed with the same properties. -The constituents of the globe, indeed, appear in almost endless -combinations; but each combination bears its fixed character, and -when resolved is found to be the compound of definite substances. -Misapprehensions must continually occur, owing to the limited extent -of our experience. We can never have examined and registered possible -existences so thoroughly as to be sure that no new ones will occur and -frustrate our calculations. The same outward appearances may cover -any amount of hidden differences which we have not yet suspected. To -the variety of substances and powers diffused through nature at its -creation, we should not suppose that our brief experience can assign -a limit, and the necessary imperfection of our knowledge must be ever -borne in mind. - -Yet there is much to give us confidence in Science. The wider our -experience, the more minute our examination of the globe, the -greater the accumulation of well-reasoned knowledge,--the fewer in -all probability will be the failures of inference compared with the -successes. Exceptions to the prevalence of Law are gradually reduced -to Law themselves. Certain deep similarities have been detected among -the objects around us, and have never yet been found wanting. As the -means of examining distant parts of the universe have been acquired, -those similarities have been traced there as here. Other worlds and -stellar systems may be almost incomprehensively different from ours in -magnitude, condition and disposition of parts, and yet we detect there -the same elements of which our own limbs are composed. The same natural -laws can be detected in operation in every part of the universe within -the scope of our instruments; and doubtless these laws are obeyed -irrespective of distance, time, and circumstance. - -It is the prerogative of Intellect to discover what is uniform and -unchanging in the phenomena around us. So far as object is different -from object, knowledge is useless and inference impossible. But so -far as object resembles object, we can pass from one to the other. In -proportion as resemblance is deeper and more general, the commanding -powers of knowledge become more wonderful. Identity in one or other -of its phases is thus always the bridge by which we pass in inference -from case to case; and it is my purpose in this treatise to trace out -the various forms in which the one same process of reasoning presents -itself in the ever-growing achievements of Scientific Method. - - -*The Powers of Mind concerned in the Creation of Science.* - -It is no part of the purpose of this work to investigate the nature -of mind. People not uncommonly suppose that logic is a branch of -psychology, because reasoning is a mental operation. On the same -ground, however, we might argue that all the sciences are branches -of psychology. As will be further explained, I adopt the opinion of -Mr. Herbert Spencer, that logic is really an objective science, like -mathematics or mechanics. Only in an incidental manner, then, need -I point out that the mental powers employed in the acquisition of -knowledge are probably three in number. They are substantially as -Professor Bain has stated them[24]:-- - - [24] *The Senses and the Intellect*, Second Ed., pp. 5, 325, &c. - - 1. The Power of Discrimination. - 2. The Power of Detecting Identity. - 3. The Power of Retention. - - -We exert the first power in every act of perception. Hardly can we have -a sensation or feeling unless we discriminate it from something else -which preceded. Consciousness would almost seem to consist in the break -between one state of mind and the next, just as an induced current of -electricity arises from the beginning or the ending of the primary -current. We are always engaged in discrimination; and the rudiment of -thought which exists in the lower animals probably consists in their -power of feeling difference and being agitated by it. - -Yet had we the power of discrimination only, Science could not be -created. To know that one feeling differs from another gives purely -negative information. It cannot teach us what will happen. In such a -state of intellect each sensation would stand out distinct from every -other; there would be no tie, no bridge of affinity between them. We -want a unifying power by which the present and the future may be linked -to the past; and this seems to be accomplished by a different power of -mind. Lord Bacon has pointed out that different men possess in very -different degrees the powers of discrimination and identification. It -may be said indeed that discrimination necessarily implies the action -of the opposite process of identification; and so it doubtless does in -negative points. But there is a rare property of mind which consists -in penetrating the disguise of variety and seizing the common elements -of sameness; and it is this property which furnishes the true measure -of intellect. The name of “intellect” expresses the interlacing of the -general and the single, which is the peculiar province of mind.[25] To -*cogitate* is the Latin *coagitare*, resting on a like metaphor. Logic, -also, is but another name for the same process, the peculiar work of -reason; for λογος is derived from λεγειν, which like the Latin *legere* -meant originally to gather. Plato said of this unifying power, that if -he met the man who could detect *the one in the many*, he would follow -him as a god. - - [25] Max Müller, *Lectures on the Science of Language*, Second - Series, vol. ii. p. 63; or Sixth Edition, vol. ii. p. 67. The view - of the etymological meaning of “intellect” is given above on the - authority of Professor Max Müller. It seems to be opposed to the - ordinary opinion, according to which the Latin *intelligere* means to - choose between, to see a difference between, to discriminate, instead - of to unite. - - -*Laws of Identity and Difference.* - -At the base of all thought and science must lie the laws which express -the very nature and conditions of the discriminating and identifying -powers of mind. These are the so-called Fundamental Laws of Thought, -usually stated as follows:-- - - 1. The Law of Identity. *Whatever is, is.* - - 2. The Law of Contradiction. *A thing cannot both be and not be.* - - 3. The Law of Duality. *A thing must either be or not be.* - -The first of these statements may perhaps be regarded as a description -of identity itself, if so fundamental a notion can admit of -description. A thing at any moment is perfectly identical with itself, -and, if any person were unaware of the meaning of the word “identity,” -we could not better describe it than by such an example. - -The second law points out that contradictory attributes can never be -joined together. The same object may vary in its different parts; -here it may be black, and there white; at one time it may be hard and -at another time soft; but at the same time and place an attribute -cannot be both present and absent. Aristotle truly described this law -as the first of all axioms--one of which we need not seek for any -demonstration. All truths cannot be proved, otherwise there would be an -endless chain of demonstration; and it is in self-evident truths like -this that we find the simplest foundations. - -The third of these laws completes the other two. It asserts that at -every step there are two possible alternatives--presence or absence, -affirmation or negation. Hence I propose to name this law the Law of -Duality, for it gives to all the formulæ of reasoning a dual character. -It asserts also that between presence and absence, existence and -non-existence, affirmation and negation, there is no third alternative. -As Aristotle said, there can be no mean between opposite assertions: we -must either affirm or deny. Hence the inconvenient name by which it has -been known--The Law of Excluded Middle. - -It may be allowed that these laws are not three independent and -distinct laws; they rather express three different aspects of the -same truth, and each law doubtless presupposes and implies the other -two. But it has not hitherto been found possible to state these -characters of identity and difference in less than the threefold -formula. The reader may perhaps desire some information as to the -mode in which these laws have been stated, or the way in which they -have been regarded, by philosophers in different ages of the world. -Abundant information on this and many other points of logical history -will be found in Ueberweg’s *System of Logic*, of which an excellent -translation has been published by Professor T. M. Lindsay (see -pp. 228–281). - - -*The Nature of the Laws of Identity and Difference.* - -I must at least allude to the profoundly difficult question concerning -the nature and authority of these Laws of Identity and Difference. -Are they Laws of Thought or Laws of Things? Do they belong to mind or -to material nature? On the one hand it may be said that science is a -purely mental existence, and must therefore conform to the laws of -that which formed it. Science is in the mind and not in the things, -and the properties of mind are therefore all important. It is true -that these laws are verified in the observation of the exterior world; -and it would seem that they might have been gathered and proved by -generalisation, had they not already been in our possession. But -on the other hand, it may well be urged that we cannot prove these -laws by any process of reasoning or observation, because the laws -themselves are presupposed, as Leibnitz acutely remarked, in the very -notion of a proof. They are the prior conditions of all thought and -all knowledge, and even to question their truth is to allow them true. -Hartley ingeniously refined upon this argument, remarking that if the -fundamental laws of logic be not certain, there must exist a logic of -a second order whereby we may determine the degree of uncertainty: if -the second logic be not certain, there must be a third; and so on *ad -infinitum*. Thus we must suppose either that absolutely certain laws of -thought exist, or that there is no such thing as certainty whatever.[26] - - [26] Hartley on Man, vol. i. p. 359. - -Logicians, indeed, appear to me to have paid insufficient attention to -the fact that mistakes in reasoning are always possible, and of not -unfrequent occurrence. The Laws of Thought are often called necessary -laws, that is, laws which cannot but be obeyed. Yet as a matter of -fact, who is there that does not often fail to obey them? They are -the laws which the mind ought to obey rather than what it always does -obey. Our thoughts cannot be the criterion of truth, for we often -have to acknowledge mistakes in arguments of moderate complexity, -and we sometimes only discover our mistakes by collision between our -expectations and the events of objective nature. - -Mr. Herbert Spencer holds that the laws of logic are objective -laws,[27] and he regards the mind as being in a state of constant -education, each act of false reasoning or miscalculation leading to -results which are likely to prevent similar mistakes from being again -committed. I am quite inclined to accept such ingenious views; but at -the same time it is necessary to distinguish between the accumulation -of knowledge, and the constitution of the mind which allows of the -acquisition of knowledge. Before the mind can perceive or reason at -all it must have the conditions of thought impressed upon it. Before -a mistake can be committed, the mind must clearly distinguish the -mistaken conclusion from all other assertions. Are not the Laws of -Identity and Difference the prior conditions of all consciousness and -all existence? Must they not hold true, alike of things material and -immaterial? and if so, can we say that they are only subjectively true -or objectively true? I am inclined, in short, to regard them as true -both “in the nature of thought and things,” as I expressed it in my -first logical essay;[28] and I hold that they belong to the common -basis of all existence. But this is one of the most difficult questions -of psychology and metaphysics which can be raised, and it is hardly -one for the logician to decide. As the mathematician does not inquire -into the nature of unity and plurality, but develops the formal laws of -plurality, so the logician, as I conceive, must assume the truth of the -Laws of Identity and Difference, and occupy himself in developing the -variety of forms of reasoning in which their truth may be manifested. - - [27] *Principles of Psychology*, Second Ed., vol. ii. p. 86. - - [28] *Pure Logic, or the Logic of Quality apart from Quantity*, 1864, - pp. 10, 16, 22, 29, 36, &c. - -Again, I need hardly dwell upon the question whether logic treats of -language, notions, or things. As reasonably might we debate whether a -mathematician treats of symbols, quantities, or things. A mathematician -certainly does treat of symbols, but only as the instruments whereby -to facilitate his reasoning concerning quantities; and as the axioms -and rules of mathematical science must be verified in concrete objects -in order that the calculations founded upon them may have any validity -or utility, it follows that the ultimate objects of mathematical -science are the things themselves. In like manner I conceive that -the logician treats of language so far as it is essential for the -embodiment and exhibition of thought. Even if reasoning can take place -in the inner consciousness of man without the use of any signs, which -is doubtful, at any rate it cannot become the subject of discussion -until by some system of material signs it is manifested to other -persons. The logician then uses words and symbols as instruments of -reasoning, and leaves the nature and peculiarities of language to the -grammarian. But signs again must correspond to the thoughts and things -expressed, in order that they shall serve their intended purpose. We -may therefore say that logic treats ultimately of thoughts and things, -and immediately of the signs which stand for them. Signs, thoughts, and -exterior objects may be regarded as parallel and analogous series of -phenomena, and to treat any one of the three series is equivalent to -treating either of the other series. - - -*The Process of Inference.* - -The fundamental action of our reasoning faculties consists in inferring -or carrying to a new instance of a phenomenon whatever we have -previously known of its like, analogue, equivalent or equal. Sameness -or identity presents itself in all degrees, and is known under various -names; but the great rule of inference embraces all degrees, and -affirms that *so far as there exists sameness, identity or likeness, -what is true of one thing will be true of the other*. The great -difficulty doubtless consists in ascertaining that there does exist -a sufficient degree of likeness or sameness to warrant an intended -inference; and it will be our main task to investigate the conditions -under which reasoning is valid. In this place I wish to point out that -there is something common to all acts of inference, however different -their apparent forms. The one same rule lends itself to the most -diverse applications. - -The simplest possible case of inference, perhaps, occurs in the use of -a *pattern*, *example*, or, as it is commonly called, a *sample*. To -prove the exact similarity of two portions of commodity, we need not -bring one portion beside the other. It is sufficient that we take a -sample which exactly represents the texture, appearance, and general -nature of one portion, and according as this sample agrees or not with -the other, so will the two portions of commodity agree or differ. -Whatever is true as regards the colour, texture, density, material of -the sample will be true of the goods themselves. In such cases likeness -of quality is the condition of inference. - -Exactly the same mode of reasoning holds true of magnitude and figure. -To compare the sizes of two objects, we need not lay them beside each -other. A staff, string, or other kind of measure may be employed to -represent the length of one object, and according as it agrees or not -with the other, so must the two objects agree or differ. In this case -the proxy or sample represents length; but the fact that lengths can -be added and multiplied renders it unnecessary that the proxy should -always be as large as the object. Any standard of convenient size, such -as a common foot-rule, may be made the medium of comparison. The height -of a church in one town may be carried to that in another, and objects -existing immovably at opposite sides of the earth may be vicariously -measured against each other. We obviously employ the axiom that -whatever is true of a thing as regards its length, is true of its equal. - -To every other simple phenomenon in nature the same principle of -substitution is applicable. We may compare weights, densities, degrees -of hardness, and degrees of all other qualities, in like manner. To -ascertain whether two sounds are in unison we need not compare them -directly, but a third sound may be the go-between. If a tuning-fork is -in unison with the middle C of York Minster organ, and we afterwards -find it to be in unison with the same note of the organ in Westminster -Abbey, then it follows that the two organs are tuned in unison. The -rule of inference now is, that what is true of the tuning-fork as -regards the tone or pitch of its sound, is true of any sound in unison -with it. - -The skilful employment of this substitutive process enables us to -make measurements beyond the powers of our senses. No one can count -the vibrations, for instance, of an organ-pipe. But we can construct -an instrument called the *siren*, so that, while producing a sound of -any pitch, it shall register the number of vibrations constituting the -sound. Adjusting the sound of the siren in unison with an organ-pipe, -we measure indirectly the number of vibrations belonging to a sound -of that pitch. To measure a sound of the same pitch is as good as to -measure the sound itself. - -Sir David Brewster, in a somewhat similar manner, succeeded in -measuring the refractive indices of irregular fragments of transparent -minerals. It was a troublesome, and sometimes impracticable work to -grind the minerals into prisms, so that the power of refracting light -could be directly observed; but he fell upon the ingenious device -of compounding a liquid possessing the same refractive power as the -transparent fragment under examination. The moment when this equality -was attained could be known by the fragments ceasing to reflect or -refract light when immersed in the liquid, so that they became almost -invisible in it. The refractive power of the liquid being then measured -gave that of the solid. A more beautiful instance of representative -measurement, depending immediately upon the principle of inference, -could not be found.[29] - - [29] Brewster, *Treatise on New Philosophical Instruments*, p. 273. - Concerning this method see also Whewell, *Philosophy of the Inductive - Sciences*, vol. ii. p. 355; Tomlinson, *Philosophical Magazine*, - Fourth Series, vol. xl. p. 328; Tyndall, in Youmans’ *Modern - Culture*, p. 16. - -Throughout the various logical processes which we are about -to consider--Deduction, Induction, Generalisation, Analogy, -Classification, Quantitative Reasoning--we shall find the one same -principle operating in a more or less disguised form. - - -*Deduction and Induction.* - -The processes of inference always depend on the one same principle of -substitution; but they may nevertheless be distinguished according as -the results are inductive or deductive. As generally stated, deduction -consists in passing from more general to less general truths; induction -is the contrary process from less to more general truths. We may -however describe the difference in another manner. In deduction we are -engaged in developing the consequences of a law. We learn the meaning, -contents, results or inferences, which attach to any given proposition. -Induction is the exactly inverse process. Given certain results or -consequences, we are required to discover the general law from which -they flow. - -In a certain sense all knowledge is inductive. We can only learn the -laws and relations of things in nature by observing those things. But -the knowledge gained from the senses is knowledge only of particular -facts, and we require some process of reasoning by which we may -collect out of the facts the laws obeyed by them. Experience gives -us the materials of knowledge: induction digests those materials, and -yields us general knowledge. When we possess such knowledge, in the -form of general propositions and natural laws, we can usefully apply -the reverse process of deduction to ascertain the exact information -required at any moment. In its ultimate foundation, then, all knowledge -is inductive--in the sense that it is derived by a certain inductive -reasoning from the facts of experience. - -It is nevertheless true,--and this is a point to which insufficient -attention has been paid, that all reasoning is founded on the -principles of deduction. I call in question the existence of any method -of reasoning which can be carried on without a knowledge of deductive -processes. I shall endeavour to show that *induction is really the -inverse process of deduction*. There is no mode of ascertaining the -laws which are obeyed in certain phenomena, unless we have the power -of determining what results would follow from a given law. Just as the -process of division necessitates a prior knowledge of multiplication, -or the integral calculus rests upon the observation and remembrance -of the results of the differential calculus, so induction requires a -prior knowledge of deduction. An inverse process is the undoing of -the direct process. A person who enters a maze must either trust to -chance to lead him out again, or he must carefully notice the road by -which he entered. The facts furnished to us by experience are a maze of -particular results; we might by chance observe in them the fulfilment -of a law, but this is scarcely possible, unless we thoroughly learn the -effects which would attach to any particular law. - -Accordingly, the importance of deductive reasoning is doubly supreme. -Even when we gain the results of induction they would be of no use -unless we could deductively apply them. But before we can gain them -at all we must understand deduction, since it is the inversion of -deduction which constitutes induction. Our first task in this work, -then, must be to trace out fully the nature of identity in all its -forms of occurrence. Having given any series of propositions we must be -prepared to develop deductively the whole meaning embodied in them, and -the whole of the consequences which flow from them. - - -*Symbolic Expression of Logical Inference.* - -In developing the results of the Principle of Inference we require to -use an appropriate language of signs. It would indeed be quite possible -to explain the processes of reasoning by the use of words found in the -dictionary. Special examples of reasoning, too, may seem to be more -readily apprehended than general symbolic forms. But it has been shown -in the mathematical sciences that the attainment of truth depends -greatly upon the invention of a clear, brief, and appropriate system -of symbols. Not only is such a language convenient, but it is almost -essential to the expression of those general truths which are the very -soul of science. To apprehend the truth of special cases of inference -does not constitute logic; we must apprehend them as cases of more -general truths. The object of all science is the separation of what is -common and general from what is accidental and different. In a system -of logic, if anywhere, we should esteem this generality, and strive to -exhibit clearly what is similar in very diverse cases. Hence the great -value of *general symbols* by which we can represent the form of a -reasoning process, disentangled from any consideration of the special -subject to which it is applied. - -The signs required in logic are of a very simple kind. As sameness or -difference must exist between two things or notions, we need signs to -indicate the things or notions compared, and other signs to denote the -relations between them. We need, then, (1) symbols for terms, (2) a -symbol for sameness, (3) a symbol for difference, and (4) one or two -symbols to take the place of conjunctions. - -Ordinary nouns substantive, such as *Iron*, *Metal*, *Electricity*, -*Undulation*, might serve as terms, but, for the reasons explained -above, it is better to adopt blank letters, devoid of special -signification, such as A, B, C, &c. Each letter must be understood to -represent a noun, and, so far as the conditions of the argument allow, -*any noun*. Just as in Algebra, *x*, *y*, *z*, *p*, *q*, &c. are used -for *any quantities*, undetermined or unknown, except when the special -conditions of the problem are taken into account, so will our letters -stand for undetermined or unknown things. - -These letter-terms will be used indifferently for nouns substantive -and adjective. Between these two kinds of nouns there may perhaps -be differences in a metaphysical or grammatical point of view. -But grammatical usage sanctions the conversion of adjectives into -substantives, and *vice versâ*; we may avail ourselves of this latitude -without in any way prejudging the metaphysical difficulties which may -be involved. Here, as throughout this work, I shall devote my attention -to truths which I can exhibit in a clear and formal manner, believing -that in the present condition of logical science, this course will lead -to greater advantage than discussion upon the metaphysical questions -which may underlie any part of the subject. - -Every noun or term denotes an object, and usually implies the -possession by that object of certain qualities or circumstances common -to all the objects denoted. There are certain terms, however, which -imply the absence of qualities or circumstances attaching to other -objects. It will be convenient to employ a special mode of indicating -these *negative terms*, as they are called. If the general name A -denotes an object or class of objects possessing certain defined -qualities, then the term Not A will denote any object which does not -possess the whole of those qualities; in short, Not A is the sign for -anything which differs from A in regard to any one or more of the -assigned qualities. If A denote “transparent object,” Not A will denote -“not transparent object.” Brevity and facility of expression are of -no slight importance in a system of notation, and it will therefore -be desirable to substitute for the negative term Not A a briefer -symbol. De Morgan represented negative terms by small Roman letters, -or sometimes by small italic letters;[30] as the latter seem to be -highly convenient, I shall use *a*, *b*, *c*, ... *p*, *q*, &c., as the -negative terms corresponding to A, B, C, ... P, Q, &c. Thus if A means -“fluid,” *a* will mean “not fluid.” - - [30] *Formal Logic*, p. 38. - - -*Expression of Identity and Difference.* - -To denote the relation of sameness or identity I unhesitatingly adopt -the sign =, so long used by mathematicians to denote equality. This -symbol was originally appropriated by Robert Recorde in his *Whetstone -of Wit*, to avoid the tedious repetition of the words “is equal to;” -and he chose a pair of parallel lines, because no two things can be -more equal.[31] The meaning of the sign has however been gradually -extended beyond that of equality of quantities; mathematicians have -themselves used it to indicate equivalence of operations. The force -of analogy has been so great that writers in most other branches -of science have employed the same sign. The philologist uses it to -indicate the equivalence of meaning of words: chemists adopt it to -signify identity in kind and equality in weight of the elements which -form two different compounds. Not a few logicians, for instance -Lambert, Drobitsch, George Bentham,[32] Boole,[33] have employed it -as the copula of propositions. De Morgan declined to use it for this -purpose, but still further extended its meaning so as to include the -equivalence of a proposition with the premises from which it can be -inferred;[34] and Herbert Spencer has applied it in a like manner.[35] - - [31] Hallam’s *Literature of Europe*, First Ed., vol. ii. p. 444. - - [32] *Outline of a New System of Logic*, London, 1827, pp. 133, &c. - - [33] *An Investigation of the Laws of Thought*, pp. 27, &c. - - [34] *Formal Logic*, pp. 82, 106. In his later work, *The Syllabus of - a New System of Logic*, he discontinued the use of the sign. - - [35] *Principles of Psychology*, Second Ed., vol. ii. pp. 54, 55. - -Many persons may think that the choice of a symbol is a matter of -slight importance or of mere convenience; but I hold that the common -use of this sign = in so many different meanings is really founded -upon a generalisation of the widest character and of the greatest -importance--one indeed which it is a principal purpose of this work to -explain. The employment of the same sign in different cases would be -unphilosophical unless there were some real analogy between its diverse -meanings. If such analogy exists, it is not only allowable, but highly -desirable and even imperative, to use the symbol of equivalence with a -generality of meaning corresponding to the generality of the principles -involved. Accordingly De Morgan’s refusal to use the symbol in logical -propositions indicated his opinion that there was a want of analogy -between logical propositions and mathematical equations. I use the sign -because I hold the contrary opinion. - -I conceive that the sign = as commonly employed, always denotes -some form or degree of sameness, and the particular form is usually -indicated by the nature of the terms joined by it. Thus “6,720 pounds = -3 tons” is evidently an equation of quantities. The formula - × - = + -expresses the equivalence of operations. “Exogens = Dicotyledons” is a -logical identity expressing a profound truth concerning the character -and origin of a most important group of plants. - -We have great need in logic of a distinct sign for the copula, because -the little verb *is* (or *are*), hitherto used both in logic and -ordinary discourse, is thoroughly ambiguous. It sometimes denotes -identity, as in “St. Paul’s is the *chef-d’œuvre* of Sir Christopher -Wren;” but it more commonly indicates inclusion of class within class, -or partial identity, as in “Bishops are members of the House of -Lords.” This latter relation involves identity, but requires careful -discrimination from simple identity, as will be shown further on. - -When with this sign of equality we join two nouns or logical terms, as -in - - Hydrogen = The least dense element, - -we signify that the object or group of objects denoted by one term is -identical with that denoted by the other, in everything except the -names. The general formula - - A = B - -must be taken to mean that A and B are symbols for the same object -or group of objects. This identity may sometimes arise from the mere -imposition of names, but it may also arise from the deepest laws of the -constitution of nature; as when we say - - Gravitating matter = Matter possessing inertia, - Exogenous plants = Dicotyledonous plants, - Plagihedral quartz crystals = Quartz crystals causing - the plane of polarisation of light to rotate. - -We shall need carefully to distinguish between relations of terms which -can be modified at our own will and those which are fixed as expressing -the laws of nature; but at present we are considering only the mode of -expression which may be the same in either case. - -Sometimes, but much less frequently, we require a symbol to indicate -difference or the absence of complete sameness. For this purpose we -may generalise in like manner the symbol ~, which was introduced by -Wallis to signify difference between quantities. The general formula - - B ~ C - -denotes that B and C are the names of two objects or groups which are -not identical with each other. Thus we may say - - Acrogens ~ Flowering plants. - Snowdon ~ The highest mountain in Great Britain. - -I shall also occasionally use the sign ᔕ to signify in the most general -manner the existence of any relation between the two terms connected by -it. Thus ᔕ might mean not only the relations of equality or inequality, -sameness or difference, but any special relation of time, place, size, -causation, &c. in which one thing may stand to another. By A ᔕ B I -mean, then, any two objects of thought related to each other in any -conceivable manner. - - -*General Formula of Logical Inference.* - -The one supreme rule of inference consists, as I have said, in the -direction to affirm of anything whatever is known of its like, equal -or equivalent. The *Substitution of Similars* is a phrase which seems -aptly to express the capacity of mutual replacement existing in any -two objects which are like or equivalent to a sufficient degree. -It is matter for further investigation to ascertain when and for -what purposes a degree of similarity less than complete identity is -sufficient to warrant substitution. For the present we think only of -the exact sameness expressed in the form - - A = B. - -Now if we take the letter C to denote any third conceivable object, and -use the sign ᔕ in its stated meaning of *indefinite relation*, then the -general formula of all inference may be thus exhibited:-- - - From A = B ᔕ C - we may infer A ᔕ C - -or, in words--*In whatever relation a thing stands to a second thing, -in the same relation it stands to the like or equivalent of that second -thing.* The identity between A and B allows us indifferently to place -A where B was, or B where A was; and there is no limit to the variety -of special meanings which we can bestow upon the signs used in this -formula consistently with its truth. Thus if we first specify only the -meaning of the sign ᔕ, we may say that if *C is the weight of B*, then -*C is also the weight of A*. Similarly - - If C is the father of B, C is the father of A; - If C is a fragment of B, C is a fragment of A; - If C is a quality of B, C is a quality of A; - If C is a species of B, C is a species of A; - If C is the equal of B, C is the equal of A; - -and so on *ad infinitum*. - -We may also endow with special meanings the letter-terms A, B, and C, -and the process of inference will never be false. Thus let the sign -ᔕ mean “is height of,” and let - - A = Snowdon, - B = Highest mountain in England or Wales, - C = 3,590 feet; - -then it obviously follows since “3,590 feet is the height of Snowdon,” -and “Snowdon = the highest mountain in England or Wales,” that, “3,590 -feet is the height of the highest mountain in England or Wales.” - -One result of this general process of inference is that we may in any -aggregate or complex whole replace any part by its equivalent without -altering the whole. To alter is to make a difference; but if in -replacing a part I make no difference, there is no alteration of the -whole. Many inferences which have been very imperfectly included in -logical formulas at once follow. I remember the late Prof. De Morgan -remarking that all Aristotle’s logic could not prove that “Because a -horse is an animal, the head of a horse is the head of an animal.” I -conceive that this amounts merely to replacing in the complete notion -*head of a horse*, the term “horse,” by its equivalent *some animal* or -*an animal*. Similarly, since - - The Lord Chancellor = The Speaker of the House of Lords, - -it follows that - - The death of the Lord Chancellor = The death of the Speaker of the - House of Lords; - -and any event, circumstance or thing, which stands in a certain -relation to the one will stand in like relation to the other. Milton -reasons in this way when he says, in his Areopagitica, “Who kills a -man, kills a reasonable creature, God’s image.” If we may suppose him -to mean - - God’s image = man = some reasonable creature, - -it follows that “The killer of a man is the killer of some reasonable -creature,” and also “The killer of God’s image.” - -This replacement of equivalents may be repeated over and over again to -any extent. Thus if *person* is identical in meaning with *individual*, -it follows that - - Meeting of persons = meeting of individuals; - -and if *assemblage* = *meeting*, we may make a new replacement and show -that - - Meeting of persons = assemblage of individuals. - -We may in fact found upon this principle of substitution a most general -axiom in the following terms[36]:-- - - [36] *Pure Logic, or the Logic of Quality*, p. 14. - - *Same parts samely related make same wholes.* - -If, for instance, exactly similar bricks and other materials be used -to build two houses, and they be similarly placed in each house, the -two houses must be similar. There are millions of cells in a human -body, but if each cell of one person were represented by an exactly -similar cell similarly placed in another body, the two persons would -be undistinguishable, and would be only *numerically* different. It -is upon this principle, as we shall see, that all accurate processes -of measurement depend. If for a weight in a scale of a balance we -substitute another weight, and the equilibrium remains entirely -unchanged, then the weights must be exactly equal. The general test of -equality is substitution. Objects are equally bright when on replacing -one by the other the eye perceives no difference. Objects are equal in -dimensions when tested by the same gauge they fit in the same manner. -Generally speaking, two objects are alike so far as when substituted -one for another no alteration is produced, and *vice versâ* when alike -no alteration is produced by the substitution. - - -*The Propagating Power of Similarity.* - -The relation of similarity in all its degrees is reciprocal. So far -as things are alike, either may be substituted for the other; and -this may perhaps be considered the very meaning of the relation. But -it is well worth notice that there is in similarity a peculiar power -of extending itself among all the things which are similar. To render -a number of things similar to each other we need only render them -similar to one standard object. Each coin struck from a pair of dies -not only resembles the matrix or original pattern from which the dies -were struck, but resembles every other coin manufactured from the same -original pattern. Among a million such coins there are not less than -499,999,500,000 *pairs of coins* resembling each other. Similars to -the same are similars to all. It is one great advantage of printing -that all copies of a document struck from the same type are necessarily -identical each with each, and whatever is true of one copy will be true -of every copy. Similarly, if fifty rows of pipes in an organ be tuned -in perfect unison with one row, usually the Principal, they must be -in unison with each other. Similarity can also reproduce or propagate -itself *ad infinitum*: for if a number of tuning-forks be adjusted in -perfect unison with one standard fork, all instruments tuned to any one -fork will agree with any instrument tuned to any other fork. Standard -measures of length, capacity, weight, or any other measurable quality, -are propagated in the same manner. So far as copies of the original -standard, or copies of copies, or copies again of those copies, are -accurately executed, they must all agree each with every other. - -It is the capability of mutual substitution which gives such great -value to the modern methods of mechanical construction, according -to which all the parts of a machine are exact facsimiles of a fixed -pattern. The rifles used in the British army are constructed on the -American interchangeable system, so that any part of any rifle can be -substituted for the same part of another. A bullet fitting one rifle -will fit all others of the same bore. Sir J. Whitworth has extended -the same system to the screws and screw-bolts used in connecting -together the parts of machines, by establishing a series of standard -screws. - - -*Anticipations of the Principle of Substitution.* - -In such a subject as logic it is hardly possible to put forth any -opinions which have not been in some degree previously entertained. The -germ at least of every doctrine will be found in earlier writers, and -novelty must arise chiefly in the mode of harmonising and developing -ideas. When I first employed the process and name of *substitution* -in logic,[37] I was led to do so from analogy with the familiar -mathematical process of substituting for a symbol its value as given in -an equation. In writing my first logical essay I had a most imperfect -conception of the importance and generality of the process, and I -described, as if they were of equal importance, a number of other laws -which now seem to be but particular cases of the one general rule of -substitution. - - [37] *Pure Logic*, pp. 18, 19. - -My second essay, “The Substitution of Similars,” was written shortly -after I had become aware of the great simplification which may be -effected by a proper application of the principle of substitution. I -was not then acquainted with the fact that the German logician Beneke -had employed the principle of substitution, and had used the word -itself in forming a theory of the syllogism. My imperfect acquaintance -with the German language had prevented me from acquiring a complete -knowledge of Beneke’s views; but there is no doubt that Professor -Lindsay is right in saying that he, and probably other logicians, -were in some degree familiar with the principle.[38] Even Aristotle’s -dictum may be regarded as an imperfect statement of the principle of -substitution; and, as I have pointed out, we have only to modify that -dictum in accordance with the quantification of the predicate in order -to arrive at the complete process of substitution.[39] The Port-Royal -logicians appear to have entertained nearly equivalent views, for -they considered that all moods of the syllogism might be reduced -under one general principle.[40] Of two premises they regard one as -the *containing proposition* (propositio continens), and the other as -the *applicative proposition*. The latter proposition must always be -affirmative, and represents that by which a substitution is made; the -former may or may not be negative, and is that in which a substitution -is effected. They also show that this method will embrace certain cases -of complex reasoning which had no place in the Aristotelian syllogism. -Their views probably constitute the greatest improvement in logical -doctrine made up to that time since the days of Aristotle. But a true -reform in logic must consist, not in explaining the syllogism in one -way or another, but in doing away with all the narrow restrictions of -the Aristotelian system, and in showing that there exists an infinite -variety of logical arguments immediately deducible from the principle -of substitution of which the ancient syllogism forms but a small and -not even the most important part. - - [38] Ueberweg’s *System of Logic*, transl. by Lindsay, pp. 442–446, - 571, 572. The anticipations of the principle of substitution to be - found in the works of Leibnitz, Reusch, and perhaps other German - logicians, will be noticed in the preface to this second edition. - - [39] *Substitution of Similars* (1869), p. 9. - - [40] *Port-Royal Logic*, transl. by Spencer Baynes, pp. 212–219. Part - III. chap. x. and xi. - - -*The Logic of Relatives.* - -There is a difficult and important branch of logic which may be -called the Logic of Relatives. If I argue, for instance, that because -Daniel Bernoulli was the son of John, and John the brother of James, -therefore Daniel was the nephew of James, it is not possible to prove -this conclusion by any simple logical process. We require at any rate -to assume that the son of a brother is a nephew. A simple logical -relation is that which exists between properties and circumstances of -the same object or class. But objects and classes of objects may also -be related according to all the properties of time and space. I believe -it may be shown, indeed, that where an inference concerning such -relations is drawn, a process of substitution is really employed and an -identity must exist; but I will not undertake to prove the assertion -in this work. The relations of time and space are logical relations -of a complicated character demanding much abstract and difficult -investigation. The subject has been treated with such great ability by -Peirce,[41] De Morgan,[42] Ellis,[43] and Harley, that I will not in -the present work attempt any review of their writings, but merely refer -the reader to the publications in which they are to be found. - - [41] *Description of a Notation for the Logic of Relatives, resulting - from an Amplification of the Conceptions of Boole’s Calculus of - Logic.* By C. S. Peirce. *Memoirs of the American Academy*, vol. ix. - Cambridge, U.S., 1870. - - [42] *On the Syllogism No IV., and on the Logic of Relations.* By - Augustus De Morgan. *Transactions of the Cambridge Philosophical - Society*, vol. x. part ii., 1860. - - [43] *Observations on Boole’s Laws of Thought.* By the late R. Leslie - Ellis; communicated by the Rev. Robert Harley, F.R.S. *Report of the - British Association*, 1870. *Report of Sections*, p. 12. Also, *On - Boole’s Laws of Thought*. By the Rev. Robert Harley, F.R.S., *ibid.* - p. 14. - - - - -CHAPTER II. - -TERMS. - - -Every proposition expresses the resemblance or difference of the things -denoted by its terms. As inference treats of the relation between two -or more propositions, so a proposition expresses a relation between two -or more terms. In the portion of this work which treats of deduction -it will be convenient to follow the usual order of exposition. We will -consider in succession the various kinds of terms, propositions, and -arguments, and we commence in this chapter with terms. - -The simplest and most palpable meaning which can belong to a term -consists of some single material object, such as Westminster Abbey, -Stonehenge, the Sun, Sirius, &c. It is probable that in early stages of -intellect only concrete and palpable things are the objects of thought. -The youngest child knows the difference between a hot and a cold body. -The dog can recognise his master among a hundred other persons, and -animals of much lower intelligence know and discriminate their haunts. -In all such acts there is judgment concerning the likeness of physical -objects, but there is little or no power of analysing each object and -regarding it as a group of qualities. - -The dignity of intellect begins with the power of separating points of -agreement from those of difference. Comparison of two objects may lead -us to perceive that they are at once like and unlike. Two fragments of -rock may differ entirely in outward form, yet they may have the same -colour, hardness, and texture. Flowers which agree in colour may differ -in odour. The mind learns to regard each object as an aggregate of -qualities, and acquires the power of dwelling at will upon one or other -of those qualities to the exclusion of the rest. Logical abstraction, -in short, comes into play, and the mind becomes capable of reasoning, -not merely about objects which are physically complete and concrete, -but about things which may be thought of separately in the mind though -they exist not separately in nature. We can think of the hardness of -a rock, or the colour of a flower, and thus produce abstract notions, -denoted by abstract terms, which will form a subject for further -consideration. - -At the same time arise general notions and classes of objects. We -cannot fail to observe that the quality *hardness* exists in many -objects, for instance in many fragments of rock; mentally joining these -together, we create the class *hard object*, which will include, not -only the actual objects examined, but all others which may happen to -agree with them, as they agree with each other. As our senses cannot -possibly report to us all the contents of space, we cannot usually -set any limits to the number of objects which may fall into any such -class. At this point we begin to perceive the power and generality of -thought, which enables us in a single act to treat of indefinitely or -even infinitely numerous objects. We can safely assert that whatever is -true of any one object coming under a class is true of any of the other -objects so far as they possess the common qualities implied in their -belonging to the class. We must not place a thing in a class unless -we are prepared to believe of it all that is believed of the class in -general; but it remains a matter of important consideration to decide -how far and in what manner we can safely undertake thus to assign -the place of objects in that general system of classification which -constitutes the body of science. - - -*Twofold Meaning of General Names.* - -Etymologically the *meaning* of a name is that which we are caused -to think of when the name is used. Now every general name causes us -to think of some one or more of the objects belonging to a class; it -may also cause us to think of the common qualities possessed by those -objects. A name is said to *denote* the object of thought to which it -may be applied; it *implies* at the same time the possession of certain -qualities or circumstances. The objects denoted form the *extent* -of meaning of the term; the qualities implied form the *intent* of -meaning. Crystal is the name of any substance of which the molecules -are arranged in a regular geometrical manner. The substances or objects -in question form the extent of meaning; the circumstance of having the -molecules so arranged forms the intent of meaning. - -When we compare general terms together, it may often be found that -the meaning of one is included in the meaning of another. Thus -all *crystals* are included among *material substances*, and all -*opaque crystals* are included among *crystals*; here the inclusion -is in extension. We may also have inclusion of meaning in regard to -intension. For, as all crystals are material substances, the qualities -implied by the term material substance must be among those implied by -crystal. Again, it is obvious that while in extension of meaning opaque -crystals are but a part of crystals, in intension of meaning crystal -is but part of opaque crystal. We increase the intent of meaning of a -term by joining to it adjectives, or phrases equivalent to adjectives, -and the removal of such adjectives of course decreases the intensive -meaning. Now, concerning such changes of meaning, the following -all-important law holds universally true:--*When the intent of meaning -of a term is increased the extent is decreased; and* vice versâ, *when -the extent is increased the intent is decreased*. In short, as one is -increased the other is decreased. - -This law refers only to logical changes. The number of steam-engines -in the world may be undergoing a rapid increase without the intensive -meaning of the name being altered. The law will only be verified, -again, when there is a real change in the intensive meaning, and an -adjective may often be joined to a noun without making a change. -*Elementary metal* is identical with *metal*; *mortal man* with *man*; -it being a *property* of all metals to be elements, and of all men to -be mortals. - -There is no limit to the amount of meaning which a term may have. A -term may denote one object, or many, or an infinite number; it may -imply a single quality, if such there be, or a group of any number -of qualities, and yet the law connecting the extension and intension -will infallibly apply. Taking the general name *planet*, we increase -its intension and decrease its extension by prefixing the adjective -*exterior*; and if we further add *nearest to the earth*, there remains -but one planet, *Mars*, to which the name can then be applied. Singular -terms, which denote a single individual only, come under the same law -of meaning as general names. They may be regarded as general names of -which the meaning in extension is reduced to a minimum. Logicians have -erroneously asserted, as it seems to me, that singular terms are devoid -of meaning in intension, the fact being that they exceed all other -terms in that kind of meaning, as I have elsewhere tried to show.[44] - - [44] Jevons’ *Elementary Lessons in Logic*, pp. 41–43; *Pure Logic*, - p. 6. See also J. S. Mill, *System of Logic*, Book I. chap. ii. - section 5, and Shedden’s *Elements of Logic*, London, 1864, pp. 14, - &c. Professor Robertson objects (*Mind*, vol. i. p. 210) that I - confuse *singular* and *proper* names; if so, it is because I hold - that the same remarks apply to proper names, which do not seem to me - to differ logically from singular names. - - -*Abstract Terms.* - -Comparison of objects, and analysis of the complex resemblances and -differences which they present, lead us to the conception of *abstract -qualities*. We learn to think of one object as not only different from -another, but as differing in some particular point, such as colour, or -weight, or size. We may then convert points of agreement or difference -into separate objects of thought which we call qualities and denote by -*abstract terms*. Thus the term *redness* means something in which a -number of objects agree as to colour, and in virtue of which they are -called red. Redness forms, in fact, the intensive meaning of the term -red. - -Abstract terms are strongly distinguished from general terms by -possessing only one kind of meaning; for as they denote qualities -there is nothing which they cannot in addition imply. The adjective -“red” is the name of red objects, but it implies the possession by -them of the quality *redness*; but this latter term has one single -meaning--the quality alone. Thus it arises that abstract terms are -incapable of plurality. Red objects are numerically distinct each -from each, and there are multitudes of such objects; but redness is a -single quality which runs through all those objects, and is the same in -one as it is in another. It is true that we may speak of *rednesses*, -meaning different kinds or tints of redness, just as we may speak of -*colours*, meaning different kinds of colours. But in distinguishing -kinds, degrees, or other differences, we render the terms so far -concrete. In that they are merely red there is but a single nature in -red objects, and so far as things are merely coloured, colour is a -single indivisible quality. Redness, so far as it is redness merely, -is one and the same everywhere, and possesses absolute oneness. In -virtue of this unity we acquire the power of treating all instances of -such quality as we may treat any one. We possess, in short, general -knowledge. - - -*Substantial Terms.* - -Logicians appear to have taken little notice of a class of terms which -partake in certain respects of the character of abstract terms and yet -are undoubtedly the names of concrete existing things. These terms are -the names of substances, such as gold, carbonate of lime, nitrogen, &c. -We cannot speak of two golds, twenty carbonates of lime, or a hundred -nitrogens. There is no such distinction between the parts of a uniform -substance as will allow of a discrimination of numerous individuals. -The qualities of colour, lustre, malleability, density, &c., by which -we recognise gold, extend through its substance irrespective of -particular size or shape. So far as a substance is gold, it is one and -the same everywhere; so that terms of this kind, which I propose to -call *substantial terms*, possess the peculiar unity of abstract terms. -Yet they are not abstract; for gold is of course a tangible visible -body, entirely concrete, and existing independently of other bodies. - -It is only when, by actual mechanical division, we break up the uniform -whole which forms the meaning of a substantial term, that we introduce -number. *Piece of gold* is a term capable of plurality; for there may -be a great many pieces discriminated either by their various shapes and -sizes, or, in the absence of such marks, by simultaneously occupying -different parts of space. In substance they are one; as regards the -properties of space they are many.[45] We need not further pursue this -question, which involves the distinction between unity and plurality, -until we consider the principles of number in a subsequent chapter. - - [45] Professor Robertson has criticised my introduction of - “Substantial Terms” (*Mind*, vol. i. p. 210), and objects, perhaps - correctly, that the distinction if valid is extra-logical. I am - inclined to think, however, that the doctrine of terms is, strictly - speaking, for the most part extra-logical. - - -*Collective Terms.* - -We must clearly distinguish between the *collective* and the *general -meanings* of terms. The same name may be used to denote the whole body -of existing objects of a certain kind, or any one of those objects -taken separately. “Man” may mean the aggregate of existing men, which -we sometimes describe as *mankind*; it is also the general name -applying to any man. The vegetable kingdom is the name of the whole -aggregate of *plants*, but “plant” itself is a general name applying -to any one or other plant. Every material object may be conceived as -divisible into parts, and is therefore collective as regards those -parts. The animal body is made up of cells and fibres, a crystal -of molecules; wherever physical division, or as it has been called -*partition*, is possible, there we deal in reality with a collective -whole. Thus the greater number of general terms are at the same time -collective as regards each individual whole which they denote. - -It need hardly be pointed out that we must not infer of a collective -whole what we know only of the parts, nor of the parts what we know -only of the whole. The relation of whole and part is not one of -identity, and does not allow of substitution. There may nevertheless be -qualities which are true alike of the whole and of its parts. A number -of organ-pipes tuned in unison produce an aggregate of sound which -is of exactly the same pitch as each separate sound. In the case of -substantial terms, certain qualities may be present equally in each -minutest part as in the whole. The chemical nature of the largest mass -of pure carbonate of lime is the same as the nature of the smallest -particle. In the case of abstract terms, again, we cannot draw a -distinction between whole and part; what is true of redness in any case -is always true of redness, so far as it is merely red. - - -*Synthesis of Terms.* - -We continually combine simple terms together so as to form new terms -of more complex meaning. Thus, to increase the intension of meaning of -a term we write it with an adjective or a phrase of adjectival nature. -By joining “brittle” to “metal,” we obtain a combined term, “brittle -metal,” which denotes a certain portion of the metals, namely, such as -are selected on account of possessing the quality of *brittleness*. -As we have already seen, “brittle metal” possesses less extension and -greater intension than metal. Nouns, prepositional phrases, participial -phrases and subordinate propositions may also be added to terms so as -to increase their intension and decrease their extension. - -In our symbolic language we need some mode of indicating this junction -of terms, and the most convenient device will be the juxtaposition of -the letter-terms. Thus if A mean brittle, and B mean metal, then AB -will mean brittle metal. Nor need there be any limit to the number of -letters thus joined together, or the complexity of the notions which -they may represent. - -Thus if we take the letters - - P = metal, - Q = white, - R = monovalent, - S = of specific gravity 10·5, - T = melting above 1000° C., - V = good conductor of heat and electricity, - -then we can form a combined term PQRSTV, which will denote “a white -monovalent metal, of specific gravity 10·5, melting above 1000° C., and -a good conductor of heat and electricity.” - -There are many grammatical usages concerning the junction of words and -phrases to which we need pay no attention in logic. We can never say -in ordinary language “of wood table,” meaning “table of wood;” but we -may consider “of wood” as logically an exact equivalent of “wooden”; so -that if - - X = of wood, - Y = table, - -there is no reason why, in our symbols, XY should not be just as -correct an expression for “table of wood ” as YX. In this case indeed -we might substitute for “of wood ” the corresponding adjective -“wooden,” but we should often fail to find any adjective answering -exactly to a phrase. There is no single word by which we could express -the notion “of specific gravity 10·5:” but logically we may consider -these words as forming an adjective; and denoting this by S and metal -by P, we may say that SP means “metal of specific gravity 10·5.” It -is one of many advantages in these blank letter-symbols that they -enable us completely to neglect all grammatical peculiarities and to -fix our attention solely on the purely logical relations involved. -Investigation will probably show that the rules of grammar are mainly -founded upon traditional usage and have little logical signification. -This indeed is sufficiently proved by the wide grammatical differences -which exist between languages, though the logical foundation must be -the same. - - -*Symbolic Expression of the Law of Contradiction.* - -The synthesis of terms is subject to the all-important Law of -Thought, described in a previous section (p. 5) and called the Law of -Contradiction, It is self-evident that no quality can be both present -and absent at the same time and place. This fundamental condition -of all thought and of all existence is expressed symbolically by a -rule that a term and its negative shall never be allowed to come -into combination. Such combined terms as A*a*, B*b*, C*c*, &c., are -self-contradictory and devoid of all intelligible meaning. If they -could represent anything, it would be what cannot exist, and cannot -even be imagined in the mind. They can therefore only enter into our -consideration to suffer immediate exclusion. The criterion of false -reasoning, as we shall find, is that it involves self-contradiction, -the affirming and denying of the same statement. We might represent -the object of all reasoning as the separation of the consistent and -possible from the inconsistent and impossible; and we cannot make any -statement except a truism without implying that certain combinations of -terms are contradictory and excluded from thought. To assert that “all -A’s are B’s” is equivalent to the assertion that “A’s which are not B’s -cannot exist.” - -It will be convenient to have the means of indicating the exclusion of -the self-contradictory, and we may use the familiar sign for *nothing*, -the cipher 0. Thus the second law of thought may be symbolised in the -forms - - A*a* = 0 AB*b* = 0 ABC*a* = 0 - -We may variously describe the meaning of 0 in logic as the -*non-existent*, the *impossible*, the *self-inconsistent*, the -*inconceivable*. Close analogy exists between this meaning and its -mathematical signification. - - -*Certain Special Conditions of Logical Symbols.* - -In order that we may argue and infer truly we must treat our logical -symbols according to the fundamental laws of Identity and Difference. -But in thus using our symbols we shall frequently meet with -combinations of which the meaning will not at first sight be apparent. -If in one case we learn that an object is “yellow and round,” and in -another case that it is “round and yellow,” there arises the question -whether these two descriptions are identical in meaning or not. Again, -if we proved that an object was “round round,” the meaning of such an -expression would be open to doubt. Accordingly we must take notice, -before proceeding further, of certain special laws which govern the -combination of logical terms. - -In the first place the combination of a logical term with itself is -without effect, just as the repetition of a statement does not alter -the meaning of the statement; “a round round object” is simply “a round -object.” What is yellow yellow is merely yellow; metallic metals cannot -differ from metals, nor circular circles from circles. In our symbolic -language we may similarly hold that AA is identical with A, or - - A = AA = AAA = &c. - -The late Professor Boole is the only logician in modern times who has -drawn attention to this remarkable property of logical terms;[46] -but in place of the name which he gave to the law, I have proposed -to call it The Law of Simplicity.[47] Its high importance will only -become apparent when we attempt to determine the relations of logical -and mathematical science. Two symbols of quantity, and only two, seem -to obey this law; we may say that 1 × 1 = 1, and 0 × 0 = 0 (taking 0 -to mean absolute zero or 1 – 1); there is apparently no other number -which combined with itself gives an unchanged result. I shall point -out, however, in the chapter upon Number, that in reality all numerical -symbols obey this logical principle. - - [46] *Mathematical Analysis of Logic*, Cambridge, 1847, p. 17. *An - Investigation of the Laws of Thought*, London, 1854, p. 31. - - [47] *Pure Logic*, p. 15. - -It is curious that this Law of Simplicity, though almost unnoticed -in modern times, was known to Boëthius, who makes a singular remark -in his treatise *De Trinitate et Unitate Dei* (p. 959). He says: “If -I should say sun, sun, sun, I should not have made three suns, but I -should have named one sun so many times.”[48] Ancient discussions about -the doctrine of the Trinity drew more attention to subtle questions -concerning the nature of unity and plurality than has ever since been -given to them. - - [48] “Velut si dicam, Sol, Sol, Sol, non tres soles effecerim, sed - uno toties prædicaverim.” - -It is a second law of logical symbols that order of combination is a -matter of indifference. “Rich and rare gems” are the same as “rare and -rich gems,” or even as “gems, rich and rare.” Grammatical, rhetorical, -or poetic usage may give considerable significance to order of -expression. The limited power of our minds prevents our grasping many -ideas at once, and thus the order of statement may produce some effect, -but not in a simply logical manner. All life proceeds in the succession -of time, and we are obliged to write, speak, or even think of things -and their qualities one after the other; but between the things and -their qualities there need be no such relation of order in time or -space. The sweetness of sugar is neither before nor after its weight -and solubility. The hardness of a metal, its colour, weight, opacity, -malleability, electric and chemical properties, are all coexistent -and coextensive, pervading the metal and every part of it in perfect -community, none before nor after the others. In our words and symbols -we cannot observe this natural condition; we must name one quality -first and another second, just as some one must be the first to sign a -petition, or to walk foremost in a procession. In nature there is no -such precedence. - -I find that the opinion here stated, to the effect that relations of -space and time do not apply to many of our ideas, is clearly adopted by -Hume in his celebrated *Treatise on Human Nature* (vol. i. p. 410). He -says:[49]--“An object may be said to be no where, when its parts are -not so situated with respect to each other, as to form any figure or -quantity; nor the whole with respect to other bodies so as to answer -to our notions of contiguity or distance. Now this is evidently the -case with all our perceptions and objects, except those of sight and -feeling. A moral reflection cannot be placed on the right hand or on -the left hand of a passion, nor can a smell or sound be either of a -circular or a square figure. These objects and perceptions, so far from -requiring any particular place, are absolutely incompatible with it, -and even the imagination cannot attribute it to them.” - - [49] Book i., Part iv., Section 5. - -A little reflection will show that knowledge in the highest perfection -would consist in the *simultaneous* possession of a multitude of -facts. To comprehend a science perfectly we should have every fact -present with every other fact. We must write a book and we must read -it successively word by word, but how infinitely higher would be -our powers of thought if we could grasp the whole in one collective -act of consciousness! Compared with the brutes we do possess some -slight approximation to such power, and it is conceivable that in the -indefinite future mind may acquire an increase of capacity, and be less -restricted to the piecemeal examination of a subject. But I wish here -to make plain that there is no logical foundation for the successive -character of thought and reasoning unavoidable under our present mental -conditions. *We are logically weak and imperfect in respect of the -fact that we are obliged to think of one thing after another.* We must -describe metal as “hard and opaque,” or “opaque and hard,” but in the -metal itself there is no such difference of order; the properties are -simultaneous and coextensive in existence. - -Setting aside all grammatical peculiarities which render a substantive -less moveable than an adjective, and disregarding any meaning indicated -by emphasis or marked order of words, we may state, as a general law of -logic, that AB is identical with BA, or AB = BA. Similarly, ABC = ACB = -BCA = &c. - -Boole first drew attention in recent years to this property of logical -terms, and he called it the property of Commutativeness.[50] He not -only stated the law with the utmost clearness, but pointed out that -it is a Law of Thought rather than a Law of Things. I shall have in -various parts of this work to show how the necessary imperfection of -our symbols expressed in this law clings to our modes of expression, -and introduces complication into the whole body of mathematical -formulæ, which are really founded on a logical basis. - - [50] *Laws of Thought*, p. 29. It is pointed out in the preface to - this Second Edition that Leibnitz was acquainted with the Laws of - Simplicity and of Commutativeness. - -It is of course apparent that the power of commutation belongs only -to terms related in the simple logical mode of synthesis. No one can -confuse “a house of bricks” with “bricks of a house,” “twelve square -feet” with “twelve feet square,” “the water of crystallization” with -“the crystallization of water.” All relations which involve differences -of time and space are inconvertible; the higher must not be made to -change places with the lower, nor the first with the last. For the -parties concerned there is all the difference in the world between A -killing B and B killing A. The law of commutativeness simply asserts -that difference of order does not attach to the connection between the -properties and circumstances of a thing--to what I call *simple logical -relation*. - - - - -CHAPTER III. - -PROPOSITIONS. - - -We now proceed to consider the variety of forms of propositions in -which the truths of science must be expressed. I shall endeavour to -show that, however diverse these forms may be, they all admit the -application of the one same principle of inference that what is true of -a thing is true of the like or same. This principle holds true whatever -be the kind or manner of the likeness, provided proper regard be had to -its nature. Propositions may assert an identity of time, space, manner, -quantity, degree, or any other circumstance in which things may agree -or differ. - -We find an instance of a proposition concerning time in the -following:--“The year in which Newton was born, was the year in which -Galileo died.” This proposition expresses an approximate identity of -time between two events; hence whatever is true of the year in which -Galileo died is true of that in which Newton was born, and *vice -versâ*. “Tower Hill is the place where Raleigh was executed” expresses -an identity of place; and whatever is true of the one spot is true -of the spot otherwise defined, but in reality the same. In ordinary -language we have many propositions obscurely expressing identities -of number, quantity, or degree. “So many men, so many minds,” is a -proposition concerning number, that is to say, an equation; whatever -is true of the number of men is true of the number of minds, and -*vice versâ*. “The density of Mars is (nearly) the same as that of -the Earth,” “The force of gravity is directly as the product of the -masses, and inversely as the square of the distance,” are propositions -concerning magnitude or degree. Logicians have not paid adequate -attention to the great variety of propositions which can be stated by -the use of the little conjunction *as*, together with *so*. “As the -home so the people,” is a proposition expressing identity of manner; -and a great number of similar propositions all indicating some kind of -resemblance might be quoted. Whatever be the special kind of identity, -all such expressions are subject to the great principle of inference; -but as we shall in later parts of this work treat more particularly of -inference in cases of number and magnitude, we will here confine our -attention to logical propositions which involve only notions of quality. - - -*Simple Identities.* - -The most important class of propositions consists of those which fall -under the formula - - A = B, - -and may be called *simple identities*. I may instance, in the first -place, those most elementary propositions which express the exact -similarity of a quality encountered in two or more objects. I may -compare the colour of the Pacific Ocean with that of the Atlantic, and -declare them identical. I may assert that “the smell of a rotten egg -is like that of hydrogen sulphide;” “the taste of silver hyposulphite -is like that of cane sugar;” “the sound of an earthquake resembles -that of distant artillery.” Such are propositions stating, accurately -or otherwise, the identity of simple physical sensations. Judgments of -this kind are necessarily pre-supposed in more complex judgments. If -I declare that “this coin is made of gold,” I must base the judgment -upon the exact likeness of the substance in several qualities to -other pieces of substance which are undoubtedly gold. I must make -judgments of the colour, the specific gravity, the hardness, and of -other mechanical and chemical properties; each of these judgments is -expressed in an elementary proposition, “the colour of this coin is the -colour of gold,” and so on. Even when we establish the identity of a -thing with itself under a different name or aspect, it is by distinct -judgments concerning single circumstances. To prove that the Homeric -χαλκός is copper we must show the identity of each quality recorded of -χαλκός with a quality of copper. To establish Deal as the landing-place -of Cæsar all material circumstances must be shown to agree. If the -modern Wroxeter is the ancient Uriconium, there must be the like -agreement of all features of the country not subject to alteration by -time. - -Such identities must be expressed in the form A = B. We may say - - Colour of Pacific Ocean = Colour of Atlantic Ocean. - Smell of rotten egg = Smell of hydrogen sulphide. - -In these and similar propositions we assert identity of single -qualities or causes of sensation. In the same form we may also express -identity of any group of qualities, as in - - χαλκός = Copper. - Deal = Landing-place of Cæsar. - -A multitude of propositions involving singular terms fall into the same -form, as in - - The Pole star = The slowest-moving star. - Jupiter = The greatest of the planets. - The ringed planet = The planet having seven satellites. - The Queen of England = The Empress of India. - The number two = The even prime number. - Honesty = The best policy. - -In mathematical and scientific theories we often meet with simple -identities capable of expression in the same form. Thus in mechanical -science “The process for finding the resultant of forces = the process -for finding the resultant of simultaneous velocities.” Theorems in -geometry often give results in this form, as - - Equilateral triangles = Equiangular triangles. - Circle = Finite plane curve of constant curvature. - Circle = Curve of least perimeter. - -The more profound and important laws of nature are often expressible in -the form of simple identities; in addition to some instances which have -already been given, I may suggest, - - Crystals of cubical system = Crystals not possessing the power of - double refraction. - -All definitions are necessarily of this form, whether the objects -defined be many, few, or singular. Thus we may say, - - Common salt = Sodium chloride. - Chlorophyl = Green colouring matter of leaves. - Square = Equal-sided rectangle. - -It is an extraordinary fact that propositions of this elementary form, -all-important and very numerous as they are, had no recognised place -in Aristotle’s system of Logic. Accordingly their importance was -overlooked until very recent times, and logic was the most deformed -of sciences. But it is impossible that Aristotle or any other person -should avoid constantly using them; not a term could be defined -without their use. In one place at least Aristotle actually notices a -proposition of the kind. He observes: “We sometimes say that that white -thing is Socrates, or that the object approaching is Callias.”[51] Here -we certainly have simple identity of terms; but he considered such -propositions purely accidental, and came to the unfortunate conclusion, -that “Singulars cannot be predicated of other terms.” - - [51] *Prior Analytics*, i. cap. xxvii. 3. - -Propositions may also express the identity of extensive groups of -objects taken collectively or in one connected whole; as when we say, - - The Queen, Lords, and Commons = The Legislature of the United Kingdom. - -When Blackstone asserts that “The only true and natural foundation of -society are the wants and fears of individuals,” we must interpret him -as meaning that the whole of the wants and fears of individuals in the -aggregate form the foundation of society. But many propositions which -might seem to be collective are but groups of singular propositions or -identities. When we say “Potassium and sodium are the metallic bases of -potash and soda,” we obviously mean, - - Potassium = Metallic base of potash; - Sodium = Metallic base of soda. - -It is the work of grammatical analysis to separate the various -propositions often combined into a single sentence. Logic cannot be -properly required to interpret the forms and devices of language, but -only to treat the meaning when clearly exhibited. - - -*Partial Identities.* - -A second highly important kind of proposition is that which I propose -to call *a partial identity*. When we say that “All mammalia are -vertebrata,” we do not mean that mammalian animals are identical with -vertebrate animals, but only that the mammalia form a *part of the -class vertebrata*. Such a proposition was regarded in the old logic as -asserting the inclusion of one class in another, or of an object in a -class. It was called a universal affirmative proposition, because the -attribute *vertebrate* was affirmed of the whole subject *mammalia*; -but the attribute was said to be *undistributed*, because not all -vertebrata were of necessity involved in the proposition. Aristotle, -overlooking the importance of simple identities, and indeed almost -denying their existence, unfortunately founded his system upon the -notion of inclusion in a class, instead of adopting the basis of -identity. He regarded inference as resting upon the rule that what is -true of the containing class is true of the contained, in place of the -vastly more general rule that what is true of a class or thing is true -of the like. Thus he not only reduced logic to a fragment of its proper -self, but destroyed the deep analogies which bind together logical and -mathematical reasoning. Hence a crowd of defects, difficulties and -errors which will long disfigure the first and simplest of the sciences. - -It is surely evident that the relation of inclusion rests upon the -relation of identity. Mammalian animals cannot be included among -vertebrates unless they be identical with part of the vertebrates. -Cabinet Ministers are included almost always in the class Members of -Parliament, because they are identical with some who sit in Parliament. -We may indicate this identity with a part of the larger class in -various ways; as for instance, - - Mammalia = part of the vertebrata. - Diatomaceæ = a class of plants. - Cabinet Ministers = some members of Parliament. - Iron = a metal. - -In ordinary language the verbs *is* and *are* express mere inclusion -more often than not. *Men are mortals*, means that *men* form a part -of the class *mortal*; but great confusion exists between this sense -of the verb and that in which it expresses identity, as in “The sun is -the centre of the planetary system.” The introduction of the indefinite -article *a* often expresses partiality; when we say “Iron is a metal” -we clearly mean that iron is *one only* of several metals. - -Certain recent logicians have proposed to avoid the indefiniteness -in question by what is called the Quantification of the Predicate, -and they have generally used the little word *some* to show that only -a part of the predicate is identical with the subject. *Some* is an -*indeterminate adjective*; it implies unknown qualities by which we -might select the part in question if the qualities were known, but -it gives no hint as to their nature. I might make use of such an -indeterminate sign to express partial identities in this work. Thus, -taking the special symbol V = Some, the general form of a partial -identity would be A = VB, and in Boole’s Logic expressions of the -kind were much used. But I believe that indeterminate symbols only -introduce complexity, and destroy the beauty and simple universality -of the system which may be created without their use. A vague word -like *some* is only used in ordinary language by *ellipsis*, and to -avoid the trouble of attaining accuracy. We can always employ more -definite expressions if we like; but when once the indefinite *some* is -introduced we cannot replace it by the special description. We do not -know whether *some* colour is red, yellow, blue, or what it is; but on -the other hand *red* colour is certainly *some* colour. - -Throughout this system of logic I shall dispense with such indefinite -expressions; and this can readily be done by substituting one of the -other terms. To express the proposition “All A’s are some B’s” I shall -not use the form A = VB, but - - A = AB. - -This formula states that the class A is identical with the class AB; -and as the latter must be a part at least of the class B, it implies -the inclusion of the class A in that of B. We might represent our -former example thus, - - Mammalia = Mammalian vertebrata. - -This proposition asserts identity between a part (or it may be the -whole) of the vertebrata and the mammalia. If it is asked What part? -the proposition affords no answer, except that it is the part which is -mammalian; but the assertion “mammalia = some vertebrata” tells us no -more. - -It is quite likely that some readers will think this mode of -representing the universal affirmative proposition artificial and -complicated. I will not undertake to convince them of the opposite -at this point of my exposition. Justification for it will be found, -not so much in the immediate treatment of this proposition, as in the -general harmony which it will enable us to disclose between all parts -of reasoning. I have no doubt that this is the critical difficulty in -the relation of logical to other forms of reasoning. Grant this mode of -denoting that “all A’s are B’s,” and I fear no further difficulties; -refuse it, and we find want of analogy and endless anomaly in every -direction. It is on general grounds that I hope to show overwhelming -reasons for seeking to reduce every kind of proposition to the form of -an identity. - -I may add that not a few logicians have accepted this view of the -universal affirmative proposition. Leibnitz, in his *Difficultates -Quædam Logicæ*, adopts it, saying, “Omne A est B; id est æquivalent AB -et A, seu A non B est nonens.” Boole employed the logical equation *x* -= *xy* concurrently with *x* = *vy*; and Spalding[52] distinctly -says that the proposition “all metals are minerals” might be described -as an assertion of *partial identity* between the two classes. Hence -the name which I have adopted for the proposition. - - [52] *Encyclopædia Britannica*, Eighth Ed. art. Logic, sect. 37, - note. 8vo. reprint, p. 79. - - -*Limited Identities.* - -An important class of propositions have the form - - AB = AC, - -expressing the identity of the class AB with the class AC. In other -words, “Within the sphere of the class A, all the B’s are all the -C’s;” or again, “The B’s and C’s, which are A’s, are identical.” But -it will be observed that nothing is asserted concerning things which -are outside of the class A; and thus the identity is of limited extent. -It is the proposition B = C limited to the sphere of things called A. -Thus we may say, with some approximation to truth, that “Large plants -are plants devoid of locomotive power.” - -A barrister may make numbers of most general statements concerning -the relations of persons and things in the course of an argument, but -it is of course to be understood that he speaks only of persons and -things under the English Law. Even mathematicians make statements which -are not true with absolute generality. They say that imaginary roots -enter into equations by pairs; but this is only true under the tacit -condition that the equations in question shall not have imaginary -coefficients.[53] The universe, in short, within which they habitually -discourse is that of equations with real coefficients. These implied -limitations form part of that great mass of tacit knowledge which -accompanies all special arguments. - - [53] De Morgan, *On the Root of any Function*. Cambridge - Philosophical Transactions, 1867, vol. xi. p. 25. - -To De Morgan is due the remark, that we do usually think and argue in -a limited universe or sphere of notions, even when it is not expressly -stated.[54] - - [54] *Syllabus of a proposed System of Logic*, §§ 122, 123. - -It is worthy of inquiry whether all identities are not really limited -to an implied sphere of meaning. When we make such a plain statement as -“Gold is malleable” we obviously speak of gold only in its solid state; -when we say that “Mercury is a liquid metal” we must be understood to -exclude the frozen condition to which it may be reduced in the Arctic -regions. Even when we take such a fundamental law of nature as “All -substances gravitate,” we must mean by substance, material substance, -not including that basis of heat, light, and electrical undulations -which occupies space and possesses many wonderful mechanical -properties, but not gravity. The proposition then is really of the form - - Material substance = Material gravitating substance. - - -*Negative Propositions.* - -In every act of intellect we are engaged with a certain identity or -difference between things or sensations compared together. Hitherto -I have treated only of identities; and yet it might seem that the -relation of difference must be infinitely more common than that of -likeness. One thing may resemble a great many other things, but then it -differs from all remaining things in the world. Diversity may almost be -said to constitute life, being to thought what motion is to a river. -The perception of an object involves its discrimination from all other -objects. But we may nevertheless be said to detect resemblance as often -as we detect difference. We cannot, in fact, assert the existence of -a difference, without at the same time implying the existence of an -agreement. - -If I compare mercury, for instance, with other metals, and decide that -it is *not solid*, here is a difference between mercury and solid -things, expressed in a negative proposition; but there must be implied, -at the same time, an agreement between mercury and the other substances -which are not solid. As it is impossible to separate the vowels of the -alphabet from the consonants without at the same time separating the -consonants from the vowels, so I cannot select as the object of thought -*solid things*, without thereby throwing together into another class -all things which are *not solid*. The very fact of not possessing a -quality, constitutes a new quality which may be the ground of judgment -and classification. In this point of view, agreement and difference are -ever the two sides of the same act of intellect, and it becomes equally -possible to express the same judgment in the one or other aspect. - -Between affirmation and negation there is accordingly a perfect -equilibrium. Every affirmative proposition implies a negative one, and -*vice versâ*. It is even a matter of indifference, in a logical point -of view, whether a positive or negative term be used to denote a given -quality and the class of things possessing it. If the ordinary state -of a man’s body be called *good health*, then in other circumstances -he is said *not to be in good health*; but we might equally describe -him in the latter state as *sickly*, and in his normal condition he -would be *not sickly*. Animal and vegetable substances are now called -*organic*, so that the other substances, forming an immensely greater -part of the globe, are described negatively as *inorganic*. But we -might, with at least equal logical correctness, have described the -preponderating class of substances as *mineral*, and then vegetable and -animal substances would have been *non-mineral*. - -It is plain that any positive term and its corresponding negative -divide between them the whole universe of thought: whatever does not -fall into one must fall into the other, by the third fundamental Law -of Thought, the Law of Duality. It follows at once that there are -two modes of representing a difference. Supposing that the things -represented by A and B are found to differ, we may indicate (see p. 17) -the result of the judgment by the notation - - A ~ B. - -We may now represent the same judgment by the assertion that A agrees -with those things which differ from B, or that A agrees with the -not-B’s. Using our notation for negative terms (see p. 14), we obtain - - A = A*b* - -as the expression of the ordinary negative proposition. Thus if we -take A to mean quicksilver, and B solid, then we have the following -proposition:-- - - Quicksilver = Quicksilver not-solid. - -There may also be several other classes of negative propositions, of -which no notice was taken in the old logic. We may have cases where -all A’s are not-B’s, and at the same time all not-B’s are A’s; there -may, in short, be a simple identity between A and not-B, which may be -expressed in the form - - A = *b*. - -An example of this form would be - - Conductors of electricity = non-electrics. - -We shall also frequently have to deal as results of deduction, with -simple, partial, or limited identities between negative terms, as in -the forms - - *a* = *b*, *a* = *a**b*, *a*C = *b*C, etc. - -It would be possible to represent affirmative propositions in the -negative form. Thus “Iron is solid,” might be expressed as “Iron is not -not-solid,” or “Iron is not fluid;” or, taking A and *b* for the terms -“iron,” and “not-solid,” the form would be A ~ *b*. - -But there are very strong reasons why we should employ all propositions -in their affirmative form. All inference proceeds by the substitution -of equivalents, and a proposition expressed in the form of an identity -is ready to yield all its consequences in the most direct manner. As -will be more fully shown, we can infer *in* a negative proposition, -but not *by* it. Difference is incapable of becoming the ground of -inference; it is only the implied agreement with other differing -objects which admits of deductive reasoning; and it will always be -found advantageous to employ propositions in the form which exhibits -clearly the implied agreements. - - -*Conversion of Propositions.* - -The old books of logic contain many rules concerning the conversion of -propositions, that is, the transposition of the subject and predicate -in such a way as to obtain a new proposition which will be true when -the original proposition is true. The reduction of every proposition to -the form of an identity renders all such rules and processes needless. -Identity is essentially reciprocal. If the colour of the Atlantic Ocean -is the same as that of the Pacific Ocean, that of the Pacific must -be the same as that of the Atlantic. Sodium chloride being identical -with common salt, common salt must be identical with sodium chloride. -If the number of windows in Salisbury Cathedral equals the number of -days in the year, the number of days in the year must equal the number -of the windows. Lord Chesterfield was not wrong when he said, “I will -give anybody their choice of these two truths, which amount to the -same thing; He who loves himself best is the honestest man; or, The -honestest man loves himself best.” Scotus Erigena exactly expresses -this reciprocal character of identity in saying, “There are not two -studies, one of philosophy and the other of religion; true philosophy -is true religion, and true religion is true philosophy.” - -A mathematician would not think it worth while to mention that if -*x* = *y* then also *y* = *x*. He would not consider these to be -two equations at all, but one equation accidentally written in two -different manners. In written symbols one of two names must come first, -and the other second, and a like succession must perhaps be observed -in our thoughts: but in the relation of identity there is no need for -succession in order (see p. 33), each is simultaneously equal and -identical to the other. These remarks will hold true both of logical -and mathematical identity; so that I shall consider the two forms - - A = B and B = A - -to express exactly the same identity differently written. All need for -rules of conversion disappears, and there will be no single proposition -in the system which may not be written with either end foremost. Thus A -= AB is the same as AB = A, *a*C = *b*C is the same as *b*C = *a*C, and -so forth. - -The same remarks are partially true of differences and inequalities, -which are also reciprocal to the extent that one thing cannot differ -from a second without the second differing from the first. Mars differs -in colour from Venus, and Venus must differ from Mars. The Earth -differs from Jupiter in density; therefore Jupiter must differ from the -Earth. Speaking generally, if A ~ B we shall also have B ~ A, and these -two forms may be considered expressions of the same difference. But -the relation of differing things is not wholly reciprocal. The density -of Jupiter does not differ from that of the Earth in the same way that -that of the Earth differs from that of Jupiter. The change of sensation -which we experience in passing from Venus to Mars is not the same as -what we experience in passing back to Venus, but just the opposite -in nature. The colour of the sky is lighter than that of the ocean; -therefore that of the ocean cannot be lighter than that of the sky, but -darker. In these and all similar cases we gain a notion of *direction* -or character of change, and results of immense importance may be shown -to rest on this notion. For the present we shall be concerned with the -mere fact of identity existing or not existing. - - -*Twofold Interpretation of Propositions.* - -Terms, as we have seen (p. 25), may have a meaning either in extension -or intension; and according as one or the other meaning is attributed -to the terms of a proposition, so may a different interpretation be -assigned to the proposition itself. When the terms are abstract we -must read them in intension, and a proposition connecting such terms -must denote the identity or non-identity of the qualities respectively -denoted by the terms. Thus if we say - - Equality = Identity of magnitude, - -the assertion means that the circumstance of being equal exactly -corresponds with the circumstance of being identical in magnitude. -Similarly in - - Opacity = Incapability of transmitting light, - -the quality of being incapable of transmitting light is declared to be -the same as the intended meaning of the word opacity. - -When general names form the terms of a proposition we may apply a -double interpretation. Thus - - Exogens = Dicotyledons - -means either that the qualities which belong to all exogens are the -same as those which belong to all dicotyledons, or else that every -individual falling under one name falls equally under the other. -Hence it may be said that there are two distinct fields of logical -thought. We may argue either by the qualitative meaning of names or -by the quantitative, that is, the extensive meaning. Every argument -involving concrete plural terms might be converted into one involving -only abstract singular terms, and *vice versâ*. But there are reasons -for believing that the intensive or qualitative form of reasoning is -the primary and fundamental one. It is sufficient to point out that the -extensive meaning of a name is a changeable and fleeting thing, while -the intensive meaning may nevertheless remain fixed. Very numerous -additions have been lately made to the extensive meanings both of -planet and element. Every iron steam-ship which is made or destroyed -adds to or subtracts from the extensive meaning of the name steam-ship, -without necessarily affecting the intensive meaning. Stage coach means -as much as ever in one way, but in extension the class is nearly -extinct. Chinese railway, on the other hand, is a term represented only -by a single instance; in twenty years it may be the name of a large -class. - - - - -CHAPTER IV. - -DEDUCTIVE REASONING. - - -The general principle of inference having been explained in the -previous chapters, and a suitable system of symbols provided, we have -now before us the comparatively easy task of tracing out the most -common and important forms of deductive reasoning. The general problem -of deduction is as follows:--*From one or more propositions called -premises to draw such other propositions as will necessarily be true -when the premises are true.* By deduction we investigate and unfold the -information contained in the premises; and this we can do by one single -rule--*For any term occurring in any proposition substitute the term -which is asserted in any premise to be identical with it.* To obtain -certain deductions, especially those involving negative conclusions, we -shall require to bring into use the second and third Laws of Thought, -and the process of reasoning will then be called *Indirect Deduction*. -In the present chapter, however, I shall confine my attention to -those results which can be obtained by the process of *Direct -Deduction*, that is, by applying to the premises themselves the rule of -substitution. It will be found that we can combine into one harmonious -system, not only the various moods of the ancient syllogism but a great -number of equally important forms of reasoning, which had no recognised -place in the old logic. We can at the same time dispense entirely with -the elaborate apparatus of logical rules and mnemonic lines, which were -requisite so long as the vital principle of reasoning was not clearly -expressed. - - -*Immediate Inference.* - -Probably the simplest of all forms of inference is that which has been -called *Immediate Inference*, because it can be performed upon a single -proposition. It consists in joining an adjective, or other qualifying -clause of the same nature, to both sides of an identity, and asserting -the equivalence of the terms thus produced. For instance, since - - Conductors of electricity = Non-electrics, - -it follows that - - Liquid conductors of electricity = Liquid non-electrics. - -If we suppose that - - Plants = Bodies decomposing carbonic acid, - -it follows that - - Microscopic plants = Microscopic bodies decomposing - carbonic acid. - -In general terms, from the identity - - A = B - -we can infer the identity - - AC = BC. - -This is but a case of plain substitution; for by the first Law of -Thought it must be admitted that - - AC = AC, - -and if, in the second side of this identity, we substitute for A its -equivalent B, we obtain - - AC = BC. - -In like manner from the partial identity - - A = AB - -we may obtain - - AC = ABC - -by an exactly similar act of substitution; and in every other case -the rule will be found capable of verification by the principle of -inference. The process when performed as here described will be quite -free from the liability to error which I have shown[55] to exist in -“Immediate Inference by added Determinants,” as described by Dr. -Thomson.[56] - - [55] *Elementary Lessons in Logic*, p. 86. - - [56] *Outline of the Laws of Thought*, § 87. - - -*Inference with Two Simple Identities.* - -One of the most common forms of inference, and one to which I shall -especially direct attention, is practised with two simple identities. -From the two statements that “London is the capital of England” and -“London is the most populous city in the world,” we instantaneously -draw the conclusion that “The capital of England is the most populous -city in the world.” Similarly, from the identities - - Hydrogen = Substance of least density, - Hydrogen = Substance of least atomic weight, - -we infer - - Substance of least density = Substance of least atomic weight. - -The general form of the argument is exhibited in the symbols - - B = A (1) - B = C (2) - hence A = C. (3) - -We may describe the result by saying that terms identical with the -same term are identical with each other; and it is impossible to -overlook the analogy to the first axiom of Euclid that “things equal -to the same thing are equal to each other.” It has been very commonly -supposed that this is a fundamental principle of thought, incapable of -reduction to anything simpler. But I entertain no doubt that this form -of reasoning is only one case of the general rule of inference. We have -two propositions, A = B and B = C, and we may for a moment consider -the second one as affirming a truth concerning B, while the former one -informs us that B is identical with A; hence by substitution we may -affirm the same truth of A. It happens in this particular case that the -truth affirmed is identity to C, and we might, if we preferred it, have -considered the substitution as made by means of the second identity in -the first. Having two identities we have a choice of the mode in which -we will make the substitution, though the result is exactly the same in -either case. - -Now compare the three following formulæ, - - (1) A = B = C, hence A = C - (2) A = B ~ C, hence A ~ C - (3) A ~ B ~ C, no inference. - -In the second formula we have an identity and a difference, and we are -able to infer a difference; in the third we have two differences and -are unable to make any inference at all. Because A and C both differ -from B, we cannot tell whether they will or will not differ from each -other. The flowers and leaves of a plant may both differ in colour from -the earth in which the plant grows, and yet they may differ from each -other; in other cases the leaves and stem may both differ from the -soil and yet agree with each other. Where we have difference only we -can make no inference; where we have identity we can infer. This fact -gives great countenance to my assertion that inference proceeds always -through identity, but may be equally well effected in propositions -asserting difference or identity. - -Deferring a more complete discussion of this point, I will only mention -now that arguments from double identity occur very frequently, and are -usually taken for granted, owing to their extreme simplicity. In regard -to the equivalence of words this form of inference must be constantly -employed. If the ancient Greek χαλκός is our *copper*, then it must -be the French *cuivre*, the German *kupfer*, the Latin *cuprum*, -because these are words, in one sense at least, equivalent to copper. -Whenever we can give two definitions or expressions for the same term, -the formula applies; thus Senior defined wealth as “All those things, -and those things only, which are transferable, are limited in supply, -and are directly or indirectly productive of pleasure or preventive -of pain.” Wealth is also equivalent to “things which have value in -exchange;” hence obviously, “things which have value in exchange = all -those things, and those things only, which are transferable, &c.” Two -expressions for the same term are often given in the same sentence, -and their equivalence implied. Thus Thomson and Tait say,[57] “The -naturalist may be content to know matter as that which can be perceived -by the senses, or as that which can be acted upon by or can exert -force.” I take this to mean-- - - Matter = what can be perceived by the senses; - Matter = what can be acted upon by or can exert force. - - [57] *Treatise on Natural Philosophy*, vol. i. p. 161. - -For the term “matter” in either of these identities we may substitute -its equivalent given in the other definition. Elsewhere they often -employ sentences of the form exemplified in the following:[58] “The -integral curvature, or whole change of direction of an arc of a plane -curve, is the angle through which the tangent has turned as we pass -from one extremity to the other.” This sentence is certainly of the -form-- - - The integral curvature = the whole change of direction, &c. = the - angle through which the tangent has turned, &c. - - [58] *Treatise on Natural Philosophy*, vol. i. p. 6. - -Disguised cases of the same kind of inference occur throughout all -sciences, and a remarkable instance is found in algebraic geometry. -Mathematicians readily show that every equation of the form *y* = *mx* -+ *c* corresponds to or represents a straight line; it is also easily -proved that the same equation is equivalent to one of the general form -A*x* + B*y* + C = 0, and *vice versâ*. Hence it follows that every -equation of the form in question, that is to say, every equation of the -first degree, corresponds to or represents a straight line.[59] - - [59] Todhunter’s *Plane Co-ordinate Geometry*, chap. ii. pp. 11–14. - - -*Inference with a Simple and a Partial Identity.* - -A form of reasoning somewhat different from that last considered -consists in inference-between a simple and a partial identity. If we -have two propositions of the forms - - A = B, - B = BC, - -we may then substitute for B in either proposition its equivalent in -the other, getting in both cases A = BC; in this we may if we like make -a second substitution for B, getting - - A = AC. - -Thus, since “The Mont Blanc is the highest mountain in Europe, and -the Mont Blanc is deeply covered with snow,” we infer by an obvious -substitution that “The highest mountain in Europe is deeply covered -with snow.” These propositions when rigorously stated fall into the -forms above exhibited. - -This mode of inference is constantly employed when for a term we -substitute its definition, or *vice versâ*. The very purpose of a -definition is to allow a single noun to be employed in place of a long -descriptive phrase. Thus, when we say “A circle is a curve of the -second degree,” we may substitute a definition of the circle, getting -“A curve, all points of which are at equal distances from one point, is -a curve of the second degree.” The real forms of the propositions here -given are exactly those shown in the symbolic statement, but in this -and many other cases it will be sufficient to state them in ordinary -elliptical language for sake of brevity. In scientific treatises a -term and its definition are often both given in the same sentence, -as in “The weight of a body in any given locality, or the force -with which the earth attracts it, is proportional to its mass.” The -conjunction *or* in this statement gives the force of equivalence to -the parenthetic phrase, so that the propositions really are - - Weight of a body = force with which the earth attracts it. - Weight of a body = weight, &c. proportional to its mass. - -A slightly different case of inference consists in substituting in a -proposition of the form A = AB, a definition of the term B. Thus from A -= AB and B = C we get A = AC. For instance, we may say that “Metals are -elements” and “Elements are incapable of decomposition.” - - Metal = metal element. - Element = what is incapable of decomposition. - -Hence - - Metal = metal incapable of decomposition. - -It is almost needless to point out that the form of these arguments -does not suffer any real modification if some of the terms happen to be -negative; indeed in the last example “incapable of decomposition” may -be treated as a negative term. Taking - - A = metal - B = element - C = capable of decomposition - *c* = incapable of decomposition; - -the propositions are of the forms - - A = AB - B = *c* - -whence, by substitution, - - A = A*c*. - - -*Inference of a Partial from Two Partial Identities.* - -However common be the cases of inference already noticed, there is -a form occurring almost more frequently, and which deserves much -attention, because it occupied a prominent place in the ancient -syllogistic system. That system strangely overlooked all the kinds of -argument we have as yet considered, and selected, as the type of all -reasoning, one which employs two partial identities as premises. Thus -from the propositions - - Sodium is a metal (1) - Metals conduct electricity, (2) - -we may conclude that - - Sodium conducts electricity. (3) - -Taking A, B, C to represent the three terms respectively, the premises -are of the forms - - A = AB (1) - B = BC. (2) - -Now for B in (1) we can substitute its expression as given in (2), -obtaining - - A = ABC, (3) - -or, in words, from - - Sodium = sodium metal, (1) - Metal = metal conducting electricity, (2) - -we infer - - Sodium = sodium metal conducting electricity, (3) - -which, in the elliptical language of common life, becomes - - “Sodium conducts electricity.” - -The above is a syllogism in the mood called Barbara[60] in the truly -barbarous language of ancient logicians; and the first figure of the -syllogism contained Barbara and three other moods which were esteemed -distinct forms of argument. But it is worthy of notice that, without -any real change in our form of inference, we readily include these -three other moods under Barbara. The negative mood Celarent will be -represented by the example - - [60] An explanation of this and other technical terms of the old - logic will be found in my *Elementary Lessons in Logic*, Sixth - Edition, 1876; Macmillan. - - Neptune is a planet, (1) - No planet has retrograde motion; (2) - Hence Neptune has not retrograde motion. (3) - -If we put A for Neptune, B for planet, and C for “having retrograde -motion,” then by the corresponding negative term c, we denote “not -having retrograde motion.” The premises now fall into the forms - - A = AB (1) - B = B*c*, (2) - -and by substitution for B, exactly as before, we obtain - - A = AB*c*. (3) - -What is called in the old logic a particular conclusion may be deduced -without any real variation in the symbols. Particular quantity is -indicated, as before mentioned (p. 41), by joining to the term an -indefinite adjective of quantity, such as *some*, *a part of*, -*certain*, &c., meaning that an unknown part of the term enters into -the proposition as subject. Considerable doubt and ambiguity arise out -of the question whether the part may not in some cases be the whole, -and in the syllogism at least it must be understood in this sense.[61] -Now, if we take a letter to represent this indefinite part, we need -make no change in our formulæ to express the syllogisms Darii and -Ferio. Consider the example-- - - [61] *Elementary Lessons in Logic*, pp. 67, 79. - - Some metals are of less density than water, (1) - - All bodies of less density than water will float - upon the surface of water; hence (2) - - Some metals will float upon the surface of - water. (3) - -Let - - A = some metals, - B = body of less density than water, - C = floating on the surface of water - -then the propositions are evidently as before, - - A = AB, (1) - B = BC; (2) - hence A = ABC, (3) - -Thus the syllogism Darii does not really differ from Barbara. If the -reader prefer it, we can readily employ a distinct symbol for the -indefinite sign of quantity. - - Let P = some, - Q = metal, - -B and C having the same meanings as before. Then the premises become - - PQ = PQB, (1) - B = BC; (2) - -hence, by substitution, as before, - - PQ = PQBC. (3) - -Except that the formulæ look a little more complicated there is no -difference whatever. - -The mood Ferio is of exactly the same character as Darii or Barbara, -except that it involves the use of a negative term. Take the example, - - Bodies which are equally elastic in all directions do not doubly - refract light; - - Some crystals are bodies equally elastic in all directions; - therefore, some crystals do not doubly refract light. - -Assigning the letters as follows:-- - - A = some crystals, - B = bodies equally elastic in all directions, - C = doubly refracting light, - *c* = not doubly refracting light. - -Our argument is of the same form as before, and may be concisely stated -in one line, - - A = AB = AB*c*. - -If it is preferred to put PQ for the indefinite *some crystals*, we have - - PQ = PQB = PQB*c*. - -The only difference is that the negative term c takes the place of C in -the mood Darii. - - -*Ellipsis of Terms in Partial Identities.* - -The reader will probably have noticed that the conclusion which we -obtain from premises is often more full than that drawn by the old -Aristotelian processes. Thus from “Sodium is a metal,” and “Metals -conduct electricity,” we inferred (p. 55) that “Sodium = sodium, metal, -conducting electricity,” whereas the old logic simply concludes that -“Sodium conducts electricity.” Symbolically, from A = AB, and B = BC, -we get A = ABC, whereas the old logic gets at the most A = AC. It is -therefore well to show that without employing any other principles of -inference than those already described, we may infer A = AC from A = -ABC, though we cannot infer the latter more full and accurate result -from the former. We may show this most simply as follows:-- - -By the first Law of Thought it is evident that - - AA = AA; - -and if we have given the proposition A = ABC, we may substitute for -both the A’s in the second side of the above, obtaining - - AA = ABC . ABC. - -But from the property of logical symbols expressed in the Law of -Simplicity (p. 33) some of the repeated letters may be made to -coalesce, and we have - - A = ABC . C. - -Substituting again for ABC its equivalent A, we obtain - - A = AC, - -the desired result. - -By a similar process of reasoning it may be shown that we can always -drop out any term appearing in one member of a proposition, provided -that we substitute for it the whole of the other member. This -process was described in my first logical Essay,[62] as *Intrinsic -Elimination*, but it might perhaps be better entitled the *Ellipsis -of Terms*. It enables us to get rid of needless terms by strict -substitutive reasoning. - - [62] *Pure Logic*, p. 19. - - -*Inference of a Simple from Two Partial Identities.* - -Two terms may be connected together by two partial identities in yet -another manner, and a case of inference then arises which is of the -highest importance. In the two premises - - A = AB (1) - B = AB (2) - -the second member of each is the same; so that we can by obvious -substitution obtain - - A = B. - -Thus, in plain geometry we readily prove that “Every equilateral -triangle is also an equiangular triangle,” and we can with equal ease -prove that “Every equiangular triangle is an equilateral triangle.” -Thence by substitution, as explained above, we pass to the simple -identity, - - Equilateral triangle = equiangular triangle. - -We thus prove that one class of triangles is entirely identical with -another class; that is to say, they differ only in our way of naming -and regarding them. - -The great importance of this process of inference arises from the -fact that the conclusion is more simple and general than either of -the premises, and contains as much information as both of them put -together. It is on this account constantly employed in inductive -investigation, as will afterwards be more fully explained, and it is -the natural mode by which we arrive at a conviction of the truth of -simple identities as existing between classes of numerous objects. - - -*Inference of a Limited from Two Partial Identities.* - -We have considered some arguments which are of the type treated by -Aristotle in the first figure of the syllogism. But there exist two -other types of argument which employ a pair of partial identities. If -our premises are as shown in these symbols, - - B = AB (1) - B = CB, (2) - -we may substitute for B either by (1) in (2) or by (2) in (1), and by -both modes we obtain the conclusion - - AB = CB, (3) - -a proposition of the kind which we have called a limited identity -(p. 42). Thus, for example, - - Potassium = potassium metal (1) - Potassium = potassium capable of floating on water; (2) - -hence - - Potassium metal = potassium capable of floating on water. (3) - -This is really a syllogism of the mood Darapti in the third figure, -except that we obtain a conclusion of a more exact character than -the old syllogism gives. From the premises “Potassium is a metal” -and “Potassium floats on water,” Aristotle would have inferred that -“Some metals float on water.” But if inquiry were made what the “some -metals” are, the answer would certainly be “Metal which is potassium.” -Hence Aristotle’s conclusion simply leaves out some of the information -afforded in the premises. It even leaves us open to interpret the -*some metals* in a wider sense than we are warranted in doing. From -these distinct defects of the old syllogism the process of substitution -is free, and the new process only incurs the possible objection of -being tediously minute and accurate. - - -*Miscellaneous Forms of Deductive Inference.* - -The more common forms of deductive reasoning having been exhibited -and demonstrated on the principle of substitution, there still remain -many, in fact an indefinite number, which may be explained with nearly -equal ease. Such as involve the use of disjunctive propositions will -be described in a later chapter, and several of the syllogistic moods -which include negative terms will be more conveniently treated after -we have introduced the symbolic use of the second and third laws of -thought. - -We sometimes meet with a chain of propositions which allow of repeated -substitution, and form an argument called in the old logic a Sorites. -Take, for instance, the premises - - Iron is a metal, (1) - Metals are good conductors of electricity, (2) - Good conductors of electricity are useful for - telegraphic purposes. (3) - -It obviously follows that - - Iron is useful for telegraphic purposes. (4) - -Now if we take our letters thus, - - A = Iron, B = metal, C = good conductor of electricity, D = useful - for telegraphic purposes, - -the premises will assume the forms - - A = AB, (1) - B = BC, (2) - C = CD. (3) - -For B in (1) we can substitute its equivalent in (2) obtaining, as -before, - - A = ABC. - -Substituting for C in this intermediate result its equivalent as given -in (3), we obtain the complete conclusion - - A = ABCD. (4) - -The full interpretation is that *Iron is iron, metal, good conductor of -electricity, useful for telegraphic purposes*, which is abridged in -common language by the ellipsis of the circumstances which are not of -immediate importance. - -Instead of all the propositions being exactly of the same kind as -in the last example, we may have a series of premises of various -character; for instance, - -Common salt is sodium chloride, (1) - -Sodium chloride crystallizes in a cubical form, (2) - -What crystallizes in a cubical form does not possess the power of -double refraction; (3) - -it will follow that - -Common salt does not possess the power of double refraction. (4) - -Taking our letter-terms thus, - - A = Common salt, - B = Sodium chloride, - C = Crystallizing in a cubical form, - D = Possessing the power of double refraction, - -we may state the premises in the forms - - A = B, (1) - B = BC, (2) - C = C*d*. (3) - -Substituting by (3) in (2) and then by (2) as thus altered in (1) we -obtain - - A = BC*d*, (4) - -which is a more precise version of the common conclusion. - -We often meet with a series of propositions describing the qualities or -circumstances of the one same thing, and we may combine them all into -one proposition by the process of substitution. This case is, in fact, -that which Dr. Thomson has called “Immediate Inference by the sum of -several predicates,” and his example will serve my purpose well.[63] -He describes copper as “A metal--of a red colour--and disagreeable -smell--and taste--all the preparations of which are poisonous--which is -highly malleable--ductile--and tenacious--with a specific gravity of -about 8.83.” If we assign the letter A to copper, and the succeeding -letters of the alphabet in succession to the series of predicates, we -have nine distinct statements, of the form A = AB (1) A = AC (2) A = AD -(3) ... A = AK (9). We can readily combine these propositions into one -by substituting for A in the second side of (1) its expression in (2). -We thus get - - [63] *An Outline of the Necessary Laws of Thought*, Fifth Ed. p. 161. - - A = ABC, - -and by repeating the process over and over again we obviously get the -single proposition - - A = ABCD ... JK. - -But Dr. Thomson is mistaken in supposing that we can obtain in -this manner a *definition* of copper. Strictly speaking, the above -proposition is only a *description* of copper, and all the ordinary descriptions -of substances in scientific works may be summed up in this form. Thus -we may assert of the organic substances called Paraffins that they are -all saturated hydrocarbons, incapable of uniting with other substances, -produced by heating the alcoholic iodides with zinc, and so on. It may -be shown that no amount of ordinary description can be equivalent to a -definition of any substance. - - -*Fallacies.* - -I have hitherto been engaged in showing that all the forms of -reasoning of the old syllogistic logic, and an indefinite number of -other forms in addition, may be readily and clearly explained on the -single principle of substitution. It is now desirable to show that the -same principle will prevent us falling into fallacies. So long as we -exactly observe the one rule of substitution of equivalents it will -be impossible to commit a *paralogism*, that is to break any one of -the elaborate rules of the ancient system. The one new rule is thus -proved to be as powerful as the six, eight, or more rules by which the -correctness of syllogistic reasoning was guarded. - -It was a fundamental rule, for instance, that two negative premises -could give no conclusion. If we take the propositions - - Granite is not a sedimentary rock, (1) - Basalt is not a sedimentary rock, (2) - -we ought not to be able to draw any inference concerning the relation -between granite and basalt. Taking our letter-terms thus: - - A = granite, B = sedimentary rock, C = basalt, - -the premises may be expressed in the forms - - A ~ B, (1) - C ~ B. (2) - -We have in this form two statements of difference; but the principle -of inference can only work with a statement of agreement or identity -(p. 63). Thus our rule gives us no power whatever of drawing any -inference; this is exactly in accordance with the fifth rule of the -syllogism. - -It is to be remembered, indeed, that we claim the power of always -turning a negative proposition into an affirmative one (p. 45); and it -might seem that the old rule against negative premises would thus be -circumvented. Let us try. The premises (1) and (2) when affirmatively -stated take the forms - - A = A*b* (1) - C = C*b*. (2) - -The reader will find it impossible by the rule of substitution to -discover a relation between A and C. Three terms occur in the above -premises, namely A, *b*, and C; but they are so combined that no term -occurring in one has its exact equivalent stated in the other. No -substitution can therefore be made, and the principle of the fifth rule -of the syllogism holds true. Fallacy is impossible. - -It would be a mistake, however, to suppose that the mere occurrence of -negative terms in both premises of a syllogism renders them incapable -of yielding a conclusion. The old rule informed us that from two -negative premises no conclusion could be drawn, but it is a fact that -the rule in this bare form does not hold universally true; and I am not -aware that any precise explanation has been given of the conditions -under which it is or is not imperative. Consider the following example: - - Whatever is not metallic is not capable of powerful - magnetic influence, (1) - Carbon is not metallic, (2) - Therefore, carbon is not capable of powerful magnetic - influence. (3) - -Here we have two distinctly negative premises (1) and (2), and yet they -yield a perfectly valid negative conclusion (3). The syllogistic rule -is actually falsified in its bare and general statement. In this and -many other cases we can convert the propositions into affirmative ones -which will yield a conclusion by substitution without any difficulty. -To show this let - - A = carbon, B = metallic, - C = capable of powerful magnetic influence. - -The premises readily take the forms - - *b* = *bc*, (1) - A = A*b*, (2) - -and substitution for *b* in (2) by means of (1) gives the conclusion - - A = A*bc*. (3) - -Our principle of inference then includes the rule of negative premises -whenever it is true, and discriminates correctly between the cases -where it does and does not hold true. - -The paralogism, anciently called *the Fallacy of Undistributed Middle*, -is also easily exhibited and infallibly avoided by our system. Let the -premises be - - Hydrogen is an element, (1) - All metals are elements. (2) - -According to the syllogistic rules the middle term “element” is here -undistributed, and no conclusion can be obtained; we cannot tell then -whether hydrogen is or is not a metal. Represent the terms as follows - - A = hydrogen, - B = element, - C = metal. - -The premises then become - - A = AB, (1) - C = CB. (2) - -The reader will here, as in a former page (p. 62), find it impossible -to make any substitution. The only term which occurs in both premises -is B, but it is differently combined in the two premises. For B we -must not substitute A, which is equivalent to AB, not to B. Nor must -we confuse together CB and AB, which, though they contain one common -letter, are different aggregate terms. The rule of substitution gives -us no right to decompose combinations; and if we adhere rigidly to the -rule, that if two terms are stated to be equivalent we may substitute -one for the other, we cannot commit the fallacy. It is apparent that -the form of premises stated above is the same as that which we obtained -by translating two negative premises into the affirmative form. - -The old fallacy, technically called the *Illicit Process of the Major -Term*, is more easy to commit and more difficult to detect than any -other breach of the syllogistic rules. In our system it could hardly -occur. From the premises - - All planets are subject to gravity, (1) - Fixed stars are not planets, (2) - -we might inadvertently but fallaciously infer that, “Fixed stars are -not subject to gravity.” To reduce the premises to symbolic form, let - - A = planet - B = fixed star - C = subject to gravity; - -then we have the propositions - - A = AC (1) - B = B*a*. (2) - -The reader will try in vain to produce from these premises by -legitimate substitution any relation between B and C; he could not then -commit the fallacy of asserting that B is not C. - -There remain two other kinds of paralogism, commonly known as the -fallacy of Four Terms and the Illicit Process of the Minor Term. They -are so evidently impossible while we obey the rule of the substitution -of equivalents, that it is not necessary to give any illustrations. -When there are four distinct terms in two propositions as in A = B -and C = D, there could evidently be no opening for substitution. As -to the Illicit Process of the Minor Term it consists in a flagrant -substitution for a term of another wider term which is not known to be -equivalent to it, and which is therefore not allowed by our rule to be -substituted for it. - - - - -CHAPTER V. - -DISJUNCTIVE PROPOSITIONS. - - -In the previous chapter I have exhibited various cases of deductive -reasoning by the process of substitution, avoiding the introduction of -disjunctive propositions; but we cannot long defer the consideration of -this more complex class of identities. General terms arise, as we have -seen (p. 24), from classifying or mentally uniting together all objects -which agree in certain qualities, the value of this union consisting in -the fact that the power of knowledge is multiplied thereby. In forming -such classes or general notions, we overlook or abstract the points of -difference which exist between the objects joined together, and fix our -attention only on the points of agreement. But every process of thought -may be said to have its inverse process, which consists in undoing the -effects of the direct process. Just as division undoes multiplication, -and evolution undoes involution, so we must have a process which undoes -generalization, or the operation of forming general notions. This -inverse process will consist in distinguishing the separate objects or -minor classes which are the constituent parts of any wider class. If -we mentally unite together certain objects visible in the sky and call -them planets, we shall afterwards need to distinguish the contents of -this general notion, which we do in the disjunctive proposition-- - - A planet is either Mercury or Venus or the Earth or ... or Neptune. - -Having formed the very wide class “vertebrate animal,” we may specify -its subordinate classes thus:--“A vertebrate animal is either a -mammal, bird, reptile, or fish.” Nor is there any limit to the -number of possible alternatives. “An exogenous plant is either a -ranunculus, a poppy, a crucifer, a rose, or it belongs to some one -of the other seventy natural orders of exogens at present recognized -by botanists.” A cathedral church in England must be either that of -London, Canterbury, Winchester, Salisbury, Manchester, or of one of -about twenty-four cities possessing such churches. And if we were to -attempt to specify the meaning of the term “star,” we should require -to enumerate as alternatives, not only the many thousands of stars -recorded in catalogues, but the many millions unnamed. - -Whenever we thus distinguish the parts of a general notion we employ -a disjunctive proposition, in at least one side of which are several -alternatives joined by the so-called disjunctive conjunction or, -a contracted form of *other*. There must be some relation between -the parts thus connected in one proposition; we may call it the -*disjunctive* or *alternative* relation, and we must carefully inquire -into its nature. This relation is that of ignorance and doubt, giving -rise to choice. Whenever we classify and abstract we must open the way -to such uncertainty. By fixing our attention on certain attributes to -the exclusion of others, we necessarily leave it doubtful what those -other attributes are. The term “molar tooth” bears upon the face of -it that it is a part of the wider term “tooth.” But if we meet with -the simple term “tooth” there is nothing to indicate whether it is -an incisor, a canine, or a molar tooth. This doubt, however, may be -resolved by further information, and we have to consider what are the -appropriate logical processes for treating disjunctive propositions in -connection with other propositions disjunctive or otherwise. - - -*Expression of the Alternative Relation.* - -In order to represent disjunctive propositions with convenience we -require a sign of the alternative relation, equivalent to one meaning -at least of the little conjunction *or* so frequently used in common -language. I propose to use for this purpose the symbol ꖌ. In my first -logical essay I followed the practice of Boole and adopted the sign -+; but this sign should not be employed unless there exists exact -analogy between mathematical addition and logical alternation. We shall -find that the analogy is imperfect, and that there is such profound -difference between logical and mathematical terms as should prevent -our uniting them by the same symbol. Accordingly I have chosen a sign -ꖌ, which seems aptly to suggest whatever degree of analogy may exist -without implying more. The exact meaning of the symbol we will now -proceed to investigate. - - -*Nature of the Alternative Relation.* - -Before treating disjunctive propositions it is indispensable to decide -whether the alternatives must be considered exclusive or unexclusive. -By *exclusive alternatives* we mean those which cannot contain the same -things. If we say “Arches are circular or pointed,” it is certainly to -be understood that the same arch cannot be described as both circular -and pointed. Many examples, on the other hand, can readily be suggested -in which two or more alternatives may hold true of the same object. Thus - - Luminous bodies are self-luminous or luminous by reflection. - -It is undoubtedly possible, by the laws of optics, that the same -surface may at one and the same moment give off light of its own and -reflect light from other bodies. We speak familiarly of *deaf or dumb* -persons, knowing that the majority of those who are deaf from birth are -also dumb. - -There can be no doubt that in a great many cases, perhaps the greater -number of cases, alternatives are exclusive as a matter of fact. Any -one number is incompatible with any other; one point of time or place -is exclusive of all others. Roger Bacon died either in 1284 or 1292; it -is certain that he could not die in both years. Henry Fielding was born -either in Dublin or Somersetshire; he could not be born in both places. -There is so much more precision and clearness in the use of exclusive -alternatives that we ought doubtless to select them when possible. Old -works on logic accordingly contained a rule directing that the *Membra -dividentia*, the parts of a division or the constituent species of a -genus, should be exclusive of each other. - -It is no doubt owing to the great prevalence and convenience of -exclusive divisions that the majority of logicians have held it -necessary to make every alternative in a disjunctive proposition -exclusive of every other one. Aquinas considered that when this was not -the case the proposition was actually *false*, and Kant adopted the -same opinion.[64] A multitude of statements to the same effect might -readily be quoted, and if the question were to be determined by the -weight of historical evidence, it would certainly go against my view. -Among recent logicians Hamilton, as well as Boole, took the exclusive -side. But there are authorities to the opposite effect. Whately, -Mansel, and J. S. Mill have all pointed out that we may often treat -alternatives as *Compossible*, or true at the same time. Whately gives -us an example,[65] “Virtue tends to procure us either the esteem of -mankind, or the favour of God,” and he adds--“Here both members are -true, and consequently from one being affirmed we are not authorized to -deny the other. Of course we are left to conjecture in each case, from -the context, whether it is meant to be implied that the members are -or are not exclusive.” Mansel says,[66] “*We may happen to know* that -two alternatives cannot be true together, so that the affirmation of -the second necessitates the denial of the first; but this, as Boethius -observes, is a *material*, not a *formal* consequence.” Mill has also -pointed out the absurdities which would arise from always interpreting -alternatives as exclusive. “If we assert,” he says,[67] “that a man -who has acted in some particular way must be either a knave or a fool, -we by no means assert, or intend to assert, that he cannot be both.” -Again, “to make an entirely unselfish use of despotic power, a man must -be either a saint or a philosopher.... Does the disjunctive premise -necessarily imply, or must it be construed as supposing, that the same -person cannot be both a saint and a philosopher? Such a construction -would be ridiculous.” - - [64] Mansel’s *Aldrich*, p. 103, and *Prolegomena Logica*, p. 221. - - [65] *Elements of Logic*, Book II. chap. iv. sect. 4. - - [66] Aldrich, *Artis Logicæ Rudimenta*, p. 104. - - [67] *Examination of Sir W. Hamilton’s Philosophy*, pp. 452–454. - -I discuss this subject fully because it is really the point which -separates my logical system from that of Boole. In his *Laws of -Thought* (p. 32) he expressly says, “In strictness, the words ‘and,’ -‘or,’ interposed between the terms descriptive of two or more classes -of objects, imply that those classes are quite distinct, so that no -member of one is found in another.” This I altogether dispute. In the -ordinary use of these conjunctions we do not join distinct terms only; -and when terms so joined do prove to be logically distinct, it is by -virtue of a *tacit premise*, something in the meaning of the names and -our knowledge of them, which teaches us that they are distinct. If our -knowledge of the meanings of the words joined is defective it will -often be impossible to decide whether terms joined by conjunctions are -exclusive or not. - -In the sentence “Repentance is not a single act, but a habit or -virtue,” it cannot be implied that a virtue is not a habit; by -Aristotle’s definition it is. Milton has the expression in one of -his sonnets, “Unstain’d by gold or fee,” where it is obvious that if -the fee is not always gold, the gold is meant to be a fee or bribe. -Tennyson has the expression “wreath or anadem.” Most readers would -be quite uncertain whether a wreath may be an anadem, or an anadem a -wreath, or whether they are quite distinct or quite the same. From -Darwin’s *Origin of Species*, I take the expression, “When we see any -*part or organ* developed in a remarkable *degree or manner*.” In this, -*or* is used twice, and neither time exclusively. For if *part* and -*organ* are not synonymous, at any rate an organ is a part. And it -is obvious that a part may be developed at the same time both in an -extraordinary degree and an extraordinary manner, although such cases -may be comparatively rare. - -From a careful examination of ordinary writings, it will thus be found -that the meanings of terms joined by “and,” “or” vary from absolute -identity up to absolute contrariety. There is no logical condition of -distinctness at all, and when we do choose exclusive alternatives, it -is because our subject demands it. The matter, not the form of an -expression, points out whether terms are exclusive or not.[68] In -bills, policies, and other kinds of legal documents, it is sometimes -necessary to express very distinctly that alternatives are not -exclusive. The form and/or is then used, and, as Mr. J. J. Murphy has -remarked, this form coincides exactly in meaning with the symbol ꖌ. - - [68] *Pure Logic*, pp 76, 77. - -In the first edition of this work (vol. i., p. 81), I took the -disjunctive proposition “Matter is solid, or liquid, or gaseous,” and -treated it as an instance of exclusive alternatives, remarking that the -same portion of matter cannot be at once solid and liquid, properly -speaking, and that still less can we suppose it to be solid and -gaseous, or solid, liquid, and gaseous all at the same time. But the -experiments of Professor Andrews show that, under certain conditions -of temperature and pressure, there is no abrupt change from the liquid -to the gaseous state. The same substance may be in such a state as to -be indifferently described as liquid and gaseous. In many cases, too, -the transition from solid to liquid is gradual, so that the properties -of solidity are at least partially joined with those of liquidity. -The proposition then, instead of being an instance of exclusive -alternatives, seems to afford an excellent instance to the opposite -effect. When such doubts can arise, it is evidently impossible to treat -alternatives as absolutely exclusive by the logical nature of the -relation. It becomes purely a question of the matter of the proposition. - -The question, as we shall afterwards see more fully, is one of -the greatest theoretical importance, because it concerns the true -distinction between the sciences of Logic and Mathematics. It is the -foundation of number that every unit shall be distinct from every other -unit; but Boole imported the conditions of number into the science of -Logic, and produced a system which, though wonderful in its results, -was not a system of logic at all. - - -*Laws of the Disjunctive Relation.* - -In considering the combination or synthesis of terms (p. 30), we found -that certain laws, those of Simplicity and Commutativeness, must be -observed. In uniting terms by the disjunctive symbol we shall find that -the same or closely similar laws hold true. The alternatives of either -member of a disjunctive proposition are certainly commutative. Just as -we cannot properly distinguish between *rich and rare gems* and *rare -and rich gems*, so we must consider as identical the expression *rich -or rare gems*, and *rare or rich gems*. In our symbolic language we may -say - - A ꖌ B = B ꖌ A. - -The order of statement, in short, has no effect upon the meaning of an -aggregate of alternatives, so that the Law of Commutativeness holds -true of the disjunctive symbol. - -As we have admitted the possibility of joining as alternatives terms -which are not really different, the question arises, How shall we treat -two or more alternatives when they are clearly shown to be the same? -If we have it asserted that P is Q or R, and it is afterwards proved -that Q is but another name for R, the result is that P is either R or -R. How shall we interpret such a statement? What would be the meaning, -for instance, of “wreath or anadem” if, on referring to a dictionary, -we found *anadem* described as a wreath? I take it to be self-evident -that the meaning would then become simply “wreath.” Accordingly we may -affirm the general law - - A ꖌ A = A. - -Any number of identical alternatives may always be reduced to, and are -logically equivalent to, any one of those alternatives. This is a law -which distinguishes mathematical terms from logical terms, because -it obviously does not apply to the former. I propose to call it the -*Law of Unity*, because it must really be involved in any definition -of a mathematical unit. This law is closely analogous to the Law of -Simplicity, AA = A; and the nature of the connection is worthy of -attention. - -Few or no logicians except De Morgan have adequately noticed the close -relation between combined and disjunctive terms, namely, that every -disjunctive term is the negative of a corresponding combined term, and -*vice versâ*. Consider the term - - Malleable dense metal. - -How shall we describe the class of things which are not -malleable-dense-metals? Whatever is included under that term must have -all the qualities of malleability, denseness, and metallicity. Wherever -any one or more of the qualities is wanting, the combined term will not -apply. Hence the negative of the whole term is - - Not-malleable or not-dense or not-metallic. - -In the above the conjunction *or* must clearly be interpreted -as unexclusive; for there may readily be objects which are both -not-malleable, and not-dense, and perhaps not-metallic at the same -time. If in fact we were required to use *or* in a strictly exclusive -manner, it would be requisite to specify seven distinct alternatives -in order to describe the negative of a combination of three terms. The -negatives of four or five terms would consist of fifteen or thirty-one -alternatives. This consideration alone is sufficient to prove that the -meaning of *or* cannot be always exclusive in common language. - -Expressed symbolically, we may say that the negative of - - ABC - is not-A or not-B or not-C; - that is, *a* ꖌ *b* ꖌ *c*. - -Reciprocally the negative of - - P ꖌ Q ꖌ R - is *pqr*. - -Every disjunctive term, then, is the negative of a combined term, and -*vice versâ*. - -Apply this result to the combined term AAA, and its negative is - - *a* ꖌ *a* ꖌ *a*. - -Since AAA is by the Law of Simplicity equivalent to A, so *a* ꖌ *a* ꖌ -*a* must be equivalent to *a*, and the Law of Unity holds true. Each -law thus necessarily presupposes the other. - - -*Symbolic expression of the Law of Duality.* - -We may now employ our symbol of alternation to express in a clear and -formal manner the third Fundamental Law of Thought, which I have called -the Law of Duality (p. 6). Taking A to represent any class or object -or quality, and B any other class, object or quality, we may always -assert that A either agrees with B, or does not agree. Thus we may say - - A = AB ꖌ A*b*. - -This is a formula which will henceforth be constantly employed, and it -lies at the basis of reasoning. - -The reader may perhaps wish to know why A is inserted in both -alternatives of the second member of the identity, and why the law is -not stated in the form - - A = B ꖌ *b*. - -But if he will consider the contents of the last section (p. 73), he -will see that the latter expression cannot be correct, otherwise no -term could have a corresponding negative term. For the negative of B -ꖌ *b* is *b*B, or a self-contradictory term; thus if A were identical -with B ꖌ *b*, its negative *a* would be non-existent. To say the least, -this result would in most cases be an absurd one, and I see much reason -to think that in a strictly logical point of view it would always be -absurd. In all probability we ought to assume as a fundamental logical -axiom that *every term has its negative in thought*. We cannot think at -all without separating what we think about from other things, and these -things necessarily form the negative notion.[69] It follows that any -proposition of the form A = B ꖌ *b* is just as self-contradictory as -one of the form A = B*b*. - - [69] *Pure Logic*, p. 65. See also the criticism of this point by De - Morgan in the *Athenæum*, No. 1892, 30th January, 1864; p. 155. - -It is convenient to recapitulate in this place the three Laws of -Thought in their symbolic form, thus - - Law of Identity A = A. - Law of Contradiction A*a* = 0. - Law of Duality A = AB ꖌ A*b*. - - -*Various Forms of the Disjunctive Proposition.* - -Disjunctive propositions may occur in a great variety of forms, of -which the old logicians took insufficient notice. There may be any -number of alternatives, each of which may be a combination of any -number of simple terms. A proposition, again, may be disjunctive in one -or both members. The proposition - - Solids or liquids or gases are electrics or conductors of electricity - -is an example of the doubly disjunctive form. The meaning of such a -proposition is that whatever falls under any one or more alternatives -on one side must fall under one or more alternatives on the other side. -From what has been said before, it is apparent that the proposition - - A ꖌ B = C ꖌ D - -will correspond to - - *ab* = *cd*, - -each member of the latter being the negative of a member of the former -proposition. - -As an instance of a complex disjunctive proposition I may give Senior’s -definition of wealth, which, briefly stated, amounts to the proposition -“Wealth is what is transferable, limited in supply, and either -productive of pleasure or preventive of pain.”[70] - - [70] Boole’s *Laws of Thought*, p. 106. Jevons’ *Pure Logic*, p. 69. - - Let A = wealth - B = transferable - C = limited in supply - D = productive of pleasure - E = preventive of pain. - -The definition takes the form - - A = BC(D ꖌ E); - -but if we develop the alternatives by a method to be afterwards more -fully considered, it becomes - - A = BCDE ꖌ BCD*e* ꖌ BC*d*E. - -An example of a still more complex proposition is found in De Morgan’s -writings,[71] as follows:--“He must have been rich, and if not -absolutely mad was weakness itself, subjected either to bad advice or -to most unfavourable circumstances.” - - [71] *On the Syllogism*, No. iii. p. 12. Camb. Phil. Trans. vol. x, - part i. - -If we assign the letters of the alphabet in succession, thus, - - A = he - B = rich - C = absolutely mad - D = weakness itself - E = subjected to bad advice - F = subjected to most unfavourable circumstances, - the proposition will take the form - - A = AB{C ꖌ D (E ꖌ F)}, - -and if we develop the alternatives, expressing some of the different -cases which may happen, we obtain - - A = ABC ꖌ AB*c*DEF ꖌ AB*c*DE*f* ꖌ AB*c*D*e*F. - -The above gives the strict logical interpretation of the sentence, and -the first alternative ABC is capable of development into eight cases, -according as D, E and F are or are not present. Although from our -knowledge of the matter, we may infer that weakness of character cannot -be asserted of a person absolutely mad, there is no explicit statement -to this effect. - - -*Inference by Disjunctive Propositions.* - -Before we can make a free use of disjunctive propositions in the -processes of inference we must consider how disjunctive terms can be -combined together or with simple terms. In the first place, to combine -a simple term with a disjunctive one, we must combine it with every -alternative of the disjunctive term. A vegetable, for instance, is -either a herb, a shrub, or a tree. Hence an exogenous vegetable is -either an exogenous herb, or an exogenous shrub, or an exogenous tree. -Symbolically stated, this process of combination is as follows, - - A(B ꖌ C) = AB ꖌ AC. - -Secondly, to combine two disjunctive terms with each other, combine -each alternative of one with each alternative of the other. Since -flowering plants are either exogens or endogens, and are at the -same time either herbs, shrubs or trees, it follows that there are -altogether six alternatives--namely, exogenous herbs, exogenous shrubs, -exogenous trees, endogenous herbs, endogenous shrubs, endogenous trees. -This process of combination is shown in the general form - - (A ꖌ B) (C ꖌ D ꖌ E) = AC ꖌ AD ꖌ AE ꖌ BC ꖌ BD ꖌ BE. - -It is hardly necessary to point out that, however numerous the -terms combined, or the alternatives in those terms, we may effect -the combination, provided each alternative is combined with each -alternative of the other terms, as in the algebraic process of -multiplication. - -Some processes of deduction may be at once exhibited. We may -always, for instance, unite the same qualifying term to each side -of an identity even though one or both members of the identity be -disjunctive. Thus let - - A = B ꖌ C. - -Now it is self-evident that - - AD = AD, - -and in one side of this identity we may for A substitute its equivalent -B ꖌ C, obtaining - - AD = BD ꖌ CD. - -Since “a gaseous element is either hydrogen, or oxygen, or nitrogen, -or chlorine, or fluorine,” it follows that “a free gaseous element -is either free hydrogen, or free oxygen, or free nitrogen, or free -chlorine, or free fluorine.” - -This process of combination will lead to most useful inferences when -the qualifying adjective combined with both sides of the proposition is -a negative of one or more alternatives. Since chlorine is a coloured -gas, we may infer that “a colourless gaseous element is either -(colourless) hydrogen, oxygen, nitrogen, or fluorine.” The alternative -chlorine disappears because colourless chlorine does not exist. Again, -since “a tooth is either an incisor, canine, bicuspid, or molar,” -it follows that “a not-incisor tooth is either canine, bicuspid, -or molar.” The general rule is that from the denial of any of the -alternatives the affirmation of the remainder can be inferred. Now this -result clearly follows from our process of substitution; for if we have -the proposition - - A = B ꖌ C ꖌ D, - -and we insert this expression for A on one side of the self-evident -identity - - A*b* = A*b*, - -we obtain A*b* = AB*b* ꖌ A*b*C ꖌ A*b*D; - -and, as the first of the three alternatives is self-contradictory, we -strike it out according to the law of contradiction: there remains - - A*b* = A*b*C ꖌ A*b*D. - -Thus our system fully includes and explains that mood of the -Disjunctive Syllogism technically called the *modus tollendo ponens*. - -But the reader must carefully observe that the Disjunctive Syllogism of -the mood *ponendo tollens*, which affirms one alternative, and thence -infers the denial of the rest, cannot be held true in this system. If I -say, indeed, that - - Water is either salt or fresh water, - -it seems evident that “water which is salt is not fresh.” But this -inference really proceeds from our knowledge that water cannot be at -once salt and fresh. This inconsistency of the alternatives, as I have -fully shown, will not always hold. Thus, if I say - - Gems are either rare stones or beautiful stones, (1) - -it will obviously not follow that - - A rare gem is not a beautiful stone, (2) - -nor that - - A beautiful gem is not a rare stone. (3) - -Our symbolic method gives only true conclusions; for if we take - - A = gem - B = rare stone - C = beautiful stone, - -the proposition (1) is of the form - - A = B ꖌ C - hence AB = B ꖌ BC - and AC = BC ꖌ C; - -but these inferences are not equivalent to the false ones (2) and (3). - - -We can readily represent disjunctive reasoning by the *modus ponendo -tollens*, when it is valid, by expressing the inconsistency of the -alternatives explicitly. Thus if we resort to our instance of - - Water is either salt or fresh, - -and take - - A = Water B = salt C = fresh, - -then the premise is apparently of the form - - A = AB ꖌ AC; - -but in reality there is an unexpressed condition that “what is salt -is not fresh,” from which follows, by a process of inference to be -afterwards described, that “what is fresh is not salt.” We have then, -in letter-terms, the two propositions - - B = B*c* - C = *b*C. - -If we substitute these descriptions in the original proposition, we -obtain /* A = AB*c* ꖌ A*b*C; */ - -uniting B to each side we infer - - AB = AB*c* ꖌ AB*b*C - or AB = AB*c*; - -that is, - - Water which is salt is water salt and not fresh. - -I should weary the reader if I attempted to illustrate the multitude of -forms which disjunctive reasoning may take; and as in the next chapter -we shall be constantly treating the subject, I must here restrict -myself to a single instance. A very common process of reasoning -consists in the determination of the name of a thing by the successive -exclusion of alternatives, a process called by the old name *abscissio -infiniti*. Take the case: - - Red-coloured metal is either copper or gold (1) - Copper is dissolved by nitric acid (2) - This specimen is red-coloured metal (3) - This specimen is not dissolved by nitric acid (4) - Therefore, this specimen consists of gold (5) - -Let us assign the letter-symbols thus-- - - A = this specimen - B = red-coloured metal - C = copper - D = gold - E = dissolved by nitric acid. - -Assuming that the alternatives copper or gold are intended to be -exclusive, as just explained in the case of fresh and salt water, the -premises may be stated in the forms - - B = BC*d* ꖌ B*c*D (1) - C = CE (2) - A = AB (3) - A = A*e* (4) - -Substituting for C in (1) by means of (2) we get - - B = BC*d*E ꖌ B*c*D - -From (3) and (4) we may infer likewise - - A = AB*e* - -and if in this we substitute for B its equivalent just stated, it -follows that - - A = ABC*d*E*e* ꖌ AB*c*D*e* - -The first of the alternatives being contradictory the result is - - A = AB*c*D*e* - -which contains a full description of “this specimen,” as furnished -in the premises, but by ellipsis asserts that it is gold. It will be -observed that in the symbolic expression (1) I have explicitly stated -what is certainly implied, that copper is not gold, and gold not -copper, without which condition the inference would not hold good. - - - - -CHAPTER VI. - -THE INDIRECT METHOD OF INFERENCE. - - -The forms of deductive reasoning as yet considered, are mostly cases -of Direct Deduction as distinguished from those which we are now about -to treat. The method of Indirect Deduction may be described as that -which points out what a thing is, by showing that it cannot be anything -else. We can define a certain space upon a map, either by colouring -that space, or by colouring all except the space; the first mode is -positive, the second negative. The difference, it will be readily seen, -is exactly analogous to that between the direct and indirect modes -of proof in geometry. Euclid often shows that two lines are equal, -by showing that they cannot be unequal, and the proof rests upon the -known number of alternatives, greater, equal or less, which are alone -conceivable. In other cases, as for instance in the seventh proposition -of the first book, he shows that two lines must meet in a particular -point, by showing that they cannot meet elsewhere. - -In logic we can always define with certainty the utmost number of -alternatives which are conceivable. The Law of Duality (pp. 6, 74) -enables us always to assert that any quality or circumstance whatsoever -is either present or absent. Whatever may be the meaning of the terms A -and B it is certainly true that - - A = AB ꖌ A*b* - B = AB ꖌ *a*B. - -These are universal tacit premises which may be employed in the -solution of every problem, and which are such invariable and necessary -conditions of all thought, that they need not be specially laid down. -The Law of Contradiction is a further condition of all thought and of -all logical symbols; it enables, and in fact obliges, us to reject from -further consideration all terms which imply the presence and absence -of the same quality. Now, whenever we bring both these Laws of Thought -into explicit action by the method of substitution, we employ the -Indirect Method of Inference. It will be found that we can treat not -only those arguments already exhibited according to the direct method, -but we can include an infinite multitude of other arguments which are -incapable of solution by any other means. - -Some philosophers, especially those of France, have held that the -Indirect Method of Proof has a certain inferiority to the direct -method, which should prevent our using it except when obliged. But -there are many truths which we can prove only indirectly. We can -prove that a number is a prime only by the purely indirect method of -showing that it is not any of the numbers which have divisors, and the -remarkable process known as Eratosthenes’ Sieve is the only mode by -which we can select the prime numbers.[72] It bears a strong analogy to -the indirect method here to be described. We can prove that the side -and diameter of a square are incommensurable, but only in the negative -or indirect manner, by showing that the contrary supposition inevitably -leads to contradiction.[73] Many other demonstrations in various -branches of the mathematical sciences proceed upon a like method. -Now, if there is only one important truth which must be, and can only -be, proved indirectly, we may say that the process is a necessary and -sufficient one, and the question of its comparative excellence or -usefulness is not worth discussion. As a matter of fact I believe that -nearly half our logical conclusions rest upon its employment. - - [72] See Horsley, *Philosophical Transactions*, 1772; vol. lxii. - p. 327. Montucla, *Histoire des Mathematiques*, vol. i. p. 239. - *Penny Cyclopædia*, article “Eratosthenes.” - - [73] Euclid, Book x. Prop. 117. - - -*Simple Illustrations.* - -In tracing out the powers and results of this method, we will begin -with the simplest possible instance. Let us take a proposition of the -common form, A = AB, say, - - *A Metal is an Element,* - -and let us investigate its full meaning. Any person who has had the -least logical training, is aware that we can draw from the above -proposition an apparently different one, namely, - - *A Not-element is a Not-metal.* - -While some logicians, as for instance De Morgan,[74] have considered -the relation of these two propositions to be purely self-evident, and -neither needing nor allowing analysis, a great many more persons, as -I have observed while teaching logic, are at first unable to perceive -the close connection between them. I believe that a true and complete -system of logic will furnish a clear analysis of this process, which -has been called *Contrapositive Conversion*; the full process is as -follows:-- - - [74] *Philosophical Magazine*, December 1852; Fourth Series, vol. iv. - p. 435, “On Indirect Demonstration.” - -Firstly, by the Law of Duality we know that - - *Not-element is either Metal or Not-metal.* - -If it be metal, we know that it is by the premise *an element*; we -should thus be supposing that the same thing is an element and a -not-element, which is in opposition to the Law of Contradiction. -According to the only other alternative, then, the not-element must be -a not-metal. - -To represent this process of inference symbolically we take the premise -in the form - - A = AB. (1) - -We observe that by the Law of Duality the term not-B is thus described - - *b* = A*b* ꖌ *ab*. (2) - -For A in this proposition we substitute its description as given in -(1), obtaining - - *b* = AB*b* ꖌ *ab*. - -But according to the Law of Contradiction the term AB*b* must be -excluded from thought, or - - AB*b* = 0. - -Hence it results that *b* is either nothing at all, or it is *ab*; and -the conclusion is - - *b* = *ab*. - -As it will often be necessary to refer to a conclusion of this kind I -shall call it, as is usual, the *Contrapositive Proposition* of the -original. The reader need hardly be cautioned to observe that from all -A’s are B’s it does not follow that all not-A’s are not-B’s. For by the -Law of Duality we have - - *a* = *a*B ꖌ *ab*, - -and it will not be found possible to make any substitution in this by -our original premise A = AB. It still remains doubtful, therefore, -whether not-metal is element or not-element. - -The proof of the Contrapositive Proposition given above is exactly the -same as that which Euclid applies in the case of geometrical notions. -De Morgan describes Euclid’s process as follows[75]:--“From every not-B -is not-A he produces Every A is B, thus: If it be possible, let this A -be not-B, but every not-B is not-A, therefore this A is not-A, which -is absurd: whence every A is B.” Now De Morgan thinks that this proof -is entirely needless, because common logic gives the inference without -the use of any geometrical reasoning. I conceive however that logic -gives the inference only by an indirect process. De Morgan claims “to -see identity in Every A is B and every not-B is not-A, by a process -of thought prior to syllogism.” Whether prior to syllogism or not, I -claim that it is not prior to the laws of thought and the process of -substitutive inference, by which it may be undoubtedly demonstrated. - - [75] *Philosophical Magazine*, Dec. 1852; p. 437. - - -*Employment of the Contrapositive Proposition.* - -We can frequently employ the contrapositive form of a proposition by -the method of substitution; and certain moods of the ancient syllogism, -which we have hitherto passed over, may thus be satisfactorily -comprehended in our system. Take for instance the following syllogism -in the mood Camestres:-- - - “Whales are not true fish; for they do not respire water, - whereas true fish do respire water.” - -Let us take - - A = whale - B = true fish - C = respiring water - -The premises are of the forms - - A = A*c* (1) - B = BC (2) - -Now, by the process of contraposition we obtain from the second premise - - *c* = *bc* - -and we can substitute this expression for *c* in (1), obtaining - - A = A*bc* - -or “Whales are not true fish, not respiring water.” - -The mood Cesare does not really differ from Camestres except in the -order of the premises, and it could be exhibited in an exactly similar -manner. - -The mood Baroko gave much trouble to the old logicians, who could -not *reduce* it to the first figure in the same manner as the other -moods, and were obliged to invent, specially for it and for Bokardo, a -method of Indirect Reduction closely analogous to the indirect proof -of Euclid. Now these moods require no exceptional treatment in this -system. Let us take as an instance of Baroko, the argument - - All heated solids give continuous spectra (1) - Some nebulæ do not give continuous spectra (2) - Therefore, some nebulæ are not heated solids (3) - -Treating the little word some as an indeterminate adjective of -selection, to which we assign a symbol like any other adjective, let - - A = some - B = nebulæ - C = giving continuous spectra - D = heated solids - -The premises then become - - D = DC (1) - AB = AB*c* (2) - -Now from (1) we obtain by the indirect method the contrapositive -proposition - - *c* = *cd* - -and if we substitute this expression for *c* in (2) we have - - AB = AB*cd* - -the full meaning of which is that “some nebulæ do not give continuous -spectra and are not heated solids.” - -We might similarly apply the contrapositive in many other instances. -Take the argument, “All fixed stars are self-luminous; but some of the -heavenly bodies are not self-luminous, and are therefore not fixed -stars.” Taking our terms - - A = fixed stars - B = self-luminous - C = some - D = heavenly bodies - -we have the premises - - A = AB, (1) - CD = *b*CD (2) - -Now from (1) we can draw the contrapositive - - *b* = *ab* - -and substituting this expression for *b* in (2) we obtain - - CD = *ab*CD - -which expresses the conclusion of the argument that some heavenly -bodies are not fixed stars. - - -*Contrapositive of a Simple Identity.* - -The reader should carefully note that when we apply the process of -Indirect Inference to a simple identity of the form - - A = B - -we may obtain further results. If we wish to know what is the term -not-B, we have as before, by the Law of Duality, - - *b* = A*b* ꖌ *ab* - -and substituting for A we obtain - - *b* = B*b* ꖌ *ab* = *ab*. - -But we may now also draw a second contrapositive; for we have - - *a* = *a*B ꖌ *ab*, - -and substituting for B its equivalent A we have - - *a* = *a*A ꖌ *ab* = *ab*. - -Hence from the single identity A = B we can draw the two propositions - - *a* = *ab* - *b* = *ab*, - -and observing that these propositions have a common term *ab* we can -make a new substitution, getting - - *a* = *b*. - -This result is in strict accordance with the fundamental principles of -inference, and it may be a question whether it is not a self-evident -result, independent of the steps of deduction by which we have reached -it. For where two classes are coincident like A and B, whatever is true -of the one is true of the other; what is excluded from the one must be -excluded from the other similarly. Now as *a* bears to A exactly the -same relation that *b* bears to B, the identity of either pair follows -from the identity of the other pair. In every identity, equality, or -similarity, we may argue from the negative of the one side to the -negative of the other. Thus at ordinary temperatures - - Mercury = liquid-metal, - -hence obviously - - Not-mercury = not liquid-metal; - -or since - - Sirius = brightest fixed star, - -it follows that whatever star is not the brightest is not Sirius, and -*vice versâ*. Every correct definition is of the form A = B, and may -often require to be applied in the equivalent negative form. - -Let us take as an illustration of the mode of using this result the -argument following: - - Vowels are letters which can be sounded alone, (1) - The letter *w* cannot be sounded alone; (2) - Therefore the letter *w* is not a vowel. (3) - -Here we have a definition (1), and a comparison of a thing with that -definition (2), leading to exclusion of the thing from the class -defined. - -Taking the terms - - A = vowel, - B = letter which can be sounded alone, - C = letter *w*, - -the premises are plainly of the forms - - A = B, (1) - C = *b*C. (2) - -Now by the Indirect method we obtain from (1) the Contrapositive - - *b* = *a*, - -and inserting in (2) the equivalent for *b* we have - - C = *a*C, (3) - -or “the letter *w* is not a vowel.” - - -*Miscellaneous Examples of the Method.* - -We can apply the Indirect Method of Inference however many may be the -terms involved or the premises containing those terms. As the working -of the method is best learnt from examples, I will take a case of two -premises forming the syllogism Barbara: thus - - Iron is metal (1) - Metal is element. (2) - -If we want to ascertain what inference is possible concerning the -term *Iron*, we develop the term by the Law of Duality. Iron must be -either metal or not-metal; iron which is metal must be either element -or not-element; and similarly iron which is not-metal must be either -element or not-element. There are then altogether four alternatives -among which the description of iron must be contained; thus - - Iron, metal, element, (α) - Iron, metal, not-element, (β) - Iron, not-metal, element, (γ) - Iron, not-metal, not-element. (δ) - -Our first premise informs us that iron is a metal, and if we substitute -this description in (γ) and (δ) we shall have self-contradictory -combinations. Our second premise likewise informs us that metal -is element, and applying this description to (β) we again have -self-contradiction, so that there remains only (α) as a description of -iron--our inference is - - Iron = iron, metal, element. - -To represent this process of reasoning in general symbols, let - - A = iron - B = metal - C = element, - -The premises of the problem take the forms - - A = AB (1) - B = BC. (2) - -By the Law of Duality we have - - A = AB ꖌ A*b* (3) - A = AC ꖌ A*c*. (4) - -Now, if we insert for A in the second side of (3) its description in -(4), we obtain what I shall call the *development of A with respect to -B and C*, namely - - A = ABC ꖌ AB*c* ꖌ A*b*C ꖌ A*bc*. (5) - -Wherever the letters A or B appear in the second side of (5) substitute -their equivalents given in (1) and (2), and the results stated at full -length are - - A = ABC ꖌ ABC*c* ꖌ AB*b*C ꖌ AB*b*C*c*. - -The last three alternatives break the Law of Contradiction, so that - - A = ABC ꖌ 0 ꖌ 0 ꖌ 0 = ABC. - -This conclusion is, indeed, no more than we could obtain by the -direct process of substitution, that is by substituting for B in (1), -its description in (2) as in p. 55; it is the characteristic of the -Indirect process that it gives all possible logical conclusions, both -those which we have previously obtained, and an immense number of -others or which the ancient logic took little or no account. From the -same premises, for instance, we can obtain a description of the class -*not-element* or *c*. By the Law of Duality we can develop *c* into -four alternatives, thus - - *c* = AB*c* ꖌ A*bc* ꖌ *a*B*c* ꖌ *abc*. - -If we substitute for A and B as before, we get - - *c* = ABC*c* ꖌ AB*bc* ꖌ *a*BC*c* ꖌ *abc*, - -and, striking out the terms which break the Law of Contradiction, there -remains - - *c* = *abc*, - -or what is not element is also not iron and not metal. This Indirect -Method of Inference thus furnishes a complete solution of the following -problem--*Given any number of logical premises or conditions, required -the description of any class of objects, or of any term, as governed by -those conditions.* - -The steps of the process of inference may thus be concisely stated-- - -1. By the Law of Duality develop the utmost number of alternatives -which may exist in the description of the required class or term as -regards the terms involved in the premises. - -2. For each term in these alternatives substitute its description as -given in the premises. - -3. Strike out every alternative which is then found to break the Law of -Contradiction. - -4. The remaining terms may be equated to the term in question as the -desired description. - - -*Mr. Venn’s Problem.* - -The need of some logical method more powerful and comprehensive than -the old logic of Aristotle is strikingly illustrated by Mr. Venn -in his most interesting and able article on Boole’s logic.[76] An -easy example, originally got, as he says, by the aid of my method as -simply described in the *Elementary Lessons in Logic*, was proposed in -examination and lecture-rooms to some hundred and fifty students as a -problem in ordinary logic. It was answered by, at most, five or six -of them. It was afterwards set, as an example on Boole’s method, to -a small class who had attended a few lectures on the nature of these -symbolic methods. It was readily answered by half or more of their -number. - - [76] *Mind*; a Quarterly Review of Psychology and Philosophy; - October, 1876, vol. i. p. 487. - -The problem was as follows:--“The members of a board were all of them -either bondholders, or shareholders, but not both; and the bondholders -as it happened, were all on the board. What conclusion can be drawn?” -The conclusion wanted is, “No shareholders are bondholders.” Now, as -Mr. Venn says, nothing can look simpler than the following reasoning, -*when stated*:--“There can be no bondholders who are shareholders; for -if there were they must be either on the board, or off it. But they -are not on it, by the first of the given statements; nor off it, by -the second.” Yet from the want of any systematic mode of treating such -a question only five or six of some hundred and fifty students could -succeed in so simple a problem. - -By symbolic statement the problem is instantly solved. Taking - - A = member of board - B = bondholder - C = shareholder - -the premises are evidently - - A = AB*c* ꖌ A*b*C - B = AB. - -The class C or shareholders may in respect of A and B be developed into -four alternatives, - - C = ABC ꖌ A*b*C ꖌ *a*BC ꖌ *ab*C. - -But substituting for A in the first and for B in the third alternative -we get - - C = ABC*c* ꖌ AB*b*C ꖌ A*b*C ꖌ *a*ABC ꖌ *ab*C. - -The first, second, and fourth alternatives in the above are -self-contradictory combinations, and only these; striking them out -there remain - - C = A*b*C ꖌ *ab*C = *b*C, - -the required answer. This symbolic reasoning is, I believe, the exact -equivalent of Mr. Venn’s reasoning, and I do not believe that the -result can be attained in a simpler manner. Mr. Venn adds that he -could adduce other similar instances, that is, instances showing the -necessity of a better logical method. - - -*Abbreviation of the Process.* - -Before proceeding to further illustrations of the use of this method, -I must point out how much its practical employment can be simplified, -and how much more easy it is than would appear from the description. -When we want to effect at all a thorough solution of a logical problem -it is best to form, in the first place, a complete series of all the -combinations of terms involved in it. If there be two terms A and B, -the utmost variety of combinations in which they can appear are - - AB *a*B - A*b* *ab*. - -The term A appears in the first and second; B in the first and third; -*a* in the third and fourth; and *b* in the second and fourth. Now if -we have any premise, say - - A = B, - -we must ascertain which of these combinations will be rendered -self-contradictory by substitution; the second and third will have to -be struck out, and there will remain only - - AB - *ba*. - -Hence we draw the following inferences - - A = AB, B = AB, *a* = *ab*, *b* = *ab*. - -Exactly the same method must be followed when a question involves a -greater number of terms. Thus by the Law of Duality the three terms A, -B, C, give rise to eight conceivable combinations, namely - - ABC (α) *a*BC (ε) - AB*c* (β) *a*B*c* (ζ) - A*b*C (γ) *ab*C (η) - A*bc* (δ) *abc*. (θ) - -The development of the term A is formed by the first four of these; for -B we must select (α), (β), (ε), (ζ); C consists of (α), (γ), (ε), (η); -*b* of (γ), (δ), (η), (θ), and so on. - -Now if we want to investigate completely the meaning of the premises - - A = AB (1) - B = BC (2) - -we examine each of the eight combinations as regards each premise; (γ) -and (δ) are contradicted by (1), and (β) and (ζ) by (2), so that there -remain only - - ABC (α) - *a*BC (ε) - *ab*C (η) - *abc*. (θ) - -To describe any term under the conditions of the premises (1) and (2), -we have simply to draw out the proper combinations from this list; -thus, A is represented only by ABC, that is to say - - A = ABC, - similarly *c* = *abc*. - -For B we have two alternatives thus stated, - - B = ABC ꖌ *a*BC; - -and for *b* we have - - *b* = *ab*C ꖌ *abc*. - -When we have a problem involving four distinct terms we need to -double the number of combinations, and as we add each new term the -combinations become twice as numerous. Thus - - A, B produce four combinations - A, B, C, " eight " - A, B, C, D " sixteen " - A, B, C, D, E " thirty-two " - A, B, C, D, E, F " sixty-four " - -and so on. - -I propose to call any such series of combinations the *Logical -Alphabet*. It holds in logical science a position the importance -of which cannot be exaggerated, and as we proceed from logical to -mathematical considerations, it will become apparent that there is -a close connection between these combinations and the fundamental -theorems of mathematical science. For the convenience of the reader -who may wish to employ the *Alphabet* in logical questions, I have -had printed on the next page a complete series of the combinations up -to those of six terms. At the very commencement, in the first column, -is placed a single letter X, which might seem to be superfluous. This -letter serves to denote that it is always some higher class which is -divided up. Thus the combination AB really means ABX, or that part of -some larger class, say X, which has the qualities of A and B present. -The letter X is omitted in the greater part of the table merely for -the sake of brevity and clearness. In a later chapter on Combinations -it will become apparent that the introduction of this unit class is -requisite in order to complete the analogy with the Arithmetical -Triangle there described. - -The reader ought to bear in mind that though the Logical Alphabet seems -to give mere lists of combinations, these combinations are intended in -every case to constitute the development of a term of a proposition. -Thus the four combinations AB, A*b*, *a*B, *ab* really mean that any -class X is described by the following proposition, - - X = XAB ꖌ XA*b* ꖌ X*a*B ꖌ X*ab*. - -If we select the A’s, we obtain the following proposition - - AX = XAB ꖌ XA*b*. - -Thus whatever group of combinations we treat must be conceived as part -of a higher class, *summum genus* or universe symbolised in the term -X; but, bearing this in mind, it is needless to complicate our formulæ -by always introducing the letter. All inference consists in passing -from propositions to propositions, and combinations *per se* have no -meaning. They are consequently to be regarded in all cases as forming -parts of propositions. - - -THE LOGICAL ALPHABET. - - I. II. III. IV. V. VI. VII. - X AX AB ABC ABCD ABCDE ABCDEF - *a*X A*b* AB*c* ABC*d* ABCD*e* ABCDE*f* - *a*B A*b*C AB*c*D ABC*d*E ABCD*e*F - *ab* A*bc* AB*cd* ABC*de* ABCD*ef* - *a*BC A*b*CD AB*c*DE ABC*d*EF - *a*B*c* A*b*C*d* AB*c*D*e* ABC*d*E*f* - *ab*C A*bc*D AB*cd*E ABC*de*F - *abc* Ab*cd* AB*cde* ABC*def* - *a*BCD A*b*CDE AB*c*DEF - *a*BC*d* A*b*CD*e* AB*c*DE*f* - *a*B*c*D A*b*C*d*E AB*c*D*e*F - *a*B*cd* A*b*C*de* AB*c*D*ef* - *ab*CD A*bc*DE AB*cd*EF - *ab*C*d* A*bc*D*e* AB*cd*E*f* - *abc*D A*bcd*E AB*cde*F - *abcd* A*bcde* AB*cdef* - *a*BCDE A*b*CDEF - *a*BCD*e* A*b*CDE*f* - *a*BC*d*E A*b*CD*e*F - *a*BC*de* A*b*CD*ef* - *a*B*c*DE A*b*C*d*EF - *a*B*c*D*e* A*b*C*d*E*f* - *a*B*cd*E A*b*C*de*F - *a*B*cde* A*b*C*def* - *ab*CDE A*bc*DEF - *ab*CD*e* A*bc*DE*f* - *ab*C*d*E A*bc*D*e*F - *ab*Cd*e* A*bc*D*ef* - *abc*DE A*bcd*EF - *abc*D*e* A*bcd*E*f* - *abcd*E A*bcde*F - *abcde* A*bcdef* - *a*BCDEF - *a*BCDE*f* - *a*BCD*e*F - *a*BCD*ef* - *a*BC*d*EF - *a*BC*d*E*f* - *a*BC*de*F - *a*BC*def* - *a*B*c*DEF - *a*B*c*DE*f* - *a*B*c*D*e*F - *a*B*c*D*ef* - *a*B*cd*EF - *a*B*cd*E*f* - *a*B*cde*F - *a*B*cdef* - *ab*CDEF - *ab*CDE*f* - *ab*CD*e*F - *ab*CD*ef* - *ab*C*d*EF - *ab*C*d*E*f* - *ab*C*de*F - *ab*C*def* - *abc*DEF - *abc*DE*f* - *abc*D*e*F - *abc*D*ef* - *abcd*EF - *abcd*E*f* - *abcde*F - *abcdef* - -In a theoretical point of view we may conceive that the Logical -Alphabet is infinitely extended. Every new quality or circumstance -which can belong to an object, subdivides each combination or class, -so that the number of such combinations, when unrestricted by logical -conditions, is represented by an infinitely high power of two. The -extremely rapid increase in the number of subdivisions obliges us to -confine our attention to a few qualities at a time. - -When contemplating the properties of this Alphabet I am often inclined -to think that Pythagoras perceived the deep logical importance of -duality; for while unity was the symbol of identity and harmony, he -described the number two as the origin of contrasts, or the symbol -of diversity, division and separation. The number four, or the -*Tetractys*, was also regarded by him as one of the chief elements of -existence, for it represented the generating virtue whence come all -combinations. In one of the golden verses ascribed to Pythagoras, he -conjures his pupil to be virtuous:[77] - - “By him who stampt *The Four* upon the Mind, - *The Four*, the fount of Nature’s endless stream.” - - [77] Whewell, *History of the Inductive Sciences*, vol. i. p. 222. - -Now four and the higher powers of duality do represent in this logical -system the numbers of combinations which can be generated in the -absence of logical restrictions. The followers of Pythagoras may have -shrouded their master’s doctrines in mysterious and superstitious -notions, but in many points these doctrines seem to have some basis in -logical philosophy. - - -*The Logical Slate.* - -To a person who has once comprehended the extreme significance and -utility of the Logical Alphabet the indirect process of inference -becomes reduced to the repetition of a few uniform operations of -classification, selection, and elimination of contradictories. Logical -deduction, even in the most complicated questions, becomes a matter -of mere routine, and the amount of labour required is the only -impediment, when once the meaning of the premises is rendered clear. -But the amount of labour is often found to be considerable. The mere -writing down of sixty-four combinations of six letters each is no small -task, and, if we had a problem of five premises, each of the sixty-four -combinations would have to be examined in connection with each premise. -The requisite comparison is often of a very tedious character, and -considerable chance of error intervenes. - -I have given much attention, therefore, to lessening both the manual -and mental labour of the process, and I shall describe several devices -which may be adopted for saving trouble and risk of mistake. - -In the first place, as the same sets of combinations occur over and -over again in different problems, we may avoid the labour of writing -them out by having the sets of letters ready printed upon small sheets -of writing-paper. It has also been suggested by a correspondent that, -if any one series of combinations were marked upon the margin of a -sheet of paper, and a slit cut between each pair of combinations, it -would be easy to fold down any particular combination, and thus strike -it out of view. The combinations consistent with the premises would -then remain in a broken series. This method answers sufficiently well -for occasional use. - -A more convenient mode, however, is to have the series of letters shown -on p. 94, engraved upon a common school writing slate, of such a size, -that the letters may occupy only about a third of the space on the -left hand side of the slate. The conditions of the problem can then be -written down on the unoccupied part of the slate, and the proper series -of combinations being chosen, the contradictory combinations can be -struck out with the pencil. I have used a slate of this kind, which I -call a *Logical Slate*, for more than twelve years, and it has saved me -much trouble. It is hardly possible to apply this process to problems -of more than six terms, owing to the large number of combinations which -would require examination. - - -*Abstraction of Indifferent Circumstances.* - -There is a simple but highly important process of inference which -enables us to abstract, eliminate or disregard all circumstances -indifferently present and absent. Thus if I were to state that “a -triangle is a three-sided rectilinear figure, either large or not -large,” these two alternatives would be superfluous, because, by the -Law of Duality, I know that everything must be either large or not -large. To add the qualification gives no new knowledge, since the -existence of the two alternatives will be understood in the absence of -any information to the contrary. Accordingly, when two alternatives -differ only as regards a single component term which is positive in one -and negative in the other, we may reduce them to one term by striking -out their indifferent part. It is really a process of substitution -which enables us to do this; for having any proposition of the form - - A = ABC ꖌ AB*c*, (1) - -we know by the Law of Duality that - - AB = ABC ꖌ AB*c*. (2) - -As the second member of this is identical with the second member of (1) -we may substitute, obtaining - - A = AB. - -This process of reducing useless alternatives may be applied again and -again; for it is plain that - - A = AB (CD ꖌ C*d* ꖌ *c*D ꖌ *cd*) - -communicates no more information than that A is B. Abstraction -of indifferent terms is in fact the converse process to that of -development described in p. 89; and it is one of the most important -operations in the whole sphere of reasoning. - -The reader should observe that in the proposition - - AC = BC - -we cannot abstract C and infer - - A = B; - -but from - - AC ꖌ A*c* = BC ꖌ B*c* - -we may abstract all reference to the term C. - -It ought to be carefully remarked, however, that alternatives which -seem to be without meaning often imply important knowledge. Thus if -I say that “a triangle is a three-sided rectilinear figure, with or -without three equal angles,” the last alternatives really express a -property of triangles, namely, that some triangles have three equal -angles, and some do not have them. If we put P = “Some,” meaning by the -indefinite adjective “Some,” one or more of the undefined properties of -triangles with three equal angles, and take - - A = triangle - B = three-sided rectilinear figure - C = with three equal angles, - -then the knowledge implied is expressed in the two propositions - - PA = PBC - *p*A = *p*B*c*. - -These may also be thrown into the form of one proposition, namely, - - A = PBC ꖌ *p*B*c*; - -but these alternatives cannot be reduced, and the proposition is quite -different from - - A = BC ꖌ B*c*. - - -*Illustrations of the Indirect Method.* - -A great variety of arguments and logical problems might be introduced -here to show the comprehensive character and powers of the Indirect -Method. We can treat either a single premise or a series of premises. - -Take in the first place a simple definition, such as “a triangle is a -three-sided rectilinear figure.” Let - - A = triangle - B = three-sided - C = rectilinear figure, - -then the definition is of the form - - A = BC. - -If we take the series of eight combinations of three letters in the -Logical Alphabet (p. 94) and strike out those which are inconsistent -with the definition, we have the following result:-- - - ABC - *a*B*c* - *ab*C - *abc.* - -For the description of the class C we have - - C = ABC ꖌ *ab*C, - -that is, “a rectilinear figure is either a triangle and three-sided, or -not a triangle and not three-sided.” - -For the class *b* we have - - *b* = *ab*C ꖌ *abc*. - -To the second side of this we may apply the process of simplification -by abstraction described in the last section; for by the Law of Duality - - *ab* = *ab*C ꖌ *abc*; - -and as we have two propositions identical in the second side of each we -may substitute, getting - - *b* = *ab*, - -or what is not three-sided is not a triangle (whether it be rectilinear -or not). - - -*Second Example.* - -Let us treat by this method the following argument:-- - - “Blende is not an elementary substance; elementary substances - are those which are undecomposable; blende, therefore, is - decomposable.” - -Taking our letters thus-- - - A = blende, - B = elementary substance, - C = undecomposable, - -the premises are of the forms - - A = A*b*, (1) - B = C. (2) - -No immediate substitution can be made; but if we take the -contrapositive of (2) (see p. 86), namely - - *b* = *c*, (3) - -we can substitute in (1) obtaining the conclusion - - A = A*c*. - -But the same result may be obtained by taking the eight combinations -of A, B, C, of the Logical Alphabet; it will be found that only three -combinations, namely, - - A*bc* - *a*BC - *abc*, - -are consistent with the premises, whence it results that - - A = A*bc*, - -or by the process of Ellipsis before described (p. 57) - - A = A*c*. - - -*Third Example.* - -As a somewhat more complex example I take the argument thus stated, one -which could not be thrown into the syllogistic form:-- - - “All metals except gold and silver are opaque; therefore what is not - opaque is either gold or silver or is not-metal.” - -There is more implied in this statement than is distinctly asserted, -the full meaning being as follows: - - All metals not gold or silver are opaque, (1) - Gold is not opaque but is a metal, (2) - Silver is not opaque but is a metal, (3) - Gold is not silver. (4) - -Taking our letters thus-- - - A = metal C = silver - B = gold D = opaque, - -we may state the premises in the forms - - A*bc* = A*bc*D (1) - B = AB*d* (2) - C = AC*d* (3) - B = B*c*. (4) - -To obtain a complete solution of the question we take the sixteen -combinations of A, B, C, D, and striking out those which are -inconsistent with the premises, there remain only - - AB*cd* - A*b*C*d* - A*bc*D - *abc*D - *abcd*. - -The expression for not-opaque things consists of the three combinations -containing *d*, thus - - *d* = AB*cd* ꖌ A*b*C*d* ꖌ *abcd*, - or *d* = A*d* (B*c* ꖌ *b*C) ꖌ *abcd*. - -In ordinary language, what is not-opaque is either metal which is -gold, and then not-silver, or silver and then not-gold, or else it is -not-metal and neither gold nor silver. - - -*Fourth Example.* - -A good example for the illustration of the Indirect Method is to be -found in De Morgan’s *Formal Logic* (p. 123), the premises being -substantially as follows:-- - -From A follows B, and from C follows D; but B and D are inconsistent -with each other; therefore A and C are inconsistent. - -The meaning no doubt is that where A is, B will be found, or that -every A is a B, and similarly every C is a D; but B and D cannot occur -together. The premises therefore appear to be of the forms - - A = AB, (1) - C = CD, (2) - B = B*d*. (3) - -On examining the series of sixteen combinations, only five are found to -be consistent with the above conditions, namely, - - AB*cd* - *a*B*cd* - *ab*CD - *abc*D - *abcd*. - -In these combinations the only A which appears is joined to *c*, and -similarly C is joined to *a*, or A is inconsistent with C. - - -*Fifth Example.* - -A more complex argument, also given by De Morgan,[78] contains five -terms, and is as stated below, except that the letters are altered. - - Every A is one only of the two B or C; D is both B and C, except - when B is E, and then it is neither; therefore no A is D. - - [78] *Formal Logic*, p. 124. As Professor Croom Robertson has pointed - out to me, the second and third premises may be thrown into a single - proposition, D = D*e*BC ꖌ DE*bc*. - -The meaning of the above premises is difficult to interpret, but seems -to be capable of expression in the following symbolic forms-- - - A = AB*c* ꖌ A*b*C, (1) - De = D*e*BC, (2) - DE = DE*bc*. (3) - -As five terms enter into these premises it is requisite to treat their -thirty-two combinations, and it will be found that fourteen of them -remain consistent with the premises, namely - - AB*cd*E *a*BCD*e* *ab*C*d*E - AB*cde* *a*BC*d*E *ab*C*de* - A*b*C*d*E *a*BC*de* *abc*DE - A*b*C*de* *a*B*cd*E *abcd*E - *a*B*cde* *abcde*. - -If we examine the first four combinations, all of which contain A, we -find that they none of them contain D; or again, if we select those -which contain D, we have only two, thus-- - - D = *a*BCD*e* ꖌ *abc*DE. - -Hence it is clear that no A is D, and *vice versâ* no D is A. We might -draw many other conclusions from the same premises; for instance-- - - DE = *abc*DE, - -or D and E never meet but in the absence of A, B, and C. - - -*Fallacies analysed by the Indirect Method.* - -It has been sufficiently shown, perhaps, that we can by the Indirect -Method of Inference extract the whole truth from a series of -propositions, and exhibit it anew in any required form of conclusion. -But it may also need to be shown by examples that so long as we follow -correctly the almost mechanical rules of the method, we cannot fall -into any of the fallacies or paralogisms which are often committed in -ordinary discussion. Let us take the example of a fallacious argument, -previously treated by the Method of Direct Inference (p. 62), - - Granite is not a sedimentary rock, (1) - Basalt is not a sedimentary rock, (2) - -and let us ascertain whether any precise conclusion can be drawn -concerning the relation of granite and basalt. Taking as before - - A = granite, - B = sedimentary rock, - C = basalt, - -the premises become - - A = A*b*, (1) - C = C*b*. (2) - -Of the eight conceivable combinations of A, B, C, five agree with these -conditions, namely - - A*b*C *a*B*c* - A*bc* *ab*C - *abc*. - -Selecting the combinations which contain A, we find the description of -granite to be - - A = A*b*C ꖌ A*bc* = A*b*(C ꖌ *c*), - -that is, granite is not a sedimentary rock, and is either basalt or -not-basalt. If we want a description of basalt the answer is of like -form - - C = A*b*C ꖌ *ab*C = *b*C(A ꖌ *a*), - -that is basalt is not a sedimentary rock, and is either granite or -not-granite. As it is already perfectly evident that basalt must be -either granite or not, and *vice versâ*, the premises fail to give us -any information on the point, that is to say the Method of Indirect -Inference saves us from falling into any fallacious conclusions. This -example sufficiently illustrates both the fallacy of Negative premises -and that of Undistributed Middle of the old logic. - -The fallacy called the Illicit Process of the Major Term is also -incapable of commission in following the rules of the method. Our -example was (p. 65) - - All planets are subject to gravity, (1) - Fixed stars are not planets. (2) - -The false conclusion is that “fixed stars are not subject to gravity.” -The terms are - - A = planet - B = fixed star - C = subject to gravity. - -And the premises are A = AC, (1) B = *a*B. (2) - -The combinations which remain uncontradicted on comparison with these -premises are - - A*b*C *a*B*c* - *a*BC *ab*C - *abc*. - -For fixed star we have the description - - B = *a*BC ꖌ *a*B*c*, - -that is, “a fixed star is not a planet, but is either subject or not, -as the case may be, to gravity.” Here we have no conclusion concerning -the connection of fixed stars and gravity. - - -*The Logical Abacus.* - -The Indirect Method of Inference has now been sufficiently described, -and a careful examination of its powers will show that it is capable of -giving a full analysis and solution of every question involving only -logical relations. The chief difficulty of the method consists in the -great number of combinations which may have to be examined; not only -may the requisite labour become formidable, but a considerable chance -of mistake arises. I have therefore given much attention to modes -of facilitating the work, and have succeeded in reducing the method -to an almost mechanical form. It soon appeared obvious that if the -conceivable combinations of the Logical Alphabet, for any number of -letters, instead of being printed in fixed order on a piece of paper -or slate, were marked upon light movable pieces of wood, mechanical -arrangements could readily be devised for selecting any required class -of the combinations. The labour of comparison and rejection might thus -be immensely reduced. This idea was first carried out in the Logical -Abacus, which I have found useful in the lecture-room for exhibiting -the complete solution of logical problems. A minute description of -the construction and use of the Abacus, together with figures of the -parts, has already been given in my essay called *The Substitution of -Similars*,[79] and I will here give only a general description. - - [79] Pp. 55–59, 81–86. - -The Logical Abacus consists of a common school black-board placed in a -sloping position and furnished with four horizontal and equi-distant -ledges. The combinations of the letters shown in the first four columns -of the Logical Alphabet are printed in somewhat large type, so that -each letter is about an inch from the neighbouring one, but the letters -are placed one above the other instead of being in horizontal lines -as in p. 94. Each combination of letters is separately fixed to the -surface of a thin slip of wood one inch broad and about one-eighth -inch thick. Short steel pins are then driven in an inclined position -into the wood. When a letter is a large capital representing a positive -term, the pin is fixed in the upper part of its space; when the letter -is a small italic representing a negative term, the pin is fixed in -the lower part of the space. Now, if one of the series of combinations -be ranged upon a ledge of the black-board, the sharp edge of a flat -rule can be inserted beneath the pins belonging to any one letter--say -A, so that all the combinations marked A can be lifted out and placed -upon a separate ledge. Thus we have represented the act of thought -which separates the class A from what is not-A. The operation can be -repeated; out of the A’s we can in like manner select those which are -B’s, obtaining the AB’s; and in like manner we may select any other -classes such as the *a*B’s, the *ab*’s, or the *abc*’s. - -If now we take the series of eight combinations of the letters A, B, -C, *a*, *b*, *c*, and wish to analyse the argument anciently called -Barbara, having the premises - - A = AB (1) - B = BC, (2) - -we proceed as follows--We raise the combinations marked *a*, leaving -the A’s behind; out of these A’s we move to a lower ledge such as -are *b*’s, and to the remaining AB’s we join the *a*’s which have -been raised. The result is that we have divided all the combinations -into two classes, namely, the A*b*’s which are incapable of existing -consistently with premise (1), and the combinations which are -consistent with the premise. Turning now to the second premise, we -raise out of those which agree with (1) the *b*’s, then we lower -the B*c*’s; lastly we join the *b*’s to the BC’s. We now find our -combinations arranged as below. - - +---+-----+-----+-----+-----+-----+-----+-----+ - | A | | | | *a* | | *a* | *a* | - | B | | | | B | | *b* | *b* | - | C | | | | C | | C | *c* | - +---+-----+-----+-----+-----+-----+-----+-----+ - | | A | A | A | | *a* | | | - | | B | *b* | *b* | | B | | | - | | *c* | C | *c* | | *c* | | | - +---+-----+-----+-----+-----+-----+-----+-----+ - -The lower line contains all the combinations which are inconsistent -with either premise; we have carried out in a mechanical manner that -exclusion of self-contradictories which was formerly done upon the -slate or upon paper. Accordingly, from the combinations remaining in -the upper line we can draw any inference which the premises yield. If -we raise the A’s we find only one, and that is C, so that A must be C. -If we select the *c*’s we again find only one, which is *a* and also -*b*; thus we prove that not-C is not-A and not-B. - -When a disjunctive proposition occurs among the premises the requisite -movements become rather more complicated. Take the disjunctive argument - - A is either B or C or D, - A is not C and not D, - Therefore A is B. - -The premises are represented accurately as follows:-- - - A = AB ꖌ AC ꖌ AD (1) - A = A*c* (2) - A = A*d*. (3) - -As there are four terms, we choose the series of sixteen combinations -and place them on the highest ledge of the board but one. We raise -the *a*’s and out of the A’s, which remain, we lower the *b*’s. But -we are not to reject all the A*b*’s as contradictory, because by the -first premise A’s may be either B’s or C’s or D’s. Accordingly out -of the A*b*’s we must select the *c*’s, and out of these again the -*d*’s, so that only A*bcd* will remain to be rejected finally. Joining -all the other fifteen combinations together again, and proceeding to -premise (2), we raise the *a*’s and lower the AC’s, and thus reject -the combinations inconsistent with (2); similarly we reject the AD’s -which are inconsistent with (3). It will be found that there remain, -in addition to all the eight combinations containing *a*, only one -containing A, namely - - AB*cd*, - -whence it is apparent that A must be B, the ordinary conclusion of the -argument. - -In my “Substitution of Similars” (pp. 56–59) I have described the -working upon the Abacus of two other logical problems, which it would -be tedious to repeat in this place. - - -*The Logical Machine.* - -Although the Logical Abacus considerably reduced the labour of using -the Indirect Method, it was not free from the possibility of error. -I thought moreover that it would afford a conspicuous proof of the -generality and power of the method if I could reduce it to a purely -mechanical form. Logicians had long been accustomed to speak of Logic -as an Organon or Instrument, and even Lord Bacon, while he rejected -the old syllogistic logic, had insisted, in the second aphorism of his -“New Instrument,” that the mind required some kind of systematic aid. -In the kindred science of mathematics mechanical assistance of one kind -or another had long been employed. Orreries, globes, mechanical clocks, -and such like instruments, are really aids to calculation and are of -considerable antiquity. The Arithmetical Abacus is still in common use -in Russia and China. The calculating machine of Pascal is more than two -centuries old, having been constructed in 1642–45. M. Thomas of Colmar -manufactures an arithmetical machine on Pascal’s principles which -is employed by engineers and others who need frequently to multiply -or divide. To Babbage and Scheutz is due the merit of embodying the -Calculus of Differences in a machine, which thus became capable of -calculating the most complicated tables of figures. It seemed strange -that in the more intricate science of quantity mechanism should be -applicable, whereas in the simple science of qualitative reasoning, the -syllogism was only called an instrument by a figure of speech. It is -true that Swift satirically described the Professors of Laputa as in -possession of a thinking machine, and in 1851 Mr. Alfred Smee actually -proposed the construction of a Relational machine and a Differential -machine, the first of which would be a mechanical dictionary and the -second a mode of comparing ideas; but with these exceptions I have not -yet met with so much as a suggestion of a reasoning machine. It may be -added that Mr. Smee’s designs, though highly ingenious, appear to be -impracticable, and in any case they do not attempt the performance of -logical inference.[80] - - [80] See his work called *The Process of Thought adapted to Words - and Language, together with a Description of the Relational and - Differential Machines*. Also *Philosophical Transactions*, [1870] - vol. 160, p. 518. - -The Logical Abacus soon suggested the notion of a Logical Machine, -which, after two unsuccessful attempts, I succeeded in constructing -in a comparatively simple and effective form. The details of the -Logical Machine have been fully described by the aid of plates in the -Philosophical Transactions,[81] and it would be needless to repeat the -account of the somewhat intricate movements of the machine in this -place. - - [81] *Philosophical Transactions* [1870], vol. 160, p. 497. - *Proceedings of the Royal Society*, vol. xviii. p. 166, Jan. 20, - 1870. *Nature*, vol, i. p. 343. - -The general appearance of the machine is shown in a plate facing -the title-page of this volume. It somewhat resembles a very small -upright piano or organ, and has a keyboard containing twenty-one keys. -These keys are of two kinds, sixteen of them representing the terms -or letters A, *a*, B, *b*, C, *c*, D, *d*, which have so often been -employed in our logical notation. When letters occur on the left-hand -side of a proposition, formerly called the subject, each is represented -by a key on the left-hand half of the keyboard; but when they occur on -the right-hand side, or as it used to be called the predicate of the -proposition, the letter-keys on the right-hand side of the keyboard are -the proper representatives. The five other keys may be called operation -keys, to distinguish them from the letter or term keys. They stand for -the stops, copula, and disjunctive conjunctions of a proposition. The -middle key of all is the copula, to be pressed when the verb *is* or -the sign = is met. The key to the extreme right-hand is called the Full -Stop, because it should be pressed when a proposition is completed, -in fact in the proper place of the full stop. The key to the extreme -left-hand is used to terminate an argument or to restore the machine to -its initial condition; it is called the Finis key. The last keys but -one on the right and left complete the whole series, and represent the -conjunction *or* in its unexclusive meaning, or the sign ꖌ which I have -employed, according as it occurs in the right or left hand side of the -proposition. The whole keyboard is arranged as shown on the next page-- - - +-+-----------------------------------+-+-----------------------------------+---+ - | | |C| | | - |F| Left-hand side of Proposition. |o| Right-hand side of Proposition. |F S| - |i| |p| |u t| - |n+---+---+---+---+---+---+---+---+---+u+---+---+---+---+---+---+---+---+---+l o| - |i| | | | | | | | | |l| | | | | | | | | |l p| - |s|ꖌ|*d*| D |*c*| C |*b*| B |*a*| A |a| A |*a*| B |*b*| C |*c*| D |*d*|ꖌ| .| - |.|Or | | | | | | | | |.| | | | | | | | | Or| | - +-+---+---+---+---+---+---+---+---+---+-+---+---+---+---+---+---+---+---+---+---+ - -To work the machine it is only requisite to press the keys in -succession as indicated by the letters and signs of a symbolical -proposition. All the premises of an argument are supposed to be reduced -to the simple notation which has been employed in the previous pages. -Taking then such a simple proposition as - - A = AB, - -we press the keys A (left), copula, A (right), B (right), and full stop. - -If there be a second premise, for instance - - B = BC, - -we press in like manner the keys-- - - B (left), copula, B (right), C (right), full stop. - -The process is exactly the same however numerous the premises may be. -When they are completed the operator will see indicated on the face of -the machine the exact combinations of letters which are consistent with -the premises according to the principles of thought. - -As shown in the figure opposite the title-page, the machine exhibits in -front a Logical Alphabet of sixteen combinations, exactly like that of -the Abacus, except that the letters of each combination are separated -by a certain interval. After the above problem has been worked upon the -machine the Logical Alphabet will have been modified so as to present -the following appearance-- - - +-------------------------------------------------------+ - | | - +---+---+--+--+--+--+--+--+---+---+--+--+---+---+---+---+ - | A | A | | | | | | |*a*|*a*| | |*a*|*a*|*a*|*a*| - +---+---+--+--+--+--+--+--+---+---+--+--+---+---+---+---+ - | | - +---+---+--+--+--+--+--+--+---+---+--+--+---+---+---+---+ - | B | B | | | | | | | B | B | | |*b*|*b*|*b*|*b*| - +---+---+--+--+--+--+--+--+---+---+--+--+---+---+---+---+ - | | - +---+---+--+--+--+--+--+--+---+---+--+--+---+---+---+---+ - | C | C | | | | | | | C | C | | | C | C |*c*|*c*| - +---+---+--+--+--+--+--+--+---+---+--+--+---+---+---+---+ - | | - +---+---+--+--+--+--+--+--+---+---+--+--+---+---+---+---+ - | D |*d*| | | | | | | D |*d*| | | D |*d*| D |*d*| - +---+---+--+--+--+--+--+--+---+---+--+--+---+---+---+---+ - | | - +-------------------------------------------------------+ - -The operator will readily collect the various conclusions in the manner -described in previous pages, as, for instance that A is always C, that -not-C is not-B and not-A; and not-B is not-A but either C or not-C. The -results are thus to be read off exactly as in the case of the Logical -Slate, or the Logical Abacus. - -Disjunctive propositions are to be treated in an exactly similar -manner. Thus, to work the premises - - A = AB ꖌ AC - B ꖌ C = BD ꖌ CD, - -it is only necessary to press in succession the keys - - A (left), copula, A (right), B, ꖌ, A, C, full stop. - B (left), ꖌ, C, copula, B (right), D, ꖌ, C, D, full stop. - -The combinations then remaining will be as follows - - ABCD *a*BCD *abc*D - AB*c*D *a*B*c*D *abcd.* - A*c*CD *ab*CD - -On pressing the left-hand key A, all the possible combinations which do -not contain A will disappear, and the description of A may be gathered -from what remain, namely that it is always D. The full-stop key -restores all combinations consistent with the premises and any other -selection may be made, as say not-D, which will be found to be always -not-A, not-B, and not-C. - -At the end of every problem, when no further questions need be -addressed to the machine, we press the Finis key, which has the effect -of bringing into view the whole of the conceivable combinations of -the alphabet. This key in fact obliterates the conditions impressed -upon the machine by moving back into their ordinary places those -combinations which had been rejected as inconsistent with the premises. -Before beginning any new problem it is requisite to observe that the -whole sixteen combinations are visible. After the Finis key has been -used the machine represents a mind endowed with powers of thought, -but wholly devoid of knowledge. It would not in that condition give -any answer but such as would consist in the primary laws of thought -themselves. But when any proposition is worked upon the keys, the -machine analyses and digests the meaning of it and becomes charged with -the knowledge embodied in that proposition. Accordingly it is able -to return as an answer any description of a term or class so far as -furnished by that proposition in accordance with the Laws of Thought. -The machine is thus the embodiment of a true logical system. The -combinations are classified, selected or rejected, just as they should -be by a reasoning mind, so that at each step in a problem, the Logical -Alphabet represents the proper condition of a mind exempt from mistake. -It cannot be asserted indeed that the machine entirely supersedes the -agency of conscious thought; mental labour is required in interpreting -the meaning of grammatical expressions, and in correctly impressing -that meaning on the machine; it is further required in gathering -the conclusion from the remaining combinations. Nevertheless the -true process of logical inference is really accomplished in a purely -mechanical manner. - -It is worthy of remark that the machine can detect any -self-contradiction existing between the premises presented to it; -should the premises be self-contradictory it will be found that one or -more of the letter-terms disappears entirely from the Logical Alphabet. -Thus if we work the two propositions, A is B, and A is not-B, and then -inquire for a description of A, the machine will refuse to give it -by exhibiting no combination at all containing A. This result is in -agreement with the law, which I have explained, that every term must -have its negative (p. 74). Accordingly, whenever any one of the letters -A, B, C, D, *a*, *b*, *c*, *d*, wholly disappears from the alphabet, -it may be safely inferred that some act of self-contradiction has been -committed. - -It ought to be carefully observed that the logical machine cannot -receive a simple identity of the form A = B except in the double form -of A = B and B = A. To work the proposition A = B, it is therefore -necessary to press the keys-- - - A (left), copula, B (right), full stop; - B (left), copula, A (right), full stop. - -The same double operation will be necessary whenever the proposition is -not of the kind called a partial identity (p. 40). Thus AB = CD, AB = -AC, A = B ꖌ C, A ꖌ B = C ꖌ D, all require to be read from both ends -separately. - -The proper rule for using the machine may in fact be given in the -following way:--(1) *Read each proposition as it stands, and play -the corresponding keys*: (2) *Convert the proposition and read and -play the keys again in the transposed order of the terms.* So long -as this rule is observed the true result must always be obtained. -There can be no mistake. But it will be found that in the case of -partial identities, and some other similar forms of propositions, the -transposed reading has no effect upon the combinations of the Logical -Alphabet. One reading is in such cases all that is practically needful. -After some experience has been gained in the use of the machine, the -worker naturally saves himself the trouble of the second reading when -possible. - -It is no doubt a remarkable fact that a simple identity cannot -be impressed upon the machine except in the form of two partial -identities, and this may be thought by some logicians to militate -against the equational mode of representing propositions. - -Before leaving the subject I may remark that these mechanical devices -are not likely to possess much practical utility. We do not require in -common life to be constantly solving complex logical questions. Even in -mathematical calculation the ordinary rules of arithmetic are generally -sufficient, and a calculating machine can only be used with advantage -in peculiar cases. But the machine and abacus have nevertheless two -important uses. - -In the first place I hope that the time is not very far distant when -the predominance of the ancient Aristotelian Logic will be a matter -of history only, and when the teaching of logic will be placed on a -footing more worthy of its supreme importance. It will then be found -that the solution of logical questions is an exercise of mind at least -as valuable and necessary as mathematical calculation. I believe that -these mechanical devices, or something of the same kind, will then -become useful for exhibiting to a class of students a clear and visible -analysis of logical problems of any degree of complexity, the nature -of each step being rendered plain to the eyes of the students. I often -used the machine or abacus for this purpose in my class lectures while -I was Professor of Logic at Owens College. - -Secondly, the more immediate importance of the machine seems to consist -in the unquestionable proof which it affords that correct views of the -fundamental principles of reasoning have now been attained, although -they were unknown to Aristotle and his followers. The time must come -when the inevitable results of the admirable investigations of the late -Dr. Boole must be recognised at their true value, and the plain and -palpable form in which the machine presents those results will, I hope, -hasten the time. Undoubtedly Boole’s life marks an era in the science -of human reason. It may seem strange that it had remained for him first -to set forth in its full extent the problem of logic, but I am not -aware that anyone before him had treated logic as a symbolic method -for evolving from any premises the description of any class whatsoever -as defined by those premises. In spite of several serious errors into -which he fell, it will probably be allowed that Boole discovered the -true and general form of logic, and put the science substantially into -the form which it must hold for evermore. He thus effected a reform -with which there is hardly anything comparable in the history of logic -between his time and the remote age of Aristotle. - -Nevertheless, Boole’s quasi-mathematical system could hardly be -regarded as a final and unexceptionable solution of the problem. Not -only did it require the manipulation of mathematical symbols in a very -intricate and perplexing manner, but the results when obtained were -devoid of demonstrative force, because they turned upon the employment -of unintelligible symbols, acquiring meaning only by analogy. I -have also pointed out that he imported into his system a condition -concerning the exclusive nature of alternatives (p. 70), which is -not necessarily true of logical terms. I shall have to show in the -next chapter that logic is really the basis of the whole science of -mathematical reasoning, so that Boole inverted the true order of proof -when he proposed to infer logical truths by algebraic processes. It is -wonderful evidence of his mental power that by methods fundamentally -false he should have succeeded in reaching true conclusions and -widening the sphere of reason. - -The mechanical performance of logical inference affords a demonstration -both of the truth of Boole’s results and of the mistaken nature of his -mode of deducing them. Conclusions which he could obtain only by pages -of intricate calculation, are exhibited by the machine after one or -two minutes of manipulation. And not only are those conclusions easily -reached, but they are demonstratively true, because every step of the -process involves nothing more obscure than the three fundamental Laws -of Thought. - - -*The Order of Premises.* - -Before quitting the subject of deductive reasoning, I may remark that -the order in which the premises of an argument are placed is a matter -of logical indifference. Much discussion has taken place at various -times concerning the arrangement of the premises of a syllogism; and it -has been generally held, in accordance with the opinion of Aristotle, -that the so-called major premise, containing the major term, or the -predicate of the conclusion, should stand first. This distinction -however falls to the ground in our system, since the proposition is -reduced to an identical form, in which there is no distinction of -subject and predicate. In a strictly logical point of view the order -of statement is wholly devoid of significance. The premises are -simultaneously coexistent, and are not related to each other according -to the properties of space and time. Just as the qualities of the same -object are neither before nor after each other in nature (p. 33), and -are only thought of in some one order owing to the limited capacity of -mind, so the premises of an argument are neither before nor after each -other, and are only thought of in succession because the mind cannot -grasp many ideas at once. The combinations of the logical alphabet -are exactly the same in whatever order the premises be treated on -the logical slate or machine. Some difference may doubtless exist as -regards convenience to human memory. The mind may take in the results -of an argument more easily in one mode of statement than another, -although there is no real difference in the logical results. But in -this point of view I think that Aristotle and the old logicians were -clearly wrong. It is more easy to gather the conclusion that “all A’s -are C’s” from “all A’s are B’s and all B’s are C’s,” than from the same -propositions in inverted order, “all B’s are C’s and all A’s are B’s.” - - -*The Equivalence of Propositions*. - -One great advantage which arises from the study of this Indirect -Method of Inference consists in the clear notion which we gain of -the Equivalence of Propositions. The older logicians showed how from -certain simple premises we might draw an inference, but they failed -to point out whether that inference contained the whole, or only a -part, of the information embodied in the premises. Any one proposition -or group of propositions may be classed with respect to another -proposition or group of propositions, as - - 1. Equivalent, - 2. Inferrible, - 3. Consistent, - 4. Contradictory. - -Taking the proposition “All men are mortals” as the original, then -“All immortals are not men” is its equivalent; “Some mortals are men” -is inferrible, or capable of inference, but is not equivalent; “All -not-men are not mortals” cannot be inferred, but is consistent, that -is, may be true at the same time; “All men are immortals” is of course -contradictory. - -One sufficient test of equivalence is capability of mutual inference. -Thus from - - All electrics = all non-conductors, - -I can infer - - All non-electrics = all conductors, - -and *vice versâ* from the latter I can pass back to the former. In -short, A = B is equivalent to *a* = *b*. Again, from the union of the -two propositions, A = AB and B = AB, I get A = B, and from this I -might as easily deduce the two with which I started. In this case one -proposition is equivalent to two other propositions. There are in fact -no less than four modes in which we may express the identity of two -classes A and B, namely, - - FIRST MODE. SECOND MODE. THIRD MODE. FOURTH MODE. - - A = B *a* = *b* A = AB } *a* = *ab* } - B = AB } *b* = *ab* } - -The Indirect Method of Inference furnishes a universal and clear -criterion as to the relationship of propositions. The import of a -statement is always to be measured by the combinations of terms which -it destroys. Hence two propositions are equivalent when they remove -the same combinations from the Logical Alphabet, and neither more nor -less. A proposition is inferrible but not equivalent to another when -it removes some but not all the combinations which the other removes, -and none except what this other removes. Again, propositions are -consistent provided that they jointly allow each term and the negative -of each term to remain somewhere in the Logical Alphabet. If after all -the combinations inconsistent with two propositions are struck out, -there still appears each of the letters A, *a*, B, *b*, C, *c*, D, *d*, -which were there before, then no inconsistency between the propositions -exists, although they may not be equivalent or even inferrible. -Finally, contradictory propositions are those which taken together -remove any one or more letter-terms from the Logical Alphabet. - -What is true of single propositions applies also to groups of -propositions, however large or complicated; that is to say, one group -may be equivalent, inferrible, consistent, or contradictory as regards -another, and we may similarly compare one proposition with a group of -propositions. - -To give in this place illustrations of all the four kinds of relation -would require much space: as the examples given in previous sections or -chapters may serve more or less to explain the relations of inference, -consistency, and contradiction, I will only add a few instances of -equivalent propositions or groups. - -In the following list each proposition or group of propositions is -exactly equivalent in meaning to the corresponding one in the other -column, and the truth of this statement may be tested by working out -the combinations of the alphabet, which ought to be found exactly the -same in the case of each pair of equivalents. - - A = A*b* . . . . . . . B = *a*B - A = *b* . . . . . . . . *a* = B - A = BC . . . . . . . . *a* = *b* ꖌ *c* - A = AB ꖌ AC . . . . . . *b* = *ab* ꖌ A*b*C - A ꖌB = C ꖌ D . . . . . . . *ab* = *cd* - A ꖌ *c* = B ꖌ *d* . . . . . . *a*C = *b*D - A = AB*c* ꖌ A*b*C . . .{ A = AB ꖌ AC - { AB = AB*c* - - A = B } { A = B - B = C } . . . . . . . . . { A = C - - A = AB } { A = AC - B = BC }. . . . . . . . . { B = A ꖌ *a*BC - -Although in these and many other cases the equivalents of certain -propositions can readily be given, yet I believe that no uniform and -infallible process can be pointed out by which the exact equivalents -of premises can be ascertained. Ordinary deductive inference usually -gives us only a portion of the contained information. It is true that -the combinations consistent with a set of premises may always be thrown -into the form of a proposition which must be logically equivalent to -those premises; but the difficulty consists in detecting the other -forms of propositions which will be equivalent to the premises. The -task is here of a different character from any which we have yet -attempted. It is in reality an inverse process, and is just as much -more troublesome and uncertain than the direct process, as seeking is -compared with hiding. Not only may several different answers equally -apply, but there is no method of discovering any of those answers -except by repeated trial. The problem which we have here met is really -that of induction, the inverse of deduction; and, as I shall soon show, -induction is always tentative, and, unless conducted with peculiar -skill and insight, must be exceedingly laborious in cases of complexity. - -De Morgan was unfortunately led by this equivalence of propositions -into the most serious error of his ingenious system of Logic. He held -that because the proposition “All A’s are all B’s,” is but another -expression for the two propositions “All A’s are B’s” and “All B’s -are A’s,” it must be a composite and not really an elementary form -of proposition.[82] But on taking a general view of the equivalence -of propositions such an objection seems to have no weight. Logicians -have, with few exceptions, persistently upheld the original error of -Aristotle in rejecting from their science the one simple relation of -identity on which all more complex logical relations must really rest. - - [82] *Syllabus of a proposed system of Logic*, §§ 57, 121, &c. - *Formal Logic*, p. 66. - - -*The Nature of Inference.* - -The question, What is Inference? is involved, even to the present day, -in as much uncertainty as that ancient question, What is Truth? I shall -in more than one part of this work endeavour to show that inference -never does more than explicate, unfold, or develop the information -contained in certain premises or facts. Neither in deductive nor -inductive reasoning can we add a tittle to our implicit knowledge, -which is like that contained in an unread book or a sealed letter. Sir -W. Hamilton has well said, “Reasoning is the showing out explicitly -that a proposition not granted or supposed, is implicitly contained in -something different, which is granted or supposed.”[83] - - [83] Lectures on Metaphysics, vol. iv. p. 369. - -Professor Bowen has explained[84] with much clearness that the -conclusion of an argument states explicitly what is virtually or -implicitly thought. “The process of reasoning is not so much a mode of -evolving a new truth, as it is of establishing or proving an old one, -by showing how much was admitted in the concession of the two premises -taken together.” It is true that the whole meaning of these statements -rests upon that of such words as “explicit,” “implicit,” “virtual.” -That is implicit which is wrapped up, and we render it explicit when -we unfold it. Just as the conception of a circle involves a hundred -important geometrical properties, all following from what we know, if -we have acuteness to unfold the results, so every fact and statement -involves more meaning than seems at first sight. Reasoning explicates -or brings to conscious possession what was before unconscious. It does -not create, nor does it destroy, but it transmutes and throws the same -matter into a new form. - - [84] Bowen, *Treatise on Logic*, Cambridge, U.S., 1866; p. 362. - -The difficult question still remains, Where does novelty of form begin? -Is it a case of inference when we pass from “Sincerity is the parent of -truth” to “The parent of truth is sincerity?” The old logicians would -have called this change *conversion*, one case of immediate inference. -But as all identity is necessarily reciprocal, and the very meaning -of such a proposition is that the two terms are identical in their -signification, I fail to see any difference between the statements -whatever. As well might we say that *x* = *y* and *y* = *x* are -different equations. - -Another point of difficulty is to decide when a change is merely -grammatical and when it involves a real logical transformation. Between -a *table of wood* and a *wooden table* there is no logical difference -(p. 31), the adjective being merely a convenient substitute for the -prepositional phrase. But it is uncertain to my mind whether the -change from “All men are mortal” to “No men are not mortal” is purely -grammatical. Logical change may perhaps be best described as consisting -in the determination of a relation between certain classes of objects -from a relation between certain other classes. Thus I consider it a -truly logical inference when we pass from “All men are mortal” to “All -immortals are not-men,” because the classes *immortals* and *not-men* -are different from *mortals* and *men*, and yet the propositions -contain at the bottom the very same truth, as shown in the combinations -of the Logical Alphabet. - -The passage from the qualitative to the quantitative mode of expressing -a proposition is another kind of change which we must discriminate -from true logical inference. We state the same truth when we say that -“mortality belongs to all men,” as when we assert that “all men are -mortals.” Here we do not pass from class to class, but from one kind -of term, the abstract, to another kind, the concrete. But inference -probably enters when we pass from either of the above propositions to -the assertion that the class of immortal men is zero, or contains no -objects. - -It is of course a question of words to what processes we shall or shall -not apply the name “inference,” and I have no wish to continue the -trifling discussions which have already taken place upon the subject. -What we need to do is to define accurately the sense in which we use -the word “inference,” and to distinguish the relation of inferrible -propositions from other possible relations. It seems to be sufficient -to recognise four modes in which two apparently different propositions -may be related. Thus two propositions may be-- - -1. *Tautologous* or *identical*, involving the same relation between -the same terms and classes, and only differing in the order of -statement; thus “Victoria is the Queen of England” is tautologous with -“The Queen of England is Victoria.” - -2. *Grammatically related*, when the classes or objects are the same -and similarly related, and the only difference is in the words; thus -“Victoria is the Queen of England” is grammatically equivalent to -“Victoria is England’s Queen.” - -3. *Equivalents* in qualitative and quantitative form, the classes -being the same, but viewed in a different manner. - -4. *Logically inferrible*, one from the other, or it may be -*equivalent*, when the classes and relations are different, but involve -the same knowledge of the possible combinations. - - - - -CHAPTER VII. - -INDUCTION. - - -We enter in this chapter upon the second great department of logical -method, that of Induction or the Inference of general from particular -truths. It cannot be said that the Inductive process is of greater -importance than the Deductive process already considered, because the -latter process is absolutely essential to the existence of the former. -Each is the complement and counterpart of the other. The principles -of thought and existence which underlie them are at the bottom the -same, just as subtraction of numbers necessarily rests upon the same -principles as addition. Induction is, in fact, the inverse operation of -deduction, and cannot be conceived to exist without the corresponding -operation, so that the question of relative importance cannot arise. -Who thinks of asking whether addition or subtraction is the more -important process in arithmetic? But at the same time much difference -in difficulty may exist between a direct and inverse operation; the -integral calculus, for instance, is infinitely more difficult than the -differential calculus of which it is the inverse. Similarly, it must -be allowed that inductive investigations are of a far higher degree of -difficulty and complexity than any questions of deduction; and it is -this fact no doubt which led some logicians, such as Francis Bacon, -Locke, and J. S. Mill, to erroneous opinions concerning the exclusive -importance of induction. - -Hitherto we have been engaged in considering how from certain -conditions, laws, or identities governing the combinations of -qualities, we may deduce the nature of the combinations agreeing -with those conditions. Our work has been to unfold the results of -what is contained in any statements, and the process has been one of -*Synthesis*. The terms or combinations of which the character has been -determined have usually, though by no means always, involved more -qualities, and therefore, by the relation of extension and intension, -fewer objects than the terms in which they were described. The truths -inferred were thus usually less general than the truths from which they -were inferred. - -In induction all is inverted. The truths to be ascertained are more -general than the data from which they are drawn. The process by which -they are reached is *analytical*, and consists in separating the -complex combinations in which natural phenomena are presented to us, -and determining the relations of separate qualities. Given events -obeying certain unknown laws, we have to discover the laws obeyed. -Instead of the comparatively easy task of finding what effects will -follow from a given law, the effects are now given and the law is -required. We have to interpret the will by which the conditions of -creation were laid down. - - -*Induction an Inverse Operation* - -I have already asserted that induction is the inverse operation of -deduction, but the difference is one of such great importance that I -must dwell upon it. There are many cases in which we can easily and -infallibly do a certain thing but may have much trouble in undoing -it. A person may walk into the most complicated labyrinth or the -most extensive catacombs, and turn hither and thither at his will; -it is when he wishes to return that doubt and difficulty commence. -In entering, any path served him; in leaving, he must select certain -definite paths, and in this selection he must either trust to memory -of the way he entered or else make an exhaustive trial of all possible -ways. The explorer entering a new country makes sure his line of return -by barking the trees. - -The same difficulty arises in many scientific processes. Given any -two numbers, we may by a simple and infallible process obtain their -product; but when a large number is given it is quite another -matter to determine its factors. Can the reader say what two numbers -multiplied together will produce the number 8,616,460,799? I think it -unlikely that anyone but myself will ever know; for they are two large -prime numbers, and can only be rediscovered by trying in succession -a long series of prime divisors until the right one be fallen upon. -The work would probably occupy a good computer for many weeks, but it -did not occupy me many minutes to multiply the two factors together. -Similarly there is no direct process for discovering whether any number -is a prime or not; it is only by exhaustively trying all inferior -numbers which could be divisors, that we can show there is none, and -the labour of the process would be intolerable were it not performed -systematically once for all in the process known as the Sieve of -Eratosthenes, the results being registered in tables of prime numbers. - -The immense difficulties which are encountered in the solution of -algebraic equations afford another illustration. Given any algebraic -factors, we can easily and infallibly arrive at the product; but given -a product it is a matter of infinite difficulty to resolve it into -factors. Given any series of quantities however numerous, there is very -little trouble in making an equation which shall have those quantities -as roots. Let *a*, *b*, *c*, *d*, &c., be the quantities; then - - (*x* - *a*)(*x* - *b*)(*x* - *c*)(*x* - d) ... = 0 - -is the equation required, and we only need to multiply out the -expression on the left hand by ordinary rules. But having given a -complex algebraic expression equated to zero, it is a matter of -exceeding difficulty to discover all the roots. Mathematicians have -exhausted their highest powers in carrying the complete solution up to -the fourth degree. In every other mathematical operation the inverse -process is far more difficult than the direct process, subtraction than -addition, division than multiplication, evolution than involution; -but the difficulty increases vastly as the process becomes more -complex. Differentiation, the direct process, is always capable of -performance by fixed rules, but as these rules produce considerable -variety of results, the inverse process of integration presents -immense difficulties, and in an infinite majority of cases surpasses -the present resources of mathematicians. There are no infallible and -general rules for its accomplishment; it must be done by trial, by -guesswork, or by remembering the results of differentiation, and using -them as a guide. - -Coming more nearly to our own immediate subject, exactly the same -difficulty exists in determining the law which certain things obey. -Given a general mathematical expression, we can infallibly ascertain -its value for any required value of the variable. But I am not aware -that mathematicians have ever attempted to lay down the rules of a -process by which, having given certain numbers, one might discover a -rational or precise formula from which they proceed. The reader may -test his power of detecting a law, by contemplation of its results, if -he, not being a mathematician, will attempt to point out the law obeyed -by the following numbers: - - 1/6, 1/30, 1/42, 1/30, 5/66, 691/2730, 7/6, 3617/510, 43867/798, etc. - -These numbers are sometimes in low terms, but unexpectedly spring up -to high terms; in absolute magnitude they are very variable. They seem -to set all regularity and method at defiance, and it is hardly to be -supposed that anyone could, from contemplation of the numbers, have -detected the relations between them. Yet they are derived from the -most regular and symmetrical laws of relation, and are of the highest -importance in mathematical analysis, being known as the numbers of -Bernoulli. - -Compare again the difficulty of decyphering with that of cyphering. -Anyone can invent a secret language, and with a little steady labour -can translate the longest letter into the character. But to decypher -the letter, having no key to the signs adopted, is a wholly different -matter. As the possible modes of secret writing are infinite in number -and exceedingly various in kind, there is no direct mode of discovery -whatever. Repeated trial, guided more or less by knowledge of the -customary form of cypher, and resting entirely on the principles of -probability and logical induction, is the only resource. A peculiar -tact or skill is requisite for the process, and a few men, such as -Wallis or Wheatstone, have attained great success. - -Induction is the decyphering of the hidden meaning of natural -phenomena. Given events which happen in certain definite combinations, -we are required to point out the laws which govern those combinations. -Any laws being supposed, we can, with ease and certainty, decide -whether the phenomena obey those laws. But the laws which may exist -are infinite in variety, so that the chances are immensely against -mere random guessing. The difficulty is much increased by the fact -that several laws will usually be in operation at the same time, the -effects of which are complicated together. The only modes of discovery -consist either in exhaustively trying a great number of supposed laws, -a process which is exhaustive in more senses than one, or else in -carefully contemplating the effects, endeavouring to remember cases -in which like effects followed from known laws. In whatever manner we -accomplish the discovery, it must be done by the more or less conscious -application of the direct process of deduction. - -The Logical Alphabet illustrates induction as well as deduction. In -considering the Indirect Process of Inference we found that from -certain propositions we could infallibly determine the combinations -of terms agreeing with those premises. The inductive problem is just -the inverse. Having given certain combinations of terms, we need to -ascertain the propositions with which the combinations are consistent, -and from which they may have proceeded. Now, if the reader contemplates -the following combinations, - - ABC *ab*C - *a*BC *abc*, - -he will probably remember at once that they belong to the premises -A = AB, B = BC (p. 92). If not, he will require a few trials before he -meets with the right answer, and every trial will consist in assuming -certain laws and observing whether the deduced results agree with the -data. To test the facility with which he can solve this inductive -problem, let him casually strike out any of the combinations of the -fourth column of the Logical Alphabet, (p. 94), and say what laws -the remaining combinations obey, observing that every one of the -letter-terms and their negatives ought to appear in order to avoid -self-contradiction in the premises (pp. 74, 111). Let him say, for -instance, what laws are embodied in the combinations - - ABC *a*BC - A*bc* *ab*C. - -The difficulty becomes much greater when more terms enter into the -combinations. It would require some little examination to ascertain the -complete conditions fulfilled in the combinations - - AC*e* *ab*C*e* - *a*BC*e* *abc*E. - *a*B*cd*E - -The reader may discover easily enough that the principal laws are -C = *e*, and A = A*e*; but he would hardly discover without some -trouble the remaining law, namely, that BD = BD*e*. - -The difficulties encountered in the inductive investigations of -nature, are of an exactly similar kind. We seldom observe any law in -uninterrupted and undisguised operation. The acuteness of Aristotle and -the ancient Greeks did not enable them to detect that all terrestrial -bodies tend to fall towards the centre of the earth. A few nights of -observation might have convinced an astronomer viewing the solar system -from its centre, that the planets travelled round the sun; but the -fact that our place of observation is one of the travelling planets, -so complicates the apparent motions of the other bodies, that it -required all the sagacity of Copernicus to prove the real simplicity of -the planetary system. It is the same throughout nature; the laws may -be simple, but their combined effects are not simple, and we have no -clue to guide us through their intricacies. “It is the glory of God,” -said Solomon, “to conceal a thing, but the glory of a king to search -it out.” The laws of nature are the invaluable secrets which God has -hidden, and it is the kingly prerogative of the philosopher to search -them out by industry and sagacity. - - -*Inductive Problems for Solution by the Reader.* - -In the first edition (vol. ii. p. 370) I gave a logical problem -involving six terms, and requested readers to discover the laws -governing the combinations given. I received satisfactory replies -from readers both in the United States and in England. I formed -the combinations deductively from four laws of correction, but my -correspondents found that three simpler laws, equivalent to the four -more complex ones, were the best answer; these laws are as follows: -*a* = *ac*, *b* = *cd*, *d* = E*f*. - -In case other readers should like to test their skill in the inductive -or inverse problem, I give below several series of combinations forming -problems of graduated difficulty. - - PROBLEM I. - - A B *c* - A *b* C - *a* B C - - PROBLEM II. - - A B C - A *b* C - *a* B C - *a* B *c* - - PROBLEM III. - - A B C - A *b* C - *a* B C - *a* B *c* - *a* *b* *c* - - PROBLEM IV. - - A B C D - A *b* *c* D - *a* B *c* *d* - *a* *b* C *d* - - PROBLEM V. - - A B C D - A B C *d* - A B *c* *d* - A *b* C D - A *b* *c* D - *a* B C D - *a* B *c* D - *a* B *c* *d* - *a* *b* C *d* - - PROBLEM VI. - - A B C D E - A B C *d* *e* - A B *c* D E - A B *c* *d* *e* - A *b* C D E - *a* B C D E - *a* B C *d* *e* - *a* *b* C D E - *a* *b* *c* *d* *e* - - PROBLEM VII. - - A *b* *c* D *e* - *a* B C *d* E - *a* *b* C *d* E - - PROBLEM VIII. - - A B C D E - A B C D *e* - A B C *d* *e* - A B *c* *d* *e* - A *b* C D E - A *b* *c* *d* E - A *b* *c* *d* *e* - *a* B C D *e* - *a* B C *d* *e* - *a* B *c* D *e* - *a* *b* C D *e* - *a* *b* C *d* E - *a* *b* *c* D *e* - *a* *b* *c* *d* E - - PROBLEM IX. - - A B *c* D E F - A B *c* D *e* F - A *b* C D *e* *f* - A *b* *c* D E *f* - A *b* *c* D *e* *f* - A *b* *c* *d* E F - A *b* *c* *d* *e* F - *a* B *c* D E F - *a* B *c* D *e* F - *a* B *c* *d* E F - *a* *b* C D E F - *a* *b* C D *e* F - *a* *b* C D *e* *f* - *a* *b* *c* D *e* *f* - *a* *b* *c* D E *f* - *a* *b* *c* *d* *e* F - - PROBLEM X. - - A B C D *e* F - A B *c* D E *f* - A *b* C D E F - A *b* C D *e* F - A *b* *c* D *e* F - *a* B C D E *f* - *a* B *c* D E *f* - *a* *b* C D *e* F - *a* *b* C *d* *e* F - *a* *b* *c* D *e* *f* - *a* *b* *c* *d* *e* *f* - - -*Induction of Simple Identities*. - -Many important laws of nature are expressible in the form of simple -identities, and I can at once adduce them as examples to illustrate -what I have said of the difficulty of the inverse process of induction. -Two phenomena are conjoined. Thus all gravitating matter is exactly -coincident with all matter possessing inertia; where one property -appears, the other likewise appears. All crystals of the cubical -system, are all the crystals which do not doubly refract light. All -exogenous plants are, with some exceptions, those which have two -cotyledons or seed-leaves. - -A little reflection will show that there is no direct and infallible -process by which such complete coincidences may be discovered. -Natural objects are aggregates of many qualities, and any one of -those qualities may prove to be in close connection with some others. -If each of a numerous group of objects is endowed with a hundred -distinct physical or chemical qualities, there will be no less than -(1/2)(100 × 99) or 4950 pairs of qualities, which may be connected, -and it will evidently be a matter of great intricacy and labour to -ascertain exactly which qualities are connected by any simple law. - -One principal source of difficulty is that the finite powers of the -human mind are not sufficient to compare by a single act any large -group of objects with another large group. We cannot hold in the -conscious possession of the mind at any one moment more than five or -six different ideas. Hence we must treat any more complex group by -successive acts of attention. The reader will perceive by an almost -individual act of comparison that the words *Roma* and *Mora* contain -the same letters. He may perhaps see at a glance whether the same is -true of *Causal* and *Casual*, and of *Logica* and *Caligo*. To assure -himself that the letters in *Astronomers* make *No more stars*, that -*Serpens in akuleo* is an anagram of *Joannes Keplerus*, or *Great gun -do us a sum* an anagram of *Augustus de Morgan*, it will certainly be -necessary to break up the act of comparison into several successive -acts. The process will acquire a double character, and will consist in -ascertaining that each letter of the first group is among the letters -of the second group, and *vice versâ*, that each letter of the second -is among those of the first group. In the same way we can only prove -that two long lists of names are identical, by showing that each name -in one list occurs in the other, and *vice versâ*. - -This process of comparison really consists in establishing two partial -identities, which are, as already shown (p. 58), equivalent in -conjunction to one simple identity. We first ascertain the truth of the -two propositions A = AB, B = AB, and we then rise by substitution to -the single law A = B. - -There is another process, it is true, by which we may get to exactly -the same result; for the two propositions A = AB, *a* = *ab* are also -equivalent to the simple identity A = B. If then we can show that -all objects included under A are included under B, and also that all -objects not included under A are not included under B, our purpose is -effected. By this process we should usually compare two lists if we are -allowed to mark them. For each name in the first list we should strike -off one in the second, and if, when the first list is exhausted, the -second list is also exhausted, it follows that all names absent from -the first must be absent from the second, and the coincidence must be -complete. - -These two modes of proving an identity are so closely allied that it -is doubtful how far we can detect any difference in their powers and -instances of application. The first method is perhaps more convenient -when the phenomena to be compared are rare. Thus we prove that all the -musical concords coincide with all the more simple numerical ratios, by -showing that each concord arises from a simple ratio of undulations, -and then showing that each simple ratio gives rise to one of the -concords. To examine all the possible cases of discord or complex ratio -of undulation would be impossible. By a happy stroke of induction Sir -John Herschel discovered that all crystals of quartz which cause the -plane of polarization of light to rotate are precisely those crystals -which have plagihedral faces, that is, oblique faces on the corners of -the prism unsymmetrical with the ordinary faces. This singular relation -would be proved by observing that all plagihedral crystals possessed -the power of rotation, and *vice versâ* all crystals possessing this -power were plagihedral. But it might at the same time be noticed that -all ordinary crystals were devoid of the power. There is no reason -why we should not detect any of the four propositions A = AB, B = AB, -*a* = *ab*, *b* = *ab*, all of which follow from A = B (p. 115). - -Sometimes the terms of the identity may be singular objects; thus we -observe that diamond is a combustible gem, and being unable to discover -any other that is, we affirm-- - - Diamond = combustible gem. - -In a similar manner we ascertain that - - Mercury = metal liquid at ordinary temperatures, - Substance of least density = substance of least atomic weight. - -Two or three objects may occasionally enter into the induction, as when -we learn that - - Sodium ꖌ potassium = metal of less density than water, - - Venus ꖌ Mercury ꖌ Mars = major planet devoid of satellites. - - -*Induction of Partial Identities*. - -We found in the last section that the complete identity of two classes -is almost always discovered not by direct observation of the fact, -but by first establishing two partial identities. There are also a -multitude of cases in which the partial identity of one class with -another is the only relation to be discovered. Thus the most common -of all inductive inferences consists in establishing the fact that -all objects having the properties of A have also those of B, or that -A = AB. To ascertain the truth of a proposition of this kind it is -merely necessary to assemble together, mentally or physically, all the -objects included under A, and then observe whether B is present in -each of them, or, which is the same, whether it would be impossible -to select from among them any not-B. Thus, if we mentally assemble -together all the heavenly bodies which move with apparent rapidity, -that is to say, the planets, we find that they all possess the property -of not scintillating. We cannot analyse any vegetable substance without -discovering that it contains carbon and hydrogen, but it is not true -that all substances containing carbon and hydrogen are vegetable -substances. - -The great mass of scientific truths consists of propositions of -this form A = AB. Thus in astronomy we learn that all the planets -are spheroidal bodies; that they all revolve in one direction round -the sun; that they all shine by reflected light; that they all obey -the law of gravitation. But of course it is not to be asserted that -all bodies obeying the law of gravitation, or shining by reflected -light, or revolving in a particular direction, or being spheroidal -in form, are planets. In other sciences we have immense numbers of -propositions of the same form, as, for instance, all substances in -becoming gaseous absorb heat; all metals are elements; they are all -good conductors of heat and electricity; all the alkaline metals are -monad elements; all foraminifera are marine organisms; all parasitic -animals are non-mammalian; lightning never issues from stratous clouds; -pumice never occurs where only Labrador felspar is present; milkmaids -do not suffer from small-pox; and, in the works of Darwin, scientific -importance may attach even to such an apparently trifling observation -as that “white tom-cats having blue eyes are deaf.” - -The process of inference by which all such truths are obtained may -readily be exhibited in a precise symbolic form. We must have one -premise specifying in a disjunctive form all the possible individuals -which belong to a class; we resolve the class, in short, into its -constituents. We then need a number of propositions, each of which -affirms that one of the individuals possesses a certain property. Thus -the premises must be of the forms - - A = B ꖌ C ꖌ D ꖌ .... ꖌ P ꖌ Q - B = BX - C = CX - ... ... - ... ... - Q = QX. - -Now, if we substitute for each alternative of the first premise its -description as found among the succeeding premises, we obtain - - A = BX ꖌ CX ꖌ .... ꖌ PX ꖌ QX - -or - - A = (B ꖌ C ꖌ .... ꖌ Q)X - -But for the aggregate of alternatives we may now substitute their -equivalent as given in the first premise, namely A, so that we get the -required result: - - A = AX. - -We should have reached the same result if the first premise had been of -the form - - A = AB ꖌ AC ꖌ .... ꖌ AQ. - -We can always prove a proposition, if we find it more convenient, by -proving its equivalent. To assert that all not-B’s are not-A’s, is -exactly the same as to assert that all A’s are B’s. Accordingly we may -ascertain that A = AB by first ascertaining that *b* = *ab*. If we -observe, for instance, that all substances which are not solids are -also not capable of double refraction, it follows necessarily that all -double refracting substances are solids. We may convince ourselves that -all electric substances are nonconductors of electricity, by reflecting -that all good conductors do not, and in fact cannot, retain electric -excitation. When we come to questions of probability it will be found -desirable to prove, as far as possible, both the original proposition -and its equivalent, as there is then an increased area of observation. - -The number of alternatives which may arise in the division of a class -varies greatly, and may be any number from two upwards. Thus it is -probable that every substance is either magnetic or diamagnetic, and no -substance can be both at the same time. The division then must be made -in the form - - A = AB*c* ꖌ A*b*C. - -If now we can prove that all magnetic substances are capable of -polarity, say B = BD, and also that all diamagnetic substances are -capable of polarity, C = CD, it follows by substitution that all -substances are capable of polarity, or A = AD. We commonly divide -the class substance into the three subclasses, solid, liquid, and -gas; and if we can show that in each of these forms it obeys Carnot’s -thermodynamic law, it follows that all substances obey that law. -Similarly we may show that all vertebrate animals possess red blood, -if we can show separately that fish, reptiles, birds, marsupials, and -mammals possess red blood, there being, as far as is known, only five -principal subclasses of vertebrata. - -Our inductions will often be embarrassed by exceptions, real or -apparent. We might affirm that all gems are incombustible were not -diamonds undoubtedly combustible. Nothing seems more evident than that -all the metals are opaque until we examine them in fine films, when -gold and silver are found to be transparent. All plants absorb carbonic -acid except certain fungi; all the bodies of the planetary system -have a progressive motion from west to east, except the satellites of -Uranus and Neptune. Even some of the profoundest laws of matter are not -quite universal; all solids expand by heat except india-rubber, and -possibly a few other substances; all liquids which have been tested -expand by heat except water below 4° C. and fused bismuth; all gases -have a coefficient of expansion increasing with the temperature, except -hydrogen. In a later chapter I shall consider how such anomalous cases -may be regarded and classified; here we have only to express them in a -consistent manner by our notation. - -Let us take the case of the transparency of metals, and assign the -terms thus:-- - - A = metal D = iron - B = gold E, F, &c. = copper, lead, &c. - C = silver X = opaque. - -Our premises will be - - A = B ꖌ C ꖌ D ꖌ E, &c. - B = B*x* - C = C*x* - D = DX - E = EX, - -and so on for the rest of the metals. Now evidently - - A*bc* = (D ꖌ E ꖌ F ꖌ ...)*bc*, - -and by substitution as before we shall obtain - - A*bc* = A*bc*X, - -or in words, “All metals not gold nor silver are opaque;” at the same -time we have - - A(B ꖌ C) = AB ꖌ AC = AB*x* ꖌ AC*x* = A(B ꖌ C)*x*, - -or “Metals which are either gold or silver are not opaque.” - -In some cases the problem of induction assumes a much higher degree of -complexity. If we examine the properties of crystallized substances -we may find some properties which are common to all, as cleavage or -fracture in definite planes; but it would soon become requisite to -break up the class into several minor ones. We should divide crystals -according to the seven accepted systems--and we should then find -that crystals of each system possess many common properties. Thus -crystals of the Regular or Cubical system expand equally by heat, -conduct heat and electricity with uniform rapidity, and are of like -elasticity in all directions; they have but one index of refraction -for light; and every facet is repeated in like relation to each of -the three axes. Crystals of the system having one principal axis -will be found to possess the various physical powers of conduction, -refraction, elasticity, &c., uniformly in directions perpendicular -to the principal axis; in other directions their properties vary -according to complicated laws. The remaining systems in which the -crystals possess three unequal axes, or have inclined axes, exhibit -still more complicated results, the effects of the crystal upon light, -heat, electricity, &c., varying in all directions. But when we pursue -induction into the intricacies of its application to nature we really -enter upon the subject of classification, which we must take up again -in a later part of this work. - - -*Solution of the Inverse or Inductive Problem, involving Two Classes*. - -It is now plain that Induction consists in passing back from a series -of combinations to the laws by which such combinations are governed. -The natural law that all metals are conductors of electricity really -means that in nature we find three classes of objects, namely-- - - 1. Metals, conductors; - 2. Not-metals, conductors; - 3. Not-metals, not-conductors. - -It comes to the same thing if we say that it excludes the existence -of the class, “metals not-conductors.” In the same way every other -law or group of laws will really mean the exclusion from existence -of certain combinations of the things, circumstances or phenomena -governed by those laws. Now in logic, strictly speaking, we treat not -the phenomena, nor the laws, but the general forms of the laws; and a -little consideration will show that for a finite number of things the -possible number of forms or kinds of law governing them must also be -finite. Using general terms, we know that A and B can be present or -absent in four ways and no more--thus: - - AB, A*b*, *a*B, *ab*; - -therefore every possible law which can exist concerning the relation -of A and B must be marked by the exclusion of one or more of the above -combinations. The number of possible laws then cannot exceed the -number of selections which we can make from these four combinations. -Since each combination may be present or absent, the number of cases -to be considered is 2 × 2 × 2 × 2, or sixteen; and these cases are all -shown in the following table, in which the sign 0 indicates absence or -non-existence of the combination shown at the left-hand column in the -same line, and the mark 1 its presence:-- - - 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 - * * * * * * * - AB 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 - A*b* 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 - *a*B 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 - *ab* 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 - -Thus in column sixteen we find that all the conceivable combinations -are present, which means that there are no special laws in existence -in such a case, and that the combinations are governed only by the -universal Laws of Identity and Difference. The example of metals and -conductors of electricity would be represented by the twelfth column; -and every other mode in which two things or qualities might present -themselves is shown in one or other of the columns. More than half -the cases may indeed be at once rejected, because they involve the -entire absence of a term or its negative. It has been shown to be a -logical principle that every term must have its negative (p. 111), -and when this is not the case, inconsistency between the conditions -of combination must exist. Thus if we laid down the two following -propositions, “Graphite conducts electricity,” and “Graphite does not -conduct electricity,” it would amount to asserting the impossibility -of graphite existing at all; or in general terms, A is B and A is -not B result in destroying altogether the combinations containing A, -a case shown in the fourth column of the above table. We therefore -restrict our attention to those cases which may be represented in -natural phenomena when at least two combinations are present, and which -correspond to those columns of the table in which each of A, *a*, -B, *b* appears. These cases are shown in the columns marked with an -asterisk. - -We find that seven cases remain for examination, thus characterised-- - - Four cases exhibiting three combinations, - Two cases exhibiting two combinations, - One case exhibiting four combinations. - -It has already been pointed out that a proposition of the form A = -AB destroys one combination, A*b*, so that this is the form of law -applying to the twelfth column. But by changing one or more of the -terms in A = AB into its negative, or by interchanging A and B, *a* and -*b*, we obtain no less than eight different varieties of the one form; -thus-- - - 12th case. 8th case. 15th case. 14th case. - A = AB A = A*b* *a* = *a*B *a* = *ab* - *b* = *ab* B = *a*B *b* = A*b* B = AB - -The reader of the preceding sections will see that each proposition -in the lower line is logically equivalent to, and is in fact the -contrapositive of, that above it (p. 83). Thus the propositions -A = A*b* and B = *a*B both give the same combinations, shown in the -eighth column of the table, and trial shows that the twelfth, eighth, -fifteenth and fourteenth columns are thus accounted for. We come to -this conclusion then--*The general form of proposition* A = AB *admits -of four logically distinct varieties, each capable of expression in two -modes*. - -In two columns of the table, namely the seventh and tenth, we observe -that two combinations are missing. Now a simple identity A = B renders -impossible both A*b* and *a*B, accounting for the tenth case; and if we -change B into *b* the identity A = *b* accounts for the seventh case. -There may indeed be two other varieties of the simple identity, namely -*a* = *b* and *a* = B; but it has already been shown repeatedly that -these are equivalent respectively to A = B and A = *b* (p. 115). As -the sixteenth column has already been accounted for as governed by no -special conditions, we come to the following general conclusion:--The -laws governing the combinations of two terms must be capable of -expression either in a partial identity or a simple identity; the -partial identity is capable of only four logically distinct varieties, -and the simple identity of two. Every logical relation between two -terms must be expressed in one of these six forms of law, or must be -logically equivalent to one of them. - -In short, we may conclude that in treating of partial and complete -identity, we have exhaustively treated the modes in which two terms or -classes of objects can be related. Of any two classes it can be said -that one must either be included in the other, or must be identical -with it, or a like relation must exist between one class and the -negative of the other. We have thus completely solved the inverse -logical problem concerning two terms.[85] - - [85] The contents of this and the following section nearly correspond - with those of a paper read before the Manchester Literary and - Philosophical Society on December 26th, 1871. See Proceedings of the - Society, vol. xi. pp. 65–68, and Memoirs, Third Series, vol. v. pp. - 119–130. - - -*The Inverse Logical Problem involving Three Classes.* - -No sooner do we introduce into the problem a third term C, than the -investigation assumes a far more complex character, so that some -readers may prefer to pass over this section. Three terms and their -negatives may be combined, as we have frequently seen, in eight -different combinations, and the effect of laws or logical conditions -is to destroy any one or more of these combinations. Now we may make -selections from eight things in 2^{8} or 256 ways; so that we have no -less than 256 different cases to treat, and the complete solution is -at least fifty times as troublesome as with two terms. Many series of -combinations, indeed, are contradictory, as in the simpler problem, -and may be passed over, the test of consistency being that each of the -letters A, B, C, *a*, *b*, *c*, shall appear somewhere in the series of -combinations. - -My mode of solving the problem was as follows:--Having written out the -whole of the 256 series of combinations, I examined them separately and -struck out such as did not fulfil the test of consistency. I then chose -some form of proposition involving two or three terms, and varied it -in every possible manner, both by the circular interchange of letters -(A, B, C into B, C, A and then into C, A, B), and by the substitution -for any one or more of the terms of the corresponding negative terms. -For instance, the proposition AB = ABC can be first varied by circular -interchange so as to give BC = BCA and then CA = CAB. Each of these -three can then be thrown into eight varieties by negative change. Thus -AB = ABC gives *a*B = *a*BC, A*b* = A*b*C, AB = AB*c*, *ab* = *ab*C, -and so on. Thus there may possibly exist no less than twenty-four -varieties of the law having the general form AB = ABC, meaning that -whatever has the properties of A and B has those also of C. It by no -means follows that some of the varieties may not be equivalent to -others; and trial shows, in fact, that AB = ABC is exactly the same -in meaning as A*c* = A*bc* or B*c* = B*ca*. Thus the law in question -has but eight varieties of distinct logical meaning. I now ascertain -by actual deductive reasoning which of the 256 series of combinations -result from each of these distinct laws, and mark them off as soon as -found. I then proceed to some other form of law, for instance A = ABC, -meaning that whatever has the qualities of A has those also of B and -C. I find that it admits of twenty-four variations, all of which are -found to be logically distinct; the combinations being worked out, I am -able to mark off twenty-four more of the list of 256 series. I proceed -in this way to work out the results of every form of law which I can -find or invent. If in the course of this work I obtain any series of -combinations which had been previously marked off, I learn at once that -the law giving these combinations is logically equivalent to some law -previously treated. It may be safely inferred that every variety of the -apparently new law will coincide in meaning with some variety of the -former expression of the same law. I have sufficiently verified this -assumption in some cases, and have never found it lead to error. Thus -as AB = ABC is equivalent to A*c* = A*bc*, so we find that *ab* = *ab*C -is equivalent to *ac* = *ac*B. - -Among the laws treated were the two A = AB and A = B which involve only -two terms, because it may of course happen that among three things two -only are in special logical relation, and the third independent; and -the series of combinations representing such cases of relation are sure -to occur in the complete enumeration. All single propositions which -I could invent having been treated, pairs of propositions were next -investigated. Thus we have the relations, “All A’s are B’s, and all -B’s are C’s,” of which the old logical syllogism is the development. -We may also have “all A’s are all B’s, and all B’s are C’s,” or even -“all A’s are all B’s, and all B’s are all C’s.” All such premises admit -of variations, greater or less in number, the logical distinctness -of which can only be determined by trial in detail. Disjunctive -propositions either singly or in pairs were also treated, but were -often found to be equivalent to other propositions of a simpler form; -thus A = ABC ꖌ A*bc* is exactly the same in meaning as AB = AC. - -This mode of exhaustive trial bears some analogy to that ancient -mathematical process called the Sieve of Eratosthenes. Having taken -a long series of the natural numbers, Eratosthenes is said to have -calculated out in succession all the multiples of every number, and -to have marked them off, so that at last the prime numbers alone -remained, and the factors of every number were exhaustively discovered. -My problem of 256 series of combinations is the logical analogue, the -chief points of difference being that there is a limit to the number of -cases, and that prime numbers have no analogue in logic, since every -series of combinations corresponds to a law or group of conditions. -But the analogy is perfect in the point that they are both inverse -processes. There is no mode of ascertaining that a number is prime but -by showing that it is not the product of any assignable factors. So -there is no mode of ascertaining what laws are embodied in any series -of combinations but trying exhaustively the laws which would give them. -Just as the results of Eratosthenes’ method have been worked out to -a great extent and registered in tables for the convenience of other -mathematicians, I have endeavoured to work out the inverse logical -problem to the utmost extent which is at present practicable or useful. - -I have thus found that there are altogether fifteen conditions or -series of conditions which may govern the combinations of three -terms, forming the premises of fifteen essentially different kinds -of arguments. The following table contains a statement of these -conditions, together with the numbers of combinations which are -contradicted or destroyed by each, and the numbers of logically -distinct variations of which the law is capable. There might be -also added, as a sixteenth case, that case where no special logical -condition exists, so that all the eight combinations remain. - - +---------+-------------------------------+-----------+------------+ - | | | Number of | Number of | - |Reference| Propositions expressing the | distinct |combinations| - | Number. | general type of the logical | logical |contradicted| - | | conditions. |variations.| by each. | - +---------+-------------------------------+-----------+------------+ - | I. | A = B | 6 | 4 | - | II. | A = AB | 12 | 2 | - | III. | A = B, B = C | 4 | 6 | - | IV. | A = B, B = BC | 24 | 5 | - | V. | A = AB, B = BC | 24 | 4 | - | VI. | A = BC | 24 | 4 | - | VII. | A = ABC | 24 | 3 | - | VIII. | AB = ABC | 8 | 1 | - | IX. | A = AB, *a*B = *a*B*c* | 24 | 3 | - | X. | A = ABC, *ab* = *ab*C | 8 | 4 | - | XI. | AB = ABC, *ab* = *abc* | 4 | 2 | - | XII. | AB = AC | 12 | 2 | - | XIII. | A = BC ꖌ A*bc* | 8 | 3 | - | XIV. | A = BC ꖌ *bc* | 2 | 4 | - | XV. | A = ABC, *a* = B*c* ꖌ *b*C | 8 | 5 | - +---------+-------------------------------+-----------+------------+ - -There are sixty-three series of combinations derived from -self-contradictory premises, which with 192, the sum of the numbers of -distinct logical variations stated in the third column of the table, -and with the one case where there are no conditions or laws at all, -make up the whole conceivable number of 256 series. - -We learn from this table, for instance, that two propositions of -the form A = AB, B = BC, which are such as constitute the premises -of the old syllogism Barbara, exclude as impossible four of the -eight combinations in which three terms may be united, and that -these propositions are capable of taking twenty-four variations by -transpositions of the terms or the introduction of negatives. This -table then presents the results of a complete analysis of all the -possible logical relations arising in the case of three terms, and the -old syllogism forms but one out of fifteen typical forms. Generally -speaking, every form can be converted into apparently different -propositions; thus the fourth type A = B, B = BC may appear in the -form A = ABC, *a* = *ab*, or again in the form of three propositions -A = AB, B = BC, *a*B = *a*B*c*; but all these sets of premises yield -identically the same series of combinations, and are therefore of -equivalent logical meaning. The fifth type, or Barbara, can also be -thrown into the equivalent forms A = ABC, *a*B = *a*BC and A = AC, -B = A ꖌ *a*BC. In other cases I have obtained the very same logical -conditions in four modes of statements. As regards mere appearance and -form of statement, the number of possible premises would be very great, -and difficult to exhibit exhaustively. - -The most remarkable of all the types of logical condition is the -fourteenth, namely, A = BC ꖌ *bc*. It is that which expresses the -division of a genus into two doubly marked species, and might be -illustrated by the example--“Component of the physical universe = -matter, gravitating, or not-matter (ether), not-gravitating.” It is -capable of only two distinct logical variations, namely, A = BC ꖌ *bc* -and A = B*c* ꖌ *b*C. By transposition or negative change of the letters -we can indeed obtain six different expressions of each of these -propositions; but when their meanings are analysed, by working out the -combinations, they are found to be logically equivalent to one or other -of the above two. Thus the proposition A = BC ꖌ *bc* can be written in -any of the following five other modes, - - *a* = *b*C ꖌ B*c*, B = CA ꖌ *ca*, *b* = *c*A ꖌ C*a*, - C = AB ꖌ *ab*, *c* = *a*B ꖌ A*b*. - -I do not think it needful to publish at present the complete table of -193 series of combinations and the premises corresponding to each. Such -a table enables us by mere inspection to learn the laws obeyed by any -set of combinations of three things, and is to logic what a table of -factors and prime numbers is to the theory of numbers, or a table of -integrals to the higher mathematics. The table already given (p. 140) -would enable a person with but little labour to discover the law of any -combinations. If there be seven combinations (one contradicted) the law -must be of the eighth type, and the proper variety will be apparent. -If there be six combinations (two contradicted), either the second, -eleventh, or twelfth type applies, and a certain number of trials will -disclose the proper type and variety. If there be but two combinations -the law must be of the third type, and so on. - -The above investigations are complete as regards the possible logical -relations of two or three terms. But when we attempt to apply the -same kind of method to the relations of four or more terms, the labour -becomes impracticably great. Four terms give sixteen combinations -compatible with the laws of thought, and the number of possible -selections of combinations is no less than 2^{16} or 65,536. The -following table shows the extraordinary manner in which the number of -possible logical relations increases with the number of terms involved. - - +---------+-------------+---------------------------------------+ - |Number of| Number of |Number of possible selections of combi-| - | terms. | possible | nations corresponding to consistent | - | |combinations.| or inconsistent logical relations. | - +-----------------------+---------------------------------------+ - | 2 | 4 | 16 | - | 3 | 8 | 256 | - | 4 | 16 | 65,536 | - | 5 | 32 | 4,294,967,296 | - | 6 | 64 | 18,446,744,073,709,551,616 | - +---------+-------------+---------------------------------------+ - -Some years of continuous labour would be required to ascertain the -types of laws which may govern the combinations of only four things, -and but a small part of such laws would be exemplified or capable of -practical application in science. The purely logical inverse problem, -whereby we pass from combinations to their laws, is solved in the -preceding pages, as far as it is likely to be for a long time to come; -and it is almost impossible that it should ever be carried more than a -single step further. - -In the first edition, vol i. p. 158, I stated that I had not been -able to discover any mode of calculating the number of cases in which -inconsistency would be implied in the selection of combinations from -the Logical Alphabet. The logical complexity of the problem appeared -to be so great that the ordinary modes of calculating numbers of -combinations failed, in my opinion, to give any aid, and exhaustive -examination of the combinations in detail seemed to be the only method -applicable. This opinion, however, was mistaken, for both Mr. R. B. -Hayward, of Harrow, and Mr. W. H. Brewer have calculated the numbers -of inconsistent cases both for three and for four terms, without much -difficulty. In the case of four terms they find that there are 1761 -inconsistent selections and 63,774 consistent, which with one case -where no condition exists, make up the total of 65,536 possible -selections. - -The inconsistent cases are distributed in the manner shown in the -following table:-- - - +--------------+---------------------------------------------------+ - | Number of | | - | Combinations | 0 1 2 3 4 5 6 7 8 9 10, &c. | - | remaining. | | - +--------------+---------------------------------------------------+ - | Number of | | - | Inconsistent | 1 16 112 352 536 448 224 64 8 0 0, &c. | - | Cases. | | - +--------------+---------------------------------------------------+ - -When more than eight combinations of the Logical Alphabet (p. 94, -column V.) remain unexcluded, there cannot be inconsistency. The whole -numbers of ways of selecting 0, 1, 2, &c., combinations out of 16 are -given in the 17th line of the Arithmetical Triangle given further on in -the Chapter on Combinations and Permutations, the sum of the numbers in -that line being 65,536. - - -*Professor Clifford on the Types of Compound Statement involving Four -Classes.* - -In the first edition (vol. i. p. 163), I asserted that some years of -labour would be required to ascertain even the precise number of types -of law governing the combinations of four classes of things. Though I -still believe that some years’ labour would be required to work out the -types themselves, it is clearly a mistake to suppose that the *numbers* -of such types cannot be calculated with a reasonable amount of labour, -Professor W. K. Clifford having actually accomplished the task. His -solution of the numerical problem involves the use of a complete new -system of nomenclature and is far too intricate to be fully described -here. I can only give a brief abstract of the results, and refer -readers, who wish to follow out the reasoning, to the Proceedings of -the Literary and Philosophical Society of Manchester, for the 9th -January, 1877, vol. xvi., p. 88, where Professor Clifford’s paper is -printed in full. - -By a *simple statement* Professor Clifford means the denial of the -existence of any single combination or *cross-division*, of the -classes, as in ABCD = 0, or A*b*C*d* = 0. The denial of two or more -such combinations is called a *compound statement*, and is further said -to be *twofold*, *threefold*, &c., according to the number denied. Thus -ABC = 0 is a twofold compound statement in regard to four classes, -because it involves both ABCD = 0 and ABC*d* = 0. When two compound -statements can be converted into one another by interchange of the -classes, A, B, C, D, with each other or with their complementary -classes, *a*, *b*, *c*, *d*, they are called *similar*, and all similar -statements are said to belong to the same *type*. - -Two statements are called *complementary* when they deny between them -all the sixteen combinations without both denying any one; or, which -is the same thing, when each denies just those combinations which -the other permits to exist. It is obvious that when two statements -are similar, the complementary statements will also be similar, -and consequently for every type of *n*-fold statement, there is a -complementary type of (16--*n*)-fold statement. It follows that we need -only enumerate the types as far as the eighth order; for the types -of more-than-eight-fold statement will already have been given as -complementary to types of lower orders. - -One combination, ABCD, may be converted into another A*b*C*d* by -interchanging one or more of the classes with the complementary -classes. The number of such changes is called the *distance*, which in -the above case is 2. In two similar compound statements the distances -of the combinations denied must be the same; but it does not follow -that when all the distances are the same, the statements are similar. -There is, however, only one example of two dissimilar statements having -the same distances. When the distance is 4, the two combinations -are said to be *obverse* to one another, and the statements denying -them are called *obverse statements*, as in ABCD = 0 and *abcd* = 0 -or again A*b*C*d* = 0 and *a*B*c*D = 0. When any one combination is -given, called the *origin*, all the others may be grouped in respect -of their relations to it as follows:--Four are at distance *one* from -it, and may be called *proximates*; six are at distance *two*, and may -be called *mediates*; four are at distance *three*, and may be called -*ultimates*; finally the obverse is at distance *four*. - - Origin and Six Obverse and - four proximates. mediates. four ultimates. - *ab*CD - | - *a*BCD A*bc*D | A*b*C*d* A*bcd* - | \ | / | - | \ | / | - | \|/ | - ABC*d*--ABCD--A*b*CD + *abc*D--*abcd*--*a*B*cd* - | /|\ | - | / | \ | - | / | \ | - AB*c*D *a*B*c*D | *a*BC*d* *ab*C*d*. - | - AB*cd* - -It will be seen that the four proximates are respectively obverse to -the four ultimates, and that the mediates form three pairs of obverses. -Every proximate or ultimate is distant 1 and 3 respectively from such a -pair of mediates. - -Aided by this system of nomenclature Professor Clifford proceeds to an -exhaustive enumeration of types, in which it is impossible to follow -him. The results are as follows:-- - - 1-fold statements 1 type } - 2 " " 4 types} - 3 " " 6 " } - 4 " " 19 " } 159 - 5 " " 27 " } - 6 " " 47 " } - 7 " " 55 " } - 8-fold statements 78 " - -Now as each seven-fold or less-than-seven-fold statement is -complementary to a nine-fold or more-than-nine-fold statement, it -follows that the complete number of types will be 159 × 2 + 78 = 396. - -It appears then that the types of statement concerning four classes -are only about 26 times as numerous as those concerning three classes, -fifteen in number, although the number of possible combinations is 256 -times as great. - -Professor Clifford informs me that the knowledge of the possible -groupings of subdivisions of classes which he obtained by this inquiry -has been of service to him in some applications of hyper-elliptic -functions to which he has subsequently been led. Professor Cayley has -since expressed his opinion that this line of investigation should -be followed out, owing to the bearing of the theory of compound -combinations upon the higher geometry.[86] It seems likely that many -unexpected points of connection will in time be disclosed between the -sciences of logic and mathematics. - - [86] *Proceedings of the Manchester Literary and Philosophical - Society*, 6th February, 1877, vol. xvi., p. 113. - - -*Distinction between Perfect and Imperfect Induction.* - -We cannot proceed with advantage before noticing the extreme difference -which exists between cases of perfect and those of imperfect induction. -We call an induction *perfect* when all the objects or events which -can possibly come under the class treated have been examined. But in -the majority of cases it is impossible to collect together, or in any -way to investigate, the properties of all portions of a substance or -of all the individuals of a race. The number of objects would often -be practically infinite, and the greater part of them might be beyond -our reach, in the interior of the earth, or in the most distant parts -of the Universe. In all such cases induction is *imperfect*, and is -affected by more or less uncertainty. As some writers have fallen into -much error concerning the functions and relative importance of these -two branches of reasoning, I shall have to point out that-- - - 1. Perfect Induction is a process absolutely requisite, both in the - performance of imperfect induction and in the treatment of large - bodies of facts of which our knowledge is complete. - - 2. Imperfect Induction is founded on Perfect Induction, but involves - another process of inference of a widely different character. - -It is certain that if I can draw any inference at all concerning -objects not examined, it must be done on the data afforded by the -objects which have been examined. If I judge that a distant star obeys -the law of gravity, it must be because all other material objects -sufficiently known to me obey that law. If I venture to assert that -all ruminant animals have cloven hoofs, it is because all ruminant -animals which have come under my notice have cloven hoofs. On the -other hand, I cannot safely say that all cryptogamous plants possess -a purely cellular structure, because some cryptogamous plants, which -have been examined by botanists, have a partially vascular structure. -The probability that a new cryptogam will be cellular only can be -estimated, if at all, on the ground of the comparative numbers of -known cryptogams which are and are not cellular. Thus the first step -in every induction will consist in accurately summing up the number -of instances of a particular phenomenon which have fallen under our -observation. Adams and Leverrier, for instance, must have inferred -that the undiscovered planet Neptune would obey Bode’s law, because -*all the planets known at that time obeyed it*. On what principles the -passage from the known to the apparently unknown is warranted, must be -carefully discussed in the next section, and in various parts of this -work. - -It would be a great mistake, however, to suppose that Perfect Induction -is in itself useless. Even when the enumeration of objects belonging -to any class is complete, and admits of no inference to unexamined -objects, the statement of our knowledge in a general proposition is a -process of so much importance that we may consider it necessary. In -many cases we may render our investigations exhaustive; all the teeth -or bones of an animal; all the cells in a minute vegetable organ; all -the caves in a mountain side; all the strata in a geological section; -all the coins in a newly found hoard, may be so completely scrutinized -that we may make some general assertion concerning them without fear -of mistake. Every bone might be proved to contain phosphate of lime; -every cell to enclose a nucleus; every cave to hide remains of extinct -animals; every stratum to exhibit signs of marine origin; every coin -to be of Roman manufacture. These are cases where our investigation -is limited to a definite portion of matter, or a definite area on the -earth’s surface. - -There is another class of cases where induction is naturally and -necessarily limited to a definite number of alternatives. Of the -regular solids we can say without the least doubt that no one has -more than twenty faces, thirty edges, and twenty corners; for by the -principles of geometry we learn that there cannot exist more than five -regular solids, of each of which we easily observe that the above -statements are true. In the theory of numbers, an endless variety of -perfect inductions might be made; we can show that no number less than -sixty possesses so many divisors, and the like is true of 360; for it -does not require a great amount of labour to ascertain and count all -the divisors of numbers up to sixty or 360. I can assert that between -60,041 and 60,077 no prime number occurs, because the exhaustive -examination of those who have constructed tables of prime numbers -proves it to be so. - -In matters of human appointment or history, we can frequently have -a complete limitation of the number of instances to be included in -an induction. We might show that the propositions of the third book -of Euclid treat only of circles; that no part of the works of Galen -mentions the fourth figure of the syllogism; that none of the other -kings of England reigned so long as George III.; that Magna Charta has -not been repealed by any subsequent statute; that the price of corn in -England has never been so high since 1847 as it was in that year; that -the price of the English funds has never been lower than it was on the -23rd of January, 1798, when it fell to 47-1/4. - -It has been urged against this process of Perfect Induction that it -gives no new information, and is merely a summing up in a brief form -of a multitude of particulars. But mere abbreviation of mental labour -is one of the most important aids we can enjoy in the acquisition -of knowledge. The powers of the human mind are so limited that -multiplicity of detail is alone sufficient to prevent its progress -in many directions. Thought would be practically impossible if every -separate fact had to be separately thought and treated. Economy of -mental power may be considered one of the main conditions on which our -elevated intellectual position depends. Mathematical processes are for -the most part but abbreviations of the simpler acts of addition and -subtraction. The invention of logarithms was one of the most striking -additions ever made to human power: yet it was a mere abbreviation of -operations which could have been done before had a sufficient amount -of labour been available. Similar additions to our power will, it -is hoped, be made from time to time; for the number of mathematical -problems hitherto solved is but an indefinitely small fraction of those -which await solution, because the labour they have hitherto demanded -renders them impracticable. So it is throughout all regions of thought. -The amount of our knowledge depends upon our power of bringing it -within practicable compass. Unless we arrange and classify facts and -condense them into general truths, they soon surpass our powers of -memory, and serve but to confuse. Hence Perfect Induction, even as a -process of abbreviation, is absolutely essential to any high degree of -mental achievement. - - -*Transition from Perfect to Imperfect Induction.* - -It is a question of profound difficulty on what grounds we are -warranted in inferring the future from the present, or the nature -of undiscovered objects from those which we have examined with our -senses. We pass from Perfect to Imperfect Induction when once we -allow our conclusion to apply, at all events apparently, beyond the -data on which it was founded. In making such a step we seem to gain -a net addition to our knowledge; for we learn the nature of what was -unknown. We reap where we have never sown. We appear to possess the -divine power of creating knowledge, and reaching with our mental arms -far beyond the sphere of our own observation. I shall have, indeed, to -point out certain methods of reasoning in which we do pass altogether -beyond the sphere of the senses, and acquire accurate knowledge which -observation could never have given; but it is not imperfect induction -that accomplishes such a task. Of imperfect induction itself, I venture -to assert that it never makes any real addition to our knowledge, in -the meaning of the expression sometimes accepted. As in other cases -of inference, it merely unfolds the information contained in past -observations; it merely renders explicit what was implicit in previous -experience. It transmutes, but certainly does not create knowledge. - -There is no fact which I shall more constantly keep before the reader’s -mind in the following pages than that the results of imperfect -induction, however well authenticated and verified, are never more than -probable. We never can be sure that the future will be as the present. -We hang ever upon the will of the Creator: and it is only so far as He -has created two things alike, or maintains the framework of the world -unchanged from moment to moment, that our most careful inferences can -be fulfilled. All predictions, all inferences which reach beyond their -data, are purely hypothetical, and proceed on the assumption that new -events will conform to the conditions detected in our observation of -past events. No experience of finite duration can give an exhaustive -knowledge of the forces which are in operation. There is thus a -double uncertainty; even supposing the Universe as a whole to proceed -unchanged, we do not really know the Universe as a whole. We know only -a point in its infinite extent, and a moment in its infinite duration. -We cannot be sure, then, that our observations have not escaped some -fact, which will cause the future to be apparently different from the -past; nor can we be sure that the future really will be the outcome of -the past. We proceed then in all our inferences to unexamined objects -and times on the assumptions-- - - 1. That our past observation gives us a complete knowledge of what - exists. - - 2. That the conditions of things which did exist will continue to be - the conditions which will exist. - -We shall often need to illustrate the character of our knowledge of -nature by the simile of a ballot-box, so often employed by mathematical -writers in the theory of probability. Nature is to us like an infinite -ballot-box, the contents of which are being continually drawn, ball -after ball, and exhibited to us. Science is but the careful observation -of the succession in which balls of various character present -themselves; we register the combinations, notice those which seem to -be excluded from occurrence, and from the proportional frequency of -those which appear we infer the probable character of future drawings. -But under such circumstances certainty of prediction depends on two -conditions:-- - - 1. That we acquire a perfect knowledge of the comparative numbers of - balls of each kind within the box. - - 2. That the contents of the ballot-box remain unchanged. - -Of the latter assumption, or rather that concerning the constitution -of the world which it illustrates, the logician or physicist can -have nothing to say. As the Creation of the Universe is necessarily -an act passing all experience and all conception, so any change in -that Universe, or, it may be, a termination of it, must likewise be -infinitely beyond the bounds of our mental faculties. No science -no reasoning upon the subject, can have any validity; for without -experience we are without the basis and materials of knowledge. It -is the fundamental postulate accordingly of all inference concerning -the future, that there shall be no arbitrary change in the subject -of inference; of the probability or improbability of such a change I -conceive that our faculties can give no estimate. - -The other condition of inductive inference--that we acquire an -approximately complete knowledge of the combinations in which events -do occur, is in some degree within our power. There are branches -of science in which phenomena seem to be governed by conditions of -a most fixed and general character. We have ground in such cases -for believing that the future occurrence of such phenomena can be -calculated and predicted. But the whole question now becomes one -of probability and improbability. We seem to leave the region of -logic to enter one in which the number of events is the ground of -inference. We do not really leave the region of logic; we only leave -that where certainty, affirmative or negative, is the result, and the -agreement or disagreement of qualities the means of inference. For the -future, number and quantity will commonly enter into our processes of -reasoning; but then I hold that number and quantity are but portions -of the great logical domain. I venture to assert that number is wholly -logical, both in its fundamental nature and in its developments. -Quantity in all its forms is but a development of number. That which is -mathematical is not the less logical; if anything it is more logical, -in the sense that it presents logical results in a higher degree of -complexity and variety. - -Before proceeding then from Perfect to Imperfect Induction I must -devote a portion of this work to treating the logical conditions -of number. I shall then employ number to estimate the variety of -combinations in which natural phenomena may present themselves, and -the probability or improbability of their occurrence under definite -circumstances. It is in later parts of the work that I must endeavour -to establish the notions which I have set forth upon the subject of -Imperfect Induction, as applied in the investigation of Nature, which -notions maybe thus briefly stated:-- - - 1. Imperfect Induction entirely rests upon Perfect Induction for its - materials. - - 2. The logical process by which we seem to pass directly from - examined to unexamined cases consists in an inverse application of - deductive inference, so that all reasoning may be said to be either - directly or inversely deductive. - - 3. The result is always of a hypothetical character, and is never - more than probable. - - 4. No net addition is ever made to our knowledge by reasoning; what - we know of future events or unexamined objects is only the unfolded - contents of our previous knowledge, and it becomes less probable as - it is more boldly extended to remote cases. - - - - -BOOK II. - -NUMBER, VARIETY, AND PROBABILITY. - - - - -CHAPTER VIII. - -PRINCIPLES OF NUMBER. - - -Not without reason did Pythagoras represent the world as ruled by -number. Into almost all our acts of thought number enters, and in -proportion as we can define numerically we enjoy exact and useful -knowledge of the Universe. The science of numbers, too, has hitherto -presented the widest and most practicable training in logic. So free -and energetic has been the study of mathematical forms, compared -with the forms of logic, that mathematicians have passed far in -advance of pure logicians. Occasionally, in recent times, they have -condescended to apply their algebraic instrument to a reflex treatment -of the primary logical science. It is thus that we owe to profound -mathematicians, such as John Herschel, Whewell, De Morgan, or Boole, -the regeneration of logic in the present century. I entertain no -doubt that it is in maintaining a close alliance with quantitative -reasoning that we must look for further progress in our comprehension -of qualitative inference. - -I cannot assent, indeed, to the common notion that certainty begins and -ends with numerical determination. Nothing is more certain than logical -truth. The laws of identity and difference are the tests of all that is -certain throughout the range of thought, and mathematical reasoning is -cogent only when it conforms to these conditions, of which logic is the -first development. And if it be erroneous to suppose that all certainty -is mathematical, it is equally an error to imagine that all which is -mathematical is certain. Many processes of mathematical reasoning are -of most doubtful validity. There are points of mathematical doctrine -which must long remain matter of opinion; for instance, the best form -of the definition and axiom concerning parallel lines, or the true -nature of a limit. In the use of symbolic reasoning questions occur on -which the best mathematicians may differ, as Bernoulli and Leibnitz -differed irreconcileably concerning the existence of the logarithms of -negative quantities.[87] In fact we no sooner leave the simple logical -conditions of number, than we find ourselves involved in a mazy and -mysterious science of symbols. - - [87] Montucla. *Histoire des Mathématiques*, vol. iii. p. 373. - -Mathematical science enjoys no monopoly, and not even a supremacy, -in certainty of results. It is the boundless extent and variety of -quantitative questions that delights the mathematical student. When -simple logic can give but a bare answer Yes or No, the algebraist -raises a score of subtle questions, and brings out a crowd of curious -results. The flower and the fruit, all that is attractive and -delightful, fall to the share of the mathematician, who too often -despises the plain but necessary stem from which all has arisen. In -no region of thought can a reasoner cast himself free from the prior -conditions of logical correctness. The mathematician is only strong and -true as long as he is logical, and if number rules the world, it is -logic which rules number. - -Nearly all writers have hitherto been strangely content to look upon -numerical reasoning as something apart from logical inference. A long -divorce has existed between quality and quantity, and it has not -been uncommon to treat them as contrasted in nature and restricted -to independent branches of thought. For my own part, I believe that -all the sciences meet somewhere. No part of knowledge can stand -wholly disconnected from other parts of the universe of thought; it -is incredible, above all, that the two great branches of abstract -science, interlacing and co-operating in every discourse, should -rest upon totally distinct foundations. I assume that a connection -exists, and care only to inquire, What is its nature? Does the science -of quantity rest upon that of quality; or, *vice versâ*, does the -science of quality rest upon that of quantity? There might conceivably -be a third view, that they both rest upon some still deeper set of -principles. - -It is generally supposed that Boole adopted the second view, and -treated logic as an application of algebra, a special case of -analytical reasoning which admits only two quantities, unity and zero. -It is not easy to ascertain clearly which of these views really was -accepted by Boole. In his interesting biographical sketch of Boole,[88] -the Rev. R. Harley protests against the statement that Boole’s logical -calculus imported the conditions of number and quantity into logic. -He says: “Logic is never identified or confounded with mathematics; -the two systems of thought are kept perfectly distinct, each being -subject to its own laws and conditions. The symbols are the same for -both systems, but they have not the same interpretation.” The Rev. J. -Venn, again, in his review of Boole’s logical system,[89] holds that -Boole’s processes are at bottom logical, not mathematical, though -stated in a highly generalized form and with a mathematical dress. But -it is quite likely that readers of Boole should be misled. Not only -have his logical works an entirely mathematical appearance, but I find -on p. 12 of his *Laws of Thought* the following unequivocal statement: -“That logic, as a science, is susceptible of very wide applications -is admitted; but it is equally certain that its ultimate forms and -processes are mathematical.” A few lines below he adds, “It is not of -the essence of mathematics to be conversant with the ideas of number -and quantity.” - - [88] *British Quarterly Review*, No. lxxxvii, July 1866. - - [89] *Mind*, October 1876, vol. i. p. 484. - -The solution of the difficulty is that Boole used the term mathematics -in a wider sense than that usually attributed to it. He probably -adopted the third view, so that his mathematical *Laws of Thought* are -the common basis both of logic and of quantitative mathematics. But -I do not care to pursue the subject because I think that, in either -case Boole was wrong. In my opinion logic is the superior science, the -general basis of mathematics as well as of all other sciences. Number -is but logical discrimination, and algebra a highly developed logic. -Thus it is easy to understand the deep analogy which Boole pointed out -between the forms of algebraic and logical deduction. Logic resembles -algebra as the mould resembles that which is cast in it. Boole mistook -the cast for the mould. Considering that logic imposes its own laws -upon every branch of mathematical science, it is no wonder that we -constantly meet with the traces of logical laws in mathematical -processes. - - -*The Nature of Number.* - -Number is but another name for *diversity*. Exact identity is unity, -and with difference arises plurality. An abstract notion, as was -pointed out (p. 28), possesses a certain *oneness*. The quality of -*justice*, for instance, is one and the same in whatever just acts it -is manifested. In justice itself there are no marks of difference by -which to discriminate justice from justice. But one just act can be -discriminated from another just act by circumstances of time and place, -and we can count many acts thus discriminated each from each. In like -manner pure gold is simply pure gold, and is so far one and the same -throughout. But besides its intrinsic qualities, gold occupies space -and must have shape and size. Portions of gold are always mutually -exclusive and capable of discrimination, in respect that they must be -each without the other. Hence they may be numbered. - -Plurality arises when and only when we detect difference. For instance, -in counting a number of gold coins I must count each coin once, and not -more than once. Let C denote a coin, and the mark above it the order of -counting. Then I must count the coins - - C′ + C″ + C‴ + C″″ + .... - -If I were to count them as follows - - C′ + C″ + C‴ + C‴ + C″″ + ..., - -I should make the third coin into two, and should imply the existence -of difference where there is no difference.[90] C‴ and C‴ are but -the names of one coin named twice over. But according to one of the -conditions of logical symbols, which I have called the Law of Unity -(p. 72), the same name repeated has no effect, and - - A ꖌ A = A. - - [90] *Pure Logic*, Appendix, p. 82, § 192. - -We must apply the Law of Unity, and must reduce all identical -alternatives before we can count with certainty and use the processes -of numerical calculation. Identical alternatives are harmless in -logic, but are wholly inadmissible in number. Thus logical science -ascertains the nature of the mathematical unit, and the definition may -be given in these terms--*A unit is any object of thought which can be -discriminated from every other object treated as a unit in the same -problem.* - -It has often been said that units are units in respect of being -perfectly similar to each other; but though they may be perfectly -similar in some respects, they must be different in at least one point, -otherwise they would be incapable of plurality. If three coins were -so similar that they occupied the same space at the same time, they -would not be three coins, but one coin. It is a property of space that -every point is discriminable from every other point, and in time every -moment is necessarily distinct from any other moment before or after. -Hence we frequently count in space or time, and Locke, with some other -philosophers, has held that number arises from repetition in time. -Beats of a pendulum may be so perfectly similar that we can discover no -difference except that one beat is before and another after. Time alone -is here the ground of difference and is a sufficient foundation for the -discrimination of plurality; but it is by no means the only foundation. -Three coins are three coins, whether we count them successively or -regard them all simultaneously. In many cases neither time nor space -is the ground of difference, but pure quality alone enters. We can -discriminate the weight, inertia, and hardness of gold as three -qualities, though none of these is before nor after the other, neither -in space nor time. Every means of discrimination may be a source of -plurality. - -Our logical notation may be used to express the rise of number. -The symbol A stands for one thing or one class, and in itself must -be regarded as a unit, because no difference is specified. But the -combinations AB and A*b* are necessarily *two*, because they cannot -logically coalesce, and there is a mark B which distinguishes one -from the other. A logical definition of the number *four* is given in -the combinations ABC, AB*c*, A*b*C, A*bc*, where there is a double -difference. As Puck says-- - - “Yet but three? Come one more; - Two of both kinds makes up four.” - -I conceive that all numbers might be represented as arising out of -the combinations of the Logical Alphabet, more or less of each series -being struck out by various logical conditions. The number three, for -instance, arises from the condition that A must be either B or C, so -that the combinations are ABC, AB*c*, A*b*C. - - -*Of Numerical Abstraction.* - -There will now be little difficulty in forming a clear notion of -the nature of numerical abstraction. It consists in abstracting the -character of the difference from which plurality arises, retaining -merely the fact. When I speak of *three men* I need not at once specify -the marks by which each may be known from each. Those marks must exist -if they are really three men and not one and the same, and in speaking -of them as many I imply the existence of the requisite differences. -Abstract number, then, is *the empty form of difference*; the abstract -number *three* asserts the existence of marks without specifying their -kind. - -Numerical abstraction is thus seen to be a different process from -logical abstraction (p. 27), for in the latter process we drop out -of notice the very existence of difference and plurality. In forming -the abstract notion *hardness*, we ignore entirely the diverse -circumstances in which the quality may appear. It is the concrete -notion *three hard objects*, which asserts the existence of hardness -along with sufficient other undefined qualities, to mark out *three* -such objects. Numerical thought is indeed closely interwoven with -logical thought. We cannot use a concrete term in the plural, as -*men*, without implying that there are marks of difference. But when we -use an abstract term, we deal with unity. - -The origin of the great generality of number is now apparent. Three -sounds differ from three colours, or three riders from three horses; -but they agree in respect of the variety of marks by which they can be -discriminated. The symbols 1 + 1 + 1 are thus the empty marks asserting -the existence of discrimination. But in dropping out of sight the -character of the differences we give rise to new agreements on which -mathematical reasoning is founded. Numerical abstraction is so far from -being incompatible with logical abstraction that it is the origin of -our widest acts of generalization. - - -*Concrete and Abstract Number.* - -The common distinction between concrete and abstract number can now be -easily stated. In proportion as we specify the logical characters of -the things numbered, we render them concrete. In the abstract number -three there is no statement of the points in which the *three* objects -agree; but in *three coins*, *three men*, or *three horses*, not only -are the objects numbered but their nature is restricted. Concrete -number thus implies the same consciousness of difference as abstract -number, but it is mingled with a groundwork of similarity expressed in -the logical terms. There is identity so far as logical terms enter; -difference so far as the terms are merely numerical. - -The reason of the important Law of Homogeneity will now be apparent. -This law asserts that in every arithmetical calculation the logical -nature of the things numbered must remain unaltered. The specified -logical agreement of the things must not be affected by the unspecified -numerical differences. A calculation would be palpably absurd which, -after commencing with length, gave a result in hours. It is equally -absurd, in a purely arithmetical point of view, to deduce areas from -the calculation of lengths, masses from the combination of volume -and density, or momenta from mass and velocity. It must remain for -subsequent consideration to decide in what sense we may truly say that -two linear feet multiplied by two linear feet give four superficial -feet; arithmetically it is absurd, because there is a change of unit. - -As a general rule we treat in each calculation only objects of one -nature. We do not, and cannot properly add, in the same sum yards of -cloth and pounds of sugar. We cannot even conceive the result of adding -area to velocity, or length to density, or weight to value. The units -added must have a basis of homogeneity, or must be reducible to some -common denominator. Nevertheless it is possible, and in fact common, to -treat in one complex calculation the most heterogeneous quantities, on -the condition that each kind of object is kept distinct, and treated -numerically only in conjunction with its own kind. Different units, -so far as their logical differences are specified, must never be -substituted one for the other. Chemists continually use equations which -assert the equivalence of groups of atoms. Ordinary fermentation is -represented by the formula - - C^{6} H^{12} O^{6} = 2C^{2} H^{6} O + 2CO^{2}. - -Three kinds of units, the atoms respectively of carbon, hydrogen, and -oxygen, are here intermingled, but there is really a separate equation -in regard to each kind. Mathematicians also employ compound equations -of the same kind; for in, *a* + *b* √ - 1 = *c* + *d* √ - 1, -it is impossible by ordinary addition to add *a* to *b* √ - 1. -Hence we really have the separate equations *a* = *b*, and -*c* √ - 1 = *d* √ - 1. Similarly an equation between -two quaternions is equivalent to four equations between ordinary -quantities, whence indeed the name *quaternion*. - - -*Analogy of Logical and Numerical Terms.* - -If my assertion is correct that number arises out of logical -conditions, we ought to find number obeying all the laws of logic. -It is almost superfluous to point out that this is the case with the -fundamental laws of identity and difference, and it only remains to -show that mathematical symbols do really obey the special conditions -of logical symbols which were formerly pointed out (p. 32). Thus the -Law of Commutativeness, is equally true of quality and quantity. As in -logic we have - - AB = BA, - -so in mathematics it is familiarly known that - - 2 × 3 = 3 × 2, or *x* × *y* = *y* × *x*. - -The properties of space are as indifferent in multiplication as we -found them in pure logical thought. - -Similarly, as in logic - - triangle or square = square or triangle, - - or generally A ꖌ B = B ꖌ A, - so in quantity 2 + 3 = 3 + 2, - or generally *x* + *y* = *y* + *x*. - -The symbol ꖌ is not identical with +, but it is thus far analogous. - -How far, now, is it true that mathematical symbols obey the Law of -Simplicity expressed in the form - - AA = A, - -or the example - - Round round = round? - -Apparently there are but two numbers which obey this law; for it is -certain that - - *x* × *x* = *x* - -is true only in the two cases when *x* = 1, or *x* = 0. - -In reality all numbers obey the law, for 2 × 2 = 2 is not really -analogous to AA = A. According to the definition of a unit already -given, each unit is discriminated from each other in the same problem, -so that in 2′ × 2″, the first *two* involves a different discrimination -from the second *two*. I get four kinds of things, for instance, if I -first discriminate “heavy and light” and then “cubical and spherical,” -for we now have the following classes-- - - heavy, cubical. light, cubical. - heavy, spherical. light, spherical. - -But suppose that my two classes are in both cases discriminated by the -same difference of light and heavy, then we have - - heavy heavy = heavy, - heavy light = 0, - light heavy = 0, - light light = light. - -Thus, (heavy or light) × (heavy or light) = (heavy or light). - -In short, *twice two is two* unless we take care that the second two -has a different meaning from the first. But under similar circumstances -logical terms give the like result, and it is not true that A′A″ = A′, -when A″ is different in meaning from A′. - -In a similar manner it may be shown that the Law of Unity - - A ꖌ A = A. - -holds true alike of logical and mathematical terms. It is absurd indeed -to say that - - *x* + *x* = *x* - -except in the one case when *x* = absolute zero. But this contradiction -*x* + *x* = *x* arises from the fact that we have already defined -the units in one x as differing from those in the other. Under such -circumstances the Law of Unity does not apply. For if in - - A′ ꖌ A″ = A′ - -we mean that A″ is in any way different from A′ the assertion of -identity is evidently false. - -The contrast then which seems to exist between logical and mathematical -symbols is only apparent. It is because the Laws of Simplicity and -Unity must always be observed in the operation of counting that those -laws seem no further to apply. This is the understood condition under -which we use all numerical symbols. Whenever I write the symbol 5 I -really mean - - 1 + 1 + 1 + 1 + 1, - -and it is perfectly understood that each of these units is distinct -from each other. If requisite I might mark them thus - - 1′+ 1″ + 1‴ + 1″″ + 1″‴. - - -Were this not the case and were the units really - - 1′ + 1″ + 1″ + 1‴ + 1″″, - -the Law of Unity would, as before remarked, apply, and - - 1″ + 1″ = 1″. - -Mathematical symbols then obey all the laws of logical symbols, but -two of these laws seem to be inapplicable simply because they are -presupposed in the definition of the mathematical unit. Logic thus lays -down the conditions of number, and the science of arithmetic developed -as it is into all the wondrous branches of mathematical calculus is but -an outgrowth of logical discrimination. - - -*Principle of Mathematical Inference.* - -The universal principle of all reasoning, as I have asserted, is that -which allows us to substitute like for like. I have now to point out -how in the mathematical sciences this principle is involved in each -step of reasoning. It is in these sciences indeed that we meet with the -clearest cases of substitution, and it is the simplicity with which the -principle can be applied which probably led to the comparatively early -perfection of the sciences of geometry and arithmetic. Euclid, and -the Greek mathematicians from the first, recognised *equality* as the -fundamental relation of quantitative thought, but Aristotle rejected -the exactly analogous, but far more general relation of identity, and -thus crippled the formal science of logic as it has descended to the -present day. - -Geometrical reasoning starts from the axiom that “things equal to the -same thing are equal to each other.” Two equalities enable us to infer -a third equality; and this is true not only of lines and angles, but -of areas, volumes, numbers, intervals of time, forces, velocities, -degrees of intensity, or, in short, anything which is capable of being -equal or unequal. Two stars equally bright with the same star must be -equally bright with each other, and two forces equally intense with a -third force are equally intense with each other. It is remarkable that -Euclid has not explicitly stated two other axioms, the truth of which -is necessarily implied. The second axiom should be that “Two things of -which one is equal and the other unequal to a third common thing, are -unequal to each other.” An equality and inequality, in short, give an -inequality, and this is equally true with the first axiom of all kinds -of quantity. If Venus, for instance, agrees with Mars in density, but -Mars differs from Jupiter, then Venus differs from Jupiter. A third -axiom must exist to the effect that “Things unequal to the same thing -may or may not be equal to each other.” *Two inequalities give no -ground of inference whatever.* If we only know, for instance, that -Mercury and Jupiter differ in density from Mars, we cannot say whether -or not they agree between themselves. As a fact they do not agree; -but Venus and Mars on the other hand both differ from Jupiter and yet -closely agree with each other. The force of the axioms can be most -clearly illustrated by drawing equal and unequal lines.[91] - - [91] *Elementary Lessons in Logic* (Macmillan), p. 123. It is pointed - out in the preface to this Second Edition, that the views here given - were partially stated by Leibnitz. - -The general conclusion then must be that where there is equality there -may be inference, but where there is not equality there cannot be -inference. A plain induction will lead us to believe that *equality is -the condition of inference concerning quantity*. All the three axioms -may in fact be summed up in one, to the effect, that “*in whatever -relation one quantity stands to another, it stands in the same relation -to the equal of that other*.” - -The active power is always the substitution of equals, and it is an -accident that in a pair of equalities we can make the substitution -in two ways. From *a* = *b* = *c* we can infer *a* = *c*, either by -substituting in *a* = *b* the value of *b* as given in *b* = *c*, -or else by substituting in *b* = *c* the value of *b* as given in -*a* = *b*. In *a* = *b* ~ *d* we can make but the one substitution of -*a* for *b*. In *e* ~ *f* ~ *g* we can make no substitution and get no -inference. - -In mathematics the relations in which terms may stand to each other are -far more varied than in pure logic, yet our principle of substitution -always holds true. We may say in the most general manner that *In -whatever relation one quantity stands to another, it stands in the same -relation to the equal of that other.* In this axiom we sum up a number -of axioms which have been stated in more or less detail by algebraists. -Thus, “If equal quantities be added to equal quantities, the sums will -be equal.” To explain this, let - - *a* = *b*, *c* = *d*. - -Now *a* + *c*, whatever it means, must be identical with itself, so that - - *a* + *c* = *a* + *c*. - -In one side of this equation substitute for the quantities their -equivalents, and we have the axiom proved - - *a* + *c* = *b* + *d*. - -The similar axiom concerning subtraction is equally evident, for -whatever *a* - *c* may mean it is equal to *a* - *c*, and therefore by -substitution to *b* - *d*. Again, “if equal quantities be multiplied by -the same or equal quantities, the products will be equal,” For evidently - - *ac* = *ac*, - -and if for *c* in one side we substitute its equal *d*, we have - - *ac* = *ad*, - -and a second similar substitution gives us - - *ac* = *bd*. - -We might prove a like axiom concerning division in an exactly -similar manner. I might even extend the list of axioms and say that -“Equal powers of equal numbers are equal.” For certainly, whatever -*a* × *a* × *a* may mean, it is equal to *a* × *a* × *a*; hence by our -usual substitution it is equal to *b* × *b* × *b*. The same will be -true of roots of numbers and ^{c}√*a* = ^{d}√*b* provided that -the roots are so taken that the root of *a* shall really be related -to *a* as the root of *b* is to *b*. The ambiguity of meaning of an -operation thus fails in any way to shake the universality of the -principle. We may go further and assert that, not only the above common -relations, but all other known or conceivable mathematical relations -obey the same principle. Let Q*a* denote in the most general manner -that we do something with the quantity *a*; then if *a* = *b* it -follows that - - Q*a* = Q*b*. - -The reader will also remember that one of the most frequent operations -in mathematical reasoning is to substitute for a quantity its equal, -as known either by assumed, natural, or self-evident conditions. -Whenever a quantity appears twice over in a problem, we may apply -what we learn of its relations in one place to its relations in the -other. All reasoning in mathematics, as in other branches of science, -thus involves the principle of treating equals equally, or similars -similarly. In whatever way we employ quantitative reasoning in the -remaining parts of this work, we never can desert the simple principle -on which we first set out. - - -*Reasoning by Inequalities.* - -I have stated that all the processes of mathematical reasoning may -be deduced from the principle of substitution. Exceptions to this -assertion may seem to exist in the use of inequalities. The greater of -a greater is undoubtedly a greater, and what is less than a less is -certainly less. Snowdon is higher than the Wrekin, and Ben Nevis than -Snowdon; therefore Ben Nevis is higher than the Wrekin. But a little -consideration discloses sufficient reason for believing that even in -such cases, where equality does not apparently enter, the force of the -reasoning entirely depends upon underlying and implied equalities. - -In the first place, two statements of mere difference do not give -any ground of inference. We learn nothing concerning the comparative -heights of St. Paul’s and Westminster Abbey from the assertions that -they both differ in height from St. Peter’s at Rome. We need something -more than inequality; we require one identity in addition, namely the -identity in direction of the two differences. Thus we cannot employ -inequalities in the simple way in which we do equalities, and, when we -try to express what other conditions are requisite, we find ourselves -lapsing into the use of equalities or identities. - -In the second place, every argument by inequalities may be represented -in the form of equalities. We express that *a* is greater than *b* by -the equation - - *a* = *b* + *p*, (1) - -where *p* is an intrinsically positive quantity, denoting the -difference of *a* and *b*. Similarly we express that *b* is greater -than *c* by the equation - - *b* = *c* + *q*, (2) - -and substituting for *b* in (1) its value in (2) we have - - *a* = *c* + *q* + *p*. (3) - -Now as *p* and *q* are both positive, it follows that *a* is greater -than *c*, and we have the exact amount of excess specified. It will be -easily seen that the reasoning concerning that which is less than a -less will result in an equation of the form - - *c* = *a* - *r* - *s*. - -Every argument by inequalities may then be thrown into the form of an -equality; but the converse is not true. We cannot possibly prove that -two quantities are equal by merely asserting that they are both greater -or both less than another quantity. From *e* > *f* and *g* > *f*, or -*e* < *f* and *g* < *f*, we can infer no relation between *e* and *g*. -And if the reader take the equations *x* = *y* = 3 and attempt to prove -that therefore *x* = 3, by throwing them into inequalities, he will -find it impossible to do so. - -From these considerations I gather that reasoning in arithmetic or -algebra by so-called inequalities, is only an imperfectly expressed -reasoning by equalities, and when we want to exhibit exactly and -clearly the conditions of reasoning, we are obliged to use equalities -explicitly. Just as in pure logic a negative proposition, as expressing -mere difference, cannot be the means of inference, so inequality can -never really be the true ground of inference. I do not deny that -affirmation and negation, agreement and difference, equality and -inequality, are pairs of equally fundamental relations, but I assert -that inference is possible only where affirmation, agreement, or -equality, some species of identity in fact, is present, explicitly or -implicitly. - - -*Arithmetical Reasoning.* - -It may seem somewhat inconsistent that I assert number to arise out of -difference or discrimination, and yet hold that no reasoning can be -grounded on difference. Number, of course, opens a most wide sphere -for inference, and a little consideration shows that this is due to -the unlimited series of identities which spring up out of numerical -abstraction. If six people are sitting on six chairs, there is no -resemblance between the chairs and the people in logical character. -But if we overlook all the qualities both of a chair and a person and -merely remember that there are marks by which each of six chairs may -be discriminated from the others, and similarly with the people, then -there arises a resemblance between the chairs and the people, and this -resemblance in number may be the ground of inference. If on another -occasion the chairs are filled by people again, we may infer that these -people resemble the others in number though they need not resemble them -in any other points. - -Groups of units are what we really treat in arithmetic. The number -*five* is really 1 + 1 + 1 + 1 + 1, but for the sake of conciseness we -substitute the more compact sign 5, or the name *five*. These names -being arbitrarily imposed in any one manner, an infinite variety of -relations spring up between them which are not in the least arbitrary. -If we define *four* as 1 + 1 + 1 + 1, and *five* as 1 + 1 + 1 + 1 + 1, -then of course it follows that *five* = *four* + 1; but it would be -equally possible to take this latter equality as a definition, in -which case one of the former equalities would become an inference. It -is hardly requisite to decide how we define the names of numbers, -provided we remember that out of the infinitely numerous relations -of one number to others, some one relation expressed in an equality -must be a definition of the number in question and the other relations -immediately become necessary inferences. - -In the science of number the variety of classes which can be formed is -altogether infinite, and statements of perfect generality may be made -subject only to difficulty or exception at the lower end of the scale. -Every existing number for instance belongs to the class *m* + 7; that -is, every number must be the sum of another number and seven, except of -course the first six or seven numbers, negative quantities not being -here taken into account. Every number is the half of some other, and so -on. The subject of generalization, as exhibited in mathematical truths, -is an infinitely wide one. In number we are only at the first step of -an extensive series of generalizations. As number is general compared -with the particular things numbered, so we have general symbols for -numbers, and general symbols for relations between undetermined -numbers. There is an unlimited hierarchy of successive generalizations. - - -*Numerically Definite Reasoning.* - -It was first discovered by De Morgan that many arguments are valid -which combine logical and numerical reasoning, although they cannot be -included in the ancient logical formulas. He developed the doctrine of -the “Numerically Definite Syllogism,” fully explained in his *Formal -Logic* (pp. 141–170). Boole also devoted considerable attention to the -determination of what he called “Statistical Conditions,” meaning the -numerical conditions of logical classes. In a paper published among the -Memoirs of the Manchester Literary and Philosophical Society, Third -Series, vol. IV. p. 330 (Session 1869–70), I have pointed out that we -can apply arithmetical calculation to the Logical Alphabet. Having -given certain logical conditions and the numbers of objects in certain -classes, we can either determine the numbers of objects in other -classes governed by those conditions, or can show what further data -are required to determine them. As an example of the kind of questions -treated in numerical logic, and the mode of treatment, I give the -following problem suggested by De Morgan, with my mode of representing -its solution. - -“For every man in the house there is a person who is aged; some of the -men are not aged. It follows that some persons in the house are not -men.”[92] - - [92] *Syllabus of a Proposed System of Logic*, p. 29. - - Now let A = person in house, - B = male, - C = aged. - -By enclosing a logical symbol in brackets, let us denote the number of -objects belonging to the class indicated by the symbol. Thus let - - (A) = number of persons in house, - (AB) = number of male persons in house, - (ABC) = number of aged male persons in house, - -and so on. Now if we use *w* and *w*′ to denote unknown numbers, -the conditions of the problem may be thus stated according to my -interpretation of the words-- - - (AB) = (AC) - *w*, (1) - -that is to say, the number of persons in the house who are aged is at -least equal to, and may exceed, the number of male persons in the house; - - (AB*c*) = *w*′, (2) - -that is to say, the number of male persons in the house who are not -aged is some unknown positive quantity. - -If we develop the terms in (1) by the Law of Duality (pp. 74, 81, 89), -we obtain - - (ABC) + (AB*c*) = (ABC) + (A*b*C) - *w*. - -Subtracting the common term (ABC) from each side and substituting for -(AB*c*) its value as given in (2), we get at once - - (A*b*C) = *w* + *w*′, - -and adding (A*bc*) to each side, we have - - (A*b*) = (A*bc*) + *w* + *w*′. - -The meaning of this result is that the number of persons in the house -who are not men is at least equal to *w* + *w*′, and exceeds it by the -number of persons in the house who are neither men nor aged (A*bc*). - -It should be understood that this solution applies only to the terms of -the example quoted above, and not to the general problem for which De -Morgan intended it to serve as an illustration. - -As a second instance, let us take the following question:--The -whole number of voters in a borough is *a*; the number against whom -objections have been lodged by liberals is *b*; and the number against -whom objections have been lodged by conservatives is *c*; required the -number, if any, who have been objected to on both sides. Taking - - A = voter, - B = objected to by liberals, - C = objected to by conservatives, - -then we require the value of (ABC). Now the following equation is -identically true-- - - (ABC) = (AB) + (AC) + (A*bc*) - (A). (1) - -For if we develop all the terms on the second side we obtain - - (ABC) = (ABC) + (AB*c*) + (ABC) + (A*b*C) + (A*bc*) - - (ABC) - (AB*c*) - (A*b*C) - (A*bc*); - -and striking out the corresponding positive and negative terms, we have -left only (ABC) = (ABC). Since then (1) is necessarily true, we have -only to insert the known values, and we have - - (ABC) = *b* + *c* - *a* + (A*bc*). - -Hence the number who have received objections from both sides is equal -to the excess, if any, of the whole number of objections over the -number of voters together with the number of voters who have received -no objection (A*bc*). - -The following problem illustrates the expression for the common part of -any three classes:--The number of paupers who are blind males, is equal -to the excess, if any, of the sum of the whole number of blind persons, -added to the whole number of male persons, added to the number of those -who being paupers are neither blind nor males, above the sum of the -whole number of paupers added to the number of those who, not being -paupers, are blind, and to the number of those who, not being paupers, -are male. - -The reader is requested to prove the truth of the above statement, (1) -by his own unaided common sense; (2) by the Aristotelian Logic; (3) by -the method of numerical logic just expounded; and then to decide which -method is most satisfactory. - - -*Numerical meaning of Logical Conditions.* - -In many cases classes of objects may exist under special logical -conditions, and we must consider how these conditions can be -interpreted numerically. Every logical proposition gives rise to a -corresponding numerical equation. Sameness of qualities occasions -sameness of numbers. Hence if - - A = B - -denotes the identity of the qualities of A and B, we may conclude that - - (A) = (B). - -It is evident that exactly those objects, and those objects only, which -are comprehended under A must be comprehended under B. It follows that -wherever we can draw an equation of qualities, we can draw a similar -equation of numbers. Thus, from - - A = B = C - -we infer - - A = C; - -and similarly from - - (A) = (B) = (C), - -meaning that the numbers of A’s and C’s are equal to the number of B’s, -we can infer - - (A) = (C). - -But, curiously enough, this does not apply to negative propositions and -inequalities. For if - - A = B ~ D - -means that A is identical with B, which differs from D, it does not -follow that - - (A) = (B) ~ (D). - -Two classes of objects may differ in qualities, and yet they may agree -in number. This point strongly confirms me in the opinion which I have -already expressed, that all inference really depends upon equations, -not differences. - -The Logical Alphabet thus enables us to make a complete analysis of any -numerical problem, and though the symbolical statement may sometimes -seem prolix, I conceive that it really represents the course which the -mind must follow in solving the question. Although thought may outstrip -the rapidity with which the symbols can be written down, yet the mind -does not really follow a different course from that indicated by the -symbols. For a fuller explanation of this natural system of Numerically -Definite Reasoning, with more abundant illustrations and an analysis -of De Morgan’s Numerically Definite Syllogism, I must refer the -reader to the paper[93] in the Memoirs of the Manchester Literary and -Philosophical Society, already mentioned, portions of which, however, -have been embodied in the present section. - - [93] It has been pointed out to me by Mr. C. J. Monroe, that section - 14 (p. 339) of this paper is erroneous, and ought to be cancelled. - The problem concerning the number of paupers illustrates the answer - which should have been obtained. Mr. A. J. Ellis, F.R.S., had - previously observed that my solution in the paper of De Morgan’s - problem about “men in the house” did not answer the conditions - intended by De Morgan, and I therefore give in the text a more - satisfactory solution. - -The reader may be referred, also, to Boole’s writings upon the -subject in the *Laws of Thought*, chap. xix. p. 295, and in a paper -on “Propositions Numerically Definite,” communicated by De Morgan, in -1868, to the Cambridge Philosophical Society, and printed in their -*Transactions*, vol. xi. part ii. - - - - -CHAPTER IX. - -THE VARIETY OF NATURE, OR THE DOCTRINE OF COMBINATIONS AND PERMUTATIONS. - - -Nature may be said to be evolved from the monotony of non-existence -by the creation of diversity. It is plausibly asserted that we are -conscious only so far as we experience difference. Life is change, and -perfectly uniform existence would be no better than non-existence. -Certain it is that life demands incessant novelty, and that nature, -though it probably never fails to obey the same fixed laws, yet -presents to us an apparently unlimited series of varied combinations -of events. It is the work of science to observe and record the kinds -and comparative numbers of such combinations of phenomena, occurring -spontaneously or produced by our interference. Patient and skilful -examination of the records may then disclose the laws imposed on matter -at its creation, and enable us more or less successfully to predict, or -even to regulate, the future occurrence of any particular combination. - -The Laws of Thought are the first and most important of all the laws -which govern the combinations of phenomena, and, though they be binding -on the mind, they may also be regarded as verified in the external -world. The Logical Alphabet develops the utmost variety of things and -events which may occur, and it is evident that as each new quality is -introduced, the number of combinations is doubled. Thus four qualities -may occur in 16 combinations; five qualities in 32; six qualities in -64; and so on. In general language, if n be the number of qualities, -2^{n} is the number of varieties of things which may be formed from -them, if there be no conditions but those of logic. This number, -it need hardly be said, increases after the first few terms, in an -extraordinary manner, so that it would require 302 figures to express -the number of combinations in which 1,000 qualities might conceivably -present themselves. - -If all the combinations allowed by the Laws of Thought occurred -indifferently in nature, then science would begin and end with those -laws. To observe nature would give us no additional knowledge, because -no two qualities would in the long run be oftener associated than any -other two. We could never predict events with more certainty than we -now predict the throws of dice, and experience would be without use. -But the universe, as actually created, presents a far different and -much more interesting problem. The most superficial observation shows -that some things are constantly associated with other things. The more -mature our examination, the more we become convinced that each event -depends upon the prior occurrence of some other series of events. -Action and reaction are gradually discovered to underlie the whole -scene, and an independent or casual occurrence does not exist except -in appearance. Even dice as they fall are surely determined in their -course by prior conditions and fixed laws. Thus the combinations of -events which can really occur are found to be comparatively restricted, -and it is the work of science to detect these restricting conditions. - -In the English alphabet, for instance, we have twenty-six letters. Were -the combinations of such letters perfectly free, so that any letter -could be indifferently sounded with any other, the number of words -which could be formed without any repetition would be 2^{26} - 1, or -67,108,863, equal in number to the combinations of the twenty-seventh -column of the Logical Alphabet, excluding one for the case in which -all the letters would be absent. But the formation of our vocal organs -prevents us from using the far greater part of these conjunctions of -letters. At least one vowel must be present in each word; more than two -consonants cannot usually be brought together; and to produce words -capable of smooth utterance a number of other rules must be observed. -To determine exactly how many words might exist in the English language -under these circumstances, would be an exceedingly complex problem, -the solution of which has never been attempted. The number of existing -English words may perhaps be said not to exceed one hundred thousand, -and it is only by investigating the combinations presented in the -dictionary, that we can learn the Laws of Euphony or calculate the -possible number of words. In this example we have an epitome of the -work and method of science. The combinations of natural phenomena are -limited by a great number of conditions which are in no way brought to -our knowledge except so far as they are disclosed in the examination of -nature. - -It is often a very difficult matter to determine the numbers -of permutations or combinations which may exist under various -restrictions. Many learned men puzzled themselves in former centuries -over what were called Protean verses, or verses admitting many -variations in accordance with the Laws of Metre. The most celebrated of -these verses was that invented by Bernard Bauhusius, as follows:[94]-- - - “Tot tibi sunt dotes, Virgo, quot sidera cœlo.” - - [94] Montucla, *Histoire*, &c., vol. iii. p. 388. - -One author, Ericius Puteanus, filled forty-eight pages of a work in -reckoning up its possible transpositions, making them only 1022. Other -calculators gave 2196, 3276, 2580 as their results. Wallis assigned -3096, but without much confidence in the accuracy of his result.[95] -It required the skill of James Bernoulli to decide that the number of -transpositions was 3312, under the condition that the sense and metre -of the verse shall be perfectly preserved. - - [95] Wallis, *Of Combinations*, &c., p. 119. - -In approaching the consideration of the great Inductive problem, -it is very necessary that we should acquire correct notions as -to the comparative numbers of combinations which may exist under -different circumstances. The doctrine of combinations is that part of -mathematical science which applies numerical calculation to determine -the numbers of combinations under various conditions. It is a part of -the science which really lies at the base not only of other sciences, -but of other branches of mathematics. The forms of algebraical -expressions are determined by the principles of combination, and -Hindenburg recognised this fact in his Combinatorial Analysis. The -greatest mathematicians have, during the last three centuries, -given their best powers to the treatment of this subject; it was -the favourite study of Pascal; it early attracted the attention of -Leibnitz, who wrote his curious essay, *De Arte Combinatoria*, at -twenty years of age; James Bernoulli, one of the very profoundest -mathematicians, devoted no small part of his life to the investigation -of the subject, as connected with that of Probability; and in his -celebrated work, *De Arte Conjectandi*, he has so finely described the -importance of the doctrine of combinations, that I need offer no excuse -for quoting his remarks at full length. - -“It is easy to perceive that the prodigious variety which appears -both in the works of nature and in the actions of men, and which -constitutes the greatest part of the beauty of the universe, is owing -to the multitude of different ways in which its several parts are -mixed with, or placed near, each other. But, because the number of -causes that concur in producing a given event, or effect, is oftentimes -so immensely great, and the causes themselves are so different one -from another, that it is extremely difficult to reckon up all the -different ways in which they may be arranged or combined together, it -often happens that men, even of the best understandings and greatest -circumspection, are guilty of that fault in reasoning which the writers -on logic call *the insufficient or imperfect enumeration of parts or -cases*: insomuch that I will venture to assert, that this is the chief, -and almost the only, source of the vast number of erroneous opinions, -and those too very often in matters of great importance, which we are -apt to form on all the subjects we reflect upon, whether they relate to -the knowledge of nature, or the merits and motives of human actions. - -“It must therefore be acknowledged, that that art which affords a cure -to this weakness, or defect, of our understandings, and teaches us so -to enumerate all the possible ways in which a given number of things -may be mixed and combined together, that we may be certain that we have -not omitted any one arrangement of them that can lead to the object -of our inquiry, deserves to be considered as most eminently useful and -worthy of our highest esteem and attention. And this is the business -of *the art or doctrine of combinations*. Nor is this art or doctrine -to be considered merely as a branch of the mathematical sciences. For -it has a relation to almost every species of useful knowledge that the -mind of man can be employed upon. It proceeds indeed upon mathematical -principles, in calculating the number of the combinations of the things -proposed: but by the conclusions that are obtained by it, the sagacity -of the natural philosopher, the exactness of the historian, the skill -and judgment of the physician, and the prudence and foresight of the -politician may be assisted; because the business of all these important -professions is but *to form reasonable conjectures* concerning the -several objects which engage their attention, and all wise conjectures -are the results of a just and careful examination of the several -different effects that may possibly arise from the causes that are -capable of producing them.”[96] - - [96] James Bernoulli, *De Arte Conjectandi*, translated by Baron - Maseres. London, 1795, pp. 35, 36. - - -*Distinction of Combinations and Permutations.* - -We must first consider the deep difference which exists between -Combinations and Permutations, a difference involving important logical -principles, and influencing the form of mathematical expressions. -In *permutation* we recognise varieties of order, treating AB as a -different group from BA. In *combination* we take notice only of the -presence or absence of a certain thing, and pay no regard to its place -in order of time or space. Thus the four letters *a*, *e*, *m*, *n* -can form but one combination, but they occur in language in several -permutations, as *name*, *amen*, *mean*, *mane*. - -We have hitherto been dealing with purely logical questions, involving -only combination of qualities. I have fully pointed out in more than -one place that, though our symbols could not but be written in order -of place and read in order of time, the relations expressed had no -regard to place or time (pp. 33, 114). The Law of Commutativeness, in -fact, expresses the condition that in logic we deal with combinations, -and the same law is true of all the processes of algebra. In some -cases, order may be a matter of indifference; it makes no difference, -for instance, whether gunpowder is a mixture of sulphur, carbon, and -nitre, or carbon, nitre, and sulphur, or nitre, sulphur, and carbon, -provided that the substances are present in proper proportions and -well mixed. But this indifference of order does not usually extend to -the events of physical science or the operations of art. The change of -mechanical energy into heat is not exactly the same as the change from -heat into mechanical energy; thunder does not indifferently precede and -follow lightning; it is a matter of some importance that we load, cap, -present, and fire a rifle in this precise order. Time is the condition -of all our thoughts, space of all our actions, and therefore both in -art and science we are to a great extent concerned with permutations. -Language, for instance, treats different permutations of letters as -having different meanings. - -Permutations of things are far more numerous than combinations of those -things, for the obvious reason that each distinct thing is regarded -differently according to its place. Thus the letters A, B, C, will -make different permutations according as A stands first, second, or -third; having decided the place of A, there are two places between -which we may choose for B; and then there remains but one place for -C. Accordingly the permutations of these letters will be altogether -3 × 2 × 1 or 6 in number. With four things or letters, A, B, C, D, -we shall have four choices of place for the first letter, three for -the second, two for the third, and one for the fourth, so that there -will be altogether, 4 × 3 × 2 × 1, or 24 permutations. The same simple -rule applies in all cases; beginning with the whole number of things -we multiply at each step by a number decreased by a unit. In general -language, if *n* be the number of things in a combination, the number -of permutations is - -*n* (*n* - 1)(*n* - 2) .... 4 . 3 . 2 . 1. - -If we were to re-arrange the names of the days of the week, the -possible arrangements out of which we should have to choose the new -order, would be no less than 7 . 6 . 5 . 4 . 3 . 2 . 1, or 5040, or, -excluding the existing order, 5039. - -The reader will see that the numbers which we reach in questions -of permutation, increase in a more extraordinary manner even than -in combination. Each new object or term doubles the number of -combinations, but increases the permutations by a factor continually -growing. Instead of 2 × 2 × 2 × 2 × .... we have 2 × 3 × 4 × 5 × .... -and the products of the latter expression immensely exceed those of the -former. These products of increasing factors are frequently employed, -as we shall see, in questions both of permutation and combination. They -are technically called *factorials*, that is to say, the product of all -integer numbers, from unity up to any number *n* is the *factorial* -of *n*, and is often indicated symbolically by *n*!. I give below the -factorials up to that of twelve:-- - - 24 = 1 . 2 . 3 . 4 - 120 = 1 . 2 ... 5 - 720 = 1 . 2 ... 6 - 5,040 = 7! - 40,320 = 8! - 362,880 = 9! - 3,628,800 = 10! - 39,916,800 = 11! - 479,001,600 = 12! - -The factorials up to 36! are given in Rees’s ‘Cyclopædia,’ art. -*Cipher*, and the logarithms of factorials up to 265! are to be -found at the end of the table of logarithms published under the -superintendence of the Society for the Diffusion of Useful Knowledge -(p. 215). To express the factorial 265! would require 529 places of -figures. - -Many writers have from time to time remarked upon the extraordinary -magnitude of the numbers with which we deal in this subject. Tacquet -calculated[97] that the twenty-four [sic] letters of the alphabet may -be arranged in more than 620 thousand trillions of orders; and Schott -estimated[98] that if a thousand millions of men were employed for the -same number of years in writing out these arrangements, and each man -filled each day forty pages with forty arrangements in each, they would -not have accomplished the task, as they would have written only 584 -thousand trillions instead of 620 thousand trillions. - - [97] *Arithmeticæ Theoria.* Ed. Amsterd. 1704. p. 517. - - [98] Rees’s *Cyclopædia*, art. *Cipher*. - -In some questions the number of permutations may be restricted -and reduced by various conditions. Some things in a group may -be undistinguishable from others, so that change of order will -produce no difference. Thus if we were to permutate the letters of -the name *Ann*, according to our previous rule, we should obtain -3 × 2 × 1, or 6 orders; but half of these arrangements would be -identical with the other half, because the interchange of the two -*n*’s has no effect. The really different orders will therefore be -(3 . 2 . 1)/(1 . 2) or 3, namely *Ann*, *Nan*, *Nna*. In the word -*utility* there are two *i*’s and two *t*’s, in respect of both of -which pairs the numbers of permutations must be halved. Thus we obtain -(7 . 6 . 5 . 4 . 3 . 2 . 1)/(1 . 2 . 1 . 2) or 1260, as the number of -permutations. The simple rule evidently is--when some things or letters -are undistinguished, proceed in the first place to calculate all the -possible permutations as if all were different, and then divide by -the numbers of possible permutations of those series of things which -are not distinguished, and of which the permutations have therefore -been counted in excess. Thus since the word *Utilitarianism* contains -fourteen letters, of which four are *i*’s, two *a*’s, and two *t*’s, -the number of distinct arrangements will be found by dividing the -factorial of 14, by the factorials of 4, 2, and 2, the result being -908,107,200. From the letters of the word *Mississippi* we can get in -like manner 11!/(4! × 4! × 2!) or 34,650 permutations, which is not -the one-thousandth part of what we should obtain were all the letters -different. - - -*Calculation of Number of Combinations.* - -Although in many questions both of art and science we need to calculate -the number of permutations on account of their own interest, it far -more frequently happens in scientific subjects that they possess but -an indirect interest. As I have already pointed out, we almost always -deal in the logical and mathematical sciences with *combinations*, and -variety of order enters only through the inherent imperfections of our -symbols and modes of calculation. Signs must be used in some order, -and we must withdraw our attention from this order before the signs -correctly represent the relations of things which exist neither before -nor after each other. Now, it often happens that we cannot choose all -the combinations of things, without first choosing them subject to the -accidental variety of order, and we must then divide by the number of -possible variations of order, that we may get to the true number of -pure combinations. - -Suppose that we wish to determine the number of ways in which we can -select a group of three letters out of the alphabet, without allowing -the same letter to be repeated. At the first choice we can take any -one of 26 letters; at the next step there remain 25 letters, any one -of which may be joined with that already taken; at the third step -there will be 24 choices, so that apparently the whole number of ways -of choosing is 26 × 25 × 24. But the fact that one choice succeeded -another has caused us to obtain the same combinations of letters in -different orders; we should get, for instance, *a*, *p*, *r* at one -time, and *p*, *r*, *a* at another, and every three distinct letters -will appear six times over, because three things can be arranged in -six permutations. To get the number of combinations, then, we must -divide the whole number of ways of choosing, by six, the number of -permutations of three things, obtaining (26 × 25 × 24)/(1 × 2 × 3) or -2,600. - -It is apparent that we need the doctrine of combinations in order -that we may in many questions counteract the exaggerating effect of -successive selection. If out of a senate of 30 persons we have to -choose a committee of 5, we may choose any of 30 first, any of 29 next, -and so on, in fact there will be 30 × 29 × 28 × 27 × 26 selections; -but as the actual character of the members of the committee will not -be affected by the accidental order of their selection, we divide by -1 × 2 × 3 × 4 × 5, and the possible number of different committees will -be 142,506. Similarly if we want to calculate the number of ways in -which the eight major planets may come into conjunction, it is evident -that they may meet either two at a time or three at a time, or four or -more at a time, and as nothing is said as to the relative order or -place in the conjunction, we require the number of combinations. Now -a selection of 2 out of 8 is possible in (8 . 7)/(1 . 2) or 28 ways; -of 3 out of 8 in (8 . 7 . 6)/(1 . 2 . 3) or 56 ways; of 4 out of 8 in -(8 . 7 . 6 . 5)/(1 . 2 . 3 . 4) or 70 ways; and it may be similarly -shown that for 5, 6, 7, and 8 planets, meeting at one time, the numbers -of ways are 56, 28, 8, and 1. Thus we have solved the whole question -of the variety of conjunctions of eight planets; and adding all the -numbers together, we find that 247 is the utmost possible number of -modes of meeting. - -In general algebraic language, we may say that a group of *m* things -may be chosen out of a total number of *n* things, in a number of -combinations denoted by the formula - - (*n* . (*n*-1)(*n*-2)(*n*-3) .... (*n* - *m* + 1))/(1 . 2 . 3 . 4 .... *m*) - -The extreme importance and significance of this formula seems to have -been first adequately recognised by Pascal, although its discovery -is attributed by him to a friend, M. de Ganières.[99] We shall find -it perpetually recurring in questions both of combinations and -probability, and throughout the formulæ of mathematical analysis traces -of its influence may be noticed. - - [99] *Œuvres Complètes de Pascal* (1865), vol. iii. p. 302. Montucla - states the name as De Gruières, *Histoire des Mathématiques*, - vol. iii. p. 389. - - -*The Arithmetical Triangle.* - -The Arithmetical Triangle is a name long since given to a series -of remarkable numbers connected with the subject we are treating. -According to Montucla[100] “this triangle is in the theory of -combinations and changes of order, almost what the table of Pythagoras -is in ordinary arithmetic, that is to say, it places at once under the -eyes the numbers required in a multitude of cases of this theory.” As -early as 1544 Stifels had noticed the remarkable properties of these -numbers and the mode of their evolution. Briggs, the inventor of the -common system of logarithms, was so struck with their importance that -he called them the Abacus Panchrestus. Pascal, however, was the first -who wrote a distinct treatise on these numbers, and gave them the name -by which they are still known. But Pascal did not by any means exhaust -the subject, and it remained for James Bernoulli to demonstrate fully -the importance of the *figurate numbers*, as they are also called. In -his treatise *De Arte Conjectandi*, he points out their application -in the theory of combinations and probabilities, and remarks of the -Arithmetical Triangle, “It not only contains the clue to the mysterious -doctrine of combinations, but it is also the ground or foundation of -most of the important and abstruse discoveries that have been made in -the other branches of the mathematics.”[101] - - [100] *Histoire des Mathématiques*, vol. iii. p. 378. - - [101] Bernoulli, *De Arte Conjectandi*, translated by Francis - Maseres. London, 1795, p. 75. - -The numbers of the triangle can be calculated in a very easy manner by -successive additions. We commence with unity at the apex; in the next -line we place a second unit to the right of this; to obtain the third -line of figures we move the previous line one place to the right, and -add them to the same figures as they were before removal; we can then -repeat the same process *ad infinitum*. The fourth line of figures, -for instance, contains 1, 3, 3, 1; moving them one place and adding as -directed we obtain:-- - - Fourth line ... 1 3 3 1 - 1 3 3 1 - -------------- - Fifth line .... 1 4 6 4 1 - 1 4 6 4 1 - ---------------- - Sixth line .... 1 5 10 10 5 1 - -Carrying out this simple process through ten more steps we obtain the -first seventeen lines of the Arithmetical Triangle as printed on the -next page. Theoretically speaking the Triangle must be regarded as -infinite in extent, but the numbers increase so rapidly that it soon -becomes impracticable to continue the table. The longest table of the -numbers which I have found is in Fortia’s “Traité des Progressions” -(p. 80), where they are given up to the fortieth line and the ninth -column. - -THE ARITHMETICAL TRIANGLE. - -Line. First Column. -1 1 Second Column. -2 1 1 Third Column. -3 1 2 1 Fourth Column. -4 1 3 3 1 Fifth Column. -5 1 4 6 4 1 Sixth Column. -6 1 5 10 10 5 1 Seventh Column. -7 1 6 15 20 15 6 1 Eighth Column. -8 1 7 21 35 35 21 7 1 Ninth Column. -9 1 8 28 56 70 56 28 8 1 Tenth Column. -10 1 9 36 84 126 126 84 36 9 1 Eleventh Column. -11 1 10 45 120 210 252 210 120 45 10 1 Twelfth Column. -12 1 11 55 165 330 462 462 330 165 55 11 1 Thirteenth Column. -13 1 12 66 220 495 792 924 792 495 220 66 12 1 Fourteenth Column. -14 1 13 78 286 715 1287 1716 1716 1287 715 286 78 13 1 Fifteenth Column. -15 1 14 91 364 1001 2002 3003 3432 3003 2002 1001 364 91 14 1 Sixteenth Column. -16 1 15 105 455 1365 3003 5005 6435 6435 5005 3003 1365 455 105 15 1 Seventeenth Col. -17 1 16 120 560 1820 4368 8008 11440 12870 11440 8008 4368 1820 560 120 16 1 - -Examining these numbers, we find that they are connected by an -unlimited series of relations, a few of the more simple of which may be -noticed. Each vertical column of numbers exactly corresponds with an -oblique series descending from left to right, so that the triangle is -perfectly symmetrical in its contents. The first column contains only -*units*; the second column contains the *natural numbers*, 1, 2, 3, -&c.; the third column contains a remarkable series of numbers, 1, 3, -6, 10, 15, &c., which have long been called *the triangular numbers*, -because they correspond with the numbers of balls which may be arranged -in a triangular form, thus-- - -[Illustration] - -The fourth column contains the *pyramidal numbers*, so called because -they correspond to the numbers of equal balls which can be piled in -regular triangular pyramids. Their differences are the triangular -numbers. The numbers of the fifth column have the pyramidal numbers -for their differences, but as there is no regular figure of which -they express the contents, they have been arbitrarily called the -*trianguli-triangular numbers*. The succeeding columns have, in a -similar manner, been said to contain the *trianguli-pyramidal*, the -*pyramidi-pyramidal* numbers, and so on.[102] - - [102] Wallis’s *Algebra*, Discourse of Combinations, &c., p. 109. - -From the mode of formation of the table, it follows that the -differences of the numbers in each column will be found in the -preceding column to the left. Hence the *second differences*, or the -*differences of differences*, will be in the second column to the left -of any given column, the third differences in the third column, and so -on. Thus we may say that unity which appears in the first column is the -*first difference* of the numbers in the second column; the *second -difference* of those in the third column; the *third difference* of -those in the fourth, and so on. The triangle is seen to be a complete -classification of all numbers according as they have unity for any of -their differences. - -Since each line is formed by adding the previous line to itself, it -is evident that the sum of the numbers in each horizontal line must be -double the sum of the numbers in the line next above. Hence we know, -without making the additions, that the successive sums must be 1, 2, -4, 8, 16, 32, 64, &c., the same as the numbers of combinations in the -Logical Alphabet. Speaking generally, the sum of the numbers in the -*n*th line will be 2^{*n* - 1}. - -Again, if the whole of the numbers down to any line be added together, -we shall obtain a number less by unity than some power of 2; thus, -the first line gives 1 or 2^{1} - 1; the first two lines give 3 or -2^{2} - 1; the first three lines 7 or 2^{3} - 1; the first six lines -give 63 or 2^{6} - 1; or, speaking in general language, the sum of the -first *n* lines is 2^{*n*} - 1. It follows that the sum of the numbers -in any one line is equal to the sum of those in all the preceding -lines increased by a unit. For the sum of the *n*th line is, as -already shown, 2^{*n* - 1}, and the sum of the first *n* - 1 lines is -2^{*n* - 1} - 1, or less by a unit. - -This account of the properties of the figurate numbers does not -approach completeness; a considerable, probably an unlimited, number of -less simple and obvious relations might be traced out. Pascal, after -giving many of the properties, exclaims[103]: “Mais j’en laisse bien -plus que je n’en donne; c’est une chose étrange combien il est fertile -en propriétés! Chacun peut s’y exercer.” The arithmetical triangle may -be considered a natural classification of numbers, exhibiting, in the -most complete manner, their evolution and relations in a certain point -of view. It is obvious that in an unlimited extension of the triangle, -each number, with the single exception of the number *two*, has at -least two places. - - [103] *Œuvres Complètes*, vol. iii. p. 251. - -Though the properties above explained are highly curious, the greatest -value of the triangle arises from the fact that it contains a complete -statement of the values of the formula (p. 182), for the numbers of -combinations of *m* things out of *n*, for all possible values of *m* -and *n*. Out of seven things one may be chosen in seven ways, and -seven occurs in the eighth line of the second column. The combinations -of two things chosen out of seven are (7 × 6)/(1 × 2) or 21, which -is the third number in the eighth line. The combinations of three -things out of seven are (7 × 6 × 5)/(1 × 2 × 3) or 35, which appears -fourth in the eighth line. In a similar manner, in the fifth, sixth, -seventh, and eighth columns of the eighth line I find it stated in -how many ways I can select combinations of 4, 5, 6, and 7 things out -of 7. Proceeding to the ninth line, I find in succession the number -of ways in which I can select 1, 2, 3, 4, 5, 6, 7, and 8 things, out -of 8 things. In general language, if I wish to know in how many ways -*m* things can be selected in combinations out of *n* things, I must -look in the *n* + 1^{th} line, and take the *m* + 1^{th} number, as -the answer. In how many ways, for instance, can a subcommittee of -five be chosen out of a committee of nine. The answer is 126, and -is the sixth number in the tenth line; it will be found equal to -(9 . 8 . 7 . 6 . 5)/(1 . 2 . 3 . 4 . 5), which our formula (p. 182) -gives. - -The full utility of the figurate numbers will be more apparent when -we reach the subject of probabilities, but I may give an illustration -or two in this place. In how many ways can we arrange four pennies as -regards head and tail? The question amounts to asking in how many ways -we can select 0, 1, 2, 3, or 4 heads, out of 4 heads, and the *fifth* -line of the triangle gives us the complete answer, thus-- - - We can select No head and 4 tails in 1 way. - " 1 head and 3 tails in 4 ways. - " 2 heads and 2 tails in 6 ways. - " 3 heads and 1 tail in 4 ways. - " 4 heads and 0 tail in 1 way. - -The total number of different cases is 16, or 2^{4}, and when we come -to the next chapter, it will be found that these numbers give us the -respective probabilities of all throws with four pennies. - -I gave in p. 181 a calculation of the number of ways in which eight -planets can meet in conjunction; the reader will find all the numbers -detailed in the ninth line of the arithmetical triangle. The sum of the -whole line is 2^{8} or 256; but we must subtract a unit for the case -where no planet appears, and 8 for the 8 cases in which only one planet -appears; so that the total number of conjunctions is 2^{8} -1 - 8 -or 247. If an organ has eleven stops we find in the twelfth line the -numbers of ways in which we can draw them, 1, 2, 3, or more at a time. -Thus there are 462 ways of drawing five stops at once, and as many of -drawing six stops. The total number of ways of varying the sound is -2048, including the single case in which no stop at all is drawn. - -One of the most important scientific uses of the arithmetical triangle -consists in the information which it gives concerning the comparative -frequency of divergencies from an average. Suppose, for the sake of -argument, that all persons were naturally of the equal stature of five -feet, but enjoyed during youth seven independent chances of growing one -inch in addition. Of these seven chances, one, two, three, or more, -may happen favourably to any individual; but, as it does not matter -what the chances are, so that the inch is gained, the question really -turns upon the number of combinations of 0, 1, 2, 3, &c., things out of -seven. Hence the eighth line of the triangle gives us a complete answer -to the question, as follows:-- - -Out of every 128 people-- - - Feet Inches. -One person would have the stature of 5 0 - 7 persons " " 5 1 -21 persons " " 5 2 -35 persons " " 5 3 -35 persons " " 5 4 -21 persons " " 5 5 - 7 persons " " 5 6 - 1 person " " 5 7 - -By taking a proper line of the triangle, an answer may be had under -any more natural supposition. This theory of comparative frequency of -divergence from an average, was first adequately noticed by Quetelet, -and has lately been employed in a very interesting and bold manner by -Mr. Francis Galton,[104] in his remarkable work on “Hereditary Genius.” -We shall afterwards find that the theory of error, to which is made the -ultimate appeal in cases of quantitative investigation, is founded upon -the comparative numbers of combinations as displayed in the triangle. - - [104] See also Galton’s Lecture at the Royal Institution, 27th - February, 1874; Catalogue of the Special Loan Collection of - Scientific Instruments, South Kensington, Nos. 48, 49; and Galton, - *Philosophical Magazine*, January 1875. - - -*Connection between the Arithmetical Triangle and the Logical Alphabet.* - -There exists a close connection between the arithmetical triangle -described in the last section, and the series of combinations of -letters called the Logical Alphabet. The one is to mathematical science -what the other is to logical science. In fact the figurate numbers, or -those exhibited in the triangle, are obtained by summing up the logical -combinations. Accordingly, just as the total of the numbers in each -line of the triangle is twice as great as that for the preceding line -(p. 186), so each column of the Alphabet (p. 94) contains twice as many -combinations as the preceding one. The like correspondence also exists -between the sums of all the lines of figures down to any particular -line, and of the combinations down to any particular column. - -By examining any column of the Logical Alphabet we find that the -combinations naturally group themselves according to the figurate -numbers. Take the combinations of the letters A, B, C, D; they consist -of all the ways in which I can choose four, three, two, one, or none of -the four letters, filling up the vacant spaces with negative terms. - -There is one combination, ABCD, in which all the positive letters are -present; there are four combinations in each of which three positive -letters are present; six in which two are present; four in which only -one is present; and, finally, there is the single case, *abcd*, in -which all positive letters are absent. These numbers, 1, 4, 6, 4, 1, -are those of the fifth line of the arithmetical triangle, and a like -correspondence will be found to exist in each column of the Logical -Alphabet. - -Numerical abstraction, it has been asserted, consists in overlooking -the kind of difference, and retaining only a consciousness of its -existence (p. 158). While in logic, then, we have to deal with each -combination as a separate kind of thing, in arithmetic we distinguish -only the classes which depend upon more or less positive terms being -present, and the numbers of these classes immediately produce the -numbers of the arithmetical triangle. - -It may here be pointed out that there are two modes in which we -can calculate the whole number of combinations of certain things. -Either we may take the whole number at once as shown in the Logical -Alphabet, in which case the number will be some power of two, or else -we may calculate successively, by aid of permutations, the number of -combinations of none, one, two, three things, and so on. Hence we -arrive at a necessary identity between two series of numbers. In the -case of four things we shall have - - 2 = 1 + 4/1 + (4 . 3)/(1 . 2) + (4 . 3 . 2)/(1 . 2 . 3) + - (4 . 3 . 2 . 1)/(1 . 2 . 3 . 4). - -In a general form of expression we shall have - - 2 = 1 + *n*/1 + (*n* . (*n* - 1))/(1 . 2) + (*n* - (*n* - 1)(*n* - 2))/(1 . 2 . 3) + &c., - -the terms being continued until they cease to have any value. Thus we -arrive at a proof of simple cases of the Binomial Theorem, of which -each column of the Logical Alphabet is an exemplification. It may be -shown that all other mathematical expansions likewise arise out of -simple processes of combination, but the more complete consideration of -this subject must be deferred to another work. - - -*Possible Variety of Nature and Art.* - -We cannot adequately understand the difficulties which beset us in -certain branches of science, unless we have some clear idea of the vast -numbers of combinations or permutations which may be possible under -certain conditions. Thus only can we learn how hopeless it would be -to attempt to treat nature in detail, and exhaust the whole number of -events which might arise. It is instructive to consider, in the first -place, how immensely great are the numbers of combinations with which -we deal in many arts and amusements. - -In dealing a pack of cards, the number of hands, of thirteen cards -each, which can be produced is evidently 52 × 51 × 50 × ... × 40 -divided by 1 × 2 × 3 ... × 13. or 635,013,559,600. But in whist -four hands are simultaneously held, and the number of distinct -deals becomes so vast that it would require twenty-eight figures to -express it. If the whole population of the world, say one thousand -millions of persons, were to deal cards day and night, for a hundred -million of years, they would not in that time have exhausted one -hundred-thousandth part of the possible deals. Even with the same hands -of cards the play may be almost infinitely varied, so that the complete -variety of games at whist which may exist is almost incalculably great. -It is in the highest degree improbable that any one game of whist was -ever exactly like another, except it were intentionally so. - -The end of novelty in art might well be dreaded, did we not find that -nature at least has placed no attainable limit, and that the deficiency -will lie in our inventive faculties. It would be a cheerless time -indeed when all possible varieties of melody were exhausted, but it -is readily shown that if a peal of twenty-four bells had been rung -continuously from the so-called beginning of the world to the present -day, no approach could have been made to the completion of the possible -changes. Nay, had every single minute been prolonged to 10,000 years, -still the task would have been unaccomplished.[105] As regards ordinary -melodies, the eight notes of a single octave give more than 40,000 -permutations, and two octaves more than a million millions. If we were -to take into account the semitones, it would become apparent that it -is impossible to exhaust the variety of music. When the late Mr. J. S. -Mill, in a depressed state of mind, feared the approaching exhaustion -of musical melodies, he had certainly not bestowed sufficient study on -the subject of permutations. - - [105] Wallis, *Of Combinations*, p. 116, quoting Vossius. - -Similar considerations apply to the possible number of natural -substances, though we cannot always give precise numerical results. It -was recommended by Hatchett[106] that a systematic examination of all -alloys of metals should be carried out, proceeding from the binary ones -to more complicated ternary or quaternary ones. He can hardly have been -aware of the extent of his proposed inquiry. If we operate only upon -thirty of the known metals, the number of binary alloys would be 435, -of ternary alloys 4060, of quaternary 27,405, without paying regard -to the varying proportions of the metals, and only regarding the kind -of metal. If we varied all the ternary alloys by quantities not less -than one per cent., the number of these alloys would be 11,445,060. -An exhaustive investigation of the subject is therefore out of the -question, and unless some laws connecting the properties of the alloy -and its components can be discovered, it is not apparent how our -knowledge of them can ever be more than fragmentary. - - [106] *Philosophical Transactions* (1803), vol. xciii. p. 193. - -The possible variety of definite chemical compounds, again, is -enormously great. Chemists have already examined many thousands -of inorganic substances, and a still greater number of organic -compounds;[107] they have nevertheless made no appreciable impression -on the number which may exist. Taking the number of elements at -sixty-one, the number of compounds containing different selections of -four elements each would be more than half a million (521,855). As the -same elements often combine in many different proportions, and some of -them, especially carbon, have the power of forming an almost endless -number of compounds, it would hardly be possible to assign any limit -to the number of chemical compounds which may be formed. There are -branches of physical science, therefore, of which it is unlikely that -scientific men, with all their industry, can ever obtain a knowledge in -any appreciable degree approaching to completeness. - - [107] Hofmann’s *Introduction to Chemistry*, p. 36. - - -*Higher Orders of Variety.* - -The consideration of the facts already given in this chapter will -not produce an adequate notion of the possible variety of existence, -unless we consider the comparative numbers of combinations of different -orders. By a combination of a higher order, I mean a combination of -groups, which are themselves groups. The immense numbers of compounds -of carbon, hydrogen, and oxygen, described in organic chemistry, are -combinations of a second order, for the atoms are groups of groups. -The wave of sound produced by a musical instrument may be regarded as -a combination of motions; the body of sound proceeding from a large -orchestra is therefore a complex aggregate of sounds, each in itself -a complex combination of movements. All literature may be said to be -developed out of the difference of white paper and black ink. From the -unlimited number of marks which might be chosen we select twenty-six -conventional letters. The pronounceable combinations of letters are -probably some trillions in number. Now, as a sentence is a selection -of words, the possible sentences must be inconceivably more numerous -than the words of which it may be composed. A book is a combination -of sentences, and a library is a combination of books. A library, -therefore, may be regarded as a combination of the fifth order, and the -powers of numerical expression would be severely tasked in attempting -to express the number of distinct libraries which might be constructed. -The calculation, of course, would not be possible, because the union -of letters in words, of words in sentences, and of sentences in books, -is governed by conditions so complex as to defy analysis. I wish only -to point out that the infinite variety of literature, existing or -possible, is all developed out of one fundamental difference. Galileo -remarked that all truth is contained in the compass of the alphabet. He -ought to have said that it is all contained in the difference of ink -and paper. - -One consequence of successive combination is that the simplest marks -will suffice to express any information. Francis Bacon proposed for -secret writing a biliteral cipher, which resolves all letters of the -alphabet into permutations of the two letters *a* and *b*. Thus A -was *aaaaa*, B *aaaab*, X *babab*, and so on.[108] In a similar way, -as Bacon clearly saw, any one difference can be made the ground of a -code of signals; we can express, as he says, *omnia per omnia*. The -Morse alphabet uses only a succession of long and short marks, and -other systems of telegraphic language employ right and left strokes. -A single lamp obscured at various intervals, long or short, may be -made to spell out any words, and with two lamps, distinguished by -colour, position, or any other circumstance, we could at once represent -Bacon’s biliteral alphabet. Babbage ingeniously suggested that every -lighthouse in the world should be made to spell out its own name or -number perpetually, by flashes or obscurations of various duration -and succession. A system like that of Babbage is now being applied -to lighthouses in the United Kingdom by Sir W. Thomson and Dr. John -Hopkinson. - - [108] *Works*, edited by Shaw, vol. i. pp. 141–145, quoted in Rees’s - *Encyclopædia*, art. *Cipher*. - -Let us calculate the numbers of combinations of different orders which -may arise out of the presence or absence of a single mark, say A. In -these figures - - +---+---+ +---+---+ +---+---+ +---+---+ - | A | A | | A | | | | A | | | | - +---+---+ +---+---+ +---+---+ +---+---+ - -we have four distinct varieties. Form them into a group of a higher -order, and consider in how many ways we may vary that group by omitting -one or more of the component parts. Now, as there are four parts, -and any one may be present or absent, the possible varieties will -be 2 × 2 × 2 × 2, or 16 in number. Form these into a new whole, and -proceed again to create variety by omitting any one or more of the -sixteen. The number of possible changes will now be 2 . 2 . 2 . 2 . -2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2, or 2^{16}, and we can -repeat the process again and again. We are imagining the creation of -objects, whose numbers are represented by the successive orders of the -powers of *two*. - -At the first step we have 2; at the next 2^{2}, or 4; at the third -(2^{2})^{2}, or 16, numbers of very moderate amount. Let the reader -calculate the next term, ((2^{2})^{2})^{2}, and he will be surprised -to find it leap up to 65,536. But at the next step he has to calculate -the value of 65,536 *two*’s multiplied together, and it is so great -that we could not possibly compute it, the mere expression of the -result requiring 19,729 places of figures. But go one step more and we -pass the bounds of all reason. The sixth order of the powers of *two* -becomes so great, that we could not even express the number of figures -required in writing it down, without using about 19,729 figures for -the purpose. The successive orders of the powers of two have then the -following values so far as we can succeed in describing them:-- - - First order 2 - Second order 4 - Third order 16 - Fourth order 65,536 - Fifth order, number expressed by 19,729 figures. - Sixth order, number expressed by - figures, to express the number - of which figures would require - about 19,729 figures. - -It may give us some notion of infinity to remember that at this sixth -step`, having long surpassed all bounds of intuitive conception, we -make no approach to a limit. Nay, were we to make a hundred such steps, -we should be as far away as ever from actual infinity. - -It is well worth observing that our powers of expression rapidly -overcome the possible multitude of finite objects which may exist in -any assignable space. Archimedes showed long ago, in one of the most -remarkable writings of antiquity, the *Liber de Arcnæ Numero*, that -the grains of sand in the world could be numbered, or rather, that -if numbered, the result could readily be expressed in arithmetical -notation. Let us extend his problem, and ascertain whether we could -express the number of atoms which could exist in the visible universe. -The most distant stars which can now be seen by telescopes--those of -the sixteenth magnitude--are supposed to have a distance of about -33,900,000,000,000,000 miles. Sir W. Thomson has shown reasons for -supposing that there do not exist more than from 3 × 10^{24} to 10^{26} -molecules in a cubic centimetre of a solid or liquid substance.[109] -Assuming these data to be true, for the sake of argument, a simple -calculation enables us to show that the almost inconceivably vast -sphere of our stellar system if entirely filled with solid matter, -would not contain more than about 68 × 10^{90} atoms, that is to say, -a number requiring for its expression 92 places of figures. Now, this -number would be immensely less than the fifth order of the powers of -two. - - [109] *Nature*, vol. i. p. 553. - -In the variety of logical relations, which may exist between a certain -number of logical terms, we also meet a case of higher combinations. -We have seen (p. 142) that with only six terms the number of possible -selections of combinations is 18,446,744,073,709,551,616. Considering -that it is the most common thing in the world to use an argument -involving six objects or terms, it may excite some surprise that the -complete investigation of the relations in which six such terms may -stand to each other, should involve an almost inconceivable number of -cases. Yet these numbers of possible logical relations belong only to -the second order of combinations. - - - - -CHAPTER X. - -THE THEORY OF PROBABILITY. - - -The subject upon which we now enter must not be regarded as an isolated -and curious branch of speculation. It is the necessary basis of the -judgments we make in the prosecution of science, or the decisions we -come to in the conduct of ordinary affairs. As Butler truly said, -“Probability is the very guide of life.” Had the science of numbers -been studied for no other purpose, it must have been developed for -the calculation of probabilities. All our inferences concerning the -future are merely probable, and a due appreciation of the degree of -probability depends upon a comprehension of the principles of the -subject. I am convinced that it is impossible to expound the methods of -induction in a sound manner, without resting them upon the theory of -probability. Perfect knowledge alone can give certainty, and in nature -perfect knowledge would be infinite knowledge, which is clearly beyond -our capacities. We have, therefore, to content ourselves with partial -knowledge--knowledge mingled with ignorance, producing doubt. - -A great difficulty in this subject consists in acquiring a precise -notion of the matter treated. What is it that we number, and measure, -and calculate in the theory of probabilities? Is it belief, or opinion, -or doubt, or knowledge, or chance, or necessity, or want of art? Does -probability exist in the things which are probable, or in the mind -which regards them as such? The etymology of the name lends us no -assistance: for, curiously enough, *probable* is ultimately the same -word as *provable*, a good instance of one word becoming differentiated -to two opposite meanings. - -Chance cannot be the subject of the theory, because there is really -no such thing as chance, regarded as producing and governing events. -The word chance signifies *falling*, and the notion of falling is -continually used as a simile to express uncertainty, because we can -seldom predict how a die, a coin, or a leaf will fall, or when a bullet -will hit the mark. But everyone sees, after a little reflection, that -it is in our knowledge the deficiency lies, not in the certainty of -nature’s laws. There is no doubt in lightning as to the point it shall -strike; in the greatest storm there is nothing capricious; not a grain -of sand lies upon the beach, but infinite knowledge would account for -its lying there; and the course of every falling leaf is guided by the -principles of mechanics which rule the motions of the heavenly bodies. - -Chance then exists not in nature, and cannot coexist with knowledge; -it is merely an expression, as Laplace remarked, for our ignorance -of the causes in action, and our consequent inability to predict the -result, or to bring it about infallibly. In nature the happening of -an event has been pre-determined from the first fashioning of the -universe. *Probability belongs wholly to the mind.* This is proved by -the fact that different minds may regard the very same event at the -same time with widely different degrees of probability. A steam-vessel, -for instance, is missing and some persons believe that she has sunk -in mid-ocean; others think differently. In the event itself there can -be no such uncertainty; the steam-vessel either has sunk or has not -sunk, and no subsequent discussion of the probable nature of the event -can alter the fact. Yet the probability of the event will really vary -from day to day, and from mind to mind, according as the slightest -information is gained regarding the vessels met at sea, the weather -prevailing there, the signs of wreck picked up, or the previous -condition of the vessel. Probability thus belongs to our mental -condition, to the light in which we regard events, the occurrence -or non-occurrence of which is certain in themselves. Many writers -accordingly have asserted that probability is concerned with degree or -quantity of belief. De Morgan says,[110] “By degree of probability -we really mean or ought to mean degree of belief.” The late Professor -Donkin expressed the meaning of probability as “quantity of belief;” -but I have never felt satisfied with such definitions of probability. -The nature of *belief* is not more clear to my mind than the notion -which it is used to define. But an all-sufficient objection is, that -*the theory does not measure what the belief is, but what it ought to -be*. Few minds think in close accordance with the theory, and there -are many cases of evidence in which the belief existing is habitually -different from what it ought to be. Even if the state of belief in any -mind could be measured and expressed in figures, the results would be -worthless. The value of the theory consists in correcting and guiding -our belief, and rendering our states of mind and consequent actions -harmonious with our knowledge of exterior conditions. - - [110] *Formal Logic*, p. 172. - -This objection has been clearly perceived by some of those who still -used quantity of belief as a definition of probability. Thus De -Morgan adds--“Belief is but another name for imperfect knowledge.” -Donkin has well said that the quantity of belief is “always relative -to a particular state of knowledge or ignorance; but it must be -observed that it is absolute in the sense of not being relative to -any individual mind; since, the same information being presupposed, -all minds *ought* to distribute their belief in the same way.”[111] -Boole seemed to entertain a like view, when he described the theory as -engaged with “the equal distribution of ignorance;”[112] but we may -just as well say that it is engaged with the equal distribution of -knowledge. - - [111] *Philosophical Magazine*, 4th Series, vol. i. p. 355. - - [112] *Transactions of the Royal Society of Edinburgh*, vol. xxi. - part 4. - -I prefer to dispense altogether with this obscure word belief, and to -say that the theory of probability deals with *quantity of knowledge*, -an expression of which a precise explanation and measure can presently -be given. An event is only probable when our knowledge of it is diluted -with ignorance, and exact calculation is needed to discriminate how -much we do and do not know. The theory has been described by some -writers as professing *to evolve knowledge out of ignorance*; but as -Donkin admirably remarked, it is really “a method of avoiding the -erection of belief upon ignorance.” It defines rational expectation -by measuring the comparative amounts of knowledge and ignorance, and -teaches us to regulate our actions with regard to future events in a -way which will, in the long run, lead to the least disappointment. It -is, as Laplace happily said, *good sense reduced to calculation*. This -theory appears to me the noblest creation of intellect, and it passes -my conception how two such men as Auguste Comte and J. S. Mill could be -found depreciating it and vainly questioning its validity. To eulogise -the theory ought to be as needless as to eulogise reason itself. - - -*Fundamental Principles of the Theory.* - -The calculation of probabilities is really founded, as I conceive, upon -the principle of reasoning set forth in preceding chapters. We must -treat equals equally, and what we know of one case may be affirmed of -every case resembling it in the necessary circumstances. The theory -consists in putting similar cases on a par, and distributing equally -among them whatever knowledge we possess. Throw a penny into the air, -and consider what we know with regard to its way of falling. We know -that it will certainly fall upon a side, so that either head or tail -will be uppermost; but as to whether it will be head or tail, our -knowledge is equally divided. Whatever we know concerning head, we know -also concerning tail, so that we have no reason for expecting one more -than the other. The least predominance of belief to either side would -be irrational; it would consist in treating unequally things of which -our knowledge is equal. - -The theory does not require, as some writers have erroneously supposed, -that we should first ascertain by experiment the equal facility of -the events we are considering. So far as we can examine and measure -the causes in operation, events are removed out of the sphere of -probability. The theory comes into play where ignorance begins, and the -knowledge we possess requires to be distributed over many cases. Nor -does the theory show that the coin will fall as often on the one side -as the other. It is almost impossible that this should happen, because -some inequality in the form of the coin, or some uniform manner in -throwing it up, is almost sure to occasion a slight preponderance -in one direction. But as we do not previously know in which way a -preponderance will exist, we have no reason for expecting head more -than tail. Our state of knowledge will be changed should we throw up -the coin many times and register the results. Every throw gives us some -slight information as to the probable tendency of the coin, and in -subsequent calculations we must take this into account. In other cases -experience might show that we had been entirely mistaken; we might -expect that a die would fall as often on each of the six sides as on -each other side in the long run; trial might show that the die was a -loaded one, and falls most often on a particular face. The theory would -not have misled us: it treated correctly the information we had, which -is all that any theory can do. - -It may be asked, as Mill asks, Why spend so much trouble in calculating -from imperfect data, when a little trouble would enable us to render a -conclusion certain by actual trial? Why calculate the probability of a -measurement being correct, when we can try whether it is correct? But I -shall fully point out in later parts of this work that in measurement -we never can attain perfect coincidence. Two measurements of the -same base line in a survey may show a difference of some inches, and -there may be no means of knowing which is the better result. A third -measurement would probably agree with neither. To select any one of the -measurements, would imply that we knew it to be the most nearly correct -one, which we do not. In this state of ignorance, the only guide is -the theory of probability, which proves that in the long run the mean -of divergent results will come most nearly to the truth. In all other -scientific operations whatsoever, perfect knowledge is impossible, and -when we have exhausted all our instrumental means in the attainment of -truth, there is a margin of error which can only be safely treated by -the principles of probability. - -The method which we employ in the theory consists in calculating the -number of all the cases or events concerning which our knowledge is -equal. If we have the slightest reason for suspecting that one event -is more likely to occur than another, we should take this knowledge -into account. This being done, we must determine the whole number of -events which are, so far as we know, equally likely. Thus, if we have -no reason for supposing that a penny will fall more often one way than -another, there are two cases, head and tail, equally likely. But if -from trial or otherwise we know, or think we know, that of 100 throws -55 will give tail, then the probability is measured by the ratio of 55 -to 100. - -The mathematical formulæ of the theory are exactly the same as those -of the theory of combinations. In this latter theory we determine in -how many ways events may be joined together, and we now proceed to use -this knowledge in calculating the number of ways in which a certain -event may come about. It is the comparative numbers of ways in which -events can happen which measure their comparative probabilities. If -we throw three pennies into the air, what is the probability that two -of them will fall tail uppermost? This amounts to asking in how many -possible ways can we select two tails out of three, compared with the -whole number of ways in which the coins can be placed. Now, the fourth -line of the Arithmetical Triangle (p. 184) gives us the answer. The -whole number of ways in which we can select or leave three things is -eight, and the possible combinations of two things at a time is three; -hence the probability of two tails is the ratio of three to eight. From -the numbers in the triangle we may similarly draw all the following -probabilities:-- - - One combination gives 0 tail. Probability 1/8. - Three combinations gives 1 tail. Probability 3/8. - Three combinations give 2 tails. Probability 3/8. - One combination gives 3 tails. Probability 1/8. - -We can apply the same considerations to the imaginary causes of the -difference of stature, the combinations of which were shown in p. 188. -There are altogether 128 ways in which seven causes can be present or -absent. Now, twenty-one of these combinations give an addition of two -inches, so that the probability of a person under the circumstances -being five feet two inches is 21/128. The probability of five feet -three inches is 35/128; of five feet one inch 7/128; of five feet -1/128, and so on. Thus the eighth line of the Arithmetical Triangle -gives all the probabilities arising out of the combinations of seven -causes. - - -*Rules for the Calculation of Probabilities.* - -I will now explain as simply as possible the rules for calculating -probabilities. The principal rule is as follows:-- - -Calculate the number of events which may happen independently of each -other, and which, as far as is known, are equally probable. Make this -number the denominator of a fraction, and take for the numerator the -number of such events as imply or constitute the happening of the -event, whose probability is required. - -Thus, if the letters of the word *Roma* be thrown down casually in a -row, what is the probability that they will form a significant Latin -word? The possible arrangements of four letters are 4 × 3 × 2 × 1, -or 24 in number (p. 178), and if all the arrangements be examined, -seven of these will be found to have meaning, namely *Roma*, *ramo*, -*oram*, *mora*, *maro*, *armo*, and *amor*. Hence the probability of a -significant result is 7/24. - -We must distinguish comparative from absolute probabilities. In drawing -a card casually from a pack, there is no reason to expect any one card -more than any other. Now, there are four kings and four queens in a -pack, so that there are just as many ways of drawing one as the other, -and the probabilities are equal. But there are thirteen diamonds, so -that the probability of a king is to that of a diamond as four to -thirteen. Thus the probabilities of each are proportional to their -respective numbers of ways of happening. Again, I can draw a king in -four ways, and not draw one in forty-eight, so that the probabilities -are in this proportion, or, as is commonly said, the *odds* against -drawing a king are forty-eight to four. The odds are seven to seventeen -in favour, or seventeen to seven against the letters R,o,m,a, -accidentally forming a significant word. The odds are five to three -against two tails appearing in three throws of a penny. Conversely, -when the odds of an event are given, and the probability is required, -*take the odds in favour of the event for numerator, and the sum of the -odds for denominator*. - -It is obvious that an event is certain when all the combinations of -causes which can take place produce that event. If we represent the -probability of such event according to our rule, it gives the ratio -of some number to itself, or unity. An event is certain not to happen -when no possible combination of causes gives the event, and the ratio -by the same rule becomes that of 0 to some number. Hence it follows -that in the theory of probability certainty is expressed by 1, and -impossibility by 0; but no mystical meaning should be attached to these -symbols, as they merely express the fact that *all* or *no* possible -combinations give the event. - -By a *compound event*, we mean an event which may be decomposed into -two or more simpler events. Thus the firing of a gun may be decomposed -into pulling the trigger, the fall of the hammer, the explosion of -the cap, &c. In this example the simple events are not *independent*, -because if the trigger is pulled, the other events will under proper -conditions necessarily follow, and their probabilities are therefore -the same as that of the first event. Events are *independent* when -the happening of one does not render the other either more or less -probable than before. Thus the death of a person is neither more nor -less probable because the planet Mars happens to be visible. When -the component events are independent, a simple rule can be given for -calculating the probability of the compound event, thus--*Multiply -together the fractions expressing the probabilities of the independent -component events.* - -The probability of throwing tail twice with a penny is 1/2 × 1/2, -or 1/4; the probability of throwing it three times running is -1/2 × 1/2 × 1/2, or 1/8; a result agreeing with that obtained in -an apparently different manner (p. 202). In fact, when we multiply -together the denominators, we get the whole number of ways of happening -of the compound event, and when we multiply the numerators, we get the -number of ways favourable to the required event. - -Probabilities may be added to or subtracted from each other under the -important condition that the events in question are exclusive of each -other, so that not more than one of them can happen. It might be argued -that, since the probability of throwing head at the first trial is -1/2, and at the second trial also 1/2, the probability of throwing it -in the first two throws is 1/2 + 1/2, or certainty. Not only is this -result evidently absurd, but a repetition of the process would lead -us to a probability of 1-1/2 or of any greater number, results which -could have no meaning whatever. The probability we wish to calculate is -that of one head in two throws, but in our addition we have included -the case in which two heads appear. The true result is 1/2 + 1/2 × 1/2 -or 3/4, or the probability of head at the first throw, added to the -exclusive probability that if it does not come at the first, it will -come at the second. The greatest difficulties of the theory arise -from the confusion of exclusive and unexclusive alternatives. I may -remind the reader that the possibility of unexclusive alternatives was -a point previously discussed (p. 68), and to the reasons then given -for considering alternation as logically unexclusive, may be added -the existence of these difficulties in the theory of probability. The -erroneous result explained above really arose from overlooking the -fact that the expression “head first throw or head second throw” might -include the case of head at both throws. - - -*The Logical Alphabet in questions of Probability.* - -When the probabilities of certain simple events are given, and it is -required to deduce the probabilities of compound events, the Logical -Alphabet may give assistance, provided that there are no special -logical conditions so that all the combinations are possible. Thus, -if there be three events, A, B, C, of which the probabilities are, α, -β, γ, then the negatives of those events, expressing the absence of -the events, will have the probabilities 1 - α, 1 - β, 1 - γ. We have -only to insert these values for the letters of the combinations and -multiply, and we obtain the probability of each combination. Thus the -probability of ABC is αβγ; of A*bc*, α(1 - β)(1 - γ). - -We can now clearly distinguish between the probabilities of exclusive -and unexclusive events. Thus, if A and B are events which may happen -together like rain and high tide, or an earthquake and a storm, the -probability of A or B happening is not the sum of their separate -probabilities. For by the Laws of Thought we develop A ꖌ B into -AB ꖌ A*b* ꖌ *a*B, and substituting α and β, the probabilities of A -and B respectively, we obtain α . β + α . (1 - β) + (1 - α) . β or -α + β - α . β. But if events are *incompossible* or incapable of -happening together, like a clear sky and rain, or a new moon and a full -moon, then the events are not really A or B, but A not-B, or B not-A, -or in symbols A*b* ꖌ *a*B. Now if we take μ = probability of A*b* and ν -= probability of *a*B, then we may add simply, and the probability of -A*b* ꖌ *a*B is μ + ν. - -Let the reader carefully observe that if the combination AB cannot -exist, the probability of A*b* is not the product of the probabilities -of A and *b*. When certain combinations are logically impossible, it -is no longer allowable to substitute the probability of each term for -the term, because the multiplication of probabilities presupposes the -independence of the events. A large part of Boole’s Laws of Thought -is devoted to an attempt to overcome this difficulty and to produce -a General Method in Probabilities by which from certain logical -conditions and certain given probabilities it would be possible to -deduce the probability of any other combinations of events under those -conditions. Boole pursued his task with wonderful ingenuity and power, -but after spending much study on his work, I am compelled to adopt -the conclusion that his method is fundamentally erroneous. As pointed -out by Mr. Wilbraham,[113] Boole obtained his results by an arbitrary -assumption, which is only the most probable, and not the only possible -assumption. The answer obtained is therefore not the real probability, -which is usually indeterminate, but only, as it were, the most probable -probability. Certain problems solved by Boole are free from logical -conditions and therefore may admit of valid answers. These, as I have -shown,[114] may be solved by the combinations of the Logical Alphabet, -but the rest of the problems do not admit of a determinate answer, at -least by Boole’s method. - - [113] *Philosophical Magazine*, 4th Series, vol. vii. p. 465; - vol. viii. p. 91. - - [114] *Memoirs of the Manchester Literary and Philosophical Society*, - 3rd Series, vol. iv. p. 347. - - -*Comparison of the Theory with Experience.* - -The Laws of Probability rest upon the fundamental principles of -reasoning, and cannot be really negatived by any possible experience. -It might happen that a person should always throw a coin head -uppermost, and appear incapable of getting tail by chance. The theory -would not be falsified, because it contemplates the possibility of -the most extreme runs of luck. Our actual experience might be counter -to all that is probable; the whole course of events might seem to be -in complete contradiction to what we should expect, and yet a casual -conjunction of events might be the real explanation. It is just -possible that some regular coincidences, which we attribute to fixed -laws of nature, are due to the accidental conjunction of phenomena -in the cases to which our attention is directed. All that we can -learn from finite experience is capable, according to the theory of -probabilities, of misleading us, and it is only infinite experience -that could assure us of any inductive truths. - -At the same time, the probability that any extreme runs of luck will -occur is so excessively slight, that it would be absurd seriously -to expect their occurrence. It is almost impossible, for instance, -that any whist player should have played in any two games where the -distribution of the cards was exactly the same, by pure accident -(p. 191). Such a thing as a person always losing at a game of -pure chance, is wholly unknown. Coincidences of this kind are not -impossible, as I have said, but they are so unlikely that the lifetime -of any person, or indeed the whole duration of history, does not give -any appreciable probability of their being encountered. Whenever we -make any extensive series of trials of chance results, as in throwing -a die or coin, the probability is great that the results will agree -nearly with the predictions yielded by theory. Precise agreement must -not be expected, for that, as the theory shows, is highly improbable. -Several attempts have been made to test, in this way, the accordance of -theory and experience. Buffon caused the first trial to be made by a -young child who threw a coin many times in succession, and he obtained -1992 tails to 2048 heads. A pupil of De Morgan repeated the trial for -his own satisfaction, and obtained 2044 tails to 2048 heads. In both -cases the coincidence with theory is as close as could be expected, and -the details may be found in De Morgan’s “Formal Logic,” p. 185. - -Quetelet also tested the theory in a rather more complete manner, by -placing 20 black and 20 white balls in an urn and drawing a ball out -time after time in an indifferent manner, each ball being replaced -before a new drawing was made. He found, as might be expected, that the -greater the number of drawings made, the more nearly were the white -and black balls equal in number. At the termination of the experiment -he had registered 2066 white and 2030 black balls, the ratio being -1·02.[115] - - [115] *Letters on the Theory of Probabilities*, translated by Downes, - 1849, pp. 36, 37. - -I have made a series of experiments in a third manner, which seemed to -me even more interesting, and capable of more extensive trial. Taking -a handful of ten coins, usually shillings, I threw them up time after -time, and registered the numbers of heads which appeared each time. Now -the probability of obtaining 10, 9, 8, 7, &c., heads is proportional -to the number of combinations of 10, 9, 8, 7, &c., things out of 10 -things. Consequently the results ought to approximate to the numbers in -the eleventh line of the Arithmetical Triangle. I made altogether 2048 -throws, in two sets of 1024 throws each, and the numbers obtained are -given in the following table:-- - -+-------------------+-----------+---------+---------+----------+-----------+ -|Character of Throw.|Theoretical| First | Second | Average. |Divergence.| -| | Numbers. | Series. | Series. | | | -+-------------------+-----------+---------+---------+----------+-----------+ -| 10 Heads 0 Tail | 1 | 3 | 1 | 2 | + 1 | -| 9 " 1 " | 10 | 12 | 23 | 17-1/2 | + 7-1/2 | -| 8 " 2 " | 45 | 57 | 73 | 65 | + 20 | -| 7 " 3 " | 120 | 129 | 123 | 126 | + 6 | -| 6 " 4 " | 210 | 181 | 190 | 185-1/2 | - 25 | -| 5 " 5 " | 252 | 257 | 232 | 244-1/2 | - 7-1/2 | -| 4 " 6 " | 210 | 201 | 197 | 199 | - 11 | -| 3 " 7 " | 120 | 111 | 119 | 115 | - 5 | -| 2 " 8 " | 45 | 52 | 50 | 51 | + 6 | -| 1 " 9 " | 10 | 21 | 15 | 18 | + 8 | -| 0 " 10 " | 1 | 0 | 1 | 1/2 | - 1/2 | -+-------------------+-----------+---------+---------+----------+-----------+ -| Totals | 1024 | 1024 | 1024 | 1024 | - 1 | -+-------------------+-----------+---------+---------+----------+-----------+ - -The whole number of single throws of coins amounted to 10 × 2048, or -20,480 in all, one half of which or 10,240 should theoretically give -head. The total number of heads obtained was actually 10,353, or 5222 -in the first series, and 5131 in the second. The coincidence with -theory is pretty close, but considering the large number of throws -there is some reason to suspect a tendency in favour of heads. - -The special interest of this trial consists in the exhibition, in a -practical form, of the results of Bernoulli’s theorem, and the law -of error or divergence from the mean to be afterwards more fully -considered. It illustrates the connection between combinations and -permutations, which is exhibited in the Arithmetical Triangle, and -which underlies many important theorems of science. - - -*Probable Deductive Arguments*. - -With the aid of the theory of probabilities, we may extend the sphere -of deductive argument. Hitherto we have treated propositions as -certain, and on the hypothesis of certainty have deduced conclusions -equally certain. But the information on which we reason in ordinary -life is seldom or never certain, and almost all reasoning is really a -question of probability. We ought therefore to be fully aware of the -mode and degree in which deductive reasoning is affected by the theory -of probability, and many persons may be surprised at the results which -must be admitted. Some controversial writers appear to consider, as De -Morgan remarked,[116] that an inference from several equally probable -premises is itself as probable as any of them, but the true result is -very different. If an argument involves many propositions, and each of -them is uncertain, the conclusion will be of very little force. - - [116] *Encyclopædia Metropolitana*, art. *Probabilities*, p. 396. - -The validity of a conclusion may be regarded as a compound event, -depending upon the premises happening to be true; thus, to obtain the -probability of the conclusion, we must multiply together the fractions -expressing the probabilities of the premises. If the probability is -1/2 that A is B, and also 1/2 that B is C, the conclusion that A is -C, on the ground of these premises, is 1/2 × 1/2 or 1/4. Similarly -if there be any number of premises requisite to the establishment -of a conclusion and their probabilities be *p*, *q*, *r*, &c., the -probability of the conclusion on the ground of these premises is -*p* × *q* × *r* × ... This product has but a small value, unless each -of the quantities *p*, *q*, &c., be nearly unity. - -But it is particularly to be noticed that the probability thus -calculated is not the whole probability of the conclusion, but that -only which it derives from the premises in question. Whately’s[117] -remarks on this subject might mislead the reader into supposing that -the calculation is completed by multiplying together the probabilities -of the premises. But it has been fully explained by De Morgan[118] that -we must take into account the antecedent probability of the conclusion; -A may be C for other reasons besides its being B, and as he remarks, -“It is difficult, if not impossible, to produce a chain of argument of -which the reasoner can rest the result on those arguments only.” The -failure of one argument does not, except under special circumstances, -disprove the truth of the conclusion it is intended to uphold, -otherwise there are few truths which could survive the ill-considered -arguments adduced in their favour. As a rope does not necessarily break -because one or two strands in it fail, so a conclusion may depend upon -an endless number of considerations besides those immediately in view. -Even when we have no other information we must not consider a statement -as devoid of all probability. The true expression of complete doubt is -a ratio of equality between the chances in favour of and against it, -and this ratio is expressed in the probability 1/2. - - [117] *Elements of Logic*, Book III. sections 11 and 18. - - [118] *Encyclopædia Metropolitana*, art. *Probabilities*, p. 400. - -Now if A and C are wholly unknown things, we have no reason to believe -that A is C rather than A is not C. The antecedent probability is then -1/2. If we also have the probabilities that A is B, 1/2 and that B is -C, 1/2 we have no right to suppose that the probability of A being C -is reduced by the argument in its favour. If the conclusion is true -on its own grounds, the failure of the argument does not affect it; -thus its total probability is its antecedent probability, added to the -probability that this failing, the new argument in question establishes -it. There is a probability 1/2 that we shall not require the special -argument; a probability 1/2 that we shall, and a probability 1/4 -that the argument does in that case establish it. Thus the complete -result is 1/2 + 1/2 × 1/4, or 5/8. In general language, if *a* be the -probability founded on a particular argument, and *c* the antecedent -probability of the event, the general result is 1 - (1 - *a*)(1 - *c*), -or *a* + *c* - *ac*. - -We may put it still more generally in this way:--Let *a*, *b*, *c*, &c. -be the probabilities of a conclusion grounded on various arguments. It -is only when all the arguments fail that our conclusion proves finally -untrue; the probabilities of each failing are respectively, 1 - *a*, -1 - *b*, 1 - *c*, &c.; the probability that they will all fail is -(1 - *a*)(1 - *b*)(1 - *c*) ...; therefore the probability that the -conclusion will not fail is 1 - (1 - *a*)(1 - *b*)(1 - *c*) ... &c. It -follows that every argument in favour of a conclusion, however flimsy -and slight, adds probability to it. When it is unknown whether an -overdue vessel has foundered or not, every slight indication of a lost -vessel will add some probability to the belief of its loss, and the -disproof of any particular evidence will not disprove the event. - -We must apply these principles of evidence with great care, and -observe that in a great proportion of cases the adducing of a weak -argument does tend to the disproof of its conclusion. The assertion -may have in itself great inherent improbability as being opposed to -other evidence or to the supposed law of nature, and every reasoner -may be assumed to be dealing plainly, and putting forward the whole -force of evidence which he possesses in its favour. If he brings but -one argument, and its probability *a* is small, then in the formula -1 - (1 - *a*)(1 - *c*) both *a* and *c* are small, and the whole -expression has but little value. The whole effect of an argument thus -turns upon the question whether other arguments remain, so that we -can introduce other factors (1 - *b*), (1 - *d*), &c., into the above -expression. In a court of justice, in a publication having an express -purpose, and in many other cases, it is doubtless right to assume -that the whole evidence considered to have any value as regards the -conclusion asserted, is put forward. - -To assign the antecedent probability of any proposition, may be a -matter of difficulty or impossibility, and one with which logic and -the theory of probability have little concern. From the general body of -science in our possession, we must in each case make the best judgment -we can. But in the absence of all knowledge the probability should -be considered = 1/2, for if we make it less than this we incline to -believe it false rather than true. Thus, before we possessed any means -of estimating the magnitudes of the fixed stars, the statement that -Sirius was greater than the sun had a probability of exactly 1/2; it -was as likely that it would be greater as that it would be smaller; and -so of any other star. This was the assumption which Michell made in -his admirable speculations.[119] It might seem, indeed, that as every -proposition expresses an agreement, and the agreements or resemblances -between phenomena are infinitely fewer than the differences (p. 44), -every proposition should in the absence of other information be -infinitely improbable. But in our logical system every term may be -indifferently positive or negative, so that we express under the same -form as many differences as agreements. It is impossible therefore -that we should have any reason to disbelieve rather than to believe a -statement about things of which we know nothing. We can hardly indeed -invent a proposition concerning the truth of which we are absolutely -ignorant, except when we are entirely ignorant of the terms used. If I -ask the reader to assign the odds that a “Platythliptic Coefficient is -positive” he will hardly see his way to doing so, unless he regard them -as even. - - [119] *Philosophical Transactions* (1767). Abridg. vol. xii. p. 435. - -The assumption that complete doubt is properly expressed by 1/2 has -been called in question by Bishop Terrot,[120] who proposes instead -the indefinite symbol 0/0; and he considers that “the *à priori* -probability derived from absolute ignorance has no effect upon the -force of a subsequently admitted probability.” But if we grant that the -probability may have any value between 0 and 1, and that every separate -value is equally likely, then *n* and 1 - *n* are equally likely, and -the average is always 1/2. Or we may take *p* . *dp* to express the -probability that our estimate concerning any proposition should lie -between *p* and *p* + *dp*. The complete probability of the proposition -is then the integral taken between the limits 1 and 0, or again 1/2. - - [120] *Transactions of the Edinburgh Philosophical Society*, - vol. xxi. p. 375. - - -*Difficulties of the Theory.* - -The theory of probability, though undoubtedly true, requires very -careful application. Not only is it a branch of mathematics in which -oversights are frequently committed, but it is a matter of great -difficulty in many cases, to be sure that the formula correctly -represents the data of the problem. These difficulties often arise from -the logical complexity of the conditions, which might be, perhaps, to -some extent cleared up by constantly bearing in mind the system of -combinations as developed in the Indirect Logical Method. In the study -of probabilities, mathematicians had unconsciously employed logical -processes far in advance of those in possession of logicians, and the -Indirect Method is but the full statement of these processes. - -It is very curious how often the most acute and powerful intellects -have gone astray in the calculation of probabilities. Seldom was Pascal -mistaken, yet he inaugurated the science with a mistaken solution.[121] -Leibnitz fell into the extraordinary blunder of thinking that the -number twelve was as probable a result in the throwing of two dice as -the number eleven.[122] In not a few cases the false solution first -obtained seems more plausible to the present day than the correct -one since demonstrated. James Bernoulli candidly records two false -solutions of a problem which he at first thought self-evident; and he -adds a warning against the risk of error, especially when we attempt -to reason on this subject without a rigid adherence to methodical -rules and symbols. Montmort was not free from similar mistakes. -D’Alembert constantly fell into blunders, and could not perceive, -for instance, that the probabilities would be the same when coins -are thrown successively as when thrown simultaneously. Some men of -great reputation, such as Ancillon, Moses Mendelssohn, Garve, Auguste -Comte,[123] Poinsot, and J. S. Mill,[124] have so far misapprehended -the theory, as to question its value or even to dispute its validity. -The erroneous statements about the theory given in the earlier editions -of Mill’s *System of Logic* were partially withdrawn in the later -editions. - - [121] Montucla, *Histoire des Mathématiques*, vol. iii. p. 386. - - [122] Leibnitz *Opera*, Dutens’ Edition, vol. vi. part i. p. 217. - Todhunter’s *History of the Theory of Probability*, p. 48. To the - latter work I am indebted for many of the statements in the text. - - [123] *Positive Philosophy*, translated by Martineau, vol. ii. p. 120. - - [124] *System of Logic*, bk. iii. chap. 18, 5th Ed. vol. ii. p. 61. - -Many persons have a fallacious tendency to believe that when a chance -event has happened several times together in an unusual conjunction, it -is less likely to happen again. D’Alembert seriously held that if head -was thrown three times running with a coin, tail would more probably -appear at the next trial.[125] Bequelin adopted the same opinion, and -yet there is no reason for it whatever. If the event be really casual, -what has gone before cannot in the slightest degree influence it. As -a matter of fact, the more often a casual event takes place the more -likely it is to happen again; because there is some slight empirical -evidence of a tendency. The source of the fallacy is to be found -entirely in the feelings of surprise with which we witness an event -happening by chance, in a manner which seems to proceed from design. - - [125] Montucla, *Histoire*, vol. iii. p. 405; Todhunter, p. 263. - -Misapprehension may also arise from overlooking the difference between -permutations and combinations. To throw ten heads in succession with a -coin is no more unlikely than to throw any other particular succession -of heads and tails, but it is much less likely than five heads and five -tails without regard to their order, because there are no less than -252 different particular throws which will give this result, when we -abstract the difference of order. - -Difficulties arise in the application of the theory from our habitual -disregard of slight probabilities. We are obliged practically to accept -truths as certain which are nearly so, because it ceases to be worth -while to calculate the difference. No punishment could be inflicted -if absolutely certain evidence of guilt were required, and as Locke -remarks, “He that will not stir till he infallibly knows the business -he goes about will succeed, will have but little else to do but to -sit still and perish.”[126] There is not a moment of our lives when -we do not lie under a slight danger of death, or some most terrible -fate. There is not a single action of eating, drinking, sitting down, -or standing up, which has not proved fatal to some person. Several -philosophers have tried to assign the limit of the probabilities -which we regard as zero; Buffon named 1/10,000, because it is the -probability, practically disregarded, that a man of 56 years of age -will die the next day. Pascal remarked that a man would be esteemed a -fool for hesitating to accept death when three dice gave sixes twenty -times running, if his reward in case of a different result was to be a -crown; but as the chance of death in question is only 1 ÷ 6^{60}, or -unity divided by a number of 47 places of figures, we may be said to -incur greater risks every day for less motives. There is far greater -risk of death, for instance, in a game of cricket or a visit to the -rink. - - [126] *Essay concerning Human Understanding*, bk. iv. ch. 14. § 1. - -Nothing is more requisite than to distinguish carefully between the -truth of a theory and the truthful application of the theory to actual -circumstances. As a general rule, events in nature and art will present -a complexity of relations exceeding our powers of treatment. The -intricate action of the mind often intervenes and renders complete -analysis hopeless. If, for instance, the probability that a marksman -shall hit the target in a single shot be 1 in 10, we might seem to -have no difficulty in calculating the probability of any succession of -hits; thus the probability of three successive hits would be one in a -thousand. But, in reality, the confidence and experience derived from -the first successful shot would render a second success more probable. -The events are not really independent, and there would generally be -a far greater preponderance of runs of apparent luck, than a simple -calculation of probabilities could account for. In some persons, -however, a remarkable series of successes will produce a degree of -excitement rendering continued success almost impossible. - -Attempts to apply the theory of probability to the results of judicial -proceedings have proved of little value, simply because the conditions -are far too intricate. As Laplace said, “Tant de passions, d’intérêts -divers et de circonstances compliquent les questions relatives à ces -objets, qu’elles sont presque toujours insolubles.” Men acting on a -jury, or giving evidence before a court, are subject to so many complex -influences that no mathematical formulas can be framed to express the -real conditions. Jurymen or even judges on the bench cannot be regarded -as acting independently, with a definite probability in favour of each -delivering a correct judgment. Each man of the jury is more or less -influenced by the opinion of the others, and there are subtle effects -of character and manner and strength of mind which defy analysis. Even -in physical science we can in comparatively few cases apply the theory -in a definite manner, because the data required are too complicated and -difficult to obtain. But such failures in no way diminish the truth and -beauty of the theory itself; in reality there is no branch of science -in which our symbols can cope with the complexity of Nature. As Donkin -said,-- - -“I do not see on what ground it can be doubted that every definite -state of belief concerning a proposed hypothesis, is in itself capable -of being represented by a numerical expression, however difficult or -impracticable it may be to ascertain its actual value. It would be very -difficult to estimate in numbers the *vis viva* of all the particles of -a human body at any instant; but no one doubts that it is capable of -numerical expression.”[127] - - [127] *Philosophical Magazine*, 4th Series, vol. i. p. 354. - -The difficulty, in short, is merely relative to our knowledge and -skill, and is not absolute or inherent in the subject. We must -distinguish between what is theoretically conceivable and what is -practicable with our present mental resources. Provided that our -aspirations are pointed in a right direction, we must not allow them -to be damped by the consideration that they pass beyond what can now -be turned to immediate use. In spite of its immense difficulties of -application, and the aspersions which have been mistakenly cast upon -it, the theory of probabilities, I repeat, is the noblest, as it -will in course of time prove, perhaps the most fruitful branch of -mathematical science. It is the very guide of life, and hardly can -we take a step or make a decision of any kind without correctly or -incorrectly making an estimation of probabilities. In the next chapter -we proceed to consider how the whole cogency of inductive reasoning -rests upon probabilities. The truth or untruth of a natural law, when -carefully investigated, resolves itself into a high or low degree of -probability, and this is the case whether or not we are capable of -producing precise numerical data. - - - - -CHAPTER XI. - -PHILOSOPHY OF INDUCTIVE INFERENCE. - - -We have inquired into the nature of perfect induction, whereby we pass -backwards from certain observed combinations of events, to the logical -conditions governing such combinations. We have also investigated the -grounds of that theory of probability, which must be our guide when we -leave certainty behind, and dilute knowledge with ignorance. There is -now before us the difficult task of endeavouring to decide how, by the -aid of that theory, we can ascend from the facts to the laws of nature; -and may then with more or less success anticipate the future course -of events. All our knowledge of natural objects must be ultimately -derived from observation, and the difficult question arises--How can -we ever know anything which we have not directly observed through one -of our senses, the apertures of the mind? The utility of reasoning is -to assure ourselves that, at a determinate time and place, or under -specified conditions, a certain phenomenon will be observed. When -we can use our senses and perceive that the phenomenon does occur, -reasoning is superfluous. If the senses cannot be used, because the -event is in the future, or out of reach, how can reasoning take their -place? Apparently, at least, we must infer the unknown from the known, -and the mind must itself create an addition to the sum of knowledge. -But I hold that it is quite impossible to make any real additions to -the contents of our knowledge, except through new impressions upon -the senses, or upon some seat of feeling. I shall attempt to show -that inference, whether inductive or deductive, is never more than -an unfolding of the contents of our experience, and that it always -proceeds upon the assumption that the future and the unperceived will -be governed by the same conditions as the past and the perceived, an -assumption which will often prove to be mistaken. - -In inductive as in deductive reasoning the conclusion never passes -beyond the premises. Reasoning adds no more to the implicit contents of -our knowledge, than the arrangement of the specimens in a museum adds -to the number of those specimens. Arrangement adds to our knowledge -in a certain sense: it allows us to perceive the similarities and -peculiarities of the specimens, and on the assumption that the museum -is an adequate representation of nature, it enables us to judge of -the prevailing forms of natural objects. Bacon’s first aphorism holds -perfectly true, that man knows nothing but what he has observed, -provided that we include his whole sources of experience, and the whole -implicit contents of his knowledge. Inference but unfolds the hidden -meaning of our observations, and *the theory of probability shows how -far we go beyond our data in assuming that new specimens will resemble -the old ones*, or that the future may be regarded as proceeding -uniformly with the past. - - -*Various Classes of Inductive Truths.* - -It will be desirable, in the first place, to distinguish between the -several kinds of truths which we endeavour to establish by induction. -Although there is a certain common and universal element in all our -processes of reasoning, yet diversity arises in their application. -Similarity of condition between the events from which we argue, and -those to which we argue, must always be the ground of inference; but -this similarity may have regard either to time or place, or the simple -logical combination of events, or to any conceivable junction of -circumstances involving quality, time, and place. Having met with many -pieces of substance possessing ductility and a bright yellow colour, -and having discovered, by perfect induction, that they all possess a -high specific gravity, and a freedom from the corrosive action of -acids, we are led to expect that every piece of substance, possessing -like ductility and a similar yellow colour, will have an equally high -specific gravity, and a like freedom from corrosion by acids. This -is a case of the coexistence of qualities; for the character of the -specimens examined alters not with time nor place. - -In a second class of cases, time will enter as a principal ground of -similarity. When we hear a clock pendulum beat time after time, at -equal intervals, and with a uniform sound, we confidently expect that -the stroke will continue to be repeated uniformly. A comet having -appeared several times at nearly equal intervals, we infer that it will -probably appear again at the end of another like interval. A man who -has returned home evening after evening for many years, and found his -house standing, may, on like grounds, expect that it will be standing -the next evening, and on many succeeding evenings. Even the continuous -existence of an object in an unaltered state, or the finding again of -that which we have hidden, is but a matter of inference depending on -experience. - -A still larger and more complex class of cases involves the relations -of space, in addition to those of time and quality. Having observed -that every triangle drawn upon the diameter of a circle, with its -apex upon the circumference, apparently contains a right angle, -we may ascertain that all triangles in similar circumstances will -contain right angles. This is a case of pure space reasoning, apart -from circumstances of time or quality, and it seems to be governed by -different principles of reasoning. I shall endeavour to show, however, -that geometrical reasoning differs but in degree from that which -applies to other natural relations. - - -*The Relation of Cause and Effect.* - -In a very large part of the scientific investigations which must be -considered, we deal with events which follow from previous events, or -with existences which succeed existences. Science, indeed, might arise -even were material nature a fixed and changeless whole. Endow mind -with the power to travel about, and compare part with part, and it -could certainly draw inferences concerning the similarity of forms, the -coexistence of qualities, or the preponderance of a particular kind of -matter in a changeless world. A solid universe, in at least approximate -equilibrium, is not inconceivable, and then the relation of cause and -effect would evidently be no more than the relation of before and -after. As nature exists, however, it is a progressive existence, ever -moving and changing as time, the great independent variable, proceeds. -Hence it arises that we must continually compare what is happening now -with what happened a moment before, and a moment before that moment, -and so on, until we reach indefinite periods of past time. A comet -is seen moving in the sky, or its constituent particles illumine -the heavens with their tails of fire. We cannot explain the present -movements of such a body without supposing its prior existence, with -a definite amount of energy and a definite direction of motion; nor -can we validly suppose that our task is concluded when we find that it -came wandering to our solar system through the unmeasured vastness of -surrounding space. Every event must have a cause, and that cause again -a cause, until we are lost in the obscurity of the past, and are driven -to the belief in one First Cause, by whom the course of nature was -determined. - - -*Fallacious Use of the Term Cause.* - -The words Cause and Causation have given rise to infinite trouble -and obscurity, and have in no slight degree retarded the progress of -science. From the time of Aristotle, the work of philosophy has been -described as the discovery of the causes of things, and Francis Bacon -adopted the notion when he said “*vere scire esse per causas scire*.” -Even now it is not uncommonly supposed that the knowledge of causes is -something different from other knowledge, and consists, as it were, in -getting possession of the keys of nature. A single word may thus act -as a spell, and throw the clearest intellect into confusion, as I have -often thought that Locke was thrown into confusion when endeavouring to -find a meaning for the word *power*.[128] In Mill’s *System of Logic* -the term *cause* seems to have re-asserted its old noxious power. Not -only does Mill treat the Laws of Causation as almost coextensive with -science, but he so uses the expression as to imply that when once we -pass within the circle of causation we deal with certainties. - - [128] *Essay concerning Human Understanding*, bk. ii. chap. xxi. - -The philosophical danger which attaches to the use of this word may -be thus described. A cause is defined as the necessary or invariable -antecedent of an event, so that when the cause exists the effect will -also exist or soon follow. If then we know the cause of an event, we -know what will certainly happen; and as it is implied that science, -by a proper experimental method, may attain to a knowledge of causes, -it follows that experience may give us a certain knowledge of future -events. But nothing is more unquestionable than that finite experience -can never give us certain knowledge of the future, so that either -a cause is not an invariable antecedent, or else we can never gain -certain knowledge of causes. The first horn of this dilemma is hardly -to be accepted. Doubtless there is in nature some invariably acting -mechanism, such that from certain fixed conditions an invariable result -always emerges. But we, with our finite minds and short experience, can -never penetrate the mystery of those existences which embody the Will -of the Creator, and evolve it throughout time. We are in the position -of spectators who witness the productions of a complicated machine, but -are not allowed to examine its intimate structure. We learn what does -happen and what does appear, but if we ask for the reason, the answer -would involve an infinite depth of mystery. The simplest bit of matter, -or the most trivial incident, such as the stroke of two billiard balls, -offers infinitely more to learn than ever the human intellect can -fathom. The word cause covers just as much untold meaning as any of the -words *substance*, *matter*, *thought*, *existence*. - - -*Confusion of Two Questions.* - -The subject is much complicated, too, by the confusion of two distinct -questions. An event having happened, we may ask-- - - (1) Is there any cause for the event? - (2) Of what kind is that cause? - -No one would assert that the mind possesses any faculty capable of -inferring, prior to experience, that the occurrence of a sudden noise -with flame and smoke indicates the combustion of a black powder, formed -by the mixture of black, white, and yellow powders. The greatest -upholder of *à priori* doctrines will allow that the particular aspect, -shape, size, colour, texture, and other qualities of a cause must be -gathered through the senses. - -The question whether there is any cause at all for an event, is of -a totally different kind. If an explosion could happen without any -prior existing conditions, it must be a new creation--a distinct -addition to the universe. It may be plausibly held that we can imagine -neither the creation nor annihilation of anything. As regards matter, -this has long been held true; as regards force, it is now almost -universally assumed as an axiom that energy can neither come into -nor go out of existence without distinct acts of Creative Will. That -there exists any instinctive belief to this effect, indeed, seems -doubtful. We find Lucretius, a philosopher of the utmost intellectual -power and cultivation, gravely assuming that his raining atoms could -turn aside from their straight paths in a self-determining manner, -and by this spontaneous origination of energy determine the form of -the universe.[129] Sir George Airy, too, seriously discussed the -mathematical conditions under which a perpetual motion, that is, a -perpetual source of self-created energy, might exist.[130] The larger -part of the philosophic world has long held that in mental acts there -is free will--in short, self-causation. It is in vain to attempt to -reconcile this doctrine with that of an intuitive belief in causation, -as Sir W. Hamilton candidly allowed. - - [129] *De Rerum Natura*, bk. ii. ll. 216–293. - - [130] *Cambridge Philosophical Transactions* (1830), vol. iii. pp. - 369–372. - -It is obvious, moreover, that to assert the existence of a cause for -every event cannot do more than remove into the indefinite past the -inconceivable fact and mystery of creation. At any given moment matter -and energy were equal to what they are at present, or they were not; -if equal, we may make the same inquiry concerning any other moment, -however long prior, and we are thus obliged to accept one horn of the -dilemma--existence from infinity, or creation at some moment. This is -but one of the many cases in which we are compelled to believe in one -or other of two alternatives, both inconceivable. My present purpose, -however, is to point out that we must not confuse this supremely -difficult question with that into which inductive science inquires on -the foundation of facts. By induction we gain no certain knowledge; -but by observation, and the inverse use of deductive reasoning, we -estimate the probability that an event which has occurred was preceded -by conditions of specified character, or that such conditions will be -followed by the event. - - -*Definition of the Term Cause.* - -Clear definitions of the word cause have been given by several -philosophers. Hobbes has said, “A cause is the sum or aggregate of all -such accidents, both in the agents and the patients, as concur in the -producing of the effect propounded; all which existing together, it -cannot be understood but that the effect existeth with them; or that -it can possibly exist if any of them be absent.” Brown, in his *Essay -on Causation*, gave a nearly corresponding statement. “A cause,” he -says,[131] “may be defined to be the object or event which immediately -precedes any change, and which existing again in similar circumstances -will be always immediately followed by a similar change.” Of the -kindred word *power*, he likewise says:[132] “Power is nothing more -than that invariableness of antecedence which is implied in the belief -of causation.” - - [131] *Observations on the Nature and Tendency of the Doctrine of Mr. - Hume, concerning the Relation of Cause and Effect.* Second ed. p. 44. - - [132] Ibid. p. 97. - -These definitions may be accepted with the qualification that our -knowledge of causes in such a sense can be probable only. The work of -science consists in ascertaining the combinations in which phenomena -present themselves. Concerning every event we shall have to determine -its probable conditions, or the group of antecedents from which it -probably follows. An antecedent is anything which exists prior to -an event; a consequent is anything which exists subsequently to an -antecedent. It will not usually happen that there is any probable -connection between an antecedent and consequent. Thus nitrogen is an -antecedent to the lighting of a common fire; but it is so far from -being a cause of the lighting, that it renders the combustion less -active. Daylight is an antecedent to all fires lighted during the day, -but it probably has no appreciable effect upon their burning. But -in the case of any given event it is usually possible to discover a -certain number of antecedents which seem to be always present, and with -more or less probability we conclude that when they exist the event -will follow. - -Let it be observed that the utmost latitude is at present enjoyed in -the use of the term *cause*. Not only may a cause be an existent thing -endowed with powers, as oxygen is the cause of combustion, gunpowder -the cause of explosion, but the very absence or removal of a thing may -also be a cause. It is quite correct to speak of the dryness of the -Egyptian atmosphere, or the absence of moisture, as being the cause of -the preservation of mummies, and other remains of antiquity. The cause -of a mountain elevation, Ingleborough for instance, is the excavation -of the surrounding valleys by denudation. It is not so usual to speak -of the existence of a thing at one moment as the cause of its existence -at the next, but to me it seems the commonest case of causation which -can occur. The cause of motion of a billiard ball may be the stroke of -another ball; and recent philosophy leads us to look upon all motions -and changes, as but so many manifestations of prior existing energy. -In all probability there is no creation of energy and no destruction, -so that as regards both mechanical and molecular changes, the cause is -really the manifestation of existing energy. In the same way I see not -why the prior existence of matter is not also a cause as regards its -subsequent existence. All science tends to show us that the existence -of the universe in a particular state at one moment, is the condition -of its existence at the next moment, in an apparently different -state. When we analyse the meaning which we can attribute to the word -*cause*, it amounts to the existence of suitable portions of matter -endowed with suitable quantities of energy. If we may accept Horne -Tooke’s assertion, *cause* has etymologically the meaning of *thing -before*. Though, indeed, the origin of the word is very obscure, its -derivatives, the Italian *cosa*, and the French *chose*, mean simply -*thing*. In the German equivalent *ursache*, we have plainly the -original meaning of *thing before*, the *sache* denoting “interesting -or important object,” the English *sake*, and *ur* being the equivalent -of the English *ere*, *before*. We abandon, then, both etymology and -philosophy, when we attribute to the *laws of causation* any meaning -beyond that of the *conditions* under which an event may be expected to -happen, according to our observation of the previous course of nature. - -I have no objection to use the words cause and causation, provided they -are never allowed to lead us to imagine that our knowledge of nature -can attain to certainty. I repeat that if a cause is an invariable -and necessary condition of an event, we can never know certainly -whether the cause exists or not. To us, then, a cause is not to be -distinguished from the group of positive or negative conditions which, -with more or less probability, precede an event. In this sense, there -is no particular difference between knowledge of causes and our general -knowledge of the succession of combinations, in which the phenomena of -nature are presented to us, or found to occur in experimental inquiry. - - -*Distinction of Inductive and Deductive Results.* - -We must carefully avoid confusing together inductive investigations -which terminate in the establishment of general laws, and those which -seem to lead directly to the knowledge of future particular events. -That process only can be called induction which gives general laws, -and it is by the subsequent employment of deduction that we anticipate -particular events. If the observation of a number of cases shows that -alloys of metals fuse at lower temperatures than their constituent -metals, I may with more or less probability draw a general inference -to that effect, and may thence deductively ascertain the probability -that the next alloy examined will fuse at a lower temperature than -its constituents. It has been asserted, indeed, by Mill,[133] and -partially admitted by Mr. Fowler,[134] that we can argue directly from -case to case, so that what is true of some alloys will be true of the -next. Professor Bain has adopted the same view of reasoning. He thinks -that Mill has extricated us from the dead lock of the syllogism and -effected a total revolution in logic. He holds that reasoning from -particulars to particulars is not only the usual, the most obvious and -the most ready method, but that it is the type of reasoning which best -discloses the real process.[135] Doubtless, this is the usual result of -our reasoning, regard being had to degrees of probability; but these -logicians fail entirely to give any explanation of the process by which -we get from case to case. - - [133] *System of Logic*, bk. II. chap, iii. - - [134] *Inductive Logic*, pp. 13, 14. - - [135] Bain, *Deductive Logic*, pp. 208, 209. - -It may be allowed that the knowledge of future particular events is -the main purpose of our investigations, and if there were any process -of thought by which we could pass directly from event to event without -ascending into general truths, this method would be sufficient, and -certainly the briefest. It is true, also, that the laws of mental -association lead the mind always to expect the like again in apparently -like circumstances, and even animals of very low intelligence must -have some trace of such powers of association, serving to guide them -more or less correctly, in the absence of true reasoning faculties. -But it is the purpose of logic, according to Mill, to ascertain -whether inferences have been correctly drawn, rather than to discover -them.[136] Even if we can, then, by habit, association, or any rude -process of inference, infer the future directly from the past, it is -the work of logic to analyse the conditions on which the correctness -of this inference depends. Even Mill would admit that such analysis -involves the consideration of general truths,[137] and in this, as in -several other important points, we might controvert Mill’s own views -by his own statements. It seems to me undesirable in a systematic work -like this to enter into controversy at any length, or to attempt to -refute the views of other logicians. But I shall feel bound to state, -in a separate publication, my very deliberate opinion that many of -Mill’s innovations in logical science, and especially his doctrine of -reasoning from particulars to particulars, are entirely groundless and -false. - - [136] *System of Logic.* Introduction, § 4. Fifth ed. pp. 8, 9. - - [137] Ibid. bk. II. chap. iii. § 5, pp. 225, &c. - - -*The Grounds of Inductive Inference.* - -I hold that in all cases of inductive inference we must invent -hypotheses, until we fall upon some hypothesis which yields deductive -results in accordance with experience. Such accordance renders the -chosen hypothesis more or less probable, and we may then deduce, with -some degree of likelihood, the nature of our future experience, on the -assumption that no arbitrary change takes place in the conditions of -nature. We can only argue from the past to the future, on the general -principle set forth in this work, that what is true of a thing will -be true of the like. So far then as one object or event differs from -another, all inference is impossible, particulars as particulars can -no more make an inference than grains of sand can make a rope. We -must always rise to something which is general or same in the cases, -and assuming that sameness to be extended to new cases we learn their -nature. Hearing a clock tick five thousand times without exception or -variation, we adopt the very probable hypothesis that there is some -invariably acting machine which produces those uniform sounds, and -which will, in the absence of change, go on producing them. Meeting -twenty times with a bright yellow ductile substance, and finding it -always to be very heavy and incorrodible, I infer that there was some -natural condition which tended in the creation of things to associate -these properties together, and I expect to find them associated in the -next instance. But there always is the possibility that some unknown -change may take place between past and future cases. The clock may run -down, or be subject to a hundred accidents altering its condition. -There is no reason in the nature of things, so far as known to us, why -yellow colour, ductility, high specific gravity, and incorrodibility, -should always be associated together, and in other cases, if not in -this, men’s expectations have been deceived. Our inferences, therefore, -always retain more or less of a hypothetical character, and are so far -open to doubt. Only in proportion as our induction approximates to the -character of perfect induction, does it approximate to certainty. The -amount of uncertainty corresponds to the probability that other objects -than those examined may exist and falsity our inferences; the amount -of probability corresponds to the amount of information yielded by our -examination; and the theory of probability will be needed to prevent us -from over-estimating or under-estimating the knowledge we possess. - - -*Illustrations of the Inductive Process.* - -To illustrate the passage from the known to the apparently unknown, let -us suppose that the phenomena under investigation consist of numbers, -and that the following six numbers being exhibited to us, we are -required to infer the character of the next in the series:-- - - 5, 15, 35, 45, 65, 95. - -The question first of all arises, How may we describe this series of -numbers? What is uniformly true of them? The reader cannot fail to -perceive at the first glance that they all end in five, and the problem -is, from the properties of these six numbers, to infer the properties -of the next number ending in five. If we test their properties by the -process of perfect induction, we soon perceive that they have another -common property, namely that of being *divisible by five without -remainder*. May we then assert that the next number ending in five is -also divisible by five, and, if so, upon what grounds? Or extending -the question, Is every number ending in five divisible by five? Does -it follow that because six numbers obey a supposed law, therefore -376,685,975 or any other number, however large, obeys the law? I answer -*certainly not*. The law in question is undoubtedly true; but its truth -is not proved by any finite number of examples. All that these six -numbers can do is to suggest to my mind the possible existence of such -a law; and I then ascertain its truth, by proving deductively from the -rules of decimal numeration, that any number ending in five must be -made up of multiples of five, and must therefore be itself a multiple. - -To make this more plain, let the reader now examine the numbers-- - - 7, 17, 37, 47, 67, 97. - -They all end in 7 instead of 5, and though not at equal intervals, the -intervals are the same as in the previous case. After consideration, -the reader will perceive that these numbers all agree in being *prime -numbers*, or multiples of unity only. May we then infer that the next, -or any other number ending in 7, is a prime number? Clearly not, for -on trial we find that 27, 57, 117 are not primes. Six instances, -then, treated empirically, lead us to a true and universal law in one -case, and mislead us in another case. We ought, in fact, to have no -confidence in any law until we have treated it deductively, and have -shown that from the conditions supposed the results expected must -ensue. No one can show from the principles of number, that numbers -ending in 7 should be primes. - -From the history of the theory of numbers some good examples of false -induction can be adduced. Taking the following series of prime numbers, - - 41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, &c., - -it will be found that they all agree in being values of the general -expression *x*^{2} + *x* + 41, putting for *x* in succession the -values, 0, 1, 2, 3, 4, &c. We seem always to obtain a prime number, and -the induction is apparently strong, to the effect that this expression -always will give primes. Yet a few more trials disprove this false -conclusion. Put *x* = 40, and we obtain 40 × 40 + 40 + 41, or 41 × 41. -Such a failure could never have happened, had we shown any deductive -reason why *x*^{2} + *x* + 41 should give primes. - -There can be no doubt that what here happens with forty instances, -might happen with forty thousand or forty million instances. An -apparent law never once failing up to a certain point may then suddenly -break down, so that inductive reasoning, as it has been described by -some writers, can give no sure knowledge of what is to come. Babbage -pointed out, in his Ninth Bridgewater Treatise, that a machine could -be constructed to give a perfectly regular series of numbers through -a vast series of steps, and yet to break the law of progression -suddenly at any required point. No number of particular cases as -particulars enables us to pass by inference to any new case. It is -hardly needful to inquire here what can be inferred from an infinite -series of facts, because they are never practically within our power; -but we may unhesitatingly accept the conclusion, that no finite number -of instances can ever prove a general law, or can give us certain -knowledge of even one other instance. - -General mathematical theorems have indeed been discovered by the -observation of particular cases, and may again be so discovered. We -have Newton’s own statement, to the effect that he was thus led to -the all-important Binomial Theorem, the basis of the whole structure -of mathematical analysis. Speaking of a certain series of terms, -expressing the area of a circle or hyperbola, he says: “I reflected -that the denominators were in arithmetical progression; so that -only the numerical co-efficients of the numerators remained to be -investigated. But these, in the alternate areas, were the figures of -the powers of the number eleven, namely 11^{0}, 11^{1}, 11^{2}, 11^{3}, -11^{4}; that is, in the first 1; in the second 1, 1; in the third 1, -2, 1; in the fourth 1, 3, 3, 1; in the fifth 1, 4, 6, 4, 1.[138] I -inquired, therefore, in what manner all the remaining figures could -be found from the first two; and I found that if the first figure be -called *m*, all the rest could be found by the continual multiplication -of the terms of the formula - - ((*m* - 0)/1) × ((*m* - 1)/2) × ((*m* - 2)/3) × - ((*m* - 3)/4) × &c.”[139] - - [138] These are the figurate numbers considered in pages 183, 187, &c. - - [139] *Commercium Epistolicum.* *Epistola ad Oldenburgum*, Oct. 24, - 1676. Horsley’s *Works of Newton*, vol. iv. p. 541. See De Morgan in - *Penny Cyclopædia*, art. “Binomial Theorem,” p. 412. - -It is pretty evident, from this most interesting statement, that -Newton, having simply observed the succession of the numbers, tried -various formulæ until he found one which agreed with them all. He -was so little satisfied with this process, however, that he verified -particular results of his new theorem by comparison with the results -of common multiplication, and the rule for the extraction of the -square root. Newton, in fact, gave no demonstration of his theorem; -and the greatest mathematicians of the last century, James Bernoulli, -Maclaurin, Landen, Euler, Lagrange, &c., occupied themselves with -discovering a conclusive method of deductive proof. - -There can be no doubt that in geometry also discoveries have been -suggested by direct observation. Many of the now trivial propositions -of Euclid’s Elements were probably thus discovered, by the ancient -Greek geometers; and we have pretty clear evidence of this in the -Commentaries of Proclus.[140] Galileo was the first to examine the -remarkable properties of the cycloid, the curve described by a point in -the circumference of a wheel rolling on a plane. By direct observation -he ascertained that the area of the curve is apparently three times -that of the generating circle or wheel, but he was unable to prove this -exactly, or to verify it by strict geometrical reasoning. Sir George -Airy has recorded a curious case, in which he fell accidentally by -trial on a new geometrical property of the sphere.[141] But discovery -in such cases means nothing more than suggestion, and it is always by -pure deduction that the general law is really established. As Proclus -puts it, *we must pass from sense to consideration*. - - [140] Bk. ii. chap. iv. - - [141] *Philosophical Transactions* (1866), vol. 146, p. 334. - -[Illustration] - -Given, for instance, the series of figures in the accompanying diagram, -measurement will show that the curved lines approximate to semicircles, -and the rectilinear figures to right-angled triangles. These figures -may seem to suggest to the mind the general law that angles inscribed -in semicircles are right angles; but no number of instances, and no -possible accuracy of measurement would really establish the truth of -that general law. Availing ourselves of the suggestion furnished by -the figures, we can only investigate deductively the consequences -which flow from the definition of a circle, until we discover among -them the property of containing right angles. Persons have thought -that they had discovered a method of trisecting angles by plane -geometrical construction, because a certain complex arrangement of -lines and circles had appeared to trisect an angle in every case tried -by them, and they inferred, by a supposed act of induction, that it -would succeed in all other cases. De Morgan has recorded a proposed -mode of trisecting the angle which could not be discriminated by the -senses from a true general solution, except when it was applied to -very obtuse angles.[142] In all such cases, it has always turned out -either that the angle was not trisected at all, or that only certain -particular angles could be thus trisected. The trisectors were misled -by some apparent or special coincidence, and only deductive proof could -establish the truth and generality of the result. In this particular -case, deductive proof shows that the problem attempted is impossible, -and that angles generally cannot be trisected by common geometrical -methods. - - [142] *Budget of Paradoxes*, p. 257. - - -*Geometrical Reasoning.* - -This view of the matter is strongly supported by the further -consideration of geometrical reasoning. No skill and care could ever -enable us to verify absolutely any one geometrical proposition. -Rousseau, in his *Emile*, tells us that we should teach a child -geometry by causing him to measure and compare figures by -superposition. While a child was yet incapable of general reasoning, -this would doubtless be an instructive exercise; but it never could -teach geometry, nor prove the truth of any one proposition. All our -figures are rude approximations, and they may happen to seem unequal -when they should be equal, and equal when they should be unequal. -Moreover figures may from chance be equal in case after case, and yet -there may be no general reason why they should be so. The results of -deductive geometrical reasoning are absolutely certain, and are either -exactly true or capable of being carried to any required degree of -approximation. In a perfect triangle, the angles must be equal to one -half-revolution precisely; even an infinitesimal divergence would be -impossible; and I believe with equal confidence, that however many are -the angles of a figure, provided there are no re-entrant angles, the -sum of the angles will be precisely and absolutely equal to twice as -many right-angles as the figure has sides, less by four right-angles. -In such cases, the deductive proof is absolute and complete; empirical -verification can at the most guard against accidental oversights. - -There is a second class of geometrical truths which can only be -proved by approximation; but, as the mind sees no reason why that -approximation should not always go on, we arrive at complete -conviction. We thus learn that the surface of a sphere is equal exactly -to two-thirds of the whole surface of the circumscribing cylinder, or -to four times the area of the generating circle. The area of a parabola -is exactly two-thirds of that of the circumscribing parallelogram. -The area of the cycloid is exactly three times that of the generating -circle. These are truths that we could never ascertain, nor even verify -by observation; for any finite amount of difference, less than what the -senses can discern, would falsify them. - -There are geometrical relations again which we cannot assign -exactly, but can carry to any desirable degree of approximation. The -ratio of the circumference to the diameter of a circle is that of -3·14159265358979323846.... to 1, and the approximation may be carried -to any extent by the expenditure of sufficient labour. Mr. W. Shanks -has given the value of this natural constant, known as π, to the extent -of 707 places of decimals.[143] Some years since, I amused myself -by trying how near I could get to this ratio, by the careful use of -compasses, and I did not come nearer than 1 part in 540. We might -imagine measurements so accurately executed as to give us eight or ten -places correctly. But the power of the hands and senses must soon -stop, whereas the mental powers of deductive reasoning can proceed -to an unlimited degree of approximation. Geometrical truths, then, -are incapable of verification; and, if so, they cannot even be learnt -by observation. How can I have learnt by observation a proposition -of which I cannot even prove the truth by observation, when I am in -possession of it? All that observation or empirical trial can do is -to suggest propositions, of which the truth may afterwards be proved -deductively. - - [143] *Proceedings of the Royal Society* (1872–3), vol. xxi. p. 319. - -If Viviani’s story is to be believed, Galileo endeavoured to satisfy -himself about the area of the cycloid by cutting out several large -cycloids in pasteboard, and then comparing the areas of the curve and -the generating circle by weighing them. In every trial the curve seemed -to be rather less than three times the circle, so that Galileo, we are -told, began to suspect that the ratio was not precisely 3 to 1. It is -quite clear, however, that no process of weighing or measuring could -ever prove truths like these, and it remained for Torricelli to show -what his master Galileo had only guessed at.[144] - - [144] *Life of Galileo*, Society for the Diffusion of Useful - Knowledge, p. 102. - -Much has been said about the peculiar certainty of mathematical -reasoning, but it is only certainty of deductive reasoning, and equal -certainty attaches to all correct logical deduction. If a triangle be -right-angled, the square on the hypothenuse will undoubtedly equal the -sum of the two squares on the other sides; but I can never be sure that -a triangle is right-angled: so I can be certain that nitric acid will -not dissolve gold, provided I know that the substances employed really -correspond to those on which I tried the experiment previously. Here is -like certainty of inference, and like doubt as to the facts. - - -*Discrimination of Certainty and Probability.* - -We can never recur too often to the truth that our knowledge of the -laws and future events of the external world is only probable. The mind -itself is quite capable of possessing certain knowledge, and it is well -to discriminate carefully between what we can and cannot know with -certainty. In the first place, whatever feeling is actually present to -the mind is certainly known to that mind. If I see blue sky, I may be -quite sure that I do experience the sensation of blueness. Whatever -I do feel, I do feel beyond all doubt. We are indeed very likely to -confuse what we really feel with what we are inclined to associate with -it, or infer inductively from it; but the whole of our consciousness, -as far as it is the result of pure intuition and free from inference, -is certain knowledge beyond all doubt. - -In the second place, we may have certainty of inference; the -fundamental laws of thought, and the rule of substitution (p. 9), -are certainly true; and if my senses could inform me that A was -indistinguishable in colour from B, and B from C, then I should be -equally certain that A was indistinguishable from C. In short, whatever -truth there is in the premises, I can certainly embody in their correct -logical result. But the certainty generally assumes a hypothetical -character. I never can be quite sure that two colours are exactly -alike, that two magnitudes are exactly equal, or that two bodies -whatsoever are identical even in their apparent qualities. Almost all -our judgments involve quantitative relations, and, as will be shown in -succeeding chapters, we can never attain exactness and certainty where -continuous quantity enters. Judgments concerning discontinuous quantity -or numbers, however, allow of certainty; I may establish beyond doubt, -for instance, that the difference of the squares of 17 and 13 is the -product of 17 + 13 and 17 - 13, and is therefore 30 × 4, or 120. - -Inferences which we draw concerning natural objects are never certain -except in a hypothetical point of view. It might seem to be certain -that iron is magnetic, or that gold is incapable of solution in -nitric acid; but, if we carefully investigate the meanings of these -statements, they will be found to involve no certainty but that of -consciousness and that of hypothetical inference. For what do I mean -by iron or gold? If I choose a remarkable piece of yellow substance, -call it gold, and then immerse it in a liquid which I call nitric acid, -and find that there is no change called solution, then consciousness -has certainly informed me that, with my meaning of the terms, “Gold is -insoluble in nitric acid.” I may further be certain of something else; -for if this gold and nitric acid remain what they were, I may be sure -there will be no solution on again trying the experiment. If I take -other portions of gold and nitric acid, and am sure that they really -are identical in properties with the former portions, I can be certain -that there will be no solution. But at this point my knowledge becomes -purely hypothetical; for how can I be sure without trial that the gold -and acid are really identical in nature with what I formerly called -gold and nitric acid. How do I know gold when I see it? If I judge by -the apparent qualities--colour, ductility, specific gravity, &c., I -may be misled, because there may always exist a substance which to the -colour, ductility, specific gravity, and other specified qualities, -joins others which we do not expect. Similarly, if iron is magnetic, -as shown by an experiment with objects answering to those names, then -all iron is magnetic, meaning all pieces of matter identical with my -assumed piece. But in trying to identify iron, I am always open to -mistake. Nor is this liability to mistake a matter of speculation -only.[145] - - [145] Professor Bowen has excellently stated this view. *Treatise on - Logic.* Cambridge, U.S.A., 1866, p. 354. - -The history of chemistry shows that the most confident inferences may -have been falsified by the confusion of one substance with another. -Thus strontia was never discriminated from baryta until Klaproth -and Haüy detected differences between some of their properties. -Accordingly chemists must often have inferred concerning strontia what -was only true of baryta, and *vice versâ*. There is now no doubt that -the recently discovered substances, cæsium and rubidium, were long -mistaken for potassium.[146] Other elements have often been confused -together--for instance, tantalum and niobium; sulphur and selenium; -cerium, lanthanum, and didymium; yttrium and erbium. - - [146] Roscoe’s *Spectrum Analysis*, 1st edit., p. 98. - -Even the best known laws of physical science do not exclude false -inference. No law of nature has been better established than that of -universal gravitation, and we believe with the utmost confidence that -any body capable of affecting the senses will attract other bodies, -and fall to the earth if not prevented. Euler remarks that, although -he had never made trial of the stones which compose the church of -Magdeburg, yet he had not the least doubt that all of them were heavy, -and would fall if unsupported. But he adds, that it would be extremely -difficult to give any satisfactory explanation of this confident -belief.[147] The fact is, that the belief ought not to amount to -certainty until the experiment has been tried, and in the meantime a -slight amount of uncertainty enters, because we cannot be sure that -the stones of the Magdeburg Church resemble other stones in all their -properties. - - [147] Euler’s *Letters to a German Princess*, translated by Hunter. - 2nd ed., vol. ii. pp. 17, 18. - -In like manner, not one of the inductive truths which men have -established, or think they have established, is really safe from -exception or reversal. Lavoisier, when laying the foundations of -chemistry, met with so many instances tending to show the existence -of oxygen in all acids, that he adopted a general conclusion to that -effect, and devised the name oxygen accordingly. He entertained no -appreciable doubt that the acid existing in sea salt also contained -oxygen;[148] yet subsequent experience falsified his expectations. This -instance refers to a science in its infancy, speaking relatively to -the possible achievements of men. But all sciences are and ever will -remain in their infancy, relatively to the extent and complexity of -the universe which they undertake to investigate. Euler expresses no -more than the truth when he says that it would be impossible to fix -on any one thing really existing, of which we could have so perfect -a knowledge as to put us beyond the reach of mistake.[149] We may be -quite certain that a comet will go on moving in a similar path *if* -all circumstances remain the same as before; but if we leave out this -extensive qualification, our predictions will always be subject to the -chance of falsification by some unexpected event, such as the division -of Biela’s comet or the interference of an unknown gravitating body. - - [148] Lavoisier’s *Chemistry*, translated by Kerr. 3rd ed., pp. 114, - 121, 123. - - [149] Euler’s *Letters*, vol. ii. p. 21. - -Inductive inference might attain to certainty if our knowledge of the -agents existing throughout the universe were complete, and if we were -at the same time certain that the same Power which created the universe -would allow it to proceed without arbitrary change. There is always -a possibility of causes being in existence without our knowledge, -and these may at any moment produce an unexpected effect. Even when -by the theory of probabilities we succeed in forming some notion of -the comparative confidence with which we should receive inductive -results, it yet appears to me that we must make an assumption. Events -come out like balls from the vast ballot-box of nature, and close -observation will enable us to form some notion, as we shall see in the -next chapter, of the contents of that ballot-box. But we must still -assume that, between the time of an observation and that to which our -inferences relate, no change in the ballot-box has been made. - - - - -CHAPTER XII. - -THE INDUCTIVE OR INVERSE APPLICATION OF THE THEORY OF PROBABILITY. - - -We have hitherto considered the theory of probability only in its -simple deductive employment, in which it enables us to determine -from given conditions the probable character of events happening -under those conditions. But as deductive reasoning when inversely -applied constitutes the process of induction, so the calculation of -probabilities may be inversely applied; from the known character -of certain events we may argue backwards to the probability of a -certain law or condition governing those events. Having satisfactorily -accomplished this work, we may indeed calculate forwards to the -probable character of future events happening under the same -conditions; but this part of the process is a direct use of deductive -reasoning (p. 226). - -Now it is highly instructive to find that whether the theory of -probability be deductively or inductively applied, the calculation is -always performed according to the principles and rules of deduction. -The probability that an event has a particular condition entirely -depends upon the probability that if the condition existed the event -would follow. If we take up a pack of common playing cards, and observe -that they are arranged in perfect numerical order, we conclude beyond -all reasonable doubt that they have been thus intentionally arranged -by some person acquainted with the usual order of sequence. This -conclusion is quite irresistible, and rightly so; for there are but -two suppositions which we can make as to the reason of the cards being -in that particular order:-- - -1. They may have been intentionally arranged by some one who would -probably prefer the numerical order. - -2. They may have fallen into that order by chance, that is, by some -series of conditions which, being unknown to us, cannot be known to -lead by preference to the particular order in question. - -The latter supposition is by no means absurd, for any one order is as -likely as any other when there is no preponderating tendency. But we -can readily calculate by the doctrine of permutations the probability -that fifty-two objects would fall by chance into any one particular -order. Fifty-two objects can be arranged in 52 × 51 × ... × 3 × 2 × 1 -or about 8066 × (10)^{64} possible orders, the number obtained -requiring 68 places of figures for its full expression. Hence it is -excessively unlikely that anyone should ever meet with a pack of cards -arranged in perfect order by accident. If we do meet with a pack so -arranged, we inevitably adopt the other supposition, that some person, -having reasons for preferring that special order, has thus put them -together. - -We know that of the immense number of possible orders the numerical -order is the most remarkable; it is useful as proving the perfect -constitution of the pack, and it is the intentional result of certain -games. At any rate, the probability that intention should produce that -order is incomparably greater than the probability that chance should -produce it; and as a certain pack exists in that order, we rightly -prefer the supposition which most probably leads to the observed result. - -By a similar mode of reasoning we every day arrive, and validly arrive, -at conclusions approximating to certainty. Whenever we observe a -perfect resemblance between two objects, as, for instance, two printed -pages, two engravings, two coins, two foot-prints, we are warranted in -asserting that they proceed from the same type, the same plate, the -same pair of dies, or the same boot. And why? Because it is almost -impossible that with different types, plates, dies, or boots some -apparent distinction of form should not be produced. It is impossible -for the hand of the most skilful artist to make two objects alike, so -that mechanical repetition is the only probable explanation of exact -similarity. - -We can often establish with extreme probability that one document -is copied from another. Suppose that each document contains 10,000 -words, and that the same word is incorrectly spelt in each. There is -then a probability of less than 1 in 10,000 that the same mistake -should be made in each. If we meet with a second error occurring in -each document, the probability is less than 1 in 10,000 × 9999, that -two such coincidences should occur by chance, and the numbers grow -with extreme rapidity for more numerous coincidences. We cannot make -any precise calculations without taking into account the character of -the errors committed, concerning the conditions of which we have no -accurate means of estimating probabilities. Nevertheless, abundant -evidence may thus be obtained as to the derivation of documents from -each other. In the examination of many sets of logarithmic tables, six -remarkable errors were found to be present in all but two, and it was -proved that tables printed at Paris, Berlin, Florence, Avignon, and -even in China, besides thirteen sets printed in England between the -years 1633 and 1822, were derived directly or indirectly from some -common source.[150] With a certain amount of labour, it is possible -to establish beyond reasonable doubt the relationship or genealogy of -any number of copies of one document, proceeding possibly from parent -copies now lost. The relations between the manuscripts of the New -Testament have been elaborately investigated in this manner, and the -same work has been performed for many classical writings, especially by -German scholars. - - [150] Lardner, *Edinburgh Review*, July 1834, p. 277. - - -*Principle of the Inverse Method.* - -The inverse application of the rules of probability entirely depends -upon a proposition which may be thus stated, nearly in the words of -Laplace.[151] *If an event can be produced by any one of a certain -number of different causes, all equally probable à priori, the -probabilities of the existence of these causes as inferred from the -event, are proportional to the probabilities of the event as derived -from these causes.* In other words, the most probable cause of an -event which has happened is that which would most probably lead to the -event supposing the cause to exist; but all other possible causes are -also to be taken into account with probabilities proportional to the -probability that the event would happen if the cause existed. Suppose, -to fix our ideas clearly, that E is the event, and C_{1} C_{2} C_{3} -are the three only conceivable causes. If C_{1} exist, the probability -is *p*_{1} that E would follow; if C_{2} or C_{3} exist, the like -probabilities are respectively *p*_{2} and *p*_{3}. Then as *p*_{1} -is to *p*_{2}, so is the probability of C_{1} being the actual cause -to the probability of C_{2} being it; and, similarly, as *p*_{2} is -to *p*_{3}, so is the probability of C_{2} being the actual cause to -the probability of C_{3} being it. By a simple mathematical process we -arrive at the conclusion that the actual probability of C_{1} being the -cause is - - *p*_{1}/(*p*_{1} + *p*_{2} + *p*_{3}); - - [151] *Mémoires par divers Savans*, tom. vi.; quoted by Todhunter in - his *History of the Theory of Probability*, p. 458. - -and the similar probabilities of the existence of C_{2} and C_{3} are, - - *p*_{2}/(*p*_{1} + *p*_{2} + *p*_{3}) and - *p*_{3}/(*p*_{1} + *p*_{2} + *p*_{3}). - -The sum of these three fractions amounts to unity, which correctly -expresses the certainty that one cause or other must be in operation. - -We may thus state the result in general language. *If it is certain -that one or other of the supposed causes exists, the probability that -any one does exist is the probability that if it exists the event -happens, divided by the sum of all the similar probabilities.* There -may seem to be an intricacy in this subject which may prove distasteful -to some readers; but this intricacy is essential to the subject in -hand. No one can possibly understand the principles of inductive -reasoning, unless he will take the trouble to master the meaning of -this rule, by which we recede from an event to the probability of each -of its possible causes. - -This rule or principle of the indirect method is that which common -sense leads us to adopt almost instinctively, before we have any -comprehension of the principle in its general form. It is easy to see, -too, that it is the rule which will, out of a great multitude of cases, -lead us most often to the truth, since the most probable cause of an -event really means that cause which in the greatest number of cases -produces the event. Donkin and Boole have given demonstrations of this -principle, but the one most easy to comprehend is that of Poisson. -He imagines each possible cause of an event to be represented by a -distinct ballot-box, containing black and white balls, in such a ratio -that the probability of a white ball being drawn is equal to that of -the event happening. He further supposes that each box, as is possible, -contains the same total number of balls, black and white; then, mixing -all the contents of the boxes together, he shows that if a white ball -be drawn from the aggregate ballot-box thus formed, the probability -that it proceeded from any particular ballot-box is represented by the -number of white balls in that particular box, divided by the total -number of white balls in all the boxes. This result corresponds to that -given by the principle in question.[152] - - [152] Poisson, *Recherches sur la Probabilité des Jugements*, Paris, - 1837, pp. 82, 83. - -Thus, if there be three boxes, each containing ten balls in all, and -respectively containing seven, four, and three white balls, then on -mixing all the balls together we have fourteen white ones; and if -we draw a white ball, that is if the event happens, the probability -that it came out of the first box is 7/14; which is exactly equal to -(7/10)/(7/10 + 4/10 + 3/10), the fraction given by the rule of the -Inverse Method. - - -*Simple Applications of the Inverse Method.* - -In many cases of scientific induction we may apply the principle of the -inverse method in a simple manner. If only two, or at the most a few -hypotheses, may be made as to the origin of certain phenomena, we may -sometimes easily calculate the respective probabilities. It was thus -that Bunsen and Kirchhoff established, with a probability little short -of certainty, that iron exists in the sun. On comparing the spectra -of sunlight and of the light proceeding from the incandescent vapour -of iron, it became apparent that at least sixty bright lines in the -spectrum of iron coincided with dark lines in the sun’s spectrum. Such -coincidences could never be observed with certainty, because, even if -the lines only closely approached, the instrumental imperfections of -the spectroscope would make them apparently coincident, and if one line -came within half a millimetre of another, on the map of the spectra, -they could not be pronounced distinct. Now the average distance of the -solar lines on Kirchhoff’s map is 2 mm., and if we throw down a line, -as it were, by pure chance on such a map, the probability is about -one-half that the new line will fall within 1/2 mm. on one side or the -other of some one of the solar lines. To put it in another way, we may -suppose that each solar line, either on account of its real breadth, or -the defects of the instrument, possesses a breadth of 1/2 mm., and that -each line in the iron spectrum has a like breadth. The probability then -is just one-half that the centre of each iron line will come by chance -within 1 mm. of the centre of a solar line, so as to appear to coincide -with it. The probability of casual coincidence of each iron line with -a solar line is in like manner 1/2. Coincidence in the case of each of -the sixty iron lines is a very unlikely event if it arises casually, -for it would have a probability of only (1/2)^{60} or less than 1 -in a trillion. The odds, in short, are more than a million million -millions to unity against such casual coincidence.[153] But on the -other hypothesis, that iron exists in the sun, it is highly probable -that such coincidences would be observed; it is immensely more probable -that sixty coincidences would be observed if iron existed in the sun, -than that they should arise from chance. Hence by our principle it is -immensely probable that iron does exist in the sun. - - [153] Kirchhoff’s *Researches on the Solar Spectrum*. First part, - translated by Roscoe, pp. 18, 19. - -All the other interesting results, given by the comparison of spectra, -rest upon the same principle of probability. The almost complete -coincidence between the spectra of solar, lunar, and planetary light -renders it practically certain that the light is all of solar origin, -and is reflected from the surfaces of the moon and planets, suffering -only slight alteration from the atmospheres of some of the planets. -A fresh confirmation of the truth of the Copernican theory is thus -furnished. - -Herschel proved in this way the connection between the direction of the -oblique faces of quartz crystals, and the direction in which the same -crystals rotate the plane of polarisation of light. For if it is found -in a second crystal that the relation is the same as in the first, the -probability of this happening by chance is 1/2; the probability that -in another crystal also the direction will be the same is 1/4, and so -on. The probability that in *n* + 1 crystals there would be casual -agreement of direction is the nth power of 1/2. Thus, if in examining -fourteen crystals the same relation of the two phenomena is discovered -in each, the odds that it proceeds from uniform conditions are more -than 8000 to 1.[154] Since the first observations on this subject were -made in 1820, no exceptions have been observed, so that the probability -of invariable connection is incalculably great. - - [154] *Edinburgh Review*, No. 185, vol. xcii. July 1850, p. 32; - Herschel’s *Essays*, p. 421; *Transactions of the Cambridge - Philosophical Society*, vol. i. p. 43. - -It is exceedingly probable that the ancient Egyptians had exactly -recorded the eclipses occurring during long periods of time, for -Diogenes Laertius mentions that 373 solar and 832 lunar eclipses had -been observed, and the ratio between these numbers exactly expresses -that which would hold true of the eclipses of any long period, of say -1200 or 1300 years, as estimated on astronomical grounds. It is evident -that an agreement between small numbers, or customary numbers, such -as seven, one hundred, a myriad, &c., is much more likely to happen -from chance, and therefore gives much less presumption of dependence. -If two ancient writers spoke of the sacrifice of oxen, they would in -all probability describe it as a hecatomb, and there would be nothing -remarkable in the coincidence. But it is impossible to point out any -special reason why an old writer should select such numbers as 373 and -832, unless they had been the results of observation. - -On similar grounds, we must inevitably believe in the human origin -of the flint flakes so copiously discovered of late years. For -though the accidental stroke of one stone against another may often -produce flakes, such as are occasionally found on the sea-shore, yet -when several flakes are found in close company, and each one bears -evidence, not of a single blow only, but of several successive blows, -all conducing to form a symmetrical knife-like form, the probability -of a natural and accidental origin becomes incredibly small, and the -contrary supposition, that they are the work of intelligent beings, -approximately certain.[155] - - [155] Evans’ *Ancient Stone Implements of Great Britain*. London, - 1872 (Longmans). - - -*The Theory of Probability in Astronomy.* - -The science of astronomy, occupied with the simple relations of -distance, magnitude, and motion of the heavenly bodies, admits more -easily than almost any other science of interesting conclusions founded -on the theory of probability. More than a century ago, in 1767, Michell -showed the extreme probability of bonds connecting together systems -of stars. He was struck by the unexpected number of fixed stars -which have companions close to them. Such a conjunction might happen -casually by one star, although possibly at a great distance from the -other, happening to lie on a straight line passing near the earth. -But the probabilities are so greatly against such an optical union -happening often in the expanse of the heavens, that Michell asserted -the existence of some connection between most of the double stars. -It has since been estimated by Struve, that the odds are 9570 to 1 -against any two stars of not less than the seventh magnitude falling -within the apparent distance of four seconds of each other by chance, -and yet ninety-one such cases were known when the estimation was made, -and many more cases have since been discovered. There were also four -known triple stars, and yet the odds against the appearance of any one -such conjunction are 173,524 to 1.[156] The conclusions of Michell have -been entirely verified by the discovery that many double stars are -connected by gravitation. - - [156] Herschel, *Outlines of Astronomy*, 1849, p. 565; but Todhunter, - in his *History of the Theory of Probability*, p. 335, states that - the calculations do not agree with those published by Struve. - -Michell also investigated the probability that the six brightest stars -in the Pleiades should have come by accidents into such striking -proximity. Estimating the number of stars of equal or greater -brightness at 1500, be found the odds to be nearly 500,000 to 1 against -casual conjunction. Extending the same kind of argument to other -clusters, such as that of Præsepe, the nebula in the hilt of Perseus’ -sword, he says:[157] “We may with the highest probability conclude, -the odds against the contrary opinion being many million millions -to one, that the stars are really collected together in clusters -in some places, where they form a kind of system, while in others -there are either few or none of them, to whatever cause this may be -owing, whether to their mutual gravitation, or to some other law or -appointment of the Creator.” - - [157] *Philosophical Transactions*, 1767, vol. lvii. p. 431. - -The calculations of Michell have been called in question by the late -James D. Forbes,[158] and Mr. Todhunter vaguely countenances his -objections,[159] otherwise I should not have thought them of much -weight. Certainly Laplace accepts Michell’s views,[160] and if Michell -be in error it is in the methods of calculation, not in the general -validity of his reasoning and conclusions. - - [158] *Philosophical Magazine*, 3rd Series, vol. xxxvii. p. 401, - December 1850; also August 1849. - - [159] *History*, &c., p. 334. - - [160] *Essai Philosophique*, p. 57. - -Similar calculations might no doubt be applied to the peculiar drifting -motions which have been detected by Mr. R A. Proctor in some of the -constellations.[161] The odds are very greatly against any numerous -group of stars moving together in any one direction by chance. On like -grounds, there can be no doubt that the sun has a considerable proper -motion because on the average the fixed stars show a tendency to move -apparently from one point of the heavens towards that diametrically -opposite. The sun’s motion in the contrary direction would explain -this tendency, otherwise we must believe that thousands of stars -accidentally agree in their direction of motion, or are urged by some -common force from which the sun is exempt. It may be said that the -rotation of the earth is proved in like manner, because it is immensely -more probable that one body would revolve than that the sun, moon, -planets, comets, and the whole of the stars of the heavens should be -whirled round the earth daily, with a uniform motion superadded to -their own peculiar motions. This appears to be mainly the reason which -led Gilbert, one of the earliest English Copernicans, and in every -way an admirable physicist, to admit the rotation of the earth, while -Francis Bacon denied it. - - [161] *Proceedings of the Royal Society*; 20 January, 1870; - *Philosophical Magazine*, 4th Series, vol. xxxix. p. 381. - -In contemplating the planetary system, we are struck with the -similarity in direction of nearly all its movements. Newton remarked -upon the regularity and uniformity of these motions, and contrasted -them with the eccentricity and irregularity of the cometary -orbits.[162] Could we, in fact, look down upon the system from the -northern side, we should see all the planets moving round from west -to east, the satellites moving round their primaries, and the sun, -planets, and satellites rotating in the same direction, with some -exceptions on the verge of the system. In the time of Laplace eleven -planets were known, and the directions of rotation were known for the -sun, six planets, the satellites of Jupiter, Saturn’s ring, and one of -his satellites. Thus there were altogether 43 motions all concurring, -namely:-- - - Orbital motions of eleven planets 11 - Orbital motions of eighteen satellites 18 - Axial rotations 14 - -- - 43 - - [162] *Principia*, bk. ii. General scholium. - -The probability that 43 motions independent of each other would -coincide by chance is the 42nd power of 1/2, so that the odds are -about 4,400,000,000,000 to 1 in favour of some common cause for the -uniformity of direction. This probability, as Laplace observes,[163] -is higher than that of many historical events which we undoubtingly -believe. In the present day, the probability is much increased by the -discovery of additional planets, and the rotation of other satellites, -and it is only slightly weakened by the fact that some of the outlying -satellites are exceptional in direction, there being considerable -evidence of an accidental disturbance in the more distant parts of the -system. - - [163] *Essai Philosophique*, p. 55. Laplace appears to count the - rings of Saturn as giving two independent movements. - -Hardly less remarkable than the uniform direction of motion is the -near approximation of the orbits of the planets to a common plane. -Daniel Bernoulli roughly estimated the probability of such an agreement -arising from accident as 1 ÷ (12)^{6} the greatest inclination of any -orbit to the sun’s equator being 1-12th part of a quadrant. Laplace -devoted to this subject some of his most ingenious investigations. He -found the probability that the sum of the inclinations of the planetary -orbits would not exceed by accident the actual amount (·914187 -of a right angle for the ten planets known in 1801) to be (1/10)! -(·914187)^{10} or about ·00000011235. This probability may be combined -with that derived from the direction of motion, and it then becomes -immensely probable that the constitution of the planetary system arose -out of uniform conditions, or, as we say, from some common cause.[164] - - [164] Lubbock, *Essay on Probability*, p. 14. De Morgan, *Encyc. - Metrop.* art. *Probability*, p. 412. Todhunter’s *History of the - Theory of Probability*, p. 543. Concerning the objections raised to - these conclusions by Boole, see the *Philosophical Magazine*, 4th - Series, vol. ii. p. 98. Boole’s *Laws of Thought*, pp. 364–375. - -If the same kind of calculation be applied to the orbits of comets, -the result is very different.[165] Of the orbits which have been -determined 48·9 per cent. only are direct or in the same direction as -the planetary motions.[166] Hence it becomes apparent that comets do -not properly belong to the solar system, and it is probable that they -are stray portions of nebulous matter which have accidentally become -attached to the system by the attractive powers of the sun or Jupiter. - - [165] Laplace, *Essai Philosophique*, pp. 55, 56. - - [166] Chambers’ *Astronomy*, 2nd ed. pp. 346–49. - - -*The General Inverse Problem.* - -In the instances described in the preceding sections, we have been -occupied in receding from the occurrence of certain similar events to -the probability that there must have been a condition or cause for -such events. We have found that the theory of probability, although -never yielding a certain result, often enables us to establish an -hypothesis beyond the reach of reasonable doubt. There is, however, -another method of applying the theory, which possesses for us even -greater interest, because it illustrates, in the most complete manner, -the theory of inference adopted in this work, which theory indeed it -suggested. The problem to be solved is as follows:-- - -*An event having happened a certain number of times, and failed a -certain number of times, required the probability that it will happen -any given number of times in the future under the same circumstances.* - -All the *larger* planets hitherto discovered move in one direction -round the sun; what is the probability that, if a new planet exterior -to Neptune be discovered, it will move in the same direction? All -known permanent gases, except chlorine, are colourless; what is the -probability that, if some new permanent gas should be discovered, it -will be colourless? In the general solution of this problem, we wish to -infer the future happening of any event from the number of times that -it has already been observed to happen. Now, it is very instructive to -find that there is no known process by which we can pass directly from -the data to the conclusion. It is always requisite to recede from the -data to the probability of some hypothesis, and to make that hypothesis -the ground of our inference concerning future events. Mathematicians, -in fact, make every hypothesis which is applicable to the question in -hand; they then calculate, by the inverse method, the probability of -every such hypothesis according to the data, and the probability that -if each hypothesis be true, the required future event will happen. The -total probability that the event will happen is the sum of the separate -probabilities contributed by each distinct hypothesis. - -To illustrate more precisely the method of solving the problem, it -is desirable to adopt some concrete mode of representation, and the -ballot-box, so often employed by mathematicians, will best serve -our purpose. Let the happening of any event be represented by the -drawing of a white ball from a ballot-box, while the failure of an -event is represented by the drawing of a black ball. Now, in the -inductive problem we are supposed to be ignorant of the contents of -the ballot-box, and are required to ground all our inferences on our -experience of those contents as shown in successive drawings. Rude -common sense would guide us nearly to a true conclusion. Thus, if we -had drawn twenty balls one after another, replacing the ball after each -drawing, and the ball had in each case proved to be white, we should -believe that there was a considerable preponderance of white balls in -the urn, and a probability in favour of drawing a white ball on the -next occasion. Though we had drawn white balls for thousands of times -without fail, it would still be possible that some black balls lurked -in the urn and would at last appear, so that our inferences could never -be certain. On the other hand, if black balls came at intervals, we -should expect that after a certain number of trials the black balls -would appear again from time to time with somewhat the same frequency. - -The mathematical solution of the question consists in little more -than a close analysis of the mode in which our common sense proceeds. -If twenty white balls have been drawn and no black ball, my common -sense tells me that any hypothesis which makes the black balls in -the urn considerable compared with the white ones is improbable; a -preponderance of white balls is a more probable hypothesis, and as a -deduction from this more probable hypothesis, I expect a recurrence -of white balls. The mathematician merely reduces this process of -thought to exact numbers. Taking, for instance, the hypothesis that -there are 99 white and one black ball in the urn, he can calculate the -probability that 20 white balls would be drawn in succession in those -circumstances; he thus forms a definite estimate of the probability -of this hypothesis, and knowing at the same time the probability of a -white ball reappearing if such be the contents of the urn, he combines -these probabilities, and obtains an exact estimate that a white ball -will recur in consequence of this hypothesis. But as this hypothesis -is only one out of many possible ones, since the ratio of white and -black balls may be 98 to 2, or 97 to 3, or 96 to 4, and so on, he has -to repeat the estimate for every such possible hypothesis. To make the -method of solving the problem perfectly evident, I will describe in the -next section a very simple case of the problem, originally devised for -the purpose by Condorcet, which was also adopted by Lacroix,[167] and -has passed into the works of De Morgan, Lubbock, and others. - - [167] *Traité élémentaire du Calcul des Probabilités*, 3rd ed. - (1833), p. 148. - - -*Simple Illustration of the Inverse Problem.* - -Suppose it to be known that a ballot-box contains only four black or -white balls, the ratio of black and white balls being unknown. Four -drawings having been made with replacement, and a white ball having -appeared on each occasion but one, it is required to determine the -probability that a white ball will appear next time. Now the hypotheses -which can be made as to the contents of the urn are very limited in -number, and are at most the following five:-- - - 4 white and 0 black balls - 3 " " 1 " " - 2 " " 2 " " - 1 " " 3 " " - 0 " " 4 " " - -The actual occurrence of black and white balls in the drawings puts the -first and last hypothesis out of the question, so that we have only -three left to consider. - -If the box contains three white and one black, the probability of -drawing a white each time is 3/4, and a black 1/4; so that the compound -event observed, namely, three white and one black, has the probability -3/4 × 3/4 × 3/4 × 1/4, by the rule already given (p. 204). But as it is -indifferent in what order the balls are drawn, and the black ball might -come first, second, third, or fourth, we must multiply by four, to -obtain the probability of three white and one black in any order, thus -getting 27/64. - -Taking the next hypothesis of two white and two black balls -in the urn, we obtain for the same probability the quantity -1/2 × 1/2 × 1/2 × 1/2 × 4, or 16/64, and from the third hypothesis of -one white and three black we deduce likewise 1/4 × 1/4 × 1/4 × 3/4 × 4, -or 3/64. According, then, as we adopt the first, second, or third -hypothesis, the probability that the result actually noticed would -follow is 27/64, 16/64, and 3/64. Now it is certain that one or -other of these hypotheses must be the true one, and their absolute -probabilities are proportional to the probabilities that the observed -events would follow from them (pp. 242, 243). All we have to do, then, -in order to obtain the absolute probability of each hypothesis, is to -alter these fractions in a uniform ratio, so that their sum shall be -unity, the expression of certainty. Now, since 27 + 16 + 3 = 46, this -will be effected by dividing each fraction by 46, and multiplying by -64. Thus the probabilities of the first, second, and third hypotheses -are respectively-- - - 27/46, 16/46, 3/46. - -The inductive part of the problem is completed, since we have found -that the urn most likely contains three white and one black ball, and -have assigned the exact probability of each possible supposition. But -we are now in a position to resume deductive reasoning, and infer the -probability that the next drawing will yield, say a white ball. For if -the box contains three white and one black ball, the probability of -drawing a white one is certainly 3/4; and as the probability of the box -being so constituted is 27/46, the compound probability that the box -will be so filled and will give a white ball at the next trial, is - - 27/46 × 3/4 or 81/184. - -Again, the probability is 16/46 that the box contains two white and two -black, and under those conditions the probability is 1/2 that a white -ball will appear; hence the probability that a white ball will appear -in consequence of that condition, is - - 16/46 × 1/2 or 32/184. - -From the third supposition we get in like manner the probability - - 3/46 × 1/4 or 3/184. - -Since one and not more than one hypothesis can be true, we may add -together these separate probabilities, and we find that - - 81/184 + 32/184 + 3/184 or 116/184 - -is the complete probability that a white ball will be next drawn under -the conditions and data supposed. - - -*General Solution of the Inverse Problem.* - -In the instance of the inverse method described in the last section, -the balls supposed to be in the ballot-box were few, for the purpose of -simplifying the calculation. In order that our solution may apply to -natural phenomena, we must render our hypotheses as little arbitrary -as possible. Having no *à priori* knowledge of the conditions of the -phenomena in question, there is no limit to the variety of hypotheses -which might be suggested. Mathematicians have therefore had recourse -to the most extensive suppositions which can be made, namely, that the -ballot-box contains an infinite number of balls; they have then varied -the proportion of white to black balls continuously, from the smallest -to the greatest possible proportion, and estimated the aggregate -probability which results from this comprehensive supposition. - -To explain their procedure, let us imagine that, instead of an infinite -number, the ballot-box contains a large finite number of balls, say -1000. Then the number of white balls might be 1 or 2 or 3 or 4, and so -on, up to 999. Supposing that three white and one black ball have been -drawn from the urn as before, there is a certain very small probability -that this would have occurred in the case of a box containing one white -and 999 black balls; there is also a small probability that from such -a box the next ball would be white. Compound these probabilities, and -we have the probability that the next ball really will be white, in -consequence of the existence of that proportion of balls. If there be -two white and 998 black balls in the box, the probability is greater -and will increase until the balls are supposed to be in the proportion -of those drawn. Now 999 different hypotheses are possible, and the -calculation is to be made for each of these, and their aggregate taken -as the final result. It is apparent that as the number of balls in -the box is increased, the absolute probability of any one hypothesis -concerning the exact proportion of balls is decreased, but the -aggregate results of all the hypotheses will assume the character of a -wider average. - -When we take the step of supposing the balls within the urn to be -infinite in number, the possible proportions of white and black balls -also become infinite, and the probability of any one proportion -actually existing is infinitely small. Hence the final result that -the next ball drawn will be white is really the sum of an infinite -number of infinitely small quantities. It might seem impossible to -calculate out a problem having an infinite number of hypotheses, -but the wonderful resources of the integral calculus enable this -to be done with far greater facility than if we supposed any large -finite number of balls, and then actually computed the results. I -will not attempt to describe the processes by which Laplace finally -accomplished the complete solution of the problem. They are to be found -described in several English works, especially De Morgan’s *Treatise -on Probabilities*, in the *Encyclopædia Metropolitana*, and Mr. -Todhunter’s *History of the Theory of Probability*. The abbreviating -power of mathematical analysis was never more strikingly shown. But -I may add that though the integral calculus is employed as a means -of summing infinitely numerous results, we in no way abandon the -principles of combinations already treated. We calculate the values of -infinitely numerous factorials, not, however, obtaining their actual -products, which would lead to an infinite number of figures, but -obtaining the final answer to the problem by devices which can only be -comprehended after study of the integral calculus. - -It must be allowed that the hypothesis adopted by Laplace is in some -degree arbitrary, so that there was some opening for the doubt which -Boole has cast upon it.[168] But it may be replied, (1) that the -supposition of an infinite number of balls treated in the manner of -Laplace is less arbitrary and more comprehensive than any other that -can be suggested. (2) The result does not differ much from that -which would be obtained on the hypothesis of any large finite number -of balls. (3) The supposition leads to a series of simple formulas -which can be applied with ease in many cases, and which bear all the -appearance of truth so far as it can be independently judged by a sound -and practiced understanding. - - [168] *Laws of Thought*, pp. 368–375. - - -*Rules of the Inverse Method.* - -By the solution of the problem, as described in the last section, we -obtain the following series of simple rules. - -1. *To find the probability that an event which has not hitherto been -observed to fail will happen once more, divide the number of times the -event has been observed increased by one, by the same number increased -by two.* - -If there have been *m* occasions on which a certain event might have -been observed to happen, and it has happened on all those occasions, -then the probability that it will happen on the next occasion of the -same kind (*m* + 1)/(*m* + 2). For instance, we may say that there are -nine places in the planetary system where planets might exist obeying -Bode’s law of distance, and in every place there is a planet obeying -the law more or less exactly, although no reason is known for the -coincidence. Hence the probability that the next planet beyond Neptune -will conform to the law is 10/11. - -2. *To find the, probability that an event which has not hitherto -failed will not fail for a certain number of new occasions, divide the -number of times the event has happened increased by one, by the same -number increased by one and the number of times it is to happen.* - -An event having happened *m* times without fail, the probability that -it will happen *n* more times is (*m* + 1)/(*m* + *n* + 1). Thus the -probability that three new planets would obey Bode’s law is 10/13; but -it must be allowed that this, as well as the previous result, would be -much weakened by the fact that Neptune can barely be said to obey the -law. - -*3. An event having happened and failed a certain number of times, to -find the probability that it will happen the next time, divide the -number of times the event has happened increased by one, by the whole -number of times the event has happened or failed increased by two.* - -If an event has happened *m* times and failed *n* times, -the probability that it will happen on the next occasion is -(*m* + 1)/(*m* + *n* + 2). Thus, if we assume that of the elements -discovered up to the year 1873, 50 are metallic and 14 non-metallic, -then the probability that the next element discovered will be metallic -is 51/66. Again, since of 37 metals which have been sufficiently -examined only four, namely, sodium, potassium, lanthanum, and -lithium, are of less density than water, the probability that the -next metal examined or discovered will be less dense than water is -(4 + 1)/(37 + 2) or 5/39. - -We may state the results of the method in a more general manner -thus,[169]--If under given circumstances certain events A, B, C, &c., -have happened respectively *m*, *n*, *p*, &c., times, and one or other -of these events must happen, then the probabilities of these events are -proportional to *m* + 1, *n* + 1, *p* + 1, &c., so that the probability -of A will be (*m* + 1)/(*m* + 1 + *n* + 1 + *p* + 1 + &c.) But if new -events may happen in addition to those which have been observed, we -must assign unity for the probability of such new event. The odds then -become 1 for a new event, *m* + 1 for A, *n* + 1 for B, and so on, and -the absolute probability of A is (*m* + 1)/(1 + *m* + 1 + *n* + 1 + &c.) - - [169] De Morgan’s *Essay on Probabilities*, Cabinet Cyclopædia, p. 67. - -It is interesting to trace out the variations of probability according -to these rules. The first time a casual event happens it is 2 to 1 -that it will happen again; if it does happen it is 3 to 1 that it -will happen a third time; and on successive occasions of the like -kind the odds become 4, 5, 6, &c., to 1. The odds of course will be -discriminated from the probabilities which are successively 2/3, 3/4, -4/5, &c. Thus on the first occasion on which a person sees a shark, -and notices that it is accompanied by a little pilot fish, the odds -are 2 to 1, or the probability 2/3, that the next shark will be so -accompanied. - -When an event has happened a very great number of times, its -happening once again approaches nearly to certainty. If we suppose -the sun to have risen one thousand million times, the probability -that it will rise again, on the ground of this knowledge merely, is -(1,000,000,000 + 1)/(1,000,000,000 + 1 + 1). But then the probability -that it will continue to rise for as long a period in the future is -only (1,000,000,000 + 1)/(2,000,000,000 + 1), or almost exactly 1/2. -The probability that it will continue so rising a thousand times -as long is only about 1/1001. The lesson which we may draw from -these figures is quite that which we should adopt on other grounds, -namely, that experience never affords certain knowledge, and that -it is exceedingly improbable that events will always happen as we -observe them. Inferences pushed far beyond their data soon lose -any considerable probability. De Morgan has said,[170] “No finite -experience whatsoever can justify us in saying that the future shall -coincide with the past in all time to come, or that there is any -probability for such a conclusion.” On the other hand, we gain the -assurance that experience sufficiently extended and prolonged will -give us the knowledge of future events with an unlimited degree of -probability, provided indeed that those events are not subject to -arbitrary interference. - - [170] *Essay on Probabilities*, p. 128. - -It must be clearly understood that these probabilities are only such -as arise from the mere happening of the events, irrespective of any -knowledge derived from other sources concerning those events or the -general laws of nature. All our knowledge of nature is indeed founded -in like manner upon observation, and is therefore only probable. The -law of gravitation itself is only probably true. But when a number of -different facts, observed under the most diverse circumstances, are -found to be harmonized under a supposed law of nature, the probability -of the law approximates closely to certainty. Each science rests upon -so many observed facts, and derives so much support from analogies or -connections with other sciences, that there are comparatively few cases -where our judgment of the probability of an event depends entirely -upon a few antecedent events, disconnected from the general body of -physical science. - -Events, again, may often exhibit a regularity of succession or -preponderance of character, which the simple formula will not take into -account. For instance, the majority of the elements recently discovered -are metals, so that the probability of the next discovery being that -of a metal, is doubtless greater than we calculated (p. 258). At -the more distant parts of the planetary system, there are symptoms -of disturbance which would prevent our placing much reliance on any -inference from the prevailing order of the known planets to those -undiscovered ones which may possibly exist at great distances. These -and all like complications in no way invalidate the theoretic truth of -the formulas, but render their sound application much more difficult. - -Erroneous objections have been raised to the theory of probability, on -the ground that we ought not to trust to our *à priori* conceptions -of what is likely to happen, but should always endeavour to obtain -precise experimental data to guide us.[171] This course, however, -is perfectly in accordance with the theory, which is our best and -only guide, whatever data we possess. We ought to be always applying -the inverse method of probabilities so as to take into account all -additional information. When we throw up a coin for the first time, we -are probably quite ignorant whether it tends more to fall head or tail -upwards, and we must therefore assume the probability of each event as -1/2. But if it shows head in the first throw, we now have very slight -experimental evidence in favour of a tendency to show head. The chance -of two heads is now slightly greater than 1/4, which it appeared to -be at first,[172] and as we go on throwing the coin time after time, -the probability of head appearing next time constantly varies in a -slight degree according to the character of our previous experience. As -Laplace remarks, we ought always to have regard to such considerations -in common life. Events when closely scrutinized will hardly ever prove -to be quite independent, and the slightest preponderance one way or -the other is some evidence of connection, and in the absence of better -evidence should be taken into account. - - [171] J. S. Mill, *System of Logic*, 5th edition, bk. iii. chap. - xviii. § 3. - - [172] Todhunter’s *History*, pp. 472, 598. - -The grand object of seeking to estimate the probability of future -events from past experience, seems to have been entertained by James -Bernoulli and De Moivre, at least such was the opinion of Condorcet; -and Bernoulli may be said to have solved one case of the problem.[173] -The English writers Bayes and Price are, however, undoubtedly the first -who put forward any distinct rules on the subject.[174] Condorcet and -several other eminent mathematicians advanced the mathematical theory -of the subject; but it was reserved to the immortal Laplace to bring -to the subject the full power of his genius, and carry the solution of -the problem almost to perfection. It is instructive to observe that a -theory which arose from petty games of chance, the rules and the very -names of which are forgotten, gradually advanced, until it embraced the -most sublime problems of science, and finally undertook to measure the -value and certainty of all our inductions. - - [173] Todhunter’s *History*, pp. 378, 379. - - [174] *Philosophical Transactions*, [1763], vol. liii. p. 370, and - [1764], vol. liv. p. 296. Todhunter, pp. 294–300. - - -*Fortuitous Coincidences.* - -We should have studied the theory of probability to very little -purpose, if we thought that it would furnish us with an infallible -guide. The theory itself points out the approximate certainty, that we -shall sometimes be deceived by extraordinary fortuitous coincidences. -There is no run of luck so extreme that it may not happen, and it -may happen to us, or in our time, as well as to other persons or in -other times. We may be forced by correct calculation to refer such -coincidences to a necessary cause, and yet we may be deceived. All -that the calculus of probability pretends to give, is *the result in -the long run*, as it is called, and this really means in *an infinity -of cases*. During any finite experience, however long, chances may be -against us. Nevertheless the theory is the best guide we can have. If -we always think and act according to its well-interpreted indications, -we shall have the best chance of escaping error; and if all persons, -throughout all time to come, obey the theory in like manner, they will -undoubtedly thereby reap the greatest advantage. - -No rule can be given for discriminating between coincidences which -are casual and those which are the effects of law. By a fortuitous -or casual coincidence, we mean an agreement between events, which -nevertheless arise from wholly independent and different causes or -conditions, and which will not always so agree. It is a fortuitous -coincidence, if a penny thrown up repeatedly in various ways always -falls on the same side; but it would not be fortuitous if there were -any similarity in the motions of the hand, and the height of the -throw, so as to cause or tend to cause a uniform result. Now among the -infinitely numerous events, objects, or relations in the universe, it -is quite likely that we shall occasionally notice casual coincidences. -There are seven intervals in the octave, and there is nothing very -improbable in the colours of the spectrum happening to be apparently -divisible into the same or similar series of seven intervals. It is -hardly yet decided whether this apparent coincidence, with which Newton -was much struck, is well founded or not,[175] but the question will -probably be decided in the negative. - - [175] Newton’s *Opticks*, Bk. I., Part ii. Prop. 3; *Nature*, vol. i. - p. 286. - -It is certainly a casual coincidence which the ancients noticed between -the seven vowels, the seven strings of the lyre, the seven Pleiades, -and the seven chiefs at Thebes.[176] The accidents connected with the -number seven have misled the human intellect throughout the historical -period. Pythagoras imagined a connection between the seven planets and -the seven intervals of the monochord. The alchemists were never tired -of drawing inferences from the coincidence in numbers of the seven -planets and the seven metals, not to speak of the seven days of the -week. - - [176] Aristotle’s *Metaphysics*, xiii. 6. 3. - -A singular circumstance was pointed out concerning the dimensions -of the earth, sun, and moon; the sun’s diameter was almost exactly -110 times as great as the earth’s diameter, while in almost exactly -the same ratio the mean distance of the earth was greater than the -sun’s diameter, and the mean distance of the moon from the earth was -greater than the moon’s diameter. The agreement was so close that it -might have proved more than casual, but its fortuitous character is -now sufficiently shown by the fact, that the coincidence ceases to be -remarkable when we adopt the amended dimensions of the planetary system. - -A considerable number of the elements have atomic weights, which are -apparently exact multiples of that of hydrogen. If this be not a law to -be ultimately extended to all the elements, as supposed by Prout, it -is a most remarkable coincidence. But, as I have observed, we have no -means of absolutely discriminating accidental coincidences from those -which imply a deep producing cause. A coincidence must either be very -strong in itself, or it must be corroborated by some explanation or -connection with other laws of nature. Little attention was ever given -to the coincidence concerning the dimensions of the sun, earth, and -moon, because it was not very strong in itself, and had no apparent -connection with the principles of physical astronomy. Prout’s Law -bears more probability because it would bring the constitution of -the elements themselves in close connection with the atomic theory, -representing them as built up out of a simpler substance. - -In historical and social matters, coincidences are frequently pointed -out which are due to chance, although there is always a strong popular -tendency to regard them as the work of design, or as having some hidden -meaning. If to 1794, the number of the year in which Robespierre fell, -we add the sum of its digits, the result is 1815, the year in which -Napoleon fell; the repetition of the process gives 1830 the year -in which Charles the Tenth abdicated. Again, the French Chamber of -Deputies, in 1830, consisted of 402 members, of whom 221 formed the -party called “La queue de Robespierre,” while the remainder, 181 in -number, were named “Les honnêtes gens.” If we give to each letter a -numerical value corresponding to its place in the alphabet, it will be -found that the sum of the values of the letters in each name exactly -indicates the number of the party. - -A number of such coincidences, often of a very curious character, -might be adduced, and the probability against the occurrence of each -is enormously great. They must be attributed to chance, because they -cannot be shown to have the slightest connection with the general -laws of nature; but persons are often found to be greatly influenced -by such coincidences, regarding them as evidence of fatality, that -is of a system of causation governing human affairs independently of -the ordinary laws of nature. Let it be remembered that there are an -infinite number of opportunities in life for some strange coincidence -to present itself, so that it is quite to be expected that remarkable -conjunctions will sometimes happen. - -In all matters of judicial evidence, we must bear in mind the probable -occurrence from time to time of unaccountable coincidences. The Roman -jurists refused for this reason to invalidate a testamentary deed, the -witnesses of which had sealed it with the same seal. For witnesses -independently using their own seals might be found to possess identical -ones by accident.[177] It is well known that circumstantial evidence of -apparently overwhelming completeness will sometimes lead to a mistaken -judgment, and as absolute certainty is never really attainable, every -court must act upon probabilities of a high amount, and in a certain -small proportion of cases they must almost of necessity condemn the -innocent victims of a remarkable conjuncture of circumstances.[178] -Popular judgments usually turn upon probabilities of far less amount, -as when the palace of Nicomedia, and even the bedchamber of Diocletian, -having been on fire twice within fifteen days, the people entirely -refused to believe that it could be the result of accident. The Romans -believed that there was fatality connected with the name of Sextus. - - “Semper sub Sextis perdita Roma fuit.” - - [177] Possunt autem omnes testes et uno annulo signare testamentum - Quid enim si septem annuli una sculptura fuerint, secundum quod - Pomponio visum est?--*Justinian*, ii. tit. x. 5. - - [178] See Wills on *Circumstantial Evidence*, p. 148. - -The utmost precautions will not provide against all contingencies. -To avoid errors in important calculations, it is usual to have them -repeated by different computers; but a case is on record in which three -computers made exactly the same calculations of the place of a star, -and yet all did it wrong in precisely the same manner, for no apparent -reason.[179] - - [179] *Memoirs of the Royal Astronomical Society*, vol. iv. p. 290, - quoted by Lardner, *Edinburgh Review*, July 1834, p. 278. - - -*Summary of the Theory of Inductive Inference.* - -The theory of inductive inference stated in this and the previous -chapters, was suggested by the study of the Inverse Method of -Probability, but it also bears much resemblance to the so-called -Deductive Method described by Mill, in his celebrated *System of -Logic*. Mill’s views concerning the Deductive Method, probably form -the most original and valuable part of his treatise, and I should -have ascribed the doctrine entirely to him, had I not found that -the opinions put forward in other parts of his work are entirely -inconsistent with the theory here upheld. As this subject is the most -important and difficult one with which we have to deal, I will try to -remedy the imperfect manner in which I have treated it, by giving a -recapitulation of the views adopted. - -All inductive reasoning is but the inverse application of deductive -reasoning. Being in possession of certain particular facts or events -expressed in propositions, we imagine some more general proposition -expressing the existence of a law or cause; and, deducing the -particular results of that supposed general proposition, we observe -whether they agree with the facts in question. Hypothesis is thus -always employed, consciously or unconsciously. The sole conditions to -which we need conform in framing any hypothesis is, that we both have -and exercise the power of inferring deductively from the hypothesis to -the particular results, which are to be compared with the known facts. -Thus there are but three steps in the process of induction:-- - -(1) Framing some hypothesis as to the character of the general law. - -(2) Deducing consequences from that law. - -(3) Observing whether the consequences agree with the particular facts -under consideration. - -In very simple cases of inverse reasoning, hypothesis may seem -altogether needless. To take numbers again as a convenient -illustration, I have only to look at the series, - - 1, 2, 4, 8, 16, 32, &c., - -to know at once that the general law is that of geometrical -progression; I need no successive trial of various hypotheses, because -I am familiar with the series, and have long since learnt from what -general formula it proceeds. In the same way a mathematician becomes -acquainted with the integrals of a number of common formulas, so -that he need not go through any process of discovery. But it is none -the less true that whenever previous reasoning does not furnish the -knowledge, hypotheses must be framed and tried (p. 124). - -There naturally arise two cases, according as the nature of the -subject admits of certain or only probable deductive reasoning. -Certainty, indeed, is but a singular case of probability, and the -general principles of procedure are always the same. Nevertheless, -when certainty of inference is possible, the process is simplified. -Of several mutually inconsistent hypotheses, the results of which can -be certainly compared with fact, but one hypothesis can ultimately -be entertained. Thus in the inverse logical problem, two logically -distinct conditions could not yield the same series of possible -combinations. Accordingly, in the case of two terms we had to choose -one of six different kinds of propositions (p. 136), and in the case -of three terms, our choice lay among 192 possible distinct hypotheses -(p. 140). Natural laws, however, are often quantitative in character, -and the possible hypotheses are then infinite in variety. - -When deduction is certain, comparison with fact is needed only to -assure ourselves that we have rightly selected the hypothetical -conditions. The law establishes itself, and no number of particular -verifications can add to its probability. Having once deduced from -the principles of algebra that the difference of the squares of two -numbers is equal to the product of their sum and difference, no number -of particular trials of its truth will render it more certain. On the -other hand, no finite number of particular verifications of a supposed -law will render that law certain. In short, certainty belongs only to -the deductive process, and to the teachings of direct intuition; and -as the conditions of nature are not given by intuition, we can only be -certain that we have got a correct hypothesis when, out of a limited -number conceivably possible, we select that one which alone agrees with -the facts to be explained. - -In geometry and kindred branches of mathematics, deductive reasoning -is conspicuously certain, and it would often seem as if the -consideration of a single diagram yields us certain knowledge of a -general proposition. But in reality all this certainty is of a purely -hypothetical character. Doubtless if we could ascertain that a supposed -circle was a true and perfect circle, we could be certain concerning a -multitude of its geometrical properties. But geometrical figures are -physical objects, and the senses can never assure us as to their exact -forms. The figures really treated in Euclid’s *Elements* are imaginary, -and we never can verify in practice the conclusions which we draw with -certainty in inference; questions of degree and probability enter. - -Passing now to subjects in which deduction is only probable, it ceases -to be possible to adopt one hypothesis to the exclusion of the others. -We must entertain at the same time all conceivable hypotheses, and -regard each with the degree of esteem proportionate to its probability. -We go through the same steps as before. - -(1) We frame an hypothesis. - -(2) We deduce the probability of various series of possible -consequences. - -(3) We compare the consequences with the particular facts, and observe -the probability that such facts would happen under the hypothesis. - -The above processes must be performed for every conceivable hypothesis, -and then the absolute probability of each will be yielded by the -principle of the inverse method (p. 242). As in the case of certainty -we accept that hypothesis which certainly gives the required results, -so now we accept as most probable that hypothesis which most probably -gives the results; but we are obliged to entertain at the same time -all other hypotheses with degrees of probability proportionate to the -probabilities that they would give the same results. - -So far we have treated only of the process by which we pass from -special facts to general laws, that inverse application of deduction -which constitutes induction. But the direct employment of deduction -is often combined with the inverse. No sooner have we established -a general law, than the mind rapidly draws particular consequences -from it. In geometry we may almost seem to infer that *because* one -equilateral triangle is equiangular, therefore another is so. In -reality it is not because one is that another is, but because all -are. The geometrical conditions are perfectly general, and by what -is sometimes called *parity of reasoning* whatever is true of one -equilateral triangle, so far as it is equilateral, is true of all -equilateral triangles. - -Similarly, in all other cases of inductive inference, where we seem to -pass from some particular instances to a new instance, we go through -the same process. We form an hypothesis as to the logical conditions -under which the given instances might occur; we calculate inversely -the probability of that hypothesis, and compounding this with the -probability that a new instance would proceed from the same conditions, -we gain the absolute probability of occurrence of the new instance in -virtue of this hypothesis. But as several, or many, or even an infinite -number of mutually inconsistent hypotheses may be possible, we must -repeat the calculation for each such conceivable hypothesis, and then -the complete probability of the future instance will be the sum of the -separate probabilities. The complication of this process is often very -much reduced in practice, owing to the fact that one hypothesis may be -almost certainly true, and other hypotheses, though conceivable, may be -so improbable as to be neglected without appreciable error. - -When we possess no knowledge whatever of the conditions from which -the events proceed, we may be unable to form any probable hypotheses -as to their mode of origin. We have now to fall back upon the general -solution of the problem effected by Laplace, which consists in -admitting on an equal footing every conceivable ratio of favourable -and unfavourable chances for the production of the event, and then -accepting the aggregate result as the best which can be obtained. This -solution is only to be accepted in the absence of all better means, -but like other results of the calculus of probability, it comes to our -aid where knowledge is at an end and ignorance begins, and it prevents -us from over-estimating the knowledge we possess. The general results -of the solution are in accordance with common sense, namely, that -the more often an event has happened the more probable, as a general -rule, is its subsequent recurrence. With the extension of experience -this probability increases, but at the same time the probability is -slight that events will long continue to happen as they have previously -happened. - -We have now pursued the theory of inductive inference, as far as -can be done with regard to simple logical or numerical relations. -The laws of nature deal with time and space, which are infinitely -divisible. As we passed from pure logic to numerical logic, so we must -now pass from questions of discontinuous, to questions of continuous -quantity, encountering fresh considerations of much difficulty. Before, -therefore, we consider how the great inductions and generalisations -of physical science illustrate the views of inductive reasoning just -explained, we must break off for a time, and review the means which we -possess of measuring and comparing magnitudes of time, space, mass, -force, momentum, energy, and the various manifestations of energy in -motion, heat, electricity, chemical change, and the other phenomena of -nature. - - - - -BOOK III. - -METHODS OF MEASUREMENT. - - - - -CHAPTER XIII. - -THE EXACT MEASUREMENT OF PHENOMENA. - - -As physical science advances, it becomes more and more accurately -quantitative. Questions of simple logical fact after a time resolve -themselves into questions of degree, time, distance, or weight. Forces -hardly suspected to exist by one generation, are clearly recognised -by the next, and precisely measured by the third generation. But -one condition of this rapid advance is the invention of suitable -instruments of measurement. We need what Francis Bacon called -*Instantiæ citantes*, or *evocantes*, methods of rendering minute -phenomena perceptible to the senses; and we also require *Instantiæ -radii* or *curriculi*, that is measuring instruments. Accordingly, -the introduction of a new instrument often forms an epoch in the -history of science. As Davy said, “Nothing tends so much to the -advancement of knowledge as the application of a new instrument. The -native intellectual powers of men in different times are not so much -the causes of the different success of their labours, as the peculiar -nature of the means and artificial resources in their possession.” - -In the absence indeed of advanced theory and analytical power, a -very precise instrument would be useless. Measuring apparatus and -mathematical theory should advance *pari passu*, and with just such -precision as the theorist can anticipate results, the experimentalist -should be able to compare them with experience. The scrupulously -accurate observations of Flamsteed were the proper complement to the -intense mathematical powers of Newton. - -Every branch of knowledge commences with quantitative notions of a -very rude character. After we have far progressed, it is often amusing -to look back into the infancy of the science, and contrast present -with past methods. At Greenwich Observatory in the present day, the -hundredth part of a second is not thought an inconsiderable portion -of time. The ancient Chaldæans recorded an eclipse to the nearest -hour, and the early Alexandrian astronomers thought it superfluous to -distinguish between the edge and centre of the sun. By the introduction -of the astrolabe, Ptolemy and the later Alexandrian astronomers could -determine the places of the heavenly bodies within about ten minutes -of arc. Little progress then ensued for thirteen centuries, until -Tycho Brahe made the first great step towards accuracy, not only by -employing better instruments, but even more by ceasing to regard an -instrument as correct. Tycho, in fact, determined the errors of his -instruments, and corrected his observations. He also took notice of -the effects of atmospheric refraction, and succeeded in attaining an -accuracy often sixty times as great as that of Ptolemy. Yet Tycho and -Hevelius often erred several minutes in the determination of a star’s -place, and it was a great achievement of Rœmer and Flamsteed to reduce -this error to seconds. Bradley, the modern Hipparchus, carried on the -improvement, his errors in right ascension, according to Bessel, being -under one second of time, and those of declination under four seconds -of arc. In the present day the average error of a single observation -is probably reduced to the half or quarter of what it was in Bradley’s -time; and further extreme accuracy is attained by the multiplication -of observations, and their skilful combination according to the theory -of error. Some of the more important constants, for instance that of -nutation, have been determined within the tenth part of a second of -space.[180] - - [180] Baily, *British Association Catalogue of Stars*, pp. 7, 23. - -It would be a matter of great interest to trace out the dependence of -this progress upon the introduction of new instruments. The astrolabe -of Ptolemy, the telescope of Galileo, the pendulum of Galileo and -Huyghens, the micrometer of Horrocks, and the telescopic sights and -micrometer of Gascoygne and Picard, Rœmer’s transit instrument, -Newton’s and Hadley’s quadrant, Dollond’s achromatic lenses, Harrison’s -chronometer, and Ramsden’s dividing engine--such were some of the -principal additions to astronomical apparatus. The result is, that we -now take note of quantities, 300,000 or 400,000 times as small as in -the time of the Chaldæans. - -It would be interesting again to compare the scrupulous accuracy of a -modern trigonometrical survey with Eratosthenes’ rude but ingenious -guess at the difference of latitude between Alexandria and Syene--or -with Norwood’s measurement of a degree of latitude in 1635. “Sometimes -I measured, sometimes I paced,” said Norwood; “and I believe I am -within a scantling of the truth.” Such was the germ of those elaborate -geodesical measurements which have made the dimensions of the globe -known to us within a few hundred yards. - -In other branches of science, the invention of an instrument has -usually marked, if it has not made, an epoch. The science of heat might -be said to commence with the construction of the thermometer, and it -has recently been advanced by the introduction of the thermo-electric -pile. Chemistry has been created chiefly by the careful use of the -balance, which forms a unique instance of an instrument remaining -substantially in the form in which it was first applied to scientific -purposes by Archimedes. The balance never has been and probably never -can be improved, except in details of construction. The torsion -balance, introduced by Coulomb towards the end of last century, has -rapidly become essential in many branches of investigation. In the -hands of Cavendish and Baily, it gave a determination of the earth’s -density; applied in the galvanometer, it gave a delicate measure of -electrical forces, and is indispensable in the thermo-electric pile. -This balance is made by simply suspending any light rod by a thin wire -or thread attached to the middle point. And we owe to it almost all the -more delicate investigations in the theories of heat, electricity, and -magnetism. - -Though we can now take note of the millionth of an inch in space, -and the millionth of a second in time, we must not overlook the fact -that in other operations of science we are yet in the position of the -Chaldæans. Not many years have elapsed since the magnitudes of the -stars, meaning the amounts of light they send to the observer’s eye, -were guessed at in the rudest manner, and the astronomer adjudged a -star to this or that order of magnitude by a rough comparison with -other stars of the same order. To Sir John Herschel we owe an attempt -to introduce a uniform method of measurement and expression, bearing -some relation to the real photometric magnitudes of the stars.[181] -Previous to the researches of Bunsen and Roscoe on the chemical action -of light, we were devoid of any mode of measuring the energy of light; -even now the methods are tedious, and it is not clear that they give -the energy of light so much as one of its special effects. Many natural -phenomena have hardly yet been made the subject of measurement at all, -such as the intensity of sound, the phenomena of taste and smell, the -magnitude of atoms, the temperature of the electric spark or of the -sun’s photosphere. - - [181] *Outlines of Astronomy*, 4th ed. sect. 781, p. 522. *Results of - Observations at the Cape of Good Hope*, &c., p. 37. - -To suppose, then, that quantitative science treats only of exactly -measurable quantities, is a gross if it be a common mistake. Whenever -we are treating of an event which either happens altogether or does -not happen at all, we are engaged with a non-quantitative phenomenon, -a matter of fact, not of degree; but whenever a thing may be greater -or less, or twice or thrice as great as another, whenever, in short, -ratio enters even in the rudest manner, there science will have a -quantitative character. There can be little doubt, indeed, that -every science as it progresses will become gradually more and more -quantitative. Numerical precision is the soul of science, as Herschel -said, and as all natural objects exist in space, and involve molecular -movements, measurable in velocity and extent, there is no apparent -limit to the ultimate extension of quantitative science. But the reader -must not for a moment suppose that, because we depend more and more -upon mathematical methods, we leave logical methods behind us. Number, -as I have endeavoured to show, is logical in its origin, and quantity -is but a development of number, or analogous thereto. - - -*Division of the Subject.* - -The general subject of quantitative investigation will have to be -divided into several parts. We shall firstly consider the means at -our disposal for measuring phenomena, and thus rendering them more or -less amenable to mathematical treatment. This task will involve an -analysis of the principles on which accurate methods of measurement are -founded, forming the subject of the remainder of the present chapter. -As measurement, however, only yields ratios, we have in the next -chapter to consider the establishment of unit magnitudes, in terms of -which our results may be expressed. As every phenomenon is usually the -sum of several distinct quantities depending upon different causes, -we have next to investigate in Chapter XV. the methods by which we -may disentangle complicated effects, and refer each part of the joint -effect to its separate cause. - -It yet remains for us in subsequent chapters to treat of quantitative -induction, properly so called. We must follow out the inverse logical -method, as it presents itself in problems of a far higher degree of -difficulty than those which treat of objects related in a simple -logical manner, and incapable of merging into each other by addition -and subtraction. - - -*Continuous Quantity.* - -The phenomena of nature are for the most part manifested in quantities -which increase or decrease continuously. When we inquire into the -precise meaning of continuous quantity, we find that it can only be -described as that which is divisible without limit. We can divide -a millimetre into ten, or a hundred, or a thousand, or ten thousand -parts, and mentally at any rate we can carry on the division *ad -infinitum*. Any finite space, then, must be conceived as made up of an -infinite number of parts each infinitely small. We cannot entertain the -simplest geometrical notions without allowing this. The conception of a -square involves the conception of a side and diagonal, which, as Euclid -beautifully proves in the 117th proposition of his tenth book, have no -common measure,[182] meaning no finite common measure. Incommensurable -quantities are, in fact, those which have for their only common measure -an infinitely small quantity. It is somewhat startling to find, too, -that in theory incommensurable quantities will be infinitely more -frequent than commensurable. Let any two lines be drawn haphazard; -it is infinitely unlikely that they will be commensurable, so that -the commensurable quantities, which we are supposed to deal with in -practice, are but singular cases among an infinitely greater number of -incommensurable cases. - - [182] See De Morgan, *Study of Mathematics*, in U.K.S. Library, p. 81. - -Practically, however, we treat all quantities as made up of the -least quantities which our senses, assisted by the best measuring -instruments, can perceive. So long as microscopes were uninvented, it -was sufficient to regard an inch as made up of a thousand thousandths -of an inch; now we must treat it as composed of a million millionths. -We might apparently avoid all mention of infinitely small quantities, -by never carrying our approximations beyond quantities which the -senses can appreciate. In geometry, as thus treated, we should never -assert two quantities to be equal, but only to be *apparently* equal. -Legendre really adopts this mode of treatment in the twentieth -proposition of the first book of his Geometry; and it is practically -adopted throughout the physical sciences, as we shall afterwards -see. But though our fingers, and senses, and instruments must stop -somewhere, there is no reason why the mind should not go on. We can -see that a proof which is only carried through a few steps in fact, -might be carried on without limit, and it is this consciousness of no -stopping-place, which renders Euclid’s proof of his 117th proposition -so impressive. Try how we will to circumvent the matter, we cannot -really avoid the consideration of the infinitely small and the -infinitely great. The same methods of approximation which seem confined -to the finite, mentally extend themselves to the infinite. - -One result of these considerations is, that we cannot possibly adjust -two quantities in absolute equality. The suspension of Mahomet’s coffin -between two precisely equal magnets is theoretically conceivable but -practically impossible. The story of the *Merchant of Venice* turns -upon the infinite improbability that an exact quantity of flesh could -be cut. Unstable equilibrium cannot exist in nature, for it is that -which is destroyed by an infinitely small displacement. It might be -possible to balance an egg on its end practically, because no egg has -a surface of perfect curvature. Suppose the egg shell to be perfectly -smooth, and the feat would become impossible. - - -*The Fallacious Indications of the Senses.* - -I may briefly remind the reader how little we can trust to our -unassisted senses in estimating the degree or magnitude of any -phenomenon. The eye cannot correctly estimate the comparative -brightness of two luminous bodies which differ much in brilliancy; -for we know that the iris is constantly adjusting itself to the -intensity of the light received, and thus admits more or less light -according to circumstances. The moon which shines with almost dazzling -brightness by night, is pale and nearly imperceptible while the eye is -yet affected by the vastly more powerful light of day. Much has been -recorded concerning the comparative brightness of the zodiacal light at -different times, but it would be difficult to prove that these changes -are not due to the varying darkness at the time, or the different -acuteness of the observer’s eye. For a like reason it is exceedingly -difficult to establish the existence of any change in the form or -comparative brightness of nebulæ; the appearance of a nebula greatly -depends upon the keenness of sight of the observer, or the accidental -condition of freshness or fatigue of his eye. The same is true of -lunar observations; and even the use of the best telescope fails to -remove this difficulty. In judging of colours, again, we must remember -that light of any given colour tends to dull the sensibility of the eye -for light of the same colour. - -Nor is the eye when unassisted by instruments a much better judge -of magnitude. Our estimates of the size of minute bright points, -such as the fixed stars, are completely falsified by the effects of -irradiation. Tycho calculated from the apparent size of the star-discs, -that no one of the principal fixed stars could be contained within the -area of the earth’s orbit. Apart, however, from irradiation or other -distinct causes of error our visual estimates of sizes and shapes are -often astonishingly incorrect. Artists almost invariably draw distant -mountains in ludicrous disproportion to nearer objects, as a comparison -of a sketch with a photograph at once shows. The extraordinary apparent -difference of size of the sun or moon, according as it is high in the -heavens or near the horizon, should be sufficient to make us cautious -in accepting the plainest indications of our senses, unassisted by -instrumental measurement. As to statements concerning the height of the -aurora and the distance of meteors, they are to be utterly distrusted. -When Captain Parry says that a ray of the aurora shot suddenly -downwards between him and the land which was only 3,000 yards distant, -we must consider him subject to an illusion of sense.[183] - - [183] Loomis, *On the Aurora Borealis*. Smithsonian Transactions, - quoting Parry’s Third Voyage, p. 61. - -It is true that errors of observation are more often errors of judgment -than of sense. That which is actually seen must be so far truly seen; -and if we correctly interpret the meaning of the phenomenon, there -would be no error at all. But the weakness of the bare senses as -measuring instruments, arises from the fact that they import varying -conditions of unknown amount, and we cannot make the requisite -corrections and allowances as in the case of a solid and invariable -instrument. - -Bacon has excellently stated the insufficiency of the senses for -estimating the magnitudes of objects, or detecting the degrees in which -phenomena present themselves. “Things escape the senses,” he says, -“because the object is not sufficient in quantity to strike the sense: -as all minute bodies; because the percussion of the object is too -great to be endured by the senses: as the form of the sun when looking -directly at it in mid-day; because the time is not proportionate to -actuate the sense: as the motion of a bullet in the air, or the quick -circular motion of a firebrand, which are too fast, or the hour-hand of -a common clock, which is too slow; from the distance of the object as -to place: as the size of the celestial bodies, and the size and nature -of all distant bodies; from prepossession by another object: as one -powerful smell renders other smells in the same room imperceptible; -from the interruption of interposing bodies: as the internal parts of -animals; and because the object is unfit to make an impression upon -the sense: as the air or the invisible and untangible spirit which is -included in every living body.” - - -*Complexity of Quantitative Questions.* - -One remark which we may well make in entering upon quantitative -questions, has regard to the great variety and extent of phenomena -presented to our notice. So long as we deal only with a simply logical -question, that question is merely, Does a certain event happen? or, -Does a certain object exist? No sooner do we regard the event or object -as capable of more and less, than the question branches out into many. -We must now ask, How much is it compared with its cause? Does it change -when the amount of the cause changes? If so, does it change in the same -or opposite direction? Is the change in simple proportion to that of -the cause? If not, what more complex law of connection holds true? This -law determined satisfactorily in one series of circumstances may be -varied under new conditions, and the most complex relations of several -quantities may ultimately be established. - -In every question of physical science there is thus a series of steps -the first one or two of which are usually made with ease while the -succeeding ones demand more and more careful measurement. We cannot -lay down any invariable series of questions which must be asked from -nature. The exact character of the questions will vary according -to the nature of the case, but they will usually be of an evident -kind, and we may readily illustrate them by examples. Suppose that -we are investigating the solution of some salt in water. The first -is a purely logical question: Is there solution, or is there not? -Assuming the answer to be in the affirmative, we next inquire, Does -the solubility vary with the temperature, or not? In all probability -some variation will exist, and we must have an answer to the further -question, Does the quantity dissolved increase, or does it diminish -with the temperature? In by far the greatest number of cases salts and -substances of all kinds dissolve more freely the higher the temperature -of the water; but there are a few salts, such as calcium sulphate, -which follow the opposite rule. A considerable number of salts resemble -sodium sulphate in becoming more soluble up to a certain temperature, -and then varying in the opposite direction. We next require to assign -the amount of variation as compared with that of the temperature, -assuming at first that the increase of solubility is proportional to -the increase of temperature. Common salt is an instance of very slight -variation, and potassium nitrate of very considerable increase with -temperature. Accurate observations will probably show, however, that -the simple law of proportionate variation is only approximately true, -and some more complicated law involving the second, third, or higher -powers of the temperature may ultimately be established. All these -investigations have to be carried out for each salt separately, since -no distinct principles by which we may infer from one substance to -another have yet been detected. There is still an indefinite field -for further research open; for the solubility of salts will probably -vary with the pressure under which the medium is placed; the presence -of other salts already dissolved may have effects yet unknown. The -researches already effected as regards the solvent power of water must -be repeated with alcohol, ether, carbon bisulphide, and other media, -so that unless general laws can be detected, this one phenomenon of -solution can never be exhaustively treated. The same kind of questions -recur as regards the solution or absorption of gases in liquids, the -pressure as well as the temperature having then a most decided effect, -and Professor Roscoe’s researches on the subject present an excellent -example of the successive determination of various complicated -laws.[184] - - [184] Watts’ *Dictionary of Chemistry*, vol. ii. p. 790. - -There is hardly a branch of physical science in which similar -complications are not ultimately encountered. In the case of gravity, -indeed, we arrive at the final law, that the force is the same for -all kinds of matter, and varies only with the distance of action. -But in other subjects the laws, if simple in their ultimate nature, -are disguised and complicated in their apparent results. Thus the -effect of heat in expanding solids, and the reverse effect of forcible -extension or compression upon the temperature of a body, will vary -from one substance to another, will vary as the temperature is already -higher or lower, and, will probably follow a highly complex law, which -in some cases gives negative or exceptional results. In crystalline -substances the same researches have to be repeated in each distinct -axial direction. - -In the sciences of pure observation, such as those of astronomy, -meteorology, and terrestrial magnetism, we meet with many interesting -series of quantitative determinations. The so-called fixed stars, as -Giordano Bruno divined, are not really fixed, and may be more truly -described as vast wandering orbs, each pursuing its own path through -space. We must then determine separately for each star the following -questions:-- - -1. Does it move? - -2. In what direction? - -3. At what velocity? - -4. Is this velocity variable or uniform? - -5. If variable, according to what law? - -6. Is the direction uniform? - -7. If not, what is the form of the apparent path? - -8. Does it approach or recede? 9. What is the form of the real path? - -The successive answers to such questions in the case of certain binary -stars, have afforded a proof that the motions are due to a central -force coinciding in law with gravity, and doubtless identical with it. -In other cases the motions are usually so small that it is exceedingly -difficult to distinguish them with certainty. And the time is yet -far off when any general results as regards stellar motions can be -established. - -The variation in the brightness of stars opens an unlimited field for -curious observation. There is not a star in the heavens concerning -which we might not have to determine:-- - -1. Does it vary in brightness? - -2. Is the brightness increasing or decreasing? - -3. Is the variation uniform? - -4. If not, according to what law does it vary? - -In a majority of cases the change will probably be found to have a -periodic character, in which case several other questions will arise, -such as-- - -5. What is the length of the period? - -6. Are there minor periods? - -7. What is the law of variation within the period? - -8. Is there any change in the amount of variation? - -9. If so, is it a secular, *i.e.* a continually growing change, or does -it give evidence of a greater period? - -Already the periodic changes of a certain number of stars have been -determined with accuracy, and the lengths of the periods vary from less -than three days up to intervals of time at least 250 times as great. -Periods within periods have also been detected. - -There is, perhaps, no subject in which more complicated quantitative -conditions have to be determined than terrestrial magnetism. Since -the time when the declination of the compass was first noticed, as -some suppose by Columbus, we have had successive discoveries from -time to time of the progressive change of declination from century to -century; of the periodic character of this change; of the difference -of the declination in various parts of the earth’s surface; of the -varying laws of the change of declination; of the dip or inclination -of the needle, and the corresponding laws of its periodic changes; the -horizontal and perpendicular intensities have also been the subject of -exact measurement, and have been found to vary with place and time, -like the directions of the needle; daily and yearly periodic changes -have also been detected, and all the elements are found to be subject -to occasional storms or abnormal perturbations, in which the eleven -year period, now known to be common to many planetary relations, -is apparent. The complete solution of these motions of the compass -needle involves nothing less than a determination of its position -and oscillations in every part of the world at any epoch, the like -determination for another epoch, and so on, time after time, until the -periods of all changes are ascertained. This one subject offers to men -of science an almost inexhaustible field for interesting quantitative -research, in which we shall doubtless at some future time discover the -operation of causes now most mysterious and unaccountable. - - -*The Methods of Accurate Measurement.* - -In studying the modes by which physicists have accomplished very exact -measurements, we find that they are very various, but that they may -perhaps be reduced under the following three classes:-- - -1. The increase or decrease, in some determinate ratio, of the quantity -to be measured, so as to bring it within the scope of our senses, and -to equate it with the standard unit, or some determinate multiple or -sub-multiple of this unit. - -2. The discovery of some natural conjunction of events which will -enable us to compare directly the multiples of the quantity with those -of the unit, or a quantity related in a definite ratio to that unit. - -3. Indirect measurement, which gives us not the quantity itself, but -some other quantity connected with it by known mathematical relations. - - -*Conditions of Accurate Measurement.* - -Several conditions are requisite in order that a measurement may be -made with great accuracy, and that the results may be closely accordant -when several independent measurements are made. - -In the first place the magnitude must be exactly defined by sharp -terminations, or precise marks of inconsiderable thickness. When a -boundary is vague and graduated, like the penumbra in a lunar eclipse, -it is impossible to say where the end really is, and different people -will come to different results. We may sometimes overcome this -difficulty to a certain extent, by observations repeated in a special -manner, as we shall afterwards see; but when possible, we should choose -opportunities for measurement when precise definition is easy. The -moment of occultation of a star by the moon can be observed with great -accuracy, because the star disappears with perfect suddenness; but -there are other astronomical conjunctions, eclipses, transits, &c., -which occupy a certain length of time in happening, and thus open the -way to differences of opinion. It would be impossible to observe with -precision the movements of a body possessing no definite points of -reference. The colours of the complete spectrum shade into each other -so continuously that exact determinations of refractive indices would -have been impossible, had we not the dark lines of the solar spectrum -as precise points for measurement, or various kinds of homogeneous -light, such as that of sodium, possessing a nearly uniform length of -vibration. - -In the second place, we cannot measure accurately unless we have the -means of multiplying or dividing a quantity without considerable error, -so that we may correctly equate one magnitude with the multiple or -submultiple of the other. In some cases we operate upon the quantity -to be measured, and bring it into accurate coincidence with the actual -standard, as when in photometry we vary the distance of our luminous -body, until its illuminating power at a certain point is equal to -that of a standard lamp. In other cases we repeat the unit until it -equals the object, as in surveying land, or determining a weight by -the balance. The requisites of accuracy now are:--(1) That we can -repeat unit after unit of exactly equal magnitude; (2) That these -can be joined together so that the aggregate shall really be the sum -of the parts. The same conditions apply to subdivision, which may be -regarded as a multiplication of subordinate units. In order to measure -to the thousandth of an inch, we must be able to add thousandth after -thousandth without error in the magnitude of these spaces, or in their -conjunction. - - -*Measuring Instruments.* - -To consider the mechanical construction of scientific instruments, is -no part of my purpose in this book. I wish to point out merely the -general purpose of such instruments, and the methods adopted to carry -out that purpose with great precision. In the first place we must -distinguish between the instrument which effects a comparison between -two quantities, and the standard magnitude which often forms one of -the quantities compared. The astronomer’s clock, for instance, is -no standard of the efflux of time; it serves but to subdivide, with -approximate accuracy, the interval of successive passages of a star -across the meridian, which it may effect perhaps to the tenth part of -a second, or 1/864000 part of the whole. The moving globe itself is -the real standard clock, and the transit instrument the finger of the -clock, while the stars are the hour, minute, and second marks, none -the less accurate because they are disposed at unequal intervals. The -photometer is a simple instrument, by which we compare the relative -intensity of rays of light falling upon a given spot. The galvanometer -shows the comparative intensity of electric currents passing through a -wire. The calorimeter gauges the quantity of heat passing from a given -object. But no such instruments furnish the standard unit in terms of -which our results are to be expressed. In one peculiar case alone does -the same instrument combine the unit of measurement and the means of -comparison. A theodolite, mural circle, sextant, or other instrument -for the measurement of angular magnitudes has no need of an additional -physical unit; for the circle itself, or complete revolution, is -the natural unit to which all greater or lesser amounts of angular -magnitude are referred. - -The result of every measurement is to make known the purely numerical -ratio existing between the magnitude to be measured, and a certain -other magnitude, which should, when possible, be a fixed unit or -standard magnitude, or at least an intermediate unit of which the -value can be ascertained in terms of the ultimate standard. But though -a ratio is the required result, an equation is the mode in which the -ratio is determined and expressed. In every measurement we equate -some multiple or submultiple of one quantity, with some multiple or -submultiple of another, and equality is always the fact which we -ascertain by the senses. By the eye, the ear, or the touch, we judge -whether there is a discrepancy or not between two lights, two sounds, -two intervals of time, two bars of metal. Often indeed we substitute -one sense for the other, as when the efflux of time is judged by -the marks upon a moving slip of paper, so that equal intervals of -time are represented by equal lengths. There is a tendency to reduce -all comparisons to the comparison of space magnitudes, but in every -case one of the senses must be the ultimate judge of coincidence or -non-coincidence. - -Since the equation to be established may exist between any multiples or -submultiples of the quantities compared, there naturally arise several -different modes of comparison adapted to different cases. Let *p* be -the magnitude to be measured, and *q* that in terms of which it is to -be expressed. Then we wish to find such numbers *x* and *y*, that the -equation *p = (x/y)q* may be true. This equation may be presented in -four forms, namely:-- - - First Form. Second Form. Third Form. Fourth Form. - *p = (x/y)q* *p(y/x) = q* *py = qx* *p/x = q/y* - -Each of these modes of expressing the same equation corresponds to one -mode of effecting a measurement. - -When the standard quantity is greater than that to be measured, we -often adopt the first mode, and subdivide the unit until we get a -magnitude equal to that measured. The angles observed in surveying, -in astronomy, or in goniometry are usually smaller than a whole -revolution, and the measuring circle is divided by the use of the -screw and microscope, until we obtain an angle undistinguishable from -that observed. The dimensions of minute objects are determined by -subdividing the inch or centimetre, the screw micrometer being the most -accurate means of subdivision. Ordinary temperatures are estimated by -division of the standard interval between the freezing and boiling -points of water, as marked on a thermometer tube. - -In a still greater number of cases, perhaps, we multiply the standard -unit until we get a magnitude equal to that to be measured. Ordinary -measurement by a foot rule, a surveyor’s chain, or the excessively -careful measurements of the base line of a trigonometrical survey by -standard bars, are sufficient instances of this procedure. - -In the second case, where *p(y/x) = q*, we multiply or divide a -magnitude until we get what is equal to the unit, or to some magnitude -easily comparable with it. As a general rule the quantities which we -desire to measure in physical science are too small rather than too -great for easy determination, and the problem consists in multiplying -them without introducing error. Thus the expansion of a metallic bar -when heated from 0°C to 100° may be multiplied by a train of levers or -cog wheels. In the common thermometer the expansion of the mercury, -though slight, is rendered very apparent, and easily measurable by the -fineness of the tube, and many other cases might be quoted. There are -some phenomena, on the contrary, which are too great or rapid to come -within the easy range of our senses, and our task is then the opposite -one of diminution. Galileo found it difficult to measure the velocity -of a falling body, owing to the considerable velocity acquired in a -single second. He adopted the elegant device, therefore, of lessening -the rapidity by letting the body roll down an inclined plane, which -enables us to reduce the accelerating force in any required ratio. -The same purpose is effected in the well-known experiments performed -on Attwood’s machine, and the measurement of gravity by the pendulum -really depends on the same principle applied in a far more advantageous -manner. Wheatstone invented a beautiful method of galvanometry for -strong currents, which consists in drawing off from the main current a -certain determinate portion, which is equated by the galvanometer to a -standard current. In short, he measures not the current itself but a -known fraction of it. - -In many electrical and other experiments, we wish to measure the -movements of a needle or other body, which are not only very slight -in themselves, but the manifestations of exceedingly small forces. We -cannot even approach a delicately balanced needle without disturbing -it. Under these circumstances the only mode of proceeding with -accuracy, is to attach a very small mirror to the moving body, and -employ a ray of light reflected from the mirror as an index of its -movements. The ray may be considered quite incapable of affecting the -body, and yet by allowing the ray to pass to a sufficient distance, the -motions of the mirror may be increased to almost any extent. A ray of -light is in fact a perfectly weightless finger or index of indefinite -length, with the additional advantage that the angular deviation is -by the law of reflection double that of the mirror. This method was -introduced by Gauss, and is now of great importance; but in Wollaston’s -reflecting goniometer a ray of light had previously been employed as an -index. Lavoisier and Laplace had also used a telescope in connection -with the pyrometer. - -It is a great advantage in some instruments that they can be readily -made to manifest a phenomenon in a greater or less degree, by a very -slight change in the construction. Thus either by enlarging the bulb -or contracting the tube of the thermometer, we can make it give -more conspicuous indications of change of temperature. The ordinary -barometer, on the other hand, always gives the variations of pressure -on one scale. The torsion balance is remarkable for the extreme -delicacy which may be attained by increasing the length and lightness -of the rod, and the length and thinness of the supporting thread. -Forces so minute as the attraction of gravitation between two balls, or -the magnetic and diamagnetic attraction of common liquids and gases, -may thus be made apparent, and even measured. The common chemical -balance, too, is capable theoretically of unlimited sensibility. - -The third mode of measurement, which may be called the Method of -Repetition, is of such great importance and interest that we must -consider it in a separate section. It consists in multiplying both -magnitudes to be compared until some multiple of the first is found -to coincide very nearly with some multiple of the second. If the -multiplication can be effected to an unlimited extent, without the -introduction of countervailing errors, the accuracy with which the -required ratio can be determined is unlimited, and we thus account for -the extraordinary precision with which intervals of time in astronomy -are compared together. - -The fourth mode of measurement, in which we equate submultiples of -two magnitudes, is comparatively seldom employed, because it does not -conduce to accuracy. In the photometer, perhaps, we may be said to use -it; we compare the intensity of two sources of light, by placing them -both at such distances from a given surface, that the light falling -on the surface is tolerable to the eye, and equally intense from each -source. Since the intensity of light varies inversely as the square -of the distance, the relative intensities of the luminous bodies are -proportional to the squares of their distances. The equal intensity of -two rays of similarly coloured light may be most accurately ascertained -in the mode suggested by Arago, namely, by causing the rays to pass in -opposite directions through two nearly flat lenses pressed together. -There is an exact equation between the intensities of the beams when -Newton’s rings disappear, the ring created by one ray being exactly the -complement of that created by the other. - - -*The Method of Repetition.* - -The ratio of two quantities can be determined with unlimited accuracy, -if we can multiply both the object of measurement and the standard unit -without error, and then observe what multiple of the one coincides or -nearly coincides with some multiple of the other. Although perfect -coincidence can never be really attained, the error thus arising -may be indefinitely reduced. For if the equation *py* = *qx* be -uncertain to the amount *e*, so that *py* = *qx* ± *e*, then we have -*p* = *q(x/y)* ± *e/y* , and as we are supposed to be able to make *y* -as great as we like without increasing the error *e*, it follows that -we can make *e* ÷ *y* as small as we like, and thus approximate within -an inconsiderable quantity to the required ratio *x* ÷ *y*. - -This method of repetition is naturally employed whenever quantities -can be repeated, or repeat themselves without error of juxtaposition, -which is especially the case with the motions of the earth and heavenly -bodies. In determining the length of the sidereal day, we determine the -ratio between the earth’s revolution round the sun, and its rotation on -its own axis. We might ascertain the ratio by observing the successive -passages of a star across the zenith, and comparing the interval by a -good clock with that between two passages of the sun, the difference -being due to the angular movement of the earth round the sun. In such -observations we should have an error of a considerable part of a second -at each observation, in addition to the irregularities of the clock. -But the revolutions of the earth repeat themselves day after day, and -year after year, without the slightest interval between the end of one -period and the beginning of another. The operation of multiplication -is perfectly performed for us by nature. If, then, we can find an -observation of the passage of a star across the meridian a hundred -years ago, that is of the interval of time between the passage of the -sun and the star, the instrumental errors in measuring this interval by -a clock and telescope may be greater than in the present day, but will -be divided by about 36,524 days, and rendered excessively small. It is -thus that astronomers have been able to ascertain the ratio of the mean -solar to the sidereal day to the 8th place of decimals (1·00273791 to -1), or to the hundred millionth part, probably the most accurate result -of measurement in the whole range of science. - -The antiquity of this mode of comparison is almost as great as that of -astronomy itself. Hipparchus made the first clear application of it, -when he compared his own observations with those of Aristarchus, made -145 years previously, and thus ascertained the length of the year. -This calculation may in fact be regarded as the earliest attempt at -an exact determination of the constants of nature. The method is the -main resource of astronomers; Tycho, for instance, detected the slow -diminution of the obliquity of the earth’s axis, by the comparison of -observations at long intervals. Living astronomers use the method as -much as earlier ones; but so superior in accuracy are all observations -taken during the last hundred years to all previous ones, that it is -often found preferable to take a shorter interval, rather than incur -the risk of greater instrumental errors in the earlier observations. - -It is obvious that many of the slower changes of the heavenly bodies -must require the lapse of large intervals of time to render their -amount perceptible. Hipparchus could not possibly have discovered the -smaller inequalities of the heavenly motions, because there were no -previous observations of sufficient age or exactness to exhibit them. -And just as the observations of Hipparchus formed the starting-point -for subsequent comparisons, so a large part of the labour of present -astronomers is directed to recording the present state of the heavens -so exactly, that future generations of astronomers may detect changes, -which cannot possibly become known in the present age. - -The principle of repetition was very ingeniously employed in an -instrument first proposed by Mayer in 1767, and carried into practice -in the Repeating Circle of Borda. The exact measurement of angles -is indispensable, not only in astronomy but also in trigonometrical -surveys, and the highest skill in the mechanical execution of the -graduated circle and telescope will not prevent terminal errors of -considerable amount. If instead of one telescope, the circle be -provided with two similar telescopes, these may be alternately directed -to two distant points, say the marks in a trigonometrical survey, so -that the circle shall be turned through any multiple of the angle -subtended by those marks, before the amount of the angular revolution -is read off upon the graduated circle. Theoretically speaking, all -error arising from imperfect graduation might thus be indefinitely -reduced, being divided by the number of repetitions. In practice, the -advantage of the invention is not found to be very great, probably -because a certain error is introduced at each observation in the -changing and fixing of the telescopes. It is moreover inapplicable to -moving objects like the heavenly bodies, so that its use is confined to -important trigonometrical surveys. - -The pendulum is the most perfect of all instruments, chiefly because -it admits of almost endless repetition. Since the force of gravity -never ceases, one swing of the pendulum is no sooner ended than the -other is begun, so that the juxtaposition of successive units is -absolutely perfect. Provided that the oscillations be equal, one -thousand oscillations will occupy exactly one thousand times as great -an interval of time as one oscillation. Not only is the subdivision of -time entirely dependent on this fact, but in the accurate measurement -of gravity, and many other important determinations, it is of the -greatest service. In the deepest mine, we could not observe the -rapidity of fall of a body for more than a quarter of a minute, and -the measurement of its velocity would be difficult, and subject to -uncertain errors from resistance of air, &c. In the pendulum, we have a -body which can be kept rising and falling for many hours, in a medium -entirely under our command or if desirable in a vacuum. Moreover, the -comparative force of gravity at different points, at the top and bottom -of a mine for instance, can be determined with wonderful precision, by -comparing the oscillations of two exactly similar pendulums, with the -aid of electric clock signals. - -To ascertain the comparative times of vibration of two pendulums, it -is only requisite to swing them one in front of the other, to record -by a clock the moment when they coincide in swing, so that one hides -the other, and then count the number of vibrations until they again -come to coincidence. If one pendulum makes *m* vibrations and the other -*n*, we at once have our equation *pn* = *qm*; which gives the length -of vibration of either pendulum in terms of the other. This method of -coincidence, embodying the principle of repetition in perfection, was -employed with wonderful skill by Sir George Airy, in his experiments on -the Density of the Earth at the Harton Colliery, the pendulums above -and below being compared with clocks, which again were compared with -each other by electric signals. So exceedingly accurate was this method -of observation, as carried out by Sir George Airy, that he was able to -measure a total difference in the vibrations at the top and bottom of -the shaft, amounting to only 2·24 seconds in the twenty-four hours, -with an error of less than one hundredth part of a second, or one part -in 8,640,000 of the whole day.[185] - - [185] *Philosophical Transactions*, (1856) vol. 146, Part i. p. 297. - -The principle of repetition has been elegantly applied in observing -the motion of waves in water. If the canal in which the experiments are -made be short, say twenty feet long, the waves will pass through it -so rapidly that an observation of one length, as practised by Walker, -will be subject to much terminal error, even when the observer is very -skilful. But it is a result of the undulatory theory that a wave is -unaltered, and loses no time by complete reflection, so that it may be -allowed to travel backwards and forwards in the same canal, and its -motion, say through sixty lengths, or 1200 feet, may be observed with -the same accuracy as in a canal 1200 feet long, with the advantage of -greater uniformity in the condition of the canal and water.[186] It -is always desirable, if possible, to bring an experiment into a small -compass, so that it may be well under command, and yet we may often by -repetition enjoy at the same time the advantage of extensive trial. - - [186] Airy, *On Tides and Waves*, Encyclopædia Metropolitana, p. 345. - Scott Russell, *British Association Report*, 1837, p. 432. - -One reason of the great accuracy of weighing with a good balance is -the fact, that weights placed in the same scale are naturally added -together without the slightest error. There is no difficulty in the -precise juxtaposition of two grams, but the juxtaposition of two metre -measures can only be effected with tolerable accuracy, by the use of -microscopes and many precautions. Hence, the extreme trouble and cost -attaching to the exact measurement of a base line for a survey, the -risk of error entering at every juxtaposition of the measuring bars, -and indefatigable attention to all the requisite precautions being -necessary throughout the operation. - - -*Measurements by Natural Coincidence.* - -In certain cases a peculiar conjunction of circumstances enables us to -dispense more or less with instrumental aids, and to obtain very exact -numerical results in the simplest manner. The mere fact, for instance, -that no human being has ever seen a different face of the moon from -that familiar to us, conclusively proves that the period of rotation -of the moon on its own axis is equal to that of its revolution round -the earth. Not only have we the repetition of these movements during -1000 or 2000 years at least, but we have observations made for us -at very remote periods, free from instrumental error, no instrument -being needed. We learn that the seventh satellite of Saturn is subject -to a similar law, because its light undergoes a variation in each -revolution, owing to the existence of some dark tract of land; now -this failure of light always occurs while it is in the same position -relative to Saturn, clearly proving the equality of the axial and -revolutional periods, as Huygens perceived.[187] A like peculiarity in -the motions of Jupiter’s fourth satellite was similarly detected by -Maraldi in 1713. - - [187] *Hugenii Cosmotheoros*, pp. 117, 118. Laplace’s *Système*, - translated, vol. i. p. 67. - -Remarkable conjunctions of the planets may sometimes allow us to -compare their periods of revolution, through great intervals of time, -with much accuracy. Laplace in explaining the long inequality in the -motions of Jupiter and Saturn, was assisted by a conjunction of these -planets, observed at Cairo, towards the close of the eleventh century. -Laplace calculated that such a conjunction must have happened on the -31st of October, A.D. 1087; and the discordance between the distances -of the planets as recorded, and as assigned by theory, was less than -one-fifth part of the apparent diameter of the sun. This difference -being less than the probable error of the early record, the theory was -confirmed as far as facts were available.[188] - - [188] Grant’s *History of Physical Astronomy*, p. 129. - -Ancient astronomers often showed the highest ingenuity in turning -any opportunities of measurement which occurred to good account. -Eratosthenes, as early as 250 B.C., happening to hear that the sun at -Syene, in Upper Egypt, was visible at the summer solstice at the bottom -of a well, proving that it was in the zenith, proposed to determine -the dimensions of the earth, by measuring the length of the shadow of -a rod at Alexandria on the same day of the year. He thus learnt in a -rude manner the difference of latitude between Alexandria and Syene and -finding it to be about one fiftieth part of the whole circumference, he -ascertained the dimensions of the earth within about one sixth part -of the truth. The use of wells in astronomical observation appears to -have been occasionally practised in comparatively recent times as by -Flamsteed in 1679.[189] The Alexandrian astronomers employed the moon -as an instrument of measurement in several sagacious modes. When the -moon is exactly half full, the moon, sun, and earth, are at the angles -of a right-angled triangle. Aristarchus measured at such a time the -moon’s elongation from the sun, which gave him the two other angles of -the triangle, and enabled him to judge of the comparative distances -of the moon and sun from the earth. His result, though very rude, was -far more accurate than any notions previously entertained, and enabled -him to form some estimate of the comparative magnitudes of the bodies. -Eclipses of the moon were very useful to Hipparchus in ascertaining -the longitude of the stars, which are invisible when the sun is above -the horizon. For the moon when eclipsed must be 180° distant from the -sun; hence it is only requisite to measure the distance of a fixed star -in longitude from the eclipsed moon to obtain with ease its angular -distance from the sun. - - [189] Baily’s *Account of Flamsteed*, p. lix. - -In later times the eclipses of Jupiter have served to measure an angle; -for at the middle moment of the eclipse the satellite must be in the -same straight line with the planet and sun, so that we can learn from -the known laws of movement of the satellite the longitude of Jupiter -as seen from the sun. If at the same time we measure the elongation or -apparent angular distance of Jupiter from the sun, as seen from the -earth, we have all the angles of the triangle between Jupiter, the sun, -and the earth, and can calculate the comparative magnitudes of the -sides of the triangle by trigonometry. - -The transits of Venus over the sun’s face are other natural events -which give most accurate measurements of the sun’s parallax, or -apparent difference of position as seen from distant points of the -earth’s surface. The sun forms a kind of background on which the place -of the planet is marked, and serves as a measuring instrument free -from all the errors of construction which affect human instruments. -The rotation of the earth, too, by variously affecting the apparent -velocity of ingress or egress of Venus, as seen from different places, -discloses the amount of the parallax. It has been sufficiently shown -that by rightly choosing the moments of observation, the planetary -bodies may often be made to reveal their relative distance, to measure -their own position, to record their own movements with a high degree -of accuracy. With the improvement of astronomical instruments, such -conjunctions become less necessary to the progress of the science, -but it will always remain advantageous to choose those moments for -observation when instrumental errors enter with the least effect. - -In other sciences, exact quantitative laws can occasionally be obtained -without instrumental measurement, as when we learn the exactly equal -velocity of sounds of different pitch, by observing that a peal of -bells or a musical performance is heard harmoniously at any distance -to which the sound penetrates; this could not be the case, as Newton -remarked, if one sound overtook the other. One of the most important -principles of the atomic theory, was proved by implication before the -use of the balance was introduced into chemistry. Wenzel observed, -before 1777, that when two neutral substances decompose each other, -the resulting salts are also neutral. In mixing sodium sulphate and -barium nitrate, we obtain insoluble barium sulphate and neutral sodium -nitrate. This result could not follow unless the nitric acid, requisite -to saturate one atom of sodium, were exactly equal to that required -by one atom of barium, so that an exchange could take place without -leaving either acid or base in excess. - -An important principle of mechanics may also be established by a simple -acoustical observation. When a rod or tongue of metal fixed at one -end is set in vibration, the pitch of the sound may be observed to -be exactly the same, whether the vibrations be small or great; hence -the oscillations are isochronous, or equally rapid, independently of -their magnitude. On the ground of theory, it can be shown that such a -result only happens when the flexure is proportional to the deflecting -force. Thus the simple observation that the pitch of the sound of a -harmonium, for instance, does not change with its loudness establishes -an exact law of nature.[190] - - [190] Jamin, *Cours de Physique*, vol. i. p. 152. - -A closely similar instance is found in the proof that the intensity -of light or heat rays varies inversely as the square of the distance -increases. For the apparent magnitude certainly varies according to -this law; hence, if the intensity of light varied according to any -other law, the brightness of an object would be different at different -distances, which is not observed to be the case. Melloni applied the -same kind of reasoning, in a somewhat different form, to the radiation -of heat-rays. - - -*Modes of Indirect Measurement.* - -Some of the most conspicuously beautiful experiments in the whole range -of science, have been devised for the purpose of indirectly measuring -quantities, which in their extreme greatness or smallness surpass the -powers of sense. All that we need to do, is to discover some other -conveniently measurable phenomenon, which is related in a known ratio -or according to a known law, however complicated, with that to be -measured. Having once obtained experimental data, there is no further -difficulty beyond that of arithmetic or algebraic calculation. - -Gold is reduced by the gold-beater to leaves so thin, that the most -powerful microscope would not detect any measurable thickness. If we -laid several hundred leaves upon each other to multiply the thickness, -we should still have no more than 1/100th of an inch at the most to -measure, and the errors arising in the superposition and measurement -would be considerable. But we can readily obtain an exact result -through the connected amount of weight. Faraday weighed 2000 leaves of -gold, each 3-3/8 inch square, and found them equal to 384 grains. From -the known specific gravity of gold it was easy to calculate that the -average thickness of the leaves was 1/282,000 of an inch.[191] - - [191] Faraday, *Chemical Researches*, p. 393. - -We must ascribe to Newton the honour of leading the way in methods of -minute measurement. He did not call waves of light by their right name, -and did not understand their nature; yet he measured their length, -though it did not exceed the 2,000,000th part of a metre or the one -fifty-thousandth part of an inch. He pressed together two lenses of -large but known radii. It was easy to calculate the interval between -the lenses at any point, by measuring the distance from the central -point of contact. Now, with homogeneous rays the successive rings of -light and darkness mark the points at which the interval between the -lenses is equal to one half, or any multiple of half a vibration of -the light, so that the length of the vibration became known. In a -similar manner many phenomena of interference of rays of light admit -of the measurement of the wave lengths. Fringes of interference arise -from rays of light which cross each other at a small angle, and an -excessively minute difference in the lengths of the waves makes a very -perceptible difference in the position of the point at which two rays -will interfere and produce darkness. - -Fizeau has recently employed Newton’s rings to measure small amounts of -motion. By merely counting the number of rings of sodium monochromatic -light passing a certain point where two glass plates are in close -proximity, he is able to ascertain with the greatest accuracy and ease -the change of distance between these glasses, produced, for instance, -by the expansion of a metallic bar, connected with one of the glass -plates.[192] - - [192] *Proceedings of the Royal Society*, 30th November, 1866. - -Nothing excites more admiration than the mode in which scientific -observers can occasionally measure quantities, which seem beyond -the bounds of human observation. We know the *average* depth of the -Pacific Ocean to be 14,190 feet, not by actual sounding, which would -be impracticable in sufficient detail, but by noticing the rate of -transmission of earthquake waves from the South American to the -opposite coasts, the rate of movement being connected by theory with -the depth of the water.[193] In the same way the average depth of -the Atlantic Ocean is inferred to be no less than 22,157 feet, from -the velocity of the ordinary tidal waves. A tidal wave again gives -beautiful evidence of an effect of the law of gravity, which we could -never in any other way detect. Newton estimated that the moon’s force -in moving the ocean is only one part in 2,871,400 of the whole force of -gravity, so that even the pendulum, used with the utmost skill, would -fail to render it apparent. Yet, the immense extent of the ocean allows -the accumulation of the effect into a very palpable amount; and from -the comparative heights of the lunar and solar tides, Newton roughly -estimated the comparative forces of the moon’s and sun’s gravity at the -earth.[194] - - [193] Herschel, *Physical Geography*, § 40. - - [194] *Principia*, bk. iii. Prop. 37, *Corollaries*, 2 and 3. Motte’s - translation, vol. ii. p. 310. - -A few years ago it might have seemed impossible that we should ever -measure the velocity with which a star approaches or recedes from the -earth, since the apparent position of the star is thereby unaltered. -But the spectroscope now enables us to detect and even measure such -motions with considerable accuracy, by the alteration which it -causes in the apparent rapidity of vibration, and consequently in -the refrangibility of rays of light of definite colour. And while -our estimates of the lateral movements of stars depend upon our very -uncertain knowledge of their distances, the spectroscope gives the -motions of approach and recess irrespective of other motions excepting -that of the earth. It gives in short the motions of approach and recess -of the stars relatively to the earth.[195] - - [195] Roscoe’s *Spectrum Analysis*, 1st ed. p. 296. - -The rapidity of vibration for each musical tone, having been accurately -determined by comparison with the Syren (p. 10), we can use sounds as -indirect indications of rapid vibrations. It is now known that the -contraction of a muscle arises from the periodical contractions of each -separate fibre, and from a faint sound or susurrus which accompanies -the action of a muscle, it is inferred that each contraction lasts for -about one 300th part of a second. Minute quantities of radiant heat are -now always measured indirectly by the electricity which they produce -when falling upon a thermopile. The extreme delicacy of the method -seems to be due to the power of multiplication at several points in the -apparatus. The number of elements or junctions of different metals in -the thermopile can be increased so that the tension of the electric -current derived from the same intensity of radiation is multiplied; -the effect of the current upon the magnetic needle can be multiplied -within certain bounds, by passing the current many times round it in -a coil; the excursions of the needle can be increased by rendering it -astatic and increasing the delicacy of its suspension; lastly, the -angular divergence can be observed, with any required accuracy, by the -use of an attached mirror and distant scale viewed through a telescope -(p. 287). Such is the delicacy of this method of measuring heat, that -Dr. Joule succeeded in making a thermopile which would indicate a -difference of 0°·000114 Cent.[196] - - [196] *Philosophical Transactions* (1859), vol. cxlix. p. 94. - -A striking case of indirect measurement is furnished by the revolving -mirror of Wheatstone and Foucault, whereby a minute interval of time -is estimated in the form of an angular deviation. Wheatstone viewed an -electric spark in a mirror rotating so rapidly, that if the duration -of the spark had been more than one 72,000th part of a second, the -point of light would have appeared elongated to an angular extent -of one-half degree. In the spark, as drawn directly from a Leyden -jar, no elongation was apparent, so that the duration of the spark -was immeasurably small; but when the discharge took place through -a bad conductor, the elongation of the spark denoted a sensible -duration.[197] In the hands of Foucault the rotating mirror gave a -measure of the time occupied by light in passing through a few metres -of space. - - [197] Watts’ *Dictionary of Chemistry*, vol. ii. p. 393. - - -*Comparative Use of Measuring Instruments.* - -In almost every case a measuring instrument serves, and should serve -only as a means of comparison between two or more magnitudes. As a -general rule, we should not attempt to make the divisions of the -measuring scale exact multiples or submultiples of the unit, but, -regarding them as arbitrary marks, should determine their values by -comparison with the standard itself. The perpendicular wires in the -field of a transit telescope, are fixed at nearly equal but arbitrary -distances, and those distances are afterwards determined, as first -suggested by Malvasia, by watching the passage of star after star -across them, and noting the intervals of time by the clock. Owing -to the perfectly regular motion of the earth, these time intervals -give exact determinations of the angular intervals. In the same way, -the angular value of each turn of the screw micrometer attached to a -telescope, can be easily and accurately ascertained. - -When a thermopile is used to observe radiant heat, it would be almost -impossible to calculate on *à priori* grounds what is the value of -each division of the galvanometer circle, and still more difficult -to construct a galvanometer, so that each division should have a -given value. But this is quite unnecessary, because by placing the -thermopile before a body of known dimensions, at a known distance, with -a known temperature and radiating power, we measure a known amount -of radiant heat, and inversely measure the value of the indications -of the thermopile. In a similar way Dr. Joule ascertained the actual -temperature produced by the compression of bars of metal. For having -inserted a small thermopile composed of a single junction of copper and -iron wire, and noted the deflections of the galvanometer, he had only -to dip the bars into water of different temperatures, until he produced -a like deflection, in order to ascertain the temperature developed by -pressure.[198] - - [198] *Philosophical Transactions* (1859), vol. cxlix. p. 119, &c. - -In some cases we are obliged to accept a very carefully constructed -instrument as a standard, as in the case of a standard barometer or -thermometer. But it is then best to treat all inferior instruments -comparatively only, and determine the values of their scales by -comparison with the assumed standard. - - -*Systematic Performance of Measurements.* - -When a large number of accurate measurements have to be effected, it -is usually desirable to make a certain number of determinations with -scrupulous care, and afterwards use them as points of reference for the -remaining determinations. In the trigonometrical survey of a country, -the principal triangulation fixes the relative positions and distances -of a few points with rigid accuracy. A minor triangulation refers every -prominent hill or village to one of the principal points, and then the -details are filled in by reference to the secondary points. The survey -of the heavens is effected in a like manner. The ancient astronomers -compared the right ascensions of a few principal stars with the moon, -and thus ascertained their positions with regard to the sun; the minor -stars were afterwards referred to the principal stars. Tycho followed -the same method, except that he used the more slowly moving planet -Venus instead of the moon. Flamsteed was in the habit of using about -seven stars, favourably situated at points all round the heavens. In -his early observations the distances of the other stars from these -standard points were determined by the use of the quadrant.[199] Even -since the introduction of the transit telescope and the mural circle, -tables of standard stars are formed at Greenwich, the positions being -determined with all possible accuracy, so that they can be employed for -purposes of reference by astronomers. - - [199] Baily’s *Account of Flamsteed*, pp. 378–380. - -In ascertaining the specific gravities of substances, all gases are -referred to atmospheric air at a given temperature and pressure; -all liquids and solids are referred to water. We require to compare -the densities of water and air with great care, and the comparative -densities of any two substances whatever can then be ascertained. - -In comparing a very great with a very small magnitude, it is -usually desirable to break up the process into several steps, using -intermediate terms of comparison. We should never think of measuring -the distance from London to Edinburgh by laying down measuring rods, -throughout the whole length. A base of several miles is selected on -level ground, and compared on the one hand with the standard yard, -and on the other with the distance of London and Edinburgh, or any -other two points, by trigonometrical survey. Again, it would be -exceedingly difficult to compare the light of a star with that of the -sun, which would be about thirty thousand million times greater; but -Herschel[200] effected the comparison by using the full moon as an -intermediate unit. Wollaston ascertained that the sun gave 801,072 -times as much light as the full moon, and Herschel determined that the -light of the latter exceeded that of α Centauri 27,408 times, so that -we find the ratio between the light of the sun and star to be that of -about 22,000,000,000 to 1. - - [200] Herschel’s *Astronomy*, § 817, 4th. ed. p. 553. - - -*The Pendulum.* - -By far the most perfect and beautiful of all instruments of measurement -is the pendulum. Consisting merely of a heavy body suspended freely -at an invariable distance from a fixed point, it is most simple in -construction; yet all the highest problems of physical measurement -depend upon its careful use. Its excessive value arises from two -circumstances. - -(1) The method of repetition is eminently applicable to it, as already -described (p. 290). - -(2) Unlike other instruments, it connects together three different -quantities, those of space, time, and force. - -In most works on natural philosophy it is shown, that when the -oscillations of the pendulum are infinitely small, the square of the -time occupied by an oscillation is directly proportional to the length -of the pendulum, and indirectly proportional to the force affecting it, -of whatever kind. The whole theory of the pendulum is contained in the -formula, first given by Huygens in his *Horologium Oscillatorium*. - - Time of oscillation = 3·14159 × √(length of pendulum/force). - -The quantity 3·14159 is the constant ratio of the circumference and -radius of a circle, and is of course known with accuracy. Hence, any -two of the three quantities concerned being given, the third may be -found; or any two being maintained invariable, the third will be -invariable. Thus a pendulum of invariable length suspended at the -same place, where the force of gravity may be considered constant, -furnishes a measure of time. The same invariable pendulum being made -to vibrate at different points of the earth’s surface, and the times -of vibration being astronomically determined, the force of gravity -becomes accurately known. Finally, with a known force of gravity, and -time of vibration ascertained by reference to the stars, the length is -determinate. - -All astronomical observations depend upon the first manner of using the -pendulum, namely, in the astronomical clock. In the second employment -it has been almost equally indispensable. The primary principle that -gravity is equal in all matter was proved by Newton’s and Gauss’ -pendulum experiments. The torsion pendulum of Michell, Cavendish, and -Baily, depending upon exactly the same principles as the ordinary -pendulum, gave the density of the earth, one of the foremost natural -constants. Kater and Sabine, by pendulum observations in different -parts of the earth, ascertained the variation of gravity, whence comes -a determination of the earth’s ellipticity. The laws of electric -and magnetic attraction have also been determined by the method -of vibrations, which is in constant use in the measurement of the -horizontal force of terrestrial magnetism. - -We must not confuse with the ordinary use of the pendulum its -application by Newton, to show the absence of internal friction against -space,[201] or to ascertain the laws of motion and elasticity.[202] In -these cases the extent of vibration is the quantity measured, and the -principles of the instrument are different. - - [201] *Principia*, bk. ii. Sect. 6. Prop. 31. Motte’s Translation, - vol. ii. p. 107. - - [202] Ibid. bk. i. Law iii. Corollary 6. Motte’s Translation, vol. i. - p. 33. - - -*Attainable Accuracy of Measurement.* - -It is a matter of some interest to compare the degrees of accuracy -which can be attained in the measurement of different kinds of -magnitude. Few measurements of any kind are exact to more than six -significant figures,[203] but it is seldom that such accuracy can be -hoped for. Time is the magnitude which seems to be capable of the most -exact estimation, owing to the properties of the pendulum, and the -principle of repetition described in previous sections. As regards -short intervals of time, it has already been stated that Sir George -Airy was able to estimate one part in 8,640,000, an exactness, as he -truly remarks, “almost beyond conception.”[204] The ratio between the -mean solar and the sidereal day is known to be about one part in one -hundred millions, or to the eighth place of decimals, (p. 289). - - [203] Thomson and Tait’s *Natural Philosophy*, vol. i. p. 333. - - [204] *Philosophical Transactions*, (1856), vol. cxlvi. pp. 330, 331. - -Determinations of weight seem to come next in exactness, owing to the -fact that repetition without error is applicable to them. An ordinary -good balance should show about one part in 500,000 of the load. The -finest balance employed by M. Stas, turned with one part in 825,000 of -the load.[205] But balances have certainly been constructed to show -one part in a million,[206] and Ramsden is said to have constructed a -balance for the Royal Society, to indicate one part in seven millions, -though this is hardly credible. Professor Clerk Maxwell takes it for -granted that one part in five millions can be detected, but we ought to -discriminate between what a balance can do when first constructed, and -when in continuous use. - - [205] *First Annual Report of the Mint*, p. 106. - - [206] Jevons, in Watts’ *Dictionary of Chemistry*, vol. i. p. 483. - -Determinations of length, unless performed with extraordinary care, -are open to much error in the junction of the measuring bars. Even -in measuring the base line of a trigonometrical survey, the accuracy -generally attained is only that of about one part in 60,000, or an -inch in the mile; but it is said that in four measurements of a base -line carried out very recently at Cape Comorin, the greatest error was -0·077 inch in 1·68 mile, or one part in 1,382,400, an almost incredible -degree of accuracy. Sir J. Whitworth has shown that touch is even a -more delicate mode of measuring lengths than sight, and by means of -a splendidly executed screw, and a small cube of iron placed between -two flat-ended iron bars, so as to be suspended when touching them, he -can detect a change of dimension in a bar, amounting to no more than -one-millionth of an inch.[207] - - [207] British Association, Glasgow, 1856. *Address of the President - of the Mechanical Section*. - - - - -CHAPTER XIV. - -UNITS AND STANDARDS OF MEASUREMENT. - - -As we have seen, instruments of measurement are only means of -comparison between one magnitude and another, and as a general rule we -must assume some one arbitrary magnitude, in terms of which all results -of measurement are to be expressed. Mere ratios between any series of -objects will never tell us their absolute magnitudes; we must have at -least one ratio for each, and we must have one absolute magnitude. -The number of ratios *n* are expressible in *n* equations, which will -contain at least *n* + 1 quantities, so that if we employ them to make -known *n* magnitudes, we must have one magnitude known. Hence, whether -we are measuring time, space, density, mass, weight, energy, or any -other physical quantity, we must refer to some concrete standard, some -actual object, which if once lost and irrecoverable, all our measures -lose their absolute meaning. This concrete standard is in all cases -arbitrary in point of theory, and its selection a question of practical -convenience. - -There are two kinds of magnitude, indeed, which do not need to be -expressed in terms of arbitrary concrete units, since they pre-suppose -the existence of natural standard units. One case is that of abstract -number itself, which needs no special unit, because any object which -exists or is thought of as separate from other objects (p. 157) -furnishes us with a unit, and is the only standard required. - -Angular magnitude is the second case in which we have a natural unit -of reference, namely the whole revolution or *perigon*, as it has -been called by Mr. Sandeman.[208] It is a necessary result of the -uniform properties of space, that all complete revolutions are equal -to each other, so that we need not select any one revolution, but can -always refer anew to space itself. Whether we take the whole perigon, -its half, or its quarter, is really immaterial; Euclid took the right -angle, because the Greek geometers had never generalised their notions -of angular magnitude sufficiently to treat angles of all magnitudes, -or of unlimited *quantity of revolution*. Euclid defines a right angle -as half that made by a line with its own continuation, which is of -course equal to half a revolution, but which was not treated as an -angle by him. In mathematical analysis a different fraction of the -perigon is taken, namely, such a fraction that the arc or portion of -the circumference included within it is equal to the radius of the -circle. In this point of view angular magnitude is an abstract ratio, -namely, the ratio between the length of arc subtended and the length -of the radius. The geometrical unit is then necessarily the angle -corresponding to the ratio unity. This angle is equal to about 57°, -17′, 44″·8, or decimally 57°·295779513... .[209] It was called by De -Morgan the *arcual unit*, but a more convenient name for common use -would be *radian*, as suggested by Professor Everett. Though this -standard angle is naturally employed in mathematical analysis, and any -other unit would introduce great complexity, we must not look upon it -as a distinct unit, since its amount is connected with that of the half -perigon, by the natural constant 3·14159... usually denoted by the -letter π. - - [208] *Pelicotetics, or the Science of Quantity; an Elementary - Treatise on Algebra, and its groundwork Arithmetic.* By Archibald - Sandeman, M. A. Cambridge (Deighton, Bell, and Co.), 1868, p. 304. - - [209] De Morgan’s *Trigonometry and Double Algebra*, p. 5. - -When we pass to other species of quantity, the choice of unit is found -to be entirely arbitrary. There is absolutely no mode of defining a -length, but by selecting some physical object exhibiting that length -between certain obvious points--as, for instance, the extremities of a -bar, or marks made upon its surface. - - -*Standard Unit of Time.* - -Time is the great independent variable of all change--that which itself -flows on uninterruptedly, and brings the variety which we call motion -and life. When we reflect upon its intimate nature, Time, like every -other element of existence, proves to be an inscrutable mystery. We -can only say with St. Augustin, to one who asks us what is time, “I -know when you do not ask me.” The mind of man will ask what can never -be answered, but one result of a true and rigorous logical philosophy -must be to convince us that scientific explanation can only take place -between phenomena which have something in common, and that when we get -down to primary notions, like those of time and space, the mind must -meet a point of mystery beyond which it cannot penetrate. A definition -of time must not be looked for; if we say with Hobbes,[210] that it is -“the phantasm of before and after in motion,” or with Aristotle that it -is “the number of motion according to former and latter,” we obviously -gain nothing, because the notion of time is involved in the expressions -*before and after*, *former and latter*. Time is undoubtedly one of -those primary notions which can only be defined physically, or by -observation of phenomena which proceed in time. - - [210] *English Works of Thos. Hobbes*, Edit. by Molesworth, vol. i. - p. 95. - -If we have not advanced a step beyond Augustin’s acute reflections on -this subject,[211] it is curious to observe the wonderful advances -which have been made in the practical measurement of its efflux. In -earlier centuries the rude sun-dial or the rising of a conspicuous star -gave points of reference, while the flow of water from the clepsydra, -the burning of a candle, or, in the monastic ages, even the continuous -chanting of psalms, were the means of roughly subdividing periods, and -marking the hours of the day and night.[212] The sun and stars still -furnish the standard of time, but means of accurate subdivision have -become requisite, and this has been furnished by the pendulum and the -chronograph. By the pendulum we can accurately divide the day into -seconds of time. By the chronograph we can subdivide the second into -a hundred, a thousand, or even a million parts. Wheatstone measured -the duration of an electric spark, and found it to be no more than one -115,200th part of a second, while more recently Captain Noble has been -able to appreciate intervals of time not exceeding the millionth part -of a second. - - [211] *Confessions*, bk. xi. chapters 20–28. - - [212] Sir G. C. Lewis gives many curious particulars concerning the - measurement of time in his *Astronomy of the Ancients*, pp. 241, &c. - -When we come to inquire precisely what phenomenon it is that we thus -so minutely measure, we meet insurmountable difficulties. Newton -distinguished time according as it was *absolute* or *apparent* time, -in the following words:--“Absolute, true, and mathematical time, -of itself and from its own nature, flows equably without regard to -anything external, and by another name is called *duration*; relative, -apparent and common time, is some sensible and external measure of -duration by the means of motion.”[213] Though we are perhaps obliged to -assume the existence of a uniformly increasing quantity which we call -time, yet we cannot feel or know abstract and absolute time. Duration -must be made manifest to us by the recurrence of some phenomenon. The -succession of our own thoughts is no doubt the first and simplest -measure of time, but a very rude one, because in some persons and -circumstances the thoughts evidently flow with much greater rapidity -than in other persons and circumstances. In the absence of all -other phenomena, the interval between one thought and another would -necessarily become the unit of time, but the most cursory observations -show that there are changes in the outward world much better fitted by -their constancy to measure time than the change of thoughts within us. - - [213] *Principia*, bk. i. *Scholium to Definitions*. Translated by - Motte, vol. i. p. 9. See also p. 11. - -The earth, as I have already said, is the real clock of the astronomer, -and is practically assumed as invariable in its movements. But on -what ground is it so assumed? According to the first law of motion, -every body perseveres in its state of rest or of uniform motion in -a right line, unless it is compelled to change that state by forces -impressed thereon. Rotatory motion is subject to a like condition, -namely, that it perseveres uniformly unless disturbed by extrinsic -forces. Now uniform motion means motion through equal spaces in equal -times, so that if we have a body entirely free from all resistance -or perturbation, and can measure equal spaces of its path, we have a -perfect measure of time. But let it be remembered that this law has -never been absolutely proved by experience; for we cannot point to any -body, and say that it is wholly unresisted or undisturbed; and even if -we had such a body, we should need some independent standard of time -to ascertain whether its motion was really uniform. As it is in moving -bodies that we find the best standard of time, we cannot use them to -prove the uniformity of their own movements, which would amount to a -*petitio principii*. Our experience comes to this, that when we examine -and compare the movements of bodies which seem to us nearly free from -disturbance, we find them giving nearly harmonious measures of time. -If any one body which seems to us to move uniformly is not doing so, -but is subject to fits and starts unknown to us, because we have no -absolute standard of time, then all other bodies must be subject to the -same arbitrary fits and starts, otherwise there would be discrepancy -disclosing the irregularities. Just as in comparing together a number -of chronometers, we should soon detect bad ones by their going -irregularly, as compared with the others, so in nature we detect -disturbed movement by its discrepancy from that of other bodies which -we believe to be undisturbed, and which agree nearly among themselves. -But inasmuch as the measure of motion involves time, and the measure -of time involves motion, there must be ultimately an assumption. We -may define equal times, as times during which a moving body under the -influence of no force describes equal spaces;[214] but all we can -say in support of this definition is, that it leads us into no known -difficulties, and that to the best of our experience one freely moving -body gives the same results as any other. - - [214] Rankine, *Philosophical Magazine*, Feb. 1867, vol. xxxiii. - p. 91. - -When we inquire where the freely moving body is, no perfectly -satisfactory answer can be given. Practically the rotating globe is -sufficiently accurate, and Thomson and Tait say: “Equal times are -times during which the earth turns through equal angles.”[215] No long -time has passed since astronomers thought it impossible to detect any -inequality in its movement. Poisson was supposed to have proved that a -change in the length of the sidereal day amounting to one ten-millionth -part in 2,500 years was incompatible with an ancient eclipse recorded -by the Chaldæans, and similar calculations were made by Laplace. But -it is now known that these calculations were somewhat in error, and -that the dissipation of energy arising out of the friction of tidal -waves, and the radiation of the heat into space, has slightly decreased -the rapidity of the earth’s rotatory motion. The sidereal day is now -longer by one part in 2,700,000, than it was in 720 B.C. Even before -this discovery, it was known that invariability of rotation depended -upon the perfect maintenance of the earth’s internal heat, which is -requisite in order that the earth’s dimensions shall be unaltered. Now -the earth being superior in temperature to empty space, must cool more -or less rapidly, so that it cannot furnish an absolute measure of time. -Similar objections could be raised to all other rotating bodies within -our cognisance. - - [215] *Treatise on Natural Philosophy*, vol. i. p. 179. - -The moon’s motion round the earth, and the earth’s motion round the -sun, form the next best measure of time. They are subject, indeed, -to disturbance from other planets, but it is believed that these -perturbations must in the course of time run through their rhythmical -courses, leaving the mean distances unaffected, and consequently, by -the third Law of Kepler, the periodic times unchanged. But there is -more reason than not to believe that the earth encounters a slight -resistance in passing through space, like that which is so apparent -in Encke’s comet. There may also be dissipation of energy in the -electrical relations of the earth to the sun, possibly identical with -that which is manifested in the retardation of comets.[216] It is -probably an untrue assumption then, that the earth’s orbit remains -quite invariable. It is just possible that some other body may be found -in the course of time to furnish a better standard of time than the -earth in its annual motion. The greatly superior mass of Jupiter and -its satellites, and their greater distance from the sun, may render the -electrical dissipation of energy less considerable than in the case of -the earth. But the choice of the best measure will always be an open -one, and whatever moving body we choose may ultimately be shown to be -subject to disturbing forces. - - [216] *Proceedings of the Manchester Philosophical Society*, 28th - Nov. 1871, vol. xi. p. 33. - -The pendulum, although so admirable an instrument for subdivision of -time, fails as a standard; for though the same pendulum affected by the -same force of gravity performs equal vibrations in equal times, yet -the slightest change in the form or weight of the pendulum, the least -corrosion of any part, or the most minute displacement of the point of -suspension, falsifies the results, and there enter many other difficult -questions of temperature, friction, resistance, length of vibration, &c. - -Thomson and Tait are of opinion[217] that the ultimate standard of -chronometry must be founded on the physical properties of some body -of more constant character than the earth; for instance, a carefully -arranged metallic spring, hermetically sealed in an exhausted glass -vessel. But it is hard to see how we can be sure that the dimensions -and elasticity of a piece of wrought metal will remain perfectly -unchanged for the few millions of years contemplated by them. A nearly -perfect gas, like hydrogen, is perhaps the only kind of substance in -the unchanged elasticity of which we could have confidence. Moreover, -it is difficult to perceive how the undulations of such a spring could -be observed with the requisite accuracy. More recently Professor Clerk -Maxwell has made the novel suggestion, discussed in a subsequent -section, that undulations of light *in vacuo* would form the most -universal standard of reference, both as regards time and space. -According to this system the unit of time would be the time occupied -by one vibration of the particular kind of light whose wave length is -taken as the unit of length. - - [217] *The Elements of Natural Philosophy*, part i. p. 119. - - -*The Unit of Space and the Bar Standard.* - -Next in importance after the measurement of time is that of space. -Time comes first in theory, because phenomena, our internal thoughts -for instance, may change in time without regard to space. As to the -phenomena of outward nature, they tend more and more to resolve -themselves into motions of molecules, and motion cannot be conceived or -measured without reference both to time and space. - -Turning now to space measurement, we find it almost equally difficult -to fix and define once and for ever, a unit magnitude. There are -three different modes in which it has been proposed to attempt the -perpetuation of a standard length. - -(1) By constructing an actual specimen of the standard yard or metre, -in the form of a bar. - -(2) By assuming the globe itself to be the ultimate standard of -magnitude, the practical unit being a submultiple of some dimension of -the globe. - -(3) By adopting the length of the simple seconds pendulum, as a -standard of reference. - -At first sight it might seem that there was no great difficulty in this -matter, and that any one of these methods might serve well enough; -but the more minutely we inquire into the details, the more hopeless -appears to be the attempt to establish an invariable standard. We must -in the first place point out a principle not of an obvious character, -namely, that *the standard length must be defined by one single -object*.[218] To make two bars of exactly the same length, or even two -bars bearing a perfectly defined ratio to each other, is beyond the -power of human art. If two copies of the standard metre be made and -declared equally correct, future investigators will certainly discover -some discrepancy between them, proving of course that they cannot both -be the standard, and giving cause for dispute as to what magnitude -should then be taken as correct. - - [218] See Harris’ *Essay upon Money and Coins*, part. ii. [1758] - p. 127. - -If one invariable bar could be constructed and maintained as the -absolute standard, no such inconvenience could arise. Each successive -generation as it acquired higher powers of measurement, would detect -errors in the copies of the standard, but the standard itself would be -unimpeached, and would, as it were, become by degrees more and more -accurately known. Unfortunately to construct and preserve a metre or -yard is also a task which is either impossible, or what comes nearly -to the same thing, cannot be shown to be possible. Passing over the -practical difficulty of defining the ends of the standard length -with complete accuracy, whether by dots or lines on the surface, or -by the terminal points of the bar, we have no means of proving that -substances remain of invariable dimensions. Just as we cannot tell -whether the rotation of the earth is uniform, except by comparing it -with other moving bodies, believed to be more uniform in motion, so -we cannot detect the change of length in a bar, except by comparing -it with some other bar supposed to be invariable. But how are we to -know which is the invariable bar? It is certain that many rigid and -apparently invariable substances do change in dimensions. The bulb of -a thermometer certainly contracts by age, besides undergoing rapid -changes of dimensions when warmed or cooled through 100° Cent. Can -we be sure that even the most solid metallic bars do not slightly -contract by age, or undergo variations in their structure by change -of temperature. Fizeau was induced to try whether a quartz crystal, -subjected to several hundred alternations of temperature, would be -modified in its physical properties, and he was unable to detect any -change in the coefficient of expansion.[219] It does not follow, -however, that, because no apparent change was discovered in a quartz -crystal, newly-constructed bars of metal would undergo no change. - - [219] *Philosophical Magazine*, (1868), 4th Series, vol. xxxvi. p. 32. - -The best principle, as it seems to me, upon which the perpetuation of -a standard of length can be rested, is that, if a variation of length -occurs, it will in all probability be of different amount in different -substances. If then a great number of standard metres were constructed -of all kinds of different metals and alloys; hard rocks, such as -granite, serpentine, slate, quartz, limestone; artificial substances, -such as porcelain, glass, &c., &c., careful comparison would show from -time to time the comparative variations of length of these different -substances. The most variable substances would be the most divergent, -and the standard would be furnished by the mean length of those which -agreed most closely with each other just as uniform motion is that of -those bodies which agree most closely in indicating the efflux of time. - - -*The Terrestrial Standard.* - -The second method assumes that the globe itself is a body of invariable -dimensions and the founders of the metrical system selected the -ten-millionth part of the distance from the equator to the pole as -the definition of the metre. The first imperfection in such a method -is that the earth is certainly not invariable in size; for we know -that it is superior in temperature to surrounding space, and must be -slowly cooling and contracting. There is much reason to believe that -all earthquakes, volcanoes, mountain elevations, and changes of sea -level are evidences of this contraction as asserted by Mr. Mallet.[220] -But such is the vast bulk of the earth and the duration of its past -existence, that this contraction is perhaps less rapid in proportion -than that of any bar or other material standard which we can construct. - - [220] *Proceedings of the Royal Society*, 20th June, 1872, vol. xx. - p. 438. - -The second and chief difficulty of this method arises from the vast -size of the earth, which prevents us from making any comparison with -the ultimate standard, except by a trigonometrical survey of a most -elaborate and costly kind. The French physicists, who first proposed -the method, attempted to obviate this inconvenience by carrying out -the survey once for all, and then constructing a standard metre, which -should be exactly the one ten millionth part of the distance from the -pole to the equator. But since all measuring operations are merely -approximate, it was impossible that this operation could be perfectly -achieved. Accordingly, it was shown in 1838 that the supposed French -metre was erroneous to the considerable extent of one part in 5527. It -then became necessary either to alter the length of the assumed metre, -or to abandon its supposed relation to the earth’s dimensions. The -French Government and the International Metrical Commission have for -obvious reasons decided in favour of the latter course, and have thus -reverted to the first method of defining the metre by a given bar. As -from time to time the ratio between this assumed standard metre and the -quadrant of the earth becomes more accurately known, we have better -means of restoring that metre by reference to the globe if required. -But until lost, destroyed, or for some clear reason discredited, -the bar metre and not the globe is the standard. Thomson and Tait -remark that any of the more accurate measurements of the English -trigonometrical survey might in like manner be employed to restore our -standard yard, in terms of which the results are recorded. - - -*The Pendulum Standard.* - -The third method of defining a standard length, by reference to the -seconds pendulum, was first proposed by Huyghens, and was at one time -adopted by the English Government. From the principle of the pendulum -(p. 302) it clearly appears that if the time of oscillation and the -force actuating the pendulum be the same, the length of the pendulum -must be the same. We do not get rid of theoretical difficulties, for -we must assume the attraction of gravity at some point of the earth’s -surface, say London, to be unchanged from time to time, and the -sidereal day to be invariable, neither assumption being absolutely -correct so far as we can judge. The pendulum, in short, is only an -indirect means of making one physical quantity of space depend upon two -other physical quantities of time and force. - -The practical difficulties are, however, of a far more serious -character than the theoretical ones. The length of a pendulum is not -the ordinary length of the instrument, which might be greatly varied -without affecting the duration of a vibration, but the distance from -the centre of suspension to the centre of oscillation. There are no -direct means of determining this latter centre, which depends upon -the average momentum of all the particles of the pendulum as regards -the centre of suspension. Huyghens discovered that the centres of -suspension and oscillation are interchangeable, and Kater pointed -out that if a pendulum vibrates with exactly the same rapidity when -suspended from two different points, the distance between these points -is the true length of the equivalent simple pendulum.[221] But the -practical difficulties in employing Kater’s reversible pendulum are -considerable, and questions regarding the disturbance of the air, the -force of gravity, or even the interference of electrical attractions -have to be entertained. It has been shown that all the experiments made -under the authority of Government for determining the ratio between -the standard yard and the seconds pendulum, were vitiated by an error -in the corrections for the resisting, adherent, or buoyant power of -the air in which the pendulums were swung. Even if such corrections -were rendered unnecessary by operating in a vacuum, other difficult -questions remain.[222] Gauss’ mode of comparing the vibrations of a -wire pendulum when suspended at two different lengths is open to equal -or greater practical difficulties. Thus it is found that the pendulum -standard cannot compete in accuracy and certainty with the simple bar -standard, and the method would only be useful as an accessory mode of -restoring the bar standard if at any time again destroyed. - - [221] Kater’s *Treatise on Mechanics*, Cabinet Cyclopædia, p. 154. - - [222] Grant’s *History of Physical Astronomy*, p. 156. - - -*Unit of Density.* - -Before we can measure the phenomena of nature, we require a third -independent unit, which shall enable us to define the quantity of -matter occupying any given space. All the changes of nature, as we -shall see, are probably so many manifestations of energy; but energy -requires some substratum or material machinery of molecules, in and by -which it may be manifested. Observation shows that, as regards force, -there may be two modes of variation of matter. As Newton says in the -first definition of the Principia, “the quantity of matter is the -measure of the same, arising from its density and bulk conjunctly.” -Thus the force required to set a body in motion varies both according -to the bulk of the matter, and also according to its quality. Two cubic -inches of iron of uniform quality, will require twice as much force -as one cubic inch to produce a certain velocity in a given time; but -one cubic inch of gold will require more force than one cubic inch of -iron. There is then some new measurable quality in matter apart from -its bulk, which we may call *density*, and which is, strictly speaking, -indicated by its capacity to resist and absorb the action of force. -For the unit of density we may assume that of any substance which is -uniform in quality, and can readily be referred to from time to time. -Pure water at any definite temperature, for instance that of snow -melting under inappreciable pressure, furnishes an invariable standard -of density, and by comparing equal bulks of various substances with -a like bulk of ice-cold water, as regards the velocity produced in a -unit of time by the same force, we should ascertain the densities of -those substances as expressed in that of water. Practically the force -of gravity is used to measure density; for a beautiful experiment with -the pendulum, performed by Newton and repeated by Gauss, shows that all -kinds of matter gravitate equally. Two portions of matter then which -are in equilibrium in the balance, may be assumed to possess equal -inertia, and their densities will therefore be inversely as their cubic -dimensions. - - -*Unit of Mass.* - -Multiplying the number of units of density of a portion of matter, -by the number of units of space occupied by it, we arrive at the -quantity of matter, or, as it is usually called, the *unit of mass*, as -indicated by the inertia and gravity it possesses. To proceed in the -most simple manner, the unit of mass ought to be that of a cubic unit -of matter of the standard density; but the founders of the metrical -system took as their unit of mass, the cubic centimetre of water, at -the temperature of maximum density (about 4° Cent.). They called this -unit of mass the *gramme*, and constructed standard specimens of the -kilogram, which might be readily referred to by all who required to -employ accurate weights. Unfortunately the determination of the bulk -of a given weight of water at a certain temperature is an operation -involving many difficulties, and it cannot be performed in the present -day with a greater exactness than that of about one part in 5000, the -results of careful observers being sometimes found to differ as much as -one part in 1000.[223] - - [223] Clerk Maxwell’s *Theory of Heat*, p. 79. - -Weights, on the other hand, can be compared with each other to at least -one part in a million. Hence if different specimens of the kilogram be -prepared by direct weighing against water, they will not agree closely -with each other; the two principal standard kilograms agree neither -with each other, nor with their definition. According to Professor -Miller the so-called Kilogramme des Archives weighs 15432·34874 grains, -while the kilogram deposited at the Ministry of the Interior in Paris, -as the standard for commercial purposes, weighs 15432·344 grains. Since -a standard weight constructed of platinum, or platinum and iridium, can -be preserved free from any appreciable alteration, and since it can be -very accurately compared with other weights, we shall ultimately attain -the greatest exactness in our measurements of mass, by assuming some -single kilogram as a *provisional standard*, leaving the determination -of its actual mass in units of space and density for future -investigation. This is what is practically done at the present day, -and thus a unit of mass takes the place of the unit of density, both -in the French and English systems. The English pound is defined by a -certain lump of platinum, preserved at Westminster, and is an arbitrary -mass, chosen merely that it may agree as nearly as possible with old -English pounds. The gallon, the old English unit of cubic measurement, -is defined by the condition that it shall contain exactly ten pounds -weight of water at 62° Fahr.; and although it is stated that it has the -capacity of about 277·274 cubic inches, this ratio between the cubic -and linear systems of measurement is not legally enacted, but left open -to investigation. While the French metric system as originally designed -was theoretically perfect, it does not differ practically in this point -from the English system. - - -*Natural System of Standards.* - -Quite recently Professor Clerk Maxwell has suggested that the -vibrations of light and the atoms of matter might conceivably be -employed as the ultimate standards of length, time, and mass. We -should thus arrive at a *natural system of standards*, which, -though possessing no present practical importance, has considerable -theoretical interest. “In the present state of science,” he says, “the -most universal standard of length which we could assume would be the -wave-length in vacuum of a particular kind of light, emitted by some -widely diffused substance such as sodium, which has well-defined lines -in its spectrum. Such a standard would be independent of any changes in -the dimensions of the earth, and should be adopted by those who expect -their writings to be more permanent than that body.”[224] In the same -way we should get a universal standard unit of time, independent of -all questions about the motion of material bodies, by taking as the -unit the periodic time of vibration of that particular kind of light -whose wave-length is the unit of length. It would follow that with -these units of length and time the unit of velocity would coincide with -the velocity of light in empty space. As regards the unit of mass, -Professor Maxwell, humorously as I should think, remarks that if we -expect soon to be able to determine the mass of a single molecule of -some standard substance, we may wait for this determination before -fixing a universal standard of mass. - - [224] *Treatise on Electricity and Magnetism*, vol. i. p. 3. - -In a theoretical point of view there can be no reasonable doubt that -vibrations of light are, as far as we can tell, the most fixed in -magnitude of all phenomena. There is as usual no certainty in the -matter, for the properties of the basis of light may vary to some -extent in different parts of space. But no differences could ever -be established in the velocity of light in different parts of the -solar system, and the spectra of the stars show that the times of -vibration there do not differ perceptibly from those in this part -of the universe. Thus all presumption is in favour of the absolute -constancy of the vibrations of light--absolute, that is, so far as -regards any means of investigation we are likely to possess. Nearly -the same considerations apply to the atomic weight as the standard of -mass. It is impossible to prove that all atoms of the same substance -are of equal mass, and some physicists think that they differ, so that -the fixity of combining proportions may be due only to the approximate -constancy of the mean of countless millions of discrepant weights. But -in any case the detection of difference is probably beyond our powers. -In a theoretical point of view, then, the magnitudes suggested by -Professor Maxwell seem to be the most fixed ones of which we have any -knowledge, so that they necessarily become the natural units. - -In a practical point of view, as Professor Maxwell would be the first -to point out, they are of little or no value, because in the present -state of science we cannot measure a vibration or weigh an atom with -any approach to the accuracy which is attainable in the comparison -of standard metres and kilograms. The velocity of light is not known -probably within a thousandth part, and as we progress in the knowledge -of light, so we shall progress in the accurate fixation of other -standards. All that can be said then, is that it is very desirable -to determine the wave-lengths and periods of the principal lines of -the solar spectrum, and the absolute atomic weights of the elements, -with all attainable accuracy, in terms of our existing standards. The -numbers thus obtained would admit of the reproduction of our standards -in some future age of the world to a corresponding degree of accuracy, -were there need of such reference; but so far as we can see at present, -there is no considerable probability that this mode of reproduction -would ever be the best mode. - - -*Subsidiary Units.* - -Having once established the standard units of time, space, and density -or mass, we might employ them for the expression of all quantities -of such nature. But it is often convenient in particular branches of -science to use multiples or submultiples of the original units, for the -expression of quantities in a simple manner. We use the mile rather -than the yard when treating of the magnitude of the globe, and the -mean distance of the earth and sun is not too large a unit when we -have to describe the distances of the stars. On the other hand, when -we are occupied with microscopic objects, the inch, the line or the -millimetre, become the most convenient terms of expression. - -It is allowable for a scientific man to introduce a new unit in any -branch of knowledge, provided that it assists precise expression, -and is carefully brought into relation with the primary units. Thus -Professor A. W. Williamson has proposed as a convenient unit of -volume in chemical science, an absolute volume equal to about 11·2 -litres representing the bulk of one gram of hydrogen gas at standard -temperature and pressure, or the *equivalent* weight of any other -gas, such as 16 grams of oxygen, 14 grams of nitrogen, &c.; in short, -the bulk of that quantity of any one of those gases which weighs as -many grams as there are units in the number expressing its atomic -weight.[225] Hofmann has proposed a new unit of weight for chemists, -called a *crith*, to be defined by the weight of one litre of hydrogen -gas at 0° C. and 0°·76 mm., weighing about 0·0896 gram.[226] Both of -these units must be regarded as purely subordinate units, ultimately -defined by reference to the primary units, and not involving any new -assumption. - - [225] *Chemistry for Students*, by A. W. Williamson. Clarendon Press - Series, 2nd ed. Preface p. vi. - - [226] *Introduction to Chemistry*, p. 131. - - -*Derived Units.* - -The standard units of time, space, and mass having been once fixed, -many kinds of magnitude are naturally measured by units derived from -them. From the metre, the unit of linear magnitude follows in the most -obvious manner the centiare or square metre, the unit of superficial -magnitude, and the litre that is the cube of the tenth part of a metre, -the unit of capacity or volume. Velocity of motion is expressed by the -ratio of the space passed over, when the motion is uniform, to the time -occupied; hence the unit of velocity is that of a body which passes -over a unit of space in a unit of time. In physical science the unit of -velocity might be taken as one metre per second. Momentum is measured -by the mass moving, regard being paid both to the amount of matter and -the velocity at which it is moving. Hence the unit of momentum will be -that of a unit volume of matter of the unit density moving with the -unit velocity, or in the French system, a cubic centimetre of water of -the maximum density moving one metre per second. - -An accelerating force is measured by the ratio of the momentum -generated to the time occupied, the force being supposed to act -uniformly. The unit of force will therefore be that which generates -a unit of momentum in a unit of time, or which causes, in the French -system, one cubic centimetre of water at maximum density to acquire in -one second a velocity of one metre per second. The force of gravity is -the most familiar kind of force, and as, when acting unimpeded upon any -substance, it produces in a second a velocity of 9·80868 . . metres per -second in Paris, it follows that the absolute unit of force is about -the tenth part of the force of gravity. If we employ British weights -and measures, the absolute unit of force is represented by the gravity -of about half an ounce, since the force of gravity of any portion of -matter acting upon that matter during one second, produces a final -velocity of 32·1889 feet per second or about 32 units of velocity. -Although from its perpetual action and approximate uniformity we find -in gravity the most convenient force for reference, and thus habitually -employ it to estimate quantities of matter, we must remember that it -is only one of many instances of force. Strictly speaking, we should -express weight in terms of force, but practically we express other -forces in terms of weight. - -We still require the unit of energy, a more complex notion. The -momentum of a body expresses the quantity of motion which belongs or -would belong to the aggregate of the particles; but when we consider -how this motion is related to the action of a force producing or -removing it, we find that the effect of a force is proportional to the -mass multiplied by the square of the velocity and it is convenient to -take half this product as the expression required. But it is shown in -books upon dynamics that it will be exactly the same thing if we define -energy by a force acting through a space. The natural unit of energy -will then be that which overcomes a unit of force acting through a unit -of space; when we lift one kilogram through one metre, against gravity, -we therefore accomplish 9·80868 . . units of work, that is, we turn so -many units of potential energy existing in the muscles, into potential -energy of gravitation. In lifting one pound through one foot there is -in like manner a conversion of 32·1889 units of energy. Accordingly the -unit of energy will be in the English system, that required to lift -one pound through about the thirty-second part of a foot; in terms of -metric units, it will be that required to lift a kilogram through about -one tenth part of a metre. - -Every person is at liberty to measure and record quantities in terms of -any unit which he likes. He may use the yard for linear measurement and -the litre for cubic measurement, only there will then be a complicated -relation between his different results. The system of derived units -which we have been briefly considering, is that which gives the most -simple and natural relations between quantitative expressions of -different kinds, and therefore conduces to ease of comprehension and -saving of laborious calculation. - -It would evidently be a source of great convenience if scientific men -could agree upon some single system of units, original and derived, in -terms of which all quantities could be expressed. Statements would thus -be rendered easily comparable, a large part of scientific literature -would be made intelligible to all, and the saving of mental labour -would be immense. It seems to be generally allowed, too, that the -metric system of weights and measures presents the best basis for the -ultimate system; it is thoroughly established in Western Europe; it is -legalised in England; it is already commonly employed by scientific -men; it is in itself the most simple and scientific of systems. There -is every reason then why the metric system should be accepted at least -in its main features. - - -*Provisional Units.* - -Ultimately, as we can hardly doubt, all phenomena will be recognised -as so many manifestations of energy; and, being expressed in terms of -the unit of energy, will be referable to the primary units of space, -time, and density. To effect this reduction, however, in any particular -case, we must not only be able to compare different quantities of -the phenomenon, but to trace the whole series of steps by which it -is connected with the primary notions. We can readily observe that -the intensity of one source of light is greater than that of another; -and, knowing that the intensity of light decreases as the square of -the distance increases, we can easily determine their comparative -brilliance. Hence we can express the intensity of light falling upon -any surface, if we have a unit in which to make the expression. Light -is undoubtedly one form of energy, and the unit ought therefore to be -the unit of energy. But at present it is quite impossible to say how -much energy there is in any particular amount of light. The question -then arises,--Are we to defer the measurement of light until we can -assign its relation to other forms of energy? If we answer Yes, it -is equivalent to saying that the science of light must stand still -perhaps for a generation; and not only this science but many others. -The true course evidently is to select, as the provisional unit of -light, some light of convenient intensity, which can be reproduced from -time to time in the same intensity, and which is defined by physical -circumstances. All the phenomena of light may be experimentally -investigated relatively to this unit, for instance that obtained after -much labour by Bunsen and Roscoe.[227] In after years it will become a -matter of inquiry what is the energy exerted in such unit of light; but -it may be long before the relation is exactly determined. - - [227] *Philosophical Transactions* (1859), vol. cxlix. p. 884, &c. - -A provisional unit, then, means one which is assumed and physically -defined in a safe and reproducible manner, in order that particular -quantities may be compared *inter se* more accurately than they can -yet be referred to the primary units. In reality the great majority -of our measurements are expressed in terms of such provisionally -independent units, and even the unit of mass, as we have seen, ought to -be considered as provisional. - -The unit of heat ought to be simply the unit of energy, already -described. But a weight can be measured to the one-millionth part, and -temperature to less than the thousandth part of a degree Fahrenheit, -and to less therefore than the five-hundred thousandth part of the -absolute temperature, whereas the mechanical equivalent of heat -is probably not known to the thousandth part. Hence the need of a -provisional unit of heat, which is often taken as that requisite to -raise one gram of water through one degree Centigrade, that is from -0° to 1°. This quantity of heat is capable of approximate expression -in terms of time, space, and mass; for by the natural constant, -determined by Dr. Joule, and called the mechanical equivalent of heat, -we know that the assumed unit of heat is equal to the energy of 423·55 -gram-metres, or that energy which will raise the mass of 423·55 grams -through one metre against 9·8... absolute units of force. Heat may also -be expressed in terms of the quantity of ice at 0° Cent., which it is -capable of converting into water under inappreciable pressure. - - -*Theory of Dimensions.* - -In order to understand the relations between the quantities dealt with -in physical science, it is necessary to pay attention to the Theory of -Dimensions, first clearly stated by Joseph Fourier,[228] but in later -years developed by several physicists. This theory investigates the -manner in which each derived unit depends upon or involves one or more -of the fundamental units. The number of units in a rectangular area -is found by multiplying together the numbers of units in the sides; -thus the unit of length enters twice into the unit of area, which is -therefore said to have two dimensions with respect to length. Denoting -length by *L*, we may say that the dimensions of area are *L* × *L* or -*L*^{2}. It is obvious in the same way that the dimensions of volume or -bulk will be *L*^{3}. - - [228] *Théorie Analytique de la Chaleur*, Paris; 1822, §§ 157–162. - -The number of units of mass in a body is found by multiplying the -number of units of volume, by those of density. Hence mass is of -three dimensions as regards length, and one as regards density. -Calling density *D*, the dimensions of mass are *L*^{3}*D*. As already -explained, however, it is usual to substitute an arbitrary provisional -unit of mass, symbolised by *M*; according to the view here taken we -may say that the dimensions of *M* are *L*^{3}*D*. - -Introducing time, denoted by *T*, it is easy to see that the dimensions -of velocity will be *L/T* or *LT*^{-1}, because the number of units -in the velocity of a body is found by *dividing* the units of length -passed over by the units of time occupied in passing. The acceleration -of a body is measured by the increase of velocity in relation to the -time, that is, we must divide the units of velocity gained by the units -of time occupied in gaining it; hence its dimensions will be *LT*^{-2}. -Momentum is the product of mass and velocity, so that its dimensions -are *MLT*^{-1}. The effect of a force is measured by the acceleration -produced in a unit of mass in a unit of time; hence the dimensions of -force are *MLT*^{-2}. Work done is proportional to the force acting and -to the space through which it acts; so that it has the dimensions of -force with that of length added, giving *ML*^{2}*T*^{-2}. - -It should be particularly noticed that angular magnitude has no -dimensions at all, being measured by the ratio of the arc to the radius -(p. 305). Thus we have the dimensions *LL*^{-1} or *L*^{0}. This -agrees with the statement previously made, that no arbitrary unit of -angular magnitude is needed. Similarly, all pure numbers expressing -ratios only, such as sines and other trigonometrical functions, -logarithms, exponents, &c., are devoid of dimensions. They are absolute -numbers necessarily expressed in terms of unity itself, and are quite -unaffected by the selection of the arbitrary physical units. Angular -magnitude, however, enters into other quantities, such as angular -velocity, which has the dimensions 1/*T* or *T*^{-1}, the units of -angle being divided by the units of time occupied. The dimensions of -angular acceleration are denoted by *T*^{-2}. - -The quantities treated in the theories of heat and electricity -are numerous and complicated as regards their dimensions. Thermal -capacity has the dimensions *ML*^{-3}, thermal conductivity, -*ML*^{-1}*T*^{-1}. In Magnetism the dimensions of the strength -of pole are *M*^{1/2}*L*^{3/2}*T*^{-1}, the dimensions of -field-intensity are *M*^{1/2}*L*^{-1/2}*T*^{-1}, and the intensity -of magnetisation has the same dimensions. In the science of -electricity physicists have to deal with numerous kinds of quantity, -and their dimensions are different too in the electro-static -and the electro-magnetic systems. Thus electro-motive force has -the dimensions *M*^{1/2}*L*^{1/2}*T*^{-1}, in the former, and -*M*^{1/2}*L*^{3/2}*T*^{-2} in the latter system. Capacity simply -depends upon length in electro-statics, but upon *L*^{-1}*T*^{2} in -electro-magnetics. It is worthy of particular notice that electrical -quantities have simple dimensions when expressed in terms of density -instead of mass. The instances now given are sufficient to show the -difficulty of conceiving and following out the relations of the -quantities treated in physical science without a systematic method of -calculating and exhibiting their dimensions. It is only in quite recent -years that clear ideas about these quantities have been attained. Half -a century ago probably no one but Fourier could have explained what -he meant by temperature or capacity for heat. The notion of measuring -electricity had hardly been entertained. - -Besides affording us a clear view of the complex relations of physical -quantities, this theory is specially useful in two ways. Firstly, it -affords a test of the correctness of mathematical reasoning. According -to the *Principle of Homogeneity*, all the quantities *added* together, -and equated in any equation, must have the same dimensions. Hence if, -on estimating the dimensions of the terms in any equation, they be not -homogeneous, some blunder must have been committed. It is impossible -to add a force to a velocity, or a mass to a momentum. Even if the -numerical values of the two members of a non-homogeneous equation were -equal, this would be accidental, and any alteration in the physical -units would produce inequality and disclose the falsity of the law -expressed in the equation. - -Secondly, the theory of units enables us readily and infallibly to -deduce the change in the numerical expression of any physical quantity, -produced by a change in the fundamental units. It is of course obvious -that in order to represent the same absolute quantity, a number must -vary inversely as the magnitude of the units which are numbered. The -yard expressed in feet is 3; taking the inch as the unit instead of -the foot it becomes 36. Every quantity into which the dimension length -enters positively must be altered in like manner. Changing the unit -from the foot to the inch, numerical expressions of volume must be -multiplied by 12 × 12 × 12. When a dimension enters negatively the -opposite rule will hold. If for the minute we substitute the second -as unit of time, then we must divide all numbers expressing angular -velocities by 60, and numbers expressing angular acceleration by -60 × 60. The rule is that a numerical expression varies inversely as -the magnitude of the unit as regards each whole dimension entering -positively, and it varies directly as the magnitude of the unit for -each whole dimension entering negatively. In the case of fractional -exponents, the proper root of the ratio of change has to be taken. - -The study of this subject may be continued in Professor J. D. Everett’s -“Illustrations of the Centimetre-gramme-second System of Units,” -published by Taylor and Francis, 1875; in Professor Maxwell’s “Theory -of Heat;” or Professor Fleeming Jenkin’s “Text Book of Electricity.” - - -*Natural Constants.* - -Having acquired accurate measuring instruments, and decided upon the -units in which the results shall be expressed, there remains the -question, What use shall be made of our powers of measurement? Our -principal object must be to discover general quantitative laws of -nature; but a very large amount of preliminary labour is employed in -the accurate determination of the dimensions of existing objects, and -the numerical relations between diverse forces and phenomena. Step -by step every part of the material universe is surveyed and brought -into known relations with other parts. Each manifestation of energy is -correlated with each other kind of manifestation. Professor Tyndall has -described the care with which such operations are conducted.[229] - - [229] Tyndall’s *Sound*, 1st ed. p. 26. - -“Those who are unacquainted with the details of scientific -investigation, have no idea of the amount of labour expended on -the determination of those numbers on which important calculations -or inferences depend. They have no idea of the patience shown by a -Berzelius in determining atomic weights; by a Regnault in determining -coefficients of expansion; or by a Joule in determining the mechanical -equivalent of heat. There is a morality brought to bear upon such -matters which, in point of severity, is probably without a parallel in -any other domain of intellectual action.” - -Every new natural constant which is recorded brings many fresh -inferences within our power. For if *n* be the number of such constants -known, then 1/2 (*n*^{2}--*n*) is the number of ratios which are within -our powers of calculation, and this increases with the square of *n*. -We thus gradually piece together a map of nature, in which the lines of -inference from one phenomenon to another rapidly grow in complexity, -and the powers of scientific prediction are correspondingly augmented. - -Babbage[230] proposed the formation of a collection of the constant -numbers of nature, a work which has at last been taken in hand by the -Smithsonian Institution.[231] It is true that a complete collection of -such numbers would be almost co-extensive with scientific literature, -since almost all the numbers occurring in works on chemistry, -mineralogy, physics, astronomy, &c., would have to be included. -Still a handy volume giving all the more important numbers and their -logarithms, referred when requisite to the different units in common -use, would be very useful. A small collection of constant numbers will -be found at the end of Babbage’s, Hutton’s, and many other tables of -logarithms, and a somewhat larger collection is given in Templeton’s -*Millwright and Engineer’s Pocket Companion*. - - [230] British Association, Cambridge, 1833. Report, pp. 484–490. - - [231] *Smithsonian Miscellaneous Collections*, vol. xii., the - Constants of Nature, part. i. Specific gravities compiled by F. W. - Clarke, 8vo. Washington, 1873. - -Our present object will be to classify these constant numbers roughly, -according to their comparative generality and importance, under the -following heads:-- - - (1) Mathematical constants. - (2) Physical constants. - (3) Astronomical constants. - (4) Terrestrial numbers. - (5) Organic numbers. - (6) Social numbers. - - -*Mathematical Constants.* - -At the head of the list of natural constants must come those which -express the necessary relations of numbers to each other. The ordinary -Multiplication Table is the most familiar and the most important of -such series of constants, and is, theoretically speaking, infinite in -extent. Next we must place the Arithmetical Triangle, the significance -of which has already been pointed out (p. 182). Tables of logarithms -also contain vast series of natural constants, arising out of the -relations of pure numbers. At the base of all logarithmic theory is -the mysterious natural constant commonly denoted by *e*, or ε, being -equal to the infinite series 1 + 1/1 + 1/1.2 + 1/1.2.3 + 1/1.2.3.4 -+...., and thus consisting of the sum of the ratios between the numbers -of permutations and combinations of 0, 1, 2, 3, 4, &c. things. Tables -of prime numbers and of the factors of composite numbers must not be -forgotten. - -Another vast and in fact infinite series of numerical constants -contains those connected with the measurement of angles, and embodied -in trigonometrical tables, whether as natural or logarithmic sines, -cosines, and tangents. It should never be forgotten that though these -numbers find their chief employment in connection with trigonometry, -or the measurement of the sides of a right-angled triangle, yet the -numbers themselves arise out of numerical relations bearing no special -relation to space. Foremost among trigonometrical constants is the -well known number π, usually employed as expressing the ratio of the -circumference and the diameter of a circle; from π follows the value of -the arcual or natural unit of angular value as expressed in ordinary -degrees (p. 306). - -Among other mathematical constants not uncommonly used may be mentioned -tables of factorials (p. 179), tables of Bernoulli’s numbers, tables of -the error function,[232] which latter are indispensable not only in the -theory of probability but also in several other branches of science. - - [232] J. W. L. Glaisher, *Philosophical Magazine*, 4th Series, - vol. xlii. p. 421. - -It should be clearly understood that the mathematical constants and -tables of reference already in our possession, although very extensive, -are only an infinitely small part of what might be formed. With the -progress of science the tabulation of new functions will be continually -demanded, and it is worthy of consideration whether public money -should not be available to reward the severe, long continued, and -generally thankless labour which must be gone through in calculating -tables. Such labours are a benefit to the whole human race as long as -it shall exist, though there are few who can appreciate the extent -of this benefit. A most interesting and excellent description of -many mathematical tables will be found in De Morgan’s article on -*Tables*, in the *English Cyclopædia*, Division of Arts and Sciences, -vol. vii. p. 976. An almost exhaustive critical catalogue of extant -tables is being published by a Committee of the British Association, -two portions, drawn up chiefly by Mr. J. W. L. Glaisher and Professor -Cayley, having appeared in the Reports of the Association for 1873 and -1875. - - -*Physical Constants.* - -The second class of constants contains those which refer to the -actual constitution of matter. For the most part they depend upon -the peculiarities of the chemical substance in question, but we may -begin with those which are of the most general character. In a first -sub-class we may place the velocity of light or heat undulations, the -numbers expressing the relation between the lengths of the undulations, -and the rapidity of the undulations, these numbers depending only on -the properties of the ethereal medium, and being probably the same in -all parts of the universe. The theory of heat gives rise to several -numbers of the highest importance, especially Joule’s mechanical -equivalent of heat, the absolute zero of temperature, the mean -temperature of empty space, &c. - -Taking into account the diverse properties of the elements we must -have tables of the atomic weights, the specific heats, the specific -gravities, the refractive powers, not only of the elements, but their -almost infinitely numerous compounds. The properties of hardness, -elasticity, viscosity, expansion by heat, conducting powers for heat -and electricity, must also be determined in immense detail. There are, -however, certain of these numbers which stand out prominently because -they serve as intermediate units or terms of comparison. Such are, for -instance, the absolute coefficients of expansion of air, water and -mercury, the temperature of the maximum density of water, the latent -heats of water and steam, the boiling-point of water under standard -pressure, the melting and boiling-points of mercury, and so forth. - - -*Astronomical Constants.* - -The third great class consists of numbers possessing far less -generality because they refer not to the properties of matter, but to -the special forms and distances in which matter has been disposed in -the part of the universe open to our examination. We have, first of -all, to define the magnitude and form of the earth, its mean density, -the constant of aberration of light expressing the relation between -the earth’s mean velocity in space and the velocity of light. From -the earth, as our observatory, we then proceed to lay down the mean -distances of the sun, and of the planets from the same centre; all the -elements of the planetary orbits, the magnitudes, densities, masses, -periods of axial rotation of the several planets are by degrees -determined with growing accuracy. The same labours must be gone through -for the satellites. Catalogues of comets with the elements of their -orbits, as far as ascertainable, must not be omitted. - -From the earth’s orbit as a new base of observations, we next proceed -to survey the heavens and lay down the apparent positions, magnitudes, -motions, distances, periods of variation, &c. of the stars. All -catalogues of stars from those of Hipparchus and Tycho, are full of -numbers expressing rudely the conformation of the visible universe. -But there is obviously no limit to the labours of astronomers; not -only are millions of distant stars awaiting their first measurements, -but those already registered require endless scrutiny as regards -their movements in the three dimensions of space, their periods of -revolution, their changes of brilliance and colour. It is obvious that -though astronomical numbers are conventionally called *constant*, they -are probably in all cases subject to more or less rapid variation. - - -*Terrestrial Numbers.* - -Our knowledge of the globe we inhabit involves many numerical -determinations, which have little or no connection with astronomical -theory. The extreme heights of the principal mountains, the mean -elevations of continents, the mean or extreme depths of the oceans, -the specific gravities of rocks, the temperature of mines, the host of -numbers expressing the meteorological or magnetic conditions of every -part of the surface, must fall into this class. Many such numbers -are not to be called constant, being subject to periodic or secular -changes, but they are hardly more variable in fact than some which in -astronomical science are set down as constant. In many cases quantities -which seem most variable may go through rhythmical changes resulting -in a nearly uniform average, and it is only in the long progress of -physical investigation that we can hope to discriminate successfully -between those elemental numbers which are fixed and those which vary. -In the latter case the law of variation becomes the constant relation -which is the object of our search. - -*Organic Numbers.* - -The forms and properties of brute nature having been sufficiently -defined by the previous classes of numbers, the organic world, both -vegetable and animal, remains outstanding, and offers a higher series -of phenomena for our investigation. All exact knowledge relating to -the forms and sizes of living things, their numbers, the quantities -of various compounds which they consume, contain, or excrete, their -muscular or nervous energy, &c. must be placed apart in a class by -themselves. All such numbers are doubtless more or less subject to -variation, and but in a minor degree capable of exact determination. -Man, so far as he is an animal, and as regards his physical form, must -also be treated in this class. - - -*Social Numbers.* - -Little allusion need be made in this work to the fact that man in -his economic, sanitary, intellectual, æsthetic, or moral relations -may become the subject of sciences, the highest and most useful of -all sciences. Every one who is engaged in statistical inquiry must -acknowledge the possibility of natural laws governing such statistical -facts. Hence we must allot a distinct place to numerical information -relating to the numbers, ages, physical and sanitary condition, -mortality, &c., of different peoples, in short, to vital statistics. -Economic statistics, comprehending the quantities of commodities -produced, existing, exchanged and consumed, constitute another -extensive body of science. In the progress of time exact investigation -may possibly subdue regions of phenomena which at present defy all -scientific treatment. That scientific method can ever exhaust the -phenomena of the human mind is incredible. - - - - -CHAPTER XV. - -ANALYSIS OF QUANTITATIVE PHENOMENA. - - -In the two preceding chapters we have been engaged in considering how -a phenomenon may be accurately measured and expressed. So delicate -and complex an operation is a measurement which pretends to any -considerable degree of exactness, that no small part of the skill -and patience of physicists is usually spent upon this work. Much -of this difficulty arises from the fact that it is scarcely ever -possible to measure a single effect at a time. The ultimate object -must be to discover the mathematical equation or law connecting a -quantitative cause with its quantitative effect; this purpose usually -involves, as we shall see, the varying of one condition at a time, -the other conditions being maintained constant. The labours of the -experimentalist would be comparatively light if he could carry out -this rule of varying one circumstance at a time. He would then obtain -a series of corresponding values of the variable quantities concerned, -from which he might by proper hypothetical treatment obtain the -required law of connection. But in reality it is seldom possible to -carry out this direction except in an approximate manner. Before then -we proceed to the consideration of the actual process of quantitative -induction, it is necessary to review the several devices by which a -complicated series of effects can be disentangled. Every phenomenon -measured will usually be the sum, difference, or it may be the product -or quotient, of two or more different effects, and these must be in -some way analysed and separately measured before we possess the -materials for inductive treatment. - - -*Illustrations of the Complication of Effects.* - -It is easy to bring forward a multitude of instances to show that a -phenomenon is seldom to be observed simple and alone. A more or less -elaborate process of analysis is almost always necessary. Thus if an -experimentalist wishes to observe and measure the expansion of a liquid -by heat, he places it in a thermometer tube and registers the rise of -the column of liquid in the narrow tube. But he cannot heat the liquid -without also heating the glass, so that the change observed is really -the difference between the expansions of the liquid and the glass. More -minute investigation will show the necessity perhaps of allowing for -further minute effects, namely the compression of the liquid and the -expansion of the bulb due to the increased pressure of the column as it -becomes lengthened. - -In a great many cases an observed effect will be apparently at least -the simple sum of two separate and independent effects. The heat -evolved in the combustion of oil is partly due to the carbon and partly -to the hydrogen. A measurement of the heat yielded by the two jointly, -cannot inform us how much proceeds from the one and how much from the -other. If by some separate determination we can ascertain how much the -hydrogen yields, then by mere subtraction we learn what is due to the -carbon; and *vice versâ*. The heat conveyed by a liquid, may be partly -conveyed by true conduction, partly by convection. The light dispersed -in the interior of a liquid consists both of what is reflected by -floating particles and what is due to true fluorescence;[233] and we -must find some mode of determining one portion before we can learn the -other. The apparent motion of the spots on the sun, is the algebraic -sum of the sun’s axial rotation, and of the proper motion of the spots -upon the sun’s surface; hence the difficulty of ascertaining by direct -observations the period of the sun’s rotation. - - [233] Stokes, *Philosophical Transactions* (1852), vol. cxlii. p. 529. - -We cannot obtain the weight of a portion of liquid in a chemical -balance without weighing it with the containing vessel. Hence to have -the real weight of the liquid operated upon in an experiment, we must -make a separate weighing of the vessel, with or without the adhering -film of liquid according to circumstances. This is likewise the mode -in which a cart and its load are weighed together, the *tare* of the -cart previously ascertained being deducted. The variation in the height -of the barometer is a joint effect, partly due to the real variation -of the atmospheric pressure, partly to the expansion of the mercurial -column by heat. The effects may be discriminated, if, instead of one -barometer tube we have two tubes containing mercury placed closely side -by side, so as to have the same temperature. If one of them be closed -at the bottom so as to be unaffected by the atmospheric pressure, it -will show the changes due to temperature only, and, by subtracting -these changes from those shown in the other tube, employed as a -barometer, we get the real oscillations of atmospheric pressure. But -this correction, as it is called, of the barometric reading, is better -effected by calculation from the readings of an ordinary thermometer. - -In other cases a quantitative effect will be the difference of -two causes acting in opposite directions. Sir John Herschel -invented an instrument like a large thermometer, which he called -the Actinometer,[234] and Pouillet constructed a somewhat similar -instrument called the Pyrheliometer, for ascertaining the heating power -of the sun’s rays. In both instruments the heat of the sun was absorbed -by a reservoir containing water, and the rise of temperature of the -water was exactly observed, either by its own expansion, or by the -readings of a delicate thermometer immersed in it. But in exposing the -actinometer to the sun, we do not obtain the full effect of the heat -absorbed, because the receiving surface is at the same time radiating -heat into empty space. The observed increment of temperature is in -short the difference between what is received from the sun and lost -by radiation. The latter quantity is capable of ready determination; -we have only to shade the instrument from the direct rays of the sun, -leaving it exposed to the sky, and we can observe how much it cools -in a certain time. The total effect of the sun’s rays will obviously -be the apparent effect *plus* the cooling effect in an equal time. By -alternate exposure in sun and shade during equal intervals the desired -result may be obtained with considerable accuracy.[235] - - [234] *Admiralty Manual of Scientific Enquiry*, 2nd ed. p. 299. - - [235] Pouillet, *Taylor’s Scientific Memoirs*, vol. iv. p. 45. - -Two quantitative effects were beautifully distinguished in an -experiment of John Canton, devised in 1761 for the purpose of -demonstrating the compressibility of water. He constructed a -thermometer with a large bulb full of water and a short capillary -tube, the part of which above the water was freed from air. Under -these circumstances the water was relieved from the pressure of the -atmosphere, but the glass bulb in bearing that pressure was somewhat -contracted. He next placed the instrument under the receiver of an -air-pump, and on exhausting the air, the water sank in the tube. Having -thus obtained a measure of the effect of atmospheric pressure on the -bulb, he opened the top of the thermometer tube and admitted the air. -The level of the water now sank still more, partly from the pressure -on the bulb being now compensated, and partly from the compression of -the water by the atmospheric pressure. It is obvious that the amount of -the latter effect was approximately the difference of the two observed -depressions. - -Not uncommonly the actual phenomenon which we wish to measure is -considerably less than various disturbing effects which enter into -the question. Thus the compressibility of mercury is considerably -less than the expansion of the vessels in which it is measured under -pressure, so that the attention of the experimentalist has chiefly -to be concentrated on the change of magnitude of the vessels. Many -astronomical phenomena, such as the parallax or the proper motions of -the fixed stars, are far less than the errors caused by instrumental -imperfections, or motions arising from precession, nutation, and -aberration. We need not be surprised that astronomers have from time to -time mistaken one phenomenon for another, as when Flamsteed imagined -that he had discovered the parallax of the Pole star.[236] - - [236] Baily’s *Account of the Rev. John Flamsteed*, p. 58. - - -*Methods of Eliminating Error.* - -In any particular experiment it is the object of the experimentalist to -measure a single effect only, and he endeavours to obtain that effect -free from interfering effects. If this cannot be, as it seldom or -never can really be, he makes the effect as considerable as possible -compared with the other effects, which he reduces to a minimum, and -treats as noxious errors. Those quantities, which are called *errors* -in one case, may really be most important and interesting phenomena in -another investigation. When we speak of eliminating error we really -mean disentangling the complicated phenomena of nature. The physicist -rightly wishes to treat one thing at a time, but as this object can -seldom be rigorously carried into practice, he has to seek some mode of -counteracting the irrelevant and interfering causes. - -The general principle is that a single observation can render known -only a single quantity. Hence, if several different quantitative -effects are known to enter into any investigation, we must have at -least as many distinct observations as there are quantities to be -determined. Every complete experiment will therefore consist in general -of several operations. Guided if possible by previous knowledge of the -causes in action, we must arrange the determinations, so that by a -simple mathematical process we may distinguish the separate quantities. -There appear to be five principal methods by which we may accomplish -this object; these methods are specified below and illustrated in the -succeeding sections. - -(1) *The Method of Avoidance.* The physicist may seek for some special -mode of experiment or opportunity of observation, in which the error is -non-existent or inappreciable. - -(2) *The Differential Method.* He may find opportunities of observation -when all interfering phenomena remain constant, and only the subject -of observation is at one time present and another time absent; the -difference between two observations then gives its amount. - -(3) *The Method of Correction.* He may endeavour to estimate the amount -of the interfering effect by the best available mode, and then make a -corresponding correction in the results of observation. - -(4) *The Method of Compensation.* He may invent some mode of -neutralising the interfering cause by balancing against it an exactly -equal and opposite cause of unknown amount. - -(5) *The Method of Reversal.* He may so conduct the experiment that -the interfering cause may act in opposite directions, in alternate -observations, the mean result being free from interference. - - -I. *Method of Avoidance of Error.* - -Astronomers seek opportunities of observation when errors will -be as small as possible. In spite of elaborate observations and -long-continued theoretical investigation, it is not practicable to -assign any satisfactory law to the refractive power of the atmosphere. -Although the apparent change of place of a heavenly body produced by -refraction may be more or less accurately calculated yet the error -depends upon the temperature and pressure of the atmosphere, and, when -a ray is highly inclined to the perpendicular, the uncertainty in the -refraction becomes very considerable. Hence astronomers always make -their observations, if possible, when the object is at the highest -point of its daily course, *i.e.* on the meridian. In some kinds of -investigation, as, for instance, in the determination of the latitude -of an observatory, the astronomer is at liberty to select one or -more stars out of the countless number visible. There is an evident -advantage in such a case, in selecting a star which passes close to -the zenith, so that it may be observed almost entirely free from -atmospheric refraction, as was done by Hooke. - -Astronomers endeavour to render their clocks as accurate as -possible, by removing the source of variation. The pendulum is -perfectly isochronous so long as its length remains invariable, and -the vibrations are exactly of equal length. They render it nearly -invariable in length, that is in the distance between the centres of -suspension and oscillation, by a compensatory arrangement for the -change of temperature. But as this compensation may not be perfectly -accomplished, some astronomers place their chief controlling clock in -a cellar, or other apartment, where the changes of temperature may -be as slight as possible. At the Paris Observatory a clock has been -placed in the caves beneath the building, where there is no appreciable -difference between the summer and winter temperature. - -To avoid the effect of unequal oscillations Huyghens made his beautiful -investigations, which resulted in the discovery that a pendulum, of -which the centre of oscillation moved upon a cycloidal path, would -be perfectly isochronous, whatever the variation in the length of -oscillations. But though a pendulum may be easily rendered in some -degree cycloidal by the use of a steel suspension spring, it is found -that the mechanical arrangements requisite to produce a truly cycloidal -motion introduce more error than they remove. Hence astronomers seek -to reduce the error to the smallest amount by maintaining their clock -pendulums in uniform movement; in fact, while a clock is in good order -and has the same weights, there need be little change in the length -of oscillation. When a pendulum cannot be made to swing uniformly, -as in experiments upon the force of gravity, it becomes requisite to -resort to the third method, and a correction is introduced, calculated -on theoretical grounds from the amount of the observed change in the -length of vibration. - -It has been mentioned that the apparent expansion of a liquid by heat, -when contained in a thermometer tube or other vessel, is the difference -between the real expansion of the liquid and that of the containing -vessel. The effects can be accurately distinguished provided that we -can learn the real expansion by heat of any one convenient liquid; -for by observing the apparent expansion of the same liquid in any -required vessel we can by difference learn the amount of expansion of -the vessel due to any given change of temperature. When we once know -the change of dimensions of the vessel, we can of course determine the -absolute expansion of any other liquid tested in it. Thus it became an -all-important object in scientific research to measure with accuracy -the absolute dilatation by heat of some one liquid, and mercury owing -to several circumstances was by far the most suitable. Dulong and -Petit devised a beautiful mode of effecting this by simply avoiding -altogether the effect of the change of size of the vessel. Two upright -tubes full of mercury were connected by a fine tube at the bottom, and -were maintained at two different temperatures. As mercury was free to -flow from one tube to the other by the connecting tube, the two columns -necessarily exerted equal pressures by the principles of hydrostatics. -Hence it was only necessary to measure very accurately by a -cathetometer the difference of level of the surfaces of the two columns -of mercury, to learn the difference of length of columns of equal -hydrostatic pressure, which at once gives the difference of density of -the mercury, and the dilatation by heat. The changes of dimension in -the containing tubes became a matter of entire indifference, and the -length of a column of mercury at different temperatures was measured -as easily as if it had formed a solid bar. The experiment was carried -out by Regnault with many improvements of detail, and the absolute -dilatation of mercury, at temperatures between 0° Cent. and 350°, was -determined almost as accurately as was needful.[237] - - [237] Jamin, *Cours de Physique*, vol. ii. pp. 15–28. - -The presence of a large and uncertain amount of error may render a -method of experiment valueless. Foucault devised a beautiful experiment -with the pendulum for demonstrating popularly the rotation of the -earth, but it could be of no use for measuring the rotation exactly. It -is impossible to make the pendulum swing in a perfect plane, and the -slightest lateral motion gives it an elliptic path with a progressive -motion of the axis of the ellipse, which disguises and often entirely -overpowers that due to the rotation of the earth.[238] - - [238] *Philosophical Magazine*, 1851, 4th Series, vol. ii. *passim*. - -Faraday’s laborious experiments on the relation of gravity and -electricity were much obstructed by the fact that it is impossible -to move a large weight of metal without generating currents of -electricity, either by friction or induction. To distinguish the -electricity, if any, directly due to the action of gravity from the -greater quantities indirectly produced was a problem of excessive -difficulty. Baily in his experiments on the density of the earth was -aware of the existence of inexplicable disturbances which have since -been referred with much probability to the action of electricity.[239] -The skill and ingenuity of the experimentalist are often exhausted -in trying to devise a form of apparatus in which such causes of error -shall be reduced to a minimum. - - [239] Hearn, *Philosophical Transactions*, 1847, vol. cxxxvii. - pp. 217–221. - -In some rudimentary experiments we wish merely to establish the -existence of a quantitative effect without precisely measuring its -amount; if there exist causes of error of which we can neither -render the amount known or inappreciable, the best way is to make -them all negative so that the quantitative effects will be less than -the truth rather than greater. Grove, for instance, in proving that -the magnetisation or demagnetisation of a piece of iron raises its -temperature, took care to maintain the electro-magnet by which the iron -was magnetised at a lower temperature than the iron, so that it would -cool rather than warm the iron by radiation or conduction.[240] - - [240] *The Correlation of Physical Forces*, 3rd ed. p. 159. - -Rumford’s celebrated experiment to prove that heat was generated out -of mechanical force in the boring of a cannon was subject to the -difficulty that heat might be brought to the cannon by conduction -from neighbouring bodies. It was an ingenious device of Davy to -produce friction by a piece of clock-work resting upon a block of -ice in an exhausted receiver; as the machine rose in temperature -above 32°, it was certain that no heat was received by conduction -from the support.[241] In many other experiments ice may be employed -to prevent the access of heat by conduction, and this device, first -put in practice by Murray,[242] is beautifully employed in Bunsen’s -calorimeter. - - [241] *Collected Works of Sir H. Davy*, vol. ii. pp. 12–14. *Elements - of Chemical Philosophy*, p. 94. - - [242] *Nicholson’s Journal*, vol. i. p. 241; quoted in *Treatise on - Heat*, Useful Knowledge Society, p. 24. - -To observe the true temperature of the air, though apparently so easy, -is really a very difficult matter, because the thermometer is sure to -be affected either by the sun’s rays, the radiation from neighbouring -objects, or the escape of heat into space. These sources of error are -too fluctuating to allow of correction, so that the only accurate -mode of procedure is that devised by Dr. Joule, of surrounding the -thermometer with a copper cylinder ingeniously adjusted to the -temperature of the air, as described by him, so that the effect of -radiation shall be nullified.[243] - - [243] Clerk Maxwell, *Theory of Heat*, p. 228. *Proceedings of the - Manchester Philosophical Society*, Nov. 26, 1867, vol. vii. p. 35. - -When the avoidance of error is not practicable, it will yet be -desirable to reduce the absolute amount of the interfering error as -much as possible before employing the succeeding methods to correct -the result. As a general rule we can determine a quantity with less -inaccuracy as it is smaller, so that if the error itself be small -the error in determining that error will be of a still lower order -of magnitude. But in some cases the absolute amount of an error is -of no consequence, as in the index error of a divided circle, or the -difference between a chronometer and astronomical time. Even the rate -at which a clock gains or loses is a matter of little importance -provided it remain constant, so that a sure calculation of its amount -can be made. - - -2. *Differential Method.* - -When we cannot avoid the existence of error, we can often resort -with success to the second mode by measuring phenomena under such -circumstances that the error shall remain very nearly the same in all -the observations, and neutralise itself as regards the purposes in -view. This mode is available whenever we want a difference between -quantities and not the absolute quantity of either. The determination -of the parallax of the fixed stars is exceedingly difficult, because -the amount of parallax is far less than most of the corrections for -atmospheric refraction, nutation, aberration, precession, instrumental -irregularities, &c., and can with difficulty be detected among these -phenomena of various magnitude. But, as Galileo long ago suggested, -all such difficulties would be avoided by the differential observation -of stars, which, though apparently close together, are really far -separated on the line of sight. Two such stars in close apparent -proximity will be subject to almost exactly equal errors, so that all -we need do is to observe the apparent change of place of the nearer -star as referred to the more distant one. A good telescope furnished -with an accurate micrometer is alone needed for the application of -the method. Huyghens appears to have been the first observer who -actually tried to employ the method practically, but it was not until -1835 that the improvement of telescopes and micrometers enabled -Struve to detect in this way the parallax of the star α Lyræ. It is -one of the many advantages of the observation of transits of Venus -for the determination of the solar parallax that the refraction of -the atmosphere affects in an exactly equal degree the planet and -the portion of the sun’s face over which it is passing. Thus the -observations are strictly of a differential nature. - -By the process of substitutive weighing it is possible to ascertain -the equality or inequality of two weights with almost perfect freedom -from error. If two weights A and B be placed in the scales of the -best balance we cannot be sure that the equilibrium of the beam -indicates exact equality, because the arms of the beam may be unequal -or unbalanced. But if we take B out and put another weight C in, and -equilibrium still exists, it is apparent that the same causes of -erroneous weighing exist in both cases, supposing that the balance has -not been disarranged; B then must be exactly equal to C, since it has -exactly the same effect under the same circumstances. In like manner it -is a general rule that, if by any uniform mechanical process we get a -copy of an object, it is unlikely that this copy will be precisely the -same as the original in magnitude and form, but two copies will equally -diverge from the original, and will therefore almost exactly resemble -each other. - -Leslie’s Differential Thermometer[244] was well adapted to the -experiments for which it was invented. Having two equal bulbs any -alteration in the temperature of the air will act equally by conduction -on each and produce no change in the indications of the instrument. -Only that radiant heat which is purposely thrown upon one of the bulbs -will produce any effect. This thermometer in short carries out the -principle of the differential method in a mechanical manner. - - [244] Leslie, *Inquiry into the Nature of Heat*, p. 10. - - -3. *Method of Correction.* - -Whenever the result of an experiment is affected by an interfering -cause to a calculable amount, it is sufficient to add or subtract this -amount. We are said to correct observations when we thus eliminate -what is due to extraneous causes, although of course we are only -separating the correct effects of several agents. The variation in the -height of the barometer is partly due to the change of temperature, -but since the coefficient of absolute dilatation of mercury has been -exactly determined, as already described (p. 341), we have only to make -calculations of a simple character, or, what is better still, tabulate -a series of such calculations for general use, and the correction for -temperature can be made with all desired accuracy. The height of the -mercury in the barometer is also affected by capillary attraction, -which depresses it by a constant amount depending mainly on the -diameter of the tube. The requisite corrections can be estimated with -accuracy sufficient for most purposes, more especially as we can check -the correctness of the reading of a barometer by comparison with a -standard barometer, and introduce if need be an index error including -both the error in the affixing of the scale and the effect due to -capillarity. But in constructing the standard barometer itself we must -take greater precautions; the capillary depression depends somewhat -upon the quality of the glass, the absence of air, and the perfect -cleanliness of the mercury, so that we cannot assign the exact amount -of the effect. Hence a standard barometer is constructed with a wide -tube, sometimes even an inch in diameter, so that the capillary effect -may be rendered almost zero.[245] Gay-Lussac made barometers in the -form of a uniform siphon tube, so that the capillary forces acting at -the upper and lower surfaces should balance and destroy each other; -but the method fails in practice because the lower surface, being -open to the air, becomes sullied and subject to a different force of -capillarity. - - [245] Jevons, Watts’ *Dictionary of Chemistry*, vol. i. pp. 513–515. - -In mechanical experiments friction is an interfering condition, and -drains away a portion of the energy intended to be operated upon in a -definite manner. We should of course reduce the friction in the first -place to the lowest possible amount, but as it cannot be altogether -prevented, and is not calculable with certainty from any general -laws, we must determine it separately for each apparatus by suitable -experiments. Thus Smeaton, in his admirable but almost forgotten -researches concerning water-wheels, eliminated friction in the most -simple manner by determining by trial what weight, acting by a cord -and roller upon his model water-wheel, would make it turn without -water as rapidly as the water made it turn. In short, he ascertained -what weight concurring with the water would exactly compensate for the -friction.[246] In Dr. Joule’s experiments to determine the mechanical -equivalent of heat by the condensation of air, a considerable amount -of heat was produced by friction of the condensing pump, and a small -portion by stirring the water employed to absorb the heat. This heat of -friction was measured by simply repeating the experiment in an exactly -similar manner except that no condensation was effected, and observing -the change of temperature then produced.[247] - - [246] *Philosophical Transactions*, vol. li. p. 100. - - [247] *Philosophical Magazine*, 3rd Series, vol. xxvi. p. 372. - -We may describe as *test experiments* any in which we perform -operations not intended to give the quantity of the principal -phenomenon, but some quantity which would otherwise remain as an -error in the result. Thus in astronomical observations almost every -instrumental error may be avoided by increasing the number of -observations and distributing them in such a manner as to produce in -the final mean as much error in one way as in the other. But there -is one source of error, first discovered by Maskelyne, which cannot -be thus avoided, because it affects all observations in the same -direction and to the same average amount, namely the Personal Error of -the observer or the inclination to record the passage of a star across -the wires of the telescope a little too soon or a little too late. -This personal error was first carefully described in the *Edinburgh -Journal of Science*, vol. i. p. 178. The difference between the -judgment of observers at the Greenwich Observatory usually varies from -1/100 to 1/3 of a second, and remains pretty constant for the same -observers.[248] One practised observer in Sir George Airy’s pendulum -experiments recorded all his time observations half a second too early -on the average as compared with the chief observer.[249] In some -observers it has amounted to seven or eight-tenths of a second.[250] -De Morgan appears to have entertained the opinion that this source of -error was essentially incapable of elimination or correction.[251] -But it seems clear, as I suggested without knowing what had been -done,[252] that this personal error might be determined absolutely with -any desirable degree of accuracy by test experiments, consisting in -making an artificial star move at a considerable distance and recording -by electricity the exact moment of its passage over the wire. This -method has in fact been successfully employed in Leyden, Paris, and -Neuchatel.[253] More recently, observers were trained for the Transit -of Venus Expeditions by means of a mechanical model representing the -motion of Venus over the sun, this model being placed at a little -distance and viewed through a telescope, so that differences in the -judgments of different observers would become apparent. It seems likely -that tests of this nature might be employed with advantage in other -cases. - - [248] *Greenwich Observations for* 1866, p. xlix. - - [249] *Philosophical Transactions*, 1856, p. 309. - - [250] Penny *Cyclopædia*, art. *Transit*, vol. xxv. pp. 129, 130. - - [251] Ibid. art. *Observation*, p. 390. - - [252] *Nature*, vol. i. p. 85. - - [253] *Nature*, vol. i. p 337. See references to the Memoirs - describing the method. - -Newton employed the pendulum for making experiments on the impact of -balls. Two balls were hung in contact, and one of them, being drawn -aside through a measured arc, was then allowed to strike the other, -the arcs of vibration giving sufficient data for calculating the -distribution of energy at the moment of impact. The resistance of the -air was an interfering cause which he estimated very simply by causing -one of the balls to make several complete vibrations without impact -and then marking the reduction in the lengths of the arcs, a proper -fraction of which reduction was added to each of the other arcs of -vibration when impact took place.[254] - - [254] *Principia*, Book I. Law III. Corollary VI. Scholium. Motte’s - translation, vol. i. p. 33. - -The exact definition of the standard of length is one of the most -important, as it is one of the most difficult questions in physical -science, and the different practice of different nations introduces -needless confusion. Were all standards constructed so as to give -the true length at a fixed uniform temperature, for instance the -freezing-point, then any two standards could be compared without the -interference of temperature by bringing them both to exactly the same -fixed temperature. Unfortunately the French metre was defined by a -bar of platinum at 0°C, while our yard was defined by a bronze bar at -62°F. It is quite impossible, then, to make a comparison of the yard -and metre without the introduction of a correction, either for the -expansion of platinum or bronze, or both. Bars of metal differ too so -much in their rates of expansion according to their molecular condition -that it is dangerous to infer from one bar to another. - -When we come to use instruments with great accuracy there are many -minute sources of error which must be guarded against. If a thermometer -has been graduated when perpendicular, it will read somewhat -differently when laid flat, as the pressure of a column of mercury is -removed from the bulb. The reading may also be somewhat altered if it -has recently been raised to a higher temperature than usual, if it be -placed under a vacuous receiver, or if the tube be unequally heated -as compared with the bulb. For these minute causes of error we may -have to introduce troublesome corrections, unless we adopt the simple -precaution of using the thermometer in circumstances of position, &c., -exactly similar to those in which it was graduated. There is no end to -the number of minute corrections which may ultimately be required. A -large number of experiments on gases, standard weights and measures, -&c., depend upon the height of the barometer; but when experiments -in different parts of the world are compared together we ought as a -further refinement to take into account the varying force of gravity, -which even between London and Paris makes a difference of ·008 inch of -mercury. - -The measurement of quantities of heat is a matter of great difficulty, -because there is no known substance impervious to heat, and the problem -is therefore as difficult as to measure liquids in porous vessels. -To determine the latent heat of steam we must condense a certain -amount of the steam in a known weight of water, and then observe the -rise of temperature of the water. But while we are carrying out the -experiment, part of the heat will escape by radiation and conduction -from the condensing vessel or calorimeter. We may indeed reduce the -loss of heat by using vessels with double sides and bright surfaces, -surrounded with swans-down wool or other non-conducting materials; and -we may also avoid raising the temperature of the water much above that -of the surrounding air. Yet we cannot by any such means render the -loss of heat inconsiderable. Rumford ingeniously proposed to reduce -the loss to zero by commencing the experiment when the temperature -of the calorimeter is as much below that of the air as it is at the -end of the experiment above it. Thus the vessel will first gain and -then lose by radiation and conduction, and these opposite errors will -approximately balance each other. But Regnault has shown that the loss -and gain do not proceed by exactly the same laws, so that in very -accurate investigations Rumford’s method is not sufficient. There -remains the method of correction which was beautifully carried out by -Regnault in his determination of the latent heat of steam. He employed -two calorimeters, made in exactly the same way and alternately used to -condense a certain amount of steam, so that while one was measuring -the latent heat, the other calorimeter was engaged in determining -the corrections to be applied, whether on account of radiation and -conduction from the vessel or on account of heat reaching the vessel by -means of the connecting pipes.[255] - - [255] Graham’s *Chemical Reports and Memoirs*, Cavendish Society, pp. - 247, 268, &c. - - -4. *Method of Compensation.* - -There are many cases in which a cause of error cannot conveniently be -rendered null, and is yet beyond the reach of the third method, that -of calculating the requisite correction from independent observations. -The magnitude of an error may be subject to continual variations, on -account of change of weather, or other fickle circumstances beyond -our control. It may either be impracticable to observe the variation -of those circumstances in sufficient detail, or, if observed, the -calculation of the amount of error may be subject to doubt. In these -cases, and only in these cases, it will be desirable to invent some -artificial mode of counterpoising the variable error against an equal -error subject to exactly the same variation. - -We cannot weigh an object with great accuracy unless we make a -correction for the weight of the air displaced by the object, and add -this to the apparent weight. In very accurate investigations relating -to standard weights, it is usual to note the barometer and thermometer -at the time of making a weighing, and, from the measured bulks of -the objects compared, to calculate the weight of air displaced; the -third method in fact is adopted. To make these calculations in the -frequent weighings requisite in chemical analysis would be exceedingly -laborious, hence the correction is usually neglected. But when the -chemist wishes to weigh gas contained in a large glass globe for the -purpose of determining its specific gravity, the correction becomes of -much importance. Hence chemists avoid at once the error, and the labour -of correcting it, by attaching to the opposite scale of the balance a -dummy sealed glass globe of equal capacity to that containing the gas -to be weighed, noting only the difference of weight when the operating -globe is full and empty. The correction, being the same for both -globes, may be entirely neglected.[256] - - [256] Regnault’s *Cours Elémentaire de Chimie*, 1851, vol i. p. 141. - -A device of nearly the same kind is employed in the construction of -galvanometers which measure the force of an electric current by the -deflection of a suspended magnetic needle. The resistance of the needle -is partly due to the directive influence of the earth’s magnetism, and -partly to the torsion of the thread. But the former force may often be -inconveniently great as well as troublesome to determine for different -inclinations. Hence it is customary to connect together two equally -magnetised needles, with their poles pointing in opposite directions, -one needle being within and another without the coil of wire. As -regards the earth’s magnetism, the needles are now *astatic* or -indifferent, the tendency of one needle towards the pole being balanced -by that of the other. - -An elegant instance of the elimination of a disturbing force by -compensation is found in Faraday’s researches upon the magnetism of -gases. To observe the magnetic attraction or repulsion of a gas seems -impossible unless we enclose the gas in an envelope, probably best made -of glass. But any such envelope is sure to be more or less affected -by the magnet, so that it becomes difficult to distinguish between -three forces which enter into the problem, namely, the magnetism of -the gas in question, that of the envelope, and that of the surrounding -atmospheric air. Faraday avoided all difficulties by employing two -equal and similar glass tubes connected together, and so suspended from -the arm of a torsion balance that the tubes were in similar parts of -the magnetic field. One tube being filled with nitrogen and the other -with oxygen, it was found that the oxygen seemed to be attracted and -the nitrogen repelled. The suspending thread of the balance was then -turned until the force of torsion restored the tubes to their original -places, where the magnetism of the tubes as well as that of the -surrounding air, being the same and in the opposite directions upon the -two tubes, could not produce any interference. The force required to -restore the tubes was measured by the amount of torsion of the thread, -and it indicated correctly the difference between the attractive powers -of oxygen and nitrogen. The oxygen was then withdrawn from one of the -tubes, and a second experiment made, so as to compare a vacuum with -nitrogen. No force was now required to maintain the tubes in their -places, so that nitrogen was found to be, approximately speaking, -indifferent to the magnet, that is, neither magnetic nor diamagnetic, -while oxygen was proved to be positively magnetic.[257] It required -the highest experimental skill on the part of Faraday and Tyndall, to -distinguish between what is apparent and real in magnetic attraction -and repulsion. - - [257] Tyndall’s *Faraday*, pp. 114, 115. - -Experience alone can finally decide when a compensating arrangement -is conducive to accuracy. As a general rule mechanical compensation -is the last resource, and in the more accurate observations it is -likely to introduce more uncertainty than it removes. A multitude -of instruments involving mechanical compensation have been devised, -but they are usually of an unscientific character,[258] because the -errors compensated can be more accurately determined and allowed for. -But there are exceptions to this rule, and it seems to be proved that -in the delicate and tiresome operation of measuring a base line, -invariable bars, compensated for expansion by heat, give the most -accurate results. This arises from the fact that it is very difficult -to determine accurately the temperature of the measuring bars under -varying conditions of weather and manipulation.[259] Again, the last -refinement in the measurement of time at Greenwich Observatory depends -upon mechanical compensation. Sir George Airy, observing that the -standard clock increased its losing rate 0·30 second for an increase of -one inch in atmospheric pressure, placed a magnet moved by a barometer -in such a position below the pendulum, as almost entirely to neutralise -this cause of irregularity. The thorough remedy, however, would be to -remove the cause of error altogether by placing the clock in a vacuous -case. - - [258] See, for instance, the Compensated Sympiesometer, - *Philosophical Magazine*, 4th Series, vol. xxxix. p. 371. - - [259] Grant, *History of Physical Astronomy*, pp. 146, 147. - -We thus see that the choice of one or other mode of eliminating an -error depends entirely upon circumstances and the object in view; but -we may safely lay down the following conclusions. First of all, seek -to avoid the source of error altogether if it can be conveniently -done; if not, make the experiment so that the error may be as small, -but more especially as constant, as possible. If the means are at hand -for determining its amount by calculation from other experiments and -principles of science, allow the error to exist and make a correction -in the result. If this cannot be accurately done or involves too -much labour for the purposes in view, then throw in a counteracting -error which shall as nearly as possible be of equal amount in all -circumstances with that to be eliminated. There yet remains, however, -one important method, that of Reversal, which will form an appropriate -transition to the succeeding chapters on the Method of Mean Results and -the Law of Error. - - -5. *Method of Reversal.* - -The fifth method of eliminating error is most potent and satisfactory -when it can be applied, but it requires that we shall be able to -reverse the apparatus and mode of procedure, so as to make the -interfering cause act alternately in opposite directions. If we can -get two experimental results, one of which is as much too great as -the other is too small, the error is equal to half the difference, -and the true result is the mean of the two apparent results. It is an -unavoidable defect of the chemical balance, for instance, that the -points of suspension of the pans cannot be fixed at exactly equal -distances from the centre of suspension of the beam. Hence two weights -which seem to balance each other will never be quite equal in reality. -The difference is detected by reversing the weights, and it may be -estimated by adding small weights to the deficient side to restore -equilibrium, and then taking as the true weight the geometric mean -of the two apparent weights of the same object. If the difference is -small, the arithmetic mean, that is half the sum, may be substituted -for the geometric mean, from which it will not appreciably differ. - -This method of reversal is most extensively employed in practical -astronomy. The apparent elevation of a heavenly body is observed by a -telescope moving upon a divided circle, upon which the inclination of -the telescope is read off. Now this reading will be erroneous if the -circle and the telescope have not accurately the same centre. But if we -read off at the same time both ends of the telescope, the one reading -will be about as much too small as the other is too great, and the -mean will be nearly free from error. In practice the observation is -differently conducted, but the principle is the same; the telescope is -fixed to the circle, which moves with it, and the angle through which -it moves is read off at three, six, or more points, disposed at equal -intervals round the circle. The older astronomers, down even to the -time of Flamsteed, were accustomed to use portions only of a divided -circle, generally quadrants, and Römer made a vast improvement when he -introduced the complete circle. - -The transit circle, employed to determine the meridian passage of -heavenly bodies, is so constructed that the telescope and the axis -bearing it, in fact the whole moving part of the instrument, can be -taken out of the bearing sockets and turned over, so that what was -formerly the western pivot becomes the eastern one, and *vice versâ*. -It is impossible that the instrument could have been so perfectly -constructed, mounted, and adjusted that the telescope should point -exactly to the meridian, but the effect of the reversal is that it will -point as much to the west in one position as it does to the east in the -other, and the mean result of observations in the two positions must be -free from such cause of error. - -The accuracy with which the inclination of the compass needle can be -determined depends almost entirely on the method of reversal. The -dip needle consists of a bar of magnetised steel, suspended somewhat -like the beam of a delicate balance on a slender axis passing through -the centre of gravity of the bar, so that it is at liberty to rest -in that exact degree of inclination in the magnetic meridian which -the magnetism of the earth induces. The inclination is read off -upon a vertical divided circle, but to avoid error arising from the -centring of the needle and circle, both ends are read, and the mean -of the results is taken. The whole instrument is now turned carefully -round through 180°, which causes the needle to assume a new position -relatively to the circle and gives two new readings, in which any error -due to the wrong position of the zero of the division will be reversed. -As the axis of the needle may not be exactly horizontal, it is now -reversed in the same manner as the transit instrument, the end of the -axis which formerly pointed east being made to point west, and a new -set of four readings is taken. - -Finally, error may arise from the axis not passing accurately through -the centre of gravity of the bar, and this error can only be detected -and eliminated on changing the magnetic poles of the bar by the -application of a strong magnet. The error is thus made to act in -opposite directions. To ensure all possible accuracy each reversal -ought to be combined with each other reversal, so that the needle will -be observed in eight different positions by sixteen readings, the mean -of the whole of which will give the required inclination free from all -eliminable errors.[260] - - [260] Quetelet, *Sur la Physique du Globe*, p. 174. Jamin, *Cours de - Physique*, vol. i. p. 504. - -There are certain cases in which a disturbing cause can with ease -be made to act in opposite directions, in alternate observations, -so that the mean of the results will be free from disturbance. Thus -in direct experiments upon the velocity of sound in passing through -the air between stations two or three miles apart, the wind is a -cause of error. It will be well, in the first place, to choose a -time for the experiment when the air is very nearly at rest, and the -disturbance slight, but if at the same moment signal sounds be made -at each station and observed at the other, two sounds will be passing -in opposite directions through the same body of air and the wind will -accelerate one sound almost exactly as it retards the other. Again, in -trigonometrical surveys the apparent height of a point will be affected -by atmospheric refraction and the curvature of the earth. But if in -the case of two points the apparent elevation of each as seen from -the other be observed, the corrections will be the same in amount, -but reversed in direction, and the mean between the two apparent -differences of altitude will give the true difference of level. - -In the next two chapters we really pursue the Method of Reversal into -more complicated applications. - - - - -CHAPTER XVI. - -THE METHOD OF MEANS. - - -All results of the measurement of continuous quantity can be only -approximately true. Were this assertion doubted, it could readily be -proved by direct experience. If any person, using an instrument of -the greatest precision, makes and registers successive observations -in an unbiassed manner, it will almost invariably be found that the -results differ from each other. When we operate with sufficient care -we cannot perform so simple an experiment as weighing an object in a -good balance without getting discrepant numbers. Only the rough and -careless experimenter will think that his observations agree, but -in reality he will be found to overlook the differences. The most -elaborate researches, such as those undertaken in connection with -standard weights and measures, always render it apparent that complete -coincidence is out of the question, and that the more accurate our -modes of observation are rendered, the more numerous are the sources -of minute error which become apparent. We may look upon the existence -of error in all measurements as the normal state of things. It is -absolutely impossible to eliminate separately the multitude of small -disturbing influences, except by balancing them off against each other. -Even in drawing a mean it is to be expected that we shall come near -the truth rather than exactly to it. In the measurement of continuous -quantity, absolute coincidence, if it seems to occur, must be only -apparent, and is no indication of precision. It is one of the most -embarrassing things we can meet when experimental results agree -too closely. Such coincidences should raise our suspicion that the -apparatus in use is in some way restricted in its operation, so as -not really to give the true result at all, or that the actual results -have not been faithfully recorded by the assistant in charge of the -apparatus. - -If then we cannot get twice over exactly the same result, the question -arises, How can we ever attain the truth or select the result which may -be supposed to approach most nearly to it? The quantity of a certain -phenomenon is expressed in several numbers which differ from each -other; no more than one of them at the most can be true, and it is more -probable that they are all false. It may be suggested, perhaps, that -the observer should select the one observation which he judged to be -the best made, and there will often doubtless be a feeling that one -or more results were satisfactory, and the others less trustworthy. -This seems to have been the course adopted by the early astronomers. -Flamsteed, when he had made several observations of a star, probably -chose in an arbitrary manner that which seemed to him nearest to the -truth.[261] - - [261] Baily’s *Account of Flamsteed*, p. 376. - -When Horrocks selected for his estimate of the sun’s semi-diameter a -mean between the results of Kepler and Tycho, he professed not to do -it from any regard to the idle adage, “Medio tutissimus ibis,” but -because he thought it from his own observations to be correct.[262] But -this method will not apply at all when the observer has made a number -of measurements which are equally good in his opinion, and it is quite -apparent that in using an instrument or apparatus of considerable -complication the observer will not necessarily be able to judge whether -slight causes have affected its operation or not. - - [262] *The Transit of Venus across the Sun*, by Horrocks, London, - 1859, p. 146. - -In this question, as indeed throughout inductive logic, we deal only -with probabilities. There is no infallible mode of arriving at the -absolute truth, which lies beyond the reach of human intellect, and -can only be the distant object of our long-continued and painful -approximations. Nevertheless there is a mode pointed out alike by -common sense and the highest mathematical reasoning, which is more -likely than any other, as a general rule, to bring us near the truth. -The ἄριστον μέτρον, or the *aurea mediocritas*, was highly esteemed -in the ancient philosophy of Greece and Rome; but it is not probable -that any of the ancients should have been able clearly to analyse and -express the reasons why they advocated the *mean* as the safest course. -But in the last two centuries this apparently simple question of the -mean has been found to afford a field for the exercise of the utmost -mathematical skill. Roger Cotes, the editor of the *Principia*, appears -to have had some insight into the value of the mean; but profound -mathematicians such as De Moivre, Daniel Bernoulli, Laplace, Lagrange, -Gauss, Quetelet, De Morgan, Airy, Leslie Ellis, Boole, Glaisher, and -others, have hardly exhausted the subject. - - -*Several uses of the Mean Result.* - -The elimination of errors of unknown sources, is almost always -accomplished by the simple arithmetical process of taking the *mean*, -or, as it is often called, the *average* of several discrepant numbers. -To take an average is to add the several quantities together, and -divide by the number of quantities thus added, which gives a quotient -lying among, or in the *middle* of, the several quantities. Before -however inquiring fully into the grounds of this procedure, it is -essential to observe that this one arithmetical process is really -applied in at least three different cases, for different purposes, and -upon different principles, and we must take great care not to confuse -one application of the process with another. A *mean result*, then, may -have any one of the following significations. - -(1) It may give a merely representative number, expressing the general -magnitude of a series of quantities, and serving as a convenient mode -of comparing them with other series of quantities. Such a number is -properly called *The fictitious mean* or *The average result*. - -(2) It may give a result approximately free from disturbing quantities, -which are known to affect some results in one direction, and other -results equally in the opposite direction. We may say that in this case -we get a *Precise mean result*. - -(3) It may give a result more or less free from unknown and uncertain -errors; this we may call the *Probable mean result*. - -Of these three uses of the mean the first is entirely different in -nature from the two last, since it does not yield an approximation -to any natural quantity, but furnishes us with an arithmetic result -comparing the aggregate of certain quantities with their number. The -third use of the mean rests entirely upon the theory of probability, -and will be more fully considered in a later part of this chapter. The -second use is closely connected, or even identical with, the Method of -Reversal already described, but it will be desirable to enter somewhat -fully into all the three employments of the same arithmetical process. - - -*The Mean and the Average.* - -Much confusion exists in the popular, or even the scientific employment -of the terms *mean* and *average*, and they are commonly taken as -synonymous. It is necessary to ascertain carefully what significations -we ought to attach to them. The English word *mean* is equivalent to -*medium*, being derived, perhaps through the French *moyen*, from the -Latin *medius*, which again is undoubtedly kindred with the Greek -μεσος. Etymologists believe, too, that this Greek word is connected -with the preposition μετα, the German *mitte*, and the true English -*mid* or *middle*; so that after all the *mean* is a technical term -identical in its root with the more popular equivalent *middle*. - -If we inquire what is the mean in a mathematical point of view, the -true answer is that there are several or many kinds of means. The old -arithmeticians recognised ten kinds, which are stated by Boethius, and -an eleventh was added by Jordanus.[263] - - [263] De Morgan, Supplement to the *Penny Cyclopædia*, art. *Old - Appellations of Numbers*. - -The *arithmetic mean* is the one by far the most commonly denoted by -the term, and that which we may understand it to signify in the absence -of any qualification. It is the sum of a series of quantities divided -by their number, and may be represented by the formula 1/2(*a* + *b*). -But there is also the *geometric mean*, which is the square root of the -product, √(*a* × *b*), or that quantity the logarithm of which -is the arithmetic mean of the logarithms of the quantities. There is -also the *harmonic mean*, which is the reciprocal of the arithmetic -mean of the reciprocals of the quantities. Thus if *a* and *b* be the -quantities, as before, their reciprocals are 1/*a* and 1/*b*, the -mean of which is 1/2 (1/*a* + 1/*b*), and the reciprocal again is -(2*ab*)/(*a* + *b*), which is the harmonic mean. Other kinds of means -might no doubt be invented for particular purposes, and we might apply -the term, as De Morgan pointed out,[264] to any quantity a function of -which is equal to a function of two or more other quantities, and is -such that the interchange of these latter quantities among themselves -will make no alteration in the value of the function. Symbolically, if -Φ(*y*, *y*, *y* ....) = Φ(*x*_{1}, *x*_{2}, *x*_{3} ....), then *y* is -a kind of mean of the quantities, *x*_{1}, *x*_{2}, &c. - - [264] *Penny Cyclopædia*, art. *Mean*. - -The geometric mean is necessarily adopted in certain cases. When we -estimate the work done against a force which varies inversely as -the square of the distance from a fixed point, the mean force is -the geometric mean between the forces at the beginning and end of -the path. When in an imperfect balance, we reverse the weights to -eliminate error, the true weight will be the geometric mean of the two -apparent weights. In almost all the calculations of statistics and -commerce the geometric mean ought, strictly speaking, to be used. If a -commodity rises in price 100 per cent. and another remains unaltered, -the mean rise of a price is not 50 per cent. because the ratio -150 : 200 is not the same as 100 : 150. The mean ratio is as unity to -√(1·00 × 2·00) or 1 to 1·41. The difference between the three -kinds of means in such a case[265] is very considerable; while the -rise of price estimated by the Arithmetic mean would be 50 per cent. -it would be only 41 and 33 per cent. respectively according to the -Geometric and Harmonic means. - - [265] Jevons, *Journal of the Statistical Society*, June 1865, - vol. xxviii, p. 296. - -In all calculations concerning the average rate of progress of a -community, or any of its operations, the geometric mean should be -employed. For if a quantity increases 100 per cent. in 100 years, it -would not on the average increase 10 per cent. in each ten years, as -the 10 per cent. would at the end of each decade be calculated upon -larger and larger quantities, and give at the end of 100 years much -more than 100 per cent., in fact as much as 159 per cent. The true mean -rate in each decade would be ^{10}√2 or about 1·07, that is, the -increase would be about 7 per cent. in each ten years. But when the -quantities differ very little, the arithmetic and geometric means are -approximately the same. Thus the arithmetic mean of 1·000 and 1·001 -is 1·0005, and the geometric mean is about 1·0004998, the difference -being of an order inappreciable in almost all scientific and practical -matters. Even in the comparison of standard weights by Gauss’ method -of reversal, the arithmetic mean may usually be substituted for the -geometric mean which is the true result. - -Regarding the mean in the absence of express qualification to the -contrary as the common arithmetic mean, we must still distinguish -between its two uses where it gives with more or less accuracy and -probability a really existing quantity, and where it acts as a mere -representative of other quantities. If I make many experiments to -determine the atomic weight of an element, there is a certain number -which I wish to approximate to, and the mean of my separate results -will, in the absence of any reasons to the contrary, be the most -probable approximate result. When we determine the mean density of -the earth, it is not because any part of the earth is of that exact -density; there may be no part exactly corresponding to the mean -density, and as the crust of the earth has only about half the mean -density, the internal matter of the globe must of course be above the -mean. Even the density of a homogeneous substance like carbon or gold -must be regarded as a mean between the real density of its atoms, and -the zero density of the intervening vacuous space. - -The very different signification of the word “mean” in these two uses -was fully explained by Quetelet,[266] and the importance of the -distinction was pointed out by Sir John Herschel in reviewing his -work.[267] It is much to be desired that scientific men would mark the -difference by using the word *mean* only in the former sense when it -denotes approximation to a definite existing quantity; and *average*, -when the mean is only a fictitious quantity, used for convenience -of thought and expression. The etymology of this word “average” is -somewhat obscure; but according to De Morgan[268] it comes from -*averia*, “havings or possessions,” especially applied to farm stock. -By the accidents of language *averagium* came to mean the labour of -farm horses to which the lord was entitled, and it probably acquired in -this manner the notion of distributing a whole into parts, a sense in -which it was early applied to maritime averages or contributions of the -other owners of cargo to those whose goods have been thrown overboard -or used for the safety of the vessel. - - [266] *Letters on the Theory of Probabilities*, transl. by Downes, - Part ii. - - [267] Herschel’s *Essays*, &c. pp. 404, 405. - - [268] *On the Theory of Errors of Observations, Cambridge - Philosophical Transactions*, vol. x. Part ii. 416. - - -*On the Average or Fictitious Mean.* - -Although the average when employed in its proper sense of a fictitious -mean, represents no really existing quantity, it is yet of the -highest scientific importance, as enabling us to conceive in a single -result a multitude of details. It enables us to make a hypothetical -simplification of a problem, and avoid complexity without committing -error. The weight of a body is the sum of the weights of infinitely -small particles, each acting at a different place, so that a mechanical -problem resolves itself, strictly speaking, into an infinite number -of distinct problems. We owe to Archimedes the first introduction of -the beautiful idea that one point may be discovered in a gravitating -body such that the weight of all the particles may be regarded as -concentrated in that point, and yet the behaviour of the whole body -will be exactly represented by the behaviour of this heavy point. This -Centre of Gravity may be within the body, as in the case of a sphere, -or it may be in empty space, as in the case of a ring. Any two bodies, -whether connected or separate, may be conceived as having a centre of -gravity, that of the sun and earth lying within the sun and only 267 -miles from its centre. - -Although we most commonly use the notion of a centre or average point -with regard to gravity, the same notion is applicable to other cases. -Terrestrial gravity is a case of approximately parallel forces, and -the centre of gravity is but a special case of the more general Centre -of Parallel Forces. Wherever a number of forces of whatever amount -act in parallel lines, it is possible to discover a point at which -the algebraic sum of the forces may be imagined to act with exactly -the same effect. Water in a cistern presses against the side with a -pressure varying according to the depth, but always in a direction -perpendicular to the side. We may then conceive the whole pressure -as exerted on one point, which will be one-third from the bottom of -the cistern, and may be called the Centre of Pressure. The Centre -of Oscillation of a pendulum, discovered by Huyghens, is that point -at which the whole weight of the pendulum may be considered as -concentrated, without altering the time of oscillation (p. 315). When -one body strikes another the Centre of Percussion is that point in -the striking body at which all its mass might be concentrated without -altering the effect of the stroke. In position the Centre of Percussion -does not differ from the Centre of Oscillation. Mathematicians have -also described the Centre of Gyration, the Centre of Conversion, the -Centre of Friction, &c. - -We ought carefully to distinguish between those cases in which an -*invariable* centre can be assigned, and those in which it cannot. -In perfect strictness, there is no such thing as a true invariable -centre of gravity. As a general rule a body is capable of possessing an -invariable centre only for perfectly parallel forces, and gravity never -does act in absolutely parallel lines. Thus, as usual, we find that our -conceptions are only hypothetically correct, and only approximately -applicable to real circumstances. There are indeed certain geometrical -forms called *Centrobaric*,[269] such that a body of that shape would -attract another exactly as if the mass were concentrated at the centre -of gravity, whether the forces act in a parallel manner or not. -Newton showed that uniform spheres of matter have this property, -and this truth proved of the greatest importance in simplifying his -calculations. But it is after all a purely hypothetical truth, because -we can nowhere meet with, nor can we construct, a perfectly spherical -and homogeneous body. The slightest irregularity or protrusion from -the surface will destroy the rigorous correctness of the assumption. -The spheroid, on the other hand, has no invariable centre at which its -mass may always be regarded as concentrated. The point from which its -resultant attraction acts will move about according to the distance -and position of the other attracting body, and it will only coincide -with the centre as regards an infinitely distant body whose attractive -forces may be considered as acting in parallel lines. - - [269] Thomson and Tait, *Treatise on Natural Philosophy*, vol. i. - p. 394. - -Physicists speak familiarly of the poles of a magnet, and the term -may be used with convenience. But, if we attach any definite meaning -to the word, the poles are not the ends of the magnet, nor any fixed -points within, but the variable points from which the resultants of -all the forces exerted by the particles in the bar upon exterior -magnetic particles may be considered as acting. The poles are, in -short, Centres of Magnetic Forces; but as those forces are never -really parallel, these centres will vary in position according to -the relative place of the object attracted. Only when we regard the -magnet as attracting a very distant, or, strictly speaking, infinitely -distant particle, do its centres become fixed points, situated in short -magnets approximately at one-sixth of the whole length from each end of -the bar. We have in the above instances of centres or poles of force -sufficient examples of the mode in which the Fictitious Mean or Average -is employed in physical science. - - -*The Precise Mean Result.* - -We now turn to that mode of employing the mean result which is -analogous to the method of reversal, but which is brought into practice -in a most extensive manner throughout many branches of physical -science. We find the simplest possible case in the determination of -the latitude of a place by observations of the Pole-star. Tycho Brahe -suggested that if the elevation of any circumpolar star were observed -at its higher and lower passages across the meridian, half the sum of -the elevations would be the latitude of the place, which is equal to -the height of the pole. Such a star is as much above the pole at its -highest passage, as it is below at its lowest, so that the mean must -necessarily give the height of the pole itself free from doubt, except -as regards incidental errors. The Pole-star is usually selected for the -purpose of such observations because it describes the smallest circle, -and is thus on the whole least affected by atmospheric refraction. - -Whenever several causes are in action, each of which at one time -increases and at another time decreases the joint effect by equal -quantities, we may apply this method and disentangle the effects. Thus -the solar and lunar tides roll on in almost complete independence of -each other. When the moon is new or full the solar tide coincides, or -nearly so, with that caused by the moon, and the joint effect is the -sum of the separate effects. When the moon is in quadrature, or half -full, the two tides are acting in opposition, one raising and the other -depressing the water, so that we observe only the difference of the -effects. We have in fact-- - - Spring tide = lunar tide + solar tide; - Neap tide = lunar tide - solar tide. - -We have only then to add together the heights of the maximum spring -tide and the minimum neap tide, and half the sum is the true height of -the lunar tide. Half the difference of the spring and neap tides on the -other hand gives the solar tide. - -Effects of very small amount may be detected with great approach to -certainty among much greater fluctuations, provided that we have -a series of observations sufficiently numerous and long continued -to enable us to balance all the larger effects against each other. -For this purpose the observations should be continued over at least -one complete cycle, in which the effects run through all their -variations, and return exactly to the same relative positions as at the -commencement. If casual or irregular disturbing causes exist, we should -probably require many such cycles of results to render their effect -inappreciable. We obtain the desired result by taking the mean of all -the observations in which a cause acts positively, and the mean of all -in which it acts negatively. Half the difference of these means will -give the effect of the cause in question, provided that no other effect -happens to vary in the same period or nearly so. - -Since the moon causes a movement of the ocean, it is evident that -its attraction must have some effect upon the atmosphere. The laws -of atmospheric tides were investigated by Laplace, but as it would -be impracticable by theory to calculate their amounts we can only -determine them by observation, as Laplace predicted that they would -one day be determined.[270] But the oscillations of the barometer thus -caused are far smaller than the oscillations due to several other -causes. Storms, hurricanes, or changes of weather produce movements -of the barometer sometimes as much as a thousand times as great as -the tides in question. There are also regular daily, yearly, or other -fluctuations, all greater than the desired quantity. To detect and -measure the atmospheric tide it was desirable that observations should -be made in a place as free as possible from irregular disturbances. -On this account several long series of observations were made at St. -Helena, where the barometer is far more regular in its movements than -in a continental climate. The effect of the moon’s attraction was -then detected by taking the mean of all the readings when the moon -was on the meridian and the similar mean when she was on the horizon. -The difference of these means was found to be only ·00365, yet it was -possible to discover even the variation of this tide according as the -moon was nearer to or further from the earth, though this difference -was only ·00056 inch.[271] It is quite evident that such minute effects -could never be discovered in a purely empirical manner. Having no -information but the series of observations before us, we could have -no clue as to the mode of grouping them which would give so small a -difference. In applying this method of means in an extensive manner -we must generally then have *à priori* knowledge as to the periods at -which a cause will act in one direction or the other. - - [270] *Essai Philosophique sur les Probabilités*, pp. 49, 50. - - [271] Grant, *History of Physical Astronomy*, p. 163. - -We are sometimes able to eliminate fluctuations and take a mean result -by purely mechanical arrangements. The daily variations of temperature, -for instance, become imperceptible one or two feet below the surface -of the earth, so that a thermometer placed with its bulb at that -depth gives very nearly the true daily mean temperature. At a depth -of twenty feet even the yearly fluctuations are nearly effaced, and -the thermometer stands a little above the true mean temperature of the -locality. In registering the rise and fall of the tide by a tide-gauge, -it is desirable to avoid the oscillations arising from surface waves, -which is very readily accomplished by placing the float in a cistern -communicating by a small hole with the sea. Only a general rise or -fall of the level is then perceptible, just as in the marine barometer -the narrow tube prevents any casual fluctuations and allows only a -continued change of pressure to manifest itself. - - -*Determination of the Zero point.* - -In many important observations the chief difficulty consists in -defining exactly the zero point from which we are to measure. We can -point a telescope with great precision to a star and can measure to -a second of arc the angle through which the telescope is raised or -lowered; but all this precision will be useless unless we know exactly -the centre point of the heavens from which we measure, or, what comes -to the same thing, the horizontal line 90° distant from it. Since the -true horizon has reference to the figure of the earth at the place of -observation, we can only determine it by the direction of gravity, -as marked either by the plumb-line or the surface of a liquid. The -question resolves itself then into the most accurate mode of observing -the direction of gravity, and as the plumb-line has long been found -hopelessly inaccurate, astronomers generally employ the surface of -mercury in repose as the criterion of horizontality. They ingeniously -observe the direction of the surface by making a star the index. From -the laws of reflection it follows that the angle between the direct -ray from a star and that reflected from a surface of mercury will -be exactly double the angle between the surface and the direct ray -from the star. Hence the horizontal or zero point is the mean between -the apparent place of any star or other very distant object and its -reflection in mercury. - -A plumb-line is perpendicular, or a liquid surface is horizontal only -in an approximate sense; for any irregularity of the surface of the -earth, a mountain, or even a house must cause some deviation by its -attracting power. To detect such deviation might seem very difficult, -because every other plumb-line or liquid surface would be equally -affected by gravity. Nevertheless it can be detected; for if we place -one plumb-line to the north of a mountain, and another to the south, -they will be about equally deflected in opposite directions, and if -by observations of the same star we can measure the angle between the -plumb-lines, half the inclination will be the deviation of either, -after allowance has been made for the inclination due to the difference -of latitude of the two places of observation. By this mode of -observation applied to the mountain Schiehallion the deviation of the -plumb-line was accurately measured by Maskelyne, and thus a comparison -instituted between the attractive forces of the mountain and the whole -globe, which led to a probable estimate of the earth’s density. - -In some cases it is actually better to determine the zero point by the -average of equally diverging quantities than by direct observation. In -delicate weighings by a chemical balance it is requisite to ascertain -exactly the point at which the beam comes to rest, and when standard -weights are being compared the position of the beam is ascertained -by a carefully divided scale viewed through a microscope. But when -the beam is just coming to rest, friction, small impediments or other -accidental causes may readily obstruct it, because it is near the -point at which the force of stability becomes infinitely small. Hence -it is found better to let the beam vibrate and observe the terminal -points of the vibrations. The mean between two extreme points will -nearly indicate the position of rest. Friction and the resistance of -air tend to reduce the vibrations, so that this mean will be erroneous -by half the amount of this effect during a half vibration. But by -taking several observations we may determine this retardation and -allow for it. Thus if *a*, *b*, *c* be the readings of the terminal -points of three excursions of the beam from the zero of the scale, -then 1/2(*a* + *b*) will be about as much erroneous in one direction -as 1/2(*b* + *c*) in the other, so that the mean of these two means, -or 1/4(*a* + 2*b* + *c*), will be exceedingly near to the point of -rest.[272] A still closer approximation may be made by taking four -readings and reducing them by the formula 1/6(*a* + 2*b* + 2*c* + *d*). - - [272] Gauss, Taylor’s *Scientific Memoirs*, vol. ii. p. 43, &c. - -The accuracy of Baily’s experiments, directed to determine the -density of the earth, entirely depended upon this mode of observing -oscillations. The balls whose gravitation was measured were so -delicately suspended by a torsion balance that they never came to -rest. The extreme points of the oscillations were observed both when -the heavy leaden attracting ball was on one side and on the other. The -difference of the mean points when the leaden ball was on the right -hand and that when it was on the left hand gave double the amount of -the deflection. - -A beautiful instance of avoiding the use of a zero point is found in -Mr. E. J. Stone’s observations on the radiant heat of the fixed stars. -The difficulty of these observations arose from the comparatively great -amounts of heat which were sent into the telescope from the atmosphere, -and which were sufficient to disguise almost entirely the feeble heat -rays of a star. But Mr. Stone fixed at the focus of his telescope a -double thermo-electric pile of which the two parts were reversed in -order. Now any disturbance of temperature which acted uniformly upon -both piles produced no effect upon the galvanometer needle, and when -the rays of the star were made to fall alternately upon one pile and -the other, the total amount of the deflection represented double the -heating power of the star. Thus Mr. Stone was able to detect with -much certainty a heating effect of the star Arcturus, which even when -concentrated by the telescope amounted only to 0°·02 Fahr., and which -represents a heating effect of the direct ray of only about 0°·00000137 -Fahr., equivalent to the heat which would be received from a three-inch -cubic vessel full of boiling water at the distance of 400 yards.[273] -It is probable that Mr. Stone’s arrangement of the pile might be -usefully employed in other delicate thermometric experiments subject to -considerable disturbing influences. - - [273] *Proceedings of the Royal Society*, vol. xviii. p. 159 (Jan. - 13, 1870). *Philosophical Magazine* (4th Series), vol. xxxix. p. 376. - - -*Determination of Maximum Points.* - -We employ the method of means in a certain number of observations -directed to determine the moment at which a phenomenon reaches its -highest point in quantity. In noting the place of a fixed star at a -given time there is no difficulty in ascertaining the point to be -observed, for a star in a good telescope presents an exceedingly small -disc. In observing a nebulous body which from a bright centre fades -gradually away on all sides, it will not be possible to select with -certainty the middle point. In many such cases the best method is not -to select arbitrarily the supposed middle point, but points of equal -brightness on either side, and then take the mean of the observations -of these two points for the centre. As a general rule, a variable -quantity in reaching its maximum increases at a less and less rate, -and after passing the highest point begins to decrease by insensible -degrees. The maximum may indeed be defined as that point at which -the increase or decrease is null. Hence it will usually be the most -indefinite point, and if we can accurately measure the phenomenon we -shall best determine the place of the maximum by determining points on -either side at which the ordinates are equal. There is moreover this -advantage in the method that several points may be determined with the -corresponding ones on the other side, and the mean of the whole taken -as the true place of the maximum. But this method entirely depends upon -the existence of symmetry in the curve, so that of two equal ordinates -one shall be as far on one side of the maximum as the other is on the -other side. The method fails when other laws of variation prevail. - -In tidal observations great difficulty is encountered in fixing -the moment of high water, because the rate at which the water is -then rising or falling, is almost imperceptible. Whewell proposed, -therefore, to note the time at which the water passes a fixed point -somewhat below the maximum both in rising and falling, and take -the mean time as that of high water. But this mode of proceeding -unfortunately does not give a correct result, because the tide follows -different laws in rising and in falling. There is a difficulty again in -selecting the highest spring tide, another object of much importance in -tidology. Laplace discovered that the tide of the second day preceding -the conjunction of the sun and moon is nearly equal to that of the -fifth day following; and, believing that the increase and decrease of -the tides proceeded in a nearly symmetrical manner, he decided that the -highest tide would occur about thirty-six hours after the conjunction, -that is half-way between the second day before and the fifth day -after.[274] - - [274] Airy *On Tides and Waves*, Encycl. Metrop. pp. 364*-366*. - -This method is also employed in determining the time of passage of the -middle or densest point of a stream of meteors. The earth takes two -or three days in passing completely through the November stream; but -astronomers need for their calculations to have some definite point -fixed within a few minutes if possible. When near to the middle they -observe the numbers of meteors which come within the sphere of vision -in each half hour, or quarter hour, and then, assuming that the law of -variation is symmetrical, they select a moment for the passage of the -centre equidistant between times of equal frequency. - -The eclipses of Jupiter’s satellites are not only of great interest as -regards the motions of the satellites themselves, but were, and perhaps -still are, of use in determining longitudes, because they are events -occurring at fixed moments of absolute time, and visible in all parts -of the planetary system at the same time, allowance being made for the -interval occupied by the light in travelling. But, as is explained by -Herschel,[275] the moment of the event is wanting in definiteness, -partly because the long cone of Jupiter’s shadow is surrounded by -a penumbra, and partly because the satellite has itself a sensible -disc, and takes time in entering the shadow. Different observers using -different telescopes would usually select different moments for that -of the eclipse. But the increase of light in the emersion will proceed -according to a law the reverse of that observed in the immersion, -so that if an observer notes the time of both events with the same -telescope, he will be as much too soon in one observation as he is too -late in the other, and the mean moment of the two observations will -represent with considerable accuracy the time when the satellite is in -the middle of the shadow. Error of judgment of the observer is thus -eliminated, provided that he takes care to act at the emersion as he -did at the immersion. - - [275] *Outlines of Astronomy*, 4th edition, § 538. - - - - -CHAPTER XVII. - -THE LAW OF ERROR. - - -To bring error itself under law might seem beyond human power. He who -errs surely diverges from law, and it might be deemed hopeless out of -error to draw truth. One of the most remarkable achievements of the -human intellect is the establishment of a general theory which not -only enables us among discrepant results to approximate to the truth, -but to assign the degree of probability which fairly attaches to this -conclusion. It would be a mistake indeed to suppose that this law is -necessarily the best guide under all circumstances. Every measuring -instrument and every form of experiment may have its own special law -of error; there may in one instrument be a tendency in one direction -and in another in the opposite direction. Every process has its -peculiar liabilities to disturbance, and we are never relieved from the -necessity of providing against special difficulties. The general Law of -Error is the best guide only when we have exhausted all other means of -approximation, and still find discrepancies, which are due to unknown -causes. We must treat such residual differences in some way or other, -since they will occur in all accurate experiments, and as their origin -is assumed to be unknown, there is no reason why we should treat them -differently in different cases. Accordingly the ultimate Law of Error -must be a uniform and general one. - -It is perfectly recognised by mathematicians that in each case a -special Law of Error may exist, and should be discovered if possible. -“Nothing can be more unlikely than that the errors committed in all -classes of observations should follow the same law,”[276] and the -special Laws of Error which will apply to certain instruments, as for -instance the repeating circle, have been investigated by Bravais.[277] -He concludes that every distinct cause of error gives rise to a curve -of possibility of errors, which may have any form,--a curve which -we may either be able or unable to discover, and which in the first -case may be determined by *à priori* considerations on the peculiar -nature of this cause, or which may be determined *à posteriori* by -observation. Whenever it is practicable and worth the labour, we ought -to investigate these special conditions of error; nevertheless, when -there are a great number of different sources of minute error, the -general resultant will always tend to obey that general law which we -are about to consider. - - [276] *Philosophical Magazine*, 3rd Series, vol. xxxvii. p. 324. - - [277] *Letters on the Theory of Probabilities*, by Quetelet, - translated by O. G. Downes, Notes to Letter XXVI. pp. 286–295. - - -*Establishment of the Law of Error.* - -Mathematicians agree far better as to the form of the Law of Error -than they do as to the manner in which it can be deduced and proved. -They agree that among a number of discrepant results of observation, -that mean quantity is probably the best approximation to the truth -which makes the sum of the squares of the errors as small as possible. -But there are three principal ways in which this law has been arrived -at respectively by Gauss, by Laplace and Quetelet, and by Sir John -Herschel. Gauss proceeds much upon assumption; Herschel rests upon -geometrical considerations; while Laplace and Quetelet regard the Law -of Error as a development of the doctrine of combinations. A number -of other mathematicians, such as Adrain of New Brunswick, Bessel, -Ivory, Donkin, Leslie Ellis, Tait, and Crofton have either attempted -independent proofs or have modified or commented on those here to be -described. For full accounts of the literature of the subject the -reader should refer either to Mr. Todhunter’s *History of the Theory of -Probability* or to the able memoir of Mr. J. W. L. Glaisher.[278] - - [278] *On the Law of Facility of Errors of Observations, and on the - Method of Least Squares*, Memoirs of the Royal Astronomical Society, - vol. xxxix. p. 75. - -According to Gauss the Law of Error expresses the comparative -probability of errors of various magnitude, and partly from experience, -partly from *à priori* considerations, we may readily lay down certain -conditions to which the law will certainly conform. It may fairly -be assumed as a first principle to guide us in the selection of the -law, that large errors will be far less frequent and probable than -small ones. We know that very large errors are almost impossible, so -that the probability must rapidly decrease as the amount of the error -increases. A second principle is that positive and negative errors -shall be equally probable, which may certainly be assumed, because we -are supposed to be devoid of any knowledge as to the causes of the -residual errors. It follows that the probability of the error must be a -function of an even power of the magnitude, that is of the square, or -the fourth power, or the sixth power, otherwise the probability of the -same amount of error would vary according as the error was positive or -negative. The even powers *x*^{2}, *x*^{4}, *x*^{6}, &c., are always -intrinsically positive, whether *x* be positive or negative. There is -no *à priori* reason why one rather than another of these even powers -should be selected. Gauss himself allows that the fourth or sixth power -would fulfil the conditions as well as the second;[279] but in the -absence of any theoretical reasons we should prefer the second power, -because it leads to formulæ of great comparative simplicity. Did the -Law of Error necessitate the use of the higher powers of the error, the -complexity of the necessary calculations would much reduce the utility -of the theory. - - [279] *Méthode des Moindres Carrés. Mémoires sur la Combinaison - des Observations, par Ch. Fr. Gauss. Traduit en Français par J. - Bertrand*, Paris, 1855, pp. 6, 133, &c. - -By mathematical reasoning which it would be undesirable to attempt -to follow in this book, it is shown that under these conditions, the -facility of occurrence, or in other, words, the probability of error -is expressed by a function of the general form ε^{–*h*^{2} *x*^{2}}, in -which *x* represents the variable amount of errors. From this law, to -be more fully described in the following sections, it at once follows -that the most probable result of any observations is that which makes -the sum of the squares of the consequent errors the least possible. -Let *a*, *b*, *c*, &c., be the results of observation, and *x* the -quantity selected as the most probable, that is the most free from -unknown errors: then we must determine *x* so that (*a* - *x*)^{2} + -(*b* - *x*)^{2} + (*c* - *x*)^{2} + ... shall be the least possible -quantity. Thus we arrive at the celebrated *Method of Least Squares*, -as it is usually called, which appears to have been first distinctly -put in practice by Gauss in 1795, while Legendre first published in -1806 an account of the process in his work, entitled, *Nouvelles -Méthodes pour la Détermination des Orbites des Comètes*. It is worthy -of notice, however, that Roger Cotes had long previously recommended a -method of equivalent nature in his tract, “Estimatio Erroris in Mixta -Mathesi.”[280] - - [280] De Morgan, *Penny Cyclopædia*, art. *Least Squares*. - - -*Herschel’s Geometrical Proof.* - -A second way of arriving at the Law of Error was proposed by Herschel, -and although only applicable to geometrical cases, it is remarkable as -showing that from whatever point of view we regard the subject, the -same principle will be detected. After assuming that some general law -must exist, and that it is subject to the principles of probability, -he supposes that a ball is dropped from a high point with the -intention that it shall strike a given mark on a horizontal plane. In -the absence of any known causes of deviation it will either strike -that mark, or, as is infinitely more probable, diverge from it by an -amount which we must regard as error of unknown origin. Now, to quote -the words of Herschel,[281] “the probability of that error is the -unknown function of its square, *i.e.* of the sum of the squares of -its deviations in any two rectangular directions. Now, the probability -of any deviation depending solely on its magnitude, and not on its -direction, it follows that the probability of each of these rectangular -deviations must be the same function of *its* square. And since the -observed oblique deviation is equivalent to the two rectangular ones, -supposed concurrent, and which are essentially independent of one -another, and is, therefore, a compound event of which they are the -simple independent constituents, therefore its probability will be the -product of their separate probabilities. Thus the form of our unknown -function comes to be determined from this condition, viz., that the -product of such functions of two independent elements is equal to the -same function of their sum. But it is shown in every work on algebra -that this property is the peculiar characteristic of, and belongs only -to, the exponential or antilogarithmic function. This, then, is the -function of the square of the error, which expresses the probability -of committing that error. That probability decreases, therefore, in -geometrical progression, as the square of the error increases in -arithmetical.” - - [281] *Edinburgh Review*, July 1850, vol. xcii. p. 17. Reprinted - *Essays*, p. 399. This method of demonstration is discussed by Boole, - *Transactions of Royal Society of Edinburgh*, vol. xxi. pp. 627–630. - - -*Laplace’s and Quetelet’s Proof of the Law.* - -However much presumption the modes of determining the Law of Error, -already described, may give in favour of the law usually adopted, it is -difficult to feel that the arguments are satisfactory. The law adopted -is chosen rather on the grounds of convenience and plausibility, than -because it can be seen to be the necessary law. We can however approach -the subject from an entirely different point of view, and yet get to -the same result. - -Let us assume that a particular observation is subject to four chances -of error, each of which will increase the result one inch if it occurs. -Each of these errors is to be regarded as an event independent of the -rest and we can therefore assign, by the theory of probability, the -comparative probability and frequency of each conjunction of errors. -From the Arithmetical Triangle (pp. 182–188) we learn that no error -at all can happen only in one way; an error of one inch can happen -in 4 ways; and the ways of happening of errors of 2, 3 and 4 inches -respectively, will be 6, 4 and 1 in number. - -We may infer that the error of two inches is the most likely to occur, -and will occur in the long run in six cases out of sixteen. Errors -of one and three inches will be equally likely, but will occur less -frequently; while no error at all, or one of four inches will be a -comparatively rare occurrence. If we now suppose the errors to act -as often in one direction as the other, the effect will be to alter -the average error by the amount of two inches, and we shall have the -following results:-- - - Negative error of 2 inches 1 way. - Negative error of 1 inch 4 ways. - No error at all 6 ways. - Positive error of 1 inch 4 ways. - Positive error of 2 inches 1 way. - -We may now imagine the number of causes of error increased and the -amount of each error decreased, and the arithmetical triangle will -give us the frequency of the resulting errors. Thus if there be five -positive causes of error and five negative causes, the following -table shows the numbers of errors of various amount which will be the -result:-- - - +----------------------+-------------------+---+-------------------+ - | Direction of Error. | Positive Error. | | Negative Error. | - +----------------------+-------------------+---+-------------------+ - | Amount of Error. |5, 4, 3, 2, 1| 0 | 1, 2, 3, 4, 5| - +----------------------+-------------------+---+-------------------+ - |Number of such Errors.|1, 10, 45, 120, 210|252|210, 120, 45, 10, 1| - +----------------------+-------------------+---+-------------------+ - -It is plain that from such numbers I can ascertain the probability -of any particular amount of error under the conditions supposed. The -probability of a positive error of exactly one inch is 210/1024, in -which fraction the numerator is the number of combinations giving -one inch positive error, and the denominator the whole number of -possible errors of all magnitudes. I can also, by adding together the -appropriate numbers get the probability of an error not exceeding a -certain amount. Thus the probability of an error of three inches or -less, positive or negative, is a fraction whose numerator is the sum of -45 + 120 + 210 + 252 + 210 + 120 + 45, and the denominator, as before, -giving the result 1002/1024. We may see at once that, according to -these principles, the probability of small errors is far greater than -of large ones: the odds are 1002 to 22, or more than 45 to 1, that the -error will not exceed three inches; and the odds are 1022 to 2 against -the occurrence of the greatest possible error of five inches. - -If any case should arise in which the observer knows the number and -magnitude of the chief errors which may occur, he ought certainly to -calculate from the Arithmetical Triangle the special Law of Error -which would apply. But the general law, of which we are in search, -is to be used in the dark, when we have no knowledge whatever of the -sources of error. To assume any special number of causes of error -is then an arbitrary proceeding, and mathematicians have chosen the -least arbitrary course of imagining the existence of an infinite -number of infinitely small errors, just as, in the inverse method of -probabilities, an infinite number of infinitely improbable hypotheses -were submitted to calculation (p. 255). - -The reasons in favour of this choice are of several different kinds. - -1. It cannot be denied that there may exist infinitely numerous causes -of error in any act of observation. - -2. The law resulting from the hypothesis of a moderate number of causes -of error, does not appreciably differ from that given by the hypothesis -of an infinite number of causes of error. - -3. We gain by the hypothesis of infinity a general law capable of ready -calculation, and applicable by uniform rules to all problems. - -4. This law, when tested by comparison with extensive series of -observations, is strikingly verified, as will be shown in a later -section. - -When we imagine the existence of any large number of causes of -error, for instance one hundred, the numbers of combinations become -impracticably large, as may be seen to be the case from a glance at -the Arithmetical Triangle, which proceeds only up to the seventeenth -line. Quetelet, by suitable abbreviating processes, calculated out -a table of probability of errors on the hypothesis of one thousand -distinct causes;[282] but mathematicians have generally proceeded on -the hypothesis of infinity, and then, by the devices of analysis, -have substituted a general law of easy treatment. In mathematical -works upon the subject, it is shown that the standard Law of Error is -expressed in the formula - - *y* = *Y*ε^{-*cx*^{2}}, - - [282] *Letters on the Theory of Probabilities*, Letter XV. and - Appendix, note pp. 256–266. - -in which *x* is the amount of the error, *Y* the maximum ordinate -of the curve of error, and *c* a number constant for each series of -observations, and expressing the amount of the tendency to error, -varying between one series of observations and another. The letter ε -is the mathematical constant, the sum of ratios between the numbers of -permutations and combinations, previously referred to (p. 330). - -[Illustration] - -To show the close correspondence of this general law with the special -law which might be derived from the supposition of a moderate number -of causes of error, I have in the accompanying figure drawn a curved -line representing accurately the variation of *y* when *x* in the above -formula is taken equal 0, 1/2, 1, 3/2, 2, &c., positive or negative, -the arbitrary quantities *Y* and *c* being each assumed equal to unity, -in order to simplify the calculations. In the same figure are inserted -eleven dots, whose heights above the base line are proportional to -the numbers in the eleventh line of the Arithmetical Triangle, thus -representing the comparative probabilities of errors of various amounts -arising from ten equal causes of error. The correspondence of the -general and the special Law of Error is almost as close as can be -exhibited in the figure, and the assumption of a greater number of -equal causes of error would render the correspondence far more close. - -It may be explained that the ordinates NM, *nm*, *n′m′*, represent -values of *y* in the equation expressing the Law of Error. The -occurrence of any one definite amount of error is infinitely -improbable, because an infinite number of such ordinates might be -drawn. But the probability of an error occurring between certain limits -is finite, and is represented by a portion of the *area* of the curve. -Thus the probability that an error, positive or negative, not exceeding -unity will occur, is represented by the area M*mnn′m′*, in short, by -the area standing upon the line *nn′*. Since every observation must -either have some definite error or none at all, it follows that the -whole area of the curve should be considered as the unit expressing -certainty, and the probability of an error falling between particular -limits will then be expressed by the ratio which the area of the curve -between those limits bears to the whole area of the curve. - -The mere fact that the Law of Error allows of the possible existence of -errors of every assignable amount shows that it is only approximately -true. We may fairly say that in measuring a mile it would be impossible -to commit an error of a hundred miles, and the length of life -would never allow of our committing an error of one million miles. -Nevertheless the general Law of Error would assign a probability for -an error of that amount or more, but so small a probability as to be -utterly inconsiderable and almost inconceivable. All that can, or in -fact need, be said in defence of the law is, that it may be made to -represent the errors in any special case to a very close approximation, -and that the probability of large and practically impossible errors, as -given by the law, will be so small as to be entirely inconsiderable. -And as we are dealing with error itself, and our results pretend to -nothing more than approximation and probability, an indefinitely small -error in our process of approximation is of no importance whatever. - - -*Logical Origin of the Law of Error.* - -It is worthy of notice that this Law of Error, abstruse though the -subject may seem, is really founded upon the simplest principles. -It arises entirely out of the difference between permutations and -combinations, a subject upon which I may seem to have dwelt with -unnecessary prolixity in previous pages (pp. 170, 189). The order in -which we add quantities together does not affect the amount of the sum, -so that if there be three positive and five negative causes of error -in operation, it does not matter in which order they are considered as -acting. They may be intermixed in any arrangement, and yet the result -will be the same. The reader should not fail to notice how laws or -principles which appeared to be absurdly simple and evident when first -noticed, reappear in the most complicated and mysterious processes of -scientific method. The fundamental Laws of Identity and Difference gave -rise to the Logical Alphabet which, after abstracting the character of -the differences, led to the Arithmetical Triangle. The Law of Error is -defined by an infinitely high line of that triangle, and the law proves -that the mean is the most probable result, and that divergencies from -the mean become much less probable as they increase in amount. Now -the comparative greatness of the numbers towards the middle of each -line of the Arithmetical Triangle is entirely due to the indifference -of order in space or time, which was first prominently pointed out -as a condition of logical relations, and the symbols indicating them -(pp. 32–35), and which was afterwards shown to attach equally to -numerical symbols, the derivatives of logical terms (p. 160). - - -*Verification of the Law of Error.* - -The theory of error which we have been considering rests entirely -upon an assumption, namely that when known sources of disturbances -are allowed for, there yet remain an indefinite, possibly an infinite -number of other minute sources of error, which will as often produce -excess as deficiency. Granting this assumption, the Law of Error must -be as it is usually taken to be, and there is no more need to verify -it empirically than to test the truth of one of Euclid’s propositions -mechanically. Nevertheless, it is an interesting occupation to verify -even the propositions of geometry, and it is still more instructive to -try whether a large number of observations will justify our assumption -of the Law of Error. - -Encke has given an excellent instance of the correspondence of theory -with experience, in the case of observations of the differences of -Right Ascension of the sun and two stars, namely α Aquilæ and α Canis -minoris. The observations were 470 in number, and were made by Bradley -and reduced by Bessel, who found the probable error of the final result -to be only about one-fourth part of a second (0·2637). He then compared -the numbers of errors of each magnitude from 0·1 second upwards, as -actually given by the observations, with what should occur according to -the Law of Error. - -The results were as follow:--[283] - - +-------------------------+--------------------------+ - | | Number of errors of each | - | Magnitude of the errors | magnitude according to | - | in parts of a second. +-------------+------------+ - | | Observation.| Theory. | - +-------------------------+-------------+------------+ - | 0·0 to 0·1 | 94 | 95 | - | ·1 " ·2 | 88 | 89 | - | ·2 " ·3 | 78 | 78 | - | ·3 " ·4 | 58 | 64 | - | ·4 " ·5 | 51 | 50 | - | ·5 " ·6 | 36 | 36 | - | ·6 " ·7 | 26 | 24 | - | ·7 " ·8 | 14 | 15 | - | ·8 " ·9 | 10 | 9 | - | ·9 " 1·0 | 7 | 5 | - | above 1·0 | 8 | 5 | - +-------------------------+-------------+------------+ - - [283] Encke, *On the Method of Least Squares*, Taylor’s *Scientific - Memoirs*, vol. ii. pp. 338, 339. - -The reader will remark that the correspondence is very close, except -as regards larger errors, which are excessive in practice. It is one -objection, indeed, to the theory of error, that, being expressed in -a continuous mathematical function, it contemplates the existence of -errors of every magnitude, such as could not practically occur; yet -in this case the theory seems to under-estimate the number of large -errors. - -Another comparison of the law with observation was made by Quetelet, -who investigated the errors of 487 determinations in time of the Right -Ascension of the Pole-Star made at Greenwich during the four years -1836–39. These observations, although carefully corrected for all known -causes of error, as well as for nutation, precession, &c., are yet of -course found to differ, and being classified as regards intervals of -one-half second of time, and then proportionately increased in number, -so that their sum may be one thousand, give the following results as -compared with what Quetelet’s theory would lead us to expect:--[284] - - +------------+--------------------+------------+--------------------+ - |Magnitude of| Number of Errors |Magnitude of| Number of Errors | - | error +------------+-------+ error +------------+-------+ - | in tenths | by | by | in tenths | by | by | - |of a second.|Observation.|Theory.|of a second.|Observation.|Theory.| - +------------+------------+-------+------------+------------+-------+ - | 0·0 | 168 | 163 | -- | -- | -- | - | +0·5 | 148 | 147 | -0·5 | 150 | 152 | - | +1·0 | 129 | 112 | -1·0 | 126 | 121 | - | +1·5 | 78 | 72 | -1·5 | 74 | 82 | - | +2·0 | 33 | 40 | -2·0 | 43 | 46 | - | +2·5 | 10 | 19 | -2·5 | 25 | 22 | - | +3·0 | 2 | 10 | -3·0 | 12 | 10 | - | -- | -- | -- | -3·5 | 2 | 4 | - +------------+------------+-------+------------+------------+-------+ - - [284] Quetelet, *Letters on the Theory of Probabilities*, translated - by Downes, Letter XIX. p. 88. See also Galton’s *Hereditary Genius*, - p. 379. - -In this instance also the correspondence is satisfactory, but the -divergence between theory and fact is in the opposite direction to -that discovered in the former comparison, the larger errors being less -frequent than theory would indicate. It will be noticed that Quetelet’s -theoretical results are not symmetrical. - - -*The Probable Mean Result.* - -One immediate result of the Law of Error, as thus stated, is that the -mean result is the most probable one; and when there is only a single -variable this mean is found by the familiar arithmetical process. An -unfortunate error has crept into several works which allude to this -subject. Mill, in treating of the “Elimination of Chance,” remarks in -a note[285] that “the mean is spoken of as if it were exactly the -same thing as the average. But the mean, for purposes of inductive -inquiry, is not the average, or arithmetical mean, though in a familiar -illustration of the theory the difference may be disregarded.” He goes -on to say that, according to mathematical principles, the most probable -result is that for which the sums of the squares of the deviations is -the least possible. It seems probable that Mill and other writers were -misled by Whewell, who says[286] that “The method of least squares is -in fact a method of means, but with some peculiar characters.... The -method proceeds upon this supposition: that all errors are not equally -probable, but that small errors are more probable than large ones.” He -adds that this method “removes much that is arbitrary in the method -of means.” It is strange to find a mathematician like Whewell making -such remarks, when there is no doubt whatever that the Method of Means -is only an application of the Method of Least Squares. They are, in -fact, the same method, except that the latter method may be applied to -cases where two or more quantities have to be determined at the same -time. Lubbock and Drinkwater say,[287] “If only one quantity has to -be determined, this method evidently resolves itself into taking the -mean of all the values given by observation.” Encke says,[288] that the -expression for the probability of an error “not only contains in itself -the principle of the arithmetical mean, but depends so immediately upon -it, that for all those magnitudes for which the arithmetical mean holds -good in the simple cases in which it is principally applied, no other -law of probability can be assumed than that which is expressed by this -formula.” - - [285] *System of Logic*, bk. iii. chap. 17, § 3. 5th ed. vol. ii. - p. 56. - - [286] *Philosophy of the Inductive Sciences*, 2nd ed. vol. ii. - pp. 408, 409. - - [287] *Essay on Probability*, Useful Knowledge Society, 1833, p. 41. - - [288] Taylor’s *Scientific Memoirs*, vol. ii. p. 333. - - -*The Probable Error of Results.* - -When we draw a conclusion from the numerical results of observations we -ought not to consider it sufficient, in cases of importance, to content -ourselves with finding the simple mean and treating it as true. We -ought also to ascertain what is the degree of confidence we may place -in this mean, and our confidence should be measured by the degree of -concurrence of the observations from which it is derived. In some cases -the mean may be approximately certain and accurate. In other cases it -may really be worth little or nothing. The Law of Error enables us to -give exact expression to the degree of confidence proper in any case; -for it shows how to calculate the probability of a divergence of any -amount from the mean, and we can thence ascertain the probability that -the mean in question is within a certain distance from the true number. -The *probable error* is taken by mathematicians to mean the limits -within which it is as likely as not that the truth will fall. Thus -if 5·45 be the mean of all the determinations of the density of the -earth, and ·20 be approximately the probable error, the meaning is that -the probability of the real density of the earth falling between 5·25 -and 5·65 is 1/2. Any other limits might have been selected at will. -We might calculate the limits within which it was one hundred or one -thousand to one that the truth would fall; but there is a convention to -take the even odds one to one, as the quantity of probability of which -the limits are to be estimated. - -Many books on probability give rules for making the calculations, -but as, in the progress of science, persons ought to become more -familiar with these processes, I propose to repeat the rules here and -illustrate their use. The calculations, when made in accordance with -the directions, involve none but arithmetic or logarithmic operations. - -The following are the rules for treating a mean result, so as -thoroughly to ascertain its trustworthiness. - -1. Draw the mean of all the observed results. - -2. Find the excess or defect, that is, the error of each result from -the mean. - -3. Square each of these reputed errors. - -4. Add together all these squares of the errors, which are of course -all positive. - -5. Divide by one less than the number of observations. This gives the -*square of the mean error*. - -6. Take the square root of the last result; it is the *mean error of a -single observation*. - -7. Divide now by the square root of the number of observations, and we -get the *mean error of the mean result*. - -8. Lastly, multiply by the natural constant O·6745 (or approximately -by 0·674, or even by 2/3), and we arrive at the *probable error of the -mean result*. - -Suppose, for instance, that five measurements of the height of a -hill, by the barometer or otherwise, have given the numbers of feet -as 293, 301, 306, 307, 313; we want to know the probable error of the -mean, namely 304. Now the differences between this mean and the above -numbers, *paying no regard to direction*, are 11, 3, 2, 3, 9; their -squares are 121, 9, 4, 9, 81, and the sum of the squares of the errors -consequently 224. The number of observations being 5, we divide by -1 less, or 4, getting 56. This is the square of the mean error, and -taking its square root we have 7·48 (say 7-1/2), the mean error of -a single observation. Dividing by 2·236, the square root of 5, the -number of observations, we find the mean error of the *mean* result to -be 3·35, or say 3-1/3, and lastly, multiplying by ·6745, we arrive at -the *probable error of the mean result*, which is found to be 2·259, -or say 2-1/4. The meaning of this is that the probability is one half, -or the odds are even that the true height of the mountain lies between -301-3/4 and 306-1/4 feet. We have thus an exact measure of the degree -of credibility of our mean result, which mean indicates the most likely -point for the truth to fall upon. - -The reader should observe that as the object in these calculations -is only to gain a notion of the degree of confidence with which we -view the mean, there is no real use in carrying the calculations to -any great degree of precision; and whenever the neglect of decimal -fractions, or even the slight alteration of a number, will much -abbreviate the computations, it may be fearlessly done, except in -cases of high importance and precision. Brodie has shown how the law -of error may be usefully applied in chemical investigations, and some -illustrations of its employment may be found in his paper.[289] - - [289] *Philosophical Transactions*, 1873, p. 83. - -The experiments of Benzenberg to detect the revolution of the earth, by -the deviation of a ball from the perpendicular line in falling down a -deep pit, have been cited by Encke[290] as an interesting illustration -of the Law of Error. The mean deviation was 5·086 lines, and its -probable error was calculated by Encke to be not more than ·950 line, -that is, the odds were even that the true result lay between 4·136 and -6·036. As the deviation, according to astronomical theory, should be -4·6 lines, which lies well within the limits, we may consider that the -experiments are consistent with the Copernican system of the universe. - - [290] Taylor’s *Scientific Memoirs*, vol. ii. pp. 330, 347, &c. - -It will of course be understood that the probable error has regard only -to those causes of errors which in the long run act as much in one -direction as another; it takes no account of constant errors. The true -result accordingly will often fall far beyond the limits of probable -error, owing to some considerable constant error or errors, of the -existence of which we are unaware. - - -*Rejection of the Mean Result.* - -We ought always to bear in mind that the mean of any series of -observations is the best, that is, the most probable approximation -to the truth, only in the absence of knowledge to the contrary. The -selection of the mean rests entirely upon the probability that unknown -causes of error will in the long run fall as often in one direction as -the opposite, so that in drawing the mean they will balance each other. -If we have any reason to suppose that there exists a tendency to error -in one direction rather than the other, then to choose the mean would -be to ignore that tendency. We may certainly approximate to the length -of the circumference of a circle, by taking the mean of the perimeters -of inscribed and circumscribed polygons of an equal and large number -of sides. The length of the circular line undoubtedly lies between the -lengths of the two perimeters, but it does not follow that the mean is -the best approximation. It may in fact be shown that the circumference -of the circle is *very nearly* equal to the perimeter of the inscribed -polygon, together with one-third part of the difference between the -inscribed and circumscribed polygons of the same number of sides. -Having this knowledge, we ought of course to act upon it, instead of -trusting to probability. - -We may often perceive that a series of measurements tends towards an -extreme limit rather than towards a mean. In endeavouring to obtain -a correct estimate of the apparent diameter of the brightest fixed -stars, we find a continuous diminution in estimates as the powers of -observation increased. Kepler assigned to Sirius an apparent diameter -of 240 seconds; Tycho Brahe made it 126; Gassendi 10 seconds; Galileo, -Hevelius, and J. Cassini, 5 or 6 seconds. Halley, Michell, and -subsequently Sir W. Herschel came to the conclusion that the brightest -stars in the heavens could not have real discs of a second, and were -probably much less in diameter. It would of course be absurd to take -the mean of quantities which differ more than 240 times; and as the -tendency has always been to smaller estimates, there is a considerable -presumption in favour of the smallest.[291] - - [291] Quetelet, *Letters*, &c. p. 116. - -In many experiments and measurements we know that there is a -preponderating tendency to error in one direction. The readings of a -thermometer tend to rise as the age of the instrument increases, and -no drawing of means will correct this result. Barometers, on the other -hand, are likely to read too low instead of too high, owing to the -imperfection of the vacuum and the action of capillary attraction. If -the mercury be perfectly pure and no appreciable error be due to the -measuring apparatus, the best barometer will be that which gives the -highest result. In determining the specific gravity of a solid body the -chief danger of error arises from bubbles of air adhering to the body, -which would tend to make the specific gravity too small. Much attention -must always be given to one-sided errors of this kind, since the -multiplication of experiments does not remove the error. In such cases -one very careful experiment is better than any number of careless ones. - -When we have reasonable grounds for supposing that certain experimental -results are liable to grave errors, we should exclude them in drawing -a mean. If we want to find the most probable approximation to the -velocity of sound in air, it would be absurd to go back to the old -experiments which made the velocity from 1200 to 1474 feet per second; -for we know that the old observers did not guard against errors arising -from wind and other causes. Old chemical experiments are valueless as -regards quantitative results. The old chemists found the atmosphere -in different places to differ in composition nearly ten per cent., -whereas modern accurate experimenters find very slight variations. -Any method of measurement which we know to avoid a source of error is -far to be preferred to others which trust to probabilities for the -elimination of the error. As Flamsteed says,[292] “One good instrument -is of as much worth as a hundred indifferent ones.” But an instrument -is good or bad only in a comparative sense, and no instrument gives -invariable and truthful results. Hence we must always ultimately fall -back upon probabilities for the selection of the final mean, when other -precautions are exhausted. - - [292] Baily, *Account of Flamsteed*, p. 56. - -Legendre, the discoverer of the method of Least Squares, -recommended that observations differing very much from the results -of his method should be rejected. The subject has been carefully -investigated by Professor Pierce, who has proposed a criterion -for the rejection of doubtful observations based on the following -principle:[293]′“--observations should be rejected when the probability -of the system of errors obtained by retaining them is less than that -of the system of errors obtained by their rejection multiplied by the -probability of making so many and no more abnormal observations.” -Professor Pierce’s investigation is given nearly in his own words in -Professor W. Chauvenet’s “Manual of Spherical and Practical Astronomy,” -which contains a full and excellent discussion of the methods of -treating numerical observations.[294] - - [293] Gould’s *Astronomical Journal*, Cambridge, Mass., vol. ii. - p. 161. - - [294] Philadelphia (London, Trübner) 1863. Appendix, vol. ii. p. 558. - -Very difficult questions sometimes arise when one or more results of -a method of experiment diverge widely from the mean of the rest. Are -we or are we not to exclude them in adopting the supposed true mean -result of the method? The drawing of a mean result rests, as I have -frequently explained, upon the assumption that every error acting in -one direction will probably be balanced by other errors acting in an -opposite direction. If then we know or can possibly discover any causes -of error not agreeing with this assumption, we shall be justified in -excluding results which seem to be affected by this cause. - -In reducing large series of astronomical observations, it is not -uncommon to meet with numbers differing from others by a whole degree -or half a degree, or some considerable integral quantity. These are -errors which could hardly arise in the act of observation or in -instrumental irregularity; but they might readily be accounted for -by misreading of figures or mistaking of division marks. It would be -absurd to trust to chance that such mistakes would balance each other -in the long run, and it is therefore better to correct arbitrarily -the supposed mistake, or better still, if new observations can be -made, to strike out the divergent numbers altogether. When results -come sometimes too great or too small in a regular manner, we should -suspect that some part of the instrument slips through a definite -space, or that a definite cause of error enters at times, and not at -others. We should then make it a point of prime importance to discover -the exact nature and amount of such an error, and either prevent its -occurrence for the future or else introduce a corresponding correction. -In many researches the whole difficulty will consist in this detection -and avoidance of sources of error. Professor Roscoe found that the -presence of phosphorus caused serious and almost unavoidable errors in -the determination of the atomic weight of vanadium.[295] Herschel, in -reducing his observations of double stars at the Cape of Good Hope, -was perplexed by an unaccountable difference of the angles of position -as measured by the seven-feet equatorial and the twenty-feet reflector -telescopes, and after a careful investigation was obliged to be -contented with introducing a correction experimentally determined.[296] - - [295] Bakerian Lecture, *Philosophical Transactions* (1868), - vol. clviii. p. 6. - - [296] *Results of Observations at the Cape of Good Hope*, p. 283. - -When observations are sufficiently numerous it seems desirable to -project the apparent errors into a curve, and then to observe whether -this curve exhibits the symmetrical and characteristic form of the -curve of error. If so, it may be inferred that the errors arise from -many minute independent sources, and probably compensate each other -in the mean result. Any considerable irregularity will indicate the -existence of one-sided or large causes of error, which should be made -the subject of investigation. - -Even the most patient and exhaustive investigations will sometimes -fail to disclose any reason why some results diverge from others. -The question again recurs--Are we arbitrarily to exclude them? The -answer should be in the negative as a general rule. The mere fact -of divergence ought not to be taken as conclusive against a result, -and the exertion of arbitrary choice would open the way to the fatal -influence of bias, and what is commonly known as the “cooking” of -figures. It would amount to judging fact by theory instead of theory -by fact. The apparently divergent number may prove in time to be the -true one. It may be an exception of that valuable kind which upsets our -false theories, a real exception, exploding apparent coincidences, and -opening a way to a new view of the subject. To establish this position -for the divergent fact will require additional research; but in the -meantime we should give it some weight in our mean conclusions, and -should bear in mind the discrepancy as one demanding attention. To -neglect a divergent result is to neglect the possible clue to a great -discovery. - - -*Method of Least Squares.* - -When two or more unknown quantities are so involved that they cannot -be separately determined by the Simple Method of Means, we can yet -obtain their most probable values by the Method of Least Squares, -without more difficulty than arises from the length of the arithmetical -computations. If the result of each observation gives an equation -between two unknown quantities of the form - - *ax* + *by* = *c* - -then, if the observations were free from error, we should need only two -observations giving two equations; but for the attainment of greater -accuracy, we may take many observations, and reduce the equations -so as to give only a pair with mean coefficients. This reduction is -effected by (1.), multiplying the coefficients of each equation by the -first coefficient, and adding together all the similar coefficients -thus resulting for the coefficients of a new equation; and (2.), by -repeating this process, and multiplying the coefficients of each -equation by the coefficient of the second term. Meaning by (sum of -*a*^{2}) the sum of all quantities of the same kind, and having the -same place in the equations as *a*^{2}, we may briefly describe the two -resulting mean equations as follows:-- - - (sum of *a*^{2}) . *x* + (sum of *ab*) . *y* = (sum of *ac*), - (sum of *ab*) . *x* + (sum of *b*^{2}) . *y* = (sum of *bc*). - -When there are three or more unknown quantities the process is exactly -the same in nature, and we get additional mean equations by multiplying -by the third, fourth, &c., coefficients. As the numbers are in any -case approximate, it is usually unnecessary to make the computations -with accuracy, and places of decimals may be freely cut off to save -arithmetical work. The mean equations having been computed, their -solution by the ordinary methods of algebra gives the most probable -values of the unknown quantities. - - -*Works upon the Theory of Probability.* - -Regarding the Theory of Probability and the Law of Error as most -important subjects of study for any one who desires to obtain a -complete comprehension of scientific method as actually applied in -physical investigations, I will briefly indicate the works in one or -other of which the reader will best pursue the study. - -The best popular, and at the same time profound English work on the -subject is De Morgan’s “Essay on Probabilities and on their Application -to Life Contingencies and Insurance Offices,” published in the *Cabinet -Cyclopædia*, and to be obtained (in print) from Messrs. Longman. Mr. -Venn’s work on *The Logic of Chance* can now be procured in a greatly -enlarged second edition;[297] it contains a most interesting and able -discussion of the metaphysical basis of probability and of related -questions concerning causation, belief, design, testimony, &c.; but I -cannot always agree with Mr. Venn’s opinions. No mathematical knowledge -beyond that of common arithmetic is required in reading these works. -Quetelet’s *Letters* form a good introduction to the subject, and the -mathematical notes are of value. Sir George Airy’s brief treatise *On -the Algebraical and Numerical Theory of Errors of Observations and -the Combination of Observations*, contains a complete explanation of -the Law of Error and its practical applications. De Morgan’s treatise -“On the Theory of Probabilities” in the *Encyclopædia Metropolitana*, -presents an abstract of the more abstruse investigations of -Laplace, together with a multitude of profound and original remarks -concerning the theory generally. In Lubbock and Drinkwater’s work on -*Probability*, in the Library of Useful Knowledge, we have a concise -but good statement of a number of important problems. The Rev. -W. A. Whitworth has given, in a work entitled *Choice and Chance*, -a number of good illustrations of calculations both in combinations -and probabilities. In Mr. Todhunter’s admirable History we have an -exhaustive critical account of almost all writings upon the subject -of probability down to the culmination of the theory in Laplace’s -works. The Memoir of Mr. J. W. L. Glaisher has already been mentioned -(p. 375). In spite of the existence of these and some other good -English works, there seems to be a want of an easy and yet pretty -complete mathematical introduction to the study of the theory. - - [297] *The Logic of Chance*, an Essay on the Foundations and Province - of the Theory of Probability, with especial reference to its - Logical Bearings and its Application to Moral and Social Science. - (Macmillan), 1876. - -Among French works the Traité *Élémentaire du Calcul des Probabilités*, -by S. E. Lacroix, of which several editions have been published, and -which is not difficult to obtain, forms probably the best elementary -treatise. Poisson’s *Recherches sur la Probabilité des Jugements* -(Paris 1837), commence with an admirable investigation of the grounds -and methods of the theory. While Laplace’s great *Théorie Analytique -des Probabilités* is of course the “Principia” of the subject; his -*Essai Philosophique sur les Probabilités* is a popular discourse, and -is one of the most profound and interesting essays ever published. It -should be familiar to every student of logical method, and has lost -little or none of its importance by lapse of time. - - -*Detection of Constant Errors.* - -The Method of Means is absolutely incapable of eliminating any error -which is always the same, or which always lies in one direction. We -sometimes require to be roused from a false feeling of security, and -to be urged to take suitable precautions against such occult errors. -“It is to the observer,” says Gauss,[298] “that belongs the task of -carefully removing the causes of constant errors,” and this is quite -true when the error is absolutely constant. When we have made a number -of determinations with a certain apparatus or method of measurement, -there is a great advantage in altering the arrangement, or even -devising some entirely different method of getting estimates of the -same quantity. The reason obviously consists in the improbability that -the same error will affect two or more different methods of experiment. -If a discrepancy is found to exist, we shall at least be aware of the -existence of error, and can take measures for finding in which way it -lies. If we can try a considerable number of methods, the probability -becomes great that errors constant in one method will be balanced or -nearly so by errors of an opposite effect in the others. Suppose that -there be three different methods each affected by an error of equal -amount. The probability that this error will in all fall in the same -direction is only 1/4; and with four methods similarly 1/8. If each -method be affected, as is always the case, by several independent -sources of error, the probability becomes much greater that in the mean -result of all the methods some of the errors will partially compensate -the others. In this case as in all others, when human vigilance has -exhausted itself, we must trust the theory of probability. - - [298] Gauss, translated by Bertrand, p. 25. - -In the determination of a zero point, of the magnitude of the -fundamental standards of time and space, in the personal equation of -an astronomical observer, we have instances of fixed errors; but as a -general rule a change of procedure is likely to reverse the character -of the error, and many instances may be given of the value of this -precaution. If we measure over and over again the same angular -magnitude by the same divided circle, maintained in exactly the same -position, it is evident that the same mark in the circle will be the -criterion in each case, and any error in the position of that mark will -equally affect all our results. But if in each measurement we use a -different part of the circle, a new mark will come into use, and as the -error of each mark cannot be in the same direction, the average result -will be nearly free from errors of division. It will be better still to -use more than one divided circle. - -Even when we have no perception of the points at which error is -likely to enter, we may with advantage vary the construction of our -apparatus in the hope that we shall accidentally detect some latent -cause of error. Baily’s purpose in repeating the experiments of -Michell and Cavendish on the density of the earth was not merely to -follow the same course and verify the previous numbers, but to try -whether variations in the size and substance of the attracting balls, -the mode of suspension, the temperature of the surrounding air, &c., -would yield different results. He performed no less than 62 distinct -series, comprising 2153 experiments, and he carefully classified and -discussed the results so as to disclose the utmost differences. Again, -in experimenting upon the resistance of the air to the motion of a -pendulum, Baily employed no less than 80 pendulums of various forms -and materials, in order to ascertain exactly upon what conditions -the resistance depends. Regnault, in his exact researches upon the -dilatation of gases, made arbitrary changes in the magnitude of parts -of his apparatus. He thinks that if, in spite of such modification, -the results are unchanged, the errors are probably of inconsiderable -amount;[299] but in reality it is always possible, and usually likely, -that we overlook sources of error which a future generation will -detect. Thus the pendulum experiments of Baily and Sabine were directed -to ascertain the nature and amount of a correction for air resistance, -which had been entirely misunderstood in the experiments by means of -the seconds pendulum, upon which was founded the definition of the -standard yard, in the Act of 5th George IV. c. 74. It has already been -mentioned that a considerable error was discovered in the determination -of the standard metre as the ten-millionth part of the distance from -the pole to the equator (p. 314). - - [299] Jamin, *Cours de Physique*, vol. ii. p. 60. - -We shall return in Chapter XXV. to the further consideration of the -methods by which we may as far as possible secure ourselves against -permanent and undetected sources of error. In the meantime, having -completed the consideration of the special methods requisite for -treating quantitative phenomena, we must pursue our principal subject, -and endeavour to trace out the course by which the physicist, from -observation and experiment, collects the materials of knowledge, and -then proceeds by hypothesis and inverse calculation to induce from them -the laws of nature. - - - - -Book IV. - -INDUCTIVE INVESTIGATION. - - - - -CHAPTER XVIII. - -OBSERVATION. - - -ALL knowledge proceeds originally from experience. Using the name in a -wide sense, we may say that experience comprehends all that we *feel*, -externally or internally--the aggregate of the impressions which we -receive through the various apertures of perception--the aggregate -consequently of what is in the mind, except so far as some portions -of knowledge may be the reasoned equivalents of other portions. As -the word experience expresses, we *go through* much in life, and the -impressions gathered intentionally or unintentionally afford the -materials from which the active powers of the mind evolve science. - -No small part of the experience actually employed in science is -acquired without any distinct purpose. We cannot use the eyes without -gathering some facts which may prove useful. A great science has in -many cases risen from an accidental observation. Erasmus Bartholinus -thus first discovered double refraction in Iceland spar; Galvani -noticed the twitching of a frog’s leg; Oken was struck by the form of -a vertebra; Malus accidentally examined light reflected from distant -windows with a double refracting substance; and Sir John Herschel’s -attention was drawn to the peculiar appearance of a solution of quinine -sulphate. In earlier times there must have been some one who first -noticed the strange behaviour of a loadstone, or the unaccountable -motions produced by amber. As a general rule we shall not know in what -direction to look for a great body of phenomena widely different from -those familiar to us. Chance then must give us the starting point; but -one accidental observation well used may lead us to make thousands of -observations in an intentional and organised manner, and thus a science -may be gradually worked out from the smallest opening. - - -*Distinction of Observation and Experiment.* - -It is usual to say that the two sources of experience are Observation -and Experiment. When we merely note and record the phenomena which -occur around us in the ordinary course of nature we are said *to -observe*. When we change the course of nature by the intervention -of our muscular powers, and thus produce unusual combinations and -conditions of phenomena, we are said *to experiment*. Herschel justly -remarked[300] that we might properly call these two modes of experience -*passive and active observation*. In both cases we must certainly -employ our senses to observe, and an experiment differs from a mere -observation in the fact that we more or less influence the character -of the events which we observe. Experiment is thus observation *plus* -alteration of conditions. - - [300] *Preliminary Discourse on the Study of Natural Philosophy*, - p. 77. - -It may readily be seen that we pass upwards by insensible gradations -from pure observation to determinate experiment. When the earliest -astronomers simply noticed the ordinary motions of the sun, moon, and -planets upon the face of the starry heavens, they were pure observers. -But astronomers now select precise times and places for important -observations of stellar parallax, or the transits of planets. They make -the earth’s orbit the basis of a well arranged *natural experiment*, -as it were, and take well considered advantage of motions which they -cannot control. Meteorology might seem to be a science of pure -observation, because we cannot possibly govern the changes of weather -which we record. Nevertheless we may ascend mountains or rise in -balloons, like Gay-Lussac and Glaisher, and may thus so vary the points -of observation as to render our procedure experimental. We are wholly -unable either to produce or prevent earth-currents of electricity, -but when we construct long lines of telegraph, we gather such strong -currents during periods of disturbance as to render them capable of -easy observation. - -The best arranged systems of observation, however, would fail to give -us a large part of the facts which we now possess. Many processes -continually going on in nature are so slow and gentle as to escape -our powers of observation. Lavoisier remarked that the decomposition -of water must have been constantly proceeding in nature, although its -possibility was unknown till his time.[301] No substance is wholly -destitute of magnetic or diamagnetic powers; but it required all the -experimental skill of Faraday to prove that iron and a few other -metals had no monopoly of these powers. Accidental observation long -ago impressed upon men’s minds the phenomena of lightning, and the -attractive properties of amber. Experiment only could have shown that -phenomena so diverse in magnitude and character were manifestations of -the same agent. To observe with accuracy and convenience we must have -agents under our control, so as to raise or lower their intensity, -to stop or set them in action at will. Just as Smeaton found it -requisite to create an artificial and governable supply of wind for -his investigation of windmills, so we must have governable supplies of -light, heat, electricity, muscular force, or whatever other agents we -are examining. - - [301] Lavoisier’s *Elements of Chemistry*, translated by Kerr, 3rd - ed. p. 148. - -It is hardly needful to point out too that on the earth’s surface we -live under nearly constant conditions of gravity, temperature, and -atmospheric pressure, so that if we are to extend our inferences to -other parts of the universe where conditions are widely different, we -must be prepared to imitate those conditions on a small scale here. -We must have intensely high and low temperatures; we must vary the -density of gases from approximate vacuum upwards; we must subject -liquids and solids to pressures or strains of almost unlimited amount. - - -*Mental Conditions of Correct Observation.* - -Every observation must in a certain sense be true, for the observing -and recording of an event is in itself an event. But before we proceed -to deal with the supposed meaning of the record, and draw inferences -concerning the course of nature, we must take care to ascertain that -the character and feelings of the observer are not to a great extent -the phenomena recorded. The mind of man, as Francis Bacon said, is -like an uneven mirror, and does not reflect the events of nature -without distortion. We need hardly take notice of intentionally false -observations, nor of mistakes arising from defective memory, deficient -light, and so forth. Even where the utmost fidelity and care are used -in observing and recording, tendencies to error exist, and fallacious -opinions arise in consequence. - -It is difficult to find persons who can with perfect fairness register -facts for and against their own peculiar views. Among uncultivated -observers the tendency to remark favourable and forget unfavourable -events is so great, that no reliance can be placed upon their supposed -observations. Thus arises the enduring fallacy that the changes of the -weather coincide in some way with the changes of the moon, although -exact and impartial registers give no countenance to the fact. The -whole race of prophets and quacks live on the overwhelming effect of -one success, compared with hundreds of failures which are unmentioned -and forgotten. As Bacon says, “Men mark when they hit, and never mark -when they miss.” And we should do well to bear in mind the ancient -story, quoted by Bacon, of one who in Pagan times was shown a temple -with a picture of all the persons who had been saved from shipwreck, -after paying their vows. When asked whether he did not now acknowledge -the power of the gods, “Ay,” he answered; “but where are they painted -that were drowned after their vows?” - -If indeed we could estimate the amount of *bias* existing in any -particular observations, it might be treated like one of the forces -of the problem, and the true course of external nature might still be -rendered apparent. But the feelings of an observer are usually too -indeterminate, so that when there is reason to suspect considerable -bias, rejection is the only safe course. As regards facts casually -registered in past times, the capacity and impartiality of the observer -are so little known that we should spare no pains to replace these -statements by a new appeal to nature. An indiscriminate medley of -truth and absurdity, such as Francis Bacon collected in his *Natural -History*, is wholly unsuited to the purposes of science. But of course -when records relate to past events like eclipses, conjunctions, -meteoric phenomena, earthquakes, volcanic eruptions, changes of sea -margins, the existence of now extinct animals, the migrations of -tribes, remarkable customs, &c., we must make use of statements however -unsatisfactory, and must endeavour to verify them by the comparison of -independent records or traditions. - -When extensive series of observations have to be made, as in -astronomical, meteorological, or magnetical observatories, -trigonometrical surveys, and extensive chemical or physical researches, -it is an advantage that the numerical work should be executed by -assistants who are not interested in, and are perhaps unaware of, the -expected results. The record is thus rendered perfectly impartial. -It may even be desirable that those who perform the purely routine -work of measurement and computation should be unacquainted with the -principles of the subject. The great table of logarithms of the -French Revolutionary Government was worked out by a staff of sixty or -eighty computers, most of whom were acquainted only with the rules of -arithmetic, and worked under the direction of skilled mathematicians; -yet their calculations were usually found more correct than those of -persons more deeply versed in mathematics.[302] In the Indian Ordnance -Survey the actual measurers were selected so that they should not have -sufficient skill to falsify their results without detection. - - [302] Babbage, *Economy of Manufactures*, p. 194. - -Both passive observation and experimentation must, however, be -generally conducted by persons who know for what they are to look. It -is only when excited and guided by the hope of verifying a theory that -the observer will notice many of the most important points; and, where -the work is not of a routine character, no assistant can supersede the -mind-directed observations of the philosopher. Thus the successful -investigator must combine diverse qualities; he must have clear notions -of the result he expects and confidence in the truth of his theories, -and yet he must have that candour and flexibility of mind which enable -him to accept unfavourable results and abandon mistaken views. - - -*Instrumental and Sensual Conditions of Observation.* - -In every observation one or more of the senses must be employed, and -we should ever bear in mind that the extent of our knowledge may be -limited by the power of the sense concerned. What we learn of the world -only forms the lower limit of what is to be learned, and, for all that -we can tell, the processes of nature may infinitely surpass in variety -and complexity those which are capable of coming within our means of -observation. In some cases inference from observed phenomena may make -us indirectly aware of what cannot be directly felt, but we can never -be sure that we thus acquire any appreciable fraction of the knowledge -that might be acquired. - -It is a strange reflection that space may be filled with dark wandering -stars, whose existence could not have yet become in any way known to -us. The planets have already cooled so far as to be no longer luminous, -and it may well be that other stellar bodies of various size have -fallen into the same condition. From the consideration, indeed, of -variable and extinguished stars, Laplace inferred that there probably -exist opaque bodies as great and perhaps as numerous as those we -see.[303] Some of these dark stars might ultimately become known to -us, either by reflecting light, or more probably by their gravitating -effects upon luminous stars. Thus if one member of a double star -were dark, we could readily detect its existence, and even estimate -its size, position, and motions, by observing those of its visible -companion. It was a favourite notion of Huyghens that there may exist -stars and vast universes so distant that their light has never yet -had time to reach our eyes; and we must also bear in mind that light -may possibly suffer slow extinction in space, so that there is more -than one way in which an absolute limit to the powers of telescopic -discovery may exist. - - [303] *System of the World*, translated by Harte, vol. ii. p. 335. - -There are natural limits again to the power of our senses in detecting -undulations of various kinds. It is commonly said that vibrations of -more than 38,000 strokes per second are not audible as sound; and -as some ears actually do hear sounds of much higher pitch, even two -octaves higher than what other ears can detect, it is exceedingly -probable that there are incessant vibrations which we cannot call sound -because they are never heard. Insects may communicate by such acute -sounds, constituting a language inaudible to us; and the remarkable -agreement apparent among bodies of ants or bees might thus perhaps be -explained. Nay, as Fontenelle long ago suggested in his scientific -romance, there may exist unlimited numbers of senses or modes of -perception which we can never feel, though Darwin’s theory would render -it probable that any useful means of knowledge in an ancestor would -be developed and improved in the descendants. We might doubtless have -been endowed with a sense capable of feeling electric phenomena with -acuteness, so that the positive or negative state of charge of a body -could be at once estimated. The absence of such a sense is probably due -to its comparative uselessness. - -Heat undulations are subject to the same considerations. It is now -apparent that what we call light is the affection of the eye by certain -vibrations, the less rapid of which are invisible and constitute the -dark rays of radiant heat, in detecting which we must substitute -the thermometer or the thermopile for the eye. At the other end of -the spectrum, again, the ultra-violet rays are invisible, and only -indirectly brought to our knowledge in the phenomena of fluorescence or -photo-chemical action. There is no reason to believe that at either end -of the spectrum an absolute limit has yet been reached. - -Just as our knowledge of the stellar universe is limited by the -power of the telescope and other conditions, so our knowledge of the -minute world has its limit in the powers and optical conditions of -the microscope. There was a time when it would have been a reasonable -induction that vegetables are motionless, and animals alone endowed -with power of locomotion. We are astonished to discover by the -microscope that minute plants are if anything more active than -minute animals. We even find that mineral substances seem to lose -their inactive character and dance about with incessant motion when -reduced to sufficiently minute particles, at least when suspended -in a non-conducting medium.[304] Microscopists will meet a natural -limit to observation when the minuteness of the objects examined -becomes comparable to the length of light undulations, and the extreme -difficulty already encountered in determining the forms of minute marks -on Diatoms appears to be due to this cause. According to Helmholtz the -smallest distance which can be accurately defined depends upon the -interference of light passing through the centres of the bright spaces. -With a theoretically perfect microscope and a dry lense the smallest -visible object would not be less than one 80,000th part of an inch in -red light. - - [304] This curious phenomenon, which I propose to call *pedesis*, or - the *pedetic movement*, from πηδόω, to jump, is carefully described - in my paper published in the *Quarterly Journal of Science* for - April, 1878, vol. viii. (N.S.) p. 167. See also *Proceedings of the - Literary and Philosophical Society of Manchester*, 25th January, - 1870, vol. ix. p. 78, *Nature*, 22nd August, 1878, vol. xviii. - p. 440, or the *Quarterly Journal of Science*, vol. viii. (N.S.) - p. 514. - -Of the errors likely to arise in estimating quantities by the senses I -have already spoken, but there are some cases in which we actually see -things differently from what they are. A jet of water appears to be a -continuous thread, when it is really a wonderfully organised succession -of small and large drops, oscillating in form. The drops fall so -rapidly that their impressions upon the eye run into each other, and in -order to see the separate drops we require some device for giving an -instantaneous view. - -One insuperable limit to our powers of observation arises from the -impossibility of following and identifying the ultimate atoms of -matter. One atom of oxygen is probably undistinguishable from another -atom; only by keeping a certain volume of oxygen safely inclosed in a -bottle can we assure ourselves of its identity; allow it to mix with -other oxygen, and we lose all power of identification. Accordingly -we seem to have no means of directly proving that every gas is in a -constant state of diffusion of every part into every part. We can only -infer this to be the case from observing the behaviour of distinct -gases which we can distinguish in their course, and by reasoning on the -grounds of molecular theory.[305] - - [305] Maxwell, *Theory of Heat*, p. 301. - - -*External Conditions of Correct Observation.* - -Before we proceed to draw inferences from any series of recorded facts, -we must take care to ascertain perfectly, if possible, the external -conditions under which the facts are brought to our notice. Not only -may the observing mind be prejudiced and the senses defective, but -there may be circumstances which cause one kind of event to come more -frequently to our notice than another. The comparative numbers of -objects of different kinds existing may in any degree differ from the -numbers which come to our notice. This difference must if possible be -taken into account before we make any inferences. - -There long appeared to be a strong presumption that all comets moved -in elliptic orbits, because no comet had been proved to move in any -other kind of path. The theory of gravitation admitted of the existence -of comets moving in hyperbolic orbits, and the question arose whether -they were really non-existent or were only beyond the bounds of easy -observation. From reasonable suppositions Laplace calculated that -the probability was at least 6000 to 1 against a comet which comes -within the planetary system sufficiently to be visible at the earth’s -surface, presenting an orbit which could be discriminated from a very -elongated ellipse or parabola in the part of its orbit within the reach -of our telescopes.[306] In short, the chances are very much in favour -of our seeing elliptic rather than hyperbolic comets. Laplace’s views -have been confirmed by the discovery of six hyperbolic comets, which -appeared in the years 1729, 1771, 1774, 1818, 1840, and 1843,[307] and -as only about 800 comets altogether have been recorded, the proportion -of hyperbolic ones is quite as large as should be expected. - - [306] Laplace, *Essai Philosophique*, p. 59. Todhunter’s *History*, - pp. 491–494. - - [307] Chambers’ *Astronomy*, 1st ed. p. 203. - -When we attempt to estimate the numbers of objects which may have -existed, we must make large allowances for the limited sphere of our -observations. Probably not more than 4000 or 5000 comets have been seen -in historical times, but making allowance for the absence of observers -in the southern hemisphere, and for the small probability that we see -any considerable fraction of those which are in the neighbourhood of -our system, we must accept Kepler’s opinion, that there are more comets -in the regions of space than fishes in the depths of the ocean. When -like calculations are made concerning the numbers of meteors visible to -us, it is astonishing to find that the number of meteors entering the -earth’s atmosphere in every twenty-four hours is probably not less than -400,000,000, of which 13,000 exist in every portion of space equal to -that filled by the earth. - -Serious fallacies may arise from overlooking the inevitable conditions -under which the records of past events are brought to our notice. -Thus it is only the durable objects manufactured by former races of -men, such as flint implements, which can have come to our notice as a -general rule. The comparative abundance of iron and bronze articles -used by an ancient nation must not be supposed to be coincident with -their comparative abundance in our museums, because bronze is far the -more durable. There is a prevailing fallacy that our ancestors built -more strongly than we do, arising from the fact that the more fragile -structures have long since crumbled away. We have few or no relics of -the habitations of the poorer classes among the Greeks or Romans, or in -fact of any past race; for the temples, tombs, public buildings, and -mansions of the wealthier classes alone endure. There is an immense -expanse of past events necessarily lost to us for ever, and we must -generally look upon records or relics as exceptional in their character. - -The same considerations apply to geological relics. We could not -generally expect that animals would be preserved unless as regards the -bones, shells, strong integuments, or other hard and durable parts. All -the infusoria and animals devoid of mineral framework have probably -perished entirely, distilled perhaps into oils. It has been pointed -out that the peculiar character of some extinct floras may be due to -the unequal preservation of different families of plants. By various -accidents, however, we gain glimpses of a world that is usually lost -to us--as by insects embedded in amber, the great mammoth preserved in -ice, mummies, casts in solid material like that of the Roman soldier at -Pompeii, and so forth. - -We should also remember, that just as there may be conjunctions of the -heavenly bodies that can have happened only once or twice in the period -of history, so remarkable terrestrial conjunctions may take place. -Great storms, earthquakes, volcanic eruptions, landslips, floods, -irruptions of the sea, may, or rather must, have occurred, events of -such unusual magnitude and such extreme rarity that we can neither -expect to witness them nor readily to comprehend their effects. It is -a great advantage of the study of probabilities, as Laplace himself -remarked, to make us mistrust the extent of our knowledge, and pay -proper regard to the probability that events would come within the -sphere of our observations. - - -*Apparent Sequence of Events.* - -De Morgan has excellently pointed out[308] that there are no less than -four modes in which one event may seem to follow or be connected with -another, without being really so. These involve mental, sensual, and -external causes of error, and I will briefly state and illustrate them. - - [308] *Essay on Probabilities*, Cabinet Cyclopædia, p. 121. - -Instead of A causing B, it may be *our perception of A that causes B*. -Thus it is that prophecies, presentiments, and the devices of sorcery -and witchcraft often work their own ends. A man dies on the day which -he has always regarded as his last, from his own fears of the day. An -incantation effects its purpose, because care is taken to frighten the -intended victim, by letting him know his fate. In all such cases the -mental condition is the cause of apparent coincidence. - -In a second class of cases, *the event A may make our perception of -B follow, which would otherwise happen without being perceived*. -Thus it was believed to be the result of investigation that more -comets appeared in hot than cold summers. No account was taken of -the fact that hot summers would be comparatively cloudless, and -afford better opportunities for the discovery of comets. Here the -disturbing condition is of a purely external character. Certain ancient -philosophers held that the moon’s rays were cold-producing, mistaking -the cold caused by radiation into space for an effect of the moon, -which is more likely to be visible at a time when the absence of clouds -permits radiation to proceed. - -In a third class of cases, *our perception of A may make our perception -of B follow*. The event B may be constantly happening, but our -attention may not be drawn to it except by our observing A. This case -seems to be illustrated by the fallacy of the moon’s influence on -clouds. The origin of this fallacy is somewhat complicated. In the -first place, when the sky is densely clouded the moon would not be -visible at all; it would be necessary for us to see the full moon in -order that our attention should be strongly drawn to the fact, and this -would happen most often on those nights when the sky is cloudless. Mr. -W. Ellis,[309] moreover, has ingeniously pointed out that there is a -general tendency for clouds to disperse at the commencement of night, -which is the time when the full moon rises. Thus the change of the sky -and the rise of the full moon are likely to attract attention mutually, -and the coincidence in time suggests the relation of cause and effect. -Mr. Ellis proves from the results of observations at the Greenwich -Observatory that the moon possesses no appreciable power of the kind -supposed, and yet it is remarkable that so sound an observer as Sir -John Herschel was convinced of the connection. In his “Results of -Observations at the Cape of Good Hope,”[310] he mentions many evenings -when a full moon occurred with a peculiarly clear sky. - - [309] *Philosophical Magazine*, 4th Series (1867), vol. xxxiv. p. 64. - - [310] See *Notes to Measures of Double Stars*, 1204, 1336, 1477, - 1686, 1786, 1816, 1835, 1929, 2081, 2186, pp. 265, &c. See also - Herschel’s *Familiar Lectures on Scientific Subjects*, p. 147, and - *Outlines of Astronomy*, 7th ed. p. 285. - -There is yet a fourth class of cases, in which *B is really the -antecedent event, but our perception of A, which is a consequence -of B, may be necessary to bring about our perception of B*. There -can be no doubt, for instance, that upward and downward currents are -continually circulating in the lowest stratum of the atmosphere during -the day-time; but owing to the transparency of the atmosphere we have -no evidence of their existence until we perceive cumulous clouds, which -are the consequence of such currents. In like manner an interfiltration -of bodies of air in the higher parts of the atmosphere is probably in -nearly constant progress, but unless threads of cirrous cloud indicate -these motions we remain ignorant of their occurrence.[311] The highest -strata of the atmosphere are wholly imperceptible to us, except when -rendered luminous by auroral currents of electricity, or by the passage -of meteoric stones. Most of the visible phenomena of comets probably -arise from some substance which, existing previously invisible, becomes -condensed or electrified suddenly into a visible form. Sir John -Herschel attempted to explain the production of comet tails in this -manner by evaporation and condensation.[312] - - [311] Jevons, *On the Cirrous Form of Cloud*, Philosophical Magazine, - July, 1857, 4th Series, vol. xiv. p. 22. - - [312] *Astronomy*, 4th ed. p. 358. - - -*Negative Arguments from Non-observation.* - -From what has been suggested in preceding sections, it will plainly -appear that the non-observation of a phenomenon is not generally to -be taken as proving its non-occurrence. As there are sounds which we -cannot hear, rays of heat which we cannot feel, multitudes of worlds -which we cannot see, and myriads of minute organisms of which not the -most powerful microscope can give us a view, we must as a general rule -interpret our experience in an affirmative sense only. Accordingly -when inferences have been drawn from the non-occurrence of particular -facts or objects, more extended and careful examination has often -proved their falsity. Not many years since it was quite a well credited -conclusion in geology that no remains of man were found in connection -with those of extinct animals, or in any deposit not actually at -present in course of formation. Even Babbage accepted this conclusion -as strongly confirmatory of the Mosaic accounts.[313] While the opinion -was yet universally held, flint implements had been found disproving -such a conclusion, and overwhelming evidence of man’s long-continued -existence has since been forthcoming. At the end of the last century, -when Herschel had searched the heavens with his powerful telescopes, -there seemed little probability that planets yet remained unseen -within the orbit of Jupiter. But on the first day of this century such -an opinion was overturned by the discovery of Ceres, and more than a -hundred other small planets have since been added to the lists of the -planetary system. - - [313] Babbage, *Ninth Bridgewater Treatise*, p. 67. - -The discovery of the Eozoön Canadense in strata of much greater age -than any previously known to contain organic remains, has given a shock -to groundless opinions concerning the origin of organic forms; and -the oceanic dredging expeditions under Dr. Carpenter and Sir Wyville -Thomson have modified some opinions of geologists by disclosing the -continued existence of forms long supposed to be extinct. These and -many other cases which might be quoted show the extremely unsafe -character of negative inductions. - -But it must not be supposed that negative arguments are of no force and -value. The earth’s surface has been sufficiently searched to render it -highly improbable that any terrestrial animals of the size of a camel -remain to be discovered. It is believed that no new large animal has -been encountered in the last eighteen or twenty centuries,[314] and -the probability that if existent they would have been seen, increases -the probability that they do not exist. We may with somewhat less -confidence discredit the existence of any large unrecognised fish, or -sea animals, such as the alleged sea-serpent. But, as we descend to -forms of smaller size negative evidence loses weight from the less -probability of our seeing smaller objects. Even the strong induction in -favour of the four-fold division of the animal kingdom into Vertebrata, -Annulosa, Mollusca, and Cœlenterata, may break down by the discovery -of intermediate or anomalous forms. As civilisation spreads over the -surface of the earth, and unexplored tracts are gradually diminished, -negative conclusions will increase in force; but we have much to learn -yet concerning the depths of the ocean, almost wholly unexamined as -they are, and covering three-fourths of the earth’s surface. - - [314] Cuvier, *Essay on the Theory of the Earth*, translation, p. 61, - &c. - -In geology there are many statements to which considerable probability -attaches on account of the large extent of the investigations already -made, as, for instance, that true coal is found only in rocks of a -particular geological epoch; that gold occurs in secondary and tertiary -strata only in exceedingly small quantities,[315] probably derived -from the disintegration of earlier rocks. In natural history negative -conclusions are exceedingly treacherous and unsatisfactory. The utmost -patience will not enable a microscopist or the observer of any living -thing to watch the behaviour of the organism under all circumstances -continuously for a great length of time. There is always a chance -therefore that the critical act or change may take place when the -observer’s eyes are withdrawn. This certainly happens in some cases; -for though the fertilisation of orchids by agency of insects is proved -as well as any fact in natural history, Mr. Darwin has never been -able by the closest watching to detect an insect in the performance -of the operation. Mr. Darwin has himself adopted one conclusion on -negative evidence, namely, that the *Orchis pyramidalis* and certain -other orchidaceous flowers secrete no nectar. But his caution and -unwearying patience in verifying the conclusion give an impressive -lesson to the observer. For twenty-three consecutive days, as he tells -us, he examined flowers in all states of the weather, at all hours, in -various localities. As the secretion in other flowers sometimes takes -place rapidly and might happen at early dawn, that inconvenient hour -of observation was specially adopted. Flowers of different ages were -subjected to irritating vapours, to moisture, and to every condition -likely to bring on the secretion; and only after invariable failure of -this exhaustive inquiry was the barrenness of the nectaries assumed to -be proved.[316] - - [315] Murchison’s *Siluria*, 1st ed. p. 432. - - [316] Darwin’s *Fertilisation of Orchids*, p. 48. - -In order that a negative argument founded on the non-observation of -an object shall have any considerable force, it must be shown to be -probable that the object if existent would have been observed, and it -is this probability which defines the value of the negative conclusion. -The failure of astronomers to see the planet Vulcan, supposed by some -to exist within Mercury’s orbit, is no sufficient disproof of its -existence. Similarly it would be very difficult, or even impossible, to -disprove the existence of a second satellite of small size revolving -round the earth. But if any person make a particular assertion, -assigning place and time, then observation will either prove or -disprove the alleged fact. If it is true that when a French observer -professed to have seen a planet on the sun’s face, an observer in -Brazil was carefully scrutinising the sun and failed to see it, we have -a negative proof. False facts in science, it has been well said, are -more mischievous than false theories. A false theory is open to every -person’s criticism, and is ever liable to be judged by its accordance -with facts. But a false or grossly erroneous assertion of a fact -often stands in the way of science for a long time, because it may be -extremely difficult or even impossible to prove the falsity of what has -been once recorded. - -In other sciences the force of a negative argument will often depend -upon the number of possible alternatives which may exist. It was long -believed that the quality of a musical sound as distinguished from -its pitch, must depend upon the form of the undulation, because no -other cause of it had ever been suggested or was apparently possible. -The truth of the conclusion was proved by Helmholtz, who applied a -microscope to luminous points attached to the strings of various -instruments, and thus actually observed the different modes of -undulation. In mathematics negative inductive arguments have seldom -much force, because the possible forms of expression, or the possible -combinations of lines and circles in geometry, are quite unlimited in -number. An enormous number of attempts were made to trisect the angle -by the ordinary methods of Euclid’s geometry, but their invariable -failure did not establish the impossibility of the task. This was shown -in a totally different manner, by proving that the problem involves an -irreducible cubic equation to which there could be no corresponding -plane geometrical solution.[317] This is a case of *reductio ad -absurdum*, a form of argument of a totally different character. -Similarly no number of failures to obtain a general solution of -equations of the fifth degree would establish the impossibility of the -task, but in an indirect mode, equivalent to a *reductio ad absurdum*, -the impossibility is considered to be proved.[318] - - [317] Peacock, *Algebre*, vol. ii. p. 344. - - [318] Ibid, p. 359. Serret, *Algèbre Supérieure*, 2nd ed. p. 304. - - - - -CHAPTER XIX. - -EXPERIMENT. - - -We may now consider the great advantages which we enjoy in examining -the combinations of phenomena when things are within our reach and -capable of being experimented on. We are said *to experiment* when we -bring substances together under various conditions of temperature, -pressure, electric disturbance, chemical action, &c., and then record -the changes observed. Our object in inductive investigation is to -ascertain exactly the group of circumstances or conditions which being -present, a certain other group of phenomena will follow. If we denote -by A the antecedent group, and by X subsequent phenomena, our object -will usually be to discover a law of the form A = AX, the meaning of -which is that where A is X will happen. - -The circumstances which might be enumerated as present in the simplest -experiment are very numerous, in fact almost infinite. Rub two sticks -together and consider what would be an exhaustive statement of the -conditions. There are the form, hardness, organic structure, and all -the chemical qualities of the wood; the pressure and velocity of the -rubbing; the temperature, pressure, and all the chemical qualities of -the surrounding air; the proximity of the earth with its attractive -and electric powers; the temperature and other properties of the -persons producing motion; the radiation from the sun, and to and from -the sky; the electric excitement possibly existing in any overhanging -cloud; even the positions of the heavenly bodies must be mentioned. -On *à priori* grounds it is unsafe to assume that any one of these -circumstances is without effect, and it is only by experience that we -can single out those precise conditions from which the observed heat of -friction proceeds. - -The great method of experiment consists in removing, one at a time, -each of those conditions which may be imagined to have an influence -on the result. Our object in the experiment of rubbing sticks is to -discover the exact circumstances under which heat appears. Now the -presence of air may be requisite; therefore prepare a vacuum, and -rub the sticks in every respect as before, except that it is done -*in vacuo*. If heat still appears we may say that air is not, in -the presence of the other circumstances, a requisite condition. The -conduction of heat from neighbouring bodies may be a condition. Prevent -this by making all the surrounding bodies ice cold, which is what Davy -aimed at in rubbing two pieces of ice together. If heat still appears -we have eliminated another condition, and so we may go on until it -becomes apparent that the expenditure of energy in the friction of two -bodies is the sole condition of the production of heat. - -The great difficulty of experiment arises from the fact that we must -not assume the conditions to be independent. Previous to experiment we -have no right to say that the rubbing of two sticks will produce heat -in the same way when air is absent as before. We may have heat produced -in one way when air is present, and in another when air is absent. -The inquiry branches out into two lines, and we ought to try in both -cases whether cutting off a supply of heat by conduction prevents its -evolution in friction. The same branching out of the inquiry occurs -with regard to every circumstance which enters into the experiment. - -Regarding only four circumstances, say A, B, C, D, we ought to test -not only the combinations ABCD, ABC*d*, AB*c*D, A*b*CD, *a*BCD, but -we ought really to go through the whole of the combinations given in -the fifth column of the Logical Alphabet. The effect of the absence -of each condition should be tried both in the presence and absence of -every other condition, and every selection of those conditions. Perfect -and exhaustive experimentation would, in short, consist in examining -natural phenomena in all their possible combinations and registering -all relations between conditions and results which are found capable -of existence. It would thus resemble the exclusion of contradictory -combinations carried out in the Indirect Method of Inference, except -that the exclusion of combinations is grounded not on prior logical -premises, but on *à posteriori* results of actual trial. - -The reader will perceive, however, that such exhaustive investigation -is practically impossible, because the number of requisite experiments -would be immensely great. Four antecedents only would require sixteen -experiments; twelve antecedents would require 4096, and the number -increases as the powers of two. The result is that the experimenter -has to fall back upon his own tact and experience in selecting those -experiments which are most likely to yield him significant facts. It -is at this point that logical rules and forms begin to fail in giving -aid. The logical rule is--Try all possible combinations; but this being -impracticable, the experimentalist necessarily abandons strict logical -method, and trusts to his own insight. Analogy, as we shall see, gives -some assistance, and attention should be concentrated on those kinds -of conditions which have been found important in like cases. But we -are now entirely in the region of probability, and the experimenter, -while he is confidently pursuing what he thinks the right clue, may -be overlooking the one condition of importance. It is an impressive -lesson, for instance, that Newton pursued all his exquisite researches -on the spectrum unsuspicious of the fact that if he reduced the hole in -the shutter to a narrow slit, all the mysteries of the bright and dark -lines were within his grasp, provided of course that his prisms were -sufficiently good to define the rays. In like manner we know not what -slight alteration in the most familiar experiments may not open the way -to realms of new discovery. - -Practical difficulties, also, encumber the progress of the physicist. -It is often impossible to alter one condition without altering others -at the same time; and thus we may not get the pure effect of the -condition in question. Some conditions may be absolutely incapable of -alteration; others may be with great difficulty, or only in a certain -degree, removable. A very treacherous source of error is the existence -of unknown conditions, which of course we cannot remove except by -accident. These difficulties we will shortly consider in succession. - -It is beautiful to observe how the alteration of a single circumstance -sometimes conclusively explains a phenomenon. An instance is found in -Faraday’s investigation of the behaviour of Lycopodium spores scattered -on a vibrating plate. It was observed that these minute spores -collected together at the points of greatest motion, whereas sand and -all heavy particles collected at the nodes, where the motion was least. -It happily occurred to Faraday to try the experiment in the exhausted -receiver of an air-pump, and it was then found that the light powder -behaved exactly like heavy powder. A conclusive proof was thus obtained -that the presence of air was the condition of importance, doubtless -because it was thrown into eddies by the motion of the plate, and -carried the Lycopodium to the points of greatest agitation. Sand was -too heavy to be carried by the air. - - -*Exclusion of Indifferent Circumstances.* - -From what has been already said it will be apparent that the detection -and exclusion of indifferent circumstances is a work of importance, -because it allows the concentration of attention upon circumstances -which contain the principal condition. Many beautiful instances may be -given where all the most obvious antecedents have been shown to have no -part in the production of a phenomenon. A person might suppose that the -peculiar colours of mother-of-pearl were due to the chemical qualities -of the substance. Much trouble might have been spent in following out -that notion by comparing the chemical qualities of various iridescent -substances. But Brewster accidentally took an impression from a piece -of mother-of-pearl in a cement of resin and bees’-wax, and finding -the colours repeated upon the surface of the wax, he proceeded to -take other impressions in balsam, fusible metal, lead, gum arabic, -isinglass, &c., and always found the iridescent colours the same. He -thus proved that the chemical nature of the substance is a matter of -indifference, and that the form of the surface is the real condition -of such colours.[319] Nearly the same may be said of the colours -exhibited by thin plates and films. The rings and lines of colour will -be nearly the same in character whatever may be the nature of the -substance; nay, a void space, such as a crack in glass, would produce -them even though the air were withdrawn by an air-pump. The conditions -are simply the existence of two reflecting surfaces separated by a very -small space, though it should be added that the refractive index of the -intervening substance has some influence on the exact nature of the -colour produced. - - [319] *Treatise on Optics*, by Brewster, Cab. Cyclo. p. 117. - -When a ray of light passes close to the edge of an opaque body, a -portion of the light appears to be bent towards it, and produces -coloured fringes within the shadow of the body. Newton attributed -this inflexion of light to the attraction of the opaque body for the -supposed particles of light, although he was aware that the nature -of the surrounding medium, whether air or other pellucid substance, -exercised no apparent influence on the phenomena. Gravesande proved, -however, that the character of the fringes is exactly the same, whether -the body be dense or rare, compound or elementary. A wire produces -exactly the same fringes as a hair of the same thickness. Even the -form of the obstructing edge was subsequently shown to be a matter of -indifference by Fresnel, and the interference spectrum, or the spectrum -seen when light passes through a fine grating, is absolutely the same -whatever be the form or chemical nature of the bars making the grating. -Thus it appears that the stoppage of a portion of a beam of light is -the sole necessary condition for the diffraction or inflexion of light, -and the phenomenon is shown to bear no analogy the refraction of light, -in which the form and nature of the substance are all important. - -It is interesting to observe how carefully Newton, in his researches -on the spectrum, ascertained the indifference of many circumstances by -actual trial. He says:[320] “Now the different magnitude of the hole -in the window-shut, and different thickness of the prism where the -rays passed through it, and different inclinations of the prism to the -horizon, made no sensible changes in the length of the image. Neither -did the different matter of the prisms make any: for in a vessel made -of polished plates of glass cemented together in the shape of a prism, -and filled with water, there is the like success of the experiment -according to the quantity of the refraction.” But in the latter -statement, as I shall afterwards remark (p. 432), Newton assumed an -indifference which does not exist, and fell into an unfortunate mistake. - - [320] *Opticks*, 3rd. ed. p. 25. - -In the science of sound it is shown that the pitch of a sound depends -solely upon the number of impulses in a second, and the material -exciting those impulses is a matter of indifference. Whatever fluid, -air or water, gas or liquid, be forced into the Siren, the sound -produced is the same; and the material of which an organ-pipe is -constructed does not at all affect the pitch of its sound. In the -science of statical electricity it is an important principle that -the nature of the interior of a conducting body is a matter of no -importance. The electrical charge is confined to the conducting -surface, and the interior remains in a neutral state. A hollow copper -sphere takes exactly the same charge as a solid sphere of the same -metal. - -Some of Faraday’s most elegant and successful researches were devoted -to the exclusion of conditions which previous experimenters had -thought essential for the production of electrical phenomena. Davy -asserted that no known fluids, except such as contain water, could be -made the medium of connexion between the poles of a battery; and some -chemists believed that water was an essential agent in electro-chemical -decomposition. Faraday gave abundant experiments to show that other -fluids allowed of electrolysis, and he attributed the erroneous opinion -to the very general use of water as a solvent, and its presence in most -natural bodies.[321] It was, in fact, upon the weakest kind of negative -evidence that the opinion had been founded. - -Many experimenters attributed peculiar powers to the poles of a -battery, likening them to magnets, which, by their attractive powers, -tear apart the elements of a substance. By a beautiful series of -experiments,[322] Faraday proved conclusively that, on the contrary, -the substance of the poles is of no importance, being merely the path -through which the electric force reaches the liquid acted upon. Poles -of water, charcoal, and many diverse substances, even air itself, -produced similar results; if the chemical nature of the pole entered at -all into the question, it was as a disturbing agent. - - [321] *Experimental Researches in Electricity*, vol. i. pp. 133, 134. - - [322] Ibid. vol i. pp. 127, 162, &c. - -It is an essential part of the theory of gravitation that the proximity -of other attracting particles is without effect upon the attraction -existing between any two molecules. Two pound weights weigh as much -together as they do separately. Every pair of molecules in the world -have, as it were, a private communication, apart from their relations -to all other molecules. Another undoubted result of experience pointed -out by Newton[323] is that the weight of a body does not in the least -depend upon its form or texture. It may be added that the temperature, -electric condition, pressure, state of motion, chemical qualities, -and all other circumstances concerning matter, except its mass, are -indifferent as regards its gravitating power. - - [323] *Principia*, bk. iii. Prop. vi. Corollary i. - -As natural science progresses, physicists gain a kind of insight -and tact in judging what qualities of a substance are likely to be -concerned in any class of phenomena. The physical astronomer treats -matter in one point of view, the chemist in another, and the students -of physical optics, sound, mechanics, electricity, &c., make a fair -division of the qualities among them. But errors will arise if too -much confidence be placed in this independence of various kinds of -phenomena, so that it is desirable from time to time, especially -when any unexplained discrepancies come into notice, to question the -indifference which is assumed to exist, and to test its real existence -by appropriate experiments. - - -*Simplification of Experiments.* - -One of the most requisite precautions in experimentation is to -vary only one circumstance at a time, and to maintain all other -circumstances rigidly unchanged. There are two distinct reasons for -this rule, the first and most obvious being that if we vary two -conditions at a time, and find some effect, we cannot tell whether -the effect is due to one or the other condition, or to both jointly. -A second reason is that if no effect ensues we cannot safely conclude -that either of them is indifferent; for the one may have neutralised -the effect of the other. In our symbolic logic AB ꖌ A*b* was shown to -be identical with A (p. 97), so that B denotes a circumstance which is -indifferently present or absent. But if B always goes together with -another antecedent C, we cannot show the same independence, for ABC ꖌ -A*bc* is not identical with A and none of our logical processes enables -us to reduce it to A. - -If we want to prove that oxygen is necessary to life, we must not -put a rabbit into a vessel from which the oxygen has been exhausted -by a burning candle. We should then have not only an absence of -oxygen, but an addition of carbonic acid, which may have been the -destructive agent. For a similar reason Lavoisier avoided the use of -atmospheric air in experiments on combustion, because air was not a -simple substance, and the presence of nitrogen might impede or even -alter the effect of oxygen. As Lavoisier remarks,[324] “In performing -experiments, it is a necessary principle, which ought never to be -deviated from, that they be simplified as much as possible, and that -every circumstance capable of rendering their results complicated be -carefully removed.” It has also been well said by Cuvier[325] that -the method of physical inquiry consists in isolating bodies, reducing -them to their utmost simplicity, and bringing each of their properties -separately into action, either mentally or by experiment. - - [324] Lavoisier’s *Chemistry*, translated by Kerr, p. 103. - - [325] Cuvier’s *Animal Kingdom*, introduction, pp. 1, 2. - -The electro-magnet has been of the utmost service in the investigation -of the magnetic properties of matter, by allowing of the production -or removal of a most powerful magnetic force without disturbing any -of the other arrangements of the experiment. Many of Faraday’s most -valuable experiments would have been impossible had it been necessary -to introduce a heavy permanent magnet, which could not be suddenly -moved without shaking the whole apparatus, disturbing the air, -producing currents by changes of temperature, &c. The electro-magnet -is perfectly under control, and its influence can be brought into -action, reversed, or stopped by merely touching a button. Thus Faraday -was enabled to prove the rotation of the plane of circularly polarised -light by the fact that certain light ceased to be visible when the -electric current of the magnet was cut off, and re-appeared when the -current was made. “These phenomena,” he says, “could be reversed at -pleasure, and at any instant of time, and upon any occasion, showing a -perfect dependence of cause and effect.”[326] - - [326] *Experimental Researches in Electricity*, vol. iii. p. 4. - -It was Newton’s omission to obtain the solar spectrum under the -simplest conditions which prevented him from discovering the dark -lines. Using a broad beam of light which had passed through a round -hole or a triangular slit, he obtained a brilliant spectrum, but one -in which many different coloured rays overlapped each other. In the -recent history of the science of the spectrum, one main difficulty has -consisted in the mixture of the lines of several different substances, -which are usually to be found in the light of any flame or spark. It -is seldom possible to obtain the light of any element in a perfectly -simple manner. Angström greatly advanced this branch of science by -examining the light of the electric spark when formed between poles of -various metals, and in the presence of various gases. By varying the -pole alone, or the gaseous medium alone, he was able to discriminate -correctly between the lines due to the metal and those due to the -surrounding gas.[327] - - [327] *Philosophical Magazine*, 4th Series, vol. ix. p. 327. - - -*Failure in the Simplification of Experiments.* - -In some cases it seems to be impossible to carry out the rule of -varying one circumstance at a time. When we attempt to obtain two -instances or two forms of experiment in which a single circumstance -shall be present in one case and absent in another, it may be found -that this single circumstance entails others. Benjamin Franklin’s -experiment concerning the comparative absorbing powers of different -colours is well known. “I took,” he says, “a number of little square -pieces of broadcloth from a tailor’s pattern card, of various colours. -They were black, deep blue, lighter blue, green, purple, red, yellow, -white, and other colours and shades of colour. I laid them all out upon -the snow on a bright sunshiny morning. In a few hours the black, being -most warmed by the sun, was sunk so low as to be below the stroke of -the sun’s rays; the dark blue was almost as low; the lighter blue not -quite so much as the dark; the other colours less as they were lighter. -The white remained on the surface of the snow, not having entered it at -all.” This is a very elegant and apparently simple experiment; but when -Leslie had completed his series of researches upon the nature of heat, -he came to the conclusion that the colour of a surface has very little -effect upon the radiating power, the mechanical nature of the surface -appearing to be more influential. He remarks[328] that “the question -is incapable of being positively resolved, since no substance can be -made to assume different colours without at the same time changing its -internal structure.” Recent investigation has shown that the subject -is one of considerable complication, because the absorptive power of a -surface may be different according to the character of the rays which -fall upon it; but there can be no doubt as to the acuteness with which -Leslie points out the difficulty. In Well’s investigations concerning -the nature of dew, we have, again, very complicated conditions. If we -expose plates of various material, such as rough iron, glass, polished -metal, to the midnight sky, they will be dewed in various degrees; but -since these plates differ both in the nature of the surface and the -conducting power of the material, it would not be plain whether one -or both circumstances were of importance. We avoid this difficulty by -exposing the same material polished or varnished, so as to present -different conditions of surface;[329] and again by exposing different -substances with the same kind of surface. - - [328] *Inquiry into the Nature of Heat*, p. 95. - - [329] Herschel, *Preliminary Discourse*, p. 161. - -When we are quite unable to isolate circumstances we must resort to -the procedure described by Mill under the name of the Joint Method -of Agreement and Difference. We must collect as many instances as -possible in which a given circumstance produces a given result, and -as many as possible in which the absence of the circumstance is -followed by the absence of the result. To adduce his example, we -cannot experiment upon the cause of double refraction in Iceland spar, -because we cannot alter its crystalline condition without altering -it altogether, nor can we find substances exactly like calc spar in -every circumstance except one. We resort therefore to the method -of comparing together all known substances which have the property -of doubly-refracting light, and we find that they agree in being -crystalline.[330] This indeed is nothing but an ordinary process of -perfect or probable induction, already partially described, and to -be further discussed under Classification. It may be added that the -subject does admit of perfect experimental treatment, since glass, -when compressed in one direction, becomes capable of doubly-refracting -light, and as there is probably no alteration in the glass but change -of elasticity, we learn that the power of double refraction is probably -due to a difference of elasticity in different directions. - - [330] *System of Logic*, bk. iii. chap. viii. § 4, 5th ed. vol. i. - p. 433. - - -*Removal of Usual Conditions.* - -One of the great objects of experiment is to enable us to judge -of the behaviour of substances under conditions widely different -from those which prevail upon the surface of the earth. We live in -an atmosphere which does not vary beyond certain narrow limits in -temperature or pressure. Many of the powers of nature, such as gravity, -which constantly act upon us, are of almost fixed amount. Now it -will afterwards be shown that we cannot apply a quantitative law to -circumstances much differing from those in which it was observed. In -the other planets, the sun, the stars, or remote parts of the Universe, -the conditions of existence must often be widely different from what -we commonly experience here. Hence our knowledge of nature must remain -restricted and hypothetical, unless we can subject substances to -unusual conditions by suitable experiments. - -The electric arc is an invaluable means of exposing metals or other -conducting substances to the highest known temperature. By its aid we -learn not only that all the metals can be vaporised, but that they -all give off distinctive rays of light. At the other extremity of the -scale, the intensely powerful freezing mixture devised by Faraday, -consisting of solid carbonic acid and ether mixed *in vacuo*, enables -us to observe the nature of substances at temperatures immensely below -any we meet with naturally on the earth’s surface. - -We can hardly realise now the importance of the invention of the -air-pump, previous to which invention it was exceedingly difficult -to experiment except under the ordinary pressure of the atmosphere. -The Torricellian vacuum had been employed by the philosophers of the -Accademia del Cimento to show the behaviour of water, smoke, sound, -magnets, electric substances, &c., *in vacuo*, but their experiments -were often unsuccessful from the difficulty of excluding air.[331] - - [331] *Essayes of Natural Experiments made in the Accademia del - Cimento.* Englished by Richard Waller, 1684, p. 40, &c. - -Among the most constant circumstances under which we live is the force -of gravity, which does not vary, except by a slight fraction of its -amount, in any part of the earth’s crust or atmosphere to which we -can attain. This force is sufficient to overbear and disguise various -actions, for instance, the mutual gravitation of small bodies. It -was an interesting experiment of Plateau to neutralise the action of -gravity by placing substances in liquids of exactly the same specific -gravity. Thus a quantity of oil poured into the middle of a suitable -mixture of alcohol and water assumes a spherical shape; on being made -to rotate it becomes spheroidal, and then successively separates into a -ring and a group of spherules. Thus we have an illustration of the mode -in which the planetary system may have been produced,[332] though the -extreme difference of scale prevents our arguing with confidence from -the experiment to the conditions of the nebular theory. - - [332] Plateau, *Taylor’s Scientific Memoirs*, vol. iv. pp. 16–43. - -It is possible that the so-called elements are elementary only to us, -because we are restricted to temperatures at which they are fixed. -Lavoisier carefully defined an element as a substance which cannot be -decomposed *by any known means*; but it seems almost certain that some -series of elements, for instance Iodine, Bromine, and Chlorine, are -really compounds of a simpler substance. We must look to the production -of intensely high temperatures, yet quite beyond our means, for the -decomposition of these so-called elements. Possibly in this age and -part of the universe the dissipation of energy has so far proceeded -that there are no sources of heat sufficiently intense to effect the -decomposition. - - -*Interference of Unsuspected Conditions.* - -It may happen that we are not aware of all the conditions under which -our researches are made. Some substance may be present or some power -may be in action, which escapes the most vigilant examination. Not -being aware of its existence, we are unable to take proper measures -to exclude it, and thus determine the share which it has in the -results of our experiments. There can be no doubt that the alchemists -were misled and encouraged in their vain attempts by the unsuspected -presence of traces of gold and silver in the substances they proposed -to transmute. Lead, as drawn from the smelting furnace, almost always -contains some silver, and gold is associated with many other metals. -Thus small quantities of noble metal would often appear as the result -of experiment and raise delusive hopes. - -In more than one case the unsuspected presence of common salt in the -air has caused great trouble. In the early experiments on electrolysis -it was found that when water was decomposed, an acid and an alkali -were produced at the poles, together with oxygen and hydrogen. In -the absence of any other explanation, some chemists rushed to the -conclusion that electricity must have the power of *generating* -acids and alkalies, and one chemist thought he had discovered a new -substance called *electric acid*. But Davy proceeded to a systematic -investigation of the circumstances, by varying the conditions. Changing -the glass vessel for one of agate or gold, he found that far less -alkali was produced; excluding impurities by the use of carefully -distilled water, he found that the quantities of acid and alkali were -still further diminished; and having thus obtained a clue to the cause, -he completed the exclusion of impurities by avoiding contact with his -fingers, and by placing the apparatus under an exhausted receiver, -no acid or alkali being then detected. It would be difficult to meet -with a more elegant case of the detection of a condition previously -unsuspected.[333] - - [333] *Philosophical Transactions* [1826], vol. cxvi. pp. 388, 389. - Works of Sir Humphry Davy, vol. v. pp. 1–12. - -It is remarkable that the presence of common salt in the air, proved -to exist by Davy, nevertheless continued a stumbling-block in the -science of spectrum analysis, and probably prevented men, such as -Brewster, Herschel, and Talbot, from anticipating by thirty years -the discoveries of Bunsen and Kirchhoff. As I pointed out,[334] the -utility of the spectrum was known in the middle of the last century -to Thomas Melvill, a talented Scotch physicist, who died at the early -age of 27 years.[335] But Melvill was struck in his examination of -coloured flames by the extraordinary predominance of homogeneous yellow -light, which was due to some circumstance escaping his attention. -Wollaston and Fraunhofer were equally struck by the prominence of the -yellow line in the spectrum of nearly every kind of light. Talbot -expressly recommended the use of the prism for detecting the presence -of substances by what we now call spectrum analysis, but he found that -all substances, however different the light they yielded in other -respects, were identical as regards the production of yellow light. -Talbot knew that the salts of soda gave this coloured light, but in -spite of Davy’s previous difficulties with salt in electrolysis, it -did not occur to him to assert that where the light is, there sodium -must be. He suggested water as the most likely source of the yellow -light, because of its frequent presence; but even substances which were -apparently devoid of water gave the same yellow light.[336] Brewster -and Herschel both experimented upon flames almost at the same time as -Talbot, and Herschel unequivocally enounced the principle of spectrum -analysis.[337] Nevertheless Brewster, after numerous experiments -attended with great trouble and disappointment, found that yellow light -might be obtained from the combustion of almost any substance. It -was not until 1856 that Swan discovered that an almost infinitesimal -quantity of sodium chloride, say a millionth part of a grain, was -sufficient to tinge a flame of a bright yellow colour. The universal -diffusion of the salts of sodium, joined to this unique light-producing -power, was thus shown to be the unsuspected condition which had -destroyed the confidence of all previous experimenters in the use of -the prism. Some references concerning the history of this curious point -are given below.[338] - - [334] *National Review*, July, 1861, p. 13. - - [335] His published works are contained in *The Edinburgh Physical - and Literary Essays*, vol. ii. p. 34; *Philosophical Transactions* - [1753], vol. xlviii. p. 261; see also Morgan’s Papers in - *Philosophical Transactions* [1785], vol. lxxv. p. 190. - - [336] *Edinburgh Journal of Science*, vol. v. p. 79. - - [337] *Encyclopædia Metropolitana*, art. *Light*, § 524; Herschel’s - *Familiar Lectures*, p. 266. - - [338] Talbot, *Philosophical Magazine*, 3rd Series, vol. ix. p. 1 - (1836); Brewster, *Transactions of the Royal Society of Edinburgh* - [1823], vol. ix. pp. 433, 455; Swan, ibid. [1856] vol. xxi. p. 411; - *Philosophical Magazine*, 4th Series, vol. xx. p. 173 [Sept. 1860]; - Roscoe, *Spectrum Analysis*, Lecture III. - -In the science of radiant heat, early inquirers were led to the -conclusion that radiation proceeded only from the surface of a solid, -or from a very small depth below it. But they happened to experiment -upon surfaces covered by coats of varnish, which is highly athermanous -or opaque to heat. Had they properly varied the character of the -surface, using a highly diathermanous substance like rock salt, they -would have obtained very different results.[339] - -One of the most extraordinary instances of an erroneous opinion due -to overlooking interfering agents is that concerning the increase of -rainfall near to the earth’s surface. More than a century ago it was -observed that rain-gauges placed upon church steeples, house tops, and -other elevated places, gave considerably less rain than if they were -on the ground, and it has been recently shown that the variation is -most rapid in the close neighbourhood of the ground.[340] All kinds -of theories have been started to explain this phenomenon; but I have -shown[341] that it is simply due to the interference of wind, which -deflects more or less rain from all the gauges which are exposed to it. - - [339] Balfour Stewart, *Elementary Treatise on Heat*, p. 192. - - [340] British Association, Liverpool, 1870. *Report on Rainfall*, - p. 176. - - [341] *Philosophical Magazine.*, Dec. 1861. 4th Series, vol. xxii. - p. 421. - -The great magnetic power of iron renders it a source of disturbance in -magnetic experiments. In building a magnetic observatory great care -must therefore be taken that no iron is employed in the construction, -and that no masses of iron are near at hand. In some cases magnetic -observations have been seriously disturbed by the existence of masses -of iron ore in the neighbourhood. In Faraday’s experiments upon feebly -magnetic or diamagnetic substances he took the greatest precautions -against the presence of disturbing substances in the copper wire, wax, -paper, and other articles used in suspending the test objects. It was -his custom to try the effect of the magnet upon the apparatus in the -absence of the object of experiment, and without this preliminary trial -no confidence could be placed in the results.[342] Tyndall has also -employed the same mode for testing the freedom of electro-magnetic -coils from iron, and was thus enabled to obtain them devoid of any -cause of disturbance.[343] It is worthy of notice that in the very -infancy of the science of magnetism, the acute experimentalist Gilbert -correctly accounted for the opinion existing in his day that magnets -would attract silver, by pointing out that the silver contained iron. - - [342] *Experimental Researches in Electricity*, vol. iii. p. 84, &c. - - [343] *Lectures on Heat*, p. 21. - -Even when we are not aware by previous experience of the probable -presence of a special disturbing agent, we ought not to assume the -absence of unsuspected interference. If an experiment is of really -high importance, so that any considerable branch of science rests -upon it, we ought to try it again and again, in as varied conditions -as possible. We should intentionally disturb the apparatus in various -ways, so as if possible to hit by accident upon any weak point. -Especially when our results are more regular than we have fair grounds -for anticipating, ought we to suspect some peculiarity in the apparatus -which causes it to measure some other phenomenon than that in question, -just as Foucault’s pendulum almost always indicates the movement of the -axes of its own elliptic path instead of the rotation of the globe. - -It was in this cautious spirit that Baily acted in his experiments on -the density of the earth. The accuracy of his results depended upon the -elimination of all disturbing influences, so that the oscillation of -his torsion balance should measure gravity alone. Hence he varied the -apparatus in many ways, changing the small balls subject to attraction, -changing the connecting rod, and the means of suspension. He observed -the effect of disturbances, such as the presence of visitors, the -occurrence of violent storms, &c., and as no real alteration was -produced in the results, he confidently attributed them to gravity.[344] - - [344] Baily, *Memoirs of the Royal Astronomical Society*, vol. xiv. - pp. 29, 30. - -Newton would probably have discovered the mode of constructing -achromatic lenses, but for the unsuspected effect of some sugar of -lead which he is supposed to have dissolved in the water of a prism. -He tried, by means of a glass prism combined with a water prism, to -produce dispersion of light without refraction, and if he had succeeded -there would have been an obvious mode of producing refraction without -dispersion. His failure is attributed to his adding lead acetate to -the water for the purpose of increasing its refractive power, the lead -having a high dispersive power which frustrated his purpose.[345] -Judging from Newton’s remarks, in the *Philosophical Transactions*, -it would appear as if he had not, without many unsuccessful trials, -despaired of the construction of achromatic glasses.[346] - - [345] Grant, *History of Physical Astronomy*, p. 531. - - [346] *Philosophical Transactions*, abridged by Lowthorp, 4th - edition, vol. i. p. 202. - -The Academicians of Cimento, in their early and ingenious experiments -upon the vacuum, were often misled by the mechanical imperfections -of their apparatus. They concluded that the air had nothing to do -with the production of sounds, evidently because their vacuum was not -sufficiently perfect. Otto von Guericke fell into a like mistake in the -use of his newly-constructed air-pump, doubtless from the unsuspected -presence of air sufficiently dense to convey the sound of the bell. - -It is hardly requisite to point out that the doctrine of spontaneous -generation is due to the unsuspected presence of germs, even after the -most careful efforts to exclude them, and in the case of many diseases, -both of animals and plants, germs which we have no means as yet of -detecting are doubtless the active cause. It has long been a subject -of dispute, again, whether the plants which spring from newly turned -land grow from seeds long buried in that land, or from seeds brought -by the wind. Argument is unphilosophical when direct trial can readily -be applied; for by turning up some old ground, and covering a portion -of it with a glass case, the conveyance of seeds by the wind can be -entirely prevented, and if the same plants appear within and without -the case, it will become clear that the seeds are in the earth. By -gross oversight some experimenters have thought before now that crops -of rye had sprung up where oats had been sown. - - -*Blind or Test Experiments.* - -Every conclusive experiment necessarily consists in the comparison of -results between two different combinations of circumstances. To give a -fair probability that A is the cause of X, we must maintain invariable -all surrounding objects and conditions, and we must then show that -where A is X is, and where A is not X is not. This cannot really be -accomplished in a single trial. If, for instance, a chemist places -a certain suspected substance in Marsh’s test apparatus, and finds -that it gives a small deposit of metallic arsenic, he cannot be sure -that the arsenic really proceeds from the suspected substance; the -impurity of the zinc or sulphuric acid may have been the cause of its -appearance. It is therefore the practice of chemists to make what they -call a *blind experiment*, that is to try whether arsenic appears in -the absence of the suspected substance. The same precaution ought to be -taken in all important analytical operations. Indeed, it is not merely -a precaution, it is an essential part of any experiment. If the blind -trial be not made, the chemist merely assumes that he knows what would -happen. Whenever we assert that because A and X are found together A -is the cause of X, we assume that if A were absent X would be absent. -But wherever it is possible, we ought not to take this as a mere -assumption, or even as a matter of inference. Experience is ultimately -the basis of all our inferences, but if we can bring immediate -experience to bear upon the point in question we should not trust to -anything more remote and liable to error. When Faraday examined the -magnetic properties of the bearing apparatus, in the absence of the -substance to be experimented on, he really made a blind experiment -(p. 431). - -We ought, also, to test the accuracy of a method of experiment whenever -we can, by introducing known amounts of the substance or force to be -detected. A new analytical process for the quantitative estimation of -an element should be tested by performing it upon a mixture compounded -so as to contain a known quantity of that element. The accuracy of the -gold assay process greatly depends upon the precaution of assaying -alloys of gold of exactly known composition.[347] Gabriel Plattes’ -works give evidence of much scientific spirit, and when discussing the -supposed merits of the divining rod for the discovery of subterranean -treasure, he sensibly suggests that the rod should be tried in places -where veins of metal are known to exist.[348] - - [347] Jevons in Watts’ *Dictionary of Chemistry*, vol. ii. pp. 936, - 937. - - [348] *Discovery of Subterraneal Treasure.* London, 1639, p. 48. - - -*Negative Results of Experiment.* - -When we pay proper regard to the imperfection of all measuring -instruments and the possible minuteness of effects, we shall see -much reason for interpreting with caution the negative results of -experiments. We may fail to discover the existence of an expected -effect, not because that effect is really non-existent, but because it -is of a magnitude inappreciable to our senses, or confounded with other -effects of much greater amount. As there is no limit on *à priori* -grounds to the smallness of a phenomenon, we can never, by a single -experiment, prove the non-existence of a supposed effect. We are always -at liberty to assume that a certain amount of effect might have been -detected by greater delicacy of measurement. We cannot safely affirm -that the moon has no atmosphere at all. We may doubtless show that the -atmosphere, if present, is less dense than the air in the so-called -vacuum of an air-pump, as did Du Sejour. It is equally impossible to -prove that gravity occupies *no time* in transmission. Laplace indeed -ascertained that the velocity of propagation of the influence was at -least fifty million times greater than that of light;[349] but it does -not really follow that it is instantaneous; and were there any means -of detecting the action of one star upon another exceedingly distant -star, we might possibly find an appreciable interval occupied in the -transmission of the gravitating impulse. Newton could not demonstrate -the absence of all resistance to matter moving through empty space; but -he ascertained by an experiment with the pendulum (p. 443), that if -such resistance existed, it was in amount less than one five-thousandth -part of the external resistance of the air.[350] - - [349] Laplace, *System of the World*, translated by Harte, vol. ii. - p. 322. - - [350] *Principia*, bk. ii. sect. 6, Prop. xxxi. Motte’s translation, - vol. ii. p. 108. - -A curious instance of false negative inference is furnished by -experiments on light. Euler rejected the corpuscular theory on the -ground that particles of matter moving with the immense velocity of -light would possess momentum, of which there was no evidence. Bennet -had attempted to detect the momentum of light by concentrating the rays -of the sun upon a delicately balanced body. Observing no result, it -was considered to be proved that light had no momentum. Mr. Crookes, -however, having suspended thin vanes, blacked on one side, in a nearly -vacuous globe, found that they move under the influence of light. It -is now allowed that this effect can be explained in accordance with -the undulatory theory of light, and the molecular theory of gases. It -comes to this--that Bennet failed to detect an effect which he might -have detected with a better method of experimenting; but if he had -found it, the phenomenon would have confirmed, not the corpuscular -theory of light, as was expected, but the rival undulatory theory. The -conclusion drawn from Bennet’s experiment was falsely drawn, but it was -nevertheless true in matter. - -Many incidents in the history of science tend to show that phenomena, -which one generation has failed to discover, may become accurately -known to a succeeding generation. The compressibility of water which -the Academicians of Florence could not detect, because at a low -pressure the effect was too small to perceive, and at a high pressure -the water oozed through their silver vessel,[351] has now become the -subject of exact measurement and precise calculation. Independently of -Newton, Hooke entertained very remarkable notions concerning the nature -of gravitation. In this and other subjects he showed, indeed, a genius -for experimental investigation which would have placed him in the first -rank in any other age than that of Newton. He correctly conceived that -the force of gravity would decrease as we recede from the centre of -the earth, and he boldly attempted to prove it by experiment. Having -exactly counterpoised two weights in the scales of a balance, or rather -one weight against another weight and a long piece of fine cord, he -removed his balance to the top of the dome of St. Paul’s, and tried -whether the balance remained in equilibrium after one weight was -allowed to hang down to a depth of 240 feet. No difference could be -perceived when the weights were at the same and at different levels, -but Hooke rightly held that the failure arose from the insufficient -elevation. He says, “Yet I am apt to think some difference might be -discovered in greater heights.”[352] The radius of the earth being -about 20,922,000 feet, we can now readily calculate from the law of -gravity that a height of 240 would not make a greater difference than -one part in 40,000 of the weight. Such a difference would doubtless -be inappreciable in the balances of that day, though it could readily -be detected by balances now frequently constructed. Again, the -mutual gravitation of bodies at the earth’s surface is so small that -Newton appears to have made no attempt to demonstrate its existence -experimentally, merely remarking that it was too small to fall under -the observation of our senses.[353] It has since been successfully -detected and measured by Cavendish, Baily, and others. - - [351] *Essayes of Natural Experiments*, &c. p. 117. - - [352] Hooke’s *Posthumous Works*, p. 182. - - [353] *Principia*, bk. iii. Prop. vii. Corollary 1. - -The smallness of the quantities which we can sometimes observe is -astonishing. A balance will weigh to one millionth part of the load. -Whitworth can measure to the millionth part of an inch. A rise of -temperature of the 8800th part of a degree centigrade has been -detected by Dr. Joule. The spectroscope has revealed the presence of -the 10,000,000th part of a gram. It is said that the eye can observe -the colour produced in a drop of water by the 50,000,000th part of a -gram of fuschine, and about the same quantity of cyanine. By the sense -of smell we can probably feel still smaller quantities of odorous -matter.[354] We must nevertheless remember that quantitative effects of -far less amount than these must exist, and we should state our negative -results with corresponding caution. We can only disprove the existence -of a quantitative phenomenon by showing deductively from the laws of -nature, that if present it would amount to a perceptible quantity. As -in the case of other negative arguments (p. 414), we must demonstrate -that the effect would appear, where it is by experiment found not to -appear. - - [354] Keill’s *Introduction to Natural Philosophy*, 3rd ed., London, - 1733, pp. 48–54. - - -*Limits of Experiment.* - -It will be obvious that there are many operations of nature which we -are quite incapable of imitating in our experiments. Our object is to -study the conditions under which a certain effect is produced; but -one of those conditions may involve a great length of time. There -are instances on record of experiments extending over five or ten -years, and even over a large part of a lifetime; but such intervals -of time are almost nothing to the time during which nature may have -been at work. The contents of a mineral vein in Cornwall may have been -undergoing gradual change for a hundred million years. All metamorphic -rocks have doubtless endured high temperature and enormous, pressure -for inconceivable periods of time, so that chemical geology is -generally beyond the scope of experiment. - -Arguments have been brought against Darwin’s theory, founded upon the -absence of any clear instance of the production of a new species. -During an historical interval of perhaps four thousand years, no -animal, it is said, has been so much domesticated as to become -different in species. It might as well be argued that no geological -changes are taking place, because no new mountain has risen in Great -Britain within the memory of man. Our actual experience of geological -changes is like a point in the infinite progression of time. When we -know that rain water falling on limestone will carry away a minute -portion of the rock in solution, we do not hesitate to multiply that -quantity by millions, and infer that in course of time a mountain may -be dissolved away. We have actual experience concerning the rise of -land in some parts of the globe and its fall in others to the extent -of some feet. Do we hesitate to infer what may thus be done in course -of geological ages? As Gabriel Plattes long ago remarked, “The sea -never resting, but perpetually winning land in one place and losing in -another, doth show what may be done in length of time by a continual -operation, not subject unto ceasing or intermission.”[355] The action -of physical circumstances upon the forms and characters of animals by -natural selection is subject to exactly the same remarks. As regards -animals living in a state of nature, the change of circumstances which -can be ascertained to have occurred is so slight, that we could not -expect to observe any change in those animals whatever. Nature has made -no experiment at all for us within historical times. Man, however, by -taming and domesticating dogs, horses, oxen, pigeons, &c., has made -considerable change in their circumstances, and we find considerable -change also in their forms and characters. Supposing the state of -domestication to continue unchanged, these new forms would continue -permanent so far as we know, and in this sense they are permanent. Thus -the arguments against Darwin’s theory, founded on the non-observation -of natural changes within the historical period, are of the weakest -character, being purely negative. - - [355] *Discovery of Subterraneal Treasure*, 1639, p. 52. - - - - -CHAPTER XX. - -METHOD OF VARIATIONS. - - -Experiments may be of two kinds, experiments of simple fact, and -experiments of quantity. In the first class of experiments we combine -certain conditions, and wish to ascertain whether or not a certain -effect of any quantity exists. Hooke wished to ascertain whether or not -there was any difference in the force of gravity at the top and bottom -of St. Paul’s Cathedral. The chemist continually performs analyses for -the purpose of ascertaining whether or not a given element exists in a -particular mineral or mixture; all such experiments and analyses are -qualitative rather than quantitative, because though the result may be -more or less, the particular amount of the result is not the object of -the inquiry. - -So soon, however, as a result is known to be discoverable, the -scientific man ought to proceed to the quantitative inquiry, how great -a result follows from a certain amount of the conditions which are -supposed to constitute the cause? The possible numbers of experiments -are now infinitely great, for every variation in a quantitative -condition will usually produce a variation in the amount of the effect. -The method of variation which thus arises is no narrow or special -method, but it is the general application of experiment to phenomena -capable of continuous variation. As Mr. Fowler has well remarked,[356] -the observation of variations is really an integration of a supposed -infinite number of applications of the so-called method of difference, -that is of experiment in its perfect form. - - [356] *Elements of Inductive Logic*, 1st edit. p. 175. - -In induction we aim at establishing a general law, and if we deal -with quantities that law must really be expressed more or less -obviously in the form of an equation, or equations. We treat as -before of conditions, and of what happens under those conditions. But -the conditions will now vary, not in quality, but quantity, and the -effect will also vary in quantity, so that the result of quantitative -induction is always to arrive at some mathematical expression involving -the quantity of each condition, and expressing the quantity of the -result. In other words, we wish to know what function the effect is -of its conditions. We shall find that it is one thing to obtain the -numerical results, and quite another thing to detect the law obeyed -by those results, the latter being an operation of an inverse and -tentative character. - - -*The Variable and the Variant.* - -Almost every series of quantitative experiments is directed to obtain -the relation between the different values of one quantity which is -varied at will, and another quantity which is caused thereby to vary. -We may conveniently distinguish these as respectively the *variable* -and the *variant*. When we are examining the effect of heat in -expanding bodies, heat, or one of its dimensions, temperature, is the -variable, length the variant. If we compress a body to observe how much -it is thereby heated, pressure, or it may be the dimensions of the -body, forms the variable, heat the variant. In the thermo-electric pile -we make heat the variable and measure electricity as the variant. That -one of the two measured quantities which is an antecedent condition of -the other will be the variable. - -It is always convenient to have the variable entirely under our -command. Experiments may indeed be made with accuracy, provided we -can exactly measure the variable at the moment when the quantity of -the effect is determined. But if we have to trust to the action of -some capricious force, there may be great difficulty in making exact -measurements, and those results may not be disposed over the whole -range of quantity in a convenient manner. It is one prime object of the -experimenter, therefore, to obtain a regular and governable supply -of the force which he is investigating. To determine correctly the -efficiency of windmills, when the natural winds were constantly varying -in force, would be exceedingly difficult. Smeaton, therefore, in his -experiments on the subject, created a uniform wind of the required -force by moving his models against the air on the extremity of a -revolving arm.[357] The velocity of the wind could thus be rendered -greater or less, it could be maintained uniform for any length of -time, and its amount could be exactly ascertained. In determining the -laws of the chemical action of light it would be out of the question -to employ the rays of the sun, which vary in intensity with the -clearness of the atmosphere, and with every passing cloud. One great -difficulty in photometry and the investigation of the chemical action -of light consists in obtaining a uniform and governable source of light -rays.[358] - - [357] *Philosophical Transactions*, vol. li. p. 138; abridgment, - vol. xi. p. 355. - - [358] See Bunsen and Roscoe’s researches, in *Philosophical - Transactions* (1859), vol. cxlix. p. 880, &c., where they describe a - constant flame of carbon monoxide gas. - -Fizeau’s method of measuring the velocity of light enabled him -to appreciate the time occupied by light in travelling through a -distance of eight or nine thousand metres. But the revolving mirror -of Wheatstone subsequently enabled Foucault and Fizeau to measure the -velocity in a space of four metres. In this latter method there was -the advantage that various media could be substituted for air, and the -temperature, density, and other conditions of the experiment could be -accurately governed and measured. - - -*Measurement of the Variable.* - -There is little use in obtaining exact measurements of an effect unless -we can also exactly measure its conditions. - -It is absurd to measure the electrical resistance of a piece of metal, -its elasticity, tenacity, density, or other physical qualities, if -these vary, not only with the minute impurities of the metal, but also -with its physical condition. If the same bar changes its properties -by being heated and cooled, and we cannot exactly define the state -in which it is at any moment, our care in measuring will be wasted, -because it can lead to no law. It is of little use to determine very -exactly the electric conductibility of carbon, which as graphite or gas -carbon conducts like a metal, as diamond is almost a non-conductor, -and in several other forms possesses variable and intermediate -powers of conduction. It will be of use only for immediate practical -applications. Before measuring these we ought to have something to -measure of which the conditions are capable of exact definition, and -to which at a future time we can recur. Similarly the accuracy of our -measurement need not much surpass the accuracy with which we can define -the conditions of the object treated. - -The speed of electricity in passing through a conductor mainly depends -upon the inductive capacity of the surrounding substances, and, except -for technical or special purposes, there is little use in measuring -velocities which in some cases are one hundred times as great as in -other cases. But the maximum speed of electric conduction is probably -a constant quantity of great scientific importance, and according -to Prof. Clerk Maxwell’s determination in 1868 is 174,800 miles per -second, or little less than that of light. The true boiling point of -water is a point on which practical thermometry depends, and it is -highly important to determine that point in relation to the absolute -thermometric scale. But when water free from air and impurity is -heated there seems to be no definite limit to the temperature it may -reach, a temperature of 180° Cent. having been actually observed. -Such temperatures, therefore, do not require accurate measurement. -All meteorological measurements depending on the accidental condition -of the sky are of far less importance than physical measurements in -which such accidental conditions do not intervene. Many profound -investigations depend upon our knowledge of the radiant energy -continually poured upon the earth by the sun; but this must be measured -when the sky is perfectly clear, and the absorption of the atmosphere -at its minimum. The slightest interference of cloud destroys the value -of such a measurement, except for meteorological purposes, which are of -vastly less generality and importance. It is seldom useful, again, to -measure the height of a snow-covered mountain within a foot, when the -thickness of the snow alone may cause it to vary 25 feet or more, when -in short the height itself is indefinite to that extent.[359] - - [359] Humboldt’s *Cosmos* (Bohn), vol. i. p. 7. - - -*Maintenance of Similar Conditions.* - -Our ultimate object in induction must be to obtain the complete -relation between the conditions and the effect, but this relation -will generally be so complex that we can only attack it in detail. -We must, as far as possible, confine the variation to one condition -at a time, and establish a separate relation between each condition -and the effect. This is at any rate the first step in approximating -to the complete law, and it will be a subsequent question how far the -simultaneous variation of several conditions modifies their separate -actions. In many experiments, indeed, it is only one condition which -we wish to study, and the others are interfering forces which we would -avoid if possible. One of the conditions of the motion of a pendulum is -the resistance of the air, or other medium in which it swings; but when -Newton was desirous of proving the equal gravitation of all substances, -he had no interest in the air. His object was to observe a single force -only, and so it is in a great many other experiments. Accordingly, -one of the most important precautions in investigation consists in -maintaining all conditions constant except that which is to be studied. -As that admirable experimental philosopher, Gilbert, expressed it,[360] -“There is always need of similar preparation, of similar figure, and -of equal magnitude, for in dissimilar and unequal circumstances the -experiment is doubtful.” - - [360] Gilbert, *De Magnete*, p. 109. - -In Newton’s decisive experiment similar conditions were provided -for, with the simplicity which characterises the highest art. The -pendulums of which the oscillations were compared consisted of equal -boxes of wood, hanging by equal threads, and filled with different -substances, so that the total weights should be equal and the centres -of oscillation at the same distance from the points of suspension. -Hence the resistance of the air became approximately a matter of -indifference; for the outward size and shape of the pendulums being -the same, the absolute force of resistance would be the same, so long -as the pendulums vibrated with equal velocity; and the weights being -equal the resistance would diminish the velocity equally. Hence if any -inequality were observed in the vibrations of the two pendulums, it -must arise from the only circumstance which was different, namely the -chemical nature of the matter within the boxes. No inequality being -observed, the chemical nature of substances can have no appreciable -influence upon the force of gravitation.[361] - - [361] *Principia*, bk. iii. Prop. vi. - -A beautiful experiment was devised by Dr. Joule for the purpose of -showing that the gain or loss of heat by a gas is connected, not -with the mere change of its volume and density, but with the energy -received or given out by the gas. Two strong vessels, connected by -a tube and stopcock, were placed in water after the air had been -exhausted from one vessel and condensed in the other to the extent -of twenty atmospheres. The whole apparatus having been brought to a -uniform temperature by agitating the water, and the temperature having -been exactly observed, the stopcock was opened, so that the air at -once expanded and filled the two vessels uniformly. The temperature -of the water being again noted was found to be almost unchanged. The -experiment was then repeated in an exactly similar manner, except that -the strong vessels were placed in separate portions of the water. Now -cold was produced in the vessel from which the air rushed, and an -almost exactly equal quantity of heat appeared in that to which it was -conducted. Thus Dr. Joule clearly proved that rarefaction produces -as much heat as cold, and that only when there is disappearance of -mechanical energy will there be production of heat.[362] What we have -to notice, however, is not so much the result of the experiment, as the -simple manner in which a single change in the apparatus, the separation -of the portions of water surrounding the air vessels, is made to give -indications of the utmost significance. - - [362] *Philosophical Magazine*, 3rd Series, vol. xxvi. p. 375. - - -*Collective Experiments.* - -There is an interesting class of experiments which enable us to observe -a number of quantitative results in one act. Generally speaking, each -experiment yields us but one number, and before we can approach the -real processes of reasoning we must laboriously repeat measurement -after measurement, until we can lay out a curve of the variation of -one quantity as depending on another. We can sometimes abbreviate -this labour, by making a quantity vary in different parts of the same -apparatus through every required amount. In observing the height to -which water rises by the capillary attraction of a glass vessel, we may -take a series of glass tubes of different bore, and measure the height -through which it rises in each. But if we take two glass plates, and -place them vertically in water, so as to be in contact at one vertical -side, and slightly separated at the other side, the interval between -the plates varies through every intermediate width, and the water rises -to a corresponding height, producing at its upper surface a hyperbolic -curve. - -The absorption of light in passing through a coloured liquid may be -beautifully shown by enclosing the liquid in a wedge-shaped glass, so -that we have at a single glance an infinite variety of thicknesses in -view. As Newton himself remarked, a red liquid viewed in this manner is -found to have a pale yellow colour at the thinnest part, and it passes -through orange into red, which gradually becomes of a deeper and darker -tint.[363] The effect may be noticed in a conical wine-glass. The -prismatic analysis of light from such a wedge-shaped vessel discloses -the reason, by exhibiting the progressive absorption of different rays -of the spectrum as investigated by Dr. J. H. Gladstone.[364] - - [363] *Opticks*, 3rd edit. p. 159. - - [364] Watts, *Dictionary of Chemistry*, vol. iii. p. 637. - -A moving body may sometimes be made to mark out its own course, like -a shooting star which leaves a tail behind it. Thus an inclined jet -of water exhibits in the clearest manner the parabolic path of a -projectile. In Wheatstone’s Kaleidophone the curves produced by the -combination of vibrations of different ratios are shown by placing -bright reflective buttons on the tops of wires of various forms. The -motions are performed so quickly that the eye receives the impression -of the path as a complete whole, just as a burning stick whirled round -produces a continuous circle. The laws of electric induction are -beautifully shown when iron filings are brought under the influence of -a magnet, and fall into curves corresponding to what Faraday called -the Lines of Magnetic Force. When Faraday tried to define what he -meant by his lines of force, he was obliged to refer to the filings. -“By magnetic curves,” he says,[365] “I mean lines of magnetic forces -which would be depicted by iron filings.” Robison had previously -produced similar curves by the action of frictional electricity, and -from a mathematical investigation of the forms of such curves we may -infer that magnetic and electric attractions obey the general law of -emanation, that of the inverse square of the distance. In the electric -brush we have a similar exhibition of the laws of electric attraction. - - [365] *Faraday’s Life*, by Bence Jones, vol. ii. p. 5. - -There are several branches of science in which collective experiments -have been used with great advantage. Lichtenberg’s electric figures, -produced by scattering electrified powder on an electrified resin cake, -so as to show the condition of the latter, suggested to Chladni the -notion of discovering the state of vibration of plates by strewing sand -upon them. The sand collects at the points where the motion is least, -and we gain at a glance a comprehension of the undulations of the -plate. To this method of experiment we owe the beautiful observations -of Savart. The exquisite coloured figures exhibited by plates of -crystal, when examined by polarised light, afford a more complicated -example of the same kind of investigation. They led Brewster and -Fresnel to an explanation of the properties of the optic axes of -crystals. The unequal conduction of heat in crystalline substances has -also been shown in a similar manner, by spreading a thin layer of wax -over the plate of crystal, and applying heat to a single point. The -wax then melts in a circular or elliptic area according as the rate of -conduction is uniform or not. Nor should we forget that Newton’s rings -were an early and most important instance of investigations of the -same kind, showing the effects of interference of light undulations -of all magnitudes at a single view. Herschel gave to all such -opportunities of observing directly the results of a general law, the -name of *Collective Instances*,[366] and I propose to adopt the name -*Collective Experiments*. - - [366] *Preliminary Discourse*, &c., p. 185. - -Such experiments will in many subjects only give the first hint of -the nature of the law in question, but will not admit of any exact -measurements. The parabolic form of a jet of water may well have -suggested to Galileo his views concerning the path of a projectile; -but it would not serve now for the exact investigation of the laws of -gravity. It is unlikely that capillary attraction could be exactly -measured by the use of inclined plates of glass, and tubes would -probably be better for precise investigation. As a general rule, these -collective experiments would be most useful for popular illustration. -But when the curves are of a precise and permanent character, as in -the coloured figures produced by crystalline plates, they may admit of -exact measurement. Newton’s rings and diffraction fringes allow of very -accurate measurements. - -Under collective experiments we may perhaps place those in which we -render visible the motions of gas or liquid by diffusing some opaque -substance in it. The behaviour of a body of air may often be studied -in a beautiful way by the use of smoke, as in the production of smoke -rings and jets. In the case of liquids lycopodium powder is sometimes -employed. To detect the mixture of currents or strata of liquid, I -employed very dilute solutions of common salt and silver nitrate, -which produce a visible cloud wherever they come into contact.[367] -Atmospheric clouds often reveal to us the movements of great volumes of -air which would otherwise be quite unapparent. - - [367] *Philosophical Magazine*, July, 1857, 4th Series, vol. xiv. - p. 24. - - -*Periodic Variations.* - -A large class of investigations is concerned with Periodic Variations. -We may define a periodic phenomenon as one which, with the uniform -change of the variable, returns time after time to the same value. -If we strike a pendulum it presently returns to the point from which -we disturbed it, and while time, the variable, progresses uniformly, -it goes on making excursions and returning, until stopped by the -dissipation of its energy. If one body in space approaches by gravity -towards another, they will revolve round each other in elliptic -orbits, and return for an indefinite number of times to the same -relative positions. On the other hand a single body projected into -empty space, free from the action of any extraneous force, would go -on moving for ever in a straight line, according to the first law of -motion. In the latter case the variation is called *secular*, because -it proceeds during ages in a similar manner, and suffers no περίοδος -or going round. It may be doubted whether there really is any motion -in the universe which is not periodic. Mr. Herbert Spencer long since -adopted the doctrine that all motion is ultimately rhythmical,[368] and -abundance of evidence may be adduced in favour of his view. - - [368] *First Principles*, 3rd edit. chap. x. p. 253. - -The so-called secular acceleration of the moon’s motion is certainly -periodic, and as, so far as we can tell, no body is beyond the -attractive power of other bodies, rectilinear motion becomes purely -hypothetical, or at least infinitely improbable. All the motions of all -the stars must tend to become periodic. Though certain disturbances -in the planetary system seem to be uniformly progressive, Laplace is -considered to have proved that they really have their limits, so that -after an immense time, all the planetary bodies might return to the -same places, and the stability of the system be established. Such a -theory of periodic stability is really hypothetical, and does not -take into account phenomena resulting in the dissipation of energy, -which may be a really secular process. For our present purposes we -need not attempt to form an opinion on such questions. Any change -which does not present the appearance of a periodic character will be -empirically regarded as a secular change, so that there will be plenty -of non-periodic variations. - -The variations which we produce experimentally will often be -non-periodic. When we communicate heat to a gas it increases in -bulk or pressure, and as far as we can go the higher the temperature -the higher the pressure. Our experiments are of course restricted -in temperature both above and below, but there is every reason to -believe that the bulk being the same, the pressure would never return -to the same point at any two different temperatures. We may of course -repeatedly raise and lower the temperature at regular or irregular -intervals entirely at our will, and the pressure of the gas will vary -in like manner and exactly at the same intervals, but such an arbitrary -series of changes would not constitute Periodic Variation. It would -constitute a succession of distinct experiments, which would place -beyond reasonable doubt the connexion of cause and effect. - -Whenever a phenomenon recurs at equal or nearly equal intervals, there -is, according to the theory of probability, considerable evidence -of connexion, because if the recurrences were entirely casual it is -unlikely that they would happen at equal intervals. The fact that a -brilliant comet had appeared in the years 1301, 1378, 1456, 1531, 1607, -and 1682 gave considerable presumption in favour of the identity of the -body, apart from similarity of the orbit. There is nothing which so -fascinates the attention of men as the recurrence time after time of -some unusual event. Things and appearances which remain ever the same, -like mountains and valleys, fail to excite the curiosity of a primitive -people. It has been remarked by Laplace that even in his day the rising -of Venus in its brightest phase never failed to excite surprise and -interest. So there is little doubt that the first germ of science -arose in the attention given by Eastern people to the changes of the -moon and the motions of the planets. Perhaps the earliest astronomical -discovery consisted in proving the identity of the morning and evening -stars, on the grounds of their similarity of aspect and invariable -alternation.[369] Periodical changes of a somewhat complicated kind -must have been understood by the Chaldeans, because they were aware -of the cycle of 6585 days or 19 years which brings round the new and -full moon upon the same days, hours, and even minutes of the year. -The earliest efforts of scientific prophecy were founded upon this -knowledge, and if at present we cannot help wondering at the precise -anticipations of the nautical almanack, we may imagine the wonder -excited by such predictions in early times. - - [369] Laplace, *System of the World*, vol. i. pp. 50, 54, &c. - - -*Combined Periodic Changes.* - -We shall seldom find a body subject to a single periodic variation, -and free from other disturbances. We may expect the periodic variation -itself to undergo variation, which may possibly be secular, but is more -likely to prove periodic; nor is there any limit to the complication of -periods beyond periods, or periods within periods, which may ultimately -be disclosed. In studying a phenomenon of rhythmical character we have -a succession of questions to ask. Is the periodic variation uniform? If -not, is the change uniform? If not, is the change itself periodic? Is -that new period uniform, or subject to any other change, or not? and so -on *ad infinitum*. - -In some cases there may be many distinct causes of periodic variations, -and according to the principle of the superposition of small effects, -to be afterwards considered, these periodic effects will be simply -added together, or at least approximately so, and the joint result may -present a very complicated subject of investigation. The tides of the -ocean consist of a series of superimposed undulations. Not only are -there the ordinary semi-diurnal tides caused by sun and moon, but a -series of minor tides, such as the lunar diurnal, the solar diurnal, -the lunar monthly, the lunar fortnightly, the solar annual and solar -semi-annual are gradually being disentangled by the labours of Sir W. -Thomson, Professor Haughton and others. - -Variable stars present interesting periodic phenomena; while some -stars, δ Cephei for instance, are subject to very regular variations, -others, like Mira Ceti, are less constant in the degrees of brilliancy -which they attain or the rapidity of the changes, possibly on account -of some longer periodic variation.[370] The star β Lyræ presents a -double maximum and minimum in each of its periods of nearly 13 days, -and since the discovery of this variation the period in a period has -probably been on the increase. “At first the variability was more -rapid, then it became gradually slower; and this decrease in the length -of time reached its limit between the years 1840 and 1844. During that -time its period was nearly invariable; at present it is again decidedly -on the decrease.”[371] The tracing out of such complicated variations -presents an unlimited field for interesting investigation. The number -of such variable stars already known is considerable, and there is no -reason to suppose that any appreciable fraction of the whole number has -yet been detected. - - [370] Herschel’s *Outlines of Astronomy*, 4th edit. pp. 555–557. - - [371] Humboldt’s *Cosmos* (Bohn), vol. iii. p. 229. - - -*Principle of Forced Vibrations.* - -Investigations of the connection of periodic causes and effects -rest upon a principle, which has been demonstrated by Sir John -Herschel for some special cases, and clearly explained by him in -several of his works.[372] The principle may be formally stated in -the following manner: “If one part of any system connected together -either by material ties, or by the mutual attractions of its members, -be continually maintained by any cause, whether inherent in the -constitution of the system or external to it, in a state of regular -periodic motion, that motion will be propagated throughout the whole -system, and will give rise, in every member of it, and in every part -of each member, to periodic movements executed in equal periods, with -that to which they owe their origin, though not necessarily synchronous -with them in their maxima and minima.” The meaning of the proposition -is that the effect of a periodic cause will be periodic, and will recur -at intervals equal to those of the cause. Accordingly when we find two -phenomena which do proceed, time after time, through changes of the -same period, there is much probability that they are connected. In -this manner, doubtless, Pliny correctly inferred that the cause of the -tides lies in the sun and the moon, the intervals between successive -high tides being equal to the intervals between the moon’s passage -across the meridian. Kepler and Descartes too admitted the connection -previous to Newton’s demonstration of its precise nature. When Bradley -discovered the apparent motion of the stars arising from the aberration -of light, he was soon able to attribute it to the earth’s annual -motion, because it went through its phases in a year. - - [372] *Encyclopædia Metropolitana*, art. *Sound*, § 323; *Outlines - of Astronomy*, 4th edit., § 650. pp. 410, 487–88; *Meteorology, - Encyclopædia Britannica*, Reprint, p. 197. - -The most beautiful instance of induction concerning periodic changes -which can be cited, is the discovery of an eleven-year period in -various meteorological phenomena. It would be difficult to mention any -two things apparently more disconnected than the spots upon the sun and -auroras. As long ago as 1826, Schwabe commenced a regular series of -observations of the spots upon the sun, which has been continued to the -present time, and he was able to show that at intervals of about eleven -years the spots increased much in size and number. Hardly was this -discovery made known, when Lamont pointed out a nearly equal period of -variation in the declination of the magnetic needle. Magnetic storms or -sudden disturbances of the needle were next shown to take place most -frequently at the times when sun-spots were prevalent, and as auroras -are generally coincident with magnetic storms, these phenomena were -brought into the cycle. It has since been shown by Professor Piazzi -Smyth and Mr. E. J. Stone, that the temperature of the earth’s surface -as indicated by sunken thermometers gives some evidence of a like -period. The existence of a periodic cause having once been established, -it is quite to be expected, according to the principle of forced -vibrations, that its influence will be detected in all meteorological -phenomena. - - -*Integrated Variations.* - -In considering the various modes in which one effect may depend upon -another, we must set in a distinct class those which arise from the -accumulated effects of a constantly acting cause. When water runs out -of a cistern, the velocity of motion depends, according to Torricelli’s -theorem, on the height of the surface of the water above the vent; -but the amount of water which leaves the cistern in a given time -depends upon the aggregate result of that velocity, and is only to -be ascertained by the mathematical process of integration. When one -gravitating body falls towards another, the force of gravity varies -according to the inverse square of the distance; to obtain the velocity -produced we must integrate or sum the effects of that law; and to -obtain the space passed over by the body in a given time, we must -integrate again. - -In periodic variations the same distinction must be drawn. The heating -power of the sun’s rays at any place on the earth varies every day with -the height attained, and is greatest about noon; but the temperature of -the air will not be greatest at the same time. This temperature is an -integrated effect of the sun’s heating power, and as long as the sun -is able to give more heat to the air than the air loses in other ways, -the temperature continues to rise, so that the maximum is deferred -until about 3 P.M. Similarly the hottest day of the year falls, on an -average, about one month later than the summer solstice, and all the -seasons lag about a month behind the motions of the sun. In the case -of the tides, too, the effect of the moon’s attractive power is never -greatest when the power is greatest; the effect always lags more or -less behind the cause. Yet the intervals between successive tides are -equal, in the absence of disturbance, to the intervals between the -passages of the moon across the meridian. Thus the principle of forced -vibrations holds true. - -In periodic phenomena, however, curious results sometimes follow from -the integration of effects. If we strike a pendulum, and then repeat -the stroke time after time at the same part of the vibration, all the -strokes concur in adding to the momentum, and we can thus increase the -extent and violence of the vibrations to any degree. We can stop the -pendulum again by strokes applied when it is moving in the opposite -direction, and the effects being added together will soon bring it to -rest. Now if we alter the intervals of the strokes so that each two -successive strokes act in opposite manners they will neutralise each -other, and the energy expended will be turned into heat or sound at the -point of percussion. Similar effects occur in all cases of rhythmical -motion. If a musical note is sounded in a room containing a piano, -the string corresponding to it will be thrown into vibration, because -every successive stroke of the air-waves upon the string finds it in -like position as regards the vibration, and thus adds to its energy of -motion. But the other strings being incapable of vibrating with the -same rapidity are struck at various points of their vibrations, and one -stroke will soon be opposed by one contrary in effect. All phenomena of -*resonance* arise from this coincidence in time of undulation. The air -in a pipe closed at one end, and about 12 inches in length, is capable -of vibrating 512 times in a second. If, then, the note C is sounded in -front of the open end of the pipe, every successive vibration of the -air is treasured up as it were in the motion of the air. In a pipe of -different length the pulses of air would strike each other, and the -mechanical energy being transmuted into heat would become no longer -perceptible as sound. - -Accumulated vibrations sometimes become so intense as to lead to -unexpected results. A glass vessel if touched with a violin bow at -a suitable point may be fractured with the violence of vibration. A -suspension bridge may be broken down if a company of soldiers walk -across it in steps the intervals of which agree with the vibrations -of the bridge itself. But if they break the step or march in either -quicker or slower pace, they may have no perceptible effect upon the -bridge. In fact if the impulses communicated to any vibrating body are -synchronous with its vibrations, the energy of those vibrations will be -unlimited, and may fracture any body. - -Let us now consider what will happen if the strokes be not exactly at -the same intervals as the vibrations of the body, but, say, a little -slower. Then a succession of strokes will meet the body in nearly but -not quite the same position, and their efforts will be accumulated. -Afterwards the strokes will begin to fall when the body is in the -opposite phase. Imagine that one pendulum moving from one extreme -point to another in a second, should be struck by another pendulum -which makes 61 beats in a minute; then, if the pendulums commence -together, they will at the end of 30-1/2 beats be moving in opposite -directions. Hence whatever energy was communicated in the first half -minute will be neutralised by the opposite effect of that given in the -second half. The effect of the strokes of the second pendulum will -therefore be alternately to increase and decrease the vibrations of -the first, so that a new kind of vibration will be produced running -through its phases in 61 seconds. An effect of this kind was actually -observed by Ellicott, a member of the Royal Society, in the case of two -clocks.[373] He found that through the wood-work by which the clocks -were connected a slight impulse was transmitted, and each pendulum -alternately lost and gained momentum. Each clock, in fact, tended to -stop the other at regular intervals, and in the intermediate times to -be stopped by the other. - - [373] *Philosophical Transactions*, (1739), vol. xli. p. 126. - -Many disturbances in the planetary system depend upon the same -principle; for if one planet happens always to pull another in the -same direction in similar parts of their orbits, the effects, however -slight, will be accumulated, and a disturbance of large ultimate -amount and of long period will be produced. The long inequality in -the motions of Jupiter and Saturn is thus due to the fact that five -times the mean motion of Saturn is very nearly equal to twice the mean -motion of Jupiter, causing a coincidence in their relative positions -and disturbing powers. The rolling of ships depends mainly upon the -question whether the period of vibration of the ship corresponds or -not with the intervals at which the waves strike her. Much which seems -at first sight unaccountable in the behaviour of vessels is thus -explained, and the loss of the *Captain* is a sad case in point. - - - - -CHAPTER XXI. - -THEORY OF APPROXIMATION. - - -In order that we may gain a true understanding of the kind, degree, and -value of the knowledge which we acquire by experimental investigation, -it is requisite that we should be fully conscious of its approximate -character. We must learn to distinguish between what we can know -and cannot know--between the questions which admit of solution, and -those which only seem to be solved. Many persons may be misled by the -expression *exact science*, and may think that the knowledge acquired -by scientific methods admits of our reaching absolutely true laws, -exact to the last degree. There is even a prevailing impression that -when once mathematical formulæ have been successfully applied to a -branch of science, this portion of knowledge assumes a new nature, and -admits of reasoning of a higher character than those sciences which are -still unmathematical. - -The very satisfactory degree of accuracy attained in the science of -astronomy gives a certain plausibility to erroneous notions of this -kind. Some persons no doubt consider it to be *proved* that planets -move in ellipses, in such a manner that all Kepler’s laws hold exactly -true; but there is a double error in any such notions. In the first -place, Kepler’s laws are *not proved*, if by proof we mean certain -demonstration of their exact truth. In the next place, even assuming -Kepler’s laws to be exactly true in a theoretical point of view, the -planets never move according to those laws. Even if we could observe -the motions of a planet, of a perfect globular form, free from all -perturbing or retarding forces, we could never prove that it moved -in a perfect ellipse. To prove the elliptical form we should have to -measure infinitely small angles, and infinitely small fractions of a -second; we should have to perform impossibilities. All we can do is to -show that the motion of an unperturbed planet approaches *very nearly* -to the form of an ellipse, and more nearly the more accurately our -observations are made. But if we go on to assert that the path *is* an -ellipse we pass beyond our data, and make an assumption which cannot be -verified by observation. - -But, secondly, as a matter of fact no planet does move in a perfect -ellipse, or manifest the truth of Kepler’s laws exactly. The law of -gravity prevents its own results from being clearly exhibited, because -the mutual perturbations of the planets distort the elliptical paths. -Those laws, again, hold exactly true only of infinitely small bodies, -and when two great globes, like the sun and Jupiter, attract each -other, the law must be modified. The periodic time is then shortened -in the ratio of the square root of the number expressing the sun’s -mass, to that of the sum of the numbers expressing the masses of the -sun and planet, as was shown by Newton.[374] Even at the present day -discrepancies exist between the observed dimensions of the planetary -orbits and their theoretical magnitudes, after making allowance for -all disturbing causes.[375] Nothing is more certain in scientific -method than that approximate coincidence alone can be expected. In -the measurement of continuous quantity perfect correspondence must -be accidental, and should give rise to suspicion rather than to -satisfaction. - - [374] *Principia*, bk. iii. Prop. 15. - - [375] Lockyer’s *Lessons in Elementary Astronomy*, p. 301. - -One remarkable result of the approximate character of our observations -is that we could never prove the existence of perfectly circular or -parabolic movement, even if it existed. The circle is a singular case -of the ellipse, for which the eccentricity is zero; it is infinitely -improbable that any planet, even if undisturbed by other bodies, would -have a circle for its orbit; but if the orbit were a circle we could -never prove the entire absence of eccentricity. All that we could -do would be to declare the divergence from the circular form to be -inappreciable. Delambre was unable to detect the slightest ellipticity -in the orbit of Jupiter’s first satellite, but he could only infer -that the orbit was *nearly* circular. The parabola is the singular -limit between the ellipse and the hyperbola. As there are elliptic and -hyperbolic comets, so we might conceive the existence of a parabolic -comet. Indeed if an undisturbed comet fell towards the sun from an -infinite distance it would move in a parabola; but we could never prove -that it so moved. - - -*Substitution of Simple Hypotheses.* - -In truth men never can solve problems fulfilling the complex -circumstances of nature. All laws and explanations are in a certain -sense hypothetical, and apply exactly to nothing which we can know -to exist. In place of the actual objects which we see and feel, the -mathematician substitutes imaginary objects, only partially resembling -those represented, but so devised that the discrepancies are not of -an amount to alter seriously the character of the solution. When we -probe the matter to the bottom physical astronomy is as hypothetical as -Euclid’s elements. There may exist in nature perfect straight lines, -triangles, circles, and other regular geometrical figures; to our -science it is a matter of indifference whether they do or do not exist, -because in any case they must be beyond our powers of perception. If -we submitted a perfect circle to the most rigorous scrutiny, it is -impossible that we should discover whether it were perfect or not. -Nevertheless in geometry we argue concerning perfect curves, and -rectilinear figures, and the conclusions apply to existing objects so -far as we can assure ourselves that they agree with the hypothetical -conditions of our reasoning. This is in reality all that we can do in -the most perfect of the sciences. - -Doubtless in astronomy we meet with the nearest approximation to -actual conditions. The law of gravity is not a complex one in itself, -and we believe it with much probability to be exactly true; but we -cannot calculate out in any real case its accurate results. The law -asserts that every particle of matter in the universe attracts every -other particle, with a force depending on the masses of the particles -and their distances. We cannot know the force acting on any particle -unless we know the masses and distances and positions of all other -particles in the universe. The physical astronomer has made a sweeping -assumption, namely, that all the millions of existing systems exert no -perturbing effects on our planetary system, that is to say, no effects -in the least appreciable. The problem at once becomes hypothetical, -because there is little doubt that gravitation between our sun and -planets and other systems does exist. Even when they consider the -relations of our planetary bodies *inter se*, all their processes are -only approximate. In the first place they assume that each of the -planets is a perfect ellipsoid, with a smooth surface and a homogeneous -interior. That this assumption is untrue every mountain and valley, -every sea, every mine affords conclusive evidence. If astronomers are -to make their calculations perfect, they must not only take account -of the Himalayas and the Andes, but must calculate separately the -attraction of every hill, nay, of every ant-hill. So far are they -from having considered any local inequality of the surface, that they -have not yet decided upon the general form of the earth; it is still -a matter of speculation whether or not the earth is an ellipsoid with -three unequal axes. If, as is probable, the globe is irregularly -compressed in some directions, the calculations of astronomers will -have to be repeated and refined, in order that they may approximate -to the attractive power of such a body. If we cannot accurately learn -the form of our own earth, how can we expect to ascertain that of -the moon, the sun, and other planets, in some of which probably are -irregularities of greater proportional amount? - -In a further way the science of physical astronomy is merely -approximate and hypothetical. Given homogeneous ellipsoids acting upon -each other according to the law of gravity, the best mathematicians -have never and perhaps never will determine exactly the resulting -movements. Even when three bodies simultaneously attract each other the -complication of effects is so great that only approximate calculations -can be made. Astronomers have not even attempted the general problem -of the simultaneous attractions of four, five, six, or more bodies; -they resolve the general problem into so many different problems of -three bodies. The principle upon which the calculations of physical -astronomy proceed, is to neglect every quantity which does not seem -likely to lead to an effect appreciable in observation, and the -quantities rejected are far more numerous and complex than the few -larger terms which are retained. All then is merely approximate. - -Concerning other branches of physical science the same statements are -even more evidently true. We speak and calculate about inflexible -bars, inextensible lines, heavy points, homogeneous substances, -uniform spheres, perfect fluids and gases, and we deduce a great -number of beautiful theorems; but all is hypothetical. There is no -such thing as an inflexible bar, an inextensible line, nor any one -of the other perfect objects of mechanical science; they are to be -classed with those mythical existences, the straight line, triangle, -circle, &c., about which Euclid so freely reasoned. Take the simplest -operation considered in statics--the use of a crowbar in raising a -heavy stone, and we shall find, as Thomson and Tait have pointed -out, that we neglect far more than we observe.[376] If we suppose -the bar to be quite rigid, the fulcrum and stone perfectly hard, and -the points of contact real points, we may give the true relation of -the forces. But in reality the bar must bend, and the extension and -compression of different parts involve us in difficulties. Even if -the bar be homogeneous in all its parts, there is no mathematical -theory capable of determining with accuracy all that goes on; if, as -is infinitely more probable, the bar is not homogeneous, the complete -solution will be immensely more complicated, but hardly more hopeless. -No sooner had we determined the change of form according to simple -mechanical principles, than we should discover the interference of -thermodynamic principles. Compression produces heat and extension -cold, and thus the conditions of the problem are modified throughout. -In attempting a fourth approximation we should have to allow for the -conduction of heat from one part of the bar to another. All these -effects are utterly inappreciable in a practical point of view, if -the bar be a good stout one; but in a theoretical point of view they -entirely prevent our saying that we have solved a natural problem. The -faculties of the human mind, even when aided by the wonderful powers -of abbreviation conferred by analytical methods, are utterly unable to -cope with the complications of any real problem. And had we exhausted -all the known phenomena of a mechanical problem, how can we tell that -hidden phenomena, as yet undetected, do not intervene in the commonest -actions? It is plain that no phenomenon comes within the sphere of -our senses unless it possesses a momentum capable of irritating the -appropriate nerves. There may then be worlds of phenomena too slight to -rise within the scope of our consciousness. - - [376] *Treatise on Natural Philosophy*, vol. i. pp. 337, &c. - -All the instruments with which we perform our measurements are faulty. -We assume that a plumb-line gives a vertical line; but this is never -true in an absolute sense, owing to the attraction of mountains -and other inequalities in the surface of the earth. In an accurate -trigonometrical survey, the divergencies of the plumb-line must be -approximately determined and allowed for. We assume a surface of -mercury to be a perfect plane, but even in the breadth of 5 inches -there is a calculable divergence from a true plane of about one -ten-millionth part of an inch; and this surface further diverges from -true horizontality as the plumb-line does from true verticality. -That most perfect instrument, the pendulum, is not theoretically -perfect, except for infinitely small arcs of vibration, and the -delicate experiments performed with the torsion balance proceed on the -assumption that the force of torsion of a wire is proportional to the -angle of torsion, which again is only true for infinitely small angles. - -Such is the purely approximate character of all our operations that it -is not uncommon to find the theoretically worse method giving truer -results than the theoretically perfect method. The common pendulum -which is not isochronous is better for practical purposes than the -cycloidal pendulum, which is isochronous in theory but subject to -mechanical difficulties. The spherical form is not the correct form for -a speculum or lense, but it differs so slightly from the true form, and -is so much more easily produced mechanically, that it is generally -best to rest content with the spherical surface. Even in a six-feet -mirror the difference between the parabola and the sphere is only about -one ten-thousandth part of an inch, a thickness which would be taken -off in a few rubs of the polisher. Watts’ ingenious parallel motion was -intended to produce rectilinear movement of the piston-rod. In reality -the motion was always curvilinear, but for his purposes a certain part -of the curve approximated sufficiently to a straight line. - - -*Approximation to Exact Laws.* - -Though we can not prove numerical laws with perfect accuracy, it would -be a great mistake to suppose that there is any inexactness in the laws -of nature. We may even discover a law which we believe to represent -the action of forces with perfect exactness. The mind may seem to pass -in advance of its data, and choose out certain numerical results as -absolutely true. We can never really pass beyond our data, and so far -as assumption enters in, so far want of certainty will attach to our -conclusions; nevertheless we may sometimes rightly prefer a probable -assumption of a precise law to numerical results, which are at the best -only approximate. We must accordingly draw a strong distinction between -the laws of nature which we believe to be accurately stated in our -formulas, and those to which our statements only make an approximation, -so that at a future time the law will be differently stated. - -The law of gravitation is expressed in the form F = Mm/D^{2}, -meaning that gravity is proportional directly to the product of -the gravitating masses, and indirectly to the square of their -distance. The latent heat of steam is expressed by the equation -log F = *a* + *b*α^{t} + *c*β^{t}, in which are five quantities *a*, -*b*, *c*, α, β, to be determined by experiment. There is every reason -to believe that in the progress of science the law of gravity will -remain entirely unaltered, and the only effect of further inquiry will -be to render it a more and more probable expression of the absolute -truth. The law of the latent heat of steam on the other hand, will be -modified by every new series of experiments, and it may not improbably -be shown that the assumed law can never be made to agree exactly with -the results of experiment. - -Philosophers have not always supposed that the law of gravity was -exactly true. Newton, though he had the highest confidence in its -truth, admitted that there were motions in the planetary system which -he could not reconcile with the law. Euler and Clairaut who were, with -D’Alembert, the first to apply the full powers of mathematical analysis -to the theory of gravitation as explaining the perturbations of the -planets, did not think the law sufficiently established to attribute -all discrepancies to the errors of calculation and observation. They -did not feel certain that the force of gravity exactly obeyed the -well-known rule. The law might involve other powers of the distance. It -might be expressed in the form - - F = ... + *a*/D + *b*/D^{2} + *c*/D^{3} + ... - -and the coefficients *a* and *c* might be so small that those terms -would become apparent only in very accurate comparisons with fact. -Attempts have been made to account for difficulties, by attributing -value to such neglected terms. Gauss at one time thought the even more -fundamental principle of gravity, that the force is dependent only on -mass and distance, might not be exactly true, and he undertook accurate -pendulum experiments to test this opinion. Only as repeated doubts have -time after time been resolved in favour of the law of Newton, has it -been assumed as precisely correct. But this belief does not rest on -experiment or observation only. The calculations of physical astronomy, -however accurate, could never show that the other terms of the above -expression were absolutely devoid of value. It could only be shown that -they had such slight value as never to become apparent. - -There are, however, other reasons why the law is probably complete -and true as commonly stated. Whatever influence spreads from a point, -and expands uniformly through space, will doubtless vary inversely -in intensity as the square of the distance, because the area over -which it is spread increases as the square of the radius. This part -of the law of gravity may be considered as due to the properties of -space, and there is a perfect analogy in this respect between gravity -and all other *emanating* forces, as was pointed out by Keill.[377] -Thus the undulations of light, heat, and sound, and the attractions -of electricity and magnetism obey the very same law so far as we -can ascertain. If the molecules of a gas or the particles of matter -constituting odour were to start from a point and spread uniformly, -their distances would increase and their density decrease according to -the same principle. - - [377] *An Introduction to Natural Philosophy*, 3rd edit. 1733, p. 5. - -Other laws of nature stand in a similar position. Dalton’s laws of -definite combining proportions never have been, and never can be, -exactly proved; but chemists having shown, to a considerable degree of -approximation, that the elements combine together as if each element -had atoms of an invariable mass, assume that this is exactly true. They -go even further. Prout pointed out in 1815 that the equivalent weights -of the elements appeared to be simple numbers; and the researches of -Dumas, Pelouze, Marignac, Erdmann, Stas, and others have gradually -rendered it likely that the atomic weights of hydrogen, carbon, oxygen, -nitrogen, chlorine, and silver, are in the ratios of the numbers 1, -12, 16, 14, 35·5, and 108. Chemists then step beyond their data; they -throw aside their actual experimental numbers, and assume that the true -ratios are not those exactly indicated by any weighings, but the simple -ratios of these numbers. They boldly assume that the discrepancies are -due to experimental errors, and they are justified by the fact that -the more elaborate and skilful the researches on the subject, the more -nearly their assumption is verified. Potassium is the only element -whose atomic weight has been determined with great care, but which has -not shown an approach to a simple ratio with the other elements. This -exception may be due to some unsuspected cause of error.[378] A similar -assumption is made in the law of definite combining volumes of gases, -and Brodie has clearly pointed out the line of argument by which the -chemist, observing that the discrepancies between the law and fact are -within the limits of experimental error, assumes that they are due to -error.[379] - - [378] Watts, *Dictionary of Chemistry*, vol. i. p. 455. - - [379] *Philosophical Transactions*, (1866) vol. clvi. p. 809. - -Faraday, in one of his researches, expressly makes an assumption -of the same kind. Having shown, with some degree of experimental -precision, that there exists a simple proportion between quantities of -electrical energy and the quantities of chemical substances which it -can decompose, so that for every atom dissolved in the battery cell an -atom ought theoretically, that is without regard to dissipation of some -of the energy, to be decomposed in the electrolytic cell, he does not -stop at his numerical results. “I have not hesitated,” he says,[380] -“to apply the more strict results of chemical analysis to correct the -numbers obtained as electrolytic results. This, it is evident, may be -done in a great number of cases, without using too much liberty towards -the due severity of scientific research.” - - [380] *Experimental Researches in Electricity*, vol. i. p. 246. - -The law of the conservation of energy, one of the widest of all -physical generalisations, rests upon the same footing. The most that -we can do by experiment is to show that the energy entering into any -experimental combination is almost equal to what comes out of it, -and more nearly so the more accurately we perform the measurements. -Absolute equality is always a matter of assumption. We cannot even -prove the indestructibility of matter; for were an exceedingly minute -fraction of existing matter to vanish in any experiment, say one part -in ten millions, we could never detect the loss. - - -*Successive Approximations to Natural Conditions.* - -When we examine the history of scientific problems, we find that one -man or one generation is usually able to make but a single step at -a time. A problem is solved for the first time by making some bold -hypothetical simplification, upon which the next investigator makes -hypothetical modifications approaching more nearly to the truth. Errors -are successively pointed out in previous solutions, until at last there -might seem little more to be desired. Careful examination, however, -will show that a series of minor inaccuracies remain to be corrected -and explained, were our powers of reasoning sufficiently great, and the -purpose adequate in importance. - -Newton’s successful solution of the problem of the planetary movements -entirely depended at first upon a great simplification. The law of -gravity only applies directly to two infinitely small particles, so -that when we deal with vast globes like the earth, Jupiter, and the -sun, we have an immense aggregate of separate attractions to deal with, -and the law of the aggregate need not coincide with the law of the -elementary particles. But Newton, by a great effort of mathematical -reasoning, was able to show that two homogeneous spheres of matter act -as if the whole of their masses were concentrated at the centres; in -short, that such spheres are centrobaric bodies (p. 364). He was then -able with comparative ease to calculate the motions of the planets on -the hypothesis of their being spheres, and to show that the results -roughly agreed with observation. Newton, indeed, was one of the few men -who could make two great steps at once. He did not rest contented with -the spherical hypothesis; having reason to believe that the earth was -really a spheroid with a protuberance around the equator, he proceeded -to a second approximation, and proved that the attraction of the -protuberant matter upon the moon accounted for the precession of the -equinoxes, and led to various complicated effects. But, (p. 459), even -the spheroidal hypothesis is far from the truth. It takes no account -of the irregularities of surface, the great protuberance of land in -Central Asia and South America, and the deficiency in the bed of the -Atlantic. - -To determine the law according to which a projectile, such as a cannon -ball, moves through the atmosphere is a problem very imperfectly -solved at the present day, but in which many successive advances have -been made. So little was known concerning the subject three or four -centuries ago that a cannon ball was supposed to move at first in a -straight line, and after a time to be deflected into a curve. Tartaglia -ventured to maintain that the path was curved throughout, as by the -principle of continuity it should be; but the ingenuity of Galileo -was required to prove this opinion, and to show that the curve was -approximately a parabola. It is only, however, under forced hypotheses -that we can assert the path of a projectile to be truly a parabola: the -path must be through a perfect vacuum, where there is no resisting -medium of any kind; the force of gravity must be uniform and act in -parallel lines; or else the moving body must be either a mere point, or -a perfect centrobaric body, that is a body possessing a definite centre -of gravity. These conditions cannot be really fulfilled in practice. -The next great step in the problem was made by Newton and Huyghens, the -latter of whom asserted that the atmosphere would offer a resistance -proportional to the velocity of the moving body, and concluded that -the path would have in consequence a logarithmic character. Newton -investigated in a general manner the subject of resisting media, and -came to the conclusion that the resistance is more nearly proportional -to the square of the velocity. The subject then fell into the hands -of Daniel Bernoulli, who pointed out the enormous resistance of the -air in cases of rapid movement, and calculated that a cannon ball, if -fired vertically in a vacuum, would rise eight times as high as in the -atmosphere. In recent times an immense amount both of theoretical and -experimental investigation has been spent upon the subject, since it is -one of importance in the art of war. Successive approximations to the -true law have been made, but nothing like a complete and final solution -has been achieved or even hoped for.[381] - - [381] Hutton’s *Mathematical Dictionary*, vol. ii. pp. 287–292. - -It is quite to be expected that the earliest experimenters in any -branch of science will overlook errors which afterwards become most -apparent. The Arabian astronomers determined the meridian by taking the -middle point between the places of the sun when at equal altitudes on -the same day. They overlooked the fact that the sun has its own motion -in the time between the observations. Newton thought that the mutual -disturbances of the planets might be disregarded, excepting perhaps the -effect of the mutual attraction of the greater planets, Jupiter and -Saturn, near their conjunction.[382] The expansion of quicksilver was -long used as the measure of temperature, no clear idea being possessed -of temperature apart from some of its more obvious effects. Rumford, -in the first experiment leading to a determination of the mechanical -equivalent of heat, disregarded the heat absorbed by the apparatus, -otherwise he would, in Dr. Joule’s opinion, have come nearly to the -correct result. - - [382] *Principia*, bk. iii. Prop. 13. - -It is surprising to learn the number of causes of error which enter -into the simplest experiment, when we strive to attain rigid accuracy. -We cannot accurately perform the simple experiment of compressing gas -in a bent tube by a column of mercury, in order to test the truth -of Boyle’s Law, without paying regard to--(1) the variations of -atmospheric pressure, which are communicated to the gas through the -mercury; (2) the compressibility of mercury, which causes the column -of mercury to vary in density; (3) the temperature of the mercury -throughout the column; (4) the temperature of the gas, which is with -difficulty maintained invariable; (5) the expansion of the glass tube -containing the gas. Although Regnault took all these circumstances -into account in his examination of the law,[383] there is no reason to -suppose that he exhausted the sources of inaccuracy. - - [383] Jamin, *Cours de Physique*, vol. i. pp. 282, 283. - -The early investigations concerning the nature of waves in elastic -media proceeded upon the assumption that waves of different lengths -would travel with equal speed. Newton’s theory of sound led him to -this conclusion, and observation (p. 295) had verified the inference. -When the undulatory theory came to be applied at the commencement of -this century to explain the phenomena of light, a great difficulty was -encountered. The angle at which a ray of light is refracted in entering -a denser medium depends, according to that theory, on the velocity -with which the wave travels, so that if all waves of light were to -travel with equal velocity in the same medium, the dispersion of mixed -light by the prism and the production of the spectrum could not take -place. Some most striking phenomena were thus in direct conflict with -the theory. Cauchy first pointed out the explanation, namely, that all -previous investigators had made an arbitrary assumption for the sake -of simplifying the calculations. They had assumed that the particles -of the vibrating medium are so close together that the intervals are -inconsiderable compared with the length of the wave. This hypothesis -happened to be approximately true in the case of air, so that no error -was discovered in experiments on sound. Had it not been so, the earlier -analysts would probably have failed to give any solution, and the -progress of the subject might have been retarded. Cauchy was able to -make a new approximation under the more difficult supposition, that -the particles of the vibrating medium are situated at considerable -distances, and act and react upon the neighbouring particles by -attractive and repulsive forces. To calculate the rate of propagation -of disturbance in such a medium is a work of excessive difficulty. The -complete solution of the problem appears indeed to be beyond human -power, so that we must be content, as in the case of the planetary -motions, to look forward to successive approximations. All that Cauchy -could do was to show that certain quantities, neglected in previous -theories, became of considerable amount under the new conditions of -the problem, so that there will exist a relation between the length -of the wave, and the velocity at which it travels. To remove, then, -the difficulties in the way of the undulatory theory of light, a new -approach to probable conditions was needed.[384] - - [384] Lloyd’s *Lectures on the Wave Theory*, pp. 22, 23. - -In a similar manner Fourier’s theory of the conduction and radiation -of heat was based upon the hypothesis that the quantity of heat -passing along any line is simply proportional to the rate of change -of temperature. But it has since been shown by Forbes that the -conductivity of a body diminishes as its temperature increases. All -the details of Fourier’s solution therefore require modification, and -the results are in the meantime to be regarded as only approximately -true.[385] - - [385] Tait’s *Thermodynamics*, p. 10. - -We ought to distinguish between those problems which are physically and -those which are merely mathematically incomplete. In the latter case -the physical law is correctly seized, but the mathematician neglects, -or is more often unable to follow out the law in all its results. -The law of gravitation and the principles of harmonic or undulatory -movement, even supposing the data to be correct, can never be followed -into all their ultimate results. Young explained the production of -Newton’s rings by supposing that the rays reflected from the upper and -lower surfaces of a thin film of a certain thickness were in opposite -phases, and thus neutralised each other. It was pointed out, however, -that as the light reflected from the nearer surface must be undoubtedly -a little brighter than that from the further surface, the two rays -ought not to neutralise each other so completely as they are observed -to do. It was finally shown by Poisson that the discrepancy arose only -from incomplete solution of the problem; for the light which has once -got into the film must be to a certain extent reflected backwards and -forwards *ad infinitum*; and if we follow out this course of the light -by perfect mathematical analysis, absolute darkness may be shown to -result from the interference of the rays.[386] In this case the natural -laws concerned, those of reflection and refraction, are accurately -known, and the only difficulty consists in developing their full -consequences. - - [386] Lloyd’s *Lectures on the Wave Theory*, pp. 82, 83. - - -*Discovery of Hypothetically Simple Laws.* - -In some branches of science we meet with natural laws of a simple -character which are in a certain point of view exactly true and yet can -never be manifested as exactly true in natural phenomena. Such, for -instance, are the laws concerning what is called a *perfect gas*. The -gaseous state of matter is that in which the properties of matter are -exhibited in the simplest manner. There is much advantage accordingly -in approaching the question of molecular mechanics from this side. -But when we ask the question--What is a gas? the answer must be a -hypothetical one. Finding that gases *nearly* obey the law of Boyle -and Mariotte; that they *nearly* expand by heat at the uniform rate of -one part in 272·9 of their volume at 0° for each degree centigrade; -and that they *more nearly* fulfil these conditions the more distant -the point of temperature at which we examine them from the liquefying -point, we pass by the principle of continuity to the conception of a -perfect gas. Such a gas would probably consist of atoms of matter at -so great a distance from each other as to exert no attractive forces -upon each other; but for this condition to be fulfilled the distances -must be infinite, so that an absolutely perfect gas cannot exist. But -the perfect gas is not merely a limit to which we may approach, it is a -limit passed by at least one real gas. It has been shown by Despretz, -Pouillet, Dulong, Arago, and finally Regnault, that all gases diverge -from the Boylean law, and in nearly all cases the density of the gas -increases in a somewhat greater ratio than the pressure, indicating -a tendency on the part of the molecules to approximate of their own -accord. In the more condensable gases such as sulphurous acid, ammonia, -and cyanogen, this tendency is strongly apparent near the liquefying -point. Hydrogen, on the contrary, diverges from the law of a perfect -gas in the opposite direction, that is, the density increases less than -in the ratio of the pressure.[387] This is a singular exception, the -bearing of which I am unable to comprehend. - - [387] Jamin, *Cours de Physique*, vol. i. pp. 283–288. - -All gases diverge again from the law of uniform expansion by heat, but -the divergence is less as the gas in question is less condensable, or -examined at a temperature more removed from its liquefying point. Thus -the perfect gas must have an infinitely high temperature. According -to Dalton’s law each gas in a mixture retains its own properties -unaffected by the presence of any other gas.[388] This law is probably -true only by approximation, but it is obvious that it would be true of -the perfect gas with infinitely distant particles.[389] - - [388] Joule and Thomson, *Philosophical Transactions*, 1854, - vol. cxliv. p. 337. - - [389] The properties of a perfect gas have been described by Rankine, - *Transactions of the Royal Society of Edinburgh*, vol. xxv. p. 561. - - -*Mathematical Principles of Approximation.* - -The approximate character of physical science will be rendered more -plain if we consider it from a mathematical point of view. Throughout -quantitative investigations we deal with the relation of one quantity -to other quantities, of which it is a function; but the subject is -sufficiently complicated if we view one quantity as a function of -one other. Now, as a general rule, a function can be developed or -expressed as the sum of quantities, the values of which depend upon the -successive powers of the variable quantity. If *y* be a function of *x* -then we may say that - - *y* = A + B*x* + C*x*^{2} + D*x*^{3} + E*x*^{4} .... - -In this equation, A, B, C, D, &c., are fixed quantities, of different -values in different cases. The terms may be infinite in number or -after a time may cease to have any value. Any of the coefficients A, -B, C, &c., may be zero or negative; but whatever they be they are -fixed. The quantity *x* on the other hand may be made what we like, -being variable. Suppose, in the first place, that *x* and *y* are both -lengths. Let us assume that 1/10,000 part of an inch is the least that -we can take note of. Then when *x* is one hundredth of an inch, we have -*x*^{2} = 1/10,000, and if C be less than unity, the term C*x*^{2} will -be inappreciable, being less than we can measure. Unless any of the -quantities D, E, &c., should happen to be very great, it is evident -that all the succeeding terms will also be inappreciable, because the -powers of *x* become rapidly smaller in geometrical ratio. Thus when -*x* is made small enough the quantity *y* seems to obey the equation - - *y* = A + B*x*. - -If *x* should be still less, if it should become as small, for -instance, as 1/1,000,000 of an inch, and B should not be very great, -then *y* would appear to be the fixed quantity A, and would not seem to -vary with *x* at all. On the other hand, were x to grow greater, say -equal to 1/10 inch, and C not be very small, the term C*x*^{2} would -become appreciable, and the law would now be more complicated. - -We can invert the mode of viewing this question, and suppose that -while the quantity *y* undergoes variations depending on many powers -of *x*, our power of detecting the changes of value is more or less -acute. While our powers of observation remain very rude we may be -unable to detect any change in the quantity at all, that is to say, -B*x* may always be too small to come within our notice, just as in -former days the fixed stars were so called because they remained at -apparently fixed distances from each other. With the use of telescopes -and micrometers we become able to detect the existence of some motion, -so that the distance of one star from another may be expressed by -A + B*x*, the term including *x*^{2} being still inappreciable. Under -these circumstances the star will seem to move uniformly, or in simple -proportion to the time *x*. With much improved means of measurement it -will probably be found that this uniformity of motion is only apparent, -and that there exists some acceleration or retardation. More careful -investigation will show the law to be more and more complicated than -was previously supposed. - -There is yet another way of explaining the apparent results of a -complicated law. If we take any curve and regard a portion of it free -from any kind of discontinuity, we may represent the character of such -portion by an equation of the form - - *y* = A + B*x* + C*x*^{2} + D*x*^{3} + .... - -Restrict the attention to a very small portion of the curve, and the -eye will be unable to distinguish its difference from a straight -line, which amounts to saying that in the portion examined the term -C*x*^{2} has no value appreciable by the eye. Take a larger portion of -the curve and it will be apparent that it possesses curvature, but it -will be possible to draw a parabola or ellipse so that the curve shall -apparently coincide with a portion of that parabola or ellipse. In the -same way if we take larger and larger arcs of the curve it will assume -the character successively of a curve of the third, fourth, and perhaps -higher degrees; that is to say, it corresponds to equations involving -the third, fourth, and higher powers of the variable quantity. - -We have arrived then at the conclusion that every phenomenon, when its -amount can only be rudely measured, will either be of fixed amount, -or will seem to vary uniformly like the distance between two inclined -straight lines. More exact measurement may show the error of this first -assumption, and the variation will then appear to be like that of the -distance between a straight line and a parabola or ellipse. We may -afterwards find that a curve of the third or higher degrees is really -required to represent the variation. I propose to call the variation -of a quantity *linear*, *elliptic*, *cubic*, *quartic*, *quintic*, &c., -according as it is discovered to involve the first, second, third, -fourth, fifth, or higher powers of the variable. It is a general -rule in quantitative investigation that we commence by discovering -linear, and afterwards proceed to elliptic or more complicated laws of -variation. The approximate curves which we employ are all, according -to De Morgan’s use of the name, parabolas of some order or other; and -since the common parabola of the second order is approximately the same -as a very elongated ellipse, and is in fact an infinitely elongated -ellipse, it is convenient and proper to call variation of the second -order *elliptic*. It might also be called *quadric* variation. - -As regards many important phenomena we are yet only in the first stage -of approximation. We know that the sun and many so-called fixed stars, -especially 61 Cygni, have a proper motion through space, and the -direction of this motion at the present time is known with some degree -of accuracy. But it is hardly consistent with the theory of gravity -that the path of any body should really be a straight line. Hence, we -must regard a rectilinear path as only a provisional description of -the motion, and look forward to the time when its curvature will be -detected, though centuries perhaps must first elapse. - -We are accustomed to assume that on the surface of the earth the force -of gravity is uniform, because the variation is of so slight an amount -that we are scarcely able to detect it. But supposing we could measure -the variation, we should find it simply proportional to the height. -Taking the earth’s radius to be unity, let *h* be the height at which -we measure the force of gravity. Then by the well-known law of the -inverse square, that force will be proportional to - - *g*/(1 + *h*)^{2}, or to *g*(1 - 2*h* + 3*h*^{2} - 4*h*^{3} + ...). - -But at all heights to which we can attain *h* will be so small a -fraction of the earth’s radius that 3*h*^{2} will be inappreciable, and -the force of gravity will seem to follow the law of linear variation, -being proportional to 1 - 2*h*. - -When the circumstances of an experiment are much altered, different -powers of the variable may become prominent. The resistance of a liquid -to a body moving through it may be approximately expressed as the sum -of two terms respectively involving the first and second powers of the -velocity. At very low velocities the first power is of most importance, -and the resistance, as Professor Stokes has shown, is nearly in simple -proportion to the velocity. When the motion is rapid the resistance -increases in a still greater degree, and is more nearly proportional to -the square of the velocity. - - -*Approximate Independence of Small Effects.* - -One result of the theory of approximation possesses such importance -in physical science, and is so often applied, that we may consider -it separately. The investigation of causes and effects is immensely -simplified when we may consider each cause as producing its own effect -invariably, whether other causes are acting or not. Thus, if the -body P produces *x*, and Q produces *y*, the question is whether P -and Q acting together will produce the sum of the separate effects, -*x* + *y*. It is under this supposition that we treated the methods of -eliminating error (Chap. XV.), and errors of a less amount would still -remain if the supposition was a forced one. There are probably some -parts of science in which the supposition of independence of effects -holds rigidly true. The mutual gravity of two bodies is entirely -unaffected by the presence of other gravitating bodies. People do not -usually consider that this important principle is involved in such a -simple thing as putting two pound weights in the scale of a balance. -How do we know that two pounds together will weigh twice as much as -one? Do we know it to be exactly so? Like other results founded on -induction we cannot prove it absolutely, but all the calculations of -physical astronomy proceed upon the assumption, so that we may consider -it proved to a very high degree of approximation. Had not this been -true, the calculations of physical astronomy would have been infinitely -more complex than they actually are, and the progress of knowledge -would have been much slower. - -It is a general principle of scientific method that if effects be of -small amount, comparatively to our means of observation, all joint -effects will be of a higher order of smallness, and may therefore -be rejected in a first approximation. This principle was employed -by Daniel Bernoulli in the theory of sound, under the title of *The -Principle of the Coexistence of Small Vibrations*. He showed that if -a string is affected by two kinds of vibrations, we may consider each -to be going on as if the other did not exist. We cannot perceive that -the sounding of one musical instrument prevents or even modifies the -sound of another, so that all sounds would seem to travel through the -air, and act upon the ear in independence of each other. A similar -assumption is made in the theory of tides, which are great waves. One -wave is produced by the attraction of the moon, and another by the -attraction of the sun, and the question arises, whether when these -waves coincide, as at the time of spring tides, the joint wave will be -simply the sum of the separate waves. On the principle of Bernoulli -this will be so, because the tides on the ocean are very small compared -with the depth of the ocean. - -The principle of Bernoulli, however, is only approximately true. A wave -never is exactly the same when another wave is interfering with it, -but the less the displacement of particles due to each wave, the less -in a still higher degree is the effect of one wave upon the other. In -recent years Helmholtz was led to suspect that some of the phenomena -of sound might after all be due to resultant effects overlooked by the -assumption of previous physicists. He investigated the secondary waves -which would arise from the interference of considerable disturbances, -and was able to show that certain summation of resultant tones ought to -be heard, and experiments subsequently devised for the purpose showed -that they might be heard. - -[Illustration] - -Throughout the mechanical sciences the *Principle of the Superposition -of Small Motions* is of fundamental importance,[390] and it may be -thus explained. Suppose that two forces, acting from the points B and -C, are simultaneously moving a body A. Let the force acting from B be -such that in one second it would move A to *p*, and similarly let the -second force, acting alone, move A to *r*. The question arises, then, -whether their joint action will urge A to *q* along the diagonal of the -parallelogram. May we say that A will move the distance A*p* in the -direction AB, and A*r* in the direction AC, or, what is the same thing, -along the parallel line *pq*? In strictness we cannot say so; for when -A has moved towards *p*, the force from C will no longer act along the -line AC, and similarly the motion of A towards *r* will modify the -action of the force from B. This interference of one force with the -line of action of the other will evidently be greater the larger is -the extent of motion considered; on the other hand, as we reduce the -parallelogram A*pqr*, compared with the distances AB and AC, the less -will be the interference of the forces. Accordingly mathematicians -avoid all error by considering the motions as infinitely small, so that -the interference becomes of a still higher order of infinite smallness, -and may be entirely neglected. By the resources of the differential -calculus it is possible to calculate the motion of the particle A, as -if it went through an infinite number of infinitely small diagonals -of parallelograms. The great discoveries of Newton really arose from -applying this method of calculation to the movements of the moon round -the earth, which, while constantly tending to move onward in a straight -line, is also deflected towards the earth by gravity, and moves -through an elliptic curve, composed as it were of the infinitely small -diagonals of infinitely numerous parallelograms. The mathematician, -in his investigation of a curve, always treats it as made up of a -great number of straight lines, and it may be doubted whether he could -treat it in any other manner. There is no error in the final results, -because having obtained the formulæ flowing from this supposition, each -straight line is then regarded as becoming infinitely small, and the -polygonal line becomes undistinguishable from a perfect curve.[391] - - [390] Thomson and Tait’s *Natural Philosophy*, vol. i. p. 60. - - [391] Challis, *Notes on the Principles of Pure and Applied - Calculation*, 1869, p. 83. - -In abstract mathematical theorems the approximation to absolute truth -is perfect, because we can treat of infinitesimals. In physical -science, on the contrary, we treat of the least quantities which are -perceptible. Nevertheless, while carefully distinguishing between these -two different cases, we may fearlessly apply to both the principle -of the superposition of small effects. In physical science we have -only to take care that the effects really are so small that any joint -effect will be unquestionably imperceptible. Suppose, for instance, -that there is some cause which alters the dimensions of a body in the -ratio of 1 to 1 + α, and another cause which produces an alteration in -the ratio of 1 to 1 + β. If they both act at once the change will be -in the ratio of 1 to (1 + α)(1 + β), or as 1 to 1 + α + β + αβ. But if -α and β be both very small fractions of the total dimensions, αβ will -be yet far smaller and may be disregarded; the ratio of change is then -approximately that of 1 to 1 + α + β, or the joint effect is the sum of -the separate effects. Thus if a body were subjected to three strains, -at right angles to each other, the total change in the volume of the -body would be approximately equal to the sum of the changes produced -by the separate strains, provided that these are very small. In like -manner not only is the expansion of every solid and liquid substance -by heat approximately proportional to the change of temperature, when -this change is very small in amount, but the cubic expansion may also -be considered as being three times as great as the linear expansion. -For if the increase of temperature expands a bar of metal in the ratio -of 1 to 1 + α, and the expansion be equal in all directions, then a -cube of the same metal would expand as 1 to (1 + α)^{3}, or as 1 to -1 + 3α + 3α^{2} + α^{3}. When α is a very small quantity the third term -3α^{2} will be imperceptible, and still more so the fourth term α^{3}. -The coefficients of expansion of solids are in fact so small, and so -imperfectly determined, that physicists seldom take into account their -second and higher powers. - -It is a result of these principles that all small errors may be assumed -to vary in simple proportion to their causes--a new reason why, in -eliminating errors, we should first of all make them as small as -possible. Let us suppose that there is a right-angled triangle of which -the two sides containing the right angle are really of the lengths 3 -and 4, so that the hypothenuse is √(3^{2} + 4^{2}) or 5. Now, -if in two measurements of the first side we commit slight errors, -making it successively 4·001 and 4·002, then calculation will give the -lengths of the hypothenuse as almost exactly 5·0008 and 5·0016, so that -the error in the hypothenuse will seem to vary in simple proportion -to that of the side, although it does not really do so with perfect -exactness. The logarithm of a number does not vary in proportion to -that number--nevertheless we find the difference between the logarithms -of the numbers 100000 and 100001 to be almost exactly equal to that -between the numbers 100001 and 100002. It is thus a general rule that -very small differences between successive values of a function are -approximately proportional to the small differences of the variable -quantity. - -On these principles it is easy to draw up a series of rules such as -those given by Kohlrausch[392] for performing calculations in an -abbreviated form when the variable quantity is very small compared with -unity. Thus for 1 ÷ (1 + α) we may substitute 1 - α; for 1 ÷ (1 - α) we -may put 1 + α; 1 ÷ √(1 + α) becomes 1 - (1/2)α, and so forth. - - [392] *An Introduction to Physical Measurements*, translated by - Waller and Procter, 1873, p. 10. - - -*Four Meanings of Equality.* - -Although it might seem that there are few terms more free from -ambiguity than the term *equal*, yet scientific men do employ it with -at least four meanings, which it is desirable to distinguish. These -meanings I may describe as - - (1) Absolute Equality. - (2) Sub-equality. - (3) Apparent Equality. - (4) Probable Equality. - -By *absolute equality* we signify that which is complete and perfect -to the last degree; but it is obvious that we can only know such -equality in a theoretical or hypothetical manner. The areas of two -triangles standing upon the same base and between the same parallels -are absolutely equal. Hippocrates beautifully proved that the area -of a lunula or figure contained between two segments of circles was -absolutely equal to that of a certain right-angled triangle. As a -general rule all geometrical and other elementary mathematical theorems -involve absolute equality. - -De Morgan proposed to describe as *sub-equal* those quantities which -are equal within an infinitely small quantity, so that *x* is sub-equal -to *x* + *dx*. The differential calculus may be said to arise out -of the neglect of infinitely small quantities, and in mathematical -science other subtle distinctions may have to be drawn between kinds of -equality, as De Morgan has shown in a remarkable memoir “On Infinity; -and on the sign of Equality.”[393] - - [393] *Cambridge Philosophical Transactions* (1865), vol. xi. Part I. - -*Apparent equality* is that with which physical science deals. Those -magnitudes are apparently equal which differ only by an imperceptible -quantity. To the carpenter anything less than the hundredth part of -an inch is non-existent; there are few arts or artists to which the -hundred-thousandth of an inch is of any account. Since all coincidence -between physical magnitudes is judged by one or other sense, we must be -restricted to a knowledge of apparent equality. - -In reality even apparent equality is rarely to be expected. More -commonly experiments will give only *probable equality*, that is -results will come so near to each other that the difference may be -ascribed to unimportant disturbing causes. Physicists often assume -quantities to be equal provided that they fall within the limits of -probable error of the processes employed. We cannot expect observations -to agree with theory more closely than they agree with each other, as -Newton remarked of his investigations concerning Halley’s Comet. - - -*Arithmetic of Approximate Quantities.* - -Considering that almost all the quantities which we treat in physical -and social science are approximate only, it seems desirable that -attention should be paid in the teaching of arithmetic to the correct -interpretation and treatment of approximate numerical statements. We -seem to need notation for expressing the approximateness or exactness -of decimal numbers. The fraction ·025 may mean either precisely one -40th part, or it may mean anything between ·0245 and ·0255. I propose -that when a decimal fraction is completely and exactly given, a *small -cipher* or circle should be added to indicate that there is nothing -more to come, as in ·025◦. When the first figure of the decimals -rejected is 5 or more, the first figure retained should be raised by -a unit, according to a rule approved by De Morgan, and now generally -recognised. To indicate that the fraction thus retained is more than -the truth, a point has been placed over the last figure in some tables -of logarithms; but a similar point is used to denote the period of a -repeating decimal, and I should therefore propose to employ a colon -*after* the figure; thus ·025: would mean that the true quantity lies -between ·0245° and ·025° inclusive of the lower but not the higher -limit. When the fraction is less than the truth, two dots might be -placed horizontally as in 025.. which would mean anything between ·025° -and ·0255° not inclusive. - -When approximate numbers are added, subtracted, multiplied, or divided, -it becomes a matter of some complexity to determine the degree of -accuracy of the result. There are few persons who could assert off-hand -that the sum of the approximate numbers 34·70, 52·693, 80·1, is 167·5 -*within less than* ·07. Mr. Sandeman has traced out the rules of -approximate arithmetic in a very thorough manner, and his directions -are worthy of careful attention.[394] The third part of Sonnenschein -and Nesbitt’s excellent book on arithmetic[395] describes fully all -kinds of approximate calculations, and shows both how to avoid needless -labour and how to take proper account of inaccuracy in operating with -approximate decimal fractions. A simple investigation of the subject is -to be found in Sonnet’s *Algèbre Elémentaire* (Paris, 1848) chap. xiv., -“Des Approximations Absolues et Relatives.” There is also an American -work on the subject.[396] - - [394] Sandeman, *Pelicotetics*, p. 214. - - [395] *The Science and Art of Arithmetic for the Use of Schools.* - (Whitaker and Co.) - - [396] *Principles of Approximate Calculations*, by J. J. Skinner, - C.E. (New York, Henry Holt), 1876. - -Although the accuracy of measurement has so much advanced since the -time of Leslie, it is not superfluous to repeat his protest against -the unfairness of affecting by a display of decimal fractions a -greater degree of accuracy than the nature of the case requires and -admits.[397] I have known a scientific man to register the barometer -to a second of time when the nearest quarter of an hour would have -been amply sufficient. Chemists often publish results of analysis to -the ten-thousandth or even the millionth part of the whole, when in -all probability the processes employed cannot be depended on beyond -the hundredth part. It is seldom desirable to give more than one -place of figures of uncertain amount; but it must be allowed that a -nice perception of the degree of accuracy possible and desirable is -requisite to save misapprehension and needless computation on the one -hand, and to secure all attainable exactness on the other hand. - - [397] Leslie, *Inquiry into the Nature of Heat*, p. 505. - - - - -CHAPTER XXII. - -QUANTITATIVE INDUCTION. - - -We have not yet formally considered any processes of reasoning -which have for their object to disclose laws of nature expressed in -quantitative equations. We have been inquiring into the modes by which -a phenomenon may be measured, and, if it be a composite phenomenon, may -be resolved, by the aid of several measurements, into its component -parts. We have also considered the precautions to be taken in the -performance of observations and experiments in order that we may know -what phenomena we really do measure, but we must remember that, no -number of facts and observations can by themselves constitute science. -Numerical facts, like other facts, are but the raw materials of -knowledge, upon which our reasoning faculties must be exerted in order -to draw forth the principles of nature. It is by an inverse process of -reasoning that we can alone discover the mathematical laws to which -varying quantities conform. By well-conducted experiments we gain a -series of values of a variable, and a corresponding series of values -of a variant, and we now want to know what mathematical function the -variant is as regards the variable. In the usual progress of a science -three questions will have to be answered as regards every important -quantitative phenomenon:-- - -(1) Is there any constant relation between a variable and a variant? - -(2) What is the empirical formula expressing this relation? - -(3) What is the rational formula expressing the law of nature involved? - - -*Probable Connection of Varying Quantities.* - -We find it stated by Mill,[398] that “Whatever phenomenon varies in any -manner whenever another phenomenon varies in some particular manner, is -either a cause or an effect of that phenomenon, or is connected with -it through some fact of causation.” This assertion may be considered -true when it is interpreted with sufficient caution; but it might -otherwise lead us into error. There is nothing whatever in the nature -of things to prevent the existence of two variations which should -apparently follow the same law, and yet have no connection with each -other. One binary star might be going through a revolution which, so -far as we could tell, was of equal period with that of another binary -star, and according to the above rule the motion of one would be the -cause of the motion of the other, which would not be really the case. -Two astronomical clocks might conceivably be made so nearly perfect -that, for several years, no difference could be detected, and we might -then infer that the motion of one clock was the cause or effect of the -motion of the other. This matter requires careful discrimination. We -must bear in mind that the continuous quantities of space, time, force, -&c., which we measure, are made up of an infinite number of infinitely -small units. We may then meet with two variable phenomena which follow -laws so nearly the same, that in no part of the variations open to -our observation can any discrepancy be discovered. I grant that if -two clocks could be shown to have kept *exactly* the same time during -any finite interval, the probability would become infinitely high -that there was a connection between their motions. But we can never -absolutely prove such coincidences to exist. Allow that we may observe -a difference of one-tenth of a second in their time, yet it is possible -that they were independently regulated so as to go together within -less than that quantity of time. In short, it would require either an -infinitely long time of observation, or infinitely acute powers of -measuring discrepancy, to decide positively whether two clocks were or -were not in relation with each other. - - [398] *System of Logic*, bk. iii. chap. viii § 6. - -A similar question actually occurs in the case of the moon’s motion. We -have no record that any other portion of the moon was ever visible to -men than such as we now see. This fact sufficiently proves that within -the historical period the rotation of the moon on its own axis has -coincided with its revolutions round the earth. Does this coincidence -prove a relation of cause and effect to exist? The answer must be in -the negative, because there might have been so slight a discrepancy -between the motions that there has not yet been time to produce any -appreciable effect. There may nevertheless be a high probability of -connection. - -The whole question of the relation of quantities thus resolves itself -into one of probability. When we can only rudely measure a quantitative -result, we can assign but slight importance to any correspondence. -Because the brightness of two stars seems to vary in the same manner, -there is no considerable probability that they have any relation with -each other. Could it be shown that their periods of variation were -the same to infinitely small quantities it would be certain, that -is infinitely probable, that they were connected, however unlikely -this might be on other grounds. The general mode of estimating such -probabilities is identical with that applied to other inductive -problems. That any two periods of variation should by chance become -*absolutely equal* is infinitely improbable; hence if, in the case of -the moon or other moving bodies, we could prove absolute coincidence -we should have certainty of connection.[399] With approximate -measurements, which alone are within our power, we must hope for -approximate certainty at the most. - - [399] Laplace, *System of the World*, translated by Harte, vol. ii. - p. 366. - -The principles of inference and probability, according to which we -treat causes and effects varying in amount, are exactly the same as -those by which we treated simple experiments. Continuous quantity, -however, affords us an infinitely more extensive sphere of observation, -because every different amount of cause, however little different, -ought to be followed by a different amount of effect. If we can measure -temperature to the one-hundredth part of a degree centigrade, then -between 0° and 100° we have 10,000 possible trials. If the precision -of our measurements is increased, so that the one-thousandth part of -a degree can be appreciated, our trials may be increased tenfold. The -probability of connection will be proportional to the accuracy of our -measurements. - -When we can vary the quantity of a cause at will it is easy to discover -whether a certain effect is due to that cause or not. We can then -make as many irregular changes as we like, and it is quite incredible -that the supposed effect should by chance go through exactly the -corresponding series of changes except by dependence. If we have a -bell ringing *in vacuo*, the sound increases as we let in the air, and -it decreases again as we exhaust the air. Tyndall’s singing flames -evidently obeyed the directions of his own voice; and Faraday when he -discovered the relation of magnetism and light found that, by making -or breaking or reversing the current of the electro-magnet, he had -complete command over a ray of light, proving beyond all reasonable -doubt the dependence of cause and effect. In such cases it is the -perfect coincidence in time between the change in the effect and that -in the cause which raises a high improbability of casual coincidence. - -It is by a simple case of variation that we infer the existence of -a material connection between two bodies moving with exactly equal -velocity, such as the locomotive engine and the train which follows -it. Elaborate observations were requisite before astronomers could all -be convinced that the red hydrogen flames seen during solar eclipses -belonged to the sun, and not to the moon’s atmosphere as Flamsteed -assumed. As early as 1706, Stannyan noticed a blood-red streak in -an eclipse which he witnessed at Berne, and he asserted that it -belonged to the sun; but his opinion was not finally established until -photographs of the eclipse in 1860, taken by Mr. De la Rue, showed that -the moon’s dark body gradually covered the red prominences on one side, -and uncovered those on the other; in short, that these prominences -moved precisely as the sun moved, and not as the moon moved. - -Even when we have no means of accurately measuring the variable -quantities we may yet be convinced of their connection, if one always -varies perceptibly at the same time as the other. Fatigue increases -with exertion; hunger with abstinence from food; desire and degree of -utility decrease with the quantity of commodity consumed. We know that -the sun’s heating power depends upon his height of the sky; that the -temperature of the air falls in ascending a mountain; that the earth’s -crust is found to be perceptibly warmer as we sink mines into it; we -infer the direction in which a sound comes from the change of loudness -as we approach or recede. The facility with which we can time after -time observe the increase or decrease of one quantity with another -sufficiently shows the connection, although we may be unable to assign -any precise law of relation. The probability in such cases depends upon -frequent coincidence in time. - - -*Empirical Mathematical Laws.* - -It is important to acquire a clear comprehension of the part which -is played in scientific investigation by empirical formulæ and laws. -If we have a table containing certain values of a variable and the -corresponding values of the variant, there are mathematical processes -by which we can infallibly discover a mathematical formula yielding -numbers in more or less exact agreement with the table. We may -generally assume that the quantities will approximately conform to a -law of the form - - *y* = A + B*x* + C*x*^{2}, - -in which *x* is the variable and *y* the variant. We can then select -from the table three values of *y*, and the corresponding values of -*x*; inserting them in the equation, we obtain three equations by the -solution of which we gain the values of A, B, and C. It will be found -as a general rule that the formula thus obtained yields the other -numbers of the table to a considerable degree of approximation. - -In many cases even the second power of the variable will be -unnecessary; Regnault found that the results of his elaborate inquiry -into the latent heat of steam at different pressures were represented -with sufficient accuracy by the empirical formula - - λ = 606·5 + 0·305 *t*, - -in which λ is the total heat of the steam, and *t* the -temperature.[400] In other cases it may be requisite to include the -third power of the variable. Thus physicists assume the law of the -dilatation of liquids to be of the form - - δ_{t} = *at* + *bt*^{2} + *ct*^{3}, - - [400] *Chemical Reports and Memoirs*, Cavendish Society, p. 294. - -and they calculate from results of observation the values of the -three constants *a*, *b*, *c*, which are usually small quantities -not exceeding one-hundredth part of a unit, but requiring to be -determined with great accuracy.[401] Theoretically speaking, this -process of empirical representation might be applied with any degree of -accuracy; we might include still higher powers in the formula, and with -sufficient labour obtain the values of the constants, by using an equal -number of experimental results. The method of least squares may also be -employed to obtain the most probable values of the constants. - - [401] Jamin, *Cours de Physique*, vol. ii. p. 38. - -In a similar manner all periodic variations may be represented with any -required degree of accuracy by formulæ involving the sines and cosines -of angles and their multiples. The form of any tidal or other wave may -thus be expressed, as Sir G. B. Airy has explained.[402] Almost all -the phenomena registered by meteorologists are periodic in character, -and when freed from disturbing causes may be embodied in empirical -formulæ. Bessel has given a rule by which from any regular series of -observations we may, on the principle of the method of least squares, -calculate out with a moderate amount of labour a formula expressing the -variation of the quantity observed, in the most probable manner. In -meteorology three or four terms are usually sufficient for representing -any periodic phenomenon, but the calculation might be carried to any -higher degree of accuracy. As the details of the process have been -described by Herschel in his treatise on Meteorology,[403] I need not -further enter into them. - - [402] *On Tides and Waves*, Encyclopædia Metropolitana, p. 366*. - - [403] *Encyclopædia Britannica*, art. *Meteorology*. Reprint, §§ - 152–156. - -The reader might be tempted to think that in these processes of -calculation we have an infallible method of discovering inductive -laws, and that my previous statements (Chap. VII.) as to the purely -tentative and inverse character of the inductive process are negatived. -Were there indeed any general method of inferring laws from facts it -would overturn my statement, but it must be carefully observed that -these empirical formulæ do not coincide with natural laws. They are -only approximations to the results of natural laws founded upon the -general principles of approximation. It has already been pointed out -that however complicated be the nature of a curve, we may examine so -small a portion of it, or we may examine it with such rude means of -measurement, that its divergence from an elliptic curve will not be -apparent. As a still ruder approximation a portion of a straight line -will always serve our purpose; but if we need higher precision a curve -of the third or fourth degree will almost certainly be sufficient. Now -empirical formulæ really represent these approximate curves, but they -give us no information as to the precise nature of the curve itself to -which we are approximating. We do not learn what function the variant -is of the variable, but we obtain another function which, within the -bounds of observation, gives nearly the same values. - - -*Discovery of Rational Formulæ.* - -Let us now proceed to consider the modes in which from numerical -results we can establish the actual relation between the quantity of -the cause and that of the effect. What we want is a *rational* formula -or function, which will exhibit the *reason* or exact nature and -origin of the law in question. There is no word more frequently used -by mathematicians than the word *function*, and yet it is difficult -to define its meaning with perfect accuracy. Originally it meant -performance or execution, being equivalent to the Greek λειτουργία -or τέλεσμα. Mathematicians at first used it to mean *any power of a -quantity*, but afterwards generalised it so as to include “any quantity -formed in any manner whatsoever from another quantity.”[404] Any -quantity, then, which depends upon and varies with another quantity may -be called a function of it, and either may be considered a function of -the other. - - [404] Lagrange, *Leçons sur le Calcul des Fonctions*, 1806, p. 4. - -Given the quantities, we want the function of which they are the -values. Simple inspection of the numbers cannot as a general rule -disclose the function. In an earlier chapter (p. 124) I put before the -reader certain numbers, and requested him to point out the law which -they obey, and the same question will have to be asked in every case of -quantitative induction. There are perhaps three methods, more or less -distinct, by which we may hope to obtain an answer: - -(1) By purely haphazard trial. - -(2) By noting the general character of the variation of the quantities, -and trying by preference functions which give a similar form of -variation. - -(3) By deducing from previous knowledge the form of the function which -is most likely to suit. - -Having numerical results we are always at liberty to invent any kind -of mathematical formula we like, and then try whether, by the suitable -selection of values for the unknown constant quantities, we can make it -give the required results. If ever we fall upon a formula which does -so, to a fair degree of approximation, there is a presumption in favour -of its being the true function, although there is no certainty whatever -in the matter. In this way I discovered a simple mathematical law which -closely agreed with the results of my experiments on muscular exertion. -This law was afterwards shown by Professor Haughton to be the true -rational law according to his theory of muscular action.[405] - - [405] Haughton, *Principles of Animal Mechanics*, 1873, pp. 444–450. - Jevons, *Nature*, 30th of June, 1870, vol. ii. p. 158. See also the - experiments of Professor Nipher, of Washington University, St. Louis, - in *American Journal of Science*, vol. ix. p. 130, vol. x. p. 1; - *Nature*, vol. xi. pp. 256, 276. - -But the chance of succeeding in this manner is small. The number of -possible functions is infinite, and even the number of comparatively -simple functions is so large that the probability of falling upon the -correct one by mere chance is very slight. Even when we obtain the law -it is by a deductive process, not by showing that the numbers give the -law, but that the law gives the numbers. - -In the second way, we may, by a survey of the numbers, gain a general -notion of the kind of law they are likely to obey, and we may be much -assisted in this process by drawing them out in the form of a curve. -We can in this way ascertain with some probability whether the curve -is likely to return into itself, or whether it has infinite branches; -whether such branches are asymptotic, that is, approach infinitely -towards straight lines; whether it is logarithmic in character, or -trigonometric. This indeed we can only do if we remember the results of -previous investigations. The process is still inversely deductive, and -consists in noting what laws give particular curves, and then inferring -inversely that such curves belong to such laws. If we can in this way -discover the class of functions to which the required law belongs, -our chances of success are much increased, because our haphazard -trials are now reduced within a narrower sphere. But, unless we have -almost the whole curve before us, the identification of its character -must be a matter of great uncertainty; and if, as in most physical -investigations, we have a mere fragment of the curve, the assistance -given would be quite illusory. Curves of almost any character can be -made to approximate to each other for a limited extent, so that it is -only by a kind of *divination* that we fall upon the actual function, -unless we have theoretical knowledge of the kind of function applicable -to the case. - -When we have once obtained what we believe to be the correct form of -function, the remainder of the work is mere mathematical computation to -be performed infallibly according to fixed rules,[406] which include -those employed in the determination of empirical formulæ (p. 487). -The function will involve two or three or more unknown constants, the -values of which we need to determine by our experimental results. -Selecting some of our results widely apart and nearly equidistant, -we form by means of them as many equations as there are constant -quantities to be determined. The solution of these equations will then -give us the constants required, and having now the actual function we -can try whether it gives with sufficient accuracy the remainder of -our experimental results. If not, we must either make a new selection -of results to give a new set of equations, and thus obtain a new set -of values for the constants, or we must acknowledge that our form -of function has been wrongly chosen. If it appears that the form of -function has been correctly ascertained, we may regard the constants -as only approximately accurate and may proceed by the Method of Least -Squares (p. 393) to determine the most probable values as given by the -whole of the experimental results. - - [406] Jamin, *Cours de Physique*, vol. ii. p. 50. - -In most cases we shall find ourselves obliged to fall back upon the -third mode, that is, anticipation of the form of the law to be expected -on the ground of previous knowledge. Theory and analogical reasoning -must be our guides. The general nature of the phenomenon will often -indicate the kind of law to be looked for. If one form of energy or one -kind of substance is being converted into another, we may expect the -law of direct simple proportion. In one distinct class of cases the -effect already produced influences the amount of the ensuing effect, -as for instance in the cooling of a heated body, when the law will be -of an exponential form. When the direction of a force influences its -action, trigonometrical functions enter. Any influence which spreads -freely through tridimensional space will be subject to the law of -the inverse square of the distance. From such considerations we may -sometimes arrive deductively and analogically at the general nature of -the mathematical law required. - - -*The Graphical Method.* - -In endeavouring to discover the mathematical law obeyed by -experimental results it is often desirable to call in the aid of -space-representations. Every equation involving two variable quantities -corresponds to some kind of plane curve, and every plane curve may -be represented symbolically in an equation containing two unknown -quantities. Now in an experimental research we obtain a number of -values of the variant corresponding to an equal number of values of the -variable; but all the numbers are affected by more or less error, and -the values of the variable will often be irregularly disposed. Even if -the numbers were absolutely correct and disposed at regular intervals, -there is, as we have seen, no direct mode of discovering the law, but -the difficulty of discovery is much increased by the uncertainty and -irregularity of the results. - -Under such circumstances, the best mode of proceeding is to prepare a -paper divided into equal rectangular spaces, a convenient size for the -spaces being one-tenth of an inch square. The values of the variable -being marked off on the lowest horizontal line, a point is marked for -each corresponding value of the variant perpendicularly above that of -the variable, and at such a height as corresponds to the value of the -variant. - -The exact scale of the drawing is not of much importance, but it may -require to be adjusted according to circumstances, and different values -must often be attributed to the upright and horizontal divisions, so -as to make the variations conspicuous but not excessive. If a curved -line be drawn through all the points or ends of the ordinates, it will -probably exhibit irregular inflections, owing to the errors which -affect the numbers. But, when the results are numerous, it becomes -apparent which results are more divergent than others, and guided by a -so-called *sense of continuity*, it is possible to trace a line among -the points which will approximate to the true law more nearly than the -points themselves. The accompanying figure sufficiently explains itself. - -[Illustration] - -Perkins employed this graphical method with much care in exhibiting -the results of his experiments on the compression of water.[407] The -numerical results were marked upon a sheet of paper very exactly ruled -at intervals of one-tenth of an inch, and the original marks were left -in order that the reader might judge of the correctness of the curve -drawn, or choose another for himself. Regnault carried the method to -perfection by laying off the points with a screw dividing engine;[408] -and he then formed a table of results by drawing a continuous curve, -and measuring its height for equidistant values of the variable. Not -only does a curve drawn in this manner enable us to infer numerical -results more free from accidental errors than any of the numbers -obtained directly from experiment, but the form of the curve sometimes -indicates the class of functions to which our results belong. - - [407] *Philosophical Transactions*, 1826, p. 544. - - [408] Jamin, *Cours de Physique*, vol. ii. p. 24, &c. - -Engraved sheets of paper prepared for the drawing of curves may be -obtained from Mr. Stanford at Charing Cross, Messrs. W. and A. K. -Johnston, of London and Edinburgh, Waterlow and Sons, Letts and Co., -and probably other publishers. When we do not require great accuracy, -paper ruled by the common machine-ruler into equal squares of about -one-fifth or one-sixth of an inch square will serve well enough. I -have met with engineers’ and surveyors’ memorandum books ruled with -one-twelfth inch squares. When a number of curves have to be drawn, I -have found it best to rule a good sheet of drawing paper with lines -carefully adjusted at the most convenient distances, and then to -prick the points of the curve through it upon another sheet fixed -underneath. In this way we obtain an accurate curve upon a blank sheet, -and need only introduce such division lines as are requisite to the -understanding of the curve. - -In some cases our numerical results will correspond, not to the -height of single ordinates, but to the area of the curve between two -ordinates, or the average height of ordinates between certain limits. -If we measure, for instance, the quantities of heat absorbed by water -when raised in temperature from 0° to 5°, from 5° to 10°, and so on, -these quantities will really be represented by *areas* of the curve -denoting the specific heat of water; and since the specific heat varies -continuously between every two points of temperature, we shall not get -the correct curve by simply laying off the quantities of heat at the -mean temperatures, namely 2-1/2°, and 7-1/2°, and so on. Lord Rayleigh -has shown that if we have drawn such an incorrect curve, we can with -little trouble correct it by a simple geometrical process, and obtain -to a close approximation the true ordinates instead of those denoting -areas.[409] - - [409] J. W. Strutt, *On a correction sometimes required in curves - professing to represent the connexion between two physical - magnitudes*. Philosophical Magazine, 4th Series, vol. xlii. p. 441. - - -*Interpolation and Extrapolation.* - -When we have by experiment obtained two or more numerical results, -and endeavour, without further experiment, to calculate intermediate -results, we are said to *interpolate*. If we wish to assign by -reasoning results lying beyond the limits of experiment, we may be -said, using an expression of Sir George Airy, to *extrapolate*. These -two operations are the same in principle, but differ in practicability. -It is a matter of great scientific importance to apprehend precisely -how far we can practise interpolation or extrapolation, and on what -grounds we proceed. - -In the first place, if the interpolation is to be more than empirical, -we must have not only the experimental results, but the laws which they -obey--we must in fact go through the complete process of scientific -investigation. Having discovered the laws of nature applying to the -case, and verified them by showing that they agree with the experiments -in question, we are then in a position to anticipate the results of -similar experiments. Our knowledge even now is not certain, because we -cannot completely prove the truth of any assumed law, and we cannot -possibly exhaust all the circumstances which may affect the result. At -the best then our interpolations will partake of the want of certainty -and precision attaching to all our knowledge of nature. Yet, having the -supposed laws, our results will be as sure and accurate as any we can -attain to. But such a complete procedure is more than we commonly mean -by interpolation, which usually denotes some method of estimating in a -merely approximate manner the results which might have been expected -independently of a theoretical investigation. - -Regarded in this light, interpolation is in reality an indeterminate -problem. From given values of a function it is impossible to determine -that function; for we can invent an infinite number of functions which -will give those values if we are not restricted by any conditions, just -as through a given series of points we can draw an infinite number of -curves, if we may diverge between or beyond the points into bends and -cusps as we think fit.[410] In interpolation we must in fact be guided -more or less by *à priori* considerations; we must know, for instance, -whether or not periodical fluctuations are to be expected. Supposing -that the phenomenon is non-periodic, we proceed to assume that the -function can be expressed in a limited series of the powers of the -variable. The number of powers which can be included depends upon the -number of experimental results available, and must be at least one less -than this number. By processes of calculation, which have been already -alluded to in the section on empirical formulæ, we then calculate the -coefficients of the powers, and obtain an empirical formula which will -give the required intermediate results. In reality, then, we return -to the methods treated under the head of approximation and empirical -formulæ; and interpolation, as commonly understood, consists in -assuming that a curve of simple character is to pass through certain -determined points. If we have, for instance, two experimental results, -and only two, we assume that the curve is a straight line; for the -parabolas which can be passed through two points are infinitely various -in magnitude, and quite indeterminate. One straight line alone can -pass through two points, and it will have an equation of the form, -*y* = *mx* + *n*, the constant quantities of which can be determined -from two results. Thus, if the two values for *x*, 7 and 11, give -the values for *y*, 35 and 53, the solution of two equations gives -*y* = 4·5 × *x* + 3·5 as the equation, and for any other value of *x*, -for instance 10, we get a value of *y*, that is 48·5. When we take a -mean value of *x*, namely 9, this process yields a simple mean result, -namely 44. Three experimental results being given, we assume that they -fall upon a portion of a parabola and algebraic calculation gives the -position of any intermediate point upon the parabola. Concerning the -process of interpolation as practised in the science of meteorology -the reader will find some directions in the French edition of Kaëmtz’s -Meteorology.[411] - - [410] Herschel: Lacroix’ *Differential Calculus*, p. 551. - - [411] *Cours complet de Météorologie*, Note A, p. 449. - -When we have, either by direct experiment or by the use of a curve, -a series of values of the variant for equidistant values of the -variable, it is instructive to take the differences between each -value of the variant and the next, and then the differences between -those differences, and so on. If any series of differences approaches -closely to zero it is an indication that the numbers may be correctly -represented by a finite empirical formula; if the *n*th differences -are zero, then the formula will contain only the first *n* - 1 powers -of the variable. Indeed we may sometimes obtain by the calculus of -differences a correct empirical formula; for if *p* be the first term -of the series of values, and Δ*p*, Δ^{2}*p*, Δ^{3}*p*, be the first -number in each column of differences, then the *m*th term of the series -of values will be - - *p* + *m*Δ*p* + *m*[(*m* - 1)/2]Δ^{2}*p* + - *m*[(*m* - 1)/2][(*m* - 2)/3]Δ^{3}*p* + &c. - -A closely equivalent but more practicable formula for interpolation -by differences, as devised by Lagrange, will be found in Thomson and -Tait’s *Elements of Natural Philosophy*, p. 115. - -If no column of differences shows any tendency to become zero -throughout, it is an indication that the law is of a more complicated, -for instance of an exponential character, so that it requires different -treatment. Dr. J. Hopkinson has suggested a method of arithmetical -interpolation,[412] which is intended to avoid much that is arbitrary -in the graphical method. His process will yield the same results in all -hands. - - [412] *On the Calculation of Empirical Formulæ. The Messenger of - Mathematics*, New Series, No. 17, 1872. - -So far as we can infer the results likely to be obtained by variations -beyond the limits of experiment, we must proceed upon the same -principles. If possible we must detect the exact laws in action, -and then trust to them as a guide when we have no experience. If -not, an empirical formula of the same character as those employed in -interpolation is our only resource. But to extend our inference far -beyond the limits of experience is exceedingly unsafe. Our knowledge is -at the best only approximate, and takes no account of small tendencies. -Now it usually happens that tendencies small within our limits of -observation become perceptible or great under extreme circumstances. -When the variable in our empirical formula is small, we are justified -in overlooking the higher powers, and taking only two or three lower -powers. But as the variable increases, the higher powers gain in -importance, and in time yield the principal part of the value of the -function. - -This is no mere theoretical inference. Excepting the few primary -laws of nature, such as the law of gravity, of the conservation of -energy, &c., there is hardly any natural law which we can trust in -circumstances widely different from those with which we are practically -acquainted. From the expansion or contraction, fusion or vaporisation -of substances by heat at the surface of the earth, we can form a most -imperfect notion of what would happen near the centre of the earth, -where the pressure almost infinitely exceeds anything possible in our -experiments. The physics of the earth give us a feeble, and probably a -misleading, notion of a body like the sun, in which an inconceivably -high temperature is united with an inconceivably high pressure. If -there are in the realms of space nebulæ consisting of incandescent -and unoxidised vapours of metals and other elements, so highly heated -perhaps that chemical composition is out of the question, we are -hardly able to treat them as subjects of scientific inference. Hence -arises the great importance of experiments in which we investigate the -properties of substances under extreme circumstances of cold or heat, -density or rarity, intense electric excitation, &c. This insecurity -in extending our inferences arises from the approximate character -of our measurements. Had we the power of appreciating infinitely -small quantities, we should by the principle of continuity discover -some trace of every change which a substance could undergo under -unattainable circumstances. By observing, for instance, the tension of -aqueous vapour between 0° and 100° C., we ought theoretically to be -able to infer its tension at every other temperature; but this is out -of the question practically because we cannot really ascertain the law -precisely between those temperatures. - -Many instances might be given to show that laws which appear to -represent correctly the results of experiments within certain limits -altogether fail beyond those limits. The experiments of Roscoe and -Dittmar, on the absorption of gases in water[413] afford interesting -illustrations, especially in the case of hydrochloric acid, the -quantity of which dissolved in water under different pressures -follows very closely a linear law of variation, from which however it -diverges widely at low pressures.[414] Herschel, having deduced from -observations of the double star γ Virginis an elliptic orbit for the -motion of one component round the centre of gravity of both, found that -for a time the motion of the star agreed very well with this orbit. -Nevertheless divergence began to appear and after a time became so -great that an entirely new orbit, of more than double the dimensions of -the old one, had ultimately to be adopted.[415] - - [413] Watts’ *Dictionary of Chemistry*, vol. ii. p. 790. - - [414] *Quarterly Journal of the Chemical Society*, vol. viii. p. 15. - - [415] *Results of Observations at the Cape of Good Hope*, p. 293. - - -*Illustrations of Empirical Quantitative Laws.* - -Although our object in quantitative inquiry is to discover the exact or -rational formulæ, expressing the laws which apply to the subject, it -is instructive to observe in how many important branches of science, -no precise laws have yet been detected. The tension of aqueous vapour -at different temperatures has been determined by a succession of -eminent experimentalists--Dalton, Kaëmtz, Dulong, Arago, Magnus, and -Regnault--and by the last mentioned the measurements were conducted -with extraordinary care. Yet no incontestable general law has been -established. Several functions have been proposed to express the -elastic force of the vapour as depending on the temperature. The first -form is that of Young, namely F = (*a* + *b t*)^{m}, in which *a*, -*b*, and *m* are unknown quantities to be determined by observation. -Roche proposed, on theoretical grounds, a complicated formula of an -exponential form, and a third form of function is that of Biot,[416] as -follows--log F = *a* + *b*α^{t} + *c*β^{t}. I mention these formulæ, -because they well illustrate the feeble powers of empirical inquiry. -None of the formulæ can be made to correspond closely with experimental -results, and the two last forms correspond almost equally well. There -is very little probability that the real law has been reached, and -it is unlikely that it will be discovered except by deduction from -mechanical theory. - - [416] Jamin, *Cours de Physique*, vol. ii. p. 138. - -Much ingenious labour has been spent upon the discovery of some general -law of atmospheric refraction. Tycho Brahe and Kepler commenced the -inquiry: Cassini first formed a table of refractions, calculated on -theoretical grounds: Newton entered into some profound investigations -upon the subject: Brooke Taylor, Bouguer, Simpson, Bradley, Mayer, -and Kramp successively attacked the question, which is of the highest -practical importance as regards the correction of astronomical -observations. Laplace next laboured on the subject without exhausting -it, and Brinkley and Ivory have also treated it. The true law is -yet undiscovered. A closely connected problem, that regarding the -relation between the pressure and elevation in different strata of -the atmosphere, has received the attention of a long succession of -physicists and was most carefully investigated by Laplace. Yet no -invariable and general law has been detected. The same may be said -concerning the law of human mortality; abundant statistics on this -subject are available, and many hypotheses more or less satisfactory -have been put forward as to the form of the curve of mortality, but it -seems to be impossible to discover more than an approximate law. - -It may perhaps be urged that in such subjects no single invariable -law can be expected. The atmosphere may be divided into several -variable strata which by their unconnected changes frustrate the exact -calculations of astronomers. Human life may be subject at different -ages to a succession of different influences incapable of reduction -under any one law. The results observed may in fact be aggregates of an -immense number of separate results each governed by its own separate -laws, so that the subjects may be complicated beyond the possibility of -complete resolution by empirical methods. This is certainly true of the -mathematical functions which must some time or other be introduced into -the science of political economy. - - -*Simple Proportional Variation.* - -When we first treat numerical results in any novel kind of -investigation, our impression will probably be that one quantity -varies in *simple proportion* to another, so as to obey the law -*y* = *mx* + *n*. We must learn to distinguish carefully between the -cases where this proportionality is really, and where it is only -apparently true. In considering the principles of approximation we -found that a small portion of any curve will appear to be a straight -line. When our modes of measurement are comparatively rude, we must -expect to be unable to detect the curvature. Kepler made meritorious -attempts to discover the law of refraction, and he approximated to it -when he observed that the angles of incidence and refraction *if small* -bear a constant ratio to each other. Angles when small are nearly as -their sines, so that he reached an approximate result of the true law. -Cardan assumed, probably as a mere guess, that the force required to -sustain a body on an inclined plane was simply proportional to the -angle of elevation of the plane. This is approximately the case when -the angle is small, but in reality the law is much more complicated, -the power required being proportional to the sine of the angle. The -early thermometer-makers were unaware whether the expansion of mercury -was proportional or not to the heat communicated to it, and it is only -in the present century that we have learnt it to be not so. We now -know that even gases obey the law of uniform expansion by heat only -in an approximate manner. Until reason to the contrary is shown, we -should do well to look upon every law of simple proportion as only -provisionally true. - -Nevertheless many important laws of nature are in the form of simple -proportions. Wherever a cause acts in independence of its previous -effects, we may expect this relation. An accelerating force acts -equally upon a moving and a motionless body. Hence the velocity -produced is in simple proportion to the force, and to the duration of -its uniform action. As gravitating bodies never interfere with each -other’s gravity, this force is in direct simple proportion to the -mass of each of the attracting bodies, the mass being measured by, or -proportional to inertia. Similarly, in all cases of “direct unimpeded -action,” as Herschel has remarked,[417] we may expect simple proportion -to manifest itself. In such cases the equation expressing the relation -may have the simple form *y* = *mx*. - - [417] *Preliminary Discourse*, &c., p. 152. - -A similar relation holds true when there is conversion of one substance -or form of energy into another. The quantity of a compound is equal -to the quantity of the elements which combine. The heat produced in -friction is exactly proportional to the mechanical energy absorbed. -It was experimentally proved by Faraday that “the chemical power of -the current of electricity is in direct proportion to the quantity of -electricity which passes.” When an electric current is produced, the -quantity of electric energy is simply proportional to the weight of -metal dissolved. If electricity is turned into heat, there is again -simple proportion. Wherever, in fact, one thing is but another thing -with a new aspect, we may expect to find the law of simple proportion. -But it is only in the most elementary cases that this simple relation -will hold true. Simple conditions do not, generally speaking, produce -simple results. The planets move in approximate circles round the sun, -but the apparent motions, as seen from the earth, are very various. All -those motions, again, are summed up in the law of gravity, of no great -complexity; yet men never have been, and never will be, able to exhaust -the complications of action and reaction arising from that law, even -among a small number of planets. We should be on our guard against a -tendency to assume that the connection of cause and effect is one of -direct proportion. Bacon reminds us of the woman in Æsop’s fable, who -expected that her hen, with a double measure of barley, would lay two -eggs a day instead of one, whereas it grew fat, and ceased to lay any -eggs at all. It is a wise maxim that the half is often better than the -whole. - - - - -CHAPTER XXIII. - -THE USE OF HYPOTHESIS. - - -If the views upheld in this work be correct, all inductive -investigation consists in the marriage of hypothesis and experiment. -When facts are in our possession, we frame an hypothesis to explain -their relations, and by the success of this explanation is the value -of the hypothesis to be judged. In the invention and treatment of -such hypotheses, we must avail ourselves of the whole body of science -already accumulated, and when once we have obtained a probable -hypothesis, we must not rest until we have verified it by comparison -with new facts. We must endeavour by deductive reasoning to anticipate -such phenomena, especially those of a singular and exceptional nature, -as would happen if the hypothesis be true. Out of the infinite number -of experiments which are possible, theory must lead us to select those -critical ones which are suitable for confirming or negativing our -anticipations. - -This work of inductive investigation cannot be guided by any system of -precise and infallible rules, like those of deductive reasoning. There -is, in fact, nothing to which we can apply rules of method, because -the laws of nature must be in our possession before we can treat them. -If there were any rule of inductive method, it would direct us to make -an exhaustive arrangement of facts in all possible orders. Given the -specimens in a museum, we might arrive at the best classification by -going systematically through all possible classifications, and, were -we endowed with infinite time and patience, this would be an effective -method. It is the method by which the first simple steps are taken in -an incipient branch of science. Before the dignified name of science is -applicable, some coincidences will force themselves upon the attention. -Before there was a science of meteorology observant persons learned -to associate clearness of the atmosphere with coming rain, and a -colourless sunset with fine weather. Knowledge of this kind is called -*empirical*, as seeming to come directly from experience; and there is -a considerable portion of knowledge which bears this character. - -We may be obliged to trust to the casual detection of coincidences -in those branches of knowledge where we are deprived of the aid of -any guiding notions; but a little reflection will show the utter -insufficiency of haphazard experiment, when applied to investigations -of a complicated nature. At the best, it will be the simple identity, -or partial identity, of classes, as illustrated in pages 127 or 134, -which can be thus detected. It was pointed out that, even when a law -of nature involves only two circumstances, and there are one hundred -distinct circumstances which may possibly be connected, there will be -no less than 4,950 pairs of circumstances between which coincidence may -exist. When a law involves three or more circumstances, the possible -number of relations becomes vastly greater. When considering the -subject of combinations and permutations, it became apparent that we -could never cope with the possible variety of nature. An exhaustive -examination of the possible metallic alloys, or chemical compounds, was -found to be out of the question (p. 191). - -It is on such considerations that we can explain the very small -additions made to our knowledge by the alchemists. Many of them were -men of the greatest acuteness, and their indefatigable labours were -pursued through many centuries. A few things were discovered by them, -but a true insight into nature, now enables chemists to discover more -useful facts in a year than were yielded by the alchemists during -many centuries. There can be no doubt that Newton was an alchemist, -and that he often laboured night and day at alchemical experiments. -But in trying to discover the secret by which gross metals might be -rendered noble, his lofty powers of deductive investigation were wholly -useless. Deprived of all guiding clues, his experiments were like -those of all the alchemists, purely tentative and haphazard. While his -hypothetical and deductive investigations have given us the true system -of the Universe, and opened the way in almost all the great branches of -natural philosophy, the whole results of his tentative experiments are -comprehended in a few happy guesses, given in his celebrated “Queries.” - -Even when we are engaged in apparently passive observation of a -phenomenon, which we cannot modify experimentally, it is advantageous -that our attention should be guided by theoretical anticipations. A -phenomenon which seems simple is, in all probability, really complex, -and unless the mind is actively engaged in looking for particular -details, it is likely that the critical circumstances will be passed -over. Bessel regretted that no distinct theory of the constitution -of comets had guided his observations of Halley’s comet;[418] in -attempting to verify or refute a hypothesis, not only would there be a -chance of establishing a true theory, but if confuted, the confutation -would involve a store of useful observations. - - [418] Tyndall, *On Cometary Theory*, Philosophical Magazine, April - 1869. 4th Series, vol. xxxvii. p. 243. - -It would be an interesting work, but one which I cannot undertake, to -trace out the gradual reaction which has taken place in recent times -against the purely empirical or Baconian theory of induction. Francis -Bacon, seeing the futility of the scholastic logic, which had long -been predominant, asserted that the accumulation of facts and the -orderly abstraction of axioms, or general laws from them, constituted -the true method of induction. Even Bacon was not wholly unaware of -the value of hypothetical anticipation. In one or two places he -incidentally acknowledges it, as when he remarks that the subtlety of -nature surpasses that of reason, adding that “axioms abstracted from -particular facts in a careful and orderly manner, readily suggest and -mark out new particulars.” - -Nevertheless Bacon’s method, as far as we can gather the meaning of -the main portions of his writings, would correspond to the process of -empirically collecting facts and exhaustively classifying them, to -which I alluded. The value of this method may be estimated historically -by the fact that it has not been followed by any of the great masters -of science. Whether we look to Galileo, who preceded Bacon, to Gilbert, -his contemporary, or to Newton and Descartes, Leibnitz and Huyghens, -his successors, we find that discovery was achieved by the opposite -method to that advocated by Bacon. Throughout Newton’s works, as -I shall show, we find deductive reasoning wholly predominant, and -experiments are employed, as they should be, to confirm or refute -hypothetical anticipations of nature. In my “Elementary Lessons in -Logic” (p. 258), I stated my belief that there was no kind of reference -to Bacon in Newton’s works. I have since found that Newton does once or -twice employ the expression *experimentum crucis* in his “Opticks,” but -this is the only expression, so far as I am aware, which could indicate -on the part of Newton direct or indirect acquaintance with Bacon’s -writings.[419] - - [419] See *Philosophical Transactions*, abridged by Lowthorp. 4th - edit. vol. i. p. 130. I find that opinions similar to those in the - text have been briefly expressed by De Morgan in his remarkable - preface to *From Matter to Spirit*, by C.D., pp. xxi. xxii. - -Other great physicists of the same age were equally prone to the use -of hypotheses rather than the blind accumulation of facts in the -Baconian manner. Hooke emphatically asserts in his posthumous work -on Philosophical Method, that the first requisite of the Natural -Philosopher is readiness at guessing the solution of phenomena and -making queries. “He ought to be very well skilled in those several -kinds of philosophy already known, to understand their several -hypotheses, suppositions, collections, observations, &c., their various -ways of ratiocinations and proceedings, the several failings and -defects, both in their way of raising and in their way of managing -their several theories: for by this means the mind will be somewhat -more ready at guessing at the solution of many phenomena almost at -first sight, and thereby be much more prompt at making queries, and at -tracing the subtlety of Nature, and in discovering and searching into -the true reason of things.” - -We find Horrocks, again, than whom no one was more filled with the -scientific spirit, telling us how he tried theory after theory in order -to discover one which was in accordance with the motions of Mars.[420] -Huyghens, who possessed one of the most perfect philosophical -intellects, followed the deductive process combined with continual -appeal to experiment, with a skill closely analogous to that of Newton. -As to Descartes and Leibnitz, they fell into excess in the use of -hypothesis, since they sometimes adopted hypothetical reasoning to -the exclusion of experimental verification. Throughout the eighteenth -century science was supposed to be advancing by the pursuance of the -Baconian method, but in reality hypothetical investigation was the -main instrument of progress. It is only in the present century that -physicists began to recognise this truth. So much opprobrium had been -attached by Bacon to the use of hypotheses, that we find Young speaking -of them in an apologetic tone. “The practice of advancing general -principles and applying them to particular instances is so far from -being fatal to truth in all sciences, that when those principles are -advanced on sufficient grounds, it constitutes the essence of true -philosophy;”[421] and he quotes cases in which Davy trusted to his -theories rather than his experiments. - - [420] Horrocks, *Opera Posthuma* (1673), p. 276. - - [421] Young’s *Works*, vol. i. p. 593. - -Herschel, who was both a practical physicist and an abstract logician, -entertained the deepest respect for Bacon, and made the “Novum -Organum” as far as possible the basis of his own admirable *Discourse -on the Study of Natural Philosophy*. Yet we find him in Chapter VII. -recognising the part which the formation and verification of theories -takes in the higher and more general investigations of physical -science. J. S. Mill carried on the reaction by describing the Deductive -Method in which ratiocination, that is deductive reasoning, is employed -for the discovery of new opportunities of testing and verifying an -hypothesis. Nevertheless throughout the other parts of his system -he inveighed against the value of the deductive process, and even -asserted that empirical inference from particulars to particulars is -the true type of reasoning. The irony of fate will probably decide -that the most original and valuable part of Mill’s System of Logic is -irreconcilable with those views of the syllogism and of the nature of -inference which occupy the main part of the treatise, and are said to -have effected a revolution in logical science. Mill would have been -saved from much confusion of thought had he not failed to observe that -the inverse use of deduction constitutes induction. In later years -Professor Huxley has strongly insisted upon the value of hypothesis. -When he advocates the use of “working hypotheses” he means no doubt -that any hypothesis is better that none, and that we cannot avoid being -guided in our observations by some hypothesis or other. Professor -Tyndall’s views as to the use of the Imagination in the pursuit of -Science put the same truth in another light. - -It ought to be pointed out that Neil in his *Art of Reasoning*, a -popular but able exposition of the principles of Logic, published -in 1853, fully recognises in Chapter XI. the value and position of -hypothesis in the discovery of truth. He endeavours to show, too -(p. 109), that Francis Bacon did not object to the use of hypothesis. - -The true course of inductive procedure is that which has yielded -all the more lofty results of science. It consists in *Anticipating -Nature*, in the sense of forming hypotheses as to the laws which are -probably in operation; and then observing whether the combinations -of phenomena are such as would follow from the laws supposed. The -investigator begins with facts and ends with them. He uses facts to -suggest probable hypotheses; deducing other facts which would happen if -a particular hypothesis is true, he proceeds to test the truth of his -notion by fresh observations. If any result prove different from what -he expects, it leads him to modify or to abandon his hypothesis; but -every new fact may give some new suggestion as to the laws in action. -Even if the result in any case agrees with his anticipations, he does -not regard it as finally confirmatory of his theory, but proceeds to -test the truth of the theory by new deductions and new trials. - -In such a process the investigator is assisted by the whole body of -science previously accumulated. He may employ analogy, as I shall -point out, to guide him in the choice of hypotheses. The manifold -connections between one science and another give him clues to the kind -of laws to be expected, and out of the infinite number of possible -hypotheses he selects those which are, as far as can be foreseen at the -moment, most probable. Each experiment, therefore, which he performs -is that most likely to throw light upon his subject, and even if it -frustrate his first views, it tends to put him in possession of the -correct clue. - - -*Requisites of a good Hypothesis.* - -There is little difficulty in pointing out to what condition an -hypothesis must conform in order to be accepted as probable and -valid. That condition, as I conceive, is the single one of enabling -us to infer the existence of phenomena which occur in our experience. -*Agreement with fact is the sole and sufficient test of a true -hypothesis.* - -Hobbes has named two conditions which he considers requisite in an -hypothesis, namely (1) That it should be conceivable and not absurd; -(2) That it should allow of phenomena being necessarily inferred. -Boyle, in noticing Hobbes’ views, proposed to add a third condition, -to the effect that the hypothesis should not be inconsistent with any -other truth on phenomenon of nature.[422] I think that of these three -conditions, the first cannot be accepted, unless by *inconceivable* and -*absurd* we mean self-contradictory or inconsistent with the laws of -thought and nature. I shall have to point out that some satisfactory -theories involve suppositions which are wholly *inconceivable* in -a certain sense of the word, because the mind cannot sufficiently -extend its ideas to frame a notion of the actions supposed to take -place. That the force of gravity should act instantaneously between -the most distant parts of the planetary system, or that a ray of -violet light should consist of about 700 billions of vibrations in a -second, are statements of an inconceivable and absurd character in -one sense; but they are so far from being opposed to fact that we -cannot on any other suppositions account for phenomena observed. But -if an hypothesis involve self-contradiction, or is inconsistent with -known laws of nature, it is self-condemned. We cannot even apply -deductive reasoning to a self-contradictory notion; and being opposed -to the most general and certain laws known to us, the primary laws of -thought, it thereby conspicuously fails to agree with facts. Since -nature, again, is never self-contradictory, we cannot at the same time -accept two theories which lead to contradictory results. If the one -agrees with nature, the other cannot. Hence if there be a law which we -believe with high probability to be verified by observation, we must -not frame an hypothesis in conflict with it, otherwise the hypothesis -will necessarily be in disagreement with observation. Since no law or -hypothesis is proved, indeed, with absolute certainty, there is always -a chance, however slight, that the new hypothesis may displace the -old one; but the greater the probability which we assign to that old -hypothesis, the greater must be the evidence required in favour of the -new and conflicting one. - - [422] Boyle’s *Physical Examen*, p. 84. - -I assert, then, that there is but one test of a good hypothesis, -namely, *its conformity with observed facts*; but this condition may -be said to involve three constituent conditions, nearly equivalent to -those suggested by Hobbes and Boyle, namely:-- - -(1) That it allow of the application of deductive reasoning and the -inference of consequences capable of comparison with the results of -observation. - -(2) That it do not conflict with any laws of nature, or of mind, which -we hold to be true. - -(3) That the consequences inferred do agree with facts of observation. - - -*Possibility of Deductive Reasoning.* - -As the truth of an hypothesis is to be proved by its conformity with -fact, the first condition is that we be able to apply methods of -deductive reasoning, and learn what should happen according to such -an hypothesis. Even if we could imagine an object acting according to -laws hitherto wholly unknown it would be useless to do so, because we -could never decide whether it existed or not. We can only infer what -would happen under supposed conditions by applying the knowledge of -nature we possess to those conditions. Hence, as Boscovich truly said, -we are to understand by hypotheses “not fictions altogether arbitrary, -but suppositions conformable to experience or analogy.” It follows that -every hypothesis worthy of consideration must suggest some likeness, -analogy, or common law, acting in two or more things. If, in order to -explain certain facts, *a*, *a′*, *a″*, &c., we invent a cause A, then -we must in some degree appeal to experience as to the mode in which A -will act. As the laws of nature are not known to the mind intuitively, -we must point out some other cause, B, which supplies the requisite -notions, and all we do is to invent a fourth term to an analogy. As B -is to its effects *b*, *b′*, *b″*, &c., so is A to its effects *a*, -*a′*, *a″*, &c. When we attempt to explain the passage of light and -heat radiations through space unoccupied by matter, we imagine the -existence of the so-called *ether*. But if this ether were wholly -different from anything else known to us, we should in vain try to -reason about it. We must apply to it at least the laws of motion, that -is we must so far liken it to matter. And as, when applying those laws -to the elastic medium air, we are able to infer the phenomena of sound, -so by arguing in a similar manner concerning ether we are able to infer -the existence of light phenomena corresponding to what do occur. All -that we do is to take an elastic substance, increase its elasticity -immensely, and denude it of gravity and some other properties of -matter, but we must retain sufficient likeness to matter to allow of -deductive calculations. - -The force of gravity is in some respects an incomprehensible existence, -but in other respects entirely conformable to experience. We observe -that the force is proportional to mass, and that it acts in entire -independence of other matter which may be present or intervening. -The law of the decrease of intensity, as the square of the distance -increases, is observed to hold true of light, sound, and other -influences emanating from a point, and spreading uniformly through -space. The law is doubtless connected with the properties of space, and -is so far in agreement with our necessary ideas. - -It may be said, however, that no hypothesis can be so much as framed -in the mind unless it be more or less conformable to experience. As -the material of our ideas is derived from sensation we cannot figure -to ourselves any agent, but as endowed with some of the properties of -matter. All that the mind can do in the creation of new existences is -to alter combinations, or the intensity of sensuous properties. The -phenomenon of motion is familiar to sight and touch, and different -degrees of rapidity are also familiar; we can pass beyond the limits of -sense, and imagine the existence of rapid motion, such as our senses -could not observe. We know what is elasticity, and we can therefore -in a way figure to ourselves elasticity a thousand or a million times -greater than any which is sensuously known to us. The waves of the -ocean are many times higher than our own bodies; other waves, are -many times less; continue the proportion, and we ultimately arrive -at waves as small as those of light. Thus it is that the powers of -mind enable us from a sensuous basis to reason concerning agents and -phenomena different in an unlimited degree. If no hypothesis then can -be absolutely opposed to sense, accordance with experience must always -be a question of degree. - -In order that an hypothesis may allow of satisfactory comparison with -experience, it must possess definiteness and in many cases mathematical -exactness allowing of the precise calculation of results. We must -be able to ascertain whether it does or does not agree with facts. -The theory of vortices is an instance to the contrary, for it did -not present any mode of calculating the exact relations between the -distances and periods of the planets and satellites; it could not, -therefore, undergo that rigorous testing to which Newton scrupulously -submitted his theory of gravity before its promulgation. Vagueness -and incapability of precise proof or disproof often enable a false -theory to live; but with those who love truth, vagueness should -excite suspicion. The upholders of the ancient doctrine of Nature’s -abhorrence of a vacuum, had been unable to anticipate the important -fact that water would not rise more than 33 feet in a common suction -pump. Nor when the fact was pointed out could they explain it, except -by introducing a special alteration of the theory to the effect that -Nature’s abhorrence of a vacuum was limited to 33 feet. - - -*Consistency with the Laws of Nature.* - -In the second place an hypothesis must not be contradictory to -what we believe to be true concerning Nature. It must not involve -self-inconsistency which is opposed to the highest and simplest laws, -namely, those of Logic. Neither ought it to be irreconcilable with -the simple laws of motion, of gravity, of the conservation of energy, -nor any parts of physical science which we consider to be established -beyond reasonable doubt. Not that we are absolutely forbidden to -entertain such an hypothesis, but if we do so we must be prepared to -disprove some of the best demonstrated truths in the possession of -mankind. The fact that conflict exists means that the consequences of -the theory are not verified if previous discoveries are correct, and we -must therefore show that previous discoveries are incorrect before we -can verify our theory. - -An hypothesis will be exceedingly improbable, not to say absurd, if -it supposes a substance to act in a manner unknown in other cases; -for it then fails to be verified in our knowledge of that substance. -Several physicists, especially Euler and Grove, have supposed that -we might dispense with an ethereal basis of light, and infer from -the interstellar passage of rays that there was a kind of rare gas -occupying space. But if so, that gas must be excessively rare, as we -may infer from the apparent absence of an atmosphere around the moon, -and from other facts known to us concerning gases and the atmosphere; -yet it must possess an elastic force at least a billion times as great -as atmospheric air at the earth’s surface, in order to account for the -extreme rapidity of light rays. Such an hypothesis then is inconsistent -with our knowledge concerning gases. - -Provided that there be no clear and absolute conflict with known -laws of nature, there is no hypothesis so improbable or apparently -inconceivable that it may not be rendered probable, or even -approximately certain, by a sufficient number of concordances. In -fact the two best founded and most successful theories in physical -science involve the most absurd suppositions. Gravity is a force which -appears to act between bodies through vacuous space; it is in positive -contradiction to the old dictum that nothing can act but through -some medium. It is even more puzzling that the force acts in perfect -indifference to intervening obstacles. Light in spite of its extreme -velocity shows much respect to matter, for it is almost instantaneously -stopped by opaque substances, and to a considerable extent absorbed -and deflected by transparent ones. But to gravity all media are, as -it were, absolutely transparent, nay non-existent; and two particles -at opposite points of the earth affect each other exactly as if the -globe were not between. The action is, so far as we can observe, -instantaneous, so that every particle of the universe is at every -moment in separate cognisance, as it were, of the relative position of -every other particle throughout the universe at that same moment of -time. Compared with such incomprehensible conditions, the theory of -vortices deals with commonplace realities. Newton’s celebrated saying -*hypotheses non fingo*, bears the appearance of irony; and it was not -without apparent grounds that Leibnitz and the continental philosophers -charged Newton with re-introducing occult powers and qualities. - -The undulatory theory of light presents almost equal difficulties -of conception. We are asked by physical philosophers to give up our -prepossessions, and to believe that interstellar space which seems -empty is not empty at all, but filled with *something* immensely -more solid and elastic than steel. As Young himself remarked,[423] -“the luminiferous ether, pervading all space, and penetrating almost -all substances, is not only highly elastic, but absolutely solid!!!” -Herschel calculated the force which may be supposed, according to the -undulatory theory of light, to be constantly exerted at each point in -space, and finds it to be 1,148,000,000,000 times the elastic force -of ordinary air at the earth’s surface, so that the pressure of ether -per square inch must be about seventeen billions of pounds.[424] Yet -we live and move without appreciable resistance through this medium, -immensely harder and more elastic than adamant. All our ordinary -notions must be laid aside in contemplating such an hypothesis; yet it -is no more than the observed phenomena of light and heat force us to -accept. We cannot deny even the strange suggestion of Young, that there -may be independent worlds, some possibly existing in different parts -of space, but others perhaps pervading each other unseen and unknown -in the same space.[425] For if we are bound to admit the conception -of this adamantine firmament, it is equally easy to admit a plurality -of such. We see, then, that mere difficulties of conception must not -discredit a theory which otherwise agrees with facts, and we must only -reject hypotheses which are inconceivable in the sense of breaking -distinctly the primary laws of thought and nature. - - [423] Young’s *Works*, vol. i. p. 415. - - [424] *Familiar Lectures on Scientific Subjects*, p. 282. - - [425] Young’s *Works*, vol. i. p. 417. - - -*Conformity with Facts.* - -Before we accept a new hypothesis it must be shown to agree not only -with the previously known laws of nature, but also with the particular -facts which it is framed to explain. Assuming that these facts are -properly established, it must agree with all of them. A single absolute -conflict between fact and hypothesis, is fatal to the hypothesis; -*falsa in uno, falsa in omnibus*. - -Seldom, indeed, shall we have a theory free from difficulties and -apparent inconsistency with facts. Though one real inconsistency -would overturn the most plausible theory, yet there is usually some -probability that the fact may be misinterpreted, or that some supposed -law of nature, on which we are relying, may not be true. It may be -expected, moreover, that a good hypothesis, besides agreeing with -facts already noticed, will furnish us with distinct credentials by -enabling us to anticipate deductively series of facts which are not -already connected and accounted for by any equally probable hypothesis. -We cannot lay down any precise rule as to the number of accordances -which can establish the truth of an hypothesis, because the accordances -will vary much in value. While, on the one hand, no finite number -of accordances will give entire certainty, the probability of the -hypothesis will increase very rapidly with the number of accordances. -Almost every problem in science thus takes the form of a balance of -probabilities. It is only when difficulty after difficulty has been -successfully explained away, and decisive *experimenta crucis* have, -time after time, resulted in favour of our theory, that we can venture -to assert the falsity of all objections. - -The sole real test of an hypothesis is its accordance with fact. -Descartes’ celebrated system of vortices is exploded, not because it -was intrinsically absurd and inconceivable, but because it could not -give results in accordance with the actual motions of the heavenly -bodies. The difficulties of conception involved in the apparatus of -vortices, are child’s play compared with those of gravitation and -the undulatory theory already described. Vortices are on the whole -plausible suppositions; for planets and satellites bear at first sight -much resemblance to objects carried round in whirlpools, an analogy -which doubtless suggested the theory. The failure was in the first and -third requisites; for, as already remarked, the theory did not allow -of precise calculation of planetary motions, and was thus incapable of -rigorous verification. But so far as we can institute a comparison, -facts are entirely against the vortices. Newton did not ridicule the -theory as absurd, but showed[426] that it was “pressed with many -difficulties.” He carefully pointed out that the Cartesian theory was -inconsistent with the laws of Kepler, and would represent the planets -as moving more rapidly at their aphelia than at their perihelia.[427] -The rotatory motion of the sun and planets on their own axes is in -striking conflict with the revolutions of the satellites carried round -them; and comets, the most flimsy of bodies, calmly pursue their -courses in elliptic paths, irrespective of the vortices which they pass -through. We may now also point to the interlacing orbits of the minor -planets as a new and insuperable difficulty in the way of the Cartesian -ideas. - - [426] *Principia*, bk. iii. Prop. 43. General Scholium. - - [427] Ibid. bk. ii. Sect. ix. Prop. 53. - -Newton, though he established the best of theories, was also capable -of proposing one of the worst; and if we want an instance of a theory -decisively contradicted by facts, we have only to turn to his views -concerning the origin of natural colours. Having analysed, with -incomparable skill, the origin of the colours of thin plates, he -suggests that the colours of all bodies are determined in like manner -by the size of their ultimate particles. A thin plate of a definite -thickness will reflect a definite colour; hence, if broken up into -fragments it will form a powder of the same colour. But, if this be -a sufficient explanation of coloured substances, then every coloured -fluid ought to reflect the complementary colour of that which it -transmits. Colourless transparency arises, according to Newton, from -particles being too minute to reflect light; but if so, every black -substance should be transparent. Newton himself so acutely felt this -last difficulty as to suggest that true blackness is due to some -internal refraction of the rays to and fro, and an ultimate stifling -of them, which he did not attempt to explain further. Unless some -other process comes into operation, neither refraction nor reflection, -however often repeated, will destroy the energy of light. The theory -therefore gives no account, as Brewster shows, of 24 parts out of 25 of -the light which falls upon a black coal, and the remaining part which -is reflected from the lustrous surface is equally inconsistent with the -theory, because fine coal-dust is almost entirely devoid of reflective -power.[428] It is now generally believed that the colours of natural -bodies are due to the unequal absorption of rays of light of different -refrangibility. - - [428] Brewster’s *Life of Newton*, 1st edit. chap. vii. - - -*Experimentum Crucis.* - -As we deduce more and more conclusions from a theory, and find them -verified by trial, the probability of the theory increases in a rapid -manner; but we never escape the risk of error altogether. Absolute -certainty is beyond the powers of inductive investigation, and the -most plausible supposition may ultimately be proved false. Such is the -groundwork of similarity in nature, that two very different conditions -may often give closely similar results. We sometimes find ourselves -therefore in possession of two or more hypotheses which both agree -with so many experimental facts as to have great appearance of truth. -Under such circumstances we have need of some new experiment, which -shall give results agreeing with one hypothesis but not with the other. - -Any such experiment which decides between two rival theories may be -called an *Experimentum Crucis*, an Experiment of the Finger Post. -Whenever the mind stands, as it were, at cross-roads and knows not -which way to select, it needs some decisive guide, and Bacon therefore -assigned great importance and authority to instances which serve in -this capacity. The name given by Bacon has become familiar; it is -almost the only one of Bacon’s figurative expressions which has passed -into common use. Even Newton, as I have mentioned (p. 507), used the -name. - -I do not think, indeed, that the common use of the word at all agrees -with that intended by Bacon. Herschel says that “we make an experiment -of the crucial kind when we form combinations, and put in action -causes from which some particular one shall be deliberately excluded, -and some other purposely admitted.”[429] This, however, seems to be -the description of any special experiment not made at haphazard. -Pascal’s experiment of causing a barometer to be carried to the top of -the Puy-de-Dôme has often been considered as a perfect *experimentum -crucis*, if not the first distinct one on record;[430] but if so, we -must dignify the doctrine of Nature’s abhorrence of a vacuum with -the position of a rival theory. A crucial experiment must not simply -confirm one theory, but must negative another; it must decide a mind -which is in equilibrium, as Bacon says,[431] between two equally -plausible views. “When in search of any nature, the understanding comes -to an equilibrium, as it were, or stands suspended as to which of two -or more natures the cause of nature inquired after should be attributed -or assigned, by reason of the frequent and common occurrence of several -natures, then these Crucial Instances show the true and inviolable -association of one of these natures to the nature sought, and the -uncertain and separable alliance of the other, whereby the question -is decided, the former nature admitted for the cause, and the other -rejected. These instances, therefore, afford great light, and have a -kind of overruling authority, so that the course of interpretation will -sometimes terminate in them, or be finished by them.” - - [429] *Discourse on the Study of Natural Philosophy*, p. 151. - - [430] Ibid. p. 229. - - [431] *Novum Organum*, bk. ii. Aphorism 36. - -The long-continued strife between the Corpuscular and Undulatory -theories of light forms the best possible illustration of an -Experimentum Crucis. It is remarkable in how plausible a manner both -these theories agreed with the ordinary laws of geometrical optics, -relating to reflection and refraction. According to the first law of -motion a moving particle proceeds in a perfectly straight line, when -undisturbed by extraneous forces. If the particle being perfectly -elastic, strike a perfectly elastic plane, it will bound off in such -a path that the angles of incidence and reflection will be equal. -Now a ray of light proceeds in a straight line, or appears to do -so, until it meets a reflecting body, when its path is altered in a -manner exactly similar to that of the elastic particle. Here is a -remarkable correspondence which probably suggested to Newton’s mind -the hypothesis that light consists of minute elastic particles moving -with excessive rapidity in straight lines. The correspondence was found -to extend also to the law of simple refraction; for if particles of -light be supposed capable of attracting matter, and being attracted -by it at insensibly small distances, then a ray of light, falling on -the surface of a transparent medium, will suffer an increase in its -velocity perpendicular to the surface, and the law of sines is the -consequence. This remarkable explanation of the law of refraction had -doubtless a very strong effect in leading Newton to entertain the -corpuscular theory, and he appears to have thought that the analogy -between the propagation of rays of light and the motion of bodies was -perfectly exact, whatever might be the actual nature of light.[432] -It is highly remarkable, again, that Newton was able to give by his -corpuscular theory, a plausible explanation of the inflection of light -as discovered by Grimaldi. The theory would indeed have been a very -probable one could Newton’s own law of gravity have applied; but this -was out of the question, because the particles of light, in order that -they may move in straight lines, must be devoid of any influence upon -each other. - - [432] *Principia*, bk. i. Sect. xiv. Prop. 96. Scholium. *Opticks*, - Prop. vi. 3rd edit. p. 70. - -The Huyghenian or Undulatory theory of light was also able to explain -the same phenomena, but with one remarkable difference. If the -undulatory theory be true, light must move more slowly in a dense -refracting medium than in a rarer one; but the Newtonian theory assumed -that the attraction of the dense medium caused the particles of light -to move more rapidly than in the rare medium. On this point, then, -there was complete discrepancy between the theories, and observation -was required to show which theory was to be preferred. Now by simply -cutting a uniform plate of glass into two pieces, and slightly -inclining one piece so as to increase the length of the path of a ray -passing through it, experimenters were able to show that light does -move more slowly in glass than in air.[433] More recently Fizeau and -Foucault independently measured the velocity of light in air and in -water, and found that the velocity is greater in air.[434] - - [433] Airy’s *Mathematical Tracts*, 3rd edit. pp. 286–288. - - [434] Jamin, *Cours de Physique*, vol. iii. p. 372. - -There are a number of other points at which experience decides against -Newton, and in favour of Huyghens and Young. Laplace pointed out that -the attraction supposed to exist between matter and the corpuscular -particles of light would cause the velocity of light to vary with the -size of the emitting body, so that if a star were 250 times as great -in diameter as our sun, its attraction would prevent the emanation of -light altogether.[435] But experience shows that the velocity of light -is uniform, and independent of the magnitude of the emitting body, -as it should be according to the undulatory theory. Lastly, Newton’s -explanation of diffraction or inflection fringes of colours was only -*plausible*, and not true; for Fresnel ascertained that the dimensions -of the fringes are not what they would be according to Newton’s theory. - - [435] Young’s *Lectures on Natural Philosophy* (1845), vol. i. p. 361. - -Although the Science of Light presents us with the most beautiful -examples of crucial experiments and observations, instances are -not wanting in other branches of science. Copernicus asserted, in -opposition to the ancient Ptolemaic theory, that the earth moved round -the sun, and he predicted that if ever the sense of sight could be -rendered sufficiently acute and powerful, we should see phases in -Mercury and Venus. Galileo with his telescope was able, in 1610 to -verify the prediction as regards Venus, and subsequent observations -of Mercury led to a like conclusion. The discovery of the aberration -of light added a new proof, still further strengthened by the more -recent determination of the parallax of fixed stars. Hooke proposed -to prove the existence of the earth’s diurnal motion by observing the -deviation of a falling body, an experiment successfully accomplished by -Benzenberg; and Foucault’s pendulum has since furnished an additional -indication of the same motion, which is indeed also apparent in the -trade winds. All these are crucial facts in favour of the Copernican -theory. - - -*Descriptive Hypotheses.* - -There are hypotheses which we may call *descriptive hypotheses*, -and which serve for little else than to furnish convenient names. -When a phenomenon is of an unusual kind, we cannot even speak of -it without using some analogy. Every word implies some resemblance -between the thing to which it is applied, and some other thing, which -fixes the meaning of the word. If we are to speak of what constitutes -electricity, we must search for the nearest analogy, and as electricity -is characterised by the rapidity and facility of its movements, the -notion of a fluid of a very subtle character presents itself as -appropriate. There is the single-fluid and the double-fluid theory of -electricity, and a great deal of discussion has been uselessly spent -upon them. The fact is, that if these theories be understood as more -than convenient modes of describing the phenomena, they are altogether -invalid. The analogy extends only to the rapidity of motion, or rather -the fact that a phenomenon occurs successively at different points of -the body. The so-called electric fluid adds nothing to the weight of -the conductor, and to suppose that it really consists of particles of -matter is even more absurd than to reinstate the corpuscular theory -of light. A far closer analogy exists between electricity and light -undulations, which are about equally rapid in propagation. We shall -probably continue for a long time to talk of the *electric fluid*, but -there can be no doubt that this expression represents merely a phase of -molecular motion, a wave of disturbance. The invalidity of these fluid -theories is shown moreover in the fact that they have not led to the -invention of a single new experiment. - -Among these merely descriptive hypotheses I should place Newton’s -theory of Fits of Easy Reflection and Refraction. That theory did not -do more than describe what took place. It involved no analogy to other -phenomena of nature, for Newton could not point to any other substance -which went through these extraordinary fits. We now know that the true -analogy would have been waves of sound, of which Newton had acquired in -other respects so complete a comprehension. But though the notion of -interference of waves had distinctly occurred to Hooke, Newton failed -to see how the periodic phenomena of light could be connected with the -periodic character of waves. His hypothesis fell because it was out -of analogy with everything else in nature, and it therefore did not -allow him, as in other cases, to descend by mathematical deduction to -consequences which could be verified or refuted. - -We are at freedom to imagine the existence of a new agent, and to -give it an appropriate name, provided there are phenomena incapable -of explanation from known causes. We may speak of *vital force* as -occasioning life, provided that we do not take it to be more than a -name for an undefined something giving rise to inexplicable facts, just -as the French chemists called Iodine the Substance X, so long as they -were unaware of its real character and place in chemistry.[436] Encke -was quite justified in speaking of the *resisting medium* in space so -long as the retardation of his comet could not be otherwise accounted -for. But such hypotheses will do much harm whenever they divert us from -attempts to reconcile the facts with known laws, or when they lead us -to mix up discrete things. Because we speak of vital force we must not -assume that it is a really existing physical force like electricity; we -do not know what it is. We have no right to confuse Encke’s supposed -resisting medium with the basis of light without distinct evidence of -identity. The name protoplasm, now so familiarly used by physiologists, -is doubtless legitimate so long as we do not mix up different -substances under it, or imagine that the name gives us any knowledge -of the obscure origin of life. To name a substance protoplasm no more -explains the infinite variety of forms of life which spring out of the -substance, than does the *vital force* which may be supposed to reside -in the protoplasm. Both expressions are mere names for an inexplicable -series of causes which out of apparently similar conditions produce the -most diverse results. - - [436] Paris, *Life of Davy*, p. 274. - -Hardly to be distinguished from descriptive hypotheses are certain -imaginary objects which we frame for the ready comprehension of -a subject. The mathematician, in treating abstract questions of -probability, finds it convenient to represent the conditions by a -concrete hypothesis in the shape of a ballot-box. Poisson proved the -principle of the inverse method of probabilities by imagining a number -of ballot-boxes to have their contents mixed in one great ballot-box -(p. 244). Many such devices are used by mathematicians. The Ptolemaic -theory of *cycles* and *epi-cycles* was no grotesque and useless work -of the imagination, but a perfectly valid mode of analysing the motions -of the heavenly bodies; in reality it is used by mathematicians at the -present day. Newton employed the pendulum as a means of representing -the nature of an undulation. Centres of gravity, oscillation, &c., -poles of the magnet, lines of force, are other imaginary existences -employed to assist our thoughts (p. 364). Such devices may be called -*Representative Hypotheses*, and they are only permissible so far as -they embody analogies. Their further consideration belongs either to -the subject of Analogy, or to that of language and representation, -founded upon analogy. - - - - -CHAPTER XXIV. - -EMPIRICAL KNOWLEDGE, EXPLANATION, AND PREDICTION. - - -Inductive investigation, as we have seen, consists in the union of -hypothesis and experiment, deductive reasoning being the link by which -experimental results are made to confirm or confute the hypothesis. Now -when we consider this relation between hypothesis and experiment it is -obvious that we may classify our knowledge under four heads. - -(1) We may be acquainted with facts which have not yet been brought -into accordance with any hypothesis. Such facts constitute what is -called *Empirical Knowledge*. - -(2) Another extensive portion of our knowledge consists of facts which -having been first observed empirically, have afterwards been brought -into accordance with other facts by an hypothesis concerning the -general laws applying to them. This portion of our knowledge may be -said to be *explained*, *reasoned*, or *generalised*. - -(3) In the third place comes the collection of facts, minor in number, -but most important as regards their scientific interest, which have -been anticipated by theory and afterwards verified by experiment. - -(4) Lastly, there exists knowledge which is accepted solely on the -ground of theory, and is incapable of experimental confirmation, at -least with the instrumental means in our possession. - -It is a work of much interest to compare and illustrate the relative -extent and value of these four groups of knowledge. We shall observe -that as a general rule a great branch of science originates in facts -observed accidentally, or without distinct consciousness of what is to -be expected. As a science progresses, its power of foresight rapidly -increases, until the mathematician in his library acquires the power of -anticipating nature, and predicting what will happen in circumstances -which the eye of man has never examined. - - -*Empirical Knowledge.* - -By empirical knowledge we mean such as is derived directly from the -examination of detached facts, and rests entirely on those facts, -without corroboration from other branches of knowledge. It is -contrasted with generalised and theoretical knowledge, which embraces -many series of facts under a few comprehensive principles, so that each -series serves to throw light upon each other series of facts. Just -as, in the map of a half-explored country, we see detached bits of -rivers, isolated mountains, and undefined plains, not connected into -any complete plan, so a new branch of knowledge consists of groups of -facts, each group standing apart, so as not to allow us to reason from -one to another. - -Before the time of Descartes, and Newton, and Huyghens, there was -much empirical knowledge of the phenomena of light. The rainbow had -always struck the attention of the most careless observers, and there -was no difficulty in perceiving that its conditions of occurrence -consisted in rays of the sun shining upon falling drops of rain. It -was impossible to overlook the resemblance of the ordinary rainbow -to the comparatively rare lunar rainbow, to the bow which appears -upon the spray of a waterfall, or even upon beads of dew suspended on -grass and spiders’ webs. In all these cases the uniform conditions -are rays of light and round drops of water. Roger Bacon had noticed -these conditions, as well as the analogy of the rainbow colours to -those produced by crystals.[437] But the knowledge was empirical until -Descartes and Newton showed how the phenomena were connected with facts -concerning the refraction of light. - - [437] *Opus Majus.* Edit. 1733. Cap. x. p. 460. - -There can be no better instance of an empirical truth than that -detected by Newton concerning the high refractive powers of combustible -substances. Newton’s chemical notions were almost as vague as those -prevalent in his day, but he observed that certain “fat, sulphureous, -unctuous bodies,” as he calls them, such as camphor, oils spirit of -turpentine, amber, &c., have refractive powers two or three times -greater than might be anticipated from their densities.[438] The -enormous refractive index of diamond, led him with great sagacity to -regard this substance as of the same unctuous or inflammable nature, -so that he may be regarded as predicting the combustibility of the -diamond, afterwards demonstrated by the Florentine Academicians -in 1694. Brewster having entered into a long investigation of the -refractive powers of different substances, confirmed Newton’s -assertions, and found that the three elementary combustible substances, -diamond, phosphorus, and sulphur, have, in comparison with their -densities, by far the highest known refractive indices,[439] and -there are only a few substances, such as chromate of lead or glass of -antimony, which exceed them in absolute power of refraction. The oils -and hydrocarbons generally possess excessive indices. But all this -knowledge remains to the present day purely empirical, no connection -having been pointed out between this coincidence of inflammability -and high refractive power, with other laws of chemistry or optics. -It is worth notice, as pointed out by Brewster, that if Newton had -argued concerning two minerals, Greenockite and Octahedrite, as he did -concerning diamond, his predictions would have proved false, showing -sufficiently that he did not make any sure induction on the subject. In -the present day, the relation of the refractive index to the density -and atomic weight of a substance is becoming a matter of theory; yet -there remain specific differences of refracting power known only on -empirical grounds, and it is curious that in hydrogen an abnormally -high refractive power has been found to be joined to inflammability. - - [438] Newton’s *Opticks*. Third edit. p. 249. - - [439] Brewster. *Treatise on New Philosophical Instruments*, p. 266, - &c. - -The science of chemistry, however much its theory may have progressed, -still presents us with a vast body of empirical knowledge. Not only -is it as yet hopeless to attempt to account for the particular group -of qualities belonging to each element, but there are multitudes of -particular facts of which no further account can be given. Why should -the sulphides of many metals be intensely black? Why should a slight -amount of phosphoric acid have so great a power of interference with -the crystallisation of vanadic acid?[440] Why should the compound -silicates of alkalies and alkaline metals be transparent? Why should -gold be so highly ductile, and gold and silver the only two sensibly -translucent metals? Why should sulphur be capable of so many peculiar -changes into allotropic modifications? - - [440] Roscoe, Bakerian Lecture, *Philosophical Transactions* (1868), - vol. clviii. p. 6. - -There are whole branches of chemical knowledge which are mere -collections of disconnected facts. The properties of alloys are -often remarkable; but no laws have yet been detected, and the laws -of combining proportions seem to have no clear application.[441] Not -the slightest explanation can be given of the wonderful variations of -the qualities of iron, according as it contains more or less carbon -and silicon, nay, even the facts of the case are often involved in -uncertainty. Why, again, should the properties of steel be remarkably -affected by the presence of a little tungsten or manganese? All -that was determined by Matthiessen concerning the conducting powers -of copper, was of a purely empirical character.[442] Many animal -substances cannot be shown to obey the laws of combining proportions. -Thus for the most part chemistry is yet an empirical science occupied -with the registration of immense numbers of disconnected facts, which -may at some future time become the basis of a greatly extended theory. - - [441] *Life of Faraday*, vol. ii. p. 104. - - [442] Watts, *Dictionary of Chemistry*, vol. ii, p. 39, &c. - -We must not indeed suppose that any science will ever entirely cease -to be empirical. Multitudes of phenomena have been explained by the -undulatory theory of light; but there yet remain many facts to be -treated. The natural colours of bodies and the rays given off by them -when heated, are unexplained, and yield few empirical coincidences. -The theory of electricity is partially understood, but the conditions -of the production of frictional electricity defy explanation, although -they have been studied for two centuries. I shall subsequently point -out that even the establishment of a wide and true law of nature is -but the starting-point for the discovery of exceptions and divergences -giving a new scope to empirical discovery. - -There is probably no science, I have said, which is entirely free -from empirical and unexplained facts. Logic approaches most nearly to -this position, as it is merely a deductive development of the laws -of thought and the principle of substitution. Yet some of the facts -established in the investigation of the inverse logical problem may -be considered empirical. That a proposition of the form A = BC ꖌ *b -c* possesses the least number of distinct logical variations, and -the greatest number of logical equivalents of the same form among -propositions involving three classes (p. 141), is a case in point. So -also is the fact discovered by Professor Clifford that in regard to -statements involving four classes, there is only one example of two -dissimilar statements having the same distances (p. 144). Mathematical -science often yields empirical truths. Why, for instance, should the -value of π, when expressed to a great number of figures, contain the -digit 7 much less frequently than any other digit?[443] Even geometry -may allow of empirical truths, when the matter does not involve -quantities of space, but numerical results and the positive or negative -character of quantities, as in De Morgan’s theorem concerning negative -areas. - - [443] De Morgan’s *Budget of Paradoxes*, p. 291. - - -*Accidental Discovery.* - -There are not a few cases where almost pure accident has determined -the moment when a new branch of knowledge was to be created. The laws -of the structure of crystals were not discovered until Haüy happened -to drop a beautiful crystal of calc-spar upon a stone pavement. His -momentary regret at destroying a choice specimen was quickly removed -when, in attempting to join the fragments together, he observed -regular geometrical faces, which did not correspond with the external -facets of the crystals. A great many more crystals were soon broken -intentionally, to observe the planes of cleavage, and the discovery of -the internal structure of crystalline substances was the result. Here -we see how much more was due to the reasoning power of the philosopher, -than to an accident which must often have happened to other persons. - -In a similar manner, a fortuitous occurrence led Malus to discover -the polarisation of light by reflection. The phenomena of double -refraction had been long known, and when engaged in Paris in 1808, in -investigating the character of light thus polarised, Malus chanced to -look through a double refracting prism at the light of the setting -sun, reflected from the windows of the Luxembourg Palace. In turning -the prism round, he was surprised to find that the ordinary image -disappeared at two opposite positions of the prism. He remarked that -the reflected light behaved like light which had been polarised by -passing through another prism. He was induced to test the character of -light reflected under other circumstances, and it was eventually proved -that polarisation is invariably connected with reflection. Some of the -general laws of optics, previously unsuspected, were thus discovered by -pure accident. In the history of electricity, accident has had a large -part. For centuries some of the more common effects of magnetism and -of frictional electricity had presented themselves as unaccountable -deviations from the ordinary course of Nature. Accident must have -first directed attention to such phenomena, but how few of those who -witnessed them had any conception of the all-pervading character of the -power manifested. The very existence of galvanism, or electricity of -low tension, was unsuspected until Galvani accidentally touched the leg -of a frog with pieces of metal. The decomposition of water by voltaic -electricity also was accidentally discovered by Nicholson in 1801, and -Davy speaks of this discovery as the foundation of all that had since -been done in electro-chemical science. - -It is otherwise with the discovery of electro-magnetism. Oersted, in -common with many others, had suspected the existence of some relation -between the magnet and electricity, and he appears to have tried to -detect its exact nature. Once, as we are told by Hansteen, he had -employed a strong galvanic battery during a lecture, and at the close -it occurred to him to try the effect of placing the conducting wire -parallel to a magnetic needle, instead of at right angles, as he had -previously done. The needle immediately moved and took up a position -nearly at right angles to the wire; he inverted the direction of the -current, and the needle deviated in a contrary direction. The great -discovery was made, and if by accident, it was such an accident as -happens, as Lagrange remarked of Newton, only to those who deserve -it.[444] There was, in fact, nothing accidental, except that, as in -all totally new discoveries, Oersted did not know what to look for. He -could not infer from previous knowledge the nature of the relation, -and it was only repeated trial in different modes which could lead him -to the right combination. High and happy powers of inference, and not -accident, subsequently led Faraday to reverse the process, and to show -that the motion of the magnet would occasion an electric current in the -wire. - - [444] *Life of Faraday*, vol. ii p. 396. - -Sufficient investigation would probably show that almost every branch -of art and science had an accidental beginning. In historical times -almost every important new instrument as the telescope, the microscope, -or the compass, was probably suggested by some accidental occurrence. -In pre-historic times the germs of the arts must have arisen still more -exclusively in the same way. Cultivation of plants probably arose, in -Mr. Darwin’s opinion, from some such accident as the seeds of a fruit -falling upon a heap of refuse, and producing an unusually fine variety. -Even the use of fire must, some time or other, have been discovered in -an accidental manner. - -With the progress of a branch of science, the element of chance becomes -much reduced. Not only are laws discovered which enable results to be -predicted, as we shall see, but the systematic examination of phenomena -and substances often leads to discoveries which can in no sense be said -to be accidental. It has been asserted that the anæsthetic properties -of chloroform were disclosed by a little dog smelling at a saucerful -of the liquid in a chemist’s shop in Linlithgow, the singular effects -upon the dog being reported to Simpson, who turned the incident to -good account. This story, however, has been shown to be a fabrication, -the fact being that Simpson had for many years been endeavouring to -discover a better anæsthetic than those previously employed, and that -he tested the properties of chloroform, among other substances, at -the suggestion of Waldie, a Liverpool chemist. The valuable powers -of chloral hydrate have since been discovered in a like manner, and -systematic inquiries are continually being made into the therapeutic or -economic values of new chemical compounds. - -If we must attempt to draw a conclusion concerning the part which -chance plays in scientific discovery, it must be allowed that it more -or less affects the success of all inductive investigation, but becomes -less important with the progress of science. Accident may bring a new -and valuable combination to the notice of some person who had never -expressly searched for a discovery of the kind, and the probabilities -are certainly in favour of a discovery being occasionally made in this -manner. But the greater the tact and industry with which a physicist -applies himself to the study of nature, the greater is the probability -that he will meet with fortunate accidents, and will turn them to good -account. Thus it comes to pass that, in the refined investigations -of the present day, genius united to extensive knowledge, cultivated -powers, and indomitable industry, constitute the characteristics of the -successful discoverer. - - -*Empirical Observations subsequently Explained.* - -The second great portion of scientific knowledge consists of facts -which have been first learnt in a purely empirical manner, but have -afterwards been shown to follow from some law of nature, that is, -from some highly probable hypothesis. Facts are said to be explained -when they are thus brought into harmony with other facts, or bodies -of general knowledge. There are few words more familiarly used in -scientific phraseology than this word *explanation*, and it is -necessary to decide exactly what we mean by it, since the question -touches the deepest points concerning the nature of science. Like most -terms referring to mental actions, the verbs *to explain*, or *to -explicate*, involve material similes. The action is *ex plicis plana -reddere*, to take out the folds, and render a thing plain or even. -Explanation thus renders a thing clearly comprehensible in all its -points, so that there is nothing left outstanding or obscure. - -Every act of explanation consists in pointing out a resemblance -between facts, or in showing that similarity exists between apparently -diverse phenomena. This similarity may be of any extent and depth; it -may be a general law of nature, which harmonises the motions of all -the heavenly bodies by showing that there is a similar force which -governs all those motions, or the explanation may involve nothing more -than a single identity, as when we explain the appearance of shooting -stars by showing that they are identical with portions of a comet. -Wherever we detect resemblance, there is a more or less explanation. -The mind is disquieted when it meets a novel phenomenon, one which is -*sui generis*; it seeks at once for parallels which may be found in -the memory of past sensations. The so-called sulphurous smell which -attends a stroke of lightning often excited attention, and it was not -explained until the exact similarity of the smell to that of ozone was -pointed out. The marks upon a flagstone are explained when they are -shown to correspond with the feet of an extinct animal, whose bones -are elsewhere found. Explanation, in fact, generally commences by the -discovery of some simple resemblance; the theory of the rainbow began -as soon as Antonio de Dominis pointed out the resemblance between its -colours and those presented by a ray of sunlight passing through a -glass globe full of water. - -The nature and limits of explanation can only be fully considered, -after we have entered upon the subjects of generalisation and analogy. -It must suffice to remark, in this place, that the most important -process of explanation consists in showing that an observed fact is one -case of a general law or tendency. Iron is always found combined with -sulphur, when it is in contact with coal, whereas in other parts of -the carboniferous strata it always occurs as a carbonate. We explain -this empirical fact as being due to the reducing power of carbon and -hydrogen, which prevents the iron from combining with oxygen, and -leaves it open to the affinity of sulphur. The uniform strength and -direction of the trade-winds were long familiar to mariners, before -they were explained by Halley on hydrostatical principles. The winds -were found to arise from the action of gravity, which causes a heavier -body to displace a lighter one, while the direction from east to west -was explained as a result of the earth’s rotation. Whatever body in -the northern hemisphere changes its latitude, whether it be a bird, or -a railway train, or a body of air, must tend towards the right hand. -Dove’s law of the winds is that the winds tend to veer in the northern -hemisphere in the direction N.E.S.W., and in the southern hemisphere -in the direction N.W.S.E. This tendency was shown by him to be the -necessary effect of the same conditions which apply to the trade winds. -Whenever, then, any fact is connected by resemblance, law, theory, or -hypothesis, with other facts, it is explained. - -Although the great mass of recorded facts must be empirical, and -awaiting explanation, such knowledge is of minor value, because it does -not admit of safe and extensive inference. Each recorded result informs -us exactly what will be experienced again in the same circumstances, -but has no bearing upon what will happen in other circumstances. - - -*Overlooked Results of Theory.* - -We must by no means suppose that, when a scientific truth is in our -possession, all its consequences will be foreseen. Deduction is certain -and infallible, in the sense that each step in deductive reasoning will -lead us to some result, as certain as the law itself. But it does not -follow that deduction will lead the reasoner to every result of a law -or combination of laws. Whatever road a traveller takes, he is sure to -arrive somewhere, but unless he proceeds in a systematic manner, it is -unlikely that he will reach every place to which a network of roads -will conduct him. - -In like manner there are many phenomena which were virtually within -the reach of philosophers by inference from their previous knowledge, -but were never discovered until accident or systematic empirical -observation disclosed their existence. - -That light travels with a uniform high velocity was proved by Roemer -from observations of the eclipses of Jupiter’s satellites. Corrections -were thenceforward made in all astronomical observations requiring it, -for the difference of absolute time at which an event happened, and -that at which it would be seen on the earth. But no person happened -to remark that the motion of light compounded with that of the earth -in its orbit would occasion a small apparent displacement of the -greater part of the heavenly bodies. Fifty years elapsed before Bradley -empirically discovered this effect, called by him aberration, when -reducing his observations of the fixed stars. - -When once the relation between an electric current and a magnet had -been detected by Oersted and Faraday, it ought to have been possible -for them to foresee the diverse results which must ensue in different -circumstances. If, for instance, a plate of copper were placed beneath -an oscillating magnetic needle, it should have been seen that the -needle would induce currents in the copper, but as this could not take -place without a certain reaction against the needle, it ought to have -been seen that the needle would come to rest more rapidly than in the -absence of the copper. This peculiar effect was accidentally discovered -by Gambey in 1824. Arago acutely inferred from Gambey’s experiment that -if the copper were set in rotation while the needle was stationary the -motion would gradually be communicated to the needle. The phenomenon -nevertheless puzzled the whole scientific world, and it required -the deductive genius of Faraday to show that it was a result of the -principles of electro-magnetism.[445] - - [445] *Experimental Researches in Electricity*, 1st Series, pp. 24–44. - -Many other curious facts might be mentioned which when once noticed -were explained as the effects of well-known laws. It was accidentally -discovered that the navigation of canals of small depth could be -facilitated by increasing the speed of the boats, the resistance being -actually reduced by this increase of speed, which enables the boat -to ride as it were upon its own forced wave. Now mathematical theory -might have predicted this result had the right application of the -formulæ occurred to any one.[446] Giffard’s injector for supplying -steam boilers with water by the force of their own steam, was, I -believe, accidentally discovered, but no new principles of mechanics -are involved in it, so that it might have been theoretically invented. -The same may be said of the curious experiment in which a stream of -air or steam issuing from a pipe is made to hold a free disc upon the -end of the pipe and thus obstruct its own outlet. The possession then -of a true theory does not by any means imply the foreseeing of all the -results. The effects of even a few simple laws may be manifold, and -some of the most curious and useful effects may remain undetected until -accidental observation brings them to our notice. - - [446] Airy, *On Tides and Waves*, Encyclopædia Metropolitana, p. 348*. - - -*Predicted Discoveries.* - -The most interesting of the four classes of facts specified in p. 525, -is probably the third, containing those the occurrence of which has -been first predicted by theory and then verified by observation. There -is no more convincing proof of the soundness of knowledge than that it -confers the gift of foresight. Auguste Comte said that “Prevision is -the test of true theory;” I should say that it is *one test* of true -theory, and that which is most likely to strike the public attention. -Coincidence with fact is the test of true theory, but when the result -of theory is announced before-hand, there can be no doubt as to the -unprejudiced spirit in which the theorist interprets the results of his -own theory. - -The earliest instance of scientific prophecy is naturally furnished -by the science of Astronomy, which was the earliest in development. -Herodotus[447] narrates that, in the midst of a battle between the -Medes and Lydians, the day was suddenly turned into night, and the -event had been foretold by Thales, the Father of Philosophy. A -cessation of the combat and peace confirmed by marriages were the -consequences of this happy scientific effort. Much controversy has -taken place concerning the date of this occurrence, Baily assigning -the year 610 B.C., but Airy has calculated that the exact day was -the 28th of May, 584 B.C. There can be no doubt that this and other -predictions of eclipses attributed to ancient philosophers were due to -a knowledge of the Metonic Cycle, a period of 6,585 days, or 223 lunar -months, or about 19 years, after which a nearly perfect recurrence of -the phases and eclipses of the moon takes place; but if so, Thales must -have had access to long series of astronomical records of the Egyptians -or the Chaldeans. There is a well-known story as to the happy use which -Columbus made of the power of predicting eclipses in overawing the -islanders of Jamaica who refused him necessary supplies of food for his -fleet. He threatened to deprive them of the moon’s light. “His threat -was treated at first with indifference, but when the eclipse actually -commenced, the barbarians vied with each other in the production of the -necessary supplies for the Spanish fleet.” - - [447] Lib. i. cap. 74. - -Exactly the same kind of awe which the ancients experienced at the -prediction of eclipses, has been felt in modern times concerning the -return of comets. Seneca asserted in distinct terms that comets would -be found to revolve in periodic orbits and return to sight. The ancient -Chaldeans and the Pythagoreans are also said to have entertained a -like opinion. But it was not until the age of Newton and Halley that -it became possible to calculate the path of a comet in future years. A -great comet appeared in 1682, a few years before the first publication -of the *Principia*, and Halley showed that its orbit corresponded -with that of remarkable comets recorded to have appeared in the years -1531 and 1607. The intervals of time were not quite equal, but Halley -conceived the bold idea that this difference might be due to the -disturbing power of Jupiter, near which the comet had passed in the -interval 1607–1682. He predicted that the comet would return about -the end of 1758 or the beginning of 1759, and though Halley did not -live to enjoy the sight, it was actually detected on the night of -Christmas-day, 1758. A second return of the comet was witnessed in 1835 -nearly at the anticipated time. - -In recent times the discovery of Neptune has been the most remarkable -instance of prevision in astronomical science. A full account of this -discovery may be found in several works, as for instance Herschel’s -*Outlines of Astronomy*, and *Grant’s History of Physical Astronomy*, -Chapters XII and XIII. - - -*Predictions in the Science of Light.* - -Next after astronomy the science of physical optics has furnished the -most beautiful instances of the prophetic power of correct theory. -These cases are the more striking because they proceed from the -profound application of mathematical analysis and show an insight -into the mysterious workings of matter which is surprising to all, -but especially to those who are unable to comprehend the methods -of research employed. By its power of prevision the truth of the -undulatory theory of light has been conspicuously proved, and the -contrast in this respect between the undulatory and Corpuscular -theories is remarkable. Even Newton could get no aid from his -corpuscular theory in the invention of new experiments, and to his -followers who embraced that theory we owe little or nothing in the -science of light. Laplace did not derive from the theory a single -discovery. As Fresnel remarks:[448] - - [448] Taylor’s *Scientific Memoirs*, vol. v. p. 241. - -“The assistance to be derived from a good theory is not to be confined -to the calculation of the forces when the laws of the phenomena -are known. There are certain laws so complicated and so singular, -that observation alone, aided by analogy, could never lead to their -discovery. To divine these enigmas we must be guided by theoretical -ideas founded on a *true* hypothesis. The theory of luminous vibrations -presents this character, and these precious advantages; for to it -we owe the discovery of optical laws the most complicated and most -difficult to divine.” - -Physicists who embraced the corpuscular theory had nothing but their -own quickness of observation to rely upon. Fresnel having once seized -the conditions of the true undulatory theory, as previously stated -by Young, was enabled by the mere manipulation of his mathematical -symbols to foresee many of the complicated phenomena of light. Who -could possibly suppose, that by stopping a portion of the rays -passing through a circular aperture, the illumination of a point upon -a screen behind the aperture might be many times multiplied. Yet this -paradoxical effect was predicted by Fresnel, and verified both by -himself, and in a careful repetition of the experiment, by Billet. -Few persons are aware that in the middle of the shadow of an opaque -circular disc is a point of light sensibly as bright as if no disc -had been interposed. This startling fact was deduced from Fresnel’s -theory by Poisson, and was then verified experimentally by Arago. -Airy, again, was led by pure theory to predict that Newton’s rings -would present a modified appearance if produced between a lens of -glass and a plate of metal. This effect happened to have been observed -fifteen years before by Arago, unknown to Airy. Another prediction of -Airy, that there would be a further modification of the rings when -made between two substances of very different refractive indices, was -verified by subsequent trial with a diamond. A reversal of the rings -takes place when the space intervening between the plates is filled -with a substance of intermediate refractive power, another phenomenon -predicted by theory and verified by experiment. There is hardly a limit -to the number of other complicated effects of the interference of rays -of light under different circumstances which might be deduced from -the mathematical expressions, if it were worth while, or which, being -previously observed, can be explained. An interesting case was observed -by Herschel and explained by Airy.[449] - - [449] Airy’s *Mathematical Tracts*, 3rd edit. p. 312. - -By a somewhat different effort of scientific foresight, Fresnel -discovered that any solid transparent medium might be endowed with the -power of double refraction by mere compression. As he attributed the -double refracting power of crystals to unequal elasticity in different -directions, he inferred that unequal elasticity, if artificially -produced, would give similar phenomena. With a powerful screw and a -piece of glass, he then produced not only the colours due to double -refraction, but the actual duplication of images. Thus, by a great -scientific generalisation, are the remarkable properties of Iceland -spar shown to belong to all transparent substances under certain -conditions.[450] - - [450] Young’s *Works*, vol. i. p. 412. - -All other predictions in optical science are, however, thrown into the -shade by the theoretical discovery of conical refraction by the late -Sir W. R. Hamilton, of Dublin. In investigating the passage of light -through certain crystals, Hamilton found that Fresnel had slightly -misinterpreted his own formulæ, and that, when rightly understood, -they indicated a phenomenon of a kind never witnessed. A small ray of -light sent into a crystal of arragonite in a particular direction, -becomes spread out into an infinite number of rays, which form a hollow -cone within the crystal, and a hollow cylinder when emerging from the -opposite side. In another case, a different, but equally strange, -effect is produced, a ray of light being spread out into a hollow cone -at the point where it quits the crystal. These phenomena are peculiarly -interesting, because cones and cylinders of light are not produced -in any other cases. They are opposed to all analogy, and constitute -singular exceptions, of a kind which we shall afterwards consider more -fully. Their strangeness rendered them peculiarly fitted to test the -truth of the theory by which they were discovered; and when Professor -Lloyd, at Hamilton’s request, succeeded, after considerable difficulty, -in witnessing the new appearances, no further doubt could remain of -the validity of the wave theory which we owe to Huyghens, Young, and -Fresnel.[451] - - [451] Lloyd’s *Wave Theory*, Part ii. pp. 52–58. Babbage, *Ninth - Bridgewater Treatise*, p. 104, quoting Lloyd, *Transactions of the - Royal Irish Academy*, vol. xvii. Clifton, *Quarterly Journal of Pure - and Applied Mathematics*, January 1860. - - -*Predictions from the Theory of Undulations.* - -It is curious that the undulations of light, although inconceivably -rapid and small, admit of more accurate measurement than waves of any -other kind. But so far as we can carry out exact experiments on other -kinds of waves, we find the phenomena of interference repeated, and -analogy gives considerable power of prediction. Herschel was perhaps -the first to suggest that two sounds might be made to destroy each -other by interference.[452] For if one-half of a wave travelling -through a tube could be separated, and conducted by a longer passage, -so as, on rejoining the other half, to be one-quarter of a vibration -behind-hand, the two portions would exactly neutralise each other. -This experiment has been performed with success. The interference -arising between the waves from the two prongs of a tuning-fork was -also predicted by theory, and proved to exist by Weber; indeed it may -be observed by merely holding a vibrating fork close to the ear and -turning it round.[453] - - [452] *Encyclopædia Metropolitana*, art. *Sound*, p. 753. - - [453] Tyndall’s *Sound*, pp. 261, 273. - -It is a result of the theory of sound that, if we move rapidly towards -a sounding body, or if it move rapidly towards us, the pitch of the -sound will be a little more acute; and, *vice versâ*, when the relative -motion is in the opposite direction, the pitch will be more grave. This -arises from the less or greater intervals of time elapsing between the -successive strokes of waves upon the auditory nerve, according as the -ear moves towards or from the source of sound relatively speaking. -This effect was predicted by theory, and afterwards verified by the -experiments of Buys Ballot, on Dutch railways, and of Scott Russell, in -England. Whenever one railway train passes another, on the locomotive -of which the whistle is being sounded, the drop in the acuteness of -the sound may be noticed at the moment of passing. This change gives -the sound a peculiar howling character, which many persons must have -noticed. I have calculated that with two trains travelling thirty miles -an hour, the effect would amount to rather more than half a tone, and -with some express trains it would amount to a tone. A corresponding -effect is produced in the case of light undulations, when the eye and -the luminous body approach or recede from each other. It is shown by -a slight change in the refrangibility of the rays of light, and a -consequent change in the place of the lines of the spectrum, which has -been made to give important and unexpected information concerning the -relative approach or recession of stars. - -Tides are vast waves, and were the earth’s surface entirely covered -by an ocean of uniform depth, they would admit of exact theoretical -investigation. The irregular form of the seas introduces unknown -quantities and complexities with which theory cannot cope. -Nevertheless, Whewell, observing that the tides of the German Ocean -consist of interfering waves, which arrive partly round the North of -Scotland and partly through the British Channel, was enabled to predict -that at a point about midway between Brill on the coast of Holland, -and Lowestoft no tides would be found to exist. At that point the two -waves would be of the same amount, but in opposite phases, so as to -neutralise each other. This prediction was verified by a surveying -vessel of the British navy.[454] - - [454] Whewell’s *History of the Inductive Sciences*, vol. ii. p. 471. - Herschel’s *Physical Geography*, § 77. - - -*Prediction in other Sciences.* - -Generations, or even centuries, may elapse before mankind are -in possession of a mathematical theory of the constitution of -matter as complete as the theory of gravitation. Nevertheless, -mathematical physicists have in recent years acquired a hold of some -of the relations of the physical forces, and the proof is found in -anticipations of curious phenomena which had never been observed. -Professor James Thomson deduced from Carnot’s theory of heat that the -application of pressure would lower the melting-point of ice. He even -ventured to assign the amount of this effect, and his statement was -afterwards verified by Sir W. Thomson.[455] “In this very remarkable -speculation, an entirely novel physical phenomenon was *predicted*, in -anticipation of any direct experiments on the subject; and the actual -observation of the phenomenon was pointed out as a highly interesting -object for experimental research.” Just as liquids which expand in -solidifying will have the temperature of solidification lowered by -pressure, so liquids which contract in solidifying will exhibit the -reverse effect. They will be assisted in solidifying, as it were, -by pressure, so as to become solid at a higher temperature, as the -pressure is greater. This latter result was verified by Bunsen and -Hopkins, in the case of paraffin, spermaceti, wax, and stearin. The -effect upon water has more recently been carried to such an extent by -Mousson, that under the vast pressure of 1300 atmospheres, water did -not freeze until cooled down to -18°C. Another remarkable prediction -of Professor Thomson was to the effect that, if a metallic spring -be weakened by a rise of temperature, work done against the spring -in bending it will cause a cooling effect. Although the effect to -be expected in a certain apparatus was only about four-thousandths -of a degree Centigrade, Dr. Joule[456] succeeded in measuring it to -the extent of three-thousandths of a degree, such is the delicacy of -modern heat measurements. I cannot refrain from quoting Dr. Joule’s -reflections upon this fact. “Thus even in the above delicate case,” he -says, “is the formula of Professor Thomson completely verified. The -mathematical investigation of the thermo-elastic qualities of metals -has enabled my illustrious friend to predict with certainty a whole -class of highly interesting phenomena. To him especially do we owe -the important advance which has been recently made to a new era in -the history of science, when the famous philosophical system of Bacon -will be to a great extent superseded, and when, instead of arriving at -discovery by induction from experiment, we shall obtain our largest -accessions of new facts by reasoning deductively from fundamental -principles.” - - [455] Maxwell’s *Theory of Heat*, p. 174. *Philosophical Magazine*, - August 1850. Third Series, vol. xxxvii. p. 123. - - [456] *Philosophical Transactions*, 1858, vol. cxlviii. p. 127. - -The theory of electricity is a necessary part of the general theory of -matter, and is rapidly acquiring the power of prevision. As soon as -Wheatstone had proved experimentally that the conduction of electricity -occupies time, Faraday remarked in 1838, with wonderful sagacity, -that if the conducting wires were connected with the coatings of a -large Leyden jar, the rapidity of conduction would be lessened. This -prediction remained unverified for sixteen years, until the submarine -cable was laid beneath the Channel. A considerable retardation of the -electric spark was then detected, and Faraday at once pointed out that -the wire surrounded by water resembles a Leyden jar on a large scale, -so that each message sent through the cable verified his remark of -1838.[457] - - [457] Tyndall’s *Faraday*, pp. 73, 74; *Life of Faraday*, vol. ii. - pp. 82, 83. - -The joint relations of heat and electricity to the metals constitute a -new science of thermo-electricity by which Sir W. Thomson was enabled -to anticipate the following curious effect, namely, that an electric -current passing in an iron bar from a hot to a cold part produces a -cooling effect, but in a copper bar the effect is exactly opposite in -character, that is, the bar becomes heated.[458] The action of crystals -with regard to heat and electricity was partly foreseen on the grounds -of theory by Poisson. - - [458] Tait’s *Thermodynamics*, p. 77. - -Chemistry, although to a great extent an empirical science, has not -been without prophetic triumphs. The existence of the metals potassium -and sodium was foreseen by Lavoisier, and their elimination by Davy was -one of the chief *experimenta crucis* which established Lavoisier’s -system. The existence of many other metals which eye had never seen -was a natural inference, and theory has not been at fault. In the -above cases the compounds of the metal were well known, and it was the -result of decomposition that was foretold. The discovery in 1876 of the -metal gallium is peculiarly interesting because the existence of this -metal, previously wholly unknown, had been inferred from theoretical -considerations by M. Mendelief, and some of its properties had been -correctly predicted. No sooner, too, had a theory of organic compounds -been conceived by Professor A. W. Williamson than he foretold the -formation of a complex substance consisting of water in which both -atoms of hydrogen are replaced by atoms of acetyle. This substance, -known as the acetic anhydride, was afterwards produced by Gerhardt. In -the subsequent progress of organic chemistry occurrences of this kind -have become common. The theoretical chemist by the classification of -his specimens and the manipulation of his formulæ can plan out whole -series of unknown oils, acids, and alcohols, just as a designer might -draw out a multitude of patterns. Professor Cayley has even calculated -for certain cases the possible numbers of chemical compounds.[459] The -formation of many such substances is a matter of course; but there is -an interesting prediction given by Hofmann, concerning the possible -existence of new compounds of sulphur and selenium, and even oxides of -ammonium, which it remains for chemists to verify.[460] - - [459] *On the Analytical Forms called Trees, with Application to the - Theory of Chemical Combinations.* Report of the British Association, - 1875, p. 257. - - [460] Hofmann’s *Introduction to Chemistry*, pp. 224, 225. - - -*Prediction by Inversion of Cause and Effect.* - -There is one process of experiment which has so often led to important -discoveries as to deserve separate illustration--I mean the inversion -of Cause and Effect. Thus if A and B in one experiment produce C as a -consequent, then antecedents of the nature of B and C may usually be -made to produce a consequent of the nature of A inverted in direction. -When we apply heat to a gas it tends to expand; hence if we allow the -gas to expand by its own elastic force, cold is the result; that is, -B (air) and C (expansion) produce the negative of A (heat). Again, B -(air) and compression, the negative of C, produce A (heat). Similar -results may be expected in a multitude of cases. It is a familiar -law that heat expands iron. What may be expected, then, if instead -of increasing the length of an iron bar by heat we use mechanical -force and stretch the bar? Having the bar and the former consequent, -expansion, we should expect the negative of the former antecedent, -namely cold. The truth of this inference was proved by Dr. Joule, who -investigated the amount of the effect with his usual skill.[461] - - [461] *Philosophical Transactions* (1855), vol. cxlv. pp. 100, &c. - -This inversion of cause and effect in the case of heat may be itself -inverted in a highly curious manner. It happens that there are a -few substances which are unexplained exceptions to the general law -of expansion by heat. India-rubber especially is remarkable for -*contracting* when heated. Since, then, iron and india-rubber are -oppositely related to heat, we may expect that as distension of the -iron produced cold, distension of the india-rubber will produce heat. -This is actually found to be the case, and anyone may detect the effect -by suddenly stretching an india-rubber band while the middle part is -in the mouth. When being stretched it grows slightly warm, and when -relaxed cold. - -The reader will see that some of the scientific predictions mentioned -in preceding sections were due to the principle of inversion; for -instance, Thomson’s speculations on the relation between pressure and -the melting-point. But many other illustrations could be adduced. -The usual agent by which we melt a substance is heat; but if we can -melt a substance without heat, then we may expect the negative of -heat as an effect. This is the foundation of all freezing mixtures. -The affinity of salt for water causes it to melt ice, and we may thus -reduce the temperature to Fahrenheit’s zero. Calcium chloride has so -much higher an attraction for water that a temperature of -45° C. may -be attained by its use. Even the solution of a certain alloy of lead, -tin, and bismuth in mercury, may be made to reduce the temperature -through 27° C. All the other modes of producing cold are inversions of -more familiar uses of heat. Carré’s freezing machine is an inverted -distilling apparatus, the distillation being occasioned by chemical -affinity instead of heat. Another kind of freezing machine is the exact -inverse of the steam-engine. - -A very paradoxical effect is due to another inversion. It is hard to -believe that a current of steam at 100° C. can raise a body of liquid -to a higher temperature than the steam itself possesses. But Mr. Spence -has pointed out that if the boiling-point of a saline solution be -above 100°, it will continue, on account of its affinity for water, to -condense steam when above 100° in temperature. It will condense the -steam until heated to the point at which the tension of its vapour is -equal to that of the atmosphere, that is, its own boiling-point.[462] -Again, since heat melts ice, we might expect to produce heat by the -inverse change from water into ice. This is accomplished in the -phenomenon of suspended freezing. Water may be cooled in a clean glass -vessel many degrees below the freezing-point, and yet retained in the -liquid condition. But if disturbed, and especially if brought into -contact with a small particle of ice, it instantly solidifies and rises -in temperature to 0° C. The effect is still better displayed in the -lecture-room experiment of the suspended crystallisation of a solution -of sodium sulphate, in which a sudden rise of temperature of 15° or -20° C. is often manifested. - - [462] *Proceedings of the Manchester Philosophical Society*, Feb. - 1870. - -The science of electricity is full of most interesting cases of -inversion. As Professor Tyndall has remarked, Faraday had a profound -belief in the reciprocal relations of the physical forces. The great -starting-point of his researches, the discovery of electro-magnetism, -was clearly an inversion. Oersted and Ampère had proved that with -an electric current and a magnet in a particular position as -antecedents, motion is the consequent. If then a magnet, a wire and -motion be the antecedents, an *opposite* electric current will be the -consequent. It would be an endless task to trace out the results of -this fertile relationship. Another part of Faraday’s researches was -occupied in ascertaining the direct and inverse relations of magnetic -and diamagnetic, amorphous and crystalline substances in various -circumstances. In all other relations of electricity the principle of -inversion holds. The voltameter or the electro-plating cell is the -inverse of the galvanic battery. As heat applied to a junction of -antimony and bismuth bars produces electricity, it follows that an -electric current passed through such a junction will produce cold. But -it is now sufficiently apparent that inversion of cause and effect is a -most fertile means of discovery and prediction. - - -*Facts known only by Theory.* - -Of the four classes of facts enumerated in p. 525 the last remains -unconsidered. It includes the unverified predictions of science. -Scientific prophecy arrests the attention of the world when it refers -to such striking events as an eclipse, the appearance of a great comet, -or any phenomenon which people can verify with their own eyes. But -it is surely a matter for greater wonder that a physicist describes -and measures phenomena which eye cannot see, nor sense of any kind -detect. In most cases this arises from the effect being too small in -amount to affect our organs of sense, or come within the powers of our -instruments as at present constructed. But there is a class of yet more -remarkable cases, in which a phenomenon cannot possibly be observed, -and yet we can say what it would be if it were observed. - -In astronomy, systematic aberration is an effect of the sun’s proper -motion almost certainly known to exist, but which we have no hope of -detecting by observation in the present age of the world. As the -earth’s motion round the sun combined with the motion of light causes -the stars to deviate apparently from their true positions to the extent -of about 18″ at the most, so the motion of the whole planetary system -through space must occasion a similar displacement of at most 5″. The -ordinary aberration can be readily detected with modern astronomical -instruments, because it goes through a yearly change in direction or -amount; but systematic aberration is constant so long as the planetary -system moves uniformly in a sensibly straight line. Only then in the -course of ages, when the curvature of the sun’s path becomes apparent, -can we hope to verify the existence of this kind of aberration. A -curious effect must also be produced by the sun’s proper motion upon -the apparent periods of revolution of the binary stars. - -To my mind, some of the most interesting truths in the whole range of -science are those which have not been, and in many cases probably never -can be, verified by trial. Thus the chemist assigns, with a very high -degree of probability, the vapour densities of such elements as carbon -and silicon, which have never been observed separately in a state of -vapour. The chemist is also familiar with the vapour densities of -elements at temperatures at which the elements in question never have -been, and probably never can be, submitted to experiment in the form of -vapour. - -Joule and others have calculated the actual velocity of the molecules -of a gas, and even the number of collisions which must take place per -second during their constant circulation. Physicists have not yet given -us the exact magnitudes of the particles of matter, but they have -ascertained by several methods the limits within which their magnitudes -must lie. Such scientific results must be for ever beyond the power of -verification by the senses. I have elsewhere had occasion to remark -that waves of light, the intimate processes of electrical changes, -the properties of the ether which is the base of all phenomena, are -necessarily determined in a hypothetical, but not therefore a less -certain manner. - -Though only two of the metals, gold and silver, have ever been -observed to be transparent, we know on the grounds of theory that -they are all more or less so; we can even estimate by theory their -refractive indices, and prove that they are exceedingly high. The -phenomena of elliptic polarisation, and perhaps also those of internal -radiation,[463] depend upon the refractive index, and thus, even when -we cannot observe any refracted rays, we can indirectly learn how they -would be refracted. - - [463] Balfour Stewart, *Elementary Treatise on Heat*, 1st edit. - p. 198. - -In many cases large quantities of electricity must be produced, which -we cannot observe because it is instantly discharged. In the common -electric machine the cylinder and rubber are made of non-conductors, -so that we can separate and accumulate the electricity. But a little -damp, by serving as a conductor, prevents this separation from enduring -any sensible time. Hence there is no doubt that when we rub two good -conductors against each other, for instance two pieces of metals, -much electricity is produced, but instantaneously converted into some -other form of energy. Joule believes that all the heat of friction is -transmuted electricity. - -As regards phenomena of insensible amount, nature is absolutely -full of them. We must regard those changes which we can observe as -the comparatively rare aggregates of minuter changes. On a little -reflection we must allow that no object known to us remains for two -instants of exactly the same temperature. If so, the dimensions of -objects must be in a perpetual state of variation. The minor planetary -and lunar perturbations are infinitely numerous, but usually too -small to be detected by observation, although their amounts may be -assigned by theory. There is every reason to believe that chemical -and electric actions of small amount are constantly in progress. The -hardest substances, if reduced to extremely small particles, and -diffused in pure water, manifest oscillatory movements which must be -due to chemical and electric changes, so slight that they go on for -years without affecting appreciably the weight of the particles.[464] -The earth’s magnetism must more or less affect every object which we -handle. As Tyndall remarks, “An upright iron stone influenced by the -earth’s magnetism becomes a magnet, with its bottom a north and its -top a south pole. Doubtless, though in an immensely feebler degree, -every erect marble statue is a true diamagnet, with its head a north -pole and its feet a south pole. The same is certainly true of man as -he stands upon the earth’s surface, for all the tissues of the human -body are diamagnetic.”[465] The sun’s light produces a very quick and -perceptible effect upon the photographic plate; in all probability it -has a less effect upon a great variety of substances. We may regard -every phenomenon as an exaggerated and conspicuous case of a process -which is, in infinitely numerous cases, beyond the means of observation. - - [464] Jevons, *Proceedings of the Manchester Literary and - Philosophical Society*, 25th January, 1870, vol. ix. p. 78. - - [465] *Philosophical Transactions*, vol. cxlvi. p. 249. - - - - -CHAPTER XXV. - -ACCORDANCE OF QUANTITATIVE THEORIES. - - -In the preceding chapter we found that facts may be classed under -four heads as regards their connection with theory, and our powers -of explanation or prediction. The facts hitherto considered were -generally of a qualitative rather than a quantitative nature; but when -we look exclusively to the quantity of a phenomenon, and the various -modes in which we may determine its amount, nearly the same system of -classification will hold good. There will, however, be five possible -cases:-- - -(1) We may directly and empirically measure a phenomenon, without being -able to explain why it should have any particular quantity, or to -connect it by theory with other quantities. - -(2) In a considerable number of cases we can theoretically predict the -existence of a phenomenon, but are unable to assign its amount, except -by direct measurement, or to explain the amount theoretically when thus -ascertained. - -(3) We may measure a quantity, and afterwards explain it as related to -other quantities, or as governed by known quantitative laws. - -(4) We may predict the quantity of an effect on theoretical grounds, -and afterwards confirm the prediction by direct measurement. - -(5) We may indirectly determine the quantity of an effect without being -able to verity it by experiment. - -These classes of quantitative facts might be illustrated by an immense -number of interesting points in the history of physical science. Only -a few instances of each class can be given here. - - -*Empirical Measurements.* - -Under the first head of purely empirical measurements, which have -not been brought under any theoretical system, may be placed the -great bulk of quantitative facts recorded by scientific observers. -The tables of numerical results which abound in books on chemistry -and physics, the huge quartos containing the observations of public -observatories, the multitudinous tables of meteorological observations, -which are continually being published, the more abstruse results -concerning terrestrial magnetism--such results of measurement, for -the most part, remain empirical, either because theory is defective, -or the labour of calculation and comparison is too formidable. In -the Greenwich Observatory, indeed, the salutary practice has been -maintained by the present Astronomer Royal, of always reducing the -observations, and comparing them with the theories of the several -bodies. The divergences from theory thus afford material for the -discovery of errors or of new phenomena; in short, the observations -have been turned to the use for which they were intended. But it is to -be feared that other establishments are too often engaged in merely -recording numbers of which no real use is made, because the labour -of reduction and comparison with theory is too great for private -inquirers to undertake. In meteorology, especially, great waste of -labour and money is taking place, only a small fraction of the results -recorded being ever used for the advancement of the science. For one -meteorologist like Quetelet, Dove, or Baxendell, who devotes himself to -the truly useful labour of reducing other people’s observations, there -are hundreds who labour under the delusion that they are advancing -science by loading our book-shelves with numerical tables. It is to -be feared, in like manner, that almost the whole bulk of statistical -numbers, whether commercial, vital, or moral, is of little scientific -value. Purely empirical measurements may have a direct practical value, -as when tables of the specific gravity, or strength of materials, -assist the engineer; the specific gravities of mixtures of water with -acids, alcohols, salts, &c., are useful in chemical manufactories, -custom-house gauging, &c.; observations of rainfall are requisite for -questions of water supply; the refractive index of various kinds of -glass must be known in making achromatic lenses; but in all such cases -the use made of the measurements is not scientific but practical. It -may be asserted, that no number which remains isolated, and uncompared -by theory with other numbers, is of scientific value. Having tried -the tensile strength of a piece of iron in a particular condition, we -know what will be the strength of the same kind of iron in a similar -condition, provided we can ever meet with that exact kind of iron -again; but we cannot argue from piece to piece, nor lay down any laws -exactly connecting the strength of iron with the quantity of its -impurities. - - -*Quantities indicated by Theory, but Empirically Measured.* - -In many cases we are able to foresee the existence of a quantitative -effect, on the ground of general principles, but are unable, either -from the want of numerical data, or from the entire absence of any -mathematical theory, to assign the amount of such effect. We then -have recourse to direct experiment to determine its amount. Whether -we argued from the oceanic tides by analogy, or deductively from the -theory of gravitation, there could be no doubt that atmospheric tides -of some amount must occur in the atmosphere. Theory, however, even -in the hands of Laplace, was not able to overcome the complicated -mechanical conditions of the atmosphere, and predict the amounts of -such tides; and, on the other hand, these amounts were so small, and -were so masked by far larger undulations arising from the heating power -of the sun, and from other meteorological disturbances, that they would -probably have never been discovered by purely empirical observations. -Theory having, however, indicated their existence and their periods, it -was easy to make series of barometrical observations in places selected -so as to be as free as possible from casual fluctuations, and then, by -the suitable application of the method of means, to detect the small -effects in question. The principal lunar atmospheric tide was thus -proved to amount to between ·003 and ·004 inch.[466] - - [466] Grant’s *History of Physical Astronomy*, p. 162. - -Theory yields the greatest possible assistance in applying the method -of means. For if we have a great number of empirical measurements, each -representing the joint effect of a number of causes, our object will -be to take the mean of all those in which the effect to be measured is -present, and compare it with the mean of the remainder in which the -effect is absent, or acts in the opposite direction. The difference -will then represent the amount of the effect, or double the amount -respectively. Thus, in the case of the atmospheric tides, we take -the mean of all the observations when the moon was on the meridian, -and compare it with the mean of all observations when she was on the -horizon. In this case we trust to chance that all other effects will -lie about as often in one direction as the other, and will neutralise -themselves in the drawing of each mean. It is a great advantage, -however, to be able to decide by theory when each principal disturbing -effect is present or absent; for the means may then be drawn so as to -separate each such effect, leaving only minor and casual divergences -to the law of error. Thus, if there be three principal effects, and -we draw means giving respectively the sum of all three, the sum of -the first two, and the sum of the last two, then we gain three simple -equations, by the solution of which each quantity is determined. - - -*Explained Results of Measurement.* - -The second class of measured phenomena contains those which, after -being determined in a direct and purely empirical application of -measuring instruments, are afterwards shown to agree with some -hypothetical explanation. Such results are turned to their proper -use, and several advantages may arise from the comparison. The -correspondence with theory will seldom or never be precise; and, even -if it be so, the coincidence must be regarded as accidental. - -If the divergences between theory and experiment be comparatively -small, and variable in amount and direction, they may often be safely -attributed to inconsiderable sources of error in the experimental -processes. The strict method of procedure is to calculate the probable -error of the mean of the observed results (p. 387), and then observe -whether the theoretical result falls within the limits of probable -error. If it does, and if the experimental results agree as well -with theory as they agree with each other, then the probability of -the theory is much increased, and we may employ the theory with more -confidence in the anticipation of further results. The probable error, -it should be remembered, gives a measure only of the effects of -incidental and variable sources of error, but in no degree indicates -the amount of fixed causes of error. Thus, if the mean results of -two modes of determining a quantity are so far apart that the limits -of probable error do not overlap, we may infer the existence of some -overlooked source of fixed error in one or both modes. We will further -consider in a subsequent section the discordance of measurements. - - -*Quantities determined by Theory and verified by Measurement.* - -One of the most satisfactory tests of a theory consists in its -application not only to predict the nature of a phenomenon, and -the circumstances in which it may be observed, but also to assign -the precise quantity of the phenomenon. If we can subsequently -apply accurate instruments and measure the amount of the phenomenon -witnessed, we have an excellent opportunity of verifying or negativing -the theory. It was in this manner that Newton first attempted to verify -his theory of gravitation. He knew approximately the velocity produced -in falling bodies at the earth’s surface, and if the law of the inverse -square of the distance held true, and the reputed distance of the -moon was correct, he could infer that the moon ought to fall towards -the earth at the rate of fifteen feet in one minute. Now, the actual -divergence of the moon from the tangent of its orbit appeared to amount -only to thirteen feet in one minute, and there was a discrepancy of -two feet in fifteen, which caused Newton to lay “aside at that time -any further thoughts of this matter.” Many years afterwards, probably -fifteen or sixteen years, Newton obtained more precise data from which -he could calculate the size of the moon’s orbit, and he then found the -discrepancy to be inconsiderable. - -His theory of gravitation was thus verified as far as the moon was -concerned; but this was to him only the beginning of a long course of -deductive calculations, each ending in a verification. If the earth -and moon attract each other, and also the sun and the earth, there -is reason to expect that the sun and moon should attract each other. -Newton followed out the consequences of this inference, and showed -that the moon would not move as if attracted by the earth only, but -sometimes faster and sometimes slower. Comparison with Flamsteed’s -observations of the moon showed that such was the case. Newton argued -again, that as the waters of the ocean are not rigidly attached to -the earth, they might attract the moon, and be attracted in return, -independently of the rest of the earth. Certain daily motions -resembling the tides would then be caused, and there were the tides to -verify the reasoning. It was the extraordinary power with which Newton -traced out geometrically the consequences of his theory, and submitted -them to repeated comparison with experience, which constitutes his -pre-eminence over all physicists. - - -*Quantities determined by Theory and not verified.* - -It will continually happen that we are able, from certain measured -phenomena and a correct theory, to determine the amount of some other -phenomenon which we may either be unable to measure at all, or to -measure with an accuracy corresponding to that required to verify the -prediction. Thus Laplace having worked out a theory of the motions of -Jupiter’s satellites on the hypothesis of gravitation, found that these -motions were greatly affected by the spheroidal form of Jupiter. The -motions of the satellites can be observed with great accuracy owing to -their frequent eclipses and transits, and from these motions he was -able to argue inversely, and assign the ellipticity of the planet. The -ratio of the polar and equatorial axes thus determined was very nearly -that of 13 to 14; and it agrees well with such direct micrometrical -measurements of the planet as have been made; but Laplace believed -that the theory gave a more accurate result than direct observation -could yield, so that the theory could hardly be said to admit of direct -verification. - -The specific heat of air was believed on the grounds of direct -experiment to amount to 0·2669, the specific heat of water being taken -as unity; but the methods of experiment were open to considerable -causes of error. Rankine showed in 1850 that it was possible -to calculate from the mechanical equivalent of heat and other -thermodynamic data, what this number should be, and he found it to -be 0·2378. This determination was at the time accepted as the most -satisfactory result, although not verified; subsequently in 1853 -Regnault obtained by direct experiment the number 0·2377, proving that -the prediction had been well grounded. - -It is readily seen that in quantitative questions verification is a -matter of degree and probability. A less accurate method of measurement -cannot verify the results of a more accurate method, so that if we -arrive at a determination of the same physical quantity in several -distinct modes it is often a delicate matter to decide which result -is most reliable, and should be used for the indirect determination -of other quantities. For instance, Joule’s and Thomson’s ingenious -experiments upon the thermal phenomena of fluids in motion[467] -involved, as one physical constant, the mechanical equivalent of -heat; if requisite, then, they might have been used to determine that -important constant. But if more direct methods of experiment give -the mechanical equivalent of heat with superior accuracy, then the -experiments on fluids will be turned to a better use in determining -various quantities relating to the theory of fluids. We will further -consider questions of this kind in succeeding sections. - - [467] *Philosophical Transactions* (1854), vol. cxliv. p. 364. - -There are of course many quantities assigned on theoretical grounds -which we are quite unable to verify with corresponding accuracy. The -thickness of a film of gold leaf, the average depths of the oceans, -the velocity of a star’s approach to or regression from the earth as -inferred from spectroscopic data (pp. 296–99), are cases in point; but -many others might be quoted where direct verification seems impossible. -Newton and subsequent physicists have measured light undulations, and -by several methods we learn the velocity with which light travels. -Since an undulation of the middle green is about five ten-millionths -of a metre in length, and travels at the rate of nearly 300,000,000 -of metres per second, it follows that about 600,000,000,000,000 -undulations must strike in one second the retina of an eye which -perceives such light. But how are we to verify such an astounding -calculation by directly counting pulses which recur six hundred -billions of times in a second? - - -*Discordance of Theory and Experiment.* - -When a distinct want of accordance is found to exist between the -results of theory and direct measurement, interesting questions arise -as to the mode in which we can account for this discordance. The -ultimate explanation of the discrepancy may be accomplished in at least -four ways as follows:-- - -(1) The direct measurement may be erroneous owing to various sources of -casual error. - -(2) The theory may be correct as far as regards the general form of the -supposed laws, but some of the constant numbers or other quantitative -data employed in the theoretical calculations may be inaccurate. - -(3) The theory may be false, in the sense that the forms of the -mathematical equations assumed to express the laws of nature are -incorrect. - -(4) The theory and the involved quantities may be approximately -accurate, but some regular unknown cause may have interfered, so that -the divergence may be regarded as a *residual effect* representing -possibly a new and interesting phenomenon. - -No precise rules can be laid down as to the best mode of proceeding to -explain the divergence, and the experimentalist will have to depend -upon his own insight and knowledge; but the following recommendations -may be made. - -If the experimental measurements are not numerous, repeat them and -take a more extensive mean result, the probable accuracy of which, -as regards casual errors, will increase as the square root of the -number of experiments. Supposing that no considerable modification -of the result is thus effected, we may suspect the existence of more -deep-seated sources of error in our method of measurement. The next -resource will be to change the size and form of the apparatus employed, -and to introduce various modifications in the materials employed or the -course of procedure, in the hope (p. 396) that some cause of constant -error may thus be removed. If the inconsistency with theory still -remains unreduced we may attempt to invent some widely different mode -of arriving at the same physical quantity, so that we may be almost -sure that the same cause of error will not affect both the new and old -results. In some cases it is possible to find five or six essentially -different modes of arriving at the same determination. - -Supposing that the discrepancy still exists we may begin to suspect -that our direct measurements are correct, and that the data employed -in the theoretical calculations are inaccurate. We must now review the -grounds on which these data depend, consisting as they must ultimately -do of direct measurements. A comparison of the recorded data will -show the degree of probability attaching to the mean result employed; -and if there is any ground for imagining the existence of error, we -should repeat the observations, and vary the forms of experiment just -as in the case of the previous direct measurements. The continued -existence of the discrepancy must show that we have not attained to -a complete acquaintance with the theory of the causes in action, but -two different cases still remain. We may have misunderstood the action -of those causes which we know to exist, or we may have overlooked the -existence of one or more other causes. In the first case our hypothesis -appears to be wrongly chosen and inapplicable; but whether we are to -reject it will depend upon whether we can form another hypothesis which -yields a more accurate accordance. The probability of an hypothesis, -it will be remembered (p. 243), is to be judged, in the absence of *à -priori* grounds of judgment, by the probability that if the supposed -causes exist the observed result follows; but as there is now little -probability of reconciling the original hypothesis with our direct -measurements the field is open for new hypotheses, and any one which -gives a closer accordance with measurement will so far have better -claims to attention. Of course we must never estimate the probability -of an hypothesis merely by its accordance with a few results only. Its -general analogy and accordance with other known laws of nature, and -the fact that it does not conflict with other probable theories, must -be taken into account, as we shall see in the next book. The requisite -condition of a good hypothesis, that it must admit of the deduction -of facts verified in observation, must be interpreted in the widest -manner, as including all ways in which there may be accordance or -discordance. All our attempts at reconciliation having failed, the only -conclusion we can come to is that some unknown cause of a new character -exists. If the measurements be accurate and the theory probable, -then there remains a *residual phenomenon*, which, being devoid of -theoretical explanation, must be set down as a new empirical fact -worthy of further investigation. Outstanding residual discrepancies -have often been found to involve new discoveries of the greatest -importance. - - -*Accordance of Measurements of Astronomical Distances.* - -One of the most instructive instances which we can meet, of the -manner in which different measurements confirm or check each other, -is furnished by the determination of the velocity of light, and the -dimensions of the planetary system. Roemer first discovered that light -requires time to travel, by observing that the eclipses of Jupiter’s -satellites, although they occur at fixed moments of absolute time, are -visible at different moments in different parts of the earth’s orbit, -according to the distance between the earth and Jupiter. The time -occupied by light in traversing the mean semi-diameter of the earth’s -orbit is found to be about eight minutes. The mean distance of the sun -and earth was long assumed by astronomers as being about 95,274,000 -miles, this result being deduced by Bessel from the observations of the -transit of Venus, which occurred in 1769, and which were found to give -the solar parallax, or which is the same thing, the apparent angular -magnitude of the earth seen from the sun, as equal to 8″·578. Dividing -the mean distance of the sun and earth by the number of seconds in -8^{m}. 13^{s}.3 we find the velocity of light to be about 192,000 miles -per second. - -Nearly the same result was obtained in what seems a different manner. -The aberration of light is the apparent change in the direction of a -ray of light owing to the composition of its motion with that of the -earth’s motion round the sun. If we know the amount of aberration and -the mean velocity of the earth, we can estimate that of light, which -is thus found to be 191,100 miles per second. Now this determination -depends upon a new physical quantity, that of aberration, which is -ascertained by direct observation of the stars, so that the close -accordance of the estimates of the velocity of light as thus arrived -at by different methods might seem to leave little room for doubt, the -difference being less than one per cent. - -Nevertheless, experimentalists were not satisfied until they had -succeeded in measuring the velocity of light by direct experiments -performed upon the earth’s surface. Fizeau, by a rapidly revolving -toothed wheel, estimated the velocity at 195,920 miles per second. -As this result differed by about one part in sixty from estimates -previously accepted, there was thought to be room for further -investigation. The revolving mirror, used by Wheatstone in measuring -the velocity of electricity, was now applied in a more refined manner -by Fizeau and by Foucault to determine the velocity of light. The -latter physicist came to the startling conclusion that the velocity -was not really more than 185,172 miles per second. No repetition of -the experiment would shake this result, and there was accordingly a -discrepancy between the astronomical and the experimental results -of about 7,000 miles per second. The latest experiments, those of -M. Cornu, only slightly raise the estimate, giving 186,660 miles -per second. A little consideration shows that both the astronomical -determinations involve the magnitude of the earth’s orbit as one datum, -because our estimate of the earth’s velocity in its orbit depends upon -our estimate of the sun’s mean distance. Accordingly as regards this -quantity the two astronomical results count only for one. Though the -transit of Venus had been considered to give the best data for the -calculation of the sun’s parallax, yet astronomers had not neglected -less favourable opportunities. Hansen, calculating from certain -inequalities in the moon’s motion, had estimated it at 8″·916; Winneke, -from observations of Mars, at 8″·964; Leverrier, from the motions -of Mars, Venus, and the moon, at 8″·950. These independent results -agree much better with each other than with that of Bessel (8″·578) -previously received, or that of Encke (8″·58) deduced from the transits -of Venus in 1761 and 1769, and though each separately might be worthy -of less credit, yet their close accordance renders their mean result -(8″·943) comparable in probability with that of Bessel. It was further -found that if Foucault’s value for the velocity of light were assumed -to be correct, and the sun’s distance were inversely calculated from -that, the sun’s parallax would be 8″·960, which closely agreed with -the above mean result. This further correspondence of independent -results threw the balance of probability strongly against the results -of the transit of Venus, and rendered it desirable to reconsider -the observations made on that occasion. Mr. E. J. Stone, having -re-discussed those observations,[468] found that grave oversights had -been made in the calculations, which being corrected would alter the -estimate of parallax to 8″·91, a quantity in such comparatively close -accordance with the other results that astronomers did not hesitate -at once to reduce their estimate of the sun’s mean distance from -95,274,000 to 91,771,000, miles, although this alteration involved a -corresponding correction in the assumed magnitudes and distances of -most of the heavenly bodies. The solar parallax is now (1875) believed -to be about 8″·878, the number deduced from Cornu’s experiments on the -velocity of light. This result agrees very closely with 8″·879, the -estimate obtained from new observations on the transit of Venus, by the -French observers, and with 8″·873, the result of Galle’s observations -of the planet Flora. When all the observations of the late transit of -Venus are fully discussed the sun’s distance will probably be known to -less than one part in a thousand, if not one part in ten thousand.[469] - - [468] *Monthly Notices of the Royal Astronomical Society*, - vol. xxviii. p. 264. - - [469] It would seem to be absurd to repeat the profuse expenditure of - 1874 at the approaching transit in 1882. The aggregate sum spent in - 1874 by various governments and individuals can hardly be less than - £200,000, a sum which, wisely expended on scientific investigations, - would give a hundred important results. - -In this question the theoretical relations between the velocity of -light, the constant of aberration, the sun’s parallax, and the sun’s -mean distance, are of the simplest character, and can hardly be -open to any doubt, so that the only doubt was as to which result of -observation was the most reliable. Eventually the chief discrepancy was -found to arise from misapprehension in the reduction of observations, -but we have a satisfactory example of the value of different methods -of estimation in leading to the detection of a serious error. Is it -not surprising that Foucault by measuring the velocity of light when -passing through the space of a few yards, should lead the way to a -change in our estimates of the magnitudes of the whole universe? - - -*Selection of the best Mode of Measurement.* - -When we once obtain command over a question of physical science -by comprehending the theory of the subject, we often have a wide -choice opened to us as regards the methods of measurement, which may -thenceforth be made to give the most accurate results. If we can -measure one fundamental quantity very precisely we may be able by -theory to determine accurately many other quantitative results. Thus, -if we determine satisfactorily the atomic weights of certain elements, -we do not need to determine with equal accuracy the composition and -atomic weights of their several compounds. Having learnt the relative -atomic weights of oxygen and sulphur, we can calculate the composition -by weight of the several oxides of sulphur. Chemists accordingly select -with the greatest care that compound of two elements which seems to -allow of the most accurate analysis, so as to give the ratio of their -atomic weights. It is obvious that we only need the ratio of the atomic -weight of each element to that of some common element, in order to -calculate, that of each to each. Moreover the atomic weight stands -in simple relation to other quantitative facts. The weights of equal -volumes of elementary gases at equal temperature and pressure have -the same ratios as the atomic weights; now, as nitrogen under such -circumstances weighs 14·06 times as much as hydrogen, we may infer that -the atomic weight of nitrogen is about 14·06, or more probably 14·00, -that of hydrogen being unity. There is much evidence, again, that the -specific heats of elements are inversely as their atomic weights, so -that these two classes of quantitative data throw light mutually upon -each other. In fact the atomic weight, the atomic volume, and the -atomic heat of an element, are quantities so closely connected that -the determination of one will lead to that of the others. The chemist -has to solve a complicated problem in deciding in the case of each -of 60 or 70 elements which mode of determination is most accurate. -Modern chemistry presents us with an almost infinitely extensive web of -numerical ratios developed out of a few fundamental ratios. - -In hygrometry we have a choice among at least four modes of measuring -the quantity of aqueous vapour contained in a given bulk of air. We -can extract the vapour by absorption in sulphuric acid, and directly -weigh its amount; we can place the air in a barometer tube and observe -how much the absorption of the vapour alters the elastic force of the -air; we can observe the dew-point of the air, that is the temperature -at which the vapour becomes saturated; or, lastly, we can insert a dry -and wet bulb thermometer and observe the temperature of an evaporating -surface. The results of each mode can be connected by theory with -those of the other modes, and we can select for each experiment that -mode which is most accurate or most convenient. The chemical method -of direct measurement is capable of the greatest accuracy, but is -troublesome; the dry and wet bulb thermometer is sufficiently exact for -meteorological purposes and is most easy to use. - - -*Agreement of Distinct Modes of Measurement.* - -Many illustrations might be given of the accordance which has been -found to exist in some cases between the results of entirely different -methods of arriving at the measurement of a physical quantity. While -such accordance must, in the absence of information to the contrary, -be regarded as the best possible proof of the approximate correctness -of the mean result, yet instances have occurred to show that we can -never take too much trouble in confirming results of great importance. -When three or even more distinct methods have given nearly coincident -numbers, a new method has sometimes disclosed a discrepancy which it is -yet impossible to explain. - -The ellipticity of the earth is known with considerable approach to -certainty and accuracy, for it has been estimated in three independent -ways. The most direct mode is to measure long arcs extending north and -south upon the earth’s surface, by means of trigonometrical surveys, -and then to compare the lengths of these arcs with their curvature as -determined by observations of the altitude of certain stars at the -terminal points. The most probable ellipticity of the earth deduced -from all measurements of this kind was estimated by Bessel at 1/300, -though subsequent measurements might lead to a slightly different -estimate. The divergence from a globular form causes a small variation -in the force of gravity at different parts of the earth’s surface, -so that exact pendulum observations give the data for an independent -estimate of the ellipticity, which is thus found to be 1/320. In the -third place the spheroidal protuberance about the earth’s equator leads -to a certain inequality in the moon’s motion, as shown by Laplace; -and from the amount of that inequality, as given by observations, -Laplace was enabled to calculate back to the amount of its cause. -He thus inferred that the ellipticity is 1/305, which lies between -the two numbers previously given, and was considered by him the most -satisfactory determination. In this case the accordance is undisturbed -by subsequent results, so that we are obliged to accept Laplace’s -result as a highly probable one. - -The mean density of the earth is a constant of high importance, because -it is necessary for the determination of the masses of all the other -heavenly bodies. Astronomers and physicists accordingly have bestowed -a great deal of labour upon the exact estimation of this constant. -The method of procedure consists in comparing the gravitation of the -globe with that of some body of matter of which the mass is known in -terms of the assumed unit of mass. This body of matter, serving as -an intermediate term of comparison, may be variously chosen; it may -consist of a mountain, or a portion of the earth’s crust, or a heavy -ball of metal. The method of experiment varies so much according as -we select one body or the other, that we may be said to have three -independent modes of arriving at the desired result. - -The mutual gravitation of two balls is so exceedingly small compared -with their gravitation towards the immense mass of the earth, that it -is usually quite imperceptible, and although asserted by Newton to -exist, on the ground of theory, was never observed until the end of the -18th century. Michell attached two small balls to the extremities of -a delicately suspended torsion balance, and then bringing heavy balls -of lead alternately to either side of these small balls was able to -detect a slight deflection of the torsion balance. He thus furnished a -new verification of the theory of gravitation. Cavendish carried out -the experiment with more care, and estimated the gravitation of the -balls by treating the torsion balance as a pendulum; then taking into -account the respective distances of the balls from each other and from -the centre of the earth, he was able to assign 5·48 (or as re-computed -by Baily, 5·448) as the probable mean density of the earth. Newton’s -sagacious guess to the effect that the density of the earth was between -five and six times that of water, was thus remarkably confirmed. The -same kind of experiment repeated by Reich gave 5·438. Baily having -again performed the experiment with every possible refinement obtained -a slightly higher number, 5·660. - -A different method of procedure consisted in ascertaining the effect -of a mountain mass in deflecting the plumb-line; for, assuming that -we can determine the dimensions and mean density of the mountain, -the plumb-line enables us to compare its mass with that of the whole -earth. The mountain Schehallien was selected for the experiment, and -observations and calculations performed by Maskelyne, Hutton, and -Playfair, gave as the most probable result 4·713. The difference from -the experimental results already mentioned is considerable and is -important, because the instrumental operations are of an entirely -different character from those of Cavendish and Baily’s experiments. -Sir Henry James’ similar determination from the attraction of Arthur’s -Seat gave 5·14. - -A third distinct method consists in determining the force of gravity -at points elevated above the surface of the earth on mountain ranges, -or sunk below it in mines. Carlini experimented with a pendulum at the -hospice of Mont Cenis, 6,375 feet above the sea, and by comparing the -attractive forces of the earth and the Alps, found the density to be -still smaller, namely, 4·39, or as corrected by Giulio, 4·950. Lastly, -the Astronomer Royal has on two occasions adopted the opposite method -of observing a pendulum at the bottom of a deep mine, so as to compare -the density of the strata penetrated with the density of the whole -earth. On the second occasion he carried his method into effect at the -Harton Colliery, 1,260 feet deep; all that could be done by skill in -measurement and careful consideration of all the causes of error, was -accomplished in this elaborate series of observations[470] (p. 291). -No doubt Sir George Airy was much perplexed when he found that his -new result considerably exceeded that obtained by any other method, -being no less than 6·566, or 6·623 as finally corrected. In this -case we learn an impressive lesson concerning the value of repeated -determinations by distinct methods in disabusing our minds of the -reliance which we are only too apt to place in results which show a -certain degree of coincidence. - - [470] *Philosophical Transactions* (1856), vol. cxlvi. p. 342. - -In 1844 Herschel remarked in his memoir of Francis Baily,[471] -“that the mean specific gravity of this our planet is, in all -human probability, quite as well determined as that of an ordinary -hand-specimen in a mineralogical cabinet,--a marvellous result, which -should teach us to despair of nothing which lies within the compass of -number, weight and measure.” But at the same time he pointed out that -Baily’s final result, of which the probable error was only 0·0032, was -the highest of all determinations then known, and Airy’s investigation -has since given a much higher result, quite beyond the limits of -probable error of any of the previous experiments. If we treat all -determinations yet made as of equal weight, the simple mean is about -5·45, the mean error nearly 0·5, and the probable error almost 0·2, so -that it is as likely as not that the truth lies between 5·65 and 5·25 -on this view of the matter. But it is remarkable that the two most -recent and careful series of observations by Baily and Airy,[472] lie -beyond these limits, and as with the increase of care the estimate -rises, it seems requisite to reject the earlier results, and look upon -the question as still requiring further investigation. Physicists -often take 5-2/3 or 5·67 as the best guess at the truth, but it is -evident that new experiments are much required. I cannot help thinking -that a portion of the great sums of money which many governments and -private individuals spent upon the transit of Venus expeditions in -1874, and which they will probably spend again in 1882 (p. 562), would -be better appropriated to new determinations of the earth’s density. -It seems desirable to repeat Baily’s experiment in a vacuous case, -and with the greater mechanical refinements which the progress of the -last forty years places at the disposal of the experimentalist. It -would be desirable, also, to renew the pendulum experiments of Airy -in some other deep mine. It might even be well to repeat upon some -suitable mountain the observations performed at Schehallien. All these -operations might be carried out for the cost of one of the superfluous -transit expeditions. - - [471] *Monthly Notices of the Royal Astronomical Society*, for 8th - Nov. 1844, No. X. vol. vi. p. 89. - - [472] *Philosophical Magazine*, 2nd Series, vol. xxvi. p. 61. - -Since the establishment of the dynamical theory of heat it has become -a matter of the greatest importance to determine with accuracy the -mechanical equivalent of heat, or the quantity of energy which must -be given, or received, in a definite change of temperature effected -in a definite quantity of a standard substance, such as water. No -less than seven almost entirely distinct modes of determining this -constant have been tried. Dr. Joule first ascertained by the friction -of water that to raise the temperature of one kilogram of water through -one degree centigrade, we must employ energy sufficient to raise -424 kilograms through the height of one metre against the force of -gravity at the earth’s surface. Joule, Mayer, Clausius,[473] Favre -and other experimentalists have made determinations by less direct -methods. Experiments on the mechanical properties of gases give 426 -kilogrammetres as the constant; the work done by a steam-engine -gives 413; from the heat evolved in electrical experiments several -determinations have been obtained; thus from induced electric currents -we get 452; from the electro-magnetic engine 443; from the circuit of a -battery 420; and, from an electric current, the lowest result of all, -namely, 400.[474] - - [473] Clausius in *Philosophical Magazine*, 4th Series, vol. ii. - p. 119. - - [474] Watts’ *Dictionary of Chemistry*, vol. iii. p. 129. - -Considering the diverse and in many cases difficult methods of -observation, these results exhibit satisfactory accordance, and their -mean (423·9) comes very close to the number derived by Dr. Joule from -the apparently most accurate method. The constant generally assumed as -the most probable result is 423·55 kilogrammetres. - - -*Residual Phenomena.* - -Even when the experimental data employed in the verification of a -theory are sufficiently accurate, and the theory itself is sound, there -may exist discrepancies demanding further investigation. Herschel -pointed out the importance of such outstanding quantities, and called -them *residual phenomena*.[475] Now if the observations and the theory -be really correct, such discrepancies must be due to the incompleteness -of our knowledge of the causes in action, and the ultimate explanation -must consist in showing that there is in action, either - - [475] *Preliminary Discourse*, §§ 158, 174. *Outlines of Astronomy*, - 4th edit. § 856. - -(1) Some agent of known nature whose presence was not suspected; - -Or (2) Some new agent of unknown nature. - -In the first case we can hardly be said to make a new discovery, for -our ultimate success consists merely in reconciling the theory with -known facts when our investigation is more comprehensive. But in -the second case we meet with a totally new fact, which may lead us -to realms of new discovery. Take the instance adduced by Herschel. -The theory of Newton and Halley concerning comets was that they -were gravitating bodies revolving round the sun in elliptic orbits, -and the return of Halley’s Comet, in 1758, verified this theory. -But, when accurate observations of Encke’s Comet came to be made, -the verification was not found to be exact. Encke’s Comet returned -each time a little sooner than it ought to do, the period regularly -decreasing from 1212·79 days, between 1786 and 1789, to 1210·44 between -1855 and 1858; and the hypothesis has been started that there is a -resisting medium filling the space through which the comet passes. -This hypothesis is a *deus ex machinâ* for explaining this solitary -phenomenon, and cannot possess much probability unless it can be -shown that other phenomena are deducible from it. Many persons have -identified this medium with that through which light undulations pass, -but I am not aware that there is anything in the undulatory theory of -light to show that the medium would offer resistance to a moving body. -If Professor Balfour Stewart can prove that a rotating disc would -experience resistance in a vacuous receiver, here is an experimental -fact which distinctly supports the hypothesis. But in the mean time -it is open to question whether other known agents, for instance -electricity, may not be brought in, and I have tried to show that if, -as is believed, the tail of a comet is an electrical phenomenon, it -is a necessary result of the conservation of energy that the comet -shall exhibit a loss of energy manifested in a diminution of its mean -distance from the sun and its period of revolution.[476] It should -be added that if Professor Tait’s theory be correct, as seems very -probable, and comets consist of swarms of small meteors, there is no -difficulty in accounting for the retardation. It has long been known -that a collection of small bodies travelling together in an orbit round -a central body will tend to fall towards it. In either case, then, this -residual phenomenon seems likely to be reconciled with known laws of -nature. - - [476] *Proceedings of the Manchester Literary and Philosophical - Society*, 28th November, 1871, vol. xi. p. 33. Since the above - remarks were written, Professor Balfour Stewart has pointed out to - me his paper in the *Proceedings of the Manchester Literary and - Philosophical Society* for 15th November, 1870 (vol. x. p. 32), - in which he shows that a body moving in an enclosure of uniform - temperature would probably experience resistance independently - of the presence of a ponderable medium, such as gas, between the - moving body and the enclosure. The proof is founded on the theory - of the dissipation of energy, and this view is said to be accepted - by Professors Thomson and Tait. The enclosure is used in this case - by Professor Stewart simply as a means of obtaining a proof, just - as it was used by him on a previous occasion to obtain a proof of - certain consequences of the Theory of Exchanges. He is of opinion - that in both of these cases when once the proof has been obtained, - the enclosure may be dispensed with. We know, for instance, that - the relation between the inductive and absorptive powers of - bodies--although this relation may have been proved by means of an - enclosure, does not depend upon its presence, and Professor Stewart - thinks that in like manner two bodies, or at least two bodies - possessing heat such as the sun and the earth in motion relative to - each other, will have the differential motion retarded until perhaps - it is ultimately destroyed. - -In other cases residual phenomena have involved important inferences -not recognised at the time. Newton showed how the velocity of sound in -the atmosphere could be calculated by a theory of pulses or undulations -from the observed tension and density of the air. He inferred that -the velocity in the ordinary state of the atmosphere at the earth’s -surface would be 968 feet per second, and rude experiments made by -him in the cloisters of Trinity College seemed to show that this was -not far from the truth. Subsequently it was ascertained by other -experimentalists that the velocity of sound was more nearly 1,142 feet, -and the discrepancy being one-sixth part of the whole was far too much -to attribute to casual errors in the numerical data. Newton attempted -to explain away this discrepancy by hypotheses as to the reactions of -the molecules of air, but without success. - -New investigations having been made from time to time concerning the -velocity of sound, both as observed experimentally and as calculated -from theory, it was found that each of Newton’s results was inaccurate, -the theoretical velocity being 916 feet per second, and the real -velocity about 1,090 feet. The discrepancy, nevertheless, remained -as serious as ever, and it was not until the year 1816 that Laplace -showed it to be due to the heat developed by the sudden compression -of the air in the passage of the wave, this heat having the effect of -increasing the elasticity of the air and accelerating the impulse. It -is now perceived that this discrepancy really involves the doctrine -of the equivalence of heat and energy, and it was applied by Mayer, -at least by implication, to give an estimate of the mechanical -equivalent of heat. The estimate thus derived agrees satisfactorily -with direct determinations by Dr. Joule and other physicists, so that -the explanation of the residual phenomenon which exercised Newton’s -ingenuity is now complete, and forms an important part of the new -science of thermodynamics. - -As Herschel observed, almost all great astronomical discoveries -have been disclosed in the form of residual differences. It is the -practice at well-conducted observatories to compare the positions of -the heavenly bodies as actually observed with what might have been -expected theoretically. This practice was introduced by Halley when -Astronomer Royal, and his reduction of the lunar observations gave a -series of residual errors from 1722 to 1739, by the examination of -which the lunar theory was improved. Most of the greater astronomical -variations arising from nutation, aberration, planetary perturbation -were discovered in the same manner. The precession of the equinox was -perhaps the earliest residual difference observed; the systematic -divergence of Uranus from its calculated places was one of the latest, -and was the clue to the remarkable discovery of Neptune. We may also -class under residual phenomena all the so-called *proper motions* of -the stars. A complete star catalogue, such as that of the British -Association, gives a greater or less amount of proper motion for almost -every star, consisting in the apparent difference of position of the -star as derived from the earliest and latest good observations. But -these apparent motions are often due, as explained by Baily,[477] the -author of the catalogue, to errors of observation and reduction. In -many cases the best astronomical authorities have differed as to the -very direction of the supposed proper motion of stars, and as regards -the amount of the motion, for instance of α Polaris, the most different -estimates have been formed. Residual quantities will often be so small -that their very existence is doubtful. Only the gradual progress of -theory and of measurement will show clearly whether a discrepancy -is to be referred to casual errors of observation or to some new -phenomenon. But nothing is more requisite for the progress of science -than the careful recording and investigation of such discrepancies. -In no part of physical science can we be free from exceptions and -outstanding facts, of which our present knowledge can give no account. -It is among such anomalies that we must look for the clues to new -realms of facts worthy of discovery. They are like the floating waifs -which led Columbus to suspect the existence of the new world. - - [477] *British Association Catalogue of Stars*, p. 49. - - - - -CHAPTER XXVI. - -CHARACTER OF THE EXPERIMENTALIST. - - -In the present age there seems to be a tendency to believe that the -importance of individual genius is less than it was-- - - “The individual withers, and the world is more and more.” - -Society, it is supposed, has now assumed so highly developed a form, -that what was accomplished in past times by the solitary exertions of a -great intellect, may now be worked out by the united labours of an army -of investigators. Just as the well-organised power of a modern army -supersedes the single-handed bravery of the mediæval knights, so we are -to believe that the combination of intellectual labour has superseded -the genius of an Archimedes, a Newton, or a Laplace. So-called original -research is now regarded as a profession, adopted by hundreds of -men, and communicated by a system of training. All that we need to -secure additions to our knowledge of nature is the erection of great -laboratories, museums, and observatories, and the offering of pecuniary -rewards to those who can invent new chemical compounds, detect new -species, or discover new comets. Doubtless this is not the real meaning -of the eminent men who are now urging upon Government the endowment of -physical research. They can only mean that the greater the pecuniary -and material assistance given to men of science, the greater the result -which the available genius of the country may be expected to produce. -Money and opportunities of study can no more produce genius than -sunshine and moisture can generate living beings; the inexplicable -germ is wanting in both cases. But as, when the germ is present, the -plant will grow more or less vigorously according to the circumstances -in which it is placed, so it may be allowed that pecuniary assistance -may favour development of intellect. Public opinion however is not -discriminating, and is likely to interpret the agitation for the -endowment of science as meaning that science can be had for money. - -All such notions are erroneous. In no branch of human affairs, -neither in politics, war, literature, industry, nor science, is the -influence of genius less considerable than it was. It is possible -that the extension and organisation of scientific study, assisted by -the printing-press and the accelerated means of communication, has -increased the rapidity with which new discoveries are made known, and -their details worked out by many heads and hands. A Darwin now no -sooner propounds original ideas concerning the evolution of living -creatures, than those ideas are discussed and illustrated, and -applied by naturalists in every part of the world. In former days his -discoveries would have been hidden for decades of years in scarce -manuscripts, and generations would have passed away before his theory -had enjoyed the same amount of criticism and corroboration as it has -already received. The result is that the genius of Darwin is more -valuable, not less valuable, than it would formerly have been. The -advance of military science and the organisation of enormous armies -has not decreased the value of a skilful general; on the contrary, -the rank and file are still more in need than they used to be of the -guiding power of a far-seeing intellect. The swift destruction of the -French military power was not due alone to the perfection of the German -army, nor to the genius of Moltke; it was due to the combination of a -well-disciplined multitude with a leader of the highest powers. So in -every branch of human affairs the influence of the individual is not -withering, but is growing with the extent of the material resources -which are at his command. - -Turning to our own subject, it is a work of undiminished interest to -reflect upon those qualities of mind which lead to great advances in -natural knowledge. Nothing, indeed, is less amenable than genius to -scientific analysis and explanation. Even definition is out of the -question. Buffon said that “genius is patience,” and certainly patience -is one of its most requisite components. But no one can suppose that -patient labour alone will invariably lead to those conspicuous results -which we attribute to genius. In every branch of science, literature, -art, or industry, there are thousands of men and women who work with -unceasing patience, and thereby ensure moderate success; but it would -be absurd to suppose that equal amounts of intellectual labour yield -equal results. A Newton may modestly attribute his discoveries to -industry and patient thought, and there is reason to believe that -genius is unconscious and unable to account for its own peculiar -powers. As genius is essentially creative, and consists in divergence -from the ordinary grooves of thought and action, it must necessarily be -a phenomenon beyond the domain of the laws of nature. Nevertheless, it -is always an interesting and instructive work to trace out, as far as -possible, the characteristics of mind by which great discoveries have -been achieved, and we shall find in the analysis much to illustrate the -principles of scientific method. - - -*Error of the Baconian Method.* - -Hundreds of investigators may be constantly engaged in experimental -inquiry; they may compile numberless note-books full of scientific -facts, and endless tables of numerical results; but, if the views of -induction here maintained be true, they can never by such work alone -rise to new and great discoveries. By a system of research they may -work out deductively the details of a previous discovery, but to arrive -at a new principle of nature is another matter. Francis Bacon spread -abroad the notion that to advance science we must begin by accumulating -facts, and then draw from them, by a process of digestion, successive -laws of higher and higher generality. In protesting against the false -method of the scholastic logicians, he exaggerated a partially true -philosophy, until it became as false as that which preceded it. His -notion of scientific method was a kind of scientific bookkeeping. -Facts were to be indiscriminately gathered from every source, and -posted in a ledger, from which would emerge in time a balance of -truth. It is difficult to imagine a less likely way of arriving at -great discoveries. The greater the array of facts, the less is the -probability that they will by any routine system of classification -disclose the laws of nature they embody. Exhaustive classification in -all possible orders is out of the question, because the possible orders -are practically infinite in number. - -It is before the glance of the philosophic mind that facts must display -their meaning, and fall into logical order. The natural philosopher -must therefore have, in the first place, a mind of impressionable -character, which is affected by the slightest exceptional phenomenon. -His associating and identifying powers must be great, that is, a -strange fact must suggest to his mind whatever of like nature has -previously come within his experience. His imagination must be active, -and bring before his mind multitudes of relations in which the -unexplained facts may possibly stand with regard to each other, or to -more common facts. Sure and vigorous powers of deductive reasoning -must then come into play, and enable him to infer what will happen -under each supposed condition. Lastly, and above all, there must be the -love of certainty leading him diligently and with perfect candour, to -compare his speculations with the test of fact and experiment. - - -*Freedom of Theorising.* - -It would be an error to suppose that the great discoverer seizes at -once upon the truth, or has any unerring method of divining it. In all -probability the errors of the great mind exceed in number those of the -less vigorous one. Fertility of imagination and abundance of guesses at -truth are among the first requisites of discovery; but the erroneous -guesses must be many times as numerous as those which prove well -founded. The weakest analogies, the most whimsical notions, the most -apparently absurd theories, may pass through the teeming brain, and no -record remain of more than the hundredth part. There is nothing really -absurd except that which proves contrary to logic and experience. The -truest theories involve suppositions which are inconceivable, and no -limit can really be placed to the freedom of hypothesis. - -Kepler is an extraordinary instance to this effect. No minor laws -of nature are more firmly established than those which he detected -concerning the orbits and motions of planetary masses, and on these -empirical laws the theory of gravitation was founded. Did we not learn -from his own writings the multitude of errors into which he fell, we -might have imagined that he had some special faculty of seizing on the -truth. But, as is well known, he was full of chimerical notions; his -favourite and long-studied theory was founded on a fanciful analogy -between the planetary orbits and the regular solids. His celebrated -laws were the outcome of a lifetime of speculation, for the most part -vain and groundless. We know this because he had a curious pleasure -in dwelling upon erroneous and futile trains of reasoning, which -most persons consign to oblivion. But Kepler’s name was destined to -be immortal, on account of the patience with which he submitted his -hypotheses to comparison with observation, the candour with which he -acknowledged failure after failure, and the perseverance and ingenuity -with which he renewed his attack upon the riddles of nature. - -Next after Kepler perhaps Faraday is the physical philosopher who has -given us the best insight into the progress of discovery, by recording -erroneous as well as successful speculations. The recorded notions, -indeed, are probably but a tithe of the fancies which arose in his -active brain. As Faraday himself said--“The world little knows how -many of the thoughts and theories which have passed through the mind -of a scientific investigator, have been crushed in silence and secrecy -by his own severe criticism and adverse examination; that in the most -successful instances not a tenth of the suggestions, the hopes, the -wishes, the preliminary conclusions have been realised.” - -Nevertheless, in Faraday’s researches, published in the *Philosophical -Transactions*, in minor papers, in manuscript note-books, or in other -materials, made known in his interesting life by Dr. Bence Jones, we -find invaluable lessons for the experimentalist. These writings are -full of speculations which we must not judge by the light of subsequent -discovery. It may perhaps be said that Faraday committed to the -printing press crude ideas which a friend would have counselled him to -keep back. There was occasionally even a wildness and vagueness in his -notions, which in a less careful experimentalist would have been fatal -to the attainment of truth. This is especially apparent in a curious -paper concerning Ray-vibrations; but fortunately Faraday was aware of -the shadowy character of his speculations, and expressed the feeling in -words which must be quoted. “I think it likely,” he says,[478] “that -I have made many mistakes in the preceding pages, for even to myself -my ideas on this point appear only as the shadow of a speculation, or -as one of those impressions upon the mind, which are allowable for a -time as guides to thought and research. He who labours in experimental -inquiries knows how numerous these are, and how often their apparent -fitness and beauty vanish before the progress and development of real -natural truth.” If, then, the experimentalist has no royal road to the -discovery of the truth, it is an interesting matter to consider by what -logical procedure he attains the truth. - - [478] *Experimental Researches in Chemistry and Physics*, p. 372. - *Philosophical Magazine*, 3rd Series, May 1846, vol. xxviii. p. 350. - -If I have taken a correct view of logical method, there is really no -such thing as a distinct process of induction. The probability is -infinitely small that a collection of complicated facts will fall -into an arrangement capable of exhibiting directly the laws obeyed -by them. The mathematician might as well expect to integrate his -functions by a ballot-box, as the experimentalist to draw deep truths -from haphazard trials. All induction is but the inverse application -of deduction, and it is by the inexplicable action of a gifted mind -that a multitude of heterogeneous facts are ranged in luminous order -as the results of some uniformly acting law. So different, indeed, -are the qualities of mind required in different branches of science, -that it would be absurd to attempt to give an exhaustive description -of the character of mind which leads to discovery. The labours of -Newton could not have been accomplished except by a mind of the utmost -mathematical genius; Faraday, on the other hand, has made the most -extensive additions to human knowledge without passing beyond common -arithmetic. I do not remember meeting in Faraday’s writings with a -single algebraic formula or mathematical problem of any complexity. -Professor Clerk Maxwell, indeed, in the preface to his new *Treatise -on Electricity*, has strongly recommended the reading of Faraday’s -researches by all students of science, and has given his opinion -that though Faraday seldom or never employed mathematical formulæ, -his methods and conceptions were not the less mathematical in their -nature.[479] I have myself protested against the prevailing confusion -between a mathematical and an exact science,[480] yet I certainly think -that Faraday’s experiments were for the most part qualitative, and that -his mathematical ideas were of a rudimentary character. It is true that -he could not possibly investigate such a subject as magne-crystallic -action without involving himself in geometrical relations of some -complexity. Nevertheless I think that he was deficient in mathematical -deductive power, that power which is so highly developed by the modern -system of mathematical training at Cambridge. - - [479] See also *Nature*, September 18, 1873; vol. viii. p. 398. - - [480] *Theory of Political Economy*, pp. 3–14. - -Faraday was acquainted with the forms of his celebrated lines of force, -but I am not aware that he ever entered into the algebraic nature -of those curves, and I feel sure that he could not have explained -their forms as depending on the resultant attractions of all the -magnetic particles. There are even occasional indications that he did -not understand some of the simpler mathematical doctrines of modern -physical science. Although he so clearly foresaw the correlation of the -physical forces, and laboured so hard with his own hands to connect -gravity with other forces, it is doubtful whether he understood the -doctrine of the conservation of energy as applied to gravitation. -Faraday was probably equal to Newton in experimental skill, and in that -peculiar kind of deductive power which leads to the invention of simple -qualitative experiments; but it must be allowed that he exhibited -little of that mathematical power which enabled Newton to follow out -intuitively the quantitative results of a complicated problem with such -wonderful facility. Two instances, Newton and Faraday, are sufficient -to show that minds of widely different conformation will meet with -suitable regions of research. Nevertheless, there are certain traits -which we may discover in all the highest scientific minds. - - -*The Newtonian Method, the True Organum.* - -Laplace was of opinion that the *Principia* and the *Opticks* of -Newton furnished the best models then available of the delicate art -of experimental and theoretical investigation. In these, as he says, -we meet with the most happy illustrations of the way in which, from -a series of inductions, we may rise to the causes of phenomena, and -thence descend again to all the resulting details. - -The popular notion concerning Newton’s discoveries is that in early -life, when driven into the country by the Great Plague, a falling apple -accidentally suggested to him the existence of gravitation, and that, -availing himself of this hint, he was led to the discovery of the law -of gravitation, the explanation of which constitutes the *Principia*. -It is difficult to imagine a more ludicrous and inadequate picture of -Newton’s labours. No originality, or at least priority, was claimed -by Newton as regards the discovery of the law of the inverse square, -so closely associated with his name. In a well-known Scholium[481] -he acknowledges that Sir Christopher Wren, Hooke, and Halley, had -severally observed the accordance of Kepler’s third law of motion with -the principle of the inverse square. - - [481] *Principia*, bk. i. Prop. iv. - -Newton’s work was really that of developing the methods of deductive -reasoning and experimental verification, by which alone great -hypotheses can be brought to the touchstone of fact. Archimedes was the -greatest of ancient philosophers, for he showed how mathematical theory -could be wedded to physical experiments; and his works are the first -true Organum. Newton is the modern Archimedes, and the *Principia* -forms the true Novum Organum of scientific method. The laws which he -established are great, but his example of the manner of establishing -them is greater still. Excepting perhaps chemistry and electricity, -there is hardly a progressive branch of physical and mathematical -science, which has not been developed from the germs of true scientific -procedure which he disclosed in the *Principia* or the *Opticks*. -Overcome by the success of his theory of universal gravitation, we -are apt to forget that in his theory of sound he originated the -mathematical investigation of waves and the mutual action of particles; -that in his corpuscular theory of light, however mistaken, he first -ventured to apply mathematical calculation to molecular attractions -and repulsions; that in his prismatic experiments he showed how far -experimental verification could be pushed; that in his examination -of the coloured rings named after him, he accomplished the most -remarkable instance of minute measurement yet known, a mere practical -application of which by Fizeau was recently deemed worthy of a medal -by the Royal Society. We only learn by degrees how complete was his -scientific insight; a few words in his third law of motion display his -acquaintance with the fundamental principles of modern thermodynamics -and the conservation of energy, while manuscripts long overlooked prove -that in his inquiries concerning atmospheric refraction he had overcome -the main difficulties of applying theory to one of the most complex of -physical problems. - -After all, it is only by examining the way in which he effected -discoveries, that we can rightly appreciate his greatness. The -*Principia* treats not of gravity so much as of forces in general, and -the methods of reasoning about them. He investigates not one hypothesis -only, but mechanical hypotheses in general. Nothing so much strikes -the reader of the work as the exhaustiveness of his treatment, and the -unbounded power of his insight. If he treats of central forces, it -is not one law of force which he discusses, but many, or almost all -imaginable laws, the results of each of which he sketches out in a few -pregnant words. If his subject is a resisting medium, it is not air or -water alone, but resisting media in general. We have a good example -of his method in the scholium to the twenty-second proposition of the -second book, in which he runs rapidly over many suppositions as to -the laws of the compressing forces which might conceivably act in an -atmosphere of gas, a consequence being drawn from each case, and that -one hypothesis ultimately selected which yields results agreeing with -experiments upon the pressure and density of the terrestrial atmosphere. - -Newton said that he did not frame hypotheses, but, in reality, the -greater part of the *Principia* is purely hypothetical, endless -varieties of causes and laws being imagined which have no counterpart -in nature. The most grotesque hypotheses of Kepler or Descartes were -not more imaginary. But Newton’s comprehension of logical method was -perfect; no hypothesis was entertained unless it was definite in -conditions, and admitted of unquestionable deductive reasoning; and the -value of each hypothesis was entirely decided by the comparison of its -consequences with facts. I do not entertain a doubt that the general -course of his procedure is identical with that view of the nature of -induction, as the inverse application of deduction, which I advocate -throughout this book. Francis Bacon held that science should be founded -on experience, but he mistook the true mode of using experience, -and, in attempting to apply his method, ludicrously failed. Newton -did not less found his method on experience, but he seized the true -method of treating it, and applied it with a power and success never -since equalled. It is a great mistake to say that modern science is -the result of the Baconian philosophy; it is the Newtonian philosophy -and the Newtonian method which have led to all the great triumphs of -physical science, and I repeat that the *Principia* forms the true -“Novum Organum.” - -In bringing his theories to a decisive experimental verification, -Newton showed, as a general rule, exquisite skill and ingenuity. -In his hands a few simple pieces of apparatus were made to give -results involving an unsuspected depth of meaning. His most beautiful -experimental inquiry was that by which he proved the differing -refrangibility of rays of light. To suppose that he originally -discovered the power of a prism to break up a beam of white light -would be a mistake, for he speaks of procuring a glass prism to -try the “celebrated phenomena of colours.” But we certainly owe to -him the theory that white light is a mixture of rays differing in -refrangibility, and that lights which differ in colour, differ also in -refrangibility. Other persons might have conceived this theory; in -fact, any person regarding refraction as a quantitative effect must see -that different parts of the spectrum have suffered different amounts -of refraction. But the power of Newton is shown in the tenacity with -which he followed his theory into every consequence, and tested each -result by a simple but conclusive experiment. He first shows that -different coloured spots are displaced by different amounts when viewed -through a prism, and that their images come to a focus at different -distances from the lense, as they should do, if the refrangibility -differed. After excluding by many experiments a variety of indifferent -circumstances, he fixes his attention upon the question whether the -rays are merely shattered, disturbed, and spread out in a chance -manner, as Grimaldi supposed, or whether there is a constant relation -between the colour and the refrangibility. - -If Grimaldi was right, it might be expected that a part of the spectrum -taken separately, and subjected to a second refraction, would suffer a -new breaking up, and produce some new spectrum. Newton inferred from -his own theory that a particular ray of the spectrum would have a -constant refrangibility, so that a second prism would merely bend it -more or less, but not further disperse it in any considerable degree. -By simply cutting off most of the rays of the spectrum by a screen, -and allowing the remaining narrow ray to fall on a second prism, he -proved the truth of this conclusion; and then slowly turning the first -prism, so as to vary the colour of the ray falling on the second -one, he found that the spot of light formed by the twice-refracted -ray travelled up and down, a palpable proof that the amount of -refrangibility varies with the colour. For his further satisfaction, -he sometimes refracted the light a third or fourth time, and he found -that it might be refracted upwards or downwards or sideways, and yet -for each colour there was a definite amount of refraction through each -prism. He completed the proof by showing that the separated rays may -again be gathered together into white light by an inverted prism, so -that no number of refractions alters the character of the light. The -conclusion thus obtained serves to explain the confusion arising in the -use of a common lense; he shows that with homogeneous light there is -one distinct focus, with mixed light an infinite number of foci, which -prevent a clear view from being obtained at any point. - -What astonishes the reader of the *Opticks* is the persistence with -which Newton follows out the consequences of a preconceived theory, and -tests the one notion by a wonderful variety of simple comparisons with -fact. The ease with which he invents new combinations, and foresees -the results, subsequently verified, produces an insuperable conviction -in the reader that he has possession of the truth. And it is certainly -the theory which leads him to the experiments, most of which could -hardly be devised by accident. Newton actually remarks that it was by -mathematically determining all kinds of phenomena of colours which -could be produced by refraction that he had “invented” almost all the -experiments in the book, and he promises that others who shall “argue -truly,” and try the experiments with care, will not be disappointed in -the results.[482] - - [482] *Opticks*, bk. i. part ii. Prop. 3. 3rd ed. p. 115. - -The philosophic method of Huyghens was the same as that of Newton, -and Huyghens’ investigation of double refraction furnishes almost -equally beautiful instances of theory guiding experiment. So far as -we know double refraction was first discovered by accident, and was -described by Erasmus Bartholinus in 1669. The phenomenon then appeared -to be entirely exceptional, and the laws governing the two paths of -the refracted rays were so unapparent and complicated, that Newton -altogether misunderstood the phenomenon, and it was only at the latter -end of the last century that scientific men began to comprehend its -laws. - -Nevertheless, Huyghens had, with rare genius, arrived at the true -theory as early as 1678. He regarded light as an undulatory motion of -some medium, and in his *Traité de la Lumière* he pointed out that, -in ordinary refraction, the velocity of propagation of the wave is -equal in all directions, so that the front of an advancing wave is -spherical, and reaches equal distances in equal times. But in crystals, -as he supposed, the medium would be of unequal elasticity in different -directions, so that a disturbance would reach unequal distances in -equal times, and the wave produced would have a spheroidal form. -Huyghens was not satisfied with an unverified theory. He calculated -what might be expected to happen when a crystal of calc-spar was cut -in various directions, and he says: “I have examined in detail the -properties of the extraordinary refraction of this crystal, to see if -each phenomenon which is deduced from theory would agree with what is -really observed. And this being so, it is no slight proof of the truth -of our suppositions and principles; but what I am going to add here -confirms them still more wonderfully; that is, the different modes -of cutting this crystal, in which the surfaces produced give rise to -refraction exactly such as they ought to be, and as I had foreseen -them, according to the preceding theory.” - -Newton’s mistaken corpuscular theory of light caused the theories and -experiments of Huyghens to be disregarded for more than a century; but -it is not easy to imagine a more beautiful or successful application of -the true method of inductive investigation, theory guiding experiment, -and yet wholly relying on experiment for confirmation. - - -*Candour and Courage of the Philosophic Mind.* - -Perfect readiness to reject a theory inconsistent with fact is a -primary requisite of the philosophic mind. But it would be a mistake -to suppose that this candour has anything akin to fickleness; on the -contrary, readiness to reject a false theory may be combined with a -peculiar pertinacity and courage in maintaining an hypothesis as long -as its falsity is not actually apparent. There must, indeed, be no -prejudice or bias distorting the mind, and causing it to pass over the -unwelcome results of experiment. There must be that scrupulous honesty -and flexibility of mind, which assigns adequate value to all evidence; -indeed, the more a man loves his theory, the more scrupulous should be -his attention to its faults. It is common in life to meet with some -theorist, who, by long cogitation over a single theory, has allowed it -to mould his mind, and render him incapable of receiving anything but -as a contribution to the truth of his one theory. A narrow and intense -course of thought may sometimes lead to great results, but the adoption -of a wrong theory at the outset is in such a mind irretrievable. -The man of one idea has but a single chance of truth. The fertile -discoverer, on the contrary, chooses between many theories, and is -never wedded to any one, unless impartial and repeated comparison has -convinced him of its validity. He does not choose and then compare; but -he compares time after time, and then chooses. - -Having once deliberately chosen, the philosopher may rightly entertain -his theory with the strongest fidelity. He will neglect no objection; -for he may chance at any time to meet a fatal one; but he will bear -in mind the inconsiderable powers of the human mind compared with the -tasks it has to undertake. He will see that no theory can at first be -reconciled with all objections, because there may be many interfering -causes, and the very consequences of the theory may have a complexity -which prolonged investigation by successive generations of men may not -exhaust. If, then, a theory exhibit a number of striking coincidences -with fact, it must not be thrown aside until at least one *conclusive -discordance* is proved, regard being had to possible error in -establishing that discordance. In science and philosophy something must -be risked. He who quails at the least difficulty will never establish a -new truth, and it was not unphilosophic in Leslie to remark concerning -his own inquiries into the nature of heat-- - -“In the course of investigation, I have found myself compelled to -relinquish some preconceived notions; but I have not abandoned them -hastily, nor, till after a warm and obstinate defence, I was driven -from every post.”[483] - - [483] *Experimental Inquiry into the Nature of Heat.* Preface, p. xv. - -Faraday’s life, again, furnishes most interesting illustrations of this -tenacity of the philosophic mind. Though so candid in rejecting some -theories, there were others to which he clung through everything. One -of his favourite notions resulted in a brilliant discovery; another -remains in doubt to the present day. - - -*The Philosophic Character of Faraday.* - -In Faraday’s researches concerning the connection of magnetism and -light, we find an excellent instance of the pertinacity with which a -favourite theory may be pursued, so long as the results of experiment -do not clearly negative the notions entertained. In purely quantitative -questions, as we have seen, the absence of apparent effect can seldom -be regarded as proving the absence of all effect. Now Faraday was -convinced that some mutual relation must exist between magnetism and -light. As early as 1822, he attempted to produce an effect upon a ray -of polarised light, by passing it through water placed between the -poles of a voltaic battery; but he was obliged to record that not the -slightest effect was observable. During many years the subject, we are -told,[484] rose again and again to his mind, and no failure could make -him relinquish his search after this unknown relation. It was in the -year 1845 that he gained the first success; on August 30th he began -to work with common electricity, vainly trying glass, quartz, Iceland -spar, &c. Several days of labour gave no result; yet he did not desist. -Heavy glass, a transparent medium of great refractive powers, composed -of borate of lead, was now tried, being placed between the poles of a -powerful electro-magnet while a ray of polarised light was transmitted -through it. When the poles of the electro-magnet were arranged in -certain positions with regard to the substance under trial, no effects -were apparent; but at last Faraday happened fortunately to place a -piece of heavy glass so that contrary magnetic poles were on the same -side, and now an effect was witnessed. The glass was found to have the -power of twisting the plane of polarisation of the ray of light. - - [484] Bence Jones, *Life of Faraday*, vol. i. p. 362. - -All Faraday’s recorded thoughts upon this great experiment are replete -with curious interest. He attributes his success to the opinion, -almost amounting to a conviction, that the various forms, under which -the forces of matter are made manifest, have one common origin, -and are so directly related and mutually dependent that they are -convertible. “This strong persuasion,” he says,[485] “extended to the -powers of light, and led to many exertions having for their object -the discovery of the direct relation of light and electricity. These -ineffectual exertions could not remove my strong persuasion, and I -have at last succeeded.” He describes the phenomenon in somewhat -figurative language as *the magnetisation of a ray of light*, and -also as *the illumination of a magnetic curve or line of force*. He -has no sooner got the effect in one case, than he proceeds, with his -characteristic comprehensiveness of research, to test the existence -of a like phenomenon in all the substances available. He finds that -not only heavy glass, but solids and liquids, acids and alkalis, oils, -water, alcohol, ether, all possess this power; but he was not able to -detect its existence in any gaseous substance. His thoughts cannot be -restrained from running into curious speculations as to the possible -results of the power in certain cases. “What effect,” he says, “does -this force have in the earth where the magnetic curves of the earth -traverse its substance? Also what effect in a magnet?” And then he -falls upon the strange notion that perhaps this force tends to make -iron and oxide of iron transparent, a phenomenon never observed. We can -meet with nothing more instructive as to the course of mind by which -great discoveries are made, than these records of Faraday’s patient -labours, and his varied success and failure. Nor are his unsuccessful -experiments upon the relation of gravity and electricity less -interesting, or less worthy of study. - - [485] Ibid. vol. ii. p. 199. - -Throughout a large part of his life, Faraday was possessed by the idea -that gravity cannot be unconnected with the other forces of nature. On -March 19th, 1849, he wrote in his laboratory book,--“Gravity. Surely -this force must be capable of an experimental relation to electricity, -magnetism, and the other forces, so as to bind it up with them in -reciprocal action and equivalent effect?”[486] He filled twenty -paragraphs or more with reflections and suggestions, as to the mode -of treating the subject by experiment. He anticipated that the mutual -approach of two bodies would develop electricity in them, or that a -body falling through a conducting helix would excite a current changing -in direction as the motion was reversed. “*All this is a dream*,” -he remarks; “still examine it by a few experiments. Nothing is too -wonderful to be true, if it be consistent with the laws of nature; -and in such things as these, experiment is the best test of such -consistency.” - - [486] See also his more formal statement in the *Experimental - Researches in Electricity*, 24th Series, § 2702, vol. iii. p. 161. - -He executed many difficult and tedious experiments, which are described -in the 24th Series of Experimental Researches. The result was *nil*, -and yet he concludes: “Here end my trials for the present. The results -are negative; they do not shake my strong feeling of the existence of -a relation between gravity and electricity, though they give no proof -that such a relation exists.” - -He returned to the work when he was ten years older, and in 1858–9 -recorded many remarkable reflections and experiments. He was much -struck by the fact that electricity is essentially a *dual force*, -and it had always been a conviction of Faraday that no body could be -electrified positively without some other body becoming electrified -negatively; some of his researches had been simple developments of -this relation. But observing that between two mutually gravitating -bodies there was no apparent circumstance to determine which should be -positive and which negative, he does not hesitate to call in question -an old opinion. “The evolution of *one* electricity would be a new -and very remarkable thing. The idea throws a doubt on the whole; but -still try, for who knows what is possible in dealing with gravity?” -We cannot but notice the candour with which he thus acknowledges in -his laboratory book the doubtfulness of the whole thing, and is yet -prepared as a forlorn hope to frame experiments in opposition to -all his previous experience of the course of nature. For a time his -thoughts flow on as if the strange detection were already made, and he -had only to trace out its consequences throughout the universe. “Let us -encourage ourselves by a little more imagination prior to experiment,” -he says; and then he reflects upon the infinity of actions in nature, -in which the mutual relations of electricity and gravity would come -into play; he pictures to himself the planets and the comets charging -themselves as they approach the sun; cascades, rain, rising vapour, -circulating currents of the atmosphere, the fumes of a volcano, the -smoke in a chimney become so many electrical machines. A multitude of -events and changes in the atmosphere seem to be at once elucidated by -such actions; for a moment his reveries have the vividness of fact. -“I think we have been dull and blind not to have suspected some such -results,” and he sums up rapidly the consequences of his great but -imaginary theory; an entirely new mode of exciting heat or electricity, -an entirely new relation of the natural forces, an analysis of -gravitation, and a justification of the conservation of force. - -Such were Faraday’s fondest dreams of what might be, and to many a -philosopher they would have been sufficient basis for the writing of -a great book. But Faraday’s imagination was within his full control; -as he himself says, “Let the imagination go, guarding it by judgment -and principle, and holding it in and directing it by experiment.” His -dreams soon took a very practical form, and for many days he laboured -with ceaseless energy, on the staircase of the Royal Institution, in -the clock tower of the Houses of Parliament, or at the top of the Shot -Tower in Southwark, raising and lowering heavy weights, and combining -electrical helices and wires in every conceivable way. His skill and -long experience in experiment were severely taxed to eliminate the -effects of the earth’s magnetism, and time after time he saved himself -from accepting mistaken indications, which to another man might have -seemed conclusive verifications of his theory. When all was done there -remained absolutely no results. “The experiments,” he says, “were well -made, but the results are negative;” and yet, he adds, “I cannot accept -them as conclusive.” In this position the question remains to the -present day; it may be that the effect was too slight to be detected, -or it may be that the arrangements adopted were not suited to develop -the particular relation which exists, just as Oersted could not detect -electro-magnetism, so long as his wire was perpendicular to the plane -of motion of his needle. But these are not matters which concern us -further here. We have only to notice the profound conviction in the -unity of natural laws, the active powers of inference and imagination, -the unbounded licence of theorising, combined above all with the utmost -diligence in experimental verification which this remarkable research -exhibits. - - -*Reservation of Judgment.* - -There is yet another characteristic needed in the philosophic mind; it -is that of suspending judgment when the data are insufficient. Many -people will express a confident opinion on almost any question which is -put before them, but they thereby manifest not strength, but narrowness -of mind. To see all sides of a complicated subject, and to weigh all -the different facts and probabilities correctly, require no ordinary -powers of comprehension. Hence it is most frequently the philosophic -mind which is in doubt, and the ignorant mind which is ready with a -positive decision. Faraday has himself said, in a very interesting -lecture:[487] “Occasionally and frequently the exercise of the judgment -ought to end in *absolute reservation*. It may be very distasteful, and -great fatigue, to suspend a conclusion; but as we are not infallible, -so we ought to be cautious; we shall eventually find our advantage, for -the man who rests in his position is not so far from right as he who, -proceeding in a wrong direction, is ever increasing his distance.” - - [487] Printed in *Modern Culture*, edited by Youmans, p. 219. - -Arago presented a conspicuous example of this high quality of mind, as -Faraday remarks; for when he made known his curious discovery of the -relation of a magnetic needle to a revolving copper plate, a number -of supposed men of science in different countries gave immediate and -confident explanations of it, which were all wrong. But Arago, who -had both discovered the phenomenon and personally investigated its -conditions, declined to put forward publicly any theory at all. - -At the same time we must not suppose that the truly philosophic mind -can tolerate a state of doubt, while a chance of decision remains open. -In science nothing like compromise is possible, and truth must be one. -Hence, doubt is the confession of ignorance, and involves a painful -feeling of incapacity. But doubt lies between error and truth, so that -if we choose wrongly we are further away than ever from our goal. - -Summing up, then, it would seem as if the mind of the great discoverer -must combine contradictory attributes. He must be fertile in -theories and hypotheses, and yet full of facts and precise results of -experience. He must entertain the feeblest analogies, and the merest -guesses at truth, and yet he must hold them as worthless till they are -verified in experiment. When there are any grounds of probability he -must hold tenaciously to an old opinion, and yet he must be prepared -at any moment to relinquish it when a clearly contradictory fact is -encountered. “The philosopher,” says Faraday,[488] “should be a man -willing to listen to every suggestion, but determined to judge for -himself. He should not be biased by appearances; have no favourite -hypothesis; be of no school; and in doctrine have no master. He should -not be a respecter of persons, but of things. Truth should be his -primary object. If to these qualities be added industry, he may indeed -hope to walk within the veil of the temple of nature.” - - [488] *Life of Faraday*, vol. i. p. 225. - - - - -BOOK V. - -GENERALISATION, ANALOGY, AND CLASSIFICATION. - - - - -CHAPTER XXVII. - -GENERALISATION. - - -I have endeavoured to show in preceding chapters that all inductive -reasoning is an inverse application of deductive reasoning, and -consists in demonstrating that the consequences of certain assumed laws -agree with facts of nature gathered by active or passive observation. -The fundamental process of reasoning, as stated in the outset, consists -in inferring of a thing what we know of similar objects, and it is on -this principle that the whole of deductive reasoning, whether simply -logical or mathematico-logical, is founded. All inductive reasoning -must be founded on the same principle. It might seem that by a plain -use of this principle we could avoid the complicated processes of -induction and deduction, and argue directly from one particular case -to another, as Mill proposed. If the Earth, Venus, Mars, Jupiter, -and other planets move in elliptic orbits, cannot we dispense with -elaborate precautions, and assert that Neptune, Ceres, and the last -discovered planet must do so likewise? Do we not know that Mr. -Gladstone must die, because he is like other men? May we not argue -that because some men die therefore he must? Is it requisite to ascend -by induction to the general proposition “all men must die,” and then -descend by deduction from that general proposition to the case of Mr. -Gladstone? My answer undoubtedly is that we must ascend to general -propositions. The fundamental principle of the substitution of similars -gives us no warrant in affirming of Mr. Gladstone what we know of -other men, because we cannot be sure that Mr. Gladstone is exactly -similar to other men. Until his death we cannot be perfectly sure that -he possesses all the attributes of other men; it is a question of -probability, and I have endeavoured to explain the mode in which the -theory of probability is applied to calculate the probability that from -a series of similar events we may infer the recurrence of like events -under identical circumstances. There is then no such process as that -of inferring from particulars to particulars. A careful analysis of -the conditions under which such an inference appears to be made, shows -that the process is really a general one, and that what is inferred of -a particular case might be inferred of all similar cases. All reasoning -is essentially general, and all science implies generalisation. In -the very birth-time of philosophy this was held to be so: “Nulla -scientia est de individuis, sed de solis universalibus,” was the -doctrine of Plato, delivered by Porphyry. And Aristotle[489] held a -like opinion--Οὐδεμία δὲ τέχνη σκοπεȋ τὸ καθ’ ἕκαστον ... τὸ δὲ καθ’ -ἕκαστον ἄπειρον καὶ οὐκ ἐπιστητόν. “No art treats of particular cases; -for particulars are infinite and cannot be known.” No one who holds the -doctrine that reasoning may be from particulars to particulars, can -be supposed to have the most rudimentary notion of what constitutes -reasoning and scíence. - - [489] Aristotle’s *Rhetoric*, Liber I. 2. 11. - -At the same time there can be no doubt that practically what we find -to be true of many similar objects will probably be true of the next -similar object. This is the result to which an analysis of the Inverse -Method of Probabilities leads us, and, in the absence of precise data -from which we may calculate probabilities, we are usually obliged to -make a rough assumption that similars in some respects are similars -in other respects. Thus it comes to pass that a large part of the -reasoning processes in which scientific men are engaged, consists in -detecting similarities between objects, and then rudely assuming that -the like similarities will be detected in other cases. - - -*Distinction of Generalisation and Analogy.* - -There is no distinction but that of degree between what is known as -reasoning by *generalisation* and reasoning by *analogy*. In both -cases from certain observed resemblances we infer, with more or less -probability, the existence of other resemblances. In generalisation -the resemblances have great extension and usually little intension, -whereas in analogy we rely upon the great intension, the extension -being of small amount (p. 26). If we find that the qualities A and B -are associated together in a great many instances, and have never been -found separate, it is highly probable that on the next occasion when we -meet with A, B will also be present, and *vice versâ*. Thus wherever we -meet with an object possessing gravity, it is found to possess inertia -also, nor have we met with any material objects possessing inertia -without discovering that they also possess gravity. The probability -has therefore become very great, as indicated by the rules founded on -the Inverse Method of Probabilities (p. 257), that whenever in the -future we meet an object possessing either of the properties of gravity -and inertia, it will be found on examination to possess the other -of these properties. This is a clear instance of the employment of -generalisation. - -In analogy, on the other hand, we reason from likeness in many points -to likeness in other points. The qualities or points of resemblance are -now numerous, not the objects. At the poles of Mars are two white spots -which resemble in many respects the white regions of ice and snow at -the poles of the earth. There probably exist no other similar objects -with which to compare these, yet the exactness of the resemblance -enables us to infer, with high probability, that the spots on Mars -consist of ice and snow. In short, many points of resemblance imply -many more. From the appearance and behaviour of those white spots we -infer that they have all the chemical and physical properties of -frozen water. The inference is of course only probable, and based upon -the improbability that aggregates of many qualities should be formed in -a like manner in two or more cases, without being due to some uniform -condition or cause. - -In reasoning by analogy, then, we observe that two objects ABCDE.... -and A′B′C′D′E′.... have many like qualities, as indicated by the -identity of the letters, and we infer that, since the first has another -quality, X, we shall discover this quality in the second case by -sufficiently close examination. As Laplace says,--“Analogy is founded -on the probability that similar things have causes of the same kind, -and produce the same effects. The more perfect this similarity, the -greater is this probability.”[490] The nature of analogical inference -is aptly described in the work on Logic attributed to Kant, where the -rule of ordinary induction is stated in the words, “*Eines in vielen, -also in allen*,” one quality in many things, therefore in all; and -the rule of analogy is “*Vieles in einem, also auch das übrige in -demselben*,”[491] many (qualities) in one, therefore also the remainder -in the same. It is evident that there may be intermediate cases in -which, from the identity of a moderate number of objects in several -properties, we may infer to other objects. Probability must rest either -upon the number of instances or the depth of resemblance, or upon the -occurrence of both in sufficient degrees. What there is wanting in -extension must be made up by intension, and *vice versâ*. - - [490] *Essai Philosophique sur les Probabilités*, p. 86. - - [491] Kant’s *Logik*, § 84, Königsberg, 1800, p. 207. - - -*Two Meanings of Generalisation.* - -The term generalisation, as commonly used, includes two processes which -are of different character, but are often closely associated together. -In the first place, we generalise when we recognise even in two objects -a common nature. We cannot detect the slightest similarity without -opening the way to inference from one case to the other. If we compare -a cubical crystal with a regular octahedron, there is little apparent -similarity; but, as soon as we perceive that either can be produced -by the symmetrical modification of the other, we discover a groundwork -of similarity in the crystals, which enables us to infer many things -of one, because they are true of the other. Our knowledge of ozone -took its rise from the time when the similarity of smell, attending -electric sparks, strokes of lightning, and the slow combustion of -phosphorus, was noticed by Schönbein. There was a time when the rainbow -was an inexplicable phenomenon--a portent, like a comet, and a cause of -superstitious hopes and fears. But we find the true spirit of science -in Roger Bacon, who desires us to consider the objects which present -the same colours as the rainbow; he mentions hexagonal crystals from -Ireland and India, but he bids us not suppose that the hexagonal form -is essential, for similar colours may be detected in many transparent -stones. Drops of water scattered by the oar in the sun, the spray from -a water-wheel, the dewdrops lying on the grass in the summer morning, -all display a similar phenomenon. No sooner have we grouped together -these apparently diverse instances, than we have begun to generalise, -and have acquired a power of applying to one instance what we can -detect of others. Even when we do not apply the knowledge gained to -new objects, our comprehension of those already observed is greatly -strengthened and deepened by learning to view them as particular cases -of a more general property. - -A second process, to which the name of generalisation is often -given, consists in passing from a fact or partial law to a multitude -of unexamined cases, which we believe to be subject to the same -conditions. Instead of merely recognising similarity as it is brought -before us, we predict its existence before our senses can detect it, so -that generalisation of this kind endows us with a prophetic power of -more or less probability. Having observed that many substances assume, -like water and mercury, the three states of solid, liquid, and gas, and -having assured ourselves by frequent trial that the greater the means -we possess of heating and cooling, the more substances we can vaporise -and freeze, we pass confidently in advance of fact, and assume that -all substances are capable of these three forms. Such a generalisation -was accepted by Lavoisier and Laplace before many of the corroborative -facts now in our possession were known. The reduction of a single -comet beneath the sway of gravity was considered sufficient indication -that all comets obey the same power. Few persons doubted that the law -of gravity extended over the whole heavens; certainly the fact that -a few stars out of many millions manifest the action of gravity, is -now held to be sufficient evidence of its general extension over the -visible universe. - - -*Value of Generalisation.* - -It might seem that if we know particular facts, there can be little -use in connecting them together by a general law. The particulars -must be more full of useful information than an abstract general -statement. If we know, for instance, the properties of an ellipse, a -circle, a parabola, and hyperbola, what is the use of learning all -these properties over again in the general theory of curves of the -second degree? If we understand the phenomena of sound and light and -water-waves separately, what is the need of erecting a general theory -of waves, which, after all, is inapplicable to practice until resolved -again into particular cases? But, in reality, we never do obtain an -adequate knowledge of particulars until we regard them as cases of -the general. Not only is there a singular delight in discovering the -many in the one, and the one in the many, but there is a constant -interchange of light and knowledge. Properties which are unapparent -in the hyperbola may be readily observed in the ellipse. Most of the -complex relations which old geometers discovered in the circle will -be reproduced *mutatis mutandis* in the other conic sections. The -undulatory theory of light might have been unknown at the present day, -had not the theory of sound supplied hints by analogy. The study of -light has made known many phenomena of interference and polarisation, -the existence of which had hardly been suspected in the case of -sound, but which may now be sought out, and perhaps found to possess -unexpected interest. The careful study of water-waves shows how waves -alter in form and velocity with varying depth of water. Analogous -changes may some time be detected in sound waves. Thus there is mutual -interchange of aid. - -“Every study of a generalisation or extension,” De Morgan has well -said,[492] “gives additional power over the particular form by which -the generalisation is suggested. Nobody who has ever returned to -quadratic equations after the study of equations of all degrees, -or who has done the like, will deny my assertion that οὐ βλέπει -βλέπων may be predicated of any one who studies a branch or a case, -without afterwards making it part of a larger whole. Accordingly -it is always worth while to generalise, were it only to give power -over the *particular*. This principle, of daily familiarity to the -mathematician, is almost unknown to the logician.” - - [492] *Syllabus of a Proposed System of Logic*, p. 34. - - -*Comparative Generality of Properties.* - -Much of the value of science depends upon the knowledge which we -gradually acquire of the different degrees of generality of properties -and phenomena of various kinds. The use of science consists in enabling -us to act with confidence, because we can foresee the result. Now this -foresight must rest upon the knowledge of the powers which will come -into play. That knowledge, indeed, can never be certain, because it -rests upon imperfect induction, and the most confident beliefs and -predictions of the physicist may be falsified. Nevertheless, if we -always estimate the probability of each belief according to the due -teaching of the data, and bear in mind that probability when forming -our anticipations, we shall ensure the minimum of disappointment. Even -when he cannot exactly apply the theory of probabilities, the physicist -may acquire the habit of making judgments in general agreement with its -principles and results. - -Such is the constitution of nature, that the physicist learns to -distinguish those properties which have wide and uniform extension, -from those which vary between case and case. Not only are certain laws -distinctly laid down, with their extension carefully defined, but a -scientific training gives a kind of tact in judging how far other laws -are likely to apply under any particular circumstances. We learn by -degrees that crystals exhibit phenomena depending upon the directions -of the axes of elasticity, which we must not expect in uniform solids. -Liquids, compared even with non-crystalline solids, exhibit laws of -far less complexity and variety; and gases assume, in many respects, -an aspect of nearly complete uniformity. To trace out the branches of -science in which varying degrees of generality prevail, would be an -inquiry of great interest and importance; but want of space, if there -were no other reason, would forbid me to attempt it, except in a very -slight manner. - -Gases, so far as they are really gaseous, not only have exactly the -same properties in all directions of space, but one gas exactly -resembles other gases in many qualities. All gases expand by heat, -according to the same law, and by nearly the same amount; the specific -heats of equivalent weights are equal, and the densities are exactly -proportional to the atomic weights. All such gases obey the general -law, that the volume multiplied by the pressure, and divided by the -absolute temperature, is constant or nearly so. The laws of diffusion -and transpiration are the same in all cases, and, generally speaking, -all physical laws, as distinguished from chemical laws, apply -equally to all gases. Even when gases differ in chemical or physical -properties, the differences are minor in degree. Thus the differences -of viscosity are far less marked than in the liquid and solid states. -Nearly all gases, again, are colourless, the exceptions being chlorine, -the vapours of iodine, bromine, and a few other substances. - -Only in one single point, so far as I am aware, do gases present -distinguishing marks unknown or nearly so, in the solid and liquid -states. I mean as regards the light given off when incandescent. -Each gas when sufficiently heated, yields its own peculiar series -of rays, arising from the free vibrations of the constituent parts -of the molecules. Hence the possibility of distinguishing gases by -the spectroscope. But the molecules of solids and liquids appear to -be continually in conflict with each other, so that only a confused -*noise* of atoms is produced, instead of a definite series of luminous -chords. At the same temperature, accordingly, all solids and liquids -give off nearly the same rays when strongly heated, and we have in this -case an exception to the greater generality of properties in gases. - -Liquids are in many ways intermediate in character between gases -and solids. While incapable of possessing different elasticity in -different directions, and thus denuded of the rich geometrical -complexity of solids, they retain the variety of density, colour -degrees of transparency, great diversity in surface tension, viscosity, -coefficients of expansion, compressibility, and many other properties -which we observe in solids, but not for the most part in gases. Though -our knowledge of the physical properties of liquids is much wanting in -generality at present, there is ground to hope that by degrees laws -connecting and explaining the variations may be traced out. - -Solids are in every way contrasted to gases. Each solid substance -has its own peculiar degree of density, hardness, compressibility, -transparency, tenacity, elasticity, power of conducting heat and -electricity, magnetic properties, capability of producing frictional -electricity, and so forth. Even different specimens of the same kind -of substance will differ widely, according to the accidental treatment -received. And not only has each substance its own specific properties, -but, when crystallised, its properties vary in each direction with -regard to the axes of crystallisation. The velocity of radiation, the -rate of conduction of heat, the coefficients of expansibility and -compressibility, the thermo-electric properties, all vary in different -crystallographic directions. - -It is probable that many apparent differences between liquids, and -even between solids, will be explained when we learn to regard them -under exactly corresponding circumstances. The extreme generality of -the properties of gases is in reality only true at an infinitely high -temperature, when they are all equally remote from their condensing -points. Now, it is found that if we compare liquids--for instance, -different kinds of alcohols--not at equal temperatures, but at points -equally distant from their respective boiling points, the laws and -coefficients of expansion are nearly equal. The vapour-tensions of -liquids also are more nearly equal, when compared at corresponding -points, and the boiling-points appear in many cases to be simply -related to the chemical composition. No doubt the progress of -investigation will enable us to discover generality, where at present -we only see variety and puzzling complexity. - -In some cases substances exhibit the same physical properties in -the liquid as in the solid state. Lead has a high refractive power, -whether in solution, or in solid salts, crystallised or vitreous. -The magnetic power of iron is conspicuous, whatever be its chemical -condition; indeed, the magnetic properties of substances, though -varying with temperature, seem not to be greatly affected by other -physical changes. Colour, absorptive power for heat or light rays, -and a few other properties are also often the same in liquids and -gases. Iodine and bromine possess a deep colour whenever they are -chemically uncombined. Nevertheless, we can seldom argue safely from -the properties of a substance in one condition to those in another -condition. Ice is an insulator, water a conductor of electricity, and -the same contrast exists in most other substances. The conducting power -of a liquid for electricity increases with the temperature, while that -of a solid decreases. By degrees we may learn to distinguish between -those properties of matter which depend upon the intimate construction -of the chemical molecule, and those which depend upon the contact, -conflict, mutual attraction, or other relations of distinct molecules. -The properties of a substance with respect to light seem generally to -depend upon the molecule; thus, the power of certain substances to -cause the plane of polarisation of a ray of light to rotate, is exactly -the same whatever be its degree of density, or the diluteness of the -solution in which it is contained. Taken as a whole, the physical -properties of substances and their quantitative laws, present a problem -of infinite complexity, and centuries must elapse before any moderately -complete generalisations on the subject become possible. - - -*Uniform Properties of all Matter.* - -Some laws are held to be true of all matter in the universe absolutely, -without exception, no instance to the contrary having ever been -noticed. This is the case with the laws of motion, as laid down -by Galileo and Newton. It is also conspicuously true of the law -of universal gravitation. The rise of modern physical science may -perhaps be considered as beginning at the time when Galileo showed, -in opposition to the Aristotelians, that matter is equally affected -by gravity, irrespective of its form, magnitude, or texture. All -objects fall with equal rapidity, when disturbing causes, such as the -resistance of the air, are removed or allowed for. That which was -rudely demonstrated by Galileo from the leaning tower of Pisa, was -proved by Newton to a high degree of approximation, in an experiment -which has been mentioned (p. 443). - -Newton formed two pendulums, as nearly as possible the same in outward -shape and size by taking two equal round wooden boxes, and suspending -them by equal threads, eleven feet long. The pendulums were therefore -equally subject to the resistance of the air. He filled one box with -wood, and in the centre of oscillation of the other he placed an equal -weight of gold. The pendulums were then equal in weight as well as in -size; and, on setting them simultaneously in motion, Newton found that -they vibrated for a length of time with equal vibrations. He tried the -same experiment with silver, lead, glass, sand, common salt, water, -and wheat, in place of the gold, and ascertained that the motion of -his pendulum was exactly the same whatever was the kind of matter -inside.[493] He considered that a difference of a thousandth part -would have been apparent. The reader must observe that the pendulums -were made of equal weight only in order that they might suffer equal -retardation from the air. The meaning of the experiment is that all -substances manifest exactly equal acceleration from the force of -gravity, and that therefore the inertia or resistance of matter to -force, which is the only independent measure of mass known to us, is -always proportional to gravity. - - [493] *Principia*, bk. iii. Prop. VI. Motte’s translation, vol. ii. - p. 220. - -These experiments of Newton were considered conclusive up to very -recent times, when certain discordances between the theory and -observations of the movements of planets led Nicolai, in 1826, to -suggest that the equal gravitation of different kinds of matter might -not be absolutely exact. It is perfectly philosophical thus to call -in question, from time to time, some of the best accepted laws. On -this occasion Bessel carefully repeated the experiments of Newton with -pendulums composed of ivory, glass, marble, quartz, meteoric stones, -&c., but was unable to detect the least difference. This conclusion -is also confirmed by the ultimate agreement of all the calculations -of physical astronomy based upon it. Whether the mass of Jupiter be -calculated from the motion of its own satellites, from the effect -upon the small planets, Vesta, Juno, &c., or from the perturbation -of Encke’s Comet, the results are closely accordant, showing that -precisely the same law of gravity applies to the most different bodies -which we can observe. The gravity of a body, again, appears to be -entirely independent of its other physical conditions, being totally -unaffected by any alteration in the temperature, density, electric or -magnetic condition, or other physical properties of the substance. - -One paradoxical result of the law of equal gravitation is the theorem -of Torricelli, to the effect that all liquids of whatever density -fall or flow with equal rapidity. If there be two equal cisterns -respectively filled with mercury and water, the mercury, though -thirteen times as heavy, would flow from an aperture neither more -rapidly nor more slowly than the water, and the same would be true of -ether, alcohol, and other liquids, allowance being made, however, for -the resistance of the air, and the differing viscosities of the liquids. - -In its exact equality and its perfect independence of all -circumstances, except mass and distance, the force of gravity stands -apart from all the other forces and phenomena of nature, and has -not yet been brought into any relation with them except through the -general principle of the conservation of energy. Magnetic attraction, -as remarked by Newton, follows very different laws, depending upon the -chemical quality and molecular structure of each particular substance. - -We must remember that in saying “all matter gravitates,” we exclude -from the term matter the basis of light-undulations, which is -immensely more extensive in amount, and obeys in many respects the -laws of mechanics. This adamantine substance appears, so far as can be -ascertained, to be perfectly uniform in its properties when existing -in space unoccupied by matter. Light and heat are conveyed by it with -equal velocity in all directions, and in all parts of space so far as -observation informs us. But the presence of gravitating matter modifies -the density and mechanical properties of the so-called ether in a way -which is yet quite unexplained.[494] - -Leaving gravity, it is somewhat difficult to discover other laws -which are equally true of all matter. Boerhaave was considered to -have established that all bodies expand by heat; but not only is the -expansion very different in different substances, but we now know -positive exceptions. Many liquids and a few solids contract by heat -at certain temperatures. There are indeed other relations of heat to -matter which seem to be universal and uniform; all substances begin -to give off rays of light at the same temperature, according to the -law of Draper; and gases will not be an exception if sufficiently -condensed, as in the experiments of Frankland. Grove considers it to -be universally true that all bodies in combining produce heat; with -the doubtful exception of sulphur and selenium, all solids in becoming -liquids, and all liquids in becoming gases, absorb heat; but the -quantities of heat absorbed vary with the chemical qualities of the -matter. Carnot’s Thermodynamic Law is held to be exactly true of all -matter without distinction; it expresses the fact that the amount of -mechanical energy which might be theoretically obtained from a certain -amount of heat energy depends only upon the change of the temperatures, -so that whether an engine be worked by water, air, alcohol, ammonia, or -any other substance, the result would theoretically be the same, if the -boiler and condenser were maintained at similar temperatures. - - [494] Professor Lovering has pointed out how obscure and uncertain - the ideas of scientific men about this ether are, in his interesting - Presidential Address before the American Association at Hartford, - 1874. *Silliman’s Journal*, October 1874, p. 297. *Philosophical - Magazine*, vol. xlviii. p. 493. - - -*Variable Properties of Matter.* - -I have enumerated some of the few properties of matter, which are -manifested in exactly the same manner by all substances, whatever be -their differences of chemical or physical constitution. But by far -the greater number of qualities vary in degree; substances are more -or less dense, more or less transparent, more or less compressible, -more or less magnetic, and so on. One common result of the progress of -science is to show that qualities supposed to be entirely absent from -many substances are present only in so low a degree of intensity that -the means of detection were insufficient. Newton believed that most -bodies were quite unaffected by the magnet; Faraday and Tyndall have -rendered it very doubtful whether any substance whatever is wholly -devoid of magnetism, including under that term diamagnetism. We are -rapidly learning to believe that there are no substances absolutely -opaque, or non-conducting, non-electric, non-elastic, non-viscous, -non-compressible, insoluble, infusible, or non-volatile. All tends -to become a matter of degree, or sometimes of direction. There may -be some substances oppositely affected to others, as ferro-magnetic -substances are oppositely affected to diamagnetics, or as substances -which contract by heat are opposed to those which expand; but the -tendency is certainly for every affection of one kind of matter to be -represented by something similar in other kinds. On this account one of -Newton’s rules of philosophising seems to lose all validity; he said, -“Those qualities of bodies which are not capable of being heightened, -and remitted, and which are found in all bodies on which experiment can -be made, must be considered as universal qualities of all bodies.” As -far as I can see, the contrary is more probable, namely, that qualities -variable in degree will be found in every substance in a greater or -less degree. - -It is remarkable that Newton whose method of investigation was -logically perfect, seemed incapable of generalising and describing -his own procedure. His celebrated “Rules of Reasoning in Philosophy,” -described at the commencement of the third book of the *Principia*, are -of questionable truth, and still more questionable value. - - -*Extreme Instances of Properties.* - -Although substances usually differ only in degree, great interest -may attach to particular substances which manifest a property in a -conspicuous and intense manner. Every branch of physical science -has usually been developed from the attention forcibly drawn to some -singular substance. Just as the loadstone disclosed magnetism and -amber frictional electricity, so did Iceland spar show the existence -of double refraction, and sulphate of quinine the phenomenon of -fluorescence. When one such startling instance has drawn the attention -of the scientific world, numerous less remarkable cases of the -phenomenon will be detected, and it will probably prove that the -property in question is actually universal to all matter. Nevertheless, -the extreme instances retain their interest, partly in a historical -point of view, partly because they furnish the most convenient -substances for experiment. - -Francis Bacon was fully aware of the value of such examples, which he -called *Ostensive Instances* or Light-giving, Free and Predominant -Instances. “They are those,” he says,[495] “which show the nature -under investigation naked, in an exalted condition, or in the highest -degree of power; freed from impediments, or at least by its strength -predominating over and suppressing them.” He mentions quicksilver as -an ostensive instance of weight or density, thinking it not much less -dense than gold, and more remarkable than gold as joining density -to liquidity. The magnet is mentioned as an ostensive instance of -attraction. It would not be easy to distinguish clearly between these -ostensive instances and those which he calls *Instantiae Monodicae*, -or *Irregulares*, or *Heteroclitae*, under which he places whatever -is extravagant in its properties or magnitude, or exhibits least -similarity to other things, such as the sun and moon among the heavenly -bodies, the elephant among animals, the letter *s* among letters, or -the magnet among stones.[496] - - [495] *Novum Organum*, bk. ii. Aphorisms, 24, 25. - - [496] Ibid. Aph. 28. - -In optical science great use has been made of the high dispersive -power of the transparent compounds of lead, that is, the power of -giving a long spectrum (p. 432). Dollond, having noticed this peculiar -dispersive power in lenses made of flint glass, employed them to -produce an achromatic arrangement. The element strontium presents a -contrast to lead in this respect, being characterised by a remarkably -low dispersive power; but I am not aware that this property has yet -been turned to account. - -Compounds of lead have both a high dispersive and a high refractive -index, and in the latter respect they proved very useful to Faraday. -Having spent much labour in preparing various kinds of optical glass, -Faraday happened to form a compound of lead, silica, and boracic -acid, now known as *heavy glass*, which possessed an intensely high -refracting power. Many years afterwards in attempting to discover the -action of magnetism upon light he failed to detect any effect, as has -been already mentioned, (p. 588), until he happened to test a piece of -the heavy glass. The peculiar refractive power of this medium caused -the magnetic strain to be apparent, and the rotation of the plane of -polarisation was discovered. - -In almost every part of physical science there is some substance -of powers pre-eminent for the special purpose to which it is put. -Rock-salt is invaluable for its extreme diathermancy or transparency to -the least refrangible rays of the spectrum. Quartz is equally valuable -for its transparency, as regards the ultra-violet or most refrangible -rays. Diamond is the most highly refracting substance which is at the -same time transparent; were it more abundant and easily worked it would -be of great optical importance. Cinnabar is distinguished by possessing -a power of rotating the plane of polarisation of light, from 15 to 17 -times as much as quartz. In electric experiments copper is employed for -its high conducting powers and exceedingly low magnetic properties; -iron is of course indispensable for its enormous magnetic powers; while -bismuth holds a like place as regards its diamagnetic powers, and was -of much importance in Tyndall’s decisive researches upon the polar -character of the diamagnetic force.[497] In regard to magne-crystallic -action the mineral cyanite is highly remarkable, being so powerfully -affected by the earth’s magnetism, that, when delicately suspended, it -assumes a constant position with regard to the magnetic meridian, and -may almost be used like the compass needle. Sodium is distinguished -by its unique light-giving powers, which are so extraordinary that -probably one half of the whole number of stars in the heavens have a -yellow tinge in consequence. - - [497] *Philosophical Transactions* (1856) vol. cxlvi. p. 246. - -It is remarkable that water, though the most common of all fluids, -is distinguished in almost every respect by extreme qualities. Of -all known substances water has the highest specific heat, being thus -peculiarly fitted for the purpose of warming and cooling, to which -it is often put. It rises by capillary attraction to a height more -than twice that of any other liquid. In the state of ice it is nearly -twice as dilatable by heat as any other known solid substance.[498] In -proportion to its density it has a far higher surface tension than any -other substance, being surpassed in absolute tension only by mercury; -and it would not be difficult to extend considerably the list of its -remarkable and useful properties. - - [498] *Philosophical Magazine*, 4th Series, January 1870, vol. xxxix. - p. 2. - -Under extreme instances we may include cases of remarkably low powers -or qualities. Such cases seem to correspond to what Bacon calls -*Clandestine Instances*, which exhibit a given nature in the least -intensity, and as it were in a rudimentary state.[499] They may often -be important, he thinks, as allowing the detection of the cause of -the property by difference. I may add that in some cases they may be -of use in experiments. Thus hydrogen is the least dense of all known -substances, and has the least atomic weight. Liquefied nitrous oxide -has the lowest refractive index of all known fluids.[500] The compounds -of strontium have the lowest dispersive power. It is obvious that -a property of very low degree may prove as curious and valuable a -phenomenon as a property of very high degree. - - [499] *Novum Organum*, bk. ii. Aphorism 25. - - [500] Faraday’s *Experimental Researches in Chemistry and Physics*, - p. 93. - - -*The Detection of Continuity.* - -We should bear in mind that phenomena which are in reality of a -closely similar or even identical nature, may present to the senses -very different appearances. Without a careful analysis of the changes -which take place, we may often be in danger of widely separating facts -and processes, which are actually instances of the same law. Extreme -difference of degree or magnitude is a frequent cause of error. It is -truly difficult at the first moment to recognise any similarity between -the gradual rusting of a piece of iron, and the rapid combustion of a -heap of straw. Yet Lavoisier’s chemical theory was founded upon the -similarity of the oxydising process in one case and the other. We have -only to divide the iron into excessively small particles to discover -that it is really the more combustible of the two, and that it actually -takes fire spontaneously and burns like tinder. It is the excessive -slowness of the process in the case of a massive piece of iron which -disguises its real character. - -If Xenophon reports truly, Socrates was misled by not making sufficient -allowance for extreme differences of degree and quantity. Anaxagoras -held that the sun is a fire, but Socrates rejected this opinion, on the -ground that we can look at a fire, but not at the sun, and that plants -grow by sunshine while they are killed by fire. He also pointed out -that a stone heated in a fire is not luminous, and soon cools, whereas -the sun ever remains equally luminous and hot.[501] All such mistakes -evidently arise from not perceiving that difference of quantity may be -so extreme as to assume the appearance of difference of quality. It is -the least creditable thing we know of Socrates, that after pointing -out these supposed mistakes of earlier philosophers, he advised his -followers not to study astronomy. - - [501] *Memorabilia*, iv. 7. - -Masses of matter of very different size may be expected to exhibit -apparent differences of conduct, arising from the various intensity of -the forces brought into play. Many persons have thought it requisite -to imagine occult forces producing the suspension of the clouds, and -there have even been absurd theories representing cloud particles as -minute water-balloons buoyed up by the warm air within them. But we -have only to take proper account of the enormous comparative resistance -which the air opposes to the fall of minute particles, to see that -all cloud particles are probably constantly falling through the air, -but so slowly that there is no apparent effect. Mineral matter again -is always regarded as inert and incapable of spontaneous movement. We -are struck by astonishment on observing in a powerful microscope, that -every kind of solid matter suspended in extremely minute particles -in pure water, acquires an oscillatory movement, often so marked as -to resemble dancing or skipping. I conceive that this movement is due -to the comparatively vast intensity of chemical action when exerted -upon minute particles, the effect being 5,000 or 10,000 greater in -proportion to the mass than in fragments of an inch diameter (p. 406). - -Much that was formerly obscure in the science of electricity arose from -the extreme differences of intensity and quantity in which this form of -energy manifests itself. Between the brilliant explosive discharge of a -thunder-cloud and the gentle continuous current produced by two pieces -of metal and some dilute acid, there is no apparent analogy whatever. -It was therefore a work of great importance when Faraday demonstrated -the identity of the forces in action, showing that common frictional -electricity would decompose water like that from the voltaic battery. -The relation of the phenomena became plain when he succeeded in showing -that it would require 800,000 discharges of his large Leyden battery -to decompose one single grain of water. Lightning was now seen to be -electricity of excessively high tension, but extremely small quantity, -the difference being somewhat analogous to that between the force of -one million gallons of water falling through one foot, and one gallon -of water falling through one million feet. Faraday estimated that one -grain of water acting on four grains of zinc, would yield electricity -enough for a great thunderstorm. - -It was long believed that electrical conductors and insulators belonged -to two opposed classes of substances. Between the inconceivable -rapidity with which the current passes through pure copper wire, -and the apparently complete manner in which it is stopped by a -thin partition of gutta-percha or gum-lac, there seemed to be no -resemblance. Faraday again laboured successfully to show that these -were but the extreme cases of a chain of substances varying in all -degrees in their powers of conduction. Even the best conductors, such -as pure copper or silver, offer resistance to the electric current. The -other metals have considerably higher powers of resistance, and we pass -gradually down through oxides and sulphides. The best insulators, on -the other hand, allow of an atomic induction which is the necessary -antecedent of conduction. Hence Faraday inferred that whether we can -measure the effect or not, all substances discharge electricity more -or less.[502] One consequence of this doctrine must be, that every -discharge of electricity produces an induced current. In the case of -the common galvanic current we can readily detect the induced current -in any parallel wire or other neighbouring conductor, and can separate -the opposite currents which arise at the moments when the original -current begins and ends. But a discharge of high tension electricity -like lightning, though it certainly occupies time and has a beginning -and an end, yet lasts so minute a fraction of a second, that it would -be hopeless to attempt to detect and separate the two opposite induced -currents, which are nearly simultaneous and exactly neutralise each -other. Thus an apparent failure of analogy is explained away, and -we are furnished with another instance of a phenomenon incapable of -observation and yet theoretically known to exist.[503] - - [502] *Experimental Researches in Electricity*, Series xii. vol. i. - p. 420. - - [503] *Life of Faraday*, vol. ii. p. 7. - -Perhaps the most extraordinary case of the detection of unsuspected -continuity is found in the discovery of Cagniard de la Tour and -Professor Andrews, that the liquid and gaseous conditions of matter -are only remote points in a continuous course of change. Nothing is -at first sight more apparently distinct than the physical condition -of water and aqueous vapour. At the boiling-point there is an -entire breach of continuity, and the gas produced is subject to -laws incomparably more simple than the liquid from which it arose. -But Cagniard de la Tour showed that if we maintain a liquid under -sufficient pressure its boiling point may be indefinitely raised, and -yet the liquid will ultimately assume the gaseous condition with but -a small increase of volume. Professor Andrews, recently following out -this course of inquiry, has shown that liquid carbonic acid may, at -a particular temperature (30°·92 C.), and under the pressure of 74 -atmospheres, be at the same time in a state indistinguishable from -that of liquid and gas. At higher pressures carbonic acid may be -made to pass from a palpably liquid state to a truly gaseous state -without any abrupt change whatever. As the pressure is greater the -abruptness of the change from liquid to gas gradually decreases, and -finally vanishes. Similar phenomena or an approximation to them have -been observed in other liquids, and there is little doubt that we may -make a wide generalisation, and assert that, under adequate pressure, -every liquid might be made to pass into a gas without breach of -continuity.[504] The liquid state, moreover, is considered by Professor -Andrews to be but an intermediate step between the solid and gaseous -conditions. There are various indications that the process of melting -is not perfectly abrupt; and could experiments be made under adequate -pressures, it is believed that every solid could be made to pass by -insensible degrees into the state of liquid, and subsequently into that -of gas. - - [504] *Nature*, vol. ii. p. 278. - -These discoveries appear to open the way to most important and -fundamental generalisations, but it is probable that in many other -cases phenomena now regarded as discrete may be shown to be different -degrees of the same process. Graham was of opinion that chemical -affinity differs but in degree from the ordinary attraction which -holds different particles of a body together. He found that sulphuric -acid continued to evolve heat when mixed even with the fiftieth -equivalent of water, so that there seemed to be no distinct limit to -chemical affinity. He concludes, “There is reason to believe that -chemical affinity passes in its lowest degree into the attraction of -aggregation.”[505] - - [505] *Journal of the Chemical Society*, vol. viii. p. 51. - -The atomic theory is well established, but its limits are not marked -out. As Grove points out, we may by selecting sufficiently high -multipliers express any combination or mixture of elements in terms -of their equivalent weights.[506] Sir W. Thomson has suggested that -the power which vegetable fibre, oatmeal, and other substances possess -of attracting and condensing aqueous vapour is probably continuous, -or, in fact, identical with capillary attraction, which is capable -of interfering with the pressure of aqueous vapour and aiding its -condensation.[507] There are many cases of so-called catalytic or -surface action, such as the extraordinary power of animal charcoal -for attracting organic matter, or of spongy platinum for condensing -hydrogen, which can only be considered as exalted cases of a more -general power of attraction. The number of substances which are -decomposed by light in a striking manner is very limited; but many -other substances, such as vegetable colours, are affected by long -exposure; on the principle of continuity we might expect to find that -all kinds of matter are more or less susceptible of change by the -incidence of light rays.[508] It is the opinion of Grove that wherever -an electric current passes there is a tendency to decomposition, a -strain on the molecules, which when sufficiently intense leads to -disruption. Even a metallic conducting wire may be regarded as tending -to decomposition. Davy was probably correct in describing electricity -as chemical affinity acting on masses, or rather, as Grove suggests, -creating a disturbance through a chain of particles.[509] Laplace went -so far as to suggest that all chemical phenomena may be results of -the Newtonian law of attraction, applied to atoms of various mass and -position; but the time is probably far distant when the progress of -molecular philosophy and of mathematical methods will enable such a -generalisation to be verified or refuted. - - [506] *Correlation of Physical Forces*, 3rd edit. p. 184. - - [507] *Philosophical Magazine*, 4th Series, vol. xlii. p. 451. - - [508] Grove, *Correlation of Physical Forces*, 3rd edit. p. 118. - - [509] Ibid. pp. 166, 199, &c. - - -*The Law of Continuity.* - -Under the title of the Law of Continuity we may place many applications -of the general principle of reasoning, that what is true of one case -will be true of similar cases, and probably true of what are probably -similar. Whenever we find that a law or similarity is rigorously -fulfilled up to a certain point in time or space, we expect with a high -degree of probability that it will continue to be fulfilled at least a -little further. If we see part only of a circle, we naturally expect -that the circular form will be continued in the part hidden from us. If -a body has moved uniformly over a certain space, we expect that it will -continue to move uniformly. The ground of such inferences is doubtless -identical with that of other inductive inferences. In continuous -motion every infinitely small space passed over constitutes a separate -constituent fact, and had we perfect powers of observation the smallest -finite motion would include an infinity of information, which, by the -principles of the inverse method of probabilities, would enable us to -infer with certainty to the next infinitely small portion of the path. -But when we attempt to infer from one finite portion of a path to -another finite portion, inference will be only more or less probable, -according to the comparative lengths of the portions and the accuracy -of observation; the longer our experience is, the more probable our -inference will be; the greater the length of time or space over which -the inference extends, the less probable. - -This principle of continuity presents itself in nature in a great -variety of forms and cases. It is familiarly expressed in the dictum -*Natura non agit per saltum*. As Graham expressed the maxim, there are -in nature no abrupt transitions, and the distinctions of class are -never absolute.[510] There is always some notice--some forewarning -of every phenomenon, and every change begins by insensible degrees, -could we observe it with perfect accuracy. The cannon ball, indeed, is -forced from the cannon in an inappreciable portion of time; the trigger -is pulled, the fuze fired, the powder inflamed, the ball expelled, -all simultaneously to our senses. But there is no doubt that time is -occupied by every part of the process, and that the ball begins to -move at first with infinite slowness. Captain Noble is able to measure -by his chronoscope the progress of the shot in a 300-pounder gun, and -finds that the whole motion within the barrel takes place in something -less than one 200th part of a second. It is certain that no finite -force can produce motion, except in a finite space of time. The amount -of momentum communicated to a body is proportional to the accelerating -force multiplied by the time during which it acts uniformly. Thus a -slight force produces a great velocity only by long-continued action. -In a powerful shock, like that of a railway collision, the stroke of a -hammer on an anvil, or the discharge of a gun, the time is very short, -and therefore the accelerating forces brought into play are exceedingly -great, but never infinite. In the case of a large gun the powder in -exploding is said to exert for a moment a force equivalent to at least -2,800,000 horses. - - [510] *Philosophical Transactions*, 1861. *Chemical and Physical - Researches*, p. 598. - -Our belief in some of the fundamental laws of nature rests upon the -principle of continuity. Galileo is held to be the first philosopher -who consciously employed this principle in his arguments concerning -the nature of motion, and it is certain that we can never by mere -experience assure ourselves of the truth even of the first law of -motion. *A material particle*, we are told, *when not acted on by -extraneous forces will continue in the same state of rest or motion.* -This may be true, but as we can find no body which is free from the -action of extraneous causes, how are we to prove it? Only by observing -that the less the amount of those forces the more nearly is the law -found to be true. A ball rolled along rough ground is soon stopped; -along a smooth pavement it continues longer in movement. A delicately -suspended pendulum is almost free from friction against its supports, -but it is gradually stopped by the resistance of the air; place it -in the vacuous receiver of an air-pump and we find the motion much -prolonged. A large planet like Jupiter experiences almost infinitely -less friction, in comparison to its vast momentum, than we can produce -experimentally, and we find in such a case that there is not the least -evidence of the falsity of the law. Experience, then, informs us that -we may approximate indefinitely to a uniform motion by sufficiently -decreasing the disturbing forces. It is an act of inference which -enables us to travel on beyond experience, and assert that, in the -total absence of any extraneous force, motion would be absolutely -uniform. The state of rest, again, is a limiting case in which motion -is infinitely small or zero, to which we may attain, on the principle -of continuity, by successively considering cases of slower and slower -motion. There are many classes of phenomena, in which, by gradually -passing from the apparent to the obscure, we can assure ourselves of -the nature of phenomena which would otherwise be a matter of great -doubt. Thus we can sufficiently prove in the manner of Galileo, that -a musical sound consists of rapid uniform pulses, by causing strokes -to be made at intervals which we gradually diminish until the separate -strokes coalesce into a uniform hum or note. With great advantage we -approach, as Tyndall says, the sonorous through the grossly mechanical. -In listening to a great organ we cannot fail to perceive that the -longest pipes, or their partial tones, produce a tremor and fluttering -of the building. At the other extremity of the scale, there is no fixed -limit to the acuteness of sounds which we can hear; some individuals -can hear sounds too shrill for other ears, and as there is nothing in -the nature of the atmosphere to prevent the existence of undulations -far more rapid than any of which we are conscious, we may infer, by the -principle of continuity, that such undulations probably exist. - -There are many habitual actions which we perform we know not how. So -rapidly are acts of minds accomplished that analysis seems impossible. -We can only investigate them when in process of formation, observing -that the best formed habit is slowly and continuously acquired, and -it is in the early stages that we can perceive the rationale of the -process. - -Let it be observed that this principle of continuity must be held of -much weight only in exact physical laws, those which doubtless repose -ultimately upon the simple laws of motion. If we fearlessly apply -the principle to all kinds of phenomena, we may often be right in -our inferences, but also often wrong. Thus, before the development -of spectrum analysis, astronomers had observed that the more they -increased the powers of their telescopes the more nebulæ they could -resolve into distinct stars. This result had been so often found -true that they almost irresistibly assumed that all nebulæ would be -ultimately resolved by telescopes of sufficient power; yet Huggins has -in recent years proved by the spectroscope, that certain nebulæ are -actually gaseous, and in a truly nebulous state. - -The principle of continuity must have been continually employed in the -inquiries of Galileo, Newton, and other experimental philosophers, -but it appears to have been distinctly formulated for the first time -by Leibnitz. He at least claims to have first spoken of “the law of -continuity” in a letter to Bayle, printed in the *Nouvelles de la -République des Lettres*, an extract from which is given in Erdmann’s -edition of Leibnitz’s works, p. 104, under the title “Sur un Principe -Général utile à l’explication des Lois de la Nature.”[511] It has -indeed been asserted that the doctrine of the *latens processus* of -Francis Bacon involves the principle of continuity,[512] but I think -that this doctrine, like that of the *natures* of substances, is merely -a vague statement of the principle of causation. - - [511] *Life of Sir W. Hamilton*, p. 439. - - [512] Powell’s *History of Natural Philosophy*, p. 201. *Novum - Organum*, bk. ii. Aphorisms 5–7. - - -*Failure of the Law of Continuity.* - -There are certain cautions which must be given as to the application of -the principle of continuity. In the first place, where this principle -really holds true, it may seem to fail owing to our imperfect means -of observation. Though a physical law may not admit of perfectly -abrupt change, there is no limit to the approach which it may make to -abruptness. When we warm a piece of very cold ice, the absorption of -heat, the temperature, and the dilatation of the ice vary according -to apparently simple laws until we come to the zero of the Centigrade -scale. Everything is then changed; an enormous absorption of heat -takes place without any rise of temperature, and the volume of the ice -decreases as it changes into water. Unless carefully investigated, this -change appears to be perfectly abrupt; but accurate observation seems -to show that there is a certain forewarning; the ice does not turn into -water all at once, but through a small fraction of a degree the change -is gradual. All the phenomena concerned, if measured very exactly, -would be represented not by angular lines, but continuous curves, -undergoing rapid flexures; and we may probably assert with safety that -between whatever points of temperature we examine ice, there would be -found some indication, though almost infinitesimally small, of the -apparently abrupt change which was to occur at a higher temperature. -It might also be pointed out that the important and apparently simple -physical laws, such as those of Boyle and Mariotte, Dalton and -Gay-Lussac, &c., are only approximately true, and the divergences -from the simple laws are forewarnings of abrupt changes, which would -otherwise break the law of continuity. - -Secondly, it must be remembered that mathematical laws of some -complexity will probably present singular cases or negative results, -which may bear the appearance of discontinuity, as when the law of -retraction suddenly yields us with perfect abruptness the phenomenon of -total internal reflection. In the undulatory theory, however, there is -no real change of law between refraction and reflection. Faraday in the -earlier part of his career found so many substances possessing magnetic -power, that he ventured on a great generalisation, and asserted that -all bodies shared in the magnetic property of iron. His mistake, as he -afterwards discovered, consisted in overlooking the fact that though -magnetic in a certain sense, some substances have negative magnetism, -and are repelled instead of being attracted by the magnet. - -Thirdly, where we might expect to find a uniform mathematical law -prevailing, the law may undergo abrupt change at singular points, -and actual discontinuity may arise. We may sometimes be in danger of -treating under one law phenomena which really belong to different laws. -For instance, a spherical shell of uniform matter attracts an external -particle of matter with a force varying inversely as the square of the -distance from the centre of the sphere. But this law only holds true -so long as the particle is external to the shell. Within the shell -the law is wholly different, and the aggregate gravity of the sphere -becomes zero, the force in every direction being neutralised by an -exactly equal opposite force. If an infinitely small particle be in the -superficies of a sphere, the law is again different, and the attractive -power of the shell is half what it would be with regard to particles -infinitely close to the surface of the shell. Thus in approaching the -centre of a shell from a distance, the force of gravity shows double -discontinuity in passing through the shell.[513] - - [513] Thomson and Tait, *Treatise on Natural Philosophy*, vol. i. pp. - 346–351. - -It may admit of question, too, whether discontinuity is really unknown -in nature. We perpetually do meet with events which are real breaks -upon the previous law, though the discontinuity may be a sign that -some independent cause has come into operation. If the ordinary -course of the tides is interrupted by an enormous irregular wave, we -attribute it to an earthquake, or some gigantic natural disturbance. -If a meteoric stone falls upon a person and kills him, it is clearly a -discontinuity in his life, of which he could have had no anticipation. -A sudden sound may pass through the air neither preceded nor followed -by any continuous effect. Although, then, we may regard the Law of -Continuity as a principle of nature holding rigorously true in many of -the relations of natural forces, it seems to be a matter of difficulty -to assign the limits within which the law is verified. Much caution is -required in its application. - - -*Negative Arguments on the Principle of Continuity.* - -Upon the principle of continuity we may sometimes found arguments of -great force which prove an hypothesis to be impossible, because it -would involve a continual repetition of a process *ad infinitum*, or -else a purely arbitrary breach at some point. Bonnet’s famous theory -of reproduction represented every living creature as containing -germs which were perfect representatives of the next generation, so -that on the same principle they necessarily included germs of the -next generation, and so on indefinitely. The theory was sufficiently -refuted when once clearly stated, as in the following poem called the -Universe,[514] by Henry Baker:-- - - “Each seed includes a plant: that plant, again, - Has other seeds, which other plants contain: - Those other plants have all their seeds, and those - More plants again, successively inclose. - - “Thus, ev’ry single berry that we find, - Has, really, in itself whole forests of its kind, - Empire and wealth one acorn may dispense, - By fleets to sail a thousand ages hence.” - - [514] *Philosophical Transactions* (1740), vol. xli. p. 454. - -The general principle of inference, that what we know of one case must -be true of similar cases, so far as they are similar, prevents our -asserting anything which we cannot apply time after time under the same -circumstances. On this principle Stevinus beautifully demonstrated -that weights resting on two inclined planes and balancing each other -must be proportional to the lengths of the planes between their apex -and a horizontal plane. He imagined a uniform endless chain to be hung -over the planes, and to hang below in a symmetrical festoon. If the -chain were ever to move by gravity, there would be the same reason -for its moving on for ever, and thus producing a perpetual motion. As -this is absurd, the portions of the chain lying on the planes, and -equal in length to the planes, must balance each other. On similar -grounds we may disprove the existence of any *self-moving machine*; -for if it could once alter its own state of motion or rest, in however -small a degree, there is no reason why it should not do the like -time after time *ad infinitum*. Newton’s proof of his third law of -motion, in the case of gravity, is of this character. For he remarks -that if two gravitating bodies do not exert exactly equal forces in -opposite directions, the one exerting the strongest pull will carry -both away, and the two bodies will move off into space together with -velocity increasing *ad infinitum*. But though the argument might -seem sufficiently convincing, Newton in his characteristic way made -an experiment with a loadstone and iron floated upon the surface of -water.[515] In recent years the very foundation of the principle of -conservation of energy has been placed on the assumption that it is -impossible by any combination of natural bodies to produce force -continually from nothing.[516] The principle admits of application in -various subtle forms. - - [515] *Principia*, bk. i. Law iii. Corollary 6. - - [516] Helmholtz, Taylor’s *Scientific Memoirs* (1853), vol. vi. - p. 118. - -Lucretius attempted to prove, by a most ingenious argument of this -kind, that matter must be indestructible. For if a finite quantity, -however small, were to fall out of existence in any finite time, an -equal quantity might be supposed to lapse in every equal interval of -time, so that in the infinity of past time the universe must have -ceased to exist.[517] But the argument, however ingenious, seems to -fail at several points. If past time be infinite, why may not matter -have been created infinite also? It would be most reasonable, again, -to suppose the matter destroyed in any time to be proportional to the -matter then remaining, and not to the original quantity; under this -hypothesis even a finite quantity of original matter could never wholly -disappear from the universe. For like reasons we cannot hold that the -doctrine of the conservation of energy is really proved, or can ever be -proved to be absolutely true, however probable it may be regarded. - - [517] *Lucretius*, bk. i. lines 232–264. - - -*Tendency to Hasty Generalisation.* - -In spite of all the powers and advantages of generalisation, men -require no incitement to generalise; they are too apt to draw hasty and -ill-considered inferences. As Francis Bacon said, our intellects want -not wings, but rather weights of lead to moderate their course.[518] -The process is inevitable to the human mind; it begins with childhood -and lasts through the second childhood. The child that has once been -hurt fears the like result on all similar occasions, and can with -difficulty be made to distinguish between case and case. It is caution -and discrimination in the adoption of conclusions that we have chiefly -to learn, and the whole experience of life is one continued lesson -to this effect. Baden Powell has excellently described this strong -natural propensity to hasty inference, and the fondness of the human -mind for tracing resemblances real or fanciful. “Our first inductions,” -he says,[519] “are always imperfect and inconclusive; we advance -towards real evidence by successive approximations; and accordingly we -find false generalisation the besetting error of most first attempts -at scientific research. The faculty to generalise accurately and -philosophically requires large caution and long training, and is not -fully attained, especially in reference to more general views, even -by some who may properly claim the title of very accurate scientific -observers in a more limited field. It is an intellectual habit which -acquires immense and accumulating force from the contemplation of wider -analogies.” - - [518] *Novum Organum*, bk. 1 Aphorism 104. - - [519] *The Unity of Worlds and of Nature*, 2nd edit. p. 116. - -Hasty and superficial generalisations have always been the bane -of science, and there would be no difficulty in finding endless -illustrations. Between things which are the same in number there is a -certain resemblance, namely in number; but in the infancy of science -men could not be persuaded that there was not a deeper resemblance -implied in that of number. Pythagoras was not the inventor of a -mystical science of number. In the ancient Oriental religions the seven -metals were connected with the seven planets, and in the seven days -of the week we still have, and probably always shall have, a relic of -the septiform system ascribed by Dio Cassius to the ancient Egyptians. -The disciples of Pythagoras carried the doctrine of the number seven -into great detail. Seven days are mentioned in Genesis; infants acquire -their teeth at the end of seven months; they change them at the end -of seven years; seven feet was the limit of man’s height; every -seventh year was a climacteric or critical year, at which a change -of disposition took place. Then again there were the seven sages of -Greece, the seven wonders of the world, the seven rites of the Grecian -games, the seven gates of Thebes, and the seven generals destined to -conquer that city. - -In natural science there were not only the seven planets, and the seven -metals, but also the seven primitive colours, and the seven tones -of music. So deep a hold did this doctrine take that we still have -its results in many customs, not only in the seven days of the week, -but the seven years’ apprenticeship, puberty at fourteen years, the -second climacteric, and legal majority at twenty-one years, the third -climacteric. The idea was reproduced in the seven sacraments of the -Roman Catholic Church, and the seven year periods of Comte’s grotesque -system of domestic worship. Even in scientific matters the loftiest -intellects have occasionally yielded, as when Newton was misled by the -analogy between the seven tones of music and the seven colours of his -spectrum. Other numerical analogies, though rejected by Galileo, held -Kepler in thraldom; no small part of Kepler’s labours during seventeen -years was spent upon numerical and geometrical analogies of the most -baseless character; and he gravely held that there could not be more -than six planets, because there were not more than five regular solids. -Even the genius of Huyghens did not prevent him from inferring that -but one satellite could belong to Saturn, because, with those of -Jupiter and the Earth, it completed the perfect number of six. A whole -series of other superstitions and fallacies attach to the numbers six -and nine. - -It is by false generalisation, again, that the laws of nature have -been supposed to possess that perfection which we attribute to simple -forms and relations. The heavenly bodies, it was held, must move in -circles, for the circle was the perfect figure. Newton seemed to adopt -the questionable axiom that nature always proceeds in the simplest way; -in stating his first rule of philosophising, he adds:[520] “To this -purpose the philosophers say, that nature does nothing in vain, when -less will serve; for nature is pleased with simplicity, and affects -not the pomp of superfluous causes.” Keill lays down[521] as an axiom -that “The causes of natural things are such, as are the most simple, -and are sufficient to explain the phenomena: for nature always proceeds -in the simplest and most expeditious method; because by this manner of -operating the Divine Wisdom displays itself the more.” If this axiom -had any clear grounds of truth, it would not apply to proximate laws; -for even when the ultimate law is simple the results may be infinitely -diverse, as in the various elliptic, hyperbolic, parabolic, or circular -orbits of the heavenly bodies. Simplicity is naturally agreeable to a -mind of limited powers, but to an infinite mind all things are simple. - - [520] *Principia*, bk. iii, *ad initium*. - - [521] Keill, *Introduction to Natural Philosophy*, p. 89. - -Every great advance in science consists in a great generalisation, -pointing out deep and subtle resemblances. The Copernican system was -a generalisation, in that it classed the earth among the planets; it -was, as Bishop Wilkins expressed it, “the discovery of a new planet,” -but it was opposed by a more shallow generalisation. Those who argued -from the condition of things upon the earth’s surface, thought that -every object must be attached to and rest upon something else. Shall -the earth, they said, alone be free? Accustomed to certain special -results of gravity they could not conceive its action under widely -different circumstances.[522] No hasty thinker could seize the deep -analogy pointed out by Horrocks between a pendulum and a planet, -true in substance though mistaken in some details. All the advances -of modern science rise from the conception of Galileo, that in the -heavenly bodies, however apparently different their condition, we shall -ultimately recognise the same fundamental principles of mechanical -science which are true on earth. - - [522] Jeremiæ Horroccii *Opera Posthuma* (1673), pp. 26, 27. - -Generalisation is the great prerogative of the intellect, but it -is a power only to be exercised safely with much caution and after -long training. Every mind must generalise, but there are the widest -differences in the depth of the resemblances discovered and the care -with which the discovery is verified. There seems to be an innate -power of insight which a few men have possessed pre-eminently, and -which enabled them, with no exemption indeed from labour or temporary -error, to discover the one in the many. Minds of excessive acuteness -may exist, which have yet only the powers of minute discrimination, -and of storing up, in the treasure-house of memory, vast accumulations -of words and incidents. But the power of discovery belongs to a -more restricted class of minds. Laplace said that, of all inventors -who had contributed the most to the advancement of human knowledge, -Newton and Lagrange appeared to possess in the highest degree the -happy tact of distinguishing general principles among a multitude of -objects enveloping them, and this tact he conceived to be the true -characteristic of scientific genius.[523] - - [523] Young’s *Works*, vol. ii. p. 564. - - - - -CHAPTER XXVIII. - -ANALOGY. - - -As we have seen in the previous chapter, generalisation passes -insensibly into reasoning by analogy, and the difference is one of -degree. We are said to generalise when we view many objects as agreeing -in a few properties, so that the resemblance is extensive rather -than deep. When we have only a few objects of thought, but are able -to discover many points of resemblance, we argue by analogy that the -correspondence will be even deeper than appears. It may not be true -that the words are always used in such distinct senses, and there is -great vagueness in the employment of these and many logical terms; but -if any clear discrimination can be drawn between generalisation and -analogy, it is as indicated above. - -It has been said, indeed, that analogy denotes not a resemblance -between things, but between the relations of things. A pilot is a very -different man from a prime minister, but he bears the same relation to -a ship that the minister does to the state, so that we may analogically -describe the prime minister as the pilot of the state. A man differs -still more from a horse, nevertheless four men bear to three men the -same relation as four horses bear to three horses. There is a real -analogy between the tones of the Monochord, the Sages of Greece, and -the Gates of Thebes, but it does not extend beyond the fact that they -were all seven in number. Between the most discrete notions, as, for -instance, those of time and space, analogy may exist, arising from -the fact that the mathematical conditions of the lapse of time and -of motion along a line are similar. There is no identity of nature -between a word and the thing it signifies; the substance *iron* is -a heavy solid, the word *iron* is either a momentary disturbance of -the air, or a film of black pigment on white paper; but there is -analogy between words and their significates. The substance iron is -to the substance iron-carbonate, as the name iron is to the name -iron-carbonate, when these names are used according to their scientific -definitions. The whole structure of language and the whole utility -of signs, marks, symbols, pictures, and representations of various -kinds, rest upon analogy. I may hope perhaps to enter more fully upon -this important subject at some future time, and to attempt to show how -the invention of signs enables us to express, guide, and register our -thoughts. It will be sufficient to observe here that the use of words -constantly involves analogies of a subtle kind; we should often be at -a loss how to describe a notion, were we not at liberty to employ in -a metaphorical sense the name of anything sufficiently resembling it. -There would be no expression for the sweetness of a melody, or the -brilliancy of an harangue, unless it were furnished by the taste of -honey and the brightness of a torch. - -A cursory examination of the way in which we popularly use the -word analogy, shows that it includes all degrees of resemblance or -similarity. The analogy may consist only in similarity of number or -ratio, or in like relations of time and space. It may also consist in -simple resemblance between physical properties. We should not be using -the word inconsistently with custom, if we said that there was an -analogy between iron, nickel, and cobalt, manifested in the strength of -their magnetic powers. There is a still more perfect analogy between -iodine and chlorine; not that every property of iodine is identical -with the corresponding property of chlorine; for then they would -be one and the same kind of substance, and not two substances; but -every property of iodine resembles in all but degree some property of -chlorine. For almost every substance in which iodine forms a component, -a corresponding substance may be discovered containing chlorine, so -that we may confidently infer from the compounds of the one to the -compounds of the other substance. Potassium iodide crystallises in -cubes; therefore it is to be expected that potassium chloride will -also crystallise in cubes. The science of chemistry as now developed -rests almost entirely upon a careful and extensive comparison of the -properties of substances, bringing deep-lying analogies to light. -When any new substance is encountered, the chemist is guided in his -treatment of it by the analogies which it seems to present with -previously known substances. - -In this chapter I cannot hope to illustrate the all-pervading influence -of analogy in human thought and science. All science, it has been said, -at the outset, arises from the discovery of identity, and analogy is -but one name by which we denote the deeper-lying cases of resemblance. -I shall only try to point out at present how analogy between apparently -diverse classes of phenomena often serves as a guide in discovery. We -thus commonly gain the first insight into the nature of an apparently -unique object, and thus, in the progress of a science, we often -discover that we are treating over again, in a new form, phenomena -which were well known to us in another form. - - -*Analogy as a Guide in Discovery.* - -There can be no doubt that discovery is most frequently accomplished -by following up hints received from analogy, as Jeremy Bentham -remarked.[524] Whenever a phenomenon is perceived, the first impulse -of the mind is to connect it with the most nearly similar phenomenon. -If we could ever meet a thing wholly *sui generis*, presenting no -analogy to anything else, we should be incapable of investigating its -nature, except by purely haphazard trial. The probability of success -by such a process is so slight, that it is preferable to follow up the -faintest clue. As I have pointed out already (p. 418), the possible -experiments are almost infinite in number, and very numerous also are -the hypotheses upon which we may proceed. Now it is self-evident that, -however slightly superior the probability of success by one course of -procedure may be over another, the most probable one should always be -adopted first. - - [524] *Essay on Logic*, *Works*, vol. viii. p. 276. - -The chemist having discovered what he believes to be a new element, -will have before him an infinite variety of modes of treating and -investigating it. If in any of its qualities the substance displays -a resemblance to an alkaline metal, for instance, he will naturally -proceed to try whether it possesses other properties of the alkaline -metals. Even the simplest phenomenon presents so many points for notice -that we have a choice from among many hypotheses. - -It would be difficult to find a more instructive instance of the way in -which the mind is guided by analogy than in the description by Sir John -Herschel of the course of thought by which he was led to anticipate in -theory one of Faraday’s greatest discoveries. Herschel noticed that -a screw-like form, technically called helicoidal dissymmetry, was -observed in three cases, namely, in electrical helices, plagihedral -quartz crystals, and the rotation of the plane of polarisation of -light. As he said,[525] “I reasoned thus: Here are three phenomena -agreeing in a *very strange peculiarity*. Probably, this peculiarity -is a connecting link, physically speaking, among them. Now, in the -case of the crystals and the light, this probability has been turned -into certainty by my own experiments. Therefore, induction led me to -conclude that a similar connection exists, and must turn up, somehow -or other, between the electric current and polarised light, and that -the plane of polarisation would be deflected by magneto-electricity.” -By this course of analogical thought Herschel had actually been led -to anticipate Faraday’s great discovery of the influence of magnetic -strain upon polarised light. He had tried in 1822–25 to discover the -influence of electricity on light, by sending a ray of polarised light -through a helix, or near a long wire conveying an electric current. -Such a course of inquiry, followed up with the persistency of Faraday, -and with his experimental resources, would doubtless have effected the -discovery. Herschel also suggests that the plagihedral form of quartz -crystals must be due to a screw-like strain during crystallisation; but -the notion remains unverified by experiment. - - [525] *Life of Faraday*, by Bence Jones, vol. ii. p. 206. - - -*Analogy in the Mathematical Sciences.* - -Whoever wishes to acquire a deep acquaintance with Nature must observe -that there are analogies which connect whole branches of science in a -parallel manner, and enable us to infer of one class of phenomena what -we know of another. It has thus happened on several occasions that the -discovery of an unsuspected analogy between two branches of knowledge -has been the starting-point for a rapid course of discovery. The truths -readily observed in the one may be of a different character from those -which present themselves in the other. The analogy, once pointed out, -leads us to discover regions of one science yet undeveloped, to which -the key is furnished by the corresponding truths in the other science. -An interchange of aid most wonderful in its results may thus take -place, and at the same time the mind rises to a higher generalisation, -and a more comprehensive view of nature. - -No two sciences might seem at first sight more different in their -subject matter than geometry and algebra. The first deals with circles, -squares, parallelograms, and other forms in space; the latter with -mere symbols of number. Prior to the time of Descartes, the sciences -were developed slowly and painfully in almost entire independence of -each other. The Greek philosophers indeed could not avoid noticing -occasional analogies, as when Plato in the Thæetetus describes a square -number as *equally equal*, and a number produced by multiplying two -unequal factors as *oblong*. Euclid, in the 7th and 8th books of his -Elements, continually uses expressions displaying a consciousness -of the same analogies, as when he calls a number of two factors a -*plane number*, ἐπίπεδος ἀριθμός, and distinguishes a square number of -which the two factors are equal as an equal-sided and plane number, -ἰσόπλευρος καὶ ἐπίπεδος ἀριθμός. He also calls the root of a cubic -number its side, πλευρά. In the Diophantine algebra many problems of a -geometrical character were solved by algebraic or numerical processes; -but there was no general system, so that the solutions were of an -isolated character. In general the ancients were far more advanced in -geometric than symbolic methods; thus Euclid in his 4th book gives the -means of dividing a circle by purely geometric means into 2, 3, 4, 5, -6, 8, 10, 12, 15, 16, 20, 24, 30 parts, but he was totally unacquainted -with the theory of the roots of unity exactly corresponding to this -division of the circle. - -During the middle ages, on the contrary, algebra advanced beyond -geometry, and modes of solving equations were gradually discovered -by those who had no notion that at every step they were implicitly -solving geometric problems. It is true that Regiomontanus, Tartaglia, -Bombelli, and possibly other early algebraists, solved isolated -geometrical problems by the aid of algebra, but particular numbers were -always used, and no consciousness of a general method was displayed. -Vieta in some degree anticipated the final discovery, and occasionally -represented the roots of an equation geometrically, but it was -reserved for Descartes to show, in the most general manner, that every -equation may be represented by a curve or figure in space, and that -every bend, point, cusp, or other peculiarity in the curve indicates -some peculiarity in the equation. It is impossible to describe in any -adequate manner the importance of this discovery. The advantage was -two-fold: algebra aided geometry, and geometry gave reciprocal aid -to algebra. Curves such as the well-known sections of the cone were -found to correspond to quadratic equations; and it was impossible -to manipulate the equations without discovering properties of those -all-important curves. The way was thus opened for the algebraic -treatment of motions and forces, without which Newton’s *Principia* -could never have been worked out. Newton indeed was possessed by a -strong infatuation in favour of the ancient geometrical methods; but -it is well known that he employed symbolic methods to discover his -theorems, and he now and then, by some accidental use of algebraic -expression, confessed its greater power and generality. - -Geometry, on the other hand, gave great assistance to algebra, by -affording concrete representations of relations which would otherwise -be too abstract for easy comprehension. A curve of no great complexity -may give the whole history of the variations of value of a troublesome -mathematical expression. As soon as we know, too, that every regular -geometrical curve represents some algebraic equation, we are presented -by observation of mechanical movements with abundant suggestions -towards the discovery of mathematical problems. Every particle of a -carriage-wheel when moving on a level road is constantly describing -a cycloidal curve, the curious properties of which exercised the -ingenuity of all the most skilful mathematicians of the seventeenth -century, and led to important advancements in algebraic power. It may -be held that the discovery of the Differential Calculus was mainly -due to geometrical analogy, because mathematicians, in attempting to -treat algebraically the tangent of a curve, were obliged to entertain -the notion of infinitely small quantities.[526] There can be no doubt -that Newton’s fluxional, that is, geometrical mode of stating the -differential calculus, however much it subsequently retarded its -progress in England, facilitated its apprehension at first, and I -should think it almost certain that Newton discovered the principles of -the calculus geometrically. - - [526] Lacroix, *Traité Élémentaire de Calcul Différentiel et de - Calcul Intégral*, 5^{me} édit. p. 699. - -We may accordingly look upon this discovery of analogy, this happy -alliance, as Bossut calls it,[527] between geometry and algebra, -as the chief source of discoveries which have been made for three -centuries past in mathematical methods. This is certainly the opinion -of Lagrange, who says, “So long as algebra and geometry have been -separate, their progress was slow, and their employment limited; but -since these two sciences have been united, they have lent each other -mutual strength, and have marched together with a rapid step towards -perfection.” - - [527] *Histoire des Mathématiques*, vol. i. p. 298. - -The advancement of mechanical science has also been greatly aided by -analogy. An abstract and intangible existence like force demands much -power of conception, but it has a perfect concrete representative in -a line, the end of which may denote the point of application, and -the direction the line of action of the force, while the length can -be made arbitrarily to denote the amount of the force. Nor does the -analogy end here; for the moment of the force about any point, or its -product into the perpendicular distance of its line of action from the -point, is found to be represented by an area, namely twice the area -of the triangle contained between the point and the ends of the line -representing the force. Of late years a great generalisation has been -effected; the Double Algebra of De Morgan is true not only of space -relations, but of forces, so that the triangle of forces is reduced -to a case of pure geometrical addition. Nay, the triangle of lines, -the triangle of velocities, the triangle of forces, the triangle of -couples, and perhaps other cognate theorems, are reduced by analogy to -one simple theorem, which amounts to this, that there are two ways of -getting from one angular point of a triangle to another, which ways, -though different in length, are identical in their final results.[528] -In the system of quaternions of the late Sir W. R. Hamilton, these -analogies are embodied and carried out in the most general manner, so -that whatever problem involves the threefold dimensions of space, or -relations analogous to those of space, is treated by a symbolic method -of the most comprehensive simplicity. - - [528] See Goodwin, *Cambridge Philosophical Transactions* (1845), - vol. viii. p. 269. O’Brien, “On Symbolical Statics,” *Philosophical - Magazine*, 4th Series, vol. i. pp. 491, &c. See also Professor - Clerk Maxwell’s delightful *Manual of Elementary Science*, called - *Matter and Motion*, published by the Society for Promoting Christian - Knowledge. In this admirable little work some of the most advanced - results of mechanical and physical science are explained according - to the method of quaternions, but with hardly any use of algebraic - symbols. - -It ought to be added that to the discovery of analogy between the forms -of mathematical and logical expressions, we owe the greatest advance -in logical science. Boole based his extension of logical processes -upon the notion that logic is an algebra of two quantities 0 and 1. -His profound genius for symbolic investigation led him to perceive by -analogy that there must exist a general system of logical deduction, of -which the old logicians had seized only a few fragments. Mistaken as -he was in placing algebra as a higher science than logic, no one can -deny that the development of the more complex and dependent science -had advanced far beyond that of the simpler science, and that Boole, -in drawing attention to the connection, made one of the most important -discoveries in the history of science. As Descartes had wedded algebra -and geometry, so did Boole accomplish the marriage of logic and -algebra. - - -*Analogy in the Theory of Undulations.* - -There is no class of phenomena which more thoroughly illustrates alike -the power and weakness of analogy than the waves which agitate every -kind of medium. All waves, whatsoever be the matter through which -they pass, obey the principles of rhythmical or harmonic motion, -and the subject therefore presents a fine field for mathematical -generalisation. Each kind of medium may allow of waves peculiar in -their conditions, so that it is a beautiful exercise in analogical -reasoning to decide how, in making inferences from one kind of medium -to another, we must make allowance for difference of circumstances. The -waves of the ocean are large and visible, and there are the yet greater -tidal waves which extend around the globe. From such palpable cases of -rhythmical movement we pass to waves of sound, varying in length from -about 32 feet to a small fraction of an inch. We have but to imagine, -if we can, the fortieth octave of the middle C of a piano, and we reach -the undulations of yellow light, the ultra-violet being about the -forty-first octave. Thus we pass from the palpable and evident to that -which is obscure, if not incomprehensible. Yet the same phenomena of -reflection, interference, and refraction, which we find in some kinds -of waves, may be expected to occur, *mutatis mutandis*, in other kinds. - -From the great to the small, from the evident to the obscure, is not -only the natural order of inference, but it is the historical order of -discovery. The physical science of the Greek philosophers must have -remained incomplete, and their theories groundless, because they did -not understand the nature of undulations. Their systems were based -upon the notion of movement of translation from place to place. Modern -science tends to the opposite notion that all motion is alternating or -rhythmical, energy flowing onwards but matter remaining comparatively -fixed in position. Diogenes Laertius indeed correctly compared the -propagation of sound with the spreading of waves on the surface of -water when disturbed by a stone, and Vitruvius displayed a more -complete comprehension of the same analogy. It remained for Newton -to create the theory of undulatory motion in showing by mathematical -deductive reasoning that the particles of an elastic fluid by vibrating -backwards and forwards, might carry a pulse or wave moving from -the source of disturbance, while the disturbed particles return to -their place of rest. He was even able to make a first approximation -by theoretical calculation to the velocity of sound-waves in the -atmosphere. His theory of sound formed a hardly less important epoch in -science than his far more celebrated theory of gravitation. It opened -the way to all the subsequent applications of mechanical principles -to the insensible motion of molecules. He seems to have been, too, -upon the brink of another application of the same principles which -would have advanced science by a century of progress, and made him the -undisputed founder of all the theories of matter. He expressed opinions -at various times that light might be due to undulatory movements of -a medium occupying space, and in one intensely interesting sentence -remarks[529] that colours are probably vibrations of different lengths, -“much after the manner that, in the sense of hearing, nature makes -use of aërial vibrations of several bignesses to generate sounds -of divers tones, for the analogy of nature is to be observed.” He -correctly foresaw that red and yellow light would consist of the longer -undulations, and blue and violet of the shorter, while white light -would be composed of an indiscriminate mixture of waves of various -lengths. Newton almost overcame the strongest apparent difficulty of -the undulatory theory of light, namely, the propagation of light in -straight lines. For he observed that though waves of sound bend round -an obstacle to some extent, they do not do so in the same degree as -water-waves.[530] He had but to extend the analogy proportionally to -light-waves, and not only would the difficulty have vanished, but the -true theory of diffraction would have been open to him. Unfortunately -he had a preconceived theory that rays of light are bent from and -not towards the shadow of a body, a theory which for once he did not -sufficiently compare with observation to detect its falsity. I am -not aware, too, that Newton has, in any of his works, displayed an -understanding of the phenomena of interference without which his notion -of waves must have been imperfect. - - [529] Birch, *History of the Royal Society*, vol. iii. p. 262, quoted - by Young, *Works*, vol. i. p. 246. - - [530] *Opticks*, Query 28, 3rd edit. p. 337. - -While the general principles of undulatory motion will be the same -in whatever medium the motion takes place, the circumstances may be -excessively different. Between light travelling 186,000 miles per -second and sound travelling in air only about 1,100 feet in the same -time, or almost 900,000 times as slowly, we cannot expect a close -outward resemblance. There are great differences, too, in the character -of the vibrations. Gases scarcely admit of transverse vibration, so -that sound travelling in air is a longitudinal wave, the particles of -air moving backwards and forwards in the same line in which the wave -moves onwards. Light, on the other hand, appears to consist entirely -in the movement of points of force transversely to the direction of -propagation of the ray. The light-wave is partially analogous to the -bending of a rod or of a stretched cord agitated at one end. Now this -bending motion may take place in any one of an infinite number of -planes, and waves of which the planes are perpendicular to each other -cannot interfere any more than two perpendicular forces can interfere. -The complicated phenomena of polarised light arise out of this -transverse character of the luminous wave, and we must not expect to -meet analogous phenomena in atmospheric sound-waves. It is conceivable -that in solids we might produce transverse sound undulations, in which -phenomena of polarisation might be reproduced. But it would appear that -even between transverse sound and light-waves the analogy holds true -rather of the principles of harmonic motion than the circumstances of -the vibrating medium; from experiment and theory it is inferred that -the plane of polarisation in plane polarised light is perpendicular -to instead of being coincident with the direction of vibration, as it -would be in the case of transverse sound undulations. If so the laws -of elastic forces are essentially different in application to the -luminiferous ether and to ordinary solid bodies.[531] - - [531] Rankine, *Philosophical Transactions* (1856), vol. cxlvi. - p. 282. - - -*Analogy in Astronomy.* - -We shall be much assisted in gaining a true appreciation of the value -of analogy in its feebler degrees, by considering how much it has -contributed to the progress of astronomical science. Our point of -observation is so fixed with regard to the universe, and our means of -examining distant bodies are so restricted, that we are necessarily -guided by limited and apparently feeble resemblances. In many cases the -result has been confirmed by subsequent direct evidence of the most -forcible character. - -While the scientific world was divided in opinion between the -Copernican and Ptolemaic systems, it was analogy which furnished the -most satisfactory argument. Galileo discovered, by the use of his new -telescope, the four small satellites which circulate round Jupiter, -and make a miniature planetary world. These four Medicean Stars, as -they were called, were plainly seen to revolve round Jupiter in various -periods, but approximately in one plane, and astronomers irresistibly -inferred that what might happen on the smaller scale might also be -found true of the greater planetary system. This discovery gave “the -holding turn,” as Herschel expressed it, to the opinions of mankind. -Even Francis Bacon, who, little to the credit of his scientific -sagacity, had previously opposed the Copernican views, now became -convinced, saying “We affirm the solisequium of Venus and Mercury; -since it has been found by Galileo that Jupiter also has attendants.” -Nor did Huyghens think it superfluous to adopt the analogy as a valid -argument.[532] Even in an advanced stage of physical astronomy, the -Jovian system has not lost its analogical interest; for the mutual -perturbations of the four satellites pass through all their phases -within a few centuries, and thus enable us to verify in a miniature -case the principles of stability, which Laplace established for the -great planetary system. Oscillations or disturbances which in the -motions of the planets appear to be secular, because their periods -extend over millions of years, can be watched, in the case of Jupiter’s -satellites, through complete revolutions within the historical period -of astronomy.[533] - - [532] *Cosmotheoros* (1699), p. 16. - - [533] Laplace, *System of the World*, vol. ii. p. 316. - -In obtaining a knowledge of the stellar universe we must sometimes -depend upon precarious analogies. We still hold upon this ground the -opinion, entertained by Bruno as long ago as 1591, that the stars may -be suns attended by planets like our earth. This is the most probable -first assumption, and it is supported by spectrum observations, which -show the similarity of light derived from many stars with that of the -sun. But at the same time we learn by the prism that there are nebulæ -and stars in conditions widely different from anything known in our -system. In the course of time the analogy may perhaps be restored to -comparative completeness by the discovery of suns in various stages -of nebulous condensation. The history of the evolution of our own -world may be traced back in bodies less developed, or traced forwards -in systems more advanced towards the dissipation of energy, and the -extinction of life. As in a great workshop, we may perhaps see the -material work of Creation as it has progressed through thousands of -millions of years. - -In speculations concerning the physical condition of the planets and -their satellites, we depend upon analogies of a weak character. We may -be said to know that the moon has mountains and valleys, plains and -ridges, volcanoes and streams of lava, and, in spite of the absence of -air and water, the rocky surface of the moon presents so many familiar -appearances that we do not hesitate to compare them with the features -of our globe. We infer with high probability that Mars has polar snow -and an atmosphere absorbing blue rays like our own; Jupiter undoubtedly -possesses a cloudy atmosphere, possibly not unlike a magnified copy -of that surrounding the earth, but our tendency to adopt analogies -receives a salutary correction in the recently discovered fact that the -atmosphere of Uranus contains hydrogen. - -Philosophers have not stopped at these comparatively safe inferences, -but have speculated on the existence of living creatures in other -planets. Huyghens remarked that as we infer by analogy from the -dissected body of a dog to that of a pig and ox or other animal of -the same general form, and as we expect to find the same viscera, the -heart, stomach, lungs, intestines, &c., in corresponding positions, -so when we notice the similarity of the planets in many respects, -we must expect to find them alike in other respects.[534] He even -enters into an inquiry whether the inhabitants of other planets would -possess reason and knowledge of the same sort as ours, concluding in -the affirmative. Although the power of intellect might be different, -he considers that they would have the same geometry if they had any -at all, and that what is true with us would be true with them.[535] -As regards the sun, he wisely observes that every conjecture fails. -Laplace entertained a strong belief in the existence of inhabitants on -other planets. The benign influence of the sun gives birth to animals -and plants upon the surface of the earth, and analogy induces us to -believe that his rays would tend to have a similar effect elsewhere. -It is not probable that matter which is here so fruitful of life would -be sterile upon so great a globe as Jupiter, which, like the earth, -has its days and nights and years, and changes which indicate active -forces. Man indeed is formed for the temperature and atmosphere in -which he lives, and, so far as appears, could not live upon the other -planets. But there might be an infinity of organisations relative to -the diverse constitutions of the bodies of the universe. The most -active imagination cannot form any idea of such various creatures, but -their existence is not unlikely.[536] - - [534] *Cosmotheoros* (1699), p. 17. - - [535] Ibid. p. 36. - - [536] *System of the World*, vol. ii. p. 326. *Essai Philosophique*, - p. 87. - -We now know that many metals and other elements never found in organic -structures are yet capable of forming compounds with substances of -vegetable or animal origin. It is therefore just possible that at -different temperatures creatures formed of different yet analogous -compounds might exist, but it would seem indispensable that carbon -should form the basis of organic structures. We have no analogies to -lead us to suppose that in the absence of that complex element life -can exist. Could we find globes surrounded by atmospheres resembling -our own in temperature and composition, we should be almost forced to -believe them inhabited, but the probability of any analogical argument -decreases rapidly as the condition of a globe diverges from that of our -own. The Cardinal Nicholas de Cusa held long ago that the moon was -inhabited, but the absence of any appreciable atmosphere renders the -existence of inhabitants highly improbable. Speculations resting upon -weak analogies hardly belong to the scope of true science, and can only -be tolerated as an antidote to the far worse dogmas which assert that -the thousand million of persons on earth, or rather a small fraction -of them, are the sole objects of care of the Power which designed this -limitless Universe. - - -*Failures of Analogy.* - -So constant is the aid which we derive from the use of analogy in all -attempts at discovery or explanation, that it is most important to -observe in what cases it may lead us into difficulties. That which we -expect by analogy to exist - -(1) May be found to exist; - -(2) May seem not to exist, but nevertheless may really exist; - -(3) May actually be non-existent. - -In the second case the failure is only apparent, and arises from -our obtuseness of perception, the smallness of the phenomenon to be -noticed, or the disguised character in which it appears. I have already -pointed out that the analogy of sound and light seems to fail because -light does not apparently bend round a corner, the fact being that it -does so bend in the phenomena of diffraction, which present the effect, -however, in such an unexpected and minute form, that even Newton was -misled, and turned from the correct hypothesis of undulations which he -had partially entertained. - -In the third class of cases analogy fails us altogether, and we -expect that to exist which really does not exist. Thus we fail to -discover the phenomena of polarisation in sound travelling through the -atmosphere, since air is not capable of any appreciable transverse -undulations. These failures of analogy are of peculiar interest, -because they make the mind aware of its superior powers. There have -been many philosophers who said that we can conceive nothing in the -intellect which we have not previously received through the senses. -This is true in the sense that we cannot *image* them to the mind in -the concrete form of a shape or a colour; but we can speak of them -and reason concerning them; in short, we often know them in everything -but a sensuous manner. Accurate investigation shows that all material -substances retard the motion of bodies through them by subtracting -energy by impact. By the law of continuity we can frame the notion of -a vacuous space in which there is no resistance whatever, nor need we -stop there; for we have only to proceed by analogy to the case where -a medium should accelerate the motion of bodies passing through it, -somewhat in the mode which Aristotelians attributed falsely to the air. -Thus we can frame the notion of *negative density*, and Newton could -reason exactly concerning it, although no such thing exists.[537] - - [537] *Principia*, bk. ii. Section ii. Prop. x. - -In every direction of thought we may meet ultimately with similar -failures of analogy. A moving point generates a line, a moving line -generates a surface, a moving surface generates a solid, but what does -a moving solid generate? When we compare a polyhedron, or many-sided -solid, with a polygon, or plane figure of many sides, the volume of the -first is analogous to the area of the second; the face of the solid -answers to the side of the polygon; the edge of the solid to the point -of the figure; but the corner, or junction of edges in the polyhedron, -is left wholly unrepresented in the plane of the polygon. Even if -we attempted to draw the analogies in some other manner, we should -still find a geometrical notion embodied in the solid which has no -representative in the figure of two dimensions.[538] - - [538] De Morgan, *Cambridge Philosophical Transactions*, vol. xi. - Part ii. p. 246. - -Faraday was able to frame some notion of matter in a fourth condition, -which should be to gas what gas is to liquid.[539] Such substance, -he thought, would not fall far short of *radiant matter*, by which -apparently he meant the supposed caloric or matter assumed to -constitute heat, according to the corpuscular theory. Even if we could -frame the notion, matter in such a state cannot be known to exist, and -recent discoveries concerning the continuity of the solid, liquid, and -gaseous states remove the basis of the speculation. - - [539] *Life of Faraday*, vol. i. p. 216. - -From these and many other instances which might be adduced, we learn -that analogical reasoning leads us to the conception of many things -which, so far as we can ascertain, do not exist. In this way great -perplexities have arisen in the use of language and mathematical -symbols. All language depends upon analogy; for we join and arrange -words so that they may represent the corresponding junctions or -arrangements of things and their equalities. But in the use of language -we are obviously capable of forming many combinations of words to which -no corresponding meaning apparently exists. The same difficulty arises -in the use of mathematical signs, and mathematicians have needlessly -puzzled themselves about the square root of a negative quantity, which -is, in many applications of algebraic calculation, simply a sign -without any analogous meaning, there being a failure of analogy. - - - - -CHAPTER XXIX. - -EXCEPTIONAL PHENOMENA. - - -If science consists in the detection of identity and the recognition -of uniformity existing in many objects, it follows that the progress -of science depends upon the study of exceptional phenomena. Such new -phenomena are the raw material upon which we exert our faculties of -observation and reasoning, in order to reduce the new facts beneath -the sway of the laws of nature, either those laws already well known, -or those to be discovered. Not only are strange and inexplicable -facts those which are on the whole most likely to lead us to some -novel and important discovery, but they are also best fitted to -arouse our attention. So long as events happen in accordance with -our anticipations, and the routine of every-day observation is -unvaried, there is nothing to impress upon the mind the smallness of -its knowledge, and the depth of mystery, which may be hidden in the -commonest sights and objects. In early times the myriads of stars which -remained in apparently fixed relative positions upon the heavenly -sphere, received less notice from astronomers than those few planets -whose wandering and inexplicable motions formed a riddle. Hipparchus -was induced to prepare the first catalogue of stars, because a single -new star had been added to those nightly visible; and in the middle -ages two brilliant but temporary stars caused more popular interest -in astronomy than any other events, and to one of them we owe all the -observations of Tycho Brahe, the mediæval Hipparchus. - -In other sciences, as well as in that of the heavens, exceptional -events are commonly the points from which we start to explore new -regions of knowledge. It has been beautifully said that Wonder is the -daughter of Ignorance, but the mother of Invention; and though the most -familiar and slight events, if fully examined, will afford endless food -for wonder and for wisdom, yet it is the few peculiar and unlooked-for -events which most often lead to a course of discovery. It is true, -indeed, that it requires much philosophy to observe things which are -too near to us. - -The high scientific importance attaching, then, to exceptions, renders -it desirable that we should carefully consider the various modes in -which an exception may be disposed of; while some new facts will be -found to confirm the very laws to which they seem at first sight -clearly opposed, others will cause us to limit the generality of our -previous statements. In some cases the exception may be proved to be -no exception; occasionally it will prove fatal to our previous most -confident speculations; and there are some new phenomena which, without -really destroying any of our former theories, open to us wholly new -fields of scientific investigation. The study of this subject is -especially interesting and important, because, as I have before said -(p. 587), no important theory can be built up complete and perfect all -at once. When unexplained phenomena present themselves as objections -to the theory, it will often demand the utmost judgment and sagacity -to assign to them their proper place and force. The acceptance -or rejection of a theory will depend upon discriminating the one -insuperable contradictory fact from many, which, however singular and -inexplicable at first sight, may afterwards be shown to be results of -different causes, or possibly the most striking results of the very law -with which they stand in apparent conflict. - -I can enumerate at least eight classes or kinds of exceptional -phenomena, to one or other of which any supposed exception to the known -laws of nature can usually be referred; they may be briefly described -as below, and will be sufficiently illustrated in the succeeding -sections. - -(1) Imaginary, or false exceptions, that is, facts, objects, or events -which are not really what they are supposed to be. - -(2) Apparent, but congruent exceptions, which, though apparently in -conflict with a law of nature, are really in agreement with it. - -(3) Singular exceptions, which really agree with a law of nature, but -exhibit remarkable and unique results of it. - -(4) Divergent exceptions, which really proceed from the ordinary action -of known processes of nature, but which are excessive in amount or -monstrous in character. - -(5) Accidental exceptions, arising from the interference of some -entirely distinct but known law of nature. - -(6) Novel and unexplained exceptions, which lead to the discovery of a -new series of laws and phenomena, modifying or disguising the effects -of previously known laws, without being inconsistent with them. - -(7) Limiting exceptions showing the falsity of a supposed law in some -cases to which it had been extended, but not affecting its truth in -other cases. - -(8) Contradictory or real exceptions which lead us to the conclusion -that a supposed hypothesis or theory is in opposition to the phenomena -of nature, and must therefore be abandoned. - -It ought to be clearly understood that in no case is a law of nature -really thwarted or prevented from being fulfilled. The effects of a -law may be disguised and hidden from our view in some instances: in -others the law itself may be rendered inapplicable altogether; but if -a law is applicable it must be carried out. Every law of nature must -therefore be stated with the utmost generality of all the instances -really coming under it. Babbage proposed to distinguish between -*universal principles*, which do not admit of a single exception, such -as that every number ending in 5 is divisible by five, and *general -principles* which are more frequently obeyed than violated, as that -“men will be governed by what they believe to be their interest.”[540] -But in a scientific point of view general principles must be universal -as regards some distinct class of objects, or they are not principles -at all. If a law to which exceptions exist is stated without allusion -to those exceptions, the statement is erroneous. I have no right to -say that “All liquids expand by heat,” if I know that water below -4° C. does not; I ought to say, “All liquids, except water below 4° C., -expand by heat;” and every new exception discovered will falsify -the statement until inserted in it. To speak of some laws as being -*generally* true, meaning not universally but in the majority of cases, -is a hurtful abuse of the word, but is quite usual. *General* should -mean that which is true of a whole *genus* or class, and every true -statement must be true of some assigned or assignable class. - - [540] Babbage, *The Exposition of 1851*, p. 1. - - -*Imaginary or False Exceptions.* - -When a supposed exception to a law of nature is brought to our notice, -the first inquiry ought properly to be--Is there any breach of the law -at all? It may be that the supposed exceptional fact is not a fact at -all, but a mere figment of the imagination. When King Charles requested -the Royal Society to investigate the curious fact that a live fish put -into a bucket of water does not increase the weight of the bucket and -its contents, the Royal Society wisely commenced their deliberations -by inquiring whether the fact was so or not. Every statement, however -false, must have some cause or prior condition, and the real question -for the Royal Society to investigate was, how the King came to think -that the fact was so. Mental conditions, as we have seen, enter into -all acts of observation, and are often a worthy subject of inquiry. But -there are many instances in the history of science, in which trouble -and error have been caused by false assertions carelessly made, and -carelessly accepted without verification. - -The reception of the Copernican theory was much impeded by the -objection, that if the earth were moving, a stone dropped from the -top of a high tower should be left behind, and should appear to move -towards the west, just as a stone dropped from the mast-head of a -moving ship would fall behind, owing to the motion of the ship. The -Copernicans attempted to meet this grave objection in every way but -the true one, namely, showing by trial that the asserted facts are not -correct. In the first place, if a stone had been dropped with suitable -precautions from the mast-head of a moving ship, it would have fallen -close to the foot of the mast, because, by the first law of motion, it -would remain in the same state of horizontal motion communicated to it -by the mast. As the anti-Copernicans had assumed the contrary result as -certain to ensue, their argument would of course have fallen through. -Had the Copernicans next proceeded to test with great care the other -assertion involved, they would have become still better convinced of -the truth of their own theory. A stone dropped from the top of a high -tower, or into a deep well, would certainly not have been deflected -from the vertical direction in the considerable degree required to -agree with the supposed consequences of the Copernican views; but, with -very accurate observation, they might have discovered, as Benzenberg -subsequently did, a very small deflection towards the east, showing -that the eastward velocity is greater at the top than the bottom. Had -the Copernicans then been able to detect and interpret the meaning -of the small divergence thus arising, they would have found in it -corroboration of their own views. - -Multitudes of cases might be cited in which laws of nature seem to -be evidently broken, but in which the apparent breach arises from a -misapprehension of the case. It is a general law, absolutely true -of all crystals yet submitted to examination, that no crystal has -a re-entrant angle, that is an angle which towards the axis of the -crystal is greater than two right angles. Wherever the faces of a -crystal meet they produce a projecting edge, and wherever edges -meet they produce a corner. Many crystals, however, when carelessly -examined, present exceptions to this law, but closer observation -always shows that the apparently re-entrant angle really arises from -the oblique union of two distinct crystals. Other crystals seem to -possess faces contradicting all the principles of crystallography; -but careful examination shows that the supposed faces are not true -faces, but surfaces produced by the orderly junction of an immense -number of distinct thin crystalline plates, each plate being in fact -a separate crystal, in which the laws of crystallography are strictly -observed. The roughness of the supposed face, the striæ detected by -the microscope, or inference by continuity from other specimens where -the true faces of the plates are clearly seen, prove the mistaken -character of the supposed exceptions. Again, four of the faces of a -regular octahedron may become so enlarged in the crystallisation -of iron pyrites and some other substances, that the other four -faces become imperceptible and a regular tetrahedron appears to be -produced, contrary to the laws of crystallographic symmetry. Many other -crystalline forms are similarly modified, so as to produce a series of -what are called *hemihedral* forms. - -In tracing out the isomorphic relations of the elements, great -perplexity has often been caused by mistaking one substance for -another. It was pointed out that though arsenic was supposed to be -isomorphous with phosphorus, the arseniate of soda crystallised in a -form distinct from that of the corresponding phosphate. Some chemists -held this to be a fatal objection to the doctrine of isomorphism; -but it was afterwards pointed out by Clarke, that the arseniate and -phosphate in question were not corresponding compounds, as they -differed in regard to the water of crystallisation.[541] Vanadium again -appeared to be an exception to the laws of isomorphism, until it was -proved by Professor Roscoe, that what Berzelius supposed to be metallic -vanadium was really an oxide of vanadium.[542] - - [541] Daubeny’s *Atomic Theory*, p. 76. - - [542] *Bakerian Lecture, Philosophical Transactions* (1868), - vol. clviii. p. 2. - - -*Apparent but Congruent Exceptions.* - -Not unfrequently a law of nature will present results in certain -circumstances which appear to be entirely in conflict with the law -itself. Not only may the action of the law be much complicated and -disguised, but it may in various ways be reversed or inverted, so that -careless observers are misled. Ancient philosophers generally believed -that while some bodies were heavy by nature, others, such as flame, -smoke, bubbles, clouds, &c., were essentially light, or possessed a -tendency to move upwards. So acute an inquirer as Aristotle failed to -perceive the true nature of buoyancy, and the doctrine of intrinsic -lightness, expounded in his works, became the accepted view for many -centuries. It is true that Lucretius was aware why flame tends to rise, -holding that-- - - “The flame has weight, though highly rare, - Nor mounts but when compelled by heavier air.” - -Archimedes also was so perfectly acquainted with the buoyancy of bodies -immersed in water, that he could not fail to perceive the existence -of a parallel effect in air. Yet throughout the early middle ages -the light of true science could not contend with the glare of the -Peripatetic doctrine. The genius of Galileo and Newton was required to -convince people of the simple truth that all matter is heavy, but that -the gravity of one substance may be overborne by that of another, as -one scale of a balance is carried up by the preponderating weight in -the opposite scale. It is curious to find Newton gravely explaining -the difference of absolute and relative gravity, as if it were a new -discovery proceeding from his theory.[543] More than a century elapsed -before other apparent exceptions to the Newtonian philosophy were -explained away. - - [543] *Principia*, bk. ii. Prop. 20. Corollaries, 5 and 6. - -Newton himself allowed that the motion of the apsides of the moon’s -orbit appeared to be irreconcilable with the law of gravity, and -it remained for Clairaut to remove the difficulty by more complete -mathematical analysis. There must always remain, in the motions of -the heavenly bodies, discrepancies of some amount between theory and -observation; but such discrepancies have so often yielded in past times -to prolonged investigation that physicists now regard them as merely -apparent exceptions, which will afterwards be found to agree with the -law of gravity. - -The most beautiful instance of an apparent exception, is found in the -total reflection of light, which occurs when a beam of light within -a medium falls very obliquely upon the boundary separating it from a -rarer medium. The general law is that when a ray strikes the limit -between two media of different refractive indices, part of the light -is reflected and part is refracted; but when the obliquity of the ray -within the denser medium passes beyond a certain point, there is a -sudden apparent breach of continuity, and the whole of the light is -reflected. A clear reason can be given for this exceptional conduct -of the light. According to the law of refraction, the sine of the -angle of incidence bears a fixed ratio to the sine of the angle of -refraction, so that the greater of the two angles, which is always that -in the less dense medium, may increase up to a right angle; but when -the media differ in refractive power, the less angle cannot become a -right angle, as this would require the sine of an angle to be greater -than the radius. It might seem that this is an exception of the kind -described below as a limiting exception, by which a law is shown to -be inapplicable beyond certain limits; but in the explanation of the -exception according to the undulatory theory, we find that there is -really no breach of the general law. When an undulation strikes a point -in a bounding surface, spherical waves are produced and spread from -the point. The refracted ray is the resultant of an infinite number of -such spherical waves, and the bending of the ray at the common surface -of two media depends upon the comparative velocities of propagation of -the undulations in those media. But if a ray falls very obliquely upon -the surface of a rarer medium, the waves proceeding from successive -points of the surface spread so rapidly as never to intersect, and -no resultant wave will then be produced. We thus perceive that from -similar mathematical conditions arise distinct apparent effects. - -There occur from time to time failures in our best grounded -predictions. A comet, of which the orbit has been well determined, may -fail, like Lexell’s Comet, to appear at the appointed time and place in -the heavens. In the present day we should not allow such an exception -to our successful predictions to weigh against our belief in the theory -of gravitation, but should assume that some unknown body had through -the action of gravitation deflected the comet. As Clairaut remarked, -in publishing his calculations concerning the expected reappearance of -Halley’s Comet, a body which passes into regions so remote, and which -is hidden from our view during such long periods, might be exposed to -the influence of forces totally unknown to us, such as the attraction -of other comets, or of planets too far removed from the sun to be -ever perceived by us. In the case of Lexell’s Comet it was afterwards -shown, curiously enough, that its appearance was not one of a regular -series of periodical returns within the sphere of our vision, but a -single exceptional visit never to be repeated, and probably due to -the perturbing powers of Jupiter. This solitary visit became a strong -confirmation of the law of gravity with which it seemed to be in -conflict. - - -*Singular Exceptions.* - -Among the most interesting of apparent exceptions are those which I -call *singular exceptions*, because they are more or less analogous to -the singular cases or solutions which occur in mathematical science. -A general mathematical law embraces an infinite multitude of cases -which perfectly agree with each other in a certain respect. It may -nevertheless happen that a single case, while really obeying the -general law, stands out as apparently different from all the rest. -The rotation of the earth upon its axis gives to all the stars an -apparent motion of rotation from east to west; but while countless -thousands obey the rule, the Pole Star alone seems to break it. Exact -observations indeed show that it also revolves in a small circle, but -a star might happen for a short time to exist so close to the pole -that no appreciable change of place would be caused by the earth’s -rotation. It would then constitute a perfect singular exception; while -really obeying the law, it would break the terms in which it is usually -stated. In the same way the poles of every revolving body are singular -points. - -Whenever the laws of nature are reduced to a mathematical form we -may expect to meet with singular cases, and, as all the physical -sciences will meet in the mathematical principles of mechanics, there -is no part of nature where we may not encounter them. In mechanical -science the motion of rotation may be considered an exception to the -motion of translation. It is a general law that any number of parallel -forces, whether acting in the same or opposite directions, will have -a resultant which may be substituted for them with like effect. This -resultant will be equal to the algebraic sum of the forces, or the -difference of those acting in one direction and the other; it will pass -through a point which is determined by a simple formula, and which may -be described as the mean point of all the points of application of the -parallel forces (p. 364). Thus we readily determine the resultant of -parallel forces except in one peculiar case, namely, when two forces -are equal and opposite but not in the same straight line. Being equal -and opposite the amount of the resultant is nothing, yet, as the forces -are not in the same straight line, they do not balance each other. -Examining the formula for the point of application of the resultant, -we find that it gives an infinitely great magnitude, so that the -resultant is nothing at all, and acts at an infinite distance, which -is practically the same as to say that there is no resultant. Two such -forces constitute what is known in mechanical science as a *couple*, -which occasions rotatory instead of rectilinear motion, and can only be -neutralised by an equal and opposite couple of forces. - -The best instances of singular exceptions are furnished by the science -of optics. It is a general law that in passing through transparent -media the plane of vibration of polarised light remains unchanged. But -in certain liquids, some peculiar crystals of quartz, and transparent -solid media subjected to a magnetic strain, as in Faraday’s experiment -(pp. 588, 630), the plane of polarisation is rotated in a screw-like -manner. This effect is so entirely *sui generis*, so unlike any other -phenomena in nature, as to appear truly exceptional; yet mathematical -analysis shows it to be only a single case of much more general laws. -As stated by Thomson and Tait,[544] it arises from the composition -of two uniform circular motions. If while a point is moving round a -circle, the centre of that circle move upon another circle, a great -variety of curious curves will be produced according as we vary the -dimensions of the circles, the rapidity or the direction of the -motions. When the two circles are exactly equal, the rapidities nearly -so, and the directions opposite, the point will be found to move -gradually round the centre of the stationary circle, and describe a -curious star-like figure connected with the molecular motions out of -which the rotational power of the media rises. Among other singular -exceptions in optics may be placed the conical refraction of light, -already noticed (p. 540), arising from the peculiar form assumed by a -wave of light when passing through certain double-refracting crystals. -The laws obeyed by the wave are exactly the same as in other cases, -yet the results are entirely *sui generis*. So far are such cases from -contradicting the law of ordinary cases, that they afford the best -opportunities for verification. - - [544] *Treatise on Natural Philosophy*, vol. i. p. 50. - -In astronomy singular exceptions might occur, and in an approximate -manner they do occur. We may point to the rings of Saturn as objects -which, though undoubtedly obeying the law of gravity, are yet unique, -as far as our observation of the universe has gone. They agree, indeed, -with the other bodies of the planetary system in the stability of their -movements, which never diverge far from the mean position. There seems -to be little doubt that these rings are composed of swarms of small -meteoric stones; formerly they were thought to be solid continuous -rings, and mathematicians proved that if so constituted an entirely -exceptional event might have happened under certain circumstances. -Had the rings been exactly uniform all round, and with a centre of -gravity coinciding for a moment with that of Saturn, a singular case of -unstable equilibrium would have arisen, necessarily resulting in the -sudden collapse of the rings, and the fall of their debris upon the -surface of the planet. Thus in one single case the theory of gravity -would give a result wholly unlike anything else known in the mechanism -of the heavens. - -It is possible that we might meet with singular exceptions in -crystallography. If a crystal of the second or dimetric system, in -which the third axis is usually unequal to either of the other two, -happened to have the three axes equal, it might be mistaken for a -crystal of the cubic system, but would exhibit different faces and -dissimilar properties. There is, again, a possible class of diclinic -crystals in which two axes are at right angles and the third axis -inclined to the other two. This class is chiefly remarkable for its -non-existence, since no crystals have yet been proved to have such -axes. It seems likely that the class would constitute only a singular -case of the more general triclinic system, in which all three axes are -inclined to each other at various angles. Now if the diclinic form were -merely accidental, and not produced by any general law of molecular -constitution, its actual occurrence would be infinitely improbable, -just as it is infinitely improbable that any star should indicate the -North Pole with perfect exactness. - -In the curves denoting the relation between the temperature and -pressure of water there is, as shown by Professor J. Thomson, one very -remarkable point entirely unique, at which alone water can remain in -the three conditions of gas, liquid, and solid in the same vessel. It -is the triple point at which three lines meet, namely (1) the steam -line, which shows at what temperatures and pressures water is just -upon the point of becoming gaseous; (2) the ice line, showing when ice -is just about to melt; and (3) the hoar-frost line, which similarly -indicates the pressures and temperatures at which ice is capable of -passing directly into the state of gaseous vapour.[545] - - [545] Maxwell’s *Theory of Heat*, (1871), p. 175. - - -*Divergent Exceptions.* - -Closely analogous to singular exceptions are those divergent -exceptions, in which a phenomenon manifests itself in unusual magnitude -or character, without becoming subject to peculiar laws. Thus in -throwing ten coins, it happened in four cases out of 2,048 throws, -that all the coins fell with heads uppermost (p. 208); these would -usually be regarded as very singular events, and, according to the -theory of probabilities, they would be rare; yet they proceed only -from an unusual conjunction of accidental events, and from no really -exceptional causes. In all classes of natural phenomena we may expect -to meet with similar divergencies from the average, sometimes due -merely to the principles of probability, sometimes to deeper reasons. -Among every large collection of persons, we shall probably find some -persons who are remarkably large or remarkably small, giants or dwarfs, -whether in bodily or mental conformation. Such cases appear to be -not mere *lusus naturæ*, since they occur with a frequency closely -accordant with the law of error or divergence from an average, as shown -by Quetelet and Mr. Galton.[546] The rise of genius, and the occurrence -of extraordinary musical or mathematical faculties, are attributed by -Mr. Galton to the same principle of divergence. - - [546] Galton, on the Height and Weight of Boys. *Journal of the - Anthropological Institute*, 1875, p. 174. - -When several distinct forces happen to concur together, we may have -surprising or alarming results. Great storms, floods, droughts, and -other extreme deviations from the average condition of the atmosphere -thus arise. They must be expected to happen from time to time, and will -yet be very infrequent compared with minor disturbances. They are not -anomalous but only extreme events, analogous to extreme runs of luck. -There seems, indeed, to be a fallacious impression in the minds of many -persons, that the theory of probabilities necessitates uniformity in -the happening of events, so that in the same space of time there will -always be nearly the same number of railway accidents and murders. -Buckle has superficially remarked upon the constancy of such events -as ascertained by Quetelet, and some of his readers acquire the false -notion that there is a mysterious inexorable law producing uniformity -in human affairs. But nothing can be more opposed to the teachings of -the theory of probability, which always contemplates the occurrence of -unusual runs of luck. That theory shows the great improbability that -the number of railway accidents per month should be always equal, or -nearly so. The public attention is strongly attracted to any unusual -conjunction of events, and there is a fallacious tendency to suppose -that such conjunction must be due to a peculiar new cause coming -into operation. Unless it can be clearly shown that such unusual -conjunctions occur more frequently than they should do according to -the theory of probabilities, we should regard them as merely divergent -exceptions. - -Eclipses and remarkable conjunctions of the heavenly bodies may also -be regarded as results of ordinary laws which nevertheless appear to -break the regular course of nature, and never fail to excite surprise. -Such events vary greatly in frequency. One or other of the satellites -of Jupiter is eclipsed almost every day, but the simultaneous eclipse -of three satellites can only take place, according to the calculations -of Wargentin, after the lapse of 1,317,900 years. The relations of the -four satellites are so remarkable, that it is actually impossible, -according to the theory of gravity, that they should all suffer eclipse -simultaneously. But it may happen that while some of the satellites are -really eclipsed by entering Jupiter’s shadow, the others are either -occulted or rendered invisible by passing over his disk. Thus on four -occasions, in 1681, 1802, 1826, and 1843, Jupiter has been witnessed in -the singular condition of being apparently deprived of satellites. A -close conjunction of two planets always excites admiration, though such -conjunctions must occur at intervals in the ordinary course of their -motions. We cannot wonder that when three or four planets approach -each other closely, the event is long remembered. A most remarkable -conjunction of Mars, Jupiter, Saturn, and Mercury, which took place in -the year 2446 B.C., was adopted by the Chinese Emperor, Chuen Hio, as a -new epoch for the chronology of his Empire, though there is some doubt -whether the conjunction was really observed, or was calculated from the -supposed laws of motion of the planets. It is certain that on the 11th -November, 1524, the planets Venus, Jupiter, Mars, and Saturn were seen -very close together, while Mercury was only distant by about 16° or -thirty apparent diameters of the sun, this conjunction being probably -the most remarkable which has occurred in historical times. - -Among the perturbations of the planets we find divergent exceptions -arising from the peculiar accumulation of effects, as in the case of -the long inequality of Jupiter and Saturn (p. 455). Leverrier has shown -that there is one place between the orbits of Mercury and Venus, and -another between those of Mars and Jupiter, in either of which, if a -small planet happened to exist, it would suffer comparatively immense -disturbance in the elements of its orbit. Now between Mars and Jupiter -there do occur the minor planets, the orbits of which are in many cases -exceptionally divergent.[547] - - [547] Grant’s *History of Physical Astronomy*, p. 116. - -Under divergent exceptions we might place all or nearly all the -instances of substances possessing physical properties in a very high -or low degree, which were described in the chapter on Generalisation -(p. 607). Quicksilver is divergent among metals as regards its -melting point, and potassium and sodium as regards their specific -gravities. Monstrous productions and variations, whether in the animal -or vegetable kingdoms, should probably be assigned to this class of -exceptions. - -It is worthy of notice that even in such a subject as formal logic, -divergent exceptions seem to occur, not of course due to chance, -but exhibiting in an unusual degree a phenomenon which is more or -less manifested in all other cases. I pointed out in p. 141 that -propositions of the general type A = BC ꖌ *bc* are capable of -expression in six equivalent logical forms, so that they manifest in a -higher degree than any other proposition yet discovered the phenomenon -of logical equivalence. - - -*Accidental Exceptions.* - -The third and largest class of exceptions contains those which arise -from the casual interference of extraneous causes. A law may be in -operation, and, if so, must be perfectly fulfilled; but, while we -conceive that we are examining its results, we may have before us -the effects of a different cause, possessing no connexion with the -subject of our inquiry. The law is not really broken, but at the same -time the supposed exception is not illusory. It may be a phenomenon -which cannot occur but under the condition of the law in question, yet -there has been such interference that there is an apparent failure of -science. There is, for instance, no subject in which more rigorous and -invariable laws have been established than in crystallography. As a -general rule, each chemical substance possesses its own definite form, -by which it can be infallibly recognised; but the mineralogist has to -be on his guard against what are called *pseudomorphic* crystals. In -some circumstances a substance, having assumed its proper crystalline -form, may afterwards undergo chemical change; a new ingredient may be -added, a former one removed, or one element may be substituted for -another. In calcium carbonate the carbonic acid is sometimes replaced -by sulphuric acid, so that we find gypsum in the form of calcite; -other cases are known where the change is inverted and calcite is -found in the form of gypsum. Mica, talc, steatite, hematite, are -other minerals subject to these curious transmutations. Sometimes a -crystal embedded in a matrix is entirely dissolved away, and a new -mineral is subsequently deposited in the cavity as in a mould. Quartz -is thus found cast in many forms wholly unnatural to it. A still -more perplexing case sometimes occurs. Calcium carbonate is capable -of assuming two distinct forms of crystallisation, in which it bears -respectively the names of calcite and arragonite. Now arragonite, while -retaining its outward form unchanged, may undergo an internal molecular -change into calcite, as indicated by the altered cleavage. Thus we may -come across crystals apparently of arragonite, which seem to break all -the laws of crystallography, by possessing the cleavage of a different -system of crystallisation. - -Some of the most invariable laws of nature are disguised by -interference of unlooked-for causes. While the barometer was yet a -new and curious subject of investigation, its theory, as stated by -Torricelli and Pascal, seemed to be contradicted by the fact that -in a well-constructed instrument the mercury would often stand far -above 31 inches in height. Boyle showed[548] that mercury could be -made to stand as high as 75 inches in a perfectly cleansed tube, or -about two and a half times as high as could be due to the pressure of -the atmosphere. Many theories about the pressure of imaginary fluids -were in consequence put forth,[549] and the subject was involved in -much confusion until the adhesive or cohesive force between glass and -mercury, when brought into perfect contact, was pointed out as the real -interfering cause. It seems to me, however, that the phenomenon is not -thoroughly understood as yet. - - [548] *Discourse to the Royal Society*, 28th May, 1684. - - [549] Robert Hooke’s *Posthumous Works*, p. 365. - -Gay-Lussac observed that the temperature of boiling water was very -different in some kinds of vessels from what it was in others. It is -only when in contact with metallic surfaces or sharply broken edges -that the temperature is fixed at 100° C. The suspended freezing of -liquids is another case where the action of a law of nature appears -to be interrupted. Spheroidal ebullition was at first sight a most -anomalous phenomenon; it was almost incredible that water should not -boil in a red-hot vessel, or that ice could actually be produced in a -red-hot crucible. These paradoxical results are now fully explained as -due to the interposition of a non-conducting film of vapour between the -globule of liquid and the sides of the vessel. The feats of conjurors -who handle liquid metals are accounted for in the same manner. At one -time the *passive state* of steel was regarded as entirely anomalous. -It may be assumed as a general law that when pieces of electro-negative -and electro-positive metal are placed in nitric acid, and made to touch -each other, the electro-negative metal will undergo rapid solution. But -when iron is the electro-negative and platinum the electro-positive, -the solution of the iron entirely and abruptly ceases. Faraday -ingeniously proved that this effect is due to a thin film of oxide of -iron, which forms upon the surface of the iron and protects it.[550] - - [550] *Experimental Researches in Electricity*, vol. ii. pp. 240–245. - -The law of gravity is so simple, and disconnected from the other laws -of nature, that it never suffers any disturbance, and is in no way -disguised, but by the complication of its own effects. It is otherwise -with those secondary laws of the planetary system which have only -an empirical basis. The fact that all the long known planets and -satellites have a similar motion from west to east is not necessitated -by any principles of mechanics, but points to some common condition -existing in the nebulous mass from which our system has been evolved. -The retrograde motions of the satellites of Uranus constituted a -distinct breach in this law of uniform direction, which became all -the more interesting when the single satellite of Neptune was also -found to be retrograde. It now became probable, as Baden Powell well -observed, that the anomaly would cease to be singular, and become a -case of another law, pointing to some general interference which has -taken place on the bounds of the planetary system. Not only have the -satellites suffered from this perturbance, but Uranus is also anomalous -in having an axis of rotation lying nearly in the ecliptic; and Neptune -constitutes a partial exception to the empirical law of Bode concerning -the distances of the planets, which circumstance may possibly be due to -the same disturbance. - -Geology is a science in which accidental exceptions are likely to -occur. Only when we find strata in their original relative positions -can we surely infer that the order of succession is the order of -time. But it not uncommonly happens that strata are inverted by the -bending and doubling action of extreme pressure. Landslips may carry -one body of rock into proximity with an unrelated series, and produce -results apparently inexplicable.[551] Floods, streams, icebergs, and -other casual agents, may lodge remains in places where they would be -wholly unexpected. Though such interfering causes have been sometimes -wrongly supposed to explain important discoveries, the geologist must -bear the possibility of interference in mind. Scarcely more than a -century ago it was held that fossils were accidental productions of -nature, mere forms into which minerals had been shaped by no peculiar -cause. Voltaire appears not to have accepted such an explanation; but -fearing that the occurrence of fossil fishes on the Alps would support -the Mosaic account of the deluge, he did not hesitate to attribute -them to the remains of fishes accidentally brought there by pilgrims. -In archæological investigations the greatest caution is requisite -in allowing for secondary burials in ancient tombs and tumuli, for -imitations, forgeries, casual coincidences, disturbance by subsequent -races or by other archæologists. In common life extraordinary events -will happen from time to time, as when a shepherdess in France was -astonished at an iron chain falling out of the sky close to her, the -fact being that Gay-Lussac had thrown it out of his balloon, which was -passing over her head at the time. - - [551] Murchison’s *Silurian System*, vol. ii. p. 733, &c. - - -*Novel and Unexplained Exceptions.* - -When a law of nature appears to fail because some other law has -interfered with its action, two cases may present themselves;--the -interfering law may be a known one, or it may have been previously -undetected. In the first case, which we have sufficiently considered in -the preceding section, we have nothing to do but calculate as exactly -as possible the amount of interference, and make allowance for it; the -apparent failure of the law under examination should then disappear. -But in the second case the results may be much more important. A -phenomenon which cannot be explained by any known laws may indicate the -interference of undiscovered natural forces. The ancients could not -help perceiving that the general tendency of bodies downwards failed -in the case of the loadstone, nor would the doctrine of essential -lightness explain the exception, since the substance drawn upwards by -the loadstone is a heavy metal. We now see that there was no breach in -the perfect generality of the law of gravity, but that a new form of -energy manifested itself in the loadstone for the first time. - -Other sciences show us that laws of nature, rigorously true and exact, -may be developed by those who are ignorant of more complex phenomena -involved in their application. Newton’s comprehension of geometrical -optics was sufficient to explain all the ordinary refractions and -reflections of light. The simple laws of the bending of rays apply -to all rays, whatever the character of the undulations composing -them. Newton suspected the existence of other classes of phenomena -when he spoke of rays as *having sides*; but it remained for later -experimentalists to show that light is a transverse undulation, like -the bending of a rod or cord. - -Dalton’s atomic theory is doubtless true of all chemical compounds, -and the essence of it is that the same compound will always be found -to contain the same elements in the same definite proportions. Pure -calcium carbonate contains 48 parts by weight of oxygen to 40 of -calcium and 12 of carbon. But when careful analyses were made of a -great many minerals, this law appeared to fail. What was unquestionably -the same mineral, judging by its crystalline form and physical -properties, would give varying proportions of its components, and -would sometimes contain unusual elements which yet could not be set -down as mere impurities. Dolomite, for instance, is a compound of the -carbonates of magnesia and lime, but specimens from different places -do not exhibit any fixed ratio between the lime and magnesia. Such -facts could be reconciled with the laws of Dalton only by supposing the -interference of a new law, that of Isomorphism. - -It is now established that certain elements are related to each other, -so that they can, as it were, step into each other’s places without -apparently altering the shapes of the crystals which they constitute. -The carbonates of iron, calcium, and magnesium, are nearly identical -in their crystalline forms, hence they may crystallise together in -harmony, producing mixed minerals of considerable complexity, which -nevertheless perfectly verify the laws of equivalent proportions. This -principle of isomorphism once established, not only explains what -was formerly a stumbling-block, but gives valuable aid to chemists -in deciding upon the constitution of new salts, since compounds of -isomorphous elements which have identical crystalline forms must -possess corresponding chemical formulæ. - -We may expect that from time to time extraordinary phenomena will -be discovered, and will lead to new views of nature. The recent -observation, for instance, that the resistance of a bar of selenium to -a current of electricity is affected in an extraordinary degree by rays -of light falling upon the selenium, points to a new relation between -light and electricity. The allotropic changes which sulphur, selenium, -and phosphorus undergo by an alteration in the amount of latent heat -which they contain, will probably lead at some future time to important -inferences concerning the molecular constitution of solids and liquids. -The curious substance ozone has perplexed many chemists, and Andrews -and Tait thought that it afforded evidence of the decomposition of -oxygen by the electric discharge. The researches of Sir B. C. Brodie -negative this notion, and afford evidence of the real constitution of -the substance,[552] which still, however, remains exceptional in its -properties and relations, and affords a hope of important discoveries -in chemical theory. - - [552] *Philosophical Transactions* (1872), vol. clxii. No. 23. - - -*Limiting Exceptions.* - -We pass to cases where exceptional phenomena are actually -irreconcilable with a law of nature previously regarded as true. Error -must now be allowed to have been committed, but the error may be more -or less extensive. It may happen that a law holding rigorously true of -the facts actually under notice had been extended by generalisation to -other series of facts then unexamined. Subsequent investigation may -show the falsity of this generalisation, and the result must be to -limit the law for the future to those objects of which it is really -true. The contradiction to our previous opinions is partial and not -total. - -Newton laid down as a result of experiment that every ray of -homogeneous light has a definite refrangibility, which it preserves -throughout its course until extinguished. This is one case of the -general principle of undulatory movement, which Herschel stated under -the title “Principle of Forced Vibrations” (p. 451), and asserted to -be absolutely without exception. But Herschel himself described in -the *Philosophical Transactions* for 1845 a curious appearance in -a solution of quinine; as viewed by transmitted light the solution -appeared colourless, but in certain aspects it exhibited a beautiful -celestial blue tint. Curiously enough the colour is seen only in the -first portion of liquid which the light enters. Similar phenomena in -fluor-spar had been described by Brewster in 1838. Professor Stokes, -having minutely investigated the phenomena, discovered that they were -more or less present in almost all vegetable infusions, and in a number -of mineral substances. He came to the conclusion that this phenomenon, -called by him Fluorescence, could only be explained by an alteration -in the refrangibility of the rays of light; he asserts that light-rays -of very short length of vibration in falling upon certain atoms excite -undulations of greater length, in opposition to the principle of -forced vibrations. No complete explanation of the mode of change is -yet possible, because it depends upon the intimate constitution of -the atoms of the substances concerned; but Professor Stokes believes -that the principle of forced vibrations is true only so long as the -excursions of an atom are very small compared with the magnitude of the -complex molecules.[553] - - [553] *Philosophical Transactions* (1852), vol. cxlii. pp. 465, 548, - &c. - -It is well known that in Calorescence the refrangibility of rays is -increased and the wave-length diminished. Rays of obscure heat and low -refrangibility may be concentrated so as to heat a solid substance, and -make it give out rays belonging to any part of the spectrum, and it -seems probable that this effect arises from the impact of distinct but -conflicting atoms. Nor is it in light only that we discover limiting -exceptions to the law of forced vibrations; for if we notice gentle -waves lapping upon the stones at the edge of a lake we shall see that -each larger wave in breaking upon a stone gives rise to a series of -smaller waves. Thus there is constantly in progress a degradation in -the magnitude of water-waves. The principle of forced vibrations seems -then to be too generally stated by Herschel, but it must be a difficult -question of mechanical theory to discriminate the circumstances in -which it does and does not hold true. - -We sometimes foresee the possible existence of exceptions yet unknown -by experience, and limit the statement of our discoveries accordingly. -Extensive inquiries have shown that all substances yet examined fall -into one of two classes; they are all either ferro-magnetic, that is, -magnetic in the same way as iron, or they are diamagnetic like bismuth. -But it does not follow that every substance must be ferro-magnetic or -diamagnetic. The magnetic properties are shown by Sir W. Thomson[554] -to depend upon the specific inductive capacities of the substance -in three rectangular directions. If these inductive capacities are -all positive, we have a ferro-magnetic substance; if negative, a -diamagnetic substance; but if the specific inductive capacity were -positive in one direction and negative in the others, we should have -an exception to previous experience, and could not place the substance -under either of the present recognised classes. - - [554] *Philosophical Magazine*, 4th Series, vol. i. p. 182. - -So many gases have been reduced to the liquid state, and so many solids -fused, that scientific men rather hastily adopted the generalisation -that all substances could exist in all three states. A certain number -of gases, such as oxygen, hydrogen, and nitrogen, have resisted all -efforts to liquefy them, and it now seems probable from the experiments -of Dr. Andrews that they are limiting exceptions. He finds that above -31° C. carbonic acid cannot be liquefied by any pressure he could -apply, whereas below this temperature liquefaction is always possible. -By analogy it becomes probable that even hydrogen might be liquefied if -cooled to a very low temperature. We must modify our previous views, -and either assert that *below a certain critical temperature* every -gas may be liquefied, or else we must assume that a highly condensed -gas is, when above the critical temperature, undistinguishable from -a liquid. At the same time we have an explanation of a remarkable -exception presented by liquid carbonic acid to the general rule that -gases expand more by heat than liquids. Liquid carbonic acid was found -by Thilorier in 1835 to expand more than four times as much as air; -but by the light of Andrews’ experiments we learn to regard the liquid -as rather a highly condensed gas than an ordinary liquid, and it is -actually possible to reduce the gas to the apparently liquid condition -without any abrupt condensation.[555] - - [555] Maxwell, *Theory of Heat*, p. 123. - -Limiting exceptions occur most frequently in the natural sciences -of Botany, Zoology, Geology, &c., the laws of which are empirical. -In innumerable instances the confident belief of one generation -has been falsified by the wider observation of a succeeding one. -Aristotle confidently held that all swans are white,[556] and the -proposition seemed true until not a hundred years ago black swans were -discovered in Western Australia. In zoology and physiology we may -expect a fundamental identity to exist in the vital processes, but -continual discoveries show that there is no limit to the apparently -anomalous expedients by which life is reproduced. Alternate generation, -fertilisation for several successive generations, hermaphroditism, are -opposed to all we should expect from induction founded upon the higher -animals. But such phenomena are only limiting exceptions showing that -what is true of one class is not true of another. In certain of the -cephalopoda we meet the extraordinary fact that an arm of the male is -cast off and lives independently until it encounters the female. - - [556] *Prior Analytics*, ii. 2, 8, and elsewhere. - - -*Real Exceptions to Supposed Laws.* - -The exceptions which we have lastly to consider are the most important -of all, since they lead to the entire rejection of a law or theory -before accepted. No law of nature can fail; there are no such things -as real exceptions to real laws. Where contradiction exists it must -be in the mind of the experimentalist. Either the law is imaginary -or the phenomena which conflict with it; if, then, by our senses we -satisfy ourselves of the actual occurrence of the phenomena, the law -must be rejected as illusory. The followers of Aristotle held that -nature abhors a vacuum, and thus accounted for the rise of water -in a pump. When Torricelli pointed out the visible fact that water -would not rise more than 33 feet in a pump, nor mercury more than -about 30 inches in a glass tube, they attempted to represent these -facts as limiting exceptions, saying that nature abhorred a vacuum to -a certain extent and no further. But the Academicians del Cimento -completed their discomfiture by showing that if we remove the pressure -of the surrounding air, and in proportion as we remove it, nature’s -feelings of abhorrence decrease and finally disappear altogether. Even -Aristotelian doctrines could not stand such direct contradiction. - -Lavoisier’s ideas concerning the constitution of acids received -complete refutation. He named oxygen the *acid generator*, because he -believed that all acids were compounds of oxygen, a generalisation -based on insufficient data. Berthollet, as early as 1789, proved by -analysis that hydrogen sulphide and prussic acid, both clearly acting -the part of acids, were devoid of oxygen; the former might perhaps have -been interpreted as a limiting exception, but when so powerful an acid -as hydrogen chloride (muriatic acid) was found to contain no oxygen the -theory had to be relinquished. Berzelius’ theory of the dual formation -of chemical compounds met a similar fate. - -It is obvious that all conclusive *experimenta crucis* constitute real -exceptions to the supposed laws of the theory which is overthrown. -Newton’s corpuscular theory of light was not rejected on account of its -absurdity or inconceivability, for in these respects it is, as we have -seen, far superior to the undulatory theory. It was rejected because -certain small fringes of colour did not appear in the exact place -and of the exact size in which calculation showed that they ought to -appear according to the theory (pp. 516–521). One single fact clearly -irreconcilable with a theory involves its rejection. In the greater -number of cases, what appears to be a fatal exception may be afterwards -explained away as a singular or disguised result of the laws with which -it seems to conflict, or as due to the interference of extraneous -causes; but if we fail thus to reduce the fact to congruity, it remains -more powerful than any theories or any dogmas. - -Of late years not a few of the favourite doctrines of geologists have -been rudely destroyed. It was the general belief that human remains -were to be found only in those deposits which are actually in progress -at the present day, so that the creation of man appeared to have taken -place in this geological age. The discovery of a single worked flint -in older strata and in connexion with the remains of extinct mammals -was sufficient to explode such a doctrine. Similarly, the opinions -of geologists have been altered by the discovery of the Eozoön in the -Laurentian rocks of Canada; it was previously held that no remains of -life occurred in any older strata than those of the Cambrian system. As -the examination of the strata of the globe becomes more complete, our -views of the origin and succession of life upon the globe must undergo -many changes. - - -*Unclassed Exceptions.* - -At every period of scientific progress there will exist a multitude -of unexplained phenomena which we know not how to regard. They are -the outstanding facts upon which the labours of investigators must -be exerted,--the ore from which the gold of future discovery is to -be extracted. It might be thought that, as our knowledge of the laws -of nature increases, the number of such exceptions should decrease; -but, on the contrary, the more we know the more there is yet to -explain. This arises from several reasons; in the first place, the -principal laws and forces in nature are numerous, so that he who bears -in mind the wonderfully large numbers developed in the doctrine of -combinations, will anticipate the existence of immensely numerous -relations of one law to another. When we are once in possession of a -law, we are potentially in possession of all its consequences; but -it does not follow that the mind of man, so limited in its powers -and capacities, can actually work them all out in detail. Just as -the aberration of light was discovered empirically, though it should -have been foreseen, so there are multitudes of unexplained facts, the -connexion of which with laws of nature already known to us, we should -perceive, were we not hindered by the imperfection of our deductive -powers. But, in the second place, as will be more fully pointed out, -it is not to be supposed that we have approximated to an exhaustive -knowledge of nature’s powers. The most familiar facts may teem with -indications of forces, now secrets hidden from us, because we have -not mind-directed eyes to discriminate them. The progress of science -will consist in the discovery from time to time of new exceptional -phenomena, and their assignment by degrees to one or other of the -heads already described. When a new fact proves to be merely a false, -apparent, singular, divergent, or accidental exception, we gain a more -minute and accurate acquaintance with the effects of laws already known -to exist. We have indeed no addition to what was implicitly in our -possession, but there is much difference between knowing the laws of -nature and perceiving all their complicated effects. Should a new fact -prove to be a limiting or real exception, we have to alter, in part or -in whole, our views of nature, and are saved from errors into which we -had fallen. Lastly, the new fact may come under the sixth class, and -may eventually prove to be a novel phenomenon, indicating the existence -of new laws and forces, complicating but not otherwise interfering with -the effects of laws and forces previously known. - -The best instance which I can find of an unresolved exceptional -phenomenon, consists in the anomalous vapour-densities of phosphorus, -arsenic, mercury, and cadmium. It is one of the most important -laws of chemistry, discovered by Gay-Lussac, that equal volumes of -gases exactly correspond to equivalent weights of the substances. -Nevertheless phosphorus and arsenic give vapours exactly twice as -dense as they should do by analogy, and mercury and cadmium diverge -in the other direction, giving vapours half as dense as we should -expect. We cannot treat these anomalies as limiting exceptions, and -say that the law holds true of substances generally but not of these; -for the properties of gases (p. 601), usually admit of the widest -generalisations. Besides, the preciseness of the ratio of divergence -points to the real observance of the law in a modified manner. We might -endeavour to reduce the exceptions by doubling the atomic weights of -phosphorus and arsenic, and halving those of mercury and cadmium. But -this step has been maturely considered by chemists, and is found to -conflict with all the other analogies of the substances and with the -principle of isomorphism. One of the most probable explanations is, -that phosphorus and arsenic produce vapour in an allotropic condition, -which might perhaps by intense heat be resolved into a simpler gas of -half the density; but facts are wanting to support this hypothesis, and -it cannot be applied to the other two exceptions without supposing that -gases and vapours generally are capable of resolution into something -simpler. In short, chemists can at present make nothing of these -anomalies. As Hofmann says, “Their philosophical interpretation belongs -to the future.... They may turn out to be typical facts, round which -many others of the like kind may come hereafter to be grouped; and -they may prove to be allied with special properties, or dependent on -particular conditions as yet unsuspected.”[557] - - [557] Hofmann’s *Introduction to Chemistry*, p. 198. - -It would be easy to point out a great number of other unexplained -anomalies. Physicists assert, as an absolutely universal law, that -in liquefaction heat is absorbed;[558] yet sulphur is at least an -apparent exception. The two substances, sulphur and selenium, are, in -fact, very anomalous in their relations to heat. Sulphur may be said -to have two melting points, for, though liquid like water at 120° C., -it becomes quite thick and tenacious between 221° and 249°, and melts -again at a higher temperature. Both sulphur and selenium may be thrown -into several curious states, which chemists conveniently dispose of by -calling them *allotropic*, a term freely used when they are puzzled -to know what has happened. The chemical and physical history of iron, -again, is full of anomalies; not only does it undergo inexplicable -changes of hardness and texture in its alloys with carbon and other -elements, but it is almost the only substance which conveys sound with -greater velocity at a higher than at a lower temperature, the velocity -increasing from 20° to 100° C., and then decreasing. Silver also is -anomalous in regard to sound. These are instances of inexplicable -exceptions, the bearing of which must be ascertained in the future -progress of science. - - [558] Stewart’s *Elementary Treatise on Heat*, p. 80. - -When the discovery of new and peculiar phenomena conflicting with -our theories of the constitution of nature is reported to us, it -becomes no easy task to steer a philosophically correct course between -credulity and scepticism. We are not to assume, on the one hand, that -there is any limit to the wonders which nature can present to us. -Nothing except the contradictory is really impossible, and many things -which we now regard as common-place were considered as little short -of the miraculous when first perceived. The electric telegraph was -a visionary dream among mediæval physicists;[559] it has hardly yet -ceased to excite our wonder; to our descendants centuries hence it -will probably appear inferior in ingenuity to some inventions which -they will possess. Now every strange phenomenon may be a secret spring -which, if rightly touched, will open the door to new chambers in the -palace of nature. To refuse to believe in the occurrence of anything -strange would be to neglect the most precious chances of discovery. -We may say with Hooke, that “the believing strange things possible -may perhaps be an occasion of taking notice of such things as another -would pass by without regard as useless.” We are not, therefore, -to shut our ears even to such apparently absurd stories as those -concerning second-sight, clairvoyance, animal magnetism, ode force, -table-turning, or any of the popular delusions which from time to time -are current. The facts recorded concerning these matters are facts in -some sense or other, and they demand explanation, either as new natural -phenomena, or as the results of credulity and imposture. Most of the -supposed phenomena referred to have been, or by careful investigation -would doubtless be, referred to the latter head, and the absence of -scientific ability in many of those who describe them is sufficient to -cast a doubt upon their value. - - [559] Jevons, *Proceedings of the Manchester Literary and - Philosophical Society*, 6th March, 1877, vol. xvi. p. 164. See also - Mr. W. E. A. Axon’s note on the same subject, ibid. p. 166. - -It is to be remembered that according to the principle of the inverse -method of probability, the probability of any hypothetical explanation -is affected by the probability of each other possible explanation. If -no other reasonable explanation could be suggested, we should be forced -to look upon spiritualist manifestations as indicating mysterious -causes. But as soon as it is shown that fraud has been committed in -several important cases, and that in other cases persons in a credulous -and excited state of mind have deceived themselves, the probability -becomes very considerable that similar explanations may apply to most -like manifestations. The performances of conjurors sufficiently prove -that it requires no very great skill to perform tricks the *modus -operandi* of which shall entirely escape the notice of spectators. It -is on these grounds of probability that we should reject the so-called -spiritualist stories, and not simply because they are strange. - -Certainly in the obscure phenomena of mind, those relating to memory, -dreams, somnambulism, and other peculiar states of the nervous -system, there are many inexplicable and almost incredible facts, and -it is equally unphilosophical to believe or to disbelieve without -clear evidence. There are many facts, too, concerning the instincts -of animals, and the mode in which they find their way from place to -place, which are at present quite inexplicable. No doubt there are many -strange things not dreamt of in our philosophy, but this is no reason -why we should believe in every strange thing which is reported to have -happened. - - - - -CHAPTER XXX. - -CLASSIFICATION. - - -The extensive subject of Classification has been deferred to a late -part of this treatise, because it involves questions of difficulty, and -did not seem naturally to fall into an earlier place. But it must not -be supposed that, in now formally taking up the subject, we are for -the first time entertaining the notion of classification. All logical -inference involves classification, which is indeed the necessary -accompaniment of the action of judgment. It is impossible to detect -similarity between objects without thereby joining them together in -thought, and forming an incipient class. Nor can we bestow a common -name upon objects without implying the existence of a class. Every -common name is the name of a class, and every name of a class is a -common name. It is evident also that to speak of a general notion or -concept is but another way of speaking of a class. Usage leads us to -employ the word classification in some cases and not in others. We -are said to form the *general notion* parallelogram when we regard -an infinite number of possible four-sided rectilinear figures as -resembling each other in the common property of possessing parallel -sides. We should be said to form a *class*, Trilobite, when we place -together in a museum a number of specimens resembling each other in -certain defined characters. But the logical nature of the operation -is the same in both cases. We form a *class* of figures called -parallelograms and we form a *general notion* of trilobites. - -Science, it was said at the outset, is the detection of identify, and -classification is the placing together, either in thought or in actual -proximity of space, those objects between which identity has been -detected. Accordingly, the value of classification is co-extensive with -the value of science and general reasoning. Whenever we form a class -we reduce multiplicity to unity, and detect, as Plato said, the one in -the many. The result of such classification is to yield generalised -knowledge, as distinguished from the direct and sensuous knowledge of -particular facts. Of every class, so far as it is correctly formed, the -principle of substitution is true, and whatever we know of one object -in a class we know of the other objects, so far as identity has been -detected between them. The facilitation and abbreviation of mental -labour is at the bottom of all mental progress. The reasoning faculties -of Newton were not different in nature from those of a ploughman; the -difference lay in the extent to which they were exerted, and the number -of facts which could be treated. Every thinking being generalises more -or less, but it is the depth and extent of his generalisations which -distinguish the philosopher. Now it is the exertion of the classifying -and generalising powers which enables the intellect of man to cope -in some degree with the infinite number of natural phenomena. In the -chapters upon combinations and permutations it was made evident, that -from a few elementary differences immense numbers of combinations -can be produced. The process of classification enables us to resolve -these combinations, and refer each one to its place according to one -or other of the elementary circumstances out of which it was produced. -We restore nature to the simple conditions out of which its endless -variety was developed. As Professor Bowen has said,[560] “The first -necessity which is imposed upon us by the constitution of the mind -itself, is to break up the infinite wealth of Nature into groups and -classes of things, with reference to their resemblances and affinities, -and thus to enlarge the grasp of our mental faculties, even at the -expense of sacrificing the minuteness of information which can be -acquired only by studying objects in detail. The first efforts in -the pursuit of knowledge, then, must be directed to the business -of classification. Perhaps it will be found in the sequel, that -classification is not only the beginning, but the culmination and the -end, of human knowledge.” - - [560] *A Treatise on Logic, or, the Laws of Pure Thought*, by Francis - Bowen, Professor of Moral Philosophy in Harvard College, Cambridge, - United States, 1866, p. 315. - - -*Classification Involving Induction.* - -The purpose of classification is the detection of the laws of nature. -However much the process may in some cases be disguised, classification -is not really distinct from the process of perfect induction, whereby -we endeavour to ascertain the connexions existing between properties of -the objects under treatment. There can be no use in placing an object -in a class unless something more than the fact of being in the class is -implied. If we arbitrarily formed a class of metals and placed therein -a selection from the list of known metals made by ballot, we should -have no reason to expect that the metals in question would resemble -each other in any points except that they are metals, and have been -selected by the ballot. But when chemists select from the list the five -metals, potassium, sodium, cæsium, rubidium, and lithium and call them -the Alkaline metals, a great deal is implied in this classification. -On comparing the qualities of these substances they are all found to -combine very energetically with oxygen, to decompose water at all -temperatures, and to form strongly basic oxides, which are highly -soluble in water, yielding powerfully caustic and alkaline hydrates -from which water cannot be expelled by heat. Their carbonates are also -soluble in water, and each metal forms only one chloride. It may also -be expected that each salt of one of the metals will correspond to a -salt of each other metal, there being a general analogy between the -compounds of these metals and their properties. - -Now in forming this class of alkaline metals, we have done more than -merely select a convenient order of statement. We have arrived at a -discovery of certain empirical laws of nature, the probability being -very considerable that a metal which exhibits some of the properties -of alkaline metals will also possess the others. If we discovered -another metal whose carbonate was soluble in water, and which -energetically combined with water at all temperatures, producing a -strongly basic oxide, we should infer that it would form only a single -chloride, and that generally speaking, it would enter into a series -of compounds corresponding to the salts of the other alkaline metals. -The formation of this class of alkaline metals then, is no mere matter -of convenience; it is an important and successful act of inductive -discovery, enabling us to register many undoubted propositions as -results of perfect induction, and to make a great number of inferences -depending upon the principles of imperfect induction. - -An excellent instance as to what classification can do, is found in -Mr. Lockyer’s researches on the sun.[561] Wanting some guide as to -what more elements to look for in the sun’s photosphere, he prepared -a classification of the elements according as they had or had not -been traced in the sun, together with a detailed statement of the -chief chemical characters of each element. He was then able to observe -that the elements found in the sun were for the most part those -forming stable compounds with oxygen. He then inferred that other -elements forming stable oxides would probably exist in the sun, and -he was rewarded by the discovery of five such metals. Here we have -empirical and tentative classification leading to the detection of the -correlation between existence in the sun, and the power of forming -stable oxides and then leading by imperfect induction to the discovery -of more coincidences between these properties. - - [561] *Proceedings of the Royal Society*, November, 1873, vol. xxi. - p. 512. - -Professor Huxley has defined the process of classification in the -following terms.[562] “By the classification of any series of objects, -is meant the actual or ideal arrangement together of those which are -like and the separation of those which are unlike; the purpose of this -arrangement being to facilitate the operations of the mind in clearly -conceiving and retaining in the memory the characters of the objects in -question.” - - [562] *Lectures on the Elements of Comparative Anatomy*, 1864, p. 1. - -This statement is doubtless correct, so far as it goes, but it does -not include all that Professor Huxley himself implicitly treats under -classification. He is fully aware that deep correlations, or in -other terms deep uniformities or laws of nature, will be disclosed -by any well chosen and profound system of classification. I should -therefore propose to modify the above statement, as follows:--“By -the classification of any series of objects, is meant the actual or -ideal arrangement together of those which are like and the separation -of those which are unlike, the purpose of this arrangement being, -primarily, to disclose the correlations or laws of union of properties -and circumstances, and, secondarily, to facilitate the operations -of the mind in clearly conceiving and retaining in the memory the -characters of the objects in question.” - - -*Multiplicity of Modes of Classification.* - -In approaching the question how any given group of objects may be -best classified, let it be remarked that there must generally be an -unlimited number of modes of classifying a group of objects. Misled, as -we shall see, by the problem of classification in the natural sciences, -philosophers seem to think that in each subject there must be one -essentially natural system of classification which is to be selected, -to the exclusion of all others. This erroneous notion probably arises -also in part from the limited powers of thought and the inconvenient -mechanical conditions under which we labour. If we arrange the books -in a library catalogue, we must arrange them in some one order; if we -compose a treatise on mineralogy, the minerals must be successively -described in some one arrangement; if we treat such simple things as -geometrical figures, they must be taken in some fixed order. We shall -naturally select that arrangement which appears to be most convenient -and instructive for our principal purpose. But it does not follow -that this method of arrangement possesses any exclusive excellence, -and there will be usually many other possible arrangements, each -valuable in its own way. A perfect intellect would not confine itself -to one order of thought, but would simultaneously regard a group of -objects as classified in all the ways of which they are capable. Thus -the elements may be classified according to their atomicity into the -groups of monads, dyads, triads, tetrads, pentads, and hexads, and -this is probably the most instructive classification; but it does not -prevent us from also classifying them according as they are metallic -or non-metallic, solid, liquid or gaseous at ordinary temperatures, -useful or useless, abundant or scarce, ferro-magnetic or diamagnetic, -and so on. - -Mineralogists have spent a great deal of labour in trying to discover -the supposed natural system of classification for minerals. They have -constantly encountered the difficulty that the chemical composition -does not run together with the crystallographic form, and the various -physical properties of the mineral. Substances identical in the -forms of their crystals, especially those belonging to the first or -cubical system of crystals, are often found to have no resemblance -in chemical composition. The same substance, again, is occasionally -found crystallised in two essentially different crystallographic -forms; calcium carbonate, for instance, appearing as calc-spar and -arragonite. The simple truth is that if we are unable to discover -any correspondence, or, as we may call it, any *correlation* between -the properties of minerals, we cannot make any one arrangement which -will enable us to treat all these properties in a single system of -classification. We must classify minerals in as many different ways -as there are different groups of unrelated properties of sufficient -importance. Even if, for the purpose of describing minerals -successively in a treatise, we select one chief system, that, for -instance, having regard to chemical composition, we ought mentally to -regard the minerals as classified in all other useful modes. - -Exactly the same may be said of the classification of plants. An -immense number of different modes of classifying plants have been -proposed at one time or other, an exhaustive account of which will be -found in the article on classification in Rees’s “Cyclopædia,” or in -the introduction to Lindley’s “Vegetable Kingdom.” There have been the -Fructists, such as Cæsalpinus, Morison, Hermann, Boerhaave or Gaertner, -who arranged plants according to the form of the fruit. The Corollists, -Rivinus, Ludwig, and Tournefort, paid attention chiefly to the number -and arrangement of the parts of the corolla. Magnol selected the calyx -as the critical part, while Sauvage arranged plants according to their -leaves; nor are these instances more than a small selection from the -actual variety of modes of classification which have been tried. Of -such attempts it may be said that every system will probably yield some -information concerning the relations of plants, and it is only after -trying many modes that it is possible to approximate to the best. - - -*Natural and Artificial Systems of Classification.* - -It has been usual to distinguish systems of classification as natural -and artificial, those being called natural which seemed to express the -order of existing things as determined by nature. Artificial methods of -classification, on the other hand, included those formed for the mere -convenience of men in remembering or treating natural objects. - -The difference, as it is commonly regarded, has been well described -by Ampére,[563] as follows: “We can distinguish two kinds of -classifications, the natural and the artificial. In the latter kind, -some characters, arbitrarily chosen, serve to determine the place of -each object; we abstract all other characters, and the objects are thus -found to be brought near to or to be separated from each other, often -in the most bizarre manner. In natural systems of classification, on -the contrary, we employ concurrently all the characters essential to -the objects with which we are occupied, discussing the importance of -each of them; and the results of this labour are not adopted unless -the objects which present the closest analogy are brought most near -together, and the groups of the several orders which are formed from -them are also approximated in proportion as they offer more similar -characters. In this way it arises that there is always a kind of -connexion, more or less marked, between each group and the group which -follows it.” - - [563] *Essai sur la Philosophie des Sciences*, p. 9. - -There is much, however, that is vague and logically false in this -and other definitions which have been proposed by naturalists to -express their notion of a natural system. We are not informed how the -*importance* of a resemblance is to be determined, nor what is the -measure of the *closeness* of analogy. Until all the words employed -in a definition are made clear in meaning, the definition itself is -worse than useless. Now if the views concerning classification here -upheld are true, there can be no sharp and precise distinction between -natural and artificial systems. All arrangements which serve any -purpose at all must be more or less natural, because, if closely enough -scrutinised, they will involve more resemblances than those whereby the -class was defined. - -It is true that in the biological sciences there would be one -arrangement of plants or animals which would be conspicuously -instructive, and in a certain sense natural, if it could be attained, -and it is that after which naturalists have been in reality striving -for nearly two centuries, namely, that *arrangement which would display -the genealogical descent of every form from the original life germ*. -Those morphological resemblances upon which the classification of -living beings is almost always based are inherited resemblances, and -it is evident that descendants will usually resemble their parents and -each other in a great many points. - -I have said that a natural is distinguished from an arbitrary or -artificial system only in degree. It will be found almost impossible -to arrange objects according to any circumstance without finding that -some correlation of other circumstances is thus made apparent. No -arrangement could seem more arbitrary than the common alphabetical -arrangement according to the initial letter of the name. But we cannot -scrutinise a list of names of persons without noticing a predominance -of Evans’s and Jones’s, under the letters E and J, and of names -beginning with Mac under the letter M. The predominance is so great -that we could not attribute it to chance, and inquiry would of course -show that it arose from important facts concerning the nationality -of the persons. It would appear that the Evans’s and Jones’s were of -Welsh descent, and those whose names bear the prefix Mac of Keltic -descent. With the nationality would be more or less strictly correlated -many peculiarities of physical constitution, language, habits, or -mental character. In other cases I have been interested in noticing -the empirical inferences which are displayed in the most arbitrary -arrangements. If a large register of the names of ships be examined -it will often be found that a number of ships bearing the same name -were built about the same time, a correlation due to the occurrence of -some striking incident shortly previous to the building of the ships. -The age of ships or other structures is usually correlated with their -general form, nature of materials, &c., so that ships of the same name -will often resemble each other in many points. - -It is impossible to examine the details of some of the so-called -artificial systems of classification of plants, without finding that -many of the classes are natural in character. Thus in Tournefort’s -arrangement, depending almost entirely on the formation of the corolla, -we find the natural orders of the Labiatæ, Cruciferæ, Rosaceæ, -Umbelliferæ, Liliaceæ, and Papilionaceæ, recognised in his 4th, 5th, -6th, 7th, 9th, and 10th classes. Many of the classes in Linnæus’ -celebrated sexual system also approximate to natural classes. - - -*Correlation of Properties.* - -Habits and usages of language are apt to lead us into the error of -imagining that when we employ different words we always mean different -things. In introducing the subject of classification nominally I was -careful to draw the reader’s attention to the fact that all reasoning -and all operations of scientific method really involve classification, -though we are accustomed to use the name in some cases and not in -others. The name *correlation* requires to be used with the same -qualification. Things are correlated (*con*, *relata*) when they are -so related or bound to each other that *where one is the other is, and -where one is not the other is not*. Throughout this work we have then -been dealing with correlations. In geometry the occurrence of three -equal angles in a triangle is correlated with the existence of three -equal sides; in physics gravity is correlated with inertia; in botany -exogenous growth is correlated with the possession of two cotyledons, -or the production of flowers with that of spiral vessels. Wherever a -proposition of the form A = B is true there correlation exists. But it -is in the classificatory sciences especially that the word correlation -has been employed. - -We find it stated that in the class Mammalia the possession of two -occipital condyles, with a well-ossified basi-occipital, is correlated -with the possession of mandibles, each ramus of which is composed of a -single piece of bone, articulated with the squamosal element of the -skull, and also with the possession of mammæ and non-nucleated red -blood-corpuscles. Professor Huxley remarks[564] that this statement of -the character of the class mammalia is something more than an arbitrary -definition; it is a statement of a law of correlation or co-existence -of animal structures, from which most important conclusions are -deducible. It involves a generalisation to the effect that in nature -the structures mentioned are always found associated together. This -amounts to saying that the formation of the class mammalia involves an -act of inductive discovery, and results in the establishment of certain -empirical laws of nature. Professor Huxley has excellently expressed -the mode in which discoveries of this kind enable naturalists to make -deductions or predictions with considerable confidence, but he has also -pointed out that such inferences are likely from time to time to prove -mistaken. I will quote his own words: - - [564] *Lectures on the Elements of Comparative Anatomy, and on the - Classification of Animals*, 1864, p. 3. - -“If a fragmentary fossil be discovered, consisting of no more than -a ramus of a mandible, and that part of the skull with which it -articulated, a knowledge of this law may enable the palæontologist to -affirm, with great confidence, that the animal of which it formed a -part suckled its young, and had non-nucleated red blood-corpuscles; and -to predict that should the back part of that skull be discovered, it -will exhibit two occipital condyles and a well-ossified basi-occipital -bone. - -“Deductions of this kind, such as that made by Cuvier in the famous -case of the fossil opossum of Montmartre, have often been verified, -and are well calculated to impress the vulgar imagination; so that -they have taken rank as the triumphs of the anatomist. But it should -carefully be borne in mind, that, like all merely empirical laws, which -rest upon a comparatively narrow observational basis, the reasoning -from them may at any time break down. If Cuvier, for example, had had -to do with a fossil Thylacinus instead of a fossil Opossum, he would -not have found the marsupial bones, though the inflected angle of the -jaw would have been obvious enough. And so, though, practically, -any one who met with a characteristically mammalian jaw would be -justified in expecting to find the characteristically mammalian occiput -associated with it; yet, he would be a bold man indeed, who should -strictly assert the belief which is implied in this expectation, viz., -that at no period of the world’s history did animals exist which -combined a mammalian occiput with a reptilian jaw, or *vice versâ*.” - -One of the most distinct and remarkable instances of correlation in -the animal world is that which occurs in ruminating animals, and which -could not be better stated than in the following extract from the -classical work of Cuvier:[565] - - [565] *Ossemens Fossiles*, 4th edit. vol. i. p. 164. Quoted by - Huxley, *Lectures*, &c., p. 5. - -“I doubt if any one would have divined, if untaught by observation, -that all ruminants have the foot cleft, and that they alone have it. I -doubt if any one would have divined that there are frontal horns only -in this class: that those among them which have sharp canines for the -most part lack horns. - -“However, since these relations are constant, they must have some -sufficient cause; but since we are ignorant of it, we must make good -the defect of the theory by means of observation: it enables us to -establish empirical laws which become almost as certain as rational -laws when they rest on sufficiently repeated observations; so that -now whoso sees merely the print of a cleft foot may conclude that the -animal which left this impression ruminated, and this conclusion is as -certain as any other in physics or morals. This footprint alone then, -yields, to him who observes it, the form of the teeth, the form of the -jaws, the form of the vertebræ, the form of all the bones of the legs, -of the thighs, of the shoulders, and of the pelvis of the animal which -has passed by: it is a surer mark than all those of Zadig.” - -We meet with a good instance of the purely empirical correlation -of circumstances when we classify the planets according to their -densities and periods of axial rotation.[566] If we examine a table -specifying the usual astronomical elements of the solar system, we find -that four planets resemble each other very closely in the period of -axial rotation, and the same four planets are all found to have high -densities, thus:-- - - [566] Chambers, *Descriptive Astronomy*, 1st edit. p. 23. - - Name of Period of Axial - Planet. Rotation. Density. - - Mercury 24 hours 5 minutes 7·94 - Venus 23 " 21 " 5·33 - Earth 23 " 56 " 5·67 - Mars 24 " 37 " 5·84 - -A similar table for the other larger planets, is as follows:-- - - Jupiter 9 hours 55 minutes 1·36 - Saturn 10 " 29 " ·74 - Uranus 9 " 30 " ·97 - Neptune -- " -- 1·02 - -It will be observed that in neither group is the equality of the -rotational period or the density more than rudely approximate; -nevertheless the difference of the numbers in the first and second -group is so very well marked, the periods of the first being at least -double and the densities four or five times those of the second, that -the coincidence cannot be attributed to accident. The reader will -also notice that the first group consists of the planets nearest to -the sun; that with the exception of the earth none of them possess -satellites; and that they are all comparatively small. The second group -are furthest from the sun, and all of them possess several satellites, -and are comparatively great. Therefore, with but slight exceptions, the -following correlations hold true:-- - -Interior planets. Long period. Small size. High Density. No satellites. -Exterior " Short " Great " Low " Many " - -These coincidences point with much probability to a difference in the -origin of the two groups, but no further explanation of the matter is -yet possible. - -The classification of comets according to their periods by Mr. -Hind and Mr. A. S. Davies, tends to establish the conclusion that -distinct groups of comets have been brought into the solar system -by the attractive powers of Jupiter, Uranus, or other planets.[567] -The classification of nebulæ as commenced by the two Herschels, and -continued by Lord Rosse, Mr. Huggins, and others, will probably lead -at some future time to the discovery of important empirical laws -concerning the constitution of the universe. The minute examination and -classification of meteorites, as carried on by Mr. Sorby and others, -seems likely to afford us an insight into the formation of the heavenly -bodies. - - [567] *Philosophical Magazine*, 4th Series, vol. xxxix. p. 396; - vol. xl. p. 183; vol. xli. p. 44. See also Proctor, *Popular Science - Review*, October 1874, p. 350. - -We should never fail to remember the slightest and most inexplicable -correlations, for they may prove of importance in the future. -Discoveries begin when we are least expecting them. It is a significant -fact, for instance, that the greater number of variable stars are of -a reddish colour. Not all variable stars are red, nor all red stars -variable; but considering that only a small fraction of the observed -stars are known to be variable, and only a small fraction are red, the -number which fall into both classes is too great to be accidental.[568] -It is also remarkable that the greater number of stars possessing great -proper motion are double stars, the star 61 Cygni being especially -noticeable in this respect.[569] The correlation in these cases is -not without exception, but the preponderance is so great as to point -to some natural connexion, the exact nature of which must be a matter -for future investigation. Herschel remarked that the two double stars -61 Cygni and α Centauri of which the orbits were well ascertained, -evidently belonged to the same family or genus.[570] - - [568] Humboldt, *Cosmos* (Bohn), vol. iii. p. 224. - - [569] Baily, British *Association Catalogue*, p. 48. - - [570] *Outlines of Astronomy*, § 850, 4th edit. p. 578. - - -*Classification in Crystallography.* - -Perhaps the most perfect and instructive instance of classification -which we can find is furnished by the science of crystallography -(p. 133). The system of arrangement now generally adopted is -conspicuously natural, and is even mathematically perfect. A crystal -consists in every part of similar molecules similarly related to the -adjoining molecules, and connected with them by forces the nature of -which we can only learn by their apparent effects. But these forces -are exerted in space of three dimensions, so that there is a limited -number of suppositions which can be entertained as to the relations of -these forces. In one case each molecule will be similarly related to -all those which are next to it; in a second case, it will be similarly -related to those in a certain plane, but differently related to those -not in that plane. In the simpler cases the arrangement of molecules is -rectangular; in the remaining cases oblique either in one or two planes. - -In order to simplify the explanation and conception of the complicated -phenomena which crystals exhibit, an hypothesis has been invented which -is an excellent instance of the Descriptive Hypotheses before mentioned -(p. 522). Crystallographers imagine that there are within each crystal -certain axes, or lines of direction, by the comparative length and the -mutual inclination of which the nature of the crystal is determined. -In one class of crystals there are three such axes lying in one plane, -and a fourth perpendicular to that plane; but in all the other classes -there are imagined to be only three axes. Now these axes can be varied -in three ways as regards length: they may be (1) all equal, or (2) two -equal and one unequal, or (3) all unequal. They may also be varied in -four ways as regards direction: (1) they may be all at right angles -to each other; (2) two axes may be oblique to each other and at right -angles to the third; (3) two axes may be at right angles to each other -and the third oblique to both; (4) the three axes may be all oblique. -Now, if all the variations as regards length were combined with those -regarding direction, it would seem to be possible to have twelve -classes of crystals in all, the enumeration being then logically and -geometrically complete. But as a matter of empirical observation, many -of these classes are not found to occur, oblique axes being seldom or -never equal. There remain seven recognised classes of crystals, but -even of these one class is not positively known to be represented in -nature. - -The first class of crystals is defined by possessing three equal -rectangular axes, and equal elasticity in all directions. The primary -or simple form of the crystals is the cube, but by the removal of the -corners of the cube by planes variously inclined to the axes, we have -the regular octohedron, the dodecahedron, and various combinations of -these forms. Now it is a law of this class of crystals that as each -axis is exactly like each other axis, every modification of any corner -of a crystal must be repeated symmetrically with regard to the other -axes; thus the forms produced are symmetrical or regular, and the -class is called the *Regular System* of crystals. It includes a great -variety of substances, some of them being elements, such as carbon in -the form of diamond, others more or less complex compounds, such as -rock-salt, potassium iodide and bromide, the several kinds of alum, -fluor-spar, iron bisulphide, garnet, spinelle, &c. No correlation -then is apparent between the form of crystallisation and the chemical -composition. But what we have to notice is that the physical properties -of the crystallised substances with regard to light, heat, electricity, -&c., are closely similar. Light and heat undulations, wherever they -enter a crystal of the regular system, spread with equal rapidity in -all directions, just as they would in a uniform fluid. Crystals of the -regular system accordingly do not in any case exhibit the phenomena -of double refraction, unless by mechanical compression we alter the -conditions of elasticity. These crystals, again, expand equally in all -directions when heated, and if we could cut a sufficiently large plate -from a cubical crystal, and examine the sound vibrations of which it -is capable, we should find that they indicated an equal elasticity -in every direction. Thus we see that a great number of important -properties are correlated with that of crystallisation in the regular -system, and as soon as we know that the primary form of a substance -is the cube, we are able to infer with approximate certainty that it -possesses all these properties. The class of regular crystals is then -an eminently natural class, one disclosing many general laws connecting -together the physical and mechanical properties of the substances -classified. - -In the second class of crystals, called the dimetric, square prismatic, -or pyramidal system, there are also three axes at right angles to each -other; two of the axes are equal, but the third or principal axis is -unequal, being either greater or less than either of the other two. In -such crystals accordingly the elasticity and other properties are alike -in all directions perpendicular to the principal axis, but vary in all -other directions. If a point within a crystal of this system be heated, -the heat spreads with equal rapidity in planes perpendicular to the -principal axis, but more or less rapidly in the direction of this axis, -so that the isothermal surface is an ellipsoid of revolution round that -axis. - -Nearly the same statement may be made concerning the third or hexagonal -or rhombohedral system of crystals, in which there are three axes lying -in one plane and meeting at angles of 60°, while the fourth axis is -perpendicular to the other three. The hexagonal prism and rhombohedron -are the commonest forms assumed by crystals of this system, and in -ice, quartz, and calc-spar, we have abundance of beautiful specimens -of the various shapes produced by the modification of the primitive -form. Calc-spar alone is said to crystallise in at least 700 varieties -of form. Now of all the crystals belonging both to this and the -dimetric class, we know that a ray of light passing in the direction -of the principal axis will be refracted singly as in a crystal of the -regular system; but in every other direction the light will suffer -double refraction being separated into two rays, one of which obeys -the ordinary law of refraction, but the other a much more complicated -law. The other physical properties vary in an analogous manner. Thus -calc-spar expands by heat in the direction of the principal axis, but -contracts a little in directions perpendicular to it. So closely are -the physical properties correlated that Mitscherlich, having observed -the law of expansion in calc-spar, was enabled to predict that the -double refracting power of the substance would be decreased by a rise -of temperature, as was proved by experiment to be the case. - -In the fourth system, called the trimetric, rhombic, or right prismatic -system, there are three axes, at right angles, but all unequal in -length. It may be asserted in general terms that the mechanical -properties vary in such crystals in every direction, and heat spreads -so that the isothermal surface is an ellipsoid with three unequal axes. - -In the remaining three classes, called the monoclinic, diclinic, and -triclinic, the axes are more or less oblique, and at the same time -unequal. The complication of phenomena is therefore greatly increased, -and it need only be stated that there are always two directions in -which a ray is singly refracted, but that in all other directions -double refraction takes place. The conduction of heat is unequal in -all directions, the isothermal surface being an ellipsoid of three -unequal axes. The relations of such crystals to other phenomena are -often very complicated, and hardly yet reduced to law. Some crystals, -called pyro-electric, manifest vitreous electricity at some points of -their surface, and resinous electricity at other points when rising in -temperature, the character of the electricity being changed when the -temperature sinks again. This production of electricity is believed to -be connected with the hemihedral character of the crystals exhibiting -it. The crystalline structure of a substance again influences its -magnetic behaviour, the general law being that the direction in which -the molecules of a crystal are most approximated tends to place itself -axially or equatorially between the poles of a magnet, respectively -as the body is magnetic or diamagnetic. Further questions arise if we -apply pressure to crystals. Thus doubly refracting crystals with one -principal axis acquire two axes when the pressure is perpendicular in -direction to the principal axis. - -All the phenomena peculiar to crystalline bodies are thus closely -correlated with the formation of the crystal, or will almost -certainly be found to be so as investigation proceeds. It is upon -empirical observation indeed that the laws of connexion are in the -first place founded, but the simple hypothesis that the elasticity -and approximation of the particles vary in the directions of the -crystalline axes allows of the application of deductive reasoning. The -whole of the phenomena are gradually being proved to be consistent with -this hypothesis, so that we have in this subject of crystallography -a beautiful instance of successful classification, connected with -a nearly perfect physical hypothesis. Moreover this hypothesis was -verified experimentally as regards the mechanical vibrations of sound -by Savart, who found that the vibrations in a plate of biaxial crystal -indicated the existence of varying elasticity in varying directions. - - -*Classification an Inverse and Tentative Operation.* - -If attempts at so-called natural classification are really attempts -at perfect induction, it follows that they are subject to the remarks -which were made upon the inverse character of the inductive process, -and upon the difficulty of every inverse operation (pp. 11, 12, 122, -&c.). There will be no royal road to the discovery of the best system, -and it will even be impossible to lay down rules of procedure to -assist those who are in search of a good arrangement. The only logical -rule would be as follows:--Having given certain objects, group them -in every way in which they can be grouped, and then observe in which -method of grouping the correlation of properties is most conspicuously -manifested. But this method of exhaustive classification will in almost -every case be impracticable, owing to the immensely great number of -modes in which a comparatively small number of objects may be grouped -together. About sixty-three elements have been classified by chemists -in six principal groups as monad, dyad, triad, &c., elements, the -numbers in the classes varying from three to twenty elements. Now if we -were to calculate the whole number of ways in which sixty-three objects -can be arranged in six groups, we should find the number to be so great -that the life of the longest lived man would be wholly inadequate -to enable him to go through these possible groupings. The rule of -exhaustive arrangement, then, is absolutely impracticable. It follows -that mere haphazard trial cannot as a general rule give any useful -result. If we were to write the names of the elements in succession -upon sixty-three cards, throw them into a ballot-box, and draw them -out haphazard in six handfuls time after time, the probability is -excessively small that we should take them out in a specified order, -that for instance at present adopted by chemists. - -The usual mode in which an investigator proceeds to form a -classification of a new group of objects seems to consist in -tentatively arranging them according to their most obvious -similarities. Any two objects which present a close resemblance to -each other will be joined and formed into the rudiment of a class, the -definition of which will at first include all the apparent points of -resemblance. Other objects as they come to our notice will be gradually -assigned to those groups with which they present the greatest number -of points of resemblance, and the definition of a class will often -have to be altered in order to admit them. The early chemists could -hardly avoid classing together the common metals, gold, silver, copper, -lead, and iron, which present such conspicuous points of similarity as -regards density, metallic lustre, malleability, &c. With the progress -of discovery, however, difficulties began to present themselves in such -a grouping. Antimony, bismuth, and arsenic are distinctly metallic -as regards lustre, density, and some chemical properties, but are -wanting in malleability. The recently discovered tellurium presents -greater difficulties, for it has many of the physical properties of -metal, and yet all its chemical properties are analogous to those of -sulphur and selenium, which have never been regarded as metals. Great -chemical differences again are discovered by degrees between the -five metals mentioned; and the class, if it is to have any chemical -validity, must be made to include other elements, having none of the -original properties on which the class was founded. Hydrogen is a -transparent colourless gas, and the least dense of all substances; yet -in its chemical analogies it is a metal, as suggested by Faraday[571] -in 1838, and almost proved by Graham;[572] it must be placed in the -same class as silver. In this way it comes to pass that almost every -classification which is proposed in the early stages of a science will -be found to break down as the deeper similarities of the objects come -to be detected. The most obvious points of difference will have to be -neglected. Chlorine is a gas, bromine a liquid, and iodine a solid, -and at first sight these might have seemed formidable circumstances to -overlook; but in chemical analogy the substances are closely united. -The progress of organic chemistry, again, has yielded wholly new ideas -of the similarities of compounds. Who, for instance, would recognise -without extensive research a close similarity between glycerine and -alcohol, or between fatty substances and ether? The class of paraffins -contains three substances gaseous at ordinary temperatures, several -liquids, and some crystalline solids. It required much insight to -detect the analogy which exists between such apparently different -substances. - - [571] *Life of Faraday*, vol. ii. p. 87. - - [572] *Proceedings of the Royal Society*, vol. xvii. p. 212. - *Chemical and Physical Researches*, reprint, by Young and Angus - Smith, p. 290. - -The science of chemistry now depends to a great extent on a correct -classification of the elements, as will be learnt by consulting -the able article on Classification by Professor G. C. Foster in -Watts’ *Dictionary of Chemistry*. But the present system of chemical -classification was not reached until at least three previous false -systems had been long entertained. And though there is much reason to -believe that the present mode of classification according to atomicity -is substantially correct, errors may yet be discovered in the details -of the grouping. - - -*Symbolic Statement of the Theory of Classification.* - -The theory of classification can be explained in the most complete -and general manner, by reverting for a time to the use of the Logical -Alphabet, which was found to be of supreme importance in Formal Logic. -That form expresses the necessary classification of all objects and -ideas as depending on the laws of thought, and there is no point -concerning the purpose and methods of classification which may not -be stated precisely by the use of letter combinations, the only -inconvenience being the abstract form in which the subject is thus -represented. - -If we pay regard only to three qualities in which things may resemble -each other, namely, the qualities A, B, C, there are according to -the laws of thought eight possible classes of objects, shown in the -fourth column of the Logical Alphabet (p. 94). If there exist objects -belonging to all these eight classes, it follows that the qualities A, -B, C, are subject to no conditions except the primary laws of thought -and things (p. 5). There is then no special law of nature to discover, -and, if we arrange the objects in any one order rather than another, it -must be for the purpose of showing that the combinations are logically -complete. - -Suppose, however, that there are but four kinds of objects possessing -the qualities A, B, C, and that these kinds are represented by the -combinations ABC, A*b*C, *a*B*c*, *abc*. The order of arrangement will -now be of importance; for if we place them in the order - - { ABC { A*b*C - { *a*B*c* { *abc* - -placing the B’s first and those which are *b*’s last, we shall perhaps -overlook the law of correlation of properties involved. But if we -arrange the combinations as follows - - { ABC { *a*B*c* - { A*b*C { *abc* - -it becomes apparent at once that where A is, and only where A is, the -property C is to be found, B being indifferently present and absent. -The second arrangement then would be called a natural one, as rendering -manifest the conditions under which the combinations exist. - -As a further instance, let us suppose that eight objects are presented -to us for classification, which exhibit combinations of the five -properties, A, B, C, D, E, in the following manner:-- - - ABC*d*E *a*BC*d*E - AB*cde* *a*B*cde* - A*b*CDE *ab*CDE - A*bc*D*e* *abc*D*e* - -They are now classified, so that those containing A stand first, and -those devoid of A second, but no other property seems to be correlated -with A. Let us alter this arrangement and group the combinations thus:-- - - ABC*d*E A*b*CDE - AB*cde* A*bc*D*e* - *a*BC*d*E *ab*CDE - *a*B*cde* *abc*D*e* - -It requires little examination to discover that in the first group B is -always present and D absent, whereas in the second group, B is always -absent and D present. This is the result which follows from a law of -the form B = d (p. 136), so that in this mode of arrangement we readily -discover correlation between two letters. Altering the groups again as -follows:-- - - ABC*d*E AB*cde* - *a*BC*d*E *a*B*cde* - A*b*CDE A*bc*D*e* - *ab*CDE *abc*D*e*, - -we discover another evident correlation between C and E. Between A and -the other letters, or between the two pairs of letters B, D and C, E, -there is no logical connexion. - -This example may seem tedious, but it will be found instructive in this -way. We are classifying only eight objects or combinations, in each -of which only five qualities are considered. There are only two laws -of correlation between four of those five qualities, and those laws -are of the simplest logical character. Yet the reader would hardly -discover what those laws are, and confidently assign them by rapid -contemplation of the combinations, as given in the first group. Several -tentative classifications must probably be made before we can resolve -the question. Let us now suppose that instead of eight objects and five -qualities, we have, say, five hundred objects and fifty qualities. If -we were to attempt the same method of exhaustive grouping which we -before employed, we should have to arrange the five hundred objects in -fifty different ways, before we could be sure that we had discovered -even the simpler laws of correlation. But even the successive -grouping of all those possessing each of the fifty properties would -not necessarily give us all the laws. There might exist complicated -relations between several properties simultaneously, for the detection -of which no rule of procedure whatever can be given. - - -*Bifurcate Classification.* - -Every system of classification ought to be formed on the principles of -the Logical Alphabet. Each superior class should be divided into two -inferior classes, distinguished by the possession and non-possession -of a single specified difference. Each of these minor classes, again, -is divisible by any other quality whatever which can be suggested, -and thus every classification logically consists of an infinitely -extended series of subaltern genera and species. The classifications -which we form are in reality very small fragments of those which would -correctly and fully represent the relations of existing things. But if -we take more than four or five qualities into account, the number of -subdivisions grows impracticably large. Our finite minds are unable to -treat any complex group exhaustively, and we are obliged to simplify -and generalise scientific problems, often at the risk of overlooking -particular conditions and exceptions. - -Every system of classes displayed in the manner of the Logical Alphabet -may be called *bifurcate*, because every class branches out at each -step into two minor classes, existent or imaginary. It would be a -great mistake to regard this arrangement as in any way a peculiar or -special method; it is not only a natural and important one, but it is -the inevitable and only system which is logically perfect, according -to the fundamental laws of thought. All other arrangements of classes -correspond to the bifurcate arrangement, with the implication that -some of the minor classes are not represented among existing things. -If we take the genus A and divide it into the species AB and AC, we -imply two propositions, namely that in the class A, the properties of B -and C never occur together, and that they are never both absent; these -propositions are logically equivalent to one, namely AB = A*c*. Our -classification is then identical with the following bifurcate one:-- - - A - | - +----------+----------+ - | | - AB A*b* - | | - +------+------+ +------+------+ - | | | | - ABC = 0 AB*c* A*b*C A*bc* = 0 - -If, again, we divide the genus A into three species, AB, AC, AD, we -are either logically in error, or else we must be understood to imply -that, as regards the other letters, there exist only three combinations -containing A, namely AB*cd*, A*b*C*d*, and A*bc*D. - -The logical necessity of bifurcate classification has been clearly and -correctly stated in the *Outline of a New System of Logic* by George -Bentham, the eminent botanist, a work of which the logical value has -been quite overlooked until lately. Mr. Bentham points out, in p. 113, -that every classification must be essentially bifurcate, and takes, as -an example, the division of vertebrate animals into four sub-classes, -as follows:-- - - Mammifera--endowed with mammæ and lungs. - Birds without mammæ but with lungs and wings. - Fish deprived of lungs. - Reptiles deprived of mammæ and wings but with lungs. - -We have, then, as Mr. Bentham says, three bifid divisions, thus -represented:-- - - Vertebrata - | - +-----------+-----------+ - | | - Endowed with lungs deprived of lungs - | = Fish. - +--------+----------------+ - | | - Endowed with deprived of - mammæ mammæ - = Mammifera. | - +------+------+ - | | - with wings without wings - = Birds. = Reptiles. - -It is quite evident that according to the laws of thought even this -arrangement is incomplete. The sub-class mammifera must either have -wings or be deprived of them; we must either subdivide this class, or -assume that none of the mammifera have wings, which is, as a matter of -fact, the case, the wings of bats not being true wings in the meaning -of the term as applied to birds. Fish, again, ought to be considered -with regard to the possession of mammæ and wings; and in leaving them -undivided we really imply that they never have mammæ nor wings, the -wings of the flying-fish, again, being no exception. If we resort to -the use of our letters and define them as follows-- - - A = vertebrata, - B = having lungs, - C = having mammæ, - D = having wings, - -then there are four existent classes of vertebrata which appear to be -thus described-- - - ABC AB*c*D AB*cd* A*b*. - -But in reality the combinations are implied to be - - ABC*d* = Mammifera, - AB*c*D = Birds, - AB*cd* = Reptiles, - A*bcd* = Fish, - -and we imply at the same time that the other four conceivable -combinations containing B, C, or D, namely ABCD, A*b*CD, A*b*C*d*, and -A*bc*D, do not exist in nature. - -Mr. Bentham points out[573] that it is really this method of -classification which was employed by Lamarck and De Candolle in their -so-called analytical arrangement of the French Flora. He gives as an -example a table of the principal classes of De Candolle’s system, as -also a bifurcate arrangement of animals after the method proposed -by Duméril in his *Zoologie Analytique*, this naturalist being -distinguished by his clear perception of the logical importance of the -method. A bifurcate classification of the animal kingdom may also be -found in Professor Reay Greene’s *Manual of the Cœlenterata*, p. 18. - - [573] *Essai sur la Nomenclature et la Classification*, Paris, 1823, - pp. 107, 108. - -The bifurcate form of classification seems to be needless when the -quality according to which we classify any group of things admits -of numerical discrimination. It would seem absurd to arrange things -according as they have one degree of the quality or not one degree, -two degrees or not two degrees, and so on. The elements are classified -according as the atom of each saturates one, two, three, or more atoms -of a monad element, such as chlorine, and they are called accordingly -monad, dyad, triad, tetrad elements, and so on. It would be useless to -apply the bifid arrangement, thus:-- - - Element - | - +-----+-------+ - | | - Monad not-Monad - | - +---------+---------+ - | | - Dyad not-Dyad - | - +---------+---------+ - | | - Triad not-Triad - | - +---------+--------+ - | | - Tetrad not-Tetrad. - -The reason of this is that, by the nature of number (p. 157) every -number is logically discriminated from every other number. There can -thus be no logical confusion in a numerical arrangement, and the series -of numbers indefinitely extended is also exhaustive. Every thing -admitting of a quality expressible in numbers must find its place -somewhere in the series of numbers. The chords in music correspond to -the simpler numerical ratios and must admit of complete exhaustive -classification in respect to the complexity of the ratios forming -them. Plane rectilinear figures may be classified according to the -numbers of their sides, as triangles, quadrilateral figures, pentagons, -hexagons, heptagons, &c. The bifurcate arrangement is not false when -applied to such series of objects; it is even necessarily involved in -the arrangement which we do apply, so that its formal statement is -needless and tedious. The same may be said of the division of portions -of space. Reid and Kames endeavoured to cast ridicule on the bifurcate -arrangement[574] by proposing to classify the parts of England into -Middlesex and what is not Middlesex, dividing the latter again into -Kent and what is not Kent, Sussex and what is not Sussex; and so on. -This is so far, however, from being an absurd proceeding that it is -requisite to assure us that we have made an exhaustive enumeration of -the parts of England. - - [574] George Bentham, *Outline of a New System of Logic*, p. 115. - - -*The Five Predicables.* - -As a rule it is highly desirable to consign to oblivion the ancient -logical names and expressions, which have infested the science for -many centuries past. If logic is ever to be a useful and progressive -science, logicians must distinguish between logic and the history of -logic. As in the case of any other science it may be desirable to -examine the course of thought by which logic has, before or since the -time of Aristotle, been brought to its present state; the history of a -science is always instructive as giving instances of the mode in which -discoveries take place. But at the same time we ought carefully to -disencumber the statement of the science itself of all names and other -vestiges of antiquity which are not actually useful at the present day. - -Among the ancient expressions which may well be excepted from such -considerations and retained in use, are the “Five Words” or “Five -Predicables” which were described by Porphyry in his introduction to -Aristotle’s Organum. Two of them, *Genus* and *Species*, are the most -venerable names in philosophy, having probably been first employed -in their present logical meanings by Socrates. In the present day it -requires some mental effort, as remarked by Grote, to see anything -important in the invention of notions now so familiar as those of Genus -and Species. But in reality the introduction of such terms showed the -rise of the first germs of logic and scientific method; it showed that -men were beginning to analyse their processes of thought. - -The Five Predicables are Genus, Species, Difference, Property, and -Accident, or in the original Greek, γένος, εἶδος, διαφορά, ἴδιον, -συμβεβηκός. Of these, Genus may be taken to mean any class of objects -which is regarded as broken up into two minor classes, which form -Species of it. The genus is defined by a certain number of qualities or -circumstances which belong to all objects included in the class, and -which are sufficient to mark out these objects from all others which -we do not intend to include. Interpreted as regards intension, then, -the genus is a group of qualities; interpreted as regards extension, it -is a group of objects possessing those qualities. If another quality -be taken into account which is possessed by some of the objects and -not by the others, this quality becomes a difference which divides -the genus into two species. We may interpret the species either in -intension or extension; in the former respect it is more than the genus -as containing one more quality, the difference: in the latter respect -it is less than the genus as containing only a portion of the group -constituting the genus. We may say, then, with Aristotle, that in one -sense the genus is in the species, namely in intension, and in another -sense the species is in the genus, namely in extension. The difference, -it is evident, can be interpreted in intension only. - -A Property is a quality which belongs to the whole of a class, but does -not enter into the definition of that class. A generic property belongs -to every individual object contained in the genus. It is a property -of the genus parallelogram that the opposite angles are equal. If we -regard a rectangle as a species of parallelogram, the difference being -that *one* angle is a right angle, it follows as a specific property -that all the angles are right angles. Though a property in the strict -logical sense must belong to each of the objects included in the class -of which it is a property, it may or may not belong to other objects. -The property of having the opposite angles equal may belong to many -figures besides parallelograms, for instance, regular hexagons. It is a -property of the circle that all triangles constructed upon the diameter -with the apex upon the circumference are right-angled triangles, and -*vice versâ*, all curves of which this is true must be circles. A -property which thus belongs to the whole of a class and only to that -class, corresponds to the ἴδιον of Aristotle and Porphyry; we might -conveniently call it *a peculiar property*. Every such property enables -us to make a statement in the form of a simple identity (p. 37). Thus -we know it to be a peculiar property of the circle that for a given -length of perimeter it encloses a greater area than any other possible -curve; hence we may say-- - - Curve of equal curvature = curve of greatest area. - -It is a peculiar property of equilateral triangles that they are -equiangular, and *vice versâ*, it is a peculiar property of equiangular -triangles that they are equilateral. It is a property of crystals -of the regular system that they are devoid of the power of double -refraction, but this is not a property peculiar to them, because -liquids and gases are devoid of the same property. - -An Accident, the fifth and last of the Predicables, is any quality -which may or may not belong to certain objects, and which has no -connexion with the classification adopted. The particular size of -a crystal does not in the slightest degree affect the form of the -crystal, nor does the manner in which it is grouped with other -crystals; these, then, are accidents as regards a crystallographic -classification. With respect to the chemical composition of a -substance, again, it is an accident whether the substance be -crystallised or not, or whether it be organised or not. As regards -botanical classification the absolute size of a plant is an accident. -Thus we see that a logical accident is any quality or circumstance -which is not known to be correlated with those qualities or -circumstances forming the definition of the species. - -The meanings of the Predicables can be clearly explained by our -symbols. Let A be any definite group of qualities and B another quality -or group of qualities; then A will constitute a genus, and AB, A*b* -will be species of it, B being the difference. Let C, D and E be other -qualities or groups of qualities, and on examining the combinations in -which A, B, C, D, E occur let them be as follows:-- - - ABCDE A*b*C*d*E - ABCD*e* A*b*C*de*. - -Here we see that wherever A is we also find C, so that C is a generic -property; D occurs always with B, so that it constitutes a specific -property, while E is indifferently present and absent, so as not to be -related to any other letter; it represents, therefore, an accident. It -will now be seen that the Logical Alphabet represents an interminable -series of subordinate genera and species; it is but a concise symbolic -statement of what was involved in the ancient doctrine of the -Predicables. - - -*Summum Genus and Infima Species.* - -As a genus means any class whatever which is regarded as composed -of minor classes or species, it follows that the same class will be -a genus in one point of view and a species in another. Metal is a -genus as regards alkaline metal, a species as regards element, and -any extensive system of classes consists of a series of subordinate, -or as they are technically called, *subaltern* genera and species. -The question, however, arises, whether such a chain of classes -has a definite termination at either end. The doctrine of the old -logicians was to the effect that it terminated upwards in a *genus -generalissimum* or *summum genus*, which was not a species of any -wider class. Some very general notion, such as substance, object, or -thing, was supposed to be so comprehensive as to include all thinkable -objects, and for all practical purposes this might be so. But as I -have already explained (p. 74), we cannot really think of any object -or class without thereby separating it from what is not that object or -class. All thinking is relative, and implies discrimination, so that -every class and every logical notion must have its negative. If so, -there is no such thing as a *summum genus*; for we cannot frame the -requisite notion of a class forming it without implying the existence -of another class discriminated from it; add this new negative class to -the supposed *summum genus*, and we form a still higher genus, which is -absurd. - -Although there is no absolute summum genus, nevertheless relatively to -any branch of knowledge or any particular argument, there is always -some class or notion which bounds our horizon as it were. The chemist -restricts his view to material substances and the forces manifested -in them; the mathematician extends his view so as to comprehend all -notions capable of numerical discrimination. The biologist, on the -other hand, has a narrower sphere containing only organised bodies, and -of these the botanist and the zoologist take parts. In other subjects -there may be a still narrower summum genus, as when the lawyer regards -only reasoning beings of his own country together with their property. - -In the description of the Logical Alphabet it was pointed out (p. 93) -that every series of combinations is really the development of a -single class, denoted by X, which letter was accordingly placed in the -first column of the table on p. 94. This is the formal acknowledgment -of the principle clearly stated by De Morgan, that all reasoning -proceeds within an assumed summum genus. But at the same time the fact -that X as a logical term must have its negative *x*, shows that it -cannot be an absolute summum genus. - -There arises, again, the question whether there be any such thing as -an *infima species*, which cannot be divided into minor species. The -ancient logicians were of opinion that there always was some assignable -class which could only be divided into individuals, but this doctrine -appears to be theoretically incorrect, as Mr. George Bentham long ago -stated.[575] We may put an arbitrary limit to the subdivision of our -classes at any point convenient to our purpose. The crystallographer -would not generally treat as different species crystalline forms which -differ only in the degree of development of the faces. The naturalist -overlooks innumerable slight differences between animals which he -refers to the same species. But in a strictly logical point of view -classification might be carried on as long as there is a difference, -however minute, between two objects, and we might thus go on until we -arrive at individual objects which are numerically distinct in the -logical sense attributed to that expression in the chapter upon Number. -Either, then, we must call the individual the *infima species* or allow -that there is no such thing at all. - - [575] *Outline of a New System of Logic*, 1827, p. 117. - - -*The Tree of Porphyry.* - -Both Aristotle and Plato were acquainted with the value of bifurcate -classification, which they occasionally employed in an explicit manner. -It is impossible too that Aristotle should state the laws of thought, -and employ the predicables without implicitly recognising the logical -necessity of that method. It is, however, in Porphyry’s remarkable -and in many respects excellent *Introduction to the Categories of -Aristotle* that we find the most distinct account of it. Porphyry not -only fully and accurately describes the Predicables, but incidentally -introduces an example for illustrating those predicables, which -constitutes a good specimen of bifurcate classification. Translating -his words[576] freely we may say that he takes Substance as the genus -to be divided, under which are successively placed as Species--Body, -Animated Body, Animal, Rational Animal, and Man. Under Man, again, -come Socrates, Plato, and other particular men. Now of these notions -Substance is the genus generalissimum, and is a genus only, not a -species. Man, on the other hand, is the species specialissima (infima -species), and is a species only, not a genus. Body is a species of -substance, but a genus of animated body, which, again, is a species of -body but a genus of animal. Animal is a species of animated body, but -a genus of rational animal, which, again, is a species of animal, but -a genus of man. Finally, man is a species of rational animal, but is a -species merely and not a genus, being divisible only into particular -men. - - [576] *Porphyrii Isagoge*, Caput ii. 24. - -Porphyry proceeds at some length to employ his example in further -illustration of the predicables. We do not find in Porphyry’s own -work any scheme or diagram exhibiting this curious specimen of -classification, but some of the earlier commentators and epitome -writers drew what has long been called the Tree of Porphyry. This -diagram, which may be found in most elementary works on Logic,[577] is -also called the Ramean Tree, because Ramus insisted much upon the value -of Dichotomy. With the exception of Jeremy Bentham[578] and George -Bentham, hardly any modern logicians have shown an appreciation of the -value of bifurcate classification. The latter author has treated the -subject, both in his *Outline of a New System of Logic* (pp. 105–118), -and in his earlier work entitled *Essai sur la Nomenclature et la -Classification des Principales Branches d’Art-et-Science* (Paris, -1823), which consists of a free translation or improved version of his -uncle’s Essay on Classification in the *Chrestomathia*. Some interest -attaches to the history of the Tree of Porphyry and Ramus, because -it is the prototype of the Logical Alphabet which lies at the basis -of logical method. Jeremy Bentham speaks truly of “the matchless -beauty of the Ramean Tree.” After fully showing its logical value as -an exhaustive method of classification, and refuting the objections -of Reid and Kames, on a wrong ground, as I think, he proceeds to -inquire to what length it may be carried. He correctly points out two -objections to the extensive use of bifid arrangements, (1) that they -soon become impracticably extensive and unwieldy, and (2) that they -are uneconomical. In his day the recorded number of different species -of plants was 40,000, and he leaves the reader to estimate the immense -number of branches and the enormous area of a bifurcate table which -should exhibit all these species in one scheme. He also points out the -apparent loss of labour in making any large bifurcate classification; -but this he considers to be fully recompensed by the logical value of -the result, and the logical training acquired in its execution. Jeremy -Bentham, then, fully recognises the value of the Logical Alphabet under -another name, though he apprehends also the limit to its use placed by -the finiteness of our mental and manual powers. - - [577] Jevons, *Elementary Lessons in Logic*, p. 104. - - [578] *Chrestomathia; being a Collection of Papers, &c.* London, - 1816, Appendix V. - - -*Does Abstraction imply Generalisation?* - -Before we can acquire a sound comprehension of the subject of -classification we must answer the very difficult question whether -logical abstraction does or does not imply generalisation. It comes to -exactly the same thing if we ask whether a species may be coextensive -with its genus, or whether, on the other hand, the genus must contain -more than the species. To abstract logically is (p. 27), to overlook or -withdraw our notice from some point of difference. Whenever we form a -class we abstract, for the time being, the differences of the objects -so united in respect of some common quality. If we class together a -great number of objects as dwelling-houses, we overlook the fact that -some dwelling-houses are constructed of stone, others of brick, wood, -iron, &c. Often at least the abstraction of a circumstance increases -the number of objects included under a class according to the law of -the inverse relation of the quantities of extension and intension -(p. 26). Dwelling-house is a wider term than brick-dwelling-house. -House is more general than dwelling-house. But the question before -us is, whether abstraction *always* increases the number of objects -included in a class, which amounts to asking whether the law of -the inverse relation of logical quantities is *always* true. The -interest of the question partly arises from the fact, that so high -a philosophical authority as Mr. Herbert Spencer has denied that -generalisation is implied in abstraction,[579] making this doctrine -the ground for rejecting previous methods of classifying the sciences, -and for forming an ingenious but peculiar method of his own. The -question is also a fundamental one of the highest logical importance, -and involves subtle difficulties which have made me long hesitate in -forming a decisive opinion. - - [579] *The Classification of the Sciences*, &c., 3rd edit. p. 7. - *Essays: Scientific, Political, and Speculative*, vol. iii. p. 13. - -Let us attempt to answer the question by examination of a few examples. -Compare the two classes *gun* and *iron gun*. It is certain that there -are many guns which are not made of iron, so that abstraction of the -circumstance “made of iron” increases the extent of the notion. Next -compare *gun* and *metallic gun*. All guns made at the present day -consist of metal, so that the two notions seem to be coextensive; -but guns were at first made of pieces of wood bound together like -a tub, and as the logical term gun takes no account of time, it -must include all guns that have ever existed. Here again extension -increases as intension decreases. Compare once more “steam-locomotive -engine” and “locomotive engine.” In the present day, as far as I am -aware, all locomotives are worked by steam, so that the omission of -that qualification might seem not to widen the term; but it is quite -possible that in some future age a different motive power may be used -in locomotives; and as there is no limitation of time in the use of -logical terms, we must certainly assume that there is a class of -locomotives not worked by steam, as well as a class that is worked by -steam. When the natural class of Euphorbiaceæ was originally formed, -all the plants known to belong to it were devoid of corollas; it -would have seemed therefore that the two classes “Euphorbiaceæ,” and -“Euphorbiaceæ devoid of Corollas,” were of equal extent. Subsequently -a number of plants plainly belonging to the same class were found in -tropical countries, and they possessed bright coloured corollas. -Naturalists believe with the utmost confidence that “Ruminants” and -“Ruminants with cleft feet” are identical terms, because no ruminant -has yet been discovered without cleft feet. But we can see no -impossibility in the conjunction of rumination with uncleft feet, and -it would be too great an assumption to say that we are certain that an -example of it will never be met with. Instances can be quoted, without -end, of objects being ultimately discovered combining properties which -had never before been seen together. In the animal kingdom the Black -Swan, the Ornithorhynchus Paradoxus, and more recently the singular -fish called Ceratodus Forsteri, all discovered in Australia, have -united characters never previously known to coexist. At the present -time deep-sea dredging is bringing to light many animals of an -unprecedented nature. Singular exceptional discoveries may certainly -occur in other branches of science. When Davy first discovered metallic -potassium, it was a well established empirical law that all metallic -substances possessed a high specific gravity, the least dense of the -metals then known being zinc, of which the specific gravity is 7·1. Yet -to the surprise of chemists, potassium was found to be an undoubted -metal of less density than water, its specific gravity being 0·865. - -It is hardly requisite to prove by further examples that our knowledge -of nature is incomplete, so that we cannot safely assume the -non-existence of new combinations. Logically speaking, we ought to -leave a place open for animals which ruminate but are without cleft -feet, and for every possible intermediate form of animal, plant, or -mineral. A purely logical classification must take account not only of -what certainly does exist, but of what may in after ages be found to -exist. - -I will go a step further, and say that we must have places in our -scientific classifications for purely imaginary existences. A large -proportion of the mathematical functions which are conceivable have no -application to the circumstances of this world. Physicists certainly do -investigate the nature and consequences of forces which nowhere exist. -Newton’s *Principia* is full of such investigations. In one chapter of -his *Mécanique Céleste* Laplace indulges in a remarkable speculation -as to what the laws of motion would have been if momentum, instead of -varying simply as the velocity, had been a more complicated function -of it. I have already mentioned (p. 223) that Airy contemplated the -existence of a world in which the laws of force should be such that -a perpetual motion would be possible, and the Law of Conservation of -Energy would not hold true. - -Thought is not bound down to the limits of what is materially existent, -but is circumscribed only by those Fundamental Laws of Identity, -Contradiction and Duality, which were laid down at the outset. This -is the point at which I should differ from Mr. Spencer. He appears -to suppose that a classification is complete if it has a place for -every existing object, and this may perhaps seem to be practically -sufficient; but it is subject to two profound objections. Firstly, we -do not know all that exists, and therefore in limiting our classes we -are erroneously omitting multitudes of objects of unknown form and -nature which may exist either on this earth or in other parts of space. -Secondly, as I have explained, the powers of thought are not limited by -material existences, and we may, or, for some purposes, must imagine -objects which probably do not exist, and if we imagine them we ought to -find places for them in the classifications of science. - -The chief difficulty of this subject, however, consists in the fact -that mathematical or other certain laws may entirely forbid the -existence of some combinations. The circle may be defined as a plane -curve of equal curvature, and it is a property of the circle that it -contains the greatest area within the least possible perimeter. May we -then contemplate mentally a circle not a figure of greatest possible -area? Or, to take a still simpler example, a parallelogram possesses -the property of having the opposite angles equal. May we then mentally -divide parallelograms into two classes according as they do or do -not have their opposite angles equal? It might seem absurd to do so, -because we know that one of the two species of parallelogram would be -non-existent. But, then, unless the student had previously contemplated -the existence of both species as possible, what is the meaning of the -thirty-fourth proposition of Euclid’s first book? We cannot deny or -disprove the existence of a certain combination without thereby in a -certain way recognising that combination as an object of thought. - -The conclusion at which I arrive is in opposition to that of Mr. -Spencer. I think that whenever we abstract a quality or circumstance we -do generalise or widen the notion from which we abstract. Whatever the -terms A, B, and C may be, I hold that in strict logic AB is mentally -a wider term than ABC, because AB includes the two species ABC and -AB*c*. The term A is wider still, for it includes the four species -ABC, AB*c*, A*b*C, A*bc*. The Logical Alphabet, in short, is the only -limit of the classes of objects which we must contemplate in a purely -logical point of view. Whatever notions be brought before us, we must -mentally combine them in all the ways sanctioned by the laws of thought -and exhibited in the Logical Alphabet, and it is a matter for after -consideration to determine how many of these combinations exist in -outward nature, or how many are actually forbidden by the conditions of -space. A classification is essentially a mental, not a material thing. - - -*Discovery of Marks or Characteristics.* - -Although the chief purpose of classification is to disclose the -deepest and most general resemblances of the objects classified, yet -the practical value of a system will depend partly upon the ease with -which we can refer an object to its proper class, and thus infer -concerning it all that is known generally of that class. This operation -of discovering to which class of a system a certain specimen or case -belongs, is generally called *Diagnosis*, a technical term familiarly -used by physicians, who constantly require to diagnose or determine -the nature of the disease from which a patient is suffering. Now every -class is defined by certain specified qualities or circumstances, the -whole of which are present in every object contained in the class, -and *not all present* in any object excluded from it. These defining -circumstances ought to consist of the deepest and most important -circumstances, by which we vaguely mean those probably forming the -conditions with which the minor circumstances are correlated. But it -will often happen that the so-called important points of an object -are not those which can most readily be observed. Thus the two great -classes of phanerogamous plants are defined respectively by the -possession of two cotyledons or seed-leaves, and one cotyledon. But -when a plant comes to our notice and we want to refer it to the right -class, it will often happen that we have no seed at all to examine, in -order to discover whether there be one seed-leaf or two in the germ. -Even if we have a seed it will often be small, and a careful dissection -under the microscope will be requisite to ascertain the number of -cotyledons. Occasionally the examination of the germ would mislead us, -for the cotyledons may be obsolete, as in Cuscuta, or united together, -as in Clintonia. Botanists therefore seldom actually refer to the -seed for such information. Certain other characters of a plant are -correlated with the number of seed-leaves; thus monocotyledonous plants -almost always possess leaves with parallel veins like those of grass, -while dicotyledonous plants have leaves with reticulated veins like -those of an oak leaf. In monocotyledonous plants, too, the parts of the -flower are most often three or some multiple of three in number, while -in dicotyledonous plants the numbers four and five and their multiples -prevail. Botanists, therefore, by a glance at the leaves and flowers -can almost certainly refer a plant to its right class, and can infer -not only the number of cotyledons which would be found in the seed -or young plant, but also the structure of the stem and other general -characters. - -Any conspicuous and easily discriminated property which we thus -select for the purpose of deciding to which class an object belongs, -may be called a *characteristic*. The logical conditions of a good -characteristic mark are very simple, namely, that it should be -possessed by all objects entering into a certain class, and by none -others. Every characteristic should enable us to assert a simple -identity; if A is a characteristic, and B, viewed intensively, the -class of objects of which it is the mark, then A = B ought to be -true. The characteristic may consist either of a single quality or -circumstance, or of a group of such, provided that they all be constant -and easily detected. Thus in the classification of mammals the teeth -are of the greatest assistance, not because a slight variation in the -number and form of the teeth is of importance in the general economy -of the animal, but because such variations are proved by empirical -observation to coincide with most important differences in the general -affinities. It is found that the minor classes and genera of mammals -can be discriminated accurately by their teeth, especially by the -foremost molars and the hindmost pre-molars. Some teeth, indeed, are -occasionally missing, so that zoologists prefer to trust to those -characteristic teeth which are most constant,[580] and to infer from -them not only the arrangement of the other teeth, but the whole -conformation of the animal. - - [580] Owen, *Essay on the Classification and Geographical - Distribution of the Mammalia*, p. 20. - -It is a very difficult matter to mark out a boundary-line between the -animal and vegetable kingdoms, and it may even be doubted whether -a rigorous boundary can be established. The most fundamental and -important difference of a vegetable as compared with an animal -substance probably consists in the absence of nitrogen from the -constituent membranes. Supposing this to be the case, the difficulty -arises that in examining minute organisms we cannot ascertain directly -whether they contain nitrogen or not. Some minor but easily detected -circumstance is therefore needed to discriminate between animals and -vegetables, and this is furnished to some extent by the fact that the -production of starch granules is restricted to the vegetable kingdom. -Thus the Desmidiaceæ may be safely assigned to the vegetable kingdom, -because they contain starch. But we must not employ this characteristic -negatively; the Diatomaceæ are probably vegetables, though they do not -produce starch. - - -*Diagnostic Systems of Classification.* - -We have seen that diagnosis is the process of discovering the place in -any system of classes, to which an object has been referred by some -previous investigation, the object being to avail ourselves of the -information relating to such an object which has been accumulated and -recorded. It is obvious that this is a matter of great importance, -for, unless we can recognise, from time to time, objects or substances -which have been investigated, recorded discoveries would lose their -value. Even a single investigator must have means of recording and -systematising his observations of any large groups of objects like the -vegetable and animal kingdoms. - -Now whenever a class has been properly formed, a definition must have -been laid down, stating the qualities and circumstances possessed by -all the objects which are intended to be included in the class, and -not possessed *completely* by any other objects. Diagnosis, therefore, -consists in comparing the qualities of a certain object with the -definitions of a series of classes; the absence in the object of any -one quality stated in the definition excludes it from the class thus -defined; whereas, if we find every point of a definition exactly -fulfilled in the specimen, we may at once assign it to the class in -question. It is of course by no means certain that everything which has -been affirmed of a class is true of all objects afterwards referred -to the class; for this would be a case of imperfect inference, which -is never more than matter of probability. A definition can only make -known a finite number of the qualities of an object, and it always -remains possible that objects agreeing in those assigned qualities will -differ in others. *An individual cannot be defined*, and can only be -made known by the exhibition of the individual itself, or by a material -specimen exactly representing it. But this and other questions relating -to definition must be treated when I am able to take up the subject of -language in another work. - -Diagnostic systems of classification should, as a general rule, be -arranged on the bifurcate method explicitly. Any quality may be chosen -which divides the whole group of objects into two distinct parts, -and each part may be sub-divided successively by any prominent and -well-marked circumstance which is present in a large part of the genus -and not in the other. To refer an object to its proper place in such an -arrangement we have only to note whether it does or does not possess -the successive critical differentiæ. Dana devised a classification of -this kind[581] by which to refer a crystal to its place in the series -of six or seven classes already described. If a crystal has all its -edges modified alike or the angles replaced by three or six similar -planes, it belongs to the monometric system; if not, we observe -whether the number of similar planes at the extremity of the crystal -is three or some multiple of three, in which case it is a crystal -of the hexagonal system; and so we proceed with further successive -discriminations. To ascertain the name of a mineral by examination with -the blow-pipe, an arrangement more or less evidently on the bifurcate -plan, has been laid down by Von Kobell.[582] Minerals are divided -according as they possess or do not possess metallic lustre; as they -are fusible or not fusible, according as they do or do not on charcoal -give a metallic bead, and so on. - - [581] Dana’s *Mineralogy*, vol. i. p. 123; quoted in Watts’ - *Dictionary of Chemistry*, vol. ii. p. 166. - - [582] *Instructions for the Discrimination of Minerals by Simple - Chemical Experiments*, by Franz von Kobell, translated from the - German by R. C. Campbell. Glasgow, 1841. - -Perhaps the best example to be found of an arrangement devised simply -for the purpose of diagnosis, is Mr. George Bentham’s *Analytical Key -to the Natural Orders and Anomalous Genera of the British Flora*, given -in his *Handbook of the British Flora*.[583] In this scheme, the great -composite family of plants, together with the closely approximate -genus Jasione, are first separated from all other flowering plants -by the compound character of their flowers. The remaining plants are -sub-divided according as the perianth is double or single. Since no -plants are yet known in which the perianth can be said to have three -or more distinct rings, this division becomes practically the same as -one into double and not-double. Flowers with a double perianth are -next discriminated according as the corolla does or does not consist -of one piece; according as the ovary is free or not free; as it is -simple or not simple; as the corolla is regular or irregular; and so -on. On looking over this arrangement, it will be found that numerical -discriminations often occur, the numbers of petals, stamens, capsules, -or other parts being the criteria, in which cases, as already explained -(p. 697), the actual exhibition of the bifid division would be tedious. - - [583] Edition of 1866, p. lxiii. - -Linnæus appears to have been perfectly acquainted with the nature and -uses of diagnostic classification, which he describes under the name -of Synopsis, saying:[584]--“Synopsis tradit Divisiones arbitrarias, -longiores aut breviores, plures aut pauciores: a Botanicis in genere -non agnoscenda. Synopsis est dichotomia arbitraria, quæ instar viæ ad -Botanicem ducit. Limites autem non determinat.” - - [584] *Philosophia Botanica* (1770), § 154, p. 98. - -The rules and tables drawn out by chemists to facilitate the discovery -of the nature of a substance in qualitative analysis are usually -arranged on the bifurcate method, and form excellent examples of -diagnostic classification, the qualities of the substances produced -in testing being in most cases merely characteristic properties of -little importance in other respects. The chemist does not detect -potassium by reducing it to the state of metallic potassium, and -then observing whether it has all the principal qualities belonging -to potassium. He selects from among the whole number of compounds of -potassium that salt, namely the compound of platinum tetra-chloride, -and potassium chloride, which has the most distinctive appearance, -as it is comparatively insoluble and produces a peculiar yellow and -highly crystalline precipitate. Accordingly, potassium is present -whenever this precipitate can be produced by adding platinum chloride -to a solution. The fine purple or violet colour which potassium -salts communicate to the blowpipe flame, had long been used as a -characteristic mark. Some other elements were readily detected by the -colouring of the blowpipe flame, barium giving a pale yellowish green, -and salts of strontium a bright red. By the use of the spectroscope -the coloured light given off by an incandescent vapour is made to give -perfectly characteristic marks of the elements contained in the vapour. - -Diagnosis seems to be identical with the process termed by the ancient -logicians *abscissio infiniti*, the cutting off of the infinite or -negative part of a genus when we discover by observation that an -object possesses a particular difference. At every step in a bifurcate -division, some objects possessing the difference will fall into -the affirmative part or species; all the remaining objects in the -world fall into the negative part, which will be infinite in extent. -Diagnosis consists in the successive rejection from further notice of -those infinite classes with which the specimen in question does not -agree. - - -*Index Classifications.* - -Under classification we may include all arrangements of objects or -names, which we make for saving labour in the discovery of an object. -Even alphabetical indices are real classifications. No such arrangement -can be of use unless it involves some correlation of circumstances, so -that knowing one thing we learn another. If we merely arrange letters -in the pigeon-holes of a secretaire we establish a correlation, for all -letters in the first hole will be written by persons, for instance, -whose names begin with A, and so on. Knowing then the initial letter of -the writer’s name, we know also the place of the letter, and the labour -of search is thus reduced to one twenty-sixth part of what it would be -without arrangement. - -Now the purpose of a catalogue is to discover the place in which an -object is to be found; but the art of cataloguing involves logical -considerations of some importance. We want to establish a correlation -between the place of an object and some circumstance about the object -which shall enable us readily to refer to it; this circumstance -therefore should be that which will most readily dwell in the memory -of the searcher. A piece of poetry will be best remembered by the -first line of the piece, and the name of the author will be the next -most definite circumstance; a catalogue of poetry should therefore be -arranged alphabetically according to the first word of the piece, or -the name of the author, or, still better, in both ways. It would be -impossible to arrange poems according to their subjects, so vague and -mixed are these found to be when the attempt is made. - -It is a matter of considerable literary importance to decide upon the -best mode of cataloguing books, so that any required book in a library -shall be most readily found. Books may be classified in a great number -of ways, according to subject, language, date, or place of publication, -size, the initial words of the text or title-page, or colophon, the -author’s name, the publisher’s name, the printer’s name, the character -of the type, and so on. Every one of these modes of arrangement may be -useful, for we may happen to remember one circumstance about a book -when we have forgotten all others; but as we cannot usually go to the -expense of forming more than two or three indices, we must select -those circumstances which will lead to the discovery of a book most -frequently. Many of the criteria mentioned are evidently inapplicable. - -The language in which a book is written is definite enough, provided -that the whole book is written in the same language; but it is obvious -that language gives no means for the subdivision and arrangement of -the literature of any one people. Classification by subjects would be -an exceedingly useful method if it were practicable, but experience -shows it to be a logical absurdity. It is a very difficult matter to -classify the sciences, so complicated are the relations between them. -But with books the complication is vastly greater, since the same book -may treat of different sciences, or it may discuss a problem involving -many branches of knowledge. A good account of the steam-engine will be -antiquarian, so far as it traces out the earliest efforts at discovery; -purely scientific, as regards the principles of thermodynamics -involved; technical, as regards the mechanical means of applying -those principles; economical, as regards the industrial results of -the invention; biographical, as regards the lives of the inventors. -A history of Westminster Abbey might belong either to the history of -architecture, the history of the Church, or the history of England. -If we abandon the attempt to carry out an arrangement according to -the natural classification of the sciences, and form comprehensive -practical groups, we shall be continually perplexed by the occurrence -of intermediate cases, and opinions will differ *ad infinitum* as to -the details. If, to avoid the difficulty about Westminster Abbey, we -form a class of books devoted to the History of Buildings, the question -will then arise whether Stonehenge is a building, and if so, whether -cromlechs, mounds, and monoliths are so. We shall be uncertain whether -to include lighthouses, monuments, bridges, &c. In regard to literary -works, rigorous classification is still less possible. The same work -may partake of the nature of poetry, biography, history, philosophy, -or if we form a comprehensive class of Belles-lettres, nobody can say -exactly what does or does not come under the term. - -My own experience entirely bears out the opinion of De Morgan, that -classification according to the name of the author is the only one -practicable in a large library, and this method has been admirably -carried out in the great catalogue of the British Museum. The name -of the author is the most precise circumstance concerning a book, -which usually dwells in the memory. It is a better characteristic of -the book than anything else. In an alphabetical arrangement we have -an exhaustive classification, including a place for every name. The -following remarks[585] of De Morgan seem therefore to be entirely -correct. “From much, almost daily use, of catalogues for many years, -I am perfectly satisfied that a classed catalogue is more difficult -to use than to make. It is one man’s theory of the subdivision of -knowledge, and the chances are against its suiting any other man. Even -if all doubtful works were entered under several different heads, the -frontier of the dubious region would itself be a mere matter of doubt. -I never turn from a classed catalogue to an alphabetical one without -a feeling of relief and security. With the latter I can always, by -taking proper pains, make a library yield its utmost; with the former -I can never be satisfied that I have taken proper pains, until I have -made it, in fact, as many different catalogues as there are different -headings, with separate trouble for each. Those to whom bibliographical -research is familiar, know that they have much more frequently to -hunt an author than a subject: they know also that in searching for a -subject, it is never safe to take another person’s view, however good, -of the limits of that subject with reference to their own particular -purposes.” - - [585] *Philosophical Magazine*, 3rd Series (1845), vol. xxvi. p. 522. - See also De Morgan’s evidence before the Royal Commission on the - British Museum in 1849, Report (1850), Questions, 5704*-5815*, - 6481–6513. This evidence should be studied by every person who wishes - to understand the elements of Bibliography. - -It is often desirable, however, that a name catalogue should be -accompanied by a subordinate subject catalogue, but in this case -no attempt should be made to devise a theoretically complete -classification. Every principal subject treated in a book should -be entered separately in an alphabetical list, under the name most -likely to occur to the searcher, or under several names. This method -was partially carried out in Watts’ *Bibliotheca Britannica*, but -it was excellently applied in the admirable subject index to the -*British Catalogue of Books*, and equally well in the *Catalogue -of the Manchester Free Library* at Campfield, drawn up under the -direction of Mr. Crestadoro, this latter being the most perfect model -of a printed catalogue with which I am acquainted. The Catalogue of -the London Library is also in the right form, and has a useful index -of subjects, though it is too much condensed and abbreviated. The -public catalogue of the British Museum is arranged as far as possible -according to the alphabetical order of the authors’ names, but in -writing the titles for this catalogue several copies are simultaneously -produced by a manifold writer, so that a catalogue according to the -order of the books on the shelves, and another according to the first -words of the title-page, are created by a mere rearrangement of the -spare copies. In the *English Cyclopædia* it is suggested that twenty -copies of the book titles might readily have been utilised in forming -additional catalogues, arranged according to the place of publication, -the language of the book, the general nature of the subject, and so -forth.[586] An excellent suggestion has also been made to the effect -that each book when published should have a fly-leaf containing half -a dozen printed copies of the title, drawn up in a form suitable for -insertion in catalogues. Every owner of a library could then easily -make accurate printed catalogues to suit his own purposes, by merely -cutting out these titles and pasting them in books in any desirable -order. - - [586] *English Cyclopædia, Arts and Sciences*, vol. v. p. 233. - -It will hardly be a digression to point out the enormous saving of -labour, or, what comes to the same thing, the enormous increase in our -available knowledge, both literary and scientific, which arises from -the formation of extensive indices. The “State Papers,” containing -the whole history of the nation, were practically sealed to literary -inquirers until the Government undertook the task of calendaring and -indexing them. The British Museum Catalogue is another national work, -of which the importance in advancing knowledge cannot be overrated. -The Royal Society is doing great service in publishing a complete -catalogue of memoirs upon physical science. The time will perhaps -come when our views upon this subject will be extended, and either -Government or some public society will undertake the systematic -cataloguing and indexing of masses of historical and scientific -information which are now almost closed against inquiry. - - -*Classification in the Biological Sciences.* - -The great generalisations established in the works of Herbert Spencer -and Charles Darwin have thrown much light upon other sciences, and -have removed several difficulties out of the way of the logician. The -subject of classification has long been studied in almost exclusive -reference to the arrangement of animals and plants. Systematic botany -and zoology have been commonly known as the Classificatory Sciences, -and scientific men seemed to suppose that the methods of arrangement, -which were suitable for living creatures, must be the best for all -other classes of objects. Several mineralogists, especially Mohs, have -attempted to arrange minerals in genera and species, just as if they -had been animals capable of reproducing their kind with variations. -This confusion of ideas between the relationship of living forms and -the logical relationship of things in general prevailed from the -earliest times, as manifested in the etymology of words. We familiarly -speak of a *kind* of things meaning a class of things, and the kind -consists of those things which are *akin*, or come of the same race. -When Socrates and his followers wanted a name for a class regarded in a -philosophical light, they adopted the analogy in question, and called -it a γένος, or race, the root γεν- being connected with the notion of -generation. - -So long as species of plants and animals were believed to proceed from -distinct acts of Creation, there was no apparent reason why methods of -classification suitable to them should not be treated as a guide to -the classification of other objects generally. But when once we regard -these resemblances as hereditary in their origin, we see that the -sciences of systematic botany and zoology have a special character of -their own. There is no reason to suppose that the same kind of natural -classification which is best in biology will apply also in mineralogy, -in chemistry, or in astronomy. The logical principles which underlie -all classification are of course the same in natural history as in the -sciences of lifeless matter, but the special resemblances which arise -from the relation of parent and offspring will not be found to prevail -between different kinds of crystals or mineral bodies. - -The genealogical view of the relations of animals and plants leads us -to discard all notions of a regular progression of living forms, or -any theory as to their symmetrical relations. It was at one time a -question whether the ultimate scheme of natural classification would -lead to arrangement in a simple line, or a circle, or a combination -of circles. Macleay’s once celebrated system was a circular one, and -each class-circle was composed of five order-circles, each of which was -composed again of five tribe-circles, and so on, the subdivision being -at each step into five minor circles. Macleay held that in the animal -kingdom there are five sub-kingdoms--the Vertebrata, Annulosa, Radiata, -Acrita, and Mollusca. Each of these was again divided into five--the -Vertebrata, consisting of Mammalia, Reptilia, Pisces, Amphibia, and -Aves.[587] It is evident that in such a symmetrical system the animals -were made to suit themselves to the classes instead of the classes -being suited to the animals. - - [587] Swainson, “Treatise on the Geography and Classification of - Animals,” *Cabinet Cyclopædia*, p. 201. - -We now perceive that the ultimate system will have the form of an -immensely extended genealogical tree, which will be capable of -representation by lines on a plane surface of sufficient extent. -Strictly speaking, this genealogical tree ought to represent the -descent of each individual living form now existing or which has -existed. It should be as personal and minute in its detail of -relations, as the Stemma of the Kings of England. We must not assume -that any two forms are exactly alike, and in any case they are -numerically distinct. Every parent then must be represented at the apex -of a series of divergent lines, representing the generation of so many -children. Any complete system of classification must regard individuals -as the infimæ species. But as in the lower races of animals and -plants the differences between individuals are slight and apparently -unimportant, while the numbers of such individuals are immensely -great, beyond all possibility of separate treatment, scientific men -have always stopped at some convenient but arbitrary point, and have -assumed that forms so closely resembling each other as to present no -constant difference were all of one kind. They have, in short, fixed -their attention entirely upon the main features of family difference. -In the genealogical tree which they have been unconsciously aiming to -construct, diverging lines meant races diverging in character, and the -purpose of all efforts at so-called natural classification was to trace -out the descents between existing groups of plants or animals. - -Now it is evident that hereditary descent may have in different -cases produced very different results as regards the problem of -classification. In some cases the differentiation of characters may -have been very frequent, and specimens of all the characters produced -may have been transmitted to the present time. A living form will then -have, as it were, an almost infinite number of cousins of various -degrees, and there will be an immense number of forms finely graduated -in their resemblances. Exact and distinct classification will then -be almost impossible, and the wisest course will be not to attempt -arbitrarily to distinguish forms closely related in nature, but to -allow that there exist transitional forms of every degree, to mark -out if possible the extreme limits of the family relationship, and -perhaps to select the most generalised form, or that which presents the -greatest number of close resemblances to others of the family, as the -*type* of the whole. - -Mr. Darwin, in his most interesting work upon Orchids, points out -that the tribe of Malaxeæ are distinguished from Epidendreæ by the -absence of a caudicle to the pollinia; but as some of the Malaxeæ -have a minute caudicle, the division really breaks down in the most -essential point. “This is a misfortune,” he remarks,[588] “which every -naturalist encounters in attempting to classify a largely developed -or so-called natural group, in which, relatively to other groups, -there has been little extinction. In order that the naturalist may be -enabled to give precise and clear definitions of his divisions, whole -ranks of intermediate or gradational forms must have been utterly swept -away: if here and there a member of the intermediate ranks has escaped -annihilation, it puts an effectual bar to any absolutely distinct -definition.” - - [588] Darwin, *Fertilisation of Orchids*, p. 159. - -In other cases a particular plant or animal may perhaps have -transmitted its form from generation to generation almost unchanged, -or, what comes to the same result, those forms which diverged in -character from the parent stock may have proved unsuitable to their -circumstances, and perished. We shall then find a particular form -standing apart from all others, and marked by many distinct characters. -Occasionally we may meet with specimens of a race which was formerly -far more common but is now undergoing extinction, and is nearly the -last of its kind. Thus we explain the occurrence of exceptional forms -such as are found in the Amphioxus. The Equisetaceæ perplex botanists -by their want of affinity to other orders of Acrogenous plants. This -doubtless indicates that their genealogical connection with other -plants must be sought for in the most distant ages of geological -development. - -Constancy of character, as Mr. Darwin has said,[589] is what is chiefly -valued and sought after by naturalists; that is to say, naturalists -wish to find some distinct family mark, or group of characters, by -which they may clearly recognise the relationship of descent between a -large group of living forms. It is accordingly a great relief to the -mind of the naturalist when he comes upon a definitely marked group, -such as the Diatomaceæ, which are clearly separated from their nearest -neighbours the Desmidiaceæ by their siliceous framework and the absence -of chlorophyll. But we must no longer think that because we fail in -detecting constancy of character the fault is in our classificatory -sciences. Where gradation of character really exists, we must devote -ourselves to defining and registering the degrees and limits of that -gradation. The ultimate natural arrangement will often be devoid of -strong lines of demarcation. - - [589] *Descent of Man*, vol. i. p. 214. - -Let naturalists, too, form their systems of natural classification -with all care they can, yet it will certainly happen from time to -time that new and exceptional forms of animals or vegetables will be -discovered and will require the modification of the system. A natural -system is directed, as we have seen, to the discovery of empirical laws -of correlation, but these laws being purely empirical will frequently -be falsified by more extensive investigation. From time to time the -notions of naturalists have been greatly widened, especially in the -case of Australian animals and plants, by the discovery of unexpected -combinations of organs, and such events must often happen in the -future. If indeed the time shall come when all the forms of plants are -discovered and accurately described, the science of Systematic Botany -will then be placed in a new and more favourable position, as remarked -by Alphonse Decandolle.[590] - - [590] *Laws of Botanical Nomenclature*, p. 16. - -It ought to be remembered that though the genealogical classification -of plants or animals is doubtless the most instructive of all, it is -not necessarily the best for all purposes. There may be correlations -of properties important for medicinal, or other practical purposes, -which do not correspond to the correlations of descent. We must regard -the bamboo as a tree rather than a grass, although it is botanically -a grass. For legal purposes we may continue with advantage to treat -the whale, seal, and other cetaceæ, as fish. We must also class plants -according as they belong to arctic, alpine, temperate, sub-tropical or -tropical regions. There are causes of likeness apart from hereditary -relationship, and *we must not attribute exclusive excellence to any -one method of classification*. - - -*Classification by Types.* - -Perplexed by the difficulties arising in natural history from the -discovery of intermediate forms, naturalists have resorted to what they -call classification by types. Instead of forming one distinct class -defined by the invariable possession of certain assigned properties, -and rigidly including or excluding objects according as they do or -do not possess all these properties, naturalists select a typical -specimen, and they group around it all other specimens which resemble -this type more than any other selected type. “The type of each genus,” -we are told,[591] “should be that species in which the characters -of its group are best exhibited and most evenly balanced.” It would -usually consist of those descendants of a form which had undergone -little alteration, while other descendants had suffered slight -differentiation in various directions. - - [591] Waterhouse, quoted by Woodward in his *Rudimentary Treatise of - Recent and Fossil Shells*, p. 61. - -It would be a great mistake to suppose that this classification by -types is a logically distinct method. It is either not a real method -of classification at all, or it is merely an abbreviated mode of -representing a complicated system of arrangement. A class must be -defined by the invariable presence of certain common properties. If, -then, we include an individual in which one of these properties does -not appear, we either fall into logical contradiction, or else we form -a new class with a new definition. Even a single exception constitutes -a new class by itself, and by calling it an exception we merely imply -that this new class closely resembles that from which it diverges in -one or two points only. Thus in the definition of the natural order -of Rosaceæ, we find that the seeds are one or two in each carpel, but -that in the genus Spiræa there are three or four; this must mean either -that the number of seeds is not a part of the fixed definition of the -class, or else that Spiræa does not belong to that class, though it -may closely approximate to it. Naturalists continually find themselves -between two horns of a dilemma; if they restrict the number of marks -specified in a definition so that every form intended to come within -the class shall possess all those marks, it will then be usually found -to include too many forms; if the definition be made more particular, -the result is to produce so-called anomalous genera, which, while they -are held to belong to the class, do not in all respects conform to its -definition. The practice has hence arisen of allowing considerable -latitude in the definition of natural orders. The family of Cruciferæ, -for instance, forms an exceedingly well-marked natural order, and among -its characters we find it specified that the fruit is a pod, divided -into two cells by a thin partition, from which the valves generally -separate at maturity; but we are also informed that, in a few genera, -the pod is one-celled, or indehiscent, or separates transversely into -several joints.[592] Now this must either mean that the formation of -the pod is not an essential point in the definition of the family, or -that there are several closely associated families. - - [592] Bentham’s *Handbook of the British Flora* (1866), p. 25. - -The same holds true of typical classification. The type itself is -an individual, not a class, and no other object can be exactly like -the type. But as soon as we abstract the individual peculiarities -of the type and thus specify a finite number of qualities in which -other objects may resemble the type, we immediately constitute -a class. If some objects resemble the type in some points, and -others in other points, then each definite collection of points of -resemblance constitutes intensively a separate class. The very notion -of classification by types is in fact erroneous in a logical point of -view. The naturalist is constantly occupied in endeavouring to mark -out definite groups of living forms, where the forms themselves do not -in many cases admit of such rigorous lines of demarcation. A certain -laxity of logical method is thus apt to creep in, the only remedy for -which will be the frank recognition of the fact, that, according to the -theory of hereditary descent, gradation of characters is probably the -rule, and precise demarcation between groups the exception. - - -*Natural Genera and Species.* - -One important result of the establishment of the theory of evolution -is to explode all notions about natural groups constituting separate -creations. Naturalists long held that every plant belongs to some -species, marked out by invariable characters, which do not change by -difference of soil, climate, cross-breeding, or other circumstances. -They were unable to deny the existence of such things as sub-species, -varieties, and hybrids, so that a species of plants was often -subdivided and classified within itself. But then the differences upon -which this sub-classification depended were supposed to be variable, -and thus distinguished from the invariable characters imposed upon the -whole species at its creation. Similarly a natural genus was a group of -species, and was marked out from other genera by eternal differences of -still greater importance. - -We now, however, perceive that the existence of any such groups as -genera and species is an arbitrary creation of the naturalist’s -mind. All resemblances of plants are natural so far as they express -hereditary affinities; but this applies as well to the variations -within the species as to the species itself, or to the larger groups. -All is a matter of degree. The deeper differences between plants have -been produced by the differentiating action of circumstances during -millions of years, so that it would naturally require millions of -years to undo this result, and prove experimentally that the forms can -be approximated again. Sub-species may sometimes have arisen within -historical times, and varieties approaching to sub-species may often -be produced by the horticulturist in a few years. Such varieties can -easily be brought back to their original forms, or, if placed in the -original circumstances, will themselves revert to those forms; but -according to Darwin’s views all forms are capable of unlimited change, -and it might possibly be, unlimited reversion if suitable circumstances -and sufficient time be granted. - -Many fruitless attempts have been made to establish a rigorous -criterion of specific and generic difference, so that these classes -might have a definite value and rank in all branches of biology. -Linnæus adopted the view that the species was to be defined as a -distinct creation, saying,[593] “Species tot numeramus, quot diversæ -formæ in principio sunt creatæ;” or again, “Species tot sunt, quot -diversas formas ab initio produxit Infinitum Ens; quæ formæ, secundum -generationis inditas leges, produxere plures, at sibi semper similes.” -Of genera he also says,[594] “Genus omne est naturale, in primordio -tale creatum.” It was a common doctrine added to and essential to that -of distinct creation that these species could not produce intermediate -and variable forms, so that we find Linnæus obliged by the ascertained -existence of hybrids to take a different view in another work; he -says,[595] “Novas species immo et genera ex copula diversarum specierum -in regno vegetabilium oriri primo intuitu paradoxum videtur; interim -observationes sic fieri non ita dissuadent.” Even supposing in the -present day that we could assent to the notion of a certain number of -distinct creational acts, this notion would not help us in the theory -of classification. Naturalists have never pointed out any method of -deciding what are the results of distinct creations, and what are -not. As Darwin says,[596] “the definition must not include an element -which cannot possibly be ascertained, such as an act of creation.” -It is, in fact, by investigation of forms and classification that we -should ascertain what were distinct creations and what were not; this -information would be a result and not a means of classification. - - [593] *Philosophia Botanica* (1770), § 157, p. 99. - - [594] *Ibid.* § 159, p. 100. - - [595] *Amœnitates Academicæ* (1744), vol. i. p. 70. Quoted in - *Edinburgh Review*, October 1868, vol. cxxviii. pp. 416, 417. - - [596] *Descent of Man*, vol. i. p. 228. - -Agassiz seemed to consider that he had discovered an important -principle, to the effect that general plan or structure is the true -ground for the discrimination of the great classes of animals, which -may be called branches of the animal kingdom.[597] He also thought that -genera are definite and natural groups. “Genera,” he says,[598] “are -most closely allied groups of animals, differing neither in form, nor -in complication of structure, but simply in the ultimate structural -peculiarities of some of their parts; and this is, I believe, the best -definition which can be given of genera.” But it is surely apparent -that there are endless degrees both of structural peculiarity and of -complication of structure. It is impossible to define the amount of -structural peculiarity which constitutes the genus as distinguished -from the species. - - [597] Agassiz, *Essay on Classification*, p. 219. - - [598] *Ibid.* p. 249. - -The form which any classification of plants or animals tends to take is -that of an unlimited series of subaltern classes. Originally botanists -confined themselves for the most part to a small number of such -classes. Linnæus adopted Class, Order, Genus, Species, and Variety, and -even seemed to think that there was something essentially natural in a -five-fold arrangement of groups.[599] - - [599] *Philosophia Botanica*, § 155, p. 98. - -With the progress of botany intermediate and additional groups -have gradually been introduced. According to the Laws of Botanical -Nomenclature adopted by the International Botanical Congress, held at -Paris[600] in August 1867, no less than twenty-one names of classes -are recognised--namely, Kingdom, Division, Sub-division, Class, -Sub-class, Cohort, Sub-cohort, Order, Sub-order, Tribe, Sub-tribe, -Genus, Sub-genus, Section, Sub-section, Species, Sub-species, Variety, -Sub-variety, Variation, Sub-variation. It is allowed by the authors of -this scheme, that the rank or degree of importance to be attributed -to any of these divisions may vary in a certain degree according to -individual opinion. The only point on which botanists are not allowed -discretion is as to the order of the successive sub-divisions; any -inversion of the arrangement, such as division of a genus into tribes, -or of a tribe into orders, is quite inadmissible. There is no reason -to suppose that even the above list is complete and inextensible. The -Botanical Congress itself recognised the distinction between variations -according as they are Seedlings, Half-breeds, or *Lusus Naturæ*. -The complication of the inferior classes is increased again by the -existence of *hybrids*, arising from the fertilisation of one species -by another deemed a distinct species, nor can we place any limit to the -minuteness of discrimination of degrees of breeding short of an actual -pedigree of individuals. - - [600] *Laws of Botanical Nomenclature*, by Alphonse Decandolle, - translated from the French, 1868, p. 19. - -It will be evident to the reader that in the remarks upon -classification as applied to the Natural Sciences, given in this -and the preceding sections, I have not in the least attempted to -treat the subject in a manner adequate to its extent and importance. -A volume would be insufficient for tracing out the principles of -scientific method specially applicable to these branches of science. -What more I may be able to say upon the subject will be better said, -if ever, when I am able to take up the closely-connected subjects of -Scientific Nomenclature, Terminology, and Descriptive Representation. -In the meantime, I have wished to show, in a negative point of -view, that natural classification in the animal and vegetable -kingdoms is a special problem, and that the particular methods and -difficulties to which it gives rise are not those common to all cases -of classification, as so many physicists have supposed. Genealogical -resemblances are only a special case of resemblances in general. - - -*Unique or Exceptional Objects.* - -In framing a system of classification in almost any branch of science, -we must expect to meet with unique or peculiar objects, which stand -alone, having comparatively few analogies with other objects. They may -also be said to be *sui generis*, each unique object forming, as it -were, a genus by itself; or they are called *nondescript*, because from -thus standing apart it is difficult to find terms in which to describe -their properties. The rings of Saturn, for instance, form a unique -object among the celestial bodies. We have indeed considered this and -many other instances of unique objects in the preceding chapter on -Exceptional Phenomena. Apparent, Singular, and Divergent Exceptions -especially, are analogous to unique objects. - -In the classification of the elements, Carbon stands apart as a -substance entirely unique in its powers of producing compounds. It is -considered to be a quadrivalent element, and it obeys all the ordinary -laws of chemical combination. Yet it manifests powers of affinity in -such an exalted degree that the substances in which it appears are -more numerous than all the other compounds known to chemists. Almost -the whole of the substances which have been called organic contain -carbon, and are probably held together by the carbon atoms, so that -many chemists are now inclined to abandon the name Organic Chemistry, -and substitute the name Chemistry of the Carbon Compounds. It used to -be believed that the production of organic compounds could be effected -only by the action of vital force, or of some inexplicable cause -involved in the phenomena of life; but it is now found that chemists -are able to commence with the elementary materials, pure carbon, -hydrogen, and oxygen, and by strictly chemical operations to combine -these so as to form complicated organic compounds. So many substances -have already been formed that we might be inclined to generalise and -infer that all organic compounds might ultimately be produced without -the agency of living beings. Thus the distinction between the organic -and the inorganic kingdoms seems to be breaking down, but our wonder at -the peculiar powers of carbon must increase at the same time. - -In considering generalisation, the law of continuity was applied -chiefly to physical properties capable of mathematical treatment. But -in the classificatory sciences, also, the same important principle -is often beautifully exemplified. Many objects or events seem to be -entirely exceptional and abnormal, and in regard to degree or magnitude -they may be so termed; but it is often easy to show that they are -connected by intermediate links with ordinary cases. In the organic -kingdoms there is a common groundwork of similarity running through -all classes, but particular actions and processes present themselves -conspicuously in particular families and classes. Tenacity of life -is most marked in the Rotifera, and some other kinds of microscopic -organisms, which can be dried and boiled without loss of life. Reptiles -are distinguished by torpidity, and the length of time they can live -without food. Birds, on the contrary, exhibit ceaseless activity and -high muscular power. The ant is as conspicuous for intelligence and -size of brain among insects as the quadrumana and man among vertebrata. -Among plants the Leguminosæ are distinguished by a tendency to sleep, -folding their leaves at the approach of night. In the genus Mimosa, -especially the Mimosa pudica, commonly called the sensitive plant, -the same tendency is magnified into an extreme irritability, almost -resembling voluntary motion. More or less of the same irritability -probably belongs to vegetable forms of every kind, but it is of course -to be investigated with special ease in such an extreme case. In the -Gymnotus and Torpedo, we find that organic structures can act like -galvanic batteries. Are we to suppose that such animals are entirely -anomalous exceptions; or may we not justly expect to find less intense -manifestations of electric action in all animals? - -Some extraordinary differences between the modes of reproduction -of animals have been shown to be far less than was at first sight -apparent. The lower animals seem to differ entirely from the higher -ones in the power of reproducing lost limbs. A kind of crab has the -habit of casting portions of its claws when much frightened, but -they soon grow again. There are multitudes of smaller animals which, -like the Hydra, may be cut in two and yet live and develop into new -complete individuals. No mammalian animal can reproduce a limb, and -in appearance there is no analogy. But it was suggested by Blumenbach -that the healing of a wound in the higher animals really represents in -a lower degree the power of reproducing a limb. That this is true may -be shown by adducing a multitude of intermediate cases, each adjoining -pair of which are clearly analogous, so that we pass gradually from one -extreme to the other. Darwin holds, moreover, that any such restoration -of parts is closely connected with that perpetual replacement of -the particles which causes every organised body to be after a time -entirely new as regards its constituent substance. In short, we -approach to a great generalisation under which all the phenomena of -growth, restoration, and maintenance of organs are effects of one and -the same power.[601] It is perhaps still more surprising to find that -the complicated process of reproduction in the higher animals may be -gradually traced down to a simpler and simpler form, which at last -becomes undistinguishable from the budding out of one plant from the -stem of another. By a great generalisation we may regard all the modes -of reproduction of organic life as alike in their nature, and varying -only in complexity of development.[602] - - [601] Darwin, *The Variation of Animals and Plants*, vol. ii. - pp. 293, 359, &c.; quoting Paget, *Lectures on Pathology*, 1853, - pp. 152, 164. - - [602] *Ibid.* vol. ii. p. 372. - - -*Limits of Classification.* - -Science can extend only so far as the power of accurate classification -extends. If we cannot detect resemblances, and assign their exact -character and amount, we cannot have that generalised knowledge which -constitutes science; we cannot infer from case to case. Classification -is the opposite process to discrimination. If we feel that two tastes -differ, the tastes of two kinds of wine for instance, the mere fact of -difference existing prevents inference. The detection of the difference -saves us, indeed, from false inference, because so far as difference -exists, inference is impossible. But classification consists in -detecting resemblances of all degrees of generality, and ascertaining -exactly how far such resemblances extend, while assigning precisely the -points at which difference begins. It enables us, then, to generalise, -and make inferences where it is possible, and it saves us at the same -time from going too far. A full classification constitutes a complete -record of all our knowledge of the objects or events classified, -and the limits of exact knowledge are identical with the limits of -classification. - -It must by no means be supposed that every group of natural objects -will be found capable of rigorous classification. There may be -substances which vary by insensible degrees, consisting, for instance, -in varying mixtures of simpler substances. Granite is a mixture of -quartz, felspar, and mica, but there are hardly two specimens in which -the proportions of these three constituents are alike, and it would -be impossible to lay down definitions of distinct species of granite -without finding an infinite variety of intermediate species. The -only true classification of granites, then, would be founded on the -proportions of the constituents present, and a chemical or microscopic -analysis would be requisite, in order that we might assign a specimen -to its true position in the series. Granites vary, again, by insensible -degrees, as regards the magnitude of the crystals of felspar and mica. -Precisely similar remarks might be made concerning the classification -of other plutonic rocks, such as syenite, basalt, pumice-stone, lava. - -The nature of a ray of homogeneous light is strictly defined, either -by its place in the spectrum or by the corresponding wave-length, but -a ray of mixed light admits of no simple classification; any of the -infinitely numerous rays of the continuous spectrum may be present or -absent, or present in various intensities, so that we can only class -and define a mixed colour by defining the intensity and wave-length -of each ray of homogeneous light which is present in it. Complete -spectroscopic analysis and the determination of the intensity of -every part of the spectrum yielded by a mixed ray is requisite for -its accurate classification. Nearly the same may be said of complex -sounds. A simple sound undulation, if we could meet with such a sound, -would admit of precise and exhaustive classification as regards pitch, -the length of wave, or the number of waves reaching the ear per -second being a sufficient criterion. But almost all ordinary sounds, -even those of musical instruments, consist of complex aggregates of -undulations of different pitches, and in order to classify the sound -we should have to measure the intensities of each of the constituent -sounds, a work which has been partially accomplished by Helmholtz, as -regards the vowel sounds. The different tones of voice distinctive -of different individuals must also be due to the intermixture of -minute waves of various pitch, which are yet quite beyond the range -of experimental investigation. We cannot, then, at present attempt to -classify the different kinds or *timbres* of sound. - -The difficulties of classification are still greater when a varying -phenomenon cannot be shown to be a mixture of simpler phenomena. If -we attempt to classify tastes, we may rudely group them according as -they are sweet, bitter, saline, alkaline, acid, astringent or fiery; -but it is evident that these groups are bounded by no sharp lines -of definition. Tastes of mixed or intermediate character may exist -almost *ad infinitum*, and what is still more troublesome, the tastes -clearly united within one class may differ more or less from each -other, without our being able to arrange them in subordinate genera and -species. The same remarks may be made concerning the classification of -odours, which may be roughly grouped according to the arrangement of -Linnæus as, aromatic, fragrant, ambrosiac, alliaceous, fetid, virulent, -nauseous. Within each of these vague classes, however, there would be -infinite shades of variety, and each class would graduate into other -classes. The odours which can be discriminated by an acute nose are -infinite; every rock, stone, plant, or animal has some slight smell, -and it is well known that dogs, or even blind men, can discriminate -persons by a slight distinctive odour which usually passes unnoticed. - -Similar remarks may be made concerning the feelings of the human mind, -called emotions. We know what is anger, grief, fear, hatred, love; -and many systems for classifying these feelings have been proposed. -They may be roughly distinguished according as they are pleasurable -or painful, prospective or retrospective, selfish or sympathetic, -active or passive, and possibly in many other ways; but each mode of -arrangement will be indefinite and unsatisfactory when followed into -details. As a general rule, the emotional state of the mind at any -moment will be neither pure anger nor pure fear, nor any one pure -feeling, but an indefinite and complex aggregate of feelings. It may -be that the state of mind is really a sum of several distinct modes -of agitation, just as a mixed colour is the sum of the several rays -of the spectrum. In this case there may be more hope of some method -of analysis being successfully applied at a future time. But it may -be found that states of mind really graduate into each other so that -rigorous classification would be hopeless. - -A little reflection will show that there are whole worlds of -existences which in like manner are incapable of logical analysis and -classification. One friend may be able to single out and identify -another friend by his countenance among a million other countenances. -Faces are capable of infinite discrimination, but who shall classify -and define them, or say by what particular shades of feature he does -judge? There are of course certain distinct types of face, but each -type is connected with each other type by infinite intermediate -specimens. We may classify melodies according to the major or minor -key, the character of the time, and some other distinct points; -but every melody has, independently of such circumstances, its -own distinctive character and effect upon the mind. We can detect -differences between the styles of literary, musical, or artistic -compositions. We can even in some cases assign a picture to its -painter, or a symphony to its composer, by a subtle feeling of -resemblances or differences which may be felt, but cannot be described. - -Finally, it is apparent that in human character there is unfathomable -and inexhaustible diversity. Every mind is more or less like every -other mind; there is always a basis of similarity, but there is a -superstructure of feelings, impulses, and motives which is distinctive -for each person. We can sometimes predict the general character of the -feelings and actions which will be produced by a given external event -in an individual well known to us; but we also know that we are often -inexplicably at fault in our inferences. No one can safely generalise -upon the subtle variations of temper and emotion which may arise even -in a person of ordinary character. As human knowledge and civilisation -progress, these characteristic differences tend to develop and multiply -themselves, rather than decrease. Character grows more many-sided. Two -well educated Englishmen are far better distinguished from each other -than two common labourers, and these are better distinguished than two -Australian aborigines. The complexities of existing phenomena probably -develop themselves more rapidly than scientific method can overtake -them. In spite of all the boasted powers of science, we cannot really -apply scientific method to our own minds and characters, which are more -important to us than all the stars and nebulæ. - - - - -BOOK VI. - - - - -CHAPTER XXXI. - -REFLECTIONS ON THE RESULTS AND LIMITS OF SCIENTIFIC METHOD. - - -Before concluding a work on the Principles of Science, it will not be -inappropriate to add some remarks upon the limits and ultimate bearings -of the knowledge which we may acquire by the employment of scientific -method. All science consists, it has several times been stated, in the -detection of identities in the action of natural agents. The purpose of -inductive inquiry is to ascertain the apparent existence of necessary -connection between causes and effects, expressed in the form of natural -laws. Now so far as we thus learn the invariable course of nature, the -future becomes the necessary sequel of the present, and we are brought -beneath the sway of powers with which nothing can interfere. - -By degrees it is found, too, that the chemistry of organised substances -is not entirely separated from, but is continuous with, that of earth -and stones. Life seems to be nothing but a special form of energy -which is manifested in heat and electricity and mechanical force. The -time may come, it almost seems, when the tender mechanism of the brain -will be traced out, and every thought reduced to the expenditure of -a determinate weight of nitrogen and phosphorus. No apparent limit -exists to the success of scientific method in weighing and measuring, -and reducing beneath the sway of law, the phenomena both of matter -and of mind. And if mental phenomena be thus capable of treatment by -the balance and the micrometer, can we any longer hold that mind is -distinct from matter? Must not the same inexorable reign of law which -is apparent in the motions of brute matter be extended to the subtle -feelings of the human heart? Are not plants and animals, and ultimately -man himself, merely crystals, as it were, of a complicated form? If -so, our boasted free will becomes a delusion, moral responsibility a -fiction, spirit a mere name for the more curious manifestations of -material energy. All that happens, whether right or wrong, pleasurable -or painful, is but the outcome of the necessary relations of time and -space and force. - -Materialism seems, then, to be the coming religion, and resignation -to the nonentity of human will the only duty. Such may not generally -be the reflections of men of science, but I believe that we may thus -describe the secret feelings of fear which the constant advance of -scientific investigation excites in the minds of many. Is science, -then, essentially atheistic and materialistic in its tendency? -Does the uniform action of material causes, which we learn with an -ever-increasing approximation to certainty, preclude the hypothesis of -a benevolent Creator, who has not only designed the existing universe, -but who still retains the power to alter its course from time to time? - -To enter upon actual theological discussions would be evidently beyond -the scope of this work. It is with the scientific method common to all -the sciences, and not with any of the separate sciences, that we are -concerned. Theology therefore would be at least as much beyond my scope -as chemistry or geology. But I believe that grave misapprehensions -exist as regards the very nature of scientific method. There are -scientific men who assert that the interposition of Providence is -impossible, and prayer an absurdity, because the laws of nature are -inductively proved to be invariable. Inferences are drawn not so -much from particular sciences as from the logical nature of science -itself, to negative the impulses and hopes of men. Now I may state -that my own studies in logic lead me to call in question such negative -inferences. Laws of nature are uniformities observed to exist in the -action of certain material agents, but it is logically impossible to -show that all other agents must behave as these do. The too exclusive -study of particular branches of physical science seems to generate an -over-confident and dogmatic spirit. Rejoicing in the success with which -a few groups of facts are brought beneath the apparent sway of laws, -the investigator hastily assumes that he is close upon the ultimate -springs of being. A particle of gelatinous matter is found to obey -the ordinary laws of chemistry; yet it moves and lives. The world is -therefore asked to believe that chemistry can resolve the mysteries of -existence. - - -*The Meaning of Natural Law.* - -Pindar speaks of Law as the Ruler of the Mortals and the Immortals, and -it seems to be commonly supposed that the so-called Laws of Nature, in -like manner, rule man and his Creator. The course of nature is regarded -as being determined by invariable principles of mechanics which have -acted since the world began, and will act for evermore. Even if the -origin of all things is attributed to an intelligent creative mind, -that Being is regarded as having yielded up arbitrary power, and as -being subject like a human legislator to the laws which he has himself -enacted. Such notions I should describe as superficial and erroneous, -being derived, as I think, from false views of the nature of scientific -inference, and the degree of certainty of the knowledge which we -acquire by inductive investigation. - -A law of nature, as I regard the meaning of the expression, is not a -uniformity which must be obeyed by all objects, but merely a uniformity -which is as a matter of fact obeyed by those objects which have come -beneath our observation. There is nothing whatever incompatible with -logic in the discovery of objects which should prove exceptions to any -law of nature. Perhaps the best established law is that which asserts -an invariable correlation to exist between gravity and inertia, so that -all gravitating bodies are found to possess inertia, and all bodies -possessing inertia are found to gravitate. But it would be no reproach -to our scientific method, if something were ultimately discovered -to possess gravity without inertia. Strictly defined and correctly -interpreted, the law itself would acknowledge the possibility; for with -the statement of every law we ought properly to join an estimate of the -number of instances in which it has been observed to hold true, and the -probability thence calculated, that it will hold true in the next case. -Now, as we found (p. 259), no finite number of instances can warrant -us in expecting with certainty that the next instance will be of like -nature; in the formulas yielded by the inverse method of probabilities -a unit always appears to represent the probability that our inference -will be mistaken. I demur to the assumption that there is any necessary -truth even in such fundamental laws of nature as the Indestructibility -of Matter, the Conservation of Energy, or the Laws of Motion. Certain -it is that men of science have recognised the conceivability of -other laws, and even investigated their mathematical consequences. -Airy investigated the mathematical conditions of a perpetual motion -(p. 223), and Laplace and Newton discussed imaginary laws of forces -inconsistent with those observed to operate in the universe (pp. 642, -706). - -The laws of nature, as I venture to regard them, are simply general -propositions concerning the correlation of properties which have been -observed to hold true of bodies hitherto observed. On the assumption -that our experience is of adequate extent, and that no arbitrary -interference takes place, we are then able to assign the probability, -always less than certainty, that the next object of the same apparent -nature will conform to the same laws. - - -*Infiniteness of the Universe.* - -We may safely accept as a satisfactory scientific hypothesis the -doctrine so grandly put forth by Laplace, who asserted that a perfect -knowledge of the universe, as it existed at any given moment, would -give a perfect knowledge of what was to happen thenceforth and for -ever after. Scientific inference is impossible, unless we may regard -the present as the outcome of what is past, and the cause of what is -to come. To the view of perfect intelligence nothing is uncertain. The -astronomer can calculate the positions of the heavenly bodies when -thousands of generations of men shall have passed away, and in this -fact we have some illustration, as Laplace remarks, of the power which -scientific prescience may attain. Doubtless, too, all efforts in the -investigation of nature tend to bring us nearer to the possession of -that ideally perfect power of intelligence. Nevertheless, as Laplace -with profound wisdom adds,[603] we must ever remain at an infinite -distance from the goal of our aspirations. - - [603] *Théorie Analytique des Probabilités*, quoted by Babbage, - *Ninth Bridgewater Treatise*, p. 173. - -Let us assume, for a time at least, as a highly probable hypothesis, -that whatever is to happen must be the outcome of what is; there then -arises the question, What is? Now our knowledge of what exists must -ever remain imperfect and fallible in two respects. Firstly, we do -not know all the matter that has been created, nor the exact manner -in which it has been distributed through space. Secondly, assuming -that we had that knowledge, we should still be wanting in a perfect -knowledge of the way in which the particles of matter will act upon -each other. The power of scientific prediction extends at the most -to the limits of the data employed. Every conclusion is purely -hypothetical and conditional upon the non-interference of agencies -previously undetected. The law of gravity asserts that every body tends -to approach towards every other body, with a certain determinate force; -but, even supposing the law to hold true, it does not assert that the -body *will* approach. No single law of nature can warrant us in making -an absolute prediction. We must know all the laws of nature and all -the existing agents acting according to those laws before we can say -what will happen. To assume, then, that scientific method can take -everything within its cold embrace of uniformity, is to imply that the -Creator cannot outstrip the intelligence of his creatures, and that -the existing Universe is not infinite in extent and complexity, an -assumption for which I see no logical basis whatever. - - -*The Indeterminate Problem of Creation.* - -A second and very serious misapprehension concerning the import of a -law of nature may now be pointed out. It is not uncommonly supposed -that a law determines the character of the results which shall take -place, as, for instance, that the law of gravity determines what force -of gravity shall act upon a given particle. Surely a little reflection -must render it plain that a law by itself determines nothing. It is -*law plus agents obeying law which has results*, and it is no function -of law to govern or define the number and place of its own agents. -Whether a particle of matter shall gravitate, depends not only upon the -law of Newton, but also upon the distribution of surrounding particles. -The theory of gravitation may perhaps be true throughout all time -and in all parts of space, and the Creator may never find occasion -to create those possible exceptions to it which I have asserted to -be conceivable. Let this be as it may; our science cannot certainly -determine the question. Certain it is, that the law of gravity does not -alone determine the forces which may be brought to bear at any point of -space. The force of gravitation acting upon any particle depends upon -the mass, distance, and relative position of all the other particles -of matter within the bounds of space at the instant in question. -Even assuming that all matter when once distributed through space at -the Creation was thenceforth to act in an invariable manner without -subsequent interference, yet the actual configuration of matter at any -moment, and the consequent results of the law of gravitation, must have -been entirely a matter of free choice. - -Chalmers has most distinctly pointed out that the existing -*collocations* of the material world are as important as the laws -which the objects obey. He remarks that a certain class of writers -entirely overlook the distinction, and forget that mere laws without -collocations would have afforded no security against a turbid and -disorderly chaos.[604] Mill has recognised[605] the truth of Chalmers’ -statement, without drawing the proper inferences from it. He says[606] -of the distribution of matter through space, “We can discover nothing -regular in the distribution itself; we can reduce it to no uniformity, -to no law.” More lately the Duke of Argyll in his well-known work on -the *Reign of Law* has drawn attention to the profound distinction -between laws and collocations of causes. - - [604] *First Bridgewater Treatise* (1834), pp. 16–24. - - [605] *System of Logic*, 5th edit. bk. III. chap. V. § 7; chap. XVI. - § 3. - - [606] *System of Logic*, vol. i. p. 384. - -The original conformation of the material universe, as far as we can -tell, was free from all restriction. There was unlimited space in -which to frame it, and an unlimited number of material particles, -each of which could be placed in any one of an infinite number of -different positions. It should be added, that each particle might be -endowed with any one of an infinite number of quantities of *vis viva* -acting in any one of an infinite number of different directions. The -problem of Creation was, then, what a mathematician would call *an -indeterminate problem*, and it was indeterminate in a great number of -ways. Infinitely numerous and various universes might then have been -fashioned by the various distribution of the original nebulous matter, -although all the particles of matter should obey the law of gravity. - -Lucretius tells us how in the original rain of atoms some of these -little bodies diverged from the rectilinear direction, and coming into -contact with other atoms gave rise to the various combinations of -substances which exist. He omitted to tell us whence the atoms came, -or by what force some of them were caused to diverge; but surely these -omissions involve the whole question. I accept the Lucretian conception -of creation when properly supplemented. Every atom which existed in -any point of space must have existed there previously, or must have -been created there by a previously existing Power. When placed there -it must have had a definite mass and a definite energy. Now, as before -remarked, an unlimited number of atoms can be placed in unlimited space -in an unlimited number of modes of distribution. Out of infinitely -infinite choices which were open to the Creator, that one choice must -have been made which has yielded the Universe as it now exists. - -It would be a mistake, indeed, to suppose that the law of gravity, -when it holds true, is no restriction on the distribution of -force. That law is a geometrical law, and it would in many cases -be mathematically impossible, as far as we can see, that the force -of gravity acting on one particle should be small while that on a -neighbouring particle is great. We cannot conceive that even Omnipotent -Power should make the angles of a triangle greater than two right -angles. The primary laws of thought and the fundamental notions of the -mathematical sciences do not seem to admit of error or alteration. Into -the metaphysical origin and meaning of the apparent necessity attaching -to such laws I have not attempted to inquire in this work, and it is -not requisite for my present purpose. If the law of gravity were the -only law of nature and the Creator had chosen to render all matter -obedient to that law, there would doubtless be restrictions upon the -effects derivable from any one distribution of matter. - - -*Hierarchy of Natural Laws.* - -A further consideration presents itself. A natural law like that -of gravity expresses a certain uniformity in the action of agents -submitted to it, and this produces, as we have seen, certain -geometrical restrictions upon the effects which those agents may -produce. But there are other forces and laws besides gravity. One -force may override another, and two laws may each be obeyed and may -each disguise the action of the other. In the intimate constitution of -matter there may be hidden springs which, while acting in accordance -with their own fixed laws, may lead to sudden and unexpected changes. -So at least it has been found from time to time in the past, and so -there is every reason to believe it will be found in the future. To -the ancients it seemed incredible that one lifeless stone could make -another leap towards it. A piece of iron while it obeys the magnetic -force of the loadstone does not the less obey the law of gravity. -A plant gravitates downwards as regards every constituent cell or -fibre, and yet it persists in growing upwards. Life is altogether an -exception to the simpler phenomena of mineral substances, not in the -sense of disproving those laws, but in superadding forces of new and -inexplicable character. Doubtless no law of chemistry is broken by the -action of the nervous cells, and no law of physics by the pulses of the -nervous fibres, but something requires to be added to our sciences in -order that we may explain these subtle phenomena. - -Now there is absolutely nothing in science or in scientific method to -warrant us in assigning a limit to this hierarchy of laws. When in -many undoubted cases we find law overriding law, and at certain points -in our experience producing unexpected results, we cannot venture to -affirm that we have exhausted the strange phenomena which may have been -provided for in the original constitution of matter. The Universe might -have been so designed that it should go for long intervals through the -same round of unvaried existence, and yet that events of exceptional -character should be produced from time to time. Babbage showed in that -most profound and eloquent work, *The Ninth Bridgewater Treatise*, that -it was theoretically possible for human artists to design a machine, -consisting of metallic wheels and levers, which should work invariably -according to a simple law of action during any finite number of steps, -and yet at a fixed moment, however distant, should manifest a single -breach of law. Such an engine might go on counting, for instance, the -natural numbers until they would reach a number requiring for its -expression a hundred million digits. “If every letter in the volume -now before the reader’s eyes,” says Babbage,[607] “were changed into -a figure, and if all the figures contained in a thousand such volumes -were arranged in order, the whole together would yet fall far short -of the vast induction the observer would have had in favour of the -truth of the law of natural numbers.... Yet shall the engine, true to -the prediction of its director, after the lapse of myriads of ages, -fulfil its task, and give that one, the first and only exception to -that time-sanctioned law. What would have been the chances against the -appearance of the excepted case, immediately prior to its occurrence?” - - [607] *Ninth Bridgewater Treatise*, p. 140. - -As Babbage further showed,[608] a calculating engine, after proceeding -through any required number of motions according to a first law, may -be made suddenly to suffer a change, so that it shall then commence -to calculate according to a wholly new law. After giving the natural -numbers for a finite time, it might suddenly begin to give triangular, -or square, or cube numbers, and these changes might be conceived -theoretically as occurring time after time. Now if such occurrences can -be designed and foreseen by a human artist, it is surely within the -capacity of the Divine Artist to provide for analogous changes of law -in the mechanism of the atom, or the construction of the heavens. - - [608] *Ibid.* pp. 34–43. - -Physical science, so far as its highest speculations can be trusted, -gives some indication of a change of law in the past history of the -Universe. According to Sir W. Thomson’s deductions from Fourier’s -*Theory of Heat*, we can trace down the dissipation of heat by -conduction and radiation to an infinitely distant time when all -things will be uniformly cold. But we cannot similarly trace the -heat-history of the Universe to an infinite distance in the past. -For a certain negative value of the time the formulæ give impossible -values, indicating that there was some initial distribution of heat -which could not have resulted, according to known laws of nature,[609] -from any previous distribution.[610] There are other cases in which a -consideration of the dissipation of energy leads to the conception of -a limit to the antiquity of the present order of things.[611] Human -science, of course, is fallible, and some oversight or erroneous -simplification in these theoretical calculations may afterwards be -discovered; but as the present state of scientific knowledge is the -only ground on which erroneous inferences from the uniformity of nature -and the supposed reign of law are founded, I am right in appealing to -the present state of science in opposition to these inferences. Now -the theory of heat places us in the dilemma either of believing in -Creation at an assignable date in the past, or else of supposing that -some inexplicable change in the working of natural laws then took -place. Physical science gives no countenance to the notion of infinite -duration of matter in one continuous course of existence. And if in -time past there has been a discontinuity of law, why may there not be -a similar event awaiting the world in the future? Infinite ingenuity -could have implanted some agency in matter so that it might never yet -have made its tremendous powers manifest. We have a very good theory -of the conservation of energy, but the foremost physicists do not -deny that there may possibly be forms of energy, neither kinetic nor -potential, and therefore of unknown nature.[612] - - [609] Professor Clifford, in his most interesting lecture on “The - First and Last Catastrophe” (*Fortnightly Review*, April 1875, - p. 480, reprint by the Sunday Lecture Society, p. 24), objects that I - have erroneously substituted “known laws of nature” for “known laws - of conduction of heat.” I quite admit the error, without admitting - all the conclusions which Professor Clifford proceeds to draw; but I - maintain the paragraph unchanged, in order that it may be discussed - in the Preface. - - [610] Tait’s *Thermodynamics*, p. 38. *Cambridge Mathematical - Journal*, vol. iii. p. 174. - - [611] Clerk Maxwell’s *Theory of Heat*, p. 245. - - [612] Maxwell’s *Theory of Heat*, p. 92. - -We can imagine reasoning creatures dwelling in a world where the -atmosphere was a mixture of oxygen and inflammable gas like the -fire-damp of coal-mines. If devoid of fire, they might have lived -through long ages unconscious of the tremendous forces which a single -spark would call into play. In the twinkling of an eye new laws might -come into action, and the poor reasoning creatures, so confident about -their knowledge of the reign of law in their world, would have no time -to speculate upon the overthrow of all their theories. Can we with our -finite knowledge be sure that such an overthrow of our theories is -impossible? - - -*The Ambiguous Expression, “Uniformity of Nature.”* - -I have asserted that serious misconception arises from an erroneous -interpretation of the expression Uniformity of Nature. Every law of -nature is the statement of a certain uniformity observed to exist among -phenomena, and since the laws of nature are invariably obeyed, it seems -to follow that the course of nature itself is uniform, so that we can -safely judge of the future by the present. This inference is supported -by some of the results of physical astronomy. Laplace proved that -the planetary system is stable, so that no perturbation which planet -produces upon planet can become so great as to cause disruption and -permanent alteration of the planetary orbits. A full comprehension of -the law of gravity shows that all such disturbances are essentially -periodic, so that after the lapse of millions of years the planets will -return to the same relative positions, and a new cycle of disturbances -will then commence. - -As other branches of science progress, we seem to gain assurance -that no great alteration of the world’s condition is to be expected. -Conflict with a comet has long been the cause of fear, but now it is -credibly asserted that we have passed through a comet’s tail without -the fact being known at the time, or manifested by any more serious a -phenomenon than a slight luminosity of the sky. More recently still -the earth is said to have touched the comet Biela, and the only result -was a beautiful and perfectly harmless display of meteors. A decrease -in the heating power of the sun seems to be the next most probable -circumstance from which we might fear the extinction of life on the -earth. But calculations founded on reasonable physical data show -that no appreciable change can be going on, and experimental data to -indicate a change are wholly wanting. Geological investigations show -indeed that there have been extensive variations of climate in past -times; vast glaciers and icebergs have swept over the temperate regions -at one time, and tropical vegetation has flourished near the poles -at another time. But here again the vicissitudes of climate assume a -periodic character, so that the stability of the earth’s condition does -not seem to be threatened. - -All these statements may be reasonable, but they do not establish the -Uniformity of Nature in the sense that extensive alterations or sudden -catastrophes are impossible. In the first place, Laplace’s theory of -the stability of the planetary system is of an abstract character, as -paying regard to nothing but the mutual gravitation of the planetary -bodies and the sun. It overlooks several physical causes of change -and decay in the system which were not so well known in his day as at -present, and it also presupposes the absence of any interruption of the -course of things by conflict with foreign astronomical bodies. - -It is now acknowledged by astronomers that there are at least two ways -in which the *vis viva* of the planets and satellites may suffer loss. -The friction of the tides upon the earth produces a small quantity -of heat which is radiated into space, and this loss of energy must -result in a decrease of the rotational velocity, so that ultimately -the terrestrial day will become identical with the year, just as the -periods of revolution of the moon upon its axis and around the earth -have already become equal. Secondly, there can be little doubt that -certain manifestations of electricity upon the earth’s surface depend -upon the relative motions of the planets and the sun, which give -rise to periods of increased intensity. Such electrical phenomena -must result in the production and dissipation of heat, the energy -of which must be drawn, partially at least, from the moving bodies. -This effect is probably identical (p. 570) with the loss of energy of -comets attributed to the so-called resisting medium. But whatever be -the theoretical explanation of these phenomena, it is almost certain -that there exists a tendency to the dissipation of the energy of the -planetary system, which will, in the indefinite course of time, result -in the fall of the planets into the sun. - -It is hardly probable, however, that the planetary system will be left -undisturbed throughout the enormous interval of time required for the -dissipation of its energy in this way. Conflict with other bodies is -so far from being improbable, that it becomes approximately certain -when we take very long intervals of time into account. As regards -cometary conflicts, I am by no means satisfied with the negative -conclusions drawn from the remarkable display on the evening of the -27th of November, 1872. We may often have passed through the tail of a -comet, the light of which is probably an electrical manifestation no -more substantial than the aurora borealis. Every remarkable shower of -shooting stars may also be considered as proceeding from a cometary -body, so that we may be said to have passed through the thinner parts -of innumerable comets. But the earth has probably never passed, in -times of which we have any record, through the nucleus of a comet, -which consists perhaps of a dense swarm of small meteorites. We can -only speculate upon the effects which might be produced by such a -conflict, but it would probably be a much more serious event than any -yet registered in history. The probability of its occurrence, too, -cannot be assigned; for though the probability of conflict with any one -cometary nucleus is almost infinitesimal, yet the number of comets is -immensely great (p. 408). - -It is far from impossible, again, that the planetary system may be -invaded by bodies of greater mass than comets. The sun seems to be -placed in so extensive a portion of empty space that its own proper -motion would not bring it to the nearest known star (α Centauri) in -less than 139,200 years. But in order to be sure that this interval of -undisturbed life is granted to our globe, we must prove that there are -no stars moving so as to meet us, and no dark bodies of considerable -size flying through intervening space unknown to us. The intrusion -of comets into our system, and the fact that many of them have -hyperbolic paths, is sufficient to show that the surrounding parts of -space are occupied by multitudes of dark bodies of some size. It is -quite probable that small suns may have cooled sufficiently to become -non-luminous; for even if we discredit the theory that the variation of -brightness of periodic stars is due to the revolution of dark companion -stars, yet there is in our own globe an unquestionable example of a -smaller body which has cooled below the luminous point. - -Altogether, then, it is a mere assumption that the uniformity of nature -involves the unaltered existence of our own globe. There is no kind -of catastrophe which is too great or too sudden to be theoretically -consistent with the reign of law. For all that our science can tell, -human history may be closed in the next instant of time. The world may -be dashed to pieces against a wandering star; it may be involved in a -nebulous atmosphere of hydrogen to be exploded a second afterwards; it -may be scorched up or dissipated into vapour by some great explosion in -the sun; there might even be within the globe itself some secret cause -of disruption, which only needs time for its manifestation. - -There are some indications, as already noticed (p. 660), that violent -disturbances have actually occurred in the history of the solar system. -Olbers sought for the minor planets on the supposition that they were -fragments of an exploded planet, and he was rewarded with the discovery -of some of them. The retrograde motion of the satellites of the more -distant planets, the abnormal position of the poles of Uranus and the -excessive distance of Neptune, are other indications of some violent -event, of which we have no other evidence. I adduce all these facts -and arguments, not to show that there is any considerable probability, -as far as we can judge, of interruption within the scope of human -history, but to prove that the Uniformity of Nature is theoretically -consistent with the most unexpected events of which we can form a -conception. - - -*Possible States of the Universe.* - -When we give the rein to scientific imagination, it becomes apparent -that conflict of body with body must not be regarded as the rare -exception, but as the general rule and the inevitable fate of each -star system. So far as we can trace out the results of the law of -gravitation, and of the dissipation of energy, the universe must be -regarded as undergoing gradual condensation into a single cold solid -body of gigantic dimensions. Those who so frequently use the expression -Uniformity of Nature seem to forget that the Universe might exist -consistently with the laws of nature in the most diverse conditions. It -might consist, on the one hand, of a glowing nebulous mass of gaseous -substances. The heat might be so intense that all elements, even carbon -and silicon, would be in the state of gas, and all atoms, of whatever -nature, would be flying about in chemical independence, diffusing -themselves almost uniformly in the neighbouring parts of space. There -would then be no life, unless we can apply that name to the passage -through each part of space of similar average trains of atoms, the -particular succession of atoms being governed only by the theory of -probability, and the law of divergence from a mean exhibited in the -Arithmetical Triangle. Such a universe would correspond partially to -the Lucretian rain of atoms, and to that nebular hypothesis out of -which Laplace proposed philosophically to explain the evolution of the -planetary system. - -According to another extreme supposition, the intense heat-energy of -this nebulous mass might be radiated away into the unknown regions -of outer space. The attraction of gravity would exert itself between -each two particles, and the energy of motion thence arising would, -by incessant conflicts, be resolved into heat and dissipated. -Inconceivable ages might be required for the completion of this -process, but the dissipation of energy thus proceeding could end only -in the production of a cold and motionless universe. The relation of -cause and effect, as we see it manifested in life and growth, would -degenerate into the constant existence of every particle in a fixed -position relative to every other particle. Logical and geometrical -resemblances would still exist between atoms, and between groups of -atoms crystallised in their appropriate forms for evermore. But time, -the great variable, would bring no variation, and as to human hopes and -troubles, they would have gone to eternal rest. - -Science is not really adequate to proving that such is the inevitable -fate of the universe, for we can seldom trust our best-established -theories far from their data. Nevertheless, the most probable -speculations which we can form as to the history, especially of our own -planetary system, is that it originated in a heated revolving nebulous -mass of gas, and is in a state of excessively slow progress towards -the cold and stony condition. Other speculative hypotheses might -doubtless be entertained. Every hypothesis is pressed by difficulties. -If the whole universe be cooling, whither does the heat go? If we are -to get rid of it entirely, outer space must be infinite in extent, so -that it shall never be stopped and reflected back. But not to speak -of metaphysical difficulties, if the medium of heat undulations be -infinite in extent, why should not the material bodies placed in -it be infinite also in number and aggregate mass? It is apparent -that we are venturing into speculations which surpass our powers of -scientific inference. But then I am arguing negatively; I wish to show -that those who speak of the uniformity of nature, and the reign of -law, misinterpret the meaning involved in those expressions. Law is -not inconsistent with extreme diversity, and, so far as we can read -the history of this planetary system, it did probably originate in -heated nebulous matter, and man’s history forms but a brief span in -its progress towards the cold and stony condition. It is by doubtful -and speculative hypotheses alone that we can avoid such a conclusion, -and I depart least from undoubted facts and well-established laws -when I assert that, whatever uniformities may underlie the phenomena -of nature, constant variety and ever-progressing change is the real -outcome. - - -*Speculations on the Reconcentration of Energy.* - -There are unequivocal indications, as I have said, that the material -universe, as we at present see it, is progressing from some act of -creation, or some discontinuity of existence of which the date may be -approximately fixed by scientific inference. It is progressing towards -a state in which the available energy of matter will be dissipated -through infinite surrounding space, and all matter will become cold -and lifeless. This constitutes, as it were, the historical period of -physical science, that over which our scientific foresight may more -or less extend. But in this, as in other cases, we have no right to -interpret our experience negatively, so as to infer that because the -present state of things began at a particular time, there was no -previous existence. It may be that the present period of material -existence is but one of an indefinite series of like periods. All that -we can see, and feel, and infer, and reason about may be, as it were, -but a part of one single pulsation in the existence of the universe. - -After Sir W. Thomson had pointed out the preponderating tendency -which now seems to exist towards the conversion of all energy into -heat-energy, and its equal diffusion by radiation throughout space, -the late Professor Rankine put forth a remarkable speculation.[613] He -suggested that the ethereal, or, as I have called it, the *adamantine* -medium in which all the stars exist, and all radiation takes place, may -have bounds, beyond which only empty space exists. All heat undulations -reaching this boundary will be totally reflected, according to the -theory of undulations, and will be reconcentrated into foci situated in -various parts of the medium. Whenever a cold and extinct star happens -to pass through one of these foci, it will be instantly ignited and -resolved by intense heat into its constituent elements. Discontinuity -will occur in the history of that portion of matter, and the star will -begin its history afresh with a renewed store of energy. - - [613] *Report of the British Association* (1852), Report of Sections, - p. 12. - -This is doubtless a mere speculation, practically incapable of -verification by observation, and almost free from restrictions afforded -by present knowledge. We might attribute various shapes to the -adamantine medium, and the consequences would be various. But there -is this value in such speculations, that they draw attention to the -finiteness of our knowledge. We cannot deny the possible truth of such -an hypothesis, nor can we place a limit to the scientific imagination -in the framing of other like hypotheses. It is impossible, indeed, to -follow out our scientific inferences without falling into speculation. -If heat be radiated into outward space, it must either proceed *ad -infinitum*, or it must be stopped somewhere. In the latter case we fall -upon Rankine’s hypothesis. But if the material universe consist of a -finite collection of heated matter situated in a finite portion of an -infinite adamantine medium, then either this universe must have existed -for a finite time, or else it must have cooled down during the infinity -of past time indefinitely near to the absolute zero of temperature. I -objected to Lucretius’ argument against the destructibility of matter, -that we have no knowledge whatever of the laws according to which -it would undergo destruction. But we do know the laws according to -which the dissipation of heat appears to proceed, and the conclusion -inevitably is that a finite heated material body placed in a perfectly -cold infinitely extended medium would in an infinite time sink to -zero of temperature. Now our own world is not yet cooled down near to -zero, so that physical science seems to place us in the dilemma of -admitting either the finiteness of past duration of the world, or else -the finiteness of the portion of medium in which we exist. In either -case we become involved in metaphysical and mechanical difficulties -surpassing our mental powers. - - -*The Divergent Scope for New Discovery.* - -In the writings of some recent philosophers, especially of Auguste -Comte, and in some degree John Stuart Mill, there is an erroneous -and hurtful tendency to represent our knowledge as assuming an -approximately complete character. At least these and many other -writers fail to impress upon their readers a truth which cannot be -too constantly borne in mind, namely, that the utmost successes which -our scientific method can accomplish will not enable us to comprehend -more than an infinitesimal fraction of what there doubtless is to -comprehend.[614] Professor Tyndall seems to me open to the same charge -in a less degree. He remarks[615] that we can probably never bring -natural phenomena completely under mathematical laws, because the -approach of our sciences towards completeness may be asymptotic, so -that however far we may go, there may still remain some facts not -subject to scientific explanation. He thus likens the supply of novel -phenomena to a convergent series, the earlier and larger terms of which -have been successfully disposed of, so that comparatively minor groups -of phenomena alone remain for future investigators to occupy themselves -upon. - - [614] Mr. C. J. Monroe objects that in this statement I do injustice - to Comte, who, he thinks, did impress upon his readers the inadequacy - of our mental powers compared with the vastness of the subject matter - of science. The error of Comte, he holds, was in maintaining that - science had been carried about as far as it is worth while to carry - it, which is a different matter. In either case, Comte’s position is - so untenable that I am content to leave the question undecided. - - [615] *Fragments of Science*, p. 362. - -On the contrary, as it appears to me, the supply of new and unexplained -facts is divergent in extent, so that the more we have explained, the -more there is to explain. The further we advance in any generalisation, -the more numerous and intricate are the exceptional cases still -demanding further treatment. The experiments of Boyle, Mariotte, -Dalton, Gay-Lussac, and others, upon the physical properties of gases, -might seem to have exhausted that subject by showing that all gases -obey the same laws as regards temperature, pressure, and volume. But -in reality these laws are only approximately true, and the divergences -afford a wide and quite unexhausted field for further generalisation. -The recent discoveries of Professor Andrews have summed up some of -these exceptional facts under a wider generalisation, but in reality -they have opened to us vast new regions of interesting inquiry, -and they leave wholly untouched the question why one gas behaves -differently from another. - -The science of crystallography is that perhaps in which the most -precise and general laws have been detected, but it would be untrue -to assert that it has lessened the area of future discovery. We can -show that each one of the seven or eight hundred forms of calcite is -derivable by geometrical modifications from an hexagonal prism; but -who has attempted to explain the molecular forces producing these -modifications, or the chemical conditions in which they arise? The law -of isomorphism is an important generalisation, for it establishes a -general resemblance between the forms of crystallisation of natural -classes of elements. But if we examine a little more closely we find -that these forms are only approximately alike, and the divergence -peculiar to each substance is an unexplained exception. - -By many similar illustrations it might readily be shown that in -whatever direction we extend our investigations and successfully -harmonise a few facts, the result is only to raise up a host of other -unexplained facts. Can any scientific man venture to state that there -is less opening now for new discoveries than there was three centuries -ago? Is it not rather true that we have but to open a scientific book -and read a page or two, and we shall come to some recorded phenomenon -of which no explanation can yet be given? In every such fact there is a -possible opening for new discoveries, and it can only be the fault of -the investigator’s mind if he can look around him and find no scope for -the exercise of his faculties. - - -*Infinite Incompleteness of the Mathematical Sciences.* - -There is one privilege which a certain amount of knowledge should -confer; it is that of becoming aware of the weakness of our powers -compared with the tasks which they might undertake if stronger. To the -poor savage who cannot count twenty the arithmetical accomplishments of -the schoolboy are miraculously great. The schoolboy cannot comprehend -the vastly greater powers of the student, who has acquired facility in -algebraic processes. The student can but look with feelings of surprise -and reverence at the powers of a Newton or a Laplace. But the question -at once suggests itself, Do the powers of the highest human intellect -bear a finite ratio to the things which are to be understood and -calculated? How many further steps must we take in the rise of mental -ability and the extension of mathematical methods before we begin to -exhaust the knowable? - -I am inclined to find fault with mathematical writers because they -often exult in what they can accomplish, and omit to point out that -what they do is but an infinitely small part of what might be done. -They exhibit a general inclination, with few exceptions, not to do -so much as mention the existence of problems of an impracticable -character. This may be excusable as far as the immediate practical -result of their researches is in question, but the custom has the -effect of misleading the general public into the fallacious notion -that mathematics is a *perfect* science, which accomplishes what it -undertakes in a complete manner. On the contrary, it may be said that -if a mathematical problem were selected by chance out of the whole -number which might be proposed, the probability is infinitely slight -that a human mathematician could solve it. Just as the numbers we can -count are nothing compared with the numbers which might exist, so the -accomplishments of a Laplace or a Lagrange are, as it were, the little -corner of the multiplication-table, which has really an infinite extent. - -I have pointed out that the rude character of our observations prevents -us from being aware of the greater number of effects and actions -in nature. It must be added that, if we perceive them, we should -usually be incapable of including them in our theories from want of -mathematical power. Some persons may be surprised that though nearly -two centuries have elapsed since the time of Newton’s discoveries, we -have yet no general theory of molecular action. Some approximations -have been made towards such a theory. Joule and Clausius have measured -the velocity of gaseous atoms, or even determined the average distance -between the collisions of atom and atom. Thomson has approximated to -the number of atoms in a given bulk of substance. Rankine has formed -some reasonable hypotheses as to the actual constitution of atoms. It -would be a mistake to suppose that these ingenious results of theory -and experiment form any appreciable approach to a complete solution -of molecular motions. There is every reason to believe, judging from -the spectra of the elements, their atomic weights and other data, that -chemical atoms are very complicated structures. An atom of pure iron -is probably a far more complicated system than that of the planets -and their satellites. A compound atom may perhaps be compared with -a stellar system, each star a minor system in itself. The smallest -particle of solid substance will consist of a great number of such -stellar systems united in regular order, each bounded by the other, -communicating with it in some manner yet wholly incomprehensible. What -are our mathematical powers in comparison with this problem? - -After two centuries of continuous labour, the most gifted men have -succeeded in calculating the mutual effects of three bodies each -upon the other, under the simple hypothesis of the law of gravity. -Concerning these calculations we must further remember that they are -purely approximate, and that the methods would not apply where four or -more bodies are acting, and all produce considerable effects upon each -other. There is reason to believe that each constituent of a chemical -atom goes through an orbit in the millionth part of the twinkling of -an eye. In each revolution it is successively or simultaneously under -the influence of many other constituents, or possibly comes into -collision with them. It is no exaggeration to say that mathematicians -have the least notion of the way in which they could successfully -attack so difficult a problem of forces and motions. As Herschel has -remarked,[616] each of these particles is for ever solving differential -equations, which, if written out in full, might belt the earth. - - [616] *Familiar Lectures on Scientific Subjects*, p. 458. - -Some of the most extensive calculations ever made were those required -for the reduction of the measurements executed in the course of the -Trigonometrical Survey of Great Britain. The calculations arising out -of the principal triangulation occupied twenty calculators during -three or four years, in the course of which the computers had to solve -simultaneous equations involving seventy-seven unknown quantities. -The reduction of the levellings required the solution of a system of -ninety-one equations. But these vast calculations present no approach -whatever to what would be requisite for the complete treatment of -any one physical problem. The motion of glaciers is supposed to -be moderately well understood in the present day. A glacier is a -viscid, slowly yielding mass, neither absolutely solid nor absolutely -rigid, but it is expressly remarked by Forbes,[617] that not even an -approximate solution of the mathematical conditions of such a moving -mass can yet be possible. “Every one knows,” he says, “that such -problems are beyond the compass of exact mathematics;” but though -mathematicians may know this, they do not often enough impress that -knowledge on other people. - - [617] *Philosophical Magazine*, 3rd Series, vol. xxvi. p. 406. - -The problems which are solved in our mathematical books consist of -a small selection of those which happen from peculiar conditions to -be solvable. But the very simplest problem in appearance will often -give rise to impracticable calculations. Mr. Todhunter[618] seems to -blame Condorcet, because in one of his memoirs he mentions a problem -to solve which would require a great and impracticable number of -successive integrations. Now, if our mathematical sciences are to cope -with the problems which await solution, we must be prepared to effect -an unlimited number of successive integrations; yet at present, and -almost beyond doubt for ever, the probability that an integration taken -haphazard will come within our powers is exceedingly small. - - [618] *History of the Theory of Probability*, p. 398. - -In some passages of that remarkable work, the *Ninth Bridgewater -Treatise* (pp. 113–115), Babbage has pointed out that if we had power -to follow and detect the minutest effects of any disturbance, each -particle of existing matter would furnish a register of all that has -happened. “The track of every canoe--of every vessel that has yet -disturbed the surface of the ocean, whether impelled by manual force or -elemental power, remains for ever registered in the future movement of -all succeeding particles which may occupy its place. The furrow which -it left is, indeed, instantly filled up by the closing waters; but they -draw after them other and larger portions of the surrounding element, -and these again, once moved, communicate motion to others in endless -succession.” We may even say that “The air itself is one vast library, -on whose pages are for ever written all that man has ever said or -even whispered. There, in their mutable but unerring characters, mixed -with the earliest as well as the latest sighs of mortality, stand for -ever recorded, vows unredeemed, promises unfulfilled, perpetuating in -the united movements of each particle the testimony of man’s changeful -will.” - -When we read reflections such as these, we may congratulate ourselves -that we have been endowed with minds which, rightly employed, can -form some estimate of their incapacity to trace out and account for -all that proceeds in the simpler actions of material nature. It ought -to be added that, wonderful as is the extent of physical phenomena -open to our investigation, intellectual phenomena are yet vastly more -extensive. Of this I might present one satisfactory proof were space -available by pointing out that the mathematical functions employed -in the calculations of physical science form an infinitely small -fraction of the functions which might be invented. Common trigonometry -consists of a great series of useful formulæ, all of which arise out -of the relation of the sine and cosine expressed in one equation, -sin ^{2}*x* + cos ^{2}*x* = 1. But this is not the only trigonometry -which may exist; mathematicians also recognise hyperbolic trigonometry, -of which the fundamental equation is cos ^{2}*x* - sin ^{2}*x* = 1. De -Morgan has pointed out that the symbols of ordinary algebra form but -three of an interminable series of conceivable systems.[619] As the -logarithmic operation is to addition or addition to multiplication, so -is the latter to a higher operation, and so on without limit. - - [619] *Trigonometry and Double Algebra*, chap. ix. - -We may rely upon it that immense, and to us inconceivable, advances -will be made by the human intellect, in the absence of any catastrophe -to the species or the globe. Within historical periods we can trace -the rise of mathematical science from its simplest germs. We can -prove our descent from ancestors who counted only on their fingers. -How infinitely is a Newton or a Laplace above those simple savages. -Pythagoras is said to have sacrificed a hecatomb when he discovered the -forty-seventh proposition of Euclid, and the occasion was worthy of -the sacrifice. Archimedes was beside himself when he first perceived -his beautiful mode of determining specific gravities. Yet these great -discoveries are the commonplaces of our school books. Step by step we -can trace upwards the acquirement of new mental powers. What could be -more wonderful than Napier’s discovery of logarithms, a new mode of -calculation which has multiplied perhaps a hundredfold the working -powers of every computer, and has rendered easy calculations which were -before impracticable? Since the time of Newton and Leibnitz worlds of -problems have been solved which before were hardly conceived as matters -of inquiry. In our own day extended methods of mathematical reasoning, -such as the system of quaternions, have been brought into existence. -What intelligent man will doubt that the recondite speculations of a -Cayley, a Sylvester, or a Clifford may lead to some new development of -new mathematical power, at the simplicity of which a future age will -wonder, and yet wonder more that to us they were so dark and difficult. -May we not repeat the words of Seneca: “Veniet tempus, quo ista quæ -nunc latent, in lucem dies extrahat, et longioris ævi diligentia: -ad inquisitionem tantorum ætas una non sufficit. Veniet tempus, quo -posteri nostri tam aperta nos nescisse mirentur.” - - -*The Reign of Law in Mental and Social Phenomena.* - -After we pass from the so-called physical sciences to those which -attempt to investigate mental and social phenomena, the same general -conclusions will hold true. No one will be found to deny that there -are certain uniformities of thinking and acting which can be detected -in reasoning beings, and so far as we detect such laws we successfully -apply scientific method. But those who attempt to establish social or -moral sciences soon become aware that they are dealing with subjects -of enormous perplexity. Take as an instance the science of political -economy. If a science at all, it must be a mathematical science, -because it deals with quantities of commodities. But as soon as we -attempt to draw out the equations expressing the laws of demand and -supply, we discover that they have a complexity entirely surpassing -our powers of mathematical treatment. We may lay down the general form -of the equations, expressing the demand and supply for two or three -commodities among two or three trading bodies, but all the functions -involved are so complicated in character that there is not much fear -of scientific method making rapid progress in this direction. If such -be the prospects of a comparatively formal science, like political -economy, what shall we say of moral science? Any complete theory of -morals must deal with quantities of pleasure and pain, as Bentham -pointed out, and must sum up the general tendency of each kind of -action upon the good of the community. If we are to apply scientific -method to morals, we must have a calculus of moral effects, a kind -of physical astronomy investigating the mutual perturbations of -individuals. But as astronomers have not yet fully solved the problem -of three gravitating bodies, when shall we have a solution of the -problem of three moral bodies? - -The sciences of political economy and morality are comparatively -abstract and general, treating mankind from simple points of view, -and attempting to detect general principles of action. They are to -social phenomena what the abstract sciences of chemistry, heat, and -electricity are to the concrete science of meteorology. Before we -can investigate the actions of any aggregate of men, we must have -fairly mastered all the more abstract sciences applying to them, -somewhat in the way that we have acquired a fair comprehension of -the simpler truths of chemistry and physics. But all our physical -sciences do not enable us to predict the weather two days hence with -any great probability, and the general problem of meteorology is almost -unattempted as yet. What shall we say then of the general problem of -social science, which shall enable us to predict the course of events -in a nation? - -Several writers have proposed to lay the foundations of the science -of history. Buckle undertook to write the *History of Civilisation -in England*, and to show how the character of a nation could be -explained by the nature of the climate and the fertility of the soil. -He omitted to explain the contrast between the ancient Greek nation -and the present one; there must have been an extraordinary revolution -in the climate or the soil. Auguste Comte detected the simple laws -of the course of development through which nations pass. There are -always three phases of intellectual condition,--the theological, the -metaphysical, and the positive; applying this general law of progress -to concrete cases, Comte was enabled to predict that in the hierarchy -of European nations, Spain would necessarily hold the highest place. -Such are the parodies of science offered to us by the *positive* -philosophers. - -A science of history in the true sense of the term is an absurd notion. -A nation is not a mere sum of individuals whom we can treat by the -method of averages; it is an organic whole, held together by ties of -infinite complexity. Each individual acts and re-acts upon his smaller -or greater circle of friends, and those who acquire a public position -exert an influence on much larger sections of the nation. There will -always be a few great leaders of exceptional genius or opportunities, -the unaccountable phases of whose opinions and inclinations sway the -whole body. From time to time arise critical situations, battles, -delicate negotiations, internal disturbances, in which the slightest -incidents may change the course of history. A rainy day may hinder a -forced march, and change the course of a campaign; a few injudicious -words in a despatch may irritate the national pride; the accidental -discharge of a gun may precipitate a collision the effects of which -will last for centuries. It is said that the history of Europe depended -at one moment upon the question whether the look-out man upon Nelson’s -vessel would or would not descry a ship of Napoleon’s expedition -to Egypt which was passing not far off. In human affairs, then, -the smallest causes may produce the greatest effects, and the real -application of scientific method is out of the question. - - -*The Theory of Evolution.* - -Profound philosophers have lately generalised concerning the production -of living forms and the mental and moral phenomena regarded as their -highest development. Herbert Spencer’s theory of evolution purports -to explain the origin of all specific differences, so that not even -the rise of a Homer or a Beethoven would escape from his broad -theories. The homogeneous is unstable and must differentiate itself, -says Spencer, and hence comes the variety of human institutions -and characters. In order that a living form shall continue to exist -and propagate its kind, says Darwin, it must be suitable to its -circumstances, and the most suitable forms will prevail over and -extirpate those which are less suitable. From these fruitful ideas -are developed theories of evolution and natural selection which go -far towards accounting for the existence of immense numbers of living -creatures--plants, and animals. Apparent adaptations of organs to -useful purposes, which Paley regarded as distinct products of creative -intelligence, are now seen to follow as natural effects of a constantly -acting tendency. Even man, according to these theories, is no distinct -creation, but rather an extreme case of brain development. His nearest -cousins are the apes, and his pedigree extends backwards until it joins -that of the lowliest zoophytes. - -The theories of Darwin and Spencer are doubtless not demonstrated; they -are to some extent hypothetical, just as all the theories of physical -science are to some extent hypothetical, and open to doubt. Judging -from the immense numbers of diverse facts which they harmonise and -explain, I venture to look upon the theories of evolution and natural -selection in their main features as two of the most probable hypotheses -ever proposed. I question whether any scientific works which have -appeared since the *Principia* of Newton are comparable in importance -with those of Darwin and Spencer, revolutionising as they do all our -views of the origin of bodily, mental, moral, and social phenomena. - -Granting all this, I cannot for a moment admit that the theory of -evolution will destroy theology. That theory embraces several laws -or uniformities which are observed to be true in the production of -living forms; but these laws do not determine the size and figure of -living creatures, any more than the law of gravitation determines the -magnitudes and distances of the planets. Suppose that Darwin is correct -in saying that man is descended from the Ascidians: yet the precise -form of the human body must have been influenced by an infinite train -of circumstances affecting the reproduction, growth, and health of the -whole chain of intermediate beings. No doubt, the circumstances being -what they were, man could not be otherwise than he is, and if in any -other part of the universe an exactly similar earth, furnished with -exactly similar germs of life, existed, a race must have grown up there -exactly similar to the human race. - -By a different distribution of atoms in the primeval world a different -series of living forms on this earth would have been produced. From the -same causes acting according to the same laws, the same results will -follow; but from different causes acting according to the same laws, -different results will follow. So far as we can see, then, infinitely -diverse living creatures might have been created consistently with the -theory of evolution, and the precise reason why we have a backbone, -two hands with opposable thumbs, an erect stature, a complex brain, -about 223 bones, and many other peculiarities, is only to be found in -the original act of creation. I do not, any less than Paley, believe -that the eye of man manifests design. I believe that the eye was -gradually developed, and we can in fact trace its gradual development -from the first germ of a nerve affected by light-rays in some simple -zoophyte. In proportion as the eye became a more accurate instrument -of vision, it enabled its possessor the better to escape destruction, -but the ultimate result must have been contained in the aggregate of -the causes, and these causes, as far as we can see, were subject to the -arbitrary choice of the Creator. - -Although Agassiz was clearly wrong in holding that every species of -living creature appeared on earth by the immediate intervention of -the Creator, which would amount to saying that no laws of connection -between forms are discoverable, yet he seems to be right in asserting -that living forms are distinct from those produced by purely physical -causes. “The products of what are commonly called physical agents,” -he says,[620] “are everywhere the same (*i.e.* upon the whole surface -of the earth), and have always been the same (*i.e.* during all -geological periods); while organised beings are everywhere different -and have differed in all ages. Between two such series of phenomena -there can be no causal or genetic connection.” Living forms as we now -regard them are essentially variable, but from constant mechanical -causes constant effects would ensue. If vegetable cells are formed -on geometrical principles being first spherical, and then by mutual -compression dodecahedral, then all cells should have similar forms. In -the Foraminifera and some other lowly organisms, we seem to observe the -production of complex forms on geometrical principles. But from similar -causes acting according to similar laws only similar results could -be produced. If the original life germ of each creature is a simple -particle of protoplasm, unendowed with any distinctive forces, then the -whole of the complex phenomena of animal and vegetable life are effects -without causes. Protoplasm may be chemically the same substance, and -the germ-cell of a man and of a fish may be apparently the same, so far -as the microscope can decide; but if certain cells produce men, and -others as uniformly produce a species of fish, there must be a hidden -constitution determining the extremely different results. If this were -not so, the generation of every living creature from the uniform germ -would have to be regarded as a distinct act of creation. - - [620] Agassiz, *Essay on Classification*, p. 75. - -Theologians have dreaded the establishment of the theories of Darwin -and Huxley and Spencer, as if they thought that those theories could -explain everything upon the purest mechanical and material principles, -and exclude all notions of design. They do not see that those theories -have opened up more questions than they have closed. The doctrine of -evolution gives a complete explanation of no single living form. While -showing the general principles which prevail in the variation of living -creatures, it only points out the infinite complexity of the causes -and circumstances which have led to the present state of things. Any -one of Mr. Darwin’s books, admirable though they all are, consists -but in the setting forth of a multitude of indeterminate problems. He -proves in the most beautiful manner that each flower of an orchid is -adapted to some insect which frequents and fertilises it, and these -adaptations are but a few cases of those immensely numerous ones which -have occurred in the lives of plants and animals. But why orchids -should have been formed so differently from other plants, why anything, -indeed, should be as it is, rather than in some of the other infinitely -numerous possible modes of existence, he can never show. The origin -of everything that exists is wrapped up in the past history of the -universe. At some one or more points in past time there must have been -arbitrary determinations which led to the production of things as they -are. - - -*Possibility of Divine Interference.* - -I will now draw the reader’s attention to pages 149 to 152. I there -pointed out that all inductive inference involves the assumption that -our knowledge of what exists is complete, and that the conditions of -things remain unaltered between the time of our experience and the -time to which our inferences refer. Recurring to the illustration -of a ballot-box, employed in the chapter on the inverse method of -probabilities, we assume when predicting the probable nature of -the next drawing, firstly, that our previous drawings have been -sufficiently numerous to give us knowledge of the contents of the -box; and, secondly, that no interference with the ballot-box takes -place between the previous and the next drawings. The results yielded -by the theory of probability are quite plain. No finite number of -casual drawings can give us sure knowledge of the contents of the -box, so that, even in the absence of all disturbance, our inferences -are merely the best which can be made, and do not approach to -infallibility. If, however, interference be possible, even the theory -of probability ceases to be applicable, for, the amount and nature -of that interference being arbitrary and unknown, there ceases to -be any connection between premises and conclusion. Many years of -reflection have not enabled me to see the way of avoiding this hiatus -in scientific certainty. The conclusions of scientific inference appear -to be always of a hypothetical and provisional nature. Given certain -experience, the theory of probability yields us the true interpretation -of that experience and is the surest guide open to us. But the best -calculated results which it can give are never absolute probabilities; -they are purely relative to the extent of our information. It seems to -be impossible for us to judge how far our experience gives us adequate -information of the universe as a whole, and of all the forces and -phenomena which can have place therein. - -I feel that I cannot in the space remaining at my command in the -present volume, sufficiently follow out the lines of thought suggested, -or define with precision my own conclusions. This chapter contains -merely *Reflections* upon subjects of so weighty a character that I -should myself wish for many years--nay for more than a lifetime of -further reflection. My purpose, as I have repeatedly said, is the -purely negative one of showing that atheism and materialism are no -necessary results of scientific method. From the preceding reviews of -the value of our scientific knowledge, I draw one distinct conclusion, -that we cannot disprove the possibility of Divine interference in -the course of nature. Such interference might arise, so far as our -knowledge extends, in two ways. It might consist in the disclosure -of the existence of some agent or spring of energy previously -unknown, but which effects a given purpose at a given moment. Like -the pre-arranged change of law in Babbage’s imaginary calculating -machine, there may exist pre-arranged surprises in the order of -nature, as it presents itself to us. Secondly, the same Power, which -created material nature, might, so far as I can see, create additions -to it, or annihilate portions which do exist. Such events are in a -certain sense inconceivable to us; yet they are no more inconceivable -than the existence of the world as it is. The indestructibility of -matter, and the conservation of energy, are very probable scientific -hypotheses, which accord satisfactorily with experiments of scientific -men during a few years past, but it would be gross misconception of -scientific inference to suppose that they are certain in the sense -that a proposition in geometry is certain. Philosophers no doubt hold -that *de nihilo nihil fit*, that is to say, their senses give them no -means of imagining to the mind how creation can take place. But we are -on the horns of a trilemma; we must either deny that anything exists, -or we must allow that it was created out of nothing at some moment of -past time, or that it existed from eternity. The first alternative is -absurd; the other two seem to me equally conceivable. - - -*Conclusion.* - -It may seem that there is one point where our speculations must end, -namely where contradiction begins. The laws of Identity and Difference -and Duality were the foundations from which we started, and they are, -so far as I can see, the foundations which we can never quit without -tottering. Scientific Method must begin and end with the laws of -thought, but it does not follow that it will save us from encountering -inexplicable, and at least apparently contradictory results. The nature -of continuous quantity leads us into extreme difficulties. Any finite -space is composed of an infinite number of infinitely small spaces, -each of which, again, is composed of an infinite number of spaces of -a second order of smallness; these spaces of the second order are -composed, again, of infinitely small spaces of the third order. Even -these spaces of the third order are not absolute geometrical points -answering to Euclid’s definition of a point, as position without -magnitude. Go on as far as we will, in the subdivision of continuous -quantity, yet we never get down to the absolute point. Thus scientific -method leads us to the inevitable conception of an infinite series -of successive orders of infinitely small quantities. If so, there is -nothing impossible in the existence of a myriad universes within the -compass of a needle’s point, each with its stellar systems, and its -suns and planets, in number and variety unlimited. Science does nothing -to reduce the number of strange things that we may believe. When fairly -pursued it makes absurd drafts upon our powers of comprehension and -belief. - -Some of the most precise and beautiful theorems in mathematical science -seem to me to involve apparent contradiction. Can we imagine that a -point moving along a perfectly straight line towards the west would -ever get round to the east and come back again, having performed, as -it were, a circuit through infinite space, yet without ever diverging -from a perfectly straight direction? Yet this is what happens to the -intersecting point of two straight lines in the same plane, when one -line revolves. The same paradox is exhibited in the hyperbola regarded -as an infinite ellipse, one extremity of which has passed to an -infinite distance and come back in the opposite direction. A varying -quantity may change its sign by passing either through zero or through -infinity. In the latter case there must be one intermediate value of -the variable for which the variant is indifferently negative infinity -and positive infinity. Professor Clifford tells me that he has found -a mathematical function which approaches infinity as the variable -approaches a certain limit; yet at the limit the function is finite! -Mathematicians may shirk difficulties, but they cannot make such -results of mathematical principles appear otherwise than contradictory -to our common notions of space. - -The hypothesis that there is a Creator at once all-powerful -and all-benevolent is pressed, as it must seem to every candid -investigator, with difficulties verging closely upon logical -contradiction. The existence of the smallest amount of pain and evil -would seem to show that He is either not perfectly benevolent, or -not all-powerful. No one can have lived long without experiencing -sorrowful events of which the significance is inexplicable. But if we -cannot succeed in avoiding contradiction in our notions of elementary -geometry, can we expect that the ultimate purposes of existence shall -present themselves to us with perfect clearness? I can see nothing to -forbid the notion that in a higher state of intelligence much that is -now obscure may become clear. We perpetually find ourselves in the -position of finite minds attempting infinite problems, and can we be -sure that where we see contradiction, an infinite intelligence might -not discover perfect logical harmony? - -From science, modestly pursued, with a due consciousness of the extreme -finitude of our intellectual powers, there can arise only nobler and -wider notions of the purpose of Creation. Our philosophy will be an -affirmative one, not the false and negative dogmas of Auguste Comte, -which have usurped the name, and misrepresented the tendencies of a -true *positive philosophy*. True science will not deny the existence of -things because they cannot be weighed and measured. It will rather lead -us to believe that the wonders and subtleties of possible existence -surpass all that our mental powers allow us clearly to perceive. The -study of logical and mathematical forms has convinced me that even -space itself is no requisite condition of conceivable existence. -Everything, we are told by materialists, must be here or there, -nearer or further, before or after. I deny this, and point to logical -relations as my proof. - -There formerly seemed to me to be something mysterious in the -denominators of the binomial expansion (p. 190), which are reproduced -in the natural constant ε, or - - 1 + 1/1 + 1/(1 . 2) + 1/(1 . 2 . 3) + ... - -and in many results of mathematical analysis. I now perceive, as -already explained (pp. 33, 160, 383), that they arise out of the fact -that the relations of space do not apply to the logical conditions -governing the numbers of combinations as contrasted to those of -permutations. So far am I from accepting Kant’s doctrine that space is -a necessary form of thought, that I regard it as an accident, and an -impediment to pure logical reasoning. Material existences must exist in -space, no doubt, but intellectual existences may be neither in space -nor out of space; they may have no relation to space at all, just as -space itself has no relation to time. For all that I can see, then, -there may be intellectual existences to which both time and space are -nullities. - -Now among the most unquestionable rules of scientific method is -that first law that *whatever phenomenon is, is*. We must ignore no -existence whatever; we may variously interpret or explain its meaning -and origin, but, if a phenomenon does exist, it demands some kind -of explanation. If then there is to be competition for scientific -recognition, the world without us must yield to the undoubted existence -of the spirit within. Our own hopes and wishes and determinations are -the most undoubted phenomena within the sphere of consciousness. If men -do act, feel, and live as if they were not merely the brief products of -a casual conjunction of atoms, but the instruments of a far-reaching -purpose, are we to record all other phenomena and pass over these? We -investigate the instincts of the ant and the bee and the beaver, and -discover that they are led by an inscrutable agency to work towards -a distant purpose. Let us be faithful to our scientific method, and -investigate also those instincts of the human mind by which man is led -to work as if the approval of a Higher Being were the aim of life. - - - - -INDEX. - - - Abacus, logical, 104; - arithmetical, 107; - Panchrestus, 182. - - Aberration of light, 561; - systematic, 547. - - Abscissio infiniti, 79, 713. - - Abstract terms, 27; - number, 159. - - Abstraction, 704; - logical, 25; - numerical, 158; - of indifferent circumstances, 97. - - Accademia del Cimento, 427, 432, 436, 527. - - Accident, logical, 700. - - Accidental discovery, 529. - - Achromatic lenses, 432. - - Actinometer, 337. - - Adamantine medium, 605, 751. - - Adjectives, 14, 30, 31, 35; - indeterminate, 41. - - Adrain, of New Brunswick, 375. - - Affirmation, 44. - - Agassiz, on genera, 726; - on creation of species, 763. - - Agreement, 44. - - Airy, Sir George Biddell, on perpetual motion, 223; - new property of sphere, 232; - pendulum experiments, 291, 304, 348, 567; - standard clock, 353; - book on *Errors of Observation*, 395; - tides, 488; - extra-polation, 495; - Thales’ eclipse, 537; - interference of light, 539; - density of earth, 291. - - Alchemists, 505; - how misled, 428. - - Algebra, 123, 155, 164; - Diophantine, 631. - - Algebraic, equations, 123; - geometry, 633. - - Allotropic state, 663, 670. - - Alloys, possible number, 191; - properties, 528. - - Alphabet, the Logical, 93, 104, 125; - Morse, 193. - - Alphabet, permutations of letters of the, 174, 179. - - Alphabetic indexes, 714. - - Alternative relations, 67; - exclusive and unexclusive, 205. - - Ampère, electricity, 547; - classification, 679. - - Anagrams, 128. - - Analogy, 627; - of logical and numerical terms, 160; - and generalisation, 596; - in mathematical sciences, 631; - in theory of undulations, 635; - in astronomy, 638; - failure of, 641. - - Analysis, logical, 122. - - Andrews, Prof. Thomas, experiments on gaseous state, 71, 613, 665, 753. - - Angström, on spectrum, 424. - - Angular magnitude, 305, 306, 326. - - Antecedent defined, 225. - - Anticipation of Nature, 509. - - Anticipations, of Principle of Substitution, 21; - of electric telegraph, 671. - - Apparent, equality, 275; - sequence of events, 409. - - Approximation, theory of, 456; - to exact laws, 462; - mathematical principles of, 471; - arithmetic of, 481. - - Aqueous vapour, 500. - - Aquinas, on disjunctive propositions, 69. - - Arago, photometer, 288; - rotating disc, 535; - his philosophic character, 592. - - Archimedes, *De Arenæ Numero*, 195; - centre of gravity, 363. - - Arcual unit, 306, 330. - - Argyll, Duke of, 741. - - Aristarchus on sun’s and moon’s distances, 294. - - Aristotelian doctrines, 666. - - Aristotle, dictum, 21; - singular terms, 39; - overlooked simple identities, 40; - order of premises, 114; - logical error, 117; - definition of time, 307; - on science, 595; - on white swans, 666. - - Arithmetic, reasoning in, 167; - of approximate quantities, 481. - - Arithmetical triangle, 93, 143, 182, 202, 378, 383; - diagram of, 184; - connection with Logical Alphabet, 189; - in probability, 208. - - Asteroids, discovery of, 412, 748. - - Astronomy, physical, 459. - - Atmospheric tides, 553. - - Atomic theory, 662. - - Atomic weights, 563. - - Atoms, size of, 195; - impossibility of observing, 406. - - Augustin on time, 307. - - Average, 359, 360; - divergence from, 188; - etymology of, 363. - - Axes of crystals, 686. - - Axioms of algebra, 164. - - - Babbage, Charles, calculating machine, 107, 231, 743; - lighthouse signals, 194; - natural constants, 329; - Mosaic history, 412; - universal and general truths, 646; - change of law, 230; - persistence of effects, 757. - - Bacon, Francis Lord, *Novum Organum*, 107; - on induction, 121; - biliteral cipher, 193; - First Aphorism, 219; - on causes, 221; - Copernican system, 249, 638; - deficient powers of senses, 278; - observation, 402; - Natural History, 403; - use of hypothesis, 506; - his method, 507; - *experimentum crucis*, 519; - error of his method, 576; - ostensive, clandestine instances, &c., 608, 610; - *latens precessus*, 619. - - Bacon, Roger, on the rainbow, 526, 598. - - Baily, Francis, 272; - density of earth, 342, 566; - experiments with torsion balance, 370, 397, 432, 567–8; - motions of stars, 572. - - Bain, Alexander, on powers of mind, 4; - Mill’s reform of logic, 227. - - Baker’s poem, *The Universe*, 621. - - Balance, use of the chemical, 292, 351, 354, 369; - delicacy of, 304; - vibrations of, 369. - - Ballot, Buys, experiment on sound, 541. - - Ballot-box, simile of, 150, 251–6, 765. - - Barbara, 55, 57, 88, 105, 141. - - Baroko, 85. - - Barometer, 659; - Gay Lussac’s standard, 346; - variations, 337, 346, 349. - - Bartholinus on double refraction, 585. - - Base-line, measurement of, 304. - - Bauhusius, verses of, 175. - - Baxendell, Joseph, 552. - - Beneke, on substitution, 21. - - Bennet, momentum of light, 435. - - Bentham, George, 15; - bifurcate classification, 695; - infima species, 702; - works on classification, 703; - analytical key to flora, 712. - - Bentham, Jeremy, on analogy, 629; - bifurcate classification, 703. - - Benzenberg’s experiment, 388. - - Bernoulli, Daniel, planetary orbits, 250; - resisting media and projectiles, 467; - vibrations, 476. - - Bernoulli, James, 154; - numbers of, 124; - Protean verses, 175; - *De Arte Conjectandi* quoted, 176, 183; - on figurate numbers, 183; - theorem of, 209; - false solution in probability, 213; - solution of inverse problem, 261. - - Bessel, F. W., 375; - law of error, 384; - formula for periodic variations, 488; - use of hypothesis, 506; - solar parallax, 560–2; - ellipticity of earth, 565; - pendulum experiments, 604. - - Bias, 393, 402. - - Biela’s comet, 746. - - Bifurcate classification, 694. - - Binomial theorem, 190; - discovery of, 231. - - Biot, on tension of vapour, 500. - - Blind experiments, 433. - - Bode’s law, 147, 257, 660. - - Boethius, quoted, 33; - on kinds of mean, 360. - - Boiling point, 442, 659. - - Bonnet’s theory of reproduction, 621. - - Boole, George, on sign of equality, 15; - his calculus of logic, 23, 113, 634; - on logical terms, 33; - law of commutativeness, 35; - use of *some*, 41–2; - disjunctive propositions, 70; - Venn on his method, 90; - *Laws of Thought*, 155; - statistical conditions, 168; - propositions numerically definite, 172; - on probability, 199; - general method in probabilities, 206; - Laplace’s solution of inverse problem, 256; - law of error, 377. - - Borda, his repeating circle, 290. - - Boscovich’s hypothesis, 512. - - Botany, 666, 678, 681; - modes of classification, 678; - systematic, 722; - nomenclature of, 727. - - Bowen, Prof. Francis, on inference, 118; - classification, 674. - - Boyle’s, Robert, law of gaseous pressure, 468, 470, 619; - on hypothesis, 510; - barometer, 659. - - Bradley, his observations, 384; - accuracy of, 271; - aberration of light, 535. - - Bravais, on law of error, 375. - - Brewer, W. H., 142. - - Brewster, Sir David, iridescent colours, 419; - spectrum, 429; - Newton’s theory of colours, 518; - refractive indices, 10, 527; - optic axes, 446. - - British Museum, catalogue of, 717. - - Brodie, Sir B. C., on errors of experiment, 388, 464; - ozone, 663. - - Brown, Thomas, on cause, 224. - - Buckle, Thomas, on constancy of average, 656; - science of history, 760. - - Buffon, on probability, 215; - definition of genius, 576. - - Bunsen, Robert, spectrum, 244; - photometrical researches, 273, 324, 441; - calorimeter, 343. - - Butler, Bishop, on probability, 197. - - - Calorescence, 664. - - Camestres, 84. - - Canton, on compressibility of water, 338. - - Carbon, 640, 728; - conductibility of, 442. - - Cardan, on inclined plane, 501. - - Cards, combinations of, 190. - - Carlini, pendulum experiments, 567. - - Carnot’s law, 606. - - Carpenter, Dr. W. B., 412. - - Catalogues, art of making, 714. - - Cauchy, undulatory theory, 468. - - Cause, 220; - definition of, 224. - - Cavendish’s experiment, 272, 566. - - Cayley, Professor, 145; - on mathematical tables, 331; - numbers of chemical compounds, 544. - - Celarent, 55. - - Centre of gravity, 363, 524; - of oscillation, gyration, &c., 364. - - Centrobaric bodies, 364. - - Certainty, 235, 266. - - Cesare, 85. - - Chalmers, on collocations, 740. - - Chance, 198. - - Character, human, 733. - - Characteristics, 708. - - Chauvenet, Professor W., on treatment of observations, 391. - - Chemical affinity, 614; - analysis, 713. - - Chladni, 446. - - Chloroform, discovery of, 531. - - Chronoscope, 616. - - Cipher, 32; - Bacon’s, 193. - - Circle, circumference of, 389. - - Circumstances, indifferent, 419. - - Circumstantial evidence, 264. - - Clairaut, 650, 651; - on gravity, 463. - - Classes, 25; - problem of common part of three, 170. - - Classification, 673; - involving induction, 675; - multiplicity of modes, 677; - natural and artificial systems, 679; - in crystallography, 685; - symbolic statement of, 692; - bifurcate, 694; - an inverse and tentative operation, 689; - diagnostic, 710; - by indexes, 714; - of books, 715; - in biological sciences, 718; - genealogical, 719; - by types, 722; - limits of, 730. - - Clifford, Professor, on types of compound statements, 143, 529; - first and last catastrophe, 744; - mathematical function, 768. - - Clocks, astronomical, 340, 353. - - Clouds, 447; - cirrous, 411. - - Coincidences, 128; - fortuitous, 261; - measurement by, 292; - method of, 291. - - Collective terms, 29, 39. - - Collocations of matter, 740. - - Colours, iridescent, 419; - natural, 518; - perception of, 437; - of spectrum, 584. - - Combinations, 135, 142; - doctrine of, 173; - of letters of alphabet, 174; - calculations of, 180; - higher orders of, 194. - - Combinatorial analysis, 176. - - Comets, 449; - number of, 408; - hyperbolic, 407; - classification of, 684; - conflict with, 746–7; - Halley’s comet, 537; - Lexell’s comet, 651. - - Commutativeness, law of, 35, 72, 177. - - Comparative use of instruments, 299. - - Compass, variations of, 281. - - Complementary statements, 144. - - Compossible alternatives, 69. - - Compound statements, 144; - events, 204. - - Compounds, chemical, 192. - - Comte, Auguste, on probability, 200, 214; - on prevision, 536; - his positive philosophy, 752, 760, 768. - - Concrete number, 159. - - Conditions, of logical symbols, 32; - removal of usual, 426; - interference of unsuspected, 428; - maintenance of similar, 443; - approximation to natural, 465. - - Condorcet, 2; - his problem, 253. - - Confusion of elements, 237. - - Conical refraction, 653. - - Conjunction of planets, 293, 657. - - Consequent, definition of, 225. - - Conservation of energy, 738. - - Constant numbers of nature, 328; - mathematical, 330; - physical, 331; - astronomical, 332; - terrestrial, 333; - organic, 333; - social, 334. - - Continuity, law of, 615, 729; - sense of, 493; - detection of, 610; - failure of, 619. - - Continuous quantity, 274, 485. - - Contradiction, law of, 31, 74. - - Contrapositive, proposition, 84, 136; - conversion, 83. - - Conversion of propositions, 46, 118. - - Copernican theory, 522, 625, 638, 647. - - Copula, 16. - - Cornu, velocity of light, 561. - - Corpuscular theory, 520, 538, 667. - - Correction, method of, 346. - - Correlation, 678, 681. - - Cotes, Roger, use of mean, 359; - method of least squares, 377. - - Coulomb, 272. - - Couple, mechanical, 653. - - Creation, problem of, 740. - - Crookes’ radiometer, 435. - - Cross divisions, 144. - - Crystallography, 648, 654, 658, 678, 754; - systems of, 133; - classification in, 685. - - Crystals, 602; - Dana’s classification of, 711; - pseudomorphic, 658. - - Curves, use of, 392, 491, 496; - of various degrees, 473. - - Cuvier, on experiment, 423; - on inferences, 682. - - Cyanite, 609. - - Cycloid, 633. - - Cycloidal pendulum, 461. - - Cypher, 124. - - - D’Alembert, blunders in probability, 213, 214; - on gravity, 463. - - Dalton, laws of, 464, 471; - atomic theory, 662. - - Darapti, 59. - - Darii, 56. - - Darwin, Charles, his works, 131; - negative results of observation, 413; - arguments against his theory, 437; - cultivated plants, 531; - his influence, 575; - classification, 718; - constancy of character in classification, 720–1; - on definition, 726; - restoration of limbs, 730; - tendency of his theory, 762, 764. - - Davy, Sir H., on new instruments, 270; - nature of heat, 343, 417; - detection of salt in electrolysis, 428. - - Day, sidereal, 310; - length of, 289. - - Decandolle, on classification, 696. - - Decyphering, 124. - - Deduction, 11, 49. - - Deductive reasoning, 534; - miscellaneous forms of, 60; - probable, 209. - - Definition, 39, 62, 711, 723; - purpose of, 54; - of cause and power, 224. - - De Morgan, Augustus, negative terms, 14; - Aristotle’s logic, 18; - relatives, 23; - logical universe, 43; - complex propositions, 75; - contraposition, 83; - formal logic quoted, 101; - error of his system, 117; - anagram of his name, 128; - numerically definite reasoning, 168–172; - probability, 198; - belief, 199; - experiments in probability, 207; - probable deductive arguments, 209–210; - trisection of angle, 233; - probability of inference, 259; - arcual unit, 306; - mathematical tables, 331; - personal error, 348; - average, 363; - his works on probability, 394–395; - apparent sequence, 409; - sub-equality, 480; - rule of approximation, 481; - negative areas, 529; - generalisation, 600; - double algebra, 634; - bibliography, 716; - catalogues, 716; - extensions of algebra, 758. - - Density, unit of, 316; - of earth, 387; - negative, 642. - - Descartes, vortices, 517; - geometry, 632. - - Description, 62. - - Design, 762–763. - - Determinants, inference by, 50. - - Development, logical, 89, 97. - - Diagnosis, 708. - - Dichotomy, 703. - - Difference, 44; - law of, 5; - sign of, 17; - representation of, 45; - inference with, 52, 166; - form of, 158. - - Differences of numbers, 185. - - Differential calculus, 477. - - Differential thermometer, 345. - - Diffraction of light, 420. - - Dimensions, theory of, 325. - - Dip-needle, observation of, 355. - - Direct deduction, 49. - - Direction of motion, 47. - - Discontinuity, 620. - - Discordance, of theory and experiment, 558; - of theories, 587. - - Discoveries, accidental, 529; - predicted, 536; - scope for, 752. - - Discrimination, 24; - power of, 4. - - Disjunctive, terms, 66; - conjunction, 67; - propositions, 66; - syllogism, 77; - argument, 106. - - Dissipation of energy, 310. - - Distance of statements, 144. - - Divergence from average, 188. - - Diversity, 156. - - Divine interference, 765. - - Dollond, achromatic lenses, 608. - - Donkin, Professor, 375; - on probability, 199, 216; - principle of inverse method, 244. - - Double refraction, 426. - - Dove’s law of winds, 534. - - Draper’s law, 606. - - Drobitsch, 15. - - Duality, 73, 81; - law of, 5, 45, 92, 97. - - Dulong and Petit, 341, 471. - - Duration, 308. - - - ε, 330, 769. - - Earth, density of, 387; - ellipticity, 565. - - Eclipses, 656; - Egyptian records of, 246; - of Jupiter’s satellites, 294, 372; - solar, 486. - - Electric, sense, 405; - acid, 428; - fluid, 523. - - Electric telegraph, anticipations of, 671. - - Electricity, theories of, 522; - duality of, 590. - - Electrolysis, 428, 530. - - Electro-magnet, use of, 423. - - Elements, confusion of, 237; - definition, 427; - classification, 676, 677, 690. - - Elimination, 58. - - Ellicott, observation on clocks, 455. - - Ellipsis, 41; - of terms, 57. - - Elliptic variation, 474. - - Ellipticity of earth, 565. - - Ellis, A. J., contributions to formal logic, 172. - - Ellie, Leslie, 23, 375. - - Ellis, W., on moon’s influence, 410. - - Emanation, law of, 463. - - Emotions, 732. - - Empirical, knowledge, 505, 525–526; - measurement, 552. - - Encke, on mean, 386, 389; - his comet, 570, 605; - on resisting medium, 523; - solar parallax, 562. - - Energy, unit of, 322; - conservation of, 465; - reconcentration of, 751. - - English language, words in, 175. - - Eözoon canadense, 412, 668. - - Equality, sign of, 14; - axiom, 163; - four meanings of, 479. - - Equations, 46, 53, 160; - solution of, 123. - - Equilibrium, unstable, 276, 654. - - Equisetaceæ, 721. - - Equivalence of propositions, 115, 120, 132; - remarkable case of, 529, 657. - - Eratosthenes, sieve of, 82, 123, 139; - measurement of degree, 293. - - Error, function, 330, 376, 381; - elimination of, 339, 353; - personal, 347; - law of, 374; - origin of law, 383; - verification of law, 383; - probable, 386; - mean, 387; - constant, 396; - variation of small errors, 479. - - Ether, luminiferous, 512, 514, 605. - - Euclid, axioms, 51, 163; - indirect proof, 84; - 10th book, 117th proposition, 275; - on analogy, 631. - - Euler, on certainty of inference, 238; - corpuscular theory, 435; - gravity, 463; - on ether, 514. - - Everett, Professor, unit of angle, 306; - metric system, 328. - - Evolution, theory of, 761. - - Exact science, 456. - - Exceptions, 132, 644, 728; - classification of, 645; - imaginary, 647; - apparent, 649; - singular, 652; - divergent, 655; - accidental, 658; - novel, 661; - limiting, 663; - real, 666; - unclassed, 668. - - Excluded middle, law of, 6. - - Exclusive alternatives, 68. - - Exhaustive investigation, 418. - - Expansion, of bodies, 478; - of liquids, 488. - - Experiment, 400, 416; - in probability, 208; - test or blind, 433; - negative results of, 434; - limits of, 437; - collective, 445; - simplification of, 422; - failure in simplification, 424. - - Experimentalist, character of, 574, 592. - - Experimentum crucis, 518, 667. - - Explanation, 532. - - Extent of meaning, 26; - of terms, 48. - - Extrapolation, 495. - - - Factorials, 179. - - Facts, importance of false, 414; - conformity with, 516. - - Fallacies, 62; - analysed by indirect method, 102; - of observation, 408. - - Faraday, Michael, measurement of gold-leaf, 296; - on gravity, 342, 589; - magnetism of gases, 352; - vibrating plate, 419; - electric poles, 421; - circularly polarised light, 424, 588, 630; - freezing mixtures, 427; - magnetic experiments, 431, 434; - lines of magnetic force, 446, 580; - errors of experiment, 465; - electrolysis, 502; - velocity of light, 520; - prediction, 543; - relations of physical forces, 547; - character of, 578, 587; - ray vibrations, 579; - mathematical power, 580; - philosophic reservation of opinion, 592; - use of heavy glass, 609; - electricity, 612; - radiant matter, 642; - hydrogen, 691. - - Fatality, belief in, 264. - - Ferio, 56. - - Figurate numbers, 183, 186. - - Figure of earth, 459, 565. - - Fizeau, use of Newton’s rings, 297, 582; - fixity of properties, 313; - velocity of light, 441, 561. - - Flamsteed, use of wells, 294; - standard stars, 301; - parallax of pole-star, 338; - selection of observations, 358; - astronomical instruments, 391; - solar eclipses, 486. - - Fluorescence, 664. - - Fontenelle on the senses, 405. - - Forbes, J. D., 248. - - Force, unit of, 322, 326; - emanating, 464; - representation of, 633. - - Formulæ, empirical, 487; - rational, 489. - - Fortia, *Traité des Progressions*, 183. - - Fortuitous coincidences, 261. - - Fossils, 661. - - Foster, G. C., on classification, 691. - - Foucault, rotating mirror, 299; - pendulum, 342, 431, 522; - on velocity of light, 441, 521, 561. - - Fourier, Joseph, theory of dimensions, 325; - theory of heat, 469, 744. - - Fowler, Thomas, on method of difference, 439; - reasoning from case to case, 227. - - Frankland, Professor Edward, on spectrum of gases, 606. - - Franklin’s experiments on heat, 424. - - Fraunhofer, dark lines of spectrum, 429. - - Freezing-point, 546. - - Freezing mixtures, 546. - - Fresnel, inflexion of light, 420; - corpuscular theory, 521; - on use of hypothesis, 538; - double refraction, 539. - - Friction, 417; - determination of, 347. - - Function, definitions of, 489. - - Functions, discovery of, 496. - - - Galileo, 626; - on cycloid, 232, 235; - differential method of observation, 344; - projectiles, 447, 466; - use of telescope, 522; - gravity, 604; - principle of continuity, 617. - - Gallon, definition of, 318. - - Galton, Francis, divergence from mean, 188; - works by, 188, 655; - on hereditary genius, 385, 655. - - Galvanometer, 351. - - Ganières, de, 182. - - Gases, 613; - properties of, 601, 602; - perfect, 470; - liquefiable, 665. - - Gauss, pendulum experiments, 316; - law of error, 375–6; - detection of error, 396; - on gravity, 463. - - Gay Lussac, on boiling point, 659; - law of, 669. - - Genealogical classification, 680, 719. - - General, terms, 29; - truths, 647; - notions, 673. - - Generalisation, 2, 594, 704; - mathematical, 168; - two meanings of, 597; - value of, 599; - hasty, 623. - - Genius, nature of, 575. - - Genus, 433, 698; - generalissimum, 701; - natural, 724. - - Geology, 667; - records in, 408; - slowness of changes, 438; - exceptions in, 660. - - Geometric mean, 361. - - Geometric reasoning, 458; - certainty of, 267. - - Giffard’s injector, 536. - - Gilbert, on rotation of earth, 249; - magnetism of silver, 431; - experimentation, 443. - - Gladstone, J. H., 445. - - Glaisher, J. W. L., on mathematical tables, 331; - law of error, 375, 395. - - Gold, discovery of, 413. - - Gold-assay process, 434. - - Gold-leaf, thickness of, 296. - - Graham, Professor Thomas, on chemical affinity, 614; - continuity, 616; - nature of hydrogen, 691. - - Grammar, 39; - rules of, 31. - - Grammatical, change, 119; - equivalence, 120. - - Gramme, 317. - - Graphical method, 492. - - Gravesande, on inflection of light, 420. - - Gravity, 422, 512, 514, 604, 740; - determination of, 302; - elimination of, 427; - law of, 458, 462, 474; - inconceivability of, 510; - Newton’s theory, 555; - variation of, 565; - discovery of law, 581; - Faraday on, 589; - discontinuity in, 620; - Aristotle on, 649; - Hooke’s experiment, 436. - - Grimaldi on the spectrum, 584. - - Grove, Mr. Justice, on ether, 514; - electricity, 615. - - Guericke, Otto von, 432. - - - Habit, formation of, 618. - - Halley, trade-winds, 534. - - Halley’s comet, 537, 570. - - Hamilton, Sir William, disjunctive propositions, 69; - inference, 118; - free-will, 223. - - Hamilton, Sir W. Rowan, on conical refraction, 540; - quaternions, 634. - - Harley, Rev. Robert, on Boole’s logic, 23, 155. - - Harris, standards of length, 312. - - Hartley, on logic, 7. - - Hatchett, on alloys, 191. - - Haughton, Professor, on tides, 450; - muscular exertion, 490. - - Haüy, on crystallography, 529. - - Hayward, R. B., 142. - - Heat, unit of, 324; - measurement of, 349; - experiments on, 444; - mechanical equivalent of, 568. - - Heavy glass, 588, 609. - - Helmholtz, on microscopy, 406; - undulations, 414; - sound, 476. - - Hemihedral crystals, 649. - - Herschel, Sir John, on rotation of plane of polarisation of light, 129, 630; - quartz crystals, 246; - numerical precision, 273; - photometry, 273; - light of stars, 302; - actinometer, 337; - mean and average, 363; - eclipses of Jupiter’s satellites, 372; - law of error, 377; - error in observations, 392; - on observation, 400; - moon’s influence on clouds, 410; - comets, 411; - spectrum analysis, 429; - collective instances, 447; - principle of forced vibrations, 451, 663; - meteorological variations, 489; - double stars, 499, 685; - direct action, 502; - use of theory, 508; - ether, 515; - *experimentum crucis*, 519; - interference of light, 539; - interference of sound, 540; - density of earth, 567; - residual phenomena, 569; - helicoidal dissymmetry, 630; - fluorescence, 664. - - Hindenburg, on combinatorial analysis, 176. - - Hipparchus, used method of repetition, 289; - longitudes of stars, 294. - - Hippocrates, area of lunule, 480. - - History, science of, 760. - - Hobbes, Thomas, definition of cause, 224; - definition of time, 307; - on hypothesis, 510. - - Hofmann, unit called crith, 321; - on prediction, 544; - on anomalies, 670. - - Homogeneity, law of, 159, 327. - - Hooke, on gravitation, 436, 581; - philosophical method, 507; - on strange things, 671. - - Hopkinson, John, 194; - method of interpolation, 497. - - Horrocks, use of mean, 358; - use of hypothesis, 507. - - Hume on perception, 34. - - Hutton, density of earth, 566. - - Huxley, Professor Thomas, 764; - on hypothesis, 509; - classification, 676; - mammalia, 682; - palæontology, 682. - - Huyghens, theory of pendulum, 302; - pendulum standard, 315; - cycloidal pendulum, 341; - differential method, 344; - distant stars, 405; - use of hypothesis, 508; - philosophical method of, 585; - on analogy, 639. - - Hybrids, 727. - - Hydrogen, expansion of, 471; - refractive power, 527; - metallic nature of, 691. - - Hygrometry, 563. - - Hypotheses, use of, 265, 504; - substitution of simple hypotheses, 458; - working hypotheses, 509; - requisites of, 510; - descriptive, 522, 686; - representative, 524; - probability of, 559. - - - Identical propositions, 119. - - Identities, simple, 37; - partial, 40; - limited, 42; - simple and partial, 111; - inference from, 51, 55. - - Identity, law of, 5, 6, 74; - expression of, 14; - propagating power, 20; - reciprocal, 46. - - Illicit process, of major term, 65, 103; - of minor term, 65. - - Immediate inference, 50, 61. - - Imperfect induction, 146, 149. - - Inclusion, relation of, 40. - - Incommensurable quantities, 275. - - Incompossible events, 205. - - Independence of small effects, 475. - - Independent events, 204. - - Indestructibility of matter, 465. - - Indexes, classification by, 714; - formation of, 717. - - India-rubber, properties of, 545. - - Indirect method of deduction, 49, 81; - illustrations of, 98; - fallacies analysed by, 102; - the test of equivalence, 115. - - Induction, 11, 121; - symbolic statement of, 131; - perfect, 146; - imperfect, 149; - philosophy of, 218; - grounds of, 228; - illustrations of, 229; - quantitative, 483; - problem of two classes, 134; - problem of three classes, 137. - - Inductive truths, classes of, 219. - - Inequalities, reasoning by, 47, 163, 165–166. - - Inference, 9; - general formula of, 17; - immediate, 50; - with two simple identities, 51; - from simple and partial identity, 53; - with partial identities, 55; - by sum of predicates, 61; - by disjunctive propositions, 76; - indirect method of, 81; - nature of, 118; - principle of mathematical, 162; - certainty of, 236. - - Infima species, 701, 702. - - Infiniteness of universe, 738. - - Inflection of light, 420. - - Instantiæ, citantes, evocantes, radii, curriculi, 270; - monodicæ, irregulares, heteroclitæ, 608; - clandestinæ, 610. - - Instruments of measurement, 284. - - Insufficient enumeration, 176. - - Integration, 123. - - Intellect, etymology of, 5. - - Intension of logical terms, 26, 48; - of propositions, 47. - - Interchangeable system, 20. - - Interpolation, 495; - in meteorology, 497. - - Inverse, process, 12; - operation, 122, 689; - problem of two classes, 134; - problem of three classes, 137; - problem of probability, 240, 251; - rules of inverse method, 257; - simple illustrations, 253; - general solution, 255. - - Iodine, the substance X, 523. - - Iron, properties of, 528, 670. - - *Is*, ambiguity of verb, 16, 41. - - Isomorphism, 662. - - Ivory, 375. - - - James, Sir H., on density of earth, 567. - - Jenkin, Professor Fleming, 328. - - Jevons, W. S., on use of mean, 361; - on pedesis or molecular movement of microscopic particles, 406, 549; - cirrous clouds, 411; - spectrum analysis, 429; - elevated rain-gauges, 430; - experiments on clouds, 447; - on muscular exertion, 490; - resisting medium, 570; - anticipations of the electric telegraph, 671. - - Jones, Dr. Bence, Life of Faraday, 578. - - Jordanus, on the mean, 360. - - Joule, 545; - on thermopile, 299, 300; - mechanical equivalent of heat, 325, 347, 568; - temperature of air, 343; - rarefaction, 444; - on Thomson’s prediction, 543; - molecular theory of gases, 548; - friction, 549; - thermal phenomena of fluids, 557. - - Jupiter, satellites of, 372, 458, 638, 656; - long inequality of, 455; - figure of, 556. - - - Kames, Lord, on bifurcate classification, 697. - - Kant, disjunctive propositions, 69; - analogy, 597; - doctrine of space, 769. - - Kater’s pendulum, 316. - - Keill, law of emanating forces, 464; - axiom of simplicity, 625. - - Kepler, on star-discs, 390; - comets, 408; - laws of, 456; - refraction, 501; - character of, 578. - - Kinds of things, 718. - - King Charles and the Royal Society, 647. - - Kirchhoff, on lines of spectrum, 245. - - Kohlrausch, rules of approximate calculation, 479. - - - Lagrange, formula for interpolation, 497; - accidental discovery, 531; - union of algebra and geometry, 633. - - Lambert, 15. - - Lamont, 452. - - Language, 8, 628, 643. - - Laplace, on probability, 200, 216; - principles of inverse method, 242; - solution of inverse problem, 256; - planetary motions, 249, 250; - conjunctions of planets, 293; - observation of tides, 372; - atmospheric tides, 367; - law of errors, 378; - dark stars, 404; - hyperbolic comets, 407; - his works on probability, 395; - velocity of gravity, 435; - stability of planetary system, 448, 746; - form of Jupiter, 556; - corpuscular theory, 521; - ellipticity of earth, 565; - velocity of sound, 571; - analogy, 597; - law of gravity, 615; - inhabitants of planets, 640; - laws of motion, 706; - power of science, 739. - - Lavoisier, mistaken inference of, 238; - pyrometer, 287; - on experiments, 423; - prediction of, 544; - theory, 611; - on acids, 667 - - Law, 3; - of simplicity, 33, 72, 161; - commutativeness, 35, 160; - disjunctive relation, 71; - unity, 72, 157, 162; - identity, 74; - contradiction, 74, 82; - duality, 73, 74, 81, 97, 169; - homogeneity, 159; - error, 374; - continuity, 615; - of Boyle, 619; - natural, 737. - - Laws, of thought, 6; - empirical mathematical, 487; - of motion, 617; - of botanical nomenclature, 727; - natural hierarchy of, 742. - - Least squares, method of, 386, 393. - - Legendre, on geometry, 275; - rejection of observations, 391; - method of least squares, 377. - - Leibnitz, 154, 163; - on substitution, 21; - propositions, 42; - blunder in probability, 213; - on Newton, 515; - continuity, 618. - - Leslie, differential thermometer, 345; - radiating power, 425; - on affectation of accuracy, 482. - - Letters, combinations of, 193. - - Leverrier, on solar parallax, 562. - - Lewis, Sir G. C., on time, 307. - - Life is change, 173. - - Light, intensity of, 296; - unit, 324; - velocity, 535, 560, 561; - science of, 538; - total reflection, 650; - waves of, 637; - classification of, 731. - - Lighthouses, Babbage on, 194. - - Limited identities, 42; - inference of 59. - - Lindsay, Prof. T. M., 6, 21. - - Linear variation, 474. - - Linnæus on synopsis, 712; - genera and species, 725. - - Liquid state, 601, 614. - - Locke, John, on induction, 121; - origin of number, 157; - on probability, 215; - the word power, 221. - - Lockyer, J. Norman, classification of elements, 676. - - Logarithms, 148; - errors in tables, 242. - - Logic, etymology of name, 5. - - Logical abacus, 104. - - Logical alphabet, 93, 116, 173, 417, 701; - table of, 94; - connection with arithmetical triangle, 189; - in probability, 205. - - Logical conditions, numerical meaning of, 171. - - Logical machine, 107. - - Logical relations, number of, 142. - - Logical slate, 95. - - Logical truths, certainty of, 153. - - Lottery, the infinite, 2. - - Lovering, Prof., on ether, 606. - - Lubbock and Drinkwater-Bethune, 386, 395. - - Lucretius, rain of atoms, 223, 741; - indestructibility of matter, 622. - - - Machine, logical, 107. - - Macleay, system of classification, 719. - - Magnetism of gases, 352. - - Mallet, on earthquakes, 314. - - Malus, polarised light, 530. - - Mammalia, characters of, 681. - - Manchester Literary and Philosophical Society, papers quoted, 137, 143, 168. - - Mansel, on disjunctive propositions, 69. - - Mars, white spots of, 596. - - Maskelyne, on personal error, 347; - deviation of plumbline, 369; - density of earth, 566. - - Mass, unit of, 317, 325. - - Mathematical science, 767; - incompleteness of, 754. - - Matter, uniform properties of, 603; - variable properties, 606. - - Matthiessen, 528. - - Maximum points, 371. - - Maxwell, Professor Clerk, on the balance, 304; - natural system of standards, 311, 319; - velocity of electricity, 442; - on Faraday, 580; - his book on *Matter and Motion*, 634. - - Mayer, proposed repeating circle, 290; - on mechanical equivalent of heat, 568, 572. - - Mean, etymology of, 359–360; - geometric, 362; - fictitious, 363; - precise, 365; - probable, 385; - rejection of, 389; - method of, 357, 554. - - Mean error, 387. - - Meaning, of names, 25; - of propositions, 47. - - Measurement, of phenomena, 270; - methods of, 282; - instruments, 284; - indirect, 296; - accuracy of, 303; - units and standards of, 305; - explained results of, 554; - agreement of modes of, 564. - - Mediate statements, 144. - - Melodies, possible number of, 191. - - Melvill, Thomas, on the spectrum, 429. - - *Membra dividentia*, 68. - - Metals, probable character of new, 258; - transparency, 548; - classification, 675; - density, 706. - - Method, indirect, 98; - of avoidance of error, 340; - differential, 344; - correction, 346; - compensation, 350; - reversal, 354; - means, 357; - least squares, 377, 386, 393; - variations, 439; - graphical, 492; - Baconian, 507. - - Meteoric streams, 372. - - Meteoric cycle, 537. - - Metre, 349; - error of, 314. - - Metric system, 318, 323. - - Michell, speculations, 212; - on double stars, 247; - Pleiades, 248; - torsion balance, 566. - - Middle term undistributed, 64. - - Mill, John Stuart, disjunctive propositions, 69; - induction, 121, 594; - music, 191; - probability, 200–201, 222; - supposed reform of logic, 227; - deductive method, 265, 508; - elimination of chance, 385; - joint method of agreement and difference, 425; - method of variations, 484; - on collocations, 740; - erroneous tendency of his philosophy, 752. - - Miller, Prof. W. H., kilogram, 318. - - Mind, powers of, 4; - phenomena of, 672. - - Minerals, classification of, 678. - - Minor term, illicit process of, 65. - - Mistakes, 7. - - *Modus, tolendo ponens*, 77; - *ponendo tollens*, 78. - - Molecular movement, or pedesis, 406. - - Molecules, number of, 195. - - Momentum, 322, 326. - - Monro, C. J., correction by, 172; - on Comte, 753. - - Monstrous productions, 657. - - Moon, supposed influence on clouds, 410; - atmosphere of, 434; - motions, 485; - fall towards earth, 555. - - Morse alphabet, 193. - - Mother of pearl, 419. - - Müller, Max, on etymology of intellect, 5. - - Multiplication in logic, 161. - - Murphy, J. J., on disjunctive relation, 71. - - Murray, introduced use of ice, 343. - - Muscular susurrus, 298. - - Music, possible combinations of, 191. - - - Names, 25; - of persons, ships, &c., 680. - - Nature, 1; - laws of, 737; - uniformity of, 745. - - Nebular theory, 427. - - Negation, 44. - - Negative arguments, 621. - - Negative density, 642. - - Negative premises, 63, 103. - - Negative propositions, 43. - - Negative results of experiment, 434. - - Negative terms, 14, 45, 54, 74. - - Neil on use of hypothesis, 509. - - Neptune, discovery of, 537, 660. - - Newton, Sir Isaac, binomial theorem, 231; - spectrum, 262, 418, 420, 424, 583; - rings of, 288, 470; - velocity of sound, 295; - wave-lengths, 297; - use of pendulum, 303; - on time, 308; - definition of matter, 316; - pendulum experiment, 348, 443, 604; - centrobaric bodies, 365; - on weight, 422; - achromatic lenses, 432; - resistance of space, 435; - absorption of light, 445; - planetary motions, 249, 457, 463, 466, 467; - infinitesimal calculus, 477; - as an alchemist, 505; - his knowledge of Bacon’s works, 507; - *hypotheses non fingo*, 515; - on vortices, 517; - theory of colours, 518; - corpuscular theory of light, 520; - fits of easy reflection, &c., 523; - combustible substances, 527; - gravity, 555, 650; - density of earth, 566; - velocity of sound, 571; - third law of motion, 622; - his rules of philosophising, 625; - fluxions, 633; - theory of sound, 636; - negative density, 642; - rays of light having sides, 662. - - Newtonian Method, 581. - - Nicholson, discovery of electrolysis, 530. - - *Ninth Bridgewater Treatise* quoted, 743, 757. - - Nipher, Professor, on muscular exertion, 490. - - Noble, Captain, chronoscope, 308, 616. - - Nomenclature, laws of botanical, 727. - - Non-observation, arguments from, 411. - - Norwood’s measurement of a degree, 272. - - Nothing, 32. - - Number, nature of, 153, 156; - concrete and abstract, 159, 305. - - Numbers, prime, 123; - of Bernoulli, 124; - figurate, 183; - triangular, &c., 185. - - Numerical abstraction, 158. - - - Observation, 399; - mental conditions, 402; - instrumental and sensual conditions, 404; - external conditions, 407. - - Obverse statements, 144. - - Ocean, depth of, 297. - - Odours, 732. - - Oersted, on electro-magnetism, 530, 535. - - *Or*, meaning of, 70. - - Order, of premises, 114; - of terms, 33. - - Orders of combinations, 194. - - Original research, 574. - - Oscillation, centre of, 364. - - Ostensive instances, 608. - - Ozone, 663. - - - π, value of, 234, 529. - - Pack of cards, arrangement of, 241. - - Paley on design, 762, 763. - - Parallax, of stars, 344; - of sun, 560. - - Parallel forces, 652. - - Paralogism, 62. - - Parity of reasoning, 268. - - Partial identities, 40, 55, 57, 111; - induction of, 130. - - Particular quantity, 56. - - Particulars, reasoning from, 227. - - Partition, 29. - - Pascal, 176; - arithmetical machine, 107; - arithmetical triangle, 182; - binomial formula, 182; - error in probabilities, 213; - barometer, 519. - - Passive state of steel, 659. - - Pedesis, or molecular movement of microscopic particles, 406, 612. - - Peirce, Professor, 23; - on rejection of observations, 391. - - Pendulum, 290, 302, 315; - faults of, 311; - vibrations, 453, 454; - cycloidal, 461. - - Perfect induction, 146, 149. - - Perigon, 306. - - Permutations, 173, 178; - distinction from combinations, 177. - - Personal error, 347. - - Photometry, 288. - - Physiology, exceptions in, 666. - - Planets, conjunctions of, 181, 187, 657; - discovery of, 412; - motions, 457; - perturbations of, 657; - classification, 683; - system of, 748. - - Plants, classification of, 678. - - Plateau’s experiments, 427. - - Plato on science, 595. - - Plattes, Gabriel, 434, 438. - - Pliny on tides, 451. - - Plumb-line, divergence of, 461. - - Plurality, 29, 156. - - Poinsot, on probability, 214. - - Poisson, on principle of the inverse method, 244; - work on Probability, 395; - Newton’s rings, 470; - simile of ballot box, 524. - - Polarisation, 653; - discovery of, 530. - - Pole-star, 652; - observations of, 366. - - Poles, of magnets, 365; - of battery, 421. - - Political economy, 760. - - Porphyry, on the Predicables, 698; - tree of, 702. - - Port Royal logic, 22. - - Positive philosophy, 760, 768. - - Pouillet’s pyrheliometer, 337. - - Powell, Baden, 623; - on planetary motions, 660. - - Power, definition of, 224. - - Predicables, 698. - - Prediction, 536, 739; - in science of light, 538; - theory of undulations, 540; - other sciences, 542; - by inversion of cause and effect, 545. - - Premises, order of, 114. - - Prime numbers, 123, 139; - formula for, 230. - - *Principia*, Newton’s, 581, 583. - - Principle, of probability, 200; - inverse method, 242; - forced vibrations, 451; - approximation, 471; - co-existence of small vibrations, 476; - superposition of small effects, 476. - - Probable error, 555. - - Probability, etymology of, 197; - theory of, 197; - principles, 200; - calculations, 203; - difficulties of theory, 213; - application of theory, 215; - in induction, 219; - in judicial proceedings, 216; - works on, 394; - results of law, 656. - - Problems, to be worked by reader, 126; - inverse problem of two classes, 135; - of three classes, 137. - - Proclus, commentaries of, 232. - - Proctor, R. A., star-drifts, 248. - - Projectiles, theory of, 466. - - Proper names, 27. - - Properties, generality of, 600; - uniform, 603; - extreme instances, 607; - correlation, 681. - - Property, logical, 699; - peculiar, 699. - - Proportion, simple, 501. - - Propositions, 36; - negative, 43; - conversion of, 46; - twofold meaning, 47; - disjunctive, 66; - equivalence of, 115; - identical, 119; - tautologous, 119. - - Protean verses, 175. - - Protoplasm, 524, 764. - - Prout’s law, 263, 464. - - Provisional units, 323. - - Proximate statements, 144. - - Pyramidal numbers, 185. - - Pythagoras, on duality, 95; - on the number seven, 262, 624. - - - Quadric variation, 474. - - Qualitative, reasoning, 48; - propositions, 119. - - Quantification of predicate, 41. - - Quantitative, reasoning, 48; - propositions, 119; - questions, 278; - induction, 483. - - Quantities, continuous, 274; - incommensurable, 275. - - Quaternions, 160, 634. - - Quetelet, 188; - experiment on probability, 208; - on mean and average, 363; - law of error, 378, 380; - verification of law of error, 385. - - - Radian, 306. - - Radiant matter, 642. - - Radiation of heat, 430. - - Radiometer, 435. - - Rainbow, theory of, 526, 533. - - Rainfall, variation of, 430. - - Ramean tree, 703, 704. - - Ramsden’s balance, 304. - - Rankine, on specific heat of air, 557; - reconcentration of energy, 751. - - Rational formulæ, 489. - - Rayleigh, Lord, on graphical method, 495. - - Reasoning, arithmetical, 167; - numerically definite, 168; - geometrical, 458. - - Recorde, Robert, 15. - - Reduction, of syllogisms, 85; - *ad absurdum*, 415; - of observations, 552, 572. - - Reflection, total, 650. - - Refraction, atmospheric, 340, 356, 500; - law of, 501; - conical, 540; - double, 585. - - Regnault, dilatation of mercury, 342; - measurement of heat, 350; - exact experiment, 397; - on Boyle’s law, 468, 471; - latent heat of steam, 487; - graphical method, 494; - specific heat of air, 557. - - Reid, on bifurcate classification, 697. - - Reign of law, 741, 759. - - Rejection of observations, 390. - - Relation, sign of, 17; - logic of, 22; - logical, 35; - axiom of, 164. - - Repetition, method of, 287, 288. - - Representative hypotheses, 524. - - Reproduction, modes of, 730. - - Reservation of judgment, 592. - - Residual effects, 558; - phenomena, 560, 569. - - Resisting medium, 310, 523, 570. - - Resonance, 453. - - Reusch, on substitution, 21. - - Reversal, method of, 354. - - Revolution, quantity of, 306. - - Robertson, Prof. Croom, 27, 101. - - Robison, electric curves, 446. - - Rock-salt, 609. - - Rœmer, divided circle, 355; - velocity of light, 535. - - Roscoe, Prof., photometrical researches, 273; - solubility of salts, 280; - constant flame, 441; - absorption of gases, 499; - vanadium, 528; - atomic weight of vanadium, 392, 649. - - Rousseau on geometry, 233. - - Rules, of inference, 9, 17; - indirect method of inference, 89; - for calculation of combinations, 180; - of probabilities, 203; - of inverse method, 257; - for elimination of error, 353. - - Rumford, Count, experiments on heat, 343, 350, 467. - - Ruminants, Cuvier on, 683. - - Russell, Scott, on sound, 541. - - - Sample, use of, 9. - - Sandeman, on perigon, 306; - approximate arithmetic, 481. - - Saturn, motions of satellites, 293; - rings, 293. - - Schehallien, attraction of, 369, 566. - - Schottus, on combinations, 179. - - Schwabe, on sun-spots, 452. - - Science, nature of, 1, 673. - - Selenium, 663, 670. - - Self-contradiction, 32. - - Senior’s definition of wealth, 75. - - Senses, fallacious indications of, 276. - - Seven, coincidences of number, 262; - fallacies of, 624. - - Sextus, fatality of name, 264. - - Sieve of Eratosthenes, 82, 123, 139. - - Similars, substitution of, 17. - - Simple identity, 37, 111; - inference of, 58; - contrapositive, 86; - induction of, 127. - - Simple statement, 143. - - Simplicity, law of, 33, 58, 72. - - Simpson, discovery of property of chloroform, 531. - - Simultaneity of knowledge, 34. - - Singular names, 27; - terms, 129. - - Siren, 10, 298, 421. - - Slate, the logical, 95. - - Smeaton’s experiments, on water-wheels, 347; - windmills, 401, 441. - - Smee, Alfred, logical machines, 107. - - Smell, delicacy of, 437. - - Smithsonian Institution, 329. - - Smyth, Prof. Piazzi, 452. - - Socrates, on the sun, 611. - - Solids, 602. - - Solubility of salts, 279. - - *Some*, the adjective, 41, 56. - - Sorites, 60. - - Sound, observations on, 356; - undulations, 405, 421; - velocity of, 571; - classification of sounds, 732. - - Space, relations of, 220. - - Species, 698; - infima, 701; - natural, 724. - - Specific gravities, 301; - heat of air, 557. - - Spence, on boiling point, 546. - - Spencer, Herbert, nature of logic, 4, 7; - sign of equality, 15; - rhythmical motion, 448; - abstraction, 705; - philosophy of, 718, 761, 762. - - Spectroscope, 437. - - Spectrum, 583. - - Spiritualism, 671. - - Spontaneous generation, 432. - - Standards of measurement, 305; - the bar, 312; - terrestrial, 314; - pendulum, 315; - provisional, 318; - natural system, 319. - - Stars, discs of, 277; - motions of, 280, 474; - variations of, 281; - approach or recess, 298; - standard stars, 301; - apparent diameter, 390; - variable, 450; - proper motions, 572; - Bruno on, 639; - new, 644; - pole-star, 652; - conflict with wandering stars, 748. - - Stas, M., his balance, 304; - on atomic weights, 464. - - Statements, kinds of, 144. - - Statistical conditions, 168. - - Stevinus, on inclined plane, 622. - - Stewart, Professor Balfour, on resisting medium, 570; - theory of exchanges, 571. - - Stifels, arithmetical triangle, 182. - - Stokes, Professor, on resistance, 475; - fluorescence, 664. - - Stone, E. J., heat of the stars, 370; - temperature of earth’s surface, 452; - transit of Venus, 562. - - Struve on double stars, 247. - - Substantial terms, 28. - - Substantives, 14. - - Substitution of similars, 17, 45, 49, 104, 106; - anticipations of, 21. - - Substitutive weighing, 345. - - *Sui generis*, 629, 728. - - Sulphur, 670. - - Summum genus, 93, 701. - - Sun, distance, 560; - variations of spots, 452. - - Superposition, of small effects, 450; - small motions, 476. - - Swan, W., on sodium light, 430. - - Syllogism, 140; - moods of, 55, 84, 85, 88, 105, 141; - numerically definite, 168. - - Symbols, use of, 13, 31, 32; - of quantity, 33. - - Synthesis, 122; - of terms, 30. - - - Table-turning, 671. - - Tacit knowledge, 43. - - Tacquet on combinations, 179. - - Tait, P. G., 375; - theory of comets, 571. - - Talbot on the spectrum, 429. - - Tartaglia on projectiles, 466. - - Tastes, classification of, 732. - - Tautologous propositions, 119. - - Teeth, use in classification, 710. - - Temperature, variations of, 453. - - Tension of aqueous vapour, 500. - - Terms, 24; - abstract, 27; - substantial, 28; - collective, 29; - synthesis of, 30; - negative, 45. - - Terrot, Bishop, on probability, 212. - - Test experiments, 347, 433. - - Tetractys, 95. - - Thales, predicted eclipse, 537. - - Theory, results of, 534; - facts known by, 547; - quantitative, 551; - of exchanges, 571; - freedom of forming, 577; - of evolution, 761. - - Thermometer, differential, 345; - reading of, 390; - change of zero, 390. - - Thermopile, 300. - - Thomas, arithmetical machine, 107. - - Thomson, Archbishop, 50, 61. - - Thomson, James, prediction by, 542; - on gaseous state, 654. - - Thomson, Sir W., lighthouse signals, 194; - size of atoms, 195; - tides, 450; - capillary attraction, 614; - magnetism, 665; - dissipation of energy, 744. - - Thomson and Tait, chronometry, 311; - standards of length, 315; - the crowbar, 460; - polarised light, 653. - - Thomson, Sir Wyville, 412. - - Thunder-cloud, 612. - - Tides, 366, 450, 476, 541; - velocity of, 298; - gauge, 368; - atmospheric, 367, 553. - - Time, 220; - definition of, 307. - - Todhunter, Isaac, *History of the Theory of Probability*, 256, 375, 395; - on insoluble problems, 757. - - Tooke, Horne, on cause, 226. - - Torricelli, cycloid, 235; - his theorem, 605; - on barometer, 666. - - Torsion balance, 272, 287. - - Transit of Venus, 294, 348, 562. - - Transit-circle, 355. - - Tree of Porphyry, 702; - of Ramus, 703. - - Triangle, arithmetical, 93, 182. - - Triangular numbers, 185. - - Trigonometrical survey, 301; - calculations of, 756. - - Trisection of angles, 414. - - Tuning-fork, 541. - - Tycho Brahe, 271; - on star discs, 277; - obliquity of earth’s axis, 289; - circumpolar stars, 366; - Sirius, 390. - - Tyndall, Professor, on natural constants, 328; - magnetism of gases, 352; - precaution in experiments, 431; - use of imagination, 509; - on Faraday, 547; - magnetism, 549, 607; - scope for discovery, 753. - - Types, of logical conditions, 140, 144; - of statements, 145; - classification by, 722. - - - Ueberweg’s logic, 6. - - Ultimate statements, 144. - - Undistributed, attribute, 40; - middle term, 64, 103. - - Undulations, of light, 558; - analogy in theory of, 635. - - Undulatory theory, 468, 520, 538, 540; - inconceivability of, 510. - - Unique objects, 728. - - Unit, definition of, 157; - groups, 167; - of measurement, 305; - arcual, 306; - of time, 307; - space, 312; - density, 316; - mass, 317; - subsidiary, 320; - derived, 321; - provisional, 323; - of heat, 325; - magnetical and electrical units, 326, 327. - - Unity, law of, 72. - - Universe, logical, 43; - infiniteness of, 738; - heat-history of, 744, 749; - possible states of, 749. - - Uranus, anomalies of, 660. - - - Vacuum, Nature’s abhorrence of, 513. - - Vapour densities, 548. - - Variable, variant, 440, 441, 483. - - Variation, linear, elliptic, &c., 474; - method of, 439. - - Variations, logical, 140; - periodic, 447; - combined, 450; - integrated, 452; - simple proportional, 501. - - Variety, of nature, 173; - of nature and art, 190; - higher orders of, 192. - - Velocity, unit of, 321. - - Venn, Rev. John, logical problem by, 90; - on Boole, 155; - his work on *Logic of Chance*, 394. - - Venus, 449; - transits of, 294. - - Verses, Protean, 175. - - Vibrations, law of, 295; - principle of forced, 451; - co-existence of small, 476. - - Vital force, 523. - - Voltaire on fossils, 661. - - Vortices, theory of, 513, 517. - - Vulcan, supposed planet, 414. - - - Wallis, 124, 175. - - Water, compressibility of, 338; - properties of, 610. - - Watt’s parallel motion, 462. - - Waves, 599, 635; - nature of, 468; - in canals, 535; - earthquake, 297. - - Weak arguments, effect of, 211. - - Wells, on dew, 425. - - Wenzel, on neutral salts, 295. - - Whately, disjunctive propositions, 69; - probable arguments, 210. - - Wheatstone, cipher, 124; - galvanometer, 286; - revolving mirror, 299, 308; - kaleidophone, 445; - velocity of electricity, 543. - - Whewell, on tides, 371, 542; - method of least squares, 386. - - Whitworth, Sir Joseph, 304, 436. - - Whitworth, Rev. W. A., on *Choice and Chance*, 395. - - Wilbraham, on Boole, 206. - - Williamson, Professor A. W., chemical unit, 321; - prediction by, 544. - - Wollaston, the goniometer, 287; - light of moon, 302; - spectrum, 429. - - Wren, Sir C., on gravity, 581. - - - X, the substance, 523. - - - Yard, standard, 397. - - Young, Dr. Thomas, tension of aqueous vapour, 500; - use of hypotheses, 508; - ethereal medium, 515. - - - Zero point, 368. - - Zodiacal light, 276. - - Zoology, 666. - - -LONDON: R. CLAY, SONS, AND TAYLOR, PRINTERS, - - - - -BY THE SAME AUTHOR. - - - ELEMENTARY LESSONS IN LOGIC: DEDUCTIVE AND INDUCTIVE. With Copious - Questions and Examples, and a Vocabulary of Logical Terms. Ninth - Edition. Fcap. 8vo. 3*s.* 6*d.* - - PRIMER OF LOGIC. With Illustrations and Questions. New Edition. 18mo. - 1*s.* - - STUDIES IN DEDUCTIVE LOGIC. A Manual for Students. Crown 8vo. 6*s.* - - THE SUBSTITUTION OF SIMILARS THE TRUE PRINCIPLE OF REASONING. 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KING AND CO. - - - - -SPELLING CORRECTIONS - - acording → according - aklaline → alkaline - an an → an - aws → laws - beween → between - BOOK III → BOOK IV - errror → error - incapadle → incapable - interpretion → interpretation - justifed → justified - longtitude → longitude - Marriotte → Mariotte - melecules → molecules - Meropolitana → Metropolitana - necesssarily → necessarily - nnmber → number - or → of - probabilty → probability - quantites → quantities - secresy → secrecy - sucession → succession - suficiently → sufficiently - telecope → telescope - verifiy → verify - - - -*** END OF THE PROJECT GUTENBERG EBOOK 74864 *** diff --git a/old/74864-h/74864-h.htm b/old/74864-h/74864-h.htm deleted file mode 100644 index 940e5c2..0000000 --- a/old/74864-h/74864-h.htm +++ /dev/null @@ -1,48486 +0,0 @@ -<!DOCTYPE html> -<html lang="en"> -<head> - <meta charset="UTF-8"> - <title> - The Principles of Science | Project Gutenberg - </title> - <link rel="icon" href="images/cover.jpg" type="image/x-cover"> -<style> - -body { - margin-left: 10%; 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- padding-left: 2em; -} - -.poetry .indent0 {text-indent: 0em} -.poetry .indent1 {text-indent: 0.5em} - - -/* Transcriber's notes */ -.transnote { - background-color: #F2F2F2; - color: black; - font-size:smaller; - padding:0.5em; - margin-bottom:5em; - font-family:sans-serif, serif; -} - - -/* Illustration classes */ -.illowp100 {width: 100%} -.illowp64 {width: 64%} -.x-ebookmaker .illowp64 {width: 100%} -.illowp75 {width: 75%} -.x-ebookmaker .illowp75 {width: 100%} -.illowp88 {width: 88%} -.x-ebookmaker .illowp88 {width: 100%} - - </style> - </head> -<body> -<div style='text-align:center'>*** START OF THE PROJECT GUTENBERG EBOOK 74864 ***</div> - -<div class="transnote"> -<p class="tn"><b><a id="Transcribers_notes"></a>Transcriber’s notes</b>:</p> - -<p class="tn">The text of this book has been preserved as closely as -practicable to its original form. However, the author used some unusual -symbols, and I have taken the liberty of using Unicode characters -with similar appearance (ꖌ ᔕ) as substitutes, disregarding their -official meaning and aware that they might not display on all devices. -An archaic symbol used by the author to indicate the mathematical -‘factorial’ function has been replaced by the modern equivalent, viz. ! -Unusual placements of some sub- and superscripted symbols remain as -in the original text.</p> - -<p class="tn">Inconsistencies of punctuation have been corrected -silently, but inconsistent spellings such as <i>Roemer, Römer, Rœmer</i> have -not been altered. A list of <a href="#Spelling_corrections">corrected -spellings</a> is appended at the end of the book.</p> - -<p class="tn">>Footnotes have been renumbered consecutively and -relocated to the end of the book. A missing footnote marker has been -inserted on p.751 after tracking down the original document. A missing -negative symbol has been added to an exponent in a formula on p.327.</p> - -<p class="tn">There is a misleading calculation on p.194 and -the table that follows, regarding progressive powers of two: -((2<sup>2</sup>)<sup>2</sup>)<sup>2</sup> is equivalent to -(16)<sup>2</sup> which equals 256 not 65,356 as stated, but -2<sup>16</sup> <i>does</i> equal 65,356.</p> - -<p class="tn">[sic] has been inserted on p.179 alongside a statement that the -alphabet contains 24 letters; however, the statement may well be -correct given that it was written in 1704 by a Flemish author and the -language is not specified.</p> - -<p class="tn">New original cover art included with this eBook is granted to the -public domain.</p> -</div> - - - -<p class="fs140 tac ">THE PRINCIPLES OF SCIENCE.</p> - -<div class="mtb10em"> -<figure class="figcenter illowp75" id="a002" style="max-width: 6.875em;"> - <img class="w100" src="images/a002.png" alt="colophon"> -</figure> -</div> - -<div class="center"> -<figure class="figcenter illowp64" id="a004" style="max-width: 29.375em;"> - <img class="w100" src="images/a004.jpg" alt=""> - <div class="caption fs75">THE LOGICAL MACHINE.</div> -</figure> -</div> - - - -<div class="titlepage"> -<h1><span class="t1">THE PRINCIPLES OF SCIENCE:</span> - -<span class="t2"><i>A TREATISE ON LOGIC</i></span> - -<span class="t3"><i>AND</i></span> - -<span class="t2"><i>SCIENTIFIC METHOD.</i></span></h1> - - -<div class="tp1">BY</div> - -<div class="tp2">W. STANLEY JEVONS,</div> - -<div class="tp3">LL.D. (EDINB.), M.A. (LOND.), F.R.S.</div> - - -<div class="tp4">London:</div> - -<div class="tp5">MACMILLAN AND CO.</div> -<div class="tp6">1883.</div> - - -<div class="tp7"><i>The Right of Translation and Reproduction is Reserved.</i></div> -</div> - - -<p class="tac fs70 mtb10em">LONDON:<br> -<span class="smcap">R. Clay, Sons, & Taylor, Printers</span>,<br> -BREAD STREET HILL.</p> - - -<p class="tac fs80"><b>Stereotyped Edition.</b></p> - - -<hr class="chap x-ebookmaker-drop"> -<div class="chapter"> -<p><span class="pagenum" id="Page_vii">vii</span></p> - -<h2 class="nobreak" id="PREFACE1">PREFACE<br> - -<span class="title"><i>TO THE FIRST EDITION</i>.</span></h2> -</div> - -<p class="ti0">It may be truly asserted that the rapid progress of the -physical sciences during the last three centuries has not -been accompanied by a corresponding advance in the -theory of reasoning. Physicists speak familiarly of -Scientific Method, but they could not readily describe -what they mean by that expression. Profoundly engaged -in the study of particular classes of natural phenomena, -they are usually too much engrossed in the immense and -ever-accumulating details of their special sciences to -generalise upon the methods of reasoning which they -unconsciously employ. Yet few will deny that these -methods of reasoning ought to be studied, especially by -those who endeavour to introduce scientific order into less -successful and methodical branches of knowledge.</p> - -<p>The application of Scientific Method cannot be restricted -to the sphere of lifeless objects. We must sooner -or later have strict sciences of those mental and social -phenomena, which, if comparison be possible, are of more -interest to us than purely material phenomena. But it -is the proper course of reasoning to proceed from the -known to the unknown—from the evident to the obscure—from -the material and palpable to the subtle and -refined. The physical sciences may therefore be properly<span class="pagenum" id="Page_viii">viii</span> -made the practice-ground of the reasoning powers, because -they furnish us with a great body-of precise and successful -investigations. In these sciences we meet with happy -instances of unquestionable deductive reasoning, of extensive -generalisation, of happy prediction, of satisfactory -verification, of nice calculation of probabilities. We can -note how the slightest analogical clue has been followed -up to a glorious discovery, how a rash generalisation has -at length been exposed, or a conclusive <i>experimentum -crucis</i> has decided the long-continued strife between two -rival theories.</p> - -<p>In following out my design of detecting the general -methods of inductive investigation, I have found that the -more elaborate and interesting processes of quantitative -induction have their necessary foundation in the simpler -science of Formal Logic. The earlier, and probably by -far the least attractive part of this work, consists, therefore, -in a statement of the so-called Fundamental Laws -of Thought, and of the all-important Principle of Substitution, -of which, as I think, all reasoning is a development. -The whole procedure of inductive inquiry, in its -most complex cases, is foreshadowed in the combinational -view of Logic, which arises directly from these fundamental -principles. Incidentally I have described the mechanical -arrangements by which the use of the important form -called the Logical Alphabet, and the whole working of -the combinational system of Formal Logic, may be rendered -evident to the eye, and easy to the mind and -hand.</p> - -<p>The study both of Formal Logic and of the Theory of -Probabilities has led me to adopt the opinion that there -is no such thing as a distinct method of induction as -contrasted with deduction, but that induction is simply -an inverse employment of deduction. Within the last -century a reaction has been setting in against the purely -empirical procedure of Francis Bacon, and physicists have<span class="pagenum" id="Page_ix">ix</span> -learnt to advocate the use of hypotheses. I take the -extreme view of holding that Francis Bacon, although he -correctly insisted upon constant reference to experience, -had no correct notions as to the logical method by which -from particular facts we educe laws of nature. I endeavour -to show that hypothetical anticipation of nature is -an essential part of inductive inquiry, and that it is the -Newtonian method of deductive reasoning combined with -elaborate experimental verification, which has led to all -the great triumphs of scientific research.</p> - -<p>In attempting to give an explanation of this view of -Scientific Method, I have first to show that the sciences -of number and quantity repose upon and spring from the -simpler and more general science of Logic. The Theory -of Probability, which enables us to estimate and calculate -quantities of knowledge, is then described, and especial -attention is drawn to the Inverse Method of Probabilities, -which involves, as I conceive, the true principle of inductive -procedure. No inductive conclusions are more -than probable, and I adopt the opinion that the theory of -probability is an essential part of logical method, so that -the logical value of every inductive result must be determined -consciously or unconsciously, according to the -principles of the inverse method of probability.</p> - -<p>The phenomena of nature are commonly manifested -in quantities of time, space, force, energy, &c., and the -observation, measurement, and analysis of the various -quantitative conditions or results involved, even in a -simple experiment, demand much employment of systematic -procedure. I devote a book, therefore, to a simple -and general description of the devices by which exact -measurement is effected, errors eliminated, a probable -mean result attained, and the probable error of that mean -ascertained. I then proceed to the principal, and probably -the most interesting, subject of the book, illustrating -successively the conditions and precautions requisite for<span class="pagenum" id="Page_x">x</span> -accurate observation, for successful experiment, and for -the sure detection of the quantitative laws of nature. -As it is impossible to comprehend aright the value of -quantitative laws without constantly bearing in mind the -degree of quantitative approximation to the truth probably -attained, I have devoted a special chapter to the Theory -of Approximation, and however imperfectly I may have -treated this subject, I must look upon it as a very essential -part of a work on Scientific Method.</p> - -<p>It then remains to illustrate the sound use of hypothesis, -to distinguish between the portions of knowledge -which we owe to empirical observation, to accidental discovery, -or to scientific prediction. Interesting questions -arise concerning the accordance of quantitative theories -and experiments, and I point out how the successive -verification of an hypothesis by distinct methods of experiment -yields conclusions approximating to but never -attaining certainty. Additional illustrations of the general -procedure of inductive investigations are given in a -chapter on the Character of the Experimentalist, in which -I endeavour to show, moreover, that the inverse use of -deduction was really the logical method of such great -masters of experimental inquiry as Newton, Huyghens, -and Faraday.</p> - -<p>In treating Generalisation and Analogy, I consider the -precautions requisite in inferring from one case to another, -or from one part of the universe to another part; the -validity of all such inferences resting ultimately upon -the inverse method of probabilities. The treatment of -Exceptional Phenomena appeared to afford an interesting -subject for a further chapter illustrating the various modes -in which an outstanding fact may eventually be explained. -The formal part of the book closes with the subject of -Classification, which is, however, very inadequately treated. -I have, in fact, almost restricted myself to showing that -all classification is fundamentally carried out upon the<span class="pagenum" id="Page_xi">xi</span> -principles of Formal Logic and the Logical Alphabet -described at the outset.</p> - -<p>In certain concluding remarks I have expressed the -conviction which the study of Logic has by degrees -forced upon my mind, that serious misconceptions are -entertained by some scientific men as to the logical value -of our knowledge of nature. We have heard much of -what has been aptly called the Reign of Law, and the -necessity and uniformity of natural forces has been not -uncommonly interpreted as involving the non-existence -of an intelligent and benevolent Power, capable of interfering -with the course of natural events. Fears have -been expressed that the progress of Scientific Method -must therefore result in dissipating the fondest beliefs -of the human heart. Even the ‘Utility of Religion’ is -seriously proposed as a subject of discussion. It seemed -to be not out of place in a work on Scientific Method to -allude to the ultimate results and limits of that method. -I fear that I have very imperfectly succeeded in expressing -my strong conviction that before a rigorous logical scrutiny -the Reign of Law will prove to be an unverified hypothesis, -the Uniformity of Nature an ambiguous expression, -the certainty of our scientific inferences to a great extent -a delusion. The value of science is of course very high, -while the conclusions are kept well within the limits of -the data on which they are founded, but it is pointed out -that our experience is of the most limited character compared -with what there is to learn, while our mental powers -seem to fall infinitely short of the task of comprehending -and explaining fully the nature of any one object. I -draw the conclusion that we must interpret the results -of Scientific Method in an affirmative sense only. Ours -must be a truly positive philosophy, not that false negative -philosophy which, building on a few material facts, -presumes to assert that it has compassed the bounds -of existence, while it nevertheless ignores the most<span class="pagenum" id="Page_xii">xii</span> -unquestionable phenomena of the human mind and feelings.</p> - -<p>It is approximately certain that in freely employing -illustrations drawn from many different sciences, I have -frequently fallen into errors of detail. In this respect I -must throw myself upon the indulgence of the reader, -who will bear in mind, as I hope, that the scientific facts -are generally mentioned purely for the purpose of illustration, -so that inaccuracies of detail will not in the -majority of cases affect the truth of the general principles -illustrated.</p> - -<p class="fs80 mt1em"><i>December 15, 1873.</i></p> - - -<hr class="chap x-ebookmaker-drop"> - -<div class="chapter"> -<p><span class="pagenum" id="Page_xiii">xiii</span></p> - -<h2 class="nobreak" id="PREFACE2">PREFACE<br> - -<span class="title"><i>TO THE SECOND EDITION</i>.</span></h2> -</div> - - -<p class="ti0">Few alterations of importance have been made in preparing -this second edition. Nevertheless, advantage has -been taken of the opportunity to revise very carefully -both the language and the matter of the book. Correspondents -and critics having pointed out inaccuracies -of more or less importance in the first edition, suitable -corrections and emendations have been made. I am under -obligations to Mr. C. J. Monro, M.A., of Barnet, and to -Mr. W. H. Brewer, M.A., one of Her Majesty’s Inspectors -of Schools, for numerous corrections.</p> - -<p>Among several additions which have been made to the -text, I may mention the abstract (p. <a href="#Page_143">143</a>) of Professor -Clifford’s remarkable investigation into the number of -types of compound statement involving four classes of -objects. This inquiry carries forward the inverse logical -problem described in the preceding sections. Again, the -need of some better logical method than the old Barbara -Celarent, &c., is strikingly shown by Mr. Venn’s logical -problem, described at p. <a href="#Page_90">90</a>. A great number of candidates -in logic and philosophy were tested by Mr. Venn with this -problem, which, though simple in reality, was solved by -very few of those who were ignorant of Boole’s Logic. -Other evidence could be adduced by Mr. Venn of the need -for some better means of logical training. To enable the<span class="pagenum" id="Page_xiv">xiv</span> -logical student to test his skill in the solution of inductive -logical problems, I have given (p. <a href="#Page_127">127</a>) a series of ten -problems graduated in difficulty.</p> - -<p>To prevent misapprehension, it should be mentioned -that, throughout this edition, I have substituted the name -<i>Logical Alphabet</i> for <i>Logical Abecedarium</i>, the name applied -in the first edition to the exhaustive series of logical -combinations represented in terms of <i>A</i>, <i>B</i>, <i>C</i>, <i>D</i> (p. <a href="#Page_94">94</a>). -It was objected by some readers that <i>Abecedarium</i> is a -long and unfamiliar name.</p> - -<p>To the chapter on Units and Standards of Measurement, -I have added two sections, one (p. <a href="#Page_325">325</a>) containing -a brief statement of the Theory of Dimensions, and the -other (p. <a href="#Page_319">319</a>) discussing Professor Clerk Maxwell’s very -original suggestion of a Natural System of Standards for -the measurement of space and time, depending upon the -length and rapidity of waves of light.</p> - -<p>In my description of the Logical Machine in the -<i>Philosophical Transactions</i> (vol. 160, p. 498), I said—“It -is rarely indeed that any invention is made without -some anticipation being sooner or later discovered; but up -to the present time I am totally unaware of even a single -previous attempt to devise or construct a machine which -should perform the operations of logical inference; and it -is only, I believe, in the satirical writings of Swift that an -allusion to an actual reasoning machine is to be found.” -Before the paper was printed, however, I was able to refer -(p. 518) to the ingenious designs of the late Mr. Alfred -Smee as attempts to represent thought mechanically. -Mr. Smee’s machines indeed were never constructed, and, -if constructed, would not have performed actual logical -inference. It has now just come to light, however, that -the celebrated Lord Stanhope actually did construct a -mechanical device, capable of representing syllogistic -inferences in a concrete form. It appears that logic was -one of the favourite studies of this truly original and -ingenious nobleman. There remain fragments of a logical<span class="pagenum" id="Page_xv">xv</span> -work, printed by the Earl at his own press, which show -that he had arrived, before the year 1800, at the principle -of the quantified predicate. He puts forward this principle -in the most explicit manner, and proposes to employ -it throughout his syllogistic system. Moreover, he converts -negative propositions into affirmative ones, and -represents these by means of the copula “is identic with.” -Thus he anticipated, probably by the force of his own -unaided insight, the main points of the logical method -originated in the works of George Bentham and George -Boole, and developed in this work. Stanhope, indeed, has -no claim to priority of discovery, because he seems never -to have published his logical writings, although they were -put into print. There is no trace of them in the British -Museum Library, nor in any other library or logical work, -so far as I am aware. Both the papers and the logical -contrivance have been placed by the present Earl Stanhope -in the hands of the Rev. Robert Harley, F.R.S., who will, -I hope, soon publish a description of them.<a id="FNanchor_1" href="#Footnote_1" class="fnanchor">1</a></p> - -<p>By the kindness of Mr. Harley, I have been able to -examine Stanhope’s logical contrivance, called by him the -Demonstrator. It consists of a square piece of bay-wood -with a square depression in the centre, across which two -slides can be pushed, one being a piece of red glass, and -the other consisting of wood coloured gray. The extent -to which each of these slides is pushed in is indicated by -scales and figures along the edges of the aperture, and the -simple rule of inference adopted by Stanhope is: “To the -gray add the red and subtract the <i>holon</i>,” meaning by -holon (ὅλον) the whole width of the aperture. This rule -of inference is a curious anticipation of De Morgan’s -numerically definite syllogism (see below, p. <a href="#Page_168">168</a>), and of -inferences founded on what Hamilton called “Ultra-total -distribution.” Another curious point about Stanhope’s<span class="pagenum" id="Page_xvi">xvi</span> -device is, that one slide can be drawn out and pushed in -again at right angles to the other, and the overlapping -part of the slides then represents the probability of a -conclusion, derived from two premises of which the probabilities -are respectively represented by the projecting -parts of the slides. Thus it appears that Stanhope had -studied the logic of probability as well as that of certainty, -here again anticipating, however obscurely, the recent -progress of logical science. It will be seen, however, that -between Stanhope’s Demonstrator and my Logical Machine -there is no resemblance beyond the fact that they both -perform logical inference.</p> - -<p>In the first edition I inserted a section (vol. i. p. 25), on -“Anticipations of the Principle of Substitution,” and I -have reprinted that section unchanged in this edition -(p. <a href="#Page_21">21</a>). I remark therein that, “In such a subject as logic -it is hardly possible to put forth any opinions which have -not been in some degree previously entertained. The -germ at least of every doctrine will be found in earlier -writings, and novelty must arise chiefly in the mode of -harmonising and developing ideas.” I point out, as -Professor T. M. Lindsay had previously done, that Beneke -had employed the name and principle of substitution, and -that doctrines closely approximating to substitution were -stated by the Port Royal Logicians more than 200 years -ago.</p> - -<p>I have not been at all surprised to learn, however, that -other logicians have more or less distinctly stated this -principle of substitution during the last two centuries. -As my friend and successor at Owens College, Professor -Adamson, has discovered, this principle can be traced back -to no less a philosopher than Leibnitz.</p> - -<p>The remarkable tract of Leibnitz,<a id="FNanchor_2" href="#Footnote_2" class="fnanchor">2</a> entitled “Non inelegans -Specimen Demonstrandi in Abstractis,” commences at once -with a definition corresponding to the principle:—</p> - -<p><span class="pagenum" id="Page_xvii">xvii</span></p> - -<p>“Eadem sunt quorum unum potest substitui alteri salva -veritate. Si sint <i>A</i> et <i>B</i>, et <i>A</i> ingrediatur aliquam propositionem -veram, et ibi in aliquo loco ipsius <i>A</i> pro ipso -substituendo <i>B</i> fiat nova propositio æque itidem vera, idque -semper succedat in quacunque tali propositione, <i>A</i> et <i>B</i> -dicuntur esse eadem; et contra, si eadem sint <i>A</i> et <i>B</i>, -procedet substitutio quam dixi.”</p> - -<p>Leibnitz, then, explicitly adopts the principle of substitution, -but he puts it in the form of a definition, saying -that those things are the same which can be substituted -one for the other, without affecting the truth of the -proposition. It is only after having thus tested the sameness -of things that we can turn round and say that <i>A</i> and -<i>B</i>, being the same, may be substituted one for the other. -It would seem as if we were here in a vicious circle; for -we are not allowed to substitute <i>A</i> for <i>B</i>, unless we have -ascertained by trial that the result is a true proposition. -The difficulty does not seem to be removed by Leibnitz’ -proviso, “idque semper succedat in quacunque tali propositione.” -How can we learn that because <i>A</i> and <i>B</i> may -be mutually substituted in some propositions, they may -therefore be substituted in others; and what is the criterion -of likeness of propositions expressed in the word “tali”? -Whether the principle of substitution is to be regarded as a -postulate, an axiom, or a definition, is just one of those fundamental -questions which it seems impossible to settle in the -present position of philosophy, but this uncertainty will not -prevent our making a considerable step in logical science.</p> - -<p>Leibnitz proceeds to establish in the form of a theorem -what is usually taken as an axiom, thus (<i>Opera</i>, p. 95): -“Theorema I. Quæ sunt eadem uni tertio, eadem sunt -inter se. Si <i>A</i> ∝ <i>B</i> et <i>B</i> ∝ <i>C</i>, erit <i>A</i> ∝ <i>C</i>. Nam si in -propositione <i>A</i> ∝ <i>B</i> (vera ea hypothesi) substituitur <i>C</i> in -locum <i>B</i> (quod facere licet per Def. I. quia <i>B</i> ∝ <i>C</i> ex -hypothesi) fiet <i>A</i> ∝ <i>C</i>. Q. E. Dem.” Thus Leibnitz -precisely anticipates the mode of treating inference with -two simple identities described at p. 51 of this work.</p> - -<p><span class="pagenum" id="Page_xviii">xviii</span></p> - -<p>Even the mathematical axiom that ‘equals added to -equals make equals,’ is deduced from the principle of -substitution. At p. 95 of Erdmann’s edition, we find: “Si -eidem addantur coincidentia fiunt coincidentia. Si <i>A</i> ∝ <i>B</i>, -erit <i>A</i> + <i>C</i> ∝ <i>B</i> + <i>C</i>. Nam si in propositione <i>A</i> + <i>C</i> ∝ <i>A</i> -+ <i>C</i> (quæ est vera per se) pro <i>A</i> semel substituas <i>B</i> (quod -facere licet per Def. I. quia <i>A</i> ∝ <i>B</i>) fiet <i>A</i> + <i>C</i> ∝ <i>B</i> + -<i>C</i> Q. E. Dem.” This is unquestionably the mode of deducing -the several axioms of mathematical reasoning from the -higher axiom of substitution, which is explained in the -section on mathematical inference (p. <a href="#Page_162">162</a>) in this work, -and which had been previously stated in my <i>Substitution -of Similars</i>, p. 16.</p> - -<p>There are one or two other brief tracts in which Leibnitz -anticipates the modern views of logic. Thus in the -eighteenth tract in Erdmann’s edition (p. 92), called -“Fundamenta Calculi Ratiocinatoris”, he says: “Inter ea -quorum unum alteri substitui potest, salvis calculi legibus, -dicetur esse æquipollentiam.” There is evidence, also, that -he had arrived at the quantification of the predicate, and -that he fully understood the reduction of the universal -affirmative proposition to the form of an equation, which is -the key to an improved view of logic. Thus, in the tract -entitled “Difficultates Quædam Logicæ,”<a id="FNanchor_3" href="#Footnote_3" class="fnanchor">3</a> he says: “Omne <i>A</i> -est <i>B</i>; id est æquivalent <i>AB</i> et <i>A</i>, seu <i>A</i> non <i>B</i> est non-ens.”</p> - -<p>It is curious to find, too, that Leibnitz was fully acquainted -with the Laws of Commutativeness and “Simplicity” -(as I have called the second law) attaching to logical -symbols. In the “Addenda ad Specimen Calculi Universalis” -we read as follows.<a id="FNanchor_4" href="#Footnote_4" class="fnanchor">4</a> “Transpositio literarum in -eodem termino nihil mutat, ut <i>ab</i> coincidet cum <i>ba</i>, seu -animal rationale et rationale animal.”</p> - -<p>“Repetitio ejusdem literæ in eodem termino est inutilis, -ut <i>b</i> est <i>aa</i>; vel <i>bb</i> est <i>a</i>; homo est animal animal, vel -homo homo est animal. Sufficit enim dici <i>a</i> est <i>b</i>, seu -homo est animal.”</p> - -<p><span class="pagenum" id="Page_xix">xix</span></p> - -<p>Comparing this with what is stated in Boole’s <i>Mathematical -Analysis of Logic</i>, pp. 17–18, in his <i>Laws of -Thought</i>, p. 29, or in this work, pp. <a href="#Page_32">32</a>–35, we find that -Leibnitz had arrived two centuries ago at a clear perception -of the bases of logical notation. When Boole pointed out -that, in logic, <i>xx</i> = <i>x</i>, this seemed to mathematicians to be -a paradox, or in any case a wholly new discovery; but -here we have it plainly stated by Leibnitz.</p> - -<p>The reader must not assume, however, that because -Leibnitz correctly apprehended the fundamental principles -of logic, he left nothing for modern logicians to do. On -the contrary, Leibnitz obtained no useful results from his -definition of substitution. When he proceeds to explain -the syllogism, as in the paper on “Definitiones Logicæ,”<a id="FNanchor_5" href="#Footnote_5" class="fnanchor">5</a> he gives up substitution altogether, and falls back upon -the notion of inclusion of class in class, saying, “Includens -includentis est includens inclusi, seu si <i>A</i> includit <i>B</i> -et <i>B</i> includit <i>C</i>, etiam <i>A</i> includet <i>C</i>.” He proceeds to -make out certain rules of the syllogism involving the -distinction of subject and predicate, and in no important -respect better than the old rules of the syllogism. -Leibnitz’ logical tracts are, in fact, little more than brief -memoranda of investigations which seem never to have -been followed out. They remain as evidence of his -wonderful sagacity, but it would be difficult to show that -they have had any influence on the progress of logical -science in recent times.</p> - -<p>I should like to explain how it happened that these -logical writings of Leibnitz were unknown to me, until -within the last twelve months. I am so slow a reader -of Latin books, indeed, that my overlooking a few pages -of Leibnitz’ works would not have been in any case -surprising. But the fact is that the copy of Leibnitz’ -works of which I made occasional use, was one of the -edition of Dutens, contained in Owens College Library. -The logical tracts in question were not printed in that<span class="pagenum" id="Page_xx">xx</span> -edition, and with one exception, they remained in manuscript -in the Royal Library at Hanover, until edited by -Erdmann, in 1839–40. The tract “Difficultates Quædam -Logicæ,” though not known to Dutens, was published by -Raspe in 1765, in his collection called <i>Œuvres Philosophiques -de feu M<sup>r.</sup> Leibnitz</i>; but this work had not -come to my notice, nor does the tract in question seem -to contain any explicit statement of the principle of -substitution.</p> - -<p>It is, I presume, the comparatively recent publication of -Leibnitz’ most remarkable logical tracts which explains -the apparent ignorance of logicians as regards their contents -and importance. The most learned logicians, such -as Hamilton and Ueberweg, ignore Leibnitz’ principle of -substitution. In the Appendix to the fourth volume of -Hamilton’s <i>Lectures on Metaphysics and Logic</i>, is given -an elaborate compendium of the views of logical writers -concerning the ultimate basis of deductive reasoning. -Leibnitz is briefly noticed on p. 319, but without any -hint of substitution. He is here quoted as saying, “What -are the same with the same third, are the same with each -other; that is, if <i>A</i> be the same with <i>B</i>, and <i>C</i> be the -same with <i>B</i>, it is necessary that <i>A</i> and <i>C</i> should also -be the same with one another. For this principle flows -immediately from the principle of contradiction, and is -the ground and basis of all logic; if that fail, there is no -longer any way of reasoning with certainty.” This view -of the matter seems to be inconsistent with that which he -adopted in his posthumous tract.</p> - -<p>Dr. Thomson, indeed, was acquainted with Leibnitz’ -tracts, and refers to them in his <i>Outline of the Necessary -Laws of Thought</i>. He calls them valuable; nevertheless, -he seems to have missed the really valuable point; for in -making two brief quotations,<a id="FNanchor_6" href="#Footnote_6" class="fnanchor">6</a> he omits all mention of the -principle of substitution.</p> - -<p>Ueberweg is probably considered the best authority<span class="pagenum" id="Page_xxi">xxi</span> -concerning the history of logic, and in his well-known -<i>System of Logic and History of Logical Doctrines</i>,<a id="FNanchor_7" href="#Footnote_7" class="fnanchor">7</a> he gives -some account of the principle of substitution, especially -as it is implicitly stated in the <i>Port Royal Logic</i>. But he -omits all reference to Leibnitz in this connection, nor does -he elsewhere, so far as I can find, supply the omission. -His English editor, Professor T. M. Lindsay, in referring to -my <i>Substitution of Similars</i>, points out how I was anticipated -by Beneke; but he also ignores Leibnitz. It is thus -apparent that the most learned logicians, even when writing -especially on the history of logic, displayed ignorance of -Leibnitz’ most valuable logical writings.</p> - -<p>It has been recently pointed out to me, however, that -the Rev. Robert Harley did draw attention, at the Nottingham -Meeting of the British Association, in 1866, to -Leibnitz’ anticipations of Boole’s laws of logical notation,<a id="FNanchor_8" href="#Footnote_8" class="fnanchor">8</a> -and I am informed that Boole, about a year after the publication -of his <i>Laws of Thought</i>, was made acquainted with -these anticipations by R. Leslie Ellis.</p> - -<p>There seems to have been at least one other German -logician who discovered, or adopted, the principle of substitution. -Reusch, in his <i>Systema Logicum</i>, published in -1734, laboured to give a broader basis to the <i>Dictum de -Omni et Nullo</i>. He argues, that “the whole business of -ordinary reasoning is accomplished by the substitution of -ideas in place of the subject or predicate of the fundamental -proposition. This some call the <i>equation of thoughts</i>.” -But, in the hands of Reusch, substitution does not seem to -lead to simplicity, since it has to be carried on according -to the rules of Equipollence, Reciprocation, Subordination, -and Co-ordination.<a id="FNanchor_9" href="#Footnote_9" class="fnanchor">9</a> Reusch is elsewhere spoken of<a id="FNanchor_10" href="#Footnote_10" class="fnanchor">10</a> as the -“celebrated Reusch”; nevertheless, I have not been able to<span class="pagenum" id="Page_xxii">xxii</span> -find a copy of his book in London, even in the British -Museum Library; it is not mentioned in the printed -catalogue of the Bodleian Library; Messrs. Asher have -failed to obtain it for me by advertisement in Germany; -and Professor Adamson has been equally unsuccessful. -From the way in which the principle of substitution is -mentioned by Reusch, it would seem likely that other -logicians of the early part of the eighteenth century were -acquainted with it; but, if so, it is still more curious that -recent historians of logical science have overlooked the -doctrine.</p> - -<p>It is a strange and discouraging fact, that true views of -logic should have been discovered and discussed from one -to two centuries ago, and yet should have remained, like -George Bentham’s work in this century, without influence -on the subsequent progress of the science. It may -be regarded as certain that none of the discoverers of -the quantification of the predicate, Bentham, Hamilton, -Thomson, De Morgan, and Boole, were in any way assisted -by the hints of the principle contained in previous writers. -As to my own views of logic, they were originally moulded -by a careful study of Boole’s works, as fully stated in my -first logical essay.<a id="FNanchor_11" href="#Footnote_11" class="fnanchor">11</a> As to the process of substitution, it -was not learnt from any work on logic, but is simply the -process of substitution perfectly familiar to mathematicians, -and with which I necessarily became familiar in the course -of my long-continued study of mathematics under the late -Professor De Morgan.</p> - -<p>I find that the Theory of Number, which I explained in -the eighth chapter of this work, is also partially anticipated -in a single scholium of Leibnitz. He first gives as an -axiom the now well-known law of Boole, as follows:—</p> - -<p>“Axioma I. Si idem secum ipso sumatur, nihil constituitur -novum, seu <i>A</i> + <i>A</i> ∝ <i>A</i>.” Then follows this<span class="pagenum" id="Page_xxiii">xxiii</span> -remarkable scholium: “Equidem in numeris 4 + 4 facit -8, seu bini nummi binis additi faciunt quatuor nummos, -sed tunc bini additi sunt alii a prioribus; si iidem essent -nihil novi prodiret et perinde esset ac si joco ex tribus -ovis facere vellemus sex numerando, primum 3 ova, deinde -uno sublato residua 2, ac denique uno rursus sublato -residuum.”</p> - -<p>Translated this would read as follows:—</p> - -<p>“Axiom I. If the same thing is taken together with -itself, nothing new arises, or <i>A</i> + <i>A</i> = <i>A</i>.</p> - -<p>“Scholium. In numbers, indeed, 4 + 4 makes 8, or two -coins added to two coins make four coins, but then the -two added are different from the former ones; if they were -the same nothing new would be produced, and it would -be just as if we tried in joke to make six eggs out of three, -by counting firstly the three eggs, then, one being removed, -counting the remaining two, and lastly, one being again -removed, counting the remaining egg.”</p> - -<p>Compare the above with pp. <a href="#Page_156">156</a> to 162 of the present -work.</p> - -<p>M. Littré has quite recently pointed out<a id="FNanchor_12" href="#Footnote_12" class="fnanchor">12</a> what he thinks -is an analogy between the system of formal logic, stated -in the following pages, and the logical devices of the -celebrated Raymond Lully. Lully’s method of invention -was described in a great number of mediæval books, but -is best stated in his <i>Ars Compendiosa Inveniendi Veritatem, -seu Ars Magna et Major</i>. This method consisted in placing -various names of things in the sectors of concentric -circles, so that when the circles were turned, every possible -combination of the things was easily produced by mechanical -means. It might, perhaps, be possible to discover in -this method a vague and rude anticipation of combinational -logic; but it is well known that the results of Lully’s -method were usually of a fanciful, if not absurd character.</p> - -<p>A much closer analogue of the Logical Alphabet is -probably to be found in the Logical Square, invented by<span class="pagenum" id="Page_xxiv">xxiv</span> -John Christian Lange, and described in a rare and unnoticed -work by him which I have recently found in the -British Museum.<a id="FNanchor_13" href="#Footnote_13" class="fnanchor">13</a> This square involved the principle of -bifurcate classification, and was an improved form of the -Ramean and Porphyrian tree (see below, p. <a href="#Page_702">702</a>). Lange -seems, indeed, to have worked out his Logical Square -into a mechanical form, and he suggests that it might be -employed somewhat in the manner of Napier’s Bones -(p. 65). There is much analogy between his Square and -my Abacus, but Lange had not arrived at a logical system -enabling him to use his invention for logical inference in -the manner of the Logical Abacus. Another work of -Lange is said to contain the first publication of the well -known Eulerian diagrams of proposition and syllogism.<a id="FNanchor_14" href="#Footnote_14" class="fnanchor">14</a></p> - -<p>Since the first edition was published, an important -work by Mr. George Lewes has appeared, namely, his -<i>Problems of Life and Mind</i>, which to a great extent treats -of scientific method, and formulates the rules of philosophising. -I should have liked to discuss the bearing -of Mr. Lewes’s views upon those here propounded, but -I have felt it to be impossible in a book already filling -nearly 800 pages, to enter upon the discussion of a -yet more extensive book. For the same reason I have -not been able to compare my own treatment of the subject -of probability with the views expressed by Mr. Venn in -his <i>Logic of Chance</i>. With Mr. J. J. Murphy’s profound -and remarkable works on <i>Habit and Intelligence</i>, and on -<i>The Scientific Basis of Faith</i>, I was unfortunately unacquainted -when I wrote the following pages. They cannot -safely be overlooked by any one who wishes to -comprehend the tendency of philosophy and scientific -method in the present day.</p> - -<p>It seems desirable that I should endeavour to answer -some of the critics who have pointed out what they<span class="pagenum" id="Page_xxv">xxv</span> -consider defects in the doctrines of this book, especially in -the first part, which treats of deduction. Some of the -notices of the work were indeed rather statements of its -contents than critiques. Thus, I am much indebted to -M. Louis Liard, Professor of Philosophy at Bordeaux, for -the very careful exposition<a id="FNanchor_15" href="#Footnote_15" class="fnanchor">15</a> of the substitutional view of -logic which he gave in the excellent <i>Revue Philosophique</i>, -edited by M. Ribot. (Mars, 1877, tom. iii. p. 277.) An -equally careful account of the system was given by -M. Riehl, Professor of Philosophy at Graz, in his article on -“Die Englische Logik der Gegenwart,” published in the -<i>Vierteljahrsschrift für wissenschaftliche Philosophie</i>. (1 Heft, -Leipzig, 1876.) I should like to acknowledge also the -careful and able manner in which my book was reviewed -by the <i>New York Daily Tribune</i> and the <i>New York Times</i>.</p> - -<p>The most serious objections which have been brought -against my treatment of logic have regard to my failure -to enter into an analysis of the ultimate nature and origin -of the Laws of Thought. The <i>Spectator</i>,<a id="FNanchor_16" href="#Footnote_16" class="fnanchor">16</a> for instance, in -the course of a careful review, says of the principle of -substitution, “Surely it is a great omission not to discuss -whence we get this great principle itself; whether it is a -pure law of the mind, or only an approximate lesson of -experience; and if a pure product of the mind, whether -there are any other products of the same kind, furnished -by our knowing faculty itself.” Professor Robertson, in -his very acute review,<a id="FNanchor_17" href="#Footnote_17" class="fnanchor">17</a> likewise objects to the want of<span class="pagenum" id="Page_xxvi">xxvi</span> -psychological and philosophical analysis. “If the book -really corresponded to its title, Mr. Jevons could hardly -have passed so lightly over the question, which he does -not omit to raise, concerning those undoubted principles -of knowledge commonly called the Laws of Thought.... -Everywhere, indeed, he appears least at ease when he -touches on questions properly philosophical; nor is he -satisfactory in his psychological references, as on pp. 4, 5, -where he cannot commit himself to a statement without -an accompaniment of ‘probably,’ ‘almost,’ or ‘hardly.’ -Reservations are often very much in place, but there are -fundamental questions on which it is proper to make up -one’s mind.”</p> - -<p>These remarks appear to me to be well founded, and I -must state why it is that I have ventured to publish an -extensive work on logic, without properly making up my -mind as to the fundamental nature of the reasoning -process. The fault after all is one of omission rather than -of commission. It is open to me on a future occasion to -supply the deficiency if I should ever feel able to undertake -the task. But I do not conceive it to be an essential -part of any treatise to enter into an ultimate analysis of -its subject matter. Analyses must always end somewhere. -There were good treatises on light which described the -laws of the phenomenon correctly before it was known -whether light consisted of undulations or of projected -particles. Now we have treatises on the Undulatory -Theory which are very valuable and satisfactory, although -they leave us in almost complete doubt as to what the -vibrating medium really is. So I think that, in the -present day, we need a correct and scientific exhibition -of the formal laws of thought, and of the forms of -reasoning based on them, although we may not be able -to enter into any complete analysis of the nature of those -laws. What would the science of geometry be like now -if the Greek geometers had decided that it was improper -to publish any propositions before they had decided on<span class="pagenum" id="Page_xxvii">xxvii</span> -the nature of an axiom? Where would the science of -arithmetic be now if an analysis of the nature of number -itself were a necessary preliminary to a development of -the results of its laws? In recent times there have been -enormous additions to the mathematical sciences, but very -few attempts at psychological analysis. In the Alexandrian -and early mediæval schools of philosophy, much -attention was given to the nature of unity and plurality -chiefly called forth by the question of the Trinity. In -the last two centuries whole sciences have been created -out of the notion of plurality, and yet speculation on the -nature of plurality has dwindled away. This present -treatise contains, in the eighth chapter, one of the few -recent attempts to analyse the notion of number itself.</p> - -<p>If further illustration is needed, I may refer to the -differential calculus. Nobody calls in question the formal -truth of the results of that calculus. All the more exact -and successful parts of physical science depend upon its -use, and yet the mathematicians who have created so -great a body of exact truths have never decided upon -the basis of the calculus. What is the nature of a limit -or the nature of an infinitesimal? Start the question -among a knot of mathematicians, and it will be found -that hardly two agree, unless it is in regarding the question -itself as a trifling one. Some hold that there are no such -things as infinitesimals, and that it is all a question of -limits. Others would argue that the infinitesimal is the -necessary outcome of the limit, but various shades of -intermediate opinion spring up.</p> - -<p>Now it is just the same with logic. If the forms of -deductive and inductive reasoning given in the earlier -part of this treatise are correct, they constitute a definite -addition to logical science, and it would have been absurd -to decline to publish such results because I could not at -the same time decide in my own mind about the psychology -and philosophy of the subject. It comes in short -to this, that my book is a book on Formal Logic and<span class="pagenum" id="Page_xxviii">xxviii</span> -Scientific Method, and not a book on psychology and -philosophy.</p> - -<p>It may be objected, indeed, as the <i>Spectator</i> objects, -that Mill’s System of Logic is particularly strong in the -discussion of the psychological foundations of reasoning, -so that Mill would appear to have successfully treated -that which I feel myself to be incapable of attempting at -present. If Mill’s analysis of knowledge is correct, then -I have nothing to say in excuse for my own deficiencies. -But it is well to do one thing at a time, and therefore -I have not occupied any considerable part of this book -with controversy and refutation. What I have to say of -Mill’s logic will be said in a separate work, in which -his analysis of knowledge will be somewhat minutely -analysed. It will then be shown, I believe, that Mill’s -psychological and philosophical treatment of logic has not -yielded such satisfactory results as some writers seem to -believe.<a id="FNanchor_18" href="#Footnote_18" class="fnanchor">18</a></p> - -<p>Various minor but still important criticisms were made -by Professor Robertson, a few of which have been noticed -in the text (pp. <a href="#Page_27">27</a>, <a href="#Page_101">101</a>). In other cases his objections -hardly admit of any other answer than such as consists -in asking the reader to judge between the work and the -criticism. Thus Mr. Robertson asserts<a id="FNanchor_19" href="#Footnote_19" class="fnanchor">19</a> that the most -complex logical problems solved in this book (up to p. 102 -of this edition) might be more easily and shortly dealt -with upon the principles and with the recognised methods -of the traditional logic. The burden of proof here lies -upon Mr. Robertson, and his only proof consists in a -single case, where he is able, as it seems to me accidentally, -to get a special conclusion by the old form of dilemma. -It would be a long labour to test the old logic upon every -result obtained by my notation, and I must leave such<span class="pagenum" id="Page_xxix">xxix</span> -readers as are well acquainted with the syllogistic logic to -pronounce upon the comparative simplicity and power of -the new and old systems. For other acute objections -brought forward by Mr. Robertson, I must refer the reader -to the article in question.</p> - -<p>One point in my last chapter, that on the Results and -Limits of Scientific Method, has been criticised by -Professor W. K. Clifford in his lecture<a id="FNanchor_20" href="#Footnote_20" class="fnanchor">20</a> on “The First -and the Last Catastrophe.” In vol. ii. p. 438 of the -first edition (p. <a href="#Page_744">744</a> of this edition) I referred to certain -inferences drawn by eminent physicists as to a limit to -the antiquity of the present order of things. “According -to Sir W. Thomson’s deductions from Fourier’s <i>theory of -heat</i>, we can trace down the dissipation of heat by conduction -and radiation to an infinitely distant time when -all things will be uniformly cold. But we cannot similarly -trace the Heat-history of the Universe to an infinite -distance in the past. For a certain negative value of the -time, the formulæ give impossible values, indicating that -there was some initial distribution of heat which could -not have resulted, according to known laws of nature, -from any previous distribution.”</p> - -<p>Now according to Professor Clifford I have here misstated -Thomson’s results. “It is not according to the -known laws of nature, it is according to the known laws -of conduction of heat, that Sir William Thomson is speaking. . . . -All these physical writers, knowing what they -were writing about, simply drew such conclusions from -the facts which were before them as could be reasonably -drawn. They say, here is a state of things which could -not have been produced by the circumstances we are at -present investigating. Then your speculator comes, he -reads a sentence and says, ‘Here is an opportunity for -me to have my fling.’ And he has his fling, and makes a -purely baseless theory about the necessary origin of the<span class="pagenum" id="Page_"></span> -present order of nature at some definite point of time, -which might be calculated.”</p> - -<p>Professor Clifford proceeds to explain that Thomson’s -formulæ only give a limit to the heat history of, say, the -earth’s crust in the solid state. We are led back to the -time when it became solidified from the fluid condition. -There is discontinuity in the history of the solid matter, -but still discontinuity which is within our comprehension. -Still further back we should come to discontinuity again, -when the liquid was formed by the condensation of heated -gaseous matter. Beyond that event, however, there is -no need to suppose further discontinuity of law, for the -gaseous matter might consist of molecules which had been -falling together from different parts of space through infinite -past time. As Professor Clifford says (p. 481) of the -bodies of the universe, “What they have actually done -is to fall together and get solid. If we should reverse -the process we should see them separating and getting -cool, and as a limit to that, we should find that all these -bodies would be resolved into molecules, and all these -would be flying away from each other. There would be -no limit to that process, and we could trace it as far back -as ever we liked to trace it.”</p> - -<p>Assuming that I have erred, I should like to point out -that I have erred in the best company, or more strictly, -being a speculator, I have been led into error by the best -physical writers. Professor Tait, in his <i>Sketch of Thermodynamics</i>, -speaking of the laws discovered by Fourier -for the motion of heat in a solid, says, “Their mathematical -expressions point also to the fact that a uniform distribution -of heat, or a distribution tending to become uniform, -must have arisen from some primitive distribution of heat -of a kind not capable of being produced by known laws -from any previous distribution.” In the latter words it -will be seen that there is no limitation to the laws of -conduction, and, although I had carefully referred to -Sir W. Thomson’s original paper, it is not unnatural<span class="pagenum" id="Page_xxxi">xxxi</span> -that I should take Professor Tait’s interpretation of its -meaning.<a id="FNanchor_21" href="#Footnote_21" class="fnanchor">21</a></p> - -<p>In his new work <i>On some Recent Advances in Physical -Science</i>, Professor Tait has recurred to the subject as -follows:<a id="FNanchor_22" href="#Footnote_22" class="fnanchor">22</a> “A profound lesson may be learned from one -of the earliest little papers of Sir W. Thomson, published -while he was an undergraduate at Cambridge, where he -shows that Fourier’s magnificent treatment of the conduction -of heat [in a solid body] leads to formulæ for its -distribution which are intelligible (and of course capable -of being fully verified by experiment) for all time future, -but which, except in particular cases, when extended to -time past, remain intelligible for a finite period only, and -<i>then</i> indicate a state of things which could not have -resulted under known laws from any conceivable previous -distribution [of heat in the body]. So far as heat is -concerned, modern investigations have shown that a -previous distribution of the <i>matter</i> involved may, by its -potential energy, be capable of producing such a state of -things at the moment of its aggregation; but the example -is now adduced not for its bearing on heat alone, but as -a simple illustration of the fact that all portions of our -Science, especially that beautiful one, the Dissipation -of Energy, point unanimously to a beginning, to a state of -things incapable of being derived by present laws [of -tangible matter and its energy] from any conceivable -previous arrangement.” As this was published nearly a -year after Professor Clifford’s lecture, it may be inferred<span class="pagenum" id="Page_xxxii">xxxii</span> -that Professor Tait adheres to his original opinion that -the theory of heat does give evidence of “a beginning.”</p> - -<p>I may add that Professor Clerk Maxwell’s words seem -to countenance the same view, for he says,<a id="FNanchor_23" href="#Footnote_23" class="fnanchor">23</a> “This is only -one of the cases in which a consideration of the dissipation -of energy leads to the determination of a superior -limit to the antiquity of the observed order of things.” -The expression “observed order of things” is open to -much ambiguity, but in the absence of qualification I -should take it to include the aggregate of the laws of -nature known to us. I should interpret Professor Maxwell -as meaning that the theory of heat indicates the occurrence -of some event of which our science cannot give any -further explanation. The physical writers thus seem not to -be so clear about the matter as Professor Clifford assumes.</p> - -<p>So far as I may venture to form an independent -opinion on the subject, it is to the effect that Professor -Clifford is right, and that the known laws of nature do -not enable us to assign a “beginning.” Science leads us -backwards into infinite past duration. But that Professor -Clifford is right on this point, is no reason why we should -suppose him to be right in his other opinions, some of -which I am sure are wrong. Nor is it a reason why other -parts of my last chapter should be wrong. The question -only affects the single paragraph on pp. <a href="#Page_744">744</a>–5 of this -book, which might, I believe, be struck out without -necessitating any alteration in the rest of the text. It -is always to be remembered that the failure of an argument -in favour of a proposition does not, generally -speaking, add much, if any, probability to the contradictory -proposition. I cannot conclude without expressing -my acknowledgments to Professor Clifford for his kind -expressions regarding my work as a whole.</p> - -<p class="sig fs90"> -<span class="smcap">2, The Chestnuts,<br> - West Heath,<br> -    Hampstead, N. W.</span> -</p> - -<p class="ml1em fs90"> -<i>August 15, 1877.</i><br> -</p> - -<hr class="chap x-ebookmaker-drop"> - -<div class="chapter"> -<p><span class="pagenum" id="Page_xxxiii">xxxiii</span></p> - -<h2 class="nobreak" id="CONTENTS">CONTENTS.</h2> -</div> - -<hr class="r15 x-ebookmaker-drop"> - -<div class="center"> -<table id="toc"> -<tr> -<td class="toc1" colspan="3">BOOK I.</td> -</tr> -<tr> -<td class="toc2" colspan="3">FORMAL LOGIC, DEDUCTIVE AND INDUCTIVE.</td> -</tr> -<tr> -<td class="toc3" colspan="3">CHAPTER I.</td> -</tr> -<tr> -<td class="toc4" colspan="3">INTRODUCTION.</td> -</tr> -<tr class="toc5"> -<td class="tal" colspan="2">SECTION</td> -<td class="tar"><div>PAGE</div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>1.</div></td> -<td class="tal pl1">Introduction</td> -<td class="tar"><div><a href="#Page_1">1</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>2.</div></td> -<td class="tal pl1">The Powers of Mind concerned in the Creation of Science</td> -<td class="tar"><div><a href="#Page_4">4</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>3.</div></td> -<td class="tal pl1">Laws of Identity and Difference</td> -<td class="tar"><div><a href="#Page_5">5</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>4.</div></td> -<td class="tal pl1">The Nature of the Laws of Identity and Difference</td> -<td class="tar"><div><a href="#Page_6">6</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>5.</div></td> -<td class="tal pl1">The Process of Inference</td> -<td class="tar"><div><a href="#Page_9">9</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>6.</div></td> -<td class="tal pl1">Deduction and Induction</td> -<td class="tar"><div><a href="#Page_11">11</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>7.</div></td> -<td class="tal pl1">Symbolic Expression of Logical Inference</td> -<td class="tar"><div><a href="#Page_13">13</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>8.</div></td> -<td class="tal pl1">Expression of Identity and Difference</td> -<td class="tar"><div><a href="#Page_14">14</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>9.</div></td> -<td class="tal pl1">General Formula of Logical Inference</td> -<td class="tar"><div><a href="#Page_17">17</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>10.</div></td> -<td class="tal pl1">The Propagating Power of Similarity</td> -<td class="tar"><div><a href="#Page_20">20</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>11.</div></td> -<td class="tal pl1">Anticipations of the Principle of Substitution</td> -<td class="tar"><div><a href="#Page_21">21</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>12.</div></td> -<td class="tal pl1">The Logic of Relatives</td> -<td class="tar"><div><a href="#Page_22">22</a></div></td> -</tr> -<tr> -<td class="toc3" colspan="3">CHAPTER II.</td> -</tr> -<tr> -<td class="toc4" colspan="3">TERMS.</td> -</tr> -<tr class="toc5"> -<td class="tar"><div>1.</div></td> -<td class="tal pl1">Terms</td> -<td class="tar"><div><a href="#Page_24">24</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>2.</div></td> -<td class="tal pl1">Twofold meaning of General Names</td> -<td class="tar"><div><a href="#Page_25">25</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>3.</div></td> -<td class="tal pl1">Abstract Terms</td> -<td class="tar"><div><a href="#Page_27">27</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>4.</div></td> -<td class="tal pl1">Substantial Terms</td> -<td class="tar"><div><a href="#Page_28">28</a><span class="pagenum" id="Page_xxxiv">xxxiv</span></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>5.</div></td> -<td class="tal pl1">Collective Terms</td> -<td class="tar"><div><a href="#Page_29">29</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>6.</div></td> -<td class="tal pl1">Synthesis of Terms</td> -<td class="tar"><div><a href="#Page_30">30</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>7.</div></td> -<td class="tal pl1">Symbolic Expression of the Law of Contradiction</td> -<td class="tar"><div><a href="#Page_31">31</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>8.</div></td> -<td class="tal pl1">Certain Special Conditions of Logical Symbols</td> -<td class="tar"><div><a href="#Page_32">32</a></div></td> -</tr> -<tr> -<td class="toc3" colspan="3">CHAPTER III.</td> -</tr> -<tr> -<td class="toc4" colspan="3">PROPOSITIONS.</td> -</tr> -<tr class="toc5"> -<td class="tar"><div>1.</div></td> -<td class="tal pl1">Propositions</td> -<td class="tar"><div><a href="#Page_36">36</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>2.</div></td> -<td class="tal pl1">Simple Identities</td> -<td class="tar"><div><a href="#Page_37">37</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>3.</div></td> -<td class="tal pl1">Partial Identities</td> -<td class="tar"><div><a href="#Page_40">40</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>4.</div></td> -<td class="tal pl1">Limited Identities</td> -<td class="tar"><div><a href="#Page_42">42</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>5.</div></td> -<td class="tal pl1">Negative Propositions</td> -<td class="tar"><div><a href="#Page_43">43</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>6.</div></td> -<td class="tal pl1">Conversion of Propositions</td> -<td class="tar"><div><a href="#Page_46">46</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>7.</div></td> -<td class="tal pl1">Twofold Interpretation of Propositions</td> -<td class="tar"><div><a href="#Page_47">47</a></div></td> -</tr> -<tr> -<td class="toc3" colspan="3">CHAPTER IV.</td> -</tr> -<tr> -<td class="toc4" colspan="3">DEDUCTIVE REASONING.</td> -</tr> -<tr class="toc5"> -<td class="tar"><div>1.</div></td> -<td class="tal pl1">Deductive Reasoning</td> -<td class="tar"><div><a href="#Page_49">49</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>2.</div></td> -<td class="tal pl1">Immediate Inference</td> -<td class="tar"><div><a href="#Page_50">50</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>3.</div></td> -<td class="tal pl1">Inference with Two Simple Identities</td> -<td class="tar"><div><a href="#Page_51">51</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>4.</div></td> -<td class="tal pl1">Inference with a Simple and a Partial Identity</td> -<td class="tar"><div><a href="#Page_53">53</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>5.</div></td> -<td class="tal pl1">Inference of a Partial from Two Partial Identities</td> -<td class="tar"><div><a href="#Page_55">55</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>6.</div></td> -<td class="tal pl1">On the Ellipsis of Terms in Partial Identities</td> -<td class="tar"><div><a href="#Page_57">57</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>7.</div></td> -<td class="tal pl1">Inference of a Simple from Two Partial Identities</td> -<td class="tar"><div><a href="#Page_58">58</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>8.</div></td> -<td class="tal pl1">Inference of a Limited from Two Partial Identities</td> -<td class="tar"><div><a href="#Page_59">59</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>9.</div></td> -<td class="tal pl1">Miscellaneous Forms of Deductive Inference</td> -<td class="tar"><div><a href="#Page_60">60</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>10.</div></td> -<td class="tal pl1">Fallacies</td> -<td class="tar"><div><a href="#Page_62">62</a></div></td> -</tr> -<tr> -<td class="toc3" colspan="3">CHAPTER V.</td> -</tr> -<tr> -<td class="toc4" colspan="3">DISJUNCTIVE PROPOSITIONS.</td> -</tr> -<tr class="toc5"> -<td class="tar"><div>1.</div></td> -<td class="tal pl1">Disjunctive Propositions</td> -<td class="tar"><div><a href="#Page_66">66</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>2.</div></td> -<td class="tal pl1">Expression of the Alternative Relation</td> -<td class="tar"><div><a href="#Page_67">67</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>3.</div></td> -<td class="tal pl1">Nature of the Alternative Relation</td> -<td class="tar"><div><a href="#Page_68">68</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>4.</div></td> -<td class="tal pl1">Laws of the Disjunctive Relation</td> -<td class="tar"><div><a href="#Page_71">71</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>5.</div></td> -<td class="tal pl1">Symbolic Expression of the Law of Duality</td> -<td class="tar"><div><a href="#Page_73">73</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>6.</div></td> -<td class="tal pl1">Various Forms of the Disjunctive Proposition</td> -<td class="tar"><div><a href="#Page_74">74</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>7.</div></td> -<td class="tal pl1">Inference by Disjunctive Propositions</td> -<td class="tar"><div><a href="#Page_76">76</a><span class="pagenum" id="Page_xxxv">xxxv</span></div></td> -</tr> -<tr> -<td class="toc3" colspan="3">CHAPTER VI.</td> -</tr> -<tr> -<td class="toc4" colspan="3">THE INDIRECT METHOD OF INFERENCE.</td> -</tr> -<tr class="toc5"> -<td class="tar"><div>1.</div></td> -<td class="tal pl1">The Indirect Method of Inference</td> -<td class="tar"><div><a href="#Page_81">81</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>2.</div></td> -<td class="tal pl1">Simple Illustrations</td> -<td class="tar"><div><a href="#Page_83">83</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>3.</div></td> -<td class="tal pl1">Employment of the Contrapositive Proposition</td> -<td class="tar"><div><a href="#Page_84">84</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>4.</div></td> -<td class="tal pl1">Contrapositive of a Simple Identity</td> -<td class="tar"><div><a href="#Page_86">86</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>5.</div></td> -<td class="tal pl1">Miscellaneous Examples of the Method</td> -<td class="tar"><div><a href="#Page_88">88</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>6.</div></td> -<td class="tal pl1">Mr. Venn’s Problem</td> -<td class="tar"><div><a href="#Page_90">90</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>7.</div></td> -<td class="tal pl1">Abbreviation of the Process</td> -<td class="tar"><div><a href="#Page_91">91</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>8.</div></td> -<td class="tal pl1">The Logical Alphabet</td> -<td class="tar"><div><a href="#Page_94">94</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>9.</div></td> -<td class="tal pl1">The Logical Slate</td> -<td class="tar"><div><a href="#Page_95">95</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>10.</div></td> -<td class="tal pl1">Abstraction of Indifferent Circumstances</td> -<td class="tar"><div><a href="#Page_97">97</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>11.</div></td> -<td class="tal pl1">Illustrations of the Indirect Method</td> -<td class="tar"><div><a href="#Page_98">98</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>12.</div></td> -<td class="tal pl1">Second Example</td> -<td class="tar"><div><a href="#Page_99">99</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>13.</div></td> -<td class="tal pl1">Third Example</td> -<td class="tar"><div><a href="#Page_100">100</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>14.</div></td> -<td class="tal pl1">Fourth Example</td> -<td class="tar"><div><a href="#Page_101">101</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>15.</div></td> -<td class="tal pl1">Fifth Example</td> -<td class="tar"><div><a href="#Page_101">101</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>16.</div></td> -<td class="tal pl1">Fallacies Analysed by the Indirect Method</td> -<td class="tar"><div><a href="#Page_102">102</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>17.</div></td> -<td class="tal pl1">The Logical Abacus</td> -<td class="tar"><div><a href="#Page_104">104</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>18.</div></td> -<td class="tal pl1">The Logical Machine</td> -<td class="tar"><div><a href="#Page_107">107</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>19.</div></td> -<td class="tal pl1">The Order of Premises</td> -<td class="tar"><div><a href="#Page_114">114</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>20.</div></td> -<td class="tal pl1">The Equivalence of Propositions</td> -<td class="tar"><div><a href="#Page_115">115</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>21.</div></td> -<td class="tal pl1">The Nature of Inference</td> -<td class="tar"><div><a href="#Page_118">118</a></div></td> -</tr> -<tr> -<td class="toc3" colspan="3">CHAPTER VII.</td> -</tr> -<tr> -<td class="toc4" colspan="3">INDUCTION.</td> -</tr> -<tr class="toc5"> -<td class="tar"><div>1.</div></td> -<td class="tal pl1">Induction</td> -<td class="tar"><div><a href="#Page_121">121</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>2.</div></td> -<td class="tal pl1">Induction an Inverse Operation</td> -<td class="tar"><div><a href="#Page_122">122</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>3.</div></td> -<td class="tal pl1">Inductive Problems for Solution by the Reader</td> -<td class="tar"><div><a href="#Page_126">126</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>4.</div></td> -<td class="tal pl1">Induction of Simple Identities</td> -<td class="tar"><div><a href="#Page_127">127</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>5.</div></td> -<td class="tal pl1">Induction of Partial Identities</td> -<td class="tar"><div><a href="#Page_130">130</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar vat"><div>6.</div></td> -<td class="tal pl1">Solution of the Inverse or Inductive Problem, involving Two Classes</td> -<td class="tar vab"><div><a href="#Page_134">134</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>7.</div></td> -<td class="tal pl1">The Inverse Logical Problem, involving Three Classes</td> -<td class="tar"><div><a href="#Page_137">137</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar vat"><div>8.</div></td> -<td class="tal pl1">Professor Clifford on the Types of Compound Statement involving Four Classes</td> -<td class="tar vab"><div><a href="#Page_143">143</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>9.</div></td> -<td class="tal pl1">Distinction between Perfect and Imperfect Induction</td> -<td class="tar"><div><a href="#Page_146">146</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>10.</div></td> -<td class="tal pl1">Transition from Perfect to Imperfect Induction</td> -<td class="tar"><div><a href="#Page_149">149</a><span class="pagenum" id="Page_xxxvi">xxxvi</span></div></td> -</tr> -<tr> -<td class="toc1" colspan="3">BOOK II.</td> -</tr> -<tr> -<td class="toc2" colspan="3">NUMBER, VARIETY, AND PROBABILITY.</td> -</tr> -<tr> -<td class="toc3" colspan="3">CHAPTER VIII.</td> -</tr> -<tr> -<td class="toc4" colspan="3">PRINCIPLES OF NUMBER.</td> -</tr> -<tr class="toc5"> -<td class="tar"><div>1.</div></td> -<td class="tal pl1">Principles of Number</td> -<td class="tar"><div><a href="#Page_153">153</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>2.</div></td> -<td class="tal pl1">The Nature of Numbe</td> -<td class="tar"><div><a href="#Page_156">156</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>3.</div></td> -<td class="tal pl1">Of Numerical Abstraction</td> -<td class="tar"><div><a href="#Page_158">158</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>4.</div></td> -<td class="tal pl1">Concrete and Abstract Number</td> -<td class="tar"><div><a href="#Page_159">159</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>5.</div></td> -<td class="tal pl1">Analogy of Logical and Numerical Terms</td> -<td class="tar"><div><a href="#Page_160">160</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>6.</div></td> -<td class="tal pl1">Principle of Mathematical Inference</td> -<td class="tar"><div><a href="#Page_162">162</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>7.</div></td> -<td class="tal pl1">Reasoning by Inequalities</td> -<td class="tar"><div><a href="#Page_165">165</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>8.</div></td> -<td class="tal pl1">Arithmetical Reasoning</td> -<td class="tar"><div><a href="#Page_167">167</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>9.</div></td> -<td class="tal pl1">Numerically Definite Reasoning</td> -<td class="tar"><div><a href="#Page_168">168</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>10.</div></td> -<td class="tal pl1">Numerical meaning of Logical Conditions</td> -<td class="tar"><div><a href="#Page_171">171</a></div></td> -</tr> -<tr> -<td class="toc3" colspan="3">CHAPTER IX.</td> -</tr> -<tr> -<td class="toc4" colspan="3">THE VARIETY OF NATURE, OR THE DOCTRINE OF COMBINATIONS AND PERMUTATIONS.</td> -</tr> -<tr class="toc5"> -<td class="tar"><div>1.</div></td> -<td class="tal pl1">The Variety of Nature</td> -<td class="tar"><div><a href="#Page_173">173</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>2.</div></td> -<td class="tal pl1">Distinction of Combinations and Permutations</td> -<td class="tar"><div><a href="#Page_177">177</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>3.</div></td> -<td class="tal pl1">Calculation of Number of Combinations</td> -<td class="tar"><div><a href="#Page_180">180</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>4.</div></td> -<td class="tal pl1">The Arithmetical Triangle</td> -<td class="tar"><div><a href="#Page_182">182</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar vat"><div>5.</div></td> -<td class="tal pl1">Connexion between the Arithmetical Triangle and the Logical Alphabet</td> -<td class="tar vab"><div><a href="#Page_189">189</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>6.</div></td> -<td class="tal pl1">Possible Variety of Nature and Art</td> -<td class="tar"><div><a href="#Page_190">190</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>7.</div></td> -<td class="tal pl1">Higher Orders of Variety</td> -<td class="tar"><div><a href="#Page_192">192</a></div></td> -</tr> -<tr> -<td class="toc3" colspan="3">CHAPTER X.</td> -</tr> -<tr> -<td class="toc4" colspan="3">THEORY OF PROBABILITY.</td> -</tr> -<tr class="toc5"> -<td class="tar"><div>1.</div></td> -<td class="tal pl1">Theory of Probability</td> -<td class="tar"><div><a href="#Page_197">197</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>2.</div></td> -<td class="tal pl1">Fundamental Principles of the Theory</td> -<td class="tar"><div><a href="#Page_200">200</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>3.</div></td> -<td class="tal pl1">Rules for the Calculation of Probabilities</td> -<td class="tar"><div><a href="#Page_203">203</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>4.</div></td> -<td class="tal pl1">The Logical Alphabet in questions of Probability</td> -<td class="tar"><div><a href="#Page_205">205</a><span class="pagenum" id="Page_xxxvii">xxxvii</span></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>5.</div></td> -<td class="tal pl1">Comparison of the Theory with Experience</td> -<td class="tar"><div><a href="#Page_206">206</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>6.</div></td> -<td class="tal pl1">Probable Deductive Arguments</td> -<td class="tar"><div><a href="#Page_209">209</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>7.</div></td> -<td class="tal pl1">Difficulties of the Theory</td> -<td class="tar"><div><a href="#Page_213">213</a></div></td> -</tr> -<tr> -<td class="toc3" colspan="3">CHAPTER XI.</td> -</tr> -<tr> -<td class="toc4" colspan="3">PHILOSOPHY OF INDUCTIVE INFERENCE.</td> -</tr> -<tr class="toc5"> -<td class="tar"><div>1.</div></td> -<td class="tal pl1">Philosophy of Inductive Inference</td> -<td class="tar"><div><a href="#Page_218">218</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>2.</div></td> -<td class="tal pl1">Various Classes of Inductive Truths</td> -<td class="tar"><div><a href="#Page_219">219</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>3.</div></td> -<td class="tal pl1">The Relation of Cause and Effect</td> -<td class="tar"><div><a href="#Page_220">220</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>4.</div></td> -<td class="tal pl1">Fallacious Use of the Term Cause</td> -<td class="tar"><div><a href="#Page_221">221</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>5.</div></td> -<td class="tal pl1">Confusion of Two Questions</td> -<td class="tar"><div><a href="#Page_222">222</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>6.</div></td> -<td class="tal pl1">Definition of the Term Cause</td> -<td class="tar"><div><a href="#Page_224">224</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>7.</div></td> -<td class="tal pl1">Distinction of Inductive and Deductive Results</td> -<td class="tar"><div><a href="#Page_226">226</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>8.</div></td> -<td class="tal pl1">The Grounds of Inductive Inference</td> -<td class="tar"><div><a href="#Page_228">228</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>9.</div></td> -<td class="tal pl1">Illustrations of the Inductive Process</td> -<td class="tar"><div><a href="#Page_229">229</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>10.</div></td> -<td class="tal pl1">Geometrical Reasoning</td> -<td class="tar"><div><a href="#Page_233">233</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>11.</div></td> -<td class="tal pl1">Discrimination of Certainty and Probability</td> -<td class="tar"><div><a href="#Page_235">235</a></div></td> -</tr> -<tr> -<td class="toc3" colspan="3">CHAPTER XII.</td> -</tr> -<tr> -<td class="toc4" colspan="3">THE INDUCTIVE OR INVERSE APPLICATION OF THE THEORY OF PROBABILITY.</td> -</tr> -<tr class="toc5"> -<td class="tar"><div>1.</div></td> -<td class="tal pl1">The Inductive or Inverse Application of the Theory</td> -<td class="tar"><div><a href="#Page_240">240</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>2.</div></td> -<td class="tal pl1">Principle of the Inverse Method</td> -<td class="tar"><div><a href="#Page_242">242</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>3.</div></td> -<td class="tal pl1">Simple Applications of the Inverse Method</td> -<td class="tar"><div><a href="#Page_244">244</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>4.</div></td> -<td class="tal pl1">The Theory of Probability in Astronomy</td> -<td class="tar"><div><a href="#Page_247">247</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>5.</div></td> -<td class="tal pl1">The General Inverse Problem</td> -<td class="tar"><div><a href="#Page_250">250</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>6.</div></td> -<td class="tal pl1">Simple Illustration of the Inverse Problem</td> -<td class="tar"><div><a href="#Page_253">253</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>7.</div></td> -<td class="tal pl1">General Solution of the Inverse Problem</td> -<td class="tar"><div><a href="#Page_255">255</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>8.</div></td> -<td class="tal pl1">Rules of the Inverse Method</td> -<td class="tar"><div><a href="#Page_257">257</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>9.</div></td> -<td class="tal pl1">Fortuitous Coincidences</td> -<td class="tar"><div><a href="#Page_261">261</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>10.</div></td> -<td class="tal pl1">Summary of the Theory of Inductive Inference</td> -<td class="tar"><div><a href="#Page_265">265</a><span class="pagenum" id="Page_xxxviii">xxxviii</span></div></td> -</tr> -<tr> -<td class="toc1" colspan="3">BOOK III.</td> -</tr> -<tr> -<td class="toc2" colspan="3">METHODS OF MEASUREMENT.</td> -</tr> -<tr> -<td class="toc3" colspan="3">CHAPTER XIII.</td> -</tr> -<tr> -<td class="toc4" colspan="3">THE EXACT MEASUREMENT OF PHENOMENA.</td> -</tr> -<tr class="toc5"> -<td class="tar"><div>1.</div></td> -<td class="tal pl1">The Exact Measurement of Phenomena</td> -<td class="tar"><div><a href="#Page_270">270</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>2.</div></td> -<td class="tal pl1">Division of the Subject</td> -<td class="tar"><div><a href="#Page_274">274</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>3.</div></td> -<td class="tal pl1">Continuous quantity</td> -<td class="tar"><div><a href="#Page_274">274</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>4.</div></td> -<td class="tal pl1">The Fallacious Indications of the Senses</td> -<td class="tar"><div><a href="#Page_276">276</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>5.</div></td> -<td class="tal pl1">Complexity of Quantitative Questions</td> -<td class="tar"><div><a href="#Page_278">278</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>6.</div></td> -<td class="tal pl1">The Methods of Accurate Measurement</td> -<td class="tar"><div><a href="#Page_282">282</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>7.</div></td> -<td class="tal pl1">Conditions of Accurate Measurement</td> -<td class="tar"><div><a href="#Page_282">282</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>8.</div></td> -<td class="tal pl1">Measuring Instruments</td> -<td class="tar"><div><a href="#Page_284">284</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>9.</div></td> -<td class="tal pl1">The Method of Repetition</td> -<td class="tar"><div><a href="#Page_288">288</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>10.</div></td> -<td class="tal pl1">Measurements by Natural Coincidence</td> -<td class="tar"><div><a href="#Page_292">292</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>11.</div></td> -<td class="tal pl1">Modes of Indirect Measurement</td> -<td class="tar"><div><a href="#Page_296">296</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>12.</div></td> -<td class="tal pl1">Comparative Use of Measuring Instruments</td> -<td class="tar"><div><a href="#Page_299">299</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>13.</div></td> -<td class="tal pl1">Systematic Performance of Measurements</td> -<td class="tar"><div><a href="#Page_300">300</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>14.</div></td> -<td class="tal pl1">The Pendulum</td> -<td class="tar"><div><a href="#Page_302">302</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>15.</div></td> -<td class="tal pl1">Attainable Accuracy of Measurement</td> -<td class="tar"><div><a href="#Page_303">303</a></div></td> -</tr> -<tr> -<td class="toc3" colspan="3">CHAPTER XIV.</td> -</tr> -<tr> -<td class="toc4" colspan="3">UNITS AND STANDARDS OF MEASUREMENT.</td> -</tr> -<tr class="toc5"> -<td class="tar"><div>1.</div></td> -<td class="tal pl1">Units and Standards of Measurement</td> -<td class="tar"><div><a href="#Page_305">305</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>2.</div></td> -<td class="tal pl1">Standard Unit of Time</td> -<td class="tar"><div><a href="#Page_307">307</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>3.</div></td> -<td class="tal pl1">The Unit of Space and the Bar Standard</td> -<td class="tar"><div><a href="#Page_312">312</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>4.</div></td> -<td class="tal pl1">The Terrestrial Standard</td> -<td class="tar"><div><a href="#Page_314">314</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>5.</div></td> -<td class="tal pl1">The Pendulum Standard</td> -<td class="tar"><div><a href="#Page_315">315</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>6.</div></td> -<td class="tal pl1">Unit of Density</td> -<td class="tar"><div><a href="#Page_316">316</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>7.</div></td> -<td class="tal pl1">Unit of Mass</td> -<td class="tar"><div><a href="#Page_317">317</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>8.</div></td> -<td class="tal pl1">Natural System of Standards</td> -<td class="tar"><div><a href="#Page_319">319</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>9.</div></td> -<td class="tal pl1">Subsidiary Units</td> -<td class="tar"><div><a href="#Page_320">320</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>10.</div></td> -<td class="tal pl1">Derived Units</td> -<td class="tar"><div><a href="#Page_321">321</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>11.</div></td> -<td class="tal pl1">Provisional Units</td> -<td class="tar"><div><a href="#Page_323">323</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>12.</div></td> -<td class="tal pl1">Theory of Dimensions</td> -<td class="tar"><div><a href="#Page_325">325</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>13.</div></td> -<td class="tal pl1">Natural Constants</td> -<td class="tar"><div><a href="#Page_328">328</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>14.</div></td> -<td class="tal pl1">Mathematical Constants</td> -<td class="tar"><div><a href="#Page_330">330</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>15.</div></td> -<td class="tal pl1">Physical Constants</td> -<td class="tar"><div><a href="#Page_331">331</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>16.</div></td> -<td class="tal pl1">Astronomical Constants</td> -<td class="tar"><div><a href="#Page_332">332</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>17.</div></td> -<td class="tal pl1">Terrestrial Numbers</td> -<td class="tar"><div><a href="#Page_333">333</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>18.</div></td> -<td class="tal pl1">Organic Numbers</td> -<td class="tar"><div><a href="#Page_333">333</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>19.</div></td> -<td class="tal pl1">Social Numbers</td> -<td class="tar"><div><a href="#Page_334">334</a><span class="pagenum" id="Page_xxxix">xxxix</span></div></td> -</tr> -<tr> -<td class="toc3" colspan="3">CHAPTER XV.</td> -</tr> -<tr> -<td class="toc4" colspan="3">ANALYSIS OF QUANTITATIVE PHENOMENA.</td> -</tr> -<tr class="toc5"> -<td class="tar"><div>1.</div></td> -<td class="tal pl1">Analysis of Quantitative Phenomena</td> -<td class="tar"><div><a href="#Page_335">335</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>2.</div></td> -<td class="tal pl1">Illustrations of the Complication of Effects</td> -<td class="tar"><div><a href="#Page_336">336</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>3.</div></td> -<td class="tal pl1">Methods of Eliminating Error</td> -<td class="tar"><div><a href="#Page_339">339</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>4.</div></td> -<td class="tal pl1">Method of Avoidance of Error</td> -<td class="tar"><div><a href="#Page_340">340</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>5.</div></td> -<td class="tal pl1">Differential Method</td> -<td class="tar"><div><a href="#Page_344">344</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>6.</div></td> -<td class="tal pl1">Method of Correction</td> -<td class="tar"><div><a href="#Page_346">346</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>7.</div></td> -<td class="tal pl1">Method of Compensation</td> -<td class="tar"><div><a href="#Page_350">350</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>8.</div></td> -<td class="tal pl1">Method of Reversal</td> -<td class="tar"><div><a href="#Page_354">354</a></div></td> -</tr> -<tr> -<td class="toc3" colspan="3">CHAPTER XVI.</td> -</tr> -<tr> -<td class="toc4" colspan="3">THE METHOD OF MEANS.</td> -</tr> -<tr class="toc5"> -<td class="tar"><div>1.</div></td> -<td class="tal pl1">The Method of Means</td> -<td class="tar"><div><a href="#Page_357">357</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>2.</div></td> -<td class="tal pl1">Several Uses of the Mean Result</td> -<td class="tar"><div><a href="#Page_359">359</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>3.</div></td> -<td class="tal pl1">The Mean and the Average</td> -<td class="tar"><div><a href="#Page_360">360</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>4.</div></td> -<td class="tal pl1">On the Average or Fictitious Mean</td> -<td class="tar"><div><a href="#Page_363">363</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>5.</div></td> -<td class="tal pl1">The Precise Mean Result</td> -<td class="tar"><div><a href="#Page_365">365</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>6.</div></td> -<td class="tal pl1">Determination of the Zero Point</td> -<td class="tar"><div><a href="#Page_368">368</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>7.</div></td> -<td class="tal pl1">Determination of Maximum Points</td> -<td class="tar"><div><a href="#Page_371">371</a></div></td> -</tr> -<tr> -<td class="toc3" colspan="3">CHAPTER XVII.</td> -</tr> -<tr> -<td class="toc4" colspan="3">THE LAW OF ERROR.</td> -</tr> -<tr class="toc5"> -<td class="tar"><div>1.</div></td> -<td class="tal pl1">The Law of Error</td> -<td class="tar"><div><a href="#Page_374">374</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>2.</div></td> -<td class="tal pl1">Establishment of the Law of Error</td> -<td class="tar"><div><a href="#Page_375">375</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>3.</div></td> -<td class="tal pl1">Herschel’s Geometrical Proof</td> -<td class="tar"><div><a href="#Page_377">377</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>4.</div></td> -<td class="tal pl1">Laplace’s and Quetelet’s Proof of the Law</td> -<td class="tar"><div><a href="#Page_378">378</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>5.</div></td> -<td class="tal pl1">Logical Origin of the Law of Error</td> -<td class="tar"><div><a href="#Page_383">383</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>6.</div></td> -<td class="tal pl1">Verification of the Law of Error</td> -<td class="tar"><div><a href="#Page_383">383</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>7.</div></td> -<td class="tal pl1">The Probable Mean Result</td> -<td class="tar"><div><a href="#Page_385">385</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>8.</div></td> -<td class="tal pl1">The Probable Error of Results</td> -<td class="tar"><div><a href="#Page_386">386</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>9.</div></td> -<td class="tal pl1">Rejection of the Mean Result</td> -<td class="tar"><div><a href="#Page_389">389</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>10.</div></td> -<td class="tal pl1">Method of Least Squares</td> -<td class="tar"><div><a href="#Page_393">393</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>11.</div></td> -<td class="tal pl1">Works upon the Theory of Probability</td> -<td class="tar"><div><a href="#Page_394">394</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>12.</div></td> -<td class="tal pl1">Detection of Constant Errors</td> -<td class="tar"><div><a href="#Page_396">396</a><span class="pagenum" id="Page_xl">xl</span></div></td> -</tr> -<tr> -<td class="toc1" colspan="3">BOOK IV.</td> -</tr> -<tr> -<td class="toc2" colspan="3">INDUCTIVE INVESTIGATION.</td> -</tr> -<tr> -<td class="toc3" colspan="3">CHAPTER XVIII.</td> -</tr> -<tr> -<td class="toc4" colspan="3">OBSERVATION.</td> -</tr> -<tr class="toc5"> -<td class="tar"><div>1.</div></td> -<td class="tal pl1">Observation</td> -<td class="tar"><div><a href="#Page_399">399</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>2.</div></td> -<td class="tal pl1">Distinction of Observation and Experiment</td> -<td class="tar"><div><a href="#Page_400">400</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>3.</div></td> -<td class="tal pl1">Mental Conditions of Correct Observation</td> -<td class="tar"><div><a href="#Page_402">402</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>4.</div></td> -<td class="tal pl1">Instrumental and Sensual Conditions of Correct Observation</td> -<td class="tar"><div><a href="#Page_404">404</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>5.</div></td> -<td class="tal pl1">External Conditions of Correct Observation</td> -<td class="tar"><div><a href="#Page_407">407</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>6.</div></td> -<td class="tal pl1">Apparent Sequence of Events</td> -<td class="tar"><div><a href="#Page_409">409</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>7.</div></td> -<td class="tal pl1">Negative Arguments from Non-Observation</td> -<td class="tar"><div><a href="#Page_411">411</a></div></td> -</tr> -<tr> -<td class="toc3" colspan="3">CHAPTER XIX.</td> -</tr> -<tr> -<td class="toc4" colspan="3">EXPERIMENT.</td> -</tr> -<tr class="toc5"> -<td class="tar"><div>1.</div></td> -<td class="tal pl1">Experiment</td> -<td class="tar"><div><a href="#Page_416">416</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>2.</div></td> -<td class="tal pl1">Exclusion of Indifferent Circumstances</td> -<td class="tar"><div><a href="#Page_419">419</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>3.</div></td> -<td class="tal pl1">Simplification of Experiments</td> -<td class="tar"><div><a href="#Page_422">422</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>4.</div></td> -<td class="tal pl1">Failure in the Simplification of Experiments</td> -<td class="tar"><div><a href="#Page_424">424</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>5.</div></td> -<td class="tal pl1">Removal of Usual Conditions</td> -<td class="tar"><div><a href="#Page_426">426</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>6.</div></td> -<td class="tal pl1">Interference of Unsuspected Conditions</td> -<td class="tar"><div><a href="#Page_428">428</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>7.</div></td> -<td class="tal pl1">Blind or Test Experiments</td> -<td class="tar"><div><a href="#Page_433">433</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>8.</div></td> -<td class="tal pl1">Negative Results of Experiment</td> -<td class="tar"><div><a href="#Page_434">434</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>9.</div></td> -<td class="tal pl1">Limits of Experiment</td> -<td class="tar"><div><a href="#Page_437">437</a></div></td> -</tr> -<tr> -<td class="toc3" colspan="3">CHAPTER XX.</td> -</tr> -<tr> -<td class="toc4" colspan="3">METHOD OF VARIATIONS.</td> -</tr> -<tr class="toc5"> -<td class="tar"><div>1.</div></td> -<td class="tal pl1">Method of Variations</td> -<td class="tar"><div><a href="#Page_439">439</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>2.</div></td> -<td class="tal pl1">The Variable and the Variant</td> -<td class="tar"><div><a href="#Page_440">440</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>3.</div></td> -<td class="tal pl1">Measurement of the Variable</td> -<td class="tar"><div><a href="#Page_441">441</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>4.</div></td> -<td class="tal pl1">Maintenance of Similar Conditions</td> -<td class="tar"><div><a href="#Page_443">443</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>5.</div></td> -<td class="tal pl1">Collective Experiments</td> -<td class="tar"><div><a href="#Page_445">445</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>6.</div></td> -<td class="tal pl1">Periodic Variations</td> -<td class="tar"><div><a href="#Page_447">447</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>7.</div></td> -<td class="tal pl1">Combined Periodic Changes</td> -<td class="tar"><div><a href="#Page_450">450</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>8.</div></td> -<td class="tal pl1">Principle of Forced Vibrations</td> -<td class="tar"><div><a href="#Page_451">451</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>9.</div></td> -<td class="tal pl1">Integrated Variations</td> -<td class="tar"><div><a href="#Page_452">452</a><span class="pagenum" id="Page_xli">xli</span></div></td> -</tr> -<tr> -<td class="toc3" colspan="3">CHAPTER XXI.</td> -</tr> -<tr> -<td class="toc4" colspan="3">THEORY OF APPROXIMATION.</td> -</tr> -<tr class="toc5"> -<td class="tar"><div>1.</div></td> -<td class="tal pl1">Theory of Approximation</td> -<td class="tar"><div><a href="#Page_456">456</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>2.</div></td> -<td class="tal pl1">Substitution of Simple Hypotheses</td> -<td class="tar"><div><a href="#Page_458">458</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>3.</div></td> -<td class="tal pl1">Approximation to Exact Laws</td> -<td class="tar"><div><a href="#Page_462">462</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>4.</div></td> -<td class="tal pl1">Successive Approximations to Natural Conditions</td> -<td class="tar"><div><a href="#Page_465">465</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>5.</div></td> -<td class="tal pl1">Discovery of Hypothetically Simple Laws</td> -<td class="tar"><div><a href="#Page_470">470</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>6.</div></td> -<td class="tal pl1">Mathematical Principles of Approximation</td> -<td class="tar"><div><a href="#Page_471">471</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>7.</div></td> -<td class="tal pl1">Approximate Independence of Small Effects</td> -<td class="tar"><div><a href="#Page_475">475</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>8.</div></td> -<td class="tal pl1">Four Meanings of Equality</td> -<td class="tar"><div><a href="#Page_479">479</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>9.</div></td> -<td class="tal pl1">Arithmetic of Approximate Quantities</td> -<td class="tar"><div><a href="#Page_481">481</a></div></td> -</tr> -<tr> -<td class="toc3" colspan="3">CHAPTER XXII.</td> -</tr> -<tr> -<td class="toc4" colspan="3">QUANTITATIVE INDUCTION.</td> -</tr> -<tr class="toc5"> -<td class="tar"><div>1.</div></td> -<td class="tal pl1">Quantitative Induction</td> -<td class="tar"><div><a href="#Page_483">483</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>2.</div></td> -<td class="tal pl1">Probable Connexion of Varying Quantities</td> -<td class="tar"><div><a href="#Page_484">484</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>3.</div></td> -<td class="tal pl1">Empirical Mathematical Laws</td> -<td class="tar"><div><a href="#Page_487">487</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>4.</div></td> -<td class="tal pl1">Discovery of Rational Formulæ</td> -<td class="tar"><div><a href="#Page_489">489</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>5.</div></td> -<td class="tal pl1">The Graphical Method</td> -<td class="tar"><div><a href="#Page_492">492</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>6.</div></td> -<td class="tal pl1">Interpolation and Extrapolation</td> -<td class="tar"><div><a href="#Page_495">495</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>7.</div></td> -<td class="tal pl1">Illustrations of Empirical Quantitative Laws</td> -<td class="tar"><div><a href="#Page_499">499</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>8.</div></td> -<td class="tal pl1">Simple Proportional Variation</td> -<td class="tar"><div><a href="#Page_501">501</a></div></td> -</tr> -<tr> -<td class="toc3" colspan="3">CHAPTER XXIII.</td> -</tr> -<tr> -<td class="toc4" colspan="3">THE USE OF HYPOTHESIS.</td> -</tr> -<tr class="toc5"> -<td class="tar"><div>1.</div></td> -<td class="tal pl1">The Use of Hypothesis</td> -<td class="tar"><div><a href="#Page_504">504</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>2.</div></td> -<td class="tal pl1">Requisites of a good Hypothesis</td> -<td class="tar"><div><a href="#Page_510">510</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>3.</div></td> -<td class="tal pl1">Possibility of Deductive Reasoning</td> -<td class="tar"><div><a href="#Page_511">511</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>4.</div></td> -<td class="tal pl1">Consistency with the Laws of Nature</td> -<td class="tar"><div><a href="#Page_514">514</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>5.</div></td> -<td class="tal pl1">Conformity with Facts</td> -<td class="tar"><div><a href="#Page_516">516</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>6.</div></td> -<td class="tal pl1">Experimentum Crucis</td> -<td class="tar"><div><a href="#Page_518">518</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>7.</div></td> -<td class="tal pl1">Descriptive Hypotheses</td> -<td class="tar"><div><a href="#Page_522">522</a><span class="pagenum" id="Page_xlii">xlii</span></div></td> -</tr> -<tr> -<td class="toc3" colspan="3">CHAPTER XXIV.</td> -</tr> -<tr> -<td class="toc4" colspan="3">EMPIRICAL KNOWLEDGE, EXPLANATION AND PREDICTION.</td> -</tr> -<tr class="toc5"> -<td class="tar"><div>1.</div></td> -<td class="tal pl1">Empirical Knowledge, Explanation and Prediction</td> -<td class="tar"><div><a href="#Page_525">525</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>2.</div></td> -<td class="tal pl1">Empirical Knowledge</td> -<td class="tar"><div><a href="#Page_526">526</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>3.</div></td> -<td class="tal pl1">Accidental Discovery</td> -<td class="tar"><div><a href="#Page_529">529</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>4.</div></td> -<td class="tal pl1">Empirical Observations subsequently Explained</td> -<td class="tar"><div><a href="#Page_532">532</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>5.</div></td> -<td class="tal pl1">Overlooked Results of Theory</td> -<td class="tar"><div><a href="#Page_534">534</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>6.</div></td> -<td class="tal pl1">Predicted Discoveries</td> -<td class="tar"><div><a href="#Page_536">536</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>7.</div></td> -<td class="tal pl1">Predictions in the Science of Light</td> -<td class="tar"><div><a href="#Page_538">538</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>8.</div></td> -<td class="tal pl1">Predictions from the Theory of Undulations</td> -<td class="tar"><div><a href="#Page_540">540</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>9.</div></td> -<td class="tal pl1">Prediction in other Sciences</td> -<td class="tar"><div><a href="#Page_542">542</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>10.</div></td> -<td class="tal pl1">Prediction by Inversion of Cause and Effect</td> -<td class="tar"><div><a href="#Page_545">545</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>11.</div></td> -<td class="tal pl1">Facts known only by Theory</td> -<td class="tar"><div><a href="#Page_547">547</a></div></td> -</tr> -<tr> -<td class="toc3" colspan="3">CHAPTER XXV.</td> -</tr> -<tr> -<td class="toc4" colspan="3">ACCORDANCE OF QUANTITATIVE THEORIES.</td> -</tr> -<tr class="toc5"> -<td class="tar"><div>1.</div></td> -<td class="tal pl1">Accordance of Quantitative Theories</td> -<td class="tar"><div><a href="#Page_551">551</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>2.</div></td> -<td class="tal pl1">Empirical Measurements</td> -<td class="tar"><div><a href="#Page_552">552</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar vat"><div>3.</div></td> -<td class="tal pl1">Quantities indicated by Theory, but Empirically Measured</td> -<td class="tar vab"><div><a href="#Page_553">553</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>4.</div></td> -<td class="tal pl1">Explained Results of Measurement</td> -<td class="tar"><div><a href="#Page_554">554</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar vat"><div>5.</div></td> -<td class="tal pl1">Quantities determined by Theory and verified by Measurement</td> -<td class="tar vab"><div><a href="#Page_555">555</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>6.</div></td> -<td class="tal pl1">Quantities determined by Theory and not verified</td> -<td class="tar"><div><a href="#Page_556">556</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>7.</div></td> -<td class="tal pl1">Discordance of Theory and Experiment</td> -<td class="tar"><div><a href="#Page_558">558</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>8.</div></td> -<td class="tal pl1">Accordance of Measurements of Astronomical Distances</td> -<td class="tar"><div><a href="#Page_560">560</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>9.</div></td> -<td class="tal pl1">Selection of the best Mode of Measurement</td> -<td class="tar"><div><a href="#Page_563">563</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>10.</div></td> -<td class="tal pl1">Agreement of Distinct Modes of Measurement</td> -<td class="tar"><div><a href="#Page_564">564</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>11.</div></td> -<td class="tal pl1">Residual Phenomena</td> -<td class="tar"><div><a href="#Page_569">569</a></div></td> -</tr> -<tr> -<td class="toc3" colspan="3">CHAPTER XXVI.</td> -</tr> -<tr> -<td class="toc4" colspan="3">CHARACTER OF THE EXPERIMENTALIST.</td> -</tr> -<tr class="toc5"> -<td class="tar"><div>1.</div></td> -<td class="tal pl1">Character of the Experimentalist</td> -<td class="tar"><div><a href="#Page_574">574</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>2.</div></td> -<td class="tal pl1">Error of the Baconian Method</td> -<td class="tar"><div><a href="#Page_576">576</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>3.</div></td> -<td class="tal pl1">Freedom of Theorising</td> -<td class="tar"><div><a href="#Page_577">577</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>4.</div></td> -<td class="tal pl1">The Newtonian Method, the True Organum</td> -<td class="tar"><div><a href="#Page_581">581</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>5.</div></td> -<td class="tal pl1">Candour and Courage of the Philosophic Mind</td> -<td class="tar"><div><a href="#Page_586">586</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>6.</div></td> -<td class="tal pl1">The Philosophic Character of Faraday</td> -<td class="tar"><div><a href="#Page_587">587</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>7.</div></td> -<td class="tal pl1">Reservation of Judgment</td> -<td class="tar"><div><a href="#Page_592">592</a><span class="pagenum" id="Page_xliii">xliii</span></div></td> -</tr> -<tr> -<td class="toc1" colspan="3">BOOK V.</td> -</tr> -<tr> -<td class="toc2" colspan="3">GENERALISATION, ANALOGY, AND CLASSIFICATION.</td> -</tr> -<tr> -<td class="toc3" colspan="3">CHAPTER XXVII.</td> -</tr> -<tr> -<td class="toc4" colspan="3">GENERALISATION.</td> -</tr> -<tr class="toc5"> -<td class="tar"><div>1.</div></td> -<td class="tal pl1">Generalisation</td> -<td class="tar"><div><a href="#Page_594">594</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>2.</div></td> -<td class="tal pl1">Distinction of Generalisation and Analogy</td> -<td class="tar"><div><a href="#Page_596">596</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>3.</div></td> -<td class="tal pl1">Two Meanings of Generalisation</td> -<td class="tar"><div><a href="#Page_597">597</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>4.</div></td> -<td class="tal pl1">Value of Generalisation</td> -<td class="tar"><div><a href="#Page_599">599</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>5.</div></td> -<td class="tal pl1">Comparative Generality of Properties</td> -<td class="tar"><div><a href="#Page_600">600</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>6.</div></td> -<td class="tal pl1">Uniform Properties of all Matter</td> -<td class="tar"><div><a href="#Page_603">603</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>7.</div></td> -<td class="tal pl1">Variable Properties of Matter</td> -<td class="tar"><div><a href="#Page_606">606</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>8.</div></td> -<td class="tal pl1">Extreme Instances of Properties</td> -<td class="tar"><div><a href="#Page_607">607</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>9.</div></td> -<td class="tal pl1">The Detection of Continuity</td> -<td class="tar"><div><a href="#Page_610">610</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>10.</div></td> -<td class="tal pl1">The Law of Continuity</td> -<td class="tar"><div><a href="#Page_615">615</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>11.</div></td> -<td class="tal pl1">Failure of the Law of Continuity</td> -<td class="tar"><div><a href="#Page_619">619</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>12.</div></td> -<td class="tal pl1">Negative Arguments on the Principle of Continuity</td> -<td class="tar"><div><a href="#Page_621">621</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>13.</div></td> -<td class="tal pl1">Tendency to Hasty Generalisation</td> -<td class="tar"><div><a href="#Page_623">623</a></div></td> -</tr> -<tr> -<td class="toc3" colspan="3">CHAPTER XXVIII.</td> -</tr> -<tr> -<td class="toc4" colspan="3">ANALOGY.</td> -</tr> -<tr class="toc5"> -<td class="tar"><div>1.</div></td> -<td class="tal pl1">Analogy</td> -<td class="tar"><div><a href="#Page_627">627</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>2.</div></td> -<td class="tal pl1">Analogy as a Guide in Discovery</td> -<td class="tar"><div><a href="#Page_629">629</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>3.</div></td> -<td class="tal pl1">Analogy in the Mathematical Sciences</td> -<td class="tar"><div><a href="#Page_631">631</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>4.</div></td> -<td class="tal pl1">Analogy in the Theory of Undulations</td> -<td class="tar"><div><a href="#Page_635">635</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>5.</div></td> -<td class="tal pl1">Analogy in Astronomy</td> -<td class="tar"><div><a href="#Page_638">638</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>6.</div></td> -<td class="tal pl1">Failures of Analogy</td> -<td class="tar"><div><a href="#Page_641">641</a></div></td> -</tr> -<tr> -<td class="toc3" colspan="3">CHAPTER XXIX.</td> -</tr> -<tr> -<td class="toc4" colspan="3">EXCEPTIONAL PHENOMENA.</td> -</tr> -<tr class="toc5"> -<td class="tar"><div>1.</div></td> -<td class="tal pl1">Exceptional Phenomena</td> -<td class="tar"><div><a href="#Page_644">644</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>2.</div></td> -<td class="tal pl1">Imaginary or False Exceptions</td> -<td class="tar"><div><a href="#Page_647">647</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>3.</div></td> -<td class="tal pl1">Apparent but Congruent Exceptions</td> -<td class="tar"><div><a href="#Page_649">649</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>4.</div></td> -<td class="tal pl1">Singular Exceptions</td> -<td class="tar"><div><a href="#Page_652">652</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>5.</div></td> -<td class="tal pl1">Divergent Exceptions</td> -<td class="tar"><div><a href="#Page_655">655</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>6.</div></td> -<td class="tal pl1">Accidental Exceptions</td> -<td class="tar"><div><a href="#Page_658">658</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>7.</div></td> -<td class="tal pl1">Novel and Unexplained Exceptions</td> -<td class="tar"><div><a href="#Page_661">661</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>8.</div></td> -<td class="tal pl1">Limiting Exceptions</td> -<td class="tar"><div><a href="#Page_663">663</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>9.</div></td> -<td class="tal pl1">Real Exceptions to Supposed Laws</td> -<td class="tar"><div><a href="#Page_666">666</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>10.</div></td> -<td class="tal pl1">Unclassed Exceptions</td> -<td class="tar"><div><a href="#Page_668">668</a><span class="pagenum" id="Page_xliv">xliv</span></div></td> -</tr> -<tr> -<td class="toc3" colspan="3">CHAPTER XXX.</td> -</tr> -<tr> -<td class="toc4" colspan="3">CLASSIFICATION.</td> -</tr> -<tr class="toc5"> -<td class="tar"><div>1.</div></td> -<td class="tal pl1">Classification</td> -<td class="tar"><div><a href="#Page_673">673</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>2.</div></td> -<td class="tal pl1">Classification involving Induction</td> -<td class="tar"><div><a href="#Page_675">675</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>3.</div></td> -<td class="tal pl1">Multiplicity of Modes of Classification</td> -<td class="tar"><div><a href="#Page_677">677</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>4.</div></td> -<td class="tal pl1">Natural and Artificial Systems of Classification</td> -<td class="tar"><div><a href="#Page_679">679</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>5.</div></td> -<td class="tal pl1">Correlation of Properties</td> -<td class="tar"><div><a href="#Page_681">681</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>6.</div></td> -<td class="tal pl1">Classification in Crystallography</td> -<td class="tar"><div><a href="#Page_685">685</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>7.</div></td> -<td class="tal pl1">Classification an Inverse and Tentative Operation</td> -<td class="tar"><div><a href="#Page_689">689</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>8.</div></td> -<td class="tal pl1">Symbolic Statement of the Theory of Classification</td> -<td class="tar"><div><a href="#Page_692">692</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>9.</div></td> -<td class="tal pl1">Bifurcate Classification</td> -<td class="tar"><div><a href="#Page_694">694</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>10.</div></td> -<td class="tal pl1">The Five Predicates</td> -<td class="tar"><div><a href="#Page_698">698</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>11.</div></td> -<td class="tal pl1">Summum Genus and Infima Species</td> -<td class="tar"><div><a href="#Page_701">701</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>12.</div></td> -<td class="tal pl1">The Tree of Porphyry</td> -<td class="tar"><div><a href="#Page_702">702</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>13.</div></td> -<td class="tal pl1">Does Abstraction imply Generalisation?</td> -<td class="tar"><div><a href="#Page_704">704</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>14.</div></td> -<td class="tal pl1">Discovery of Marks or Characteristics</td> -<td class="tar"><div><a href="#Page_708">708</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>15.</div></td> -<td class="tal pl1">Diagnostic Systems of Classification</td> -<td class="tar"><div><a href="#Page_710">710</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>16.</div></td> -<td class="tal pl1">Index Classifications</td> -<td class="tar"><div><a href="#Page_714">714</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>17.</div></td> -<td class="tal pl1">Classification in the Biological Sciences</td> -<td class="tar"><div><a href="#Page_718">718</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>18.</div></td> -<td class="tal pl1">Classification by Types</td> -<td class="tar"><div><a href="#Page_722">722</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>19.</div></td> -<td class="tal pl1">Natural Genera and Species</td> -<td class="tar"><div><a href="#Page_724">724</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>20.</div></td> -<td class="tal pl1">Unique or Exceptional Objects</td> -<td class="tar"><div><a href="#Page_728">728</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>21.</div></td> -<td class="tal pl1">Limits of Classification</td> -<td class="tar"><div><a href="#Page_730">730</a></div></td> -</tr> -<tr> -<td class="toc1" colspan="3">BOOK VI.</td> -</tr> -<tr> -<td class="toc3" colspan="3">CHAPTER XXXI.</td> -</tr> -<tr> -<td class="toc4" colspan="3">REFLECTIONS ON THE RESULTS AND LIMITS OF SCIENTIFIC METHOD.</td> -</tr> -<tr class="toc5"> -<td class="tar vat"><div>1.</div></td> -<td class="tal pl1">Reflections on the Results and Limits of Scientific Method</td> -<td class="tar vab"><div><a href="#Page_735">735</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>2.</div></td> -<td class="tal pl1">The Meaning of Natural Law</td> -<td class="tar"><div><a href="#Page_737">737</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>3.</div></td> -<td class="tal pl1">Infiniteness of the Universe</td> -<td class="tar"><div><a href="#Page_738">738</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>4.</div></td> -<td class="tal pl1">The Indeterminate Problem of Creation</td> -<td class="tar"><div><a href="#Page_740">740</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>5.</div></td> -<td class="tal pl1">Hierarchy of Natural Laws</td> -<td class="tar"><div><a href="#Page_742">742</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>6.</div></td> -<td class="tal pl1">The Ambiguous Expression—“Uniformity of Nature”</td> -<td class="tar"><div><a href="#Page_745">745</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>7.</div></td> -<td class="tal pl1">Possible States of the Universe</td> -<td class="tar"><div><a href="#Page_749">749</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>8.</div></td> -<td class="tal pl1">Speculations on the Reconcentration of Energy</td> -<td class="tar"><div><a href="#Page_751">751</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>9.</div></td> -<td class="tal pl1">The Divergent Scope for New Discovery</td> -<td class="tar"><div><a href="#Page_752">752</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>10.</div></td> -<td class="tal pl1">Infinite Incompleteness of the Mathematical Sciences</td> -<td class="tar"><div><a href="#Page_754">754</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>11.</div></td> -<td class="tal pl1">The Reign of Law in Mental and Social Phenomena</td> -<td class="tar"><div><a href="#Page_759">759</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>12.</div></td> -<td class="tal pl1">The Theory of Evolution</td> -<td class="tar"><div><a href="#Page_761">761</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>13.</div></td> -<td class="tal pl1">Possibility of Divine Interference</td> -<td class="tar"><div><a href="#Page_765">765</a></div></td> -</tr> -<tr class="toc5"> -<td class="tar"><div>14.</div></td> -<td class="tal pl1">Conclusion</td> -<td class="tar"><div><a href="#Page_766">766</a></div></td> -</tr> -<tr> -<td class="tal pt2" colspan="2">INDEX</td> -<td class="tar pt2 toc5"><a href="#Page_773">773</a></td> -</tr> -</table> -</div> - - -<hr class="chap x-ebookmaker-drop"> -<p><span class="pagenum" id="Page_1">1</span></p> - -<div class="chapter"> -<p class="ph1">THE PRINCIPLES OF SCIENCE.</p> - -<h2 class="nobreak" id="CHAPTER_I">CHAPTER I.<br> - -<span class="title">INTRODUCTION.</span></h2> -</div> - -<p class="ti0">Science arises from the discovery of Identity amidst -Diversity. The process may be described in different -words, but our language must always imply the presence -of one common and necessary element. In every act of -inference or scientific method we are engaged about a -certain identity, sameness, similarity, likeness, resemblance, -analogy, equivalence or equality apparent between two -objects. It is doubtful whether an entirely isolated -phenomenon could present itself to our notice, since there -must always be some points of similarity between object -and object. But in any case an isolated phenomenon -could be studied to no useful purpose. The whole value -of science consists in the power which it confers upon -us of applying to one object the knowledge acquired -from like objects; and it is only so far, therefore, as we can -discover and register resemblances that we can turn our -observations to account.</p> - -<p>Nature is a spectacle continually exhibited to our senses, -in which phenomena are mingled in combinations of -endless variety and novelty. Wonder fixes the mind’s -attention; memory stores up a record of each distinct -impression; the powers of association bring forth the record -when the like is felt again. By the higher faculties of -judgment and reasoning the mind compares the new with<span class="pagenum" id="Page_2">2</span> -the old, recognises essential identity, even when disguised -by diverse circumstances, and expects to find again what -was before experienced. It must be the ground of all -reasoning and inference that <i>what is true of one thing will -be true of its equivalent</i>, and that under carefully ascertained -conditions <i>Nature repeats herself</i>.</p> - -<p>Were this indeed a Chaotic Universe, the powers of mind -employed in science would be useless to us. Did Chance -wholly take the place of order, and did all phenomena -come out of an <i>Infinite Lottery</i>, to use Condorcet’s expression, -there could be no reason to expect the like result -in like circumstances. It is possible to conceive a world -in which no two things should be associated more often, in -the long run, than any other two things. The frequent -conjunction of any two events would then be purely -fortuitous, and if we expected conjunctions to recur continually, -we should be disappointed. In such a world we -might recognise the same kind of phenomenon as it appeared -from time to time, just as we might recognise a -marked ball as it was occasionally drawn and re-drawn -from a ballot-box; but the approach of any phenomenon -would be in no way indicated by what had gone before, -nor would it be a sign of what was to come after. In such -a world knowledge would be no more than the memory of -past coincidences, and the reasoning powers, if they existed -at all, would give no clue to the nature of the present, and -no presage of the future.</p> - -<p>Happily the Universe in which we dwell is not the -result of chance, and where chance seems to work it is -our own deficient faculties which prevent us from recognising -the operation of Law and of Design. In the material -framework of this world, substances and forces present -themselves in definite and stable combinations. Things -are not in perpetual flux, as ancient philosophers held. -Element remains element; iron changes not into gold. -With suitable precautions we can calculate upon finding -the same thing again endowed with the same properties. -The constituents of the globe, indeed, appear in almost -endless combinations; but each combination bears its fixed -character, and when resolved is found to be the compound -of definite substances. Misapprehensions must continually -occur, owing to the limited extent of our experience. We<span class="pagenum" id="Page_3">3</span> -can never have examined and registered possible existences -so thoroughly as to be sure that no new ones will -occur and frustrate our calculations. The same outward -appearances may cover any amount of hidden differences -which we have not yet suspected. To the variety of substances -and powers diffused through nature at its creation, -we should not suppose that our brief experience can assign -a limit, and the necessary imperfection of our knowledge -must be ever borne in mind.</p> - -<p>Yet there is much to give us confidence in Science. The -wider our experience, the more minute our examination of -the globe, the greater the accumulation of well-reasoned -knowledge,—the fewer in all probability will be the failures -of inference compared with the successes. Exceptions -to the prevalence of Law are gradually reduced to Law -themselves. Certain deep similarities have been detected -among the objects around us, and have never yet been -found wanting. As the means of examining distant parts -of the universe have been acquired, those similarities have -been traced there as here. Other worlds and stellar -systems may be almost incomprehensively different from -ours in magnitude, condition and disposition of parts, and -yet we detect there the same elements of which our own -limbs are composed. The same natural laws can be -detected in operation in every part of the universe within -the scope of our instruments; and doubtless these laws are -obeyed irrespective of distance, time, and circumstance.</p> - -<p>It is the prerogative of Intellect to discover what is uniform -and unchanging in the phenomena around us. So -far as object is different from object, knowledge is useless -and inference impossible. But so far as object resembles -object, we can pass from one to the other. In proportion -as resemblance is deeper and more general, the commanding -powers of knowledge become more wonderful. -Identity in one or other of its phases is thus always -the bridge by which we pass in inference from case to -case; and it is my purpose in this treatise to trace out the -various forms in which the one same process of reasoning -presents itself in the ever-growing achievements of Scientific -Method.</p> - -<p><span class="pagenum" id="Page_4">4</span></p> - - -<h3><i>The Powers of Mind concerned in the Creation of Science.</i></h3> - -<p>It is no part of the purpose of this work to investigate the -nature of mind. People not uncommonly suppose that -logic is a branch of psychology, because reasoning is a -mental operation. On the same ground, however, we -might argue that all the sciences are branches of psychology. -As will be further explained, I adopt the opinion -of Mr. Herbert Spencer, that logic is really an objective -science, like mathematics or mechanics. Only in an incidental -manner, then, need I point out that the mental -powers employed in the acquisition of knowledge are probably -three in number. They are substantially as Professor -Bain has stated them<a id="FNanchor_24" href="#Footnote_24" class="fnanchor">24</a>:—</p> - -<p class="ml2em">1. The Power of Discrimination.</p> -<p class="ml2em">2. The Power of Detecting Identity.</p> -<p class="ml2em">3. The Power of Retention.</p> - -<p>We exert the first power in every act of perception. -Hardly can we have a sensation or feeling unless we discriminate -it from something else which preceded. Consciousness -would almost seem to consist in the break -between one state of mind and the next, just as an induced -current of electricity arises from the beginning or the -ending of the primary current. We are always engaged in -discrimination; and the rudiment of thought which exists -in the lower animals probably consists in their power of -feeling difference and being agitated by it.</p> - -<p>Yet had we the power of discrimination only, Science -could not be created. To know that one feeling differs -from another gives purely negative information. It cannot -teach us what will happen. In such a state of intellect -each sensation would stand out distinct from every other; -there would be no tie, no bridge of affinity between them. -We want a unifying power by which the present and the -future may be linked to the past; and this seems to be -accomplished by a different power of mind. Lord Bacon -has pointed out that different men possess in very different -degrees the powers of discrimination and identification. It -may be said indeed that discrimination necessarily implies -the action of the opposite process of identification; and so<span class="pagenum" id="Page_5">5</span> -it doubtless does in negative points. But there is a rare -property of mind which consists in penetrating the disguise -of variety and seizing the common elements of -sameness; and it is this property which furnishes the true -measure of intellect. The name of “intellect” expresses the -interlacing of the general and the single, which is the -peculiar province of mind.<a id="FNanchor_25" href="#Footnote_25" class="fnanchor">25</a> To <i>cogitate</i> is the Latin <i>coagitare</i>, -resting on a like metaphor. Logic, also, is but -another name for the same process, the peculiar work of -reason; for λογος is derived from λεγειν, which like the -Latin <i>legere</i> meant originally to gather. Plato said of this -unifying power, that if he met the man who could detect -<i>the one in the many</i>, he would follow him as a god.</p> - - -<h3><i>Laws of Identity and Difference.</i></h3> - -<p>At the base of all thought and science must lie the -laws which express the very nature and conditions of the -discriminating and identifying powers of mind. These -are the so-called Fundamental Laws of Thought, usually -stated as follows:—</p> - -<p class="ml4h25">1. The Law of Identity. <i>Whatever is, is.</i></p> - -<p class="ml4h25">2. The Law of Contradiction. <i>A thing cannot both be -and not be.</i></p> - -<p class="ml4h25">3. The Law of Duality. <i>A thing must either be or -not be.</i></p> - -<p>The first of these statements may perhaps be regarded as -a description of identity itself, if so fundamental a notion -can admit of description. A thing at any moment is perfectly -identical with itself, and, if any person were unaware -of the meaning of the word “identity,” we could not better -describe it than by such an example.</p> - -<p>The second law points out that contradictory attributes -can never be joined together. The same object may vary -in its different parts; here it may be black, and there -white; at one time it may be hard and at another time<span class="pagenum" id="Page_6">6</span> -soft; but at the same time and place an attribute cannot be -both present and absent. Aristotle truly described this -law as the first of all axioms—one of which we need not -seek for any demonstration. All truths cannot be proved, -otherwise there would be an endless chain of demonstration; -and it is in self-evident truths like this that we find the -simplest foundations.</p> - -<p>The third of these laws completes the other two. It -asserts that at every step there are two possible alternatives—presence -or absence, affirmation or negation. -Hence I propose to name this law the Law of Duality, for -it gives to all the formulæ of reasoning a dual character. It -asserts also that between presence and absence, existence -and non-existence, affirmation and negation, there is no -third alternative. As Aristotle said, there can be no mean -between opposite assertions: we must either affirm or -deny. Hence the inconvenient name by which it has been -known—The Law of Excluded Middle.</p> - -<p>It may be allowed that these laws are not three independent -and distinct laws; they rather express three different -aspects of the same truth, and each law doubtless presupposes -and implies the other two. But it has not -hitherto been found possible to state these characters of -identity and difference in less than the threefold formula. -The reader may perhaps desire some information as to the -mode in which these laws have been stated, or the -way in which they have been regarded, by philosophers -in different ages of the world. Abundant information -on this and many other points of logical history will be -found in Ueberweg’s <i>System of Logic</i>, of which an excellent -translation has been published by Professor T. M. Lindsay -(see pp. 228–281).</p> - - -<h3><i>The Nature of the Laws of Identity and Difference.</i></h3> - -<p>I must at least allude to the profoundly difficult question -concerning the nature and authority of these Laws of -Identity and Difference. Are they Laws of Thought or -Laws of Things? Do they belong to mind or to material -nature? On the one hand it may be said that science is a -purely mental existence, and must therefore conform to the -laws of that which formed it. Science is in the mind and<span class="pagenum" id="Page_7">7</span> -not in the things, and the properties of mind are therefore -all important. It is true that these laws are verified in the -observation of the exterior world; and it would seem that -they might have been gathered and proved by generalisation, -had they not already been in our possession. But -on the other hand, it may well be urged that we cannot -prove these laws by any process of reasoning or observation, -because the laws themselves are presupposed, as Leibnitz -acutely remarked, in the very notion of a proof. They are -the prior conditions of all thought and all knowledge, and -even to question their truth is to allow them true. Hartley -ingeniously refined upon this argument, remarking that if -the fundamental laws of logic be not certain, there must -exist a logic of a second order whereby we may determine -the degree of uncertainty: if the second logic be not certain, -there must be a third; and so on <i>ad infinitum</i>. Thus we -must suppose either that absolutely certain laws of thought -exist, or that there is no such thing as certainty whatever.<a id="FNanchor_26" href="#Footnote_26" class="fnanchor">26</a></p> - -<p>Logicians, indeed, appear to me to have paid insufficient -attention to the fact that mistakes in reasoning are always -possible, and of not unfrequent occurrence. The Laws -of Thought are often called necessary laws, that is, laws -which cannot but be obeyed. Yet as a matter of fact, who -is there that does not often fail to obey them? They are -the laws which the mind ought to obey rather than what -it always does obey. Our thoughts cannot be the criterion -of truth, for we often have to acknowledge mistakes in -arguments of moderate complexity, and we sometimes only -discover our mistakes by collision between our expectations -and the events of objective nature.</p> - -<p>Mr. Herbert Spencer holds that the laws of logic are -objective laws,<a id="FNanchor_27" href="#Footnote_27" class="fnanchor">27</a> and he regards the mind as being in -a state of constant education, each act of false reasoning -or miscalculation leading to results which are likely to -prevent similar mistakes from being again committed. -I am quite inclined to accept such ingenious views; but -at the same time it is necessary to distinguish between the -accumulation of knowledge, and the constitution of the -mind which allows of the acquisition of knowledge. -Before the mind can perceive or reason at all it must have<span class="pagenum" id="Page_8">8</span> -the conditions of thought impressed upon it. Before a -mistake can be committed, the mind must clearly distinguish -the mistaken conclusion from all other assertions. -Are not the Laws of Identity and Difference the prior -conditions of all consciousness and all existence? Must -they not hold true, alike of things material and immaterial? -and if so, can we say that they are only subjectively true -or objectively true? I am inclined, in short, to regard -them as true both “in the nature of thought and things,” -as I expressed it in my first logical essay;<a id="FNanchor_28" href="#Footnote_28" class="fnanchor">28</a> and I hold -that they belong to the common basis of all existence. -But this is one of the most difficult questions of psychology -and metaphysics which can be raised, and it is hardly one -for the logician to decide. As the mathematician does not -inquire into the nature of unity and plurality, but develops -the formal laws of plurality, so the logician, as I conceive, -must assume the truth of the Laws of Identity and -Difference, and occupy himself in developing the variety -of forms of reasoning in which their truth may be -manifested.</p> - -<p>Again, I need hardly dwell upon the question whether -logic treats of language, notions, or things. As reasonably -might we debate whether a mathematician treats of -symbols, quantities, or things. A mathematician certainly -does treat of symbols, but only as the instruments -whereby to facilitate his reasoning concerning quantities; -and as the axioms and rules of mathematical science must -be verified in concrete objects in order that the calculations -founded upon them may have any validity or utility, -it follows that the ultimate objects of mathematical science -are the things themselves. In like manner I conceive that -the logician treats of language so far as it is essential for the -embodiment and exhibition of thought. Even if reasoning -can take place in the inner consciousness of man without -the use of any signs, which is doubtful, at any rate it -cannot become the subject of discussion until by some -system of material signs it is manifested to other persons. -The logician then uses words and symbols as instruments -of reasoning, and leaves the nature and peculiarities of -language to the grammarian. But signs again must<span class="pagenum" id="Page_9">9</span> -correspond to the thoughts and things expressed, in order -that they shall serve their intended purpose. We may -therefore say that logic treats ultimately of thoughts and -things, and immediately of the signs which stand for them. -Signs, thoughts, and exterior objects may be regarded as -parallel and analogous series of phenomena, and to treat -any one of the three series is equivalent to treating either -of the other series.</p> - - -<h3><i>The Process of Inference.</i></h3> - -<p>The fundamental action of our reasoning faculties -consists in inferring or carrying to a new instance of a -phenomenon whatever we have previously known of its -like, analogue, equivalent or equal. Sameness or identity -presents itself in all degrees, and is known under various -names; but the great rule of inference embraces all -degrees, and affirms that <i>so far as there exists sameness, -identity or likeness, what is true of one thing will be true -of the other</i>. The great difficulty doubtless consists in -ascertaining that there does exist a sufficient degree of -likeness or sameness to warrant an intended inference; -and it will be our main task to investigate the conditions -under which reasoning is valid. In this place I wish to -point out that there is something common to all acts -of inference, however different their apparent forms. The -one same rule lends itself to the most diverse applications.</p> - -<p>The simplest possible case of inference, perhaps, occurs -in the use of a <i>pattern</i>, <i>example</i>, or, as it is commonly -called, a <i>sample</i>. To prove the exact similarity of two -portions of commodity, we need not bring one portion -beside the other. It is sufficient that we take a sample -which exactly represents the texture, appearance, and -general nature of one portion, and according as this -sample agrees or not with the other, so will the two -portions of commodity agree or differ. Whatever is true -as regards the colour, texture, density, material of the -sample will be true of the goods themselves. In such -cases likeness of quality is the condition of inference.</p> - -<p>Exactly the same mode of reasoning holds true of -magnitude and figure. To compare the sizes of two -objects, we need not lay them beside each other. A<span class="pagenum" id="Page_10">10</span> -staff, string, or other kind of measure may be employed -to represent the length of one object, and according as it -agrees or not with the other, so must the two objects -agree or differ. In this case the proxy or sample represents -length; but the fact that lengths can be added and -multiplied renders it unnecessary that the proxy should -always be as large as the object. Any standard of -convenient size, such as a common foot-rule, may be made -the medium of comparison. The height of a church in -one town may be carried to that in another, and objects -existing immovably at opposite sides of the earth may be -vicariously measured against each other. We obviously -employ the axiom that whatever is true of a thing as -regards its length, is true of its equal.</p> - -<p>To every other simple phenomenon in nature the same -principle of substitution is applicable. We may compare -weights, densities, degrees of hardness, and degrees of all -other qualities, in like manner. To ascertain whether two -sounds are in unison we need not compare them directly, -but a third sound may be the go-between. If a tuning-fork -is in unison with the middle C of York Minster -organ, and we afterwards find it to be in unison with the -same note of the organ in Westminster Abbey, then it -follows that the two organs are tuned in unison. The -rule of inference now is, that what is true of the tuning-fork -as regards the tone or pitch of its sound, is true of -any sound in unison with it.</p> - -<p>The skilful employment of this substitutive process -enables us to make measurements beyond the powers of -our senses. No one can count the vibrations, for instance, -of an organ-pipe. But we can construct an instrument -called the <i>siren</i>, so that, while producing a sound of any -pitch, it shall register the number of vibrations constituting -the sound. Adjusting the sound of the siren in -unison with an organ-pipe, we measure indirectly the -number of vibrations belonging to a sound of that pitch. -To measure a sound of the same pitch is as good as to -measure the sound itself.</p> - -<p>Sir David Brewster, in a somewhat similar manner, -succeeded in measuring the refractive indices of irregular -fragments of transparent minerals. It was a troublesome, -and sometimes impracticable work to grind the minerals<span class="pagenum" id="Page_11">11</span> -into prisms, so that the power of refracting light could -be directly observed; but he fell upon the ingenious device -of compounding a liquid possessing the same refractive -power as the transparent fragment under examination. -The moment when this equality was attained could be -known by the fragments ceasing to reflect or refract light -when immersed in the liquid, so that they became almost -invisible in it. The refractive power of the liquid being -then measured gave that of the solid. A more beautiful -instance of representative measurement, depending immediately -upon the principle of inference, could not be -found.<a id="FNanchor_29" href="#Footnote_29" class="fnanchor">29</a></p> - -<p>Throughout the various logical processes which we are -about to consider—Deduction, Induction, Generalisation, -Analogy, Classification, Quantitative Reasoning—we shall -find the one same principle operating in a more or less -disguised form.</p> - - -<h3><i>Deduction and Induction.</i></h3> - -<p>The processes of inference always depend on the one -same principle of substitution; but they may nevertheless -be distinguished according as the results are inductive or -deductive. As generally stated, deduction consists in -passing from more general to less general truths; induction -is the contrary process from less to more general -truths. We may however describe the difference in -another manner. In deduction we are engaged in developing -the consequences of a law. We learn the meaning, -contents, results or inferences, which attach to any given -proposition. Induction is the exactly inverse process. -Given certain results or consequences, we are required to -discover the general law from which they flow.</p> - -<p>In a certain sense all knowledge is inductive. We can -only learn the laws and relations of things in nature by -observing those things. But the knowledge gained from -the senses is knowledge only of particular facts, and we -require some process of reasoning by which we may -collect out of the facts the laws obeyed by them.<span class="pagenum" id="Page_12">12</span> -Experience gives us the materials of knowledge: induction -digests those materials, and yields us general knowledge. -When we possess such knowledge, in the form of -general propositions and natural laws, we can usefully -apply the reverse process of deduction to ascertain the -exact information required at any moment. In its ultimate -foundation, then, all knowledge is inductive—in the sense -that it is derived by a certain inductive reasoning from -the facts of experience.</p> - -<p>It is nevertheless true,—and this is a point to which -insufficient attention has been paid, that all reasoning -is founded on the principles of deduction. I call in -question the existence of any method of reasoning which -can be carried on without a knowledge of deductive processes. -I shall endeavour to show that <i>induction is really -the inverse process of deduction</i>. There is no mode of -ascertaining the laws which are obeyed in certain phenomena, -unless we have the power of determining what -results would follow from a given law. Just as the -process of division necessitates a prior knowledge of multiplication, -or the integral calculus rests upon the observation -and remembrance of the results of the differential -calculus, so induction requires a prior knowledge of -deduction. An inverse process is the undoing of the -direct process. A person who enters a maze must either -trust to chance to lead him out again, or he must carefully -notice the road by which he entered. The facts furnished -to us by experience are a maze of particular results; we -might by chance observe in them the fulfilment of a law, -but this is scarcely possible, unless we thoroughly learn -the effects which would attach to any particular law.</p> - -<p>Accordingly, the importance of deductive reasoning is -doubly supreme. Even when we gain the results of induction -they would be of no use unless we could deductively -apply them. But before we can gain them at all -we must understand deduction, since it is the inversion of -deduction which constitutes induction. Our first task in -this work, then, must be to trace out fully the nature of -identity in all its forms of occurrence. Having given any -series of propositions we must be prepared to develop -deductively the whole meaning embodied in them, and -the whole of the consequences which flow from them.</p> - -<p><span class="pagenum" id="Page_13">13</span></p> - - -<h3><i>Symbolic Expression of Logical Inference.</i></h3> - -<p>In developing the results of the Principle of Inference -we require to use an appropriate language of signs. It -would indeed be quite possible to explain the processes of -reasoning by the use of words found in the dictionary. -Special examples of reasoning, too, may seem to be more -readily apprehended than general symbolic forms. But it -has been shown in the mathematical sciences that the -attainment of truth depends greatly upon the invention of -a clear, brief, and appropriate system of symbols. Not -only is such a language convenient, but it is almost -essential to the expression of those general truths which -are the very soul of science. To apprehend the truth of -special cases of inference does not constitute logic; we -must apprehend them as cases of more general truths. -The object of all science is the separation of what is -common and general from what is accidental and different. -In a system of logic, if anywhere, we should esteem this -generality, and strive to exhibit clearly what is similar in -very diverse cases. Hence the great value of <i>general -symbols</i> by which we can represent the form of a reasoning -process, disentangled from any consideration of the special -subject to which it is applied.</p> - -<p>The signs required in logic are of a very simple kind. -As sameness or difference must exist between two things -or notions, we need signs to indicate the things or -notions compared, and other signs to denote the relations -between them. We need, then, (1) symbols for terms, (2) -a symbol for sameness, (3) a symbol for difference, and (4) -one or two symbols to take the place of conjunctions.</p> - -<p>Ordinary nouns substantive, such as <i>Iron</i>, <i>Metal</i>, <i>Electricity</i>, -<i>Undulation</i>, might serve as terms, but, for the -reasons explained above, it is better to adopt blank letters, -devoid of special signification, such as A, B, C, &c. -Each letter must be understood to represent a noun, and, -so far as the conditions of the argument allow, <i>any noun</i>. -Just as in Algebra, <i>x</i>, <i>y</i>, <i>z</i>, <i>p</i>, <i>q</i>, &c. are used for <i>any -quantities</i>, undetermined or unknown, except when the -special conditions of the problem are taken into account, -so will our letters stand for undetermined or unknown -things.</p> - -<p><span class="pagenum" id="Page_14">14</span></p> - -<p>These letter-terms will be used indifferently for nouns -substantive and adjective. Between these two kinds of -nouns there may perhaps be differences in a metaphysical -or grammatical point of view. But grammatical usage -sanctions the conversion of adjectives into substantives, and -<i>vice versâ</i>; we may avail ourselves of this latitude without -in any way prejudging the metaphysical difficulties which -may be involved. Here, as throughout this work, I shall -devote my attention to truths which I can exhibit in a -clear and formal manner, believing that in the present -condition of logical science, this course will lead to greater -advantage than discussion upon the metaphysical questions -which may underlie any part of the subject.</p> - -<p>Every noun or term denotes an object, and usually -implies the possession by that object of certain qualities -or circumstances common to all the objects denoted. There -are certain terms, however, which imply the absence of -qualities or circumstances attaching to other objects. It -will be convenient to employ a special mode of indicating -these <i>negative terms</i>, as they are called. If the general -name A denotes an object or class of objects possessing -certain defined qualities, then the term Not A will denote -any object which does not possess the whole of those -qualities; in short, Not A is the sign for anything which -differs from A in regard to any one or more of the assigned -qualities. If A denote “transparent object,” Not A will -denote “not transparent object.” Brevity and facility of -expression are of no slight importance in a system of -notation, and it will therefore be desirable to substitute -for the negative term Not A a briefer symbol. De Morgan -represented negative terms by small Roman letters, or -sometimes by small italic letters;<a id="FNanchor_30" href="#Footnote_30" class="fnanchor">30</a> as the latter seem to -be highly convenient, I shall use <i>a</i>, <i>b</i>, <i>c</i>, . . . <i>p</i>, <i>q</i>, &c., as -the negative terms corresponding to A, B, C, . . . P, Q, &c. -Thus if A means “fluid,” <i>a</i> will mean “not fluid.”</p> - - -<h3><i>Expression of Identity and Difference.</i></h3> - -<p>To denote the relation of sameness or identity I unhesitatingly -adopt the sign =, so long used by mathematicians -to denote equality. This symbol was originally appropriated<span class="pagenum" id="Page_15">15</span> -by Robert Recorde in his <i>Whetstone of Wit</i>, to avoid the -tedious repetition of the words “is equal to;” and he -chose a pair of parallel lines, because no two things can be -more equal.<a id="FNanchor_31" href="#Footnote_31" class="fnanchor">31</a> The meaning of the sign has however been -gradually extended beyond that of equality of quantities; -mathematicians have themselves used it to indicate -equivalence of operations. The force of analogy has been -so great that writers in most other branches of science -have employed the same sign. The philologist uses it to -indicate the equivalence of meaning of words: chemists -adopt it to signify identity in kind and equality in weight -of the elements which form two different compounds. -Not a few logicians, for instance Lambert, Drobitsch, -George Bentham,<a id="FNanchor_32" href="#Footnote_32" class="fnanchor">32</a> Boole,<a id="FNanchor_33" href="#Footnote_33" class="fnanchor">33</a> have employed it as the copula -of propositions. De Morgan declined to use it for this -purpose, but still further extended its meaning so as to -include the equivalence of a proposition with the premises -from which it can be inferred;<a id="FNanchor_34" href="#Footnote_34" class="fnanchor">34</a> and Herbert Spencer has -applied it in a like manner.<a id="FNanchor_35" href="#Footnote_35" class="fnanchor">35</a></p> - -<p>Many persons may think that the choice of a symbol is -a matter of slight importance or of mere convenience; but -I hold that the common use of this sign = in so many -different meanings is really founded upon a generalisation -of the widest character and of the greatest importance—one -indeed which it is a principal purpose of this work to -explain. The employment of the same sign in different -cases would be unphilosophical unless there were some real -analogy between its diverse meanings. If such analogy -exists, it is not only allowable, but highly desirable and -even imperative, to use the symbol of equivalence with a -generality of meaning corresponding to the generality of -the principles involved. Accordingly De Morgan’s refusal -to use the symbol in logical propositions indicated his -opinion that there was a want of analogy between logical -propositions and mathematical equations. I use the sign -because I hold the contrary opinion.</p> - -<p><span class="pagenum" id="Page_16">16</span></p> - -<p>I conceive that the sign = as commonly employed, always -denotes some form or degree of sameness, and the particular -form is usually indicated by the nature of the terms joined -by it. Thus “6,720 pounds = 3 tons” is evidently an -equation of quantities. The formula — × — = + expresses -the equivalence of operations. “Exogens = Dicotyledons” -is a logical identity expressing a profound truth -concerning the character and origin of a most important -group of plants.</p> - -<p>We have great need in logic of a distinct sign for the -copula, because the little verb <i>is</i> (or <i>are</i>), hitherto used -both in logic and ordinary discourse, is thoroughly ambiguous. -It sometimes denotes identity, as in “St. Paul’s -is the <i>chef-d’œuvre</i> of Sir Christopher Wren;” but it -more commonly indicates inclusion of class within class, -or partial identity, as in “Bishops are members of the -House of Lords.” This latter relation involves identity, -but requires careful discrimination from simple identity, as -will be shown further on.</p> - -<p>When with this sign of equality we join two nouns or -logical terms, as in</p> - -<div class="ml5em"> -Hydrogen = The least dense element, -</div> - -<p class="ti0">we signify that the object or group of objects denoted by -one term is identical with that denoted by the other, in -everything except the names. The general formula</p> - -<div class="ml5em"> -A = B -</div> - -<p class="ti0">must be taken to mean that A and B are symbols for the -same object or group of objects. This identity may sometimes -arise from the mere imposition of names, but it may -also arise from the deepest laws of the constitution of -nature; as when we say</p> - -<div class="ml5em"> -<div>Gravitating matter = Matter possessing inertia,</div> -<div>Exogenous plants = Dicotyledonous plants,</div> -<div class="pl2hi">Plagihedral quartz crystals = Quartz crystals causing -the plane of polarisation of light to rotate.</div> -</div> - -<p class="ti0">We shall need carefully to distinguish between relations -of terms which can be modified at our own will and those -which are fixed as expressing the laws of nature; but at -present we are considering only the mode of expression -which may be the same in either case.</p> - -<p>Sometimes, but much less frequently, we require a -symbol to indicate difference or the absence of complete<span class="pagenum" id="Page_17">17</span> -sameness. For this purpose we may generalise in like -manner the symbol ~, which was introduced by Wallis -to signify difference between quantities. The general -formula</p> - -<div class="ml5em"> -B ~ C -</div> - -<p class="ti0">denotes that B and C are the names of two objects or -groups which are not identical with each other. Thus -we may say</p> - -<div class="ml5em"> -<div>Acrogens ~ Flowering plants.</div> -<div class="pl2hi">Snowdon ~ The highest mountain in Great Britain.</div> -</div> - -<p>I shall also occasionally use the sign <b>ᔕ</b> to signify in the -most general manner the existence of any relation between -the two terms connected by it. Thus <b>ᔕ</b> might mean not -only the relations of equality or inequality, sameness or -difference, but any special relation of time, place, size, -causation, &c. in which one thing may stand to another. -By A <b>ᔕ</b> B I mean, then, any two objects of thought -related to each other in any conceivable manner.</p> - - -<h3><i>General Formula of Logical Inference.</i></h3> - -<p>The one supreme rule of inference consists, as I have -said, in the direction to affirm of anything whatever is -known of its like, equal or equivalent. The <i>Substitution -of Similars</i> is a phrase which seems aptly to express the -capacity of mutual replacement existing in any two objects -which are like or equivalent to a sufficient degree. It is -matter for further investigation to ascertain when and for -what purposes a degree of similarity less than complete -identity is sufficient to warrant substitution. For the -present we think only of the exact sameness expressed in -the form</p> - -<div class="ml5em"> -A = B. -</div> - -<p>Now if we take the letter C to denote any third conceivable -object, and use the sign <b>ᔕ</b> in its stated meaning -of <i>indefinite relation</i>, then the general formula of all -inference may be thus exhibited:—</p> - -<div class="ml5em"> -<div>From     A = B <b>ᔕ</b> C</div> -<div> we may infer   A <b>ᔕ</b> C</div> -</div> - -<p class="ti0">or, in words—<i>In whatever relation a thing stands to a -second thing, in the same relation it stands to the like or -equivalent of that second thing.</i> The identity between A<span class="pagenum" id="Page_18">18</span> -and B allows us indifferently to place A where B was, or -B where A was; and there is no limit to the variety of -special meanings which we can bestow upon the signs -used in this formula consistently with its truth. Thus if -we first specify only the meaning of the sign <b>ᔕ</b>, we may -say that if <i>C is the weight of B</i>, then <i>C is also the weight -of A</i>. Similarly</p> - -<div class="ml5em"> -<div class="pl2hi">If C is the father of B, C is the father of A;</div> -<div class="pl2hi">If C is a fragment of B, C is a fragment of A;</div> -<div class="pl2hi">If C is a quality of B, C is a quality of A;</div> -<div class="pl2hi">If C is a species of B, C is a species of A;</div> -<div class="pl2hi">If C is the equal of B, C is the equal of A;</div> -</div> - -<p class="ti0">and so on <i>ad infinitum</i>.</p> - -<p>We may also endow with special meanings the letter-terms -A, B, and C, and the process of inference will never -be false. Thus let the sign <b>ᔕ</b> mean “is height of,” and let</p> - -<div class="ml5em"> -<div>A = Snowdon,</div> -<div class="pl2hi">B = Highest mountain in England or Wales,</div> -<div>C = 3,590 feet;</div> -</div> - -<p class="ti0">then it obviously follows since “3,590 feet is the height -of Snowdon,” and “Snowdon = the highest mountain in -England or Wales,” that, “3,590 feet is the height of the -highest mountain in England or Wales.”</p> - -<p>One result of this general process of inference is that we -may in any aggregate or complex whole replace any part -by its equivalent without altering the whole. To alter is -to make a difference; but if in replacing a part I make no -difference, there is no alteration of the whole. Many -inferences which have been very imperfectly included in -logical formulas at once follow. I remember the late Prof. -De Morgan remarking that all Aristotle’s logic could not -prove that “Because a horse is an animal, the head of a -horse is the head of an animal.” I conceive that this -amounts merely to replacing in the complete notion <i>head of -a horse</i>, the term “horse,” by its equivalent <i>some animal</i> or -<i>an animal</i>. Similarly, since</p> - -<div class="ml7h5"> -The Lord Chancellor = The Speaker of the House of Lords, -</div> - -<p class="ti0">it follows that</p> - -<div class="ml7h5"> -The death of the Lord Chancellor = The death of the -Speaker of the House of Lords; -</div> - -<p class="ti0">and any event, circumstance or thing, which stands in a<span class="pagenum" id="Page_19">19</span> -certain relation to the one will stand in like relation to the -other. Milton reasons in this way when he says, in his -Areopagitica, “Who kills a man, kills a reasonable creature, -God’s image.” If we may suppose him to mean</p> - -<div class="ml7h5"> -God’s image = man = some reasonable creature, -</div> - -<p class="ti0">it follows that “The killer of a man is the killer of some -reasonable creature,” and also “The killer of God’s image.”</p> - -<p>This replacement of equivalents may be repeated over -and over again to any extent. Thus if <i>person</i> is identical -in meaning with <i>individual</i>, it follows that</p> - -<div class="ml7h5"> -Meeting of persons = meeting of individuals; -</div> - -<p class="ti0">and if <i>assemblage</i> = <i>meeting</i>, we may make a new replacement -and show that</p> - -<div class="ml7h5"> -Meeting of persons = assemblage of individuals. -</div> - -<p class="ti0">We may in fact found upon this principle of substitution -a most general axiom in the following terms<a id="FNanchor_36" href="#Footnote_36" class="fnanchor">36</a>:—</p> - -<div class="ml7h5"> -<i>Same parts samely related make same wholes.</i> -</div> - -<p>If, for instance, exactly similar bricks and other -materials be used to build two houses, and they be similarly -placed in each house, the two houses must be similar. -There are millions of cells in a human body, but if each -cell of one person were represented by an exactly similar -cell similarly placed in another body, the two persons -would be undistinguishable, and would be only <i>numerically</i> -different. It is upon this principle, as we shall see, that -all accurate processes of measurement depend. If for a -weight in a scale of a balance we substitute another -weight, and the equilibrium remains entirely unchanged, -then the weights must be exactly equal. The general test -of equality is substitution. Objects are equally bright -when on replacing one by the other the eye perceives no -difference. Objects are equal in dimensions when tested -by the same gauge they fit in the same manner. Generally -speaking, two objects are alike so far as when substituted -one for another no alteration is produced, and <i>vice versâ</i> -when alike no alteration is produced by the substitution.</p> - -<p><span class="pagenum" id="Page_20">20</span></p> - - -<h3><i>The Propagating Power of Similarity.</i></h3> - -<p>The relation of similarity in all its degrees is reciprocal. -So far as things are alike, either may be substituted for the -other; and this may perhaps be considered the very -meaning of the relation. But it is well worth notice that -there is in similarity a peculiar power of extending itself -among all the things which are similar. To render a -number of things similar to each other we need only -render them similar to one standard object. Each coin -struck from a pair of dies not only resembles the matrix -or original pattern from which the dies were struck, but -resembles every other coin manufactured from the same -original pattern. Among a million such coins there are -not less than 499,999,500,000 <i>pairs of coins</i> resembling -each other. Similars to the same are similars to all. It -is one great advantage of printing that all copies of a -document struck from the same type are necessarily -identical each with each, and whatever is true of one copy -will be true of every copy. Similarly, if fifty rows of -pipes in an organ be tuned in perfect unison with one row, -usually the Principal, they must be in unison with each -other. Similarity can also reproduce or propagate itself -<i>ad infinitum</i>: for if a number of tuning-forks be adjusted -in perfect unison with one standard fork, all instruments -tuned to any one fork will agree with any instrument -tuned to any other fork. Standard measures of length, -capacity, weight, or any other measurable quality, are -propagated in the same manner. So far as copies of the -original standard, or copies of copies, or copies again of -those copies, are accurately executed, they must all agree -each with every other.</p> - -<p>It is the capability of mutual substitution which gives -such great value to the modern methods of mechanical -construction, according to which all the parts of a machine -are exact facsimiles of a fixed pattern. The rifles used in -the British army are constructed on the American interchangeable -system, so that any part of any rifle can be -substituted for the same part of another. A bullet fitting -one rifle will fit all others of the same bore. Sir J.<span class="pagenum" id="Page_21">21</span> -Whitworth has extended the same system to the screws -and screw-bolts used in connecting together the parts of -machines, by establishing a series of standard screws.</p> - - -<h3><i>Anticipations of the Principle of Substitution.</i></h3> - -<p>In such a subject as logic it is hardly possible to put -forth any opinions which have not been in some degree -previously entertained. The germ at least of every -doctrine will be found in earlier writers, and novelty must -arise chiefly in the mode of harmonising and developing -ideas. When I first employed the process and name of -<i>substitution</i> in logic,<a id="FNanchor_37" href="#Footnote_37" class="fnanchor">37</a> I was led to do so from analogy with -the familiar mathematical process of substituting for a -symbol its value as given in an equation. In writing my -first logical essay I had a most imperfect conception of the -importance and generality of the process, and I described, -as if they were of equal importance, a number of other -laws which now seem to be but particular cases of the one -general rule of substitution.</p> - -<p>My second essay, “The Substitution of Similars,” was -written shortly after I had become aware of the great -simplification which may be effected by a proper application -of the principle of substitution. I was not then -acquainted with the fact that the German logician -Beneke had employed the principle of substitution, and -had used the word itself in forming a theory of the -syllogism. My imperfect acquaintance with the German -language had prevented me from acquiring a complete -knowledge of Beneke’s views; but there is no doubt that -Professor Lindsay is right in saying that he, and probably -other logicians, were in some degree familiar with -the principle.<a id="FNanchor_38" href="#Footnote_38" class="fnanchor">38</a> Even Aristotle’s dictum may be regarded -as an imperfect statement of the principle of substitution; -and, as I have pointed out, we have only to -modify that dictum in accordance with the quantification -of the predicate in order to arrive at the complete<span class="pagenum" id="Page_22">22</span> -process of substitution.<a id="FNanchor_39" href="#Footnote_39" class="fnanchor">39</a> The Port-Royal logicians appear -to have entertained nearly equivalent views, for they -considered that all moods of the syllogism might be -reduced under one general principle.<a id="FNanchor_40" href="#Footnote_40" class="fnanchor">40</a> Of two premises -they regard one as the <i>containing proposition</i> (propositio -continens), and the other as the <i>applicative proposition</i>. -The latter proposition must always be affirmative, and -represents that by which a substitution is made; the -former may or may not be negative, and is that in -which a substitution is effected. They also show that -this method will embrace certain cases of complex reasoning -which had no place in the Aristotelian syllogism. -Their views probably constitute the greatest improvement -in logical doctrine made up to that time since the days -of Aristotle. But a true reform in logic must consist, -not in explaining the syllogism in one way or another, -but in doing away with all the narrow restrictions of -the Aristotelian system, and in showing that there exists -an infinite variety of logical arguments immediately -deducible from the principle of substitution of which the -ancient syllogism forms but a small and not even the -most important part.</p> - - -<h3><i>The Logic of Relatives.</i></h3> - -<p>There is a difficult and important branch of logic -which may be called the Logic of Relatives. If I argue, -for instance, that because Daniel Bernoulli was the son -of John, and John the brother of James, therefore Daniel -was the nephew of James, it is not possible to prove -this conclusion by any simple logical process. We require -at any rate to assume that the son of a brother is -a nephew. A simple logical relation is that which exists -between properties and circumstances of the same object -or class. But objects and classes of objects may also be -related according to all the properties of time and space. -I believe it may be shown, indeed, that where an inference -concerning such relations is drawn, a process of substitution -is really employed and an identity must exist;<span class="pagenum" id="Page_23">23</span> -but I will not undertake to prove the assertion in this -work. The relations of time and space are logical -relations of a complicated character demanding much -abstract and difficult investigation. The subject has been -treated with such great ability by Peirce,<a id="FNanchor_41" href="#Footnote_41" class="fnanchor">41</a> De Morgan,<a id="FNanchor_42" href="#Footnote_42" class="fnanchor">42</a> -Ellis,<a id="FNanchor_43" href="#Footnote_43" class="fnanchor">43</a> and Harley, that I will not in the present work -attempt any review of their writings, but merely refer -the reader to the publications in which they are to be -found.</p> - - -<hr class="chap x-ebookmaker-drop"> - -<div class="chapter"> -<p><span class="pagenum" id="Page_24">24</span></p> - -<h2 class="nobreak" id="CHAPTER_II">CHAPTER II.<br> - -<span class="title">TERMS.</span></h2> -</div> - -<p class="ti0">Every proposition expresses the resemblance or difference -of the things denoted by its terms. As inference -treats of the relation between two or more propositions, so -a proposition expresses a relation between two or more -terms. In the portion of this work which treats of -deduction it will be convenient to follow the usual order -of exposition. We will consider in succession the various -kinds of terms, propositions, and arguments, and we commence -in this chapter with terms.</p> - -<p>The simplest and most palpable meaning which can -belong to a term consists of some single material object, -such as Westminster Abbey, Stonehenge, the Sun, Sirius, -&c. It is probable that in early stages of intellect only -concrete and palpable things are the objects of thought. -The youngest child knows the difference between a hot and -a cold body. The dog can recognise his master among a -hundred other persons, and animals of much lower intelligence -know and discriminate their haunts. In all such -acts there is judgment concerning the likeness of physical -objects, but there is little or no power of analysing each -object and regarding it as a group of qualities.</p> - -<p>The dignity of intellect begins with the power of -separating points of agreement from those of difference. -Comparison of two objects may lead us to perceive that -they are at once like and unlike. Two fragments of rock -may differ entirely in outward form, yet they may have the -same colour, hardness, and texture. Flowers which agree -in colour may differ in odour. The mind learns to regard<span class="pagenum" id="Page_25">25</span> -each object as an aggregate of qualities, and acquires the -power of dwelling at will upon one or other of those -qualities to the exclusion of the rest. Logical abstraction, -in short, comes into play, and the mind becomes capable of -reasoning, not merely about objects which are physically -complete and concrete, but about things which may be -thought of separately in the mind though they exist not -separately in nature. We can think of the hardness of -a rock, or the colour of a flower, and thus produce -abstract notions, denoted by abstract terms, which will -form a subject for further consideration.</p> - -<p>At the same time arise general notions and classes of -objects. We cannot fail to observe that the quality <i>hardness</i> -exists in many objects, for instance in many fragments -of rock; mentally joining these together, we create the -class <i>hard object</i>, which will include, not only the actual -objects examined, but all others which may happen to -agree with them, as they agree with each other. As our -senses cannot possibly report to us all the contents of -space, we cannot usually set any limits to the number of -objects which may fall into any such class. At this point -we begin to perceive the power and generality of thought, -which enables us in a single act to treat of indefinitely -or even infinitely numerous objects. We can safely assert -that whatever is true of any one object coming under a -class is true of any of the other objects so far as they -possess the common qualities implied in their belonging to -the class. We must not place a thing in a class unless -we are prepared to believe of it all that is believed of the -class in general; but it remains a matter of important -consideration to decide how far and in what manner we -can safely undertake thus to assign the place of objects in -that general system of classification which constitutes the -body of science.</p> - - -<h3><i>Twofold Meaning of General Names.</i></h3> - -<p>Etymologically the <i>meaning</i> of a name is that which we -are caused to think of when the name is used. Now every -general name causes us to think of some one or more of -the objects belonging to a class; it may also cause us to -think of the common qualities possessed by those objects.<span class="pagenum" id="Page_26">26</span> -A name is said to <i>denote</i> the object of thought to which it -may be applied; it <i>implies</i> at the same time the possession -of certain qualities or circumstances. The objects denoted -form the <i>extent</i> of meaning of the term; the qualities -implied form the <i>intent</i> of meaning. Crystal is the name -of any substance of which the molecules are arranged in -a regular geometrical manner. The substances or objects -in question form the extent of meaning; the circumstance -of having the molecules so arranged forms the intent of -meaning.</p> - -<p>When we compare general terms together, it may often -be found that the meaning of one is included in the meaning -of another. Thus all <i>crystals</i> are included among -<i>material substances</i>, and all <i>opaque crystals</i> are included -among <i>crystals</i>; here the inclusion is in extension. We -may also have inclusion of meaning in regard to intension. -For, as all crystals are material substances, the qualities -implied by the term material substance must be among -those implied by crystal. Again, it is obvious that while -in extension of meaning opaque crystals are but a part of -crystals, in intension of meaning crystal is but part of -opaque crystal. We increase the intent of meaning of a -term by joining to it adjectives, or phrases equivalent to -adjectives, and the removal of such adjectives of course -decreases the intensive meaning. Now, concerning such -changes of meaning, the following all-important law holds -universally true:—<i>When the intent of meaning of a term is -increased the extent is decreased; and</i> vice versâ, <i>when the -extent is increased the intent is decreased</i>. In short, as one is -increased the other is decreased.</p> - -<p>This law refers only to logical changes. The number of -steam-engines in the world may be undergoing a rapid -increase without the intensive meaning of the name being -altered. The law will only be verified, again, when there -is a real change in the intensive meaning, and an adjective -may often be joined to a noun without making a change. -<i>Elementary metal</i> is identical with <i>metal</i>; <i>mortal man</i> -with <i>man</i>; it being a <i>property</i> of all metals to be elements, -and of all men to be mortals.</p> - -<p>There is no limit to the amount of meaning which a -term may have. A term may denote one object, or many, -or an infinite number; it may imply a single quality, if such<span class="pagenum" id="Page_27">27</span> -there be, or a group of any number of qualities, and yet -the law connecting the extension and intension will infallibly -apply. Taking the general name <i>planet</i>, we -increase its intension and decrease its extension by -prefixing the adjective <i>exterior</i>; and if we further add -<i>nearest to the earth</i>, there remains but one planet, <i>Mars</i>, to -which the name can then be applied. Singular terms, -which denote a single individual only, come under the -same law of meaning as general names. They may be -regarded as general names of which the meaning in extension -is reduced to a minimum. Logicians have erroneously -asserted, as it seems to me, that singular terms are devoid -of meaning in intension, the fact being that they exceed -all other terms in that kind of meaning, as I have elsewhere -tried to show.<a id="FNanchor_44" href="#Footnote_44" class="fnanchor">44</a></p> - - -<h3><i>Abstract Terms.</i></h3> - -<p>Comparison of objects, and analysis of the complex -resemblances and differences which they present, lead us -to the conception of <i>abstract qualities</i>. We learn to think -of one object as not only different from another, but as -differing in some particular point, such as colour, or -weight, or size. We may then convert points of agreement -or difference into separate objects of thought which we -call qualities and denote by <i>abstract terms</i>. Thus the term -<i>redness</i> means something in which a number of objects -agree as to colour, and in virtue of which they are called -red. Redness forms, in fact, the intensive meaning of the -term red.</p> - -<p>Abstract terms are strongly distinguished from general -terms by possessing only one kind of meaning; for as they -denote qualities there is nothing which they cannot in -addition imply. The adjective “red” is the name of red -objects, but it implies the possession by them of the quality<span class="pagenum" id="Page_28">28</span> -<i>redness</i>; but this latter term has one single meaning—the -quality alone. Thus it arises that abstract terms are incapable -of plurality. Red objects are numerically distinct -each from each, and there are multitudes of such objects; -but redness is a single quality which runs through all -those objects, and is the same in one as it is in another. -It is true that we may speak of <i>rednesses</i>, meaning different -kinds or tints of redness, just as we may speak of <i>colours</i>, -meaning different kinds of colours. But in distinguishing -kinds, degrees, or other differences, we render the terms so -far concrete. In that they are merely red there is but a -single nature in red objects, and so far as things are merely -coloured, colour is a single indivisible quality. Redness, -so far as it is redness merely, is one and the same everywhere, -and possesses absolute oneness. In virtue of this -unity we acquire the power of treating all instances of -such quality as we may treat any one. We possess, in -short, general knowledge.</p> - - -<h3><i>Substantial Terms.</i></h3> - -<p>Logicians appear to have taken little notice of a class of -terms which partake in certain respects of the character of -abstract terms and yet are undoubtedly the names of concrete -existing things. These terms are the names of -substances, such as gold, carbonate of lime, nitrogen, &c. -We cannot speak of two golds, twenty carbonates of lime, -or a hundred nitrogens. There is no such distinction -between the parts of a uniform substance as will allow of -a discrimination of numerous individuals. The qualities of -colour, lustre, malleability, density, &c., by which we -recognise gold, extend through its substance irrespective of -particular size or shape. So far as a substance is gold, it -is one and the same everywhere; so that terms of this -kind, which I propose to call <i>substantial terms</i>, possess -the peculiar unity of abstract terms. Yet they are not -abstract; for gold is of course a tangible visible body, -entirely concrete, and existing independently of other -bodies.</p> - -<p>It is only when, by actual mechanical division, we break -up the uniform whole which forms the meaning of a -substantial term, that we introduce number. <i>Piece of gold</i><span class="pagenum" id="Page_29">29</span> -is a term capable of plurality; for there may be a great -many pieces discriminated either by their various shapes -and sizes, or, in the absence of such marks, by simultaneously -occupying different parts of space. In substance -they are one; as regards the properties of space they are -many.<a id="FNanchor_45" href="#Footnote_45" class="fnanchor">45</a> We need not further pursue this question, which -involves the distinction between unity and plurality, until -we consider the principles of number in a subsequent -chapter.</p> - - -<h3><i>Collective Terms.</i></h3> - -<p>We must clearly distinguish between the <i>collective</i> and -the <i>general meanings</i> of terms. The same name may be -used to denote the whole body of existing objects of a -certain kind, or any one of those objects taken separately. -“Man” may mean the aggregate of existing men, which we -sometimes describe as <i>mankind</i>; it is also the general -name applying to any man. The vegetable kingdom is -the name of the whole aggregate of <i>plants</i>, but “plant” -itself is a general name applying to any one or other plant. -Every material object may be conceived as divisible into -parts, and is therefore collective as regards those parts. -The animal body is made up of cells and fibres, a crystal -of molecules; wherever physical division, or as it has been -called <i>partition</i>, is possible, there we deal in reality with a -collective whole. Thus the greater number of general -terms are at the same time collective as regards each -individual whole which they denote.</p> - -<p>It need hardly be pointed out that we must not infer of -a collective whole what we know only of the parts, nor of -the parts what we know only of the whole. The relation -of whole and part is not one of identity, and does not -allow of substitution. There may nevertheless be qualities -which are true alike of the whole and of its parts. A -number of organ-pipes tuned in unison produce an aggregate -of sound which is of exactly the same pitch as each<span class="pagenum" id="Page_30">30</span> -separate sound. In the case of substantial terms, certain -qualities may be present equally in each minutest part as -in the whole. The chemical nature of the largest mass of -pure carbonate of lime is the same as the nature of the -smallest particle. In the case of abstract terms, again, we -cannot draw a distinction between whole and part; what -is true of redness in any case is always true of redness, so -far as it is merely red.</p> - - -<h3><i>Synthesis of Terms.</i></h3> - -<p>We continually combine simple terms together so as to -form new terms of more complex meaning. Thus, to -increase the intension of meaning of a term we write it -with an adjective or a phrase of adjectival nature. By -joining “brittle” to “metal,” we obtain a combined term, -“brittle metal,” which denotes a certain portion of the -metals, namely, such as are selected on account of possessing -the quality of <i>brittleness</i>. As we have already -seen, “brittle metal” possesses less extension and greater -intension than metal. Nouns, prepositional phrases, participial -phrases and subordinate propositions may also be -added to terms so as to increase their intension and -decrease their extension.</p> - -<p>In our symbolic language we need some mode of indicating -this junction of terms, and the most convenient -device will be the juxtaposition of the letter-terms. Thus -if A mean brittle, and B mean metal, then AB will mean -brittle metal. Nor need there be any limit to the number -of letters thus joined together, or the complexity of the -notions which they may represent.</p> - -<p>Thus if we take the letters</p> - -<div class="ml5em"> -P = metal,<br> -Q = white,<br> -R = monovalent,<br> -S = of specific gravity 10·5,<br> -T = melting above 1000° C.,<br> -V = good conductor of heat and electricity, -</div> - -<p class="ti0">then we can form a combined term PQRSTV, which will -denote “a white monovalent metal, of specific gravity 10·5, -melting above 1000° C., and a good conductor of heat and -electricity.”</p> - -<p><span class="pagenum" id="Page_31">31</span></p> - -<p>There are many grammatical usages concerning the -junction of words and phrases to which we need pay no -attention in logic. We can never say in ordinary language -“of wood table,” meaning “table of wood;” but we may -consider “of wood” as logically an exact equivalent of -“wooden”; so that if</p> - -<div class="ml5em"> -X = of wood,<br> -Y = table, -</div> - -<p class="ti0">there is no reason why, in our symbols, XY should not be -just as correct an expression for “table of wood ” as YX. -In this case indeed we might substitute for “of wood ” the -corresponding adjective “wooden,” but we should often fail -to find any adjective answering exactly to a phrase. There -is no single word by which we could express the notion -“of specific gravity 10·5:” but logically we may consider -these words as forming an adjective; and denoting this by -S and metal by P, we may say that SP means “metal of -specific gravity 10·5.” It is one of many advantages in -these blank letter-symbols that they enable us completely -to neglect all grammatical peculiarities and to fix our -attention solely on the purely logical relations involved. -Investigation will probably show that the rules of grammar -are mainly founded upon traditional usage and have little -logical signification. This indeed is sufficiently proved by -the wide grammatical differences which exist between -languages, though the logical foundation must be the -same.</p> - - -<h3><i>Symbolic Expression of the Law of Contradiction.</i></h3> - -<p>The synthesis of terms is subject to the all-important -Law of Thought, described in a previous section (p. <a href="#Page_5">5</a>) and -called the Law of Contradiction, It is self-evident that no -quality can be both present and absent at the same time -and place. This fundamental condition of all thought and -of all existence is expressed symbolically by a rule that a -term and its negative shall never be allowed to come into -combination. Such combined terms as A<i>a</i>, B<i>b</i>, C<i>c</i>, &c., are -self-contradictory and devoid of all intelligible meaning. -If they could represent anything, it would be what cannot -exist, and cannot even be imagined in the mind. They -can therefore only enter into our consideration to suffer<span class="pagenum" id="Page_32">32</span> -immediate exclusion. The criterion of false reasoning, as we -shall find, is that it involves self-contradiction, the affirming -and denying of the same statement. We might represent -the object of all reasoning as the separation of the -consistent and possible from the inconsistent and impossible; -and we cannot make any statement except a truism -without implying that certain combinations of terms are -contradictory and excluded from thought. To assert that -“all A’s are B’s” is equivalent to the assertion that “A’s -which are not B’s cannot exist.”</p> - -<p>It will be convenient to have the means of indicating -the exclusion of the self-contradictory, and we may use the -familiar sign for <i>nothing</i>, the cipher 0. Thus the second -law of thought may be symbolised in the forms</p> - -<div class="ml5em"> -A<i>a</i> = 0  AB<i>b</i> = 0  ABC<i>a</i> = 0 -</div> - -<p class="ti0">We may variously describe the meaning of 0 in logic as -the <i>non-existent</i>, the <i>impossible</i>, the <i>self-inconsistent</i>, the -<i>inconceivable</i>. Close analogy exists between this meaning -and its mathematical signification.</p> - - -<h3><i>Certain Special Conditions of Logical Symbols.</i></h3> - -<p>In order that we may argue and infer truly we must -treat our logical symbols according to the fundamental -laws of Identity and Difference. But in thus using our -symbols we shall frequently meet with combinations of -which the meaning will not at first sight be apparent. If -in one case we learn that an object is “yellow and round,” -and in another case that it is “round and yellow,” there -arises the question whether these two descriptions are -identical in meaning or not. Again, if we proved that an -object was “round round,” the meaning of such an expression -would be open to doubt. Accordingly we must take -notice, before proceeding further, of certain special laws -which govern the combination of logical terms.</p> - -<p>In the first place the combination of a logical term with -itself is without effect, just as the repetition of a statement -does not alter the meaning of the statement; “a round -round object” is simply “a round object.” What is -yellow yellow is merely yellow; metallic metals cannot -differ from metals, nor circular circles from circles. In our<span class="pagenum" id="Page_33">33</span> -symbolic language we may similarly hold that AA is identical -with A, or</p> - -<div class="ml5em"> -A = AA = AAA = &c. -</div> - -<p>The late Professor Boole is the only logician in modern -times who has drawn attention to this remarkable property -of logical terms;<a id="FNanchor_46" href="#Footnote_46" class="fnanchor">46</a> but in place of the name which he gave -to the law, I have proposed to call it The Law of Simplicity.<a id="FNanchor_47" href="#Footnote_47" class="fnanchor">47</a> -Its high importance will only become apparent -when we attempt to determine the relations of logical and -mathematical science. Two symbols of quantity, and only -two, seem to obey this law; we may say that 1 × 1 = 1, -and 0 × 0 = 0 (taking 0 to mean absolute zero or 1 – 1); -there is apparently no other number which combined with -itself gives an unchanged result. I shall point out, however, -in the chapter upon Number, that in reality all -numerical symbols obey this logical principle.</p> - -<p>It is curious that this Law of Simplicity, though almost -unnoticed in modern times, was known to Boëthius, who -makes a singular remark in his treatise <i>De Trinitate et -Unitate Dei</i> (p. 959). He says: “If I should say sun, -sun, sun, I should not have made three suns, but I should -have named one sun so many times.”<a id="FNanchor_48" href="#Footnote_48" class="fnanchor">48</a> Ancient discussions -about the doctrine of the Trinity drew more attention -to subtle questions concerning the nature of unity and -plurality than has ever since been given to them.</p> - -<p>It is a second law of logical symbols that order of combination -is a matter of indifference. “Rich and rare gems” -are the same as “rare and rich gems,” or even as “gems, -rich and rare.” Grammatical, rhetorical, or poetic usage -may give considerable significance to order of expression. -The limited power of our minds prevents our grasping -many ideas at once, and thus the order of statement may -produce some effect, but not in a simply logical manner. -All life proceeds in the succession of time, and we are -obliged to write, speak, or even think of things and their -qualities one after the other; but between the things and -their qualities there need be no such relation of order in<span class="pagenum" id="Page_34">34</span> -time or space. The sweetness of sugar is neither before -nor after its weight and solubility. The hardness of a -metal, its colour, weight, opacity, malleability, electric and -chemical properties, are all coexistent and coextensive, pervading -the metal and every part of it in perfect community, -none before nor after the others. In our words and symbols -we cannot observe this natural condition; we must name -one quality first and another second, just as some one must -be the first to sign a petition, or to walk foremost in a procession. -In nature there is no such precedence.</p> - -<p>I find that the opinion here stated, to the effect that -relations of space and time do not apply to many of our -ideas, is clearly adopted by Hume in his celebrated <i>Treatise -on Human Nature</i> (vol. i. p. 410). He says:<a id="FNanchor_49" href="#Footnote_49" class="fnanchor">49</a>—“An -object may be said to be no where, when its parts are not so -situated with respect to each other, as to form any figure -or quantity; nor the whole with respect to other bodies so -as to answer to our notions of contiguity or distance. Now -this is evidently the case with all our perceptions and -objects, except those of sight and feeling. A moral reflection -cannot be placed on the right hand or on the left hand -of a passion, nor can a smell or sound be either of a circular -or a square figure. These objects and perceptions, so far -from requiring any particular place, are absolutely incompatible -with it, and even the imagination cannot attribute -it to them.”</p> - -<p>A little reflection will show that knowledge in the -highest perfection would consist in the <i>simultaneous</i> possession -of a multitude of facts. To comprehend a -science perfectly we should have every fact present with -every other fact. We must write a book and we must read -it successively word by word, but how infinitely higher -would be our powers of thought if we could grasp the -whole in one collective act of consciousness! Compared -with the brutes we do possess some slight approximation -to such power, and it is conceivable that in the indefinite -future mind may acquire an increase of capacity, and be -less restricted to the piecemeal examination of a subject. -But I wish here to make plain that there is no logical -foundation for the successive character of thought and -reasoning unavoidable under our present mental conditions.<span class="pagenum" id="Page_35">35</span> -<i>We are logically weak and imperfect in respect of the fact -that we are obliged to think of one thing after another.</i> We -must describe metal as “hard and opaque,” or “opaque and -hard,” but in the metal itself there is no such difference of -order; the properties are simultaneous and coextensive in -existence.</p> - -<p>Setting aside all grammatical peculiarities which render -a substantive less moveable than an adjective, and disregarding -any meaning indicated by emphasis or marked -order of words, we may state, as a general law of logic, -that AB is identical with BA, or AB = BA. Similarly, -ABC = ACB = BCA = &c.</p> - -<p>Boole first drew attention in recent years to this property -of logical terms, and he called it the property of -Commutativeness.<a id="FNanchor_50" href="#Footnote_50" class="fnanchor">50</a> He not only stated the law with the -utmost clearness, but pointed out that it is a Law of -Thought rather than a Law of Things. I shall have in -various parts of this work to show how the necessary imperfection -of our symbols expressed in this law clings to -our modes of expression, and introduces complication into -the whole body of mathematical formulæ, which are really -founded on a logical basis.</p> - -<p>It is of course apparent that the power of commutation -belongs only to terms related in the simple logical mode of -synthesis. No one can confuse “a house of bricks” with -“bricks of a house,” “twelve square feet” with “twelve feet -square,” “the water of crystallization” with “the crystallization -of water.” All relations which involve differences of time -and space are inconvertible; the higher must not be made to -change places with the lower, nor the first with the last. For -the parties concerned there is all the difference in the world -between A killing B and B killing A. The law of commutativeness -simply asserts that difference of order does -not attach to the connection between the properties and -circumstances of a thing—to what I call <i>simple logical -relation</i>.</p> - -<hr class="chap x-ebookmaker-drop"> - -<div class="chapter"> -<p><span class="pagenum" id="Page_36">36</span></p> - -<h2 class="nobreak" id="CHAPTER_III">CHAPTER III.<br> - -<span class="title">PROPOSITIONS.</span></h2> -</div> - -<p class="ti0">We now proceed to consider the variety of forms of propositions -in which the truths of science must be expressed. -I shall endeavour to show that, however diverse these -forms may be, they all admit the application of the one -same principle of inference that what is true of a thing is -true of the like or same. This principle holds true whatever -be the kind or manner of the likeness, provided -proper regard be had to its nature. Propositions may -assert an identity of time, space, manner, quantity, degree, -or any other circumstance in which things may agree or -differ.</p> - -<p>We find an instance of a proposition concerning time in -the following:—“The year in which Newton was born, -was the year in which Galileo died.” This proposition -expresses an approximate identity of time between two -events; hence whatever is true of the year in which -Galileo died is true of that in which Newton was born, -and <i>vice versâ</i>. “Tower Hill is the place where Raleigh -was executed” expresses an identity of place; and whatever -is true of the one spot is true of the spot otherwise -defined, but in reality the same. In ordinary language we -have many propositions obscurely expressing identities -of number, quantity, or degree. “So many men, so many -minds,” is a proposition concerning number, that is to say, -an equation; whatever is true of the number of men is -true of the number of minds, and <i>vice versâ</i>. “The density -of Mars is (nearly) the same as that of the Earth,” “The -force of gravity is directly as the product of the masses, and<span class="pagenum" id="Page_37">37</span> -inversely as the square of the distance,” are propositions -concerning magnitude or degree. Logicians have not paid -adequate attention to the great variety of propositions -which can be stated by the use of the little conjunction -<i>as</i>, together with <i>so</i>. “As the home so the people,” is a -proposition expressing identity of manner; and a great -number of similar propositions all indicating some kind of -resemblance might be quoted. Whatever be the special -kind of identity, all such expressions are subject to the -great principle of inference; but as we shall in later -parts of this work treat more particularly of inference in -cases of number and magnitude, we will here confine our -attention to logical propositions which involve only notions -of quality.</p> - - -<h3><i>Simple Identities.</i></h3> - -<p>The most important class of propositions consists of -those which fall under the formula</p> - -<div class="ml5em"> -A = B, -</div> - -<p class="ti0">and may be called <i>simple identities</i>. I may instance, in -the first place, those most elementary propositions which -express the exact similarity of a quality encountered in -two or more objects. I may compare the colour of the -Pacific Ocean with that of the Atlantic, and declare them -identical. I may assert that “the smell of a rotten egg is -like that of hydrogen sulphide;” “the taste of silver hyposulphite -is like that of cane sugar;” “the sound of an -earthquake resembles that of distant artillery.” Such are -propositions stating, accurately or otherwise, the identity -of simple physical sensations. Judgments of this kind -are necessarily pre-supposed in more complex judgments. -If I declare that “this coin is made of gold,” I must base -the judgment upon the exact likeness of the substance in -several qualities to other pieces of substance which are -undoubtedly gold. I must make judgments of the colour, -the specific gravity, the hardness, and of other mechanical -and chemical properties; each of these judgments is expressed -in an elementary proposition, “the colour of this -coin is the colour of gold,” and so on. Even when we -establish the identity of a thing with itself under a -different name or aspect, it is by distinct judgments<span class="pagenum" id="Page_38">38</span> -concerning single circumstances. To prove that the -Homeric χαλκός is copper we must show the identity of -each quality recorded of χαλκός with a quality of copper. -To establish Deal as the landing-place of Cæsar all material -circumstances must be shown to agree. If the modern -Wroxeter is the ancient Uriconium, there must be the like -agreement of all features of the country not subject to -alteration by time.</p> - -<p>Such identities must be expressed in the form A = B. -We may say</p> - -<div class="ml5em"> -<div class="pl2hi">Colour of Pacific Ocean = Colour of Atlantic Ocean.</div> -<div class="pl2hi">Smell of rotten egg = Smell of hydrogen sulphide.</div> -</div> - -<p class="ti0">In these and similar propositions we assert identity of -single qualities or causes of sensation. In the same form -we may also express identity of any group of qualities, as -in</p> - -<div class="ml5em"> -χαλκός = Copper.<br> -Deal = Landing-place of Cæsar. -</div> - -<p class="ti0">A multitude of propositions involving singular terms fall -into the same form, as in</p> - -<div class="ml5em"> -<div class="pl2hi">The Pole star = The slowest-moving star.</div> -<div class="pl2hi">Jupiter = The greatest of the planets.</div> -<div class="pl2hi">The ringed planet = The planet having seven satellites.</div> -<div class="pl2hi">The Queen of England = The Empress of India.</div> -<div class="pl2hi">The number two = The even prime number.</div> -Honesty = The best policy. -</div> - -<p>In mathematical and scientific theories we often meet -with simple identities capable of expression in the same -form. Thus in mechanical science “The process for finding -the resultant of forces = the process for finding the resultant -of simultaneous velocities.” Theorems in geometry -often give results in this form, as</p> - -<div class="ml5em"> -<div class="pl2hi">Equilateral triangles = Equiangular triangles.</div> -<div class="pl2hi">Circle = Finite plane curve of constant curvature.</div> -<div class="pl2hi">Circle = Curve of least perimeter.</div> -</div> - -<p>The more profound and important laws of nature are -often expressible in the form of simple identities; in -addition to some instances which have already been given, -I may suggest,</p> - -<div class="ml7h5"> -Crystals of cubical system = Crystals not possessing -the power of double refraction. -</div> - -<p><span class="pagenum" id="Page_39">39</span></p> - -<p>All definitions are necessarily of this form, whether the -objects defined be many, few, or singular. Thus we may say,</p> - -<div class="ml5em"> -<div>Common salt = Sodium chloride.</div> -<div class="pl2hi">Chlorophyl = Green colouring matter of leaves.</div> -<div>Square = Equal-sided rectangle.</div> -</div> - -<p>It is an extraordinary fact that propositions of this -elementary form, all-important and very numerous as they -are, had no recognised place in Aristotle’s system of Logic. -Accordingly their importance was overlooked until very -recent times, and logic was the most deformed of sciences. -But it is impossible that Aristotle or any other person -should avoid constantly using them; not a term could be -defined without their use. In one place at least Aristotle -actually notices a proposition of the kind. He observes: -“We sometimes say that that white thing is Socrates, or -that the object approaching is Callias.”<a id="FNanchor_51" href="#Footnote_51" class="fnanchor">51</a> Here we certainly -have simple identity of terms; but he considered such -propositions purely accidental, and came to the unfortunate -conclusion, that “Singulars cannot be predicated of other -terms.”</p> - -<p>Propositions may also express the identity of extensive -groups of objects taken collectively or in one connected -whole; as when we say,</p> - -<div class="ml7h5" style="width: 70%;"> -The Queen, Lords, and Commons = The Legislature of -the United Kingdom. -</div> - -<p>When Blackstone asserts that “The only true and natural -foundation of society are the wants and fears of individuals,” -we must interpret him as meaning that the whole of the -wants and fears of individuals in the aggregate form the -foundation of society. But many propositions which -might seem to be collective are but groups of singular -propositions or identities. When we say “Potassium and -sodium are the metallic bases of potash and soda,” we -obviously mean,</p> - -<div class="ml5em"> -Potassium = Metallic base of potash;<br> -Sodium = Metallic base of soda. -</div> - -<p>It is the work of grammatical analysis to separate the -various propositions often combined into a single sentence. -Logic cannot be properly required to interpret the forms -and devices of language, but only to treat the meaning -when clearly exhibited.</p> - -<p><span class="pagenum" id="Page_40">40</span></p> - - -<h3><i>Partial Identities.</i></h3> - -<p>A second highly important kind of proposition is that -which I propose to call <i>a partial identity</i>. When we say -that “All mammalia are vertebrata,” we do not mean that -mammalian animals are identical with vertebrate animals, -but only that the mammalia form a <i>part of the class vertebrata</i>. -Such a proposition was regarded in the old logic as -asserting the inclusion of one class in another, or of an -object in a class. It was called a universal affirmative proposition, -because the attribute <i>vertebrate</i> was affirmed of the -whole subject <i>mammalia</i>; but the attribute was said to be -<i>undistributed</i>, because not all vertebrata were of necessity -involved in the proposition. Aristotle, overlooking the importance -of simple identities, and indeed almost denying -their existence, unfortunately founded his system upon the -notion of inclusion in a class, instead of adopting the basis -of identity. He regarded inference as resting upon the rule -that what is true of the containing class is true of the -contained, in place of the vastly more general rule that -what is true of a class or thing is true of the like. Thus -he not only reduced logic to a fragment of its proper self, -but destroyed the deep analogies which bind together -logical and mathematical reasoning. Hence a crowd of -defects, difficulties and errors which will long disfigure the -first and simplest of the sciences.</p> - -<p>It is surely evident that the relation of inclusion rests -upon the relation of identity. Mammalian animals cannot -be included among vertebrates unless they be identical with -part of the vertebrates. Cabinet Ministers are included -almost always in the class Members of Parliament, because -they are identical with some who sit in Parliament. We -may indicate this identity with a part of the larger class in -various ways; as for instance,</p> - -<div class="ml5em"> -Mammalia = part of the vertebrata.<br> -Diatomaceæ = a class of plants.<br> -<div class="pl2hi">Cabinet Ministers = some members of Parliament.</div> -Iron = a metal. -</div> - -<p>In ordinary language the verbs <i>is</i> and <i>are</i> express mere -inclusion more often than not. <i>Men are mortals</i>, means<span class="pagenum" id="Page_41">41</span> -that <i>men</i> form a part of the class <i>mortal</i>; but great confusion -exists between this sense of the verb and that in -which it expresses identity, as in “The sun is the centre of -the planetary system.” The introduction of the indefinite -article <i>a</i> often expresses partiality; when we say “Iron is -a metal” we clearly mean that iron is <i>one only</i> of several -metals.</p> - -<p>Certain recent logicians have proposed to avoid the -indefiniteness in question by what is called the Quantification -of the Predicate, and they have generally used the -little word <i>some</i> to show that only a part of the predicate -is identical with the subject. <i>Some</i> is an <i>indeterminate -adjective</i>; it implies unknown qualities by which we might -select the part in question if the qualities were known, but -it gives no hint as to their nature. I might make use of -such an indeterminate sign to express partial identities in -this work. Thus, taking the special symbol V = Some, the -general form of a partial identity would be A = VB, and in -Boole’s Logic expressions of the kind were much used. -But I believe that indeterminate symbols only introduce -complexity, and destroy the beauty and simple universality -of the system which may be created without their use. A -vague word like <i>some</i> is only used in ordinary language by -<i>ellipsis</i>, and to avoid the trouble of attaining accuracy. -We can always employ more definite expressions if we -like; but when once the indefinite <i>some</i> is introduced we -cannot replace it by the special description. We do not -know whether <i>some</i> colour is red, yellow, blue, or what it -is; but on the other hand <i>red</i> colour is certainly <i>some</i> -colour.</p> - -<p>Throughout this system of logic I shall dispense with -such indefinite expressions; and this can readily be done -by substituting one of the other terms. To express the -proposition “All A’s are some B’s” I shall not use the form -A = VB, but</p> - -<div class="ml5em"> -A = AB. -</div> - -<p>This formula states that the class A is identical with the -class AB; and as the latter must be a part at least of the -class B, it implies the inclusion of the class A in that of -B. We might represent our former example thus,</p> - -<div class="ml5em"> -Mammalia = Mammalian vertebrata. -</div> - -<p class="ti0">This proposition asserts identity between a part (or it may<span class="pagenum" id="Page_42">42</span> -be the whole) of the vertebrata and the mammalia. If it is -asked What part? the proposition affords no answer, except -that it is the part which is mammalian; but the assertion -“mammalia = some vertebrata” tells us no more.</p> - -<p>It is quite likely that some readers will think this -mode of representing the universal affirmative proposition -artificial and complicated. I will not undertake to convince -them of the opposite at this point of my exposition. -Justification for it will be found, not so much in the immediate -treatment of this proposition, as in the general -harmony which it will enable us to disclose between all -parts of reasoning. I have no doubt that this is the -critical difficulty in the relation of logical to other forms of -reasoning. Grant this mode of denoting that “all A’s are -B’s,” and I fear no further difficulties; refuse it, and we find -want of analogy and endless anomaly in every direction. It -is on general grounds that I hope to show overwhelming -reasons for seeking to reduce every kind of proposition to -the form of an identity.</p> - -<p>I may add that not a few logicians have accepted this -view of the universal affirmative proposition. Leibnitz, in -his <i>Difficultates Quædam Logicæ</i>, adopts it, saying, “Omne -A est B; id est æquivalent AB et A, seu A non B est nonens.” -Boole employed the logical equation <i>x</i> = <i>x</i><i>y</i> concurrently -with <i>x</i> = <i>v</i><i>y</i>; and Spalding<a id="FNanchor_52" href="#Footnote_52" class="fnanchor">52</a> distinctly says that -the proposition “all metals are minerals” might be described -as an assertion of <i>partial identity</i> between the two -classes. Hence the name which I have adopted for the -proposition.</p> - - -<h3><i>Limited Identities.</i></h3> - -<p>An important class of propositions have the form</p> - -<div class="ml5em"> -AB = AC, -</div> - -<p class="ti0">expressing the identity of the class AB with the class AC. -In other words, “Within the sphere of the class A, all the -B’s are all the C’s;” or again, “The B’s and C’s, which are -A’s, are identical.” But it will be observed that nothing is -asserted concerning things which are outside of the class -A; and thus the identity is of limited extent. It is the -proposition B = C limited to the sphere of things called A.<span class="pagenum" id="Page_43">43</span> -Thus we may say, with some approximation to truth, that -“Large plants are plants devoid of locomotive power.”</p> - -<p>A barrister may make numbers of most general statements -concerning the relations of persons and things in the -course of an argument, but it is of course to be understood -that he speaks only of persons and things under the -English Law. Even mathematicians make statements -which are not true with absolute generality. They say -that imaginary roots enter into equations by pairs; but this -is only true under the tacit condition that the equations in -question shall not have imaginary coefficients.<a id="FNanchor_53" href="#Footnote_53" class="fnanchor">53</a> The universe, -in short, within which they habitually discourse is -that of equations with real coefficients. These implied -limitations form part of that great mass of tacit knowledge -which accompanies all special arguments.</p> - -<p>To De Morgan is due the remark, that we do usually -think and argue in a limited universe or sphere of notions, -even when it is not expressly stated.<a id="FNanchor_54" href="#Footnote_54" class="fnanchor">54</a></p> - -<p>It is worthy of inquiry whether all identities are not -really limited to an implied sphere of meaning. When we -make such a plain statement as “Gold is malleable” we -obviously speak of gold only in its solid state; when we -say that “Mercury is a liquid metal” we must be understood -to exclude the frozen condition to which it may be -reduced in the Arctic regions. Even when we take such a -fundamental law of nature as “All substances gravitate,” -we must mean by substance, material substance, not including -that basis of heat, light, and electrical undulations -which occupies space and possesses many wonderful mechanical -properties, but not gravity. The proposition then -is really of the form</p> - -<div class="ml7h5"> -Material substance = Material gravitating substance. -</div> - - -<h3><i>Negative Propositions.</i></h3> - -<p>In every act of intellect we are engaged with a certain -identity or difference between things or sensations compared -together. Hitherto I have treated only of identities; and -yet it might seem that the relation of difference must be<span class="pagenum" id="Page_44">44</span> -infinitely more common than that of likeness. One thing -may resemble a great many other things, but then it differs -from all remaining things in the world. Diversity may -almost be said to constitute life, being to thought what -motion is to a river. The perception of an object involves -its discrimination from all other objects. But we may -nevertheless be said to detect resemblance as often as we -detect difference. We cannot, in fact, assert the existence -of a difference, without at the same time implying the -existence of an agreement.</p> - -<p>If I compare mercury, for instance, with other metals, -and decide that it is <i>not solid</i>, here is a difference between -mercury and solid things, expressed in a negative proposition; -but there must be implied, at the same time, an -agreement between mercury and the other substances -which are not solid. As it is impossible to separate the -vowels of the alphabet from the consonants without at the -same time separating the consonants from the vowels, so I -cannot select as the object of thought <i>solid things</i>, without -thereby throwing together into another class all things -which are <i>not solid</i>. The very fact of not possessing a -quality, constitutes a new quality which may be the ground -of judgment and classification. In this point of view, -agreement and difference are ever the two sides of the same -act of intellect, and it becomes equally possible to express -the same judgment in the one or other aspect.</p> - -<p>Between affirmation and negation there is accordingly a -perfect equilibrium. Every affirmative proposition implies -a negative one, and <i>vice versâ</i>. It is even a matter of indifference, -in a logical point of view, whether a positive or -negative term be used to denote a given quality and the -class of things possessing it. If the ordinary state of a -man’s body be called <i>good health</i>, then in other circumstances -he is said <i>not to be in good health</i>; but we might equally -describe him in the latter state as <i>sickly</i>, and in his normal -condition he would be <i>not sickly</i>. Animal and vegetable -substances are now called <i>organic</i>, so that the other substances, -forming an immensely greater part of the globe, are -described negatively as <i>inorganic</i>. But we might, with at -least equal logical correctness, have described the preponderating -class of substances as <i>mineral</i>, and then vegetable -and animal substances would have been <i>non-mineral</i>.</p> - -<p><span class="pagenum" id="Page_45">45</span></p> - -<p>It is plain that any positive term and its corresponding -negative divide between them the whole universe of -thought: whatever does not fall into one must fall into the -other, by the third fundamental Law of Thought, the Law -of Duality. It follows at once that there are two modes -of representing a difference. Supposing that the things -represented by A and B are found to differ, we may indicate -(see p. <a href="#Page_17">17</a>) the result of the judgment by the notation</p> - -<div class="ml5em"> -A ~ B. -</div> - -<p>We may now represent the same judgment by the assertion -that A agrees with those things which differ from B, or -that A agrees with the not-B’s. Using our notation for -negative terms (see p. <a href="#Page_14">14</a>), we obtain</p> - -<div class="ml5em"> -A = A<i>b</i> -</div> - -<p class="ti0">as the expression of the ordinary negative proposition. -Thus if we take A to mean quicksilver, and B solid, then -we have the following proposition:—</p> - -<div class="ml5em"> -Quicksilver = Quicksilver not-solid. -</div> - -<p>There may also be several other classes of negative propositions, -of which no notice was taken in the old logic. -We may have cases where all A’s are not-B’s, and at the -same time all not-B’s are A’s; there may, in short, be -a simple identity between A and not-B, which may be -expressed in the form</p> - -<div class="ml5em"> -A = <i>b</i>. -</div> - -<p class="ti0">An example of this form would be</p> - -<div class="ml5em"> -Conductors of electricity = non-electrics. -</div> - -<p>We shall also frequently have to deal as results of deduction, -with simple, partial, or limited identities between -negative terms, as in the forms</p> - -<div class="ml5em"> -<i>a</i> = <i>b</i>,  <i>a</i> = <i>a</i><i>b</i>,  <i>a</i>C = <i>b</i>C, etc. -</div> - -<p>It would be possible to represent affirmative propositions -in the negative form. Thus “Iron is solid,” might be expressed -as “Iron is not not-solid,” or “Iron is not fluid;” -or, taking A and <i>b</i> for the terms “iron,” and “not-solid,” -the form would be A ~ <i>b</i>.</p> - -<p>But there are very strong reasons why we should employ -all propositions in their affirmative form. All inference -proceeds by the substitution of equivalents, and a proposition -expressed in the form of an identity is ready to yield -all its consequences in the most direct manner. As will be -more fully shown, we can infer <i>in</i> a negative proposition,<span class="pagenum" id="Page_46">46</span> -but not <i>by</i> it. Difference is incapable of becoming the -ground of inference; it is only the implied agreement with -other differing objects which admits of deductive reasoning; -and it will always be found advantageous to employ -propositions in the form which exhibits clearly the implied -agreements.</p> - - -<h3><i>Conversion of Propositions.</i></h3> - -<p>The old books of logic contain many rules concerning -the conversion of propositions, that is, the transposition of -the subject and predicate in such a way as to obtain a new -proposition which will be true when the original proposition -is true. The reduction of every proposition to the form -of an identity renders all such rules and processes needless. -Identity is essentially reciprocal. If the colour of the -Atlantic Ocean is the same as that of the Pacific Ocean, -that of the Pacific must be the same as that of the Atlantic. -Sodium chloride being identical with common salt, common -salt must be identical with sodium chloride. If the number -of windows in Salisbury Cathedral equals the number of -days in the year, the number of days in the year must -equal the number of the windows. Lord Chesterfield was -not wrong when he said, “I will give anybody their choice -of these two truths, which amount to the same thing; He -who loves himself best is the honestest man; or, The -honestest man loves himself best.” Scotus Erigena exactly -expresses this reciprocal character of identity in saying, -“There are not two studies, one of philosophy and the -other of religion; true philosophy is true religion, and true -religion is true philosophy.”</p> - -<p>A mathematician would not think it worth while to -mention that if <i>x</i> = <i>y</i> then also <i>y</i> = <i>x</i>. He would not consider -these to be two equations at all, but one equation -accidentally written in two different manners. In written -symbols one of two names must come first, and the other -second, and a like succession must perhaps be observed in -our thoughts: but in the relation of identity there is no -need for succession in order (see p. <a href="#Page_33">33</a>), each is simultaneously -equal and identical to the other. These remarks -will hold true both of logical and mathematical identity; -so that I shall consider the two forms</p> - -<p><span class="pagenum" id="Page_47">47</span></p> - -<div class="ml5em"> -A = B and B = A -</div> - -<p class="ti0">to express exactly the same identity differently written. -All need for rules of conversion disappears, and there will -be no single proposition in the system which may not be -written with either end foremost. Thus A = AB is the -same as AB = A, <i>a</i>C = <i>b</i>C is the same as <i>b</i>C = <i>a</i>C, and so -forth.</p> - -<p>The same remarks are partially true of differences and -inequalities, which are also reciprocal to the extent that -one thing cannot differ from a second without the second -differing from the first. Mars differs in colour from -Venus, and Venus must differ from Mars. The Earth differs -from Jupiter in density; therefore Jupiter must differ from -the Earth. Speaking generally, if A ~ B we shall also -have B ~ A, and these two forms may be considered expressions -of the same difference. But the relation of -differing things is not wholly reciprocal. The density of -Jupiter does not differ from that of the Earth in the same -way that that of the Earth differs from that of Jupiter. -The change of sensation which we experience in passing -from Venus to Mars is not the same as what we experience -in passing back to Venus, but just the opposite in nature. -The colour of the sky is lighter than that of the ocean; -therefore that of the ocean cannot be lighter than that of -the sky, but darker. In these and all similar cases we gain -a notion of <i>direction</i> or character of change, and results of -immense importance may be shown to rest on this notion. -For the present we shall be concerned with the mere fact -of identity existing or not existing.</p> - - -<h3><i>Twofold Interpretation of Propositions.</i></h3> - -<p>Terms, as we have seen (p. <a href="#Page_25">25</a>), may have a meaning -either in extension or intension; and according as one or -the other meaning is attributed to the terms of a proposition, -so may a different interpretation be assigned to the -proposition itself. When the terms are abstract we must -read them in intension, and a proposition connecting such -terms must denote the identity or non-identity of the -qualities respectively denoted by the terms. Thus if we -say</p> - -<div class="ml5em"> -Equality = Identity of magnitude, -</div> - -<p><span class="pagenum" id="Page_48">48</span></p> - -<p class="ti0">the assertion means that the circumstance of being equal -exactly corresponds with the circumstance of being -identical in magnitude. Similarly in</p> - -<div class="ml5em"> -Opacity = Incapability of transmitting light, -</div> - -<p class="ti0">the quality of being incapable of transmitting light is declared -to be the same as the intended meaning of the word -opacity.</p> - -<p>When general names form the terms of a proposition we -may apply a double interpretation. Thus</p> - -<div class="ml5em"> -Exogens = Dicotyledons -</div> - -<p class="ti0">means either that the qualities which belong to all exogens -are the same as those which belong to all dicotyledons, or else -that every individual falling under one name falls equally -under the other. Hence it may be said that there are two -distinct fields of logical thought. We may argue either by -the qualitative meaning of names or by the quantitative, -that is, the extensive meaning. Every argument involving -concrete plural terms might be converted into -one involving only abstract singular terms, and <i>vice -versâ</i>. But there are reasons for believing that the -intensive or qualitative form of reasoning is the primary -and fundamental one. It is sufficient to point out that the -extensive meaning of a name is a changeable and fleeting -thing, while the intensive meaning may nevertheless remain -fixed. Very numerous additions have been lately made -to the extensive meanings both of planet and element. -Every iron steam-ship which is made or destroyed adds to -or subtracts from the extensive meaning of the name -steam-ship, without necessarily affecting the intensive -meaning. Stage coach means as much as ever in one way, -but in extension the class is nearly extinct. Chinese -railway, on the other hand, is a term represented only by a -single instance; in twenty years it may be the name of a -large class.</p> - -<hr class="chap x-ebookmaker-drop"> - -<div class="chapter"> -<p><span class="pagenum" id="Page_49">49</span></p> - -<h2 class="nobreak" id="CHAPTER_IV">CHAPTER IV.<br> - -<span class="title">DEDUCTIVE REASONING.</span></h2> -</div> - -<p class="ti0">The general principle of inference having been explained -in the previous chapters, and a suitable system of symbols -provided, we have now before us the comparatively easy -task of tracing out the most common and important forms -of deductive reasoning. The general problem of deduction -is as follows:—<i>From one or more propositions called -premises to draw such other propositions as will necessarily -be true when the premises are true.</i> By deduction we investigate -and unfold the information contained in the premises; -and this we can do by one single rule—<i>For any term occurring -in any proposition substitute the term which is asserted -in any premise to be identical with it.</i> To obtain certain -deductions, especially those involving negative conclusions, -we shall require to bring into use the second and third Laws -of Thought, and the process of reasoning will then be called -<i>Indirect Deduction</i>. In the present chapter, however, I -shall confine my attention to those results which can be -obtained by the process of <i>Direct Deduction</i>, that is, by -applying to the premises themselves the rule of substitution. -It will be found that we can combine into one harmonious -system, not only the various moods of the ancient syllogism -but a great number of equally important forms of reasoning, -which had no recognised place in the old logic. We can -at the same time dispense entirely with the elaborate -apparatus of logical rules and mnemonic lines, which -were requisite so long as the vital principle of reasoning -was not clearly expressed.</p> - -<p><span class="pagenum" id="Page_50">50</span></p> - - -<h3><i>Immediate Inference.</i></h3> - -<p>Probably the simplest of all forms of inference is that -which has been called <i>Immediate Inference</i>, because it can -be performed upon a single proposition. It consists in -joining an adjective, or other qualifying clause of the same -nature, to both sides of an identity, and asserting the -equivalence of the terms thus produced. For instance, -since</p> - -<div class="ml7h5"> -Conductors of electricity = Non-electrics, -</div> - -<p class="ti0">it follows that</p> - -<div class="ml7h5"> -Liquid conductors of electricity = Liquid non-electrics. -</div> - -<p class="ti0">If we suppose that</p> - -<div class="ml7h5"> -Plants = Bodies decomposing carbonic acid, -</div> - -<p class="ti0">it follows that</p> - -<div class="ml7h5"> -Microscopic plants = Microscopic bodies decomposing -carbonic acid. -</div> - -<p class="ti0">In general terms, from the identity</p> - -<div class="ml5em"> -A = B -</div> - -<p class="ti0">we can infer the identity</p> - -<div class="ml5em"> -AC = BC. -</div> - -<p class="ti0">This is but a case of plain substitution; for by the first -Law of Thought it must be admitted that</p> - -<div class="ml5em"> -AC = AC, -</div> - -<p class="ti0">and if, in the second side of this identity, we substitute -for A its equivalent B, we obtain</p> - -<div class="ml5em"> -AC = BC. -</div> - -<p class="ti0">In like manner from the partial identity</p> - -<div class="ml5em"> -A = AB -</div> - -<p class="ti0">we may obtain</p> - -<div class="ml5em"> -AC = ABC -</div> - -<p class="ti0">by an exactly similar act of substitution; and in every -other case the rule will be found capable of verification by -the principle of inference. The process when performed as -here described will be quite free from the liability to error -which I have shown<a id="FNanchor_55" href="#Footnote_55" class="fnanchor">55</a> to exist in “Immediate Inference by -added Determinants,” as described by Dr. Thomson.<a id="FNanchor_56" href="#Footnote_56" class="fnanchor">56</a></p> - -<p><span class="pagenum" id="Page_51">51</span></p> - -<h3><i>Inference with Two Simple Identities.</i></h3> - -<p>One of the most common forms of inference, and one to -which I shall especially direct attention, is practised with -two simple identities. From the two statements that -“London is the capital of England” and “London is the -most populous city in the world,” we instantaneously draw -the conclusion that “The capital of England is the most -populous city in the world.” Similarly, from the identities</p> - -<div class="ml5em"> -<div class="pl2hi">Hydrogen = Substance of least density,</div> -<div class="pl2hi">Hydrogen = Substance of least atomic weight,</div> -</div> - -<p class="ti0">we infer</p> - -<div class="ml7h5"> -Substance of least density = Substance of least atomic weight. -</div> - -<p>The general form of the argument is exhibited in the -symbols</p> - -<table class="ml5em"> -<tr> -<td class="tal"></td> -<td class="tal">B = A  </td> -<td class="tar"><div>(1)</div></td> -</tr> -<tr> -<td class="tal"></td> -<td class="tal">B = C  </td> -<td class="tar"><div>(2)</div></td> -</tr> -<tr> -<td class="tal">hence  </td> -<td class="tal">A = C.  </td> -<td class="tar"><div>(3)</div></td> -</tr> -</table> - -<p>We may describe the result by saying that terms identical -with the same term are identical with each other; and -it is impossible to overlook the analogy to the first axiom -of Euclid that “things equal to the same thing are equal -to each other.” It has been very commonly supposed that -this is a fundamental principle of thought, incapable of -reduction to anything simpler. But I entertain no doubt -that this form of reasoning is only one case of the general -rule of inference. We have two propositions, A = B and -B = C, and we may for a moment consider the second one -as affirming a truth concerning B, while the former one -informs us that B is identical with A; hence by substitution -we may affirm the same truth of A. It happens in -this particular case that the truth affirmed is identity to -C, and we might, if we preferred it, have considered the -substitution as made by means of the second identity in -the first. Having two identities we have a choice of the -mode in which we will make the substitution, though the -result is exactly the same in either case.</p> - -<p>Now compare the three following formulæ,</p> - -<table class="ml5em"> -<tr> -<td class="tal">(1)</td> -<td class="tal pl2">A = B = C, hence A = C</td> -</tr> -<tr> -<td class="tal">(2)</td> -<td class="tal pl2">A = B ~ C, hence A ~ C</td> -</tr> -<tr> -<td class="tal">(3)</td> -<td class="tal pl2">A ~ B ~ C, no inference.</td> -</tr> -</table> - -<p><span class="pagenum" id="Page_52">52</span></p> - -<p>In the second formula we have an identity and a difference, -and we are able to infer a difference; in the third we -have two differences and are unable to make any inference -at all. Because A and C both differ from B, we cannot -tell whether they will or will not differ from each other. -The flowers and leaves of a plant may both differ in colour -from the earth in which the plant grows, and yet they may -differ from each other; in other cases the leaves and stem -may both differ from the soil and yet agree with each other. -Where we have difference only we can make no inference; -where we have identity we can infer. This fact gives great -countenance to my assertion that inference proceeds always -through identity, but may be equally well effected in propositions -asserting difference or identity.</p> - -<p>Deferring a more complete discussion of this point, I -will only mention now that arguments from double identity -occur very frequently, and are usually taken for granted, -owing to their extreme simplicity. In regard to the equivalence -of words this form of inference must be constantly -employed. If the ancient Greek χαλκός is our <i>copper</i>, then -it must be the French <i>cuivre</i>, the German <i>kupfer</i>, the Latin -<i>cuprum</i>, because these are words, in one sense at least, -equivalent to copper. Whenever we can give two definitions -or expressions for the same term, the formula applies; -thus Senior defined wealth as “All those things, and those -things only, which are transferable, are limited in supply, -and are directly or indirectly productive of pleasure or -preventive of pain.” Wealth is also equivalent to “things -which have value in exchange;” hence obviously, “things -which have value in exchange = all those things, and those -things only, which are transferable, &c.” Two expressions -for the same term are often given in the same sentence, and -their equivalence implied. Thus Thomson and Tait say,<a id="FNanchor_57" href="#Footnote_57" class="fnanchor">57</a> -“The naturalist may be content to know matter as that -which can be perceived by the senses, or as that which -can be acted upon by or can exert force.” I take this to -mean—</p> - -<div class="ml5em"> -<div class="pl2hi">Matter = what can be perceived by the senses;</div> -<div class="pl2hi">Matter = what can be acted upon by or can exert force.</div> -</div> - -<p><span class="pagenum" id="Page_53">53</span></p> - -<p>For the term “matter” in either of these identities we -may substitute its equivalent given in the other definition. -Elsewhere they often employ sentences of the form exemplified -in the following:<a id="FNanchor_58" href="#Footnote_58" class="fnanchor">58</a> “The integral curvature, or -whole change of direction of an arc of a plane curve, is the -angle through which the tangent has turned as we pass from -one extremity to the other.” This sentence is certainly of -the form—</p> - -<div class="ml7h5" style="width: 70%;"> -The integral curvature = the whole change of direction, -&c. = the angle through which the tangent -has turned, &c. -</div> - -<p>Disguised cases of the same kind of inference occur -throughout all sciences, and a remarkable instance is found -in algebraic geometry. Mathematicians readily show that -every equation of the form <i>y</i> = <i>m</i><i>x</i> + <i>c</i> corresponds to or -represents a straight line; it is also easily proved that the -same equation is equivalent to one of the general form -A<i>x</i> + B<i>y</i> + C = 0, and <i>vice versâ</i>. Hence it follows that -every equation of the form in question, that is to say, -every equation of the first degree, corresponds to or -represents a straight line.<a id="FNanchor_59" href="#Footnote_59" class="fnanchor">59</a></p> - - -<h3><i>Inference with a Simple and a Partial Identity.</i></h3> - -<p>A form of reasoning somewhat different from that last -considered consists in inference-between a simple and a -partial identity. If we have two propositions of the forms</p> - -<div class="ml5em"> -A = B,<br> -B = BC, -</div> - -<p class="ti0">we may then substitute for B in either proposition its -equivalent in the other, getting in both cases A = BC; -in this we may if we like make a second substitution for -B, getting</p> - -<div class="ml5em"> -A = AC. -</div> - -<p>Thus, since “The Mont Blanc is the highest mountain in -Europe, and the Mont Blanc is deeply covered with snow,” -we infer by an obvious substitution that “The highest -mountain in Europe is deeply covered with snow.” These -propositions when rigorously stated fall into the forms -above exhibited.</p> - -<p>This mode of inference is constantly employed when for<span class="pagenum" id="Page_54">54</span> -a term we substitute its definition, or <i>vice versâ</i>. The very -purpose of a definition is to allow a single noun to be -employed in place of a long descriptive phrase. Thus, -when we say “A circle is a curve of the second degree,” we -may substitute a definition of the circle, getting “A curve, -all points of which are at equal distances from one point, is -a curve of the second degree.” The real forms of the propositions -here given are exactly those shown in the symbolic -statement, but in this and many other cases it will be -sufficient to state them in ordinary elliptical language for -sake of brevity. In scientific treatises a term and its -definition are often both given in the same sentence, as in -“The weight of a body in any given locality, or the force -with which the earth attracts it, is proportional to its -mass.” The conjunction <i>or</i> in this statement gives the -force of equivalence to the parenthetic phrase, so that the -propositions really are</p> - -<div class="ml5em"> -<div class="pl2hi">Weight of a body = force with which the earth attracts it.</div> -<div class="pl2hi">Weight of a body = weight, &c. proportional to its mass.</div> -</div> - -<p>A slightly different case of inference consists in substituting -in a proposition of the form A = AB, a definition of the -term B. Thus from A = AB and B = C we get A = AC. -For instance, we may say that “Metals are elements” and -“Elements are incapable of decomposition.”</p> - -<div class="ml5em"> -<div>Metal = metal element.</div> -<div class="pl2hi">Element = what is incapable of decomposition.</div> -</div> - -<p class="ti0">Hence</p> - -<div class="ml7h5"> -Metal = metal incapable of decomposition. -</div> - -<p>It is almost needless to point out that the form of these -arguments does not suffer any real modification if some -of the terms happen to be negative; indeed in the last -example “incapable of decomposition” may be treated as -a negative term. Taking</p> - -<table class="ml5em"> -<tr> -<td class="tal">A = metal</td> -<td class="tal pl2">C = capable of decomposition</td> -</tr> -<tr> -<td class="tal">B = element</td> -<td class="tal pl2"><i>c</i> = incapable of decomposition;</td> -</tr> -</table> - -<p class="ti0">the propositions are of the forms</p> - -<div class="ml5em"> -A = AB<br> -B = <i>c</i> -</div> - -<p class="ti0">whence, by substitution,</p> - -<div class="ml5em"> -A = A<i>c</i>. -</div> - -<p><span class="pagenum" id="Page_55">55</span></p> - - -<h3><i>Inference of a Partial from Two Partial Identities.</i></h3> - -<p>However common be the cases of inference already -noticed, there is a form occurring almost more frequently, -and which deserves much attention, because it occupied a -prominent place in the ancient syllogistic system. That -system strangely overlooked all the kinds of argument we -have as yet considered, and selected, as the type of all -reasoning, one which employs two partial identities as -premises. Thus from the propositions</p> - -<table class="ml5em"> -<tr> -<td class="tal">Sodium is a metal</td> -<td class="tar"><div>(1)</div></td> -</tr> -<tr> -<td class="tal">Metals conduct electricity,</td> -<td class="tar pl3">(2)</td> -</tr> -</table> - -<p class="ti0">we may conclude that</p> - -<div class="ml5em"> -<div>Sodium conducts electricity. <span class="pl2">(3)</span></div> -</div> - -<p class="ti0">Taking A, B, C to represent the three terms respectively, -the premises are of the forms</p> - -<div class="ml5em"> -A = AB   (1)<br> -B = BC.  (2) -</div> - -<p class="ti0">Now for B in (1) we can substitute its expression as given -in (2), obtaining</p> - -<div class="ml5em"> -A = ABC,  (3)<br> -</div> - -<p class="ti0">or, in words, from</p> - -<table class="ml5em"> -<tr> -<td class="tal">Sodium = sodium metal,</td> -<td class="tar"><div>(1)</div></td> -</tr> -<tr> -<td class="pl2hi">Metal = metal conducting electricity,</td> -<td class="tar pl2">(2)</td> -</tr> -</table> - -<p class="ti0">we infer</p> - -<div class="ml7h5"> -Sodium = sodium metal conducting electricity,  (3) -</div> - -<p class="ti0">which, in the elliptical language of common life, becomes</p> - -<div class="ml5em"> -“Sodium conducts electricity.” -</div> - -<p>The above is a syllogism in the mood called Barbara<a id="FNanchor_60" href="#Footnote_60" class="fnanchor">60</a> in -the truly barbarous language of ancient logicians; and the -first figure of the syllogism contained Barbara and three -other moods which were esteemed distinct forms of argument. -But it is worthy of notice that, without any real -change in our form of inference, we readily include these -three other moods under Barbara. The negative mood -Celarent will be represented by the example</p> - -<table class=""> -<tr> -<td class="tal"></td> -<td class="tal pl4h2">Neptune is a planet,</td> -<td class="tar"><div>(1)</div></td> -</tr> -<tr> -<td class="tal"></td> -<td class="tal pl4h2">No planet has retrograde motion;</td> -<td class="tar"><div>(2)</div></td> -</tr> -<tr> -<td class="tal">Hence</td> -<td class="tal pl4h2">Neptune has not retrograde motion.</td> -<td class="tar pl2">(3)</td> -</tr> -</table> - -<p><span class="pagenum" id="Page_56">56</span></p> - -<p>If we put A for Neptune, B for planet, and C for “having -retrograde motion,” then by the corresponding negative -term c, we denote “not having retrograde motion.” The -premises now fall into the forms</p> - - -<table class="ml5em"> -<tr> -<td class="tal">A = AB</td> -<td class="tar pl2">(1)</td> -</tr> -<tr> -<td class="tal">B = B<i>c</i>,</td> -<td class="tar pl2">(2)</td> -</tr> -</table> - -<p class="ti0">and by substitution for B, exactly as before, we obtain</p> - -<div class="ml5em"> -A = AB<i>c</i>.  (3) -</div> - -<p>What is called in the old logic a particular conclusion -may be deduced without any real variation in the symbols. -Particular quantity is indicated, as before mentioned -(p. <a href="#Page_41">41</a>), by joining to the term an indefinite adjective of -quantity, such as <i>some</i>, <i>a part of</i>, <i>certain</i>, &c., meaning that -an unknown part of the term enters into the proposition -as subject. Considerable doubt and ambiguity arise out of -the question whether the part may not in some cases be -the whole, and in the syllogism at least it must be understood -in this sense.<a id="FNanchor_61" href="#Footnote_61" class="fnanchor">61</a> Now, if we take a letter to represent -this indefinite part, we need make no change in our -formulæ to express the syllogisms Darii and Ferio. Consider -the example—</p> - -<table class="ml3em" style="width:70%;"> -<tr> -<td class="tal pl2hi">Some metals are of less density than water,</td> -<td class="tar pl1">(1)</td> -</tr> -<tr> -<td class="tal pl2hi">All bodies of less density than water will float upon the surface of water; hence</td> -<td class="tar pl1 vab">(2)</td> -</tr> -<tr> -<td class="tal pl2hi">Some metals will float upon the surface of water.</td> -<td class="tar pl1 vab">(3)</td> -</tr> -</table> - -<table class="ml13em"> -<tr> -<td class="tal pr2">Let</td> -<td class="tal">A = some metals,</td> -</tr> -<tr> -<td class="tal"></td> -<td class="tal pl2hi">B = body of less density than water,</td> -</tr> -<tr> -<td class="tal"></td> -<td class="tal pl2hi">C = floating on the surface of water</td> -</tr> -</table> - -<p class="ti0">then the propositions are evidently as before,</p> - -<table class="ml5em"> -<tr> -<td class="tal"></td> -<td class="tal">A = AB,</td> -<td class="tar pl2">(1)</td> -</tr> -<tr> -<td class="tal"></td> -<td class="tal">B = BC;</td> -<td class="tar pl2">(2)</td> -</tr> -<tr> -<td class="tal">hence  </td> -<td class="tal">A = ABC,</td> -<td class="tal pl2">(3)</td> -</tr> -</table> - -<p class="ti0">Thus the syllogism Darii does not really differ from Barbara. -If the reader prefer it, we can readily employ a -distinct symbol for the indefinite sign of quantity.</p> - -<table class="ml5em"> -<tr> -<td class="tal">Let  </td> -<td class="tal">P = some,</td> -</tr> -<tr> -<td class="tal"></td> -<td class="tal">Q = metal,</td> -</tr> -</table> - -<p class="ti0">B and C having the same meanings as before. Then the -premises become</p> - -<p><span class="pagenum" id="Page_57">57</span></p> - -<table class="ml5em"> -<tr> -<td class="tal">PQ = PQB,</td> -<td class="tar"><div>  (1)</div></td> -</tr> -<tr> -<td class="tal"> B = BC;</td> -<td class="tar"><div>  (2)</div></td> -</tr> -</table> - -<p class="ti0">hence, by substitution, as before,</p> - -<div class="ml5em"> -PQ = PQBC.  (3) -</div> - -<p class="ti0">Except that the formulæ look a little more complicated -there is no difference whatever.</p> - -<p>The mood Ferio is of exactly the same character as -Darii or Barbara, except that it involves the use of a -negative term. Take the example,</p> - -<div class="ml5em"> -<div class="pl2hi">Bodies which are equally elastic in all directions do -not doubly refract light;</div> - -<div class="pl2hi">Some crystals are bodies equally elastic in all directions; -therefore, some crystals do not doubly -refract light.</div> -</div> - -<p>Assigning the letters as follows:—</p> - -<div class="ml5em"> -A = some crystals,<br> -B = bodies equally elastic in all directions,<br> -C = doubly refracting light,<br> - <i>c</i> = not doubly refracting light. -</div> - -<p>Our argument is of the same form as before, and may -be concisely stated in one line,</p> - -<div class="ml5em"> -A = AB = AB<i>c</i>. -</div> - -<p class="ti0">If it is preferred to put PQ for the indefinite <i>some crystals</i>, -we have</p> - -<div class="ml5em"> -PQ = PQB = PQB<i>c</i>. -</div> - -<p class="ti0">The only difference is that the negative term c takes the -place of C in the mood Darii.</p> - - -<h3><i>Ellipsis of Terms in Partial Identities.</i></h3> - -<p>The reader will probably have noticed that the conclusion -which we obtain from premises is often more full than -that drawn by the old Aristotelian processes. Thus from -“Sodium is a metal,” and “Metals conduct electricity,” we -inferred (p. <a href="#Page_55">55</a>) that “Sodium = sodium, metal, conducting -electricity,” whereas the old logic simply concludes -that “Sodium conducts electricity.” Symbolically, from -A = AB, and B = BC, we get A = ABC, whereas the old -logic gets at the most A = AC. It is therefore well to -show that without employing any other principles of -inference than those already described, we may infer -A = AC from A = ABC, though we cannot infer the latter<span class="pagenum" id="Page_58">58</span> -more full and accurate result from the former. We may -show this most simply as follows:—</p> - -<p>By the first Law of Thought it is evident that</p> - -<div class="ml5em"> -AA = AA; -</div> - -<p class="ti0">and if we have given the proposition A = ABC, we may -substitute for both the A’s in the second side of the above, -obtaining</p> - -<div class="ml5em"> -AA = ABC . ABC.<br> -</div> - -<p class="ti0">But from the property of logical symbols expressed in the -Law of Simplicity (p. <a href="#Page_33">33</a>) some of the repeated letters may -be made to coalesce, and we have</p> - -<div class="ml5em"> -A = ABC . C. -</div> - -<p>Substituting again for ABC its equivalent A, we obtain</p> - -<div class="ml5em"> -A = AC, -</div> - -<p class="ti0">the desired result.</p> - -<p>By a similar process of reasoning it may be shown that -we can always drop out any term appearing in one member -of a proposition, provided that we substitute for it the -whole of the other member. This process was described in -my first logical Essay,<a id="FNanchor_62" href="#Footnote_62" class="fnanchor">62</a> as <i>Intrinsic Elimination</i>, but it -might perhaps be better entitled the <i>Ellipsis of Terms</i>. -It enables us to get rid of needless terms by strict -substitutive reasoning.</p> - - -<h3><i>Inference of a Simple from Two Partial Identities.</i></h3> - -<p>Two terms may be connected together by two partial -identities in yet another manner, and a case of inference -then arises which is of the highest importance. In the -two premises</p> - -<div class="ml5em"> -A = AB (1)<br> -B = AB (2) -</div> - -<p class="ti0">the second member of each is the same; so that we can by -obvious substitution obtain</p> - -<div class="ml5em"> -A = B. -</div> - -<p>Thus, in plain geometry we readily prove that “Every -equilateral triangle is also an equiangular triangle,” and we -can with equal ease prove that “Every equiangular triangle -is an equilateral triangle.” Thence by substitution, as -explained above, we pass to the simple identity,</p> - -<div class="ml5em"> -Equilateral triangle = equiangular triangle. -</div> - -<p><span class="pagenum" id="Page_59">59</span></p> - -<p class="ti0">We thus prove that one class of triangles is entirely -identical with another class; that is to say, they differ -only in our way of naming and regarding them.</p> - -<p>The great importance of this process of inference arises -from the fact that the conclusion is more simple and general -than either of the premises, and contains as much information -as both of them put together. It is on this account -constantly employed in inductive investigation, as will -afterwards be more fully explained, and it is the natural -mode by which we arrive at a conviction of the truth of -simple identities as existing between classes of numerous -objects.</p> - - -<h3><i>Inference of a Limited from Two Partial Identities.</i></h3> - -<p>We have considered some arguments which are of the -type treated by Aristotle in the first figure of the syllogism. -But there exist two other types of argument which employ -a pair of partial identities. If our premises are as shown -in these symbols,</p> - -<div class="ml5em"> -B = AB   (1)<br> -B = CB,   (2) -</div> - -<p class="ti0">we may substitute for B either by (1) in (2) or by (2) in -(1), and by both modes we obtain the conclusion</p> - -<div class="ml5em"> -AB = CB,   (3) -</div> - -<p class="ti0">a proposition of the kind which we have called a limited -identity (p. <a href="#Page_42">42</a>). Thus, for example,</p> - -<table class="ml5em"> -<tr> -<td class="tal">Potassium = potassium metal</td> -<td class="tar pl2">(1)</td> -</tr> -<tr> -<td class="tal pl2hi">Potassium = potassium capable of floating on water;</td> -<td class="tar pl2 vab">(2)</td> -</tr> -</table> - -<p class="ti0">hence</p> - -<table class="ml5em"> -<tr> -<td class="tal pl2hi">Potassium metal = potassium capable of floating on water.</td> -<td class="tar pl2 vab">(3)</td> -</tr> -</table> - -<p class="ti0">This is really a syllogism of the mood Darapti in the third -figure, except that we obtain a conclusion of a more exact -character than the old syllogism gives. From the premises -“Potassium is a metal” and “Potassium floats on water,” -Aristotle would have inferred that “Some metals float on -water.” But if inquiry were made what the “some -metals” are, the answer would certainly be “Metal which -is potassium.” Hence Aristotle’s conclusion simply leaves -out some of the information afforded in the premises. It<span class="pagenum" id="Page_60">60</span> -even leaves us open to interpret the <i>some metals</i> in a wider -sense than we are warranted in doing. From these distinct -defects of the old syllogism the process of substitution is -free, and the new process only incurs the possible objection -of being tediously minute and accurate.</p> - - -<h3><i>Miscellaneous Forms of Deductive Inference.</i></h3> - -<p>The more common forms of deductive reasoning having -been exhibited and demonstrated on the principle of -substitution, there still remain many, in fact an indefinite -number, which may be explained with nearly equal ease. -Such as involve the use of disjunctive propositions will be -described in a later chapter, and several of the syllogistic -moods which include negative terms will be more conveniently -treated after we have introduced the symbolic -use of the second and third laws of thought.</p> - -<p>We sometimes meet with a chain of propositions which -allow of repeated substitution, and form an argument -called in the old logic a Sorites. Take, for instance, the -premises</p> - -<table class="ml5em"> -<tr> -<td class="tal">Iron is a metal,</td> -<td class="tar"><div>(1)</div></td> -</tr> -<tr> -<td class="tal pl2hi">Metals are good conductors of electricity,</td> -<td class="tar pl2 vab">(2)</td> -</tr> -<tr> -<td class="tal pl2hi">Good conductors ofelectricity are useful for telegraphic purposes.</td> -<td class="tar pl2 vab">(3)</td> -</tr> -</table> - -<p class="ti0">It obviously follows that</p> - -<table class="ml5em"> -<tr> -<td class="tal pl2hi">Iron is useful for telegraphic purposes.</td> -<td class="tar pl2">(4)</td> -</tr> -</table> - -<p class="ti0">Now if we take our letters thus,</p> - -<div class="ml7h5"> -A = Iron,   B = metal,   C = good conductor of -electricity,   D = useful for telegraphic purposes, -</div> - -<p class="ti0">the premises will assume the forms</p> - -<table class="ml5em"> -<tr> -<td class="tal">A = AB,</td> -<td class="tar pl2">(1)</td> -</tr> -<tr> -<td class="tal">B = BC,</td> -<td class="tar pl2">(2)</td> -</tr> -<tr> -<td class="tal">C = CD.</td> -<td class="tar pl2">(3)</td> -</tr> -</table> - -<p class="ti0">For B in (1) we can substitute its equivalent in (2) -obtaining, as before,</p> - -<div class="ml5em"> -A = ABC. -</div> - -<p class="ti0">Substituting for C in this intermediate result its equivalent -as given in (3), we obtain the complete conclusion</p> - -<table class="ml5em"> -<tr> -<td class="tal">A = ABCD.</td> -<td class="tar pl2">(4)</td> -</tr> -</table> - -<p class="ti0">The full interpretation is that <i>Iron is iron, metal, good -conductor of electricity, useful for telegraphic purposes</i>, which<span class="pagenum" id="Page_61">61</span> -is abridged in common language by the ellipsis of the -circumstances which are not of immediate importance.</p> - -<p>Instead of all the propositions being exactly of the same -kind as in the last example, we may have a series of -premises of various character; for instance,</p> - -<table class="ml5em"> -<tr> -<td class="tal">Common salt is sodium chloride,</td> -<td class="tar pl2">(1)</td> -</tr> -<tr> -<td class="tal pl2hi">Sodium chloride crystallizes in a cubical form,</td> -<td class="tar pl2 vab">(2)</td> -</tr> -<tr> -<td class="tal pl2hi">What crystallizes in a cubical form does not possess the power of double refraction;</td> -<td class="tar pl2 vab">(3)</td> -</tr> -</table> - -<p class="ti0">it will follow that</p> - -<table class="ml5em"> -<tr> -<td class="tal pl2hi">Common salt does not possess the power of double refraction.</td> -<td class="tar pl2 vab">(4)</td> -</tr> -</table> - -<p class="ti0">Taking our letter-terms thus,</p> - -<div class="ml5em"> -A = Common salt,<br> -B = Sodium chloride,<br> -C = Crystallizing in a cubical form,<br> -D = Possessing the power of double refraction, -</div> - -<p class="ti0">we may state the premises in the forms</p> - -<table class="ml5em"> -<tr> -<td class="tal">A = B,</td> -<td class="tar pl2">(1)</td> -</tr> -<tr> -<td class="tal">B = BC,</td> -<td class="tar pl2">(2)</td> -</tr> -<tr> -<td class="tal">C = C<i>d</i>.</td> -<td class="tar pl2">(3)</td> -</tr> -</table> - -<p class="ti0">Substituting by (3) in (2) and then by (2) as thus altered -in (1) we obtain</p> - -<table class="ml5em"> -<tr> -<td class="tal">A = BC<i>d</i>,</td> -<td class="tar pl2">(4)</td> -</tr> -</table> - -<p class="ti0">which is a more precise version of the common conclusion.</p> - -<p>We often meet with a series of propositions describing -the qualities or circumstances of the one same thing, and -we may combine them all into one proposition by the -process of substitution. This case is, in fact, that which -Dr. Thomson has called “Immediate Inference by the -sum of several predicates,” and his example will serve my -purpose well.<a id="FNanchor_63" href="#Footnote_63" class="fnanchor">63</a> He describes copper as “A metal—of a -red colour—and disagreeable smell—and taste—all the -preparations of which are poisonous—which is highly -malleable—ductile—and tenacious—with a specific gravity -of about 8.83.” If we assign the letter A to copper, and the -succeeding letters of the alphabet in succession to the series -of predicates, we have nine distinct statements, of the form -A = AB (1) A = AC (2) A = AD (3) . . . A = AK (9). -We can readily combine these propositions into one by<span class="pagenum" id="Page_62">62</span> -substituting for A in the second side of (1) its expression -in (2). We thus get</p> - -<div class="ml5em"> -A = ABC, -</div> - -<p class="ti0">and by repeating the process over and over again we -obviously get the single proposition</p> - -<div class="ml5em"> -A = ABCD . . . JK. -</div> - -<p class="ti0">But Dr. Thomson is mistaken in supposing that we can -obtain in this manner a <i>definition</i> of copper. Strictly -speaking, the above proposition is only a <i>description</i> of -copper, and all the ordinary descriptions of substances in -scientific works may be summed up in this form. Thus we -may assert of the organic substances called Paraffins that -they are all saturated hydrocarbons, incapable of uniting -with other substances, produced by heating the alcoholic -iodides with zinc, and so on. It may be shown that no -amount of ordinary description can be equivalent to a definition -of any substance.</p> - - -<h3><i>Fallacies.</i></h3> - -<p>I have hitherto been engaged in showing that all the -forms of reasoning of the old syllogistic logic, and an -indefinite number of other forms in addition, may be -readily and clearly explained on the single principle of -substitution. It is now desirable to show that the same -principle will prevent us falling into fallacies. So long -as we exactly observe the one rule of substitution of -equivalents it will be impossible to commit a <i>paralogism</i>, -that is to break any one of the elaborate rules of the -ancient system. The one new rule is thus proved to be as -powerful as the six, eight, or more rules by which the correctness -of syllogistic reasoning was guarded.</p> - -<p>It was a fundamental rule, for instance, that two negative -premises could give no conclusion. If we take the -propositions</p> - -<table class="ml5em"> -<tr> -<td class="tal pl2hi">Granite is not a sedimentary rock,</td> -<td class="tar pl2 vab">(1)</td> -</tr> -<tr> -<td class="tal pl2hi">Basalt is not a sedimentary rock,</td> -<td class="tar pl2 vab">(2)</td> -</tr> -</table> - -<p class="ti0">we ought not to be able to draw any inference concerning -the relation between granite and basalt. Taking our -letter-terms thus:</p> - -<div class="ml5em"> -A = granite,   B = sedimentary rock,   C = basalt,<br> -</div> - -<p class="ti0">the premises may be expressed in the forms</p> - -<p><span class="pagenum" id="Page_63">63</span></p> - -<table class="ml5em"> -<tr> -<td class="tal">A ~ B,</td> -<td class="tar pl2">(1)</td> -</tr> -<tr> -<td class="tal">C ~ B.</td> -<td class="tar pl2">(2)</td> -</tr> -</table> - -<p class="ti0">We have in this form two statements of difference; but -the principle of inference can only work with a statement -of agreement or identity (p. <a href="#Page_63">63</a>). Thus our rule gives -us no power whatever of drawing any inference; this is -exactly in accordance with the fifth rule of the syllogism.</p> - -<p>It is to be remembered, indeed, that we claim the -power of always turning a negative proposition into an -affirmative one (p. <a href="#Page_45">45</a>); and it might seem that the old rule -against negative premises would thus be circumvented. -Let us try. The premises (1) and (2) when affirmatively -stated take the forms</p> - -<table class="ml5em"> -<tr> -<td class="tal">A = A<i>b</i></td> -<td class="tar pl2">(1)</td> -</tr> -<tr> -<td class="tal">C = C<i>b</i>.</td> -<td class="tar pl2">(2)</td> -</tr> -</table> - -<p class="ti0">The reader will find it impossible by the rule of substitution -to discover a relation between A and C. Three terms -occur in the above premises, namely A, <i>b</i>, and C; but they -are so combined that no term occurring in one has its -exact equivalent stated in the other. No substitution -can therefore be made, and the principle of the fifth rule of -the syllogism holds true. Fallacy is impossible.</p> - -<p>It would be a mistake, however, to suppose that the -mere occurrence of negative terms in both premises of a -syllogism renders them incapable of yielding a conclusion. -The old rule informed us that from two negative premises -no conclusion could be drawn, but it is a fact that the rule -in this bare form does not hold universally true; and I -am not aware that any precise explanation has been given -of the conditions under which it is or is not imperative. -Consider the following example:</p> - -<table class="ml5em"> -<tr> -<td class="tal pl2hi">Whatever is not metallic is not capable of powerful magnetic influence,</td> -<td class="tar pl2 vab">(1)</td> -</tr> -<tr> -<td class="tal">Carbon is not metallic,</td> -<td class="tar pl2">(2)</td> -</tr> -<tr> -<td class="tal pl2hi">Therefore, carbon is not capable of powerful magnetic influence.</td> -<td class="tar pl2 vab">(3)</td> -</tr> -</table> - -<p class="ti0">Here we have two distinctly negative premises (1) and -(2), and yet they yield a perfectly valid negative conclusion -(3). The syllogistic rule is actually falsified in its bare -and general statement. In this and many other cases we -can convert the propositions into affirmative ones which will -yield a conclusion by substitution without any difficulty.<span class="pagenum" id="Page_64">64</span></p> - -<p class="ti0">To show this let</p> - -<div class="ml5em"> -A = carbon,<br> -B = metallic,<br> -<div class="pl2hi">C = capable of powerful magnetic influence.</div> -</div> - -<p class="ti0">The premises readily take the forms</p> - -<table class="ml5em"> -<tr> -<td class="tal"><i>b</i> = <i>bc</i>,</td> -<td class="tar pl2">(1)</td> -</tr> -<tr> -<td class="tal">A = A<i>b</i>,</td> -<td class="tar pl2">(2)</td> -</tr> -</table> - -<p class="ti0">and substitution for <i>b</i> in (2) by means of (1) gives the -conclusion</p> - -<table class="ml5em"> -<tr> -<td class="tal">A = A<i>bc</i>.</td> -<td class="tar pl2">(3)</td> -</tr> -</table> - -<p>Our principle of inference then includes the rule of -negative premises whenever it is true, and discriminates -correctly between the cases where it does and does not -hold true.</p> - -<p>The paralogism, anciently called <i>the Fallacy of Undistributed -Middle</i>, is also easily exhibited and infallibly -avoided by our system. Let the premises be</p> - -<table class="ml5em"> -<tr> -<td class="tal">Hydrogen is an element,</td> -<td class="tar pl2">(1)</td> -</tr> -<tr> -<td class="tal">All metals are elements.</td> -<td class="tar pl2">(2)</td> -</tr> -</table> - -<p class="ti0">According to the syllogistic rules the middle term “element” -is here undistributed, and no conclusion can be obtained; -we cannot tell then whether hydrogen is or is not a metal. -Represent the terms as follows</p> - -<div class="ml5em"> -A = hydrogen,<br> -B = element,<br> -C = metal. -</div> - -<p class="ti0">The premises then become</p> - -<table class="ml5em"> -<tr> -<td class="tal">A = AB,</td> -<td class="tar pl2">(1)</td> -</tr> -<tr> -<td class="tal">C = CB.</td> -<td class="tar pl2">(2)</td> -</tr> -</table> - -<p>The reader will here, as in a former page (p. <a href="#Page_62">62</a>), find it -impossible to make any substitution. The only term which -occurs in both premises is B, but it is differently combined -in the two premises. For B we must not substitute A, -which is equivalent to AB, not to B. Nor must we confuse -together CB and AB, which, though they contain one common -letter, are different aggregate terms. The rule of substitution -gives us no right to decompose combinations; -and if we adhere rigidly to the rule, that if two terms are -stated to be equivalent we may substitute one for the other, -we cannot commit the fallacy. It is apparent that the form -of premises stated above is the same as that which we -obtained by translating two negative premises into the -affirmative form.</p> - -<p><span class="pagenum" id="Page_65">65</span></p> - -<p>The old fallacy, technically called the <i>Illicit Process of -the Major Term</i>, is more easy to commit and more difficult -to detect than any other breach of the syllogistic rules. In -our system it could hardly occur. From the premises</p> - -<table class="ml5em"> -<tr> -<td class="tal pl2hi vab">All planets are subject to gravity,</td> -<td class="tar pl2">(1)</td> -</tr> -<tr> -<td class="tal pl2hi vab">Fixed stars are not planets,</td> -<td class="tar pl2">(2)</td> -</tr> -</table> - -<p class="ti0">we might inadvertently but fallaciously infer that, “Fixed -stars are not subject to gravity.” To reduce the premises -to symbolic form, let</p> - -<div class="ml5em"> -A = planet<br> -B = fixed star<br> -C = subject to gravity; -</div> - -<p class="ti0">then we have the propositions</p> - -<table class="ml5em"> -<tr> -<td class="tal">A = AC</td> -<td class="tar pl2">(1)</td> -</tr> -<tr> -<td class="tal">B = B<i>a</i>.</td> -<td class="tar pl2">(2)</td> -</tr> -</table> - -<p class="ti0">The reader will try in vain to produce from these premises -by legitimate substitution any relation between B and C; -he could not then commit the fallacy of asserting that B is -not C.</p> - -<p>There remain two other kinds of paralogism, commonly -known as the fallacy of Four Terms and the Illicit Process -of the Minor Term. They are so evidently impossible -while we obey the rule of the substitution of equivalents, -that it is not necessary to give any illustrations. When -there are four distinct terms in two propositions as in -A = B and C = D, there could evidently be no opening for -substitution. As to the Illicit Process of the Minor Term -it consists in a flagrant substitution for a term of another -wider term which is not known to be equivalent to it, -and which is therefore not allowed by our rule to be -substituted for it.</p> - -<hr class="chap x-ebookmaker-drop"> - -<div class="chapter"> -<p><span class="pagenum" id="Page_66">66</span></p> - -<h2 class="nobreak" id="CHAPTER_V">CHAPTER V.<br> - -<span class="title">DISJUNCTIVE PROPOSITIONS.</span></h2> -</div> - -<p class="ti0">In the previous chapter I have exhibited various cases -of deductive reasoning by the process of substitution, avoiding -the introduction of disjunctive propositions; but we -cannot long defer the consideration of this more complex -class of identities. General terms arise, as we have seen -(p. <a href="#Page_24">24</a>), from classifying or mentally uniting together all -objects which agree in certain qualities, the value of this -union consisting in the fact that the power of knowledge -is multiplied thereby. In forming such classes or general -notions, we overlook or abstract the points of difference -which exist between the objects joined together, and fix our -attention only on the points of agreement. But every -process of thought may be said to have its inverse process, -which consists in undoing the effects of the direct process. -Just as division undoes multiplication, and evolution undoes -involution, so we must have a process which undoes -generalization, or the operation of forming general notions. -This inverse process will consist in distinguishing the -separate objects or minor classes which are the constituent -parts of any wider class. If we mentally unite together -certain objects visible in the sky and call them planets, we -shall afterwards need to distinguish the contents of this -general notion, which we do in the disjunctive proposition—</p> - -<div class="ml7h5"> -A planet is either Mercury or Venus or the Earth or -. . . or Neptune. -</div> - -<p class="ti0">Having formed the very wide class “vertebrate animal,” -we may specify its subordinate classes thus:—“A vertebrate<span class="pagenum" id="Page_67">67</span> -animal is either a mammal, bird, reptile, or fish.” -Nor is there any limit to the number of possible alternatives. -“An exogenous plant is either a ranunculus, a -poppy, a crucifer, a rose, or it belongs to some one of the -other seventy natural orders of exogens at present recognized -by botanists.” A cathedral church in England must -be either that of London, Canterbury, Winchester, Salisbury, -Manchester, or of one of about twenty-four cities -possessing such churches. And if we were to attempt to -specify the meaning of the term “star,” we should require -to enumerate as alternatives, not only the many thousands -of stars recorded in catalogues, but the many millions unnamed.</p> - -<p>Whenever we thus distinguish the parts of a general -notion we employ a disjunctive proposition, in at least one -side of which are several alternatives joined by the so-called -disjunctive conjunction or, a contracted form of <i>other</i>. -There must be some relation between the parts thus connected -in one proposition; we may call it the <i>disjunctive</i> or -<i>alternative</i> relation, and we must carefully inquire into its -nature. This relation is that of ignorance and doubt, -giving rise to choice. Whenever we classify and abstract -we must open the way to such uncertainty. By fixing our -attention on certain attributes to the exclusion of others, -we necessarily leave it doubtful what those other attributes -are. The term “molar tooth” bears upon the face of it -that it is a part of the wider term “tooth.” But if we -meet with the simple term “tooth” there is nothing to indicate -whether it is an incisor, a canine, or a molar tooth. -This doubt, however, may be resolved by further information, -and we have to consider what are the appropriate -logical processes for treating disjunctive propositions in -connection with other propositions disjunctive or otherwise.</p> - - -<h3><i>Expression of the Alternative Relation.</i></h3> - -<p>In order to represent disjunctive propositions with convenience -we require a sign of the alternative relation, -equivalent to one meaning at least of the little conjunction -<i>or</i> so frequently used in common language. I propose -to use for this purpose the symbol ꖌ. In my first -logical essay I followed the practice of Boole and adopted<span class="pagenum" id="Page_68">68</span> -the sign +; but this sign should not be employed unless there -exists exact analogy between mathematical addition and -logical alternation. We shall find that the analogy is imperfect, -and that there is such profound difference between -logical and mathematical terms as should prevent our -uniting them by the same symbol. Accordingly I have -chosen a sign ꖌ, which seems aptly to suggest whatever -degree of analogy may exist without implying more. -The exact meaning of the symbol we will now proceed to -investigate.</p> - - -<h3><i>Nature of the Alternative Relation.</i></h3> - -<p>Before treating disjunctive propositions it is indispensable -to decide whether the alternatives must be considered -exclusive or unexclusive. By <i>exclusive alternatives</i> we -mean those which cannot contain the same things. If we -say “Arches are circular or pointed,” it is certainly to be -understood that the same arch cannot be described as both -circular and pointed. Many examples, on the other hand, -can readily be suggested in which two or more alternatives -may hold true of the same object. Thus</p> - -<div class="ml7h5"> -Luminous bodies are self-luminous or luminous by -reflection. -</div> - -<p class="ti0">It is undoubtedly possible, by the laws of optics, that the -same surface may at one and the same moment give off -light of its own and reflect light from other bodies. We -speak familiarly of <i>deaf or dumb</i> persons, knowing that the -majority of those who are deaf from birth are also dumb.</p> - -<p>There can be no doubt that in a great many cases, -perhaps the greater number of cases, alternatives are -exclusive as a matter of fact. Any one number is -incompatible with any other; one point of time or place -is exclusive of all others. Roger Bacon died either in -1284 or 1292; it is certain that he could not die in both -years. Henry Fielding was born either in Dublin or -Somersetshire; he could not be born in both places. -There is so much more precision and clearness in the use -of exclusive alternatives that we ought doubtless to select -them when possible. Old works on logic accordingly -contained a rule directing that the <i>Membra dividentia</i>, the<span class="pagenum" id="Page_69">69</span> -parts of a division or the constituent species of a genus, -should be exclusive of each other.</p> - -<p>It is no doubt owing to the great prevalence and convenience -of exclusive divisions that the majority of logicians -have held it necessary to make every alternative in -a disjunctive proposition exclusive of every other one. -Aquinas considered that when this was not the case the -proposition was actually <i>false</i>, and Kant adopted the -same opinion.<a id="FNanchor_64" href="#Footnote_64" class="fnanchor">64</a> A multitude of statements to the same -effect might readily be quoted, and if the question were -to be determined by the weight of historical evidence, -it would certainly go against my view. Among recent -logicians Hamilton, as well as Boole, took the exclusive -side. But there are authorities to the opposite effect. -Whately, Mansel, and J. S. Mill have all pointed out that -we may often treat alternatives as <i>Compossible</i>, or true at -the same time. Whately gives us an example,<a id="FNanchor_65" href="#Footnote_65" class="fnanchor">65</a> “Virtue -tends to procure us either the esteem of mankind, or the -favour of God,” and he adds—“Here both members are -true, and consequently from one being affirmed we are not -authorized to deny the other. Of course we are left to -conjecture in each case, from the context, whether it is -meant to be implied that the members are or are not -exclusive.” Mansel says,<a id="FNanchor_66" href="#Footnote_66" class="fnanchor">66</a> “<i>We may happen to know</i> that -two alternatives cannot be true together, so that the -affirmation of the second necessitates the denial of the -first; but this, as Boethius observes, is a <i>material</i>, not a -<i>formal</i> consequence.” Mill has also pointed out the -absurdities which would arise from always interpreting -alternatives as exclusive. “If we assert,” he says,<a id="FNanchor_67" href="#Footnote_67" class="fnanchor">67</a> “that -a man who has acted in some particular way must be -either a knave or a fool, we by no means assert, or intend -to assert, that he cannot be both.” Again, “to make an -entirely unselfish use of despotic power, a man must be -either a saint or a philosopher.... Does the disjunctive -premise necessarily imply, or must it be construed -as supposing, that the same person cannot be both a<span class="pagenum" id="Page_70">70</span> -saint and a philosopher? Such a construction would be -ridiculous.”</p> - -<p>I discuss this subject fully because it is really the point -which separates my logical system from that of Boole. -In his <i>Laws of Thought</i> (p. 32) he expressly says, -“In strictness, the words ‘and,’ ‘or,’ interposed between -the terms descriptive of two or more classes of objects, -imply that those classes are quite distinct, so that no -member of one is found in another.” This I altogether -dispute. In the ordinary use of these conjunctions we do -not join distinct terms only; and when terms so joined -do prove to be logically distinct, it is by virtue of a <i>tacit -premise</i>, something in the meaning of the names and -our knowledge of them, which teaches us that they are -distinct. If our knowledge of the meanings of the -words joined is defective it will often be impossible -to decide whether terms joined by conjunctions are -exclusive or not.</p> - -<p>In the sentence “Repentance is not a single act, but -a habit or virtue,” it cannot be implied that a virtue is -not a habit; by Aristotle’s definition it is. Milton has the -expression in one of his sonnets, “Unstain’d by gold or -fee,” where it is obvious that if the fee is not always gold, -the gold is meant to be a fee or bribe. Tennyson has the -expression “wreath or anadem.” Most readers would be -quite uncertain whether a wreath may be an anadem, or -an anadem a wreath, or whether they are quite distinct or -quite the same. From Darwin’s <i>Origin of Species</i>, I -take the expression, “When we see any <i>part or organ</i> -developed in a remarkable <i>degree or manner</i>.” In this, <i>or</i> -is used twice, and neither time exclusively. For if <i>part</i> -and <i>organ</i> are not synonymous, at any rate an organ is a -part. And it is obvious that a part may be developed at -the same time both in an extraordinary degree and an -extraordinary manner, although such cases may be comparatively -rare.</p> - -<p>From a careful examination of ordinary writings, it will -thus be found that the meanings of terms joined by “and,” -“or” vary from absolute identity up to absolute contrariety. -There is no logical condition of distinctness at all, and -when we do choose exclusive alternatives, it is because -our subject demands it. The matter, not the form of an<span class="pagenum" id="Page_71">71</span> -expression, points out whether terms are exclusive or not.<a id="FNanchor_68" href="#Footnote_68" class="fnanchor">68</a> -In bills, policies, and other kinds of legal documents, it -is sometimes necessary to express very distinctly that -alternatives are not exclusive. The form -<span class="nowrap"><span class="fraction2"><span class="fnum2">and</span><span class="bar">/</span><span class="fden2">or</span></span></span> -is then used, and, as Mr. J. J. Murphy has remarked, this form -coincides exactly in meaning with the symbol ꖌ.</p> - -<p>In the first edition of this work (vol. i., p. 81), I took -the disjunctive proposition “Matter is solid, or liquid, or -gaseous,” and treated it as an instance of exclusive alternatives, -remarking that the same portion of matter cannot be -at once solid and liquid, properly speaking, and that still less -can we suppose it to be solid and gaseous, or solid, liquid, -and gaseous all at the same time. But the experiments of -Professor Andrews show that, under certain conditions of -temperature and pressure, there is no abrupt change from -the liquid to the gaseous state. The same substance may be -in such a state as to be indifferently described as liquid and -gaseous. In many cases, too, the transition from solid to -liquid is gradual, so that the properties of solidity are at least -partially joined with those of liquidity. The proposition -then, instead of being an instance of exclusive alternatives, -seems to afford an excellent instance to the opposite effect. -When such doubts can arise, it is evidently impossible to -treat alternatives as absolutely exclusive by the logical -nature of the relation. It becomes purely a question of -the matter of the proposition.</p> - -<p>The question, as we shall afterwards see more fully, is -one of the greatest theoretical importance, because it -concerns the true distinction between the sciences of -Logic and Mathematics. It is the foundation of number -that every unit shall be distinct from every other unit; -but Boole imported the conditions of number into the -science of Logic, and produced a system which, though -wonderful in its results, was not a system of logic at all.</p> - - -<h3><i>Laws of the Disjunctive Relation.</i></h3> - -<p>In considering the combination or synthesis of terms -(p. <a href="#Page_30">30</a>), we found that certain laws, those of Simplicity<span class="pagenum" id="Page_72">72</span> -and Commutativeness, must be observed. In uniting -terms by the disjunctive symbol we shall find that the -same or closely similar laws hold true. The alternatives -of either member of a disjunctive proposition are certainly -commutative. Just as we cannot properly distinguish -between <i>rich and rare gems</i> and <i>rare and rich gems</i>, so we -must consider as identical the expression <i>rich or rare gems</i>, -and <i>rare or rich gems</i>. In our symbolic language we may -say</p> - -<div class="ml5em"> -A ꖌ B = B ꖌ A. -</div> - -<p>The order of statement, in short, has no effect upon the -meaning of an aggregate of alternatives, so that the -Law of Commutativeness holds true of the disjunctive -symbol.</p> - -<p>As we have admitted the possibility of joining as alternatives -terms which are not really different, the question -arises, How shall we treat two or more alternatives when -they are clearly shown to be the same? If we have it -asserted that P is Q or R, and it is afterwards proved that -Q is but another name for R, the result is that P is either -R or R. How shall we interpret such a statement? What -would be the meaning, for instance, of “wreath or anadem” -if, on referring to a dictionary, we found <i>anadem</i> described -as a wreath? I take it to be self-evident that the meaning -would then become simply “wreath.” Accordingly we -may affirm the general law</p> - -<div class="ml5em"> -A ꖌ A = A.<br> -</div> - -<p>Any number of identical alternatives may always be -reduced to, and are logically equivalent to, any one of -those alternatives. This is a law which distinguishes -mathematical terms from logical terms, because it obviously -does not apply to the former. I propose to call it the <i>Law -of Unity</i>, because it must really be involved in any -definition of a mathematical unit. This law is closely -analogous to the Law of Simplicity, AA = A; and the -nature of the connection is worthy of attention.</p> - -<p>Few or no logicians except De Morgan have adequately -noticed the close relation between combined and disjunctive -terms, namely, that every disjunctive term is the negative -of a corresponding combined term, and <i>vice versâ</i>. Consider -the term</p> - -<div class="ml5em"> -Malleable dense metal. -</div> - -<p><span class="pagenum" id="Page_73">73</span></p> - -<p>How shall we describe the class of things which are not -malleable-dense-metals? Whatever is included under that -term must have all the qualities of malleability, denseness, -and metallicity. Wherever any one or more of the qualities -is wanting, the combined term will not apply. Hence the -negative of the whole term is</p> - -<div class="ml7h5"> -Not-malleable or not-dense or not-metallic. -</div> - -<p>In the above the conjunction <i>or</i> must clearly be interpreted -as unexclusive; for there may readily be objects -which are both not-malleable, and not-dense, and perhaps -not-metallic at the same time. If in fact we were required -to use <i>or</i> in a strictly exclusive manner, it would be -requisite to specify seven distinct alternatives in order to -describe the negative of a combination of three terms. -The negatives of four or five terms would consist of fifteen -or thirty-one alternatives. This consideration alone is -sufficient to prove that the meaning of <i>or</i> cannot be -always exclusive in common language.</p> - -<p>Expressed symbolically, we may say that the negative of</p> - -<table class="ml5em"> -<tr> -<td class="tal"></td> -<td class="tac"><div>ABC</div></td> -</tr> -<tr> -<td class="tal">is</td> -<td class="tac pl2">not-A or not-B or not-C;</td> -</tr> -<tr> -<td class="tal">that is,</td> -<td class="tac"><div><i>a</i> ꖌ <i>b</i> ꖌ <i>c</i>.</div></td> -</tr> -</table> - -<p>Reciprocally the negative of</p> - -<table class="ml5em"> -<tr> -<td class="tal"></td> -<td class="tac pl2">P ꖌ Q ꖌ R</td> -</tr> -<tr> -<td class="tal">is</td> -<td class="tac pl2"><i>pqr</i>.</td> -</tr> -</table> - -<p>Every disjunctive term, then, is the negative of a -combined term, and <i>vice versâ</i>.</p> - -<p>Apply this result to the combined term AAA, and its -negative is</p> - -<div class="ml5em"> -<i>a</i> ꖌ <i>a</i> ꖌ <i>a</i>. -</div> - -<p class="ti0">Since AAA is by the Law of Simplicity equivalent to A, -so <i>a</i> ꖌ <i>a</i> ꖌ <i>a</i> must be equivalent to <i>a</i>, and the Law of -Unity holds true. Each law thus necessarily presupposes -the other.</p> - - -<h3><i>Symbolic expression of the Law of Duality.</i></h3> - -<p>We may now employ our symbol of alternation to -express in a clear and formal manner the third Fundamental -Law of Thought, which I have called the Law -of Duality (p. <a href="#Page_6">6</a>). Taking A to represent any class or<span class="pagenum" id="Page_74">74</span> -object or quality, and B any other class, object or quality, -we may always assert that A either agrees with B, or does -not agree. Thus we may say</p> - -<div class="ml5em"> -A = AB ꖌ A<i>b</i>.<br> -</div> - -<p>This is a formula which will henceforth be constantly -employed, and it lies at the basis of reasoning.</p> - -<p>The reader may perhaps wish to know why A is inserted -in both alternatives of the second member of the identity, -and why the law is not stated in the form</p> - -<div class="ml5em"> -A = B ꖌ <i>b</i>.<br> -</div> - -<p class="ti0">But if he will consider the contents of the last section -(p. <a href="#Page_73">73</a>), he will see that the latter expression cannot be -correct, otherwise no term could have a corresponding -negative term. For the negative of B ꖌ <i>b</i> is <i>b</i>B, or a self-contradictory -term; thus if A were identical with B ꖌ <i>b</i>, -its negative <i>a</i> would be non-existent. To say the least, -this result would in most cases be an absurd one, and I -see much reason to think that in a strictly logical point of -view it would always be absurd. In all probability we -ought to assume as a fundamental logical axiom that <i>every -term has its negative in thought</i>. We cannot think at all -without separating what we think about from other things, -and these things necessarily form the negative notion.<a id="FNanchor_69" href="#Footnote_69" class="fnanchor">69</a> -It follows that any proposition of the form A = B ꖌ <i>b</i> is -just as self-contradictory as one of the form A = B<i>b</i>.</p> - -<p>It is convenient to recapitulate in this place the three -Laws of Thought in their symbolic form, thus</p> - -<table class="ml5em"> -<tr> -<td class="tal">Law of Identity</td> -<td class="tal pl2"> A = A.</td> -</tr> -<tr> -<td class="tal">Law of Contradiction</td> -<td class="tal pl2">A<i>a</i> = 0.</td> -</tr> -<tr> -<td class="tal">Law of Duality</td> -<td class="tal pl2"> A = AB ꖌ A<i>b</i>.</td> -</tr> -</table> - - -<h3><i>Various Forms of the Disjunctive Proposition.</i></h3> - -<p>Disjunctive propositions may occur in a great variety of -forms, of which the old logicians took insufficient notice. -There may be any number of alternatives, each of which -may be a combination of any number of simple terms. A -proposition, again, may be disjunctive in one or both -members. The proposition</p> - -<p><span class="pagenum" id="Page_75">75</span></p> - -<div class="ml7h5"> -Solids or liquids or gases are electrics or conductors -of electricity -</div> - -<p class="ti0">is an example of the doubly disjunctive form. The meaning -of such a proposition is that whatever falls under any -one or more alternatives on one side must fall under one -or more alternatives on the other side. From what has -been said before, it is apparent that the proposition</p> - -<div class="ml5em"> -A ꖌ B = C ꖌ D -</div> - -<p class="ti0">will correspond to</p> - -<div class="ml5em"> -<i>ab</i> = <i>cd</i>, -</div> - -<p class="ti0">each member of the latter being the negative of a member -of the former proposition.</p> - -<p>As an instance of a complex disjunctive proposition I -may give Senior’s definition of wealth, which, briefly -stated, amounts to the proposition “Wealth is what is -transferable, limited in supply, and either productive of -pleasure or preventive of pain.”<a id="FNanchor_70" href="#Footnote_70" class="fnanchor">70</a></p> - -<table class="ml5em"> -<tr> -<td class="tal pr1">Let</td> -<td class="tal">A = wealth</td> -</tr> -<tr> -<td class="tal"></td> -<td class="tal">B = transferable</td> -</tr> -<tr> -<td class="tal"></td> -<td class="tal">C = limited in supply</td> -</tr> -<tr> -<td class="tal"></td> -<td class="tal">D = productive of pleasure</td> -</tr> -<tr> -<td class="tal"></td> -<td class="tal">E = preventive of pain.</td> -</tr> -</table> - -<p class="ti0">The definition takes the form</p> - -<div class="ml5em"> -A = BC(D ꖌ E);<br> -</div> - -<p class="ti0">but if we develop the alternatives by a method to be -afterwards more fully considered, it becomes</p> - -<div class="ml5em"> -A = BCDE ꖌ BCD<i>e</i> ꖌ BC<i>d</i>E.<br> -</div> - -<p>An example of a still more complex proposition is -found in De Morgan’s writings,<a id="FNanchor_71" href="#Footnote_71" class="fnanchor">71</a> as follows:—“He must -have been rich, and if not absolutely mad was weakness -itself, subjected either to bad advice or to most unfavourable -circumstances.”</p> - -<p>If we assign the letters of the alphabet in succession, -thus,</p> - -<div class="ml5em"> -A = he<br> -B = rich<br> -C = absolutely mad<br> -D = weakness itself<br> -E = subjected to bad advice<span class="pagenum" id="Page_76">76</span><br> -F = subjected to most unfavourable circumstances, -</div> - -<p class="ti0">the proposition will take the form</p> - -<div class="ml5em"> -A = AB{C ꖌ D (E ꖌ F)}, -</div> - -<p class="ti0">and if we develop the alternatives, expressing some of -the different cases which may happen, we obtain</p> - -<div class="ml7h5">A = ABC ꖌ AB<i>c</i>DEF ꖌ AB<i>c</i>DE<i>f</i> ꖌ AB<i>c</i>D<i>e</i>F.<br> -</div> - -<p>The above gives the strict logical interpretation of the -sentence, and the first alternative ABC is capable of development -into eight cases, according as D, E and F are or -are not present. Although from our knowledge of the -matter, we may infer that weakness of character cannot be -asserted of a person absolutely mad, there is no explicit -statement to this effect.</p> - - -<h3><i>Inference by Disjunctive Propositions.</i></h3> - -<p>Before we can make a free use of disjunctive propositions -in the processes of inference we must consider how -disjunctive terms can be combined together or with -simple terms. In the first place, to combine a simple term -with a disjunctive one, we must combine it with every -alternative of the disjunctive term. A vegetable, for -instance, is either a herb, a shrub, or a tree. Hence an -exogenous vegetable is either an exogenous herb, or an -exogenous shrub, or an exogenous tree. Symbolically -stated, this process of combination is as follows,</p> - -<div class="ml5em"> -A(B ꖌ C) = AB ꖌ AC.<br> -</div> - -<p>Secondly, to combine two disjunctive terms with each -other, combine each alternative of one with each alternative -of the other. Since flowering plants are either -exogens or endogens, and are at the same time either -herbs, shrubs or trees, it follows that there are altogether -six alternatives—namely, exogenous herbs, exogenous -shrubs, exogenous trees, endogenous herbs, endogenous -shrubs, endogenous trees. This process of combination is -shown in the general form</p> - -<div class="ml7h5"> -(A ꖌ B) (C ꖌ D ꖌ E) = AC ꖌ AD ꖌ AE ꖌ BC ꖌ BD ꖌ BE. -</div> - -<p>It is hardly necessary to point out that, however -numerous the terms combined, or the alternatives in those -terms, we may effect the combination, provided each alternative -is combined with each alternative of the other -terms, as in the algebraic process of multiplication.</p> - -<p><span class="pagenum" id="Page_77">77</span></p> - -<p>Some processes of deduction may be at once exhibited. -We may always, for instance, unite the same qualifying -term to each side of an identity even though one or both -members of the identity be disjunctive. Thus let</p> - -<div class="ml5em"> -A = B ꖌ C.<br> -</div> - -<p class="ti0">Now it is self-evident that</p> - -<div class="ml5em"> -AD = AD, -</div> - -<p class="ti0">and in one side of this identity we may for A substitute -its equivalent B ꖌ C, obtaining</p> - -<div class="ml5em"> -AD = BD ꖌ CD. -</div> - -<p>Since “a gaseous element is either hydrogen, or oxygen, -or nitrogen, or chlorine, or fluorine,” it follows that “a free -gaseous element is either free hydrogen, or free oxygen, -or free nitrogen, or free chlorine, or free fluorine.”</p> - -<p>This process of combination will lead to most useful inferences -when the qualifying adjective combined with both -sides of the proposition is a negative of one or more alternatives. -Since chlorine is a coloured gas, we may infer -that “a colourless gaseous element is either (colourless) -hydrogen, oxygen, nitrogen, or fluorine.” The alternative -chlorine disappears because colourless chlorine does not -exist. Again, since “a tooth is either an incisor, canine, -bicuspid, or molar,” it follows that “a not-incisor tooth is -either canine, bicuspid, or molar.” The general rule is that -from the denial of any of the alternatives the affirmation -of the remainder can be inferred. Now this result clearly -follows from our process of substitution; for if we have -the proposition</p> - -<div class="ml5em"> -A = B ꖌ C ꖌ D, -</div> - -<p class="ti0">and we insert this expression for A on one side of the self-evident -identity</p> - -<div class="ml5em"> -A<i>b</i> = A<i>b</i>, -</div> - -<p class="ti0">we obtain A<i>b</i> = AB<i>b</i> ꖌ A<i>b</i>C ꖌ A<i>b</i>D;</p> - -<p class="ti0">and, as the first of the three alternatives is self-contradictory, -we strike it out according to the law of contradiction: -there remains</p> - -<div class="ml5em"> -A<i>b</i> = A<i>b</i>C ꖌ A<i>b</i>D. -</div> - -<p class="ti0">Thus our system fully includes and explains that mood of -the Disjunctive Syllogism technically called the <i>modus -tollendo ponens</i>.</p> - -<p>But the reader must carefully observe that the Disjunctive -Syllogism of the mood <i>ponendo tollens</i>, which affirms<span class="pagenum" id="Page_78">78</span> -one alternative, and thence infers the denial of the rest, -cannot be held true in this system. If I say, indeed, that</p> - -<div class="ml5em"> -Water is either salt or fresh water, -</div> - -<p class="ti0">it seems evident that “water which is salt is not fresh.” -But this inference really proceeds from our knowledge that -water cannot be at once salt and fresh. This inconsistency -of the alternatives, as I have fully shown, will not always -hold. Thus, if I say</p> - -<table class="ml5em"> -<tr> -<td class="tal pl2hi">Gems are either rare stones or beautiful stones,</td> -<td class="tar pl2 vab">(1)</td> -</tr> -</table> - -<p class="ti0">it will obviously not follow that</p> - -<table class="ml5em"> -<tr> -<td class="tal pl2hi">A rare gem is not a beautiful stone,</td> -<td class="tar pl2 vab">(2)</td> -</tr> -</table> - -<p class="ti0">nor that</p> - -<table class="ml5em"> -<tr> -<td class="tal pl2hi">A beautiful gem is not a rare stone.</td> -<td class="tar pl2 vab">(3)</td> -</tr> -</table> - -<p class="ti0">Our symbolic method gives only true conclusions; for if -we take</p> - -<div class="ml5em"> -A = gem<br> -B = rare stone<br> -C = beautiful stone, -</div> - -<p class="ti0">the proposition (1) is of the form</p> - -<table class="ml5em"> -<tr> -<td class="tal"></td> -<td class="tar"><div>A </div></td> -<td class="tal">= B ꖌ C</td> -</tr> -<tr> -<td class="tal pr2">hence</td> -<td class="tar"><div>AB </div></td> -<td class="tal">= B ꖌ BC</td> -</tr> -<tr> -<td class="tal">and</td> -<td class="tar"><div>AC </div></td> -<td class="tal">= BC ꖌ C;</td> -</tr> -</table> - -<p class="ti0">but these inferences are not equivalent to the false ones -(2) and (3).</p> - - -<p>We can readily represent disjunctive reasoning by the -<i>modus ponendo tollens</i>, when it is valid, by expressing the -inconsistency of the alternatives explicitly. Thus if we -resort to our instance of</p> - -<div class="ml5em"> -Water is either salt or fresh, -</div> - -<p class="ti0">and take</p> - -<div class="ml5em"> -A = Water B = salt C = fresh,<br> -</div> - -<p class="ti0">then the premise is apparently of the form</p> - -<div class="ml5em"> -A = AB ꖌ AC;<br> -</div> - -<p class="ti0">but in reality there is an unexpressed condition that “what -is salt is not fresh,” from which follows, by a process of -inference to be afterwards described, that “what is fresh -is not salt.” We have then, in letter-terms, the two propositions</p> - -<div class="ml5em"> -B = B<i>c</i><br> -C = <i>b</i>C. -</div> - -<p>If we substitute these descriptions in the original proposition, -we obtain</p> - -<p><span class="pagenum" id="Page_79">79</span></p> - -<div class="ml5em"> -A = AB<i>c</i> ꖌ A<i>b</i>C; -</div> - -<p class="ti0">uniting B to each side we infer</p> - - -<table class="ml5em"> -<tr> -<td class="tal"></td> -<td class="tar"><div>AB </div></td> -<td class="tal">= AB<i>c</i> ꖌ AB<i>b</i>C</td> -</tr> -<tr> -<td class="tal pr2">or</td> -<td class="tar"><div>AB </div></td> -<td class="tal">= AB<i>c</i>;</td> -</tr> -</table> - - -<p class="ti0">that is,</p> - -<div class="ml7h5"> -Water which is salt is water salt and not fresh. -</div> - -<p>I should weary the reader if I attempted to illustrate -the multitude of forms which disjunctive reasoning may -take; and as in the next chapter we shall be constantly -treating the subject, I must here restrict myself to a single -instance. A very common process of reasoning consists in -the determination of the name of a thing by the successive -exclusion of alternatives, a process called by the old name -<i>abscissio infiniti</i>. Take the case:</p> - -<table class="ml5em"> -<tr> -<td class="tal pr2">Red-coloured metal is either copper or gold</td> -<td class="tar"><div>(1)</div></td> -</tr> -<tr> -<td class="tal">Copper is dissolved by nitric acid</td> -<td class="tar"><div>(2)</div></td> -</tr> -<tr> -<td class="tal">This specimen is red-coloured metal</td> -<td class="tar"><div>(3)</div></td> -</tr> -<tr> -<td class="tal pr2">This specimen is not dissolved by nitric acid</td> -<td class="tar"><div>(4)</div></td> -</tr> -<tr> -<td class="tal">Therefore, this specimen consists of gold</td> -<td class="tar"><div>(5)</div></td> -</tr> -</table> - -<p class="ti0">Let us assign the letter-symbols thus—</p> - -<div class="ml5em"> -A = this specimen<br> -B = red-coloured metal<br> -C = copper<br> -D = gold<br> -E = dissolved by nitric acid. -</div> - -<p>Assuming that the alternatives copper or gold are -intended to be exclusive, as just explained in the case of -fresh and salt water, the premises may be stated in the -forms</p> - -<table class="ml5em"> -<tr> -<td class="tal pr2">B = BC<i>d</i> ꖌ B<i>c</i>D</td> -<td class="tar"><div>(1)</div></td> -</tr> -<tr> -<td class="tal">C = CE</td> -<td class="tar"><div>(2)</div></td> -</tr> -<tr> -<td class="tal">A = AB</td> -<td class="tar"><div>(3)</div></td> -</tr> -<tr> -<td class="tal">A = A<i>e</i></td> -<td class="tar"><div>(4)</div></td> -</tr> -</table> - -<p class="ti0">Substituting for C in (1) by means of (2) we get</p> - -<div class="ml5em"> -B = BC<i>d</i>E ꖌ B<i>c</i>D -</div> - -<p class="ti0">From (3) and (4) we may infer likewise</p> - -<div class="ml5em"> -A = AB<i>e</i><br> -</div> - -<p class="ti0">and if in this we substitute for B its equivalent just -stated, it follows that</p> - -<div class="ml5em"> -A = ABC<i>d</i>E<i>e</i> ꖌ AB<i>c</i>D<i>e</i> -</div> - -<p class="ti0">The first of the alternatives being contradictory the result -is</p> - -<div class="ml5em"> -A = AB<i>c</i>D<i>e</i> -</div> - -<p><span class="pagenum" id="Page_80">80</span></p> - -<p class="ti0">which contains a full description of “this specimen,” as -furnished in the premises, but by ellipsis asserts that it is -gold. It will be observed that in the symbolic expression -(1) I have explicitly stated what is certainly implied, that -copper is not gold, and gold not copper, without which -condition the inference would not hold good.</p> -<hr class="chap x-ebookmaker-drop"> - -<div class="chapter"> -<p><span class="pagenum" id="Page_81">81</span></p> - -<h2 class="nobreak" id="CHAPTER_VI">CHAPTER VI.<br> - -<span class="title">THE INDIRECT METHOD OF INFERENCE.</span></h2> -</div> - -<p class="ti0">The forms of deductive reasoning as yet considered, are -mostly cases of Direct Deduction as distinguished from -those which we are now about to treat. The method of -Indirect Deduction may be described as that which points -out what a thing is, by showing that it cannot be anything -else. We can define a certain space upon a map, either by -colouring that space, or by colouring all except the space; -the first mode is positive, the second negative. The -difference, it will be readily seen, is exactly analogous to -that between the direct and indirect modes of proof in -geometry. Euclid often shows that two lines are equal, by -showing that they cannot be unequal, and the proof rests -upon the known number of alternatives, greater, equal or -less, which are alone conceivable. In other cases, as for -instance in the seventh proposition of the first book, he -shows that two lines must meet in a particular point, by -showing that they cannot meet elsewhere.</p> - -<p>In logic we can always define with certainty the utmost -number of alternatives which are conceivable. The Law -of Duality (pp. <a href="#Page_6">6</a>, <a href="#Page_74">74</a>) enables us always to assert that any -quality or circumstance whatsoever is either present or -absent. Whatever may be the meaning of the terms A -and B it is certainly true that</p> - -<div class="ml5em"> -A = AB ꖌ A<i>b</i><br> -B = AB ꖌ <i>a</i>B. -</div> - -<p>These are universal tacit premises which may be employed -in the solution of every problem, and which are -such invariable and necessary conditions of all thought,<span class="pagenum" id="Page_82">82</span> -that they need not be specially laid down. The Law of -Contradiction is a further condition of all thought and of -all logical symbols; it enables, and in fact obliges, us to -reject from further consideration all terms which imply the -presence and absence of the same quality. Now, whenever -we bring both these Laws of Thought into explicit -action by the method of substitution, we employ the -Indirect Method of Inference. It will be found that we -can treat not only those arguments already exhibited -according to the direct method, but we can include an -infinite multitude of other arguments which are incapable -of solution by any other means.</p> - -<p>Some philosophers, especially those of France, have held -that the Indirect Method of Proof has a certain inferiority -to the direct method, which should prevent our using it -except when obliged. But there are many truths which -we can prove only indirectly. We can prove that a -number is a prime only by the purely indirect method of -showing that it is not any of the numbers which have -divisors, and the remarkable process known as Eratosthenes’ -Sieve is the only mode by which we can select the -prime numbers.<a id="FNanchor_72" href="#Footnote_72" class="fnanchor">72</a> It bears a strong analogy to the indirect -method here to be described. We can prove that the side -and diameter of a square are incommensurable, but only in -the negative or indirect manner, by showing that the contrary -supposition inevitably leads to contradiction.<a id="FNanchor_73" href="#Footnote_73" class="fnanchor">73</a> Many -other demonstrations in various branches of the mathematical -sciences proceed upon a like method. Now, if -there is only one important truth which must be, and can -only be, proved indirectly, we may say that the process is a -necessary and sufficient one, and the question of its comparative -excellence or usefulness is not worth discussion. -As a matter of fact I believe that nearly half our logical -conclusions rest upon its employment.</p> - -<p><span class="pagenum" id="Page_83">83</span></p> - -<h3><i>Simple Illustrations.</i></h3> - -<p>In tracing out the powers and results of this method, we -will begin with the simplest possible instance. Let us -take a proposition of the common form, A = AB, say,</p> - -<div class="ml5em"> -<i>A Metal is an Element,</i> -</div> - -<p class="ti0">and let us investigate its full meaning. Any person who -has had the least logical training, is aware that we can -draw from the above proposition an apparently different -one, namely,</p> - -<div class="ml5em"> -<i>A Not-element is a Not-metal.</i> -</div> - -<p class="ti0">While some logicians, as for instance De Morgan,<a id="FNanchor_74" href="#Footnote_74" class="fnanchor">74</a> have -considered the relation of these two propositions to be -purely self-evident, and neither needing nor allowing -analysis, a great many more persons, as I have observed -while teaching logic, are at first unable to perceive the -close connection between them. I believe that a true and -complete system of logic will furnish a clear analysis of -this process, which has been called <i>Contrapositive Conversion</i>; -the full process is as follows:—</p> - -<p>Firstly, by the Law of Duality we know that</p> - -<div class="ml5em"> -<i>Not-element is either Metal or Not-metal.</i> -</div> - -<p class="ti0">If it be metal, we know that it is by the premise <i>an -element</i>; we should thus be supposing that the same thing -is an element and a not-element, which is in opposition -to the Law of Contradiction. According to the only -other alternative, then, the not-element must be a not-metal.</p> - -<p>To represent this process of inference symbolically we -take the premise in the form</p> - -<table class="ml5em"> -<tr> -<td class="tal pr2">A = AB.</td> -<td class="tar"><div>(1)</div></td> -</tr> -</table> - -<p class="ti0">We observe that by the Law of Duality the term not-B is -thus described</p> - -<table class="ml5em"> -<tr> -<td class="tal pr2"><i>b</i> = A<i>b</i> ꖌ <i>ab</i>.</td> -<td class="tar"><div>(2)</div></td> -</tr> -</table> - -<p class="ti0">For A in this proposition we substitute its description as -given in (1), obtaining</p> - -<div class="ml5em"> -<i>b</i> = AB<i>b</i> ꖌ <i>ab</i>. -</div> - -<p>But according to the Law of Contradiction the term -AB<i>b</i> must be excluded from thought, or</p> - -<p><span class="pagenum" id="Page_84">84</span></p> - -<div class="ml5em"> -AB<i>b</i> = 0. -</div> - -<p class="ti0">Hence it results that <i>b</i> is either nothing at all, or it is <i>ab</i>; -and the conclusion is</p> - -<div class="ml5em"> -<i>b</i> = <i>ab</i>. -</div> - -<p>As it will often be necessary to refer to a conclusion of -this kind I shall call it, as is usual, the <i>Contrapositive -Proposition</i> of the original. The reader need hardly be -cautioned to observe that from all A’s are B’s it does not -follow that all not-A’s are not-B’s. For by the Law of -Duality we have</p> - -<div class="ml5em"> -<i>a</i> = <i>a</i>B ꖌ <i>ab</i>, -</div> - -<p class="ti0">and it will not be found possible to make any substitution -in this by our original premise A = AB. It still remains -doubtful, therefore, whether not-metal is element or not-element.</p> - -<p>The proof of the Contrapositive Proposition given above -is exactly the same as that which Euclid applies in the -case of geometrical notions. De Morgan describes Euclid’s -process as follows<a id="FNanchor_75" href="#Footnote_75" class="fnanchor">75</a>:—“From every not-B is not-A he produces -Every A is B, thus: If it be possible, let this A be -not-B, but every not-B is not-A, therefore this A is not-A, -which is absurd: whence every A is B.” Now De Morgan -thinks that this proof is entirely needless, because common -logic gives the inference without the use of any geometrical -reasoning. I conceive however that logic gives -the inference only by an indirect process. De Morgan -claims “to see identity in Every A is B and every not-B -is not-A, by a process of thought prior to syllogism.” -Whether prior to syllogism or not, I claim that it is not -prior to the laws of thought and the process of substitutive -inference, by which it may be undoubtedly demonstrated.</p> - - -<h3><i>Employment of the Contrapositive Proposition.</i></h3> - -<p>We can frequently employ the contrapositive form of a -proposition by the method of substitution; and certain -moods of the ancient syllogism, which we have hitherto -passed over, may thus be satisfactorily comprehended in -our system. Take for instance the following syllogism in -the mood Camestres:—</p> - -<p><span class="pagenum" id="Page_85">85</span></p> - -<div class="ml7h5" style="width: 70%;"> -“Whales are not true fish; for they do not respire water, -whereas true fish do respire water.” -</div> - -<p class="ti0">Let us take</p> - -<div class="ml5em"> -A = whale<br> -B = true fish<br> -C = respiring water -</div> - -<p class="ti0">The premises are of the forms</p> - -<table class="ml5em"> -<tr> -<td class="tal">A = A<i>c</i></td> -<td class="tar"><div>(1)</div></td> -</tr> -<tr> -<td class="tal pr2">B = BC</td> -<td class="tar"><div>(2)</div></td> -</tr> -</table> - -<p>Now, by the process of contraposition we obtain from -the second premise</p> - -<div class="ml5em"> -<i>c</i> = <i>bc</i> -</div> - -<p class="ti0">and we can substitute this expression for <i>c</i> in (1), obtaining</p> - -<div class="ml5em"> -A = A<i>bc</i> -</div> - -<p class="ti0">or “Whales are not true fish, not respiring water.”</p> - -<p>The mood Cesare does not really differ from Camestres -except in the order of the premises, and it could be exhibited -in an exactly similar manner.</p> - -<p>The mood Baroko gave much trouble to the old logicians, -who could not <i>reduce</i> it to the first figure in the same -manner as the other moods, and were obliged to invent, -specially for it and for Bokardo, a method of Indirect -Reduction closely analogous to the indirect proof of Euclid. -Now these moods require no exceptional treatment in this -system. Let us take as an instance of Baroko, the argument</p> - -<table class="ml5em"> -<tr> -<td class="tal">All heated solids give continuous spectra</td> -<td class="tar"><div>(1)</div></td> -</tr> -<tr> -<td class="tal">Some nebulæ do not give continuous spectra</td> -<td class="tar"><div>(2)</div></td> -</tr> -<tr> -<td class="tal pr2">Therefore, some nebulæ are not heated solids</td> -<td class="tar"><div>(3)</div></td> -</tr> -</table> - -<p>Treating the little word some as an indeterminate adjective -of selection, to which we assign a symbol like any -other adjective, let</p> - -<div class="ml5em"> -A = some<br> -B = nebulæ<br> -C = giving continuous spectra<br> -D = heated solids -</div> - -<p class="ti0">The premises then become</p> - -<table class="ml5em"> -<tr> -<td class="tar"><div>D </div></td> -<td class="tal">= DC</td> -<td class="tar"><div>(1)</div></td> -</tr> -<tr> -<td class="tar"><div>AB </div></td> -<td class="tal pr2">= AB<i>c</i></td> -<td class="tar"><div>(2)</div></td> -</tr> -</table> - -<p class="ti0">Now from (1) we obtain by the indirect method the contrapositive -proposition</p> - -<p><span class="pagenum" id="Page_86">86</span></p> - -<div class="ml5em"> -<i>c</i> = <i>cd</i> -</div> - -<p class="ti0">and if we substitute this expression for <i>c</i> in (2) we have</p> - -<div class="ml5em"> -AB = AB<i>cd</i> -</div> - -<p class="ti0">the full meaning of which is that “some nebulæ do not -give continuous spectra and are not heated solids.”</p> - -<p>We might similarly apply the contrapositive in many -other instances. Take the argument, “All fixed stars are -self-luminous; but some of the heavenly bodies are not -self-luminous, and are therefore not fixed stars.” Taking -our terms</p> - -<div class="ml5em"> -A = fixed stars<br> -B = self-luminous<br> -C = some<br> -D = heavenly bodies -</div> - -<p class="ti0">we have the premises</p> - -<table class="ml5em"> -<tr> -<td class="tar"><div>A </div></td> -<td class="tal">= AB,</td> -<td class="tar"><div>(1)</div></td> -</tr> -<tr> -<td class="tar"><div>CD </div></td> -<td class="tal pr2">= <i>b</i>CD</td> -<td class="tar"><div>(2)</div></td> -</tr> -</table> - -<p class="ti0">Now from (1) we can draw the contrapositive</p> - -<div class="ml5em"> -<i>b</i> = <i>ab</i> -</div> - -<p class="ti0">and substituting this expression for <i>b</i> in (2) we obtain</p> - -<div class="ml5em"> -CD = <i>ab</i>CD -</div> - -<p class="ti0">which expresses the conclusion of the argument that some -heavenly bodies are not fixed stars.</p> - - -<h3><i>Contrapositive of a Simple Identity.</i></h3> - -<p>The reader should carefully note that when we apply -the process of Indirect Inference to a simple identity of -the form</p> - -<div class="ml5em"> -A = B -</div> - -<p class="ti0">we may obtain further results. If we wish to know what -is the term not-B, we have as before, by the Law of Duality,</p> - -<div class="ml5em"> -<i>b</i> = A<i>b</i> ꖌ <i>ab</i> -</div> - -<p class="ti0">and substituting for A we obtain</p> - -<div class="ml5em"> -<i>b</i> = B<i>b</i> ꖌ <i>ab</i> = <i>ab</i>. -</div> - -<p>But we may now also draw a second contrapositive; for -we have</p> - -<div class="ml5em"> -<i>a</i> = <i>a</i>B ꖌ <i>ab</i>, -</div> - -<p class="ti0">and substituting for B its equivalent A we have</p> - -<div class="ml5em"> -<i>a</i> = <i>a</i>A ꖌ <i>ab</i> = <i>ab</i>. -</div> - -<p>Hence from the single identity A = B we can draw -the two propositions</p> - -<p><span class="pagenum" id="Page_87">87</span></p> - -<div class="ml5em"> -<i>a</i> = <i>ab</i><br> -<i>b</i> = <i>ab</i>, -</div> - -<p class="ti0">and observing that these propositions have a common term -<i>ab</i> we can make a new substitution, getting</p> - -<div class="ml5em"> -<i>a</i> = <i>b</i>. -</div> - -<p>This result is in strict accordance with the fundamental -principles of inference, and it may be a question whether -it is not a self-evident result, independent of the steps of -deduction by which we have reached it. For where two -classes are coincident like A and B, whatever is true of -the one is true of the other; what is excluded from the one -must be excluded from the other similarly. Now as <i>a</i> -bears to A exactly the same relation that <i>b</i> bears to B, the -identity of either pair follows from the identity of the -other pair. In every identity, equality, or similarity, we -may argue from the negative of the one side to the negative -of the other. Thus at ordinary temperatures</p> - -<div class="ml5em"> -Mercury = liquid-metal, -</div> - -<p class="ti0">hence obviously</p> - -<div class="ml5em"> -Not-mercury = not liquid-metal; -</div> - -<p class="ti0">or since</p> - -<div class="ml5em"> -Sirius = brightest fixed star, -</div> - -<p class="ti0">it follows that whatever star is not the brightest is not -Sirius, and <i>vice versâ</i>. Every correct definition is of the -form A = B, and may often require to be applied in the -equivalent negative form.</p> - -<p>Let us take as an illustration of the mode of using this -result the argument following:</p> - -<table class="ml5em"> -<tr> -<td class="tal pl2hi">Vowels are letters which can be sounded alone,</td> -<td class="tar vab pl2">(1)</td> -</tr> -<tr> -<td class="tal pl2hi">The letter <i>w</i> cannot be sounded alone;</td> -<td class="tar"><div>(2)</div></td> -</tr> -<tr> -<td class="tal pl2hi">Therefore the letter <i>w</i> is not a vowel.</td> -<td class="tar"><div>(3)</div></td> -</tr> -</table> - -<p class="ti0">Here we have a definition (1), and a comparison of a -thing with that definition (2), leading to exclusion of the -thing from the class defined.</p> - -<p>Taking the terms</p> - -<div class="ml5em"> -A = vowel,<br> -B = letter which can be sounded alone,<br> -C = letter <i>w</i>, -</div> - -<p class="ti0">the premises are plainly of the forms</p> - -<table class="ml5em"> -<tr> -<td class="tal">A = B,</td> -<td class="tar"><div>(1)</div></td> -</tr> -<tr> -<td class="tal pr2">C = <i>b</i>C.</td> -<td class="tar"><div>(2)</div></td> -</tr> -</table> - -<p><span class="pagenum" id="Page_88">88</span></p> - -<p class="ti0">Now by the Indirect method we obtain from (1) the -Contrapositive</p> - -<div class="ml5em"> -<i>b</i> = <i>a</i>, -</div> - -<p class="ti0">and inserting in (2) the equivalent for <i>b</i> we have</p> - -<table class="ml5em"> -<tr> -<td class="tal pr2">C = <i>a</i>C,</td> -<td class="tar"><div>(3)</div></td> -</tr> -</table> - -<p class="ti0">or “the letter <i>w</i> is not a vowel.”</p> - - -<h3><i>Miscellaneous Examples of the Method.</i></h3> - -<p>We can apply the Indirect Method of Inference however -many may be the terms involved or the premises containing -those terms. As the working of the method is -best learnt from examples, I will take a case of two -premises forming the syllogism Barbara: thus</p> - -<table class="ml5em"> -<tr> -<td class="tal">Iron is metal</td> -<td class="tar"><div>(1)</div></td> -</tr> -<tr> -<td class="tal pr2">Metal is element.</td> -<td class="tar"><div>(2)</div></td> -</tr> -</table> - -<p class="ti0">If we want to ascertain what inference is possible concerning -the term <i>Iron</i>, we develop the term by the Law of -Duality. Iron must be either metal or not-metal; iron -which is metal must be either element or not-element; -and similarly iron which is not-metal must be either -element or not-element. There are then altogether four -alternatives among which the description of iron must be -contained; thus</p> - -<table class="ml5em"> -<tr> -<td class="tal">Iron, metal, element,</td> -<td class="tar"><div>(α)</div></td> -</tr> -<tr> -<td class="tal">Iron, metal, not-element,</td> -<td class="tar"><div>(β)</div></td> -</tr> -<tr> -<td class="tal">Iron, not-metal, element,</td> -<td class="tar"><div>(γ)</div></td> -</tr> -<tr> -<td class="tal pr2">Iron, not-metal, not-element.</td> -<td class="tar"><div>(δ)</div></td> -</tr> -</table> - -<p class="ti0">Our first premise informs us that iron is a metal, and if -we substitute this description in (γ) and (δ) we shall have -self-contradictory combinations. Our second premise likewise -informs us that metal is element, and applying this -description to (β) we again have self-contradiction, so that -there remains only (α) as a description of iron—our -inference is</p> - -<div class="ml5em"> -Iron = iron, metal, element. -</div> - -<p>To represent this process of reasoning in general symbols, -let</p> - -<div class="ml5em"> -A = iron<br> -B = metal<br> -C = element, -</div> - -<p class="ti0">The premises of the problem take the forms</p> - -<p><span class="pagenum" id="Page_89">89</span></p> - -<table class="ml5em"> -<tr> -<td class="tal">A = AB</td> -<td class="tar"><div>(1)</div></td> -</tr> -<tr> -<td class="tal pr2">B = BC.</td> -<td class="tar"><div>(2)</div></td> -</tr> -</table> - -<p class="ti0">By the Law of Duality we have</p> - -<table class="ml5em"> -<tr> -<td class="tal">A = AB ꖌ A<i>b</i></td> -<td class="tar"><div>(3)</div></td> -</tr> -<tr> -<td class="tal pr2">A = AC ꖌ A<i>c</i>.</td> -<td class="tar"><div>(4)</div></td> -</tr> -</table> - -<p class="ti0">Now, if we insert for A in the second side of (3) its -description in (4), we obtain what I shall call the <i>development -of A with respect to B and C</i>, namely</p> - -<table class="ml5em"> -<tr> -<td class="tal pr2">A = ABC ꖌ AB<i>c</i> ꖌ A<i>b</i>C ꖌ A<i>bc</i>.</td> -<td class="tar"><div>(5)</div></td> -</tr> -</table> - -<p class="ti0">Wherever the letters A or B appear in the second side of -(5) substitute their equivalents given in (1) and (2), and -the results stated at full length are</p> - -<div class="ml5em"> -A = ABC ꖌ ABC<i>c</i> ꖌ AB<i>b</i>C ꖌ AB<i>b</i>C<i>c</i>. -</div> - -<p class="ti0">The last three alternatives break the Law of Contradiction, -so that</p> - -<div class="ml5em"> -A = ABC ꖌ 0 ꖌ 0 ꖌ 0 = ABC. -</div> - -<p class="ti0">This conclusion is, indeed, no more than we could obtain -by the direct process of substitution, that is by substituting -for B in (1), its description in (2) as in p. <a href="#Page_55">55</a>; it is the -characteristic of the Indirect process that it gives all -possible logical conclusions, both those which we have -previously obtained, and an immense number of others or -which the ancient logic took little or no account. From -the same premises, for instance, we can obtain a description -of the class <i>not-element</i> or <i>c</i>. By the Law of Duality we can -develop <i>c</i> into four alternatives, thus</p> - -<div class="ml7h5"> -<i>c</i> = AB<i>c</i> ꖌ A<i>bc</i> ꖌ <i>a</i>B<i>c</i> ꖌ <i>abc</i>. -</div> - -<p class="ti0">If we substitute for A and B as before, we get</p> - -<div class="ml7h5"> -<i>c</i> = ABC<i>c</i> ꖌ AB<i>bc</i> ꖌ <i>a</i>BC<i>c</i> ꖌ <i>abc</i>, -</div> - -<p class="ti0">and, striking out the terms which break the Law of -Contradiction, there remains</p> - -<div class="ml5em"> -<i>c</i> = <i>abc</i>, -</div> - -<p class="ti0">or what is not element is also not iron and not metal. -This Indirect Method of Inference thus furnishes a -complete solution of the following problem—<i>Given any -number of logical premises or conditions, required the -description of any class of objects, or of any term, as -governed by those conditions.</i></p> - -<p>The steps of the process of inference may thus be -concisely stated—</p> - -<p>1. By the Law of Duality develop the utmost number -of alternatives which may exist in the description of the<span class="pagenum" id="Page_90">90</span> -required class or term as regards the terms involved in the -premises.</p> - -<p>2. For each term in these alternatives substitute its -description as given in the premises.</p> - -<p>3. Strike out every alternative which is then found to -break the Law of Contradiction.</p> - -<p>4. The remaining terms may be equated to the term in -question as the desired description.</p> - - -<h3><i>Mr. Venn’s Problem.</i></h3> - -<p>The need of some logical method more powerful and -comprehensive than the old logic of Aristotle is strikingly -illustrated by Mr. Venn in his most interesting and able -article on Boole’s logic.<a id="FNanchor_76" href="#Footnote_76" class="fnanchor">76</a> An easy example, originally got, -as he says, by the aid of my method as simply described -in the <i>Elementary Lessons in Logic</i>, was proposed in -examination and lecture-rooms to some hundred and fifty -students as a problem in ordinary logic. It was answered -by, at most, five or six of them. It was afterwards set, -as an example on Boole’s method, to a small class who -had attended a few lectures on the nature of these -symbolic methods. It was readily answered by half or -more of their number.</p> - -<p>The problem was as follows:—“The members of a board -were all of them either bondholders, or shareholders, but -not both; and the bondholders as it happened, were all on -the board. What conclusion can be drawn?” The conclusion -wanted is, “No shareholders are bondholders.” -Now, as Mr. Venn says, nothing can look simpler than the -following reasoning, <i>when stated</i>:—“There can be no -bondholders who are shareholders; for if there were they -must be either on the board, or off it. But they are not -on it, by the first of the given statements; nor off it, by -the second.” Yet from the want of any systematic mode -of treating such a question only five or six of some -hundred and fifty students could succeed in so simple a -problem.</p> - -<p><span class="pagenum" id="Page_91">91</span></p> - -<p>By symbolic statement the problem is instantly solved. -Taking</p> - -<div class="ml5em"> -A = member of board<br> -B = bondholder<br> -C = shareholder -</div> - -<p class="ti0">the premises are evidently</p> - -<div class="ml5em"> -A = AB<i>c</i> ꖌ A<i>b</i>C -B = AB.<br> -</div> - -<p class="ti0">The class C or shareholders may in respect of A and B be -developed into four alternatives,</p> - -<div class="ml5em"> -C = ABC ꖌ A<i>b</i>C ꖌ <i>a</i>BC ꖌ <i>ab</i>C. -</div> - -<p class="ti0">But substituting for A in the first and for B in the third -alternative we get</p> - -<div class="ml5em"> -C = ABC<i>c</i> ꖌ AB<i>b</i>C ꖌ A<i>b</i>C ꖌ <i>a</i>ABC ꖌ <i>ab</i>C. -</div> - -<p class="ti0">The first, second, and fourth alternatives in the above are -self-contradictory combinations, and only these; striking -them out there remain</p> - -<div class="ml5em"> -C = A<i>b</i>C ꖌ <i>ab</i>C = <i>b</i>C, -</div> - -<p class="ti0">the required answer. This symbolic reasoning is, I believe, -the exact equivalent of Mr. Venn’s reasoning, and I do -not believe that the result can be attained in a simpler -manner. Mr. Venn adds that he could adduce other -similar instances, that is, instances showing the necessity -of a better logical method.</p> - - -<h3><i>Abbreviation of the Process.</i></h3> - -<p>Before proceeding to further illustrations of the use of -this method, I must point out how much its practical -employment can be simplified, and how much more easy -it is than would appear from the description. When we -want to effect at all a thorough solution of a logical -problem it is best to form, in the first place, a complete -series of all the combinations of terms involved in it. If -there be two terms A and B, the utmost variety of -combinations in which they can appear are</p> - -<table class="ml5em"> -<tr> -<td class="tal pr2">AB</td> -<td class="tal"><i>a</i>B</td> -</tr> -<tr> -<td class="tal">A<i>b</i></td> -<td class="tal"><i>ab</i>.</td> -</tr> -</table> - -<p class="ti0">The term A appears in the first and second; B in the first -and third; <i>a</i> in the third and fourth; and <i>b</i> in the second -and fourth. Now if we have any premise, say</p> - -<div class="ml5em"> -A = B, -</div> - -<p><span class="pagenum" id="Page_92">92</span></p> - -<p class="ti0">we must ascertain which of these combinations will be -rendered self-contradictory by substitution; the second -and third will have to be struck out, and there will remain -only</p> - -<div class="ml5em"> -AB<br> -<i>ba</i>. -</div> - -<p class="ti0">Hence we draw the following inferences</p> - -<div class="ml5em"> -A = AB, B = AB, <i>a</i> = <i>ab</i>, <i>b</i> = <i>ab</i>. -</div> - -<p>Exactly the same method must be followed when a -question involves a greater number of terms. Thus by the -Law of Duality the three terms A, B, C, give rise to eight -conceivable combinations, namely</p> - -<table class="ml5em"> -<tr> -<td class="tal pr2">ABC</td> -<td class="tal pr3">(α)</td> -<td class="tal pr2"><i>a</i>BC</td> -<td class="tal">(ε)</td> -</tr> -<tr> -<td class="tal">AB<i>c</i></td> -<td class="tal">(β)</td> -<td class="tal"><i>a</i>B<i>c</i></td> -<td class="tal">(ζ)</td> -</tr> -<tr> -<td class="tal">A<i>b</i>C</td> -<td class="tal">(γ)</td> -<td class="tal"><i>ab</i>C</td> -<td class="tal">(η)</td> -</tr> -<tr> -<td class="tal">A<i>bc</i></td> -<td class="tal">(δ)</td> -<td class="tal"><i>abc</i>.</td> -<td class="tal">(θ)</td> -</tr> -</table> - -<p class="ti0">The development of the term A is formed by the first four -of these; for B we must select (α), (β), (ε), (ζ); C -consists of (α), (γ), (ε), (η); <i>b</i> of (γ), (δ), (η), (θ), and so on.</p> - -<p>Now if we want to investigate completely the meaning -of the premises</p> - -<table class="ml5em"> -<tr> -<td class="tal pr2">A = AB</td> -<td class="tar"><div>(1)</div></td> -</tr> -<tr> -<td class="tal">B = BC</td> -<td class="tar"><div>(2)</div></td> -</tr> -</table> - -<p class="ti0">we examine each of the eight combinations as regards each -premise; (γ) and (δ) are contradicted by (1), and (β) and -(ζ) by (2), so that there remain only</p> - -<table class="ml5em"> -<tr> -<td class="tal pr3">ABC</td> -<td class="tar"><div>(α)</div></td> -</tr> -<tr> -<td class="tal"><i>a</i>BC</td> -<td class="tar"><div>(ε)</div></td> -</tr> -<tr> -<td class="tal"><i>ab</i>C</td> -<td class="tar"><div>(η)</div></td> -</tr> -<tr> -<td class="tal"><i>abc</i>.</td> -<td class="tar"><div>(θ)</div></td> -</tr> -</table> - -<p class="ti0">To describe any term under the conditions of the premises -(1) and (2), we have simply to draw out the proper combinations -from this list; thus, A is represented only by -ABC, that is to say</p> - -<table class="ml5em"> -<tr> -<td class="tal"></td> -<td class="tar"><div>A </div></td> -<td class="tal">= ABC,</td> -</tr> -<tr> -<td class="tal pr3">similarly</td> -<td class="tar"><div><i>c</i> </div></td> -<td class="tal">= <i>abc</i>.</td> -</tr> -</table> - -<p class="ti0">For B we have two alternatives thus stated,</p> - -<div class="ml5em"> -B = ABC ꖌ <i>a</i>BC; -</div> - -<p class="ti0">and for <i>b</i> we have</p> - -<div class="ml5em"> -<i>b</i> = <i>ab</i>C ꖌ <i>abc</i>. -</div> - -<p>When we have a problem involving four distinct terms -we need to double the number of combinations, and as -we add each new term the combinations become twice -as numerous. Thus</p> - -<p><span class="pagenum" id="Page_93">93</span></p> - -<table class="ml2em"> -<tr> -<td class="tal">A, B</td> -<td class="tac"><div>produce </div></td> -<td class="tac" colspan="2">four  combinations</td> -</tr> -<tr> -<td class="tal">A, B, C,</td> -<td class="tac"><div>"</div></td> -<td class="tal">eight</td> -<td class="tal pr3">"</td> -</tr> -<tr> -<td class="tal">A, B, C, D</td> -<td class="tac"><div>"</div></td> -<td class="tal">sixteen</td> -<td class="tal pr3">"</td> -</tr> -<tr> -<td class="tal">A, B, C, D, E</td> -<td class="tac"><div>"</div></td> -<td class="tal pr1">thirty-two</td> -<td class="tal pr3">"</td> -</tr> -<tr> -<td class="tal">A, B, C, D, E, F  </td> -<td class="tac"><div>"</div></td> -<td class="tal">sixty-four</td> -<td class="tal pr3">"</td> -</tr> -</table> - -<p class="ti0">and so on.</p> - -<p>I propose to call any such series of combinations the -<i>Logical Alphabet</i>. It holds in logical science a position -the importance of which cannot be exaggerated, and as -we proceed from logical to mathematical considerations, it -will become apparent that there is a close connection -between these combinations and the fundamental theorems -of mathematical science. For the convenience of the -reader who may wish to employ the <i>Alphabet</i> in logical -questions, I have had printed on the next page a complete -series of the combinations up to those of six terms. At -the very commencement, in the first column, is placed a -single letter X, which might seem to be superfluous. This -letter serves to denote that it is always some higher class -which is divided up. Thus the combination AB really -means ABX, or that part of some larger class, say X, -which has the qualities of A and B present. The letter -X is omitted in the greater part of the table merely for the -sake of brevity and clearness. In a later chapter on Combinations -it will become apparent that the introduction of -this unit class is requisite in order to complete the -analogy with the Arithmetical Triangle there described.</p> - -<p>The reader ought to bear in mind that though the Logical -Alphabet seems to give mere lists of combinations, these -combinations are intended in every case to constitute the -development of a term of a proposition. Thus the four -combinations AB, A<i>b</i>, <i>a</i>B, <i>ab</i> really mean that any class X -is described by the following proposition,</p> - -<div class="ml5em"> -X = XAB ꖌ XA<i>b</i> ꖌ X<i>a</i>B ꖌ X<i>ab</i>. -</div> - -<p class="ti0">If we select the A’s, we obtain the following proposition</p> - -<div class="ml5em"> -AX = XAB ꖌ XA<i>b</i>. -</div> - -<p class="ti0">Thus whatever group of combinations we treat must be -conceived as part of a higher class, <i>summum genus</i> or -universe symbolised in the term X; but, bearing this in -mind, it is needless to complicate our formulæ by always -introducing the letter. All inference consists in passing -from propositions to propositions, and combinations <i>per se</i><span class="pagenum" id="Page_94">94</span> -have no meaning. They are consequently to be regarded -in all cases as forming parts of propositions.</p> - - -<h3><span class="smcap">The Logical Alphabet.</span></h3> - -<div class="center"> -<table class="fs90 mb1em" style="width:400px; font-family: monospace"> -<tr> -<td class="tac"><div>I.</div></td> -<td class="tac"><div>II.</div></td> -<td class="tac"><div>III.</div></td> -<td class="tac"><div>IV.</div></td> -<td class="tac"><div>V.</div></td> -<td class="tac"><div>VI.</div></td> -<td class="tac"><div>VII.</div></td> -</tr> -<tr> -<td class="tac"><div>X</div></td> -<td class="tac"><div>AX</div></td> -<td class="tac"><div>AB</div></td> -<td class="tac"><div>ABC</div></td> -<td class="tac"><div>ABCD</div></td> -<td class="tac"><div>ABCDE</div></td> -<td class="tac"><div>ABCDEF</div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"><div><i>a</i>X</div></td> -<td class="tac"><div>A<i>b</i></div></td> -<td class="tac"><div>AB<i>c</i></div></td> -<td class="tac"><div>ABC<i>d</i></div></td> -<td class="tac"><div>ABCD<i>e</i></div></td> -<td class="tac"><div>ABCDE<i>f</i></div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>a</i>B</div></td> -<td class="tac"><div>A<i>b</i>C</div></td> -<td class="tac"><div>AB<i>c</i>D</div></td> -<td class="tac"><div>ABC<i>d</i>E</div></td> -<td class="tac"><div>ABCD<i>e</i>F</div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>ab</i></div></td> -<td class="tac"><div>A<i>bc</i></div></td> -<td class="tac"><div>AB<i>cd</i></div></td> -<td class="tac"><div>ABC<i>de</i></div></td> -<td class="tac"><div>ABCD<i>ef</i></div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>a</i>BC</div></td> -<td class="tac"><div>A<i>b</i>CD</div></td> -<td class="tac"><div>AB<i>c</i>DE</div></td> -<td class="tac"><div>ABC<i>d</i>EF</div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>a</i>B<i>c</i></div></td> -<td class="tac"><div>A<i>b</i>C<i>d</i></div></td> -<td class="tac"><div>AB<i>c</i>D<i>e</i></div></td> -<td class="tac"><div>ABC<i>d</i>E<i>f</i></div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>ab</i>C</div></td> -<td class="tac"><div>A<i>bc</i>D</div></td> -<td class="tac"><div>AB<i>cd</i>E</div></td> -<td class="tac"><div>ABC<i>de</i>F</div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>abc</i></div></td> -<td class="tac"><div>Ab<i>cd</i></div></td> -<td class="tac"><div>AB<i>cde</i></div></td> -<td class="tac"><div>ABC<i>def</i></div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>a</i>BCD</div></td> -<td class="tac"><div>A<i>b</i>CDE</div></td> -<td class="tac"><div>AB<i>c</i>DEF</div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>a</i>BC<i>d</i></div></td> -<td class="tac"><div>A<i>b</i>CD<i>e</i></div></td> -<td class="tac"><div>AB<i>c</i>DE<i>f</i></div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>a</i>B<i>c</i>D</div></td> -<td class="tac"><div>A<i>b</i>C<i>d</i>E</div></td> -<td class="tac"><div>AB<i>c</i>D<i>e</i>F</div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>a</i>B<i>cd</i></div></td> -<td class="tac"><div>A<i>b</i>C<i>de</i></div></td> -<td class="tac"><div>AB<i>c</i>D<i>ef</i></div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>ab</i>CD</div></td> -<td class="tac"><div>A<i>bc</i>DE</div></td> -<td class="tac"><div>AB<i>cd</i>EF</div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>ab</i>C<i>d</i></div></td> -<td class="tac"><div>A<i>bc</i>D<i>e</i></div></td> -<td class="tac"><div>AB<i>cd</i>E<i>f</i></div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>abc</i>D</div></td> -<td class="tac"><div>A<i>bcd</i>E</div></td> -<td class="tac"><div>AB<i>cde</i>F</div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>abcd</i></div></td> -<td class="tac"><div>A<i>bcde</i></div></td> -<td class="tac"><div>AB<i>cdef</i></div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>a</i>BCDE</div></td> -<td class="tac"><div>A<i>b</i>CDEF</div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>a</i>BCD<i>e</i></div></td> -<td class="tac"><div>A<i>b</i>CDE<i>f</i></div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>a</i>BC<i>d</i>E</div></td> -<td class="tac"><div>A<i>b</i>CD<i>e</i>F</div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>a</i>BC<i>de</i></div></td> -<td class="tac"><div>A<i>b</i>CD<i>ef</i></div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>a</i>B<i>c</i>DE</div></td> -<td class="tac"><div>A<i>b</i>C<i>d</i>EF</div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>a</i>B<i>c</i>D<i>e</i></div></td> -<td class="tac"><div>A<i>b</i>C<i>d</i>E<i>f</i></div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>a</i>B<i>cd</i>E</div></td> -<td class="tac"><div>A<i>b</i>C<i>de</i>F</div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>a</i>B<i>cde</i></div></td> -<td class="tac"><div>A<i>b</i>C<i>def</i></div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>ab</i>CDE</div></td> -<td class="tac"><div>A<i>bc</i>DEF</div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>ab</i>CD<i>e</i></div></td> -<td class="tac"><div>A<i>bc</i>DE<i>f</i></div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>ab</i>C<i>d</i>E</div></td> -<td class="tac"><div>A<i>bc</i>D<i>e</i>F</div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>ab</i>Cd<i>e</i></div></td> -<td class="tac"><div>A<i>bc</i>D<i>ef</i></div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>abc</i>DE</div></td> -<td class="tac"><div>A<i>bcd</i>EF</div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>abc</i>D<i>e</i></div></td> -<td class="tac"><div>A<i>bcd</i>E<i>f</i></div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>abcd</i>E</div></td> -<td class="tac"><div>A<i>bcde</i>F</div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>abcde</i></div></td> -<td class="tac"><div>A<i>bcdef</i></div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>a</i>BCDEF</div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>a</i>BCDE<i>f</i></div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>a</i>BCD<i>e</i>F</div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>a</i>BCD<i>ef</i></div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>a</i>BC<i>d</i>EF</div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>a</i>BC<i>d</i>E<i>f</i></div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>a</i>BC<i>de</i>F</div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>a</i>BC<i>def</i></div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>a</i>B<i>c</i>DEF</div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>a</i>B<i>c</i>DE<i>f</i></div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>a</i>B<i>c</i>D<i>e</i>F</div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>a</i>B<i>c</i>D<i>ef</i></div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>a</i>B<i>cd</i>EF</div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>a</i>B<i>cd</i>E<i>f</i></div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>a</i>B<i>cde</i>F</div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>a</i>B<i>cdef</i></div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>ab</i>CDEF</div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>ab</i>CDE<i>f</i></div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>ab</i>CD<i>e</i>F</div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>ab</i>CD<i>ef</i></div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>ab</i>C<i>d</i>EF</div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>ab</i>C<i>d</i>E<i>f</i></div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>ab</i>C<i>de</i>F</div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>ab</i>C<i>def</i></div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>abc</i>DEF</div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>abc</i>DE<i>f</i></div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>abc</i>D<i>e</i>F</div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>abc</i>D<i>ef</i></div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>abcd</i>EF</div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>abcd</i>E<i>f</i></div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>abcde</i>F</div></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div><i>abcdef</i></div></td> -</tr> -</table> -</div> - -<p><span class="pagenum" id="Page_95">95</span></p> - -<p>In a theoretical point of view we may conceive that -the Logical Alphabet is infinitely extended. Every new -quality or circumstance which can belong to an object, -subdivides each combination or class, so that the number -of such combinations, when unrestricted by logical -conditions, is represented by an infinitely high power of -two. The extremely rapid increase in the number of -subdivisions obliges us to confine our attention to a -few qualities at a time.</p> - -<p>When contemplating the properties of this Alphabet I -am often inclined to think that Pythagoras perceived the -deep logical importance of duality; for while unity was -the symbol of identity and harmony, he described the -number two as the origin of contrasts, or the symbol of -diversity, division and separation. The number four, or -the <i>Tetractys</i>, was also regarded by him as one of the chief -elements of existence, for it represented the generating -virtue whence come all combinations. In one of the -golden verses ascribed to Pythagoras, he conjures his -pupil to be virtuous:<a id="FNanchor_77" href="#Footnote_77" class="fnanchor">77</a></p> - -<div class="center fs95 ptb05"> -“By him who stampt <i>The Four</i> upon the Mind,<br> -  <i>The Four</i>, the fount of Nature’s endless stream.” -</div> - -<p>Now four and the higher powers of duality do represent -in this logical system the numbers of combinations which -can be generated in the absence of logical restrictions. -The followers of Pythagoras may have shrouded their -master’s doctrines in mysterious and superstitious notions, -but in many points these doctrines seem to have some -basis in logical philosophy.</p> - - -<h3><i>The Logical Slate.</i></h3> - -<p>To a person who has once comprehended the extreme -significance and utility of the Logical Alphabet the -indirect process of inference becomes reduced to the -repetition of a few uniform operations of classification, -selection, and elimination of contradictories. Logical -deduction, even in the most complicated questions, -becomes a matter of mere routine, and the amount of<span class="pagenum" id="Page_96">96</span> -labour required is the only impediment, when once the -meaning of the premises is rendered clear. But the -amount of labour is often found to be considerable. The -mere writing down of sixty-four combinations of six -letters each is no small task, and, if we had a problem of -five premises, each of the sixty-four combinations would -have to be examined in connection with each premise. -The requisite comparison is often of a very tedious -character, and considerable chance of error intervenes.</p> - -<p>I have given much attention, therefore, to lessening both -the manual and mental labour of the process, and I shall -describe several devices which may be adopted for saving -trouble and risk of mistake.</p> - -<p>In the first place, as the same sets of combinations occur -over and over again in different problems, we may avoid -the labour of writing them out by having the sets of -letters ready printed upon small sheets of writing-paper. -It has also been suggested by a correspondent that, if any -one series of combinations were marked upon the margin -of a sheet of paper, and a slit cut between each pair of -combinations, it would be easy to fold down any particular -combination, and thus strike it out of view. The combinations -consistent with the premises would then remain -in a broken series. This method answers sufficiently well -for occasional use.</p> - -<p>A more convenient mode, however, is to have the series -of letters shown on p. <a href="#Page_94">94</a>, engraved upon a common school -writing slate, of such a size, that the letters may occupy -only about a third of the space on the left hand side of -the slate. The conditions of the problem can then be -written down on the unoccupied part of the slate, and the -proper series of combinations being chosen, the contradictory -combinations can be struck out with the pencil. -I have used a slate of this kind, which I call a <i>Logical -Slate</i>, for more than twelve years, and it has saved me -much trouble. It is hardly possible to apply this -process to problems of more than six terms, owing to -the large number of combinations which would require -examination.</p> - -<p><span class="pagenum" id="Page_97">97</span></p> - - -<h3><i>Abstraction of Indifferent Circumstances.</i></h3> - -<p>There is a simple but highly important process of -inference which enables us to abstract, eliminate or disregard -all circumstances indifferently present and absent. -Thus if I were to state that “a triangle is a three-sided -rectilinear figure, either large or not large,” these two -alternatives would be superfluous, because, by the Law of -Duality, I know that everything must be either large or -not large. To add the qualification gives no new knowledge, -since the existence of the two alternatives will be -understood in the absence of any information to the -contrary. Accordingly, when two alternatives differ only -as regards a single component term which is positive in -one and negative in the other, we may reduce them to one -term by striking out their indifferent part. It is really a -process of substitution which enables us to do this; for -having any proposition of the form</p> - -<table class="ml5em"> -<tr> -<td class="tal pr2">A = ABC ꖌ AB<i>c</i>,</td> -<td class="tar"><div>(1)</div></td> -</tr> -</table> - -<p class="ti0">we know by the Law of Duality that</p> - -<table class="ml5em"> -<tr> -<td class="tal pr2">AB = ABC ꖌ AB<i>c</i>.</td> -<td class="tar"><div>(2)</div></td> -</tr> -</table> - -<p class="ti0">As the second member of this is identical with the second -member of (1) we may substitute, obtaining</p> - -<div class="ml5em"> -A = AB. -</div> - -<p>This process of reducing useless alternatives may be -applied again and again; for it is plain that</p> - -<div class="ml5em"> -A = AB (CD ꖌ C<i>d</i> ꖌ <i>c</i>D ꖌ <i>cd</i>) -</div> - -<p class="ti0">communicates no more information than that A is B. -Abstraction of indifferent terms is in fact the converse -process to that of development described in p. <a href="#Page_89">89</a>; and -it is one of the most important operations in the whole -sphere of reasoning.</p> - -<p>The reader should observe that in the proposition</p> - -<div class="ml5em"> -AC = BC -</div> - -<p class="ti0">we cannot abstract C and infer</p> - -<div class="ml5em"> -A = B; -</div> - -<p class="ti0">but from</p> - -<div class="ml5em"> -AC ꖌ A<i>c</i> = BC ꖌ B<i>c</i> -</div> - -<p class="ti0">we may abstract all reference to the term C.</p> - -<p>It ought to be carefully remarked, however, that alternatives -which seem to be without meaning often imply -important knowledge. Thus if I say that “a triangle is a<span class="pagenum" id="Page_98">98</span> -three-sided rectilinear figure, with or without three equal -angles,” the last alternatives really express a property of -triangles, namely, that some triangles have three equal -angles, and some do not have them. If we put P = -“Some,” meaning by the indefinite adjective “Some,” one -or more of the undefined properties of triangles with three -equal angles, and take</p> - -<div class="ml5em"> -A = triangle<br> -B = three-sided rectilinear figure<br> -C = with three equal angles, -</div> - -<p class="ti0">then the knowledge implied is expressed in the two -propositions</p> - -<div class="ml5em"> -PA = PBC<br> -<i>p</i>A = <i>p</i>B<i>c</i>. -</div> - -<p>These may also be thrown into the form of one proposition, -namely,</p> - -<div class="ml5em"> -A = PBC ꖌ <i>p</i>B<i>c</i>; -</div> - -<p class="ti0">but these alternatives cannot be reduced, and the proposition -is quite different from</p> - -<div class="ml5em"> -A = BC ꖌ B<i>c</i>. -</div> - - -<h3><i>Illustrations of the Indirect Method.</i></h3> - -<p>A great variety of arguments and logical problems -might be introduced here to show the comprehensive -character and powers of the Indirect Method. We can -treat either a single premise or a series of premises.</p> - -<p>Take in the first place a simple definition, such as “a -triangle is a three-sided rectilinear figure.” Let</p> - -<div class="ml5em"> -A = triangle<br> -B = three-sided<br> -C = rectilinear figure, -</div> - -<p class="ti0">then the definition is of the form</p> - -<div class="ml5em"> -A = BC. -</div> - -<p>If we take the series of eight combinations of three -letters in the Logical Alphabet (p. <a href="#Page_94">94</a>) and strike out -those which are inconsistent with the definition, we have -the following result:—</p> - -<div class="ml5em"> -ABC<br> -<i>a</i>B<i>c</i><br> -<i>ab</i>C<br> -<i>abc.</i> -</div> - -<p><span class="pagenum" id="Page_99">99</span></p> - -<p>For the description of the class C we have</p> - -<div class="ml5em"> -C = ABC ꖌ <i>ab</i>C, -</div> - -<p class="ti0">that is, “a rectilinear figure is either a triangle and three-sided, -or not a triangle and not three-sided.”</p> - -<p>For the class <i>b</i> we have</p> - -<div class="ml5em"> -<i>b</i> = <i>ab</i>C ꖌ <i>abc</i>. -</div> - -<p>To the second side of this we may apply the process of -simplification by abstraction described in the last section; -for by the Law of Duality</p> - -<div class="ml5em"> -<i>ab</i> = <i>ab</i>C ꖌ <i>abc</i>; -</div> - -<p class="ti0">and as we have two propositions identical in the second -side of each we may substitute, getting</p> - -<div class="ml5em"> -<i>b</i> = <i>ab</i>, -</div> - -<p class="ti0">or what is not three-sided is not a triangle (whether it be -rectilinear or not).</p> - - -<h3><i>Second Example.</i></h3> - -<p>Let us treat by this method the following argument:—</p> - -<div class="pl4h2"> -“Blende is not an elementary substance; elementary -substances are those which are undecomposable; -blende, therefore, is decomposable.” -</div> - -<p>Taking our letters thus—</p> - -<div class="ml5em"> -A = blende,<br> -B = elementary substance,<br> -C = undecomposable, -</div> - -<p class="ti0">the premises are of the forms</p> - -<table class="ml5em"> -<tr> -<td class="tal pr2">A = A<i>b</i>,</td> -<td class="tar"><div>(1)</div></td> -</tr> -<tr> -<td class="tal">B = C.</td> -<td class="tar"><div>(2)</div></td> -</tr> -</table> - -<p>No immediate substitution can be made; but if we take -the contrapositive of (2) (see p. <a href="#Page_86">86</a>), namely</p> - -<table class="ml5em"> -<tr> -<td class="tal pr2"><i>b</i> = <i>c</i>,</td> -<td class="tar"><div>(3)</div></td> -</tr> -</table> - -<p class="ti0">we can substitute in (1) obtaining the conclusion</p> - -<div class="ml5em"> -A = A<i>c</i>. -</div> - -<p class="ti0">But the same result may be obtained by taking the eight -combinations of A, B, C, of the Logical Alphabet; it will -be found that only three combinations, namely,</p> - -<div class="ml5em"> -A<i>bc</i><br> -<i>a</i>BC<br> -<i>abc</i>, -</div> - -<p class="ti0">are consistent with the premises, whence it results that</p> - -<div class="ml5em"> -A = A<i>bc</i>, -</div> - -<p><span class="pagenum" id="Page_100">100</span></p> - -<p class="ti0">or by the process of Ellipsis before described (p. <a href="#Page_57">57</a>)</p> - -<div class="ml5em"> -A = A<i>c</i>. -</div> - - -<h3><i>Third Example.</i></h3> - -<p>As a somewhat more complex example I take the -argument thus stated, one which could not be thrown into -the syllogistic form:—</p> - -<div class="pl4h2"> -“All metals except gold and silver are opaque; therefore -what is not opaque is either gold or silver or -is not-metal.” -</div> - -<p>There is more implied in this statement than is distinctly -asserted, the full meaning being as follows:</p> - -<table class="ml3em"> -<tr> -<td class="tal pr2">All metals not gold or silver are opaque,</td> -<td class="tar"><div>(1)</div></td> -</tr> -<tr> -<td class="tal">Gold is not opaque but is a metal,</td> -<td class="tar"><div>(2)</div></td> -</tr> -<tr> -<td class="tal">Silver is not opaque but is a metal,</td> -<td class="tar"><div>(3)</div></td> -</tr> -<tr> -<td class="tal">Gold is not silver.</td> -<td class="tar"><div>(4)</div></td> -</tr> -</table> - -<p>Taking our letters thus—</p> - -<table class="ml5em"> -<tr> -<td class="tal pr3">A = metal</td> -<td class="tal">C = silver</td> -</tr> -<tr> -<td class="tal pr3">B = gold</td> -<td class="tal">D = opaque,</td> -</tr> -</table> - -<p class="ti0">we may state the premises in the forms</p> - -<table class="ml5em"> -<tr> -<td class="tar"><div>A<i>bc</i></div></td> -<td class="tal pr2"> = A<i>bc</i>D</td> -<td class="tar"><div>(1)</div></td> -</tr> -<tr> -<td class="tar"><div>B</div></td> -<td class="tal"> = AB<i>d</i></td> -<td class="tar"><div>(2)</div></td> -</tr> -<tr> -<td class="tar"><div>C</div></td> -<td class="tal"> = AC<i>d</i></td> -<td class="tar"><div>(3)</div></td> -</tr> -<tr> -<td class="tar"><div>B</div></td> -<td class="tal"> = B<i>c</i>.</td> -<td class="tar"><div>(4)</div></td> -</tr> -</table> - -<p>To obtain a complete solution of the question we take -the sixteen combinations of A, B, C, D, and striking out -those which are inconsistent with the premises, there remain -only</p> - -<div class="ml5em"> -AB<i>cd</i><br> -A<i>b</i>C<i>d</i><br> -A<i>bc</i>D<br> -<i>abc</i>D<br> -<i>abcd</i>. -</div> - -<p>The expression for not-opaque things consists of the -three combinations containing <i>d</i>, thus</p> - -<table class="ml3em"> -<tr> -<td class="tal"></td> -<td class="tar"><div><i>d</i></div></td> -<td class="tal"> = AB<i>cd</i> ꖌ A<i>b</i>C<i>d</i> ꖌ <i>abcd</i>,</td> -</tr> -<tr> -<td class="tal pr2">or</td> -<td class="tar"><div><i>d</i></div></td> -<td class="tal"> = A<i>d</i> (B<i>c</i> ꖌ <i>b</i>C) ꖌ <i>abcd</i>.</td> -</tr> -</table> - -<p>In ordinary language, what is not-opaque is either metal -which is gold, and then not-silver, or silver and then not-gold, -or else it is not-metal and neither gold nor silver.</p> - -<p><span class="pagenum" id="Page_101">101</span></p> - - -<h3><i>Fourth Example.</i></h3> - -<p>A good example for the illustration of the Indirect -Method is to be found in De Morgan’s <i>Formal Logic</i> (p. -123), the premises being substantially as follows:—</p> - -<p>From A follows B, and from C follows D; but B and D -are inconsistent with each other; therefore A and C are -inconsistent.</p> - -<p>The meaning no doubt is that where A is, B will be -found, or that every A is a B, and similarly every C is a D; -but B and D cannot occur together. The premises therefore -appear to be of the forms</p> - -<table class="ml5em"> -<tr> -<td class="tal pr2">A = AB,</td> -<td class="tar"><div>(1)</div></td> -</tr> -<tr> -<td class="tal pr2">C = CD,</td> -<td class="tar"><div>(2)</div></td> -</tr> -<tr> -<td class="tal">B = B<i>d</i>.</td> -<td class="tar"><div>(3)</div></td> -</tr> -</table> - -<p>On examining the series of sixteen combinations, only -five are found to be consistent with the above conditions, -namely,</p> - -<div class="ml5em"> -AB<i>cd</i><br> -<i>a</i>B<i>cd</i><br> -<i>ab</i>CD<br> -<i>abc</i>D<br> -<i>abcd</i>. -</div> - -<p>In these combinations the only A which appears is joined -to <i>c</i>, and similarly C is joined to <i>a</i>, or A is inconsistent -with C.</p> - - -<h3><i>Fifth Example.</i></h3> - -<p>A more complex argument, also given by De Morgan,<a id="FNanchor_78" href="#Footnote_78" class="fnanchor">78</a> -contains five terms, and is as stated below, except that -the letters are altered.</p> - -<div class="pl4h2"> -Every A is one only of the two B or C; D is both B -and C, except when B is E, and then it is -neither; therefore no A is D. -</div> - -<p>The meaning of the above premises is difficult to -interpret, but seems to be capable of expression in the -following symbolic forms—</p> - -<p><span class="pagenum" id="Page_102">102</span></p> - - -<table class="ml5em"> -<tr> -<td class="tar"><div> A</div></td> -<td class="tal pr2"> = AB<i>c</i> ꖌ A<i>b</i>C,</td> -<td class="tar"><div>(1)</div></td> -</tr> -<tr> -<td class="tar"><div>De</div></td> -<td class="tal"> = D<i>e</i>BC,</td> -<td class="tar"><div>(2)</div></td> -</tr> -<tr> -<td class="tar"><div>DE</div></td> -<td class="tal"> = DE<i>bc</i>.</td> -<td class="tar"><div>(3)</div></td> -</tr> -</table> - - -<p>As five terms enter into these premises it is requisite to -treat their thirty-two combinations, and it will be found -that fourteen of them remain consistent with the premises, -namely</p> - - -<table class="ml5em"> -<tr> -<td class="tal pr2">AB<i>cd</i>E</td> -<td class="tal pr2"><i>a</i>BCD<i>e</i></td> -<td class="tal"><i>ab</i>C<i>d</i>E</td> -</tr> -<tr> -<td class="tal">AB<i>cde</i></td> -<td class="tal"><i>a</i>BC<i>d</i>E</td> -<td class="tal"><i>ab</i>C<i>de</i></td> -</tr> -<tr> -<td class="tal">A<i>b</i>C<i>d</i>E</td> -<td class="tal"><i>a</i>BC<i>de</i></td> -<td class="tal"><i>abc</i>DE</td> -</tr> -<tr> -<td class="tal">A<i>b</i>C<i>de</i></td> -<td class="tal"><i>a</i>B<i>cd</i>E</td> -<td class="tal"><i>abcd</i>E</td> -</tr> -<tr> -<td class="tal"></td> -<td class="tal"><i>a</i>B<i>cde</i></td> -<td class="tal"><i>abcde</i>.</td> -</tr> -</table> - - -<p>If we examine the first four combinations, all of which -contain A, we find that they none of them contain D; or -again, if we select those which contain D, we have only -two, thus—</p> - -<div class="ml5em"> -D = <i>a</i>BCD<i>e</i> ꖌ <i>abc</i>DE. -</div> - -<p class="ti0">Hence it is clear that no A is D, and <i>vice versâ</i> no D is A. -We might draw many other conclusions from the same -premises; for instance—</p> - -<div class="ml5em"> -DE = <i>abc</i>DE, -</div> - -<p class="ti0">or D and E never meet but in the absence of A, B, and C.</p> - - -<h3><i>Fallacies analysed by the Indirect Method.</i></h3> - -<p>It has been sufficiently shown, perhaps, that we can by -the Indirect Method of Inference extract the whole truth -from a series of propositions, and exhibit it anew in any -required form of conclusion. But it may also need to be -shown by examples that so long as we follow correctly -the almost mechanical rules of the method, we cannot fall -into any of the fallacies or paralogisms which are often -committed in ordinary discussion. Let us take the example -of a fallacious argument, previously treated by the Method -of Direct Inference (p. <a href="#Page_62">62</a>),</p> - -<table class="ml5em"> -<tr> -<td class="tal pr2">Granite is not a sedimentary rock,</td> -<td class="tar"><div>(1)</div></td> -</tr> -<tr> -<td class="tal">Basalt is not a sedimentary rock,</td> -<td class="tar"><div>(2)</div></td> -</tr> -</table> - -<p class="ti0">and let us ascertain whether any precise conclusion can be -drawn concerning the relation of granite and basalt. -Taking as before</p> - -<div class="ml5em"> -A = granite,<br> -B = sedimentary rock,<br> -C = basalt, -</div> - -<p><span class="pagenum" id="Page_103">103</span></p> - -<p class="ti0">the premises become</p> - -<table class="ml5em"> -<tr> -<td class="tal pr2">A = A<i>b</i>,</td> -<td class="tar"><div>(1)</div></td> -</tr> -<tr> -<td class="tal pr2">C = C<i>b</i>.</td> -<td class="tar"><div>(2)</div></td> -</tr> -</table> - -<p class="ti0">Of the eight conceivable combinations of A, B, C, five -agree with these conditions, namely</p> - -<table class="ml5em"> -<tr> -<td class="tal pr2">A<i>b</i>C</td> -<td class="tar"><div><i>a</i>B<i>c</i></div></td> -</tr> -<tr> -<td class="tal pr2">A<i>bc</i></td> -<td class="tar"><div><i>ab</i>C</div></td> -</tr> -<tr> -<td class="tal"></td> -<td class="tar"><div><i>abc</i>.</div></td> -</tr> -</table> - -<p class="ti0">Selecting the combinations which contain A, we find the -description of granite to be</p> - -<div class="ml5em"> -A = A<i>b</i>C ꖌ A<i>bc</i> = A<i>b</i>(C ꖌ <i>c</i>),<br> -</div> - -<p class="ti0">that is, granite is not a sedimentary rock, and is either -basalt or not-basalt. If we want a description of basalt the -answer is of like form</p> - -<div class="ml5em"> -C = A<i>b</i>C ꖌ <i>ab</i>C = <i>b</i>C(A ꖌ <i>a</i>), -</div> - -<p class="ti0">that is basalt is not a sedimentary rock, and is either -granite or not-granite. As it is already perfectly evident -that basalt must be either granite or not, and <i>vice versâ</i>, -the premises fail to give us any information on the point, -that is to say the Method of Indirect Inference saves us -from falling into any fallacious conclusions. This -example sufficiently illustrates both the fallacy of -Negative premises and that of Undistributed Middle of -the old logic.</p> - -<p>The fallacy called the Illicit Process of the Major Term -is also incapable of commission in following the rules of -the method. Our example was (p. <a href="#Page_65">65</a>)</p> - -<table class="ml5em"> -<tr> -<td class="tal pr2">All planets are subject to gravity,</td> -<td class="tar"><div>(1)</div></td> -</tr> -<tr> -<td class="tal">Fixed stars are not planets.</td> -<td class="tar"><div>(2)</div></td> -</tr> -</table> - -<p class="ti0">The false conclusion is that “fixed stars are not subject to -gravity.” The terms are</p> - -<div class="ml5em"> -A = planet<br> -B = fixed star<br> -C = subject to gravity. -</div> - -<p class="ti0">And the premises are</p> - -<table class="ml5em"> -<tr> -<td class="tal pr2">A = AC,</td> -<td class="tar"><div>(1)</div></td> -</tr> -<tr> -<td class="tal">B = <i>a</i>B.</td> -<td class="tar"><div>(2)</div></td> -</tr> -</table> - -<p class="ti0">The combinations which remain uncontradicted on comparison -with these premises are</p> - -<table class="ml5em"> -<tr> -<td class="tal pr2">A<i>b</i>C</td> -<td class="tar"><div><i>a</i>B<i>c</i></div></td> -</tr> -<tr> -<td class="tal pr2"><i>a</i>BC</td> -<td class="tar"><div><i>ab</i>C</div></td> -</tr> -<tr> -<td class="tal"></td> -<td class="tar"><div><i>abc</i>.</div></td> -</tr> -</table> - -<p class="ti0">For fixed star we have the description</p> - -<div class="ml5em"> -B = <i>a</i>BC ꖌ <i>a</i>B<i>c</i>, -</div> - -<p><span class="pagenum" id="Page_104">104</span></p> - -<p class="ti0">that is, “a fixed star is not a planet, but is either subject -or not, as the case may be, to gravity.” Here we have no -conclusion concerning the connection of fixed stars and -gravity.</p> - - -<h3><i>The Logical Abacus.</i></h3> - -<p>The Indirect Method of Inference has now been sufficiently -described, and a careful examination of its powers -will show that it is capable of giving a full analysis and -solution of every question involving only logical relations. -The chief difficulty of the method consists in the great -number of combinations which may have to be examined; -not only may the requisite labour become formidable, but -a considerable chance of mistake arises. I have therefore -given much attention to modes of facilitating the work, -and have succeeded in reducing the method to an almost -mechanical form. It soon appeared obvious that if the -conceivable combinations of the Logical Alphabet, for any -number of letters, instead of being printed in fixed order -on a piece of paper or slate, were marked upon light -movable pieces of wood, mechanical arrangements could -readily be devised for selecting any required class of the -combinations. The labour of comparison and rejection -might thus be immensely reduced. This idea was first -carried out in the Logical Abacus, which I have found -useful in the lecture-room for exhibiting the complete -solution of logical problems. A minute description of the -construction and use of the Abacus, together with figures -of the parts, has already been given in my essay called -<i>The Substitution of Similars</i>,<a id="FNanchor_79" href="#Footnote_79" class="fnanchor">79</a> and I will here give only -a general description.</p> - -<p>The Logical Abacus consists of a common school black-board -placed in a sloping position and furnished with four -horizontal and equi-distant ledges. The combinations -of the letters shown in the first four columns of the -Logical Alphabet are printed in somewhat large type, -so that each letter is about an inch from the neighbouring -one, but the letters are placed one above the other -instead of being in horizontal lines as in p. <a href="#Page_94">94</a>. Each -combination of letters is separately fixed to the surface of<span class="pagenum" id="Page_105">105</span> -a thin slip of wood one inch broad and about one-eighth -inch thick. Short steel pins are then driven in an inclined -position into the wood. When a letter is a large capital -representing a positive term, the pin is fixed in the upper -part of its space; when the letter is a small italic representing -a negative term, the pin is fixed in the lower part -of the space. Now, if one of the series of combinations -be ranged upon a ledge of the black-board, the sharp edge -of a flat rule can be inserted beneath the pins belonging to -any one letter—say A, so that all the combinations marked -A can be lifted out and placed upon a separate ledge. -Thus we have represented the act of thought which -separates the class A from what is not-A. The operation -can be repeated; out of the A’s we can in like manner -select those which are B’s, obtaining the AB’s; and in like -manner we may select any other classes such as the <i>a</i>B’s, -the <i>ab</i>’s, or the <i>abc</i>’s.</p> - -<p>If now we take the series of eight combinations of the -letters A, B, C, <i>a</i>, <i>b</i>, <i>c</i>, and wish to analyse the argument -anciently called Barbara, having the premises</p> - -<table class="ml5em"> -<tr> -<td class="tal pr2">A = AB</td> -<td class="tar"><div>(1)</div></td> -</tr> -<tr> -<td class="tal pr2">B = BC,</td> -<td class="tar"><div>(2)</div></td> -</tr> -</table> - -<p>we proceed as follows—We raise the combinations marked -<i>a</i>, leaving the A’s behind; out of these A’s we move to a -lower ledge such as are <i>b</i>’s, and to the remaining AB’s -we join the <i>a</i>’s which have been raised. The result is that -we have divided all the combinations into two classes, -namely, the A<i>b</i>’s which are incapable of existing consistently -with premise (1), and the combinations which are -consistent with the premise. Turning now to the second -premise, we raise out of those which agree with (1) the <i>b</i>’s, -then we lower the B<i>c</i>’s; lastly we join the <i>b</i>’s to the BC’s. -We now find our combinations arranged as below.</p> - -<div class="tac"> -<table class="tac fs80 ball mtb1em" style="width: 25%;"> -<tr> -<td class="tac pt03">A</td> -<td class="tac brl"></td> -<td class="tac"></td> -<td class="tac brl"></td> -<td class="tac"><div><i>a</i></div></td> -<td class="tac brl"></td> -<td class="tac"><div><i>a</i></div></td> -<td class="tac brl"><i>a</i></td> -</tr> -<tr> -<td class="tac"><div>B</div></td> -<td class="tac brl"></td> -<td class="tac"></td> -<td class="tac brl"></td> -<td class="tac"><div>B</div></td> -<td class="tac brl"></td> -<td class="tac"><div><i>b</i></div></td> -<td class="tac brl"><i>b</i></td> -</tr> -<tr> -<td class="tac bb pb03">C</td> -<td class="tac bbrl"></td> -<td class="tac bb"></td> -<td class="tac bbrl"></td> -<td class="tac bb">C</td> -<td class="tac bbrl"></td> -<td class="tac bb">C</td> -<td class="tac bbrl"><i>c</i></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac brl pt03">A</td> -<td class="tac pt03">A</td> -<td class="tac brl pt03">A</td> -<td class="tac"></td> -<td class="tac brl"><i>a</i></td> -<td class="tac"></td> -<td class="tac brl"></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac brl">B</td> -<td class="tac"><div><i>b</i></div></td> -<td class="tac brl"><i>b</i></td> -<td class="tac"></td> -<td class="tac brl">B</td> -<td class="tac"></td> -<td class="tac brl"></td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac brl pb03"><i>c</i></td> -<td class="tac pb03">C</td> -<td class="tac brl pb03"><i>c</i></td> -<td class="tac"></td> -<td class="tac brl pb03"><i>c</i></td> -<td class="tac"></td> -<td class="tac brl"></td> -</tr> -</table> -</div> - -<p>The lower line contains all the combinations which are -inconsistent with either premise; we have carried out in a<span class="pagenum" id="Page_106">106</span> -mechanical manner that exclusion of self-contradictories -which was formerly done upon the slate or upon paper. -Accordingly, from the combinations remaining in the upper -line we can draw any inference which the premises yield. -If we raise the A’s we find only one, and that is C, so -that A must be C. If we select the <i>c</i>’s we again find only -one, which is <i>a</i> and also <i>b</i>; thus we prove that not-C is -not-A and not-B.</p> - -<p>When a disjunctive proposition occurs among the -premises the requisite movements become rather more -complicated. Take the disjunctive argument</p> - -<div class="ml5em"> -A is either B or C or D,<br> -A is not C and not D,<br> -Therefore A is B. -</div> - -<p class="ti0">The premises are represented accurately as follows:—</p> - -<table class="ml5em"> -<tr> -<td class="tal pr2">A = AB ꖌ AC ꖌ AD</td> -<td class="tar"><div>(1)</div></td> -</tr> -<tr> -<td class="tal">A = A<i>c</i></td> -<td class="tar"><div>(2)</div></td> -</tr> -<tr> -<td class="tal">A = A<i>d</i>.</td> -<td class="tar"><div>(3)</div></td> -</tr> -</table> - -<p class="ti0">As there are four terms, we choose the series of sixteen -combinations and place them on the highest ledge of the -board but one. We raise the <i>a</i>’s and out of the A’s, which -remain, we lower the <i>b</i>’s. But we are not to reject all the -A<i>b</i>’s as contradictory, because by the first premise A’s -may be either B’s or C’s or D’s. Accordingly out of the -A<i>b</i>’s we must select the <i>c</i>’s, and out of these again the <i>d</i>’s, -so that only A<i>bcd</i> will remain to be rejected finally. -Joining all the other fifteen combinations together again, -and proceeding to premise (2), we raise the <i>a</i>’s and lower -the AC’s, and thus reject the combinations inconsistent -with (2); similarly we reject the AD’s which are inconsistent -with (3). It will be found that there remain, in -addition to all the eight combinations containing <i>a</i>, only -one containing A, namely</p> - -<div class="ml5em"> -AB<i>cd</i>, -</div> - -<p class="ti0">whence it is apparent that A must be B, the ordinary -conclusion of the argument.</p> - -<p>In my “Substitution of Similars” (pp. 56–59) I have -described the working upon the Abacus of two other -logical problems, which it would be tedious to repeat in -this place.</p> - -<p><span class="pagenum" id="Page_107">107</span></p> - - -<h3><i>The Logical Machine.</i></h3> - -<p>Although the Logical Abacus considerably reduced the -labour of using the Indirect Method, it was not free from -the possibility of error. I thought moreover that it would -afford a conspicuous proof of the generality and power of -the method if I could reduce it to a purely mechanical -form. Logicians had long been accustomed to speak of -Logic as an Organon or Instrument, and even Lord Bacon, -while he rejected the old syllogistic logic, had insisted, in -the second aphorism of his “New Instrument,” that the -mind required some kind of systematic aid. In the -kindred science of mathematics mechanical assistance of -one kind or another had long been employed. Orreries, -globes, mechanical clocks, and such like instruments, -are really aids to calculation and are of considerable -antiquity. The Arithmetical Abacus is still in common -use in Russia and China. The calculating machine of -Pascal is more than two centuries old, having been constructed -in 1642–45. M. Thomas of Colmar manufactures -an arithmetical machine on Pascal’s principles which is -employed by engineers and others who need frequently -to multiply or divide. To Babbage and Scheutz is due -the merit of embodying the Calculus of Differences in a -machine, which thus became capable of calculating the -most complicated tables of figures. It seemed strange -that in the more intricate science of quantity mechanism -should be applicable, whereas in the simple science of -qualitative reasoning, the syllogism was only called an -instrument by a figure of speech. It is true that Swift -satirically described the Professors of Laputa as in possession -of a thinking machine, and in 1851 Mr. Alfred -Smee actually proposed the construction of a Relational -machine and a Differential machine, the first of which -would be a mechanical dictionary and the second a mode -of comparing ideas; but with these exceptions I have -not yet met with so much as a suggestion of a reasoning -machine. It may be added that Mr. Smee’s designs, though -highly ingenious, appear to be impracticable, and in any -case they do not attempt the performance of logical inference.<a id="FNanchor_80" href="#Footnote_80" class="fnanchor">80</a></p> -<p><span class="pagenum" id="Page_108">108</span></p> -<p>The Logical Abacus soon suggested the notion of a -Logical Machine, which, after two unsuccessful attempts, -I succeeded in constructing in a comparatively simple and -effective form. The details of the Logical Machine have -been fully described by the aid of plates in the Philosophical -Transactions,<a id="FNanchor_81" href="#Footnote_81" class="fnanchor">81</a> and it would be needless to repeat -the account of the somewhat intricate movements of the -machine in this place.</p> - -<p>The general appearance of the machine is shown in a -plate facing the title-page of this volume. It somewhat -resembles a very small upright piano or organ, and has a -keyboard containing twenty-one keys. These keys are of -two kinds, sixteen of them representing the terms or -letters A, <i>a</i>, B, <i>b</i>, C, <i>c</i>, D, <i>d</i>, which have so often been -employed in our logical notation. When letters occur on -the left-hand side of a proposition, formerly called the -subject, each is represented by a key on the left-hand half -of the keyboard; but when they occur on the right-hand -side, or as it used to be called the predicate of the proposition, -the letter-keys on the right-hand side of the -keyboard are the proper representatives. The five other -keys may be called operation keys, to distinguish them -from the letter or term keys. They stand for the stops, -copula, and disjunctive conjunctions of a proposition. -The middle key of all is the copula, to be pressed when -the verb <i>is</i> or the sign = is met. The key to the extreme -right-hand is called the Full Stop, because it should be -pressed when a proposition is completed, in fact in the -proper place of the full stop. The key to the extreme -left-hand is used to terminate an argument or to restore -the machine to its initial condition; it is called the Finis -key. The last keys but one on the right and left complete -the whole series, and represent the conjunction <i>or</i> in -its unexclusive meaning, or the sign ꖌ which I have -employed, according as it occurs in the right or left hand -side of the proposition. The whole keyboard is arranged -as shown on the next page—</p> - -<p><span class="pagenum" id="Page_109">109</span></p> - -<div class="center"> -<table id="tab109a"> -<tr class="fs70"> -<td class="tal vertical upright pl03 pr15 fs90" rowspan="2">Finis.</td> -<td class="tac prl03" colspan="9">Left-hand side of Proposition.</td> -<td class="tac vertical upright pl03 pr15 fs90" rowspan="2">Cupola.</td> -<td class="tac prl03" colspan="9">Right-hand side of Proposition.</td> -<td class="tac vertical upright pl03 pr15 ptb05 fs90" rowspan="2">Fullstop.</td> -</tr> -<tr class="fs80"> -<td class="prl05">ꖌ<br>Or</td> -<td class="prl05"><i>d</i></td> -<td class="prl05">D</td> -<td class="prl05"><i>c</i></td> -<td class="prl05">C</td> -<td class="prl05"><i>b</i></td> -<td class="prl05">B</td> -<td class="prl05"><i>a</i></td> -<td class="prl05">A</td> -<td class="prl05">A</td> -<td class="prl05"><i>a</i></td> -<td class="prl05">B</td> -<td class="prl05"><i>b</i></td> -<td class="prl05">C</td> -<td class="prl05"><i>c</i></td> -<td class="prl05">D</td> -<td class="prl05"><i>d</i></td> -<td class="prl05">ꖌ<br>Or</td> -</tr> -</table> -</div> - -<p>To work the machine it is only requisite to press the -keys in succession as indicated by the letters and signs of -a symbolical proposition. All the premises of an argument -are supposed to be reduced to the simple notation -which has been employed in the previous pages. Taking -then such a simple proposition as</p> - -<div class="ml5em"> -A = AB, -</div> - -<p class="ti0">we press the keys A (left), copula, A (right), B (right), and -full stop.</p> - -<p>If there be a second premise, for instance</p> - -<div class="ml5em"> -B = BC, -</div> - -<p class="ti0">we press in like manner the keys—</p> - -<div class="ml5em"> -B (left), copula, B (right), C (right), full stop. -</div> - -<p class="ti0">The process is exactly the same however numerous the -premises may be. When they are completed the operator -will see indicated on the face of the machine the exact -combinations of letters which are consistent with the -premises according to the principles of thought.</p> - -<p>As shown in the figure opposite the title-page, the -machine exhibits in front a Logical Alphabet of sixteen -combinations, exactly like that of the Abacus, except -that the letters of each combination are separated by a -certain interval. After the above problem has been -worked upon the machine the Logical Alphabet will have -been modified so as to present the following appearance—</p> - - - -<div class="center"> -<table id="tab109b"> -<tr> -<td colspan="16"> </td> -</tr> -<tr> -<td>A</td> -<td>A</td> -<td><div>  </div></td> -<td><div>  </div></td> -<td><div>  </div></td> -<td><div>  </div></td> -<td><div>  </div></td> -<td><div>  </div></td> -<td><i>a</i></td> -<td><i>a</i></td> -<td><div>  </div></td> -<td><div>  </div></td> -<td><i>a</i></td> -<td><i>a</i></td> -<td><i>a</i></td> -<td><i>a</i></td> -</tr> -<tr><td colspan="16"> </td></tr> -<tr> -<td>B</td> -<td>B</td> -<td></td> -<td></td> -<td></td> -<td></td> -<td></td> -<td></td> -<td>B</td> -<td>B</td> -<td></td> -<td></td> -<td><i>b</i></td> -<td><i>b</i></td> -<td><i>b</i></td> -<td><i>b</i></td> -</tr> -<tr><td colspan="16"> </td></tr> -<tr> -<td>C</td> -<td>C</td> -<td></td> -<td></td> -<td></td> -<td></td> -<td></td> -<td></td> -<td>C</td> -<td>C</td> -<td></td> -<td></td> -<td>C</td> -<td>C</td> -<td><i>c</i></td> -<td><i>c</i></td> -</tr> -<tr><td colspan="16"> </td></tr> -<tr> -<td>D</td> -<td><i>d</i></td> -<td></td> -<td></td> -<td></td> -<td></td> -<td></td> -<td></td> -<td>D</td> -<td><i>d</i></td> -<td></td> -<td></td> -<td>D</td> -<td><i>d</i></td> -<td>D</td> -<td><i>d</i></td> -</tr> -<tr> -<td colspan="16"> </td> -</tr> -</table> -</div> - -<p><span class="pagenum" id="Page_110">110</span></p> - -<p>The operator will readily collect the various conclusions -in the manner described in previous pages, as, for instance -that A is always C, that not-C is not-B and not-A; -and not-B is not-A but either C or not-C. The results -are thus to be read off exactly as in the case of the -Logical Slate, or the Logical Abacus.</p> - -<p>Disjunctive propositions are to be treated in an exactly -similar manner. Thus, to work the premises</p> - -<table class="ml5em"> -<tr> -<td class="tar"><div>A = </div></td> -<td class="tal">AB ꖌ AC</td> -</tr> -<tr> -<td class="tar"><div>B ꖌ C = </div></td> -<td class="tal">BD ꖌ CD,</td> -</tr> -</table> - -<p class="ti0">it is only necessary to press in succession the keys</p> - -<div class="ml5em"> -A (left), copula, A (right), B, ꖌ, A, C, full stop.<br> -B (left), ꖌ, C, copula, B (right), D, ꖌ, C, D, full stop. -</div> - -<p class="ti0">The combinations then remaining will be as follows</p> - -<table class="ml5em"> -<tr> -<td class="tal pr2">ABCD</td> -<td class="tal pr2"><i>a</i>BCD</td> -<td class="tal"><i>abc</i>D</td> -</tr> -<tr> -<td class="tal pr2">AB<i>c</i>D</td> -<td class="tal pr2"><i>a</i>B<i>c</i>D</td> -<td class="tal"><i>abcd.</i></td> -</tr> -<tr> -<td class="tal pr2">A<i>c</i>CD</td> -<td class="tal pr2"><i>ab</i>CD</td> -<td class="tal"></td> -</tr> -</table> - -<p>On pressing the left-hand key A, all the possible combinations -which do not contain A will disappear, and the -description of A may be gathered from what remain, -namely that it is always D. The full-stop key restores all -combinations consistent with the premises and any other -selection may be made, as say not-D, which will be found -to be always not-A, not-B, and not-C.</p> - -<p>At the end of every problem, when no further questions -need be addressed to the machine, we press the Finis -key, which has the effect of bringing into view the whole -of the conceivable combinations of the alphabet. This -key in fact obliterates the conditions impressed upon the -machine by moving back into their ordinary places those -combinations which had been rejected as inconsistent with -the premises. Before beginning any new problem it is -requisite to observe that the whole sixteen combinations -are visible. After the Finis key has been used the machine -represents a mind endowed with powers of thought, but -wholly devoid of knowledge. It would not in that condition -give any answer but such as would consist in the -primary laws of thought themselves. But when any proposition -is worked upon the keys, the machine analyses -and digests the meaning of it and becomes charged with -the knowledge embodied in that proposition. Accordingly -it is able to return as an answer any description of a term<span class="pagenum" id="Page_111">111</span> -or class so far as furnished by that proposition in accordance -with the Laws of Thought. The machine is thus the embodiment -of a true logical system. The combinations are -classified, selected or rejected, just as they should be by a -reasoning mind, so that at each step in a problem, the -Logical Alphabet represents the proper condition of a mind -exempt from mistake. It cannot be asserted indeed that -the machine entirely supersedes the agency of conscious -thought; mental labour is required in interpreting the -meaning of grammatical expressions, and in correctly impressing -that meaning on the machine; it is further required -in gathering the conclusion from the remaining combinations. -Nevertheless the true process of logical inference -is really accomplished in a purely mechanical manner.</p> - -<p>It is worthy of remark that the machine can detect any -self-contradiction existing between the premises presented -to it; should the premises be self-contradictory it will be -found that one or more of the letter-terms disappears -entirely from the Logical Alphabet. Thus if we work the -two propositions, A is B, and A is not-B, and then inquire -for a description of A, the machine will refuse to give it -by exhibiting no combination at all containing A. This -result is in agreement with the law, which I have explained, -that every term must have its negative (p. <a href="#Page_74">74</a>). -Accordingly, whenever any one of the letters A, B, C, D, <i>a</i>, -<i>b</i>, <i>c</i>, <i>d</i>, wholly disappears from the alphabet, it may be -safely inferred that some act of self-contradiction has been -committed.</p> - -<p>It ought to be carefully observed that the logical -machine cannot receive a simple identity of the form -A = B except in the double form of A = B and B = A. -To work the proposition A = B, it is therefore necessary to -press the keys—</p> - -<div class="ml5em"> -A (left), copula, B (right), full stop;<br> -B (left), copula, A (right), full stop. -</div> - -<p class="ti0">The same double operation will be necessary whenever the -proposition is not of the kind called a partial identity -(p. <a href="#Page_40">40</a>). Thus AB = CD, AB = AC, A = B ꖌ C, A ꖌ B -= C ꖌ D, all require to be read from both ends separately.</p> - -<p>The proper rule for using the machine may in fact be -given in the following way:—(1) <i>Read each proposition as -it stands, and play the corresponding keys</i>: (2) <i>Convert the<span class="pagenum" id="Page_112">112</span> -proposition and read and play the keys again in the transposed -order of the terms.</i> So long as this rule is observed -the true result must always be obtained. There can be no -mistake. But it will be found that in the case of partial -identities, and some other similar forms of propositions, -the transposed reading has no effect upon the combinations -of the Logical Alphabet. One reading is in such cases all -that is practically needful. After some experience has -been gained in the use of the machine, the worker naturally -saves himself the trouble of the second reading when -possible.</p> - -<p>It is no doubt a remarkable fact that a simple identity -cannot be impressed upon the machine except in the form -of two partial identities, and this may be thought by some -logicians to militate against the equational mode of representing -propositions.</p> - -<p>Before leaving the subject I may remark that these -mechanical devices are not likely to possess much -practical utility. We do not require in common life to be -constantly solving complex logical questions. Even in -mathematical calculation the ordinary rules of arithmetic -are generally sufficient, and a calculating machine can only -be used with advantage in peculiar cases. But the machine -and abacus have nevertheless two important uses.</p> - -<p>In the first place I hope that the time is not very far -distant when the predominance of the ancient Aristotelian -Logic will be a matter of history only, and when the -teaching of logic will be placed on a footing more worthy -of its supreme importance. It will then be found that the -solution of logical questions is an exercise of mind at least -as valuable and necessary as mathematical calculation. I -believe that these mechanical devices, or something of the -same kind, will then become useful for exhibiting to a -class of students a clear and visible analysis of logical -problems of any degree of complexity, the nature of each -step being rendered plain to the eyes of the students. I -often used the machine or abacus for this purpose in -my class lectures while I was Professor of Logic at -Owens College.</p> - -<p>Secondly, the more immediate importance of the machine -seems to consist in the unquestionable proof which it -affords that correct views of the fundamental principles of<span class="pagenum" id="Page_113">113</span> -reasoning have now been attained, although they were -unknown to Aristotle and his followers. The time must -come when the inevitable results of the admirable -investigations of the late Dr. Boole must be recognised -at their true value, and the plain and palpable form in -which the machine presents those results will, I hope, hasten -the time. Undoubtedly Boole’s life marks an era in the -science of human reason. It may seem strange that it had -remained for him first to set forth in its full extent the -problem of logic, but I am not aware that anyone before -him had treated logic as a symbolic method for evolving -from any premises the description of any class whatsoever -as defined by those premises. In spite of several serious -errors into which he fell, it will probably be allowed that -Boole discovered the true and general form of logic, and -put the science substantially into the form which it must -hold for evermore. He thus effected a reform with which -there is hardly anything comparable in the history of logic -between his time and the remote age of Aristotle.</p> - -<p>Nevertheless, Boole’s quasi-mathematical system could -hardly be regarded as a final and unexceptionable solution -of the problem. Not only did it require the manipulation -of mathematical symbols in a very intricate and perplexing -manner, but the results when obtained were devoid of -demonstrative force, because they turned upon the employment -of unintelligible symbols, acquiring meaning only by -analogy. I have also pointed out that he imported into -his system a condition concerning the exclusive nature of -alternatives (p. <a href="#Page_70">70</a>), which is not necessarily true of logical -terms. I shall have to show in the next chapter that logic -is really the basis of the whole science of mathematical -reasoning, so that Boole inverted the true order of proof -when he proposed to infer logical truths by algebraic -processes. It is wonderful evidence of his mental power -that by methods fundamentally false he should have -succeeded in reaching true conclusions and widening the -sphere of reason.</p> - -<p>The mechanical performance of logical inference affords -a demonstration both of the truth of Boole’s results and -of the mistaken nature of his mode of deducing them. -Conclusions which he could obtain only by pages of intricate -calculation, are exhibited by the machine after one or<span class="pagenum" id="Page_114">114</span> -two minutes of manipulation. And not only are those -conclusions easily reached, but they are demonstratively -true, because every step of the process involves nothing -more obscure than the three fundamental Laws of Thought.</p> - - -<h3><i>The Order of Premises.</i></h3> - -<p>Before quitting the subject of deductive reasoning, I -may remark that the order in which the premises of an -argument are placed is a matter of logical indifference. -Much discussion has taken place at various times concerning -the arrangement of the premises of a syllogism; -and it has been generally held, in accordance with the -opinion of Aristotle, that the so-called major premise, -containing the major term, or the predicate of the conclusion, -should stand first. This distinction however falls -to the ground in our system, since the proposition is -reduced to an identical form, in which there is no distinction -of subject and predicate. In a strictly logical point -of view the order of statement is wholly devoid of -significance. The premises are simultaneously coexistent, -and are not related to each other according to the properties -of space and time. Just as the qualities of the same -object are neither before nor after each other in nature -(p. <a href="#Page_33">33</a>), and are only thought of in some one order owing -to the limited capacity of mind, so the premises of an -argument are neither before nor after each other, and are -only thought of in succession because the mind cannot -grasp many ideas at once. The combinations of the -logical alphabet are exactly the same in whatever order -the premises be treated on the logical slate or machine. -Some difference may doubtless exist as regards convenience -to human memory. The mind may take in the results -of an argument more easily in one mode of statement -than another, although there is no real difference in the -logical results. But in this point of view I think that -Aristotle and the old logicians were clearly wrong. It is -more easy to gather the conclusion that “all A’s are C’s” -from “all A’s are B’s and all B’s are C’s,” than from the -same propositions in inverted order, “all B’s are C’s and -all A’s are B’s.”</p> - -<p><span class="pagenum" id="Page_115">115</span></p> - - -<h3><i>The Equivalence of Propositions</i>.</h3> - -<p>One great advantage which arises from the study of -this Indirect Method of Inference consists in the clear -notion which we gain of the Equivalence of Propositions. -The older logicians showed how from certain simple -premises we might draw an inference, but they failed to -point out whether that inference contained the whole, or -only a part, of the information embodied in the premises. -Any one proposition or group of propositions may be -classed with respect to another proposition or group of -propositions, as</p> - -<div class="ml5em"> -1. Equivalent,<br> -2. Inferrible,<br> -3. Consistent,<br> -4. Contradictory. -</div> - -<p>Taking the proposition “All men are mortals” as the -original, then “All immortals are not men” is its equivalent; -“Some mortals are men” is inferrible, or capable of -inference, but is not equivalent; “All not-men are not -mortals” cannot be inferred, but is consistent, that is, -may be true at the same time; “All men are immortals” -is of course contradictory.</p> - -<p>One sufficient test of equivalence is capability of mutual -inference. Thus from</p> - -<div class="ml5em"> -All electrics = all non-conductors, -</div> - -<p class="ti0">I can infer</p> - -<div class="ml5em"> -All non-electrics = all conductors, -</div> - -<p class="ti0">and <i>vice versâ</i> from the latter I can pass back to the -former. In short, A = B is equivalent to <i>a</i> = <i>b</i>. Again, -from the union of the two propositions, A = AB and -B = AB, I get A = B, and from this I might as easily -deduce the two with which I started. In this case one -proposition is equivalent to two other propositions. There -are in fact no less than four modes in which we may -express the identity of two classes A and B, namely,</p> - -<table class="ml5em"> -<tr class="fs70"> -<td class="tac prl1">FIRST MODE.</td> -<td class="tac prl1">SECOND MODE.</td> -<td class="tar prl1" colspan="2">THIRD MODE.</td> -<td class="tar pl1" colspan="2">FOURTH MODE.</td> -</tr> -<tr> -<td class="tac" rowspan="2">A = B</td> -<td class="tac" rowspan="2"><i>a</i> = <i>b</i></td> -<td class="tar"><div>A = AB</div></td> -<td class="tal pr1 vab" rowspan="2"><img src="images/31x8br.png" width="8" height="31" alt="" ></td> -<td class="tar"><div><i>a</i> = <i>ab</i></div></td> -<td class="tal pr05 vab" rowspan="2"><img src="images/31x8br.png" width="8" height="31" alt="" ></td> -</tr> -<tr> -<td class="tar"><div>B = AB</div></td> -<td class="tar"><div><i>b</i> = <i>ab</i></div></td> -</tr> -</table> - -<p>The Indirect Method of Inference furnishes a universal -and clear criterion as to the relationship of propositions. -The import of a statement is always to be measured by<span class="pagenum" id="Page_116">116</span> -the combinations of terms which it destroys. Hence two -propositions are equivalent when they remove the same -combinations from the Logical Alphabet, and neither more -nor less. A proposition is inferrible but not equivalent to -another when it removes some but not all the combinations -which the other removes, and none except what this -other removes. Again, propositions are consistent provided -that they jointly allow each term and the negative of -each term to remain somewhere in the Logical Alphabet. -If after all the combinations inconsistent with two propositions -are struck out, there still appears each of the letters -A, <i>a</i>, B, <i>b</i>, C, <i>c</i>, D, <i>d</i>, which were there before, then no -inconsistency between the propositions exists, although -they may not be equivalent or even inferrible. Finally, -contradictory propositions are those which taken together -remove any one or more letter-terms from the Logical -Alphabet.</p> - -<p>What is true of single propositions applies also to groups -of propositions, however large or complicated; that is to -say, one group may be equivalent, inferrible, consistent, -or contradictory as regards another, and we may similarly -compare one proposition with a group of propositions.</p> - -<p>To give in this place illustrations of all the four kinds -of relation would require much space: as the examples -given in previous sections or chapters may serve more or -less to explain the relations of inference, consistency, and -contradiction, I will only add a few instances of equivalent -propositions or groups.</p> - -<p>In the following list each proposition or group of propositions -is exactly equivalent in meaning to the corresponding -one in the other column, and the truth of this -statement may be tested by working out the combinations -of the alphabet, which ought to be found exactly the same -in the case of each pair of equivalents.</p> - -<div class="center"> -<table class="ml3em mtb1em"> -<tr> -<td class="tar"><div>A = </div></td> -<td class="tal" colspan="5">A<i>b</i></td> -<td class="tar"><div>B = </div></td> -<td class="tal"><i>a</i>B</td> -</tr> -<tr> -<td class="tar"><div>A = </div></td> -<td class="tal" colspan="5"><i>b</i></td> -<td class="tar"><div><i>a</i> = </div></td> -<td class="tal">B</td> -</tr> -<tr> -<td class="tar"><div>A = </div></td> -<td class="tal" colspan="5">BC</td> -<td class="tar"><div><i>a</i> = </div></td> -<td class="tal"><i>b</i> ꖌ <i>c</i></td> -</tr> -<tr> -<td class="tar"><div>A = </div></td> -<td class="tal" colspan="5">AB ꖌ AC</td> -<td class="tar"><div><i>b</i> = </div></td> -<td class="tal"><i>ab</i> ꖌ A<i>b</i>C</td> -</tr> -<tr> -<td class="tar"><div>A ꖌ B = </div></td> -<td class="tal" colspan="5">B ꖌ <i>d</i></td> -<td class="tar"><div><i>ab</i> = </div></td> -<td class="tal"><i>cd</i></td> -</tr> -<tr> -<td class="tar"><div>A ꖌ <i>c</i> = </div></td> -<td class="tal" colspan="5">B ꖌ <i>d</i></td> -<td class="tar"><div><i>a</i>C = </div></td> -<td class="tal"><i>b</i>D</td> -</tr> -<tr> -<td class="tar" rowspan="2">A = </td> -<td class="tal pr2" rowspan="2" colspan="4">AB<i>c</i> ꖌ A<i>b</i>C</td> -<td class="tar vab" rowspan="2"><img src="images/31x8bl.png" width="8" height="31" alt="" ></td> -<td class="tar"><div>A = </div></td> -<td class="tal">AB ꖌ AC</td> -</tr> -<tr> -<td class="tar"><div>AB = </div></td> -<td class="tal">AB<i>c</i><span class="pagenum" id="Page_117">117</span></td> -</tr> -<tr> -<td class="tar"><div>A = </div></td> -<td class="tal">B</td> -<td class="tal vab" rowspan="2"><img src="images/31x8br.png" width="8" height="31" alt="" ></td> -<td rowspan="2" colspan="2">  </td> -<td class="tar vab" rowspan="2"><img src="images/31x8bl.png" width="8" height="31" alt="" ></td> -<td class="tar"><div>A = </div></td> -<td class="tal">B</td> -</tr> -<tr> -<td class="tar"><div>B = </div></td> -<td class="tal">C</td> -<td class="tar"><div>A = </div></td> -<td class="tal">C</td> -</tr> -<tr> -<td class="tar"><div>A = </div></td> -<td class="tal" colspan="2">AB</td> -<td class="tal vab" rowspan="2"><img src="images/31x8br.png" width="8" height="31" alt="" ></td> -<td rowspan="2">  </td> -<td class="tar vab" rowspan="2"><img src="images/31x8bl.png" width="8" height="31" alt="" ></td> -<td class="tar"><div>A = </div></td> -<td class="tal">AC</td> -</tr> -<tr> -<td class="tar"><div>B = </div></td> -<td class="tal" colspan="2">BC</td> -<td class="tar"><div>B = </div></td> -<td class="tal">A ꖌ <i>a</i>BC</td> -</tr> -</table> -</div> - -<p>Although in these and many other cases the equivalents -of certain propositions can readily be given, yet I believe -that no uniform and infallible process can be pointed out -by which the exact equivalents of premises can be -ascertained. Ordinary deductive inference usually gives -us only a portion of the contained information. It is -true that the combinations consistent with a set of -premises may always be thrown into the form of a -proposition which must be logically equivalent to those -premises; but the difficulty consists in detecting the other -forms of propositions which will be equivalent to the -premises. The task is here of a different character from -any which we have yet attempted. It is in reality an -inverse process, and is just as much more troublesome and -uncertain than the direct process, as seeking is compared -with hiding. Not only may several different answers -equally apply, but there is no method of discovering any -of those answers except by repeated trial. The problem -which we have here met is really that of induction, the -inverse of deduction; and, as I shall soon show, induction -is always tentative, and, unless conducted with peculiar -skill and insight, must be exceedingly laborious in cases -of complexity.</p> - -<p>De Morgan was unfortunately led by this equivalence of -propositions into the most serious error of his ingenious -system of Logic. He held that because the proposition -“All A’s are all B’s,” is but another expression for the -two propositions “All A’s are B’s” and “All B’s are A’s,” -it must be a composite and not really an elementary form -of proposition.<a id="FNanchor_82" href="#Footnote_82" class="fnanchor">82</a> But on taking a general view of the -equivalence of propositions such an objection seems to -have no weight. Logicians have, with few exceptions, -persistently upheld the original error of Aristotle in -rejecting from their science the one simple relation of -identity on which all more complex logical relations must -really rest.</p> -<p><span class="pagenum" id="Page_118">118</span></p> - -<h3><i>The Nature of Inference.</i></h3> - -<p>The question, What is Inference? is involved, even to -the present day, in as much uncertainty as that ancient -question, What is Truth? I shall in more than one part -of this work endeavour to show that inference never does -more than explicate, unfold, or develop the information -contained in certain premises or facts. Neither in deductive -nor inductive reasoning can we add a tittle to our -implicit knowledge, which is like that contained in an -unread book or a sealed letter. Sir W. Hamilton has well -said, “Reasoning is the showing out explicitly that a -proposition not granted or supposed, is implicitly contained -in something different, which is granted or supposed.”<a id="FNanchor_83" href="#Footnote_83" class="fnanchor">83</a></p> - -<p>Professor Bowen has explained<a id="FNanchor_84" href="#Footnote_84" class="fnanchor">84</a> with much clearness -that the conclusion of an argument states explicitly what is -virtually or implicitly thought. “The process of reasoning -is not so much a mode of evolving a new truth, as it is of -establishing or proving an old one, by showing how much -was admitted in the concession of the two premises taken -together.” It is true that the whole meaning of these -statements rests upon that of such words as “explicit,” -“implicit,” “virtual.” That is implicit which is wrapped -up, and we render it explicit when we unfold it. Just as -the conception of a circle involves a hundred important -geometrical properties, all following from what we know, -if we have acuteness to unfold the results, so every fact -and statement involves more meaning than seems at first -sight. Reasoning explicates or brings to conscious possession -what was before unconscious. It does not create, nor -does it destroy, but it transmutes and throws the same -matter into a new form.</p> - -<p>The difficult question still remains, Where does novelty -of form begin? Is it a case of inference when we pass -from “Sincerity is the parent of truth” to “The parent of -truth is sincerity?” The old logicians would have called -this change <i>conversion</i>, one case of immediate inference. But -as all identity is necessarily reciprocal, and the very -meaning of such a proposition is that the two terms are<span class="pagenum" id="Page_119">119</span> -identical in their signification, I fail to see any difference -between the statements whatever. As well might we say -that <i>x</i> = <i>y</i> and <i>y</i> = <i>x</i> are different equations.</p> - -<p>Another point of difficulty is to decide when a change -is merely grammatical and when it involves a real logical -transformation. Between a <i>table of wood</i> and a <i>wooden -table</i> there is no logical difference (p. <a href="#Page_31">31</a>), the adjective -being merely a convenient substitute for the prepositional -phrase. But it is uncertain to my mind whether the -change from “All men are mortal” to “No men are not -mortal” is purely grammatical. Logical change may -perhaps be best described as consisting in the determination -of a relation between certain classes of objects from a -relation between certain other classes. Thus I consider -it a truly logical inference when we pass from “All men -are mortal” to “All immortals are not-men,” because the -classes <i>immortals</i> and <i>not-men</i> are different from <i>mortals</i> -and <i>men</i>, and yet the propositions contain at the bottom the -very same truth, as shown in the combinations of the -Logical Alphabet.</p> - -<p>The passage from the qualitative to the quantitative -mode of expressing a proposition is another kind of change -which we must discriminate from true logical inference. -We state the same truth when we say that “mortality -belongs to all men,” as when we assert that “all men are -mortals.” Here we do not pass from class to class, but -from one kind of term, the abstract, to another kind, the -concrete. But inference probably enters when we pass -from either of the above propositions to the assertion that -the class of immortal men is zero, or contains no objects.</p> - -<p>It is of course a question of words to what processes we -shall or shall not apply the name “inference,” and I have -no wish to continue the trifling discussions which have -already taken place upon the subject. What we need to -do is to define accurately the sense in which we use the -word “inference,” and to distinguish the relation of inferrible -propositions from other possible relations. It -seems to be sufficient to recognise four modes in which -two apparently different propositions may be related. -Thus two propositions may be—</p> - -<p>1. <i>Tautologous</i> or <i>identical</i>, involving the same relation -between the same terms and classes, and only differing in<span class="pagenum" id="Page_120">120</span> -the order of statement; thus “Victoria is the Queen of -England” is tautologous with “The Queen of England is -Victoria.”</p> - -<p>2. <i>Grammatically related</i>, when the classes or objects -are the same and similarly related, and the only difference -is in the words; thus “Victoria is the Queen of England” -is grammatically equivalent to “Victoria is England’s -Queen.”</p> - -<p>3. <i>Equivalents</i> in qualitative and quantitative form, the -classes being the same, but viewed in a different manner.</p> - -<p>4. <i>Logically inferrible</i>, one from the other, or it may be -<i>equivalent</i>, when the classes and relations are different, but -involve the same knowledge of the possible combinations.</p> -<hr class="chap x-ebookmaker-drop"> - -<div class="chapter"> -<p><span class="pagenum" id="Page_121">121</span></p> - -<h2 class="nobreak" id="CHAPTER_VII">CHAPTER VII.<br> - -<span class="title">INDUCTION.</span></h2> -</div> - -<p class="ti0">We enter in this chapter upon the second great department -of logical method, that of Induction or the -Inference of general from particular truths. It cannot -be said that the Inductive process is of greater importance -than the Deductive process already considered, because the -latter process is absolutely essential to the existence of -the former. Each is the complement and counterpart of -the other. The principles of thought and existence which -underlie them are at the bottom the same, just as subtraction -of numbers necessarily rests upon the same principles -as addition. Induction is, in fact, the inverse operation -of deduction, and cannot be conceived to exist without -the corresponding operation, so that the question of relative -importance cannot arise. Who thinks of asking -whether addition or subtraction is the more important -process in arithmetic? But at the same time much -difference in difficulty may exist between a direct and -inverse operation; the integral calculus, for instance, is -infinitely more difficult than the differential calculus of -which it is the inverse. Similarly, it must be allowed -that inductive investigations are of a far higher degree of -difficulty and complexity than any questions of deduction; -and it is this fact no doubt which led some logicians, such -as Francis Bacon, Locke, and J. S. Mill, to erroneous -opinions concerning the exclusive importance of induction.</p> - -<p>Hitherto we have been engaged in considering how from -certain conditions, laws, or identities governing the combinations -of qualities, we may deduce the nature of the<span class="pagenum" id="Page_122">122</span> -combinations agreeing with those conditions. Our work -has been to unfold the results of what is contained in any -statements, and the process has been one of <i>Synthesis</i>. -The terms or combinations of which the character has -been determined have usually, though by no means always, -involved more qualities, and therefore, by the relation of -extension and intension, fewer objects than the terms in -which they were described. The truths inferred were thus -usually less general than the truths from which they were -inferred.</p> - -<p>In induction all is inverted. The truths to be ascertained -are more general than the data from which they -are drawn. The process by which they are reached is -<i>analytical</i>, and consists in separating the complex combinations -in which natural phenomena are presented to -us, and determining the relations of separate qualities. -Given events obeying certain unknown laws, we have to -discover the laws obeyed. Instead of the comparatively -easy task of finding what effects will follow from a given -law, the effects are now given and the law is required. -We have to interpret the will by which the conditions -of creation were laid down.</p> - - -<h3><i>Induction an Inverse Operation</i></h3> - -<p>I have already asserted that induction is the inverse -operation of deduction, but the difference is one of such -great importance that I must dwell upon it. There are -many cases in which we can easily and infallibly do a -certain thing but may have much trouble in undoing it. -A person may walk into the most complicated labyrinth -or the most extensive catacombs, and turn hither and thither -at his will; it is when he wishes to return that doubt and -difficulty commence. In entering, any path served him; -in leaving, he must select certain definite paths, and in this -selection he must either trust to memory of the way he -entered or else make an exhaustive trial of all possible -ways. The explorer entering a new country makes sure -his line of return by barking the trees.</p> - -<p>The same difficulty arises in many scientific processes. -Given any two numbers, we may by a simple and infallible -process obtain their product; but when a large number<span class="pagenum" id="Page_123">123</span> -is given it is quite another matter to determine its factors. -Can the reader say what two numbers multiplied together -will produce the number 8,616,460,799? I think it -unlikely that anyone but myself will ever know; for -they are two large prime numbers, and can only be rediscovered -by trying in succession a long series of prime -divisors until the right one be fallen upon. The work -would probably occupy a good computer for many weeks, -but it did not occupy me many minutes to multiply the -two factors together. Similarly there is no direct process -for discovering whether any number is a prime or not; it -is only by exhaustively trying all inferior numbers which -could be divisors, that we can show there is none, and the -labour of the process would be intolerable were it not performed -systematically once for all in the process known as -the Sieve of Eratosthenes, the results being registered in -tables of prime numbers.</p> - -<p>The immense difficulties which are encountered in the -solution of algebraic equations afford another illustration. -Given any algebraic factors, we can easily and infallibly -arrive at the product; but given a product it is a matter -of infinite difficulty to resolve it into factors. Given any -series of quantities however numerous, there is very little -trouble in making an equation which shall have those -quantities as roots. Let <i>a</i>, <i>b</i>, <i>c</i>, <i>d</i>, &c., be the quantities; -then <span class="nowrap">(<i>x</i> - <i>a</i>)</span><span class="nowrap">(<i>x</i> - <i>b</i>)</span><span class="nowrap">(<i>x</i> - <i>c</i>)</span><span class="nowrap">(<i>x</i> - d)</span> . . . = 0 -is the equation required, and we only need to multiply out -the expression on the left hand by ordinary rules. But -having given a complex algebraic expression equated to -zero, it is a matter of exceeding difficulty to discover all -the roots. Mathematicians have exhausted their highest -powers in carrying the complete solution up to the fourth -degree. In every other mathematical operation the inverse -process is far more difficult than the direct process, subtraction -than addition, division than multiplication, evolution -than involution; but the difficulty increases vastly -as the process becomes more complex. Differentiation, -the direct process, is always capable of performance by -fixed rules, but as these rules produce considerable variety -of results, the inverse process of integration presents immense -difficulties, and in an infinite majority of cases -surpasses the present resources of mathematicians. There<span class="pagenum" id="Page_124">124</span> -are no infallible and general rules for its accomplishment; -it must be done by trial, by guesswork, or by remembering -the results of differentiation, and using them as a guide.</p> - -<p>Coming more nearly to our own immediate subject, -exactly the same difficulty exists in determining the law -which certain things obey. Given a general mathematical -expression, we can infallibly ascertain its value for any -required value of the variable. But I am not aware that -mathematicians have ever attempted to lay down the rules -of a process by which, having given certain numbers, one -might discover a rational or precise formula from which -they proceed. The reader may test his power of detecting -a law, by contemplation of its results, if he, not being a -mathematician, will attempt to point out the law obeyed -by the following numbers:</p> - -<div class="center mtb05em"> -<span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">6</span></span></span>,  -<span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">30</span></span></span>,  -<span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">42</span></span></span>,  -<span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">30</span></span></span>,  -<span class="nowrap"><span class="fraction"><span class="fnum">5</span><span class="bar">/</span><span class="fden">66</span></span></span>,  -<span class="nowrap"><span class="fraction"><span class="fnum">691</span><span class="bar">/</span><span class="fden">2730</span></span></span>,  -<span class="nowrap"><span class="fraction"><span class="fnum">7</span><span class="bar">/</span><span class="fden">6</span></span></span>,  -<span class="nowrap"><span class="fraction"><span class="fnum">3617</span><span class="bar">/</span><span class="fden">510</span></span></span>,  -<span class="nowrap"><span class="fraction"><span class="fnum">43867</span><span class="bar">/</span><span class="fden">798</span></span></span>,  -<span class="fs80">etc.</span> -</div> - -<p class="ti0">These numbers are sometimes in low terms, but unexpectedly -spring up to high terms; in absolute magnitude -they are very variable. They seem to set all regularity -and method at defiance, and it is hardly to be supposed -that anyone could, from contemplation of the numbers, -have detected the relations between them. Yet they are -derived from the most regular and symmetrical laws of -relation, and are of the highest importance in mathematical -analysis, being known as the numbers of Bernoulli.</p> - -<p>Compare again the difficulty of decyphering with that -of cyphering. Anyone can invent a secret language, and -with a little steady labour can translate the longest letter -into the character. But to decypher the letter, having no -key to the signs adopted, is a wholly different matter. -As the possible modes of secret writing are infinite in -number and exceedingly various in kind, there is no direct -mode of discovery whatever. Repeated trial, guided more -or less by knowledge of the customary form of cypher, and -resting entirely on the principles of probability and logical -induction, is the only resource. A peculiar tact or skill is -requisite for the process, and a few men, such as Wallis or -Wheatstone, have attained great success.</p> - -<p>Induction is the decyphering of the hidden meaning of -natural phenomena. Given events which happen in certain<span class="pagenum" id="Page_125">125</span> -definite combinations, we are required to point out the -laws which govern those combinations. Any laws being -supposed, we can, with ease and certainty, decide whether -the phenomena obey those laws. But the laws which may -exist are infinite in variety, so that the chances are immensely -against mere random guessing. The difficulty is -much increased by the fact that several laws will usually -be in operation at the same time, the effects of which -are complicated together. The only modes of discovery -consist either in exhaustively trying a great number of -supposed laws, a process which is exhaustive in more -senses than one, or else in carefully contemplating the -effects, endeavouring to remember cases in which like -effects followed from known laws. In whatever manner -we accomplish the discovery, it must be done by the more -or less conscious application of the direct process of -deduction.</p> - -<p>The Logical Alphabet illustrates induction as well as -deduction. In considering the Indirect Process of Inference -we found that from certain propositions we could infallibly -determine the combinations of terms agreeing with those -premises. The inductive problem is just the inverse. -Having given certain combinations of terms, we need to -ascertain the propositions with which the combinations are -consistent, and from which they may have proceeded. -Now, if the reader contemplates the following combinations,</p> - - -<table class="ml5em"> -<tr> -<td class="tal pr3">ABC</td> -<td class="tal pr3"><i>ab</i>C</td> -</tr> -<tr> -<td class="tal"><i>a</i>BC</td> -<td class="tal"><i>abc</i>,</td> -</tr> -</table> - - -<p class="ti0">he will probably remember at once that they belong to the -premises A = AB, B = BC (p. <a href="#Page_92">92</a>). If not, he will require -a few trials before he meets with the right answer, and -every trial will consist in assuming certain laws and -observing whether the deduced results agree with the data. -To test the facility with which he can solve this inductive -problem, let him casually strike out any of the combinations -of the fourth column of the Logical Alphabet, (p. <a href="#Page_94">94</a>), -and say what laws the remaining combinations obey, -observing that every one of the letter-terms and their -negatives ought to appear in order to avoid self-contradiction -in the premises (pp. <a href="#Page_74">74</a>, <a href="#Page_111">111</a>). Let him say, for -instance, what laws are embodied in the combinations</p> - -<p><span class="pagenum" id="Page_126">126</span></p> - -<table class="ml5em"> -<tr> -<td class="tal pr3">ABC</td> -<td class="tal"><i>a</i>BC</td> -</tr> -<tr> -<td class="tal pr3">A<i>bc</i></td> -<td class="tal"><i>ab</i>C.</td> -</tr> -</table> - -<p class="ti0">The difficulty becomes much greater when more terms -enter into the combinations. It would require some little -examination to ascertain the complete conditions fulfilled -in the combinations</p> - -<table class="ml5em"> -<tr> -<td class="tal pr3">AC<i>e</i></td> -<td class="tal"><i>ab</i>C<i>e</i></td> -</tr> -<tr> -<td class="tal pr3"><i>a</i>BC<i>e</i></td> -<td class="tal"><i>abc</i>E.</td> -</tr> -<tr> -<td class="tal"><i>a</i>B<i>cd</i>E</td> -<td class="tal"></td> -</tr> -</table> - -<p class="ti0">The reader may discover easily enough that the principal -laws are C = <i>e</i>, and A = A<i>e</i>; but he would hardly discover -without some trouble the remaining law, namely, that -BD = BD<i>e</i>.</p> - -<p>The difficulties encountered in the inductive investigations -of nature, are of an exactly similar kind. We seldom -observe any law in uninterrupted and undisguised operation. -The acuteness of Aristotle and the ancient Greeks -did not enable them to detect that all terrestrial bodies -tend to fall towards the centre of the earth. A few nights -of observation might have convinced an astronomer -viewing the solar system from its centre, that the planets -travelled round the sun; but the fact that our place of -observation is one of the travelling planets, so complicates -the apparent motions of the other bodies, that it required -all the sagacity of Copernicus to prove the real simplicity -of the planetary system. It is the same throughout -nature; the laws may be simple, but their combined -effects are not simple, and we have no clue to guide us -through their intricacies. “It is the glory of God,” said -Solomon, “to conceal a thing, but the glory of a king to -search it out.” The laws of nature are the invaluable -secrets which God has hidden, and it is the kingly prerogative -of the philosopher to search them out by industry -and sagacity.</p> - - -<h3><i>Inductive Problems for Solution by the Reader.</i></h3> - -<p>In the first edition (vol. ii. p. 370) I gave a logical -problem involving six terms, and requested readers to -discover the laws governing the combinations given. I -received satisfactory replies from readers both in the -United States and in England. I formed the combinations<span class="pagenum" id="Page_127">127</span> -deductively from four laws of correction, but my -correspondents found that three simpler laws, equivalent -to the four more complex ones, were the best answer; these -laws are as follows: <i>a</i> = <i>ac</i>, <i>b</i> = <i>cd</i>, <i>d</i> = E<i>f</i>.</p> - -<p>In case other readers should like to test their skill in the -inductive or inverse problem, I give below several series -of combinations forming problems of graduated difficulty.</p> - -<div class="container"> -<div class="problems" style="font-family: monospace; font-size: 90%;"> -<div class="ph3a"><span class="smcap">Problem I.</span></div> - -<ul> -<li>A B <i>c</i></li> -<li>A <i>b</i> C</li> -<li><i>a</i> B C</li> -</ul> - -<div class="ph3"><span class="smcap">Problem II.</span></div> - -<ul> -<li>A B C</li> -<li>A <i>b</i> C</li> -<li><i>a</i> B C</li> -<li><i>a</i> B <i>c</i></li> -</ul> - -<div class="ph3"><span class="smcap">Problem III.</span></div> - -<ul> -<li>A B C</li> -<li>A <i>b</i> C</li> -<li><i>a</i> B C</li> -<li><i>a</i> B <i>c</i></li> -<li><i>a</i> <i>b</i> <i>c</i></li> -</ul> - -<div class="ph3"><span class="smcap">Problem IV.</span></div> - -<ul> -<li>A B C D</li> -<li>A <i>b</i> <i>c</i> D</li> -<li><i>a</i> B <i>c</i> <i>f</i></li> -<li><i>a</i> <i>b</i> C <i>f</i></li> -</ul> - -<div class="ph3"><span class="smcap">Problem V.</span></div> - -<ul> -<li>A B C D</li> -<li>A B C <i>f</i></li> -<li>A B <i>c</i> <i>f</i></li> -<li>A <i>b</i> C D</li> -<li>A <i>b</i> <i>c</i> D</li> -<li><i>a</i> B C D</li> -<li><i>a</i> B <i>c</i> D</li> -<li><i>a</i> B <i>c</i> <i>f</i></li> -<li><i>a</i> <i>b</i> C <i>f</i></li> -</ul> - -<div class="ph3"><span class="smcap">Problem VI.</span></div> - -<ul> -<li>A B C D E</li> -<li>A B C <i>f</i> <i>e</i></li> -<li>A B <i>c</i> D E</li> -<li>A B <i>c</i> <i>f</i> <i>e</i></li> -<li>A <i>b</i> C D E</li> -<li><i>a</i> B C D E</li> -<li><i>a</i> B C <i>f</i> <i>e</i></li> -<li><i>a</i> <i>b</i> C D E</li> -<li><i>a</i> <i>b</i> <i>c</i> <i>f</i> <i>e</i></li> -</ul> - -<div class="ph3"><span class="smcap">Problem VII.</span></div> - -<ul> -<li>A <i>b</i> <i>c</i> D <i>e</i></li> -<li><i>a</i> B C <i>f</i> E</li> -<li><i>a</i> <i>b</i> C <i>f</i> E</li> -</ul> - -<div class="ph3"><span class="smcap">Problem VIII.</span></div> - -<ul> -<li>A B C D E</li> -<li>A B C D <i>e</i></li> -<li>A B C <i>f</i> <i>e</i></li> -<li>A B <i>c</i> <i>f</i> <i>e</i></li> -<li>A <i>b</i> C D E</li> -<li>A <i>b</i> <i>c</i> <i>f</i> E</li> -<li>A <i>b</i> <i>c</i> <i>f</i> <i>e</i></li> -<li><i>a</i> B C D <i>e</i></li> -<li><i>a</i> B C <i>f</i> <i>e</i></li> -<li><i>a</i> B <i>c</i> D <i>e</i></li> -<li><i>a</i> <i>b</i> C D <i>e</i></li> -<li><i>a</i> <i>b</i> C <i>f</i> E</li> -<li><i>a</i> <i>b</i> <i>c</i> D <i>e</i></li> -<li><i>a</i> <i>b</i> <i>c</i> <i>f</i> E</li> -</ul> - -<div class="ph3"><span class="smcap">Problem IX.</span></div> - -<ul> -<li>A B <i>c</i> D E F</li> -<li>A B <i>c</i> D <i>e</i> F</li> -<li>A <i>b</i> C D <i>e</i> <i>f</i></li> -<li>A <i>b</i> <i>c</i> D E <i>f</i></li> -<li>A <i>b</i> <i>c</i> D <i>e</i> <i>f</i></li> -<li>A <i>b</i> <i>c</i> <i>f</i> E F</li> -<li>A <i>b</i> <i>c</i> <i>f</i> <i>e</i> F</li> -<li><i>a</i> B <i>c</i> D E F</li> -<li><i>a</i> B <i>c</i> D <i>e</i> F</li> -<li><i>a</i> B <i>c</i> <i>f</i> E F</li> -<li><i>a</i> <i>b</i> C D E F</li> -<li><i>a</i> <i>b</i> C D <i>e</i> F</li> -<li><i>a</i> <i>b</i> C D <i>e</i> <i>f</i></li> -<li><i>a</i> <i>b</i> <i>c</i> D <i>e</i> <i>f</i></li> -<li><i>a</i> <i>b</i> <i>c</i> D E <i>f</i></li> -<li><i>a</i> <i>b</i> <i>c</i> <i>f</i> <i>e</i> F</li> -</ul> - -<div class="ph3"><span class="smcap">Problem X.</span></div> - -<ul> -<li>A B C D <i>e</i> F</li> -<li>A B <i>c</i> D E <i>f</i></li> -<li>A <i>b</i> C D E F</li> -<li>A <i>b</i> C D <i>e</i> F</li> -<li>A <i>b</i> <i>c</i> D <i>e</i> F</li> -<li><i>a</i> B C D E <i>f</i></li> -<li><i>a</i> B <i>c</i> D E <i>f</i></li> -<li><i>a</i> <i>b</i> C D <i>e</i> F</li> -<li><i>a</i> <i>b</i> C <i>f</i> <i>e</i> F</li> -<li><i>a</i> <i>b</i> <i>c</i> D <i>e</i> <i>f</i></li> -<li><i>a</i> <i>b</i> <i>c</i> <i>d</i> <i>e</i> <i>f</i></li> -</ul> -</div> -</div> - -<h3><i>Induction of Simple Identities</i>.</h3> - -<p>Many important laws of nature are expressible in the -form of simple identities, and I can at once adduce them -as examples to illustrate what I have said of the difficulty -of the inverse process of induction. Two phenomena are -conjoined. Thus all gravitating matter is exactly coincident -with all matter possessing inertia; where one<span class="pagenum" id="Page_128">128</span> -property appears, the other likewise appears. All crystals -of the cubical system, are all the crystals which do not -doubly refract light. All exogenous plants are, with some -exceptions, those which have two cotyledons or seed-leaves.</p> - -<p>A little reflection will show that there is no direct and -infallible process by which such complete coincidences -may be discovered. Natural objects are aggregates of -many qualities, and any one of those qualities may prove -to be in close connection with some others. If each of a -numerous group of objects is endowed with a hundred -distinct physical or chemical qualities, there will be no -less than <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span>(100 × 99) or 4950 pairs of qualities, which -may be connected, and it will evidently be a matter of -great intricacy and labour to ascertain exactly which qualities -are connected by any simple law.</p> - - - -<p>One principal source of difficulty is that the finite powers -of the human mind are not sufficient to compare by a -single act any large group of objects with another large -group. We cannot hold in the conscious possession of the -mind at any one moment more than five or six different -ideas. Hence we must treat any more complex group by -successive acts of attention. The reader will perceive by -an almost individual act of comparison that the words -<i>Roma</i> and <i>Mora</i> contain the same letters. He may -perhaps see at a glance whether the same is true of -<i>Causal</i> and <i>Casual</i>, and of <i>Logica</i> and <i>Caligo</i>. To assure -himself that the letters in <i>Astronomers</i> make <i>No more -stars</i>, that <i>Serpens in akuleo</i> is an anagram of <i>Joannes -Keplerus</i>, or <i>Great gun do us a sum</i> an anagram of <i>Augustus -de Morgan</i>, it will certainly be necessary to break -up the act of comparison into several successive acts. The -process will acquire a double character, and will consist in -ascertaining that each letter of the first group is among -the letters of the second group, and <i>vice versâ</i>, that each -letter of the second is among those of the first group. -In the same way we can only prove that two long lists of -names are identical, by showing that each name in one -list occurs in the other, and <i>vice versâ</i>.</p> - -<p>This process of comparison really consists in establishing -two partial identities, which are, as already shown (p. <a href="#Page_58">58</a>), -equivalent in conjunction to one simple identity. We -first ascertain the truth of the two propositions A = AB,<span class="pagenum" id="Page_129">129</span> -B = AB, and we then rise by substitution to the single -law A = B.</p> - -<p>There is another process, it is true, by which we may -get to exactly the same result; for the two propositions -A = AB, <i>a</i> = <i>ab</i> are also equivalent to the simple identity -A = B. If then we can show that all objects included -under A are included under B, and also that all objects -not included under A are not included under B, our purpose -is effected. By this process we should usually compare -two lists if we are allowed to mark them. For each -name in the first list we should strike off one in the second, -and if, when the first list is exhausted, the second list is -also exhausted, it follows that all names absent from the -first must be absent from the second, and the coincidence -must be complete.</p> - -<p>These two modes of proving an identity are so closely -allied that it is doubtful how far we can detect any difference -in their powers and instances of application. The -first method is perhaps more convenient when the phenomena -to be compared are rare. Thus we prove that all -the musical concords coincide with all the more simple -numerical ratios, by showing that each concord arises from -a simple ratio of undulations, and then showing that each -simple ratio gives rise to one of the concords. To examine -all the possible cases of discord or complex ratio of -undulation would be impossible. By a happy stroke of -induction Sir John Herschel discovered that all crystals -of quartz which cause the plane of polarization of light -to rotate are precisely those crystals which have plagihedral -faces, that is, oblique faces on the corners of the -prism unsymmetrical with the ordinary faces. This -singular relation would be proved by observing that all -plagihedral crystals possessed the power of rotation, and -<i>vice versâ</i> all crystals possessing this power were plagihedral. -But it might at the same time be noticed that -all ordinary crystals were devoid of the power. There is -no reason why we should not detect any of the four propositions -A = AB, B = AB, <i>a</i> = <i>ab</i>, <i>b</i> = <i>ab</i>, all of which -follow from A = B (p. <a href="#Page_115">115</a>).</p> - -<p>Sometimes the terms of the identity may be singular -objects; thus we observe that diamond is a combustible gem, -and being unable to discover any other that is, we affirm—</p> - -<p><span class="pagenum" id="Page_130">130</span></p> - -<div class="ml5em"> -Diamond = combustible gem. -</div> - -<p>In a similar manner we ascertain that</p> - -<table class="ml5em"> -<tr> -<td class="tal">Mercury = metal liquid at ordinary temperatures,</td> -</tr> -<tr> -<td class="tal pl2hi">Substance of least density = substance of least atomic weight.</td> -</tr> -</table> - -<p>Two or three objects may occasionally enter into the -induction, as when we learn that</p> - -<table class="ml5em"> -<tr> -<td class="tal pl2hi">Sodium ꖌ potassium = metal of less density than water,</td> -</tr> -<tr> -<td class="tal pl2hi">Venus ꖌ Mercury ꖌ Mars = major planet devoid of satellites.</td> -</tr> -</table> - - -<h3><i>Induction of Partial Identities</i>.</h3> - -<p>We found in the last section that the complete identity -of two classes is almost always discovered not by direct -observation of the fact, but by first establishing two -partial identities. There are also a multitude of cases in -which the partial identity of one class with another is the -only relation to be discovered. Thus the most common of -all inductive inferences consists in establishing the fact -that all objects having the properties of A have also those -of B, or that A = AB. To ascertain the truth of a proposition -of this kind it is merely necessary to assemble -together, mentally or physically, all the objects included -under A, and then observe whether B is present in each -of them, or, which is the same, whether it would be impossible -to select from among them any not-B. Thus, if -we mentally assemble together all the heavenly bodies -which move with apparent rapidity, that is to say, the -planets, we find that they all possess the property of not -scintillating. We cannot analyse any vegetable substance -without discovering that it contains carbon and hydrogen, -but it is not true that all substances containing carbon -and hydrogen are vegetable substances.</p> - -<p>The great mass of scientific truths consists of propositions -of this form A = AB. Thus in astronomy we learn -that all the planets are spheroidal bodies; that they all -revolve in one direction round the sun; that they all shine -by reflected light; that they all obey the law of gravitation. -But of course it is not to be asserted that all -bodies obeying the law of gravitation, or shining by<span class="pagenum" id="Page_131">131</span> -reflected light, or revolving in a particular direction, or -being spheroidal in form, are planets. In other sciences -we have immense numbers of propositions of the same -form, as, for instance, all substances in becoming gaseous -absorb heat; all metals are elements; they are all good -conductors of heat and electricity; all the alkaline metals -are monad elements; all foraminifera are marine organisms; -all parasitic animals are non-mammalian; lightning -never issues from stratous clouds; pumice never occurs -where only Labrador felspar is present; milkmaids do -not suffer from small-pox; and, in the works of Darwin, -scientific importance may attach even to such an apparently -trifling observation as that “white tom-cats having -blue eyes are deaf.”</p> - -<p>The process of inference by which all such truths are -obtained may readily be exhibited in a precise symbolic -form. We must have one premise specifying in a disjunctive -form all the possible individuals which belong -to a class; we resolve the class, in short, into its constituents. -We then need a number of propositions, each -of which affirms that one of the individuals possesses a -certain property. Thus the premises must be of the -forms</p> - -<table class="ml5em"> -<tr> -<td class="tac"><div>A = B ꖌ C ꖌ D ꖌ ...... ꖌ P ꖌ Q</div></td> -</tr> -<tr> -<td class="tac"><div>B = BX</div></td> -</tr> -<tr> -<td class="tac"><div>C = CX</div></td> -</tr> -<tr> -<td class="tac"><div>...  ...</div></td> -</tr> -<tr> -<td class="tac"><div>...  ...</div></td> -</tr> -<tr> -<td class="tac"><div>Q = QX.</div></td> -</tr> -</table> - -<p>Now, if we substitute for each alternative of the first -premise its description as found among the succeeding -premises, we obtain</p> - -<div class="ml5em"> -A = BX ꖌ CX ꖌ ...... ꖌ PX ꖌ QX -</div> - -<p class="ti0">or</p> - -<div class="ml5em"> -A = (B ꖌ C ꖌ ...... ꖌ Q)X -</div> - -<p class="ti0">But for the aggregate of alternatives we may now -substitute their equivalent as given in the first premise, -namely A, so that we get the required result:</p> - -<div class="ml5em"> -A = AX. -</div> - -<p>We should have reached the same result if the first -premise had been of the form</p> - -<div class="ml5em"> -A = AB ꖌ AC ꖌ ...... ꖌ AQ. -</div> - -<p><span class="pagenum" id="Page_132">132</span></p> - -<p>We can always prove a proposition, if we find it more -convenient, by proving its equivalent. To assert that all -not-B’s are not-A’s, is exactly the same as to assert that all -A’s are B’s. Accordingly we may ascertain that A = AB by -first ascertaining that <i>b</i> = <i>ab</i>. If we observe, for instance, -that all substances which are not solids are also not capable -of double refraction, it follows necessarily that all double -refracting substances are solids. We may convince ourselves -that all electric substances are nonconductors of -electricity, by reflecting that all good conductors do not, -and in fact cannot, retain electric excitation. When we -come to questions of probability it will be found desirable -to prove, as far as possible, both the original proposition -and its equivalent, as there is then an increased area of -observation.</p> - -<p>The number of alternatives which may arise in the -division of a class varies greatly, and may be any number -from two upwards. Thus it is probable that every substance -is either magnetic or diamagnetic, and no substance -can be both at the same time. The division then must be -made in the form</p> - -<div class="ml5em"> -A = AB<i>c</i> ꖌ A<i>b</i>C. -</div> - -<p>If now we can prove that all magnetic substances are -capable of polarity, say B = BD, and also that all diamagnetic -substances are capable of polarity, C = CD, it -follows by substitution that all substances are capable of -polarity, or A = AD. We commonly divide the class substance -into the three subclasses, solid, liquid, and gas; and -if we can show that in each of these forms it obeys Carnot’s -thermodynamic law, it follows that all substances obey -that law. Similarly we may show that all vertebrate -animals possess red blood, if we can show separately that -fish, reptiles, birds, marsupials, and mammals possess red -blood, there being, as far as is known, only five principal -subclasses of vertebrata.</p> - -<p>Our inductions will often be embarrassed by exceptions, -real or apparent. We might affirm that all gems are incombustible -were not diamonds undoubtedly combustible. -Nothing seems more evident than that all the metals are -opaque until we examine them in fine films, when gold and -silver are found to be transparent. All plants absorb -carbonic acid except certain fungi; all the bodies of the<span class="pagenum" id="Page_133">133</span> -planetary system have a progressive motion from west to -east, except the satellites of Uranus and Neptune. Even -some of the profoundest laws of matter are not quite -universal; all solids expand by heat except india-rubber, -and possibly a few other substances; all liquids which have -been tested expand by heat except water below 4° C. and -fused bismuth; all gases have a coefficient of expansion -increasing with the temperature, except hydrogen. In -a later chapter I shall consider how such anomalous -cases may be regarded and classified; here we have only to -express them in a consistent manner by our notation.</p> - -<p>Let us take the case of the transparency of metals, and -assign the terms thus:—</p> - -<table class="ml5em"> -<tr> -<td class="tal pr3">A = metal</td> -<td class="tal">D = iron</td> -</tr> -<tr> -<td class="tal pr3">B = gold</td> -<td class="tal">E, F, &c. = copper, lead, &c.</td> -</tr> -<tr> -<td class="tal pr3">C = silver</td> -<td class="tal">X = opaque.</td> -</tr> -</table> - -<p>Our premises will be</p> - -<div class="ml5em"> -A = B ꖌ C ꖌ D ꖌ E, &c.<br> -B = B<i>x</i><br> -C = C<i>x</i><br> -D = DX<br> -E = EX, -</div> - -<p class="ti0">and so on for the rest of the metals. Now evidently</p> - -<div class="ml5em"> -A<i>bc</i> = (D ꖌ E ꖌ F ꖌ ......)<i>bc</i>, -</div> - -<p class="ti0">and by substitution as before we shall obtain</p> - -<div class="ml5em"> -A<i>bc</i> = A<i>bc</i>X, -</div> - -<p class="ti0">or in words, “All metals not gold nor silver are opaque;” -at the same time we have</p> - -<div class="ml5em"> -A(B ꖌ C) = AB ꖌ AC = AB<i>x</i> ꖌ AC<i>x</i> = A(B ꖌ C)<i>x</i>,<br> -</div> - -<p class="ti0">or “Metals which are either gold or silver are not opaque.”</p> - -<p>In some cases the problem of induction assumes a much -higher degree of complexity. If we examine the properties -of crystallized substances we may find some properties -which are common to all, as cleavage or fracture in definite -planes; but it would soon become requisite to break up -the class into several minor ones. We should divide -crystals according to the seven accepted systems—and we -should then find that crystals of each system possess -many common properties. Thus crystals of the Regular -or Cubical system expand equally by heat, conduct heat -and electricity with uniform rapidity, and are of like -elasticity in all directions; they have but one index of<span class="pagenum" id="Page_134">134</span> -refraction for light; and every facet is repeated in like -relation to each of the three axes. Crystals of the system -having one principal axis will be found to possess the -various physical powers of conduction, refraction, elasticity, -&c., uniformly in directions perpendicular to the -principal axis; in other directions their properties vary -according to complicated laws. The remaining systems -in which the crystals possess three unequal axes, or have -inclined axes, exhibit still more complicated results, the -effects of the crystal upon light, heat, electricity, &c., -varying in all directions. But when we pursue induction -into the intricacies of its application to nature we really -enter upon the subject of classification, which we must -take up again in a later part of this work.</p> - - -<h3><i>Solution of the Inverse or Inductive Problem, involving -Two Classes</i>.</h3> - -<p>It is now plain that Induction consists in passing back -from a series of combinations to the laws by which such -combinations are governed. The natural law that all -metals are conductors of electricity really means that in -nature we find three classes of objects, namely—</p> - -<div class="ml5em"> -1. Metals, conductors;<br> -2. Not-metals, conductors;<br> -3. Not-metals, not-conductors. -</div> - -<p class="ti0">It comes to the same thing if we say that it excludes the -existence of the class, “metals not-conductors.” In the -same way every other law or group of laws will really -mean the exclusion from existence of certain combinations -of the things, circumstances or phenomena governed by -those laws. Now in logic, strictly speaking, we treat not -the phenomena, nor the laws, but the general forms of the -laws; and a little consideration will show that for a finite -number of things the possible number of forms or kinds -of law governing them must also be finite. Using general -terms, we know that A and B can be present or absent in -four ways and no more—thus:</p> - -<div class="ml5em"> -AB, A<i>b</i>, <i>a</i>B, <i>ab</i>; -</div> - -<p class="ti0">therefore every possible law which can exist concerning -the relation of A and B must be marked by the exclusion -of one or more of the above combinations. The number<span class="pagenum" id="Page_135">135</span> -of possible laws then cannot exceed the number of selections -which we can make from these four combinations. -Since each combination may be present or absent, the -number of cases to be considered is 2 × 2 × 2 × 2, or sixteen; -and these cases are all shown in the following table, in -which the sign 0 indicates absence or non-existence of the -combination shown at the left-hand column in the same -line, and the mark 1 its presence:—</p> - -<div class="center"> -<table id="tab135" class="mtb1em fs90"> -<tr> -<th></th> -<th>1</th> -<th>2</th> -<th>3</th> -<th>4</th> -<th>5</th> -<th>6</th> -<th>7<br>*</th> -<th>8<br>*</th> -<th>9</th> -<th>10<br>*</th> -<th>11</th> -<th>12<br>*</th> -<th>13</th> -<th>14<br>*</th> -<th>15<br>*</th> -<th>16<br>*</th> -</tr> -<tr> -<td>AB</td> -<td>0</td> -<td>0</td> -<td>0</td> -<td>0</td> -<td>0</td> -<td>0</td> -<td>0</td> -<td>0</td> -<td>1</td> -<td>1</td> -<td>1</td> -<td>1</td> -<td>1</td> -<td>1</td> -<td>1</td> -<td>1</td> -</tr> -<tr> -<td>A<i>b</i></td> -<td>0</td> -<td>0</td> -<td>0</td> -<td>0</td> -<td>1</td> -<td>1</td> -<td>1</td> -<td>1</td> -<td>0</td> -<td>0</td> -<td>0</td> -<td>0</td> -<td>1</td> -<td>1</td> -<td>1</td> -<td>1</td> -</tr> -<tr> -<td><i>a</i>B</td> -<td>0</td> -<td>0</td> -<td>1</td> -<td>1</td> -<td>0</td> -<td>0</td> -<td>1</td> -<td>1</td> -<td>0</td> -<td>0</td> -<td>1</td> -<td>1</td> -<td>0</td> -<td>0</td> -<td>1</td> -<td>1</td> -</tr> -<tr> -<td class="bb"><i>ab</i></td> -<td class="bb">0</td> -<td class="bb">1</td> -<td class="bb">0</td> -<td class="bb">1</td> -<td class="bb">0</td> -<td class="bb">1</td> -<td class="bb">0</td> -<td class="bb">1</td> -<td class="bb">0</td> -<td class="bb">1</td> -<td class="bb">0</td> -<td class="bb">1</td> -<td class="bb">0</td> -<td class="bb">1</td> -<td class="bb">0</td> -<td class="bb">1</td> -</tr> -</table> -</div> - -<p>Thus in column sixteen we find that all the conceivable -combinations are present, which means that there are no -special laws in existence in such a case, and that the -combinations are governed only by the universal Laws of -Identity and Difference. The example of metals and -conductors of electricity would be represented by the -twelfth column; and every other mode in which two -things or qualities might present themselves is shown in -one or other of the columns. More than half the cases -may indeed be at once rejected, because they involve the -entire absence of a term or its negative. It has been -shown to be a logical principle that every term must have -its negative (p. <a href="#Page_111">111</a>), and when this is not the case, inconsistency -between the conditions of combination must exist. -Thus if we laid down the two following propositions, -“Graphite conducts electricity,” and “Graphite does not -conduct electricity,” it would amount to asserting the -impossibility of graphite existing at all; or in general -terms, A is B and A is not B result in destroying altogether -the combinations containing A, a case shown in the -fourth column of the above table. We therefore restrict -our attention to those cases which may be represented in -natural phenomena when at least two combinations are -present, and which correspond to those columns of the<span class="pagenum" id="Page_136">136</span> -table in which each of A, <i>a</i>, B, <i>b</i> appears. These cases -are shown in the columns marked with an asterisk.</p> - -<p>We find that seven cases remain for examination, thus -characterised—</p> - -<div class="ml5em"> -Four cases exhibiting three combinations,<br> -Two cases exhibiting two combinations,<br> -One case exhibiting four combinations. -</div> - -<p class="ti0">It has already been pointed out that a proposition of the -form A = AB destroys one combination, A<i>b</i>, so that this is -the form of law applying to the twelfth column. But by -changing one or more of the terms in A = AB into its -negative, or by interchanging A and B, <i>a</i> and <i>b</i>, we obtain -no less than eight different varieties of the one form; thus—</p> - -<table class="ml5em"> -<tr class="fs80"> -<td class="tac prl15" colspan="2">12th case.</td> -<td class="tac prl15" colspan="2">8th case.</td> -<td class="tac prl15" colspan="2">15th case.</td> -<td class="tac prl15" colspan="2">14th case.</td> -</tr> -<tr> -<td class="tar"><div>A = </div></td> -<td class="tal">AB</td> -<td class="tar"><div>A = </div></td> -<td class="tal">A<i>b</i></td> -<td class="tar"><div><i>a</i> = </div></td> -<td class="tal"><i>a</i>B</td> -<td class="tar"><div><i>a</i> = </div></td> -<td class="tal"><i>ab</i></td> -</tr> -<tr> -<td class="tar"><div><i>b</i> = </div></td> -<td class="tal"><i>ab</i></td> -<td class="tar"><div>B = </div></td> -<td class="tal"><i>a</i>B</td> -<td class="tar"><div><i>b</i> = </div></td> -<td class="tal">A<i>b</i></td> -<td class="tar"><div>B = </div></td> -<td class="tal">AB</td> -</tr> -</table> - -<p class="ti0">The reader of the preceding sections will see that each -proposition in the lower line is logically equivalent to, and -is in fact the contrapositive of, that above it (p. <a href="#Page_83">83</a>). Thus -the propositions A = A<i>b</i> and B = <i>a</i>B both give the same -combinations, shown in the eighth column of the table, -and trial shows that the twelfth, eighth, fifteenth and -fourteenth columns are thus accounted for. We come to -this conclusion then—<i>The general form of proposition</i> -A = AB <i>admits of four logically distinct varieties, each -capable of expression in two modes</i>.</p> - -<p>In two columns of the table, namely the seventh and -tenth, we observe that two combinations are missing. -Now a simple identity A = B renders impossible both A<i>b</i> -and <i>a</i>B, accounting for the tenth case; and if we change -B into <i>b</i> the identity A = <i>b</i> accounts for the seventh case. -There may indeed be two other varieties of the simple -identity, namely <i>a</i> = <i>b</i> and <i>a</i> = B; but it has already -been shown repeatedly that these are equivalent respectively -to A = B and A = <i>b</i> (p. <a href="#Page_115">115</a>). As the sixteenth -column has already been accounted for as governed -by no special conditions, we come to the following general -conclusion:—The laws governing the combinations of two -terms must be capable of expression either in a partial -identity or a simple identity; the partial identity is capable -of only four logically distinct varieties, and the simple -identity of two. Every logical relation between two terms<span class="pagenum" id="Page_137">137</span> -must be expressed in one of these six forms of law, or -must be logically equivalent to one of them.</p> - -<p>In short, we may conclude that in treating of partial -and complete identity, we have exhaustively treated the -modes in which two terms or classes of objects can be -related. Of any two classes it can be said that one must -either be included in the other, or must be identical with -it, or a like relation must exist between one class and the -negative of the other. We have thus completely solved -the inverse logical problem concerning two terms.<a id="FNanchor_85" href="#Footnote_85" class="fnanchor">85</a></p> - - -<h3><i>The Inverse Logical Problem involving Three Classes.</i></h3> - -<p>No sooner do we introduce into the problem a third term -C, than the investigation assumes a far more complex -character, so that some readers may prefer to pass over -this section. Three terms and their negatives may be -combined, as we have frequently seen, in eight different -combinations, and the effect of laws or logical conditions -is to destroy any one or more of these combinations. Now -we may make selections from eight things in 2<sup>8</sup> or 256 -ways; so that we have no less than 256 different cases to -treat, and the complete solution is at least fifty times as -troublesome as with two terms. Many series of combinations, -indeed, are contradictory, as in the simpler -problem, and may be passed over, the test of consistency -being that each of the letters A, B, C, <i>a</i>, <i>b</i>, <i>c</i>, shall appear -somewhere in the series of combinations.</p> - -<p>My mode of solving the problem was as follows:—Having -written out the whole of the 256 series of combinations, -I examined them separately and struck out such -as did not fulfil the test of consistency. I then chose -some form of proposition involving two or three terms, -and varied it in every possible manner, both by the -circular interchange of letters (A, B, C into B, C, A and -then into C, A, B), and by the substitution for any one or -more of the terms of the corresponding negative terms.<span class="pagenum" id="Page_138">138</span> -For instance, the proposition AB = ABC can be first -varied by circular interchange so as to give BC = BCA and -then CA = CAB. Each of these three can then be thrown -into eight varieties by negative change. Thus AB = ABC -gives <i>a</i>B = <i>a</i>BC, A<i>b</i> = A<i>b</i>C, AB = AB<i>c</i>, <i>ab</i> = <i>ab</i>C, and -so on. Thus there may possibly exist no less than twenty-four -varieties of the law having the general form -AB = ABC, meaning that whatever has the properties of -A and B has those also of C. It by no means follows -that some of the varieties may not be equivalent to others; -and trial shows, in fact, that AB = ABC is exactly the -same in meaning as A<i>c</i> = A<i>bc</i> or B<i>c</i> = B<i>ca</i>. Thus the law -in question has but eight varieties of distinct logical meaning. -I now ascertain by actual deductive reasoning which -of the 256 series of combinations result from each of these -distinct laws, and mark them off as soon as found. I then -proceed to some other form of law, for instance A = ABC, -meaning that whatever has the qualities of A has those -also of B and C. I find that it admits of twenty-four -variations, all of which are found to be logically distinct; -the combinations being worked out, I am able to mark off -twenty-four more of the list of 256 series. I proceed in -this way to work out the results of every form of law -which I can find or invent. If in the course of this work -I obtain any series of combinations which had been previously -marked off, I learn at once that the law giving -these combinations is logically equivalent to some law -previously treated. It may be safely inferred that every -variety of the apparently new law will coincide in meaning -with some variety of the former expression of the same -law. I have sufficiently verified this assumption in some -cases, and have never found it lead to error. Thus as -AB = ABC is equivalent to A<i>c</i> = A<i>bc</i>, so we find that -<i>ab</i> = <i>ab</i>C is equivalent to <i>ac</i> = <i>ac</i>B.</p> - -<p>Among the laws treated were the two A = AB and -A = B which involve only two terms, because it may of -course happen that among three things two only are in -special logical relation, and the third independent; and -the series of combinations representing such cases of relation -are sure to occur in the complete enumeration. All -single propositions which I could invent having been -treated, pairs of propositions were next investigated. Thus<span class="pagenum" id="Page_139">139</span> -we have the relations, “All A’s are B’s, and all B’s are -C’s,” of which the old logical syllogism is the development. -We may also have “all A’s are all B’s, and all B’s are C’s,” -or even “all A’s are all B’s, and all B’s are all C’s.” All -such premises admit of variations, greater or less in -number, the logical distinctness of which can only be -determined by trial in detail. Disjunctive propositions -either singly or in pairs were also treated, but were often -found to be equivalent to other propositions of a simpler -form; thus A = ABC ꖌ A<i>bc</i> is exactly the same in meaning -as AB = AC.</p> - -<p>This mode of exhaustive trial bears some analogy to -that ancient mathematical process called the Sieve of -Eratosthenes. Having taken a long series of the natural -numbers, Eratosthenes is said to have calculated out in -succession all the multiples of every number, and to -have marked them off, so that at last the prime numbers -alone remained, and the factors of every number were -exhaustively discovered. My problem of 256 series of -combinations is the logical analogue, the chief points of -difference being that there is a limit to the number of cases, -and that prime numbers have no analogue in logic, since -every series of combinations corresponds to a law or group -of conditions. But the analogy is perfect in the point that -they are both inverse processes. There is no mode of -ascertaining that a number is prime but by showing that -it is not the product of any assignable factors. So there -is no mode of ascertaining what laws are embodied in any -series of combinations but trying exhaustively the laws -which would give them. Just as the results of Eratosthenes’ -method have been worked out to a great extent -and registered in tables for the convenience of other -mathematicians, I have endeavoured to work out the -inverse logical problem to the utmost extent which is at -present practicable or useful.</p> - -<p>I have thus found that there are altogether fifteen conditions -or series of conditions which may govern the combinations -of three terms, forming the premises of fifteen -essentially different kinds of arguments. The following -table contains a statement of these conditions, together -with the numbers of combinations which are contradicted -or destroyed by each, and the numbers of logically distinct<span class="pagenum" id="Page_140">140</span> -variations of which the law is capable. There might be -also added, as a sixteenth case, that case where no special -logical condition exists, so that all the eight combinations -remain.</p> - -<table id="tab140" class="mtb1em mrl10"> -<tr> -<th class="tac">Reference Number.</th> -<th class="tac">Propositions expressing the general type of the logical conditions.</th> -<th class="tac">Number of distinct logical variations.</th> -<th class="tac">Number of combinations contradicted by each.</th> -</tr> -<tr> -<td class="tar pt05 pr2">I.</td> -<td class="tal pt05 pl1">A = B</td> -<td class="tac pt05"> 6</td> -<td class="tac pt05">4</td> -</tr> -<tr> -<td class="tar pr2">II.</td> -<td class="tal pl1">A = AB</td> -<td class="tac"><div>12</div></td> -<td class="tac"><div>2</div></td> -</tr> -<tr> -<td class="tar pr2">III.</td> -<td class="tal pl1">A = B, B = C</td> -<td class="tac"><div> 4</div></td> -<td class="tac"><div>6</div></td> -</tr> -<tr> -<td class="tar pr2">IV.</td> -<td class="tal pl1">A = B, B = BC</td> -<td class="tac"><div>24</div></td> -<td class="tac"><div>5</div></td> -</tr> -<tr> -<td class="tar pr2">V.</td> -<td class="tal pl1">A = AB, B = BC</td> -<td class="tac"><div>24</div></td> -<td class="tac"><div>4</div></td> -</tr> -<tr> -<td class="tar pr2">VI.</td> -<td class="tal pl1">A = BC</td> -<td class="tac"><div>24</div></td> -<td class="tac"><div>4</div></td> -</tr> -<tr> -<td class="tar pr2">VII.</td> -<td class="tal pl1">A = ABC</td> -<td class="tac"><div>24</div></td> -<td class="tac"><div>3</div></td> -</tr> -<tr> -<td class="tar pr2">VIII.</td> -<td class="tal pl1">AB = ABC</td> -<td class="tac"><div> 8</div></td> -<td class="tac"><div>1</div></td> -</tr> -<tr> -<td class="tar pr2">IX.</td> -<td class="tal pl1">A = AB, <i>a</i>B = <i>a</i>B<i>c</i></td> -<td class="tac"><div>24</div></td> -<td class="tac"><div>3</div></td> -</tr> -<tr> -<td class="tar pr2">X.</td> -<td class="tal pl1">A = ABC, <i>ab</i> = <i>ab</i>C</td> -<td class="tac"><div> 8</div></td> -<td class="tac"><div>4</div></td> -</tr> -<tr> -<td class="tar pr2">XI.</td> -<td class="tal pl1">AB = ABC, <i>ab</i> = <i>abc</i></td> -<td class="tac"><div> 4</div></td> -<td class="tac"><div>2</div></td> -</tr> -<tr> -<td class="tar pr2">XII.</td> -<td class="tal pl1">AB = AC</td> -<td class="tac"><div>12</div></td> -<td class="tac"><div>2</div></td> -</tr> -<tr> -<td class="tar pr2">XIII.</td> -<td class="tal pl1">A = BC ꖌ A<i>bc</i></td> -<td class="tac"><div> 8</div></td> -<td class="tac"><div>3</div></td> -</tr> -<tr> -<td class="tar pr2">XIV.</td> -<td class="tal pl1">A = BC ꖌ <i>bc</i></td> -<td class="tac"><div> 2</div></td> -<td class="tac"><div>4</div></td> -</tr> -<tr> -<td class="tar pb05 pr2 bb">XV.</td> -<td class="tal pb05 pl1 bb">A = ABC, <i>a</i> = B<i>c</i> ꖌ <i>b</i>C</td> -<td class="tac pb05 bb"> 8</td> -<td class="tac pb05 bb">5</td> -</tr> -</table> - -<p>There are sixty-three series of combinations derived from -self-contradictory premises, which with 192, the sum of -the numbers of distinct logical variations stated in the -third column of the table, and with the one case where -there are no conditions or laws at all, make up the whole -conceivable number of 256 series.</p> - -<p>We learn from this table, for instance, that two propositions -of the form A = AB, B = BC, which are such -as constitute the premises of the old syllogism Barbara, -exclude as impossible four of the eight combinations in -which three terms may be united, and that these propositions -are capable of taking twenty-four variations by transpositions -of the terms or the introduction of negatives. -This table then presents the results of a complete analysis -of all the possible logical relations arising in the case of -three terms, and the old syllogism forms but one out of -fifteen typical forms. Generally speaking, every form can -be converted into apparently different propositions; thus -the fourth type A = B, B = BC may appear in the form -A = ABC, <i>a</i> = <i>ab</i>, or again in the form of three propositions -A = AB, B = BC, <i>a</i>B = <i>a</i>B<i>c</i>; but all these sets of -premises yield identically the same series of combinations,<span class="pagenum" id="Page_141">141</span> -and are therefore of equivalent logical meaning. The fifth -type, or Barbara, can also be thrown into the equivalent -forms A = ABC, <i>a</i>B = <i>a</i>BC and A = AC, B = A ꖌ <i>a</i>BC. -In other cases I have obtained the very same logical -conditions in four modes of statements. As regards mere -appearance and form of statement, the number of possible -premises would be very great, and difficult to exhibit -exhaustively.</p> - -<p>The most remarkable of all the types of logical condition -is the fourteenth, namely, A = BC ꖌ <i>bc</i>. It is that which -expresses the division of a genus into two doubly marked -species, and might be illustrated by the example—“Component -of the physical universe = matter, gravitating, or -not-matter (ether), not-gravitating.” It is capable of only -two distinct logical variations, namely, A = BC ꖌ <i>bc</i> and -A = B<i>c</i> ꖌ <i>b</i>C. By transposition or negative change of the -letters we can indeed obtain six different expressions of -each of these propositions; but when their meanings are -analysed, by working out the combinations, they are found -to be logically equivalent to one or other of the above two. -Thus the proposition A = BC ꖌ <i>bc</i> can be written in any -of the following five other modes,</p> - -<div class="ml5em"> -<i>a</i> = <i>b</i>C ꖌ B<i>c</i>, B = CA ꖌ <i>ca</i>, <i>b</i> = <i>c</i>A ꖌ C<i>a</i>,<br> -C = AB ꖌ <i>ab</i>, <i>c</i> = <i>a</i>B ꖌ A<i>b</i>. -</div> - -<p>I do not think it needful to publish at present the complete -table of 193 series of combinations and the premises -corresponding to each. Such a table enables us by mere -inspection to learn the laws obeyed by any set of combinations -of three things, and is to logic what a table of -factors and prime numbers is to the theory of numbers, or -a table of integrals to the higher mathematics. The table -already given (p. <a href="#Page_140">140</a>) would enable a person with but little -labour to discover the law of any combinations. If there -be seven combinations (one contradicted) the law must be -of the eighth type, and the proper variety will be apparent. -If there be six combinations (two contradicted), either the -second, eleventh, or twelfth type applies, and a certain -number of trials will disclose the proper type and variety. -If there be but two combinations the law must be of the -third type, and so on.</p> - -<p>The above investigations are complete as regards the -possible logical relations of two or three terms. But<span class="pagenum" id="Page_142">142</span> -when we attempt to apply the same kind of method to -the relations of four or more terms, the labour becomes -impracticably great. Four terms give sixteen combinations -compatible with the laws of thought, and the number of -possible selections of combinations is no less than 2<sup>16</sup> or -65,536. The following table shows the extraordinary -manner in which the number of possible logical relations -increases with the number of terms involved.</p> - -<div class="center"> -<table id="tab142" class="mtb1em"> -<tr> -<th style="width: 5em;">Number of terms.</th> -<th style="width: 8em;">Number of possible combinations.</th> -<th style="width: 15em;">Number of possible selections of combinations corresponding to consistent or inconsistent logical relations.</th> -</tr> -<tr> -<td class="tac pt05"><div>2</div></td> -<td class="tac pt05"><div> 4</div></td> -<td class="tar pt05 pr05"><div>16</div></td> -</tr> -<tr> -<td class="tac"><div><div>3</div></div></td> -<td class="tac"><div><div> 8</div></div></td> -<td class="tar pr05"><div>256</div></td> -</tr> -<tr> -<td class="tac"><div><div>4</div></div></td> -<td class="tac"><div><div>16</div></div></td> -<td class="tar pr05"><div>65,536</div></td> -</tr> -<tr> -<td class="tac"><div><div>5</div></div></td> -<td class="tac"><div><div>32</div></div></td> -<td class="tar pr05"><div>4,294,967,296</div></td> -</tr> -<tr> -<td class="tac pb05 bb"><div>6</div></td> -<td class="tac pb05 bb"><div>64</div></td> -<td class="tar pb05 pr05 bb"><div>18,446,744,073,709,551,616</div></td> -</tr> -</table> -</div> - -<p>Some years of continuous labour would be required to -ascertain the types of laws which may govern the combinations -of only four things, and but a small part of such -laws would be exemplified or capable of practical application -in science. The purely logical inverse problem, -whereby we pass from combinations to their laws, is -solved in the preceding pages, as far as it is likely to be -for a long time to come; and it is almost impossible that -it should ever be carried more than a single step -further.</p> - -<p>In the first edition, vol i. p. 158, I stated that I had not -been able to discover any mode of calculating the number -of cases in which inconsistency would be implied in the -selection of combinations from the Logical Alphabet. The -logical complexity of the problem appeared to be so great -that the ordinary modes of calculating numbers of combinations -failed, in my opinion, to give any aid, and -exhaustive examination of the combinations in detail -seemed to be the only method applicable. This opinion, -however, was mistaken, for both Mr. R. B. Hayward, of -Harrow, and Mr. W. H. Brewer have calculated the -numbers of inconsistent cases both for three and for four -terms, without much difficulty. In the case of four -terms they find that there are 1761 inconsistent selections -and 63,774 consistent, which with one case where no<span class="pagenum" id="Page_143">143</span> -condition exists, make up the total of 65,536 possible -selections.</p> - -<p>The inconsistent cases are distributed in the manner -shown in the following table:—</p> - -<div class="center"> -<table class="fs70 mtb1em"> -<tr> -<td class="tac ball pall05" style="width:6em;"><div>Number of Combinations remaining.</div></td> -<td class="tac btb prl05"><div>0</div></td> -<td class="tac btb prl05"><div>1</div></td> -<td class="tac btb prl05"><div>2</div></td> -<td class="tac btb prl05"><div>3</div></td> -<td class="tac btb prl05"><div>4</div></td> -<td class="tac btb prl05"><div>5</div></td> -<td class="tac btb prl05"><div>6</div></td> -<td class="tac btb prl05"><div>7</div></td> -<td class="tac btb prl05"><div>8</div></td> -<td class="tac btb prl05"><div>9</div></td> -<td class="tac btrb prl05"><div>10, &c.</div></td> -</tr> -<tr> -<td class="tac ball pall05"><div>Number of Inconsistent Cases.</div></td> -<td class="tac btb prl05"><div>1</div></td> -<td class="tac btb prl05"><div>16</div></td> -<td class="tac btb prl05"><div>112</div></td> -<td class="tac btb prl05"><div>352</div></td> -<td class="tac btb prl05"><div>536</div></td> -<td class="tac btb prl05"><div>448</div></td> -<td class="tac btb prl05"><div>224</div></td> -<td class="tac btb prl05"><div>64</div></td> -<td class="tac btb prl05"><div>8</div></td> -<td class="tac btb prl05"><div>0</div></td> -<td class="tac btrb prl05"><div>0, &c.</div></td> -</tr> -</table> -</div> - -<p>When more than eight combinations of the Logical -Alphabet (p. <a href="#Page_94">94</a>, column V.) remain unexcluded, there cannot -be inconsistency. The whole numbers of ways of selecting -0, 1, 2, &c., combinations out of 16 are given in the 17th -line of the Arithmetical Triangle given further on in the -Chapter on Combinations and Permutations, the sum of -the numbers in that line being 65,536.</p> - - -<h3><i>Professor Clifford on the Types of Compound Statement -involving Four Classes.</i></h3> - -<p>In the first edition (vol. i. p. 163), I asserted that some -years of labour would be required to ascertain even the -precise number of types of law governing the combinations -of four classes of things. Though I still believe that some -years’ labour would be required to work out the types -themselves, it is clearly a mistake to suppose that the -<i>numbers</i> of such types cannot be calculated with a reasonable -amount of labour, Professor W. K. Clifford having -actually accomplished the task. His solution of the -numerical problem involves the use of a complete new -system of nomenclature and is far too intricate to be fully -described here. I can only give a brief abstract of the -results, and refer readers, who wish to follow out the -reasoning, to the Proceedings of the Literary and Philosophical -Society of Manchester, for the 9th January, 1877, -vol. xvi., p. 88, where Professor Clifford’s paper is printed -in full.</p> - -<p>By a <i>simple statement</i> Professor Clifford means the denial -of the existence of any single combination or <i>cross-division<span class="pagenum" id="Page_144">144</span></i>, -of the classes, as in ABCD = 0, or A<i>b</i>C<i>d</i> = 0. -The denial of two or more such combinations is called a -<i>compound statement</i>, and is further said to be <i>twofold</i>, -<i>threefold</i>, &c., according to the number denied. Thus -ABC = 0 is a twofold compound statement in regard to -four classes, because it involves both ABCD = 0 and -ABC<i>d</i> = 0. When two compound statements can be -converted into one another by interchange of the classes, -A, B, C, D, with each other or with their complementary -classes, <i>a</i>, <i>b</i>, <i>c</i>, <i>d</i>, they are called <i>similar</i>, and all similar -statements are said to belong to the same <i>type</i>.</p> - -<p>Two statements are called <i>complementary</i> when they -deny between them all the sixteen combinations without -both denying any one; or, which is the same thing, when -each denies just those combinations which the other -permits to exist. It is obvious that when two statements -are similar, the complementary statements will also be -similar, and consequently for every type of <i>n</i>-fold statement, -there is a complementary type of (16—<i>n</i>)-fold statement. -It follows that we need only enumerate the types as far as -the eighth order; for the types of more-than-eight-fold -statement will already have been given as complementary -to types of lower orders.</p> - -<p>One combination, ABCD, may be converted into another -A<i>b</i>C<i>d</i> by interchanging one or more of the classes with -the complementary classes. The number of such changes -is called the <i>distance</i>, which in the above case is 2. In -two similar compound statements the distances of the -combinations denied must be the same; but it does not -follow that when all the distances are the same, the statements -are similar. There is, however, only one example -of two dissimilar statements having the same distances. -When the distance is 4, the two combinations are said to -be <i>obverse</i> to one another, and the statements denying them -are called <i>obverse statements</i>, as in ABCD = 0 and <i>abcd</i> = 0 -or again A<i>b</i>C<i>d</i> = 0 and <i>a</i>B<i>c</i>D = 0. When any one combination -is given, called the <i>origin</i>, all the others may be -grouped in respect of their relations to it as follows:—Four -are at distance <i>one</i> from it, and may be called <i>proximates</i>; -six are at distance <i>two</i>, and may be called <i>mediates</i>; four -are at distance <i>three</i>, and may be called <i>ultimates</i>; finally -the obverse is at distance <i>four</i>.</p> - -<p><span class="pagenum" id="Page_145">145</span></p> - -<div class="center"> -<table class="fs80 mtb1em"> -<tr><td class="tac" colspan="5">Origin and<br>four proximates.</td><td class="tac" colspan="5">Six<br>mediates.</td><td class="tac" colspan="5">Obverse and<br>four ultimates.</td></tr> -<tr><td colspan="6"> </td><td class="tac pt05" colspan="3"><i>ab</i>CD</td><td colspan="6"> </td></tr> -<tr><td colspan="6"> </td><td class="tac" colspan="3">|</td><td colspan="6"> </td></tr> -<tr><td class="tac" colspan="5"><i>a</i>BCD</td><td> A<i>bc</i>D </td><td class="tac" colspan="3">|</td><td> A<i>b</i>C<i>d</i> </td><td class="tac" colspan="5">A<i>bcd</i></td></tr> -<tr><td class="tac" colspan="5">|</td><td> </td><td class="tal"><div>╲ </div></td><td class="tac">|</td><td class="tar"> ╱</td><td> </td><td class="tac" colspan="5">|</td></tr> -<tr><td class="tac">ABC<i>d</i></td><td>—</td><td>ABCD</td><td>—</td><td>A<i>b</i>CD</td><td colspan="2"> </td><td class="tac">╳</td><td colspan="2"> </td><td class="tac"><i>abc</i>D</td><td>—</td><td><i>abcd</i></td><td>—</td><td><i>a</i>B<i>cd</i></td></tr> -<tr><td class="tac" colspan="5">|</td><td> </td><td class="tar"><div>╱ </div></td><td class="tac">|</td><td class="tal"> ╲</td><td> </td><td class="tac" colspan="5">|</td></tr> -<tr><td class="tac" colspan="5">AB<i>c</i>D</td><td> <i>a</i>B<i>c</i>D </td><td class="tac" colspan="3">|</td><td> <i>a</i>BC<i>d</i> </td><td class="tac" colspan="5"><i>ab</i>C<i>d</i>.</td></tr> -<tr><td colspan="6"> </td><td class="tac" colspan="3">|</td><td colspan="6"> </td></tr> -<tr><td colspan="6"> </td><td class="tac" colspan="3">AB<i>cd</i></td><td colspan="6"> </td></tr> -</table> -</div> - -<p>It will be seen that the four proximates are respectively -obverse to the four ultimates, and that the mediates form -three pairs of obverses. Every proximate or ultimate is -distant 1 and 3 respectively from such a pair of mediates.</p> - -<p>Aided by this system of nomenclature Professor Clifford -proceeds to an exhaustive enumeration of types, in which -it is impossible to follow him. The results are as follows:—</p> - -<div class="center"> -<table class=""> -<tr> -<td class="tal" colspan="2">1-fold</td> -<td class="tac pr2"><div> statements</div></td> -<td class="tar"><div> 1</div></td> -<td class="tal"> type</td> -<td class="tal vab" rowspan="7"><img src="images/121x6br.png" width="6" height="111" alt="" ></td> -<td class="tal" rowspan="7">159</td> -</tr> -<tr> -<td class="tal">2</td> -<td class="tal">"</td> -<td class="tal pl2">"</td> -<td class="tar"><div><div> 4</div></div></td> -<td class="tal"> types</td> -</tr> -<tr> -<td class="tal">3</td> -<td class="tal">"</td> -<td class="tal pl2">"</td> -<td class="tar"><div><div> 6</div></div></td> -<td class="tac"><div><div>"</div></div></td> -</tr> -<tr> -<td class="tal">4</td> -<td class="tal">"</td> -<td class="tal pl2">"</td> -<td class="tar"><div><div>19</div></div></td> -<td class="tac"><div><div>"</div></div></td> -</tr> -<tr> -<td class="tal">5</td> -<td class="tal">"</td> -<td class="tal pl2">"</td> -<td class="tar"><div><div>27</div></div></td> -<td class="tac"><div><div>"</div></div></td> -</tr> -<tr> -<td class="tal">6</td> -<td class="tal">"</td> -<td class="tal pl2">"</td> -<td class="tar"><div><div>47</div></div></td> -<td class="tac"><div><div>"</div></div></td> -</tr> -<tr> -<td class="tal">7</td> -<td class="tal">"</td> -<td class="tal pl2">"</td> -<td class="tar"><div><div>55</div></div></td> -<td class="tac"><div><div>"</div></div></td> -</tr> -<tr> -<td class="tal" colspan="2">8-fold</td> -<td class="tal pr2"> statements</td> -<td class="tal pl2"><div>78</div></td> -<td class="tac"><div><div>"</div></div></td> -<td class="tar"></td> -<td class="tal"></td> -</tr> -</table> -</div> - -<p class="ti0">Now as each seven-fold or less-than-seven-fold statement -is complementary to a nine-fold or more-than-nine-fold -statement, it follows that the complete number of types -will be 159 × 2 + 78 = 396.</p> - -<p>It appears then that the types of statement concerning -four classes are only about 26 times as numerous as those -concerning three classes, fifteen in number, although the -number of possible combinations is 256 times as great.</p> - -<p>Professor Clifford informs me that the knowledge of the -possible groupings of subdivisions of classes which he -obtained by this inquiry has been of service to him in -some applications of hyper-elliptic functions to which he -has subsequently been led. Professor Cayley has since -expressed his opinion that this line of investigation should -be followed out, owing to the bearing of the theory of -compound combinations upon the higher geometry.<a id="FNanchor_86" href="#Footnote_86" class="fnanchor">86</a> It -seems likely that many unexpected points of connection<span class="pagenum" id="Page_146">146</span> -will in time be disclosed between the sciences of logic -and mathematics.</p> - - -<h3><i>Distinction between Perfect and Imperfect Induction.</i></h3> - -<p>We cannot proceed with advantage before noticing the -extreme difference which exists between cases of perfect -and those of imperfect induction. We call an induction -<i>perfect</i> when all the objects or events which can possibly -come under the class treated have been examined. But -in the majority of cases it is impossible to collect together, -or in any way to investigate, the properties of all portions -of a substance or of all the individuals of a race. The -number of objects would often be practically infinite, and -the greater part of them might be beyond our reach, in -the interior of the earth, or in the most distant parts of -the Universe. In all such cases induction is <i>imperfect</i>, -and is affected by more or less uncertainty. As some -writers have fallen into much error concerning the functions -and relative importance of these two branches of -reasoning, I shall have to point out that—</p> - -<p class="ml3h2">1. Perfect Induction is a process absolutely requisite, -both in the performance of imperfect induction and -in the treatment of large bodies of facts of which -our knowledge is complete.</p> - -<p class="ml3h2">2. Imperfect Induction is founded on Perfect Induction, -but involves another process of inference of a -widely different character.</p> -<p>It is certain that if I can draw any inference at all -concerning objects not examined, it must be done on the -data afforded by the objects which have been examined. -If I judge that a distant star obeys the law of gravity, -it must be because all other material objects sufficiently -known to me obey that law. If I venture to assert that -all ruminant animals have cloven hoofs, it is because all -ruminant animals which have come under my notice have -cloven hoofs. On the other hand, I cannot safely say -that all cryptogamous plants possess a purely cellular -structure, because some cryptogamous plants, which have -been examined by botanists, have a partially vascular -structure. The probability that a new cryptogam will be -cellular only can be estimated, if at all, on the ground of<span class="pagenum" id="Page_147">147</span> -the comparative numbers of known cryptogams which -are and are not cellular. Thus the first step in every -induction will consist in accurately summing up the -number of instances of a particular phenomenon which -have fallen under our observation. Adams and Leverrier, -for instance, must have inferred that the undiscovered -planet Neptune would obey Bode’s law, because <i>all the -planets known at that time obeyed it</i>. On what principles -the passage from the known to the apparently unknown -is warranted, must be carefully discussed in the next section, -and in various parts of this work.</p> - -<p>It would be a great mistake, however, to suppose that -Perfect Induction is in itself useless. Even when the -enumeration of objects belonging to any class is complete, -and admits of no inference to unexamined objects, the -statement of our knowledge in a general proposition is a -process of so much importance that we may consider it -necessary. In many cases we may render our investigations -exhaustive; all the teeth or bones of an animal; all -the cells in a minute vegetable organ; all the caves in a -mountain side; all the strata in a geological section; all -the coins in a newly found hoard, may be so completely -scrutinized that we may make some general assertion -concerning them without fear of mistake. Every bone -might be proved to contain phosphate of lime; every cell -to enclose a nucleus; every cave to hide remains of extinct -animals; every stratum to exhibit signs of marine origin; -every coin to be of Roman manufacture. These are cases -where our investigation is limited to a definite portion of -matter, or a definite area on the earth’s surface.</p> - -<p>There is another class of cases where induction is -naturally and necessarily limited to a definite number of -alternatives. Of the regular solids we can say without the -least doubt that no one has more than twenty faces, thirty -edges, and twenty corners; for by the principles of geometry -we learn that there cannot exist more than five regular -solids, of each of which we easily observe that the above -statements are true. In the theory of numbers, an endless -variety of perfect inductions might be made; we can show -that no number less than sixty possesses so many divisors, -and the like is true of 360; for it does not require a great -amount of labour to ascertain and count all the divisors<span class="pagenum" id="Page_148">148</span> -of numbers up to sixty or 360. I can assert that between -60,041 and 60,077 no prime number occurs, because the -exhaustive examination of those who have constructed -tables of prime numbers proves it to be so.</p> - -<p>In matters of human appointment or history, we can -frequently have a complete limitation of the number of -instances to be included in an induction. We might show -that the propositions of the third book of Euclid treat only -of circles; that no part of the works of Galen mentions the -fourth figure of the syllogism; that none of the other kings -of England reigned so long as George III.; that Magna -Charta has not been repealed by any subsequent statute; -that the price of corn in England has never been so high -since 1847 as it was in that year; that the price of the -English funds has never been lower than it was on the -23rd of January, 1798, when it fell to <span class="nowrap">47 <span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">4</span></span></span>.</p> - -<p>It has been urged against this process of Perfect Induction -that it gives no new information, and is merely a -summing up in a brief form of a multitude of particulars. -But mere abbreviation of mental labour is one of the most -important aids we can enjoy in the acquisition of knowledge. -The powers of the human mind are so limited that multiplicity -of detail is alone sufficient to prevent its progress -in many directions. Thought would be practically impossible -if every separate fact had to be separately thought -and treated. Economy of mental power may be considered -one of the main conditions on which our elevated intellectual -position depends. Mathematical processes are for the most -part but abbreviations of the simpler acts of addition and -subtraction. The invention of logarithms was one of the -most striking additions ever made to human power: yet it -was a mere abbreviation of operations which could have -been done before had a sufficient amount of labour been -available. Similar additions to our power will, it is hoped, -be made from time to time; for the number of mathematical -problems hitherto solved is but an indefinitely small -fraction of those which await solution, because the labour -they have hitherto demanded renders them impracticable. -So it is throughout all regions of thought. The amount -of our knowledge depends upon our power of bringing it -within practicable compass. Unless we arrange and -classify facts and condense them into general truths, they<span class="pagenum" id="Page_149">149</span> -soon surpass our powers of memory, and serve but to -confuse. Hence Perfect Induction, even as a process of -abbreviation, is absolutely essential to any high degree of -mental achievement.</p> - - -<h3><i>Transition from Perfect to Imperfect Induction.</i></h3> - -<p>It is a question of profound difficulty on what grounds -we are warranted in inferring the future from the present, -or the nature of undiscovered objects from those which we -have examined with our senses. We pass from Perfect to -Imperfect Induction when once we allow our conclusion to -apply, at all events apparently, beyond the data on which -it was founded. In making such a step we seem to gain a -net addition to our knowledge; for we learn the nature of -what was unknown. We reap where we have never sown. -We appear to possess the divine power of creating knowledge, -and reaching with our mental arms far beyond the -sphere of our own observation. I shall have, indeed, to -point out certain methods of reasoning in which we do -pass altogether beyond the sphere of the senses, and -acquire accurate knowledge which observation could -never have given; but it is not imperfect induction that -accomplishes such a task. Of imperfect induction itself, -I venture to assert that it never makes any real addition -to our knowledge, in the meaning of the expression sometimes -accepted. As in other cases of inference, it merely -unfolds the information contained in past observations; -it merely renders explicit what was implicit in previous -experience. It transmutes, but certainly does not create -knowledge.</p> - -<p>There is no fact which I shall more constantly keep -before the reader’s mind in the following pages than that -the results of imperfect induction, however well authenticated -and verified, are never more than probable. We -never can be sure that the future will be as the present. -We hang ever upon the will of the Creator: and it is -only so far as He has created two things alike, or maintains -the framework of the world unchanged from moment to -moment, that our most careful inferences can be fulfilled. -All predictions, all inferences which reach beyond their -data, are purely hypothetical, and proceed on the assumption<span class="pagenum" id="Page_150">150</span> -that new events will conform to the conditions detected -in our observation of past events. No experience of finite -duration can give an exhaustive knowledge of the forces -which are in operation. There is thus a double uncertainty; -even supposing the Universe as a whole to proceed unchanged, -we do not really know the Universe as a whole. -We know only a point in its infinite extent, and a moment -in its infinite duration. We cannot be sure, then, that our -observations have not escaped some fact, which will cause -the future to be apparently different from the past; nor -can we be sure that the future really will be the outcome -of the past. We proceed then in all our inferences to -unexamined objects and times on the assumptions—</p> - -<p class="ml3h2">1. That our past observation gives us a complete knowledge -of what exists.</p> - -<p class="ml3h2">2. That the conditions of things which did exist -will continue to be the conditions which will -exist.</p> - -<p>We shall often need to illustrate the character of our -knowledge of nature by the simile of a ballot-box, so often -employed by mathematical writers in the theory of probability. -Nature is to us like an infinite ballot-box, the -contents of which are being continually drawn, ball after -ball, and exhibited to us. Science is but the careful -observation of the succession in which balls of various -character present themselves; we register the combinations, -notice those which seem to be excluded from occurrence, -and from the proportional frequency of those which -appear we infer the probable character of future drawings. -But under such circumstances certainty of prediction -depends on two conditions:—</p> - -<p class="ml3h2">1. That we acquire a perfect knowledge of the comparative -numbers of balls of each kind within -the box.</p> - -<p class="ml3h2">2. That the contents of the ballot-box remain unchanged.</p> - -<p>Of the latter assumption, or rather that concerning the -constitution of the world which it illustrates, the logician -or physicist can have nothing to say. As the Creation of -the Universe is necessarily an act passing all experience -and all conception, so any change in that Universe, or, it -may be, a termination of it, must likewise be infinitely beyond -the bounds of our mental faculties. No science no<span class="pagenum" id="Page_151">151</span> -reasoning upon the subject, can have any validity; for -without experience we are without the basis and materials -of knowledge. It is the fundamental postulate accordingly -of all inference concerning the future, that there shall be -no arbitrary change in the subject of inference; of the probability -or improbability of such a change I conceive that -our faculties can give no estimate.</p> - -<p>The other condition of inductive inference—that we -acquire an approximately complete knowledge of the combinations -in which events do occur, is in some degree -within our power. There are branches of science in which -phenomena seem to be governed by conditions of a most -fixed and general character. We have ground in such -cases for believing that the future occurrence of such -phenomena can be calculated and predicted. But the -whole question now becomes one of probability and improbability. -We seem to leave the region of logic to enter -one in which the number of events is the ground of inference. -We do not really leave the region of logic; we -only leave that where certainty, affirmative or negative, is -the result, and the agreement or disagreement of qualities -the means of inference. For the future, number and -quantity will commonly enter into our processes of reasoning; -but then I hold that number and quantity are but -portions of the great logical domain. I venture to assert -that number is wholly logical, both in its fundamental -nature and in its developments. Quantity in all its forms -is but a development of number. That which is mathematical -is not the less logical; if anything it is more -logical, in the sense that it presents logical results in a -higher degree of complexity and variety.</p> - -<p>Before proceeding then from Perfect to Imperfect Induction -I must devote a portion of this work to treating -the logical conditions of number. I shall then employ -number to estimate the variety of combinations in which -natural phenomena may present themselves, and the probability -or improbability of their occurrence under definite -circumstances. It is in later parts of the work that I must -endeavour to establish the notions which I have set forth -upon the subject of Imperfect Induction, as applied in the -investigation of Nature, which notions maybe thus briefly -stated:—</p> - -<p><span class="pagenum" id="Page_152">152</span></p> - -<p class="ml3h2">1. Imperfect Induction entirely rests upon Perfect Induction -for its materials.</p> - -<p class="ml3h2">2. The logical process by which we seem to pass directly -from examined to unexamined cases consists in an -inverse application of deductive inference, so that -all reasoning may be said to be either directly or -inversely deductive.</p> - -<p class="ml3h2">3. The result is always of a hypothetical character, and -is never more than probable.</p> - -<p class="ml3h2">4. No net addition is ever made to our knowledge by -reasoning; what we know of future events or unexamined -objects is only the unfolded contents of -our previous knowledge, and it becomes less probable -as it is more boldly extended to remote -cases.</p> - -<hr class="chap x-ebookmaker-drop"> - -<div class="chapter"> -<p><span class="pagenum" id="Page_153">153</span></p> - -<p class="nobreak ph2 ti0" id="BOOK_II">BOOK II.<br> - -<span class="title">NUMBER, VARIETY, AND PROBABILITY.</span></p> - -<hr class="r30"> - -<h2 class="nobreak" id="CHAPTER_VIII">CHAPTER VIII.<br> - -<span class="title">PRINCIPLES OF NUMBER.</span></h2> -</div> - -<p class="ti0">Not without reason did Pythagoras represent the world -as ruled by number. Into almost all our acts of thought -number enters, and in proportion as we can define numerically -we enjoy exact and useful knowledge of the Universe. -The science of numbers, too, has hitherto presented the -widest and most practicable training in logic. So free and -energetic has been the study of mathematical forms, compared -with the forms of logic, that mathematicians have -passed far in advance of pure logicians. Occasionally, in -recent times, they have condescended to apply their -algebraic instrument to a reflex treatment of the primary -logical science. It is thus that we owe to profound mathematicians, -such as John Herschel, Whewell, De Morgan, or -Boole, the regeneration of logic in the present century. I -entertain no doubt that it is in maintaining a close alliance -with quantitative reasoning that we must look for further -progress in our comprehension of qualitative inference.</p> - -<p>I cannot assent, indeed, to the common notion that -certainty begins and ends with numerical determination. -Nothing is more certain than logical truth. The laws of -identity and difference are the tests of all that is certain<span class="pagenum" id="Page_154">154</span> -throughout the range of thought, and mathematical reasoning -is cogent only when it conforms to these conditions, of -which logic is the first development. And if it be -erroneous to suppose that all certainty is mathematical, it -is equally an error to imagine that all which is mathematical -is certain. Many processes of mathematical -reasoning are of most doubtful validity. There are points -of mathematical doctrine which must long remain matter -of opinion; for instance, the best form of the definition and -axiom concerning parallel lines, or the true nature of a -limit. In the use of symbolic reasoning questions occur on -which the best mathematicians may differ, as Bernoulli -and Leibnitz differed irreconcileably concerning the existence -of the logarithms of negative quantities.<a id="FNanchor_87" href="#Footnote_87" class="fnanchor">87</a> In fact we -no sooner leave the simple logical conditions of number, -than we find ourselves involved in a mazy and mysterious -science of symbols.</p> - -<p>Mathematical science enjoys no monopoly, and not even -a supremacy, in certainty of results. It is the boundless -extent and variety of quantitative questions that delights -the mathematical student. When simple logic can give -but a bare answer Yes or No, the algebraist raises a score -of subtle questions, and brings out a crowd of curious -results. The flower and the fruit, all that is attractive -and delightful, fall to the share of the mathematician, who -too often despises the plain but necessary stem from which -all has arisen. In no region of thought can a reasoner -cast himself free from the prior conditions of logical correctness. -The mathematician is only strong and true as -long as he is logical, and if number rules the world, it is -logic which rules number.</p> - -<p>Nearly all writers have hitherto been strangely content -to look upon numerical reasoning as something apart from -logical inference. A long divorce has existed between -quality and quantity, and it has not been uncommon to -treat them as contrasted in nature and restricted to -independent branches of thought. For my own part, I -believe that all the sciences meet somewhere. No part of -knowledge can stand wholly disconnected from other parts -of the universe of thought; it is incredible, above all, that<span class="pagenum" id="Page_155">155</span> -the two great branches of abstract science, interlacing and -co-operating in every discourse, should rest upon totally -distinct foundations. I assume that a connection exists, -and care only to inquire, What is its nature? Does the -science of quantity rest upon that of quality; or, <i>vice -versâ</i>, does the science of quality rest upon that of -quantity? There might conceivably be a third view, -that they both rest upon some still deeper set of principles.</p> - -<p>It is generally supposed that Boole adopted the second -view, and treated logic as an application of algebra, a -special case of analytical reasoning which admits only two -quantities, unity and zero. It is not easy to ascertain -clearly which of these views really was accepted by Boole. -In his interesting biographical sketch of Boole,<a id="FNanchor_88" href="#Footnote_88" class="fnanchor">88</a> the Rev. -R. Harley protests against the statement that Boole’s -logical calculus imported the conditions of number and -quantity into logic. He says: “Logic is never identified -or confounded with mathematics; the two systems of -thought are kept perfectly distinct, each being subject to -its own laws and conditions. The symbols are the same -for both systems, but they have not the same interpretation.” -The Rev. J. Venn, again, in his review of Boole’s -logical system,<a id="FNanchor_89" href="#Footnote_89" class="fnanchor">89</a> holds that Boole’s processes are at bottom -logical, not mathematical, though stated in a highly generalized -form and with a mathematical dress. But it is -quite likely that readers of Boole should be misled. Not -only have his logical works an entirely mathematical -appearance, but I find on p. 12 of his <i>Laws of Thought</i> -the following unequivocal statement: “That logic, as a -science, is susceptible of very wide applications is -admitted; but it is equally certain that its ultimate -forms and processes are mathematical.” A few lines -below he adds, “It is not of the essence of mathematics -to be conversant with the ideas of number and quantity.”</p> - -<p>The solution of the difficulty is that Boole used the -term mathematics in a wider sense than that usually -attributed to it. He probably adopted the third view, so -that his mathematical <i>Laws of Thought</i> are the common<span class="pagenum" id="Page_156">156</span> -basis both of logic and of quantitative mathematics. But -I do not care to pursue the subject because I think that, -in either case Boole was wrong. In my opinion logic is -the superior science, the general basis of mathematics as -well as of all other sciences. Number is but logical discrimination, -and algebra a highly developed logic. Thus -it is easy to understand the deep analogy which Boole -pointed out between the forms of algebraic and logical -deduction. Logic resembles algebra as the mould -resembles that which is cast in it. Boole mistook the -cast for the mould. Considering that logic imposes its -own laws upon every branch of mathematical science, it -is no wonder that we constantly meet with the traces of -logical laws in mathematical processes.</p> - - -<h3><i>The Nature of Number.</i></h3> - -<p>Number is but another name for <i>diversity</i>. Exact identity -is unity, and with difference arises plurality. An -abstract notion, as was pointed out (p. <a href="#Page_28">28</a>), possesses a -certain <i>oneness</i>. The quality of <i>justice</i>, for instance, is one -and the same in whatever just acts it is manifested. In -justice itself there are no marks of difference by which to -discriminate justice from justice. But one just act can be -discriminated from another just act by circumstances of -time and place, and we can count many acts thus discriminated -each from each. In like manner pure gold is -simply pure gold, and is so far one and the same throughout. -But besides its intrinsic qualities, gold occupies -space and must have shape and size. Portions of gold -are always mutually exclusive and capable of discrimination, -in respect that they must be each without the other. -Hence they may be numbered.</p> - -<p>Plurality arises when and only when we detect difference. -For instance, in counting a number of gold coins -I must count each coin once, and not more than once. -Let C denote a coin, and the mark above it the order of -counting. Then I must count the coins</p> - -<div class="ml5em"> -C′ + C″ + C‴ + C″″ + . . . . . . -</div> - -<p class="ti0">If I were to count them as follows</p> - -<div class="ml5em"> -C′ + C″ + C‴ + C‴ + C″″ + . . ., -</div> - -<p class="ti0">I should make the third coin into two, and should imply<span class="pagenum" id="Page_157">157</span> -the existence of difference where there is no difference.<a id="FNanchor_90" href="#Footnote_90" class="fnanchor">90</a> -C‴ and C‴ are but the names of one coin named twice -over. But according to one of the conditions of logical -symbols, which I have called the Law of Unity (p. <a href="#Page_72">72</a>), -the same name repeated has no effect, and</p> - -<div class="ml5em"> -A ꖌ A = A.<br> -</div> - -<p class="ti0">We must apply the Law of Unity, and must reduce all -identical alternatives before we can count with certainty -and use the processes of numerical calculation. Identical -alternatives are harmless in logic, but are wholly inadmissible -in number. Thus logical science ascertains the -nature of the mathematical unit, and the definition may -be given in these terms—<i>A unit is any object of thought -which can be discriminated from every other object treated as -a unit in the same problem.</i></p> - -<p>It has often been said that units are units in respect of -being perfectly similar to each other; but though they -may be perfectly similar in some respects, they must be -different in at least one point, otherwise they would be -incapable of plurality. If three coins were so similar that -they occupied the same space at the same time, they -would not be three coins, but one coin. It is a property -of space that every point is discriminable from every other -point, and in time every moment is necessarily distinct -from any other moment before or after. Hence we -frequently count in space or time, and Locke, with some -other philosophers, has held that number arises from -repetition in time. Beats of a pendulum may be so -perfectly similar that we can discover no difference except -that one beat is before and another after. Time alone is -here the ground of difference and is a sufficient foundation -for the discrimination of plurality; but it is by no means -the only foundation. Three coins are three coins, whether -we count them successively or regard them all simultaneously. -In many cases neither time nor space is the -ground of difference, but pure quality alone enters. We -can discriminate the weight, inertia, and hardness of gold -as three qualities, though none of these is before nor after -the other, neither in space nor time. Every means of -discrimination may be a source of plurality.</p> - -<p><span class="pagenum" id="Page_158">158</span></p> - -<p>Our logical notation may be used to express the rise of -number. The symbol A stands for one thing or one class, -and in itself must be regarded as a unit, because no -difference is specified. But the combinations AB and A<i>b</i> -are necessarily <i>two</i>, because they cannot logically coalesce, -and there is a mark B which distinguishes one from the -other. A logical definition of the number <i>four</i> is given in -the combinations ABC, AB<i>c</i>, A<i>b</i>C, A<i>bc</i>, where there is a -double difference. As Puck says—</p> - -<div class="tac fs90 mtb1em"> -“Yet but three? Come one more;<br> -  Two of both kinds makes up four.”<br> -</div> - -<p>I conceive that all numbers might be represented as -arising out of the combinations of the Logical Alphabet, -more or less of each series being struck out by various -logical conditions. The number three, for instance, arises -from the condition that A must be either B or C, so that -the combinations are ABC, AB<i>c</i>, A<i>b</i>C.</p> - - -<h3><i>Of Numerical Abstraction.</i></h3> - -<p>There will now be little difficulty in forming a clear -notion of the nature of numerical abstraction. It consists -in abstracting the character of the difference from which -plurality arises, retaining merely the fact. When I speak -of <i>three men</i> I need not at once specify the marks by which -each may be known from each. Those marks must exist -if they are really three men and not one and the same, and -in speaking of them as many I imply the existence of the -requisite differences. Abstract number, then, is <i>the empty -form of difference</i>; the abstract number <i>three</i> asserts the existence -of marks without specifying their kind.</p> - -<p>Numerical abstraction is thus seen to be a different -process from logical abstraction (p. <a href="#Page_27">27</a>), for in the -latter process we drop out of notice the very existence of -difference and plurality. In forming the abstract notion -<i>hardness</i>, we ignore entirely the diverse circumstances in -which the quality may appear. It is the concrete notion -<i>three hard objects</i>, which asserts the existence of hardness -along with sufficient other undefined qualities, to mark out -<i>three</i> such objects. Numerical thought is indeed closely -interwoven with logical thought. We cannot use a concrete<span class="pagenum" id="Page_159">159</span> -term in the plural, as <i>men</i>, without implying that -there are marks of difference. But when we use an -abstract term, we deal with unity.</p> - -<p>The origin of the great generality of number is now -apparent. Three sounds differ from three colours, or three -riders from three horses; but they agree in respect of the -variety of marks by which they can be discriminated. The -symbols 1 + 1 + 1 are thus the empty marks asserting the -existence of discrimination. But in dropping out of sight -the character of the differences we give rise to new -agreements on which mathematical reasoning is founded. -Numerical abstraction is so far from being incompatible -with logical abstraction that it is the origin of our widest -acts of generalization.</p> - - -<h3><i>Concrete and Abstract Number.</i></h3> - -<p>The common distinction between concrete and abstract -number can now be easily stated. In proportion as we -specify the logical characters of the things numbered, we -render them concrete. In the abstract number three -there is no statement of the points in which the <i>three</i> -objects agree; but in <i>three coins</i>, <i>three men</i>, or <i>three horses</i>, -not only are the objects numbered but their nature is restricted. -Concrete number thus implies the same consciousness -of difference as abstract number, but it is -mingled with a groundwork of similarity expressed in the -logical terms. There is identity so far as logical terms -enter; difference so far as the terms are merely numerical.</p> - -<p>The reason of the important Law of Homogeneity will -now be apparent. This law asserts that in every arithmetical -calculation the logical nature of the things numbered -must remain unaltered. The specified logical -agreement of the things must not be affected by the unspecified -numerical differences. A calculation would be -palpably absurd which, after commencing with length, -gave a result in hours. It is equally absurd, in a purely -arithmetical point of view, to deduce areas from the -calculation of lengths, masses from the combination of -volume and density, or momenta from mass and velocity. -It must remain for subsequent consideration to decide in -what sense we may truly say that two linear feet multiplied<span class="pagenum" id="Page_160">160</span> -by two linear feet give four superficial feet; arithmetically -it is absurd, because there is a change of unit.</p> - -<p>As a general rule we treat in each calculation only -objects of one nature. We do not, and cannot properly -add, in the same sum yards of cloth and pounds of sugar. -We cannot even conceive the result of adding area to -velocity, or length to density, or weight to value. The -units added must have a basis of homogeneity, or must be -reducible to some common denominator. Nevertheless it -is possible, and in fact common, to treat in one complex -calculation the most heterogeneous quantities, on the -condition that each kind of object is kept distinct, and -treated numerically only in conjunction with its own kind. -Different units, so far as their logical differences are specified, -must never be substituted one for the other. Chemists -continually use equations which assert the equivalence of -groups of atoms. Ordinary fermentation is represented -by the formula</p> - -<div class="tac"> -C<sup>6</sup> H<sup>12</sup> O<sup>6</sup> = 2C<sup>2</sup> H<sup>6</sup> O + 2CO<sup>2</sup>.<br> -</div> - -<p>Three kinds of units, the atoms respectively of carbon, -hydrogen, and oxygen, are here intermingled, but there is -really a separate equation in regard to each kind. Mathematicians -also employ compound equations of the same -kind; for in, <i>a</i> + <i>b</i> √<span class="o"> - 1</span> = <i>c</i> + <i>d</i> √<span class="o"> - 1</span>, it is impossible -by ordinary addition to add <i>a</i> to <i>b</i> √<span class="o"> -1</span>. Hence we -really have the separate equations <i>a</i> = <i>b</i>, and <i>c</i> √<span class="o"> - 1</span> = -<i>d</i> √<span class="o"> - 1</span>. Similarly an equation between two quaternions is -equivalent to four equations between ordinary quantities, -whence indeed the name <i>quaternion</i>.</p> - - -<h3><i>Analogy of Logical and Numerical Terms.</i></h3> - -<p>If my assertion is correct that number arises out of -logical conditions, we ought to find number obeying all the -laws of logic. It is almost superfluous to point out that -this is the case with the fundamental laws of identity and -difference, and it only remains to show that mathematical -symbols do really obey the special conditions of logical -symbols which were formerly pointed out (p. <a href="#Page_32">32</a>). Thus -the Law of Commutativeness, is equally true of quality and -quantity. As in logic we have</p> - -<div class="ml5em"> -AB = BA, -</div> - -<p class="ti0">so in mathematics it is familiarly known that</p> - -<p><span class="pagenum" id="Page_161">161</span></p> - -<div class="ml5em"> -2 × 3 = 3 × 2, or <i>x</i> × <i>y</i> = <i>y</i> × <i>x</i>. -</div> - -<p class="ti0">The properties of space are as indifferent in multiplication -as we found them in pure logical thought.</p> - -<p>Similarly, as in logic</p> - -<table class=""> -<tr> -<td></td> -<td class="tar"><div><div>triangle or square =</div></div></td> -<td class="tal"> square or triangle,</td> -</tr> -<tr> -<td class="tal">or generally</td> -<td class="tar"><div><div>A ꖌ B =</div></div></td> -<td class="tal">B ꖌ A,</td> -</tr> -<tr> -<td class="tal">so in quantity</td> -<td class="tar"><div><div>2 + 3 =</div></div></td> -<td class="tal">3 + 2,</td> -</tr> -<tr> -<td class="tal">or generally</td> -<td class="tar"><div><div><i>x</i> + <i>y</i> =</div></div></td> -<td class="tal"><i>y</i> + <i>x</i>.</td> -</tr> -</table> - -<p>The symbol ꖌ is not identical with +, but it is thus far -analogous.</p> - -<p>How far, now, is it true that mathematical symbols obey -the Law of Simplicity expressed in the form</p> - -<div class="ml5em"> -AA = A, -</div> - -<p class="ti0">or the example</p> - -<div class="ml5em"> -Round round = round? -</div> - -<p>Apparently there are but two numbers which obey this -law; for it is certain that</p> - -<div class="ml5em"> -<i>x</i> × <i>x</i> = <i>x</i> -</div> - -<p class="ti0">is true only in the two cases when <i>x</i> = 1, or <i>x</i> = 0.</p> - -<p>In reality all numbers obey the law, for 2 × 2 = 2 is not -really analogous to AA = A. According to the definition -of a unit already given, each unit is discriminated from -each other in the same problem, so that in 2′ × 2″, the -first <i>two</i> involves a different discrimination from the second -<i>two</i>. I get four kinds of things, for instance, if I first discriminate -“heavy and light” and then “cubical and -spherical,” for we now have the following classes—</p> - -<table class="ml5em"> -<tr> -<td class="tal">heavy, cubical.</td> -<td class="tal">light, cubical.</td> -</tr> -<tr> -<td class="tal pr2">heavy, spherical.</td> -<td class="tal">light, spherical.</td> -</tr> -</table> - -<p>But suppose that my two classes are in both cases discriminated -by the same difference of light and heavy, then -we have</p> - -<table class="ml5em"> -<tr> -<td class="tal">heavy</td> -<td class="tar"><div><div>heavy =</div></div></td> -<td class="tal">heavy,</td> -</tr> -<tr> -<td class="tal">heavy</td> -<td class="tar"><div><div>light =</div></div></td> -<td class="tal">0,</td> -</tr> -<tr> -<td class="tal">light</td> -<td class="tar"><div><div>heavy =</div></div></td> -<td class="tal">0,</td> -</tr> -<tr> -<td class="tal">light</td> -<td class="tar"><div><div>light =</div></div></td> -<td class="tal">light.</td> -</tr> -</table> - -<p class="ti0">Thus, (heavy or light) × (heavy or light) = (heavy or light).</p> - -<p>In short, <i>twice two is two</i> unless we take care that the -second two has a different meaning from the first. But -under similar circumstances logical terms give the like -result, and it is not true that A′A″ = A′, when A″ is -different in meaning from A′.</p> - -<p><span class="pagenum" id="Page_162">162</span></p> - -<p>In a similar manner it may be shown that the Law of -Unity</p> - -<div class="ml5em"> -A ꖌ A = A. -</div> - -<p class="ti0">holds true alike of logical and mathematical terms. It is -absurd indeed to say that</p> - -<div class="ml5em"> -<i>x</i> + <i>x</i> = <i>x</i> -</div> - -<p class="ti0">except in the one case when <i>x</i> = absolute zero. But this -contradiction <i>x</i> + <i>x</i> = <i>x</i> arises from the fact that we have -already defined the units in one x as differing from those in -the other. Under such circumstances the Law of Unity -does not apply. For if in</p> - -<div class="ml5em"> -A′ ꖌ A″ = A′ -</div> - -<p class="ti0">we mean that A″ is in any way different from A′ the -assertion of identity is evidently false.</p> - -<p>The contrast then which seems to exist between logical -and mathematical symbols is only apparent. It is because -the Laws of Simplicity and Unity must always be observed -in the operation of counting that those laws seem no further -to apply. This is the understood condition under which -we use all numerical symbols. Whenever I write the -symbol 5 I really mean</p> - -<div class="ml5em">1 + 1 + 1 + 1 + 1, -</div> - -<p class="ti0">and it is perfectly understood that each of these units is -distinct from each other. If requisite I might mark them -thus</p> - -<div class="ml5em"> -1′+ 1″ + 1‴ + 1″″ + 1″‴. -</div> - -<p class="ti0">Were this not the case and were the units really</p> - -<div class="ml5em"> -1′ + 1″ + 1″ + 1‴ + 1″″, -</div> - -<p class="ti0">the Law of Unity would, as before remarked, apply, and</p> - -<div class="ml5em"> -1″ + 1″ = 1″. -</div> - -<p>Mathematical symbols then obey all the laws of logical -symbols, but two of these laws seem to be inapplicable -simply because they are presupposed in the definition of -the mathematical unit. Logic thus lays down the conditions -of number, and the science of arithmetic developed -as it is into all the wondrous branches of mathematical -calculus is but an outgrowth of logical discrimination.</p> - - -<h3><i>Principle of Mathematical Inference.</i></h3> - -<p>The universal principle of all reasoning, as I have -asserted, is that which allows us to substitute like for like. -I have now to point out how in the mathematical sciences<span class="pagenum" id="Page_163">163</span> -this principle is involved in each step of reasoning. It is -in these sciences indeed that we meet with the clearest -cases of substitution, and it is the simplicity with which -the principle can be applied which probably led to the -comparatively early perfection of the sciences of geometry -and arithmetic. Euclid, and the Greek mathematicians -from the first, recognised <i>equality</i> as the fundamental -relation of quantitative thought, but Aristotle rejected the -exactly analogous, but far more general relation of identity, -and thus crippled the formal science of logic as it has -descended to the present day.</p> - -<p>Geometrical reasoning starts from the axiom that -“things equal to the same thing are equal to each other.” -Two equalities enable us to infer a third equality; and this -is true not only of lines and angles, but of areas, volumes, -numbers, intervals of time, forces, velocities, degrees of -intensity, or, in short, anything which is capable of being -equal or unequal. Two stars equally bright with the same -star must be equally bright with each other, and two forces -equally intense with a third force are equally intense with -each other. It is remarkable that Euclid has not explicitly -stated two other axioms, the truth of which is necessarily -implied. The second axiom should be that “Two things of -which one is equal and the other unequal to a third common -thing, are unequal to each other.” An equality and -inequality, in short, give an inequality, and this is equally -true with the first axiom of all kinds of quantity. If -Venus, for instance, agrees with Mars in density, but Mars -differs from Jupiter, then Venus differs from Jupiter. A -third axiom must exist to the effect that “Things unequal -to the same thing may or may not be equal to each -other.” <i>Two inequalities give no ground of inference whatever.</i> -If we only know, for instance, that Mercury and -Jupiter differ in density from Mars, we cannot say whether -or not they agree between themselves. As a fact they do -not agree; but Venus and Mars on the other hand both -differ from Jupiter and yet closely agree with each other. -The force of the axioms can be most clearly illustrated by -drawing equal and unequal lines.<a id="FNanchor_91" href="#Footnote_91" class="fnanchor">91</a></p> -<p><span class="pagenum" id="Page_164">164</span></p> -<p>The general conclusion then must be that where there -is equality there may be inference, but where there is not -equality there cannot be inference. A plain induction -will lead us to believe that <i>equality is the condition of -inference concerning quantity</i>. All the three axioms may -in fact be summed up in one, to the effect, that “<i>in -whatever relation one quantity stands to another, it stands -in the same relation to the equal of that other</i>.”</p> - -<p>The active power is always the substitution of equals, -and it is an accident that in a pair of equalities we can -make the substitution in two ways. From <i>a</i> = <i>b</i> = <i>c</i> we -can infer <i>a</i> = <i>c</i>, either by substituting in <i>a</i> = <i>b</i> the value -of <i>b</i> as given in <i>b</i> = <i>c</i>, or else by substituting in <i>b</i> = <i>c</i> the -value of <i>b</i> as given in <i>a</i> = <i>b</i>. In <i>a</i> = <i>b</i> ~ <i>d</i> we can make -but the one substitution of <i>a</i> for <i>b</i>. In <i>e</i> ~ <i>f</i> ~ <i>g</i> we can -make no substitution and get no inference.</p> - -<p>In mathematics the relations in which terms may stand -to each other are far more varied than in pure logic, yet -our principle of substitution always holds true. We may -say in the most general manner that <i>In whatever relation -one quantity stands to another, it stands in the same relation -to the equal of that other.</i> In this axiom we sum up a -number of axioms which have been stated in more or less -detail by algebraists. Thus, “If equal quantities be added -to equal quantities, the sums will be equal.” To explain -this, let</p> - -<div class="ml5em"> -<i>a</i> = <i>b</i>,  <i>c</i> = <i>d</i>. -</div> - -<p class="ti0">Now <i>a</i> + <i>c</i>, whatever it means, must be identical with -itself, so that</p> - -<div class="ml5em"> -<i>a</i> + <i>c</i> = <i>a</i> + <i>c</i>. -</div> - -<p class="ti0">In one side of this equation substitute for the quantities -their equivalents, and we have the axiom proved</p> - -<div class="ml5em"> -<i>a</i> + <i>c</i> = <i>b</i> + <i>d</i>.<br> -</div> - -<p class="ti0">The similar axiom concerning subtraction is equally evident, -for whatever <i>a</i> - <i>c</i> may mean it is equal to <i>a</i> - <i>c</i>, -and therefore by substitution to <i>b</i> - <i>d</i>. Again, “if equal -quantities be multiplied by the same or equal quantities, -the products will be equal,” For evidently</p> - -<div class="ml5em"> -<i>ac</i> = <i>ac</i>, -</div> - -<p class="ti0">and if for <i>c</i> in one side we substitute its equal <i>d</i>, we have</p> - -<div class="ml5em"> -<i>ac</i> = <i>ad</i>, -</div> - -<p class="ti0">and a second similar substitution gives us</p> - -<p><span class="pagenum" id="Page_165">165</span></p> - -<div class="ml5em"> -<i>ac</i> = <i>bd</i>. -</div> - -<p class="ti0">We might prove a like axiom concerning division in an -exactly similar manner. I might even extend the list of -axioms and say that “Equal powers of equal numbers are -equal.” For certainly, whatever <i>a</i> × <i>a</i> × <i>a</i> may mean, it -is equal to <i>a</i> × <i>a</i> × <i>a</i>; hence by our usual substitution it -is equal to <i>b</i> × <i>b</i> × <i>b</i>. The same will be true of roots of -numbers and [c root]<i>a</i> = [d root]<i>b</i> provided that the roots are so -taken that the root of <i>a</i> shall really be related to <i>a</i> as -the root of <i>b</i> is to <i>b</i>. The ambiguity of meaning of an -operation thus fails in any way to shake the universality -of the principle. We may go further and assert that, not -only the above common relations, but all other known or -conceivable mathematical relations obey the same principle. -Let Q<i>a</i> denote in the most general manner that we -do something with the quantity <i>a</i>; then if <i>a</i> = <i>b</i> it follows -that</p> - -<div class="ml5em"> -Q<i>a</i> = Q<i>b</i>. -</div> - -<p>The reader will also remember that one of the most -frequent operations in mathematical reasoning is to substitute -for a quantity its equal, as known either by assumed, -natural, or self-evident conditions. Whenever a quantity -appears twice over in a problem, we may apply what we -learn of its relations in one place to its relations in the -other. All reasoning in mathematics, as in other branches -of science, thus involves the principle of treating equals -equally, or similars similarly. In whatever way we -employ quantitative reasoning in the remaining parts of -this work, we never can desert the simple principle on -which we first set out.</p> - - -<h3><i>Reasoning by Inequalities.</i></h3> - -<p>I have stated that all the processes of mathematical -reasoning may be deduced from the principle of substitution. -Exceptions to this assertion may seem to exist -in the use of inequalities. The greater of a greater is -undoubtedly a greater, and what is less than a less is -certainly less. Snowdon is higher than the Wrekin, and -Ben Nevis than Snowdon; therefore Ben Nevis is higher -than the Wrekin. But a little consideration discloses -sufficient reason for believing that even in such cases,<span class="pagenum" id="Page_166">166</span> -where equality does not apparently enter, the force of the -reasoning entirely depends upon underlying and implied -equalities.</p> - -<p>In the first place, two statements of mere difference do -not give any ground of inference. We learn nothing -concerning the comparative heights of St. Paul’s and -Westminster Abbey from the assertions that they both -differ in height from St. Peter’s at Rome. We need something -more than inequality; we require one identity in -addition, namely the identity in direction of the two -differences. Thus we cannot employ inequalities in the -simple way in which we do equalities, and, when we try -to express what other conditions are requisite, we find -ourselves lapsing into the use of equalities or identities.</p> - -<p>In the second place, every argument by inequalities -may be represented in the form of equalities. We express -that <i>a</i> is greater than <i>b</i> by the equation</p> - -<div class="ml5em"> -<i>a</i> = <i>b</i> + <i>p</i>,  (1) -</div> - -<p class="ti0">where <i>p</i> is an intrinsically positive quantity, denoting the -difference of <i>a</i> and <i>b</i>. Similarly we express that <i>b</i> is -greater than <i>c</i> by the equation</p> - -<div class="ml5em"> -<i>b</i> = <i>c</i> + <i>q</i>,  (2) -</div> - -<p class="ti0">and substituting for <i>b</i> in (1) its value in (2) we have</p> - -<div class="ml5em"> -<i>a</i> = <i>c</i> + <i>q</i> + <i>p</i>.  (3) -</div> - -<p class="ti0">Now as <i>p</i> and <i>q</i> are both positive, it follows that <i>a</i> is -greater than <i>c</i>, and we have the exact amount of excess -specified. It will be easily seen that the reasoning concerning -that which is less than a less will result in an -equation of the form</p> - -<div class="ml5em"> -<i>c</i> = <i>a</i> - <i>r</i> - <i>s</i>. -</div> - -<p>Every argument by inequalities may then be thrown -into the form of an equality; but the converse is not true. -We cannot possibly prove that two quantities are equal -by merely asserting that they are both greater or both less -than another quantity. From <i>e</i> > <i>f</i> and <i>g</i> > <i>f</i>, or <i>e</i> < <i>f</i> -and <i>g</i> < <i>f</i>, we can infer no relation between <i>e</i> and <i>g</i>. And -if the reader take the equations <i>x</i> = <i>y</i> = 3 and attempt to -prove that therefore <i>x</i> = 3, by throwing them into inequalities, -he will find it impossible to do so.</p> - -<p>From these considerations I gather that reasoning in -arithmetic or algebra by so-called inequalities, is only an -imperfectly expressed reasoning by equalities, and when<span class="pagenum" id="Page_167">167</span> -we want to exhibit exactly and clearly the conditions of -reasoning, we are obliged to use equalities explicitly. Just -as in pure logic a negative proposition, as expressing mere -difference, cannot be the means of inference, so inequality -can never really be the true ground of inference. I do -not deny that affirmation and negation, agreement and -difference, equality and inequality, are pairs of equally -fundamental relations, but I assert that inference is possible -only where affirmation, agreement, or equality, some -species of identity in fact, is present, explicitly or implicitly.</p> - - -<h3><i>Arithmetical Reasoning.</i></h3> - -<p>It may seem somewhat inconsistent that I assert number -to arise out of difference or discrimination, and yet hold -that no reasoning can be grounded on difference. Number, -of course, opens a most wide sphere for inference, and a -little consideration shows that this is due to the unlimited -series of identities which spring up out of numerical -abstraction. If six people are sitting on six chairs, there -is no resemblance between the chairs and the people in -logical character. But if we overlook all the qualities -both of a chair and a person and merely remember that -there are marks by which each of six chairs may be -discriminated from the others, and similarly with the -people, then there arises a resemblance between the chairs -and the people, and this resemblance in number may be -the ground of inference. If on another occasion the chairs -are filled by people again, we may infer that these people -resemble the others in number though they need not -resemble them in any other points.</p> - -<p>Groups of units are what we really treat in arithmetic. -The number <i>five</i> is really 1 + 1 + 1 + 1 + 1, but for the -sake of conciseness we substitute the more compact sign -5, or the name <i>five</i>. These names being arbitrarily imposed -in any one manner, an infinite variety of relations -spring up between them which are not in the least -arbitrary. If we define <i>four</i> as 1 + 1 + 1 + 1, and <i>five</i> -as 1 + 1 + 1 + 1 + 1, then of course it follows that -<i>five</i> = <i>four</i> + 1; but it would be equally possible to take -this latter equality as a definition, in which case one of -the former equalities would become an inference. It is<span class="pagenum" id="Page_168">168</span> -hardly requisite to decide how we define the names of -numbers, provided we remember that out of the infinitely -numerous relations of one number to others, some one -relation expressed in an equality must be a definition of -the number in question and the other relations immediately -become necessary inferences.</p> - -<p>In the science of number the variety of classes which -can be formed is altogether infinite, and statements of -perfect generality may be made subject only to difficulty -or exception at the lower end of the scale. Every existing -number for instance belongs to the class <i>m</i> + 7; that is, -every number must be the sum of another number and -seven, except of course the first six or seven numbers, -negative quantities not being here taken into account. -Every number is the half of some other, and so on. The -subject of generalization, as exhibited in mathematical -truths, is an infinitely wide one. In number we are only -at the first step of an extensive series of generalizations. -As number is general compared with the particular things -numbered, so we have general symbols for numbers, and -general symbols for relations between undetermined -numbers. There is an unlimited hierarchy of successive -generalizations.</p> - - -<h3><i>Numerically Definite Reasoning.</i></h3> - -<p>It was first discovered by De Morgan that many arguments -are valid which combine logical and numerical -reasoning, although they cannot be included in the -ancient logical formulas. He developed the doctrine of -the “Numerically Definite Syllogism,” fully explained in -his <i>Formal Logic</i> (pp. 141–170). Boole also devoted -considerable attention to the determination of what he -called “Statistical Conditions,” meaning the numerical -conditions of logical classes. In a paper published among -the Memoirs of the Manchester Literary and Philosophical -Society, Third Series, vol. IV. p. 330 (Session 1869–70), -I have pointed out that we can apply arithmetical calculation -to the Logical Alphabet. Having given certain logical -conditions and the numbers of objects in certain classes, -we can either determine the numbers of objects in other -classes governed by those conditions, or can show what<span class="pagenum" id="Page_169">169</span> -further data are required to determine them. As an -example of the kind of questions treated in numerical -logic, and the mode of treatment, I give the following -problem suggested by De Morgan, with my mode of -representing its solution.</p> - -<p>“For every man in the house there is a person who is -aged; some of the men are not aged. It follows that -some persons in the house are not men.”<a id="FNanchor_92" href="#Footnote_92" class="fnanchor">92</a></p> - -<table class="ml5em"> -<tr> -<td class="tal pr3">Now let</td> -<td class="tal">A = person in house,</td> -</tr> -<tr> -<td class="tal"></td> -<td class="tal">B = male,</td> -</tr> -<tr> -<td class="tal"></td> -<td class="tal">C = aged.</td> -</tr> -</table> - -<p class="ti0">By enclosing a logical symbol in brackets, let us denote -the number of objects belonging to the class indicated by -the symbol. Thus let</p> - -<table class="ml5em"> -<tr> -<td class="tar"><div><div>(A) =</div></div></td> -<td class="tal">number of persons in house,</td> -</tr> -<tr> -<td class="tar"><div><div>(AB) =</div></div></td> -<td class="tal">number of male persons in house,</td> -</tr> -<tr> -<td class="tar"><div><div>(ABC) =</div></div></td> -<td class="tal">number of aged male persons in house,</td> -</tr> -</table> - -<p class="ti0">and so on. Now if we use <i>w</i> and <i>w</i>′ to denote unknown -numbers, the conditions of the problem may be thus stated -according to my interpretation of the words—</p> - -<div class="ml5em"> -(AB) = (AC) - <i>w</i>,  (1) -</div> - -<p class="ti0">that is to say, the number of persons in the house who are -aged is at least equal to, and may exceed, the number of -male persons in the house;</p> - -<div class="ml5em"> -(AB<i>c</i>) = <i>w</i>′,  (2) -</div> - -<p class="ti0">that is to say, the number of male persons in the house -who are not aged is some unknown positive quantity.</p> - -<p>If we develop the terms in (1) by the Law of Duality -(pp. <a href="#Page_74">74</a>, <a href="#Page_81">81</a>, <a href="#Page_89">89</a>), we obtain</p> - -<div class="ml5em"> -(ABC) + (AB<i>c</i>) = (ABC) + (A<i>b</i>C) - <i>w</i>. -</div> - -<p class="ti0">Subtracting the common term (ABC) from each side and -substituting for (AB<i>c</i>) its value as given in (2), we get at -once</p> - -<div class="ml5em"> -(A<i>b</i>C) = <i>w</i> + <i>w</i>′, -</div> - -<p class="ti0">and adding (A<i>bc</i>) to each side, we have</p> - -<div class="ml5em"> -(A<i>b</i>) = (A<i>bc</i>) + <i>w</i> + <i>w</i>′. -</div> - -<p class="ti0">The meaning of this result is that the number of persons -in the house who are not men is at least equal to <i>w</i> + <i>w</i>′, -and exceeds it by the number of persons in the house who -are neither men nor aged (A<i>bc</i>).</p> - -<p><span class="pagenum" id="Page_170">170</span></p> - -<p>It should be understood that this solution applies only -to the terms of the example quoted above, and not to the -general problem for which De Morgan intended it to serve -as an illustration.</p> - -<p>As a second instance, let us take the following question:—The -whole number of voters in a borough is <i>a</i>; -the number against whom objections have been lodged by -liberals is <i>b</i>; and the number against whom objections -have been lodged by conservatives is <i>c</i>; required the -number, if any, who have been objected to on both sides. -Taking</p> - - -<div class="ml5em"> -A = voter,<br> -B = objected to by liberals,<br> -C = objected to by conservatives, -</div> - -<p class="ti0">then we require the value of (ABC). Now the following -equation is identically true—</p> - -<div class="ml5em"> -(ABC) = (AB) + (AC) + (A<i>bc</i>) - (A).  (1) -</div> - -<p class="ti0">For if we develop all the terms on the second side we -obtain</p> - -<p class="ml7h5"> -(ABC) = (ABC) + (AB<i>c</i>) + (ABC) + (A<i>b</i>C) + (A<i>bc</i>)<br> -- (ABC) - (AB<i>c</i>) - (A<i>b</i>C) - (A<i>bc</i>); -</p> - -<p class="ti0">and striking out the corresponding positive and negative -terms, we have left only (ABC) = (ABC). Since then -(1) is necessarily true, we have only to insert the known -values, and we have</p> - -<div class="ml5em"> -(ABC) = <i>b</i> + <i>c</i> - <i>a</i> + (A<i>bc</i>). -</div> - -<p class="ti0">Hence the number who have received objections from both -sides is equal to the excess, if any, of the whole number -of objections over the number of voters together with the -number of voters who have received no objection (A<i>bc</i>).</p> - -<p>The following problem illustrates the expression for -the common part of any three classes:—The number of -paupers who are blind males, is equal to the excess, if -any, of the sum of the whole number of blind persons, -added to the whole number of male persons, added to the -number of those who being paupers are neither blind nor -males, above the sum of the whole number of paupers -added to the number of those who, not being paupers, -are blind, and to the number of those who, not being -paupers, are male.</p> - -<p>The reader is requested to prove the truth of the above -statement, (1) by his own unaided common sense; (2) by<span class="pagenum" id="Page_171">171</span> -the Aristotelian Logic; (3) by the method of numerical -logic just expounded; and then to decide which method -is most satisfactory.</p> - - -<h3><i>Numerical meaning of Logical Conditions.</i></h3> - -<p>In many cases classes of objects may exist under special -logical conditions, and we must consider how these -conditions can be interpreted numerically. Every logical -proposition gives rise to a corresponding numerical -equation. Sameness of qualities occasions sameness of -numbers. Hence if</p> - -<div class="ml5em"> -A = B -</div> - -<p class="ti0">denotes the identity of the qualities of A and B, we may -conclude that</p> - -<div class="ml5em"> -(A) = (B). -</div> - -<p class="ti0">It is evident that exactly those objects, and those objects -only, which are comprehended under A must be comprehended -under B. It follows that wherever we can draw -an equation of qualities, we can draw a similar equation -of numbers. Thus, from</p> - -<div class="ml5em"> -A = B = C -</div> - -<p class="ti0">we infer</p> - -<div class="ml5em"> -A = C; -</div> - -<p class="ti0">and similarly from</p> - -<div class="ml5em"> -(A) = (B) = (C), -</div> - -<p class="ti0">meaning that the numbers of A’s and C’s are equal to the -number of B’s, we can infer</p> - -<div class="ml5em"> -(A) = (C). -</div> - -<p class="ti0">But, curiously enough, this does not apply to negative -propositions and inequalities. For if</p> - -<div class="ml5em"> -A = B ~ D -</div> - -<p class="ti0">means that A is identical with B, which differs from D, it -does not follow that</p> - -<div class="ml5em"> -(A) = (B) ~ (D). -</div> - -<p class="ti0">Two classes of objects may differ in qualities, and yet they -may agree in number. This point strongly confirms me -in the opinion which I have already expressed, that all -inference really depends upon equations, not differences.</p> - -<p>The Logical Alphabet thus enables us to make a complete -analysis of any numerical problem, and though the -symbolical statement may sometimes seem prolix, I conceive<span class="pagenum" id="Page_172">172</span> -that it really represents the course which the mind -must follow in solving the question. Although thought -may outstrip the rapidity with which the symbols can -be written down, yet the mind does not really follow a -different course from that indicated by the symbols. For -a fuller explanation of this natural system of Numerically -Definite Reasoning, with more abundant illustrations -and an analysis of De Morgan’s Numerically Definite -Syllogism, I must refer the reader to the paper<a id="FNanchor_93" href="#Footnote_93" class="fnanchor">93</a> in the -Memoirs of the Manchester Literary and Philosophical -Society, already mentioned, portions of which, however, -have been embodied in the present section.</p> - -<p>The reader may be referred, also, to Boole’s writings -upon the subject in the <i>Laws of Thought</i>, chap. xix. -p. 295, and in a paper on “Propositions Numerically -Definite,” communicated by De Morgan, in 1868, to the -Cambridge Philosophical Society, and printed in their -<i>Transactions</i>, vol. xi. part ii.</p> - -<hr class="chap x-ebookmaker-drop"> - -<div class="chapter"> -<p><span class="pagenum" id="Page_173">173</span></p> - -<h2 class="nobreak" id="CHAPTER_IX">CHAPTER IX. - -<span class="title">THE VARIETY OF NATURE, OR THE DOCTRINE OF -COMBINATIONS AND PERMUTATIONS.</span></h2> -</div> - -<p class="ti0">Nature may be said to be evolved from the monotony -of non-existence by the creation of diversity. It is plausibly -asserted that we are conscious only so far as we -experience difference. Life is change, and perfectly uniform -existence would be no better than non-existence. -Certain it is that life demands incessant novelty, and that -nature, though it probably never fails to obey the same -fixed laws, yet presents to us an apparently unlimited -series of varied combinations of events. It is the work of -science to observe and record the kinds and comparative -numbers of such combinations of phenomena, occurring -spontaneously or produced by our interference. Patient -and skilful examination of the records may then disclose -the laws imposed on matter at its creation, and enable us -more or less successfully to predict, or even to regulate, -the future occurrence of any particular combination.</p> - -<p>The Laws of Thought are the first and most important -of all the laws which govern the combinations of phenomena, -and, though they be binding on the mind, they -may also be regarded as verified in the external world. -The Logical Alphabet develops the utmost variety of -things and events which may occur, and it is evident that -as each new quality is introduced, the number of combinations -is doubled. Thus four qualities may occur in 16 -combinations; five qualities in 32; six qualities in 64; -and so on. In general language, if n be the number of -<span class="pagenum" id="Page_174">174</span>qualities, 2<sup>n</sup> is the number of varieties of things which -may be formed from them, if there be no conditions but -those of logic. This number, it need hardly be said, -increases after the first few terms, in an extraordinary -manner, so that it would require 302 figures to express -the number of combinations in which 1,000 qualities -might conceivably present themselves.</p> - -<p>If all the combinations allowed by the Laws of Thought -occurred indifferently in nature, then science would begin -and end with those laws. To observe nature would give -us no additional knowledge, because no two qualities -would in the long run be oftener associated than any -other two. We could never predict events with more -certainty than we now predict the throws of dice, and -experience would be without use. But the universe, as -actually created, presents a far different and much more -interesting problem. The most superficial observation -shows that some things are constantly associated with -other things. The more mature our examination, the -more we become convinced that each event depends -upon the prior occurrence of some other series of events. -Action and reaction are gradually discovered to underlie -the whole scene, and an independent or casual occurrence -does not exist except in appearance. Even dice as they -fall are surely determined in their course by prior conditions -and fixed laws. Thus the combinations of events -which can really occur are found to be comparatively -restricted, and it is the work of science to detect these -restricting conditions.</p> - -<p>In the English alphabet, for instance, we have twenty-six -letters. Were the combinations of such letters perfectly -free, so that any letter could be indifferently -sounded with any other, the number of words which -could be formed without any repetition would be 2<sup>26</sup> - 1, -or 67,108,863, equal in number to the combinations of -the twenty-seventh column of the Logical Alphabet, -excluding one for the case in which all the letters -would be absent. But the formation of our vocal -organs prevents us from using the far greater part of -these conjunctions of letters. At least one vowel must be -present in each word; more than two consonants cannot -usually be brought together; and to produce words capable -of smooth utterance a number of other rules must be<span class="pagenum" id="Page_175">175</span> -observed. To determine exactly how many words might -exist in the English language under these circumstances, -would be an exceedingly complex problem, the solution of -which has never been attempted. The number of existing -English words may perhaps be said not to exceed one -hundred thousand, and it is only by investigating the combinations -presented in the dictionary, that we can learn the -Laws of Euphony or calculate the possible number of -words. In this example we have an epitome of the work -and method of science. The combinations of natural -phenomena are limited by a great number of conditions -which are in no way brought to our knowledge except so -far as they are disclosed in the examination of nature.</p> - -<p>It is often a very difficult matter to determine the numbers -of permutations or combinations which may exist -under various restrictions. Many learned men puzzled -themselves in former centuries over what were called -Protean verses, or verses admitting many variations in -accordance with the Laws of Metre. The most celebrated -of these verses was that invented by Bernard Bauhusius, -as follows:<a id="FNanchor_94" href="#Footnote_94" class="fnanchor">94</a>—</p> - -<div class="tac fs90 mtb05em"> -“Tot tibi sunt dotes, Virgo, quot sidera cœlo.” -</div> - -<p>One author, Ericius Puteanus, filled forty-eight pages of a -work in reckoning up its possible transpositions, making -them only 1022. Other calculators gave 2196, 3276, 2580 -as their results. Wallis assigned 3096, but without much -confidence in the accuracy of his result.<a id="FNanchor_95" href="#Footnote_95" class="fnanchor">95</a> It required the -skill of James Bernoulli to decide that the number of -transpositions was 3312, under the condition that the sense -and metre of the verse shall be perfectly preserved.</p> - -<p>In approaching the consideration of the great Inductive -problem, it is very necessary that we should acquire correct -notions as to the comparative numbers of combinations -which may exist under different circumstances. The -doctrine of combinations is that part of mathematical -science which applies numerical calculation to determine -the numbers of combinations under various conditions. -It is a part of the science which really lies at the base not -only of other sciences, but of other branches of mathematics.<span class="pagenum" id="Page_176">176</span> -The forms of algebraical expressions are determined -by the principles of combination, and Hindenburg -recognised this fact in his Combinatorial Analysis. The -greatest mathematicians have, during the last three centuries, -given their best powers to the treatment of this -subject; it was the favourite study of Pascal; it early -attracted the attention of Leibnitz, who wrote his curious -essay, <i>De Arte Combinatoria</i>, at twenty years of age; James -Bernoulli, one of the very profoundest mathematicians, -devoted no small part of his life to the investigation of the -subject, as connected with that of Probability; and in his -celebrated work, <i>De Arte Conjectandi</i>, he has so finely -described the importance of the doctrine of combinations, -that I need offer no excuse for quoting his remarks at full -length.</p> - -<p>“It is easy to perceive that the prodigious variety which -appears both in the works of nature and in the actions of -men, and which constitutes the greatest part of the beauty -of the universe, is owing to the multitude of different ways -in which its several parts are mixed with, or placed near, -each other. But, because the number of causes that concur -in producing a given event, or effect, is oftentimes so immensely -great, and the causes themselves are so different -one from another, that it is extremely difficult to reckon up -all the different ways in which they may be arranged or -combined together, it often happens that men, even of the -best understandings and greatest circumspection, are guilty -of that fault in reasoning which the writers on logic call -<i>the insufficient or imperfect enumeration of parts or cases</i>: -insomuch that I will venture to assert, that this is the -chief, and almost the only, source of the vast number of -erroneous opinions, and those too very often in matters -of great importance, which we are apt to form on all the -subjects we reflect upon, whether they relate to the knowledge -of nature, or the merits and motives of human -actions.</p> - -<p>“It must therefore be acknowledged, that that art which -affords a cure to this weakness, or defect, of our understandings, -and teaches us so to enumerate all the possible -ways in which a given number of things may be mixed -and combined together, that we may be certain that we -have not omitted any one arrangement of them that can<span class="pagenum" id="Page_177">177</span> -lead to the object of our inquiry, deserves to be considered -as most eminently useful and worthy of our highest esteem -and attention. And this is the business of <i>the art or -doctrine of combinations</i>. Nor is this art or doctrine to be -considered merely as a branch of the mathematical sciences. -For it has a relation to almost every species of useful knowledge -that the mind of man can be employed upon. It -proceeds indeed upon mathematical principles, in calculating -the number of the combinations of the things proposed: -but by the conclusions that are obtained by it, the sagacity -of the natural philosopher, the exactness of the historian, -the skill and judgment of the physician, and the prudence -and foresight of the politician may be assisted; because -the business of all these important professions is but <i>to -form reasonable conjectures</i> concerning the several objects -which engage their attention, and all wise conjectures are -the results of a just and careful examination of the several -different effects that may possibly arise from the causes -that are capable of producing them.”<a id="FNanchor_96" href="#Footnote_96" class="fnanchor">96</a></p> - - -<h3><i>Distinction of Combinations and Permutations.</i></h3> - -<p>We must first consider the deep difference which exists -between Combinations and Permutations, a difference involving -important logical principles, and influencing the -form of mathematical expressions. In <i>permutation</i> we recognise -varieties of order, treating AB as a different group -from BA. In <i>combination</i> we take notice only of the -presence or absence of a certain thing, and pay no regard -to its place in order of time or space. Thus the four -letters <i>a</i>, <i>e</i>, <i>m</i>, <i>n</i> can form but one combination, but they -occur in language in several permutations, as <i>name</i>, <i>amen</i>, -<i>mean</i>, <i>mane</i>.</p> - -<p>We have hitherto been dealing with purely logical questions, -involving only combination of qualities. I have -fully pointed out in more than one place that, though our -symbols could not but be written in order of place and read -in order of time, the relations expressed had no regard to -place or time (pp. <a href="#Page_33">33</a>, <a href="#Page_114">114</a>). The Law of Commutativeness, -in fact, expresses the condition that in logic we deal with<span class="pagenum" id="Page_178">178</span> -combinations, and the same law is true of all the processes -of algebra. In some cases, order may be a matter of -indifference; it makes no difference, for instance, whether -gunpowder is a mixture of sulphur, carbon, and nitre, or -carbon, nitre, and sulphur, or nitre, sulphur, and carbon, -provided that the substances are present in proper proportions -and well mixed. But this indifference of order does -not usually extend to the events of physical science or the -operations of art. The change of mechanical energy into -heat is not exactly the same as the change from heat into -mechanical energy; thunder does not indifferently precede -and follow lightning; it is a matter of some importance -that we load, cap, present, and fire a rifle in this precise -order. Time is the condition of all our thoughts, space of -all our actions, and therefore both in art and science we -are to a great extent concerned with permutations. -Language, for instance, treats different permutations of -letters as having different meanings.</p> - -<p>Permutations of things are far more numerous than -combinations of those things, for the obvious reason that -each distinct thing is regarded differently according to -its place. Thus the letters A, B, C, will make different -permutations according as A stands first, second, or third; -having decided the place of A, there are two places -between which we may choose for B; and then there -remains but one place for C. Accordingly the permutations -of these letters will be altogether 3 × 2 × 1 or 6 in -number. With four things or letters, A, B, C, D, we -shall have four choices of place for the first letter, three -for the second, two for the third, and one for the fourth, -so that there will be altogether, 4 × 3 × 2 × 1, or 24 -permutations. The same simple rule applies in all cases; -beginning with the whole number of things we multiply -at each step by a number decreased by a unit. In general -language, if <i>n</i> be the number of things in a combination, -the number of permutations is</p> - -<div class="ml5em"> -<i>n</i> (<i>n</i> - 1)(<i>n</i> - 2) . . . . 4 . 3 . 2 . 1. -</div> - -<p>If we were to re-arrange the names of the days of -the week, the possible arrangements out of which we -should have to choose the new order, would be no less -than 7 . 6 . 5 . 4 . 3 . 2 . 1, or 5040, or, excluding the -existing order, 5039.</p> - -<p><span class="pagenum" id="Page_179">179</span></p> - -<p>The reader will see that the numbers which we reach in -questions of permutation, increase in a more extraordinary -manner even than in combination. Each new object or -term doubles the number of combinations, but increases -the permutations by a factor continually growing. Instead -of 2 × 2 × 2 × 2 × .... we have 2 × 3 × 4 × 5 × .... -and the products of the latter expression immensely -exceed those of the former. These products of increasing -factors are frequently employed, as we shall see, in questions -both of permutation and combination. They are -technically called <i>factorials</i>, that is to say, the product of -all integer numbers, from unity up to any number <i>n</i> is the -<i>factorial</i> of <i>n</i>, and is often indicated symbolically by <i>n</i>!. -I give below the factorials up to that of twelve:—</p> - -<div class="center"> -<table class=""> -<tr> -<td class="tar"><div>24 = </div></td> -<td class="tal">1 . 2 . 3 . 4</td> -</tr> -<tr> -<td class="tar"><div>120 = </div></td> -<td class="tal">1 . 2 . . . 5</td> -</tr> -<tr> -<td class="tar"><div>720 = </div></td> -<td class="tal">1 . 2 . . . 6</td> -</tr> -<tr> -<td class="tar"><div>5,040 = </div></td> -<td class="tal">7!</td> -</tr> -<tr> -<td class="tar"><div>40,320 = </div></td> -<td class="tal">8!</td> -</tr> -<tr> -<td class="tar"><div>362,880 = </div></td> -<td class="tal">9!</td> -</tr> -<tr> -<td class="tar"><div>3,628,800 = </div></td> -<td class="tal">10!</td> -</tr> -<tr> -<td class="tar"><div>39,916,800 = </div></td> -<td class="tal">11!</td> -</tr> -<tr> -<td class="tar"><div>479,001,600 = </div></td> -<td class="tal">12!</td> -</tr> -</table> -</div> - -<p>The factorials up to 36! are given in Rees’s ‘Cyclopædia,’ -art. <i>Cipher</i>, and the logarithms of factorials up to 265! -are to be found at the end of the table of logarithms -published under the superintendence of the Society for -the Diffusion of Useful Knowledge (p. 215). To express -the factorial 265! would require 529 places of figures.</p> - -<p>Many writers have from time to time remarked upon -the extraordinary magnitude of the numbers with which -we deal in this subject. Tacquet calculated<a id="FNanchor_97" href="#Footnote_97" class="fnanchor">97</a> that the -twenty-four [sic] letters of the alphabet may be arranged in -more than 620 thousand trillions of orders; and Schott -estimated<a id="FNanchor_98" href="#Footnote_98" class="fnanchor">98</a> that if a thousand millions of men were employed -for the same number of years in writing out these -arrangements, and each man filled each day forty pages -with forty arrangements in each, they would not have -accomplished the task, as they would have written only -584 thousand trillions instead of 620 thousand trillions.</p> - -<p><span class="pagenum" id="Page_180">180</span></p> - -<p>In some questions the number of permutations may be -restricted and reduced by various conditions. Some -things in a group may be undistinguishable from others, -so that change of order will produce no difference. Thus -if we were to permutate the letters of the name <i>Ann</i>, -according to our previous rule, we should obtain 3 × 2 × 1, -or 6 orders; but half of these arrangements would be -identical with the other half, because the interchange of -the two <i>n</i>’s has no effect. The really different orders will -therefore be <span class="nowrap"><span class="fraction2"><span class="fnum2">3 . 2 . 1</span><span class="bar">/</span><span class="fden2">1 . 2</span></span></span> or 3, namely <i>Ann</i>, - <i>Nan</i>, - <i>Nna</i>. In -the word <i>utility</i> there are two <i>i</i>’s and two <i>t</i>’s, in respect -of both of which pairs the numbers of permutations must -be halved. Thus we obtain <span class="nowrap"><span class="fraction2"><span class="fnum2">7 . 6 . 5 . 4 . 3 . 2 . 1</span><span class="bar">/</span><span class="fden2">1 . 2 . 1 . 2</span></span></span> - or 1260, as -the number of permutations. The simple rule evidently -is—when some things or letters are undistinguished, -proceed in the first place to calculate all the possible -permutations as if all were different, and then divide by -the numbers of possible permutations of those series of -things which are not distinguished, and of which the -permutations have therefore been counted in excess. -Thus since the word <i>Utilitarianism</i> contains fourteen -letters, of which four are <i>i</i>’s, two <i>a</i>’s, and two <i>t</i>’s, the -number of distinct arrangements will be found by -dividing the factorial of 14, by the factorials of 4, 2, -and 2, the result being 908,107,200. From the letters -of the word <i>Mississippi</i> we can get in like manner -<span class="nowrap"><span class="fraction2"><span class="fnum2">11!</span><span class="bar">/</span><span class="fden2">4! × 4! × 2!</span></span></span> - or 34,650 permutations, which is not the one-thousandth -part of what we should obtain were all the -letters different.</p> - - -<h3><i>Calculation of Number of Combinations.</i></h3> - -<p>Although in many questions both of art and science -we need to calculate the number of permutations on -account of their own interest, it far more frequently -happens in scientific subjects that they possess but an -indirect interest. As I have already pointed out, we -almost always deal in the logical and mathematical -sciences with <i>combinations</i>, and variety of order enters<span class="pagenum" id="Page_181">181</span> -only through the inherent imperfections of our symbols -and modes of calculation. Signs must be used in some -order, and we must withdraw our attention from this order -before the signs correctly represent the relations of things -which exist neither before nor after each other. Now, it -often happens that we cannot choose all the combinations -of things, without first choosing them subject to the -accidental variety of order, and we must then divide by -the number of possible variations of order, that we may -get to the true number of pure combinations.</p> - -<p>Suppose that we wish to determine the number of ways -in which we can select a group of three letters out of the -alphabet, without allowing the same letter to be repeated. -At the first choice we can take any one of 26 letters; at -the next step there remain 25 letters, any one of which -may be joined with that already taken; at the third step -there will be 24 choices, so that apparently the whole -number of ways of choosing is 26 × 25 × 24. But the -fact that one choice succeeded another has caused us to -obtain the same combinations of letters in different orders; -we should get, for instance, <i>a</i>, <i>p</i>, <i>r</i> at one time, and <i>p</i>, <i>r</i>, <i>a</i> at -another, and every three distinct letters will appear six -times over, because three things can be arranged in six -permutations. To get the number of combinations, then, -we must divide the whole number of ways of choosing, -by six, the number of permutations of three things, -obtaining <span class="nowrap"><span class="fraction2"><span class="fnum2">26 × 25 × 24</span><span class="bar">/</span><span class="fden2">1 × 2 × 3</span></span></span> or 2,600.</p> - -<p>It is apparent that we need the doctrine of combinations -in order that we may in many questions counteract -the exaggerating effect of successive selection. If out of -a senate of 30 persons we have to choose a committee of 5, -we may choose any of 30 first, any of 29 next, and so on, -in fact there will be 30 × 29 × 28 × 27 × 26 selections; -but as the actual character of the members of the committee -will not be affected by the accidental order of their selection, -we divide by 1 × 2 × 3 × 4 × 5, and the possible -number of different committees will be 142,506. Similarly -if we want to calculate the number of ways in which the -eight major planets may come into conjunction, it is evident -that they may meet either two at a time or three at -a time, or four or more at a time, and as nothing is said as to<span class="pagenum" id="Page_182">182</span> -the relative order or place in the conjunction, we require -the number of combinations. Now a selection of 2 out of 8 -is possible in <span class="nowrap"><span class="fraction2"><span class="fnum2">8 . 7</span><span class="bar">/</span><span class="fden2">1 . 2</span></span></span> - or 28 ways; of 3 out of 8 in <span class="nowrap"><span class="fraction2"><span class="fnum2">8 . 7 . 6</span><span class="bar">/</span><span class="fden2">1 . 2 . 3</span></span></span> -or 56 ways; of 4 out of 8 in <span class="nowrap"><span class="fraction2"><span class="fnum2">8 . 7 . 6 . 5</span><span class="bar">/</span><span class="fden2">1 . 2 . 3 . 4</span></span></span> - or 70 ways; and it -may be similarly shown that for 5, 6, 7, and 8 planets, -meeting at one time, the numbers of ways are 56, 28, 8, -and 1. Thus we have solved the whole question of the -variety of conjunctions of eight planets; and adding all the -numbers together, we find that 247 is the utmost possible -number of modes of meeting.</p> - -<p>In general algebraic language, we may say that a group -of <i>m</i> things may be chosen out of a total number of <i>n</i> -things, in a number of combinations denoted by the -formula</p> - -<div class="center mt05em"> -<span class="nowrap"><span class="fraction2"><span class="fnum2"><i>n</i> . (<i>n</i>-1)(<i>n</i>-2)(<i>n</i>-3) . - . . . (<i>n</i> - <i>m</i> + 1)</span><span class="bar">/</span><span class="fden2">1 . 2 . 3 . 4 . . . .  <i>m</i></span></span></span><br> -</div> - -<p>The extreme importance and significance of this formula -seems to have been first adequately recognised by Pascal, -although its discovery is attributed by him to a friend, M. -de Ganières.<a id="FNanchor_99" href="#Footnote_99" class="fnanchor">99</a> We shall find it perpetually recurring in -questions both of combinations and probability, and -throughout the formulæ of mathematical analysis traces -of its influence may be noticed.</p> - - -<h3><i>The Arithmetical Triangle.</i></h3> - -<p>The Arithmetical Triangle is a name long since given to -a series of remarkable numbers connected with the subject -we are treating. According to Montucla<a id="FNanchor_100" href="#Footnote_100" class="fnanchor">100</a> “this triangle is -in the theory of combinations and changes of order, almost -what the table of Pythagoras is in ordinary arithmetic, that -is to say, it places at once under the eyes the numbers required -in a multitude of cases of this theory.” As early -as 1544 Stifels had noticed the remarkable properties of -these numbers and the mode of their evolution. Briggs, -the inventor of the common system of logarithms, was so -struck with their importance that he called them the<span class="pagenum" id="Page_183">183</span> -Abacus Panchrestus. Pascal, however, was the first who -wrote a distinct treatise on these numbers, and gave them -the name by which they are still known. But Pascal did -not by any means exhaust the subject, and it remained for -James Bernoulli to demonstrate fully the importance of -the <i>figurate numbers</i>, as they are also called. In his -treatise <i>De Arte Conjectandi</i>, he points out their application -in the theory of combinations and probabilities, and -remarks of the Arithmetical Triangle, “It not only contains -the clue to the mysterious doctrine of combinations, -but it is also the ground or foundation of most of the important -and abstruse discoveries that have been made in -the other branches of the mathematics.”<a id="FNanchor_101" href="#Footnote_101" class="fnanchor">101</a></p> - -<p>The numbers of the triangle can be calculated in a -very easy manner by successive additions. We commence -with unity at the apex; in the next line we place a second -unit to the right of this; to obtain the third line of figures -we move the previous line one place to the right, and add -them to the same figures as they were before removal; we -can then repeat the same process <i>ad infinitum</i>. The -fourth line of figures, for instance, contains 1, 3, 3, 1; -moving them one place and adding as directed we obtain:—</p> - -<table class="ml5em mtb05em"> -<tr> -<td class="tal">Fourth line . . .</td> -<td class="tac"><div> </div></td> -<td class="tac pr1"><div>1</div></td> -<td class="tac pl05"><div>3</div></td> -<td class="tac prl05"><div> 3</div></td> -<td class="tac prl05"><div> 1</div></td> -<td> </td> -<td> </td> -</tr> -<tr> -<td class="tal"></td> -<td class="tac bb"> </td> -<td class="tac bb"></td> -<td class="tac pl05 bb"><div>1</div></td> -<td class="tac prl05 bb"><div> 3</div></td> -<td class="tac prl05 bb"><div> 3</div></td> -<td class="tac prl05 bb"><div> 1</div></td> -<td> </td> -</tr> -<tr> -<td class="tal">Fifth line . . . . .</td> -<td class="tac"><div> </div></td> -<td class="tac pr1"><div>1</div></td> -<td class="tac pl05"><div>4</div></td> -<td class="tac prl05"><div> 6</div></td> -<td class="tac prl05"><div> 4</div></td> -<td class="tac prl05"><div> 1</div></td> -<td> </td> -</tr> -<tr> -<td class="tal"></td> -<td class="tac bb"> </td> -<td class="tac bb"></td> -<td class="tac pl05 bb"><div> 1 </div></td> -<td class="tac prl05 bb"><div> 4</div></td> -<td class="tac prl05 bb"><div> 6</div></td> -<td class="tac prl05 bb"><div> 4</div></td> -<td class="tac prl05 bb"><div> 1</div></td> -</tr> -<tr> -<td class="tal">Sixth line . . . . .</td> -<td class="tac"><div> </div></td> -<td class="tac pr1"><div>1</div></td> -<td class="tac pl05"><div>5</div></td> -<td class="tac prl05"><div>10</div></td> -<td class="tac prl05"><div>10</div></td> -<td class="tac prl05"><div> 5</div></td> -<td class="tac prl05"><div> 1</div></td> -</tr> -</table> - -<p>Carrying out this simple process through ten more steps -we obtain the first seventeen lines of the Arithmetical -Triangle as printed on the next page. Theoretically -speaking the Triangle must be regarded as infinite in -extent, but the numbers increase so rapidly that it soon -becomes impracticable to continue the table. The longest -table of the numbers which I have found is in Fortia’s -“Traité des Progressions” (p. 80), where they are given up -to the fortieth line and the ninth column.</p> - -<p><span class="pagenum" id="Page_184">184</span></p> - -<p class="tac">THE ARITHMETICAL TRIANGLE.</p> - -<table class="fs70 mtb1em"> -<tr> -<td class="tar"><div>Line.</div></td> -<td class="tal pl03" colspan="3">First Column.</td> -<td colspan="14"> </td> -</tr> -<tr> -<td class="tar pr03"><div>1</div></td> -<td class="tar brlm pr03"><div>1</div></td> -<td class="tal pl03" colspan="3">Second Column.</td> -<td colspan="13"> </td> -</tr> -<tr> -<td class="tar pr03"><div>2</div></td> -<td class="tar brlm pr03"><div>1</div></td> -<td class="tar brl pr03"><div>1</div></td> -<td class="tal pl03" colspan="3">Third Column.</td> -<td colspan="12"> </td> -</tr> -<tr> -<td class="tar pr03"><div>3</div></td> -<td class="tar brlm pr03"><div>1</div></td> -<td class="tar brl pr03"><div>2</div></td> -<td class="tar brl pr03"><div>1</div></td> -<td class="tal pl03" colspan="3">Fourth Column.</td> -<td colspan="11"> </td> -</tr> -<tr> -<td class="tar pr03"><div>4</div></td> -<td class="tar brlm pr03"><div>1</div></td> -<td class="tar brl pr03"><div>3</div></td> -<td class="tar brl pr03"><div>3</div></td> -<td class="tar brl pr03"><div>1</div></td> -<td class="tal pl03" colspan="3">Fifth Column.</td> -<td colspan="10"> </td> -</tr> -<tr> -<td class="tar pr03"><div>5</div></td> -<td class="tar brlm pr03"><div>1</div></td> -<td class="tar brl pr03"><div>4</div></td> -<td class="tar brl pr03"><div>6</div></td> -<td class="tar brl pr03"><div>4</div></td> -<td class="tar brl pr03"><div>1</div></td> -<td class="tal pl03" colspan="3">Sixth Column.</td> -<td colspan="9"> </td> -</tr> -<tr> -<td class="tar pr03"><div>6</div></td> -<td class="tar brlm pr03"><div>1</div></td> -<td class="tar brl pr03"><div>5</div></td> -<td class="tar brl pr03"><div>10</div></td> -<td class="tar brl pr03"><div>10</div></td> -<td class="tar brl pr03"><div>5</div></td> -<td class="tar brl pr03"><div>1</div></td> -<td class="tal pl03" colspan="3">Seventh Column.</td> -<td colspan="8"> </td> -</tr> -<tr> -<td class="tar pr03"><div>7</div></td> -<td class="tar brlm pr03"><div>1</div></td> -<td class="tar brl pr03"><div>6</div></td> -<td class="tar brl pr03"><div>15</div></td> -<td class="tar brl pr03"><div>20</div></td> -<td class="tar brl pr03"><div>15</div></td> -<td class="tar brl pr03"><div>6</div></td> -<td class="tar brl pr03"><div>1</div></td> -<td class="tal pl03" colspan="3">Eighth Column.</td> -<td colspan="7"> </td> -</tr> -<tr> -<td class="tar pr03"><div>8</div></td> -<td class="tar brlm pr03"><div>1</div></td> -<td class="tar brl pr03"><div>7</div></td> -<td class="tar brl pr03"><div>21</div></td> -<td class="tar brl pr03"><div>35</div></td> -<td class="tar brl pr03"><div>35</div></td> -<td class="tar brl pr03"><div>21</div></td> -<td class="tar brl pr03"><div>7</div></td> -<td class="tar brl pr03"><div>1</div></td> -<td class="tal pl03" colspan="3">Ninth Column.</td> -<td colspan="6"> </td> -</tr> -<tr> -<td class="tar pr03"><div>9</div></td> -<td class="tar brlm pr03"><div>1</div></td> -<td class="tar brl pr03"><div>8</div></td> -<td class="tar brl pr03"><div>28</div></td> -<td class="tar brl pr03"><div>56</div></td> -<td class="tar brl pr03"><div>70</div></td> -<td class="tar brl pr03"><div>56</div></td> -<td class="tar brl pr03"><div>28</div></td> -<td class="tar brl pr03"><div>8</div></td> -<td class="tar brl pr03"><div>1</div></td> -<td class="tal pl03" colspan="3">Tenth Column.</td> -<td colspan="5"> </td> -</tr> -<tr> -<td class="tar pr03"><div>10</div></td> -<td class="tar brlm pr03"><div>1</div></td> -<td class="tar brl pr03"><div>9</div></td> -<td class="tar brl pr03"><div>36</div></td> -<td class="tar brl pr03"><div>84</div></td> -<td class="tar brl pr03"><div>126</div></td> -<td class="tar brl pr03"><div>126</div></td> -<td class="tar brl pr03"><div>84</div></td> -<td class="tar brl pr03"><div>36</div></td> -<td class="tar brl pr03"><div>9</div></td> -<td class="tar brl"><div>1</div></td> -<td class="tal pl03" colspan="3">Eleventh Column.</td> -<td colspan="4"> </td> -</tr> -<tr> -<td class="tar pr03"><div>11</div></td> -<td class="tar brlm pr03"><div>1</div></td> -<td class="tar brl pr03"><div>10</div></td> -<td class="tar brl pr03"><div>45</div></td> -<td class="tar brl pr03"><div>120</div></td> -<td class="tar brl pr03"><div>210</div></td> -<td class="tar brl pr03"><div>252</div></td> -<td class="tar brl pr03"><div>210</div></td> -<td class="tar brl pr03"><div>120</div></td> -<td class="tar brl pr03"><div>45</div></td> -<td class="tar brl pr03"><div>10</div></td> -<td class="tar brl pr03"><div>1</div></td> -<td class="tal pl03" colspan="3">Twelfth Column.</td> -<td colspan="3"> </td> -</tr> -<tr> -<td class="tar pr03"><div>12</div></td> -<td class="tar brlm pr03"><div>1</div></td> -<td class="tar brl pr03"><div>11</div></td> -<td class="tar brl pr03"><div>55</div></td> -<td class="tar brl pr03"><div>165</div></td> -<td class="tar brl pr03"><div>330</div></td> -<td class="tar brl pr03"><div>462</div></td> -<td class="tar brl pr03"><div>462</div></td> -<td class="tar brl pr03"><div>330</div></td> -<td class="tar brl pr03"><div>165</div></td> -<td class="tar brl pr03"><div>55</div></td> -<td class="tar brl pr03"><div>11</div></td> -<td class="tar brl pr03"><div>1</div></td> -<td class="tal pl03" colspan="4">Thirteenth Column.</td> -<td> </td> -</tr> -<tr> -<td class="tar pr03"><div>13</div></td> -<td class="tar brlm pr03"><div>1</div></td> -<td class="tar brl pr03"><div>12</div></td> -<td class="tar brl pr03"><div>66</div></td> -<td class="tar brl pr03"><div>220</div></td> -<td class="tar brl pr03"><div>495</div></td> -<td class="tar brl pr03"><div>792</div></td> -<td class="tar brl pr03"><div>924</div></td> -<td class="tar brl pr03"><div>792</div></td> -<td class="tar brl pr03"><div>495</div></td> -<td class="tar brl pr03"><div>220</div></td> -<td class="tar brl pr03"><div>66</div></td> -<td class="tar brl pr03"><div>12</div></td> -<td class="tar brl pr03"><div>1</div></td> -<td class="tal pl03" colspan="4">Fourteenth Column.</td> -</tr> -<tr> -<td class="tar pr03"><div>14</div></td> -<td class="tar brlm pr03"><div>1</div></td> -<td class="tar brl pr03"><div>13</div></td> -<td class="tar brl pr03"><div>78</div></td> -<td class="tar brl pr03"><div>286</div></td> -<td class="tar brl pr03"><div>715</div></td> -<td class="tar brl pr03"><div>1287</div></td> -<td class="tar brl pr03"><div>1716</div></td> -<td class="tar brl pr03"><div>1716</div></td> -<td class="tar brl pr03"><div>1287</div></td> -<td class="tar brl pr03"><div>715</div></td> -<td class="tar brl pr03"><div>286</div></td> -<td class="tar brl pr03"><div>78</div></td> -<td class="tar brl pr03"><div>13</div></td> -<td class="tar brl pr03"><div>1</div></td> -<td class="tal pl03" colspan="3">Fifteenth Column.</td> -</tr> -<tr> -<td class="tar pr03"><div>15</div></td> -<td class="tar brlm pr03"><div>1</div></td> -<td class="tar brl pr03"><div>14</div></td> -<td class="tar brl pr03"><div>91</div></td> -<td class="tar brl pr03"><div>364</div></td> -<td class="tar brl pr03"><div>1001</div></td> -<td class="tar brl pr03"><div>2002</div></td> -<td class="tar brl pr03"><div>3003</div></td> -<td class="tar brl pr03"><div>3432</div></td> -<td class="tar brl pr03"><div>3003</div></td> -<td class="tar brl pr03"><div>2002</div></td> -<td class="tar brl pr03"><div>1001</div></td> -<td class="tar brl pr03"><div>364</div></td> -<td class="tar brl pr03"><div>91</div></td> -<td class="tar brl pr03"><div>14</div></td> -<td class="tar brl pr03"><div>1</div></td> -<td class="tal pl03" colspan="2">Sixteenth Column.</td> -</tr> -<tr> -<td class="tar pr03"><div>16</div></td> -<td class="tar brlm pr03"><div>1</div></td> -<td class="tar brl pr03"><div>15</div></td> -<td class="tar brl pr03"><div>105</div></td> -<td class="tar brl pr03"><div>455</div></td> -<td class="tar brl pr03"><div>1365</div></td> -<td class="tar brl pr03"><div>3003</div></td> -<td class="tar brl pr03"><div>5005</div></td> -<td class="tar brl pr03"><div>6435</div></td> -<td class="tar brl pr03"><div>6435</div></td> -<td class="tar brl pr03"><div>5005</div></td> -<td class="tar brl pr03"><div>3003</div></td> -<td class="tar brl pr03"><div>1365</div></td> -<td class="tar brl pr03"><div>455</div></td> -<td class="tar brl pr03"><div>105</div></td> -<td class="tar brl pr03"><div>15</div></td> -<td class="tar brl pr03"><div>1</div></td> -<td class="tal pl03">Seventeenth Col.</td> -</tr> -<tr> -<td class="tar pr03"><div>17</div></td> -<td class="tar brlm pr03"><div>1</div></td> -<td class="tar brl pr03"><div>16</div></td> -<td class="tar brl pr03"><div>120</div></td> -<td class="tar brl pr03"><div>560</div></td> -<td class="tar brl prl03"><div>1820</div></td> -<td class="tar brl prl03"><div>4368</div></td> -<td class="tar brl prl03"><div>8008</div></td> -<td class="tar brl prl03"><div>11440</div></td> -<td class="tar brl prl03"><div>12870</div></td> -<td class="tar brl prl03"><div>11440</div></td> -<td class="tar brl prl03"><div>8008</div></td> -<td class="tar brl prl03"><div>4368</div></td> -<td class="tar brl prl03"><div>1820</div></td> -<td class="tar brl prl03"><div>560</div></td> -<td class="tar brl prl03"><div>120</div></td> -<td class="tar brl prl03"><div>16</div></td> -<td class="tal pl03">1</td> -</tr> -</table> - -<p><span class="pagenum" id="Page_185">185</span></p> - -<p>Examining these numbers, we find that they are connected -by an unlimited series of relations, a few of the -more simple of which may be noticed. Each vertical -column of numbers exactly corresponds with an oblique -series descending from left to right, so that the triangle is -perfectly symmetrical in its contents. The first column -contains only <i>units</i>; the second column contains the -<i>natural numbers</i>, 1, 2, 3, &c.; the third column contains -a remarkable series of numbers, 1, 3, 6, 10, 15, &c., which -have long been called <i>the triangular numbers</i>, because they -correspond with the numbers of balls which may be -arranged in a triangular form, thus—</p> - -<figure class="figcenter illowp100" id="p185" style="max-width: 23.75em;"> - <img class="w100" src="images/p185.jpg" alt=""> -</figure> - -<p>The fourth column contains the <i>pyramidal numbers</i>, so -called because they correspond to the numbers of equal -balls which can be piled in regular triangular pyramids. -Their differences are the triangular numbers. The numbers -of the fifth column have the pyramidal numbers for their -differences, but as there is no regular figure of which they -express the contents, they have been arbitrarily called the -<i>trianguli-triangular numbers</i>. The succeeding columns -have, in a similar manner, been said to contain the -<i>trianguli-pyramidal</i>, the <i>pyramidi-pyramidal</i> numbers, -and so on.<a id="FNanchor_102" href="#Footnote_102" class="fnanchor">102</a></p> - -<p>From the mode of formation of the table, it follows that -the differences of the numbers in each column will be -found in the preceding column to the left. Hence the -<i>second differences</i>, or the <i>differences of differences</i>, will be -in the second column to the left of any given column, the -third differences in the third column, and so on. Thus -we may say that unity which appears in the first column -is the <i>first difference</i> of the numbers in the second column; -the <i>second difference</i> of those in the third column; the <i>third -difference</i> of those in the fourth, and so on. The triangle -is seen to be a complete classification of all numbers -according as they have unity for any of their differences.</p> - -<p>Since each line is formed by adding the previous line<span class="pagenum" id="Page_186">186</span> -to itself, it is evident that the sum of the numbers in each -horizontal line must be double the sum of the numbers in -the line next above. Hence we know, without making -the additions, that the successive sums must be 1, 2, 4, -8, 16, 32, 64, &c., the same as the numbers of combinations -in the Logical Alphabet. Speaking generally, the sum of -the numbers in the <i>n</i>th line will be 2<sup><i>n</i>–1</sup>.</p> - -<p>Again, if the whole of the numbers down to any line be -added together, we shall obtain a number less by unity -than some power of 2; thus, the first line gives 1 or -2<sup>1</sup>–1; the first two lines give 3 or 2<sup>2</sup>–1; the first three -lines 7 or 2<sup>3</sup>–1; the first six lines give 63 or 2<sup>6</sup>–1; or, -speaking in general language, the sum of the first <i>n</i> lines -is 2<sup><i>n</i></sup>–1. It follows that the sum of the numbers in any -one line is equal to the sum of those in all the preceding -lines increased by a unit. For the sum of the <i>n</i>th line is, -as already shown, 2<sup><i>n</i>–1</sup>, and the sum of the first <i>n</i> - 1 lines -is 2<sup><i>n</i>–1</sup>–1, or less by a unit.</p> - -<p>This account of the properties of the figurate numbers -does not approach completeness; a considerable, probably -an unlimited, number of less simple and obvious relations -might be traced out. Pascal, after giving many of the -properties, exclaims<a id="FNanchor_103" href="#Footnote_103" class="fnanchor">103</a>: “Mais j’en laisse bien plus que je -n’en donne; c’est une chose étrange combien il est fertile -en propriétés! Chacun peut s’y exercer.” The arithmetical -triangle may be considered a natural classification -of numbers, exhibiting, in the most complete manner, -their evolution and relations in a certain point of view. -It is obvious that in an unlimited extension of the -triangle, each number, with the single exception of the -number <i>two</i>, has at least two places.</p> - -<p>Though the properties above explained are highly -curious, the greatest value of the triangle arises from the -fact that it contains a complete statement of the values of -the formula (p. <a href="#Page_182">182</a>), for the numbers of combinations of <i>m</i> -things out of <i>n</i>, for all possible values of <i>m</i> and <i>n</i>. Out -of seven things one may be chosen in seven ways, and -seven occurs in the eighth line of the second column. The -combinations of two things chosen out of seven are -<span class="nowrap"><span class="fraction2"><span class="fnum2">7 × 6</span><span class="bar">/</span><span class="fden2">1 × 2</span></span></span> - or 21, which is the third number in the eighth<span class="pagenum" id="Page_187">187</span> -line. The combinations of three things out of seven are -<span class="nowrap"><span class="fraction2"><span class="fnum2">7 × 6 × 5</span><span class="bar">/</span><span class="fden2">1 × 2 × 3</span></span></span> - or 35, which appears fourth in the eighth line. -In a similar manner, in the fifth, sixth, seventh, and eighth -columns of the eighth line I find it stated in how many -ways I can select combinations of 4, 5, 6, and 7 things out -of 7. Proceeding to the ninth line, I find in succession -the number of ways in which I can select 1, 2, 3, 4, 5, 6, -7, and 8 things, out of 8 things. In general language, if -I wish to know in how many ways <i>m</i> things can be -selected in combinations out of <i>n</i> things, I must look in -the <i>n</i> + 1<sup>th</sup> line, and take the <i>m</i> + 1<sup>th</sup> number, as the -answer. In how many ways, for instance, can a subcommittee -of five be chosen out of a committee of nine. -The answer is 126, and is the sixth number in the tenth -line; it will be found equal to <span class="nowrap"><span class="fraction2"><span class="fnum2">9 . 8 . 7 . 6 . 5</span><span class="bar">/</span><span class="fden2">1 . 2 . 3 . 4 . 5</span></span></span>, -which our formula (p. <a href="#Page_182">182</a>) gives.</p> - -<p>The full utility of the figurate numbers will be more -apparent when we reach the subject of probabilities, but I -may give an illustration or two in this place. In how -many ways can we arrange four pennies as regards head -and tail? The question amounts to asking in how many -ways we can select 0, 1, 2, 3, or 4 heads, out of 4 heads, -and the <i>fifth</i> line of the triangle gives us the complete -answer, thus—</p> - -<table class="ml15em"> -<tr> -<td class="tal">We can select</td> -<td class="tar"><div><div>No</div></div></td> -<td class="tal"> head and 4 tails in 1 way.</td> -</tr> -<tr> -<td class="tac"><div><div>"</div></div></td> -<td class="tar"><div><div>1</div></div></td> -<td class="tal"> head and 3 tails in 4 ways.</td> -</tr> -<tr> -<td class="tac"><div><div>"</div></div></td> -<td class="tar"><div><div>2</div></div></td> -<td class="tal"> heads and 2 tails in 6 ways.</td> -</tr> -<tr> -<td class="tac"><div><div>"</div></div></td> -<td class="tar"><div><div>3</div></div></td> -<td class="tal"> heads and 1 tail in 4 ways.</td> -</tr> -<tr> -<td class="tac"><div><div>"</div></div></td> -<td class="tar"><div><div>4</div></div></td> -<td class="tal"> heads and 0 tail in 1 way.</td> -</tr> -</table> - -<p>The total number of different cases is 16, or 2<sup>4</sup>, and -when we come to the next chapter, it will be found that -these numbers give us the respective probabilities of all -throws with four pennies.</p> - -<p>I gave in p. <a href="#Page_181">181</a> a calculation of the number of ways in -which eight planets can meet in conjunction; the reader -will find all the numbers detailed in the ninth line of the -arithmetical triangle. The sum of the whole line is 2<sup>8</sup> or -256; but we must subtract a unit for the case where no -planet appears, and 8 for the 8 cases in which only one -planet appears; so that the total number of conjunctions<span class="pagenum" id="Page_188">188</span> -is 2<sup>8</sup> – 1 – 8 or 247. If an organ has eleven stops we -find in the twelfth line the numbers of ways in which we -can draw them, 1, 2, 3, or more at a time. Thus there are -462 ways of drawing five stops at once, and as many of -drawing six stops. The total number of ways of varying -the sound is 2048, including the single case in which no -stop at all is drawn.</p> - -<p>One of the most important scientific uses of the arithmetical -triangle consists in the information which it gives -concerning the comparative frequency of divergencies -from an average. Suppose, for the sake of argument, that -all persons were naturally of the equal stature of five feet, -but enjoyed during youth seven independent chances of -growing one inch in addition. Of these seven chances, -one, two, three, or more, may happen favourably to any -individual; but, as it does not matter what the chances -are, so that the inch is gained, the question really turns -upon the number of combinations of 0, 1, 2, 3, &c., things -out of seven. Hence the eighth line of the triangle gives -us a complete answer to the question, as follows:—</p> - -<p>Out of every 128 people—</p> - -<table class="ml5em"> -<tr class="fs70"> -<td class="tac"></td> -<td class="tal"></td> -<td class="tac"></td> -<td class="tac"></td> -<td class="tac"><div>Feet </div></td> -<td class="tac"><div>Inches.</div></td> -</tr> -<tr> -<td class="tac"><div>One</div></td> -<td class="tal"> person</td> -<td class="tac"><div>would have</div></td> -<td class="tac"><div> the stature of </div></td> -<td class="tac"><div>5</div></td> -<td class="tac"><div>0</div></td> -</tr> -<tr> -<td class="tac"><div> 7</div></td> -<td class="tal">persons</td> -<td class="tac"><div>"</div></td> -<td class="tac"><div>"</div></td> -<td class="tac"><div>5</div></td> -<td class="tac"><div>1</div></td> -</tr> -<tr> -<td class="tac"><div>21</div></td> -<td class="tal">persons</td> -<td class="tac"><div>"</div></td> -<td class="tac"><div>"</div></td> -<td class="tac"><div>5</div></td> -<td class="tac"><div>2</div></td> -</tr> -<tr> -<td class="tac"><div>35</div></td> -<td class="tal">persons</td> -<td class="tac"><div>"</div></td> -<td class="tac"><div>"</div></td> -<td class="tac"><div>5</div></td> -<td class="tac"><div>3</div></td> -</tr> -<tr> -<td class="tac"><div>35</div></td> -<td class="tal">persons</td> -<td class="tac"><div>"</div></td> -<td class="tac"><div>"</div></td> -<td class="tac"><div>5</div></td> -<td class="tac"><div>4</div></td> -</tr> -<tr> -<td class="tac"><div>21</div></td> -<td class="tal">persons</td> -<td class="tac"><div>"</div></td> -<td class="tac"><div>"</div></td> -<td class="tac"><div>5</div></td> -<td class="tac"><div>5</div></td> -</tr> -<tr> -<td class="tac"><div> 7</div></td> -<td class="tal">persons</td> -<td class="tac"><div>"</div></td> -<td class="tac"><div>"</div></td> -<td class="tac"><div>5</div></td> -<td class="tac"><div>6</div></td> -</tr> -<tr> -<td class="tac"><div> 1</div></td> -<td class="tal">person</td> -<td class="tac"><div>"</div></td> -<td class="tac"><div>"</div></td> -<td class="tac"><div>5</div></td> -<td class="tac"><div>7</div></td> -</tr> -</table> - -<p>By taking a proper line of the triangle, an answer may -be had under any more natural supposition. This theory -of comparative frequency of divergence from an average, -was first adequately noticed by Quetelet, and has lately -been employed in a very interesting and bold manner -by Mr. Francis Galton,<a id="FNanchor_104" href="#Footnote_104" class="fnanchor">104</a> in his remarkable work on -“Hereditary Genius.” We shall afterwards find that the -theory of error, to which is made the ultimate appeal in -cases of quantitative investigation, is founded upon the<span class="pagenum" id="Page_189">189</span> -comparative numbers of combinations as displayed in the -triangle.</p> - - -<h3><i>Connection between the Arithmetical Triangle and the -Logical Alphabet.</i></h3> - -<p>There exists a close connection between the arithmetical -triangle described in the last section, and the series of -combinations of letters called the Logical Alphabet. The -one is to mathematical science what the other is to -logical science. In fact the figurate numbers, or those -exhibited in the triangle, are obtained by summing up the -logical combinations. Accordingly, just as the total of the -numbers in each line of the triangle is twice as great as -that for the preceding line (p. <a href="#Page_186">186</a>), so each column of the -Alphabet (p. <a href="#Page_94">94</a>) contains twice as many combinations as -the preceding one. The like correspondence also exists -between the sums of all the lines of figures down to any -particular line, and of the combinations down to any -particular column.</p> - -<p>By examining any column of the Logical Alphabet we -find that the combinations naturally group themselves -according to the figurate numbers. Take the combinations -of the letters A, B, C, D; they consist of all the ways in -which I can choose four, three, two, one, or none of the -four letters, filling up the vacant spaces with negative -terms.</p> - -<p>There is one combination, ABCD, in which all the -positive letters are present; there are four combinations in -each of which three positive letters are present; six in -which two are present; four in which only one is present; -and, finally, there is the single case, <i>abcd</i>, in which all -positive letters are absent. These numbers, 1, 4, 6, 4, 1, -are those of the fifth line of the arithmetical triangle, and -a like correspondence will be found to exist in each -column of the Logical Alphabet.</p> - -<p>Numerical abstraction, it has been asserted, consists in -overlooking the kind of difference, and retaining only a -consciousness of its existence (p. <a href="#Page_158">158</a>). While in logic, -then, we have to deal with each combination as a separate -kind of thing, in arithmetic we distinguish only the classes -which depend upon more or less positive terms being<span class="pagenum" id="Page_190">190</span> -present, and the numbers of these classes immediately -produce the numbers of the arithmetical triangle.</p> - -<p>It may here be pointed out that there are two modes in -which we can calculate the whole number of combinations -of certain things. Either we may take the whole number -at once as shown in the Logical Alphabet, in which case -the number will be some power of two, or else we may -calculate successively, by aid of permutations, the number -of combinations of none, one, two, three things, and so -on. Hence we arrive at a necessary identity between two -series of numbers. In the case of four things we shall -have</p> - -<div class="ml5em"> -2 = 1 + <span class="nowrap"><span class="fraction2"><span class="fnum2">4</span><span class="bar">/</span><span class="fden2">1</span></span></span> + -<span class="nowrap"><span class="fraction2"><span class="fnum2">4 . 3</span><span class="bar">/</span><span class="fden2">1 . 2</span></span></span> + <span class="nowrap"><span class="fraction2"><span class="fnum2">4 . 3 . 2</span><span class="bar">/</span><span class="fden2">1 . 2 . 3</span></span></span> + <span class="nowrap"><span class="fraction2"><span class="fnum2">4 . 3 . 2 . 1</span><span class="bar">/</span><span class="fden2">1 . 2 . 3 . 4</span></span></span>. -</div> - -<p class="ti0">In a general form of expression we shall have</p> - -<div class="ml5em mt05em"> -2 = 1 + <span class="nowrap"><span class="fraction2"><span class="fnum2"><i>n</i></span><span class="bar">/</span><span class="fden2">1</span></span></span> + <span class="nowrap"><span class="fraction2"><span class="fnum2"><i>n</i> . (<i>n</i> - 1))</span><span class="bar">/</span><span class="fden2">1 . 2</span></span></span> + -<span class="nowrap"><span class="fraction2"><span class="fnum2"><i>n</i>(<i>n</i> - 1)(<i>n</i> - 2)</span><span class="bar">/</span><span class="fden2">1 . 2 . 3</span></span></span> + &c.,<br> -</div> - -<p class="ti0">the terms being continued until they cease to have any -value. Thus we arrive at a proof of simple cases of the -Binomial Theorem, of which each column of the Logical -Alphabet is an exemplification. It may be shown that all -other mathematical expansions likewise arise out of simple -processes of combination, but the more complete consideration -of this subject must be deferred to another work.</p> - - -<h3><i>Possible Variety of Nature and Art.</i></h3> - -<p>We cannot adequately understand the difficulties -which beset us in certain branches of science, unless we -have some clear idea of the vast numbers of combinations -or permutations which may be possible under certain conditions. -Thus only can we learn how hopeless it would -be to attempt to treat nature in detail, and exhaust the -whole number of events which might arise. It is instructive -to consider, in the first place, how immensely great -are the numbers of combinations with which we deal in -many arts and amusements.</p> - -<p>In dealing a pack of cards, the number of hands, of -thirteen cards each, which can be produced is evidently -52 × 51 × 50 × ... × 40 divided by 1 × 2 × 3 ... × 13. -or 635,013,559,600. But in whist four hands are simultaneously<span class="pagenum" id="Page_191">191</span> -held, and the number of distinct deals becomes -so vast that it would require twenty-eight figures to express -it. If the whole population of the world, say one thousand -millions of persons, were to deal cards day and night, for -a hundred million of years, they would not in that time -have exhausted one hundred-thousandth part of the possible -deals. Even with the same hands of cards the play -may be almost infinitely varied, so that the complete -variety of games at whist which may exist is almost -incalculably great. It is in the highest degree improbable -that any one game of whist was ever exactly like another, -except it were intentionally so.</p> - -<p>The end of novelty in art might well be dreaded, did -we not find that nature at least has placed no attainable -limit, and that the deficiency will lie in our inventive -faculties. It would be a cheerless time indeed when all -possible varieties of melody were exhausted, but it is -readily shown that if a peal of twenty-four bells had been -rung continuously from the so-called beginning of the -world to the present day, no approach could have been -made to the completion of the possible changes. Nay, -had every single minute been prolonged to 10,000 years, -still the task would have been unaccomplished.<a id="FNanchor_105" href="#Footnote_105" class="fnanchor">105</a> As -regards ordinary melodies, the eight notes of a single -octave give more than 40,000 permutations, and two -octaves more than a million millions. If we were to take -into account the semitones, it would become apparent that -it is impossible to exhaust the variety of music. When -the late Mr. J. S. Mill, in a depressed state of mind, feared -the approaching exhaustion of musical melodies, he had -certainly not bestowed sufficient study on the subject of -permutations.</p> - -<p>Similar considerations apply to the possible number of -natural substances, though we cannot always give precise -numerical results. It was recommended by Hatchett<a id="FNanchor_106" href="#Footnote_106" class="fnanchor">106</a> -that a systematic examination of all alloys of metals -should be carried out, proceeding from the binary ones to -more complicated ternary or quaternary ones. He can -hardly have been aware of the extent of his proposed<span class="pagenum" id="Page_192">192</span> -inquiry. If we operate only upon thirty of the known -metals, the number of binary alloys would be 435, of -ternary alloys 4060, of quaternary 27,405, without paying -regard to the varying proportions of the metals, and only -regarding the kind of metal. If we varied all the ternary -alloys by quantities not less than one per cent., the -number of these alloys would be 11,445,060. An exhaustive -investigation of the subject is therefore out of -the question, and unless some laws connecting the properties -of the alloy and its components can be discovered, it -is not apparent how our knowledge of them can ever be -more than fragmentary.</p> - -<p>The possible variety of definite chemical compounds, -again, is enormously great. Chemists have already examined -many thousands of inorganic substances, and a -still greater number of organic compounds;<a id="FNanchor_107" href="#Footnote_107" class="fnanchor">107</a> they have -nevertheless made no appreciable impression on the -number which may exist. Taking the number of elements -at sixty-one, the number of compounds containing -different selections of four elements each would -be more than half a million (521,855). As the same -elements often combine in many different proportions, -and some of them, especially carbon, have the power of -forming an almost endless number of compounds, it -would hardly be possible to assign any limit to the -number of chemical compounds which may be formed. -There are branches of physical science, therefore, of which -it is unlikely that scientific men, with all their industry, -can ever obtain a knowledge in any appreciable degree -approaching to completeness.</p> - - -<h3><i>Higher Orders of Variety.</i></h3> - -<p>The consideration of the facts already given in this -chapter will not produce an adequate notion of the possible -variety of existence, unless we consider the comparative -numbers of combinations of different orders. By -a combination of a higher order, I mean a combination -of groups, which are themselves groups. The immense -numbers of compounds of carbon, hydrogen, and oxygen,<span class="pagenum" id="Page_193">193</span> -described in organic chemistry, are combinations of a -second order, for the atoms are groups of groups. The -wave of sound produced by a musical instrument may be -regarded as a combination of motions; the body of sound -proceeding from a large orchestra is therefore a complex -aggregate of sounds, each in itself a complex combination -of movements. All literature may be said to be developed -out of the difference of white paper and black ink. From -the unlimited number of marks which might be chosen we -select twenty-six conventional letters. The pronounceable -combinations of letters are probably some trillions in -number. Now, as a sentence is a selection of words, the -possible sentences must be inconceivably more numerous -than the words of which it may be composed. A book is -a combination of sentences, and a library is a combination -of books. A library, therefore, may be regarded as a combination -of the fifth order, and the powers of numerical -expression would be severely tasked in attempting to -express the number of distinct libraries which might be -constructed. The calculation, of course, would not be -possible, because the union of letters in words, of words -in sentences, and of sentences in books, is governed by -conditions so complex as to defy analysis. I wish only to -point out that the infinite variety of literature, existing or -possible, is all developed out of one fundamental difference. -Galileo remarked that all truth is contained in the -compass of the alphabet. He ought to have said that it -is all contained in the difference of ink and paper.</p> - -<p>One consequence of successive combination is that the -simplest marks will suffice to express any information. -Francis Bacon proposed for secret writing a biliteral -cipher, which resolves all letters of the alphabet into -permutations of the two letters <i>a</i> and <i>b</i>. Thus A was -<i>aaaaa</i>, B <i>aaaab</i>, X <i>babab</i>, and so on.<a id="FNanchor_108" href="#Footnote_108" class="fnanchor">108</a> In a similar way, -as Bacon clearly saw, any one difference can be made the -ground of a code of signals; we can express, as he says, -<i>omnia per omnia</i>. The Morse alphabet uses only a -succession of long and short marks, and other systems -of telegraphic language employ right and left strokes. -A single lamp obscured at various intervals, long or<span class="pagenum" id="Page_194">194</span> -short, may be made to spell out any words, and with -two lamps, distinguished by colour, position, or any -other circumstance, we could at once represent Bacon’s -biliteral alphabet. Babbage ingeniously suggested that -every lighthouse in the world should be made to spell -out its own name or number perpetually, by flashes or -obscurations of various duration and succession. A -system like that of Babbage is now being applied to -lighthouses in the United Kingdom by Sir W. Thomson -and Dr. John Hopkinson.</p> - -<p>Let us calculate the numbers of combinations of different -orders which may arise out of the presence or -absence of a single mark, say A. In these figures</p> - - - -<div class="center"> -<table class="fs95 mt05em x-ebookmaker-drop"> -<tr> -<td class="tac ball pall"><div>A</div></td> -<td class="tac ball pall"><div>A</div></td> -<td class="tac"> </td> -<td class="tac ball pall"><div>A</div></td> -<td class="tac ball"> </td> -<td class="tac"> </td> -<td class="tac ball"> </td> -<td class="tac ball pall"><div>A</div></td> -<td class="tac"> </td> -<td class="tac ball"> </td> -<td class="tac ball"> </td> -</tr> -</table> -</div> - -<div class="center"> -<table class="fs95 mt05em epubonly"> -<tr> -<td class="tac ball pall"><div>A</div></td> -<td class="tac ball pall"><div>A</div></td> -<td class="tac pall"> </td> -<td class="tac ball pall"><div>A</div></td> -<td class="tac ball pall hide">A</td> -<td class="tac pall"> </td> -<td class="tac ball pall hide">A</td> -<td class="tac ball pall"><div>A</div></td> -<td class="tac pall"> </td> -<td class="tac ball pall hide">A</td> -<td class="tac ball pall hide">A</td> -</tr> -</table> -</div> - -<p class="ti0">we have four distinct varieties. Form them into a group -of a higher order, and consider in how many ways we -may vary that group by omitting one or more of the -component parts. Now, as there are four parts, and any -one may be present or absent, the possible varieties will -be 2 × 2 × 2 × 2, or 16 in number. Form these into a new -whole, and proceed again to create variety by omitting -any one or more of the sixteen. The number of possible -changes will now be 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2, or -2<sup>16</sup>, and we can repeat the process again and again. We -are imagining the creation of objects, whose numbers are -represented by the successive orders of the powers of <i>two</i>.</p> - -<p>At the first step we have 2; at the next 2<sup>2</sup>, or 4; -at the third (2<sup>2</sup>)<sup>2</sup>, or 16, numbers of very moderate amount. -Let the reader calculate the next term, ((2<sup>2</sup>)<sup>2</sup>)<sup>2</sup>, and he will be -surprised to find it leap up to 65,536. But at the next -step he has to calculate the value of 65,536 <i>two</i>’s multiplied -together, and it is so great that we could not possibly -compute it, the mere expression of the result requiring -19,729 places of figures. But go one step more and we -pass the bounds of all reason. The sixth order of the -powers of <i>two</i> becomes so great, that we could not even -express the number of figures required in writing it down, -without using about 19,729 figures for the purpose. The -successive orders of the powers of two have then the<span class="pagenum" id="Page_195">195</span> -following values so far as we can succeed in describing -them:—</p> - -<table class="ml5em"> -<tr> -<td class="tal">First order</td> -<td class="tar"><div><div>2</div></div></td> -<td class="tal"></td> -</tr> -<tr> -<td class="tal">Second order</td> -<td class="tar"><div><div>4</div></div></td> -<td class="tal"></td> -</tr> -<tr> -<td class="tal">Third order</td> -<td class="tar"><div><div>16</div></div></td> -<td class="tal"></td> -</tr> -<tr> -<td class="tal">Fourth order</td> -<td class="tar"><div><div>65,536</div></div></td> -<td class="tal"></td> -</tr> -<tr> -<td class="tal">Fifth order, number expressed by</td> -<td class="tar"><div><div>19,729</div></div></td> -<td class="tal"> figures.</td> -</tr> -<tr> -<td class="tal">Sixth order, number expressed by<br>figures, to express the number<br>of which figures would require<br>about</td> -<td class="tar vab"><div>19,729</div></td> -<td class="tal vab"> figures.</td> -</tr> -</table> - -<p>It may give us some notion of infinity to remember -that at this sixth step, having long surpassed all bounds -of intuitive conception, we make no approach to a limit. -Nay, were we to make a hundred such steps, we should be -as far away as ever from actual infinity.</p> - -<p>It is well worth observing that our powers of expression -rapidly overcome the possible multitude of finite objects -which may exist in any assignable space. Archimedes -showed long ago, in one of the most remarkable writings -of antiquity, the <i>Liber de Arcnæ Numero</i>, that the grains of -sand in the world could be numbered, or rather, that if -numbered, the result could readily be expressed in arithmetical -notation. Let us extend his problem, and ascertain -whether we could express the number of atoms which could -exist in the visible universe. The most distant stars which -can now be seen by telescopes—those of the sixteenth -magnitude—are supposed to have a distance of about -33,900,000,000,000,000 miles. Sir W. Thomson has -shown reasons for supposing that there do not exist -more than from 3 × 10<sup>24</sup> to 10<sup>26</sup> molecules in a cubic -centimetre of a solid or liquid substance.<a id="FNanchor_109" href="#Footnote_109" class="fnanchor">109</a> Assuming -these data to be true, for the sake of argument, a simple -calculation enables us to show that the almost inconceivably -vast sphere of our stellar system if entirely filled with -solid matter, would not contain more than about 68 × 10<sup>90</sup> -atoms, that is to say, a number requiring for its expression -92 places of figures. Now, this number would be immensely -less than the fifth order of the powers of two.</p> - -<p>In the variety of logical relations, which may exist<span class="pagenum" id="Page_196">196</span> -between a certain number of logical terms, we also meet -a case of higher combinations. We have seen (p. <a href="#Page_142">142</a>) that -with only six terms the number of possible selections of -combinations is 18,446,744,073,709,551,616. Considering -that it is the most common thing in the world to use an -argument involving six objects or terms, it may excite -some surprise that the complete investigation of the -relations in which six such terms may stand to each -other, should involve an almost inconceivable number -of cases. Yet these numbers of possible logical relations -belong only to the second order of combinations.</p> -<hr class="chap x-ebookmaker-drop"> - -<div class="chapter"> -<p><span class="pagenum" id="Page_197">197</span></p> - -<h2 class="nobreak" id="CHAPTER_X">CHAPTER X.<br> - -<span class="title">THE THEORY OF PROBABILITY.</span></h2> -</div> - -<p class="ti0">The subject upon which we now enter must not be -regarded as an isolated and curious branch of speculation. -It is the necessary basis of the judgments we make in the -prosecution of science, or the decisions we come to in the -conduct of ordinary affairs. As Butler truly said, “Probability -is the very guide of life.” Had the science of -numbers been studied for no other purpose, it must have -been developed for the calculation of probabilities. All -our inferences concerning the future are merely probable, -and a due appreciation of the degree of probability depends -upon a comprehension of the principles of the subject. I -am convinced that it is impossible to expound the methods -of induction in a sound manner, without resting them upon -the theory of probability. Perfect knowledge alone can -give certainty, and in nature perfect knowledge would be -infinite knowledge, which is clearly beyond our capacities. -We have, therefore, to content ourselves with partial -knowledge—knowledge mingled with ignorance, producing -doubt.</p> - -<p>A great difficulty in this subject consists in acquiring a -precise notion of the matter treated. What is it that we -number, and measure, and calculate in the theory of probabilities? -Is it belief, or opinion, or doubt, or knowledge, -or chance, or necessity, or want of art? Does probability -exist in the things which are probable, or in the mind which -regards them as such? The etymology of the name lends -us no assistance: for, curiously enough, <i>probable</i> is ultimately -the same word as <i>provable</i>, a good instance of one word -becoming differentiated to two opposite meanings.</p> - -<p><span class="pagenum" id="Page_198">198</span></p> - -<p>Chance cannot be the subject of the theory, because -there is really no such thing as chance, regarded as producing -and governing events. The word chance signifies -<i>falling</i>, and the notion of falling is continually used as a -simile to express uncertainty, because we can seldom predict -how a die, a coin, or a leaf will fall, or when a bullet -will hit the mark. But everyone sees, after a little -reflection, that it is in our knowledge the deficiency lies, -not in the certainty of nature’s laws. There is no doubt in -lightning as to the point it shall strike; in the greatest -storm there is nothing capricious; not a grain of sand lies -upon the beach, but infinite knowledge would account for -its lying there; and the course of every falling leaf is -guided by the principles of mechanics which rule the -motions of the heavenly bodies.</p> - -<p>Chance then exists not in nature, and cannot coexist -with knowledge; it is merely an expression, as Laplace -remarked, for our ignorance of the causes in action, and -our consequent inability to predict the result, or to bring -it about infallibly. In nature the happening of an event -has been pre-determined from the first fashioning of the -universe. <i>Probability belongs wholly to the mind.</i> This is -proved by the fact that different minds may regard the -very same event at the same time with widely different -degrees of probability. A steam-vessel, for instance, is -missing and some persons believe that she has sunk in -mid-ocean; others think differently. In the event itself -there can be no such uncertainty; the steam-vessel either -has sunk or has not sunk, and no subsequent discussion of -the probable nature of the event can alter the fact. Yet -the probability of the event will really vary from day to -day, and from mind to mind, according as the slightest -information is gained regarding the vessels met at sea, the -weather prevailing there, the signs of wreck picked up, -or the previous condition of the vessel. Probability thus -belongs to our mental condition, to the light in which we -regard events, the occurrence or non-occurrence of which -is certain in themselves. Many writers accordingly have -asserted that probability is concerned with degree or -quantity of belief. De Morgan says,<a id="FNanchor_110" href="#Footnote_110" class="fnanchor">110</a> “By degree of probability<span class="pagenum" id="Page_199">199</span> -we really mean or ought to mean degree of belief.” -The late Professor Donkin expressed the meaning of -probability as “quantity of belief;” but I have never felt -satisfied with such definitions of probability. The nature -of <i>belief</i> is not more clear to my mind than the notion -which it is used to define. But an all-sufficient objection -is, that <i>the theory does not measure what the belief is, but -what it ought to be</i>. Few minds think in close accordance -with the theory, and there are many cases of evidence in -which the belief existing is habitually different from what -it ought to be. Even if the state of belief in any mind -could be measured and expressed in figures, the results -would be worthless. The value of the theory consists in -correcting and guiding our belief, and rendering our states -of mind and consequent actions harmonious with our -knowledge of exterior conditions.</p> - -<p>This objection has been clearly perceived by some of -those who still used quantity of belief as a definition of -probability. Thus De Morgan adds—“Belief is but -another name for imperfect knowledge.” Donkin has -well said that the quantity of belief is “always relative -to a particular state of knowledge or ignorance; but it -must be observed that it is absolute in the sense of not -being relative to any individual mind; since, the same -information being presupposed, all minds <i>ought</i> to distribute -their belief in the same way.”<a id="FNanchor_111" href="#Footnote_111" class="fnanchor">111</a> Boole seemed to -entertain a like view, when he described the theory as -engaged with “the equal distribution of ignorance;”<a id="FNanchor_112" href="#Footnote_112" class="fnanchor">112</a> -but we may just as well say that it is engaged with the -equal distribution of knowledge.</p> - -<p>I prefer to dispense altogether with this obscure word -belief, and to say that the theory of probability deals with -<i>quantity of knowledge</i>, an expression of which a precise -explanation and measure can presently be given. An -event is only probable when our knowledge of it is -diluted with ignorance, and exact calculation is needed -to discriminate how much we do and do not know. The -theory has been described by some writers as professing <i>to -evolve knowledge out of ignorance</i>; but as Donkin admirably -remarked, it is really “a method of avoiding the erection<span class="pagenum" id="Page_200">200</span> -of belief upon ignorance.” It defines rational expectation -by measuring the comparative amounts of knowledge and -ignorance, and teaches us to regulate our actions with -regard to future events in a way which will, in the long -run, lead to the least disappointment. It is, as Laplace -happily said, <i>good sense reduced to calculation</i>. This theory -appears to me the noblest creation of intellect, and it -passes my conception how two such men as Auguste Comte -and J. S. Mill could be found depreciating it and vainly -questioning its validity. To eulogise the theory ought to -be as needless as to eulogise reason itself.</p> - - -<h3><i>Fundamental Principles of the Theory.</i></h3> - -<p>The calculation of probabilities is really founded, as I -conceive, upon the principle of reasoning set forth in preceding -chapters. We must treat equals equally, and what -we know of one case may be affirmed of every case -resembling it in the necessary circumstances. The theory -consists in putting similar cases on a par, and distributing -equally among them whatever knowledge we possess. -Throw a penny into the air, and consider what we know -with regard to its way of falling. We know that it will -certainly fall upon a side, so that either head or tail will -be uppermost; but as to whether it will be head or tail, -our knowledge is equally divided. Whatever we know -concerning head, we know also concerning tail, so that we -have no reason for expecting one more than the other. -The least predominance of belief to either side would be -irrational; it would consist in treating unequally things -of which our knowledge is equal.</p> - -<p>The theory does not require, as some writers have -erroneously supposed, that we should first ascertain by -experiment the equal facility of the events we are considering. -So far as we can examine and measure the -causes in operation, events are removed out of the sphere -of probability. The theory comes into play where ignorance -begins, and the knowledge we possess requires to be -distributed over many cases. Nor does the theory show -that the coin will fall as often on the one side as the other. -It is almost impossible that this should happen, because -some inequality in the form of the coin, or some uniform<span class="pagenum" id="Page_201">201</span> -manner in throwing it up, is almost sure to occasion a -slight preponderance in one direction. But as we do not -previously know in which way a preponderance will exist, -we have no reason for expecting head more than tail. Our -state of knowledge will be changed should we throw up -the coin many times and register the results. Every throw -gives us some slight information as to the probable -tendency of the coin, and in subsequent calculations we -must take this into account. In other cases experience -might show that we had been entirely mistaken; we might -expect that a die would fall as often on each of the six -sides as on each other side in the long run; trial might show -that the die was a loaded one, and falls most often on a -particular face. The theory would not have misled us: it -treated correctly the information we had, which is all that -any theory can do.</p> - -<p>It may be asked, as Mill asks, Why spend so much -trouble in calculating from imperfect data, when a little -trouble would enable us to render a conclusion certain by -actual trial? Why calculate the probability of a measurement -being correct, when we can try whether it is correct? -But I shall fully point out in later parts of this work that -in measurement we never can attain perfect coincidence. -Two measurements of the same base line in a survey may -show a difference of some inches, and there may be no -means of knowing which is the better result. A third -measurement would probably agree with neither. To -select any one of the measurements, would imply that -we knew it to be the most nearly correct one, which we -do not. In this state of ignorance, the only guide is the -theory of probability, which proves that in the long run -the mean of divergent results will come most nearly to -the truth. In all other scientific operations whatsoever, -perfect knowledge is impossible, and when we have exhausted -all our instrumental means in the attainment of -truth, there is a margin of error which can only be safely -treated by the principles of probability.</p> - -<p>The method which we employ in the theory consists in -calculating the number of all the cases or events concerning -which our knowledge is equal. If we have the slightest -reason for suspecting that one event is more likely to -occur than another, we should take this knowledge into<span class="pagenum" id="Page_202">202</span> -account. This being done, we must determine the whole -number of events which are, so far as we know, equally -likely. Thus, if we have no reason for supposing that a -penny will fall more often one way than another, there are -two cases, head and tail, equally likely. But if from trial -or otherwise we know, or think we know, that of 100 -throws 55 will give tail, then the probability is measured -by the ratio of 55 to 100.</p> - -<p>The mathematical formulæ of the theory are exactly the -same as those of the theory of combinations. In this -latter theory we determine in how many ways events may -be joined together, and we now proceed to use this knowledge -in calculating the number of ways in which a certain -event may come about. It is the comparative numbers of -ways in which events can happen which measure their -comparative probabilities. If we throw three pennies -into the air, what is the probability that two of them -will fall tail uppermost? This amounts to asking in how -many possible ways can we select two tails out of three, -compared with the whole number of ways in which the -coins can be placed. Now, the fourth line of the Arithmetical -Triangle (p. <a href="#Page_184">184</a>) gives us the answer. The whole -number of ways in which we can select or leave three things -is eight, and the possible combinations of two things at a -time is three; hence the probability of two tails is the -ratio of three to eight. From the numbers in the triangle -we may similarly draw all the following probabilities:—</p> - -<div class="ml5em"> -One combination gives 0 tail. Probability <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">8</span></span></span>.<br> -Three combinations gives 1 tail. Probability <span class="nowrap"><span class="fraction"><span class="fnum">3</span><span class="bar">/</span><span class="fden">8</span></span></span>.<br> -Three combinations give 2 tails. Probability <span class="nowrap"><span class="fraction"><span class="fnum">3</span><span class="bar">/</span><span class="fden">8</span></span></span>.<br> -One combination gives 3 tails. Probability <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">8</span></span></span>. -</div> - -<p>We can apply the same considerations to the imaginary -causes of the difference of stature, the combinations of -which were shown in p. <a href="#Page_188">188</a>. There are altogether 128 -ways in which seven causes can be present or absent. -Now, twenty-one of these combinations give an addition -of two inches, so that the probability of a person under -the circumstances being five feet two inches is <span class="nowrap"><span class="fraction"><span class="fnum">21</span><span class="bar">/</span><span class="fden">128</span></span></span>. The -probability of five feet three inches is <span class="nowrap"><span class="fraction"><span class="fnum">35</span><span class="bar">/</span><span class="fden">128</span></span></span>; of five feet -one inch <span class="nowrap"><span class="fraction"><span class="fnum">7</span><span class="bar">/</span><span class="fden">128</span></span></span>; - of five feet <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">128</span></span></span>, and so on. Thus the -eighth line of the Arithmetical Triangle gives all the -probabilities arising out of the combinations of seven causes.</p> - -<p><span class="pagenum" id="Page_203">203</span></p> - - -<h3><i>Rules for the Calculation of Probabilities.</i></h3> - -<p>I will now explain as simply as possible the rules -for calculating probabilities. The principal rule is as -follows:—</p> - -<p>Calculate the number of events which may happen -independently of each other, and which, as far as is -known, are equally probable. Make this number the -denominator of a fraction, and take for the numerator -the number of such events as imply or constitute the -happening of the event, whose probability is required.</p> - -<p>Thus, if the letters of the word <i>Roma</i> be thrown down -casually in a row, what is the probability that they will -form a significant Latin word? The possible arrangements -of four letters are 4 × 3 × 2 × 1, or 24 in number -(p. <a href="#Page_178">178</a>), and if all the arrangements be examined, seven -of these will be found to have meaning, namely <i>Roma</i>, -<i>ramo</i>, <i>oram</i>, <i>mora</i>, <i>maro</i>, <i>armo</i>, and <i>amor</i>. Hence the -probability of a significant result is <span class="nowrap"><span class="fraction"><span class="fnum">7</span><span class="bar">/</span><span class="fden">24</span></span></span>.</p> - -<p>We must distinguish comparative from absolute probabilities. -In drawing a card casually from a pack, there -is no reason to expect any one card more than any other. -Now, there are four kings and four queens in a pack, so -that there are just as many ways of drawing one as the -other, and the probabilities are equal. But there are -thirteen diamonds, so that the probability of a king is to -that of a diamond as four to thirteen. Thus the probabilities -of each are proportional to their respective numbers -of ways of happening. Again, I can draw a king in four -ways, and not draw one in forty-eight, so that the probabilities -are in this proportion, or, as is commonly said, -the <i>odds</i> against drawing a king are forty-eight to four. -The odds are seven to seventeen in favour, or seventeen to -seven against the letters R,o,m,a, accidentally forming a -significant word. The odds are five to three against two -tails appearing in three throws of a penny. Conversely, -when the odds of an event are given, and the probability is -required, <i>take the odds in favour of the event for numerator, -and the sum of the odds for denominator</i>.</p> - -<p>It is obvious that an event is certain when all the combinations -of causes which can take place produce that -event. If we represent the probability of such event<span class="pagenum" id="Page_204">204</span> -according to our rule, it gives the ratio of some number to -itself, or unity. An event is certain not to happen when -no possible combination of causes gives the event, and the -ratio by the same rule becomes that of 0 to some number. -Hence it follows that in the theory of probability certainty -is expressed by 1, and impossibility by 0; but no mystical -meaning should be attached to these symbols, as they -merely express the fact that <i>all</i> or <i>no</i> possible combinations -give the event.</p> - -<p>By a <i>compound event</i>, we mean an event which may be -decomposed into two or more simpler events. Thus the -firing of a gun may be decomposed into pulling the -trigger, the fall of the hammer, the explosion of the -cap, &c. In this example the simple events are not -<i>independent</i>, because if the trigger is pulled, the other -events will under proper conditions necessarily follow, and -their probabilities are therefore the same as that of the -first event. Events are <i>independent</i> when the happening -of one does not render the other either more or less -probable than before. Thus the death of a person is -neither more nor less probable because the planet Mars -happens to be visible. When the component events are -independent, a simple rule can be given for calculating -the probability of the compound event, thus—<i>Multiply -together the fractions expressing the probabilities of the -independent component events.</i></p> - -<p>The probability of throwing tail twice with a penny is -<span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span> × <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span>, or <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">4</span></span></span>; the probability of throwing it three times -running is <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span> × <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span> × <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span>, or <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">8</span></span></span>; a result agreeing with that -obtained in an apparently different manner (p. <a href="#Page_202">202</a>). In -fact, when we multiply together the denominators, we -get the whole number of ways of happening of the compound -event, and when we multiply the numerators, we -get the number of ways favourable to the required event.</p> - -<p>Probabilities may be added to or subtracted from each -other under the important condition that the events in -question are exclusive of each other, so that not more than -one of them can happen. It might be argued that, since -the probability of throwing head at the first trial is <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span>, and -at the second trial also <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span>, the probability of throwing it -in the first two throws is <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span> + <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span>, or certainty. Not only is -this result evidently absurd, but a repetition of the process<span class="pagenum" id="Page_205">205</span> -would lead us to a probability of <span class="nowrap">1 <span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span> or of any greater -number, results which could have no meaning whatever. -The probability we wish to calculate is that of one head in -two throws, but in our addition we have included the case -in which two heads appear. The true result is <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span> + <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span> × <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span> -or <span class="nowrap"><span class="fraction"><span class="fnum">3</span><span class="bar">/</span><span class="fden">4</span></span></span>, or the probability of head at the first throw, added to -the exclusive probability that if it does not come at the -first, it will come at the second. The greatest difficulties -of the theory arise from the confusion of exclusive and -unexclusive alternatives. I may remind the reader that -the possibility of unexclusive alternatives was a point -previously discussed (p. <a href="#Page_68">68</a>), and to the reasons then given -for considering alternation as logically unexclusive, may -be added the existence of these difficulties in the theory of -probability. The erroneous result explained above really -arose from overlooking the fact that the expression “head -first throw or head second throw” might include the case -of head at both throws.</p> - - -<h3><i>The Logical Alphabet in questions of Probability.</i></h3> - -<p>When the probabilities of certain simple events are -given, and it is required to deduce the probabilities of -compound events, the Logical Alphabet may give assistance, -provided that there are no special logical conditions -so that all the combinations are possible. Thus, if there be -three events, A, B, C, of which the probabilities are, α, β, -γ, then the negatives of those events, expressing the absence -of the events, will have the probabilities 1 - α, 1 - β, 1 - γ. -We have only to insert these values for the letters of the -combinations and multiply, and we obtain the probability -of each combination. Thus the probability of ABC is -αβγ; of A<i>bc</i>, α(1 - β)(1 - γ).</p> - -<p>We can now clearly distinguish between the probabilities -of exclusive and unexclusive events. Thus, if A and B -are events which may happen together like rain and high -tide, or an earthquake and a storm, the probability of A or -B happening is not the sum of their separate probabilities. -For by the Laws of Thought we develop A ꖌ B into -AB ꖌ A<i>b</i> ꖌ <i>a</i>B, and substituting α and β, the probabilities -of A and B respectively, we obtain α . β + α . (1 - β) + -(1 - α) . β or α + β - α . β. But if events are <i>incompossible</i><span class="pagenum" id="Page_206">206</span> -or incapable of happening together, like a clear sky and -rain, or a new moon and a full moon, then the events are -not really A or B, but A not-B, or B not-A, or in symbols -A<i>b</i> ꖌ <i>a</i>B. Now if we take μ = probability of A<i>b</i> and -ν = probability of <i>a</i>B, then we may add simply, and the -probability of A<i>b</i> ꖌ <i>a</i>B is μ + ν.</p> - -<p>Let the reader carefully observe that if the combination -AB cannot exist, the probability of A<i>b</i> is not the -product of the probabilities of A and <i>b</i>. When certain -combinations are logically impossible, it is no longer -allowable to substitute the probability of each term for -the term, because the multiplication of probabilities presupposes -the independence of the events. A large part of -Boole’s Laws of Thought is devoted to an attempt to -overcome this difficulty and to produce a General Method -in Probabilities by which from certain logical conditions -and certain given probabilities it would be possible to -deduce the probability of any other combinations of -events under those conditions. Boole pursued his task -with wonderful ingenuity and power, but after spending -much study on his work, I am compelled to adopt the -conclusion that his method is fundamentally erroneous. -As pointed out by Mr. Wilbraham,<a id="FNanchor_113" href="#Footnote_113" class="fnanchor">113</a> Boole obtained his -results by an arbitrary assumption, which is only the most -probable, and not the only possible assumption. The -answer obtained is therefore not the real probability, -which is usually indeterminate, but only, as it were, the -most probable probability. Certain problems solved by -Boole are free from logical conditions and therefore may -admit of valid answers. These, as I have shown,<a id="FNanchor_114" href="#Footnote_114" class="fnanchor">114</a> may be -solved by the combinations of the Logical Alphabet, but -the rest of the problems do not admit of a determinate -answer, at least by Boole’s method.</p> - - -<h3><i>Comparison of the Theory with Experience.</i></h3> - -<p>The Laws of Probability rest upon the fundamental principles -of reasoning, and cannot be really negatived by any<span class="pagenum" id="Page_207">207</span> -possible experience. It might happen that a person -should always throw a coin head uppermost, and appear -incapable of getting tail by chance. The theory would -not be falsified, because it contemplates the possibility of -the most extreme runs of luck. Our actual experience -might be counter to all that is probable; the whole -course of events might seem to be in complete contradiction -to what we should expect, and yet a casual conjunction -of events might be the real explanation. It is -just possible that some regular coincidences, which we -attribute to fixed laws of nature, are due to the accidental -conjunction of phenomena in the cases to which our -attention is directed. All that we can learn from finite -experience is capable, according to the theory of probabilities, -of misleading us, and it is only infinite experience -that could assure us of any inductive truths.</p> - -<p>At the same time, the probability that any extreme -runs of luck will occur is so excessively slight, that it -would be absurd seriously to expect their occurrence. It -is almost impossible, for instance, that any whist player -should have played in any two games where the distribution -of the cards was exactly the same, by pure accident -(p. <a href="#Page_191">191</a>). Such a thing as a person always losing at -a game of pure chance, is wholly unknown. Coincidences -of this kind are not impossible, as I have said, but they -are so unlikely that the lifetime of any person, or indeed -the whole duration of history, does not give any appreciable -probability of their being encountered. Whenever we -make any extensive series of trials of chance results, as in -throwing a die or coin, the probability is great that the -results will agree nearly with the predictions yielded by -theory. Precise agreement must not be expected, for that, -as the theory shows, is highly improbable. Several -attempts have been made to test, in this way, the accordance -of theory and experience. Buffon caused the first -trial to be made by a young child who threw a coin many -times in succession, and he obtained 1992 tails to 2048 -heads. A pupil of De Morgan repeated the trial for his -own satisfaction, and obtained 2044 tails to 2048 heads. In -both cases the coincidence with theory is as close as could -be expected, and the details may be found in De Morgan’s -“Formal Logic,” p. 185.</p> - -<p><span class="pagenum" id="Page_208">208</span></p> - -<p>Quetelet also tested the theory in a rather more complete -manner, by placing 20 black and 20 white balls in an -urn and drawing a ball out time after time in an indifferent -manner, each ball being replaced before a new drawing was -made. He found, as might be expected, that the greater -the number of drawings made, the more nearly were the -white and black balls equal in number. At the termination -of the experiment he had registered 2066 white -and 2030 black balls, the ratio being 1·02.<a id="FNanchor_115" href="#Footnote_115" class="fnanchor">115</a></p> - -<p>I have made a series of experiments in a third manner, -which seemed to me even more interesting, and capable -of more extensive trial. Taking a handful of ten coins, -usually shillings, I threw them up time after time, and -registered the numbers of heads which appeared each -time. Now the probability of obtaining 10, 9, 8, 7, &c., -heads is proportional to the number of combinations of -10, 9, 8, 7, &c., things out of 10 things. Consequently -the results ought to approximate to the numbers in the -eleventh line of the Arithmetical Triangle. I made -altogether 2048 throws, in two sets of 1024 throws each, -and the numbers obtained are given in the following -table:—</p> - -<div class="center"> -<table class="fs70 mtb1em"> -<tr> -<td class="tar pall05 ball" colspan="4"><div>Character of Throw.</div></td> -<td class="tac pall05 ball"><div>Theoretical<br>Numbers.</div></td> -<td class="tac pall05 ball"><div>First<br>Series.</div></td> -<td class="tac pall05 ball"><div>Second<br>Series.</div></td> -<td class="tar pall05 ball"><div>Average.</div></td> -<td class="tac pall05 ball"><div>Divergence.</div></td> -</tr> -<tr> -<td class="tar pt05 bl"><div>10</div></td> -<td class="tac pt05"><div>Heads</div></td> -<td class="tar pt05"><div>0</div></td> -<td class="tac pt05"><div>Tail</div></td> -<td class="tac pt05 brl"><div> 1</div></td> -<td class="tac pt05"><div> 3</div></td> -<td class="tac pt05 brl"><div> 1</div></td> -<td class="tac pt05"> 2 </td> -<td class="tac pt05 brl"><div>+ 1 </div></td> -</tr> -<tr> -<td class="tar bl"><div>9</div></td> -<td class="tac"><div><div>"</div></div></td> -<td class="tar"><div><div>1</div></div></td> -<td class="tac"><div><div>"</div></div></td> -<td class="tac brl"><div> 10</div></td> -<td class="tac"><div><div> 12</div></div></td> -<td class="tac brl"><div> 23</div></td> -<td class="tac"><div><span class="nowrap"> 17<span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span></div></td> -<td class="tac brl"><div>+<span class="nowrap"> 7<span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span></div></td> -</tr> -<tr> -<td class="tar bl"><div>8</div></td> -<td class="tac"><div><div>"</div></div></td> -<td class="tar"><div><div>2</div></div></td> -<td class="tac"><div><div>"</div></div></td> -<td class="tac brl"><div> 45</div></td> -<td class="tac"><div><div> 57</div></div></td> -<td class="tac brl"><div> 73</div></td> -<td class="tac"><div> 65 </div></td> -<td class="tac brl"><div>+ 20 </div></td> -</tr> -<tr> -<td class="tar bl"><div>7</div></td> -<td class="tac"><div><div>"</div></div></td> -<td class="tar"><div><div>3</div></div></td> -<td class="tac"><div><div>"</div></div></td> -<td class="tac brl"><div>120</div></td> -<td class="tac"><div><div>129</div></div></td> -<td class="tac brl"><div>123</div></td> -<td class="tac"><div>126 </div></td> -<td class="tac brl"><div>+ 6 </div></td> -</tr> -<tr> -<td class="tar bl"><div>6</div></td> -<td class="tac"><div><div>"</div></div></td> -<td class="tar"><div><div>4</div></div></td> -<td class="tac"><div><div>"</div></div></td> -<td class="tac brl"><div>210</div></td> -<td class="tac"><div><div>181</div></div></td> -<td class="tac brl"><div>190</div></td> -<td class="tac"><div><span class="nowrap">185 <span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span></div></td> -<td class="tac brl"><div>– 25 </div></td> -</tr> -<tr> -<td class="tar bl"><div>5</div></td> -<td class="tac"><div><div>"</div></div></td> -<td class="tar"><div><div>5</div></div></td> -<td class="tac"><div><div>"</div></div></td> -<td class="tac brl"><div>252</div></td> -<td class="tac"><div><div>257</div></div></td> -<td class="tac brl"><div>232</div></td> -<td class="tac"><div><span class="nowrap">244 <span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span></div></td> -<td class="tac brl"><div>–<span class="nowrap"> 7<span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span></div></td> -</tr> -<tr> -<td class="tar bl"><div>4</div></td> -<td class="tac"><div><div>"</div></div></td> -<td class="tar"><div><div>6</div></div></td> -<td class="tac"><div><div>"</div></div></td> -<td class="tac brl"><div>210</div></td> -<td class="tac"><div><div>201</div></div></td> -<td class="tac brl"><div>197</div></td> -<td class="tac"><div>199 </div></td> -<td class="tac brl"><div>– 11 </div></td> -</tr> -<tr> -<td class="tar bl"><div>3</div></td> -<td class="tac"><div><div>"</div></div></td> -<td class="tar"><div><div>7</div></div></td> -<td class="tac"><div><div>"</div></div></td> -<td class="tac brl"><div>120</div></td> -<td class="tac"><div><div>111</div></div></td> -<td class="tac brl"><div>119</div></td> -<td class="tac"><div>115 </div></td> -<td class="tac brl"><div>– 5 </div></td> -</tr> -<tr> -<td class="tar bl"><div>2</div></td> -<td class="tac"><div><div>"</div></div></td> -<td class="tar"><div><div>8</div></div></td> -<td class="tac"><div><div>"</div></div></td> -<td class="tac brl"><div> 45</div></td> -<td class="tac"><div><div> 52</div></div></td> -<td class="tac brl"><div> 50</div></td> -<td class="tac"><div> 51 </div></td> -<td class="tac brl"><div>+ 6 </div></td> -</tr> -<tr> -<td class="tar bl"><div>1</div></td> -<td class="tac"><div><div>"</div></div></td> -<td class="tar"><div><div>9</div></div></td> -<td class="tac"><div><div>"</div></div></td> -<td class="tac brl"><div> 10</div></td> -<td class="tac"><div><div> 21</div></div></td> -<td class="tac brl"><div> 15</div></td> -<td class="tac"><div> 18 </div></td> -<td class="tac brl"><div>+ 8 </div></td> -</tr> -<tr> -<td class="tar pb05 bl"><div>0</div></td> -<td class="tac pb05"><div>"</div></td> -<td class="tar pb05"><div>10</div></td> -<td class="tac pb05"><div>"</div></td> -<td class="tac pb05 brl"><div> 1</div></td> -<td class="tac pb05"><div> 0</div></td> -<td class="tac pb05 brl"><div> 1</div></td> -<td class="tac pb05"> <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span></td> -<td class="tac pb05 brl"><div>– <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span></div></td> -</tr> -<tr> -<td class="tac pall05 ball" colspan="4"><div>Totals ... ...</div></td> -<td class="tac pall05 ball"><div>1024</div></td> -<td class="tac pall05 ball"><div>1024</div></td> -<td class="tac pall05 ball"><div>1024</div></td> -<td class="tac pall05 ball"><div>1024</div></td> -<td class="tac pall05 btrb"><div>– 1 </div></td> -</tr> -</table> -</div> - -<p>The whole number of single throws of coins amounted -to 10 × 2048, or 20,480 in all, one half of which or -10,240 should theoretically give head. The total number<span class="pagenum" id="Page_209">209</span> -of heads obtained was actually 10,353, or 5222 in the -first series, and 5131 in the second. The coincidence -with theory is pretty close, but considering the large -number of throws there is some reason to suspect a -tendency in favour of heads.</p> - -<p>The special interest of this trial consists in the exhibition, -in a practical form, of the results of Bernoulli’s -theorem, and the law of error or divergence from the -mean to be afterwards more fully considered. It illustrates -the connection between combinations and permutations, -which is exhibited in the Arithmetical Triangle, -and which underlies many important theorems of science.</p> - - -<h3><i>Probable Deductive Arguments</i>.</h3> - -<p>With the aid of the theory of probabilities, we may -extend the sphere of deductive argument. Hitherto we -have treated propositions as certain, and on the hypothesis -of certainty have deduced conclusions equally -certain. But the information on which we reason in -ordinary life is seldom or never certain, and almost all -reasoning is really a question of probability. We ought -therefore to be fully aware of the mode and degree in -which deductive reasoning is affected by the theory of -probability, and many persons may be surprised at the -results which must be admitted. Some controversial -writers appear to consider, as De Morgan remarked,<a id="FNanchor_116" href="#Footnote_116" class="fnanchor">116</a> that -an inference from several equally probable premises is -itself as probable as any of them, but the true result is -very different. If an argument involves many propositions, -and each of them is uncertain, the conclusion will -be of very little force.</p> - -<p>The validity of a conclusion may be regarded as a compound -event, depending upon the premises happening -to be true; thus, to obtain the probability of the conclusion, -we must multiply together the fractions expressing the -probabilities of the premises. If the probability is <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span> that -A is B, and also <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span> that B is C, the conclusion that A is C, -on the ground of these premises, is <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span> × <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span> or <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">4</span></span></span>. Similarly if -there be any number of premises requisite to the establishment<span class="pagenum" id="Page_210">210</span> -of a conclusion and their probabilities be <i>p</i>, <i>q</i>, <i>r</i>, &c., -the probability of the conclusion on the ground of these -premises is <i>p</i> × <i>q</i> × <i>r</i> × ... This product has but a small -value, unless each of the quantities <i>p</i>, <i>q</i>, &c., be nearly -unity.</p> - -<p>But it is particularly to be noticed that the probability -thus calculated is not the whole probability of the conclusion, -but that only which it derives from the premises -in question. Whately’s<a id="FNanchor_117" href="#Footnote_117" class="fnanchor">117</a> remarks on this subject might -mislead the reader into supposing that the calculation is -completed by multiplying together the probabilities of the -premises. But it has been fully explained by De Morgan<a id="FNanchor_118" href="#Footnote_118" class="fnanchor">118</a> -that we must take into account the antecedent probability -of the conclusion; A may be C for other reasons besides -its being B, and as he remarks, “It is difficult, if not -impossible, to produce a chain of argument of which the -reasoner can rest the result on those arguments only.” -The failure of one argument does not, except under special -circumstances, disprove the truth of the conclusion it is -intended to uphold, otherwise there are few truths which -could survive the ill-considered arguments adduced in their -favour. As a rope does not necessarily break because one -or two strands in it fail, so a conclusion may depend upon -an endless number of considerations besides those immediately -in view. Even when we have no other information -we must not consider a statement as devoid of all -probability. The true expression of complete doubt is a -ratio of equality between the chances in favour of and -against it, and this ratio is expressed in the probability <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span>.</p> - -<p>Now if A and C are wholly unknown things, we have -no reason to believe that A is C rather than A is not C. -The antecedent probability is then <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span>. If we also have the -probabilities that A is B, <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span> - and that B is C, <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span> we have no -right to suppose that the probability of A being C is reduced -by the argument in its favour. If the conclusion is -true on its own grounds, the failure of the argument does -not affect it; thus its total probability is its antecedent -probability, added to the probability that this failing, the -new argument in question establishes it. There is a probability<span class="pagenum" id="Page_211">211</span> -<span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span> that we shall not require the special argument; -a probability <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span> - that we shall, and a probability <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">4</span></span></span> that the -argument does in that case establish it. Thus the complete -result is <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span> + <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span> × <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">4</span></span></span>, or <span class="nowrap"><span class="fraction"><span class="fnum">5</span><span class="bar">/</span><span class="fden">8</span></span></span>. In general language, if <i>a</i> -be the probability founded on a particular argument, and -<i>c</i> the antecedent probability of the event, the general result -is 1 - (1 - <i>a</i>)(1 - <i>c</i>), or <i>a</i> + <i>c</i> - <i>ac</i>.</p> - -<p>We may put it still more generally in this way:—Let -<i>a</i>, <i>b</i>, <i>c</i>, &c. be the probabilities of a conclusion grounded -on various arguments. It is only when all the arguments -fail that our conclusion proves finally untrue; the probabilities -of each failing are respectively, 1 - <i>a</i>, 1 - <i>b</i>, 1 - <i>c</i>, -&c.; the probability that they will all fail is (1 - <i>a</i>)(1 - <i>b</i>)(1 - <i>c</i>) -...; therefore the probability that the conclusion -will not fail is 1 - (1 - <i>a</i>)(1 - <i>b</i>)(1 - <i>c</i>) ... &c. It follows -that every argument in favour of a conclusion, however -flimsy and slight, adds probability to it. When it is -unknown whether an overdue vessel has foundered or not, -every slight indication of a lost vessel will add some probability -to the belief of its loss, and the disproof of any -particular evidence will not disprove the event.</p> - -<p>We must apply these principles of evidence with great -care, and observe that in a great proportion of cases the -adducing of a weak argument does tend to the disproof -of its conclusion. The assertion may have in itself great -inherent improbability as being opposed to other evidence -or to the supposed law of nature, and every reasoner may -be assumed to be dealing plainly, and putting forward the -whole force of evidence which he possesses in its favour. -If he brings but one argument, and its probability <i>a</i> is -small, then in the formula 1 - (1 - <i>a</i>)(1 - <i>c</i>) both <i>a</i> and <i>c</i> -are small, and the whole expression has but little value. -The whole effect of an argument thus turns upon the -question whether other arguments remain, so that we can -introduce other factors (1 - <i>b</i>), (1 - <i>d</i>), &c., into the above -expression. In a court of justice, in a publication having -an express purpose, and in many other cases, it is doubtless -right to assume that the whole evidence considered to -have any value as regards the conclusion asserted, is put -forward.</p> - -<p>To assign the antecedent probability of any proposition, -may be a matter of difficulty or impossibility, and one<span class="pagenum" id="Page_212">212</span> -with which logic and the theory of probability have little -concern. From the general body of science in our possession, -we must in each case make the best judgment we -can. But in the absence of all knowledge the probability -should be considered = <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span>, for if we make it less than this -we incline to believe it false rather than true. Thus, before -we possessed any means of estimating the magnitudes of -the fixed stars, the statement that Sirius was greater than -the sun had a probability of exactly <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span>; it was as likely that -it would be greater as that it would be smaller; and so -of any other star. This was the assumption which Michell -made in his admirable speculations.<a id="FNanchor_119" href="#Footnote_119" class="fnanchor">119</a> It might seem, -indeed, that as every proposition expresses an agreement, -and the agreements or resemblances between phenomena -are infinitely fewer than the differences (p. <a href="#Page_44">44</a>), every proposition -should in the absence of other information be -infinitely improbable. But in our logical system every -term may be indifferently positive or negative, so that we -express under the same form as many differences as agreements. -It is impossible therefore that we should have -any reason to disbelieve rather than to believe a statement -about things of which we know nothing. We can hardly -indeed invent a proposition concerning the truth of which -we are absolutely ignorant, except when we are entirely -ignorant of the terms used. If I ask the reader to assign -the odds that a “Platythliptic Coefficient is positive” he -will hardly see his way to doing so, unless he regard them -as even.</p> - -<p>The assumption that complete doubt is properly expressed -by <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span> - has been called in question by Bishop Terrot,<a id="FNanchor_120" href="#Footnote_120" class="fnanchor">120</a> -who proposes instead the indefinite symbol <span class="nowrap"><span class="fraction"><span class="fnum">0</span><span class="bar">/</span><span class="fden">0</span></span></span>; and he -considers that “the <i>à priori</i> probability derived from -absolute ignorance has no effect upon the force of a -subsequently admitted probability.” But if we grant that -the probability may have any value between 0 and 1, and -that every separate value is equally likely, then <i>n</i> and -1 - <i>n</i> are equally likely, and the average is always <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span>. - Or -we may take <i>p</i> . <i>dp</i> to express the probability that our<span class="pagenum" id="Page_213">213</span> -estimate concerning any proposition should lie between -<i>p</i> and <i>p</i> + <i>dp</i>. The complete probability of the proposition -is then the integral taken between the limits 1 and 0, or -again <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span>.</p> - - -<h3><i>Difficulties of the Theory.</i></h3> - -<p>The theory of probability, though undoubtedly true, -requires very careful application. Not only is it a branch -of mathematics in which oversights are frequently committed, -but it is a matter of great difficulty in many cases, -to be sure that the formula correctly represents the data -of the problem. These difficulties often arise from the -logical complexity of the conditions, which might be, -perhaps, to some extent cleared up by constantly bearing -in mind the system of combinations as developed in the -Indirect Logical Method. In the study of probabilities, -mathematicians had unconsciously employed logical processes -far in advance of those in possession of logicians, -and the Indirect Method is but the full statement of these -processes.</p> - -<p>It is very curious how often the most acute and powerful -intellects have gone astray in the calculation of -probabilities. Seldom was Pascal mistaken, yet he inaugurated -the science with a mistaken solution.<a id="FNanchor_121" href="#Footnote_121" class="fnanchor">121</a> Leibnitz -fell into the extraordinary blunder of thinking that the -number twelve was as probable a result in the throwing -of two dice as the number eleven.<a id="FNanchor_122" href="#Footnote_122" class="fnanchor">122</a> In not a few cases the -false solution first obtained seems more plausible to the -present day than the correct one since demonstrated. -James Bernoulli candidly records two false solutions of a -problem which he at first thought self-evident; and he -adds a warning against the risk of error, especially when -we attempt to reason on this subject without a rigid -adherence to methodical rules and symbols. Montmort -was not free from similar mistakes. D’Alembert constantly -fell into blunders, and could not perceive, for -instance, that the probabilities would be the same when<span class="pagenum" id="Page_214">214</span> -coins are thrown successively as when thrown simultaneously. -Some men of great reputation, such as -Ancillon, Moses Mendelssohn, Garve, Auguste Comte,<a id="FNanchor_123" href="#Footnote_123" class="fnanchor">123</a> -Poinsot, and J. S. Mill,<a id="FNanchor_124" href="#Footnote_124" class="fnanchor">124</a> have so far misapprehended the -theory, as to question its value or even to dispute its -validity. The erroneous statements about the theory given -in the earlier editions of Mill’s <i>System of Logic</i> were partially -withdrawn in the later editions.</p> - -<p>Many persons have a fallacious tendency to believe that -when a chance event has happened several times together -in an unusual conjunction, it is less likely to happen -again. D’Alembert seriously held that if head was thrown -three times running with a coin, tail would more probably -appear at the next trial.<a id="FNanchor_125" href="#Footnote_125" class="fnanchor">125</a> Bequelin adopted the same -opinion, and yet there is no reason for it whatever. If -the event be really casual, what has gone before cannot in -the slightest degree influence it. As a matter of fact, the -more often a casual event takes place the more likely it is -to happen again; because there is some slight empirical -evidence of a tendency. The source of the fallacy is to be -found entirely in the feelings of surprise with which we -witness an event happening by chance, in a manner which -seems to proceed from design.</p> - -<p>Misapprehension may also arise from overlooking the -difference between permutations and combinations. To -throw ten heads in succession with a coin is no more -unlikely than to throw any other particular succession -of heads and tails, but it is much less likely than five -heads and five tails without regard to their order, because -there are no less than 252 different particular -throws which will give this result, when we abstract -the difference of order.</p> - -<p>Difficulties arise in the application of the theory from -our habitual disregard of slight probabilities. We are -obliged practically to accept truths as certain which are -nearly so, because it ceases to be worth while to calculate -the difference. No punishment could be inflicted if -absolutely certain evidence of guilt were required, and as<span class="pagenum" id="Page_215">215</span> -Locke remarks, “He that will not stir till he infallibly -knows the business he goes about will succeed, will -have but little else to do but to sit still and perish.”<a id="FNanchor_126" href="#Footnote_126" class="fnanchor">126</a> -There is not a moment of our lives when we do not lie -under a slight danger of death, or some most terrible fate. -There is not a single action of eating, drinking, sitting -down, or standing up, which has not proved fatal to some -person. Several philosophers have tried to assign the -limit of the probabilities which we regard as zero; Buffon -named <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">10,000</span></span></span>, because it is the probability, practically -disregarded, that a man of 56 years of age will die the next -day. Pascal remarked that a man would be esteemed a -fool for hesitating to accept death when three dice gave -sixes twenty times running, if his reward in case of a -different result was to be a crown; but as the chance of -death in question is only 1 ÷ 6<sup>60</sup>, or unity divided by -a number of 47 places of figures, we may be said to incur -greater risks every day for less motives. There is far -greater risk of death, for instance, in a game of cricket or -a visit to the rink.</p> - -<p>Nothing is more requisite than to distinguish carefully -between the truth of a theory and the truthful application -of the theory to actual circumstances. As a general rule, -events in nature and art will present a complexity of -relations exceeding our powers of treatment. The intricate -action of the mind often intervenes and renders complete -analysis hopeless. If, for instance, the probability that -a marksman shall hit the target in a single shot be 1 in -10, we might seem to have no difficulty in calculating -the probability of any succession of hits; thus the probability -of three successive hits would be one in a thousand. -But, in reality, the confidence and experience derived from -the first successful shot would render a second success -more probable. The events are not really independent, -and there would generally be a far greater preponderance -of runs of apparent luck, than a simple calculation of -probabilities could account for. In some persons, however, -a remarkable series of successes will produce a degree of -excitement rendering continued success almost impossible.</p> - -<p>Attempts to apply the theory of probability to the<span class="pagenum" id="Page_216">216</span> -results of judicial proceedings have proved of little value, -simply because the conditions are far too intricate. As -Laplace said, “Tant de passions, d’intérêts divers et de -circonstances compliquent les questions relatives à ces -objets, qu’elles sont presque toujours insolubles.” Men -acting on a jury, or giving evidence before a court, are -subject to so many complex influences that no mathematical -formulas can be framed to express the real conditions. -Jurymen or even judges on the bench cannot be regarded -as acting independently, with a definite probability in -favour of each delivering a correct judgment. Each man -of the jury is more or less influenced by the opinion of the -others, and there are subtle effects of character and manner -and strength of mind which defy analysis. Even in -physical science we can in comparatively few cases apply -the theory in a definite manner, because the data required -are too complicated and difficult to obtain. But such failures -in no way diminish the truth and beauty of the theory -itself; in reality there is no branch of science in which our -symbols can cope with the complexity of Nature. As -Donkin said,—</p> - -<p>“I do not see on what ground it can be doubted that -every definite state of belief concerning a proposed hypothesis, -is in itself capable of being represented by a numerical -expression, however difficult or impracticable it may -be to ascertain its actual value. It would be very difficult -to estimate in numbers the <i>vis viva</i> of all the particles of -a human body at any instant; but no one doubts that it is -capable of numerical expression.”<a id="FNanchor_127" href="#Footnote_127" class="fnanchor">127</a></p> - -<p>The difficulty, in short, is merely relative to our knowledge -and skill, and is not absolute or inherent in the -subject. We must distinguish between what is theoretically -conceivable and what is practicable with our -present mental resources. Provided that our aspirations -are pointed in a right direction, we must not allow them -to be damped by the consideration that they pass beyond -what can now be turned to immediate use. In spite of -its immense difficulties of application, and the aspersions -which have been mistakenly cast upon it, the theory of -probabilities, I repeat, is the noblest, as it will in course<span class="pagenum" id="Page_217">217</span> -of time prove, perhaps the most fruitful branch of mathematical -science. It is the very guide of life, and hardly -can we take a step or make a decision of any kind without -correctly or incorrectly making an estimation of probabilities. -In the next chapter we proceed to consider how -the whole cogency of inductive reasoning rests upon probabilities. -The truth or untruth of a natural law, when -carefully investigated, resolves itself into a high or low -degree of probability, and this is the case whether or not -we are capable of producing precise numerical data.</p> -<hr class="chap x-ebookmaker-drop"> - -<div class="chapter"> -<p><span class="pagenum" id="Page_218">218</span></p> - -<h2 class="nobreak" id="CHAPTER_XI">CHAPTER XI.<br> - -<span class="title">PHILOSOPHY OF INDUCTIVE INFERENCE.</span></h2> -</div> - -<p class="ti0">We have inquired into the nature of perfect induction, -whereby we pass backwards from certain observed combinations -of events, to the logical conditions governing -such combinations. We have also investigated the grounds -of that theory of probability, which must be our guide when -we leave certainty behind, and dilute knowledge with -ignorance. There is now before us the difficult task of -endeavouring to decide how, by the aid of that theory, we -can ascend from the facts to the laws of nature; and may -then with more or less success anticipate the future -course of events. All our knowledge of natural objects -must be ultimately derived from observation, and the -difficult question arises—How can we ever know anything -which we have not directly observed through one of our -senses, the apertures of the mind? The utility of reasoning -is to assure ourselves that, at a determinate time and -place, or under specified conditions, a certain phenomenon -will be observed. When we can use our senses and perceive -that the phenomenon does occur, reasoning is superfluous. -If the senses cannot be used, because the event -is in the future, or out of reach, how can reasoning take -their place? Apparently, at least, we must infer the unknown -from the known, and the mind must itself create -an addition to the sum of knowledge. But I hold that it -is quite impossible to make any real additions to the contents -of our knowledge, except through new impressions -upon the senses, or upon some seat of feeling. I shall<span class="pagenum" id="Page_219">219</span> -attempt to show that inference, whether inductive or -deductive, is never more than an unfolding of the contents -of our experience, and that it always proceeds upon the -assumption that the future and the unperceived will be -governed by the same conditions as the past and the -perceived, an assumption which will often prove to be -mistaken.</p> - -<p>In inductive as in deductive reasoning the conclusion -never passes beyond the premises. Reasoning adds no -more to the implicit contents of our knowledge, than the -arrangement of the specimens in a museum adds to the -number of those specimens. Arrangement adds to our -knowledge in a certain sense: it allows us to perceive the -similarities and peculiarities of the specimens, and on the -assumption that the museum is an adequate representation -of nature, it enables us to judge of the prevailing forms of -natural objects. Bacon’s first aphorism holds perfectly -true, that man knows nothing but what he has observed, -provided that we include his whole sources of experience, -and the whole implicit contents of his knowledge. Inference -but unfolds the hidden meaning of our observations, -and <i>the theory of probability shows how far we go beyond -our data in assuming that new specimens will resemble the -old ones</i>, or that the future may be regarded as proceeding -uniformly with the past.</p> - - -<h3><i>Various Classes of Inductive Truths.</i></h3> - -<p>It will be desirable, in the first place, to distinguish -between the several kinds of truths which we endeavour -to establish by induction. Although there is a certain -common and universal element in all our processes of -reasoning, yet diversity arises in their application. -Similarity of condition between the events from which -we argue, and those to which we argue, must always be -the ground of inference; but this similarity may have -regard either to time or place, or the simple logical -combination of events, or to any conceivable junction of -circumstances involving quality, time, and place. Having -met with many pieces of substance possessing ductility -and a bright yellow colour, and having discovered, by -perfect induction, that they all possess a high specific<span class="pagenum" id="Page_220">220</span> -gravity, and a freedom from the corrosive action of acids, -we are led to expect that every piece of substance, possessing -like ductility and a similar yellow colour, will have an -equally high specific gravity, and a like freedom from -corrosion by acids. This is a case of the coexistence of -qualities; for the character of the specimens examined -alters not with time nor place.</p> - -<p>In a second class of cases, time will enter as a principal -ground of similarity. When we hear a clock -pendulum beat time after time, at equal intervals, and -with a uniform sound, we confidently expect that the stroke -will continue to be repeated uniformly. A comet having -appeared several times at nearly equal intervals, we infer -that it will probably appear again at the end of another -like interval. A man who has returned home evening -after evening for many years, and found his house standing, -may, on like grounds, expect that it will be standing -the next evening, and on many succeeding evenings. Even -the continuous existence of an object in an unaltered state, -or the finding again of that which we have hidden, is but -a matter of inference depending on experience.</p> - -<p>A still larger and more complex class of cases involves -the relations of space, in addition to those of time and -quality. Having observed that every triangle drawn upon -the diameter of a circle, with its apex upon the circumference, -apparently contains a right angle, we may -ascertain that all triangles in similar circumstances will -contain right angles. This is a case of pure space reasoning, -apart from circumstances of time or quality, and it -seems to be governed by different principles of reasoning. -I shall endeavour to show, however, that geometrical -reasoning differs but in degree from that which applies -to other natural relations.</p> - - -<h3><i>The Relation of Cause and Effect.</i></h3> - -<p>In a very large part of the scientific investigations -which must be considered, we deal with events which -follow from previous events, or with existences which -succeed existences. Science, indeed, might arise even were -material nature a fixed and changeless whole. Endow -mind with the power to travel about, and compare part<span class="pagenum" id="Page_221">221</span> -with part, and it could certainly draw inferences concerning -the similarity of forms, the coexistence of qualities, -or the preponderance of a particular kind of matter in -a changeless world. A solid universe, in at least approximate -equilibrium, is not inconceivable, and then the relation -of cause and effect would evidently be no more than -the relation of before and after. As nature exists, however, -it is a progressive existence, ever moving and -changing as time, the great independent variable, proceeds. -Hence it arises that we must continually compare -what is happening now with what happened a moment -before, and a moment before that moment, and so on, -until we reach indefinite periods of past time. A comet -is seen moving in the sky, or its constituent particles -illumine the heavens with their tails of fire. We cannot -explain the present movements of such a body without -supposing its prior existence, with a definite amount -of energy and a definite direction of motion; nor can we -validly suppose that our task is concluded when we find -that it came wandering to our solar system through the -unmeasured vastness of surrounding space. Every event -must have a cause, and that cause again a cause, until -we are lost in the obscurity of the past, and are driven to -the belief in one First Cause, by whom the course of -nature was determined.</p> - - -<h3><i>Fallacious Use of the Term Cause.</i></h3> - -<p>The words Cause and Causation have given rise to infinite -trouble and obscurity, and have in no slight degree retarded -the progress of science. From the time of Aristotle, the -work of philosophy has been described as the discovery of -the causes of things, and Francis Bacon adopted the notion -when he said “<i>vere scire esse per causas scire</i>.” Even now -it is not uncommonly supposed that the knowledge of -causes is something different from other knowledge, and -consists, as it were, in getting possession of the keys of -nature. A single word may thus act as a spell, and throw -the clearest intellect into confusion, as I have often thought -that Locke was thrown into confusion when endeavouring -to find a meaning for the word <i>power</i>.<a id="FNanchor_128" href="#Footnote_128" class="fnanchor">128</a> In Mill’s <i>System of<span class="pagenum" id="Page_222">222</span> -Logic</i> the term <i>cause</i> seems to have re-asserted its old -noxious power. Not only does Mill treat the Laws of -Causation as almost coextensive with science, but he so -uses the expression as to imply that when once we pass -within the circle of causation we deal with certainties.</p> - -<p>The philosophical danger which attaches to the use of -this word may be thus described. A cause is defined as -the necessary or invariable antecedent of an event, so -that when the cause exists the effect will also exist or -soon follow. If then we know the cause of an event, we -know what will certainly happen; and as it is implied -that science, by a proper experimental method, may attain -to a knowledge of causes, it follows that experience may -give us a certain knowledge of future events. But nothing -is more unquestionable than that finite experience can -never give us certain knowledge of the future, so that -either a cause is not an invariable antecedent, or else we -can never gain certain knowledge of causes. The first -horn of this dilemma is hardly to be accepted. Doubtless -there is in nature some invariably acting mechanism, such -that from certain fixed conditions an invariable result -always emerges. But we, with our finite minds and -short experience, can never penetrate the mystery of -those existences which embody the Will of the Creator, -and evolve it throughout time. We are in the position -of spectators who witness the productions of a complicated -machine, but are not allowed to examine its intimate -structure. We learn what does happen and what -does appear, but if we ask for the reason, the answer -would involve an infinite depth of mystery. The simplest -bit of matter, or the most trivial incident, such as the -stroke of two billiard balls, offers infinitely more to learn -than ever the human intellect can fathom. The word -cause covers just as much untold meaning as any of the -words <i>substance</i>, <i>matter</i>, <i>thought</i>, <i>existence</i>.</p> - - -<h3><i>Confusion of Two Questions.</i></h3> - -<p>The subject is much complicated, too, by the confusion -of two distinct questions. An event having happened, we -may ask—</p> - -<p><span class="pagenum" id="Page_223">223</span></p> - -<div class="ml5em"> -(1) Is there any cause for the event?<br> -(2) Of what kind is that cause? -</div> - -<p>No one would assert that the mind possesses any -faculty capable of inferring, prior to experience, that the -occurrence of a sudden noise with flame and smoke indicates -the combustion of a black powder, formed by the -mixture of black, white, and yellow powders. The greatest -upholder of <i>à priori</i> doctrines will allow that the particular -aspect, shape, size, colour, texture, and other -qualities of a cause must be gathered through the senses.</p> - -<p>The question whether there is any cause at all for an -event, is of a totally different kind. If an explosion could -happen without any prior existing conditions, it must be -a new creation—a distinct addition to the universe. It -may be plausibly held that we can imagine neither the -creation nor annihilation of anything. As regards matter, -this has long been held true; as regards force, it is now -almost universally assumed as an axiom that energy can -neither come into nor go out of existence without distinct -acts of Creative Will. That there exists any instinctive -belief to this effect, indeed, seems doubtful. We find -Lucretius, a philosopher of the utmost intellectual power -and cultivation, gravely assuming that his raining atoms -could turn aside from their straight paths in a self-determining -manner, and by this spontaneous origination of -energy determine the form of the universe.<a id="FNanchor_129" href="#Footnote_129" class="fnanchor">129</a> Sir George -Airy, too, seriously discussed the mathematical conditions -under which a perpetual motion, that is, a perpetual -source of self-created energy, might exist.<a id="FNanchor_130" href="#Footnote_130" class="fnanchor">130</a> The larger -part of the philosophic world has long held that in mental -acts there is free will—in short, self-causation. It is in -vain to attempt to reconcile this doctrine with that of an -intuitive belief in causation, as Sir W. Hamilton candidly -allowed.</p> - -<p>It is obvious, moreover, that to assert the existence -of a cause for every event cannot do more than remove -into the indefinite past the inconceivable fact and mystery -of creation. At any given moment matter and energy<span class="pagenum" id="Page_224">224</span> -were equal to what they are at present, or they were -not; if equal, we may make the same inquiry concerning -any other moment, however long prior, and we are thus -obliged to accept one horn of the dilemma—existence -from infinity, or creation at some moment. This is but -one of the many cases in which we are compelled to believe -in one or other of two alternatives, both inconceivable. -My present purpose, however, is to point out that we must -not confuse this supremely difficult question with that -into which inductive science inquires on the foundation of -facts. By induction we gain no certain knowledge; but -by observation, and the inverse use of deductive reasoning, -we estimate the probability that an event which has -occurred was preceded by conditions of specified character, -or that such conditions will be followed by the event.</p> - - -<h3><i>Definition of the Term Cause.</i></h3> - -<p>Clear definitions of the word cause have been given by -several philosophers. Hobbes has said, “A cause is the -sum or aggregate of all such accidents, both in the agents -and the patients, as concur in the producing of the effect -propounded; all which existing together, it cannot be -understood but that the effect existeth with them; or -that it can possibly exist if any of them be absent.” -Brown, in his <i>Essay on Causation</i>, gave a nearly corresponding -statement. “A cause,” he says,<a id="FNanchor_131" href="#Footnote_131" class="fnanchor">131</a> “may be -defined to be the object or event which immediately -precedes any change, and which existing again in similar -circumstances will be always immediately followed by a -similar change.” Of the kindred word <i>power</i>, he likewise -says:<a id="FNanchor_132" href="#Footnote_132" class="fnanchor">132</a> “Power is nothing more than that invariableness -of antecedence which is implied in the belief of -causation.”</p> - -<p>These definitions may be accepted with the qualification -that our knowledge of causes in such a sense can be -probable only. The work of science consists in ascertaining -the combinations in which phenomena present themselves.<span class="pagenum" id="Page_225">225</span> -Concerning every event we shall have to determine its -probable conditions, or the group of antecedents from which -it probably follows. An antecedent is anything which -exists prior to an event; a consequent is anything which -exists subsequently to an antecedent. It will not usually -happen that there is any probable connection between an -antecedent and consequent. Thus nitrogen is an antecedent -to the lighting of a common fire; but it is so far from -being a cause of the lighting, that it renders the combustion -less active. Daylight is an antecedent to all fires lighted -during the day, but it probably has no appreciable effect -upon their burning. But in the case of any given event it -is usually possible to discover a certain number of antecedents -which seem to be always present, and with more -or less probability we conclude that when they exist the -event will follow.</p> - -<p>Let it be observed that the utmost latitude is at present -enjoyed in the use of the term <i>cause</i>. Not only may a -cause be an existent thing endowed with powers, as -oxygen is the cause of combustion, gunpowder the cause -of explosion, but the very absence or removal of a thing -may also be a cause. It is quite correct to speak of the -dryness of the Egyptian atmosphere, or the absence of -moisture, as being the cause of the preservation of -mummies, and other remains of antiquity. The cause of -a mountain elevation, Ingleborough for instance, is the -excavation of the surrounding valleys by denudation. It -is not so usual to speak of the existence of a thing at one -moment as the cause of its existence at the next, but to -me it seems the commonest case of causation which can -occur. The cause of motion of a billiard ball may be the -stroke of another ball; and recent philosophy leads us to -look upon all motions and changes, as but so many manifestations -of prior existing energy. In all probability -there is no creation of energy and no destruction, so that -as regards both mechanical and molecular changes, the -cause is really the manifestation of existing energy. In -the same way I see not why the prior existence of matter -is not also a cause as regards its subsequent existence. All -science tends to show us that the existence of the universe -in a particular state at one moment, is the condition of its -existence at the next moment, in an apparently different<span class="pagenum" id="Page_226">226</span> -state. When we analyse the meaning which we can -attribute to the word <i>cause</i>, it amounts to the existence of -suitable portions of matter endowed with suitable quantities -of energy. If we may accept Horne Tooke’s assertion, -<i>cause</i> has etymologically the meaning of <i>thing before</i>. -Though, indeed, the origin of the word is very obscure, its -derivatives, the Italian <i>cosa</i>, and the French <i>chose</i>, mean -simply <i>thing</i>. In the German equivalent <i>ursache</i>, we have -plainly the original meaning of <i>thing before</i>, the <i>sache</i> -denoting “interesting or important object,” the English -<i>sake</i>, and <i>ur</i> being the equivalent of the English <i>ere</i>, -<i>before</i>. We abandon, then, both etymology and philosophy, -when we attribute to the <i>laws of causation</i> any -meaning beyond that of the <i>conditions</i> under which an -event may be expected to happen, according to our -observation of the previous course of nature.</p> - -<p>I have no objection to use the words cause and -causation, provided they are never allowed to lead us to -imagine that our knowledge of nature can attain to certainty. -I repeat that if a cause is an invariable and -necessary condition of an event, we can never know -certainly whether the cause exists or not. To us, then, a -cause is not to be distinguished from the group of positive -or negative conditions which, with more or less probability, -precede an event. In this sense, there is no particular -difference between knowledge of causes and our general -knowledge of the succession of combinations, in which the -phenomena of nature are presented to us, or found to -occur in experimental inquiry.</p> - - -<h3><i>Distinction of Inductive and Deductive Results.</i></h3> - -<p>We must carefully avoid confusing together inductive -investigations which terminate in the establishment of -general laws, and those which seem to lead directly to -the knowledge of future particular events. That process -only can be called induction which gives general laws, -and it is by the subsequent employment of deduction that -we anticipate particular events. If the observation of a -number of cases shows that alloys of metals fuse at lower -temperatures than their constituent metals, I may with -more or less probability draw a general inference to that<span class="pagenum" id="Page_227">227</span> -effect, and may thence deductively ascertain the probability -that the next alloy examined will fuse at a lower -temperature than its constituents. It has been asserted, -indeed, by Mill,<a id="FNanchor_133" href="#Footnote_133" class="fnanchor">133</a> and partially admitted by Mr. Fowler,<a id="FNanchor_134" href="#Footnote_134" class="fnanchor">134</a> -that we can argue directly from case to case, so that what -is true of some alloys will be true of the next. Professor -Bain has adopted the same view of reasoning. He thinks -that Mill has extricated us from the dead lock of the -syllogism and effected a total revolution in logic. He -holds that reasoning from particulars to particulars is not -only the usual, the most obvious and the most ready -method, but that it is the type of reasoning which best -discloses the real process.<a id="FNanchor_135" href="#Footnote_135" class="fnanchor">135</a> Doubtless, this is the usual -result of our reasoning, regard being had to degrees of -probability; but these logicians fail entirely to give any -explanation of the process by which we get from case -to case.</p> - -<p>It may be allowed that the knowledge of future particular -events is the main purpose of our investigations, -and if there were any process of thought by which we -could pass directly from event to event without ascending -into general truths, this method would be sufficient, and -certainly the briefest. It is true, also, that the laws of -mental association lead the mind always to expect the like -again in apparently like circumstances, and even animals -of very low intelligence must have some trace of such -powers of association, serving to guide them more or less -correctly, in the absence of true reasoning faculties. But -it is the purpose of logic, according to Mill, to ascertain -whether inferences have been correctly drawn, rather than -to discover them.<a id="FNanchor_136" href="#Footnote_136" class="fnanchor">136</a> Even if we can, then, by habit, -association, or any rude process of inference, infer the -future directly from the past, it is the work of logic to -analyse the conditions on which the correctness of this -inference depends. Even Mill would admit that such -analysis involves the consideration of general truths,<a id="FNanchor_137" href="#Footnote_137" class="fnanchor">137</a> and<span class="pagenum" id="Page_228">228</span> -in this, as in several other important points, we might -controvert Mill’s own views by his own statements. It -seems to me undesirable in a systematic work like this to -enter into controversy at any length, or to attempt to refute -the views of other logicians. But I shall feel bound to -state, in a separate publication, my very deliberate opinion -that many of Mill’s innovations in logical science, and -especially his doctrine of reasoning from particulars to -particulars, are entirely groundless and false.</p> - - -<h3><i>The Grounds of Inductive Inference.</i></h3> - -<p>I hold that in all cases of inductive inference we must -invent hypotheses, until we fall upon some hypothesis -which yields deductive results in accordance with experience. -Such accordance renders the chosen hypothesis -more or less probable, and we may then deduce, with some -degree of likelihood, the nature of our future experience, -on the assumption that no arbitrary change takes place in -the conditions of nature. We can only argue from the -past to the future, on the general principle set forth in this -work, that what is true of a thing will be true of the like. -So far then as one object or event differs from another, all -inference is impossible, particulars as particulars can no -more make an inference than grains of sand can make a -rope. We must always rise to something which is general -or same in the cases, and assuming that sameness to be -extended to new cases we learn their nature. Hearing a -clock tick five thousand times without exception or variation, -we adopt the very probable hypothesis that there is -some invariably acting machine which produces those uniform -sounds, and which will, in the absence of change, go -on producing them. Meeting twenty times with a bright -yellow ductile substance, and finding it always to be very -heavy and incorrodible, I infer that there was some natural -condition which tended in the creation of things to associate -these properties together, and I expect to find them -associated in the next instance. But there always is the -possibility that some unknown change may take place -between past and future cases. The clock may run down, -or be subject to a hundred accidents altering its condition. -There is no reason in the nature of things, so far as known<span class="pagenum" id="Page_229">229</span> -to us, why yellow colour, ductility, high specific gravity, -and incorrodibility, should always be associated together, -and in other cases, if not in this, men’s expectations -have been deceived. Our inferences, therefore, always -retain more or less of a hypothetical character, and are so -far open to doubt. Only in proportion as our induction -approximates to the character of perfect induction, does -it approximate to certainty. The amount of uncertainty -corresponds to the probability that other objects than -those examined may exist and falsity our inferences; the -amount of probability corresponds to the amount of information -yielded by our examination; and the theory of -probability will be needed to prevent us from over-estimating -or under-estimating the knowledge we possess.</p> - - -<h3><i>Illustrations of the Inductive Process.</i></h3> - -<p>To illustrate the passage from the known to the apparently -unknown, let us suppose that the phenomena -under investigation consist of numbers, and that the -following six numbers being exhibited to us, we are -required to infer the character of the next in the -series:—</p> - -<div class="ml5em"> -5, 15, 35, 45, 65, 95. -</div> - -<p class="ti0">The question first of all arises, How may we describe this -series of numbers? What is uniformly true of them? -The reader cannot fail to perceive at the first glance that -they all end in five, and the problem is, from the properties -of these six numbers, to infer the properties of the -next number ending in five. If we test their properties -by the process of perfect induction, we soon perceive that -they have another common property, namely that of being -<i>divisible by five without remainder</i>. May we then assert that -the next number ending in five is also divisible by five, -and, if so, upon what grounds? Or extending the question, -Is every number ending in five divisible by five? Does it -follow that because six numbers obey a supposed law, -therefore 376,685,975 or any other number, however large, -obeys the law? I answer <i>certainly not</i>. The law in question -is undoubtedly true; but its truth is not proved by -any finite number of examples. All that these six numbers -can do is to suggest to my mind the possible existence of<span class="pagenum" id="Page_230">230</span> -such a law; and I then ascertain its truth, by proving -deductively from the rules of decimal numeration, that any -number ending in five must be made up of multiples of -five, and must therefore be itself a multiple.</p> - -<p>To make this more plain, let the reader now examine -the numbers—</p> - -<div class="ml5em"> -7, 17, 37, 47, 67, 97. -</div> - -<p>They all end in 7 instead of 5, and though not at equal -intervals, the intervals are the same as in the previous -case. After consideration, the reader will perceive that -these numbers all agree in being <i>prime numbers</i>, or multiples -of unity only. May we then infer that the next, or -any other number ending in 7, is a prime number? -Clearly not, for on trial we find that 27, 57, 117 are not -primes. Six instances, then, treated empirically, lead us -to a true and universal law in one case, and mislead us in -another case. We ought, in fact, to have no confidence in -any law until we have treated it deductively, and have -shown that from the conditions supposed the results expected -must ensue. No one can show from the principles -of number, that numbers ending in 7 should be primes.</p> - -<p>From the history of the theory of numbers some good -examples of false induction can be adduced. Taking the -following series of prime numbers,</p> - -<div class="ml5em"> -41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, &c., -</div> - -<p class="ti0">it will be found that they all agree in being values of -the general expression <i>x</i><sup>2</sup> + <i>x</i> + 41, putting for <i>x</i> in succession -the values, 0, 1, 2, 3, 4, &c. We seem always to -obtain a prime number, and the induction is apparently -strong, to the effect that this expression always will -give primes. Yet a few more trials disprove this false conclusion. -Put <i>x</i> = 40, and we obtain 40 × 40 + 40 + 41, -or 41 × 41. Such a failure could never have happened, -had we shown any deductive reason why <i>x</i><sup>2</sup> + <i>x</i> + 41 -should give primes.</p> - -<p>There can be no doubt that what here happens with -forty instances, might happen with forty thousand or -forty million instances. An apparent law never once -failing up to a certain point may then suddenly break -down, so that inductive reasoning, as it has been described -by some writers, can give no sure knowledge of what is to -come. Babbage pointed out, in his Ninth Bridgewater<span class="pagenum" id="Page_231">231</span> -Treatise, that a machine could be constructed to give a -perfectly regular series of numbers through a vast series -of steps, and yet to break the law of progression suddenly -at any required point. No number of particular cases as -particulars enables us to pass by inference to any new case. -It is hardly needful to inquire here what can be inferred -from an infinite series of facts, because they are never -practically within our power; but we may unhesitatingly -accept the conclusion, that no finite number of instances -can ever prove a general law, or can give us certain knowledge -of even one other instance.</p> - -<p>General mathematical theorems have indeed been discovered -by the observation of particular cases, and may -again be so discovered. We have Newton’s own statement, -to the effect that he was thus led to the all-important -Binomial Theorem, the basis of the whole structure -of mathematical analysis. Speaking of a certain series of -terms, expressing the area of a circle or hyperbola, he says: -“I reflected that the denominators were in arithmetical -progression; so that only the numerical co-efficients of -the numerators remained to be investigated. But these, -in the alternate areas, were the figures of the powers of -the number eleven, namely 11<sup>0</sup>, 11<sup>1</sup>, 11<sup>2</sup>, 11<sup>3</sup>, 11<sup>4</sup>; that is, -in the first 1; in the second 1, 1; in the third 1, 2, 1; in -the fourth 1, 3, 3, 1; in the fifth 1, 4, 6, 4, 1.<a id="FNanchor_138" href="#Footnote_138" class="fnanchor">138</a> I inquired, -therefore, in what manner all the remaining figures could -be found from the first two; and I found that if the first -figure be called <i>m</i>, all the rest could be found by the -continual multiplication of the terms of the formula</p> - -<div class="ml5em mt05em"> -<span class="nowrap"><span class="fraction2"><span class="fnum2"><i>m</i> - 0</span><span class="bar">/</span><span class="fden2">1</span></span></span> × - <span class="nowrap"><span class="fraction2"><span class="fnum2"><i>m</i> - 1</span><span class="bar">/</span><span class="fden2">2</span></span></span> × - <span class="nowrap"><span class="fraction2"><span class="fnum2"><i>m</i> - 2</span><span class="bar">/</span><span class="fden2">3</span></span></span> × - <span class="nowrap"><span class="fraction2"><span class="fnum2"><i>m</i> - 3</span><span class="bar">/</span><span class="fden2">4</span></span></span> × &c.”<a id="FNanchor_139" href="#Footnote_139" class="fnanchor">139</a> -</div> - -<p class="ti0">It is pretty evident, from this most interesting statement, -that Newton, having simply observed the succession of the -numbers, tried various formulæ until he found one which -agreed with them all. He was so little satisfied with this -process, however, that he verified particular results of his -new theorem by comparison with the results of common<span class="pagenum" id="Page_232">232</span> -multiplication, and the rule for the extraction of the -square root. Newton, in fact, gave no demonstration -of his theorem; and the greatest mathematicians of the -last century, James Bernoulli, Maclaurin, Landen, Euler, -Lagrange, &c., occupied themselves with discovering a conclusive -method of deductive proof.</p> - -<p>There can be no doubt that in geometry also discoveries -have been suggested by direct observation. Many of the -now trivial propositions of Euclid’s Elements were probably -thus discovered, by the ancient Greek geometers; -and we have pretty clear evidence of this in the Commentaries -of Proclus.<a id="FNanchor_140" href="#Footnote_140" class="fnanchor">140</a> Galileo was the first to examine the -remarkable properties of the cycloid, the curve described by -a point in the circumference of a wheel rolling on a plane. -By direct observation he ascertained that the area of the -curve is apparently three times that of the generating circle -or wheel, but he was unable to prove this exactly, or to -verify it by strict geometrical reasoning. Sir George Airy -has recorded a curious case, in which he fell accidentally by -trial on a new geometrical property of the sphere.<a id="FNanchor_141" href="#Footnote_141" class="fnanchor">141</a> But -discovery in such cases means nothing more than suggestion, -and it is always by pure deduction that the general -law is really established. As Proclus puts it, <i>we must -pass from sense to consideration</i>.</p> - -<figure class="figright illowp100" id="p232" style="max-width: 17.5em;"> - <img class="w100" src="images/p232.jpg" alt=""> -</figure> - -<p>Given, for instance, the series of figures in the accompanying -diagram, measurement will show that the curved -lines approximate to semicircles, and the rectilinear figures -to right-angled triangles. These figures may seem to -suggest to the mind the general law that angles inscribed<span class="pagenum" id="Page_233">233</span> -in semicircles are right angles; but no number of instances, -and no possible accuracy of measurement would really -establish the truth of that general law. Availing ourselves -of the suggestion furnished by the figures, we can only -investigate deductively the consequences which flow from -the definition of a circle, until we discover among them the -property of containing right angles. Persons have thought -that they had discovered a method of trisecting angles by -plane geometrical construction, because a certain complex -arrangement of lines and circles had appeared to trisect an -angle in every case tried by them, and they inferred, by a -supposed act of induction, that it would succeed in all -other cases. De Morgan has recorded a proposed mode of -trisecting the angle which could not be discriminated by -the senses from a true general solution, except when it was -applied to very obtuse angles.<a id="FNanchor_142" href="#Footnote_142" class="fnanchor">142</a> In all such cases, it has -always turned out either that the angle was not trisected -at all, or that only certain particular angles could be thus -trisected. The trisectors were misled by some apparent or -special coincidence, and only deductive proof could establish -the truth and generality of the result. In this particular -case, deductive proof shows that the problem -attempted is impossible, and that angles generally cannot -be trisected by common geometrical methods.</p> - - -<h3><i>Geometrical Reasoning.</i></h3> - -<p>This view of the matter is strongly supported by the -further consideration of geometrical reasoning. No skill -and care could ever enable us to verify absolutely any one -geometrical proposition. Rousseau, in his <i>Emile</i>, tells us -that we should teach a child geometry by causing him to -measure and compare figures by superposition. While a -child was yet incapable of general reasoning, this would -doubtless be an instructive exercise; but it never could -teach geometry, nor prove the truth of any one proposition. -All our figures are rude approximations, and they may -happen to seem unequal when they should be equal, -and equal when they should be unequal. Moreover -figures may from chance be equal in case after case, and<span class="pagenum" id="Page_234">234</span> -yet there may be no general reason why they should be -so. The results of deductive geometrical reasoning are -absolutely certain, and are either exactly true or capable -of being carried to any required degree of approximation. -In a perfect triangle, the angles must be equal to one half-revolution -precisely; even an infinitesimal divergence -would be impossible; and I believe with equal confidence, -that however many are the angles of a figure, provided -there are no re-entrant angles, the sum of the angles will -be precisely and absolutely equal to twice as many right-angles -as the figure has sides, less by four right-angles. -In such cases, the deductive proof is absolute and complete; -empirical verification can at the most guard against -accidental oversights.</p> - -<p>There is a second class of geometrical truths which can -only be proved by approximation; but, as the mind sees -no reason why that approximation should not always go -on, we arrive at complete conviction. We thus learn that -the surface of a sphere is equal exactly to two-thirds of -the whole surface of the circumscribing cylinder, or to four -times the area of the generating circle. The area of a -parabola is exactly two-thirds of that of the circumscribing -parallelogram. The area of the cycloid is exactly three -times that of the generating circle. These are truths that -we could never ascertain, nor even verify by observation; -for any finite amount of difference, less than what the -senses can discern, would falsify them.</p> - -<p>There are geometrical relations again which we cannot -assign exactly, but can carry to any desirable degree of approximation. -The ratio of the circumference to the diameter -of a circle is that of 3·14159265358979323846.... -to 1, and the approximation may be carried to any extent -by the expenditure of sufficient labour. Mr. W. -Shanks has given the value of this natural constant, known -as π, to the extent of 707 places of decimals.<a id="FNanchor_143" href="#Footnote_143" class="fnanchor">143</a> Some years -since, I amused myself by trying how near I could get to -this ratio, by the careful use of compasses, and I did not -come nearer than 1 part in 540. We might imagine measurements -so accurately executed as to give us eight or -ten places correctly. But the power of the hands and<span class="pagenum" id="Page_235">235</span> -senses must soon stop, whereas the mental powers of deductive -reasoning can proceed to an unlimited degree of approximation. -Geometrical truths, then, are incapable of -verification; and, if so, they cannot even be learnt by -observation. How can I have learnt by observation a proposition -of which I cannot even prove the truth by observation, -when I am in possession of it? All that observation -or empirical trial can do is to suggest propositions, of -which the truth may afterwards be proved deductively.</p> - -<p>If Viviani’s story is to be believed, Galileo endeavoured -to satisfy himself about the area of the cycloid by cutting -out several large cycloids in pasteboard, and then comparing -the areas of the curve and the generating circle by -weighing them. In every trial the curve seemed to be -rather less than three times the circle, so that Galileo, we -are told, began to suspect that the ratio was not precisely -3 to 1. It is quite clear, however, that no process of -weighing or measuring could ever prove truths like these, -and it remained for Torricelli to show what his master -Galileo had only guessed at.<a id="FNanchor_144" href="#Footnote_144" class="fnanchor">144</a></p> - -<p>Much has been said about the peculiar certainty of -mathematical reasoning, but it is only certainty of deductive -reasoning, and equal certainty attaches to all correct -logical deduction. If a triangle be right-angled, the -square on the hypothenuse will undoubtedly equal the -sum of the two squares on the other sides; but I can -never be sure that a triangle is right-angled: so I can be -certain that nitric acid will not dissolve gold, provided I -know that the substances employed really correspond to -those on which I tried the experiment previously. Here -is like certainty of inference, and like doubt as to the -facts.</p> - - -<h3><i>Discrimination of Certainty and Probability.</i></h3> - -<p>We can never recur too often to the truth that our -knowledge of the laws and future events of the external -world is only probable. The mind itself is quite capable -of possessing certain knowledge, and it is well to discriminate -carefully between what we can and cannot know<span class="pagenum" id="Page_236">236</span> -with certainty. In the first place, whatever feeling is -actually present to the mind is certainly known to that -mind. If I see blue sky, I may be quite sure that I -do experience the sensation of blueness. Whatever I do -feel, I do feel beyond all doubt. We are indeed very -likely to confuse what we really feel with what we are -inclined to associate with it, or infer inductively from -it; but the whole of our consciousness, as far as it is -the result of pure intuition and free from inference, is -certain knowledge beyond all doubt.</p> - -<p>In the second place, we may have certainty of inference; -the fundamental laws of thought, and the rule of substitution -(p. <a href="#Page_9">9</a>), are certainly true; and if my senses could inform me -that A was indistinguishable in colour from B, and B from -C, then I should be equally certain that A was indistinguishable -from C. In short, whatever truth there is in the -premises, I can certainly embody in their correct logical -result. But the certainty generally assumes a hypothetical -character. I never can be quite sure that two colours -are exactly alike, that two magnitudes are exactly equal, -or that two bodies whatsoever are identical even in their -apparent qualities. Almost all our judgments involve -quantitative relations, and, as will be shown in succeeding -chapters, we can never attain exactness and certainty -where continuous quantity enters. Judgments concerning -discontinuous quantity or numbers, however, allow of certainty; -I may establish beyond doubt, for instance, that -the difference of the squares of 17 and 13 is the product -of 17 + 13 and 17 - 13, and is therefore 30 × 4, or 120.</p> - -<p>Inferences which we draw concerning natural objects -are never certain except in a hypothetical point of -view. It might seem to be certain that iron is magnetic, -or that gold is incapable of solution in nitric acid; but, -if we carefully investigate the meanings of these statements, -they will be found to involve no certainty but -that of consciousness and that of hypothetical inference. -For what do I mean by iron or gold? If I choose a -remarkable piece of yellow substance, call it gold, and -then immerse it in a liquid which I call nitric acid, and -find that there is no change called solution, then consciousness -has certainly informed me that, with my meaning of -the terms, “Gold is insoluble in nitric acid.” I may further<span class="pagenum" id="Page_237">237</span> -be certain of something else; for if this gold and nitric -acid remain what they were, I may be sure there will be -no solution on again trying the experiment. If I take other -portions of gold and nitric acid, and am sure that they really -are identical in properties with the former portions, I can -be certain that there will be no solution. But at this point -my knowledge becomes purely hypothetical; for how can I -be sure without trial that the gold and acid are really -identical in nature with what I formerly called gold and -nitric acid. How do I know gold when I see it? If I -judge by the apparent qualities—colour, ductility, specific -gravity, &c., I may be misled, because there may always -exist a substance which to the colour, ductility, specific -gravity, and other specified qualities, joins others which we -do not expect. Similarly, if iron is magnetic, as shown by -an experiment with objects answering to those names, then -all iron is magnetic, meaning all pieces of matter identical -with my assumed piece. But in trying to identify iron, I -am always open to mistake. Nor is this liability to mistake -a matter of speculation only.<a id="FNanchor_145" href="#Footnote_145" class="fnanchor">145</a></p> - -<p>The history of chemistry shows that the most confident -inferences may have been falsified by the confusion of one -substance with another. Thus strontia was never discriminated -from baryta until Klaproth and Haüy detected -differences between some of their properties. Accordingly -chemists must often have inferred concerning strontia -what was only true of baryta, and <i>vice versâ</i>. There is -now no doubt that the recently discovered substances, -cæsium and rubidium, were long mistaken for potassium.<a id="FNanchor_146" href="#Footnote_146" class="fnanchor">146</a> -Other elements have often been confused together—for -instance, tantalum and niobium; sulphur and selenium; -cerium, lanthanum, and didymium; yttrium and erbium.</p> - -<p>Even the best known laws of physical science do -not exclude false inference. No law of nature has been -better established than that of universal gravitation, and -we believe with the utmost confidence that any body -capable of affecting the senses will attract other bodies, -and fall to the earth if not prevented. Euler remarks<span class="pagenum" id="Page_238">238</span> -that, although he had never made trial of the stones -which compose the church of Magdeburg, yet he had -not the least doubt that all of them were heavy, and -would fall if unsupported. But he adds, that it would -be extremely difficult to give any satisfactory explanation -of this confident belief.<a id="FNanchor_147" href="#Footnote_147" class="fnanchor">147</a> The fact is, that the belief ought -not to amount to certainty until the experiment has been -tried, and in the meantime a slight amount of uncertainty -enters, because we cannot be sure that the stones of -the Magdeburg Church resemble other stones in all their -properties.</p> - -<p>In like manner, not one of the inductive truths which -men have established, or think they have established, is -really safe from exception or reversal. Lavoisier, when -laying the foundations of chemistry, met with so many -instances tending to show the existence of oxygen in -all acids, that he adopted a general conclusion to that -effect, and devised the name oxygen accordingly. He -entertained no appreciable doubt that the acid existing -in sea salt also contained oxygen;<a id="FNanchor_148" href="#Footnote_148" class="fnanchor">148</a> yet subsequent experience -falsified his expectations. This instance refers -to a science in its infancy, speaking relatively to the -possible achievements of men. But all sciences are and -ever will remain in their infancy, relatively to the extent -and complexity of the universe which they undertake to -investigate. Euler expresses no more than the truth when -he says that it would be impossible to fix on any one thing -really existing, of which we could have so perfect a knowledge -as to put us beyond the reach of mistake.<a id="FNanchor_149" href="#Footnote_149" class="fnanchor">149</a> We may -be quite certain that a comet will go on moving in a -similar path <i>if</i> all circumstances remain the same as -before; but if we leave out this extensive qualification, -our predictions will always be subject to the chance of -falsification by some unexpected event, such as the division -of Biela’s comet or the interference of an unknown gravitating -body.</p> - -<p><span class="pagenum" id="Page_239">239</span></p> - -<p>Inductive inference might attain to certainty if our -knowledge of the agents existing throughout the universe -were complete, and if we were at the same time certain -that the same Power which created the universe would -allow it to proceed without arbitrary change. There is -always a possibility of causes being in existence without -our knowledge, and these may at any moment produce -an unexpected effect. Even when by the theory of probabilities -we succeed in forming some notion of the comparative -confidence with which we should receive inductive -results, it yet appears to me that we must make -an assumption. Events come out like balls from the vast -ballot-box of nature, and close observation will enable us -to form some notion, as we shall see in the next chapter, -of the contents of that ballot-box. But we must still -assume that, between the time of an observation and that -to which our inferences relate, no change in the ballot-box -has been made.</p> -<hr class="chap x-ebookmaker-drop"> - -<div class="chapter"> -<p><span class="pagenum" id="Page_240">240</span></p> - -<h2 class="nobreak" id="CHAPTER_XII">CHAPTER XII.<br> - -<span class="title">THE INDUCTIVE OR INVERSE APPLICATION OF THE -THEORY OF PROBABILITY.</span></h2> -</div> - -<p class="ti0">We have hitherto considered the theory of probability -only in its simple deductive employment, in which it -enables us to determine from given conditions the probable -character of events happening under those conditions. -But as deductive reasoning when inversely applied constitutes -the process of induction, so the calculation of -probabilities may be inversely applied; from the known -character of certain events we may argue backwards to -the probability of a certain law or condition governing -those events. Having satisfactorily accomplished this -work, we may indeed calculate forwards to the probable -character of future events happening under the same conditions; -but this part of the process is a direct use of -deductive reasoning (p. <a href="#Page_226">226</a>).</p> - -<p>Now it is highly instructive to find that whether the -theory of probability be deductively or inductively applied, -the calculation is always performed according to -the principles and rules of deduction. The probability -that an event has a particular condition entirely depends -upon the probability that if the condition existed the -event would follow. If we take up a pack of common -playing cards, and observe that they are arranged in perfect -numerical order, we conclude beyond all reasonable -doubt that they have been thus intentionally arranged -by some person acquainted with the usual order of -sequence. This conclusion is quite irresistible, and rightly<span class="pagenum" id="Page_241">241</span> -so; for there are but two suppositions which we can make -as to the reason of the cards being in that particular -order:—</p> - -<p>1. They may have been intentionally arranged by some -one who would probably prefer the numerical order.</p> - -<p>2. They may have fallen into that order by chance, that -is, by some series of conditions which, being unknown to -us, cannot be known to lead by preference to the particular -order in question.</p> - -<p>The latter supposition is by no means absurd, for any -one order is as likely as any other when there is no preponderating -tendency. But we can readily calculate by the -doctrine of permutations the probability that fifty-two -objects would fall by chance into any one particular order. -Fifty-two objects can be arranged in 52 × 51 × ... × 3 -× 2 × 1 or about 8066 × (10)<sup>64</sup> possible orders, the -number obtained requiring 68 places of figures for its -full expression. Hence it is excessively unlikely that -anyone should ever meet with a pack of cards arranged -in perfect order by accident. If we do meet with a -pack so arranged, we inevitably adopt the other supposition, -that some person, having reasons for preferring that -special order, has thus put them together.</p> - -<p>We know that of the immense number of possible -orders the numerical order is the most remarkable; it is -useful as proving the perfect constitution of the pack, and -it is the intentional result of certain games. At any rate, -the probability that intention should produce that order is -incomparably greater than the probability that chance -should produce it; and as a certain pack exists in that -order, we rightly prefer the supposition which most probably -leads to the observed result.</p> - -<p>By a similar mode of reasoning we every day arrive, -and validly arrive, at conclusions approximating to certainty. -Whenever we observe a perfect resemblance -between two objects, as, for instance, two printed pages, -two engravings, two coins, two foot-prints, we are warranted -in asserting that they proceed from the same type, -the same plate, the same pair of dies, or the same boot. -And why? Because it is almost impossible that with -different types, plates, dies, or boots some apparent distinction -of form should not be produced. It is impossible<span class="pagenum" id="Page_242">242</span> -for the hand of the most skilful artist to make two objects -alike, so that mechanical repetition is the only probable -explanation of exact similarity.</p> - -<p>We can often establish with extreme probability that -one document is copied from another. Suppose that each -document contains 10,000 words, and that the same word -is incorrectly spelt in each. There is then a probability of -less than 1 in 10,000 that the same mistake should be -made in each. If we meet with a second error occurring -in each document, the probability is less than 1 in 10,000 -× 9999, that two such coincidences should occur by chance, -and the numbers grow with extreme rapidity for more -numerous coincidences. We cannot make any precise -calculations without taking into account the character of -the errors committed, concerning the conditions of which -we have no accurate means of estimating probabilities. -Nevertheless, abundant evidence may thus be obtained -as to the derivation of documents from each other. In -the examination of many sets of logarithmic tables, six -remarkable errors were found to be present in all but -two, and it was proved that tables printed at Paris, Berlin, -Florence, Avignon, and even in China, besides thirteen -sets printed in England between the years 1633 and 1822, -were derived directly or indirectly from some common -source.<a id="FNanchor_150" href="#Footnote_150" class="fnanchor">150</a> With a certain amount of labour, it is possible -to establish beyond reasonable doubt the relationship or -genealogy of any number of copies of one document, proceeding -possibly from parent copies now lost. The relations -between the manuscripts of the New Testament have -been elaborately investigated in this manner, and the same -work has been performed for many classical writings, -especially by German scholars.</p> - - -<h3><i>Principle of the Inverse Method.</i></h3> - -<p>The inverse application of the rules of probability -entirely depends upon a proposition which may be thus -stated, nearly in the words of Laplace.<a id="FNanchor_151" href="#Footnote_151" class="fnanchor">151</a> <i>If an event can<span class="pagenum" id="Page_243">243</span> -be produced by any one of a certain number of different -causes, all equally probable à priori, the probabilities of the -existence of these causes as inferred from the event, are proportional -to the probabilities of the event as derived from these -causes.</i> In other words, the most probable cause of an -event which has happened is that which would most probably -lead to the event supposing the cause to exist; but -all other possible causes are also to be taken into account -with probabilities proportional to the probability that the -event would happen if the cause existed. Suppose, to fix -our ideas clearly, that E is the event, and C<sub>1</sub> C<sub>2</sub> C<sub>3</sub> are the -three only conceivable causes. If C<sub>1</sub> exist, the probability -is <i>p</i><sub>1</sub> that E would follow; if C<sub>2</sub> or C<sub>3</sub> exist, the like probabilities -are respectively <i>p</i><sub>2</sub> and <i>p</i><sub>3</sub>. Then as <i>p</i><sub>1</sub> is to <i>p</i><sub>2</sub>, -so is the probability of C<sub>1</sub> being the actual cause to the -probability of C<sub>2</sub> being it; and, similarly, as <i>p</i><sub>2</sub> is to <i>p</i><sub>3</sub>, so -is the probability of C<sub>2</sub> being the actual cause to the -probability of C<sub>3</sub> being it. By a simple mathematical process -we arrive at the conclusion that the actual probability -of C<sub>1</sub> being the cause is</p> - -<div class="ml5em"> -<span class="nowrap"><span class="fraction2"><span class="fnum2"><i>p</i><sub>1</sub></span><span class="bar">/</span><span class="fden2"><i>p</i><sub>1</sub> + <i>p</i><sub>2</sub> + <i>p</i><sub>3</sub></span></span></span>; -</div> - -<p class="ti0">and the similar probabilities of the existence of C<sub>2</sub> and -C<sub>3</sub> are,</p> - -<div class="ml5em"> -<span class="nowrap"><span class="fraction2"><span class="fnum2"><i>p</i><sub>2</sub></span><span class="bar">/</span><span class="fden2"><i>p</i><sub>1</sub> + <i>p</i><sub>2</sub> + <i>p</i><sub>3</sub></span></span></span> -and <span class="nowrap"><span class="fraction2"><span class="fnum2"><i>p</i><sub>3</sub></span><span class="bar">/</span><span class="fden2"><i>p</i><sub>1</sub> + <i>p</i><sub>2</sub> + <i>p</i><sub>3</sub></span></span></span>. -</div> - -<p class="ti0">The sum of these three fractions amounts to unity, which -correctly expresses the certainty that one cause or other -must be in operation.</p> - -<p>We may thus state the result in general language. -<i>If it is certain that one or other of the supposed causes -exists, the probability that any one does exist is the probability -that if it exists the event happens, divided by the sum -of all the similar probabilities.</i> There may seem to be an -intricacy in this subject which may prove distasteful to -some readers; but this intricacy is essential to the subject -in hand. No one can possibly understand the principles -of inductive reasoning, unless he will take the trouble to -master the meaning of this rule, by which we recede from -an event to the probability of each of its possible causes.</p> - -<p>This rule or principle of the indirect method is that -which common sense leads us to adopt almost instinctively,<span class="pagenum" id="Page_244">244</span> -before we have any comprehension of the principle in its -general form. It is easy to see, too, that it is the rule -which will, out of a great multitude of cases, lead us most -often to the truth, since the most probable cause of an -event really means that cause which in the greatest -number of cases produces the event. Donkin and Boole -have given demonstrations of this principle, but the one -most easy to comprehend is that of Poisson. He imagines -each possible cause of an event to be represented by a -distinct ballot-box, containing black and white balls, in -such a ratio that the probability of a white ball being -drawn is equal to that of the event happening. He further -supposes that each box, as is possible, contains the same -total number of balls, black and white; then, mixing all -the contents of the boxes together, he shows that if a -white ball be drawn from the aggregate ballot-box thus -formed, the probability that it proceeded from any particular -ballot-box is represented by the number of white -balls in that particular box, divided by the total number -of white balls in all the boxes. This result corresponds to -that given by the principle in question.<a id="FNanchor_152" href="#Footnote_152" class="fnanchor">152</a></p> - -<p>Thus, if there be three boxes, each containing ten balls -in all, and respectively containing seven, four, and three -white balls, then on mixing all the balls together we have -fourteen white ones; and if we draw a white ball, that is -if the event happens, the probability that it came out of -the first box is <span class="nowrap"><span class="fraction"><span class="fnum">7</span><span class="bar">/</span><span class="fden">14</span></span></span>; -which is exactly equal to <span class="nowrap"><span class="fraction2"><span class="fnum2"><span class="nowrap"><span class="fraction"><span class="fnum">7</span><span class="bar">/</span><span class="fden">10</span></span></span></span><span class="bar">/</span><span class="fden2"><span class="nowrap"><span class="fraction"><span class="fnum">7</span><span class="bar">/</span><span class="fden">10</span></span></span> + <span class="nowrap"><span class="fraction"><span class="fnum">4</span><span class="bar">/</span><span class="fden">10</span></span></span> + <span class="nowrap"><span class="fraction"><span class="fnum">3</span><span class="bar">/</span><span class="fden">10</span></span></span></span></span></span>, -the fraction given by the rule of the Inverse Method.</p> - - -<h3><i>Simple Applications of the Inverse Method.</i></h3> - -<p>In many cases of scientific induction we may apply the -principle of the inverse method in a simple manner. If -only two, or at the most a few hypotheses, may be made -as to the origin of certain phenomena, we may sometimes -easily calculate the respective probabilities. It was thus -that Bunsen and Kirchhoff established, with a probability -little short of certainty, that iron exists in the sun. On -comparing the spectra of sunlight and of the light proceeding<span class="pagenum" id="Page_245">245</span> -from the incandescent vapour of iron, it became apparent -that at least sixty bright lines in the spectrum of iron -coincided with dark lines in the sun’s spectrum. Such coincidences -could never be observed with certainty, because, -even if the lines only closely approached, the instrumental -imperfections of the spectroscope would make them apparently -coincident, and if one line came within half a millimetre -of another, on the map of the spectra, they could not -be pronounced distinct. Now the average distance of the -solar lines on Kirchhoff’s map is 2 mm., and if we throw -down a line, as it were, by pure chance on such a map, -the probability is about one-half that the new line will fall -within <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span> mm. on one side or the other of some one of the -solar lines. To put it in another way, we may suppose -that each solar line, either on account of its real breadth, -or the defects of the instrument, possesses a breadth of -<span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span> mm., and that each line in the iron spectrum has a like -breadth. The probability then is just one-half that the -centre of each iron line will come by chance within 1 mm. -of the centre of a solar line, so as to appear to coincide -with it. The probability of casual coincidence of each -iron line with a solar line is in like manner <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span>. Coincidence -in the case of each of the sixty iron lines is a very -unlikely event if it arises casually, for it would have a -probability of only (<span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span>)<sup>60</sup> - or less than 1 in a trillion. The -odds, in short, are more than a million million millions -to unity against such casual coincidence.<a id="FNanchor_153" href="#Footnote_153" class="fnanchor">153</a> But on the -other hypothesis, that iron exists in the sun, it is highly -probable that such coincidences would be observed; it is -immensely more probable that sixty coincidences would be -observed if iron existed in the sun, than that they should -arise from chance. Hence by our principle it is immensely -probable that iron does exist in the sun.</p> - -<p>All the other interesting results, given by the comparison -of spectra, rest upon the same principle of probability. -The almost complete coincidence between the spectra of -solar, lunar, and planetary light renders it practically -certain that the light is all of solar origin, and is reflected -from the surfaces of the moon and planets, suffering only<span class="pagenum" id="Page_246">246</span> -slight alteration from the atmospheres of some of the -planets. A fresh confirmation of the truth of the Copernican -theory is thus furnished.</p> - -<p>Herschel proved in this way the connection between the -direction of the oblique faces of quartz crystals, and -the direction in which the same crystals rotate the -plane of polarisation of light. For if it is found in a -second crystal that the relation is the same as in the first, -the probability of this happening by chance is <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span>; the -probability that in another crystal also the direction -will be the same is <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">4</span></span></span>, and so on. The probability that -in <i>n</i> + 1 crystals there would be casual agreement of direction -is the nth power of <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span>. Thus, if in examining fourteen -crystals the same relation of the two phenomena is discovered -in each, the odds that it proceeds from uniform -conditions are more than 8000 to 1.<a id="FNanchor_154" href="#Footnote_154" class="fnanchor">154</a> Since the first -observations on this subject were made in 1820, no exceptions -have been observed, so that the probability of invariable -connection is incalculably great.</p> - -<p>It is exceedingly probable that the ancient Egyptians -had exactly recorded the eclipses occurring during long -periods of time, for Diogenes Laertius mentions that 373 -solar and 832 lunar eclipses had been observed, and the -ratio between these numbers exactly expresses that which -would hold true of the eclipses of any long period, of -say 1200 or 1300 years, as estimated on astronomical -grounds. It is evident that an agreement between small -numbers, or customary numbers, such as seven, one -hundred, a myriad, &c., is much more likely to happen from -chance, and therefore gives much less presumption of dependence. -If two ancient writers spoke of the sacrifice of -oxen, they would in all probability describe it as a hecatomb, -and there would be nothing remarkable in the coincidence. -But it is impossible to point out any special -reason why an old writer should select such numbers as -373 and 832, unless they had been the results of observation.</p> - -<p>On similar grounds, we must inevitably believe in the<span class="pagenum" id="Page_247">247</span> -human origin of the flint flakes so copiously discovered of -late years. For though the accidental stroke of one stone -against another may often produce flakes, such as are -occasionally found on the sea-shore, yet when several -flakes are found in close company, and each one bears -evidence, not of a single blow only, but of several successive -blows, all conducing to form a symmetrical knife-like -form, the probability of a natural and accidental -origin becomes incredibly small, and the contrary supposition, -that they are the work of intelligent beings, -approximately certain.<a id="FNanchor_155" href="#Footnote_155" class="fnanchor">155</a></p> - - -<h3><i>The Theory of Probability in Astronomy.</i></h3> - -<p>The science of astronomy, occupied with the simple -relations of distance, magnitude, and motion of the -heavenly bodies, admits more easily than almost any -other science of interesting conclusions founded on the -theory of probability. More than a century ago, in -1767, Michell showed the extreme probability of bonds -connecting together systems of stars. He was struck -by the unexpected number of fixed stars which have -companions close to them. Such a conjunction might -happen casually by one star, although possibly at a -great distance from the other, happening to lie on a -straight line passing near the earth. But the probabilities -are so greatly against such an optical union happening -often in the expanse of the heavens, that Michell asserted -the existence of some connection between most of the -double stars. It has since been estimated by Struve, -that the odds are 9570 to 1 against any two stars of not -less than the seventh magnitude falling within the apparent -distance of four seconds of each other by chance, and -yet ninety-one such cases were known when the estimation -was made, and many more cases have since been discovered. -There were also four known triple stars, and yet the odds -against the appearance of any one such conjunction are -173,524 to 1.<a id="FNanchor_156" href="#Footnote_156" class="fnanchor">156</a> The conclusions of Michell have been<span class="pagenum" id="Page_248">248</span> -entirely verified by the discovery that many double stars -are connected by gravitation.</p> - -<p>Michell also investigated the probability that the six -brightest stars in the Pleiades should have come by -accidents into such striking proximity. Estimating the -number of stars of equal or greater brightness at 1500, be -found the odds to be nearly 500,000 to 1 against casual -conjunction. Extending the same kind of argument to -other clusters, such as that of Præsepe, the nebula in the -hilt of Perseus’ sword, he says:<a id="FNanchor_157" href="#Footnote_157" class="fnanchor">157</a> “We may with the -highest probability conclude, the odds against the contrary -opinion being many million millions to one, that the stars -are really collected together in clusters in some places, -where they form a kind of system, while in others there -are either few or none of them, to whatever cause this may -be owing, whether to their mutual gravitation, or to some -other law or appointment of the Creator.”</p> - -<p>The calculations of Michell have been called in question -by the late James D. Forbes,<a id="FNanchor_158" href="#Footnote_158" class="fnanchor">158</a> and Mr. Todhunter vaguely -countenances his objections,<a id="FNanchor_159" href="#Footnote_159" class="fnanchor">159</a> otherwise I should not have -thought them of much weight. Certainly Laplace accepts -Michell’s views,<a id="FNanchor_160" href="#Footnote_160" class="fnanchor">160</a> and if Michell be in error it is in the -methods of calculation, not in the general validity of his -reasoning and conclusions.</p> - -<p>Similar calculations might no doubt be applied to the -peculiar drifting motions which have been detected by -Mr. R A. Proctor in some of the constellations.<a id="FNanchor_161" href="#Footnote_161" class="fnanchor">161</a> The odds -are very greatly against any numerous group of stars moving -together in any one direction by chance. On like -grounds, there can be no doubt that the sun has a considerable -proper motion because on the average the fixed -stars show a tendency to move apparently from one point -of the heavens towards that diametrically opposite. The -sun’s motion in the contrary direction would explain this -tendency, otherwise we must believe that thousands of -stars accidentally agree in their direction of motion, or are<span class="pagenum" id="Page_249">249</span> -urged by some common force from which the sun is -exempt. It may be said that the rotation of the earth is -proved in like manner, because it is immensely more probable -that one body would revolve than that the sun, -moon, planets, comets, and the whole of the stars of the -heavens should be whirled round the earth daily, with a -uniform motion superadded to their own peculiar motions. -This appears to be mainly the reason which led Gilbert, -one of the earliest English Copernicans, and in every way -an admirable physicist, to admit the rotation of the earth, -while Francis Bacon denied it.</p> - -<p>In contemplating the planetary system, we are struck -with the similarity in direction of nearly all its movements. -Newton remarked upon the regularity and uniformity of -these motions, and contrasted them with the eccentricity -and irregularity of the cometary orbits.<a id="FNanchor_162" href="#Footnote_162" class="fnanchor">162</a> Could we, in -fact, look down upon the system from the northern side, -we should see all the planets moving round from west to -east, the satellites moving round their primaries, and the -sun, planets, and satellites rotating in the same direction, -with some exceptions on the verge of the system. In the -time of Laplace eleven planets were known, and the directions -of rotation were known for the sun, six planets, the -satellites of Jupiter, Saturn’s ring, and one of his satellites. -Thus there were altogether 43 motions all concurring, -namely:—</p> - -<table class="ml5em"> -<tr> -<td class="tal">Orbital motions of eleven planets</td> -<td class="tac"><div><div>11</div></div></td> -</tr> -<tr> -<td class="tal pr2">Orbital motions of eighteen satellites</td> -<td class="tac"><div><div>18</div></div></td> -</tr> -<tr> -<td class="tal">Axial rotations</td> -<td class="tac"><div><div>14</div></div></td> -</tr> -<tr> -<td class="tal"></td> -<td class="tac"><div><div>—</div></div></td> -</tr> -<tr> -<td class="tal"></td> -<td class="tac"><div><div>43</div></div></td> -</tr> -</table> - -<p>The probability that 43 motions independent of each -other would coincide by chance is the 42nd power of <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span>, so -that the odds are about 4,400,000,000,000 to 1 in favour of -some common cause for the uniformity of direction. This -probability, as Laplace observes,<a id="FNanchor_163" href="#Footnote_163" class="fnanchor">163</a> is higher than that of -many historical events which we undoubtingly believe. In -the present day, the probability is much increased by the -discovery of additional planets, and the rotation of other<span class="pagenum" id="Page_250">250</span> -satellites, and it is only slightly weakened by the fact that -some of the outlying satellites are exceptional in direction, -there being considerable evidence of an accidental disturbance -in the more distant parts of the system.</p> - -<p>Hardly less remarkable than the uniform direction of -motion is the near approximation of the orbits of the -planets to a common plane. Daniel Bernoulli roughly -estimated the probability of such an agreement arising -from accident as 1 ÷ (12)<sup>6</sup> the greatest inclination of any -orbit to the sun’s equator being 1-12th part of a quadrant. -Laplace devoted to this subject some of his most ingenious -investigations. He found the probability that the sum of -the inclinations of the planetary orbits would not exceed -by accident the actual amount (·914187 of a right angle -for the ten planets known in 1801) to be <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">10</span></span></span>! (·914187)<sup>10</sup> -or about ·00000011235. This probability may be combined -with that derived from the direction of motion, and -it then becomes immensely probable that the constitution -of the planetary system arose out of uniform conditions, -or, as we say, from some common cause.<a id="FNanchor_164" href="#Footnote_164" class="fnanchor">164</a></p> - -<p>If the same kind of calculation be applied to the orbits -of comets, the result is very different.<a id="FNanchor_165" href="#Footnote_165" class="fnanchor">165</a> Of the orbits -which have been determined 48·9 per cent. only are direct -or in the same direction as the planetary motions.<a id="FNanchor_166" href="#Footnote_166" class="fnanchor">166</a> Hence -it becomes apparent that comets do not properly belong -to the solar system, and it is probable that they are stray -portions of nebulous matter which have accidentally become -attached to the system by the attractive powers of the -sun or Jupiter.</p> - - -<h3><i>The General Inverse Problem.</i></h3> - -<p>In the instances described in the preceding sections, -we have been occupied in receding from the occurrence -of certain similar events to the probability that there<span class="pagenum" id="Page_251">251</span> -must have been a condition or cause for such events. We -have found that the theory of probability, although never -yielding a certain result, often enables us to establish an -hypothesis beyond the reach of reasonable doubt. There -is, however, another method of applying the theory, -which possesses for us even greater interest, because it -illustrates, in the most complete manner, the theory of -inference adopted in this work, which theory indeed it -suggested. The problem to be solved is as follows:—</p> - -<p><i>An event having happened a certain number of times, -and failed a certain number of times, required the probability -that it will happen any given number of times -in the future under the same circumstances.</i></p> - -<p>All the <i>larger</i> planets hitherto discovered move in one -direction round the sun; what is the probability that, if a -new planet exterior to Neptune be discovered, it will move -in the same direction? All known permanent gases, except -chlorine, are colourless; what is the probability that, -if some new permanent gas should be discovered, it will -be colourless? In the general solution of this problem, we -wish to infer the future happening of any event from the -number of times that it has already been observed to -happen. Now, it is very instructive to find that there is -no known process by which we can pass directly from the -data to the conclusion. It is always requisite to recede -from the data to the probability of some hypothesis, and -to make that hypothesis the ground of our inference -concerning future events. Mathematicians, in fact, make -every hypothesis which is applicable to the question in -hand; they then calculate, by the inverse method, the -probability of every such hypothesis according to the -data, and the probability that if each hypothesis be true, -the required future event will happen. The total probability -that the event will happen is the sum of the -separate probabilities contributed by each distinct hypothesis.</p> - -<p>To illustrate more precisely the method of solving the -problem, it is desirable to adopt some concrete mode of -representation, and the ballot-box, so often employed by -mathematicians, will best serve our purpose. Let the -happening of any event be represented by the drawing of -a white ball from a ballot-box, while the failure of an<span class="pagenum" id="Page_252">252</span> -event is represented by the drawing of a black ball. Now, -in the inductive problem we are supposed to be ignorant -of the contents of the ballot-box, and are required to -ground all our inferences on our experience of those contents -as shown in successive drawings. Rude common -sense would guide us nearly to a true conclusion. Thus, -if we had drawn twenty balls one after another, replacing -the ball after each drawing, and the ball had in each case -proved to be white, we should believe that there was a -considerable preponderance of white balls in the urn, and -a probability in favour of drawing a white ball on the next -occasion. Though we had drawn white balls for -thousands of times without fail, it would still be possible -that some black balls lurked in the urn and would at last -appear, so that our inferences could never be certain. On -the other hand, if black balls came at intervals, we should -expect that after a certain number of trials the black balls -would appear again from time to time with somewhat the -same frequency.</p> - -<p>The mathematical solution of the question consists in -little more than a close analysis of the mode in which our -common sense proceeds. If twenty white balls have been -drawn and no black ball, my common sense tells me that -any hypothesis which makes the black balls in the urn -considerable compared with the white ones is improbable; -a preponderance of white balls is a more probable hypothesis, -and as a deduction from this more probable hypothesis, -I expect a recurrence of white balls. The mathematician -merely reduces this process of thought to exact -numbers. Taking, for instance, the hypothesis that there -are 99 white and one black ball in the urn, he can calculate -the probability that 20 white balls would be drawn -in succession in those circumstances; he thus forms a -definite estimate of the probability of this hypothesis, and -knowing at the same time the probability of a white ball -reappearing if such be the contents of the urn, he combines -these probabilities, and obtains an exact estimate -that a white ball will recur in consequence of this hypothesis. -But as this hypothesis is only one out of many -possible ones, since the ratio of white and black balls may -be 98 to 2, or 97 to 3, or 96 to 4, and so on, he has to -repeat the estimate for every such possible hypothesis.<span class="pagenum" id="Page_253">253</span> -To make the method of solving the problem perfectly -evident, I will describe in the next section a very simple -case of the problem, originally devised for the purpose by -Condorcet, which was also adopted by Lacroix,<a id="FNanchor_167" href="#Footnote_167" class="fnanchor">167</a> and has -passed into the works of De Morgan, Lubbock, and others.</p> - - -<h3><i>Simple Illustration of the Inverse Problem.</i></h3> - -<p>Suppose it to be known that a ballot-box contains only -four black or white balls, the ratio of black and white balls -being unknown. Four drawings having been made with -replacement, and a white ball having appeared on each -occasion but one, it is required to determine the probability -that a white ball will appear next time. Now the -hypotheses which can be made as to the contents of the -urn are very limited in number, and are at most the -following five:—</p> - -<table class="ml5em"> -<tr> -<td class="tac"><div>4 </div></td> -<td class="tac"><div><div>white </div></div></td> -<td class="tac"><div><div>and </div></div></td> -<td class="tac"><div><div>0 </div></div></td> -<td class="tac"><div><div>black </div></div></td> -<td class="tac"><div><div>balls</div></div></td> -</tr> -<tr> -<td class="tac"><div><div>3 </div></div></td> -<td class="tac"><div><div>"</div></div></td> -<td class="tac"><div><div>"</div></div></td> -<td class="tac"><div><div>1 </div></div></td> -<td class="tac"><div><div>"</div></div></td> -<td class="tac"><div><div>"</div></div></td> -</tr> -<tr> -<td class="tac"><div><div>2 </div></div></td> -<td class="tac"><div><div>"</div></div></td> -<td class="tac"><div><div>"</div></div></td> -<td class="tac"><div><div>2 </div></div></td> -<td class="tac"><div><div>"</div></div></td> -<td class="tac"><div><div>"</div></div></td> -</tr> -<tr> -<td class="tac"><div><div>1 </div></div></td> -<td class="tac"><div><div>"</div></div></td> -<td class="tac"><div><div>"</div></div></td> -<td class="tac"><div><div>3 </div></div></td> -<td class="tac"><div><div>"</div></div></td> -<td class="tac"><div><div>"</div></div></td> -</tr> -<tr> -<td class="tac"><div><div>0 </div></div></td> -<td class="tac"><div><div>"</div></div></td> -<td class="tac"><div><div>"</div></div></td> -<td class="tac"><div><div>4 </div></div></td> -<td class="tac"><div><div>"</div></div></td> -<td class="tac"><div><div>"</div></div></td> -</tr> -</table> - -<p class="ti0">The actual occurrence of black and white balls in the -drawings puts the first and last hypothesis out of the -question, so that we have only three left to consider.</p> - -<p>If the box contains three white and one black, the -probability of drawing a white each time is <span class="nowrap"><span class="fraction"><span class="fnum">3</span><span class="bar">/</span><span class="fden">4</span></span></span>, and a black -<span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">4</span></span></span>; so that the compound event observed, namely, three -white and one black, has the probability <span class="nowrap"><span class="fraction"><span class="fnum">3</span><span class="bar">/</span><span class="fden">4</span></span></span> × <span class="nowrap"><span class="fraction"><span class="fnum">3</span><span class="bar">/</span><span class="fden">4</span></span></span> × <span class="nowrap"><span class="fraction"><span class="fnum">3</span><span class="bar">/</span><span class="fden">4</span></span></span> × <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">4</span></span></span>, by -the rule already given (p. <a href="#Page_204">204</a>). But as it is indifferent -in what order the balls are drawn, and the black ball -might come first, second, third, or fourth, we must multiply -by four, to obtain the probability of three white and -one black in any order, thus getting <span class="nowrap"><span class="fraction"><span class="fnum">27</span><span class="bar">/</span><span class="fden">64</span></span></span>.</p> - -<p>Taking the next hypothesis of two white and two -black balls in the urn, we obtain for the same probability -the quantity <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span> × <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span> × <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span> × <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span> × 4, or <span class="nowrap"><span class="fraction"><span class="fnum">16</span><span class="bar">/</span><span class="fden">64</span></span></span>, and from the -third hypothesis of one white and three black we deduce -likewise <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">4</span></span></span> × <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">4</span></span></span> × <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">4</span></span></span> × <span class="nowrap"><span class="fraction"><span class="fnum">3</span><span class="bar">/</span><span class="fden">4</span></span></span> × 4, or <span class="nowrap"><span class="fraction"><span class="fnum">3</span><span class="bar">/</span><span class="fden">64</span></span></span>. According, then, as we<span class="pagenum" id="Page_254">254</span> -adopt the first, second, or third hypothesis, the probability -that the result actually noticed would follow is <span class="nowrap"><span class="fraction"><span class="fnum">27</span><span class="bar">/</span><span class="fden">64</span></span></span>, -<span class="nowrap"><span class="fraction"><span class="fnum">16</span><span class="bar">/</span><span class="fden">64</span></span></span>, -and <span class="nowrap"><span class="fraction"><span class="fnum">3</span><span class="bar">/</span><span class="fden">64</span></span></span>. Now it is certain that one or other of these -hypotheses must be the true one, and their absolute -probabilities are proportional to the probabilities that the -observed events would follow from them (pp. <a href="#Page_242">242</a>, <a href="#Page_242">243</a>). All -we have to do, then, in order to obtain the absolute probability -of each hypothesis, is to alter these fractions in -a uniform ratio, so that their sum shall be unity, the -expression of certainty. Now, since 27 + 16 + 3 = 46, -this will be effected by dividing each fraction by 46, and -multiplying by 64. Thus the probabilities of the first, -second, and third hypotheses are respectively—</p> - -<div class="ml5em"> -<span class="nowrap"><span class="fraction2"><span class="fnum2">27</span><span class="bar">/</span><span class="fden2">46</span></span></span>, <span class="nowrap"><span class="fraction2"><span class="fnum2">16</span><span class="bar">/</span><span class="fden2">46</span></span></span>, <span class="nowrap"><span class="fraction2"><span class="fnum2">3</span><span class="bar">/</span><span class="fden2">46</span></span></span>.<br> -</div> - -<p class="ti0">The inductive part of the problem is completed, since we -have found that the urn most likely contains three white -and one black ball, and have assigned the exact probability -of each possible supposition. But we are now in a position -to resume deductive reasoning, and infer the probability -that the next drawing will yield, say a white ball. For if -the box contains three white and one black ball, the probability -of drawing a white one is certainly <span class="nowrap"><span class="fraction"><span class="fnum">3</span><span class="bar">/</span><span class="fden">4</span></span></span>; and as the -probability of the box being so constituted is <span class="nowrap"><span class="fraction"><span class="fnum">27</span><span class="bar">/</span><span class="fden">46</span></span></span>, - the compound -probability that the box will be so filled and will -give a white ball at the next trial, is</p> - -<div class="ml5em"> -<span class="nowrap"><span class="fraction2"><span class="fnum2">27</span><span class="bar">/</span><span class="fden2">46</span></span></span> × <span class="nowrap"><span class="fraction2"><span class="fnum2">3</span><span class="bar">/</span><span class="fden2">4</span></span></span> or <span class="nowrap"><span class="fraction2"><span class="fnum2">81</span><span class="bar">/</span><span class="fden2">184</span></span></span>.<br> -</div> - -<p>Again, the probability is <span class="nowrap"><span class="fraction"><span class="fnum">16</span><span class="bar">/</span><span class="fden">46</span></span></span> that the box contains two -white and two black, and under those conditions the -probability is <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span> that a white ball will appear; hence the -probability that a white ball will appear in consequence -of that condition, is</p> - -<div class="ml5em"> -<span class="nowrap"><span class="fraction2"><span class="fnum2">16</span><span class="bar">/</span><span class="fden2">46</span></span></span> × <span class="nowrap"><span class="fraction2"><span class="fnum">1</span><span class="bar">/</span><span class="fden2">2</span></span></span> or <span class="nowrap"><span class="fraction2"><span class="fnum2">32</span><span class="bar">/</span><span class="fden2">184</span></span></span>.<br> -</div> - -<p>From the third supposition we get in like manner the -probability</p> - -<div class="ml5em"> -<span class="nowrap"><span class="fraction2"><span class="fnum2">3</span><span class="bar">/</span><span class="fden2">46</span></span></span> × <span class="nowrap"><span class="fraction2"><span class="fnum">1</span><span class="bar">/</span><span class="fden2">4</span></span></span> or <span class="nowrap"><span class="fraction2"><span class="fnum2">3</span><span class="bar">/</span><span class="fden2">184</span></span></span>.<br> -</div> - -<p class="ti0">Since one and not more than one hypothesis can be true,<span class="pagenum" id="Page_255">255</span> -we may add together these separate probabilities, and we -find that</p> - -<div class="ml5em"> -<span class="nowrap"><span class="fraction2"><span class="fnum2">81</span><span class="bar">/</span><span class="fden2">184</span></span></span> + <span class="nowrap"><span class="fraction2"><span class="fnum2">32</span><span class="bar">/</span><span class="fden2">184</span></span></span> + <span class="nowrap"><span class="fraction2"><span class="fnum2">3</span><span class="bar">/</span><span class="fden2">184</span></span></span> or <span class="nowrap"><span class="fraction2"><span class="fnum2">116</span><span class="bar">/</span><span class="fden2">184</span></span></span> -</div> - -<p class="ti0">is the complete probability that a white ball will be next -drawn under the conditions and data supposed.</p> - - -<h3><i>General Solution of the Inverse Problem.</i></h3> - -<p>In the instance of the inverse method described in the -last section, the balls supposed to be in the ballot-box -were few, for the purpose of simplifying the calculation. -In order that our solution may apply to natural phenomena, -we must render our hypotheses as little arbitrary -as possible. Having no <i>à priori</i> knowledge of the conditions -of the phenomena in question, there is no limit -to the variety of hypotheses which might be suggested. -Mathematicians have therefore had recourse to the most -extensive suppositions which can be made, namely, that -the ballot-box contains an infinite number of balls; they -have then varied the proportion of white to black balls -continuously, from the smallest to the greatest possible -proportion, and estimated the aggregate probability which -results from this comprehensive supposition.</p> - -<p>To explain their procedure, let us imagine that, instead -of an infinite number, the ballot-box contains a large -finite number of balls, say 1000. Then the number of -white balls might be 1 or 2 or 3 or 4, and so on, up to -999. Supposing that three white and one black ball -have been drawn from the urn as before, there is a certain -very small probability that this would have occurred in -the case of a box containing one white and 999 black -balls; there is also a small probability that from such a -box the next ball would be white. Compound these -probabilities, and we have the probability that the next -ball really will be white, in consequence of the existence -of that proportion of balls. If there be two white and 998 -black balls in the box, the probability is greater and will -increase until the balls are supposed to be in the proportion -of those drawn. Now 999 different hypotheses are -possible, and the calculation is to be made for each of -these, and their aggregate taken as the final result. It is<span class="pagenum" id="Page_256">256</span> -apparent that as the number of balls in the box is increased, -the absolute probability of any one hypothesis concerning -the exact proportion of balls is decreased, but the aggregate -results of all the hypotheses will assume the character of -a wider average.</p> - -<p>When we take the step of supposing the balls within -the urn to be infinite in number, the possible proportions -of white and black balls also become infinite, and the -probability of any one proportion actually existing is -infinitely small. Hence the final result that the next ball -drawn will be white is really the sum of an infinite -number of infinitely small quantities. It might seem -impossible to calculate out a problem having an infinite -number of hypotheses, but the wonderful resources of the -integral calculus enable this to be done with far greater -facility than if we supposed any large finite number of -balls, and then actually computed the results. I will not -attempt to describe the processes by which Laplace finally -accomplished the complete solution of the problem. They -are to be found described in several English works, especially -De Morgan’s <i>Treatise on Probabilities</i>, in the <i>Encyclopædia -Metropolitana</i>, and Mr. Todhunter’s <i>History of -the Theory of Probability</i>. The abbreviating power of -mathematical analysis was never more strikingly shown. -But I may add that though the integral calculus is -employed as a means of summing infinitely numerous -results, we in no way abandon the principles of combinations -already treated. We calculate the values of -infinitely numerous factorials, not, however, obtaining -their actual products, which would lead to an infinite -number of figures, but obtaining the final answer to the -problem by devices which can only be comprehended -after study of the integral calculus.</p> - -<p>It must be allowed that the hypothesis adopted by -Laplace is in some degree arbitrary, so that there was some -opening for the doubt which Boole has cast upon it.<a id="FNanchor_168" href="#Footnote_168" class="fnanchor">168</a> -But it may be replied, (1) that the supposition of an -infinite number of balls treated in the manner of Laplace -is less arbitrary and more comprehensive than any other -that can be suggested. (2) The result does not differ<span class="pagenum" id="Page_257">257</span> -much from that which would be obtained on the hypothesis -of any large finite number of balls. (3) The supposition -leads to a series of simple formulas which can be applied -with ease in many cases, and which bear all the appearance -of truth so far as it can be independently judged by a -sound and practiced understanding.</p> - - -<h3><i>Rules of the Inverse Method.</i></h3> - -<p>By the solution of the problem, as described in the last -section, we obtain the following series of simple rules.</p> - -<p>1. <i>To find the probability that an event which has not -hitherto been observed to fail will happen once more, -divide the number of times the event has been observed -increased by one, by the same number increased by two.</i></p> - -<p>If there have been <i>m</i> occasions on which a certain event -might have been observed to happen, and it has happened -on all those occasions, then the probability that it will -happen on the next occasion of the same kind <span class="nowrap"><span class="fraction2"><span class="fnum2"><i>m</i> + 1</span><span class="bar">/</span><span class="fden2"><i>m</i> + 2</span></span></span>. -For instance, we may say that there are nine places in -the planetary system where planets might exist obeying -Bode’s law of distance, and in every place there is a -planet obeying the law more or less exactly, although -no reason is known for the coincidence. Hence the -probability that the next planet beyond Neptune will -conform to the law is <span class="nowrap"><span class="fraction"><span class="fnum">10</span><span class="bar">/</span><span class="fden">11</span></span></span>.</p> - -<p>2. <i>To find the, probability that an event which has not -hitherto failed will not fail for a certain number of new -occasions, divide the number of times the event has happened -increased by one, by the same number increased by -one and the number of times it is to happen.</i></p> - -<p>An event having happened <i>m</i> times without fail, the -probability that it will happen <i>n</i> - more times is <span class="nowrap"><span class="fraction2"><span class="fnum2"><i>m</i> + 1</span><span class="bar">/</span><span class="fden2"><i>m</i> + <i>n</i> + 1</span></span></span>. -Thus the probability that three new planets would obey -Bode’s law is <span class="nowrap"><span class="fraction"><span class="fnum">10</span><span class="bar">/</span><span class="fden">13</span></span></span>; but it must be allowed that this, as well -as the previous result, would be much weakened by the -fact that Neptune can barely be said to obey the law.</p> - -<p><i>3. An event having happened and failed a certain -number of times, to find the probability that it will happen -the next time, divide the number of times the event has<span class="pagenum" id="Page_258">258</span> -happened increased by one, by the whole number of times -the event has happened or failed increased by two.</i></p> - -<p>If an event has happened <i>m</i> times and failed <i>n</i> times, -the probability that it will happen on the next occasion is -<span class="nowrap"><span class="fraction2"><span class="fnum2"><i>m</i> + 1</span><span class="bar">/</span><span class="fden2"><i>m</i> + <i>n</i> + 2</span></span></span>. - Thus, if we assume that of the elements discovered -up to the year 1873, 50 are metallic and 14 non-metallic, -then the probability that the next element discovered -will be metallic is <span class="nowrap"><span class="fraction"><span class="fnum">51</span><span class="bar">/</span><span class="fden">66</span></span></span>. Again, since of 37 metals -which have been sufficiently examined only four, namely, -sodium, potassium, lanthanum, and lithium, are of less -density than water, the probability that the next metal -examined or discovered will be less dense than water is -<span class="nowrap"><span class="fraction2"><span class="fnum2">4 + 1</span><span class="bar">/</span><span class="fden2">37 + 2</span></span></span> -or <span class="nowrap"><span class="fraction2"><span class="fnum2">5</span><span class="bar">/</span><span class="fden2">39</span></span></span>.</p> - -<p>We may state the results of the method in a more -general manner thus,<a id="FNanchor_169" href="#Footnote_169" class="fnanchor">169</a>—If under given circumstances certain -events A, B, C, &c., have happened respectively <i>m</i>, <i>n</i>, -<i>p</i>, &c., times, and one or other of these events must -happen, then the probabilities of these events are proportional -to <i>m</i> + 1, <i>n</i> + 1, <i>p</i> + 1, &c., so that the probability -of A will be <span class="nowrap"><span class="fraction2"><span class="fnum2"><i>m</i> + 1</span><span class="bar">/</span><span class="fden2"><i>m</i> + 1 + <i>n</i> + 1 + <i>p</i> + 1 + &c</span></span></span>. But if new -events may happen in addition to those which have been -observed, we must assign unity for the probability of such -new event. The odds then become 1 for a new event, -<i>m</i> + 1 for A, <i>n</i> + 1 for B, and so on, and the absolute -probability of A is <span class="nowrap"><span class="fraction2"><span class="fnum2"><i>m</i> + 1</span><span class="bar">/</span><span class="fden2">1 + <i>m</i> + 1 + <i>n</i> + 1 + &c</span></span></span>.</p> - -<p>It is interesting to trace out the variations of probability -according to these rules. The first time a casual event -happens it is 2 to 1 that it will happen again; if it does -happen it is 3 to 1 that it will happen a third time; and -on successive occasions of the like kind the odds become -4, 5, 6, &c., to 1. The odds of course will be discriminated -from the probabilities which are successively <span class="nowrap"><span class="fraction"><span class="fnum">2</span><span class="bar">/</span><span class="fden">3</span></span></span>, - <span class="nowrap"><span class="fraction"><span class="fnum">3</span><span class="bar">/</span><span class="fden">4</span></span></span>, - <span class="nowrap"><span class="fraction"><span class="fnum">4</span><span class="bar">/</span><span class="fden">5</span></span></span>, &c. -Thus on the first occasion on which a person sees a shark, -and notices that it is accompanied by a little pilot fish, -the odds are 2 to 1, or the probability <span class="nowrap"><span class="fraction"><span class="fnum">2</span><span class="bar">/</span><span class="fden">3</span></span></span>, that the next -shark will be so accompanied.</p> - -<p><span class="pagenum" id="Page_259">259</span></p> - -<p>When an event has happened a very great number of -times, its happening once again approaches nearly to certainty. -If we suppose the sun to have risen one thousand -million times, the probability that it will rise again, on -the ground of this knowledge merely, is <span class="nowrap"><span class="fraction2"><span class="fnum2">1,000,000,000 + 1</span><span class="bar">/</span><span class="fden2">1,000,000,000 + 1 + 1</span></span></span>. -But then the probability that it will continue to rise for as -long a period in the future is only <span class="nowrap"><span class="fraction2"><span class="fnum2">1,000,000,000 + 1</span><span class="bar">/</span><span class="fden2">2,000,000,000 + 1</span></span></span>, or almost -exactly <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span>. The probability that it will continue so rising a -thousand times as long is only about <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">1001</span></span></span>. The lesson which -we may draw from these figures is quite that which we -should adopt on other grounds, namely, that experience -never affords certain knowledge, and that it is exceedingly -improbable that events will always happen as we observe -them. Inferences pushed far beyond their data soon lose -any considerable probability. De Morgan has said,<a id="FNanchor_170" href="#Footnote_170" class="fnanchor">170</a> “No -finite experience whatsoever can justify us in saying that -the future shall coincide with the past in all time to come, -or that there is any probability for such a conclusion.” On -the other hand, we gain the assurance that experience -sufficiently extended and prolonged will give us the -knowledge of future events with an unlimited degree of -probability, provided indeed that those events are not -subject to arbitrary interference.</p> - -<p>It must be clearly understood that these probabilities are -only such as arise from the mere happening of the events, -irrespective of any knowledge derived from other sources -concerning those events or the general laws of nature. -All our knowledge of nature is indeed founded in like -manner upon observation, and is therefore only probable. -The law of gravitation itself is only probably true. But -when a number of different facts, observed under the most -diverse circumstances, are found to be harmonized under a -supposed law of nature, the probability of the law approximates -closely to certainty. Each science rests upon so -many observed facts, and derives so much support from -analogies or connections with other sciences, that there -are comparatively few cases where our judgment of the -probability of an event depends entirely upon a few antecedent<span class="pagenum" id="Page_260">260</span> -events, disconnected from the general body of -physical science.</p> - -<p>Events, again, may often exhibit a regularity of succession -or preponderance of character, which the simple -formula will not take into account. For instance, the -majority of the elements recently discovered are metals, -so that the probability of the next discovery being that of -a metal, is doubtless greater than we calculated (p. <a href="#Page_258">258</a>). -At the more distant parts of the planetary system, there -are symptoms of disturbance which would prevent our -placing much reliance on any inference from the prevailing -order of the known planets to those undiscovered ones -which may possibly exist at great distances. These and -all like complications in no way invalidate the theoretic -truth of the formulas, but render their sound application -much more difficult.</p> - -<p>Erroneous objections have been raised to the theory of -probability, on the ground that we ought not to trust to -our <i>à priori</i> conceptions of what is likely to happen, but -should always endeavour to obtain precise experimental -data to guide us.<a id="FNanchor_171" href="#Footnote_171" class="fnanchor">171</a> This course, however, is perfectly in -accordance with the theory, which is our best and only -guide, whatever data we possess. We ought to be always -applying the inverse method of probabilities so as to take -into account all additional information. When we throw -up a coin for the first time, we are probably quite ignorant -whether it tends more to fall head or tail upwards, and -we must therefore assume the probability of each event -as <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span>. But if it shows head in the first throw, we now -have very slight experimental evidence in favour of a -tendency to show head. The chance of two heads is -now slightly greater than <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">4</span></span></span>, which it appeared to be at -first,<a id="FNanchor_172" href="#Footnote_172" class="fnanchor">172</a> and as we go on throwing the coin time after time, -the probability of head appearing next time constantly -varies in a slight degree according to the character of our -previous experience. As Laplace remarks, we ought -always to have regard to such considerations in common -life. Events when closely scrutinized will hardly ever -prove to be quite independent, and the slightest preponderance<span class="pagenum" id="Page_261">261</span> -one way or the other is some evidence of -connection, and in the absence of better evidence should -be taken into account.</p> - -<p>The grand object of seeking to estimate the probability -of future events from past experience, seems to have been -entertained by James Bernoulli and De Moivre, at least -such was the opinion of Condorcet; and Bernoulli may be -said to have solved one case of the problem.<a id="FNanchor_173" href="#Footnote_173" class="fnanchor">173</a> The English -writers Bayes and Price are, however, undoubtedly the -first who put forward any distinct rules on the subject.<a id="FNanchor_174" href="#Footnote_174" class="fnanchor">174</a> -Condorcet and several other eminent mathematicians advanced -the mathematical theory of the subject; but it was -reserved to the immortal Laplace to bring to the subject -the full power of his genius, and carry the solution of the -problem almost to perfection. It is instructive to observe -that a theory which arose from petty games of chance, the -rules and the very names of which are forgotten, gradually -advanced, until it embraced the most sublime problems of -science, and finally undertook to measure the value and -certainty of all our inductions.</p> - - -<h3><i>Fortuitous Coincidences.</i></h3> - -<p>We should have studied the theory of probability to -very little purpose, if we thought that it would furnish -us with an infallible guide. The theory itself points -out the approximate certainty, that we shall sometimes -be deceived by extraordinary fortuitous coincidences. -There is no run of luck so extreme that it may not -happen, and it may happen to us, or in our time, as -well as to other persons or in other times. We may be -forced by correct calculation to refer such coincidences -to a necessary cause, and yet we may be deceived. All -that the calculus of probability pretends to give, is <i>the -result in the long run</i>, as it is called, and this really means -in <i>an infinity of cases</i>. During any finite experience, -however long, chances may be against us. Nevertheless -the theory is the best guide we can have. If we always -think and act according to its well-interpreted indications,<span class="pagenum" id="Page_262">262</span> -we shall have the best chance of escaping error; and if all -persons, throughout all time to come, obey the theory in -like manner, they will undoubtedly thereby reap the -greatest advantage.</p> - -<p>No rule can be given for discriminating between -coincidences which are casual and those which are the -effects of law. By a fortuitous or casual coincidence, we -mean an agreement between events, which nevertheless -arise from wholly independent and different causes or conditions, -and which will not always so agree. It is a -fortuitous coincidence, if a penny thrown up repeatedly -in various ways always falls on the same side; but it -would not be fortuitous if there were any similarity -in the motions of the hand, and the height of the throw, -so as to cause or tend to cause a uniform result. Now -among the infinitely numerous events, objects, or relations -in the universe, it is quite likely that we shall occasionally -notice casual coincidences. There are seven intervals in -the octave, and there is nothing very improbable in the -colours of the spectrum happening to be apparently -divisible into the same or similar series of seven intervals. -It is hardly yet decided whether this apparent coincidence, -with which Newton was much struck, is well founded or -not,<a id="FNanchor_175" href="#Footnote_175" class="fnanchor">175</a> but the question will probably be decided in the -negative.</p> - -<p>It is certainly a casual coincidence which the ancients -noticed between the seven vowels, the seven strings of the -lyre, the seven Pleiades, and the seven chiefs at Thebes.<a id="FNanchor_176" href="#Footnote_176" class="fnanchor">176</a> -The accidents connected with the number seven have misled -the human intellect throughout the historical period. -Pythagoras imagined a connection between the seven -planets and the seven intervals of the monochord. The -alchemists were never tired of drawing inferences from -the coincidence in numbers of the seven planets and the -seven metals, not to speak of the seven days of the -week.</p> - -<p>A singular circumstance was pointed out concerning -the dimensions of the earth, sun, and moon; the sun’s -diameter was almost exactly 110 times as great as the<span class="pagenum" id="Page_263">263</span> -earth’s diameter, while in almost exactly the same ratio -the mean distance of the earth was greater than the sun’s -diameter, and the mean distance of the moon from the -earth was greater than the moon’s diameter. The agreement -was so close that it might have proved more than -casual, but its fortuitous character is now sufficiently shown -by the fact, that the coincidence ceases to be remarkable when -we adopt the amended dimensions of the planetary system.</p> - -<p>A considerable number of the elements have atomic -weights, which are apparently exact multiples of that -of hydrogen. If this be not a law to be ultimately extended -to all the elements, as supposed by Prout, it is a -most remarkable coincidence. But, as I have observed, -we have no means of absolutely discriminating accidental -coincidences from those which imply a deep producing -cause. A coincidence must either be very strong in -itself, or it must be corroborated by some explanation or -connection with other laws of nature. Little attention -was ever given to the coincidence concerning the dimensions -of the sun, earth, and moon, because it was not very -strong in itself, and had no apparent connection with the -principles of physical astronomy. Prout’s Law bears more -probability because it would bring the constitution of the -elements themselves in close connection with the atomic -theory, representing them as built up out of a simpler -substance.</p> - -<p>In historical and social matters, coincidences are frequently -pointed out which are due to chance, although -there is always a strong popular tendency to regard them -as the work of design, or as having some hidden meaning. -If to 1794, the number of the year in which Robespierre -fell, we add the sum of its digits, the result is 1815, the -year in which Napoleon fell; the repetition of the process -gives 1830 the year in which Charles the Tenth abdicated. -Again, the French Chamber of Deputies, in 1830, consisted -of 402 members, of whom 221 formed the party called -“La queue de Robespierre,” while the remainder, 181 in -number, were named “Les honnêtes gens.” If we give to -each letter a numerical value corresponding to its place in -the alphabet, it will be found that the sum of the values -of the letters in each name exactly indicates the number -of the party.</p> - -<p><span class="pagenum" id="Page_264">264</span></p> - -<p>A number of such coincidences, often of a very curious -character, might be adduced, and the probability against -the occurrence of each is enormously great. They must -be attributed to chance, because they cannot be shown -to have the slightest connection with the general laws -of nature; but persons are often found to be greatly influenced -by such coincidences, regarding them as evidence -of fatality, that is of a system of causation governing -human affairs independently of the ordinary laws of nature. -Let it be remembered that there are an infinite number of -opportunities in life for some strange coincidence to present -itself, so that it is quite to be expected that remarkable -conjunctions will sometimes happen.</p> - -<p>In all matters of judicial evidence, we must bear in -mind the probable occurrence from time to time of unaccountable -coincidences. The Roman jurists refused for -this reason to invalidate a testamentary deed, the witnesses -of which had sealed it with the same seal. For -witnesses independently using their own seals might be -found to possess identical ones by accident.<a id="FNanchor_177" href="#Footnote_177" class="fnanchor">177</a> It is well -known that circumstantial evidence of apparently overwhelming -completeness will sometimes lead to a mistaken -judgment, and as absolute certainty is never really attainable, -every court must act upon probabilities of a high -amount, and in a certain small proportion of cases they -must almost of necessity condemn the innocent victims -of a remarkable conjuncture of circumstances.<a id="FNanchor_178" href="#Footnote_178" class="fnanchor">178</a> Popular -judgments usually turn upon probabilities of far less -amount, as when the palace of Nicomedia, and even -the bedchamber of Diocletian, having been on fire twice -within fifteen days, the people entirely refused to believe -that it could be the result of accident. The Romans -believed that there was fatality connected with the name -of Sextus.</p> - -<p class="tac"> -“Semper sub Sextis perdita Roma fuit.”<br> -</p> - -<p>The utmost precautions will not provide against all -contingencies. To avoid errors in important calculations,<span class="pagenum" id="Page_265">265</span> -it is usual to have them repeated by different computers; -but a case is on record in which three computers made -exactly the same calculations of the place of a star, and -yet all did it wrong in precisely the same manner, for no -apparent reason.<a id="FNanchor_179" href="#Footnote_179" class="fnanchor">179</a></p> - - -<h3><i>Summary of the Theory of Inductive Inference.</i></h3> - -<p>The theory of inductive inference stated in this and the -previous chapters, was suggested by the study of the -Inverse Method of Probability, but it also bears much -resemblance to the so-called Deductive Method described -by Mill, in his celebrated <i>System of Logic</i>. Mill’s views -concerning the Deductive Method, probably form the most -original and valuable part of his treatise, and I should -have ascribed the doctrine entirely to him, had I not -found that the opinions put forward in other parts of his -work are entirely inconsistent with the theory here upheld. -As this subject is the most important and difficult one -with which we have to deal, I will try to remedy the -imperfect manner in which I have treated it, by giving a -recapitulation of the views adopted.</p> - -<p>All inductive reasoning is but the inverse application -of deductive reasoning. Being in possession of certain -particular facts or events expressed in propositions, we -imagine some more general proposition expressing the -existence of a law or cause; and, deducing the particular -results of that supposed general proposition, we observe -whether they agree with the facts in question. Hypothesis -is thus always employed, consciously or unconsciously. -The sole conditions to which we need conform in -framing any hypothesis is, that we both have and exercise -the power of inferring deductively from the hypothesis to -the particular results, which are to be compared with the -known facts. Thus there are but three steps in the process -of induction:—</p> - -<p>(1) Framing some hypothesis as to the character of the -general law.</p> - -<p>(2) Deducing consequences from that law.</p> - -<p><span class="pagenum" id="Page_266">266</span></p> - -<p>(3) Observing whether the consequences agree with the -particular facts under consideration.</p> - -<p>In very simple cases of inverse reasoning, hypothesis -may seem altogether needless. To take numbers again as -a convenient illustration, I have only to look at the series,</p> - -<div class="tac"> -1, 2, 4, 8, 16, 32, &c., -</div> - -<p>to know at once that the general law is that of geometrical -progression; I need no successive trial of various -hypotheses, because I am familiar with the series, and have -long since learnt from what general formula it proceeds. -In the same way a mathematician becomes acquainted -with the integrals of a number of common formulas, so -that he need not go through any process of discovery. -But it is none the less true that whenever previous reasoning -does not furnish the knowledge, hypotheses must be -framed and tried (p. <a href="#Page_124">124</a>).</p> - -<p>There naturally arise two cases, according as the nature -of the subject admits of certain or only probable deductive -reasoning. Certainty, indeed, is but a singular case of -probability, and the general principles of procedure are -always the same. Nevertheless, when certainty of inference -is possible, the process is simplified. Of several -mutually inconsistent hypotheses, the results of which -can be certainly compared with fact, but one hypothesis -can ultimately be entertained. Thus in the inverse logical -problem, two logically distinct conditions could not yield -the same series of possible combinations. Accordingly, -in the case of two terms we had to choose one of six -different kinds of propositions (p. <a href="#Page_136">136</a>), and in the case of -three terms, our choice lay among 192 possible distinct -hypotheses (p. <a href="#Page_140">140</a>). Natural laws, however, are often -quantitative in character, and the possible hypotheses are -then infinite in variety.</p> - -<p>When deduction is certain, comparison with fact is -needed only to assure ourselves that we have rightly -selected the hypothetical conditions. The law establishes -itself, and no number of particular verifications can add -to its probability. Having once deduced from the principles -of algebra that the difference of the squares of two -numbers is equal to the product of their sum and difference, -no number of particular trials of its truth will -render it more certain. On the other hand, no finite<span class="pagenum" id="Page_267">267</span> -number of particular verifications of a supposed law will -render that law certain. In short, certainty belongs only -to the deductive process, and to the teachings of direct -intuition; and as the conditions of nature are not given -by intuition, we can only be certain that we have got a -correct hypothesis when, out of a limited number conceivably -possible, we select that one which alone agrees -with the facts to be explained.</p> - -<p>In geometry and kindred branches of mathematics, -deductive reasoning is conspicuously certain, and it would -often seem as if the consideration of a single diagram -yields us certain knowledge of a general proposition. -But in reality all this certainty is of a purely hypothetical -character. Doubtless if we could ascertain that a supposed -circle was a true and perfect circle, we could be -certain concerning a multitude of its geometrical properties. -But geometrical figures are physical objects, and -the senses can never assure us as to their exact forms. -The figures really treated in Euclid’s <i>Elements</i> are -imaginary, and we never can verify in practice the -conclusions which we draw with certainty in inference; -questions of degree and probability enter.</p> - -<p>Passing now to subjects in which deduction is only -probable, it ceases to be possible to adopt one hypothesis -to the exclusion of the others. We must entertain at the -same time all conceivable hypotheses, and regard each -with the degree of esteem proportionate to its probability. -We go through the same steps as before.</p> - -<p>(1) We frame an hypothesis.</p> - -<p>(2) We deduce the probability of various series of possible -consequences.</p> - -<p>(3) We compare the consequences with the particular -facts, and observe the probability that such facts would -happen under the hypothesis.</p> - -<p>The above processes must be performed for every conceivable -hypothesis, and then the absolute probability of -each will be yielded by the principle of the inverse -method (p. <a href="#Page_242">242</a>). As in the case of certainty we accept -that hypothesis which certainly gives the required results, -so now we accept as most probable that hypothesis which -most probably gives the results; but we are obliged to -entertain at the same time all other hypotheses with<span class="pagenum" id="Page_268">268</span> -degrees of probability proportionate to the probabilities -that they would give the same results.</p> - -<p>So far we have treated only of the process by which -we pass from special facts to general laws, that inverse -application of deduction which constitutes induction. -But the direct employment of deduction is often combined -with the inverse. No sooner have we established -a general law, than the mind rapidly draws particular -consequences from it. In geometry we may almost seem -to infer that <i>because</i> one equilateral triangle is equiangular, -therefore another is so. In reality it is not because one is -that another is, but because all are. The geometrical conditions -are perfectly general, and by what is sometimes -called <i>parity of reasoning</i> whatever is true of one equilateral -triangle, so far as it is equilateral, is true of all equilateral -triangles.</p> - -<p>Similarly, in all other cases of inductive inference, -where we seem to pass from some particular instances to -a new instance, we go through the same process. We -form an hypothesis as to the logical conditions under -which the given instances might occur; we calculate -inversely the probability of that hypothesis, and compounding -this with the probability that a new instance -would proceed from the same conditions, we gain the -absolute probability of occurrence of the new instance in -virtue of this hypothesis. But as several, or many, or -even an infinite number of mutually inconsistent hypotheses -may be possible, we must repeat the calculation for -each such conceivable hypothesis, and then the complete -probability of the future instance will be the sum of the -separate probabilities. The complication of this process -is often very much reduced in practice, owing to the fact -that one hypothesis may be almost certainly true, and -other hypotheses, though conceivable, may be so improbable -as to be neglected without appreciable error.</p> - -<p>When we possess no knowledge whatever of the conditions -from which the events proceed, we may be unable -to form any probable hypotheses as to their mode of -origin. We have now to fall back upon the general -solution of the problem effected by Laplace, which consists -in admitting on an equal footing every conceivable ratio -of favourable and unfavourable chances for the production<span class="pagenum" id="Page_269">269</span> -of the event, and then accepting the aggregate result as -the best which can be obtained. This solution is only to -be accepted in the absence of all better means, but like -other results of the calculus of probability, it comes to our -aid where knowledge is at an end and ignorance begins, -and it prevents us from over-estimating the knowledge we -possess. The general results of the solution are in accordance -with common sense, namely, that the more often an -event has happened the more probable, as a general rule, -is its subsequent recurrence. With the extension of -experience this probability increases, but at the same time -the probability is slight that events will long continue to -happen as they have previously happened.</p> - -<p>We have now pursued the theory of inductive inference, -as far as can be done with regard to simple logical or -numerical relations. The laws of nature deal with time -and space, which are infinitely divisible. As we passed -from pure logic to numerical logic, so we must now pass -from questions of discontinuous, to questions of continuous -quantity, encountering fresh considerations of much difficulty. -Before, therefore, we consider how the great inductions -and generalisations of physical science illustrate -the views of inductive reasoning just explained, we must -break off for a time, and review the means which we -possess of measuring and comparing magnitudes of time, -space, mass, force, momentum, energy, and the various -manifestations of energy in motion, heat, electricity, -chemical change, and the other phenomena of nature.</p> - -<hr class="chap x-ebookmaker-drop"> - -<div class="chapter"> -<p><span class="pagenum" id="Page_270">270</span></p> - -<p class="nobreak ph2 ti0" id="BOOK_III">BOOK III.<br> - -<span class="title">METHODS OF MEASUREMENT.</span></p> -</div> - -<hr class="r30"> - -<div class="chapter"> -<h2 class="nobreak" id="CHAPTER_XIII">CHAPTER XIII.<br> - -<span class="title">THE EXACT MEASUREMENT OF PHENOMENA.</span></h2> -</div> - -<p class="ti0">As physical science advances, it becomes more and -more accurately quantitative. Questions of simple logical -fact after a time resolve themselves into questions of -degree, time, distance, or weight. Forces hardly suspected -to exist by one generation, are clearly recognised by the -next, and precisely measured by the third generation. -But one condition of this rapid advance is the invention -of suitable instruments of measurement. We need what -Francis Bacon called <i>Instantiæ citantes</i>, or <i>evocantes</i>, -methods of rendering minute phenomena perceptible to -the senses; and we also require <i>Instantiæ radii</i> or <i>curriculi</i>, -that is measuring instruments. Accordingly, the -introduction of a new instrument often forms an epoch in -the history of science. As Davy said, “Nothing tends so -much to the advancement of knowledge as the application -of a new instrument. The native intellectual powers of -men in different times are not so much the causes of the -different success of their labours, as the peculiar nature -of the means and artificial resources in their possession.”</p> - -<p>In the absence indeed of advanced theory and analytical<span class="pagenum" id="Page_271">271</span> -power, a very precise instrument would be useless. -Measuring apparatus and mathematical theory should advance -<i>pari passu</i>, and with just such precision as the theorist -can anticipate results, the experimentalist should be able -to compare them with experience. The scrupulously -accurate observations of Flamsteed were the proper -complement to the intense mathematical powers of -Newton.</p> - -<p>Every branch of knowledge commences with quantitative -notions of a very rude character. After we have far -progressed, it is often amusing to look back into the -infancy of the science, and contrast present with past -methods. At Greenwich Observatory in the present day, -the hundredth part of a second is not thought an inconsiderable -portion of time. The ancient Chaldæans -recorded an eclipse to the nearest hour, and the early -Alexandrian astronomers thought it superfluous to distinguish -between the edge and centre of the sun. By -the introduction of the astrolabe, Ptolemy and the later -Alexandrian astronomers could determine the places of -the heavenly bodies within about ten minutes of arc. -Little progress then ensued for thirteen centuries, until -Tycho Brahe made the first great step towards accuracy, -not only by employing better instruments, but even -more by ceasing to regard an instrument as correct. -Tycho, in fact, determined the errors of his instruments, -and corrected his observations. He also took notice -of the effects of atmospheric refraction, and succeeded -in attaining an accuracy often sixty times as great as -that of Ptolemy. Yet Tycho and Hevelius often erred -several minutes in the determination of a star’s place, and -it was a great achievement of Rœmer and Flamsteed to -reduce this error to seconds. Bradley, the modern Hipparchus, -carried on the improvement, his errors in right -ascension, according to Bessel, being under one second of -time, and those of declination under four seconds of arc. -In the present day the average error of a single observation -is probably reduced to the half or quarter of what it -was in Bradley’s time; and further extreme accuracy is -attained by the multiplication of observations, and their -skilful combination according to the theory of error. -Some of the more important constants, for instance that<span class="pagenum" id="Page_272">272</span> -of nutation, have been determined within the tenth part -of a second of space.<a id="FNanchor_180" href="#Footnote_180" class="fnanchor">180</a></p> - -<p>It would be a matter of great interest to trace out the -dependence of this progress upon the introduction of -new instruments. The astrolabe of Ptolemy, the telescope -of Galileo, the pendulum of Galileo and Huyghens, -the micrometer of Horrocks, and the telescopic sights and -micrometer of Gascoygne and Picard, Rœmer’s transit instrument, -Newton’s and Hadley’s quadrant, Dollond’s -achromatic lenses, Harrison’s chronometer, and Ramsden’s -dividing engine—such were some of the principal additions -to astronomical apparatus. The result is, that we -now take note of quantities, 300,000 or 400,000 times as -small as in the time of the Chaldæans.</p> - -<p>It would be interesting again to compare the scrupulous -accuracy of a modern trigonometrical survey with Eratosthenes’ -rude but ingenious guess at the difference of latitude -between Alexandria and Syene—or with Norwood’s -measurement of a degree of latitude in 1635. “Sometimes -I measured, sometimes I paced,” said Norwood; “and I -believe I am within a scantling of the truth.” Such was -the germ of those elaborate geodesical measurements -which have made the dimensions of the globe known to -us within a few hundred yards.</p> - -<p>In other branches of science, the invention of an instrument -has usually marked, if it has not made, an epoch. -The science of heat might be said to commence with the -construction of the thermometer, and it has recently been -advanced by the introduction of the thermo-electric pile. -Chemistry has been created chiefly by the careful use of -the balance, which forms a unique instance of an instrument -remaining substantially in the form in which it was -first applied to scientific purposes by Archimedes. The -balance never has been and probably never can be improved, -except in details of construction. The torsion -balance, introduced by Coulomb towards the end of last -century, has rapidly become essential in many branches -of investigation. In the hands of Cavendish and Baily, it -gave a determination of the earth’s density; applied in the -galvanometer, it gave a delicate measure of electrical<span class="pagenum" id="Page_273">273</span> -forces, and is indispensable in the thermo-electric pile. -This balance is made by simply suspending any light rod -by a thin wire or thread attached to the middle point. -And we owe to it almost all the more delicate investigations -in the theories of heat, electricity, and magnetism.</p> - -<p>Though we can now take note of the millionth of an -inch in space, and the millionth of a second in time, we -must not overlook the fact that in other operations of -science we are yet in the position of the Chaldæans. Not -many years have elapsed since the magnitudes of the -stars, meaning the amounts of light they send to the -observer’s eye, were guessed at in the rudest manner, and -the astronomer adjudged a star to this or that order of -magnitude by a rough comparison with other stars of the -same order. To Sir John Herschel we owe an attempt -to introduce a uniform method of measurement and -expression, bearing some relation to the real photometric -magnitudes of the stars.<a id="FNanchor_181" href="#Footnote_181" class="fnanchor">181</a> Previous to the researches -of Bunsen and Roscoe on the chemical action of light, -we were devoid of any mode of measuring the energy of -light; even now the methods are tedious, and it is not -clear that they give the energy of light so much as one of -its special effects. Many natural phenomena have hardly -yet been made the subject of measurement at all, such -as the intensity of sound, the phenomena of taste and -smell, the magnitude of atoms, the temperature of the -electric spark or of the sun’s photosphere.</p> - -<p>To suppose, then, that quantitative science treats only of -exactly measurable quantities, is a gross if it be a common -mistake. Whenever we are treating of an event which -either happens altogether or does not happen at all, we are -engaged with a non-quantitative phenomenon, a matter of -fact, not of degree; but whenever a thing may be greater or -less, or twice or thrice as great as another, whenever, in -short, ratio enters even in the rudest manner, there -science will have a quantitative character. There can -be little doubt, indeed, that every science as it progresses -will become gradually more and more quantitative. -Numerical precision is the soul of science, as<span class="pagenum" id="Page_274">274</span> -Herschel said, and as all natural objects exist in space, and -involve molecular movements, measurable in velocity and -extent, there is no apparent limit to the ultimate extension -of quantitative science. But the reader must not for a -moment suppose that, because we depend more and more -upon mathematical methods, we leave logical methods -behind us. Number, as I have endeavoured to show, is -logical in its origin, and quantity is but a development of -number, or analogous thereto.</p> - - -<h3><i>Division of the Subject.</i></h3> - -<p>The general subject of quantitative investigation will -have to be divided into several parts. We shall firstly -consider the means at our disposal for measuring phenomena, -and thus rendering them more or less amenable -to mathematical treatment. This task will involve an -analysis of the principles on which accurate methods of -measurement are founded, forming the subject of the -remainder of the present chapter. As measurement, however, -only yields ratios, we have in the next chapter to -consider the establishment of unit magnitudes, in terms of -which our results may be expressed. As every phenomenon -is usually the sum of several distinct quantities -depending upon different causes, we have next to investigate -in Chapter XV. the methods by which we may disentangle -complicated effects, and refer each part of the joint -effect to its separate cause.</p> - -<p>It yet remains for us in subsequent chapters to treat of -quantitative induction, properly so called. We must -follow out the inverse logical method, as it presents itself -in problems of a far higher degree of difficulty than those -which treat of objects related in a simple logical manner, -and incapable of merging into each other by addition and -subtraction.</p> - - -<h3><i>Continuous Quantity.</i></h3> - -<p>The phenomena of nature are for the most part manifested -in quantities which increase or decrease continuously. -When we inquire into the precise meaning of -continuous quantity, we find that it can only be described<span class="pagenum" id="Page_275">275</span> -as that which is divisible without limit. We can divide -a millimetre into ten, or a hundred, or a thousand, or ten -thousand parts, and mentally at any rate we can carry -on the division <i>ad infinitum</i>. Any finite space, then, -must be conceived as made up of an infinite number of -parts each infinitely small. We cannot entertain the -simplest geometrical notions without allowing this. The -conception of a square involves the conception of a side -and diagonal, which, as Euclid beautifully proves in the -117th proposition of his tenth book, have no common -measure,<a id="FNanchor_182" href="#Footnote_182" class="fnanchor">182</a> meaning no finite common measure. Incommensurable -quantities are, in fact, those which have for their -only common measure an infinitely small quantity. It is -somewhat startling to find, too, that in theory incommensurable -quantities will be infinitely more frequent than -commensurable. Let any two lines be drawn haphazard; -it is infinitely unlikely that they will be commensurable, -so that the commensurable quantities, which we are supposed -to deal with in practice, are but singular cases -among an infinitely greater number of incommensurable -cases.</p> - -<p>Practically, however, we treat all quantities as made up -of the least quantities which our senses, assisted by the -best measuring instruments, can perceive. So long as -microscopes were uninvented, it was sufficient to regard -an inch as made up of a thousand thousandths of an -inch; now we must treat it as composed of a million -millionths. We might apparently avoid all mention of -infinitely small quantities, by never carrying our approximations -beyond quantities which the senses can appreciate. -In geometry, as thus treated, we should never assert two -quantities to be equal, but only to be <i>apparently</i> equal. -Legendre really adopts this mode of treatment in the -twentieth proposition of the first book of his Geometry; -and it is practically adopted throughout the physical -sciences, as we shall afterwards see. But though our -fingers, and senses, and instruments must stop somewhere, -there is no reason why the mind should not go on. We -can see that a proof which is only carried through a few -steps in fact, might be carried on without limit, and it is<span class="pagenum" id="Page_276">276</span> -this consciousness of no stopping-place, which renders -Euclid’s proof of his 117th proposition so impressive. Try -how we will to circumvent the matter, we cannot really -avoid the consideration of the infinitely small and the -infinitely great. The same methods of approximation -which seem confined to the finite, mentally extend themselves -to the infinite.</p> - -<p>One result of these considerations is, that we cannot -possibly adjust two quantities in absolute equality. The -suspension of Mahomet’s coffin between two precisely -equal magnets is theoretically conceivable but practically -impossible. The story of the <i>Merchant of Venice</i> turns -upon the infinite improbability that an exact quantity of -flesh could be cut. Unstable equilibrium cannot exist in -nature, for it is that which is destroyed by an infinitely -small displacement. It might be possible to balance an -egg on its end practically, because no egg has a surface of -perfect curvature. Suppose the egg shell to be perfectly -smooth, and the feat would become impossible.</p> - - -<h3><i>The Fallacious Indications of the Senses.</i></h3> - -<p>I may briefly remind the reader how little we can trust -to our unassisted senses in estimating the degree or -magnitude of any phenomenon. The eye cannot correctly -estimate the comparative brightness of two luminous -bodies which differ much in brilliancy; for we know -that the iris is constantly adjusting itself to the intensity -of the light received, and thus admits more or less light -according to circumstances. The moon which shines with -almost dazzling brightness by night, is pale and nearly -imperceptible while the eye is yet affected by the vastly -more powerful light of day. Much has been recorded -concerning the comparative brightness of the zodiacal -light at different times, but it would be difficult to prove -that these changes are not due to the varying darkness -at the time, or the different acuteness of the observer’s -eye. For a like reason it is exceedingly difficult to establish -the existence of any change in the form or comparative -brightness of nebulæ; the appearance of a nebula -greatly depends upon the keenness of sight of the -observer, or the accidental condition of freshness or<span class="pagenum" id="Page_277">277</span> -fatigue of his eye. The same is true of lunar observations; -and even the use of the best telescope fails -to remove this difficulty. In judging of colours, again, -we must remember that light of any given colour tends -to dull the sensibility of the eye for light of the same -colour.</p> - -<p>Nor is the eye when unassisted by instruments a much -better judge of magnitude. Our estimates of the size of -minute bright points, such as the fixed stars, are completely -falsified by the effects of irradiation. Tycho -calculated from the apparent size of the star-discs, that -no one of the principal fixed stars could be contained -within the area of the earth’s orbit. Apart, however, from -irradiation or other distinct causes of error our visual -estimates of sizes and shapes are often astonishingly -incorrect. Artists almost invariably draw distant mountains -in ludicrous disproportion to nearer objects, as a -comparison of a sketch with a photograph at once shows. -The extraordinary apparent difference of size of the sun -or moon, according as it is high in the heavens or near -the horizon, should be sufficient to make us cautious in -accepting the plainest indications of our senses, unassisted -by instrumental measurement. As to statements concerning -the height of the aurora and the distance of meteors, -they are to be utterly distrusted. When Captain Parry -says that a ray of the aurora shot suddenly downwards -between him and the land which was only 3,000 yards -distant, we must consider him subject to an illusion of -sense.<a id="FNanchor_183" href="#Footnote_183" class="fnanchor">183</a></p> - -<p>It is true that errors of observation are more often -errors of judgment than of sense. That which is actually -seen must be so far truly seen; and if we correctly interpret -the meaning of the phenomenon, there would be no error -at all. But the weakness of the bare senses as measuring -instruments, arises from the fact that they import varying -conditions of unknown amount, and we cannot make the -requisite corrections and allowances as in the case of a -solid and invariable instrument.</p> - -<p>Bacon has excellently stated the insufficiency of the<span class="pagenum" id="Page_278">278</span> -senses for estimating the magnitudes of objects, or detecting -the degrees in which phenomena present themselves. -“Things escape the senses,” he says, “because the -object is not sufficient in quantity to strike the sense: as -all minute bodies; because the percussion of the object is -too great to be endured by the senses: as the form of the -sun when looking directly at it in mid-day; because the -time is not proportionate to actuate the sense: as the -motion of a bullet in the air, or the quick circular motion -of a firebrand, which are too fast, or the hour-hand of -a common clock, which is too slow; from the distance -of the object as to place: as the size of the celestial -bodies, and the size and nature of all distant bodies; -from prepossession by another object: as one powerful -smell renders other smells in the same room imperceptible; -from the interruption of interposing bodies: -as the internal parts of animals; and because the object -is unfit to make an impression upon the sense: as the -air or the invisible and untangible spirit which is included -in every living body.”</p> - - -<h3><i>Complexity of Quantitative Questions.</i></h3> - -<p>One remark which we may well make in entering -upon quantitative questions, has regard to the great variety -and extent of phenomena presented to our notice. So -long as we deal only with a simply logical question, that -question is merely, Does a certain event happen? or, Does -a certain object exist? No sooner do we regard the event -or object as capable of more and less, than the question -branches out into many. We must now ask, How much -is it compared with its cause? Does it change when the -amount of the cause changes? If so, does it change in -the same or opposite direction? Is the change in simple -proportion to that of the cause? If not, what more complex -law of connection holds true? This law determined -satisfactorily in one series of circumstances may be varied -under new conditions, and the most complex relations of -several quantities may ultimately be established.</p> - -<p>In every question of physical science there is thus a -series of steps the first one or two of which are usually -made with ease while the succeeding ones demand more<span class="pagenum" id="Page_279">279</span> -and more careful measurement. We cannot lay down -any invariable series of questions which must be asked -from nature. The exact character of the questions will -vary according to the nature of the case, but they will -usually be of an evident kind, and we may readily illustrate -them by examples. Suppose that we are investigating -the solution of some salt in water. The first is a -purely logical question: Is there solution, or is there not? -Assuming the answer to be in the affirmative, we next -inquire, Does the solubility vary with the temperature, or -not? In all probability some variation will exist, and we -must have an answer to the further question, Does -the quantity dissolved increase, or does it diminish with -the temperature? In by far the greatest number of -cases salts and substances of all kinds dissolve more freely -the higher the temperature of the water; but there are a -few salts, such as calcium sulphate, which follow the -opposite rule. A considerable number of salts resemble -sodium sulphate in becoming more soluble up to a certain -temperature, and then varying in the opposite direction. -We next require to assign the amount of variation as -compared with that of the temperature, assuming at first -that the increase of solubility is proportional to the increase -of temperature. Common salt is an instance of -very slight variation, and potassium nitrate of very considerable -increase with temperature. Accurate observations -will probably show, however, that the simple law -of proportionate variation is only approximately true, -and some more complicated law involving the second, -third, or higher powers of the temperature may ultimately -be established. All these investigations have to be -carried out for each salt separately, since no distinct principles -by which we may infer from one substance to -another have yet been detected. There is still an indefinite -field for further research open; for the solubility -of salts will probably vary with the pressure under -which the medium is placed; the presence of other salts -already dissolved may have effects yet unknown. The -researches already effected as regards the solvent power of -water must be repeated with alcohol, ether, carbon -bisulphide, and other media, so that unless general laws -can be detected, this one phenomenon of solution can<span class="pagenum" id="Page_280">280</span> -never be exhaustively treated. The same kind of questions -recur as regards the solution or absorption of gases in -liquids, the pressure as well as the temperature having -then a most decided effect, and Professor Roscoe’s researches -on the subject present an excellent example of -the successive determination of various complicated laws.<a id="FNanchor_184" href="#Footnote_184" class="fnanchor">184</a></p> - -<p>There is hardly a branch of physical science in which -similar complications are not ultimately encountered. -In the case of gravity, indeed, we arrive at the final -law, that the force is the same for all kinds of matter, -and varies only with the distance of action. But in -other subjects the laws, if simple in their ultimate nature, -are disguised and complicated in their apparent results. -Thus the effect of heat in expanding solids, and the reverse -effect of forcible extension or compression upon the temperature -of a body, will vary from one substance to -another, will vary as the temperature is already higher or -lower, and, will probably follow a highly complex law, -which in some cases gives negative or exceptional results. -In crystalline substances the same researches have to be -repeated in each distinct axial direction.</p> - -<p>In the sciences of pure observation, such as those of -astronomy, meteorology, and terrestrial magnetism, we -meet with many interesting series of quantitative determinations. -The so-called fixed stars, as Giordano Bruno -divined, are not really fixed, and may be more truly -described as vast wandering orbs, each pursuing its own -path through space. We must then determine separately -for each star the following questions:—</p> - -<p class="ml2em ti0">1. Does it move?<br> - -2. In what direction?<br> - -3. At what velocity?<br> - -4. Is this velocity variable or uniform?<br> - -5. If variable, according to what law?<br> - -6. Is the direction uniform?<br> - -7. If not, what is the form of the apparent path?<br> - -8. Does it approach or recede?<br> - -9. What is the form of the real path?</p> - -<p>The successive answers to such questions in the case of -certain binary stars, have afforded a proof that the<span class="pagenum" id="Page_281">281</span> -motions are due to a central force coinciding in law with -gravity, and doubtless identical with it. In other cases -the motions are usually so small that it is exceedingly -difficult to distinguish them with certainty. And the time -is yet far off when any general results as regards stellar -motions can be established.</p> - -<p>The variation in the brightness of stars opens an unlimited -field for curious observation. There is not a star -in the heavens concerning which we might not have to -determine:—</p> - -<p class="ml2em ti0">1. Does it vary in brightness?<br> - -2. Is the brightness increasing or decreasing?<br> - -3. Is the variation uniform?<br> - -4. If not, according to what law does it vary?</p> - -<p>In a majority of cases the change will probably be -found to have a periodic character, in which case several -other questions will arise, such as—</p> - -<p class="ml2em ti0">5. What is the length of the period?<br> - -6. Are there minor periods?<br> - -7. What is the law of variation within the period?<br> - -8. Is there any change in the amount of variation?<br> - -9. If so, is it a secular, <i>i.e.</i> a continually growing -change, or does it give evidence of a greater period?</p> - -<p>Already the periodic changes of a certain number of -stars have been determined with accuracy, and the lengths -of the periods vary from less than three days up to -intervals of time at least 250 times as great. Periods -within periods have also been detected.</p> - -<p>There is, perhaps, no subject in which more complicated -quantitative conditions have to be determined than terrestrial -magnetism. Since the time when the declination -of the compass was first noticed, as some suppose by -Columbus, we have had successive discoveries from time -to time of the progressive change of declination from -century to century; of the periodic character of this -change; of the difference of the declination in various -parts of the earth’s surface; of the varying laws of -the change of declination; of the dip or inclination of -the needle, and the corresponding laws of its periodic -changes; the horizontal and perpendicular intensities have -also been the subject of exact measurement, and have been -found to vary with place and time, like the directions of<span class="pagenum" id="Page_282">282</span> -the needle; daily and yearly periodic changes have also -been detected, and all the elements are found to be subject -to occasional storms or abnormal perturbations, in which -the eleven year period, now known to be common to many -planetary relations, is apparent. The complete solution -of these motions of the compass needle involves nothing -less than a determination of its position and oscillations in -every part of the world at any epoch, the like determination -for another epoch, and so on, time after time, until -the periods of all changes are ascertained. This one subject -offers to men of science an almost inexhaustible field -for interesting quantitative research, in which we shall -doubtless at some future time discover the operation of -causes now most mysterious and unaccountable.</p> - - -<h3><i>The Methods of Accurate Measurement.</i></h3> - -<p>In studying the modes by which physicists have accomplished -very exact measurements, we find that they -are very various, but that they may perhaps be reduced -under the following three classes:—</p> - -<p>1. The increase or decrease, in some determinate ratio, -of the quantity to be measured, so as to bring it within -the scope of our senses, and to equate it with the standard -unit, or some determinate multiple or sub-multiple of this -unit.</p> - -<p>2. The discovery of some natural conjunction of events -which will enable us to compare directly the multiples of -the quantity with those of the unit, or a quantity related -in a definite ratio to that unit.</p> - -<p>3. Indirect measurement, which gives us not the quantity -itself, but some other quantity connected with it by -known mathematical relations.</p> - - -<h3><i>Conditions of Accurate Measurement.</i></h3> - -<p>Several conditions are requisite in order that a measurement -may be made with great accuracy, and that -the results may be closely accordant when several independent -measurements are made.</p> - -<p>In the first place the magnitude must be exactly defined -by sharp terminations, or precise marks of inconsiderable<span class="pagenum" id="Page_283">283</span> -thickness. When a boundary is vague and graduated, -like the penumbra in a lunar eclipse, it is impossible to -say where the end really is, and different people will come -to different results. We may sometimes overcome this -difficulty to a certain extent, by observations repeated in -a special manner, as we shall afterwards see; but when -possible, we should choose opportunities for measurement -when precise definition is easy. The moment of -occultation of a star by the moon can be observed with -great accuracy, because the star disappears with perfect -suddenness; but there are other astronomical conjunctions, -eclipses, transits, &c., which occupy a certain length of -time in happening, and thus open the way to differences -of opinion. It would be impossible to observe with precision -the movements of a body possessing no definite -points of reference. The colours of the complete spectrum -shade into each other so continuously that exact determinations -of refractive indices would have been impossible, -had we not the dark lines of the solar spectrum as precise -points for measurement, or various kinds of homogeneous -light, such as that of sodium, possessing a nearly uniform -length of vibration.</p> - -<p>In the second place, we cannot measure accurately -unless we have the means of multiplying or dividing -a quantity without considerable error, so that we may -correctly equate one magnitude with the multiple or submultiple -of the other. In some cases we operate upon the -quantity to be measured, and bring it into accurate coincidence -with the actual standard, as when in photometry -we vary the distance of our luminous body, until its -illuminating power at a certain point is equal to that of a -standard lamp. In other cases we repeat the unit until it -equals the object, as in surveying land, or determining a -weight by the balance. The requisites of accuracy now -are:—(1) That we can repeat unit after unit of exactly -equal magnitude; (2) That these can be joined together -so that the aggregate shall really be the sum of the -parts. The same conditions apply to subdivision, which -may be regarded as a multiplication of subordinate units. -In order to measure to the thousandth of an inch, we must -be able to add thousandth after thousandth without error -in the magnitude of these spaces, or in their conjunction.</p> - -<p><span class="pagenum" id="Page_284">284</span></p> - - -<h3><i>Measuring Instruments.</i></h3> - -<p>To consider the mechanical construction of scientific -instruments, is no part of my purpose in this book. I -wish to point out merely the general purpose of such -instruments, and the methods adopted to carry out that -purpose with great precision. In the first place we must -distinguish between the instrument which effects a comparison -between two quantities, and the standard magnitude -which often forms one of the quantities compared. -The astronomer’s clock, for instance, is no standard of the -efflux of time; it serves but to subdivide, with approximate -accuracy, the interval of successive passages of a -star across the meridian, which it may effect perhaps to -the tenth part of a second, or <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">864000</span></span></span> part of the whole. -The moving globe itself is the real standard clock, and the -transit instrument the finger of the clock, while the stars -are the hour, minute, and second marks, none the less -accurate because they are disposed at unequal intervals. -The photometer is a simple instrument, by which we compare -the relative intensity of rays of light falling upon a -given spot. The galvanometer shows the comparative -intensity of electric currents passing through a wire. -The calorimeter gauges the quantity of heat passing from -a given object. But no such instruments furnish the -standard unit in terms of which our results are to be expressed. -In one peculiar case alone does the same instrument -combine the unit of measurement and the means of -comparison. A theodolite, mural circle, sextant, or other -instrument for the measurement of angular magnitudes -has no need of an additional physical unit; for the circle -itself, or complete revolution, is the natural unit to which -all greater or lesser amounts of angular magnitude are -referred.</p> - -<p>The result of every measurement is to make known the -purely numerical ratio existing between the magnitude -to be measured, and a certain other magnitude, which -should, when possible, be a fixed unit or standard magnitude, -or at least an intermediate unit of which the value -can be ascertained in terms of the ultimate standard. But -though a ratio is the required result, an equation is the -mode in which the ratio is determined and expressed. In<span class="pagenum" id="Page_285">285</span> -every measurement we equate some multiple or submultiple -of one quantity, with some multiple or submultiple -of another, and equality is always the fact which we -ascertain by the senses. By the eye, the ear, or the touch, -we judge whether there is a discrepancy or not between -two lights, two sounds, two intervals of time, two bars of -metal. Often indeed we substitute one sense for the other, -as when the efflux of time is judged by the marks upon -a moving slip of paper, so that equal intervals of time are -represented by equal lengths. There is a tendency to -reduce all comparisons to the comparison of space magnitudes, -but in every case one of the senses must be the -ultimate judge of coincidence or non-coincidence.</p> - -<p>Since the equation to be established may exist between -any multiples or submultiples of the quantities compared, -there naturally arise several different modes of comparison -adapted to different cases. Let <i>p</i> be the magnitude to -be measured, and <i>q</i> that in terms of which it is to be -expressed. Then we wish to find such numbers <i>x</i> and <i>y</i>, -that the equation <i>p</i> = <span class="nowrap"><span class="fraction2"><span class="fnum2"><i>x</i></span><span class="bar">/</span><span class="fden2"><i>y</i></span></span></span><i>q</i> - may be true. This equation -may be presented in four forms, namely:—</p> - -<div class="center"> -<table class="fs80 mtb05em" style="width:60%;"> -<tr> -<td class="tac"><div><div><div>First Form.</div></div></div></td> -<td class="tac"><div><div><div>Second Form.</div></div></div></td> -<td class="tac"><div><div><div>Third Form.</div></div></div></td> -<td class="tac"><div><div><div>Fourth Form.</div></div></div></td> -</tr> -<tr> -<td class="tac"><div><div><div><i>p</i> = <span class="nowrap"><span class="fraction2"><span class="fnum2"><i>x</i></span><span class="bar">/</span><span class="fden2"><i>y</i></span></span></span> <i>q</i></div></div></div></td> -<td class="tac"><div><div><div><i>p</i> - <span class="nowrap"><span class="fraction2"><span class="fnum2"><i>y</i></span><span class="bar">/</span><span class="fden2"><i>x</i></span></span></span> = <i>q</i></div></div></div></td> -<td class="tac"><div><div><div><i>py</i> = <i>qx</i></div></div></div></td> -<td class="tac"><div><div><div><span class="nowrap"><span class="fraction2"><span class="fnum2"><i>p</i></span><span class="bar">/</span><span class="fden2"><i>x</i></span></span></span> = <span class="nowrap"><span class="fraction2"><span class="fnum2"><i>q</i></span><span class="bar">/</span><span class="fden2"><i>y</i></span></span></span></div></div></div></td> -</tr> -</table> -</div> - -<p class="ti0">Each of these modes of expressing the same equation corresponds -to one mode of effecting a measurement.</p> - -<p>When the standard quantity is greater than that to be -measured, we often adopt the first mode, and subdivide -the unit until we get a magnitude equal to that measured. -The angles observed in surveying, in astronomy, or in -goniometry are usually smaller than a whole revolution, -and the measuring circle is divided by the use of the -screw and microscope, until we obtain an angle undistinguishable -from that observed. The dimensions of minute -objects are determined by subdividing the inch or centimetre, -the screw micrometer being the most accurate -means of subdivision. Ordinary temperatures are estimated -by division of the standard interval between the -freezing and boiling points of water, as marked on a -thermometer tube.</p> - -<p><span class="pagenum" id="Page_286">286</span></p> - -<p>In a still greater number of cases, perhaps, we multiply -the standard unit until we get a magnitude equal to that -to be measured. Ordinary measurement by a foot rule, -a surveyor’s chain, or the excessively careful measurements -of the base line of a trigonometrical survey by standard -bars, are sufficient instances of this procedure.</p> - -<p>In the second case, where <i>p</i> <span class="nowrap"><span class="fraction2"><span class="fnum2"><i>y</i></span><span class="bar">/</span><span class="fden2"><i>x</i></span></span></span> = <i>q</i>, we multiply or divide -a magnitude until we get what is equal to the unit, or to -some magnitude easily comparable with it. As a general -rule the quantities which we desire to measure in -physical science are too small rather than too great for -easy determination, and the problem consists in multiplying -them without introducing error. Thus the expansion -of a metallic bar when heated from 0°C to 100° may be -multiplied by a train of levers or cog wheels. In the -common thermometer the expansion of the mercury, -though slight, is rendered very apparent, and easily -measurable by the fineness of the tube, and many other -cases might be quoted. There are some phenomena, on -the contrary, which are too great or rapid to come within -the easy range of our senses, and our task is then the opposite -one of diminution. Galileo found it difficult to measure -the velocity of a falling body, owing to the considerable -velocity acquired in a single second. He adopted the -elegant device, therefore, of lessening the rapidity by -letting the body roll down an inclined plane, which -enables us to reduce the accelerating force in any required -ratio. The same purpose is effected in the well-known -experiments performed on Attwood’s machine, and the -measurement of gravity by the pendulum really depends -on the same principle applied in a far more advantageous -manner. Wheatstone invented a beautiful method of galvanometry -for strong currents, which consists in drawing -off from the main current a certain determinate portion, -which is equated by the galvanometer to a standard -current. In short, he measures not the current itself but -a known fraction of it.</p> - -<p>In many electrical and other experiments, we wish to -measure the movements of a needle or other body, which -are not only very slight in themselves, but the manifestations -of exceedingly small forces. We cannot even<span class="pagenum" id="Page_287">287</span> -approach a delicately balanced needle without disturbing -it. Under these circumstances the only mode of proceeding -with accuracy, is to attach a very small mirror to the -moving body, and employ a ray of light reflected from -the mirror as an index of its movements. The ray may -be considered quite incapable of affecting the body, and -yet by allowing the ray to pass to a sufficient distance, -the motions of the mirror may be increased to almost any -extent. A ray of light is in fact a perfectly weightless -finger or index of indefinite length, with the additional -advantage that the angular deviation is by the law of -reflection double that of the mirror. This method was -introduced by Gauss, and is now of great importance; -but in Wollaston’s reflecting goniometer a ray of light -had previously been employed as an index. Lavoisier -and Laplace had also used a telescope in connection with -the pyrometer.</p> - -<p>It is a great advantage in some instruments that they -can be readily made to manifest a phenomenon in a greater -or less degree, by a very slight change in the construction. -Thus either by enlarging the bulb or contracting the tube -of the thermometer, we can make it give more conspicuous -indications of change of temperature. The ordinary barometer, -on the other hand, always gives the variations of -pressure on one scale. The torsion balance is remarkable -for the extreme delicacy which may be attained -by increasing the length and lightness of the rod, and the -length and thinness of the supporting thread. Forces so -minute as the attraction of gravitation between two balls, -or the magnetic and diamagnetic attraction of common -liquids and gases, may thus be made apparent, and even -measured. The common chemical balance, too, is capable -theoretically of unlimited sensibility.</p> - -<p>The third mode of measurement, which may be called -the Method of Repetition, is of such great importance and -interest that we must consider it in a separate section. It -consists in multiplying both magnitudes to be compared -until some multiple of the first is found to coincide very -nearly with some multiple of the second. If the multiplication -can be effected to an unlimited extent, without the -introduction of countervailing errors, the accuracy with -which the required ratio can be determined is unlimited,<span class="pagenum" id="Page_288">288</span> -and we thus account for the extraordinary precision with -which intervals of time in astronomy are compared together.</p> - -<p>The fourth mode of measurement, in which we equate -submultiples of two magnitudes, is comparatively seldom -employed, because it does not conduce to accuracy. In -the photometer, perhaps, we may be said to use it; we -compare the intensity of two sources of light, by placing -them both at such distances from a given surface, that the -light falling on the surface is tolerable to the eye, and -equally intense from each source. Since the intensity of -light varies inversely as the square of the distance, the -relative intensities of the luminous bodies are proportional -to the squares of their distances. The equal intensity -of two rays of similarly coloured light may be -most accurately ascertained in the mode suggested by -Arago, namely, by causing the rays to pass in opposite -directions through two nearly flat lenses pressed together. -There is an exact equation between the intensities of the -beams when Newton’s rings disappear, the ring created -by one ray being exactly the complement of that created -by the other.</p> - - -<h3><i>The Method of Repetition.</i></h3> - -<p>The ratio of two quantities can be determined with -unlimited accuracy, if we can multiply both the object -of measurement and the standard unit without error, and -then observe what multiple of the one coincides or nearly -coincides with some multiple of the other. Although perfect -coincidence can never be really attained, the error -thus arising may be indefinitely reduced. For if the -equation <i>py</i> = <i>qx</i> be uncertain to the amount <i>e</i>, so -that <i>py</i> = <i>qx</i> ± <i>e</i>, - then we have <i>p</i> = <i>q</i> <span class="nowrap"><span class="fraction2"><span class="fnum2"><i>x</i></span><span class="bar">/</span><span class="fden2"><i>y</i></span></span></span> ± <span class="nowrap"><span class="fraction2"><span class="fnum2"><i>e</i></span><span class="bar">/</span><span class="fden2"><i>y</i></span></span></span> , and -as we are supposed to be able to make <i>y</i> as great as we -like without increasing the error <i>e</i>, it follows that we -can make <i>e</i> ÷ <i>y</i> as small as we like, and thus approximate -within an inconsiderable quantity to the required -ratio <i>x</i> ÷ <i>y</i>.</p> - -<p>This method of repetition is naturally employed whenever -quantities can be repeated, or repeat themselves<span class="pagenum" id="Page_289">289</span> -without error of juxtaposition, which is especially the -case with the motions of the earth and heavenly bodies. -In determining the length of the sidereal day, we determine -the ratio between the earth’s revolution round the -sun, and its rotation on its own axis. We might ascertain -the ratio by observing the successive passages of a star -across the zenith, and comparing the interval by a good -clock with that between two passages of the sun, the -difference being due to the angular movement of the -earth round the sun. In such observations we should -have an error of a considerable part of a second at each -observation, in addition to the irregularities of the clock. -But the revolutions of the earth repeat themselves day -after day, and year after year, without the slightest interval -between the end of one period and the beginning -of another. The operation of multiplication is perfectly -performed for us by nature. If, then, we can find an observation -of the passage of a star across the meridian a hundred -years ago, that is of the interval of time between -the passage of the sun and the star, the instrumental -errors in measuring this interval by a clock and telescope -may be greater than in the present day, but will be -divided by about 36,524 days, and rendered excessively -small. It is thus that astronomers have been able to -ascertain the ratio of the mean solar to the sidereal day -to the 8th place of decimals (1·00273791 to 1), or to the -hundred millionth part, probably the most accurate result -of measurement in the whole range of science.</p> - -<p>The antiquity of this mode of comparison is almost as -great as that of astronomy itself. Hipparchus made the -first clear application of it, when he compared his own -observations with those of Aristarchus, made 145 years -previously, and thus ascertained the length of the year. -This calculation may in fact be regarded as the earliest -attempt at an exact determination of the constants of -nature. The method is the main resource of astronomers; -Tycho, for instance, detected the slow diminution -of the obliquity of the earth’s axis, by the comparison -of observations at long intervals. Living astronomers -use the method as much as earlier ones; but so superior -in accuracy are all observations taken during the last -hundred years to all previous ones, that it is often<span class="pagenum" id="Page_290">290</span> -found preferable to take a shorter interval, rather than -incur the risk of greater instrumental errors in the earlier -observations.</p> - -<p>It is obvious that many of the slower changes of the -heavenly bodies must require the lapse of large intervals -of time to render their amount perceptible. Hipparchus -could not possibly have discovered the smaller inequalities -of the heavenly motions, because there were no previous -observations of sufficient age or exactness to exhibit them. -And just as the observations of Hipparchus formed the -starting-point for subsequent comparisons, so a large part -of the labour of present astronomers is directed to recording -the present state of the heavens so exactly, that future -generations of astronomers may detect changes, which -cannot possibly become known in the present age.</p> - -<p>The principle of repetition was very ingeniously employed -in an instrument first proposed by Mayer in 1767, -and carried into practice in the Repeating Circle of Borda. -The exact measurement of angles is indispensable, not -only in astronomy but also in trigonometrical surveys, and -the highest skill in the mechanical execution of the graduated -circle and telescope will not prevent terminal errors -of considerable amount. If instead of one telescope, the -circle be provided with two similar telescopes, these may -be alternately directed to two distant points, say the -marks in a trigonometrical survey, so that the circle shall -be turned through any multiple of the angle subtended -by those marks, before the amount of the angular revolution -is read off upon the graduated circle. Theoretically -speaking, all error arising from imperfect graduation might -thus be indefinitely reduced, being divided by the number -of repetitions. In practice, the advantage of the invention -is not found to be very great, probably because a certain -error is introduced at each observation in the changing -and fixing of the telescopes. It is moreover inapplicable -to moving objects like the heavenly bodies, so that its use -is confined to important trigonometrical surveys.</p> - -<p>The pendulum is the most perfect of all instruments, -chiefly because it admits of almost endless repetition. -Since the force of gravity never ceases, one swing of the -pendulum is no sooner ended than the other is begun, -so that the juxtaposition of successive units is absolutely<span class="pagenum" id="Page_291">291</span> -perfect. Provided that the oscillations be equal, one -thousand oscillations will occupy exactly one thousand -times as great an interval of time as one oscillation. -Not only is the subdivision of time entirely dependent -on this fact, but in the accurate measurement of gravity, -and many other important determinations, it is of the -greatest service. In the deepest mine, we could not -observe the rapidity of fall of a body for more than a -quarter of a minute, and the measurement of its velocity -would be difficult, and subject to uncertain errors from -resistance of air, &c. In the pendulum, we have a body -which can be kept rising and falling for many hours, in -a medium entirely under our command or if desirable in -a vacuum. Moreover, the comparative force of gravity at -different points, at the top and bottom of a mine for -instance, can be determined with wonderful precision, by -comparing the oscillations of two exactly similar pendulums, -with the aid of electric clock signals.</p> - -<p>To ascertain the comparative times of vibration of two -pendulums, it is only requisite to swing them one in -front of the other, to record by a clock the moment when -they coincide in swing, so that one hides the other, and -then count the number of vibrations until they again come -to coincidence. If one pendulum makes <i>m</i> vibrations and -the other <i>n</i>, we at once have our equation <i>pn</i> = <i>qm</i>; -which gives the length of vibration of either pendulum in -terms of the other. This method of coincidence, embodying -the principle of repetition in perfection, was employed -with wonderful skill by Sir George Airy, in his experiments -on the Density of the Earth at the Harton Colliery, -the pendulums above and below being compared with -clocks, which again were compared with each other by -electric signals. So exceedingly accurate was this method -of observation, as carried out by Sir George Airy, that he -was able to measure a total difference in the vibrations at -the top and bottom of the shaft, amounting to only 2·24 -seconds in the twenty-four hours, with an error of less -than one hundredth part of a second, or one part in -8,640,000 of the whole day.<a id="FNanchor_185" href="#Footnote_185" class="fnanchor">185</a></p> - -<p>The principle of repetition has been elegantly applied<span class="pagenum" id="Page_292">292</span> -in observing the motion of waves in water. If the canal -in which the experiments are made be short, say twenty -feet long, the waves will pass through it so rapidly that -an observation of one length, as practised by Walker, will -be subject to much terminal error, even when the observer -is very skilful. But it is a result of the undulatory theory -that a wave is unaltered, and loses no time by complete -reflection, so that it may be allowed to travel backwards -and forwards in the same canal, and its motion, say -through sixty lengths, or 1200 feet, may be observed with -the same accuracy as in a canal 1200 feet long, with the -advantage of greater uniformity in the condition of the -canal and water.<a id="FNanchor_186" href="#Footnote_186" class="fnanchor">186</a> It is always desirable, if possible, to -bring an experiment into a small compass, so that it -may be well under command, and yet we may often -by repetition enjoy at the same time the advantage of -extensive trial.</p> - -<p>One reason of the great accuracy of weighing with a -good balance is the fact, that weights placed in the same -scale are naturally added together without the slightest -error. There is no difficulty in the precise juxtaposition -of two grams, but the juxtaposition of two metre measures -can only be effected with tolerable accuracy, by the -use of microscopes and many precautions. Hence, the -extreme trouble and cost attaching to the exact measurement -of a base line for a survey, the risk of error entering -at every juxtaposition of the measuring bars, and indefatigable -attention to all the requisite precautions being -necessary throughout the operation.</p> - - -<h3><i>Measurements by Natural Coincidence.</i></h3> - -<p>In certain cases a peculiar conjunction of circumstances -enables us to dispense more or less with instrumental -aids, and to obtain very exact numerical results in the -simplest manner. The mere fact, for instance, that no -human being has ever seen a different face of the moon -from that familiar to us, conclusively proves that the -period of rotation of the moon on its own axis is equal<span class="pagenum" id="Page_293">293</span> -to that of its revolution round the earth. Not only have -we the repetition of these movements during 1000 or -2000 years at least, but we have observations made for -us at very remote periods, free from instrumental error, -no instrument being needed. We learn that the seventh -satellite of Saturn is subject to a similar law, because its -light undergoes a variation in each revolution, owing to -the existence of some dark tract of land; now this failure -of light always occurs while it is in the same position -relative to Saturn, clearly proving the equality of the -axial and revolutional periods, as Huygens perceived.<a id="FNanchor_187" href="#Footnote_187" class="fnanchor">187</a> -A like peculiarity in the motions of Jupiter’s fourth satellite -was similarly detected by Maraldi in 1713.</p> - -<p>Remarkable conjunctions of the planets may sometimes -allow us to compare their periods of revolution, through -great intervals of time, with much accuracy. Laplace in -explaining the long inequality in the motions of Jupiter -and Saturn, was assisted by a conjunction of these -planets, observed at Cairo, towards the close of the -eleventh century. Laplace calculated that such a conjunction -must have happened on the 31st of October, <span class="allsmcap">A.D.</span> -1087; and the discordance between the distances of the -planets as recorded, and as assigned by theory, was less -than one-fifth part of the apparent diameter of the sun. -This difference being less than the probable error of the -early record, the theory was confirmed as far as facts -were available.<a id="FNanchor_188" href="#Footnote_188" class="fnanchor">188</a></p> - -<p>Ancient astronomers often showed the highest ingenuity -in turning any opportunities of measurement which -occurred to good account. Eratosthenes, as early as -250 <span class="allsmcap">B.C.</span>, happening to hear that the sun at Syene, in -Upper Egypt, was visible at the summer solstice at the -bottom of a well, proving that it was in the zenith, proposed -to determine the dimensions of the earth, by measuring -the length of the shadow of a rod at Alexandria on -the same day of the year. He thus learnt in a rude -manner the difference of latitude between Alexandria and -Syene and finding it to be about one fiftieth part of the -whole circumference, he ascertained the dimensions of the<span class="pagenum" id="Page_294">294</span> -earth within about one sixth part of the truth. The use -of wells in astronomical observation appears to have been -occasionally practised in comparatively recent times as -by Flamsteed in 1679.<a id="FNanchor_189" href="#Footnote_189" class="fnanchor">189</a> The Alexandrian astronomers -employed the moon as an instrument of measurement -in several sagacious modes. When the moon is exactly -half full, the moon, sun, and earth, are at the angles of a -right-angled triangle. Aristarchus measured at such a -time the moon’s elongation from the sun, which gave him -the two other angles of the triangle, and enabled him to -judge of the comparative distances of the moon and sun -from the earth. His result, though very rude, was far -more accurate than any notions previously entertained, -and enabled him to form some estimate of the comparative -magnitudes of the bodies. Eclipses of the moon were -very useful to Hipparchus in ascertaining the longitude -of the stars, which are invisible when the sun is above -the horizon. For the moon when eclipsed must be 180° -distant from the sun; hence it is only requisite to measure -the distance of a fixed star in longitude from the eclipsed -moon to obtain with ease its angular distance from the -sun.</p> - -<p>In later times the eclipses of Jupiter have served to -measure an angle; for at the middle moment of the -eclipse the satellite must be in the same straight line with -the planet and sun, so that we can learn from the known -laws of movement of the satellite the longitude of Jupiter -as seen from the sun. If at the same time we measure -the elongation or apparent angular distance of Jupiter -from the sun, as seen from the earth, we have all the -angles of the triangle between Jupiter, the sun, and the -earth, and can calculate the comparative magnitudes of -the sides of the triangle by trigonometry.</p> - -<p>The transits of Venus over the sun’s face are other -natural events which give most accurate measurements -of the sun’s parallax, or apparent difference of position -as seen from distant points of the earth’s surface. The -sun forms a kind of background on which the place of -the planet is marked, and serves as a measuring instrument -free from all the errors of construction which affect<span class="pagenum" id="Page_295">295</span> -human instruments. The rotation of the earth, too, by -variously affecting the apparent velocity of ingress or -egress of Venus, as seen from different places, discloses -the amount of the parallax. It has been sufficiently -shown that by rightly choosing the moments of observation, -the planetary bodies may often be made to reveal -their relative distance, to measure their own position, to -record their own movements with a high degree of -accuracy. With the improvement of astronomical instruments, -such conjunctions become less necessary to the -progress of the science, but it will always remain advantageous -to choose those moments for observation when -instrumental errors enter with the least effect.</p> - -<p>In other sciences, exact quantitative laws can occasionally -be obtained without instrumental measurement, as -when we learn the exactly equal velocity of sounds of -different pitch, by observing that a peal of bells or a -musical performance is heard harmoniously at any distance -to which the sound penetrates; this could not be -the case, as Newton remarked, if one sound overtook -the other. One of the most important principles of the -atomic theory, was proved by implication before the use -of the balance was introduced into chemistry. Wenzel -observed, before 1777, that when two neutral substances -decompose each other, the resulting salts are also neutral. -In mixing sodium sulphate and barium nitrate, we -obtain insoluble barium sulphate and neutral sodium -nitrate. This result could not follow unless the nitric -acid, requisite to saturate one atom of sodium, were -exactly equal to that required by one atom of barium, -so that an exchange could take place without leaving -either acid or base in excess.</p> - -<p>An important principle of mechanics may also be -established by a simple acoustical observation. When -a rod or tongue of metal fixed at one end is set in -vibration, the pitch of the sound may be observed to -be exactly the same, whether the vibrations be small or -great; hence the oscillations are isochronous, or equally -rapid, independently of their magnitude. On the ground -of theory, it can be shown that such a result only -happens when the flexure is proportional to the deflecting -force. Thus the simple observation that the pitch of<span class="pagenum" id="Page_296">296</span> -the sound of a harmonium, for instance, does not change -with its loudness establishes an exact law of nature.<a id="FNanchor_190" href="#Footnote_190" class="fnanchor">190</a></p> - -<p>A closely similar instance is found in the proof that the -intensity of light or heat rays varies inversely as the -square of the distance increases. For the apparent magnitude -certainly varies according to this law; hence, if the -intensity of light varied according to any other law, the -brightness of an object would be different at different -distances, which is not observed to be the case. Melloni -applied the same kind of reasoning, in a somewhat -different form, to the radiation of heat-rays.</p> - - -<h3><i>Modes of Indirect Measurement.</i></h3> - -<p>Some of the most conspicuously beautiful experiments -in the whole range of science, have been devised for the -purpose of indirectly measuring quantities, which in their -extreme greatness or smallness surpass the powers of -sense. All that we need to do, is to discover some -other conveniently measurable phenomenon, which is related -in a known ratio or according to a known law, -however complicated, with that to be measured. Having -once obtained experimental data, there is no further -difficulty beyond that of arithmetic or algebraic calculation.</p> - -<p>Gold is reduced by the gold-beater to leaves so thin, -that the most powerful microscope would not detect any -measurable thickness. If we laid several hundred leaves -upon each other to multiply the thickness, we should -still have no more than <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">100</span></span></span>th of an inch at the most to -measure, and the errors arising in the superposition and -measurement would be considerable. But we can readily -obtain an exact result through the connected amount of -weight. Faraday weighed 2000 leaves of gold, each -<span class="nowrap">3 <span class="fraction"><span class="fnum">3</span><span class="bar">/</span><span class="fden">8</span></span></span> inch square, and found them equal to 384 grains. -From the known specific gravity of gold it was easy to -calculate that the average thickness of the leaves was -<span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">282,000</span></span></span> - of an inch.<a id="FNanchor_191" href="#Footnote_191" class="fnanchor">191</a></p> - -<p>We must ascribe to Newton the honour of leading the<span class="pagenum" id="Page_297">297</span> -way in methods of minute measurement. He did not -call waves of light by their right name, and did not -understand their nature; yet he measured their length, -though it did not exceed the 2,000,000th part of a metre -or the one fifty-thousandth part of an inch. He pressed -together two lenses of large but known radii. It was -easy to calculate the interval between the lenses at any -point, by measuring the distance from the central point -of contact. Now, with homogeneous rays the successive -rings of light and darkness mark the points at which the -interval between the lenses is equal to one half, or any -multiple of half a vibration of the light, so that the -length of the vibration became known. In a similar -manner many phenomena of interference of rays of light -admit of the measurement of the wave lengths. Fringes -of interference arise from rays of light which cross each -other at a small angle, and an excessively minute difference -in the lengths of the waves makes a very perceptible -difference in the position of the point at which two rays -will interfere and produce darkness.</p> - -<p>Fizeau has recently employed Newton’s rings to measure -small amounts of motion. By merely counting the number -of rings of sodium monochromatic light passing a certain -point where two glass plates are in close proximity, he is -able to ascertain with the greatest accuracy and ease the -change of distance between these glasses, produced, for -instance, by the expansion of a metallic bar, connected with -one of the glass plates.<a id="FNanchor_192" href="#Footnote_192" class="fnanchor">192</a></p> - -<p>Nothing excites more admiration than the mode in which -scientific observers can occasionally measure quantities, -which seem beyond the bounds of human observation. -We know the <i>average</i> depth of the Pacific Ocean to be -14,190 feet, not by actual sounding, which would be -impracticable in sufficient detail, but by noticing the -rate of transmission of earthquake waves from the South -American to the opposite coasts, the rate of movement -being connected by theory with the depth of the water.<a id="FNanchor_193" href="#Footnote_193" class="fnanchor">193</a> -In the same way the average depth of the Atlantic Ocean -is inferred to be no less than 22,157 feet, from the velocity<span class="pagenum" id="Page_298">298</span> -of the ordinary tidal waves. A tidal wave again gives -beautiful evidence of an effect of the law of gravity, -which we could never in any other way detect. Newton -estimated that the moon’s force in moving the ocean is -only one part in 2,871,400 of the whole force of gravity, -so that even the pendulum, used with the utmost skill, -would fail to render it apparent. Yet, the immense extent -of the ocean allows the accumulation of the effect into a -very palpable amount; and from the comparative heights -of the lunar and solar tides, Newton roughly estimated -the comparative forces of the moon’s and sun’s gravity at -the earth.<a id="FNanchor_194" href="#Footnote_194" class="fnanchor">194</a></p> - -<p>A few years ago it might have seemed impossible that -we should ever measure the velocity with which a star -approaches or recedes from the earth, since the apparent -position of the star is thereby unaltered. But the spectroscope -now enables us to detect and even measure such -motions with considerable accuracy, by the alteration which -it causes in the apparent rapidity of vibration, and consequently -in the refrangibility of rays of light of definite -colour. And while our estimates of the lateral movements -of stars depend upon our very uncertain knowledge -of their distances, the spectroscope gives the motions -of approach and recess irrespective of other motions excepting -that of the earth. It gives in short the motions of -approach and recess of the stars relatively to the earth.<a id="FNanchor_195" href="#Footnote_195" class="fnanchor">195</a></p> - -<p>The rapidity of vibration for each musical tone, having -been accurately determined by comparison with the Syren -(p. <a href="#Page_10">10</a>), we can use sounds as indirect indications of rapid -vibrations. It is now known that the contraction of a -muscle arises from the periodical contractions of each -separate fibre, and from a faint sound or susurrus which -accompanies the action of a muscle, it is inferred that each -contraction lasts for about one 300th part of a second. -Minute quantities of radiant heat are now always measured -indirectly by the electricity which they produce when falling -upon a thermopile. The extreme delicacy of the method -seems to be due to the power of multiplication at several -points in the apparatus. The number of elements or junctions<span class="pagenum" id="Page_299">299</span> -of different metals in the thermopile can be increased -so that the tension of the electric current derived from the -same intensity of radiation is multiplied; the effect of the -current upon the magnetic needle can be multiplied within -certain bounds, by passing the current many times round -it in a coil; the excursions of the needle can be increased -by rendering it astatic and increasing the delicacy of its -suspension; lastly, the angular divergence can be observed, -with any required accuracy, by the use of an attached -mirror and distant scale viewed through a telescope (p. <a href="#Page_287">287</a>). -Such is the delicacy of this method of measuring heat, that -Dr. Joule succeeded in making a thermopile which would -indicate a difference of 0°·000114 Cent.<a id="FNanchor_196" href="#Footnote_196" class="fnanchor">196</a></p> - -<p>A striking case of indirect measurement is furnished by -the revolving mirror of Wheatstone and Foucault, whereby -a minute interval of time is estimated in the form of an -angular deviation. Wheatstone viewed an electric spark -in a mirror rotating so rapidly, that if the duration of the -spark had been more than one 72,000th part of a second, -the point of light would have appeared elongated to an -angular extent of one-half degree. In the spark, as drawn -directly from a Leyden jar, no elongation was apparent, so -that the duration of the spark was immeasurably small; but -when the discharge took place through a bad conductor, -the elongation of the spark denoted a sensible duration.<a id="FNanchor_197" href="#Footnote_197" class="fnanchor">197</a> -In the hands of Foucault the rotating mirror gave a -measure of the time occupied by light in passing through -a few metres of space.</p> - - -<h3><i>Comparative Use of Measuring Instruments.</i></h3> - -<p>In almost every case a measuring instrument serves, -and should serve only as a means of comparison between -two or more magnitudes. As a general rule, we should -not attempt to make the divisions of the measuring scale -exact multiples or submultiples of the unit, but, regarding -them as arbitrary marks, should determine their values by -comparison with the standard itself. The perpendicular -wires in the field of a transit telescope, are fixed at nearly<span class="pagenum" id="Page_300">300</span> -equal but arbitrary distances, and those distances are afterwards -determined, as first suggested by Malvasia, by watching -the passage of star after star across them, and noting -the intervals of time by the clock. Owing to the perfectly -regular motion of the earth, these time intervals give exact -determinations of the angular intervals. In the same way, -the angular value of each turn of the screw micrometer -attached to a telescope, can be easily and accurately -ascertained.</p> - -<p>When a thermopile is used to observe radiant heat, it -would be almost impossible to calculate on <i>à priori</i> grounds -what is the value of each division of the galvanometer -circle, and still more difficult to construct a galvanometer, -so that each division should have a given value. But this -is quite unnecessary, because by placing the thermopile -before a body of known dimensions, at a known distance, -with a known temperature and radiating power, we measure -a known amount of radiant heat, and inversely measure -the value of the indications of the thermopile. In a -similar way Dr. Joule ascertained the actual temperature -produced by the compression of bars of metal. For having -inserted a small thermopile composed of a single junction -of copper and iron wire, and noted the deflections of the -galvanometer, he had only to dip the bars into water of -different temperatures, until he produced a like deflection, -in order to ascertain the temperature developed by -pressure.<a id="FNanchor_198" href="#Footnote_198" class="fnanchor">198</a></p> - -<p>In some cases we are obliged to accept a very carefully -constructed instrument as a standard, as in the case of a -standard barometer or thermometer. But it is then best -to treat all inferior instruments comparatively only, and -determine the values of their scales by comparison with -the assumed standard.</p> - - -<h3><i>Systematic Performance of Measurements.</i></h3> - -<p>When a large number of accurate measurements have -to be effected, it is usually desirable to make a certain -number of determinations with scrupulous care, and afterwards -use them as points of reference for the remaining<span class="pagenum" id="Page_301">301</span> -determinations. In the trigonometrical survey of a country, -the principal triangulation fixes the relative positions -and distances of a few points with rigid accuracy. A -minor triangulation refers every prominent hill or village -to one of the principal points, and then the details are -filled in by reference to the secondary points. The survey -of the heavens is effected in a like manner. The ancient -astronomers compared the right ascensions of a few principal -stars with the moon, and thus ascertained their positions -with regard to the sun; the minor stars were afterwards -referred to the principal stars. Tycho followed the same -method, except that he used the more slowly moving -planet Venus instead of the moon. Flamsteed was in the -habit of using about seven stars, favourably situated at -points all round the heavens. In his early observations -the distances of the other stars from these standard points -were determined by the use of the quadrant.<a id="FNanchor_199" href="#Footnote_199" class="fnanchor">199</a> Even since -the introduction of the transit telescope and the mural -circle, tables of standard stars are formed at Greenwich, -the positions being determined with all possible accuracy, -so that they can be employed for purposes of reference by -astronomers.</p> - -<p>In ascertaining the specific gravities of substances, all -gases are referred to atmospheric air at a given temperature -and pressure; all liquids and solids are referred to -water. We require to compare the densities of water and -air with great care, and the comparative densities of any -two substances whatever can then be ascertained.</p> - -<p>In comparing a very great with a very small magnitude, -it is usually desirable to break up the process into several -steps, using intermediate terms of comparison. We should -never think of measuring the distance from London to -Edinburgh by laying down measuring rods, throughout the -whole length. A base of several miles is selected on level -ground, and compared on the one hand with the standard -yard, and on the other with the distance of London and -Edinburgh, or any other two points, by trigonometrical -survey. Again, it would be exceedingly difficult to compare -the light of a star with that of the sun, which would -be about thirty thousand million times greater; but Herschel<span class="pagenum" id="Page_302">302</span><a id="FNanchor_200" href="#Footnote_200" class="fnanchor">200</a> -effected the comparison by using the full moon as -an intermediate unit. Wollaston ascertained that the sun -gave 801,072 times as much light as the full moon, and -Herschel determined that the light of the latter exceeded -that of α Centauri 27,408 times, so that we find the ratio -between the light of the sun and star to be that of about -22,000,000,000 to 1.</p> - - -<h3><i>The Pendulum.</i></h3> - -<p>By far the most perfect and beautiful of all instruments -of measurement is the pendulum. Consisting merely of a -heavy body suspended freely at an invariable distance from -a fixed point, it is most simple in construction; yet all the -highest problems of physical measurement depend upon its -careful use. Its excessive value arises from two circumstances.</p> - -<p>(1) The method of repetition is eminently applicable -to it, as already described (p. <a href="#Page_290">290</a>).</p> - -<p>(2) Unlike other instruments, it connects together three -different quantities, those of space, time, and force.</p> - -<p>In most works on natural philosophy it is shown, that -when the oscillations of the pendulum are infinitely small, -the square of the time occupied by an oscillation is directly -proportional to the length of the pendulum, and indirectly -proportional to the force affecting it, of whatever kind. -The whole theory of the pendulum is contained in the -formula, first given by Huygens in his <i>Horologium Oscillatorium</i>.</p> - -<div class="ml5em"> -Time of oscillation = 3·14159 × <span class="fs200 lower">√</span><span class="nowrap"><span class="fraction2"><span class="fnum2"><span class="o">length of pendulum</span></span><span class="bar">/</span><span class="fden2">force</span></span></span>. -</div> - -<p class="ti0">The quantity 3·14159 is the constant ratio of the circumference -and radius of a circle, and is of course known with -accuracy. Hence, any two of the three quantities concerned -being given, the third may be found; or any two -being maintained invariable, the third will be invariable. -Thus a pendulum of invariable length suspended at the -same place, where the force of gravity may be considered -constant, furnishes a measure of time. The same invariable -pendulum being made to vibrate at different points of<span class="pagenum" id="Page_303">303</span> -the earth’s surface, and the times of vibration being astronomically -determined, the force of gravity becomes accurately -known. Finally, with a known force of gravity, -and time of vibration ascertained by reference to the stars, -the length is determinate.</p> - -<p>All astronomical observations depend upon the first -manner of using the pendulum, namely, in the astronomical -clock. In the second employment it has been almost -equally indispensable. The primary principle that gravity -is equal in all matter was proved by Newton’s and Gauss’ -pendulum experiments. The torsion pendulum of Michell, -Cavendish, and Baily, depending upon exactly the same -principles as the ordinary pendulum, gave the density of -the earth, one of the foremost natural constants. Kater -and Sabine, by pendulum observations in different parts -of the earth, ascertained the variation of gravity, whence -comes a determination of the earth’s ellipticity. The laws -of electric and magnetic attraction have also been determined -by the method of vibrations, which is in constant -use in the measurement of the horizontal force of terrestrial -magnetism.</p> - -<p>We must not confuse with the ordinary use of the -pendulum its application by Newton, to show the absence -of internal friction against space,<a id="FNanchor_201" href="#Footnote_201" class="fnanchor">201</a> or to ascertain the laws -of motion and elasticity.<a id="FNanchor_202" href="#Footnote_202" class="fnanchor">202</a> In these cases the extent of -vibration is the quantity measured, and the principles of -the instrument are different.</p> - - -<h3><i>Attainable Accuracy of Measurement.</i></h3> - -<p>It is a matter of some interest to compare the degrees -of accuracy which can be attained in the measurement of -different kinds of magnitude. Few measurements of any -kind are exact to more than six significant figures,<a id="FNanchor_203" href="#Footnote_203" class="fnanchor">203</a> but it -is seldom that such accuracy can be hoped for. Time is -the magnitude which seems to be capable of the most exact -estimation, owing to the properties of the pendulum, and -the principle of repetition described in previous sections.<span class="pagenum" id="Page_304">304</span> -As regards short intervals of time, it has already been -stated that Sir George Airy was able to estimate one part -in 8,640,000, an exactness, as he truly remarks, “almost -beyond conception.”<a id="FNanchor_204" href="#Footnote_204" class="fnanchor">204</a> The ratio between the mean solar -and the sidereal day is known to be about one part in -one hundred millions, or to the eighth place of decimals, -(p. <a href="#Page_289">289</a>).</p> - -<p>Determinations of weight seem to come next in exactness, -owing to the fact that repetition without error is -applicable to them. An ordinary good balance should -show about one part in 500,000 of the load. The finest -balance employed by M. Stas, turned with one part in -825,000 of the load.<a id="FNanchor_205" href="#Footnote_205" class="fnanchor">205</a> But balances have certainly been -constructed to show one part in a million,<a id="FNanchor_206" href="#Footnote_206" class="fnanchor">206</a> and Ramsden is -said to have constructed a balance for the Royal Society, -to indicate one part in seven millions, though this is hardly -credible. Professor Clerk Maxwell takes it for granted that -one part in five millions can be detected, but we ought to -discriminate between what a balance can do when first -constructed, and when in continuous use.</p> - -<p>Determinations of length, unless performed with extraordinary -care, are open to much error in the junction of -the measuring bars. Even in measuring the base line of -a trigonometrical survey, the accuracy generally attained -is only that of about one part in 60,000, or an inch in the -mile; but it is said that in four measurements of a -base line carried out very recently at Cape Comorin, the -greatest error was 0·077 inch in 1·68 mile, or one part in -1,382,400, an almost incredible degree of accuracy. Sir J. -Whitworth has shown that touch is even a more delicate -mode of measuring lengths than sight, and by means of a -splendidly executed screw, and a small cube of iron placed -between two flat-ended iron bars, so as to be suspended -when touching them, he can detect a change of dimension -in a bar, amounting to no more than one-millionth of an -inch.<a id="FNanchor_207" href="#Footnote_207" class="fnanchor">207</a></p> - -<hr class="chap x-ebookmaker-drop"> - -<div class="chapter"> -<p><span class="pagenum" id="Page_305">305</span></p> -<h2 class="nobreak" id="CHAPTER_XIV">CHAPTER XIV.<br> - -<span class="title">UNITS AND STANDARDS OF MEASUREMENT.</span></h2> -</div> - -<p class="ti0">As we have seen, instruments of measurement are -only means of comparison between one magnitude and -another, and as a general rule we must assume some -one arbitrary magnitude, in terms of which all results -of measurement are to be expressed. Mere ratios between -any series of objects will never tell us their -absolute magnitudes; we must have at least one ratio -for each, and we must have one absolute magnitude. The -number of ratios <i>n</i> are expressible in <i>n</i> equations, which -will contain at least <i>n</i> + 1 quantities, so that if we -employ them to make known <i>n</i> magnitudes, we must -have one magnitude known. Hence, whether we are -measuring time, space, density, mass, weight, energy, or -any other physical quantity, we must refer to some concrete -standard, some actual object, which if once lost and -irrecoverable, all our measures lose their absolute meaning. -This concrete standard is in all cases arbitrary in -point of theory, and its selection a question of practical -convenience.</p> - -<p>There are two kinds of magnitude, indeed, which do not -need to be expressed in terms of arbitrary concrete units, -since they pre-suppose the existence of natural standard -units. One case is that of abstract number itself, which -needs no special unit, because any object which exists or -is thought of as separate from other objects (p. <a href="#Page_157">157</a>) furnishes -us with a unit, and is the only standard required.</p> - -<p>Angular magnitude is the second case in which -we have a natural unit of reference, namely the whole<span class="pagenum" id="Page_306">306</span> -revolution or <i>perigon</i>, as it has been called by Mr. Sandeman.<a id="FNanchor_208" href="#Footnote_208" class="fnanchor">208</a> -It is a necessary result of the uniform properties -of space, that all complete revolutions are equal to each -other, so that we need not select any one revolution, but -can always refer anew to space itself. Whether we take -the whole perigon, its half, or its quarter, is really immaterial; -Euclid took the right angle, because the Greek geometers -had never generalised their notions of angular -magnitude sufficiently to treat angles of all magnitudes, or -of unlimited <i>quantity of revolution</i>. Euclid defines a right -angle as half that made by a line with its own continuation, -which is of course equal to half a revolution, but which -was not treated as an angle by him. In mathematical -analysis a different fraction of the perigon is taken, namely, -such a fraction that the arc or portion of the circumference -included within it is equal to the radius of the circle. In -this point of view angular magnitude is an abstract ratio, -namely, the ratio between the length of arc subtended and -the length of the radius. The geometrical unit is then -necessarily the angle corresponding to the ratio unity. -This angle is equal to about 57°, 17′, 44″·8, or decimally -57°·295779513... .<a id="FNanchor_209" href="#Footnote_209" class="fnanchor">209</a> It was called by De Morgan the <i>arcual -unit</i>, but a more convenient name for common use would -be <i>radian</i>, as suggested by Professor Everett. Though this -standard angle is naturally employed in mathematical -analysis, and any other unit would introduce great complexity, -we must not look upon it as a distinct unit, since -its amount is connected with that of the half perigon, -by the natural constant 3·14159... usually denoted by -the letter π.</p> - -<p>When we pass to other species of quantity, the choice -of unit is found to be entirely arbitrary. There is absolutely -no mode of defining a length, but by selecting some -physical object exhibiting that length between certain -obvious points—as, for instance, the extremities of a bar, -or marks made upon its surface.</p> - -<p><span class="pagenum" id="Page_307">307</span></p> - - -<h3><i>Standard Unit of Time.</i></h3> - -<p>Time is the great independent variable of all change—that -which itself flows on uninterruptedly, and brings the -variety which we call motion and life. When we reflect -upon its intimate nature, Time, like every other element of -existence, proves to be an inscrutable mystery. We can -only say with St. Augustin, to one who asks us what is -time, “I know when you do not ask me.” The mind of -man will ask what can never be answered, but one result -of a true and rigorous logical philosophy must be to -convince us that scientific explanation can only take place -between phenomena which have something in common, -and that when we get down to primary notions, like those -of time and space, the mind must meet a point of mystery -beyond which it cannot penetrate. A definition of time -must not be looked for; if we say with Hobbes,<a id="FNanchor_210" href="#Footnote_210" class="fnanchor">210</a> that it -is “the phantasm of before and after in motion,” or with -Aristotle that it is “the number of motion according to -former and latter,” we obviously gain nothing, because -the notion of time is involved in the expressions <i>before -and after</i>, <i>former and latter</i>. Time is undoubtedly one -of those primary notions which can only be defined physically, -or by observation of phenomena which proceed in -time.</p> - -<p>If we have not advanced a step beyond Augustin’s acute -reflections on this subject,<a id="FNanchor_211" href="#Footnote_211" class="fnanchor">211</a> it is curious to observe the -wonderful advances which have been made in the practical -measurement of its efflux. In earlier centuries the rude -sun-dial or the rising of a conspicuous star gave points of -reference, while the flow of water from the clepsydra, the -burning of a candle, or, in the monastic ages, even the -continuous chanting of psalms, were the means of roughly -subdividing periods, and marking the hours of the day and -night.<a id="FNanchor_212" href="#Footnote_212" class="fnanchor">212</a> The sun and stars still furnish the standard of -time, but means of accurate subdivision have become -requisite, and this has been furnished by the pendulum<span class="pagenum" id="Page_308">308</span> -and the chronograph. By the pendulum we can accurately -divide the day into seconds of time. By the chronograph -we can subdivide the second into a hundred, a thousand, -or even a million parts. Wheatstone measured the duration -of an electric spark, and found it to be no more than -one 115,200th part of a second, while more recently -Captain Noble has been able to appreciate intervals of -time not exceeding the millionth part of a second.</p> - -<p>When we come to inquire precisely what phenomenon -it is that we thus so minutely measure, we meet insurmountable -difficulties. Newton distinguished time according -as it was <i>absolute</i> or <i>apparent</i> time, in the following -words:—“Absolute, true, and mathematical time, of itself -and from its own nature, flows equably without regard to -anything external, and by another name is called <i>duration</i>; -relative, apparent and common time, is some sensible and -external measure of duration by the means of motion.”<a id="FNanchor_213" href="#Footnote_213" class="fnanchor">213</a> -Though we are perhaps obliged to assume the existence -of a uniformly increasing quantity which we call time, -yet we cannot feel or know abstract and absolute time. -Duration must be made manifest to us by the recurrence -of some phenomenon. The succession of our own thoughts -is no doubt the first and simplest measure of time, but a -very rude one, because in some persons and circumstances -the thoughts evidently flow with much greater rapidity -than in other persons and circumstances. In the absence -of all other phenomena, the interval between one thought -and another would necessarily become the unit of time, -but the most cursory observations show that there are -changes in the outward world much better fitted by their -constancy to measure time than the change of thoughts -within us.</p> - -<p>The earth, as I have already said, is the real clock of the -astronomer, and is practically assumed as invariable in -its movements. But on what ground is it so assumed? -According to the first law of motion, every body perseveres -in its state of rest or of uniform motion in a right line, -unless it is compelled to change that state by forces impressed -thereon. Rotatory motion is subject to a like<span class="pagenum" id="Page_309">309</span> -condition, namely, that it perseveres uniformly unless disturbed -by extrinsic forces. Now uniform motion means -motion through equal spaces in equal times, so that if we -have a body entirely free from all resistance or perturbation, -and can measure equal spaces of its path, we have a -perfect measure of time. But let it be remembered that -this law has never been absolutely proved by experience; -for we cannot point to any body, and say that it is wholly -unresisted or undisturbed; and even if we had such a body, -we should need some independent standard of time to -ascertain whether its motion was really uniform. As it -is in moving bodies that we find the best standard of time, -we cannot use them to prove the uniformity of their own -movements, which would amount to a <i>petitio principii</i>. -Our experience comes to this, that when we examine and -compare the movements of bodies which seem to us nearly -free from disturbance, we find them giving nearly harmonious -measures of time. If any one body which seems -to us to move uniformly is not doing so, but is subject to -fits and starts unknown to us, because we have no absolute -standard of time, then all other bodies must be subject to -the same arbitrary fits and starts, otherwise there would be -discrepancy disclosing the irregularities. Just as in comparing -together a number of chronometers, we should soon -detect bad ones by their going irregularly, as compared -with the others, so in nature we detect disturbed movement -by its discrepancy from that of other bodies which we -believe to be undisturbed, and which agree nearly among -themselves. But inasmuch as the measure of motion -involves time, and the measure of time involves motion, -there must be ultimately an assumption. We may define -equal times, as times during which a moving body under -the influence of no force describes equal spaces;<a id="FNanchor_214" href="#Footnote_214" class="fnanchor">214</a> but all -we can say in support of this definition is, that it leads us -into no known difficulties, and that to the best of our experience -one freely moving body gives the same results as -any other.</p> - -<p>When we inquire where the freely moving body is, no -perfectly satisfactory answer can be given. Practically -the rotating globe is sufficiently accurate, and Thomson<span class="pagenum" id="Page_310">310</span> -and Tait say: “Equal times are times during which the -earth turns through equal angles.”<a id="FNanchor_215" href="#Footnote_215" class="fnanchor">215</a> No long time has -passed since astronomers thought it impossible to detect -any inequality in its movement. Poisson was supposed -to have proved that a change in the length of the sidereal -day amounting to one ten-millionth part in 2,500 years was -incompatible with an ancient eclipse recorded by the -Chaldæans, and similar calculations were made by Laplace. -But it is now known that these calculations were somewhat -in error, and that the dissipation of energy arising -out of the friction of tidal waves, and the radiation of the -heat into space, has slightly decreased the rapidity of the -earth’s rotatory motion. The sidereal day is now longer by -one part in 2,700,000, than it was in 720 <span class="allsmcap">B.C.</span> Even before -this discovery, it was known that invariability of rotation -depended upon the perfect maintenance of the earth’s -internal heat, which is requisite in order that the earth’s -dimensions shall be unaltered. Now the earth being -superior in temperature to empty space, must cool more or -less rapidly, so that it cannot furnish an absolute measure -of time. Similar objections could be raised to all other -rotating bodies within our cognisance.</p> - -<p>The moon’s motion round the earth, and the earth’s -motion round the sun, form the next best measure of -time. They are subject, indeed, to disturbance from other -planets, but it is believed that these perturbations must -in the course of time run through their rhythmical courses, -leaving the mean distances unaffected, and consequently, -by the third Law of Kepler, the periodic times unchanged. -But there is more reason than not to believe that the earth -encounters a slight resistance in passing through space, -like that which is so apparent in Encke’s comet. There -may also be dissipation of energy in the electrical relations -of the earth to the sun, possibly identical with that which -is manifested in the retardation of comets.<a id="FNanchor_216" href="#Footnote_216" class="fnanchor">216</a> It is probably -an untrue assumption then, that the earth’s orbit remains -quite invariable. It is just possible that some other body -may be found in the course of time to furnish a better<span class="pagenum" id="Page_311">311</span> -standard of time than the earth in its annual motion. -The greatly superior mass of Jupiter and its satellites, and -their greater distance from the sun, may render the -electrical dissipation of energy less considerable than in -the case of the earth. But the choice of the best measure -will always be an open one, and whatever moving body -we choose may ultimately be shown to be subject to -disturbing forces.</p> - -<p>The pendulum, although so admirable an instrument for -subdivision of time, fails as a standard; for though the -same pendulum affected by the same force of gravity performs -equal vibrations in equal times, yet the slightest -change in the form or weight of the pendulum, the least -corrosion of any part, or the most minute displacement of -the point of suspension, falsifies the results, and there enter -many other difficult questions of temperature, friction, -resistance, length of vibration, &c.</p> - -<p>Thomson and Tait are of opinion<a id="FNanchor_217" href="#Footnote_217" class="fnanchor">217</a> that the ultimate -standard of chronometry must be founded on the physical -properties of some body of more constant character than -the earth; for instance, a carefully arranged metallic -spring, hermetically sealed in an exhausted glass vessel. -But it is hard to see how we can be sure that the dimensions -and elasticity of a piece of wrought metal will -remain perfectly unchanged for the few millions of years -contemplated by them. A nearly perfect gas, like -hydrogen, is perhaps the only kind of substance in the -unchanged elasticity of which we could have confidence. -Moreover, it is difficult to perceive how the undulations of -such a spring could be observed with the requisite -accuracy. More recently Professor Clerk Maxwell has -made the novel suggestion, discussed in a subsequent -section, that undulations of light <i>in vacuo</i> would form the -most universal standard of reference, both as regards time -and space. According to this system the unit of time -would be the time occupied by one vibration of the particular -kind of light whose wave length is taken as the -unit of length.</p> -<p><span class="pagenum" id="Page_312">312</span></p> - -<h3><i>The Unit of Space and the Bar Standard.</i></h3> - -<p>Next in importance after the measurement of time is -that of space. Time comes first in theory, because phenomena, -our internal thoughts for instance, may change in -time without regard to space. As to the phenomena -of outward nature, they tend more and more to resolve -themselves into motions of molecules, and motion cannot -be conceived or measured without reference both to time -and space.</p> - -<p>Turning now to space measurement, we find it almost -equally difficult to fix and define once and for ever, a unit -magnitude. There are three different modes in which -it has been proposed to attempt the perpetuation of a -standard length.</p> - -<p>(1) By constructing an actual specimen of the standard -yard or metre, in the form of a bar.</p> - -<p>(2) By assuming the globe itself to be the ultimate -standard of magnitude, the practical unit being a submultiple -of some dimension of the globe.</p> - -<p>(3) By adopting the length of the simple seconds pendulum, -as a standard of reference.</p> - -<p>At first sight it might seem that there was no great -difficulty in this matter, and that any one of these methods -might serve well enough; but the more minutely we -inquire into the details, the more hopeless appears to be -the attempt to establish an invariable standard. We must -in the first place point out a principle not of an obvious -character, namely, that <i>the standard length must be defined -by one single object</i>.<a id="FNanchor_218" href="#Footnote_218" class="fnanchor">218</a> To make two bars of exactly the -same length, or even two bars bearing a perfectly defined -ratio to each other, is beyond the power of human art. If -two copies of the standard metre be made and declared -equally correct, future investigators will certainly discover -some discrepancy between them, proving of course that they -cannot both be the standard, and giving cause for dispute -as to what magnitude should then be taken as correct.</p> - -<p>If one invariable bar could be constructed and maintained -as the absolute standard, no such inconvenience -could arise. Each successive generation as it acquired<span class="pagenum" id="Page_313">313</span> -higher powers of measurement, would detect errors in -the copies of the standard, but the standard itself would -be unimpeached, and would, as it were, become by degrees -more and more accurately known. Unfortunately to construct -and preserve a metre or yard is also a task which -is either impossible, or what comes nearly to the same -thing, cannot be shown to be possible. Passing over the -practical difficulty of defining the ends of the standard -length with complete accuracy, whether by dots or lines -on the surface, or by the terminal points of the bar, we -have no means of proving that substances remain of invariable -dimensions. Just as we cannot tell whether the -rotation of the earth is uniform, except by comparing it -with other moving bodies, believed to be more uniform -in motion, so we cannot detect the change of length in a -bar, except by comparing it with some other bar supposed -to be invariable. But how are we to know which -is the invariable bar? It is certain that many rigid -and apparently invariable substances do change in dimensions. -The bulb of a thermometer certainly contracts -by age, besides undergoing rapid changes of dimensions -when warmed or cooled through 100° Cent. Can we -be sure that even the most solid metallic bars do not -slightly contract by age, or undergo variations in their -structure by change of temperature. Fizeau was induced -to try whether a quartz crystal, subjected to several -hundred alternations of temperature, would be modified in -its physical properties, and he was unable to detect any -change in the coefficient of expansion.<a id="FNanchor_219" href="#Footnote_219" class="fnanchor">219</a> It does not -follow, however, that, because no apparent change was -discovered in a quartz crystal, newly-constructed bars of -metal would undergo no change.</p> - -<p>The best principle, as it seems to me, upon which the -perpetuation of a standard of length can be rested, is that, -if a variation of length occurs, it will in all probability be -of different amount in different substances. If then a -great number of standard metres were constructed of all -kinds of different metals and alloys; hard rocks, such as -granite, serpentine, slate, quartz, limestone; artificial -substances, such as porcelain, glass, &c., &c., careful<span class="pagenum" id="Page_314">314</span> -comparison would show from time to time the comparative -variations of length of these different substances. The -most variable substances would be the most divergent, and -the standard would be furnished by the mean length -of those which agreed most closely with each other just -as uniform motion is that of those bodies which agree -most closely in indicating the efflux of time.</p> - - -<h3><i>The Terrestrial Standard.</i></h3> - -<p>The second method assumes that the globe itself is a -body of invariable dimensions and the founders of the metrical -system selected the ten-millionth part of the distance -from the equator to the pole as the definition of the -metre. The first imperfection in such a method is that the -earth is certainly not invariable in size; for we know -that it is superior in temperature to surrounding space, and -must be slowly cooling and contracting. There is much -reason to believe that all earthquakes, volcanoes, mountain -elevations, and changes of sea level are evidences of this -contraction as asserted by Mr. Mallet.<a id="FNanchor_220" href="#Footnote_220" class="fnanchor">220</a> But such is the -vast bulk of the earth and the duration of its past existence, -that this contraction is perhaps less rapid in proportion -than that of any bar or other material standard which -we can construct.</p> - -<p>The second and chief difficulty of this method arises -from the vast size of the earth, which prevents us from -making any comparison with the ultimate standard, except -by a trigonometrical survey of a most elaborate and -costly kind. The French physicists, who first proposed -the method, attempted to obviate this inconvenience by -carrying out the survey once for all, and then constructing -a standard metre, which should be exactly the one ten -millionth part of the distance from the pole to the -equator. But since all measuring operations are merely -approximate, it was impossible that this operation could be -perfectly achieved. Accordingly, it was shown in 1838 -that the supposed French metre was erroneous to the considerable -extent of one part in 5527. It then became -necessary either to alter the length of the assumed metre,<span class="pagenum" id="Page_315">315</span> -or to abandon its supposed relation to the earth’s dimensions. -The French Government and the International -Metrical Commission have for obvious reasons decided in -favour of the latter course, and have thus reverted to the -first method of defining the metre by a given bar. As -from time to time the ratio between this assumed standard -metre and the quadrant of the earth becomes more accurately -known, we have better means of restoring that metre -by reference to the globe if required. But until lost, destroyed, -or for some clear reason discredited, the bar metre -and not the globe is the standard. Thomson and Tait remark -that any of the more accurate measurements of the -English trigonometrical survey might in like manner be -employed to restore our standard yard, in terms of which -the results are recorded.</p> - - -<h3><i>The Pendulum Standard.</i></h3> - -<p>The third method of defining a standard length, by -reference to the seconds pendulum, was first proposed by -Huyghens, and was at one time adopted by the English -Government. From the principle of the pendulum (p. <a href="#Page_302">302</a>) -it clearly appears that if the time of oscillation and the -force actuating the pendulum be the same, the length of -the pendulum must be the same. We do not get rid of -theoretical difficulties, for we must assume the attraction -of gravity at some point of the earth’s surface, say -London, to be unchanged from time to time, and the -sidereal day to be invariable, neither assumption being -absolutely correct so far as we can judge. The pendulum, -in short, is only an indirect means of making one physical -quantity of space depend upon two other physical quantities -of time and force.</p> - -<p>The practical difficulties are, however, of a far more -serious character than the theoretical ones. The length -of a pendulum is not the ordinary length of the instrument, -which might be greatly varied without affecting the -duration of a vibration, but the distance from the centre of -suspension to the centre of oscillation. There are no -direct means of determining this latter centre, which -depends upon the average momentum of all the particles<span class="pagenum" id="Page_316">316</span> -of the pendulum as regards the centre of suspension. -Huyghens discovered that the centres of suspension -and oscillation are interchangeable, and Kater pointed out -that if a pendulum vibrates with exactly the same rapidity -when suspended from two different points, the distance -between these points is the true length of the equivalent -simple pendulum.<a id="FNanchor_221" href="#Footnote_221" class="fnanchor">221</a> But the practical difficulties in employing -Kater’s reversible pendulum are considerable, and -questions regarding the disturbance of the air, the force -of gravity, or even the interference of electrical attractions -have to be entertained. It has been shown that all the -experiments made under the authority of Government for -determining the ratio between the standard yard and the -seconds pendulum, were vitiated by an error in the corrections -for the resisting, adherent, or buoyant power of the -air in which the pendulums were swung. Even if such -corrections were rendered unnecessary by operating in a -vacuum, other difficult questions remain.<a id="FNanchor_222" href="#Footnote_222" class="fnanchor">222</a> Gauss’ mode of -comparing the vibrations of a wire pendulum when suspended -at two different lengths is open to equal or greater -practical difficulties. Thus it is found that the pendulum -standard cannot compete in accuracy and certainty with -the simple bar standard, and the method would only be -useful as an accessory mode of restoring the bar standard -if at any time again destroyed.</p> - - -<h3><i>Unit of Density.</i></h3> - -<p>Before we can measure the phenomena of nature, we -require a third independent unit, which shall enable us to -define the quantity of matter occupying any given space. -All the changes of nature, as we shall see, are probably so -many manifestations of energy; but energy requires some -substratum or material machinery of molecules, in and by -which it may be manifested. Observation shows that, as -regards force, there may be two modes of variation of -matter. As Newton says in the first definition of the -Principia, “the quantity of matter is the measure of the -same, arising from its density and bulk conjunctly.”<span class="pagenum" id="Page_317">317</span> -Thus the force required to set a body in motion varies -both according to the bulk of the matter, and also according -to its quality. Two cubic inches of iron of uniform -quality, will require twice as much force as one cubic inch -to produce a certain velocity in a given time; but one cubic -inch of gold will require more force than one cubic inch of -iron. There is then some new measurable quality in -matter apart from its bulk, which we may call <i>density</i>, and -which is, strictly speaking, indicated by its capacity to -resist and absorb the action of force. For the unit of -density we may assume that of any substance which is uniform -in quality, and can readily be referred to from time to -time. Pure water at any definite temperature, for instance -that of snow melting under inappreciable pressure, furnishes -an invariable standard of density, and by comparing -equal bulks of various substances with a like bulk of -ice-cold water, as regards the velocity produced in a unit -of time by the same force, we should ascertain the densities -of those substances as expressed in that of water. Practically -the force of gravity is used to measure density; for a -beautiful experiment with the pendulum, performed by -Newton and repeated by Gauss, shows that all kinds of -matter gravitate equally. Two portions of matter then -which are in equilibrium in the balance, may be assumed -to possess equal inertia, and their densities will therefore -be inversely as their cubic dimensions.</p> - - -<h3><i>Unit of Mass.</i></h3> - -<p>Multiplying the number of units of density of a portion -of matter, by the number of units of space occupied by it, -we arrive at the quantity of matter, or, as it is usually -called, the <i>unit of mass</i>, as indicated by the inertia and -gravity it possesses. To proceed in the most simple -manner, the unit of mass ought to be that of a cubic unit -of matter of the standard density; but the founders of -the metrical system took as their unit of mass, the cubic -centimetre of water, at the temperature of maximum -density (about 4° Cent.). They called this unit of mass -the <i>gramme</i>, and constructed standard specimens of the -kilogram, which might be readily referred to by all who -required to employ accurate weights. Unfortunately the<span class="pagenum" id="Page_318">318</span> -determination of the bulk of a given weight of water at a -certain temperature is an operation involving many difficulties, -and it cannot be performed in the present day -with a greater exactness than that of about one part in -5000, the results of careful observers being sometimes -found to differ as much as one part in 1000.<a id="FNanchor_223" href="#Footnote_223" class="fnanchor">223</a></p> - -<p>Weights, on the other hand, can be compared with -each other to at least one part in a million. Hence if -different specimens of the kilogram be prepared by direct -weighing against water, they will not agree closely with -each other; the two principal standard kilograms agree -neither with each other, nor with their definition. According -to Professor Miller the so-called Kilogramme des -Archives weighs 15432·34874 grains, while the kilogram -deposited at the Ministry of the Interior in Paris, as the -standard for commercial purposes, weighs 15432·344 grains. -Since a standard weight constructed of platinum, or platinum -and iridium, can be preserved free from any appreciable -alteration, and since it can be very accurately compared -with other weights, we shall ultimately attain the -greatest exactness in our measurements of mass, by assuming -some single kilogram as a <i>provisional standard</i>, leaving -the determination of its actual mass in units of space and -density for future investigation. This is what is practically -done at the present day, and thus a unit of mass -takes the place of the unit of density, both in the French -and English systems. The English pound is defined by a -certain lump of platinum, preserved at Westminster, and -is an arbitrary mass, chosen merely that it may agree as -nearly as possible with old English pounds. The gallon, -the old English unit of cubic measurement, is defined by -the condition that it shall contain exactly ten pounds -weight of water at 62° Fahr.; and although it is stated that -it has the capacity of about 277·274 cubic inches, this -ratio between the cubic and linear systems of measurement -is not legally enacted, but left open to investigation. -While the French metric system as originally designed -was theoretically perfect, it does not differ practically in -this point from the English system.</p> - -<p><span class="pagenum" id="Page_319">319</span></p> - - -<h3><i>Natural System of Standards.</i></h3> - -<p>Quite recently Professor Clerk Maxwell has suggested -that the vibrations of light and the atoms of matter might -conceivably be employed as the ultimate standards of -length, time, and mass. We should thus arrive at a -<i>natural system of standards</i>, which, though possessing no -present practical importance, has considerable theoretical -interest. “In the present state of science,” he says, “the -most universal standard of length which we could assume -would be the wave-length in vacuum of a particular kind -of light, emitted by some widely diffused substance such -as sodium, which has well-defined lines in its spectrum. -Such a standard would be independent of any changes in -the dimensions of the earth, and should be adopted by -those who expect their writings to be more permanent than -that body.”<a id="FNanchor_224" href="#Footnote_224" class="fnanchor">224</a> In the same way we should get a universal -standard unit of time, independent of all questions about -the motion of material bodies, by taking as the unit the -periodic time of vibration of that particular kind of light -whose wave-length is the unit of length. It would follow -that with these units of length and time the unit of -velocity would coincide with the velocity of light in empty -space. As regards the unit of mass, Professor Maxwell, -humorously as I should think, remarks that if we expect -soon to be able to determine the mass of a single molecule -of some standard substance, we may wait for this determination -before fixing a universal standard of mass.</p> - -<p>In a theoretical point of view there can be no reasonable -doubt that vibrations of light are, as far as we can tell, the -most fixed in magnitude of all phenomena. There is as -usual no certainty in the matter, for the properties of the -basis of light may vary to some extent in different parts of -space. But no differences could ever be established in the -velocity of light in different parts of the solar system, and -the spectra of the stars show that the times of vibration -there do not differ perceptibly from those in this part of -the universe. Thus all presumption is in favour of the -absolute constancy of the vibrations of light—absolute, -that is, so far as regards any means of investigation we are<span class="pagenum" id="Page_320">320</span> -likely to possess. Nearly the same considerations apply -to the atomic weight as the standard of mass. It is impossible -to prove that all atoms of the same substance are -of equal mass, and some physicists think that they differ, so -that the fixity of combining proportions may be due only -to the approximate constancy of the mean of countless -millions of discrepant weights. But in any case the detection -of difference is probably beyond our powers. In a -theoretical point of view, then, the magnitudes suggested -by Professor Maxwell seem to be the most fixed ones of -which we have any knowledge, so that they necessarily -become the natural units.</p> - -<p>In a practical point of view, as Professor Maxwell would -be the first to point out, they are of little or no value, because -in the present state of science we cannot measure a -vibration or weigh an atom with any approach to the -accuracy which is attainable in the comparison of standard -metres and kilograms. The velocity of light is not known -probably within a thousandth part, and as we progress in -the knowledge of light, so we shall progress in the accurate -fixation of other standards. All that can be said then, -is that it is very desirable to determine the wave-lengths -and periods of the principal lines of the solar spectrum, -and the absolute atomic weights of the elements, with all -attainable accuracy, in terms of our existing standards. -The numbers thus obtained would admit of the reproduction -of our standards in some future age of the world to a -corresponding degree of accuracy, were there need of such -reference; but so far as we can see at present, there is no -considerable probability that this mode of reproduction -would ever be the best mode.</p> - - -<h3><i>Subsidiary Units.</i></h3> - -<p>Having once established the standard units of time, -space, and density or mass, we might employ them for the -expression of all quantities of such nature. But it is often -convenient in particular branches of science to use multiples -or submultiples of the original units, for the expression -of quantities in a simple manner. We use the -mile rather than the yard when treating of the magnitude -of the globe, and the mean distance of the earth and<span class="pagenum" id="Page_321">321</span> -sun is not too large a unit when we have to describe -the distances of the stars. On the other hand, when we -are occupied with microscopic objects, the inch, the line -or the millimetre, become the most convenient terms of -expression.</p> - -<p>It is allowable for a scientific man to introduce a new -unit in any branch of knowledge, provided that it assists -precise expression, and is carefully brought into relation -with the primary units. Thus Professor A. W. Williamson -has proposed as a convenient unit of volume in chemical -science, an absolute volume equal to about 11·2 litres -representing the bulk of one gram of hydrogen gas at -standard temperature and pressure, or the <i>equivalent</i> weight -of any other gas, such as 16 grams of oxygen, 14 grams -of nitrogen, &c.; in short, the bulk of that quantity of -any one of those gases which weighs as many grams as -there are units in the number expressing its atomic -weight.<a id="FNanchor_225" href="#Footnote_225" class="fnanchor">225</a> Hofmann has proposed a new unit of weight for -chemists, called a <i>crith</i>, to be defined by the weight of one -litre of hydrogen gas at 0° C. and 0°·76 mm., weighing -about 0·0896 gram.<a id="FNanchor_226" href="#Footnote_226" class="fnanchor">226</a> Both of these units must be regarded -as purely subordinate units, ultimately defined by -reference to the primary units, and not involving any new -assumption.</p> - - -<h3><i>Derived Units.</i></h3> - -<p>The standard units of time, space, and mass having been -once fixed, many kinds of magnitude are naturally measured -by units derived from them. From the metre, the unit of -linear magnitude follows in the most obvious manner the -centiare or square metre, the unit of superficial magnitude, -and the litre that is the cube of the tenth part of a metre, -the unit of capacity or volume. Velocity of motion is expressed -by the ratio of the space passed over, when the -motion is uniform, to the time occupied; hence the unit -of velocity is that of a body which passes over a unit -of space in a unit of time. In physical science the -unit of velocity might be taken as one metre per second.<span class="pagenum" id="Page_322">322</span> -Momentum is measured by the mass moving, regard being -paid both to the amount of matter and the velocity at -which it is moving. Hence the unit of momentum will be -that of a unit volume of matter of the unit density moving -with the unit velocity, or in the French system, a cubic -centimetre of water of the maximum density moving one -metre per second.</p> - -<p>An accelerating force is measured by the ratio of the -momentum generated to the time occupied, the force -being supposed to act uniformly. The unit of force will -therefore be that which generates a unit of momentum -in a unit of time, or which causes, in the French system, -one cubic centimetre of water at maximum density to -acquire in one second a velocity of one metre per second. -The force of gravity is the most familiar kind of force, -and as, when acting unimpeded upon any substance, it -produces in a second a velocity of 9·80868 . . metres -per second in Paris, it follows that the absolute unit -of force is about the tenth part of the force of gravity. -If we employ British weights and measures, the absolute -unit of force is represented by the gravity of about half -an ounce, since the force of gravity of any portion of -matter acting upon that matter during one second, produces -a final velocity of 32·1889 feet per second or about -32 units of velocity. Although from its perpetual action -and approximate uniformity we find in gravity the most -convenient force for reference, and thus habitually employ -it to estimate quantities of matter, we must remember -that it is only one of many instances of force. Strictly -speaking, we should express weight in terms of force, but -practically we express other forces in terms of weight.</p> - -<p>We still require the unit of energy, a more complex -notion. The momentum of a body expresses the -quantity of motion which belongs or would belong to the -aggregate of the particles; but when we consider how this -motion is related to the action of a force producing or -removing it, we find that the effect of a force is proportional -to the mass multiplied by the square of the -velocity and it is convenient to take half this product -as the expression required. But it is shown in books -upon dynamics that it will be exactly the same thing if -we define energy by a force acting through a space. The<span class="pagenum" id="Page_323">323</span> -natural unit of energy will then be that which overcomes -a unit of force acting through a unit of space; when we -lift one kilogram through one metre, against gravity, we -therefore accomplish 9·80868... units of work, that is, we -turn so many units of potential energy existing in the -muscles, into potential energy of gravitation. In lifting -one pound through one foot there is in like manner a conversion -of 32·1889 units of energy. Accordingly the -unit of energy will be in the English system, that required -to lift one pound through about the thirty-second part of -a foot; in terms of metric units, it will be that required to -lift a kilogram through about one tenth part of a metre.</p> - -<p>Every person is at liberty to measure and record -quantities in terms of any unit which he likes. He -may use the yard for linear measurement and the litre -for cubic measurement, only there will then be a complicated -relation between his different results. The -system of derived units which we have been briefly considering, -is that which gives the most simple and natural -relations between quantitative expressions of different -kinds, and therefore conduces to ease of comprehension -and saving of laborious calculation.</p> - -<p>It would evidently be a source of great convenience if -scientific men could agree upon some single system of -units, original and derived, in terms of which all quantities -could be expressed. Statements would thus be rendered -easily comparable, a large part of scientific literature would -be made intelligible to all, and the saving of mental labour -would be immense. It seems to be generally allowed, too, -that the metric system of weights and measures presents -the best basis for the ultimate system; it is thoroughly -established in Western Europe; it is legalised in England; -it is already commonly employed by scientific men; it is -in itself the most simple and scientific of systems. There -is every reason then why the metric system should be -accepted at least in its main features.</p> - - -<h3><i>Provisional Units.</i></h3> - -<p>Ultimately, as we can hardly doubt, all phenomena -will be recognised as so many manifestations of energy; -and, being expressed in terms of the unit of energy, will<span class="pagenum" id="Page_324">324</span> -be referable to the primary units of space, time, and -density. To effect this reduction, however, in any particular -case, we must not only be able to compare different -quantities of the phenomenon, but to trace the whole -series of steps by which it is connected with the primary -notions. We can readily observe that the intensity of -one source of light is greater than that of another; and, -knowing that the intensity of light decreases as the -square of the distance increases, we can easily determine -their comparative brilliance. Hence we can express the -intensity of light falling upon any surface, if we have a -unit in which to make the expression. Light is undoubtedly -one form of energy, and the unit ought therefore -to be the unit of energy. But at present it is quite impossible -to say how much energy there is in any particular -amount of light. The question then arises,—Are we to -defer the measurement of light until we can assign its -relation to other forms of energy? If we answer Yes, it is -equivalent to saying that the science of light must stand -still perhaps for a generation; and not only this science -but many others. The true course evidently is to select, -as the provisional unit of light, some light of convenient -intensity, which can be reproduced from time to time in -the same intensity, and which is defined by physical circumstances. -All the phenomena of light may be experimentally -investigated relatively to this unit, for instance -that obtained after much labour by Bunsen and Roscoe.<a id="FNanchor_227" href="#Footnote_227" class="fnanchor">227</a> -In after years it will become a matter of inquiry what is -the energy exerted in such unit of light; but it may be -long before the relation is exactly determined.</p> - -<p>A provisional unit, then, means one which is assumed -and physically defined in a safe and reproducible manner, -in order that particular quantities may be compared <i>inter -se</i> more accurately than they can yet be referred to the -primary units. In reality the great majority of our -measurements are expressed in terms of such provisionally -independent units, and even the unit of mass, as we have -seen, ought to be considered as provisional.</p> - -<p>The unit of heat ought to be simply the unit of energy, -already described. But a weight can be measured to the<span class="pagenum" id="Page_325">325</span> -one-millionth part, and temperature to less than the -thousandth part of a degree Fahrenheit, and to less therefore -than the five-hundred thousandth part of the absolute -temperature, whereas the mechanical equivalent of heat is -probably not known to the thousandth part. Hence the -need of a provisional unit of heat, which is often taken as -that requisite to raise one gram of water through one degree -Centigrade, that is from 0° to 1°. This quantity of heat is -capable of approximate expression in terms of time, space, -and mass; for by the natural constant, determined by Dr. -Joule, and called the mechanical equivalent of heat, we -know that the assumed unit of heat is equal to the energy -of 423·55 gram-metres, or that energy which will raise -the mass of 423·55 grams through one metre against 9·8... -absolute units of force. Heat may also be expressed in -terms of the quantity of ice at 0° Cent., which it is capable -of converting into water under inappreciable pressure.</p> - - -<h3><i>Theory of Dimensions.</i></h3> - -<p>In order to understand the relations between the quantities -dealt with in physical science, it is necessary to pay -attention to the Theory of Dimensions, first clearly stated -by Joseph Fourier,<a id="FNanchor_228" href="#Footnote_228" class="fnanchor">228</a> but in later years developed by several -physicists. This theory investigates the manner in which -each derived unit depends upon or involves one or more of -the fundamental units. The number of units in a rectangular -area is found by multiplying together the numbers -of units in the sides; thus the unit of length enters twice -into the unit of area, which is therefore said to have two -dimensions with respect to length. Denoting length by <i>L</i>, -we may say that the dimensions of area are <i>L</i> × <i>L</i> or -<i>L</i><sup>2</sup>. It is obvious in the same way that the dimensions of -volume or bulk will be <i>L</i><sup>3</sup>.</p> - -<p>The number of units of mass in a body is found by multiplying -the number of units of volume, by those of density. -Hence mass is of three dimensions as regards length, -and one as regards density. Calling density <i>D</i>, the dimensions -of mass are <i>L</i><sup>3</sup><i>D</i>. As already explained, however, -it is usual to substitute an arbitrary provisional unit of<span class="pagenum" id="Page_326">326</span> -mass, symbolised by <i>M</i>; according to the view here taken -we may say that the dimensions of <i>M</i> are <i>L</i><sup>3</sup><i>D</i>.</p> - -<p>Introducing time, denoted by <i>T</i>, it is easy to see that -the dimensions of velocity will be <span class="nowrap"><span class="fraction2"><span class="fnum2"><i>L</i></span><span class="bar">/</span><span class="fden2"><i>T</i></span></span></span> - or <i>LT</i><sup>-1</sup>, because -the number of units in the velocity of a body is found -by <i>dividing</i> the units of length passed over by the units -of time occupied in passing. The acceleration of a body -is measured by the increase of velocity in relation to -the time, that is, we must divide the units of velocity -gained by the units of time occupied in gaining it; hence -its dimensions will be <i>LT</i><sup>-2</sup>. Momentum is the product -of mass and velocity, so that its dimensions are <i>MLT</i><sup>-1</sup>. -The effect of a force is measured by the acceleration -produced in a unit of mass in a unit of time; hence the -dimensions of force are <i>MLT</i><sup>-2</sup>. Work done is proportional -to the force acting and to the space through -which it acts; so that it has the dimensions of force with -that of length added, giving <i>ML</i><sup>2</sup><i>T</i><sup>-2</sup>.</p> - -<p>It should be particularly noticed that angular magnitude -has no dimensions at all, being measured by the -ratio of the arc to the radius (p. <a href="#Page_305">305</a>). Thus we have the -dimensions <i>LL</i><sup>-1</sup> or <i>L</i><sup>0</sup>. This agrees with the statement -previously made, that no arbitrary unit of angular magnitude -is needed. Similarly, all pure numbers expressing -ratios only, such as sines and other trigonometrical functions, -logarithms, exponents, &c., are devoid of dimensions. -They are absolute numbers necessarily expressed in terms -of unity itself, and are quite unaffected by the selection of -the arbitrary physical units. Angular magnitude, however, -enters into other quantities, such as angular velocity, which -has the dimensions <span class="nowrap"><span class="fraction2"><span class="fnum2">1</span><span class="bar">/</span><span class="fden2"><i>T</i></span></span></span> or <i>T</i><sup>-1</sup>, - the units of angle being -divided by the units of time occupied. The dimensions of -angular acceleration are denoted by <i>T</i><sup>-2</sup>.</p> - -<p>The quantities treated in the theories of heat and -electricity are numerous and complicated as regards -their dimensions. Thermal capacity has the dimensions -<i>ML</i><sup>-3</sup>, thermal conductivity, <i>ML</i><sup>-1</sup><i>T</i><sup>-1</sup>. In Magnetism -the dimensions of the strength of pole are <i>M</i><sup><span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span></sup><i>L</i><sup><span class="nowrap"><span class="fraction"><span class="fnum">3</span><span class="bar">/</span><span class="fden">2</span></span></span></sup><i>T</i><sup>-1</sup>, -<span class="pagenum" id="Page_327">327</span>the - dimensions of field-intensity are <i>M</i><sup><span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span></sup><i>L</i><sup>-<span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span></sup><i>T</i><sup>-1</sup>, and the -intensity of magnetisation has the same dimensions. In the -science of electricity physicists have to deal with numerous -kinds of quantity, and their dimensions are different too in -the electro-static and the electro-magnetic systems. Thus -electro-motive force has the dimensions <i>M</i><sup><span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span></sup><i>L</i><sup><span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span></sup><i>T</i><sup>-1</sup>, in -the former, and <i>M</i><sup><span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span></sup><i>L</i><sup><span class="nowrap"><span class="fraction"><span class="fnum">3</span><span class="bar">/</span><span class="fden">2</span></span></span></sup><i>T</i><sup>-2</sup> in the latter system. Capacity -simply depends upon length in electro-statics, but -upon <i>L</i><sup>-1</sup><i>T</i><sup>2</sup> in electro-magnetics. It is worthy of particular -notice that electrical quantities have simple dimensions -when expressed in terms of density instead of mass. -The instances now given are sufficient to show the difficulty -of conceiving and following out the relations of the -quantities treated in physical science without a systematic -method of calculating and exhibiting their dimensions. It -is only in quite recent years that clear ideas about these -quantities have been attained. Half a century ago probably -no one but Fourier could have explained what he -meant by temperature or capacity for heat. The notion -of measuring electricity had hardly been entertained.</p> - -<p>Besides affording us a clear view of the complex relations -of physical quantities, this theory is specially useful in -two ways. Firstly, it affords a test of the correctness of -mathematical reasoning. According to the <i>Principle of -Homogeneity</i>, all the quantities <i>added</i> together, and equated -in any equation, must have the same dimensions. Hence -if, on estimating the dimensions of the terms in any equation, -they be not homogeneous, some blunder must have -been committed. It is impossible to add a force to a velocity, -or a mass to a momentum. Even if the numerical -values of the two members of a non-homogeneous equation -were equal, this would be accidental, and any alteration in -the physical units would produce inequality and disclose -the falsity of the law expressed in the equation.</p> - -<p>Secondly, the theory of units enables us readily and -infallibly to deduce the change in the numerical expression -of any physical quantity, produced by a change in the -fundamental units. It is of course obvious that in order -to represent the same absolute quantity, a number must -vary inversely as the magnitude of the units which are -numbered. The yard expressed in feet is 3; taking the -inch as the unit instead of the foot it becomes 36. Every -quantity into which the dimension length enters positively<span class="pagenum" id="Page_328">328</span> -must be altered in like manner. Changing the unit from -the foot to the inch, numerical expressions of volume must -be multiplied by 12 × 12 × 12. When a dimension enters -negatively the opposite rule will hold. If for the minute -we substitute the second as unit of time, then we must -divide all numbers expressing angular velocities by 60, -and numbers expressing angular acceleration by 60 × 60. -The rule is that a numerical expression varies inversely as -the magnitude of the unit as regards each whole dimension -entering positively, and it varies directly as the magnitude -of the unit for each whole dimension entering negatively. -In the case of fractional exponents, the proper root of the -ratio of change has to be taken.</p> - -<p>The study of this subject may be continued in Professor -J. D. Everett’s “Illustrations of the Centimetre-gramme-second -System of Units,” published by Taylor and Francis, -1875; in Professor Maxwell’s “Theory of Heat;” or Professor -Fleeming Jenkin’s “Text Book of Electricity.”</p> - - -<h3><i>Natural Constants.</i></h3> - -<p>Having acquired accurate measuring instruments, and -decided upon the units in which the results shall be expressed, -there remains the question, What use shall be -made of our powers of measurement? Our principal -object must be to discover general quantitative laws of -nature; but a very large amount of preliminary labour is -employed in the accurate determination of the dimensions -of existing objects, and the numerical relations between -diverse forces and phenomena. Step by step every part -of the material universe is surveyed and brought into -known relations with other parts. Each manifestation of -energy is correlated with each other kind of manifestation. -Professor Tyndall has described the care with which such -operations are conducted.<a id="FNanchor_229" href="#Footnote_229" class="fnanchor">229</a></p> - -<p>“Those who are unacquainted with the details of -scientific investigation, have no idea of the amount of -labour expended on the determination of those numbers -on which important calculations or inferences depend. -They have no idea of the patience shown by a Berzelius -in determining atomic weights; by a Regnault in determining<span class="pagenum" id="Page_329">329</span> -coefficients of expansion; or by a Joule in determining -the mechanical equivalent of heat. There is a -morality brought to bear upon such matters which, in -point of severity, is probably without a parallel in any other -domain of intellectual action.”</p> - -<p>Every new natural constant which is recorded brings -many fresh inferences within our power. For if <i>n</i> be the -number of such constants known, then <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span> - (<i>n</i><sup>2</sup>—<i>n</i>) is the -number of ratios which are within our powers of calculation, -and this increases with the square of <i>n</i>. We thus -gradually piece together a map of nature, in which the -lines of inference from one phenomenon to another rapidly -grow in complexity, and the powers of scientific prediction -are correspondingly augmented.</p> - -<p>Babbage<a id="FNanchor_230" href="#Footnote_230" class="fnanchor">230</a> proposed the formation of a collection of the -constant numbers of nature, a work which has at last -been taken in hand by the Smithsonian Institution.<a id="FNanchor_231" href="#Footnote_231" class="fnanchor">231</a> It -is true that a complete collection of such numbers would -be almost co-extensive with scientific literature, since -almost all the numbers occurring in works on chemistry, -mineralogy, physics, astronomy, &c., would have to be -included. Still a handy volume giving all the more -important numbers and their logarithms, referred when -requisite to the different units in common use, would be -very useful. A small collection of constant numbers will -be found at the end of Babbage’s, Hutton’s, and many -other tables of logarithms, and a somewhat larger collection -is given in Templeton’s <i>Millwright and Engineer’s -Pocket Companion</i>.</p> - -<p>Our present object will be to classify these constant -numbers roughly, according to their comparative generality -and importance, under the following heads:—</p> - -<div class="container"> -<div class="content"> -(1) Mathematical constants.<br> -(2) Physical constants.<br> -(3) Astronomical constants.<br> -(4) Terrestrial numbers.<br> -(5) Organic numbers.<br> -(6) Social numbers.<br> -</div> -</div> - -<p><span class="pagenum" id="Page_330">330</span></p> - - -<h3><i>Mathematical Constants.</i></h3> - -<p>At the head of the list of natural constants must come -those which express the necessary relations of numbers to -each other. The ordinary Multiplication Table is the -most familiar and the most important of such series of -constants, and is, theoretically speaking, infinite in extent. -Next we must place the Arithmetical Triangle, the significance -of which has already been pointed out (p. <a href="#Page_182">182</a>). -Tables of logarithms also contain vast series of natural -constants, arising out of the relations of pure numbers. -At the base of all logarithmic theory is the mysterious -natural constant commonly denoted by <i>e</i>, or ε, being -equal to the infinite series 1 + <span class="nowrap"><span class="fraction2"><span class="fnum2">1</span><span class="bar">/</span><span class="fden2">1</span></span></span> + <span class="nowrap"><span class="fraction2"><span class="fnum2">1</span><span class="bar">/</span><span class="fden2">1.2</span></span></span> + <span class="nowrap"><span class="fraction2"><span class="fnum2">1</span><span class="bar">/</span><span class="fden2">1.2.3</span></span></span> + <span class="nowrap"><span class="fraction2"><span class="fnum2">1</span><span class="bar">/</span><span class="fden2">1.2.3.4</span></span></span> +...., -and thus consisting of the sum of the ratios between the -numbers of permutations and combinations of 0, 1, 2, 3, -4, &c. things. Tables of prime numbers and of the factors -of composite numbers must not be forgotten.</p> - -<p>Another vast and in fact infinite series of numerical -constants contains those connected with the measurement -of angles, and embodied in trigonometrical tables, -whether as natural or logarithmic sines, cosines, and -tangents. It should never be forgotten that though -these numbers find their chief employment in connection -with trigonometry, or the measurement of the sides of a -right-angled triangle, yet the numbers themselves arise -out of numerical relations bearing no special relation to -space. Foremost among trigonometrical constants is the -well known number π, usually employed as expressing -the ratio of the circumference and the diameter of a -circle; from π follows the value of the arcual or natural -unit of angular value as expressed in ordinary degrees -(p. <a href="#Page_306">306</a>).</p> - -<p>Among other mathematical constants not uncommonly -used may be mentioned tables of factorials (p. <a href="#Page_179">179</a>), tables -of Bernoulli’s numbers, tables of the error function,<a id="FNanchor_232" href="#Footnote_232" class="fnanchor">232</a> -which latter are indispensable not only in the theory of -probability but also in several other branches of science.</p> -<p><span class="pagenum" id="Page_331">331</span></p> -<p>It should be clearly understood that the mathematical -constants and tables of reference already in our possession, -although very extensive, are only an infinitely small part -of what might be formed. With the progress of science -the tabulation of new functions will be continually -demanded, and it is worthy of consideration whether -public money should not be available to reward the -severe, long continued, and generally thankless labour -which must be gone through in calculating tables. Such -labours are a benefit to the whole human race as long as -it shall exist, though there are few who can appreciate -the extent of this benefit. A most interesting and excellent -description of many mathematical tables will be -found in De Morgan’s article on <i>Tables</i>, in the <i>English -Cyclopædia</i>, Division of Arts and Sciences, vol. vii. p. 976. -An almost exhaustive critical catalogue of extant tables is -being published by a Committee of the British Association, -two portions, drawn up chiefly by Mr. J. W. L. Glaisher -and Professor Cayley, having appeared in the Reports of -the Association for 1873 and 1875.</p> - - -<h3><i>Physical Constants.</i></h3> - -<p>The second class of constants contains those which -refer to the actual constitution of matter. For the most -part they depend upon the peculiarities of the chemical -substance in question, but we may begin with those -which are of the most general character. In a first sub-class -we may place the velocity of light or heat undulations, -the numbers expressing the relation between the -lengths of the undulations, and the rapidity of the -undulations, these numbers depending only on the properties -of the ethereal medium, and being probably the -same in all parts of the universe. The theory of heat -gives rise to several numbers of the highest importance, -especially Joule’s mechanical equivalent of heat, the -absolute zero of temperature, the mean temperature of -empty space, &c.</p> - -<p>Taking into account the diverse properties of the -elements we must have tables of the atomic weights, -the specific heats, the specific gravities, the refractive -powers, not only of the elements, but their almost<span class="pagenum" id="Page_332">332</span> -infinitely numerous compounds. The properties of hardness, -elasticity, viscosity, expansion by heat, conducting powers -for heat and electricity, must also be determined in -immense detail. There are, however, certain of these -numbers which stand out prominently because they serve -as intermediate units or terms of comparison. Such are, -for instance, the absolute coefficients of expansion of air, -water and mercury, the temperature of the maximum -density of water, the latent heats of water and steam, -the boiling-point of water under standard pressure, the -melting and boiling-points of mercury, and so forth.</p> - - -<h3><i>Astronomical Constants.</i></h3> - -<p>The third great class consists of numbers possessing far -less generality because they refer not to the properties of -matter, but to the special forms and distances in which -matter has been disposed in the part of the universe open -to our examination. We have, first of all, to define the -magnitude and form of the earth, its mean density, the -constant of aberration of light expressing the relation -between the earth’s mean velocity in space and the -velocity of light. From the earth, as our observatory, we -then proceed to lay down the mean distances of the sun, -and of the planets from the same centre; all the elements -of the planetary orbits, the magnitudes, densities, masses, -periods of axial rotation of the several planets are by -degrees determined with growing accuracy. The same -labours must be gone through for the satellites. Catalogues -of comets with the elements of their orbits, as far -as ascertainable, must not be omitted.</p> - -<p>From the earth’s orbit as a new base of observations, -we next proceed to survey the heavens and lay down the -apparent positions, magnitudes, motions, distances, periods -of variation, &c. of the stars. All catalogues of stars from -those of Hipparchus and Tycho, are full of numbers expressing -rudely the conformation of the visible universe. -But there is obviously no limit to the labours of astronomers; -not only are millions of distant stars awaiting their -first measurements, but those already registered require -endless scrutiny as regards their movements in the three -dimensions of space, their periods of revolution, their<span class="pagenum" id="Page_333">333</span> -changes of brilliance and colour. It is obvious that -though astronomical numbers are conventionally called -<i>constant</i>, they are probably in all cases subject to more -or less rapid variation.</p> - - -<h3><i>Terrestrial Numbers.</i></h3> - -<p>Our knowledge of the globe we inhabit involves many -numerical determinations, which have little or no connection -with astronomical theory. The extreme heights -of the principal mountains, the mean elevations of -continents, the mean or extreme depths of the oceans, -the specific gravities of rocks, the temperature of mines, -the host of numbers expressing the meteorological or -magnetic conditions of every part of the surface, must -fall into this class. Many such numbers are not to be -called constant, being subject to periodic or secular -changes, but they are hardly more variable in fact than -some which in astronomical science are set down as -constant. In many cases quantities which seem most -variable may go through rhythmical changes resulting -in a nearly uniform average, and it is only in the long -progress of physical investigation that we can hope to -discriminate successfully between those elemental numbers -which are fixed and those which vary. In the latter -case the law of variation becomes the constant relation -which is the object of our search.</p> - -<h3><i>Organic Numbers.</i></h3> - -<p>The forms and properties of brute nature having been -sufficiently defined by the previous classes of numbers, -the organic world, both vegetable and animal, remains -outstanding, and offers a higher series of phenomena for -our investigation. All exact knowledge relating to the -forms and sizes of living things, their numbers, the -quantities of various compounds which they consume, -contain, or excrete, their muscular or nervous energy, &c. -must be placed apart in a class by themselves. All such -numbers are doubtless more or less subject to variation, -and but in a minor degree capable of exact determination. -Man, so far as he is an animal, and as regards his physical -form, must also be treated in this class.</p> - -<p><span class="pagenum" id="Page_334">334</span></p> - - -<h3><i>Social Numbers.</i></h3> - -<p>Little allusion need be made in this work to the fact -that man in his economic, sanitary, intellectual, æsthetic, -or moral relations may become the subject of sciences, -the highest and most useful of all sciences. Every one -who is engaged in statistical inquiry must acknowledge -the possibility of natural laws governing such statistical -facts. Hence we must allot a distinct place to numerical -information relating to the numbers, ages, physical and -sanitary condition, mortality, &c., of different peoples, in -short, to vital statistics. Economic statistics, comprehending -the quantities of commodities produced, existing, -exchanged and consumed, constitute another extensive -body of science. In the progress of time exact investigation -may possibly subdue regions of phenomena which -at present defy all scientific treatment. That scientific -method can ever exhaust the phenomena of the human -mind is incredible.</p> -<hr class="chap x-ebookmaker-drop"> - -<div class="chapter"> -<p><span class="pagenum" id="Page_335">335</span></p> - -<h2 class="nobreak" id="CHAPTER_XV">CHAPTER XV.<br> - -<span class="title">ANALYSIS OF QUANTITATIVE PHENOMENA.</span></h2> -</div> - -<p class="ti0">In the two preceding chapters we have been engaged -in considering how a phenomenon may be accurately -measured and expressed. So delicate and complex an -operation is a measurement which pretends to any considerable -degree of exactness, that no small part of the -skill and patience of physicists is usually spent upon this -work. Much of this difficulty arises from the fact that -it is scarcely ever possible to measure a single effect at a -time. The ultimate object must be to discover the -mathematical equation or law connecting a quantitative -cause with its quantitative effect; this purpose usually -involves, as we shall see, the varying of one condition at -a time, the other conditions being maintained constant. -The labours of the experimentalist would be comparatively -light if he could carry out this rule of varying one -circumstance at a time. He would then obtain a series of -corresponding values of the variable quantities concerned, -from which he might by proper hypothetical treatment -obtain the required law of connection. But in reality it -is seldom possible to carry out this direction except in an -approximate manner. Before then we proceed to the -consideration of the actual process of quantitative induction, -it is necessary to review the several devices by -which a complicated series of effects can be disentangled. -Every phenomenon measured will usually be the sum, -difference, or it may be the product or quotient, of -two or more different effects, and these must be in some<span class="pagenum" id="Page_336">336</span> -way analysed and separately measured before we possess -the materials for inductive treatment.</p> - - -<h3><i>Illustrations of the Complication of Effects.</i></h3> - -<p>It is easy to bring forward a multitude of instances to -show that a phenomenon is seldom to be observed simple -and alone. A more or less elaborate process of analysis -is almost always necessary. Thus if an experimentalist -wishes to observe and measure the expansion of a liquid -by heat, he places it in a thermometer tube and registers -the rise of the column of liquid in the narrow tube. But -he cannot heat the liquid without also heating the glass, -so that the change observed is really the difference between -the expansions of the liquid and the glass. More minute -investigation will show the necessity perhaps of allowing -for further minute effects, namely the compression of the -liquid and the expansion of the bulb due to the increased -pressure of the column as it becomes lengthened.</p> - -<p>In a great many cases an observed effect will be -apparently at least the simple sum of two separate and -independent effects. The heat evolved in the combustion -of oil is partly due to the carbon and partly to the -hydrogen. A measurement of the heat yielded by the two -jointly, cannot inform us how much proceeds from the -one and how much from the other. If by some separate -determination we can ascertain how much the hydrogen -yields, then by mere subtraction we learn what is due -to the carbon; and <i>vice versâ</i>. The heat conveyed by a -liquid, may be partly conveyed by true conduction, partly -by convection. The light dispersed in the interior of a -liquid consists both of what is reflected by floating -particles and what is due to true fluorescence;<a id="FNanchor_233" href="#Footnote_233" class="fnanchor">233</a> and we -must find some mode of determining one portion before -we can learn the other. The apparent motion of the spots -on the sun, is the algebraic sum of the sun’s axial -rotation, and of the proper motion of the spots upon the -sun’s surface; hence the difficulty of ascertaining by -direct observations the period of the sun’s rotation.</p> - -<p>We cannot obtain the weight of a portion of liquid -<span class="pagenum" id="Page_337">337</span>in a chemical balance without weighing it with the -containing vessel. Hence to have the real weight of -the liquid operated upon in an experiment, we must -make a separate weighing of the vessel, with or without -the adhering film of liquid according to circumstances. -This is likewise the mode in which a cart and its load -are weighed together, the <i>tare</i> of the cart previously -ascertained being deducted. The variation in the height -of the barometer is a joint effect, partly due to the real -variation of the atmospheric pressure, partly to the expansion -of the mercurial column by heat. The effects may -be discriminated, if, instead of one barometer tube we have -two tubes containing mercury placed closely side by side, -so as to have the same temperature. If one of them be -closed at the bottom so as to be unaffected by the atmospheric -pressure, it will show the changes due to temperature -only, and, by subtracting these changes from those -shown in the other tube, employed as a barometer, we -get the real oscillations of atmospheric pressure. But -this correction, as it is called, of the barometric reading, -is better effected by calculation from the readings of -an ordinary thermometer.</p> - -<p>In other cases a quantitative effect will be the difference -of two causes acting in opposite directions. Sir John -Herschel invented an instrument like a large thermometer, -which he called the Actinometer,<a id="FNanchor_234" href="#Footnote_234" class="fnanchor">234</a> and Pouillet constructed -a somewhat similar instrument called the Pyrheliometer, -for ascertaining the heating power of the sun’s rays. In -both instruments the heat of the sun was absorbed by a -reservoir containing water, and the rise of temperature -of the water was exactly observed, either by its own -expansion, or by the readings of a delicate thermometer -immersed in it. But in exposing the actinometer to the -sun, we do not obtain the full effect of the heat absorbed, -because the receiving surface is at the same time radiating -heat into empty space. The observed increment of temperature -is in short the difference between what is received -from the sun and lost by radiation. The latter quantity is -capable of ready determination; we have only to shade the -instrument from the direct rays of the sun, leaving it<span class="pagenum" id="Page_338">338</span> -exposed to the sky, and we can observe how much it cools -in a certain time. The total effect of the sun’s rays will -obviously be the apparent effect <i>plus</i> the cooling effect in -an equal time. By alternate exposure in sun and shade -during equal intervals the desired result may be obtained -with considerable accuracy.<a id="FNanchor_235" href="#Footnote_235" class="fnanchor">235</a></p> - -<p>Two quantitative effects were beautifully distinguished -in an experiment of John Canton, devised in 1761 for the -purpose of demonstrating the compressibility of water. -He constructed a thermometer with a large bulb full of -water and a short capillary tube, the part of which above -the water was freed from air. Under these circumstances -the water was relieved from the pressure of the atmosphere, -but the glass bulb in bearing that pressure was -somewhat contracted. He next placed the instrument -under the receiver of an air-pump, and on exhausting the -air, the water sank in the tube. Having thus obtained a -measure of the effect of atmospheric pressure on the bulb, -he opened the top of the thermometer tube and admitted -the air. The level of the water now sank still more, partly -from the pressure on the bulb being now compensated, and -partly from the compression of the water by the atmospheric -pressure. It is obvious that the amount of the -latter effect was approximately the difference of the two -observed depressions.</p> - -<p>Not uncommonly the actual phenomenon which we wish -to measure is considerably less than various disturbing -effects which enter into the question. Thus the compressibility -of mercury is considerably less than the expansion -of the vessels in which it is measured under pressure, so -that the attention of the experimentalist has chiefly to be -concentrated on the change of magnitude of the vessels. -Many astronomical phenomena, such as the parallax or the -proper motions of the fixed stars, are far less than the -errors caused by instrumental imperfections, or motions -arising from precession, nutation, and aberration. We -need not be surprised that astronomers have from time to -time mistaken one phenomenon for another, as when Flamsteed -imagined that he had discovered the parallax of the -Pole star.<a id="FNanchor_236" href="#Footnote_236" class="fnanchor">236</a></p> -<p><span class="pagenum" id="Page_339">339</span></p> - -<h3><i>Methods of Eliminating Error.</i></h3> - -<p>In any particular experiment it is the object of the experimentalist -to measure a single effect only, and he -endeavours to obtain that effect free from interfering -effects. If this cannot be, as it seldom or never can -really be, he makes the effect as considerable as possible -compared with the other effects, which he reduces to a -minimum, and treats as noxious errors. Those quantities, -which are called <i>errors</i> in one case, may really be most -important and interesting phenomena in another investigation. -When we speak of eliminating error we really -mean disentangling the complicated phenomena of nature. -The physicist rightly wishes to treat one thing at a time, -but as this object can seldom be rigorously carried into -practice, he has to seek some mode of counteracting the -irrelevant and interfering causes.</p> - -<p>The general principle is that a single observation can -render known only a single quantity. Hence, if several -different quantitative effects are known to enter into any -investigation, we must have at least as many distinct observations -as there are quantities to be determined. Every -complete experiment will therefore consist in general of -several operations. Guided if possible by previous knowledge -of the causes in action, we must arrange the determinations, -so that by a simple mathematical process we -may distinguish the separate quantities. There appear to -be five principal methods by which we may accomplish -this object; these methods are specified below and illustrated -in the succeeding sections.</p> - -<p>(1) <i>The Method of Avoidance.</i> The physicist may seek -for some special mode of experiment or opportunity of observation, -in which the error is non-existent or inappreciable.</p> - -<p>(2) <i>The Differential Method.</i> He may find opportunities -of observation when all interfering phenomena remain constant, -and only the subject of observation is at one time -present and another time absent; the difference between -two observations then gives its amount.</p> - -<p>(3) <i>The Method of Correction.</i> He may endeavour to -estimate the amount of the interfering effect by the best -available mode, and then make a corresponding correction -in the results of observation.</p> - -<p><span class="pagenum" id="Page_340">340</span></p> - -<p>(4) <i>The Method of Compensation.</i> He may invent some -mode of neutralising the interfering cause by balancing -against it an exactly equal and opposite cause of unknown -amount.</p> - -<p>(5) <i>The Method of Reversal.</i> He may so conduct the -experiment that the interfering cause may act in opposite -directions, in alternate observations, the mean result being -free from interference.</p> - - -<h4>I. <i>Method of Avoidance of Error.</i></h4> - -<p>Astronomers seek opportunities of observation when -errors will be as small as possible. In spite of elaborate -observations and long-continued theoretical investigation, -it is not practicable to assign any satisfactory law to the -refractive power of the atmosphere. Although the apparent -change of place of a heavenly body produced by -refraction may be more or less accurately calculated yet -the error depends upon the temperature and pressure of -the atmosphere, and, when a ray is highly inclined to the -perpendicular, the uncertainty in the refraction becomes -very considerable. Hence astronomers always make their -observations, if possible, when the object is at the highest -point of its daily course, <i>i.e.</i> on the meridian. In some -kinds of investigation, as, for instance, in the determination -of the latitude of an observatory, the astronomer is at -liberty to select one or more stars out of the countless -number visible. There is an evident advantage in such a -case, in selecting a star which passes close to the zenith, -so that it may be observed almost entirely free from atmospheric -refraction, as was done by Hooke.</p> - -<p>Astronomers endeavour to render their clocks as accurate -as possible, by removing the source of variation. The -pendulum is perfectly isochronous so long as its length -remains invariable, and the vibrations are exactly of equal -length. They render it nearly invariable in length, that -is in the distance between the centres of suspension and -oscillation, by a compensatory arrangement for the change -of temperature. But as this compensation may not be -perfectly accomplished, some astronomers place their chief -controlling clock in a cellar, or other apartment, where -the changes of temperature may be as slight as possible.<span class="pagenum" id="Page_341">341</span> -At the Paris Observatory a clock has been placed in the -caves beneath the building, where there is no appreciable -difference between the summer and winter temperature.</p> - -<p>To avoid the effect of unequal oscillations Huyghens -made his beautiful investigations, which resulted in the -discovery that a pendulum, of which the centre of oscillation -moved upon a cycloidal path, would be perfectly -isochronous, whatever the variation in the length of oscillations. -But though a pendulum may be easily rendered in -some degree cycloidal by the use of a steel suspension -spring, it is found that the mechanical arrangements requisite -to produce a truly cycloidal motion introduce more -error than they remove. Hence astronomers seek to -reduce the error to the smallest amount by maintaining -their clock pendulums in uniform movement; in fact, -while a clock is in good order and has the same weights, -there need be little change in the length of oscillation. -When a pendulum cannot be made to swing uniformly, as -in experiments upon the force of gravity, it becomes requisite -to resort to the third method, and a correction is -introduced, calculated on theoretical grounds from the -amount of the observed change in the length of vibration.</p> - -<p>It has been mentioned that the apparent expansion of a -liquid by heat, when contained in a thermometer tube or -other vessel, is the difference between the real expansion -of the liquid and that of the containing vessel. The -effects can be accurately distinguished provided that we -can learn the real expansion by heat of any one convenient -liquid; for by observing the apparent expansion of the -same liquid in any required vessel we can by difference -learn the amount of expansion of the vessel due to any -given change of temperature. When we once know the -change of dimensions of the vessel, we can of course determine -the absolute expansion of any other liquid tested in -it. Thus it became an all-important object in scientific -research to measure with accuracy the absolute dilatation -by heat of some one liquid, and mercury owing to several -circumstances was by far the most suitable. Dulong and -Petit devised a beautiful mode of effecting this by simply -avoiding altogether the effect of the change of size of the -vessel. Two upright tubes full of mercury were connected -by a fine tube at the bottom, and were maintained at two<span class="pagenum" id="Page_342">342</span> -different temperatures. As mercury was free to flow from -one tube to the other by the connecting tube, the two -columns necessarily exerted equal pressures by the principles -of hydrostatics. Hence it was only necessary to measure -very accurately by a cathetometer the difference of -level of the surfaces of the two columns of mercury, to -learn the difference of length of columns of equal hydrostatic -pressure, which at once gives the difference of density -of the mercury, and the dilatation by heat. The -changes of dimension in the containing tubes became a -matter of entire indifference, and the length of a column -of mercury at different temperatures was measured as -easily as if it had formed a solid bar. The experiment was -carried out by Regnault with many improvements of detail, -and the absolute dilatation of mercury, at temperatures -between 0° Cent. and 350°, was determined almost as -accurately as was needful.<a id="FNanchor_237" href="#Footnote_237" class="fnanchor">237</a></p> - -<p>The presence of a large and uncertain amount of error -may render a method of experiment valueless. Foucault -devised a beautiful experiment with the pendulum for -demonstrating popularly the rotation of the earth, but it -could be of no use for measuring the rotation exactly. It -is impossible to make the pendulum swing in a perfect -plane, and the slightest lateral motion gives it an elliptic -path with a progressive motion of the axis of the ellipse, -which disguises and often entirely overpowers that due to -the rotation of the earth.<a id="FNanchor_238" href="#Footnote_238" class="fnanchor">238</a></p> - -<p>Faraday’s laborious experiments on the relation of gravity -and electricity were much obstructed by the fact that it is -impossible to move a large weight of metal without generating -currents of electricity, either by friction or induction. -To distinguish the electricity, if any, directly due to the -action of gravity from the greater quantities indirectly produced -was a problem of excessive difficulty. Baily in his -experiments on the density of the earth was aware of the -existence of inexplicable disturbances which have since -been referred with much probability to the action of -electricity.<a id="FNanchor_239" href="#Footnote_239" class="fnanchor">239</a> The skill and ingenuity of the experimentalist<span class="pagenum" id="Page_343">343</span> -are often exhausted in trying to devise a form of apparatus -in which such causes of error shall be reduced to a -minimum.</p> - -<p>In some rudimentary experiments we wish merely to -establish the existence of a quantitative effect without -precisely measuring its amount; if there exist causes of -error of which we can neither render the amount known -or inappreciable, the best way is to make them all -negative so that the quantitative effects will be less than -the truth rather than greater. Grove, for instance, in -proving that the magnetisation or demagnetisation of a -piece of iron raises its temperature, took care to maintain -the electro-magnet by which the iron was magnetised at -a lower temperature than the iron, so that it would cool -rather than warm the iron by radiation or conduction.<a id="FNanchor_240" href="#Footnote_240" class="fnanchor">240</a></p> - -<p>Rumford’s celebrated experiment to prove that heat was -generated out of mechanical force in the boring of a -cannon was subject to the difficulty that heat might be -brought to the cannon by conduction from neighbouring -bodies. It was an ingenious device of Davy to produce -friction by a piece of clock-work resting upon a block -of ice in an exhausted receiver; as the machine rose in -temperature above 32°, it was certain that no heat was -received by conduction from the support.<a id="FNanchor_241" href="#Footnote_241" class="fnanchor">241</a> In many -other experiments ice may be employed to prevent the -access of heat by conduction, and this device, first put in -practice by Murray,<a id="FNanchor_242" href="#Footnote_242" class="fnanchor">242</a> is beautifully employed in Bunsen’s -calorimeter.</p> - -<p>To observe the true temperature of the air, though -apparently so easy, is really a very difficult matter, because -the thermometer is sure to be affected either by the sun’s -rays, the radiation from neighbouring objects, or the escape -of heat into space. These sources of error are too fluctuating -to allow of correction, so that the only accurate mode -of procedure is that devised by Dr. Joule, of surrounding -the thermometer with a copper cylinder ingeniously<span class="pagenum" id="Page_344">344</span> -adjusted to the temperature of the air, as described by -him, so that the effect of radiation shall be nullified.<a id="FNanchor_243" href="#Footnote_243" class="fnanchor">243</a></p> - -<p>When the avoidance of error is not practicable, it will -yet be desirable to reduce the absolute amount of the -interfering error as much as possible before employing the -succeeding methods to correct the result. As a general -rule we can determine a quantity with less inaccuracy as -it is smaller, so that if the error itself be small the error in -determining that error will be of a still lower order of -magnitude. But in some cases the absolute amount of an -error is of no consequence, as in the index error of a -divided circle, or the difference between a chronometer and -astronomical time. Even the rate at which a clock gains -or loses is a matter of little importance provided it remain -constant, so that a sure calculation of its amount can be -made.</p> - - -<h4>2. <i>Differential Method.</i></h4> - -<p>When we cannot avoid the existence of error, we can -often resort with success to the second mode by measuring -phenomena under such circumstances that the error shall -remain very nearly the same in all the observations, and -neutralise itself as regards the purposes in view. This -mode is available whenever we want a difference between -quantities and not the absolute quantity of either. The -determination of the parallax of the fixed stars is exceedingly -difficult, because the amount of parallax is far less -than most of the corrections for atmospheric refraction, -nutation, aberration, precession, instrumental irregularities, -&c., and can with difficulty be detected among these phenomena -of various magnitude. But, as Galileo long ago -suggested, all such difficulties would be avoided by the -differential observation of stars, which, though apparently -close together, are really far separated on the line of sight. -Two such stars in close apparent proximity will be subject -to almost exactly equal errors, so that all we -need do is to observe the apparent change of place of -the nearer star as referred to the more distant one.<span class="pagenum" id="Page_345">345</span> -A good telescope furnished with an accurate micrometer -is alone needed for the application of the method. -Huyghens appears to have been the first observer who -actually tried to employ the method practically, but -it was not until 1835 that the improvement of telescopes -and micrometers enabled Struve to detect in this way -the parallax of the star α Lyræ. It is one of the many -advantages of the observation of transits of Venus for the -determination of the solar parallax that the refraction of -the atmosphere affects in an exactly equal degree the planet -and the portion of the sun’s face over which it is passing. -Thus the observations are strictly of a differential nature.</p> - -<p>By the process of substitutive weighing it is possible -to ascertain the equality or inequality of two weights -with almost perfect freedom from error. If two weights -A and B be placed in the scales of the best balance -we cannot be sure that the equilibrium of the beam -indicates exact equality, because the arms of the beam -may be unequal or unbalanced. But if we take B out -and put another weight C in, and equilibrium still -exists, it is apparent that the same causes of erroneous -weighing exist in both cases, supposing that the balance -has not been disarranged; B then must be exactly equal -to C, since it has exactly the same effect under the same -circumstances. In like manner it is a general rule that, -if by any uniform mechanical process we get a copy of an -object, it is unlikely that this copy will be precisely the -same as the original in magnitude and form, but two copies -will equally diverge from the original, and will therefore -almost exactly resemble each other.</p> - -<p>Leslie’s Differential Thermometer<a id="FNanchor_244" href="#Footnote_244" class="fnanchor">244</a> was well adapted -to the experiments for which it was invented. Having -two equal bulbs any alteration in the temperature of the -air will act equally by conduction on each and produce -no change in the indications of the instrument. Only -that radiant heat which is purposely thrown upon one -of the bulbs will produce any effect. This thermometer -in short carries out the principle of the differential method -in a mechanical manner.</p> -<p><span class="pagenum" id="Page_346">346</span></p> - -<h4>3. <i>Method of Correction.</i></h4> - -<p>Whenever the result of an experiment is affected by an -interfering cause to a calculable amount, it is sufficient to -add or subtract this amount. We are said to correct -observations when we thus eliminate what is due to -extraneous causes, although of course we are only separating -the correct effects of several agents. The variation -in the height of the barometer is partly due to the change -of temperature, but since the coefficient of absolute -dilatation of mercury has been exactly determined, as -already described (p. <a href="#Page_341">341</a>), we have only to make calculations -of a simple character, or, what is better still, -tabulate a series of such calculations for general use, and -the correction for temperature can be made with all desired -accuracy. The height of the mercury in the barometer is -also affected by capillary attraction, which depresses it by -a constant amount depending mainly on the diameter of -the tube. The requisite corrections can be estimated with -accuracy sufficient for most purposes, more especially as -we can check the correctness of the reading of a barometer -by comparison with a standard barometer, and introduce -if need be an index error including both the error in the -affixing of the scale and the effect due to capillarity. But -in constructing the standard barometer itself we must take -greater precautions; the capillary depression depends -somewhat upon the quality of the glass, the absence of air, -and the perfect cleanliness of the mercury, so that we -cannot assign the exact amount of the effect. Hence a -standard barometer is constructed with a wide tube, sometimes -even an inch in diameter, so that the capillary effect -may be rendered almost zero.<a id="FNanchor_245" href="#Footnote_245" class="fnanchor">245</a> Gay-Lussac made barometers -in the form of a uniform siphon tube, so that the -capillary forces acting at the upper and lower surfaces -should balance and destroy each other; but the method -fails in practice because the lower surface, being open to -the air, becomes sullied and subject to a different force of -capillarity.</p> - -<p>In mechanical experiments friction is an interfering -condition, and drains away a portion of the energy intended<span class="pagenum" id="Page_347">347</span> -to be operated upon in a definite manner. We -should of course reduce the friction in the first place to the -lowest possible amount, but as it cannot be altogether prevented, -and is not calculable with certainty from any -general laws, we must determine it separately for each -apparatus by suitable experiments. Thus Smeaton, in -his admirable but almost forgotten researches concerning -water-wheels, eliminated friction in the most simple -manner by determining by trial what weight, acting by a -cord and roller upon his model water-wheel, would make -it turn without water as rapidly as the water made it turn. -In short, he ascertained what weight concurring with the -water would exactly compensate for the friction.<a id="FNanchor_246" href="#Footnote_246" class="fnanchor">246</a> In Dr. -Joule’s experiments to determine the mechanical equivalent -of heat by the condensation of air, a considerable -amount of heat was produced by friction of the condensing -pump, and a small portion by stirring the water employed -to absorb the heat. This heat of friction was measured by -simply repeating the experiment in an exactly similar -manner except that no condensation was effected, and observing -the change of temperature then produced.<a id="FNanchor_247" href="#Footnote_247" class="fnanchor">247</a></p> - -<p>We may describe as <i>test experiments</i> any in which we -perform operations not intended to give the quantity of -the principal phenomenon, but some quantity which would -otherwise remain as an error in the result. Thus in -astronomical observations almost every instrumental error -may be avoided by increasing the number of observations -and distributing them in such a manner as to produce -in the final mean as much error in one way as in the -other. But there is one source of error, first discovered -by Maskelyne, which cannot be thus avoided, because it -affects all observations in the same direction and to the -same average amount, namely the Personal Error of the -observer or the inclination to record the passage of a star -across the wires of the telescope a little too soon or a -little too late. This personal error was first carefully -described in the <i>Edinburgh Journal of Science</i>, vol. i. -p. 178. The difference between the judgment of observers -at the Greenwich Observatory usually varies from <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">100</span></span></span> - to <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">3</span></span></span><span class="pagenum" id="Page_348">348</span> -of a second, and remains pretty constant for the same -observers.<a id="FNanchor_248" href="#Footnote_248" class="fnanchor">248</a> One practised observer in Sir George Airy’s -pendulum experiments recorded all his time observations -half a second too early on the average as compared with -the chief observer.<a id="FNanchor_249" href="#Footnote_249" class="fnanchor">249</a> In some observers it has amounted to -seven or eight-tenths of a second.<a id="FNanchor_250" href="#Footnote_250" class="fnanchor">250</a> De Morgan appears to -have entertained the opinion that this source of error was -essentially incapable of elimination or correction.<a id="FNanchor_251" href="#Footnote_251" class="fnanchor">251</a> But it -seems clear, as I suggested without knowing what had -been done,<a id="FNanchor_252" href="#Footnote_252" class="fnanchor">252</a> that this personal error might be determined -absolutely with any desirable degree of accuracy by test -experiments, consisting in making an artificial star move -at a considerable distance and recording by electricity the -exact moment of its passage over the wire. This method -has in fact been successfully employed in Leyden, Paris, -and Neuchatel.<a id="FNanchor_253" href="#Footnote_253" class="fnanchor">253</a> More recently, observers were trained -for the Transit of Venus Expeditions by means of a -mechanical model representing the motion of Venus over -the sun, this model being placed at a little distance and -viewed through a telescope, so that differences in the -judgments of different observers would become apparent. -It seems likely that tests of this nature might be employed -with advantage in other cases.</p> - -<p>Newton employed the pendulum for making experiments -on the impact of balls. Two balls were hung in -contact, and one of them, being drawn aside through a -measured arc, was then allowed to strike the other, the -arcs of vibration giving sufficient data for calculating the -distribution of energy at the moment of impact. The -resistance of the air was an interfering cause which he -estimated very simply by causing one of the balls to -make several complete vibrations without impact and then -marking the reduction in the lengths of the arcs, a proper -fraction of which reduction was added to each of the other -arcs of vibration when impact took place.<a id="FNanchor_254" href="#Footnote_254" class="fnanchor">254</a></p> -<p><span class="pagenum" id="Page_349">349</span></p> -<p>The exact definition of the standard of length is one -of the most important, as it is one of the most difficult -questions in physical science, and the different practice of -different nations introduces needless confusion. Were -all standards constructed so as to give the true length -at a fixed uniform temperature, for instance the freezing-point, -then any two standards could be compared without -the interference of temperature by bringing them both -to exactly the same fixed temperature. Unfortunately -the French metre was defined by a bar of platinum at -0°C, while our yard was defined by a bronze bar at 62°F. -It is quite impossible, then, to make a comparison of the -yard and metre without the introduction of a correction, -either for the expansion of platinum or bronze, or both. -Bars of metal differ too so much in their rates of expansion -according to their molecular condition that it is -dangerous to infer from one bar to another.</p> - -<p>When we come to use instruments with great accuracy -there are many minute sources of error which must be -guarded against. If a thermometer has been graduated -when perpendicular, it will read somewhat differently -when laid flat, as the pressure of a column of mercury -is removed from the bulb. The reading may also be -somewhat altered if it has recently been raised to a -higher temperature than usual, if it be placed under a -vacuous receiver, or if the tube be unequally heated as -compared with the bulb. For these minute causes of -error we may have to introduce troublesome corrections, -unless we adopt the simple precaution of using the thermometer -in circumstances of position, &c., exactly similar to -those in which it was graduated. There is no end to -the number of minute corrections which may ultimately -be required. A large number of experiments on gases, -standard weights and measures, &c., depend upon the -height of the barometer; but when experiments in different -parts of the world are compared together we ought -as a further refinement to take into account the varying -force of gravity, which even between London and Paris -makes a difference of ·008 inch of mercury.</p> - -<p>The measurement of quantities of heat is a matter of -great difficulty, because there is no known substance -impervious to heat, and the problem is therefore as<span class="pagenum" id="Page_350">350</span> -difficult as to measure liquids in porous vessels. To -determine the latent heat of steam we must condense a -certain amount of the steam in a known weight of water, -and then observe the rise of temperature of the water. -But while we are carrying out the experiment, part of -the heat will escape by radiation and conduction from -the condensing vessel or calorimeter. We may indeed -reduce the loss of heat by using vessels with double sides -and bright surfaces, surrounded with swans-down wool or -other non-conducting materials; and we may also avoid -raising the temperature of the water much above that of -the surrounding air. Yet we cannot by any such means -render the loss of heat inconsiderable. Rumford ingeniously -proposed to reduce the loss to zero by commencing -the experiment when the temperature of the calorimeter -is as much below that of the air as it is at the end of the -experiment above it. Thus the vessel will first gain and -then lose by radiation and conduction, and these opposite -errors will approximately balance each other. But Regnault -has shown that the loss and gain do not proceed by -exactly the same laws, so that in very accurate investigations -Rumford’s method is not sufficient. There -remains the method of correction which was beautifully -carried out by Regnault in his determination of the latent -heat of steam. He employed two calorimeters, made in -exactly the same way and alternately used to condense a -certain amount of steam, so that while one was measuring -the latent heat, the other calorimeter was engaged in -determining the corrections to be applied, whether on -account of radiation and conduction from the vessel or -on account of heat reaching the vessel by means of the -connecting pipes.<a id="FNanchor_255" href="#Footnote_255" class="fnanchor">255</a></p> - - -<h4>4. <i>Method of Compensation.</i></h4> - -<p>There are many cases in which a cause of error cannot -conveniently be rendered null, and is yet beyond the -reach of the third method, that of calculating the requisite -correction from independent observations. The magnitude<span class="pagenum" id="Page_351">351</span> -of an error may be subject to continual variations, on -account of change of weather, or other fickle circumstances -beyond our control. It may either be impracticable to -observe the variation of those circumstances in sufficient -detail, or, if observed, the calculation of the amount of -error may be subject to doubt. In these cases, and only -in these cases, it will be desirable to invent some artificial -mode of counterpoising the variable error against an equal -error subject to exactly the same variation.</p> - -<p>We cannot weigh an object with great accuracy unless -we make a correction for the weight of the air displaced -by the object, and add this to the apparent weight. In -very accurate investigations relating to standard weights, -it is usual to note the barometer and thermometer at the -time of making a weighing, and, from the measured bulks -of the objects compared, to calculate the weight of air -displaced; the third method in fact is adopted. To make -these calculations in the frequent weighings requisite in -chemical analysis would be exceedingly laborious, hence -the correction is usually neglected. But when the chemist -wishes to weigh gas contained in a large glass globe for -the purpose of determining its specific gravity, the correction -becomes of much importance. Hence chemists avoid -at once the error, and the labour of correcting it, by -attaching to the opposite scale of the balance a dummy -sealed glass globe of equal capacity to that containing the -gas to be weighed, noting only the difference of weight -when the operating globe is full and empty. The correction, -being the same for both globes, may be entirely -neglected.<a id="FNanchor_256" href="#Footnote_256" class="fnanchor">256</a></p> - -<p>A device of nearly the same kind is employed in the -construction of galvanometers which measure the force of -an electric current by the deflection of a suspended -magnetic needle. The resistance of the needle is partly -due to the directive influence of the earth’s magnetism, -and partly to the torsion of the thread. But the former -force may often be inconveniently great as well as -troublesome to determine for different inclinations. Hence -it is customary to connect together two equally magnetised -needles, with their poles pointing in opposite directions,<span class="pagenum" id="Page_352">352</span> -one needle being within and another without the coil of -wire. As regards the earth’s magnetism, the needles are -now <i>astatic</i> or indifferent, the tendency of one needle -towards the pole being balanced by that of the other.</p> - -<p>An elegant instance of the elimination of a disturbing -force by compensation is found in Faraday’s researches -upon the magnetism of gases. To observe the magnetic -attraction or repulsion of a gas seems impossible unless we -enclose the gas in an envelope, probably best made of -glass. But any such envelope is sure to be more or less -affected by the magnet, so that it becomes difficult to -distinguish between three forces which enter into the -problem, namely, the magnetism of the gas in question, -that of the envelope, and that of the surrounding atmospheric -air. Faraday avoided all difficulties by employing -two equal and similar glass tubes connected together, and -so suspended from the arm of a torsion balance that the -tubes were in similar parts of the magnetic field. One -tube being filled with nitrogen and the other with oxygen, -it was found that the oxygen seemed to be attracted and -the nitrogen repelled. The suspending thread of the -balance was then turned until the force of torsion restored -the tubes to their original places, where the magnetism of -the tubes as well as that of the surrounding air, being -the same and in the opposite directions upon the two tubes, -could not produce any interference. The force required -to restore the tubes was measured by the amount of -torsion of the thread, and it indicated correctly the difference -between the attractive powers of oxygen and -nitrogen. The oxygen was then withdrawn from one of -the tubes, and a second experiment made, so as to compare -a vacuum with nitrogen. No force was now required to -maintain the tubes in their places, so that nitrogen was -found to be, approximately speaking, indifferent to the -magnet, that is, neither magnetic nor diamagnetic, while -oxygen was proved to be positively magnetic.<a id="FNanchor_257" href="#Footnote_257" class="fnanchor">257</a> It required -the highest experimental skill on the part of Faraday -and Tyndall, to distinguish between what is apparent and -real in magnetic attraction and repulsion.</p> - -<p>Experience alone can finally decide when a compensating<span class="pagenum" id="Page_353">353</span> -arrangement is conducive to accuracy. As a -general rule mechanical compensation is the last resource, -and in the more accurate observations it is likely to -introduce more uncertainty than it removes. A multitude -of instruments involving mechanical compensation have -been devised, but they are usually of an unscientific -character,<a id="FNanchor_258" href="#Footnote_258" class="fnanchor">258</a> because the errors compensated can be more -accurately determined and allowed for. But there are -exceptions to this rule, and it seems to be proved that in -the delicate and tiresome operation of measuring a base -line, invariable bars, compensated for expansion by heat, -give the most accurate results. This arises from the fact -that it is very difficult to determine accurately the -temperature of the measuring bars under varying conditions -of weather and manipulation.<a id="FNanchor_259" href="#Footnote_259" class="fnanchor">259</a> Again, the last -refinement in the measurement of time at Greenwich -Observatory depends upon mechanical compensation. Sir -George Airy, observing that the standard clock increased -its losing rate 0·30 second for an increase of one inch in -atmospheric pressure, placed a magnet moved by a barometer -in such a position below the pendulum, as almost -entirely to neutralise this cause of irregularity. The -thorough remedy, however, would be to remove the cause -of error altogether by placing the clock in a vacuous case.</p> - -<p>We thus see that the choice of one or other mode of -eliminating an error depends entirely upon circumstances -and the object in view; but we may safely lay down the -following conclusions. First of all, seek to avoid the -source of error altogether if it can be conveniently done; -if not, make the experiment so that the error may be as -small, but more especially as constant, as possible. If the -means are at hand for determining its amount by calculation -from other experiments and principles of science, allow -the error to exist and make a correction in the result. If -this cannot be accurately done or involves too much labour -for the purposes in view, then throw in a counteracting -error which shall as nearly as possible be of equal amount -in all circumstances with that to be eliminated. There yet -remains, however, one important method, that of Reversal,<span class="pagenum" id="Page_354">354</span> -which will form an appropriate transition to the succeeding -chapters on the Method of Mean Results and the Law of -Error.</p> - - -<h4>5. <i>Method of Reversal.</i></h4> - -<p>The fifth method of eliminating error is most potent -and satisfactory when it can be applied, but it requires -that we shall be able to reverse the apparatus and mode -of procedure, so as to make the interfering cause act -alternately in opposite directions. If we can get two -experimental results, one of which is as much too great as -the other is too small, the error is equal to half the difference, -and the true result is the mean of the two -apparent results. It is an unavoidable defect of the -chemical balance, for instance, that the points of suspension -of the pans cannot be fixed at exactly equal distances -from the centre of suspension of the beam. Hence two -weights which seem to balance each other will never be -quite equal in reality. The difference is detected by reversing -the weights, and it may be estimated by adding -small weights to the deficient side to restore equilibrium, -and then taking as the true weight the geometric mean of -the two apparent weights of the same object. If the -difference is small, the arithmetic mean, that is half the -sum, may be substituted for the geometric mean, from which -it will not appreciably differ.</p> - -<p>This method of reversal is most extensively employed -in practical astronomy. The apparent elevation of a -heavenly body is observed by a telescope moving upon -a divided circle, upon which the inclination of the -telescope is read off. Now this reading will be erroneous -if the circle and the telescope have not accurately the -same centre. But if we read off at the same time both -ends of the telescope, the one reading will be about as -much too small as the other is too great, and the mean -will be nearly free from error. In practice the observation -is differently conducted, but the principle is the same; -the telescope is fixed to the circle, which moves with it, -and the angle through which it moves is read off at three, -six, or more points, disposed at equal intervals round the -circle. The older astronomers, down even to the time of<span class="pagenum" id="Page_355">355</span> -Flamsteed, were accustomed to use portions only of a -divided circle, generally quadrants, and Römer made a -vast improvement when he introduced the complete circle.</p> - -<p>The transit circle, employed to determine the meridian -passage of heavenly bodies, is so constructed that the -telescope and the axis bearing it, in fact the whole moving -part of the instrument, can be taken out of the bearing -sockets and turned over, so that what was formerly the -western pivot becomes the eastern one, and <i>vice versâ</i>. -It is impossible that the instrument could have been -so perfectly constructed, mounted, and adjusted that the -telescope should point exactly to the meridian, but the -effect of the reversal is that it will point as much to -the west in one position as it does to the east in the -other, and the mean result of observations in the two -positions must be free from such cause of error.</p> - -<p>The accuracy with which the inclination of the compass -needle can be determined depends almost entirely on the -method of reversal. The dip needle consists of a bar -of magnetised steel, suspended somewhat like the beam of -a delicate balance on a slender axis passing through the -centre of gravity of the bar, so that it is at liberty to rest -in that exact degree of inclination in the magnetic meridian -which the magnetism of the earth induces. The inclination -is read off upon a vertical divided circle, but to avoid -error arising from the centring of the needle and circle, -both ends are read, and the mean of the results is taken. -The whole instrument is now turned carefully round -through 180°, which causes the needle to assume a new -position relatively to the circle and gives two new readings, -in which any error due to the wrong position of the zero -of the division will be reversed. As the axis of the needle -may not be exactly horizontal, it is now reversed in the -same manner as the transit instrument, the end of the axis -which formerly pointed east being made to point west, and -a new set of four readings is taken.</p> - -<p>Finally, error may arise from the axis not passing -accurately through the centre of gravity of the bar, and -this error can only be detected and eliminated on changing -the magnetic poles of the bar by the application of a -strong magnet. The error is thus made to act in opposite -directions. To ensure all possible accuracy each reversal<span class="pagenum" id="Page_356">356</span> -ought to be combined with each other reversal, so that the -needle will be observed in eight different positions by -sixteen readings, the mean of the whole of which will give -the required inclination free from all eliminable errors.<a id="FNanchor_260" href="#Footnote_260" class="fnanchor">260</a></p> - -<p>There are certain cases in which a disturbing cause can -with ease be made to act in opposite directions, in alternate -observations, so that the mean of the results will be -free from disturbance. Thus in direct experiments upon -the velocity of sound in passing through the air between -stations two or three miles apart, the wind is a cause of -error. It will be well, in the first place, to choose a time -for the experiment when the air is very nearly at rest, and -the disturbance slight, but if at the same moment signal -sounds be made at each station and observed at the other, -two sounds will be passing in opposite directions through -the same body of air and the wind will accelerate one -sound almost exactly as it retards the other. Again, in -trigonometrical surveys the apparent height of a point will -be affected by atmospheric refraction and the curvature of -the earth. But if in the case of two points the apparent -elevation of each as seen from the other be observed, the -corrections will be the same in amount, but reversed in -direction, and the mean between the two apparent differences -of altitude will give the true difference of level.</p> - -<p>In the next two chapters we really pursue the Method -of Reversal into more complicated applications.</p> -<hr class="chap x-ebookmaker-drop"> - -<div class="chapter"> -<p><span class="pagenum" id="Page_357">357</span></p> - -<h2 class="nobreak" id="CHAPTER_XVI">CHAPTER XVI.<br> - -<span class="title">THE METHOD OF MEANS.</span></h2> -</div> - -<p class="ti0">All results of the measurement of continuous quantity -can be only approximately true. Were this assertion -doubted, it could readily be proved by direct experience. -If any person, using an instrument of the greatest precision, -makes and registers successive observations in -an unbiassed manner, it will almost invariably be found -that the results differ from each other. When we operate -with sufficient care we cannot perform so simple an -experiment as weighing an object in a good balance -without getting discrepant numbers. Only the rough -and careless experimenter will think that his observations -agree, but in reality he will be found to overlook the -differences. The most elaborate researches, such as those -undertaken in connection with standard weights and -measures, always render it apparent that complete coincidence -is out of the question, and that the more accurate -our modes of observation are rendered, the more numerous -are the sources of minute error which become apparent. -We may look upon the existence of error in all measurements -as the normal state of things. It is absolutely -impossible to eliminate separately the multitude of small -disturbing influences, except by balancing them off against -each other. Even in drawing a mean it is to be expected -that we shall come near the truth rather than exactly to -it. In the measurement of continuous quantity, absolute -coincidence, if it seems to occur, must be only apparent, -and is no indication of precision. It is one of the most -embarrassing things we can meet when experimental<span class="pagenum" id="Page_358">358</span> -results agree too closely. Such coincidences should raise -our suspicion that the apparatus in use is in some way -restricted in its operation, so as not really to give the true -result at all, or that the actual results have not been faithfully -recorded by the assistant in charge of the apparatus.</p> - -<p>If then we cannot get twice over exactly the same -result, the question arises, How can we ever attain the -truth or select the result which may be supposed to -approach most nearly to it? The quantity of a certain -phenomenon is expressed in several numbers which differ -from each other; no more than one of them at the most -can be true, and it is more probable that they are all -false. It may be suggested, perhaps, that the observer -should select the one observation which he judged to be -the best made, and there will often doubtless be a feeling -that one or more results were satisfactory, and the others -less trustworthy. This seems to have been the course -adopted by the early astronomers. Flamsteed, when he -had made several observations of a star, probably chose in -an arbitrary manner that which seemed to him nearest to -the truth.<a id="FNanchor_261" href="#Footnote_261" class="fnanchor">261</a></p> - -<p>When Horrocks selected for his estimate of the sun’s -semi-diameter a mean between the results of Kepler and -Tycho, he professed not to do it from any regard to the -idle adage, “Medio tutissimus ibis,” but because he -thought it from his own observations to be correct.<a id="FNanchor_262" href="#Footnote_262" class="fnanchor">262</a> But -this method will not apply at all when the observer has -made a number of measurements which are equally good -in his opinion, and it is quite apparent that in using an -instrument or apparatus of considerable complication the -observer will not necessarily be able to judge whether -slight causes have affected its operation or not.</p> - -<p>In this question, as indeed throughout inductive logic, -we deal only with probabilities. There is no infallible -mode of arriving at the absolute truth, which lies beyond -the reach of human intellect, and can only be the distant -object of our long-continued and painful approximations. -Nevertheless there is a mode pointed out alike by common -sense and the highest mathematical reasoning, which is<span class="pagenum" id="Page_359">359</span> -more likely than any other, as a general rule, to bring us -near the truth. The ἄριστον μέτρον, or the <i>aurea mediocritas</i>, -was highly esteemed in the ancient philosophy of Greece -and Rome; but it is not probable that any of the ancients -should have been able clearly to analyse and express the -reasons why they advocated the <i>mean</i> as the safest course. -But in the last two centuries this apparently simple -question of the mean has been found to afford a field for -the exercise of the utmost mathematical skill. Roger -Cotes, the editor of the <i>Principia</i>, appears to have had -some insight into the value of the mean; but profound -mathematicians such as De Moivre, Daniel Bernoulli, -Laplace, Lagrange, Gauss, Quetelet, De Morgan, Airy, -Leslie Ellis, Boole, Glaisher, and others, have hardly exhausted -the subject.</p> - - -<h3><i>Several uses of the Mean Result.</i></h3> - -<p>The elimination of errors of unknown sources, is almost -always accomplished by the simple arithmetical process -of taking the <i>mean</i>, or, as it is often called, the <i>average</i> -of several discrepant numbers. To take an average is to -add the several quantities together, and divide by the -number of quantities thus added, which gives a quotient -lying among, or in the <i>middle</i> of, the several quantities. -Before however inquiring fully into the grounds of this -procedure, it is essential to observe that this one arithmetical -process is really applied in at least three different -cases, for different purposes, and upon different principles, -and we must take great care not to confuse one application -of the process with another. A <i>mean result</i>, then, -may have any one of the following significations.</p> - -<p>(1) It may give a merely representative number, -expressing the general magnitude of a series of quantities, -and serving as a convenient mode of comparing them -with other series of quantities. Such a number is properly -called <i>The fictitious mean</i> or <i>The average result</i>.</p> - -<p>(2) It may give a result approximately free from -disturbing quantities, which are known to affect some -results in one direction, and other results equally in the -opposite direction. We may say that in this case we get -a <i>Precise mean result</i>.</p> - -<p><span class="pagenum" id="Page_360">360</span></p> - -<p>(3) It may give a result more or less free from unknown -and uncertain errors; this we may call the <i>Probable -mean result</i>.</p> - -<p>Of these three uses of the mean the first is entirely different -in nature from the two last, since it does not yield -an approximation to any natural quantity, but furnishes -us with an arithmetic result comparing the aggregate of -certain quantities with their number. The third use of -the mean rests entirely upon the theory of probability, -and will be more fully considered in a later part of this -chapter. The second use is closely connected, or even -identical with, the Method of Reversal already described, -but it will be desirable to enter somewhat fully into all the -three employments of the same arithmetical process.</p> - - -<h3><i>The Mean and the Average.</i></h3> - -<p>Much confusion exists in the popular, or even the -scientific employment of the terms <i>mean</i> and <i>average</i>, and -they are commonly taken as synonymous. It is necessary -to ascertain carefully what significations we ought to -attach to them. The English word <i>mean</i> is equivalent to -<i>medium</i>, being derived, perhaps through the French <i>moyen</i>, -from the Latin <i>medius</i>, which again is undoubtedly kindred -with the Greek μεσος. Etymologists believe, too, that this -Greek word is connected with the preposition μετα, the -German <i>mitte</i>, and the true English <i>mid</i> or <i>middle</i>; so that -after all the <i>mean</i> is a technical term identical in its root -with the more popular equivalent <i>middle</i>.</p> - -<p>If we inquire what is the mean in a mathematical point -of view, the true answer is that there are several or many -kinds of means. The old arithmeticians recognised ten -kinds, which are stated by Boethius, and an eleventh was -added by Jordanus.<a id="FNanchor_263" href="#Footnote_263" class="fnanchor">263</a></p> - -<p>The <i>arithmetic mean</i> is the one by far the most -commonly denoted by the term, and that which we may -understand it to signify in the absence of any qualification. -It is the sum of a series of quantities divided by their -number, and may be represented by the formula <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span>(<i>a + b</i>).<span class="pagenum" id="Page_361">361</span> -But there is also the <i>geometric mean</i>, which is the square -root of the product, √<span class="o"><i>a</i> × <i>b</i></span>, or that quantity the logarithm -of which is the arithmetic mean of the logarithms -of the quantities. There is also the <i>harmonic mean</i>, -which is the reciprocal of the arithmetic mean of the -reciprocals of the quantities. Thus if <i>a</i> and <i>b</i> be the -quantities, as before, their reciprocals are <span class="nowrap"><span class="fraction2"><span class="fnum2">1</span><span class="bar">/</span><span class="fden2"><i>a</i></span></span></span> - and <span class="nowrap"><span class="fraction2"><span class="fnum2">1</span><span class="bar">/</span><span class="fden2"><i>b</i></span></span></span>, the -mean of which is <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span> <span class="fs140">(</span><span class="nowrap"><span class="fraction2"><span class="fnum2">1</span><span class="bar">/</span><span class="fden2"><i>a</i></span></span></span> + <span class="nowrap"><span class="fraction2"><span class="fnum2">1</span><span class="bar">/</span><span class="fden2"><i>b</i></span></span></span><span class="fs140">)</span>, and the reciprocal again is -<span class="nowrap"><span class="fraction2"><span class="fnum2">2<i>ab</i></span><span class="bar">/</span><span class="fden2"><i>a</i> + <i>b</i></span></span></span>, - which is the harmonic mean. Other kinds of -means might no doubt be invented for particular purposes, -and we might apply the term, as De Morgan pointed -out,<a id="FNanchor_264" href="#Footnote_264" class="fnanchor">264</a> to any quantity a function of which is equal to -a function of two or more other quantities, and is such -that the interchange of these latter quantities among themselves -will make no alteration in the value of the function. -Symbolically, if Φ (<i>y</i>, <i>y</i>, <i>y</i> ....) = Φ (<i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>, <i>x</i><sub>3</sub> ....), then <i>y</i> -is a kind of mean of the quantities, <i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>, &c.</p> - -<p>The geometric mean is necessarily adopted in certain -cases. When we estimate the work done against a force -which varies inversely as the square of the distance from a -fixed point, the mean force is the geometric mean between -the forces at the beginning and end of the path. When in -an imperfect balance, we reverse the weights to eliminate -error, the true weight will be the geometric mean of the -two apparent weights. In almost all the calculations of -statistics and commerce the geometric mean ought, strictly -speaking, to be used. If a commodity rises in price 100 -per cent. and another remains unaltered, the mean rise of -a price is not 50 per cent. because the ratio 150 : 200 is -not the same as 100 : 150. The mean ratio is as unity to -√<span class="o">1·00 × 2·00</span> or 1 to 1·41. The difference between the -three kinds of means in such a case<a id="FNanchor_265" href="#Footnote_265" class="fnanchor">265</a> is very considerable; -while the rise of price estimated by the Arithmetic mean -would be 50 per cent. it would be only 41 and 33 per cent. -respectively according to the Geometric and Harmonic -means.</p> -<p><span class="pagenum" id="Page_362">362</span></p> - -<p>In all calculations concerning the average rate of -progress of a community, or any of its operations, the -geometric mean should be employed. For if a quantity -increases 100 per cent. in 100 years, it would not on the -average increase 10 per cent. in each ten years, as the -10 per cent. would at the end of each decade be calculated -upon larger and larger quantities, and give at the end of -100 years much more than 100 per cent., in fact as much -as 159 per cent. The true mean rate in each decade -would be <sup>10</sup>√<span class="o">2</span> or about 1·07, that is, the increase would -be about 7 per cent. in each ten years. But when the -quantities differ very little, the arithmetic and geometric -means are approximately the same. Thus the arithmetic -mean of 1·000 and 1·001 is 1·0005, and the geometric mean -is about 1·0004998, the difference being of an order inappreciable -in almost all scientific and practical matters. -Even in the comparison of standard weights by Gauss’ -method of reversal, the arithmetic mean may usually be -substituted for the geometric mean which is the true result.</p> - -<p>Regarding the mean in the absence of express qualification -to the contrary as the common arithmetic mean, we -must still distinguish between its two uses where it -gives with more or less accuracy and probability a -really existing quantity, and where it acts as a mere -representative of other quantities. If I make many -experiments to determine the atomic weight of an element, -there is a certain number which I wish to approximate to, -and the mean of my separate results will, in the absence -of any reasons to the contrary, be the most probable -approximate result. When we determine the mean -density of the earth, it is not because any part of the earth -is of that exact density; there may be no part exactly -corresponding to the mean density, and as the crust of the -earth has only about half the mean density, the internal -matter of the globe must of course be above the mean. -Even the density of a homogeneous substance like carbon -or gold must be regarded as a mean between the real -density of its atoms, and the zero density of the intervening -vacuous space.</p> - -<p>The very different signification of the word “mean” in -these two uses was fully explained by Quetelet,<a id="FNanchor_266" href="#Footnote_266" class="fnanchor">266</a> and the<span class="pagenum" id="Page_363">363</span> -importance of the distinction was pointed out by Sir John -Herschel in reviewing his work.<a id="FNanchor_267" href="#Footnote_267" class="fnanchor">267</a> It is much to be desired -that scientific men would mark the difference by using the -word <i>mean</i> only in the former sense when it denotes approximation -to a definite existing quantity; and <i>average</i>, -when the mean is only a fictitious quantity, used for convenience -of thought and expression. The etymology of -this word “average” is somewhat obscure; but according -to De Morgan<a id="FNanchor_268" href="#Footnote_268" class="fnanchor">268</a> it comes from <i>averia</i>, “havings or possessions,” -especially applied to farm stock. By the accidents -of language <i>averagium</i> came to mean the labour of -farm horses to which the lord was entitled, and it probably -acquired in this manner the notion of distributing a -whole into parts, a sense in which it was early applied to -maritime averages or contributions of the other owners of -cargo to those whose goods have been thrown overboard or -used for the safety of the vessel.</p> - - -<h3><i>On the Average or Fictitious Mean.</i></h3> - -<p>Although the average when employed in its proper -sense of a fictitious mean, represents no really existing -quantity, it is yet of the highest scientific importance, as -enabling us to conceive in a single result a multitude of -details. It enables us to make a hypothetical simplification -of a problem, and avoid complexity without committing -error. The weight of a body is the sum of the weights of -infinitely small particles, each acting at a different place, -so that a mechanical problem resolves itself, strictly speaking, -into an infinite number of distinct problems. We -owe to Archimedes the first introduction of the beautiful -idea that one point may be discovered in a gravitating -body such that the weight of all the particles may be regarded -as concentrated in that point, and yet the behaviour -of the whole body will be exactly represented by the -behaviour of this heavy point. This Centre of Gravity -may be within the body, as in the case of a sphere, or it -may be in empty space, as in the case of a ring. Any two -bodies, whether connected or separate, may be conceived<span class="pagenum" id="Page_364">364</span> -as having a centre of gravity, that of the sun and earth -lying within the sun and only 267 miles from its centre.</p> - -<p>Although we most commonly use the notion of a centre -or average point with regard to gravity, the same notion -is applicable to other cases. Terrestrial gravity is a case -of approximately parallel forces, and the centre of gravity -is but a special case of the more general Centre of Parallel -Forces. Wherever a number of forces of whatever amount -act in parallel lines, it is possible to discover a point at -which the algebraic sum of the forces may be imagined to -act with exactly the same effect. Water in a cistern -presses against the side with a pressure varying according -to the depth, but always in a direction perpendicular to -the side. We may then conceive the whole pressure as -exerted on one point, which will be one-third from the -bottom of the cistern, and may be called the Centre of -Pressure. The Centre of Oscillation of a pendulum, discovered -by Huyghens, is that point at which the whole -weight of the pendulum may be considered as concentrated, -without altering the time of oscillation (p. <a href="#Page_315">315</a>). When -one body strikes another the Centre of Percussion is that -point in the striking body at which all its mass might be -concentrated without altering the effect of the stroke. In -position the Centre of Percussion does not differ from the -Centre of Oscillation. Mathematicians have also described -the Centre of Gyration, the Centre of Conversion, the -Centre of Friction, &c.</p> - -<p>We ought carefully to distinguish between those cases -in which an <i>invariable</i> centre can be assigned, and those in -which it cannot. In perfect strictness, there is no such -thing as a true invariable centre of gravity. As a general -rule a body is capable of possessing an invariable centre -only for perfectly parallel forces, and gravity never does -act in absolutely parallel lines. Thus, as usual, we find -that our conceptions are only hypothetically correct, and -only approximately applicable to real circumstances. -There are indeed certain geometrical forms called <i>Centrobaric</i>,<a id="FNanchor_269" href="#Footnote_269" class="fnanchor">269</a> -such that a body of that shape would attract another -exactly as if the mass were concentrated at the centre of -gravity, whether the forces act in a parallel manner or not.<span class="pagenum" id="Page_365">365</span> -Newton showed that uniform spheres of matter have this -property, and this truth proved of the greatest importance -in simplifying his calculations. But it is after all a purely -hypothetical truth, because we can nowhere meet with, nor -can we construct, a perfectly spherical and homogeneous -body. The slightest irregularity or protrusion from the -surface will destroy the rigorous correctness of the assumption. -The spheroid, on the other hand, has no invariable -centre at which its mass may always be regarded as concentrated. -The point from which its resultant attraction -acts will move about according to the distance and position -of the other attracting body, and it will only coincide -with the centre as regards an infinitely distant body whose -attractive forces may be considered as acting in parallel -lines.</p> - -<p>Physicists speak familiarly of the poles of a magnet, and -the term may be used with convenience. But, if we attach -any definite meaning to the word, the poles are not the -ends of the magnet, nor any fixed points within, but the -variable points from which the resultants of all the forces -exerted by the particles in the bar upon exterior magnetic -particles may be considered as acting. The poles are, in -short, Centres of Magnetic Forces; but as those forces are -never really parallel, these centres will vary in position -according to the relative place of the object attracted. -Only when we regard the magnet as attracting a very -distant, or, strictly speaking, infinitely distant particle, do -its centres become fixed points, situated in short magnets -approximately at one-sixth of the whole length from each -end of the bar. We have in the above instances of centres -or poles of force sufficient examples of the mode in which -the Fictitious Mean or Average is employed in physical -science.</p> - - -<h3><i>The Precise Mean Result.</i></h3> - -<p>We now turn to that mode of employing the mean -result which is analogous to the method of reversal, but -which is brought into practice in a most extensive manner -throughout many branches of physical science. We find -the simplest possible case in the determination of the latitude -of a place by observations of the Pole-star. Tycho<span class="pagenum" id="Page_366">366</span> -Brahe suggested that if the elevation of any circumpolar -star were observed at its higher and lower passages across -the meridian, half the sum of the elevations would be the -latitude of the place, which is equal to the height of the -pole. Such a star is as much above the pole at its highest -passage, as it is below at its lowest, so that the mean must -necessarily give the height of the pole itself free from -doubt, except as regards incidental errors. The Pole-star -is usually selected for the purpose of such observations -because it describes the smallest circle, and is thus on the -whole least affected by atmospheric refraction.</p> - -<p>Whenever several causes are in action, each of which at -one time increases and at another time decreases the joint -effect by equal quantities, we may apply this method and -disentangle the effects. Thus the solar and lunar tides -roll on in almost complete independence of each other. -When the moon is new or full the solar tide coincides, or -nearly so, with that caused by the moon, and the joint -effect is the sum of the separate effects. When the moon -is in quadrature, or half full, the two tides are acting in -opposition, one raising and the other depressing the water, -so that we observe only the difference of the effects. We -have in fact—</p> - -<div class="ml5em"> -Spring tide = lunar tide + solar tide;<br> -Neap tide = lunar tide - solar tide. -</div> - -<p class="ti0">We have only then to add together the heights of the -maximum spring tide and the minimum neap tide, and -half the sum is the true height of the lunar tide. Half -the difference of the spring and neap tides on the other -hand gives the solar tide.</p> - -<p>Effects of very small amount may be detected with -great approach to certainty among much greater fluctuations, -provided that we have a series of observations sufficiently -numerous and long continued to enable us to -balance all the larger effects against each other. For this -purpose the observations should be continued over at least -one complete cycle, in which the effects run through all -their variations, and return exactly to the same relative -positions as at the commencement. If casual or irregular -disturbing causes exist, we should probably require many -such cycles of results to render their effect inappreciable. -We obtain the desired result by taking the mean of all the<span class="pagenum" id="Page_367">367</span> -observations in which a cause acts positively, and the -mean of all in which it acts negatively. Half the difference -of these means will give the effect of the cause in -question, provided that no other effect happens to vary in -the same period or nearly so.</p> - -<p>Since the moon causes a movement of the ocean, it is -evident that its attraction must have some effect upon the -atmosphere. The laws of atmospheric tides were investigated -by Laplace, but as it would be impracticable by -theory to calculate their amounts we can only determine -them by observation, as Laplace predicted that they would -one day be determined.<a id="FNanchor_270" href="#Footnote_270" class="fnanchor">270</a> But the oscillations of the -barometer thus caused are far smaller than the oscillations -due to several other causes. Storms, hurricanes, or changes -of weather produce movements of the barometer sometimes -as much as a thousand times as great as the tides in -question. There are also regular daily, yearly, or other -fluctuations, all greater than the desired quantity. To -detect and measure the atmospheric tide it was desirable -that observations should be made in a place as free as -possible from irregular disturbances. On this account -several long series of observations were made at St. -Helena, where the barometer is far more regular in its -movements than in a continental climate. The effect of -the moon’s attraction was then detected by taking the -mean of all the readings when the moon was on the meridian -and the similar mean when she was on the horizon. -The difference of these means was found to be only -·00365, yet it was possible to discover even the variation -of this tide according as the moon was nearer to or further -from the earth, though this difference was only ·00056 -inch.<a id="FNanchor_271" href="#Footnote_271" class="fnanchor">271</a> It is quite evident that such minute effects could -never be discovered in a purely empirical manner. Having -no information but the series of observations before us, -we could have no clue as to the mode of grouping them -which would give so small a difference. In applying this -method of means in an extensive manner we must generally -then have <i>à priori</i> knowledge as to the periods at -which a cause will act in one direction or the other.</p> - -<p><span class="pagenum" id="Page_368">368</span></p> - -<p>We are sometimes able to eliminate fluctuations and -take a mean result by purely mechanical arrangements. -The daily variations of temperature, for instance, become -imperceptible one or two feet below the surface of the -earth, so that a thermometer placed with its bulb at that -depth gives very nearly the true daily mean temperature. -At a depth of twenty feet even the yearly fluctuations are -nearly effaced, and the thermometer stands a little above -the true mean temperature of the locality. In registering -the rise and fall of the tide by a tide-gauge, it is desirable -to avoid the oscillations arising from surface waves, which -is very readily accomplished by placing the float in a cistern -communicating by a small hole with the sea. Only a -general rise or fall of the level is then perceptible, just as -in the marine barometer the narrow tube prevents any -casual fluctuations and allows only a continued change of -pressure to manifest itself.</p> - - -<h3><i>Determination of the Zero point.</i></h3> - -<p>In many important observations the chief difficulty consists -in defining exactly the zero point from which we are -to measure. We can point a telescope with great precision -to a star and can measure to a second of arc the -angle through which the telescope is raised or lowered; -but all this precision will be useless unless we know -exactly the centre point of the heavens from which we -measure, or, what comes to the same thing, the horizontal -line 90° distant from it. Since the true horizon has -reference to the figure of the earth at the place of -observation, we can only determine it by the direction -of gravity, as marked either by the plumb-line or the -surface of a liquid. The question resolves itself then into -the most accurate mode of observing the direction of -gravity, and as the plumb-line has long been found -hopelessly inaccurate, astronomers generally employ the -surface of mercury in repose as the criterion of horizontality. -They ingeniously observe the direction of the -surface by making a star the index. From the laws -of reflection it follows that the angle between the -direct ray from a star and that reflected from a surface -of mercury will be exactly double the angle between the<span class="pagenum" id="Page_369">369</span> -surface and the direct ray from the star. Hence the -horizontal or zero point is the mean between the apparent -place of any star or other very distant object and its -reflection in mercury.</p> - -<p>A plumb-line is perpendicular, or a liquid surface is -horizontal only in an approximate sense; for any irregularity -of the surface of the earth, a mountain, or even a -house must cause some deviation by its attracting power. -To detect such deviation might seem very difficult, because -every other plumb-line or liquid surface would be equally -affected by gravity. Nevertheless it can be detected; for -if we place one plumb-line to the north of a mountain, and -another to the south, they will be about equally deflected -in opposite directions, and if by observations of the same -star we can measure the angle between the plumb-lines, -half the inclination will be the deviation of either, after -allowance has been made for the inclination due to the -difference of latitude of the two places of observation. By -this mode of observation applied to the mountain Schiehallion -the deviation of the plumb-line was accurately measured -by Maskelyne, and thus a comparison instituted between -the attractive forces of the mountain and the whole globe, -which led to a probable estimate of the earth’s density.</p> - -<p>In some cases it is actually better to determine the zero -point by the average of equally diverging quantities than -by direct observation. In delicate weighings by a chemical -balance it is requisite to ascertain exactly the point at -which the beam comes to rest, and when standard weights -are being compared the position of the beam is ascertained -by a carefully divided scale viewed through a microscope. -But when the beam is just coming to rest, friction, small -impediments or other accidental causes may readily obstruct -it, because it is near the point at which the force of -stability becomes infinitely small. Hence it is found better -to let the beam vibrate and observe the terminal points of -the vibrations. The mean between two extreme points -will nearly indicate the position of rest. Friction and -the resistance of air tend to reduce the vibrations, so that -this mean will be erroneous by half the amount of this -effect during a half vibration. But by taking several observations -we may determine this retardation and allow -for it. Thus if <i>a</i>, <i>b</i>, <i>c</i> be the readings of the terminal<span class="pagenum" id="Page_370">370</span> -points of three excursions of the beam from the zero of the -scale, then <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span> (<i>a</i> + <i>b</i>) - will be about as much erroneous in -one direction as <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span> (<i>b</i> + <i>c</i>) - in the other, so that the mean -of these two means, or <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">4</span></span></span> (<i>a</i> + 2<i>b</i> + <i>c</i>), - will be exceedingly -near to the point of rest.<a id="FNanchor_272" href="#Footnote_272" class="fnanchor">272</a> A still closer approximation -may be made by taking four readings and reducing them -by the formula <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">6</span></span></span> (<i>a</i> + 2<i>b</i> + 2<i>c</i> + <i>d</i>).</p> - -<p>The accuracy of Baily’s experiments, directed to determine -the density of the earth, entirely depended upon this -mode of observing oscillations. The balls whose gravitation -was measured were so delicately suspended by a -torsion balance that they never came to rest. The extreme -points of the oscillations were observed both when the -heavy leaden attracting ball was on one side and on the -other. The difference of the mean points when the leaden -ball was on the right hand and that when it was on the -left hand gave double the amount of the deflection.</p> - -<p>A beautiful instance of avoiding the use of a zero point -is found in Mr. E. J. Stone’s observations on the radiant -heat of the fixed stars. The difficulty of these observations -arose from the comparatively great amounts of heat which -were sent into the telescope from the atmosphere, and which -were sufficient to disguise almost entirely the feeble heat -rays of a star. But Mr. Stone fixed at the focus of his -telescope a double thermo-electric pile of which the two -parts were reversed in order. Now any disturbance of -temperature which acted uniformly upon both piles produced -no effect upon the galvanometer needle, and when -the rays of the star were made to fall alternately upon -one pile and the other, the total amount of the deflection -represented double the heating power of the star. Thus -Mr. Stone was able to detect with much certainty a heating -effect of the star Arcturus, which even when concentrated -by the telescope amounted only to 0°·02 Fahr., and which -represents a heating effect of the direct ray of only about -0°·00000137 Fahr., equivalent to the heat which would be -received from a three-inch cubic vessel full of boiling -water at the distance of 400 yards.<a id="FNanchor_273" href="#Footnote_273" class="fnanchor">273</a> It is probable that<span class="pagenum" id="Page_371">371</span> -Mr. Stone’s arrangement of the pile might be usefully -employed in other delicate thermometric experiments -subject to considerable disturbing influences.</p> - - -<h3><i>Determination of Maximum Points.</i></h3> - -<p>We employ the method of means in a certain number -of observations directed to determine the moment at which -a phenomenon reaches its highest point in quantity. In -noting the place of a fixed star at a given time there is no -difficulty in ascertaining the point to be observed, for a -star in a good telescope presents an exceedingly small disc. -In observing a nebulous body which from a bright centre -fades gradually away on all sides, it will not be possible -to select with certainty the middle point. In many such -cases the best method is not to select arbitrarily the supposed -middle point, but points of equal brightness on -either side, and then take the mean of the observations of -these two points for the centre. As a general rule, a -variable quantity in reaching its maximum increases at a -less and less rate, and after passing the highest point -begins to decrease by insensible degrees. The maximum -may indeed be defined as that point at which the increase -or decrease is null. Hence it will usually be the most -indefinite point, and if we can accurately measure the -phenomenon we shall best determine the place of the -maximum by determining points on either side at which -the ordinates are equal. There is moreover this advantage -in the method that several points may be determined with -the corresponding ones on the other side, and the mean of -the whole taken as the true place of the maximum. But -this method entirely depends upon the existence of symmetry -in the curve, so that of two equal ordinates one -shall be as far on one side of the maximum as the other -is on the other side. The method fails when other laws of -variation prevail.</p> - -<p>In tidal observations great difficulty is encountered in -fixing the moment of high water, because the rate at which -the water is then rising or falling, is almost imperceptible. -Whewell proposed, therefore, to note the time at -which the water passes a fixed point somewhat below the -maximum both in rising and falling, and take the mean<span class="pagenum" id="Page_372">372</span> -time as that of high water. But this mode of proceeding -unfortunately does not give a correct result, because the -tide follows different laws in rising and in falling. There -is a difficulty again in selecting the highest spring tide, -another object of much importance in tidology. Laplace -discovered that the tide of the second day preceding the -conjunction of the sun and moon is nearly equal to that of -the fifth day following; and, believing that the increase -and decrease of the tides proceeded in a nearly symmetrical -manner, he decided that the highest tide would occur about -thirty-six hours after the conjunction, that is half-way -between the second day before and the fifth day after.<a id="FNanchor_274" href="#Footnote_274" class="fnanchor">274</a></p> - -<p>This method is also employed in determining the time -of passage of the middle or densest point of a stream of -meteors. The earth takes two or three days in passing -completely through the November stream; but astronomers -need for their calculations to have some definite point fixed -within a few minutes if possible. When near to the -middle they observe the numbers of meteors which come -within the sphere of vision in each half hour, or quarter -hour, and then, assuming that the law of variation is -symmetrical, they select a moment for the passage of the -centre equidistant between times of equal frequency.</p> - -<p>The eclipses of Jupiter’s satellites are not only of great -interest as regards the motions of the satellites themselves, -but were, and perhaps still are, of use in determining -longitudes, because they are events occurring at fixed -moments of absolute time, and visible in all parts of the -planetary system at the same time, allowance being made -for the interval occupied by the light in travelling. But, -as is explained by Herschel,<a id="FNanchor_275" href="#Footnote_275" class="fnanchor">275</a> the moment of the event is -wanting in definiteness, partly because the long cone of -Jupiter’s shadow is surrounded by a penumbra, and partly -because the satellite has itself a sensible disc, and takes -time in entering the shadow. Different observers using -different telescopes would usually select different moments -for that of the eclipse. But the increase of light in the -emersion will proceed according to a law the reverse of -that observed in the immersion, so that if an observer notes<span class="pagenum" id="Page_373">373</span> -the time of both events with the same telescope, he will be -as much too soon in one observation as he is too late in the -other, and the mean moment of the two observations will -represent with considerable accuracy the time when the -satellite is in the middle of the shadow. Error of judgment -of the observer is thus eliminated, provided that -he takes care to act at the emersion as he did at the -immersion.</p> - -<hr class="chap x-ebookmaker-drop"> - -<div class="chapter"> -<p><span class="pagenum" id="Page_374">374</span></p> -<h2 class="nobreak" id="CHAPTER_XVII">CHAPTER XVII.<br> - -<span class="title">THE LAW OF ERROR.</span></h2> -</div> - -<p class="ti0">To bring error itself under law might seem beyond human -power. He who errs surely diverges from law, and it -might be deemed hopeless out of error to draw truth. One -of the most remarkable achievements of the human intellect -is the establishment of a general theory which not only -enables us among discrepant results to approximate to -the truth, but to assign the degree of probability which -fairly attaches to this conclusion. It would be a mistake -indeed to suppose that this law is necessarily the best -guide under all circumstances. Every measuring instrument -and every form of experiment may have its own -special law of error; there may in one instrument be a -tendency in one direction and in another in the opposite -direction. Every process has its peculiar liabilities to -disturbance, and we are never relieved from the necessity of -providing against special difficulties. The general Law of -Error is the best guide only when we have exhausted all -other means of approximation, and still find discrepancies, -which are due to unknown causes. We must treat such -residual differences in some way or other, since they will -occur in all accurate experiments, and as their origin is -assumed to be unknown, there is no reason why we should -treat them differently in different cases. Accordingly the -ultimate Law of Error must be a uniform and general one.</p> - -<p>It is perfectly recognised by mathematicians that in -each case a special Law of Error may exist, and should be -discovered if possible. “Nothing can be more unlikely -than that the errors committed in all classes of observations<span class="pagenum" id="Page_375">375</span> -should follow the same law,”<a id="FNanchor_276" href="#Footnote_276" class="fnanchor">276</a> and the special Laws -of Error which will apply to certain instruments, as for instance -the repeating circle, have been investigated by -Bravais.<a id="FNanchor_277" href="#Footnote_277" class="fnanchor">277</a> He concludes that every distinct cause of error -gives rise to a curve of possibility of errors, which may -have any form,—a curve which we may either be able or -unable to discover, and which in the first case may be -determined by <i>à priori</i> considerations on the peculiar -nature of this cause, or which may be determined <i>à -posteriori</i> by observation. Whenever it is practicable and -worth the labour, we ought to investigate these special -conditions of error; nevertheless, when there are a great -number of different sources of minute error, the general -resultant will always tend to obey that general law which -we are about to consider.</p> - - -<h3><i>Establishment of the Law of Error.</i></h3> - -<p>Mathematicians agree far better as to the form of the -Law of Error than they do as to the manner in which it -can be deduced and proved. They agree that among a -number of discrepant results of observation, that mean -quantity is probably the best approximation to the truth -which makes the sum of the squares of the errors as small -as possible. But there are three principal ways in which -this law has been arrived at respectively by Gauss, by -Laplace and Quetelet, and by Sir John Herschel. Gauss -proceeds much upon assumption; Herschel rests upon -geometrical considerations; while Laplace and Quetelet -regard the Law of Error as a development of the doctrine -of combinations. A number of other mathematicians, such -as Adrain of New Brunswick, Bessel, Ivory, Donkin, Leslie -Ellis, Tait, and Crofton have either attempted independent -proofs or have modified or commented on those here to be -described. For full accounts of the literature of the -subject the reader should refer either to Mr. Todhunter’s -<i>History of the Theory of Probability</i> or to the able memoir -of Mr. J. W. L. Glaisher.<a id="FNanchor_278" href="#Footnote_278" class="fnanchor">278</a></p> -<p><span class="pagenum" id="Page_376">376</span></p> - -<p>According to Gauss the Law of Error expresses the -comparative probability of errors of various magnitude, and -partly from experience, partly from <i>à priori</i> considerations, -we may readily lay down certain conditions to which -the law will certainly conform. It may fairly be assumed -as a first principle to guide us in the selection of the -law, that large errors will be far less frequent and probable -than small ones. We know that very large errors are -almost impossible, so that the probability must rapidly -decrease as the amount of the error increases. A second -principle is that positive and negative errors shall be -equally probable, which may certainly be assumed, because -we are supposed to be devoid of any knowledge as to the -causes of the residual errors. It follows that the probability -of the error must be a function of an even power of -the magnitude, that is of the square, or the fourth power, -or the sixth power, otherwise the probability of the same -amount of error would vary according as the error was -positive or negative. The even powers <i>x</i><sup>2</sup>, <i>x</i><sup>4</sup>, <i>x</i><sup>6</sup>, &c., are -always intrinsically positive, whether <i>x</i> be positive or -negative. There is no <i>à priori</i> reason why one rather than -another of these even powers should be selected. Gauss -himself allows that the fourth or sixth power would fulfil -the conditions as well as the second;<a id="FNanchor_279" href="#Footnote_279" class="fnanchor">279</a> but in the absence -of any theoretical reasons we should prefer the second -power, because it leads to formulæ of great comparative -simplicity. Did the Law of Error necessitate the use of -the higher powers of the error, the complexity of the -necessary calculations would much reduce the utility of -the theory.</p> - -<p>By mathematical reasoning which it would be undesirable -to attempt to follow in this book, it is shown -that under these conditions, the facility of occurrence, -or in other, words, the probability of error is expressed -by a function of the general form ε<sup>–<i>h</i><sup>2</sup> <i>x</i><sup>2</sup></sup>, in which <i>x</i> represents -the variable amount of errors. From this law, -to be more fully described in the following sections, it at -once follows that the most probable result of any observations<span class="pagenum" id="Page_377">377</span> -is that which makes the sum of the squares of -the consequent errors the least possible. Let <i>a</i>, <i>b</i>, <i>c</i>, -&c., be the results of observation, and <i>x</i> the quantity -selected as the most probable, that is the most free -from unknown errors: then we must determine <i>x</i> so that -(<i>a</i> - <i>x</i>)<sup>2</sup> + (<i>b</i> - <i>x</i>)<sup>2</sup> + (<i>c</i> - <i>x</i>)<sup>2</sup> + . . . shall be the least -possible quantity. Thus we arrive at the celebrated -<i>Method of Least Squares</i>, as it is usually called, which -appears to have been first distinctly put in practice by -Gauss in 1795, while Legendre first published in 1806 an -account of the process in his work, entitled, <i>Nouvelles -Méthodes pour la Détermination des Orbites des Comètes</i>. It -is worthy of notice, however, that Roger Cotes had long -previously recommended a method of equivalent nature in -his tract, “Estimatio Erroris in Mixta Mathesi.”<a id="FNanchor_280" href="#Footnote_280" class="fnanchor">280</a></p> - - -<h3><i>Herschel’s Geometrical Proof.</i></h3> - -<p>A second way of arriving at the Law of Error was -proposed by Herschel, and although only applicable to -geometrical cases, it is remarkable as showing that from -whatever point of view we regard the subject, the same -principle will be detected. After assuming that some -general law must exist, and that it is subject to the -principles of probability, he supposes that a ball is -dropped from a high point with the intention that it -shall strike a given mark on a horizontal plane. In the -absence of any known causes of deviation it will either -strike that mark, or, as is infinitely more probable, diverge -from it by an amount which we must regard as error of -unknown origin. Now, to quote the words of Herschel,<a id="FNanchor_281" href="#Footnote_281" class="fnanchor">281</a> -“the probability of that error is the unknown function of -its square, <i>i.e.</i> of the sum of the squares of its deviations in -any two rectangular directions. Now, the probability of -any deviation depending solely on its magnitude, and not -on its direction, it follows that the probability of each of -these rectangular deviations must be the same function of -<i>its</i> square. And since the observed oblique deviation is<span class="pagenum" id="Page_378">378</span> -equivalent to the two rectangular ones, supposed concurrent, -and which are essentially independent of one another, -and is, therefore, a compound event of which they are the -simple independent constituents, therefore its probability -will be the product of their separate probabilities. Thus -the form of our unknown function comes to be determined -from this condition, viz., that the product of such functions -of two independent elements is equal to the same function -of their sum. But it is shown in every work on algebra -that this property is the peculiar characteristic of, and -belongs only to, the exponential or antilogarithmic function. -This, then, is the function of the square of the error, which -expresses the probability of committing that error. That -probability decreases, therefore, in geometrical progression, -as the square of the error increases in arithmetical.”</p> - - -<h3><i>Laplace’s and Quetelet’s Proof of the Law.</i></h3> - -<p>However much presumption the modes of determining -the Law of Error, already described, may give in favour of -the law usually adopted, it is difficult to feel that the -arguments are satisfactory. The law adopted is chosen -rather on the grounds of convenience and plausibility, than -because it can be seen to be the necessary law. We can -however approach the subject from an entirely different -point of view, and yet get to the same result.</p> - -<p>Let us assume that a particular observation is subject -to four chances of error, each of which will increase the -result one inch if it occurs. Each of these errors is to be -regarded as an event independent of the rest and we can -therefore assign, by the theory of probability, the comparative -probability and frequency of each conjunction of errors. -From the Arithmetical Triangle (pp. <a href="#Page_182">182</a>–188) we learn that -no error at all can happen only in one way; an error of -one inch can happen in 4 ways; and the ways of happening -of errors of 2, 3 and 4 inches respectively, will be 6, 4 and -1 in number.</p> - -<p>We may infer that the error of two inches is the most -likely to occur, and will occur in the long run in six cases -out of sixteen. Errors of one and three inches will be -equally likely, but will occur less frequently; while no -error at all, or one of four inches will be a comparatively<span class="pagenum" id="Page_379">379</span> -rare occurrence. If we now suppose the errors to act as -often in one direction as the other, the effect will be to -alter the average error by the amount of two inches, and -we shall have the following results:—</p> - -<table class="ml5em"> -<tr> -<td class="tal pr2">Negative error of 2 inches</td> -<td class="tar"><div><div>1</div></div></td> -<td class="tal"> way.</td> -</tr> -<tr> -<td class="tal">Negative error of 1 inch</td> -<td class="tar"><div><div>4</div></div></td> -<td class="tal"> ways.</td> -</tr> -<tr> -<td class="tal">No error at all</td> -<td class="tar"><div><div>6</div></div></td> -<td class="tal"> ways.</td> -</tr> -<tr> -<td class="tal">Positive error of 1 inch</td> -<td class="tar"><div><div>4</div></div></td> -<td class="tal"> ways.</td> -</tr> -<tr> -<td class="tal">Positive error of 2 inches</td> -<td class="tar"><div><div>1</div></div></td> -<td class="tal"> way.</td> -</tr> -</table> - -<p>We may now imagine the number of causes of error -increased and the amount of each error decreased, and the -arithmetical triangle will give us the frequency of the resulting -errors. Thus if there be five positive causes of -error and five negative causes, the following table shows -the numbers of errors of various amount which will be the -result:—</p> - -<table class="ml5em fs75 mtb1em"> -<tr> -<td class="tac ball pall05">Direction of Error.</td> -<td class="tac ball pall05"><div>Positive Error.</div></td> -<td class="tac ball pall05"></td> -<td class="tac ball pall05"><div>Negative Error.</div></td> -</tr> -<tr> -<td class="tal ball pall05">Amount of Error.</td> -<td class="tac ball pall05"><div>5, 4, 3, 2, 1</div></td> -<td class="tac ball pall05"><div>0</div></td> -<td class="tac ball pall05"><div>1, 2, 3, 4, 5</div></td> -</tr> -<tr> -<td class="tal ball pall05">Number of such Errors.</td> -<td class="tac ball pall05"><div>1, 10, 45, 120, 210</div></td> -<td class="tac ball pall05"><div>252</div></td> -<td class="tac ball pall05"><div>210 120, 45, 10, 1</div></td> -</tr> -</table> - -<p>It is plain that from such numbers I can ascertain -the probability of any particular amount of error under -the conditions supposed. The probability of a positive -error of exactly one inch is <span class="nowrap"><span class="fraction2"><span class="fnum2">210</span><span class="bar">/</span><span class="fden2">1024</span></span></span>, in which fraction the -numerator is the number of combinations giving one -inch positive error, and the denominator the whole -number of possible errors of all magnitudes. I can also, -by adding together the appropriate numbers get the probability -of an error not exceeding a certain amount. Thus -the probability of an error of three inches or less, positive -or negative, is a fraction whose numerator is the sum of -45 + 120 + 210 + 252 + 210 + 120 + 45, and the denominator, -as before, giving the result <span class="nowrap"><span class="fraction2"><span class="fnum2">1002</span><span class="bar">/</span><span class="fden2">1024</span></span></span>. We may see at -once that, according to these principles, the probability of -small errors is far greater than of large ones: the odds are -1002 to 22, or more than 45 to 1, that the error will not<span class="pagenum" id="Page_380">380</span> -exceed three inches; and the odds are 1022 to 2 against -the occurrence of the greatest possible error of five inches.</p> - -<p>If any case should arise in which the observer knows -the number and magnitude of the chief errors which -may occur, he ought certainly to calculate from the Arithmetical -Triangle the special Law of Error which would -apply. But the general law, of which we are in search, -is to be used in the dark, when we have no knowledge -whatever of the sources of error. To assume any special -number of causes of error is then an arbitrary proceeding, -and mathematicians have chosen the least arbitrary course -of imagining the existence of an infinite number of infinitely -small errors, just as, in the inverse method of -probabilities, an infinite number of infinitely improbable -hypotheses were submitted to calculation (p. <a href="#Page_255">255</a>).</p> - -<p>The reasons in favour of this choice are of several -different kinds.</p> - -<p>1. It cannot be denied that there may exist infinitely -numerous causes of error in any act of observation.</p> - -<p>2. The law resulting from the hypothesis of a moderate -number of causes of error, does not appreciably differ from -that given by the hypothesis of an infinite number of -causes of error.</p> - -<p>3. We gain by the hypothesis of infinity a general law -capable of ready calculation, and applicable by uniform -rules to all problems.</p> - -<p>4. This law, when tested by comparison with extensive -series of observations, is strikingly verified, as will be -shown in a later section.</p> - -<p>When we imagine the existence of any large number of -causes of error, for instance one hundred, the numbers of -combinations become impracticably large, as may be seen -to be the case from a glance at the Arithmetical Triangle, -which proceeds only up to the seventeenth line. Quetelet, -by suitable abbreviating processes, calculated out a table -of probability of errors on the hypothesis of one thousand -distinct causes;<a id="FNanchor_282" href="#Footnote_282" class="fnanchor">282</a> but mathematicians have generally -proceeded on the hypothesis of infinity, and then, by the -devices of analysis, have substituted a general law of easy<span class="pagenum" id="Page_381">381</span> -treatment. In mathematical works upon the subject, it is -shown that the standard Law of Error is expressed in the -formula</p> - -<div class="ml5em fs110"> -<i>y</i> = <i>Y</i> ε <sup>-<i>cx</i><sup>2</sup></sup>, -</div> - -<p class="ti0">in which <i>x</i> is the amount of the error, <i>Y</i> the maximum -ordinate of the curve of error, and <i>c</i> a number constant -for each series of observations, and expressing the amount -of the tendency to error, varying between one series of -observations and another. The letter ε is the mathematical -constant, the sum of ratios between the numbers of permutations -and combinations, previously referred to (p. <a href="#Page_330">330</a>).</p> - -<figure class="figcenter illowp100" id="p381" style="max-width: 26.9375em;"> - <img class="w100" src="images/p381.jpg" alt=""> -</figure> - -<p>To show the close correspondence of this general -law with the special law which might be derived -from the supposition of a moderate number of causes -of error, I have in the accompanying figure drawn a -curved line representing accurately the variation of <i>y</i> -when <i>x</i> in the above formula is taken equal 0, <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span>, - 1, <span class="nowrap"><span class="fraction"><span class="fnum">3</span><span class="bar">/</span><span class="fden">2</span></span></span>, 2, -&c., positive or negative, the arbitrary quantities <i>Y</i> and <i>c</i> -being each assumed equal to unity, in order to simplify -the calculations. In the same figure are inserted eleven -dots, whose heights above the base line are proportional -to the numbers in the eleventh line of the Arithmetical -Triangle, thus representing the comparative probabilities -of errors of various amounts arising from ten equal causes<span class="pagenum" id="Page_382">382</span> -of error. The correspondence of the general and the -special Law of Error is almost as close as can be exhibited -in the figure, and the assumption of a greater number of -equal causes of error would render the correspondence far -more close.</p> - -<p>It may be explained that the ordinates NM, <i>nm</i>, <i>n′m′</i>, -represent values of <i>y</i> in the equation expressing the Law -of Error. The occurrence of any one definite amount of -error is infinitely improbable, because an infinite number -of such ordinates might be drawn. But the probability of -an error occurring between certain limits is finite, and is -represented by a portion of the <i>area</i> of the curve. Thus the -probability that an error, positive or negative, not exceeding -unity will occur, is represented by the area M<i>mnn′m′</i>, -in short, by the area standing upon the line <i>nn′</i>. -Since every observation must either have some definite -error or none at all, it follows that the whole area of the -curve should be considered as the unit expressing certainty, -and the probability of an error falling between particular -limits will then be expressed by the ratio which the area -of the curve between those limits bears to the whole area -of the curve.</p> - -<p>The mere fact that the Law of Error allows of the possible -existence of errors of every assignable amount shows -that it is only approximately true. We may fairly say -that in measuring a mile it would be impossible to commit -an error of a hundred miles, and the length of life would -never allow of our committing an error of one million -miles. Nevertheless the general Law of Error would assign -a probability for an error of that amount or more, but so -small a probability as to be utterly inconsiderable and -almost inconceivable. All that can, or in fact need, be -said in defence of the law is, that it may be made to represent -the errors in any special case to a very close -approximation, and that the probability of large and practically -impossible errors, as given by the law, will be so -small as to be entirely inconsiderable. And as we are -dealing with error itself, and our results pretend to nothing -more than approximation and probability, an indefinitely -small error in our process of approximation is of no importance -whatever.</p> - -<p><span class="pagenum" id="Page_383">383</span></p> - - -<h3><i>Logical Origin of the Law of Error.</i></h3> - -<p>It is worthy of notice that this Law of Error, abstruse -though the subject may seem, is really founded upon the -simplest principles. It arises entirely out of the difference -between permutations and combinations, a subject upon -which I may seem to have dwelt with unnecessary prolixity -in previous pages (pp. <a href="#Page_170">170</a>, <a href="#Page_189">189</a>). The order in which we -add quantities together does not affect the amount of the -sum, so that if there be three positive and five negative -causes of error in operation, it does not matter in which -order they are considered as acting. They may be intermixed -in any arrangement, and yet the result will be the -same. The reader should not fail to notice how laws or -principles which appeared to be absurdly simple and -evident when first noticed, reappear in the most complicated -and mysterious processes of scientific method. The fundamental -Laws of Identity and Difference gave rise to the -Logical Alphabet which, after abstracting the character of -the differences, led to the Arithmetical Triangle. The -Law of Error is defined by an infinitely high line of that -triangle, and the law proves that the mean is the most probable -result, and that divergencies from the mean become -much less probable as they increase in amount. Now the -comparative greatness of the numbers towards the middle -of each line of the Arithmetical Triangle is entirely due -to the indifference of order in space or time, which was -first prominently pointed out as a condition of logical relations, -and the symbols indicating them (pp. <a href="#Page_32">32</a>–35), and -which was afterwards shown to attach equally to numerical -symbols, the derivatives of logical terms (p. <a href="#Page_160">160</a>).</p> - - -<h3><i>Verification of the Law of Error.</i></h3> - -<p>The theory of error which we have been considering -rests entirely upon an assumption, namely that when -known sources of disturbances are allowed for, there yet -remain an indefinite, possibly an infinite number of other -minute sources of error, which will as often produce excess -as deficiency. Granting this assumption, the Law of -Error must be as it is usually taken to be, and there is -no more need to verify it empirically than to test the truth<span class="pagenum" id="Page_384">384</span> -of one of Euclid’s propositions mechanically. Nevertheless, -it is an interesting occupation to verify even the propositions -of geometry, and it is still more instructive to -try whether a large number of observations will justify our -assumption of the Law of Error.</p> - -<p>Encke has given an excellent instance of the correspondence -of theory with experience, in the case of observations -of the differences of Right Ascension of the sun and two -stars, namely α Aquilæ and α Canis minoris. The observations -were 470 in number, and were made by Bradley -and reduced by Bessel, who found the probable error of -the final result to be only about one-fourth part of a second -(0·2637). He then compared the numbers of errors of -each magnitude from 0·1 second upwards, as actually given -by the observations, with what should occur according to -the Law of Error.</p> - -<p>The results were as follow:—<a id="FNanchor_283" href="#Footnote_283" class="fnanchor">283</a></p> - -<table class="ml5em fs75 mtb1em"> -<tr> -<td class="tac ball pall05" rowspan="2" colspan="3"><div>Magnitude of the errors<br>in parts of a second.</div></td> -<td class="tac ball pall05" colspan="2"><div>Number of errors of each<br>magnitude according to</div></td> -</tr> -<tr> -<td class="tac ball pall05"><div>Observation.</div></td> -<td class="tac ball pall05"><div>Theory.</div></td> -</tr> -<tr> -<td class="tar pl1 bl pt05"><div>0·0</div></td> -<td class="tac pt05"><div>to</div></td> -<td class="tal pr1 pt05">0·1</td> -<td class="tac brl pt05"><div>94</div></td> -<td class="tac br pt05"><div>95</div></td> -</tr> -<tr> -<td class="tar bl"><div> ·1</div></td> -<td class="tac"><div>"</div></td> -<td class="tal"> ·2</td> -<td class="tac brl"><div>88</div></td> -<td class="tac br"><div>89</div></td> -</tr> -<tr> -<td class="tar bl"><div> ·2</div></td> -<td class="tac"><div>"</div></td> -<td class="tal"> ·3</td> -<td class="tac brl"><div>78</div></td> -<td class="tac br"><div>78</div></td> -</tr> -<tr> -<td class="tar bl"><div> ·3</div></td> -<td class="tac"><div>"</div></td> -<td class="tal"> ·4</td> -<td class="tac brl"><div>58</div></td> -<td class="tac br"><div>64</div></td> -</tr> -<tr> -<td class="tar bl"><div> ·4</div></td> -<td class="tac"><div>"</div></td> -<td class="tal"> ·5</td> -<td class="tac brl"><div>51</div></td> -<td class="tac br"><div>50</div></td> -</tr> -<tr> -<td class="tar bl"><div> ·5</div></td> -<td class="tac"><div>"</div></td> -<td class="tal"> ·6</td> -<td class="tac brl"><div>36</div></td> -<td class="tac br"><div>36</div></td> -</tr> -<tr> -<td class="tar bl"><div> ·6</div></td> -<td class="tac"><div>"</div></td> -<td class="tal"> ·7</td> -<td class="tac brl"><div>26</div></td> -<td class="tac br"><div>24</div></td> -</tr> -<tr> -<td class="tar bl"><div> ·7</div></td> -<td class="tac"><div>"</div></td> -<td class="tal"> ·8</td> -<td class="tac brl"><div>14</div></td> -<td class="tac br"><div>15</div></td> -</tr> -<tr> -<td class="tar bl"><div> ·8</div></td> -<td class="tac"><div>"</div></td> -<td class="tal"> ·9</td> -<td class="tac brl"><div>10</div></td> -<td class="tac br"><div> 9</div></td> -</tr> -<tr> -<td class="tar bl"><div> ·9</div></td> -<td class="tac"><div>"</div></td> -<td class="tal">1·0</td> -<td class="tac brl"><div> 7</div></td> -<td class="tac br"><div> 5</div></td> -</tr> -<tr> -<td class="tar bbl pb05" colspan="2"><div>above </div></td> -<td class="tal bb pb05">1·0</td> -<td class="tac bbrl pb05"><div> 8</div></td> -<td class="tac bbr pb05"> 5</td> -</tr> -</table> - -<p>The reader will remark that the correspondence is very -close, except as regards larger errors, which are excessive -in practice. It is one objection, indeed, to the theory of -error, that, being expressed in a continuous mathematical -function, it contemplates the existence of errors of every -magnitude, such as could not practically occur; yet in this -case the theory seems to under-estimate the number of -large errors.</p> - -<p><span class="pagenum" id="Page_385">385</span></p> - -<p>Another comparison of the law with observation was made -by Quetelet, who investigated the errors of 487 determinations -in time of the Right Ascension of the Pole-Star -made at Greenwich during the four years 1836–39. These -observations, although carefully corrected for all known -causes of error, as well as for nutation, precession, &c., -are yet of course found to differ, and being classified as -regards intervals of one-half second of time, and then proportionately -increased in number, so that their sum may -be one thousand, give the following results as compared -with what Quetelet’s theory would lead us to expect:—<a id="FNanchor_284" href="#Footnote_284" class="fnanchor">284</a></p> - -<table class="ml5em fs75 mtb1em"> -<tr> -<td class="tac ball pall05" rowspan="2"><div>Magnitude of<br>error in tenths<br>of a second.</div></td> -<td class="tac ball pall05" colspan="2"><div>Number of Errors</div></td> -<td class="tac ball pall05" rowspan="2"><div>Magnitude of<br>error in tenths<br>of a second.</div></td> -<td class="tac ball pall05" colspan="2"><div>Number of Errors</div></td> -</tr> -<tr> -<td class="tac ball pall05"><div>by<br>Observation.</div></td> -<td class="tac ball pall05"><div>by<br>Theory.</div></td> -<td class="tac ball pall05"><div>by<br>Observation.</div></td> -<td class="tac ball pall05"><div>by<br>Theory.</div></td> -</tr> -<tr> -<td class="tac brl pt05"><div> 0·0</div></td> -<td class="tac pt05"><div>168</div></td> -<td class="tac brl pt05"><div>163</div></td> -<td class="tac pt05"><div>–</div></td> -<td class="tac brl pt05"><div>–</div></td> -<td class="tac br pt05"><div>–</div></td> -</tr> -<tr> -<td class="tac brl"><div>+0·5</div></td> -<td class="tac"><div>148</div></td> -<td class="tac brl"><div>147</div></td> -<td class="tac"><div>–0·5</div></td> -<td class="tac brl"><div>150</div></td> -<td class="tac br"><div>152</div></td> -</tr> -<tr> -<td class="tac brl"><div>+1·0</div></td> -<td class="tac"><div>129</div></td> -<td class="tac brl"><div>112</div></td> -<td class="tac"><div>–1·0</div></td> -<td class="tac brl"><div>126</div></td> -<td class="tac br"><div>121</div></td> -</tr> -<tr> -<td class="tac brl"><div>+1·5</div></td> -<td class="tac"><div> 78</div></td> -<td class="tac brl"><div> 72</div></td> -<td class="tac"><div>–1·5</div></td> -<td class="tac brl"><div> 74</div></td> -<td class="tac br"><div> 82</div></td> -</tr> -<tr> -<td class="tac brl"><div>+2·0</div></td> -<td class="tac"><div> 33</div></td> -<td class="tac brl"><div> 40</div></td> -<td class="tac"><div>–2·0</div></td> -<td class="tac brl"><div> 43</div></td> -<td class="tac br"><div> 46</div></td> -</tr> -<tr> -<td class="tac brl"><div>+2·5</div></td> -<td class="tac"><div> 10</div></td> -<td class="tac brl"><div> 19</div></td> -<td class="tac"><div>–2·5</div></td> -<td class="tac brl"><div> 25</div></td> -<td class="tac br"><div> 22</div></td> -</tr> -<tr> -<td class="tac brl"><div>+3·0</div></td> -<td class="tac"><div> 2</div></td> -<td class="tac brl"><div> 10</div></td> -<td class="tac"><div>–3·0</div></td> -<td class="tac brl"><div> 12</div></td> -<td class="tac br"><div> 10</div></td> -</tr> -<tr> -<td class="tac bbrl pb05"><div> –</div></td> -<td class="tac bb pb05"><div>–</div></td> -<td class="tac bbrl pb05"><div>–</div></td> -<td class="tac bb pb05"><div>–3·5</div></td> -<td class="tac bbrl pb05"><div> 2</div></td> -<td class="tac bbr pb05"><div> 4</div></td> -</tr> -</table> - -<p>In this instance also the correspondence is satisfactory, -but the divergence between theory and fact is in the opposite -direction to that discovered in the former comparison, the -larger errors being less frequent than theory would indicate. -It will be noticed that Quetelet’s theoretical results -are not symmetrical.</p> - - -<h3><i>The Probable Mean Result.</i></h3> - -<p>One immediate result of the Law of Error, as thus -stated, is that the mean result is the most probable one; -and when there is only a single variable this mean is -found by the familiar arithmetical process. An unfortunate -error has crept into several works which allude -to this subject. Mill, in treating of the “Elimination of -Chance,” remarks in a note<a id="FNanchor_285" href="#Footnote_285" class="fnanchor">285</a> that “the mean is spoken of<span class="pagenum" id="Page_386">386</span> -as if it were exactly the same thing as the average. -But the mean, for purposes of inductive inquiry, is not the -average, or arithmetical mean, though in a familiar illustration -of the theory the difference may be disregarded.” -He goes on to say that, according to mathematical principles, -the most probable result is that for which the sums -of the squares of the deviations is the least possible. It -seems probable that Mill and other writers were misled -by Whewell, who says<a id="FNanchor_286" href="#Footnote_286" class="fnanchor">286</a> that “The method of least -squares is in fact a method of means, but with some -peculiar characters.... The method proceeds upon -this supposition: that all errors are not equally probable, -but that small errors are more probable than large ones.” -He adds that this method “removes much that is arbitrary -in the method of means.” It is strange to find a mathematician -like Whewell making such remarks, when there -is no doubt whatever that the Method of Means is only -an application of the Method of Least Squares. They are, -in fact, the same method, except that the latter method -may be applied to cases where two or more quantities have -to be determined at the same time. Lubbock and Drinkwater -say,<a id="FNanchor_287" href="#Footnote_287" class="fnanchor">287</a> “If only one quantity has to be determined, -this method evidently resolves itself into taking the mean -of all the values given by observation.” Encke says,<a id="FNanchor_288" href="#Footnote_288" class="fnanchor">288</a> that -the expression for the probability of an error “not only -contains in itself the principle of the arithmetical mean, -but depends so immediately upon it, that for all those -magnitudes for which the arithmetical mean holds good -in the simple cases in which it is principally applied, -no other law of probability can be assumed than that -which is expressed by this formula.”</p> - - -<h3><i>The Probable Error of Results.</i></h3> - -<p>When we draw a conclusion from the numerical -results of observations we ought not to consider it sufficient, -in cases of importance, to content ourselves with -finding the simple mean and treating it as true. We -ought also to ascertain what is the degree of confidence<span class="pagenum" id="Page_387">387</span> -we may place in this mean, and our confidence should be -measured by the degree of concurrence of the observations -from which it is derived. In some cases the mean may -be approximately certain and accurate. In other cases it -may really be worth little or nothing. The Law of Error -enables us to give exact expression to the degree of confidence -proper in any case; for it shows how to calculate -the probability of a divergence of any amount from the -mean, and we can thence ascertain the probability that -the mean in question is within a certain distance from the -true number. The <i>probable error</i> is taken by mathematicians -to mean the limits within which it is as likely as -not that the truth will fall. Thus if 5·45 be the mean of -all the determinations of the density of the earth, and ·20 -be approximately the probable error, the meaning is that -the probability of the real density of the earth falling between -5·25 and 5·65 is <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2.</span></span></span> Any other limits might have -been selected at will. We might calculate the limits -within which it was one hundred or one thousand to one -that the truth would fall; but there is a convention to -take the even odds one to one, as the quantity of probability -of which the limits are to be estimated.</p> - -<p>Many books on probability give rules for making the -calculations, but as, in the progress of science, persons -ought to become more familiar with these processes, -I propose to repeat the rules here and illustrate their -use. The calculations, when made in accordance with -the directions, involve none but arithmetic or logarithmic -operations.</p> - -<p>The following are the rules for treating a mean result, -so as thoroughly to ascertain its trustworthiness.</p> - -<p>1. Draw the mean of all the observed results.</p> - -<p>2. Find the excess or defect, that is, the error of each -result from the mean.</p> - -<p>3. Square each of these reputed errors.</p> - -<p>4. Add together all these squares of the errors, which -are of course all positive.</p> - -<p>5. Divide by one less than the number of observations. -This gives the <i>square of the mean error</i>.</p> - -<p>6. Take the square root of the last result; it is the <i>mean -error of a single observation</i>.</p> - -<p>7. Divide now by the square root of the number of<span class="pagenum" id="Page_388">388</span> -observations, and we get the <i>mean error of the mean -result</i>.</p> - -<p>8. Lastly, multiply by the natural constant O·6745 (or -approximately by 0·674, or even by <span class="nowrap"><span class="fraction"><span class="fnum">2</span><span class="bar">/</span><span class="fden">3</span></span></span>), and we arrive at -the <i>probable error of the mean result</i>.</p> - -<p>Suppose, for instance, that five measurements of the -height of a hill, by the barometer or otherwise, have given -the numbers of feet as 293, 301, 306, 307, 313; we want -to know the probable error of the mean, namely 304. Now -the differences between this mean and the above numbers, -<i>paying no regard to direction</i>, are 11, 3, 2, 3, 9; their -squares are 121, 9, 4, 9, 81, and the sum of the squares -of the errors consequently 224. The number of observations -being 5, we divide by 1 less, or 4, getting 56. This -is the square of the mean error, and taking its square root -we have 7·48 (say <span class="nowrap">7 <span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span>), - the mean error of a single observation. -Dividing by 2·236, the square root of 5, the -number of observations, we find the mean error of the <i>mean</i> -result to be 3·35, or say <span class="nowrap">3 <span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">3</span></span></span>, and lastly, multiplying by -·6745, we arrive at the <i>probable error of the mean result</i>, -which is found to be 2·259, or say <span class="nowrap">2 <span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">4</span></span></span>. The meaning of -this is that the probability is one half, or the odds are -even that the true height of the mountain lies between -<span class="nowrap">301 <span class="fraction"><span class="fnum">3</span><span class="bar">/</span><span class="fden">4</span></span></span> - and <span class="nowrap">306 <span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">4</span></span></span> feet. We have thus an exact measure of -the degree of credibility of our mean result, which mean -indicates the most likely point for the truth to fall -upon.</p> - -<p>The reader should observe that as the object in these -calculations is only to gain a notion of the degree of confidence -with which we view the mean, there is no real use -in carrying the calculations to any great degree of precision; -and whenever the neglect of decimal fractions, or -even the slight alteration of a number, will much abbreviate -the computations, it may be fearlessly done, except in -cases of high importance and precision. Brodie has shown -how the law of error may be usefully applied in chemical -investigations, and some illustrations of its employment -may be found in his paper.<a id="FNanchor_289" href="#Footnote_289" class="fnanchor">289</a></p> - -<p>The experiments of Benzenberg to detect the revolution -of the earth, by the deviation of a ball from the perpendicular<span class="pagenum" id="Page_389">389</span> -line in falling down a deep pit, have been cited by -Encke<a id="FNanchor_290" href="#Footnote_290" class="fnanchor">290</a> as an interesting illustration of the Law of Error. -The mean deviation was 5·086 lines, and its probable error -was calculated by Encke to be not more than ·950 line, -that is, the odds were even that the true result lay between -4·136 and 6·036. As the deviation, according to astronomical -theory, should be 4·6 lines, which lies well within -the limits, we may consider that the experiments are -consistent with the Copernican system of the universe.</p> - -<p>It will of course be understood that the probable error -has regard only to those causes of errors which in the long -run act as much in one direction as another; it takes no -account of constant errors. The true result accordingly -will often fall far beyond the limits of probable error, owing -to some considerable constant error or errors, of the existence -of which we are unaware.</p> - - -<h3><i>Rejection of the Mean Result.</i></h3> - -<p>We ought always to bear in mind that the mean of any -series of observations is the best, that is, the most probable -approximation to the truth, only in the absence of knowledge -to the contrary. The selection of the mean rests -entirely upon the probability that unknown causes of error -will in the long run fall as often in one direction as the -opposite, so that in drawing the mean they will balance -each other. If we have any reason to suppose that there -exists a tendency to error in one direction rather than the -other, then to choose the mean would be to ignore that -tendency. We may certainly approximate to the length -of the circumference of a circle, by taking the mean of the -perimeters of inscribed and circumscribed polygons of an -equal and large number of sides. The length of the circular -line undoubtedly lies between the lengths of the two -perimeters, but it does not follow that the mean is the -best approximation. It may in fact be shown that the -circumference of the circle is <i>very nearly</i> equal to the -perimeter of the inscribed polygon, together with one-third -part of the difference between the inscribed and circumscribed -polygons of the same number of sides. Having<span class="pagenum" id="Page_390">390</span> -this knowledge, we ought of course to act upon it, instead -of trusting to probability.</p> - -<p>We may often perceive that a series of measurements -tends towards an extreme limit rather than towards a -mean. In endeavouring to obtain a correct estimate -of the apparent diameter of the brightest fixed stars, we -find a continuous diminution in estimates as the powers -of observation increased. Kepler assigned to Sirius an -apparent diameter of 240 seconds; Tycho Brahe made -it 126; Gassendi 10 seconds; Galileo, Hevelius, and J. -Cassini, 5 or 6 seconds. Halley, Michell, and subsequently -Sir W. Herschel came to the conclusion that the brightest -stars in the heavens could not have real discs of a second, -and were probably much less in diameter. It would of -course be absurd to take the mean of quantities which -differ more than 240 times; and as the tendency has -always been to smaller estimates, there is a considerable -presumption in favour of the smallest.<a id="FNanchor_291" href="#Footnote_291" class="fnanchor">291</a></p> - -<p>In many experiments and measurements we know that -there is a preponderating tendency to error in one direction. -The readings of a thermometer tend to rise as -the age of the instrument increases, and no drawing of -means will correct this result. Barometers, on the other -hand, are likely to read too low instead of too high, -owing to the imperfection of the vacuum and the action of -capillary attraction. If the mercury be perfectly pure and -no appreciable error be due to the measuring apparatus, -the best barometer will be that which gives the highest -result. In determining the specific gravity of a solid -body the chief danger of error arises from bubbles of air -adhering to the body, which would tend to make the -specific gravity too small. Much attention must always -be given to one-sided errors of this kind, since the multiplication -of experiments does not remove the error. In -such cases one very careful experiment is better than any -number of careless ones.</p> - -<p>When we have reasonable grounds for supposing that -certain experimental results are liable to grave errors, we -should exclude them in drawing a mean. If we want to -find the most probable approximation to the velocity of<span class="pagenum" id="Page_391">391</span> -sound in air, it would be absurd to go back to the old -experiments which made the velocity from 1200 to 1474 -feet per second; for we know that the old observers did -not guard against errors arising from wind and other -causes. Old chemical experiments are valueless as regards -quantitative results. The old chemists found the -atmosphere in different places to differ in composition -nearly ten per cent., whereas modern accurate experimenters -find very slight variations. Any method of -measurement which we know to avoid a source of error -is far to be preferred to others which trust to probabilities -for the elimination of the error. As Flamsteed says,<a id="FNanchor_292" href="#Footnote_292" class="fnanchor">292</a> “One -good instrument is of as much worth as a hundred indifferent -ones.” But an instrument is good or bad only in -a comparative sense, and no instrument gives invariable -and truthful results. Hence we must always ultimately -fall back upon probabilities for the selection of the final -mean, when other precautions are exhausted.</p> - -<p>Legendre, the discoverer of the method of Least Squares, -recommended that observations differing very much from -the results of his method should be rejected. The subject -has been carefully investigated by Professor Pierce, who has -proposed a criterion for the rejection of doubtful observations -based on the following principle:<a id="FNanchor_293" href="#Footnote_293" class="fnanchor">293</a>′“—observations -should be rejected when the probability of the system of -errors obtained by retaining them is less than that of the -system of errors obtained by their rejection multiplied by -the probability of making so many and no more abnormal -observations.” Professor Pierce’s investigation is given -nearly in his own words in Professor W. Chauvenet’s -“Manual of Spherical and Practical Astronomy,” which -contains a full and excellent discussion of the methods of -treating numerical observations.<a id="FNanchor_294" href="#Footnote_294" class="fnanchor">294</a></p> - -<p>Very difficult questions sometimes arise when one or -more results of a method of experiment diverge widely -from the mean of the rest. Are we or are we not to exclude -them in adopting the supposed true mean result of -the method? The drawing of a mean result rests, as I<span class="pagenum" id="Page_392">392</span> -have frequently explained, upon the assumption that every -error acting in one direction will probably be balanced by -other errors acting in an opposite direction. If then we -know or can possibly discover any causes of error not -agreeing with this assumption, we shall be justified in -excluding results which seem to be affected by this cause.</p> - -<p>In reducing large series of astronomical observations, it is -not uncommon to meet with numbers differing from others -by a whole degree or half a degree, or some considerable integral -quantity. These are errors which could hardly arise -in the act of observation or in instrumental irregularity; -but they might readily be accounted for by misreading -of figures or mistaking of division marks. It would be -absurd to trust to chance that such mistakes would -balance each other in the long run, and it is therefore better -to correct arbitrarily the supposed mistake, or better still, -if new observations can be made, to strike out the divergent -numbers altogether. When results come sometimes -too great or too small in a regular manner, we should -suspect that some part of the instrument slips through a -definite space, or that a definite cause of error enters at -times, and not at others. We should then make it a point -of prime importance to discover the exact nature and -amount of such an error, and either prevent its occurrence -for the future or else introduce a corresponding correction. -In many researches the whole difficulty will consist in -this detection and avoidance of sources of error. Professor -Roscoe found that the presence of phosphorus caused -serious and almost unavoidable errors in the determination -of the atomic weight of vanadium.<a id="FNanchor_295" href="#Footnote_295" class="fnanchor">295</a> Herschel, in reducing -his observations of double stars at the Cape of Good Hope, -was perplexed by an unaccountable difference of the angles -of position as measured by the seven-feet equatorial and -the twenty-feet reflector telescopes, and after a careful investigation -was obliged to be contented with introducing -a correction experimentally determined.<a id="FNanchor_296" href="#Footnote_296" class="fnanchor">296</a></p> - -<p>When observations are sufficiently numerous it seems -desirable to project the apparent errors into a curve, and -then to observe whether this curve exhibits the symmetrical<span class="pagenum" id="Page_393">393</span> -and characteristic form of the curve of error. If so, -it may be inferred that the errors arise from many minute -independent sources, and probably compensate each other -in the mean result. Any considerable irregularity will -indicate the existence of one-sided or large causes of error, -which should be made the subject of investigation.</p> - -<p>Even the most patient and exhaustive investigations -will sometimes fail to disclose any reason why some -results diverge from others. The question again recurs—Are -we arbitrarily to exclude them? The answer should -be in the negative as a general rule. The mere fact of -divergence ought not to be taken as conclusive against a -result, and the exertion of arbitrary choice would open -the way to the fatal influence of bias, and what is commonly -known as the “cooking” of figures. It would -amount to judging fact by theory instead of theory by fact. -The apparently divergent number may prove in time to be -the true one. It may be an exception of that valuable -kind which upsets our false theories, a real exception, -exploding apparent coincidences, and opening a way to a -new view of the subject. To establish this position for -the divergent fact will require additional research; but -in the meantime we should give it some weight in our -mean conclusions, and should bear in mind the discrepancy -as one demanding attention. To neglect a divergent result -is to neglect the possible clue to a great discovery.</p> - - -<h3><i>Method of Least Squares.</i></h3> - -<p>When two or more unknown quantities are so involved -that they cannot be separately determined by the Simple -Method of Means, we can yet obtain their most probable -values by the Method of Least Squares, without more -difficulty than arises from the length of the arithmetical -computations. If the result of each observation gives an -equation between two unknown quantities of the form</p> - -<div class="ml5em"> -<i>ax</i> + <i>by</i> = <i>c</i> -</div> - -<p class="ti0">then, if the observations were free from error, we should -need only two observations giving two equations; but for -the attainment of greater accuracy, we may take many observations, -and reduce the equations so as to give only a -pair with mean coefficients. This reduction is effected by<span class="pagenum" id="Page_394">394</span> -(1.), multiplying the coefficients of each equation by the -first coefficient, and adding together all the similar coefficients -thus resulting for the coefficients of a new -equation; and (2.), by repeating this process, and multiplying -the coefficients of each equation by the coefficient -of the second term. Meaning by (sum of <i>a</i><sup>2</sup>) the sum of -all quantities of the same kind, and having the same place -in the equations as <i>a</i><sup>2</sup>, we may briefly describe the two -resulting mean equations as follows:—</p> - -<div class="ml5em"> -(sum of <i>a</i><sup>2</sup>) . <i>x</i> + (sum of <i>ab</i>) . <i>y</i> = (sum of <i>ac</i>),<br> -(sum of <i>ab</i>) . <i>x</i> + (sum of <i>b</i><sup>2</sup>) . <i>y</i> = (sum of <i>bc</i>). -</div> - -<p>When there are three or more unknown quantities -the process is exactly the same in nature, and we get -additional mean equations by multiplying by the third, -fourth, &c., coefficients. As the numbers are in any case -approximate, it is usually unnecessary to make the computations -with accuracy, and places of decimals may be -freely cut off to save arithmetical work. The mean -equations having been computed, their solution by the -ordinary methods of algebra gives the most probable -values of the unknown quantities.</p> - - -<h3><i>Works upon the Theory of Probability.</i></h3> - -<p>Regarding the Theory of Probability and the Law of -Error as most important subjects of study for any one who -desires to obtain a complete comprehension of scientific -method as actually applied in physical investigations, I -will briefly indicate the works in one or other of which -the reader will best pursue the study.</p> - -<p>The best popular, and at the same time profound English -work on the subject is De Morgan’s “Essay on Probabilities -and on their Application to Life Contingencies and -Insurance Offices,” published in the <i>Cabinet Cyclopædia</i>, -and to be obtained (in print) from Messrs. Longman. -Mr. Venn’s work on <i>The Logic of Chance</i> can now be -procured in a greatly enlarged second edition;<a id="FNanchor_297" href="#Footnote_297" class="fnanchor">297</a> it contains -a most interesting and able discussion of the metaphysical<span class="pagenum" id="Page_395">395</span> -basis of probability and of related questions concerning -causation, belief, design, testimony, &c.; but I cannot -always agree with Mr. Venn’s opinions. No mathematical -knowledge beyond that of common arithmetic is required -in reading these works. Quetelet’s <i>Letters</i> form a good -introduction to the subject, and the mathematical notes -are of value. Sir George Airy’s brief treatise <i>On the -Algebraical and Numerical Theory of Errors of Observations -and the Combination of Observations</i>, contains a -complete explanation of the Law of Error and its practical -applications. De Morgan’s treatise “On the Theory -of Probabilities” in the <i>Encyclopædia Metropolitana</i>, -presents an abstract of the more abstruse investigations -of Laplace, together with a multitude of profound and -original remarks concerning the theory generally. In -Lubbock and Drinkwater’s work on <i>Probability</i>, in the -Library of Useful Knowledge, we have a concise but -good statement of a number of important problems. The -Rev. W. A. Whitworth has given, in a work entitled -<i>Choice and Chance</i>, a number of good illustrations of -calculations both in combinations and probabilities. In -Mr. Todhunter’s admirable History we have an exhaustive -critical account of almost all writings upon the subject of -probability down to the culmination of the theory in -Laplace’s works. The Memoir of Mr. J. W. L. Glaisher -has already been mentioned (p. <a href="#Page_375">375</a>). In spite of the -existence of these and some other good English works, -there seems to be a want of an easy and yet pretty complete -mathematical introduction to the study of the theory.</p> - -<p>Among French works the Traité <i>Élémentaire du Calcul -des Probabilités</i>, by S. E. Lacroix, of which several editions -have been published, and which is not difficult to obtain, -forms probably the best elementary treatise. Poisson’s -<i>Recherches sur la Probabilité des Jugements</i> (Paris 1837), -commence with an admirable investigation of the grounds -and methods of the theory. While Laplace’s great <i>Théorie -Analytique des Probabilités</i> is of course the “Principia” -of the subject; his <i>Essai Philosophique sur les Probabilités</i> -is a popular discourse, and is one of the most profound -and interesting essays ever published. It should be -familiar to every student of logical method, and has lost -little or none of its importance by lapse of time.</p> - -<p><span class="pagenum" id="Page_396">396</span></p> - - -<h3><i>Detection of Constant Errors.</i></h3> - -<p>The Method of Means is absolutely incapable of eliminating -any error which is always the same, or which always -lies in one direction. We sometimes require to be roused -from a false feeling of security, and to be urged to take -suitable precautions against such occult errors. “It is -to the observer,” says Gauss,<a id="FNanchor_298" href="#Footnote_298" class="fnanchor">298</a> “that belongs the task of -carefully removing the causes of constant errors,” and this -is quite true when the error is absolutely constant. When -we have made a number of determinations with a certain -apparatus or method of measurement, there is a great -advantage in altering the arrangement, or even devising -some entirely different method of getting estimates of the -same quantity. The reason obviously consists in the improbability -that the same error will affect two or more -different methods of experiment. If a discrepancy is -found to exist, we shall at least be aware of the existence -of error, and can take measures for finding in which way -it lies. If we can try a considerable number of methods, -the probability becomes great that errors constant in one -method will be balanced or nearly so by errors of an opposite -effect in the others. Suppose that there be three -different methods each affected by an error of equal -amount. The probability that this error will in all fall in -the same direction is only <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">4</span></span></span>; and with four methods -similarly <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">8.</span></span></span> If each method be affected, as is always -the case, by several independent sources of error, the -probability becomes much greater that in the mean result -of all the methods some of the errors will partially -compensate the others. In this case as in all others, when -human vigilance has exhausted itself, we must trust the -theory of probability.</p> - -<p>In the determination of a zero point, of the magnitude -of the fundamental standards of time and space, in the -personal equation of an astronomical observer, we have -instances of fixed errors; but as a general rule a change of -procedure is likely to reverse the character of the error, -and many instances may be given of the value of this -precaution. If we measure over and over again the same<span class="pagenum" id="Page_397">397</span> -angular magnitude by the same divided circle, maintained -in exactly the same position, it is evident that the same -mark in the circle will be the criterion in each case, and -any error in the position of that mark will equally affect -all our results. But if in each measurement we use a -different part of the circle, a new mark will come into use, -and as the error of each mark cannot be in the same -direction, the average result will be nearly free from -errors of division. It will be better still to use more -than one divided circle.</p> - -<p>Even when we have no perception of the points at -which error is likely to enter, we may with advantage -vary the construction of our apparatus in the hope that we -shall accidentally detect some latent cause of error. Baily’s -purpose in repeating the experiments of Michell and Cavendish -on the density of the earth was not merely to follow -the same course and verify the previous numbers, but to -try whether variations in the size and substance of the -attracting balls, the mode of suspension, the temperature -of the surrounding air, &c., would yield different results. -He performed no less than 62 distinct series, comprising -2153 experiments, and he carefully classified and discussed -the results so as to disclose the utmost differences. Again, -in experimenting upon the resistance of the air to the -motion of a pendulum, Baily employed no less than 80 -pendulums of various forms and materials, in order to -ascertain exactly upon what conditions the resistance -depends. Regnault, in his exact researches upon the -dilatation of gases, made arbitrary changes in the magnitude -of parts of his apparatus. He thinks that if, in spite -of such modification, the results are unchanged, the errors -are probably of inconsiderable amount;<a id="FNanchor_299" href="#Footnote_299" class="fnanchor">299</a> but in reality it -is always possible, and usually likely, that we overlook -sources of error which a future generation will detect. -Thus the pendulum experiments of Baily and Sabine were -directed to ascertain the nature and amount of a correction -for air resistance, which had been entirely misunderstood -in the experiments by means of the seconds pendulum, -upon which was founded the definition of the standard -yard, in the Act of 5th George IV. c. 74. It has already<span class="pagenum" id="Page_398">398</span> -been mentioned that a considerable error was discovered -in the determination of the standard metre as the ten-millionth -part of the distance from the pole to the -equator (p. <a href="#Page_314">314</a>).</p> - -<p>We shall return in Chapter XXV. to the further consideration -of the methods by which we may as far as possible -secure ourselves against permanent and undetected sources -of error. In the meantime, having completed the consideration -of the special methods requisite for treating -quantitative phenomena, we must pursue our principal -subject, and endeavour to trace out the course by which -the physicist, from observation and experiment, collects -the materials of knowledge, and then proceeds by hypothesis -and inverse calculation to induce from them the -laws of nature.</p> - -<hr class="chap x-ebookmaker-drop"> - -<div class="chapter"> -<p><span class="pagenum" id="Page_399">399</span></p> - -<p class="nobreak ph2 ti0" id="BOOK_IV">BOOK IV.<br> - -<span class="title">INDUCTIVE INVESTIGATION.</span></p> -</div> - -<hr class="r30"> - -<div class="chapter"> -<h2 class="nobreak" id="CHAPTER_XVIII">CHAPTER XVIII.<br> - -<span class="title">OBSERVATION.</span></h2> -</div> - -<p class="ti0">All knowledge proceeds originally from experience. Using -the name in a wide sense, we may say that experience -comprehends all that we <i>feel</i>, externally or internally—the -aggregate of the impressions which we receive through -the various apertures of perception—the aggregate consequently -of what is in the mind, except so far as some -portions of knowledge may be the reasoned equivalents of -other portions. As the word experience expresses, we <i>go -through</i> much in life, and the impressions gathered intentionally -or unintentionally afford the materials from which -the active powers of the mind evolve science.</p> - -<p>No small part of the experience actually employed in -science is acquired without any distinct purpose. We -cannot use the eyes without gathering some facts which -may prove useful. A great science has in many cases -risen from an accidental observation. Erasmus Bartholinus -thus first discovered double refraction in Iceland spar; -Galvani noticed the twitching of a frog’s leg; Oken was -struck by the form of a vertebra; Malus accidentally -examined light reflected from distant windows with a<span class="pagenum" id="Page_400">400</span> -double refracting substance; and Sir John Herschel’s -attention was drawn to the peculiar appearance of a -solution of quinine sulphate. In earlier times there must -have been some one who first noticed the strange behaviour -of a loadstone, or the unaccountable motions produced by -amber. As a general rule we shall not know in what -direction to look for a great body of phenomena widely -different from those familiar to us. Chance then must -give us the starting point; but one accidental observation -well used may lead us to make thousands of observations -in an intentional and organised manner, and thus a science -may be gradually worked out from the smallest opening.</p> - - -<h3><i>Distinction of Observation and Experiment.</i></h3> - -<p>It is usual to say that the two sources of experience -are Observation and Experiment. When we merely note -and record the phenomena which occur around us in the -ordinary course of nature we are said <i>to observe</i>. When we -change the course of nature by the intervention of our -muscular powers, and thus produce unusual combinations -and conditions of phenomena, we are said <i>to experiment</i>. -Herschel justly remarked<a id="FNanchor_300" href="#Footnote_300" class="fnanchor">300</a> that we might properly call -these two modes of experience <i>passive and active observation</i>. -In both cases we must certainly employ our senses -to observe, and an experiment differs from a mere observation -in the fact that we more or less influence the -character of the events which we observe. Experiment is -thus observation <i>plus</i> alteration of conditions.</p> - -<p>It may readily be seen that we pass upwards by insensible -gradations from pure observation to determinate -experiment. When the earliest astronomers simply noticed -the ordinary motions of the sun, moon, and planets upon -the face of the starry heavens, they were pure observers. -But astronomers now select precise times and places for -important observations of stellar parallax, or the transits -of planets. They make the earth’s orbit the basis of a -well arranged <i>natural experiment</i>, as it were, and take well -considered advantage of motions which they cannot -control. Meteorology might seem to be a science of pure<span class="pagenum" id="Page_401">401</span> -observation, because we cannot possibly govern the changes -of weather which we record. Nevertheless we may ascend -mountains or rise in balloons, like Gay-Lussac and Glaisher, -and may thus so vary the points of observation as to render -our procedure experimental. We are wholly unable either -to produce or prevent earth-currents of electricity, but -when we construct long lines of telegraph, we gather such -strong currents during periods of disturbance as to render -them capable of easy observation.</p> - -<p>The best arranged systems of observation, however, would -fail to give us a large part of the facts which we now -possess. Many processes continually going on in nature -are so slow and gentle as to escape our powers of observation. -Lavoisier remarked that the decomposition of water -must have been constantly proceeding in nature, although -its possibility was unknown till his time.<a id="FNanchor_301" href="#Footnote_301" class="fnanchor">301</a> No substance -is wholly destitute of magnetic or diamagnetic powers; -but it required all the experimental skill of Faraday to -prove that iron and a few other metals had no monopoly -of these powers. Accidental observation long ago impressed -upon men’s minds the phenomena of lightning, -and the attractive properties of amber. Experiment only -could have shown that phenomena so diverse in magnitude -and character were manifestations of the same agent. To -observe with accuracy and convenience we must have -agents under our control, so as to raise or lower their -intensity, to stop or set them in action at will. Just as -Smeaton found it requisite to create an artificial and -governable supply of wind for his investigation of windmills, -so we must have governable supplies of light, heat, -electricity, muscular force, or whatever other agents we are -examining.</p> - -<p>It is hardly needful to point out too that on the earth’s -surface we live under nearly constant conditions of gravity, -temperature, and atmospheric pressure, so that if we are to -extend our inferences to other parts of the universe where -conditions are widely different, we must be prepared to -imitate those conditions on a small scale here. We must -have intensely high and low temperatures; we must vary<span class="pagenum" id="Page_402">402</span> -the density of gases from approximate vacuum upwards; -we must subject liquids and solids to pressures or strains -of almost unlimited amount.</p> - - -<h3><i>Mental Conditions of Correct Observation.</i></h3> - -<p>Every observation must in a certain sense be true, for -the observing and recording of an event is in itself an -event. But before we proceed to deal with the supposed -meaning of the record, and draw inferences concerning the -course of nature, we must take care to ascertain that the -character and feelings of the observer are not to a great -extent the phenomena recorded. The mind of man, as -Francis Bacon said, is like an uneven mirror, and does not -reflect the events of nature without distortion. We need -hardly take notice of intentionally false observations, nor -of mistakes arising from defective memory, deficient light, -and so forth. Even where the utmost fidelity and care -are used in observing and recording, tendencies to error -exist, and fallacious opinions arise in consequence.</p> - -<p>It is difficult to find persons who can with perfect fairness -register facts for and against their own peculiar views. -Among uncultivated observers the tendency to remark -favourable and forget unfavourable events is so great, that -no reliance can be placed upon their supposed observations. -Thus arises the enduring fallacy that the changes of the -weather coincide in some way with the changes of the -moon, although exact and impartial registers give no -countenance to the fact. The whole race of prophets and -quacks live on the overwhelming effect of one success, -compared with hundreds of failures which are unmentioned -and forgotten. As Bacon says, “Men mark when -they hit, and never mark when they miss.” And we -should do well to bear in mind the ancient story, quoted -by Bacon, of one who in Pagan times was shown a temple -with a picture of all the persons who had been saved from -shipwreck, after paying their vows. When asked whether -he did not now acknowledge the power of the gods, -“Ay,” he answered; “but where are they painted that -were drowned after their vows?”</p> - -<p>If indeed we could estimate the amount of <i>bias</i> existing -in any particular observations, it might be treated like<span class="pagenum" id="Page_403">403</span> -one of the forces of the problem, and the true course of -external nature might still be rendered apparent. But the -feelings of an observer are usually too indeterminate, so -that when there is reason to suspect considerable bias, rejection -is the only safe course. As regards facts casually -registered in past times, the capacity and impartiality of -the observer are so little known that we should spare no -pains to replace these statements by a new appeal to -nature. An indiscriminate medley of truth and absurdity, -such as Francis Bacon collected in his <i>Natural History</i>, is -wholly unsuited to the purposes of science. But of course -when records relate to past events like eclipses, conjunctions, -meteoric phenomena, earthquakes, volcanic -eruptions, changes of sea margins, the existence of now -extinct animals, the migrations of tribes, remarkable -customs, &c., we must make use of statements however -unsatisfactory, and must endeavour to verify them by the -comparison of independent records or traditions.</p> - -<p>When extensive series of observations have to be made, -as in astronomical, meteorological, or magnetical observatories, -trigonometrical surveys, and extensive chemical or -physical researches, it is an advantage that the numerical -work should be executed by assistants who are not interested -in, and are perhaps unaware of, the expected results. The -record is thus rendered perfectly impartial. It may even -be desirable that those who perform the purely routine -work of measurement and computation should be unacquainted -with the principles of the subject. The great -table of logarithms of the French Revolutionary Government -was worked out by a staff of sixty or eighty -computers, most of whom were acquainted only with the -rules of arithmetic, and worked under the direction of -skilled mathematicians; yet their calculations were usually -found more correct than those of persons more deeply -versed in mathematics.<a id="FNanchor_302" href="#Footnote_302" class="fnanchor">302</a> In the Indian Ordnance Survey -the actual measurers were selected so that they should -not have sufficient skill to falsify their results without -detection.</p> - -<p>Both passive observation and experimentation must, -however, be generally conducted by persons who know for<span class="pagenum" id="Page_404">404</span> -what they are to look. It is only when excited and guided -by the hope of verifying a theory that the observer will -notice many of the most important points; and, where the -work is not of a routine character, no assistant can supersede -the mind-directed observations of the philosopher. -Thus the successful investigator must combine diverse -qualities; he must have clear notions of the result he expects -and confidence in the truth of his theories, and yet -he must have that candour and flexibility of mind which -enable him to accept unfavourable results and abandon -mistaken views.</p> - - -<h3><i>Instrumental and Sensual Conditions of Observation.</i></h3> - -<p>In every observation one or more of the senses must be -employed, and we should ever bear in mind that the extent -of our knowledge may be limited by the power of the -sense concerned. What we learn of the world only forms -the lower limit of what is to be learned, and, for all that -we can tell, the processes of nature may infinitely surpass -in variety and complexity those which are capable of -coming within our means of observation. In some cases -inference from observed phenomena may make us indirectly -aware of what cannot be directly felt, but we -can never be sure that we thus acquire any appreciable -fraction of the knowledge that might be acquired.</p> - -<p>It is a strange reflection that space may be filled with -dark wandering stars, whose existence could not have yet -become in any way known to us. The planets have -already cooled so far as to be no longer luminous, and it -may well be that other stellar bodies of various size have -fallen into the same condition. From the consideration, -indeed, of variable and extinguished stars, Laplace inferred -that there probably exist opaque bodies as great and -perhaps as numerous as those we see.<a id="FNanchor_303" href="#Footnote_303" class="fnanchor">303</a> Some of these -dark stars might ultimately become known to us, either -by reflecting light, or more probably by their gravitating -effects upon luminous stars. Thus if one member of a -double star were dark, we could readily detect its existence, -and even estimate its size, position, and motions,<span class="pagenum" id="Page_405">405</span> -by observing those of its visible companion. It was a -favourite notion of Huyghens that there may exist stars -and vast universes so distant that their light has never -yet had time to reach our eyes; and we must also bear -in mind that light may possibly suffer slow extinction -in space, so that there is more than one way in which -an absolute limit to the powers of telescopic discovery -may exist.</p> - -<p>There are natural limits again to the power of our -senses in detecting undulations of various kinds. It is -commonly said that vibrations of more than 38,000 strokes -per second are not audible as sound; and as some ears -actually do hear sounds of much higher pitch, even two -octaves higher than what other ears can detect, it is -exceedingly probable that there are incessant vibrations -which we cannot call sound because they are never heard. -Insects may communicate by such acute sounds, constituting -a language inaudible to us; and the remarkable -agreement apparent among bodies of ants or bees might -thus perhaps be explained. Nay, as Fontenelle long ago -suggested in his scientific romance, there may exist unlimited -numbers of senses or modes of perception which -we can never feel, though Darwin’s theory would render it -probable that any useful means of knowledge in an ancestor -would be developed and improved in the descendants. -We might doubtless have been endowed with a sense -capable of feeling electric phenomena with acuteness, so -that the positive or negative state of charge of a body -could be at once estimated. The absence of such a -sense is probably due to its comparative uselessness.</p> - -<p>Heat undulations are subject to the same considerations. -It is now apparent that what we call light is the affection -of the eye by certain vibrations, the less rapid of which -are invisible and constitute the dark rays of radiant heat, -in detecting which we must substitute the thermometer -or the thermopile for the eye. At the other end of the -spectrum, again, the ultra-violet rays are invisible, and -only indirectly brought to our knowledge in the phenomena -of fluorescence or photo-chemical action. There is -no reason to believe that at either end of the spectrum an -absolute limit has yet been reached.</p> - -<p>Just as our knowledge of the stellar universe is limited<span class="pagenum" id="Page_406">406</span> -by the power of the telescope and other conditions, so our -knowledge of the minute world has its limit in the powers -and optical conditions of the microscope. There was a -time when it would have been a reasonable induction that -vegetables are motionless, and animals alone endowed -with power of locomotion. We are astonished to discover -by the microscope that minute plants are if anything -more active than minute animals. We even find -that mineral substances seem to lose their inactive -character and dance about with incessant motion when -reduced to sufficiently minute particles, at least when suspended -in a non-conducting medium.<a id="FNanchor_304" href="#Footnote_304" class="fnanchor">304</a> Microscopists will -meet a natural limit to observation when the minuteness -of the objects examined becomes comparable to the length -of light undulations, and the extreme difficulty already -encountered in determining the forms of minute marks on -Diatoms appears to be due to this cause. According to -Helmholtz the smallest distance which can be accurately -defined depends upon the interference of light passing -through the centres of the bright spaces. With a theoretically -perfect microscope and a dry lense the smallest -visible object would not be less than one 80,000th part -of an inch in red light.</p> - -<p>Of the errors likely to arise in estimating quantities by -the senses I have already spoken, but there are some cases -in which we actually see things differently from what -they are. A jet of water appears to be a continuous -thread, when it is really a wonderfully organised succession -of small and large drops, oscillating in form. The -drops fall so rapidly that their impressions upon the eye -run into each other, and in order to see the separate drops -we require some device for giving an instantaneous view.</p> - -<p>One insuperable limit to our powers of observation -arises from the impossibility of following and identifying -the ultimate atoms of matter. One atom of oxygen is -probably undistinguishable from another atom; only by<span class="pagenum" id="Page_407">407</span> -keeping a certain volume of oxygen safely inclosed in -a bottle can we assure ourselves of its identity; allow it -to mix with other oxygen, and we lose all power of identification. -Accordingly we seem to have no means of -directly proving that every gas is in a constant state of -diffusion of every part into every part. We can only -infer this to be the case from observing the behaviour -of distinct gases which we can distinguish in their course, -and by reasoning on the grounds of molecular theory.<a id="FNanchor_305" href="#Footnote_305" class="fnanchor">305</a></p> - - -<h3><i>External Conditions of Correct Observation.</i></h3> - -<p>Before we proceed to draw inferences from any series of -recorded facts, we must take care to ascertain perfectly, -if possible, the external conditions under which the facts -are brought to our notice. Not only may the observing -mind be prejudiced and the senses defective, but there -may be circumstances which cause one kind of event to -come more frequently to our notice than another. The -comparative numbers of objects of different kinds existing -may in any degree differ from the numbers which come to -our notice. This difference must if possible be taken into -account before we make any inferences.</p> - -<p>There long appeared to be a strong presumption that -all comets moved in elliptic orbits, because no comet had -been proved to move in any other kind of path. The -theory of gravitation admitted of the existence of comets -moving in hyperbolic orbits, and the question arose -whether they were really non-existent or were only -beyond the bounds of easy observation. From reasonable -suppositions Laplace calculated that the probability -was at least 6000 to 1 against a comet which comes -within the planetary system sufficiently to be visible at -the earth’s surface, presenting an orbit which could be -discriminated from a very elongated ellipse or parabola in -the part of its orbit within the reach of our telescopes.<a id="FNanchor_306" href="#Footnote_306" class="fnanchor">306</a> -In short, the chances are very much in favour of our -seeing elliptic rather than hyperbolic comets. Laplace’s -views have been confirmed by the discovery of six<span class="pagenum" id="Page_408">408</span> -hyperbolic comets, which appeared in the years 1729, -1771, 1774, 1818, 1840, and 1843,<a id="FNanchor_307" href="#Footnote_307" class="fnanchor">307</a> and as only about 800 -comets altogether have been recorded, the proportion of -hyperbolic ones is quite as large as should be expected.</p> - -<p>When we attempt to estimate the numbers of objects -which may have existed, we must make large allowances -for the limited sphere of our observations. Probably not -more than 4000 or 5000 comets have been seen in -historical times, but making allowance for the absence of -observers in the southern hemisphere, and for the small -probability that we see any considerable fraction of those -which are in the neighbourhood of our system, we must -accept Kepler’s opinion, that there are more comets in -the regions of space than fishes in the depths of the ocean. -When like calculations are made concerning the numbers -of meteors visible to us, it is astonishing to find that the -number of meteors entering the earth’s atmosphere in every -twenty-four hours is probably not less than 400,000,000, -of which 13,000 exist in every portion of space equal to -that filled by the earth.</p> - -<p>Serious fallacies may arise from overlooking the inevitable -conditions under which the records of past events are -brought to our notice. Thus it is only the durable objects -manufactured by former races of men, such as flint implements, -which can have come to our notice as a general -rule. The comparative abundance of iron and bronze -articles used by an ancient nation must not be supposed -to be coincident with their comparative abundance in our -museums, because bronze is far the more durable. There -is a prevailing fallacy that our ancestors built more -strongly than we do, arising from the fact that the more -fragile structures have long since crumbled away. We -have few or no relics of the habitations of the poorer -classes among the Greeks or Romans, or in fact of any -past race; for the temples, tombs, public buildings, and -mansions of the wealthier classes alone endure. There is -an immense expanse of past events necessarily lost to us -for ever, and we must generally look upon records or relics -as exceptional in their character.</p> - -<p>The same considerations apply to geological relics. -We could not generally expect that animals would be<span class="pagenum" id="Page_409">409</span> -preserved unless as regards the bones, shells, strong integuments, -or other hard and durable parts. All the infusoria -and animals devoid of mineral framework have probably -perished entirely, distilled perhaps into oils. It has been -pointed out that the peculiar character of some extinct -floras may be due to the unequal preservation of different -families of plants. By various accidents, however, we gain -glimpses of a world that is usually lost to us—as by -insects embedded in amber, the great mammoth preserved -in ice, mummies, casts in solid material like that of the -Roman soldier at Pompeii, and so forth.</p> - -<p>We should also remember, that just as there may be -conjunctions of the heavenly bodies that can have happened -only once or twice in the period of history, so remarkable -terrestrial conjunctions may take place. Great -storms, earthquakes, volcanic eruptions, landslips, floods, -irruptions of the sea, may, or rather must, have occurred, -events of such unusual magnitude and such extreme rarity -that we can neither expect to witness them nor readily -to comprehend their effects. It is a great advantage of -the study of probabilities, as Laplace himself remarked, to -make us mistrust the extent of our knowledge, and pay -proper regard to the probability that events would come -within the sphere of our observations.</p> - - -<h3><i>Apparent Sequence of Events.</i></h3> - -<p>De Morgan has excellently pointed out<a id="FNanchor_308" href="#Footnote_308" class="fnanchor">308</a> that there -are no less than four modes in which one event may -seem to follow or be connected with another, without -being really so. These involve mental, sensual, and external -causes of error, and I will briefly state and illustrate -them.</p> - -<p>Instead of A causing B, it may be <i>our perception of A -that causes B</i>. Thus it is that prophecies, presentiments, -and the devices of sorcery and witchcraft often work their -own ends. A man dies on the day which he has always -regarded as his last, from his own fears of the day. An -incantation effects its purpose, because care is taken to -frighten the intended victim, by letting him know his -fate. In all such cases the mental condition is the cause -of apparent coincidence.</p> - -<p><span class="pagenum" id="Page_410">410</span></p> - -<p>In a second class of cases, <i>the event A may make our -perception of B follow, which would otherwise happen -without being perceived</i>. Thus it was believed to be the -result of investigation that more comets appeared in hot -than cold summers. No account was taken of the fact -that hot summers would be comparatively cloudless, and -afford better opportunities for the discovery of comets. -Here the disturbing condition is of a purely external -character. Certain ancient philosophers held that the -moon’s rays were cold-producing, mistaking the cold -caused by radiation into space for an effect of the moon, -which is more likely to be visible at a time when the -absence of clouds permits radiation to proceed.</p> - -<p>In a third class of cases, <i>our perception of A may make -our perception of B follow</i>. The event B may be constantly -happening, but our attention may not be drawn to -it except by our observing A. This case seems to be -illustrated by the fallacy of the moon’s influence on clouds. -The origin of this fallacy is somewhat complicated. In -the first place, when the sky is densely clouded the moon -would not be visible at all; it would be necessary for us to -see the full moon in order that our attention should be -strongly drawn to the fact, and this would happen most -often on those nights when the sky is cloudless. Mr. -W. Ellis,<a id="FNanchor_309" href="#Footnote_309" class="fnanchor">309</a> moreover, has ingeniously pointed out that there -is a general tendency for clouds to disperse at the commencement -of night, which is the time when the full moon -rises. Thus the change of the sky and the rise of the full -moon are likely to attract attention mutually, and the -coincidence in time suggests the relation of cause and -effect. Mr. Ellis proves from the results of observations -at the Greenwich Observatory that the moon possesses no -appreciable power of the kind supposed, and yet it is -remarkable that so sound an observer as Sir John Herschel -was convinced of the connection. In his “Results of -Observations at the Cape of Good Hope,”<a id="FNanchor_310" href="#Footnote_310" class="fnanchor">310</a> he mentions -many evenings when a full moon occurred with a -peculiarly clear sky.</p> - -<p><span class="pagenum" id="Page_411">411</span></p> - -<p>There is yet a fourth class of cases, in which <i>B is really -the antecedent event, but our perception of A, which is a -consequence of B, may be necessary to bring about our -perception of B</i>. There can be no doubt, for instance, -that upward and downward currents are continually circulating -in the lowest stratum of the atmosphere during -the day-time; but owing to the transparency of the atmosphere -we have no evidence of their existence until we -perceive cumulous clouds, which are the consequence of -such currents. In like manner an interfiltration of bodies -of air in the higher parts of the atmosphere is probably in -nearly constant progress, but unless threads of cirrous -cloud indicate these motions we remain ignorant of their -occurrence.<a id="FNanchor_311" href="#Footnote_311" class="fnanchor">311</a> The highest strata of the atmosphere are -wholly imperceptible to us, except when rendered luminous -by auroral currents of electricity, or by the passage of -meteoric stones. Most of the visible phenomena of comets -probably arise from some substance which, existing previously -invisible, becomes condensed or electrified suddenly -into a visible form. Sir John Herschel attempted to -explain the production of comet tails in this manner by -evaporation and condensation.<a id="FNanchor_312" href="#Footnote_312" class="fnanchor">312</a></p> - - -<h3><i>Negative Arguments from Non-observation.</i></h3> - -<p>From what has been suggested in preceding sections, it -will plainly appear that the non-observation of a phenomenon -is not generally to be taken as proving its non-occurrence. -As there are sounds which we cannot hear, -rays of heat which we cannot feel, multitudes of worlds -which we cannot see, and myriads of minute organisms -of which not the most powerful microscope can give us -a view, we must as a general rule interpret our experience -in an affirmative sense only. Accordingly when inferences -have been drawn from the non-occurrence of particular -facts or objects, more extended and careful examination -has often proved their falsity. Not many years since it -was quite a well credited conclusion in geology that no -remains of man were found in connection with those of<span class="pagenum" id="Page_412">412</span> -extinct animals, or in any deposit not actually at present -in course of formation. Even Babbage accepted this conclusion -as strongly confirmatory of the Mosaic accounts.<a id="FNanchor_313" href="#Footnote_313" class="fnanchor">313</a> -While the opinion was yet universally held, flint implements -had been found disproving such a conclusion, and -overwhelming evidence of man’s long-continued existence -has since been forthcoming. At the end of the last century, -when Herschel had searched the heavens with his powerful -telescopes, there seemed little probability that planets yet -remained unseen within the orbit of Jupiter. But on the -first day of this century such an opinion was overturned -by the discovery of Ceres, and more than a hundred other -small planets have since been added to the lists of the -planetary system.</p> - -<p>The discovery of the Eozoön Canadense in strata of -much greater age than any previously known to contain -organic remains, has given a shock to groundless opinions -concerning the origin of organic forms; and the oceanic -dredging expeditions under Dr. Carpenter and Sir Wyville -Thomson have modified some opinions of geologists by -disclosing the continued existence of forms long supposed -to be extinct. These and many other cases which might -be quoted show the extremely unsafe character of negative -inductions.</p> - -<p>But it must not be supposed that negative arguments -are of no force and value. The earth’s surface has been -sufficiently searched to render it highly improbable that -any terrestrial animals of the size of a camel remain to be -discovered. It is believed that no new large animal has -been encountered in the last eighteen or twenty centuries,<a id="FNanchor_314" href="#Footnote_314" class="fnanchor">314</a> -and the probability that if existent they would have been -seen, increases the probability that they do not exist. -We may with somewhat less confidence discredit the -existence of any large unrecognised fish, or sea animals, -such as the alleged sea-serpent. But, as we descend to -forms of smaller size negative evidence loses weight from -the less probability of our seeing smaller objects. Even -the strong induction in favour of the four-fold division of -the animal kingdom into Vertebrata, Annulosa, Mollusca,<span class="pagenum" id="Page_413">413</span> -and Cœlenterata, may break down by the discovery of intermediate -or anomalous forms. As civilisation spreads -over the surface of the earth, and unexplored tracts are -gradually diminished, negative conclusions will increase -in force; but we have much to learn yet concerning the -depths of the ocean, almost wholly unexamined as they -are, and covering three-fourths of the earth’s surface.</p> - -<p>In geology there are many statements to which considerable -probability attaches on account of the large -extent of the investigations already made, as, for instance, -that true coal is found only in rocks of a particular geological -epoch; that gold occurs in secondary and tertiary -strata only in exceedingly small quantities,<a id="FNanchor_315" href="#Footnote_315" class="fnanchor">315</a> probably -derived from the disintegration of earlier rocks. In -natural history negative conclusions are exceedingly -treacherous and unsatisfactory. The utmost patience -will not enable a microscopist or the observer of any -living thing to watch the behaviour of the organism under -all circumstances continuously for a great length of time. -There is always a chance therefore that the critical act or -change may take place when the observer’s eyes are withdrawn. -This certainly happens in some cases; for though -the fertilisation of orchids by agency of insects is proved -as well as any fact in natural history, Mr. Darwin has -never been able by the closest watching to detect an insect -in the performance of the operation. Mr. Darwin has -himself adopted one conclusion on negative evidence, -namely, that the <i>Orchis pyramidalis</i> and certain other -orchidaceous flowers secrete no nectar. But his caution -and unwearying patience in verifying the conclusion give -an impressive lesson to the observer. For twenty-three -consecutive days, as he tells us, he examined flowers in all -states of the weather, at all hours, in various localities. -As the secretion in other flowers sometimes takes place -rapidly and might happen at early dawn, that inconvenient -hour of observation was specially adopted. Flowers of -different ages were subjected to irritating vapours, to moisture, -and to every condition likely to bring on the secretion; -and only after invariable failure of this exhaustive inquiry -was the barrenness of the nectaries assumed to be proved.<a id="FNanchor_316" href="#Footnote_316" class="fnanchor">316</a></p> - -<p><span class="pagenum" id="Page_414">414</span></p> - -<p>In order that a negative argument founded on the non-observation -of an object shall have any considerable force, -it must be shown to be probable that the object if existent -would have been observed, and it is this probability which -defines the value of the negative conclusion. The failure -of astronomers to see the planet Vulcan, supposed by some -to exist within Mercury’s orbit, is no sufficient disproof of -its existence. Similarly it would be very difficult, or even -impossible, to disprove the existence of a second satellite of -small size revolving round the earth. But if any person -make a particular assertion, assigning place and time, then -observation will either prove or disprove the alleged fact. -If it is true that when a French observer professed to -have seen a planet on the sun’s face, an observer in Brazil -was carefully scrutinising the sun and failed to see it, we -have a negative proof. False facts in science, it has been -well said, are more mischievous than false theories. A -false theory is open to every person’s criticism, and is ever -liable to be judged by its accordance with facts. But a -false or grossly erroneous assertion of a fact often stands -in the way of science for a long time, because it may be -extremely difficult or even impossible to prove the falsity -of what has been once recorded.</p> - -<p>In other sciences the force of a negative argument will -often depend upon the number of possible alternatives -which may exist. It was long believed that the quality -of a musical sound as distinguished from its pitch, must -depend upon the form of the undulation, because no other -cause of it had ever been suggested or was apparently -possible. The truth of the conclusion was proved by -Helmholtz, who applied a microscope to luminous points -attached to the strings of various instruments, and -thus actually observed the different modes of undulation. -In mathematics negative inductive arguments have -seldom much force, because the possible forms of expression, -or the possible combinations of lines and circles in -geometry, are quite unlimited in number. An enormous -number of attempts were made to trisect the angle by the -ordinary methods of Euclid’s geometry, but their invariable -failure did not establish the impossibility of the -task. This was shown in a totally different manner, by -proving that the problem involves an irreducible cubic<span class="pagenum" id="Page_415">415</span> -equation to which there could be no corresponding plane -geometrical solution.<a id="FNanchor_317" href="#Footnote_317" class="fnanchor">317</a> This is a case of <i>reductio ad -absurdum</i>, a form of argument of a totally different -character. Similarly no number of failures to obtain a -general solution of equations of the fifth degree would -establish the impossibility of the task, but in an indirect -mode, equivalent to a <i>reductio ad absurdum</i>, the impossibility -is considered to be proved.<a id="FNanchor_318" href="#Footnote_318" class="fnanchor">318</a></p> - - -<hr class="chap x-ebookmaker-drop"> - -<div class="chapter"> -<p><span class="pagenum" id="Page_416">416</span></p> -<h2 class="nobreak" id="CHAPTER_XIX">CHAPTER XIX.<br> - -<span class="title">EXPERIMENT.</span></h2> -</div> - -<p class="ti0">We may now consider the great advantages which we -enjoy in examining the combinations of phenomena when -things are within our reach and capable of being experimented -on. We are said <i>to experiment</i> when we bring substances -together under various conditions of temperature, -pressure, electric disturbance, chemical action, &c., and -then record the changes observed. Our object in inductive -investigation is to ascertain exactly the group of circumstances -or conditions which being present, a certain -other group of phenomena will follow. If we denote by -A the antecedent group, and by X subsequent phenomena, -our object will usually be to discover a law of the -form A = AX, the meaning of which is that where A is X -will happen.</p> - -<p>The circumstances which might be enumerated as present -in the simplest experiment are very numerous, in fact almost -infinite. Rub two sticks together and consider what -would be an exhaustive statement of the conditions. -There are the form, hardness, organic structure, and all -the chemical qualities of the wood; the pressure and -velocity of the rubbing; the temperature, pressure, and all -the chemical qualities of the surrounding air; the proximity -of the earth with its attractive and electric powers; -the temperature and other properties of the persons producing -motion; the radiation from the sun, and to and -from the sky; the electric excitement possibly existing in -any overhanging cloud; even the positions of the heavenly -bodies must be mentioned. On <i>à priori</i> grounds it is<span class="pagenum" id="Page_417">417</span> -unsafe to assume that any one of these circumstances is -without effect, and it is only by experience that we can -single out those precise conditions from which the observed -heat of friction proceeds.</p> - -<p>The great method of experiment consists in removing, -one at a time, each of those conditions which may be -imagined to have an influence on the result. Our object -in the experiment of rubbing sticks is to discover the exact -circumstances under which heat appears. Now the presence -of air may be requisite; therefore prepare a vacuum, -and rub the sticks in every respect as before, except that -it is done <i>in vacuo</i>. If heat still appears we may say that -air is not, in the presence of the other circumstances, a -requisite condition. The conduction of heat from neighbouring -bodies may be a condition. Prevent this by making -all the surrounding bodies ice cold, which is what Davy -aimed at in rubbing two pieces of ice together. If heat -still appears we have eliminated another condition, and so -we may go on until it becomes apparent that the expenditure -of energy in the friction of two bodies is the sole -condition of the production of heat.</p> - -<p>The great difficulty of experiment arises from the fact -that we must not assume the conditions to be independent. -Previous to experiment we have no right to say that the -rubbing of two sticks will produce heat in the same way -when air is absent as before. We may have heat produced -in one way when air is present, and in another when air -is absent. The inquiry branches out into two lines, and -we ought to try in both cases whether cutting off a supply -of heat by conduction prevents its evolution in friction. -The same branching out of the inquiry occurs with regard -to every circumstance which enters into the experiment.</p> - -<p>Regarding only four circumstances, say A, B, C, D, we -ought to test not only the combinations ABCD, ABC<i>d</i>, -AB<i>c</i>D, A<i>b</i>CD, <i>a</i>BCD, but we ought really to go through -the whole of the combinations given in the fifth column -of the Logical Alphabet. The effect of the absence of -each condition should be tried both in the presence and -absence of every other condition, and every selection of -those conditions. Perfect and exhaustive experimentation -would, in short, consist in examining natural phenomena -in all their possible combinations and registering all<span class="pagenum" id="Page_418">418</span> -relations between conditions and results which are found -capable of existence. It would thus resemble the exclusion -of contradictory combinations carried out in the Indirect -Method of Inference, except that the exclusion of combinations -is grounded not on prior logical premises, but -on <i>à posteriori</i> results of actual trial.</p> - -<p>The reader will perceive, however, that such exhaustive -investigation is practically impossible, because the number -of requisite experiments would be immensely great. Four -antecedents only would require sixteen experiments; twelve -antecedents would require 4096, and the number increases -as the powers of two. The result is that the experimenter -has to fall back upon his own tact and experience in selecting -those experiments which are most likely to yield him -significant facts. It is at this point that logical rules and -forms begin to fail in giving aid. The logical rule is—Try -all possible combinations; but this being impracticable, -the experimentalist necessarily abandons strict logical -method, and trusts to his own insight. Analogy, as we -shall see, gives some assistance, and attention should be -concentrated on those kinds of conditions which have been -found important in like cases. But we are now entirely -in the region of probability, and the experimenter, while -he is confidently pursuing what he thinks the right clue, -may be overlooking the one condition of importance. It is -an impressive lesson, for instance, that Newton pursued -all his exquisite researches on the spectrum unsuspicious of -the fact that if he reduced the hole in the shutter to a -narrow slit, all the mysteries of the bright and dark lines -were within his grasp, provided of course that his prisms -were sufficiently good to define the rays. In like manner -we know not what slight alteration in the most familiar -experiments may not open the way to realms of new -discovery.</p> - -<p>Practical difficulties, also, encumber the progress of the -physicist. It is often impossible to alter one condition -without altering others at the same time; and thus we -may not get the pure effect of the condition in question. -Some conditions may be absolutely incapable of alteration; -others may be with great difficulty, or only in a certain -degree, removable. A very treacherous source of error is -the existence of unknown conditions, which of course we<span class="pagenum" id="Page_419">419</span> -cannot remove except by accident. These difficulties we -will shortly consider in succession.</p> - -<p>It is beautiful to observe how the alteration of a single -circumstance sometimes conclusively explains a phenomenon. -An instance is found in Faraday’s investigation -of the behaviour of Lycopodium spores scattered on a -vibrating plate. It was observed that these minute spores -collected together at the points of greatest motion, whereas -sand and all heavy particles collected at the nodes, where -the motion was least. It happily occurred to Faraday to -try the experiment in the exhausted receiver of an air-pump, -and it was then found that the light powder behaved -exactly like heavy powder. A conclusive proof was thus -obtained that the presence of air was the condition of importance, -doubtless because it was thrown into eddies by -the motion of the plate, and carried the Lycopodium to -the points of greatest agitation. Sand was too heavy to be -carried by the air.</p> - - -<h3><i>Exclusion of Indifferent Circumstances.</i></h3> - -<p>From what has been already said it will be apparent -that the detection and exclusion of indifferent circumstances -is a work of importance, because it allows the -concentration of attention upon circumstances which contain -the principal condition. Many beautiful instances may -be given where all the most obvious antecedents have been -shown to have no part in the production of a phenomenon. -A person might suppose that the peculiar colours of mother-of-pearl -were due to the chemical qualities of the substance. -Much trouble might have been spent in following out that -notion by comparing the chemical qualities of various iridescent -substances. But Brewster accidentally took an -impression from a piece of mother-of-pearl in a cement of -resin and bees’-wax, and finding the colours repeated upon -the surface of the wax, he proceeded to take other impressions -in balsam, fusible metal, lead, gum arabic, isinglass, -&c., and always found the iridescent colours the same. He -thus proved that the chemical nature of the substance is a -matter of indifference, and that the form of the surface is -the real condition of such colours.<a id="FNanchor_319" href="#Footnote_319" class="fnanchor">319</a> Nearly the same may<span class="pagenum" id="Page_420">420</span> -be said of the colours exhibited by thin plates and films. -The rings and lines of colour will be nearly the same in -character whatever may be the nature of the substance; -nay, a void space, such as a crack in glass, would produce -them even though the air were withdrawn by an air-pump. -The conditions are simply the existence of two reflecting -surfaces separated by a very small space, though it should -be added that the refractive index of the intervening substance -has some influence on the exact nature of the colour -produced.</p> - -<p>When a ray of light passes close to the edge of an opaque -body, a portion of the light appears to be bent towards it, -and produces coloured fringes within the shadow of the -body. Newton attributed this inflexion of light to the -attraction of the opaque body for the supposed particles of -light, although he was aware that the nature of the surrounding -medium, whether air or other pellucid substance, -exercised no apparent influence on the phenomena. -Gravesande proved, however, that the character of the -fringes is exactly the same, whether the body be dense or -rare, compound or elementary. A wire produces exactly -the same fringes as a hair of the same thickness. Even the -form of the obstructing edge was subsequently shown to -be a matter of indifference by Fresnel, and the interference -spectrum, or the spectrum seen when light passes -through a fine grating, is absolutely the same whatever be -the form or chemical nature of the bars making the -grating. Thus it appears that the stoppage of a portion of -a beam of light is the sole necessary condition for the -diffraction or inflexion of light, and the phenomenon is -shown to bear no analogy the refraction of light, in -which the form and nature of the substance are all important.</p> - -<p>It is interesting to observe how carefully Newton, in his -researches on the spectrum, ascertained the indifference -of many circumstances by actual trial. He says:<a id="FNanchor_320" href="#Footnote_320" class="fnanchor">320</a> “Now -the different magnitude of the hole in the window-shut, -and different thickness of the prism where the rays passed -through it, and different inclinations of the prism to the -horizon, made no sensible changes in the length of the<span class="pagenum" id="Page_421">421</span> -image. Neither did the different matter of the prisms -make any: for in a vessel made of polished plates of glass -cemented together in the shape of a prism, and filled with -water, there is the like success of the experiment according -to the quantity of the refraction.” But in the latter statement, -as I shall afterwards remark (p. <a href="#Page_432">432</a>), Newton -assumed an indifference which does not exist, and fell -into an unfortunate mistake.</p> - -<p>In the science of sound it is shown that the pitch of a -sound depends solely upon the number of impulses in a -second, and the material exciting those impulses is a matter -of indifference. Whatever fluid, air or water, gas or liquid, -be forced into the Siren, the sound produced is the same; -and the material of which an organ-pipe is constructed -does not at all affect the pitch of its sound. In the science -of statical electricity it is an important principle that the -nature of the interior of a conducting body is a matter of -no importance. The electrical charge is confined to the -conducting surface, and the interior remains in a neutral -state. A hollow copper sphere takes exactly the same -charge as a solid sphere of the same metal.</p> - -<p>Some of Faraday’s most elegant and successful researches -were devoted to the exclusion of conditions which previous -experimenters had thought essential for the production of -electrical phenomena. Davy asserted that no known fluids, -except such as contain water, could be made the medium -of connexion between the poles of a battery; and some -chemists believed that water was an essential agent in -electro-chemical decomposition. Faraday gave abundant -experiments to show that other fluids allowed of electrolysis, -and he attributed the erroneous opinion to the very -general use of water as a solvent, and its presence in most -natural bodies.<a id="FNanchor_321" href="#Footnote_321" class="fnanchor">321</a> It was, in fact, upon the weakest kind of -negative evidence that the opinion had been founded.</p> - -<p>Many experimenters attributed peculiar powers to the -poles of a battery, likening them to magnets, which, by -their attractive powers, tear apart the elements of a substance. -By a beautiful series of experiments,<a id="FNanchor_322" href="#Footnote_322" class="fnanchor">322</a> Faraday -proved conclusively that, on the contrary, the substance of<span class="pagenum" id="Page_422">422</span> -the poles is of no importance, being merely the path -through which the electric force reaches the liquid acted -upon. Poles of water, charcoal, and many diverse substances, -even air itself, produced similar results; if the -chemical nature of the pole entered at all into the question, -it was as a disturbing agent.</p> - -<p>It is an essential part of the theory of gravitation that -the proximity of other attracting particles is without effect -upon the attraction existing between any two molecules. -Two pound weights weigh as much together as they do -separately. Every pair of molecules in the world have, as -it were, a private communication, apart from their relations -to all other molecules. Another undoubted result of -experience pointed out by Newton<a id="FNanchor_323" href="#Footnote_323" class="fnanchor">323</a> is that the weight of -a body does not in the least depend upon its form or -texture. It may be added that the temperature, electric -condition, pressure, state of motion, chemical qualities, and -all other circumstances concerning matter, except its mass, -are indifferent as regards its gravitating power.</p> - -<p>As natural science progresses, physicists gain a kind of -insight and tact in judging what qualities of a substance -are likely to be concerned in any class of phenomena. The -physical astronomer treats matter in one point of view, -the chemist in another, and the students of physical optics, -sound, mechanics, electricity, &c., make a fair division of -the qualities among them. But errors will arise if too -much confidence be placed in this independence of various -kinds of phenomena, so that it is desirable from time to -time, especially when any unexplained discrepancies come -into notice, to question the indifference which is assumed -to exist, and to test its real existence by appropriate -experiments.</p> - - -<h3><i>Simplification of Experiments.</i></h3> - -<p>One of the most requisite precautions in experimentation -is to vary only one circumstance at a time, and to maintain -all other circumstances rigidly unchanged. There are -two distinct reasons for this rule, the first and most obvious -being that if we vary two conditions at a time, and<span class="pagenum" id="Page_423">423</span> -find some effect, we cannot tell whether the effect is due -to one or the other condition, or to both jointly. A second -reason is that if no effect ensues we cannot safely conclude -that either of them is indifferent; for the one may have -neutralised the effect of the other. In our symbolic logic -AB ꖌ A<i>b</i> was shown to be identical with A (p. <a href="#Page_97">97</a>), so -that B denotes a circumstance which is indifferently -present or absent. But if B always goes together with -another antecedent C, we cannot show the same independence, -for ABC ꖌ A<i>bc</i> is not identical with A and -none of our logical processes enables us to reduce it to A.</p> - -<p>If we want to prove that oxygen is necessary to life, we -must not put a rabbit into a vessel from which the oxygen -has been exhausted by a burning candle. We should then -have not only an absence of oxygen, but an addition of -carbonic acid, which may have been the destructive agent. -For a similar reason Lavoisier avoided the use of atmospheric -air in experiments on combustion, because air was -not a simple substance, and the presence of nitrogen might -impede or even alter the effect of oxygen. As Lavoisier -remarks,<a id="FNanchor_324" href="#Footnote_324" class="fnanchor">324</a> “In performing experiments, it is a necessary -principle, which ought never to be deviated from, that -they be simplified as much as possible, and that every -circumstance capable of rendering their results complicated -be carefully removed.” It has also been well said by -Cuvier<a id="FNanchor_325" href="#Footnote_325" class="fnanchor">325</a> that the method of physical inquiry consists in -isolating bodies, reducing them to their utmost simplicity, -and bringing each of their properties separately into action, -either mentally or by experiment.</p> - -<p>The electro-magnet has been of the utmost service in -the investigation of the magnetic properties of matter, by -allowing of the production or removal of a most powerful -magnetic force without disturbing any of the other arrangements -of the experiment. Many of Faraday’s most -valuable experiments would have been impossible had it -been necessary to introduce a heavy permanent magnet, -which could not be suddenly moved without shaking the -whole apparatus, disturbing the air, producing currents -by changes of temperature, &c. The electro-magnet is<span class="pagenum" id="Page_424">424</span> -perfectly under control, and its influence can be brought -into action, reversed, or stopped by merely touching a -button. Thus Faraday was enabled to prove the rotation -of the plane of circularly polarised light by the fact that -certain light ceased to be visible when the electric current -of the magnet was cut off, and re-appeared when the -current was made. “These phenomena,” he says, “could -be reversed at pleasure, and at any instant of time, and -upon any occasion, showing a perfect dependence of cause -and effect.”<a id="FNanchor_326" href="#Footnote_326" class="fnanchor">326</a></p> - -<p>It was Newton’s omission to obtain the solar spectrum -under the simplest conditions which prevented him from -discovering the dark lines. Using a broad beam of light -which had passed through a round hole or a triangular -slit, he obtained a brilliant spectrum, but one in which -many different coloured rays overlapped each other. In -the recent history of the science of the spectrum, one -main difficulty has consisted in the mixture of the lines of -several different substances, which are usually to be found -in the light of any flame or spark. It is seldom possible -to obtain the light of any element in a perfectly simple -manner. Angström greatly advanced this branch of science -by examining the light of the electric spark when formed -between poles of various metals, and in the presence of -various gases. By varying the pole alone, or the gaseous -medium alone, he was able to discriminate correctly between -the lines due to the metal and those due to the -surrounding gas.<a id="FNanchor_327" href="#Footnote_327" class="fnanchor">327</a></p> - - -<h3><i>Failure in the Simplification of Experiments.</i></h3> - -<p>In some cases it seems to be impossible to carry out the -rule of varying one circumstance at a time. When we -attempt to obtain two instances or two forms of experiment -in which a single circumstance shall be present in -one case and absent in another, it may be found that this -single circumstance entails others. Benjamin Franklin’s -experiment concerning the comparative absorbing powers -of different colours is well known. “I took,” he says, “a<span class="pagenum" id="Page_425">425</span> -number of little square pieces of broadcloth from a tailor’s -pattern card, of various colours. They were black, deep -blue, lighter blue, green, purple, red, yellow, white, and -other colours and shades of colour. I laid them all out -upon the snow on a bright sunshiny morning. In a few -hours the black, being most warmed by the sun, was sunk -so low as to be below the stroke of the sun’s rays; the -dark blue was almost as low; the lighter blue not quite -so much as the dark; the other colours less as they were -lighter. The white remained on the surface of the snow, -not having entered it at all.” This is a very elegant and -apparently simple experiment; but when Leslie had completed -his series of researches upon the nature of heat, he -came to the conclusion that the colour of a surface has -very little effect upon the radiating power, the mechanical -nature of the surface appearing to be more influential. -He remarks<a id="FNanchor_328" href="#Footnote_328" class="fnanchor">328</a> that “the question is incapable of being positively -resolved, since no substance can be made to assume -different colours without at the same time changing its -internal structure.” Recent investigation has shown that -the subject is one of considerable complication, because -the absorptive power of a surface may be different according -to the character of the rays which fall upon it; -but there can be no doubt as to the acuteness with which -Leslie points out the difficulty. In Well’s investigations -concerning the nature of dew, we have, again, very -complicated conditions. If we expose plates of various -material, such as rough iron, glass, polished metal, to the -midnight sky, they will be dewed in various degrees; -but since these plates differ both in the nature of the -surface and the conducting power of the material, it would -not be plain whether one or both circumstances were of -importance. We avoid this difficulty by exposing the -same material polished or varnished, so as to present different -conditions of surface;<a id="FNanchor_329" href="#Footnote_329" class="fnanchor">329</a> and again by exposing -different substances with the same kind of surface.</p> - -<p>When we are quite unable to isolate circumstances we -must resort to the procedure described by Mill under the -name of the Joint Method of Agreement and Difference.<span class="pagenum" id="Page_426">426</span> -We must collect as many instances as possible in which -a given circumstance produces a given result, and as many -as possible in which the absence of the circumstance is -followed by the absence of the result. To adduce his -example, we cannot experiment upon the cause of double -refraction in Iceland spar, because we cannot alter its -crystalline condition without altering it altogether, nor can -we find substances exactly like calc spar in every circumstance -except one. We resort therefore to the method of -comparing together all known substances which have the -property of doubly-refracting light, and we find that they -agree in being crystalline.<a id="FNanchor_330" href="#Footnote_330" class="fnanchor">330</a> This indeed is nothing but an -ordinary process of perfect or probable induction, already -partially described, and to be further discussed under -Classification. It may be added that the subject does -admit of perfect experimental treatment, since glass, when -compressed in one direction, becomes capable of doubly-refracting -light, and as there is probably no alteration in -the glass but change of elasticity, we learn that the power -of double refraction is probably due to a difference of -elasticity in different directions.</p> - - -<h3><i>Removal of Usual Conditions.</i></h3> - -<p>One of the great objects of experiment is to enable us -to judge of the behaviour of substances under conditions -widely different from those which prevail upon the surface -of the earth. We live in an atmosphere which does not -vary beyond certain narrow limits in temperature or -pressure. Many of the powers of nature, such as gravity, -which constantly act upon us, are of almost fixed amount. -Now it will afterwards be shown that we cannot apply a -quantitative law to circumstances much differing from -those in which it was observed. In the other planets, the -sun, the stars, or remote parts of the Universe, the conditions -of existence must often be widely different from -what we commonly experience here. Hence our knowledge -of nature must remain restricted and hypothetical, -unless we can subject substances to unusual conditions by -suitable experiments.</p> - -<p><span class="pagenum" id="Page_427">427</span></p> - -<p>The electric arc is an invaluable means of exposing -metals or other conducting substances to the highest -known temperature. By its aid we learn not only that -all the metals can be vaporised, but that they all give off -distinctive rays of light. At the other extremity of the -scale, the intensely powerful freezing mixture devised by -Faraday, consisting of solid carbonic acid and ether mixed -<i>in vacuo</i>, enables us to observe the nature of substances at -temperatures immensely below any we meet with naturally -on the earth’s surface.</p> - -<p>We can hardly realise now the importance of the invention -of the air-pump, previous to which invention it -was exceedingly difficult to experiment except under the -ordinary pressure of the atmosphere. The Torricellian -vacuum had been employed by the philosophers of the -Accademia del Cimento to show the behaviour of water, -smoke, sound, magnets, electric substances, &c., <i>in vacuo</i>, -but their experiments were often unsuccessful from the -difficulty of excluding air.<a id="FNanchor_331" href="#Footnote_331" class="fnanchor">331</a></p> - -<p>Among the most constant circumstances under which -we live is the force of gravity, which does not vary, except -by a slight fraction of its amount, in any part of the earth’s -crust or atmosphere to which we can attain. This force is -sufficient to overbear and disguise various actions, for instance, -the mutual gravitation of small bodies. It was an -interesting experiment of Plateau to neutralise the action -of gravity by placing substances in liquids of exactly the -same specific gravity. Thus a quantity of oil poured into -the middle of a suitable mixture of alcohol and water -assumes a spherical shape; on being made to rotate it -becomes spheroidal, and then successively separates into -a ring and a group of spherules. Thus we have an -illustration of the mode in which the planetary system -may have been produced,<a id="FNanchor_332" href="#Footnote_332" class="fnanchor">332</a> though the extreme difference -of scale prevents our arguing with confidence from the -experiment to the conditions of the nebular theory.</p> - -<p>It is possible that the so-called elements are elementary -only to us, because we are restricted to temperatures at -which they are fixed. Lavoisier carefully defined an<span class="pagenum" id="Page_428">428</span> -element as a substance which cannot be decomposed <i>by -any known means</i>; but it seems almost certain that some -series of elements, for instance Iodine, Bromine, and Chlorine, -are really compounds of a simpler substance. We -must look to the production of intensely high temperatures, -yet quite beyond our means, for the decomposition of these -so-called elements. Possibly in this age and part of the -universe the dissipation of energy has so far proceeded -that there are no sources of heat sufficiently intense to -effect the decomposition.</p> - - -<h3><i>Interference of Unsuspected Conditions.</i></h3> - -<p>It may happen that we are not aware of all the conditions -under which our researches are made. Some substance -may be present or some power may be in action, which -escapes the most vigilant examination. Not being aware -of its existence, we are unable to take proper measures to -exclude it, and thus determine the share which it has in -the results of our experiments. There can be no doubt -that the alchemists were misled and encouraged in their -vain attempts by the unsuspected presence of traces of -gold and silver in the substances they proposed to transmute. -Lead, as drawn from the smelting furnace, almost -always contains some silver, and gold is associated with -many other metals. Thus small quantities of noble metal -would often appear as the result of experiment and raise -delusive hopes.</p> - -<p>In more than one case the unsuspected presence of -common salt in the air has caused great trouble. In -the early experiments on electrolysis it was found that -when water was decomposed, an acid and an alkali were -produced at the poles, together with oxygen and hydrogen. -In the absence of any other explanation, some chemists -rushed to the conclusion that electricity must have the -power of <i>generating</i> acids and alkalies, and one chemist -thought he had discovered a new substance called <i>electric -acid</i>. But Davy proceeded to a systematic investigation -of the circumstances, by varying the conditions. Changing -the glass vessel for one of agate or gold, he found that far -less alkali was produced; excluding impurities by the use -of carefully distilled water, he found that the quantities of<span class="pagenum" id="Page_429">429</span> -acid and alkali were still further diminished; and having -thus obtained a clue to the cause, he completed the exclusion -of impurities by avoiding contact with his fingers, -and by placing the apparatus under an exhausted receiver, -no acid or alkali being then detected. It would be difficult -to meet with a more elegant case of the detection of a -condition previously unsuspected.<a id="FNanchor_333" href="#Footnote_333" class="fnanchor">333</a></p> - -<p>It is remarkable that the presence of common salt in -the air, proved to exist by Davy, nevertheless continued a -stumbling-block in the science of spectrum analysis, and -probably prevented men, such as Brewster, Herschel, and -Talbot, from anticipating by thirty years the discoveries -of Bunsen and Kirchhoff. As I pointed out,<a id="FNanchor_334" href="#Footnote_334" class="fnanchor">334</a> the utility -of the spectrum was known in the middle of the last -century to Thomas Melvill, a talented Scotch physicist, -who died at the early age of 27 years.<a id="FNanchor_335" href="#Footnote_335" class="fnanchor">335</a> But Melvill -was struck in his examination of coloured flames by the -extraordinary predominance of homogeneous yellow light, -which was due to some circumstance escaping his attention. -Wollaston and Fraunhofer were equally struck by -the prominence of the yellow line in the spectrum of -nearly every kind of light. Talbot expressly recommended -the use of the prism for detecting the presence of substances -by what we now call spectrum analysis, but he found that -all substances, however different the light they yielded in -other respects, were identical as regards the production of -yellow light. Talbot knew that the salts of soda gave this -coloured light, but in spite of Davy’s previous difficulties -with salt in electrolysis, it did not occur to him to assert -that where the light is, there sodium must be. He suggested -water as the most likely source of the yellow light, -because of its frequent presence; but even substances -which were apparently devoid of water gave the same -yellow light.<a id="FNanchor_336" href="#Footnote_336" class="fnanchor">336</a> Brewster and Herschel both experimented<span class="pagenum" id="Page_430">430</span> -upon flames almost at the same time as Talbot, and -Herschel unequivocally enounced the principle of spectrum -analysis.<a id="FNanchor_337" href="#Footnote_337" class="fnanchor">337</a> Nevertheless Brewster, after numerous -experiments attended with great trouble and disappointment, -found that yellow light might be obtained from the -combustion of almost any substance. It was not until -1856 that Swan discovered that an almost infinitesimal -quantity of sodium chloride, say a millionth part of a grain, -was sufficient to tinge a flame of a bright yellow colour. -The universal diffusion of the salts of sodium, joined to -this unique light-producing power, was thus shown to be -the unsuspected condition which had destroyed the confidence -of all previous experimenters in the use of the -prism. Some references concerning the history of this -curious point are given below.<a id="FNanchor_338" href="#Footnote_338" class="fnanchor">338</a></p> - -<p>In the science of radiant heat, early inquirers were led -to the conclusion that radiation proceeded only from the -surface of a solid, or from a very small depth below it. -But they happened to experiment upon surfaces covered -by coats of varnish, which is highly athermanous or -opaque to heat. Had they properly varied the character -of the surface, using a highly diathermanous substance like -rock salt, they would have obtained very different results.<a id="FNanchor_339" href="#Footnote_339" class="fnanchor">339</a></p> - -<p>One of the most extraordinary instances of an erroneous -opinion due to overlooking interfering agents is that concerning -the increase of rainfall near to the earth’s surface. -More than a century ago it was observed that rain-gauges -placed upon church steeples, house tops, and other elevated -places, gave considerably less rain than if they were on the -ground, and it has been recently shown that the variation -is most rapid in the close neighbourhood of the ground.<a id="FNanchor_340" href="#Footnote_340" class="fnanchor">340</a> -All kinds of theories have been started to explain this -phenomenon; but I have shown<a id="FNanchor_341" href="#Footnote_341" class="fnanchor">341</a> that it is simply due to<span class="pagenum" id="Page_431">431</span> -the interference of wind, which deflects more or less rain -from all the gauges which are exposed to it.</p> - -<p>The great magnetic power of iron renders it a source of -disturbance in magnetic experiments. In building a magnetic -observatory great care must therefore be taken that -no iron is employed in the construction, and that no -masses of iron are near at hand. In some cases magnetic -observations have been seriously disturbed by the existence -of masses of iron ore in the neighbourhood. In Faraday’s -experiments upon feebly magnetic or diamagnetic substances -he took the greatest precautions against the presence of -disturbing substances in the copper wire, wax, paper, and -other articles used in suspending the test objects. It was -his custom to try the effect of the magnet upon the apparatus -in the absence of the object of experiment, and without -this preliminary trial no confidence could be placed in -the results.<a id="FNanchor_342" href="#Footnote_342" class="fnanchor">342</a> Tyndall has also employed the same mode -for testing the freedom of electro-magnetic coils from iron, -and was thus enabled to obtain them devoid of any cause -of disturbance.<a id="FNanchor_343" href="#Footnote_343" class="fnanchor">343</a> It is worthy of notice that in the very -infancy of the science of magnetism, the acute experimentalist -Gilbert correctly accounted for the opinion existing -in his day that magnets would attract silver, by pointing -out that the silver contained iron.</p> - -<p>Even when we are not aware by previous experience of -the probable presence of a special disturbing agent, we -ought not to assume the absence of unsuspected interference. -If an experiment is of really high importance, so -that any considerable branch of science rests upon it, we -ought to try it again and again, in as varied conditions as -possible. We should intentionally disturb the apparatus -in various ways, so as if possible to hit by accident upon -any weak point. Especially when our results are more -regular than we have fair grounds for anticipating, ought -we to suspect some peculiarity in the apparatus which -causes it to measure some other phenomenon than that in -question, just as Foucault’s pendulum almost always indicates -the movement of the axes of its own elliptic path -instead of the rotation of the globe.</p> - -<p><span class="pagenum" id="Page_432">432</span></p> - -<p>It was in this cautious spirit that Baily acted in his -experiments on the density of the earth. The accuracy -of his results depended upon the elimination of all disturbing -influences, so that the oscillation of his torsion balance -should measure gravity alone. Hence he varied the apparatus -in many ways, changing the small balls subject to -attraction, changing the connecting rod, and the means of -suspension. He observed the effect of disturbances, such -as the presence of visitors, the occurrence of violent storms, -&c., and as no real alteration was produced in the results, -he confidently attributed them to gravity.<a id="FNanchor_344" href="#Footnote_344" class="fnanchor">344</a></p> - -<p>Newton would probably have discovered the mode of -constructing achromatic lenses, but for the unsuspected -effect of some sugar of lead which he is supposed to have -dissolved in the water of a prism. He tried, by means of -a glass prism combined with a water prism, to produce -dispersion of light without refraction, and if he had -succeeded there would have been an obvious mode of -producing refraction without dispersion. His failure is -attributed to his adding lead acetate to the water for the -purpose of increasing its refractive power, the lead having -a high dispersive power which frustrated his purpose.<a id="FNanchor_345" href="#Footnote_345" class="fnanchor">345</a> -Judging from Newton’s remarks, in the <i>Philosophical -Transactions</i>, it would appear as if he had not, without -many unsuccessful trials, despaired of the construction of -achromatic glasses.<a id="FNanchor_346" href="#Footnote_346" class="fnanchor">346</a></p> - -<p>The Academicians of Cimento, in their early and ingenious -experiments upon the vacuum, were often misled -by the mechanical imperfections of their apparatus. They -concluded that the air had nothing to do with the production -of sounds, evidently because their vacuum was not -sufficiently perfect. Otto von Guericke fell into a like -mistake in the use of his newly-constructed air-pump, -doubtless from the unsuspected presence of air sufficiently -dense to convey the sound of the bell.</p> - -<p>It is hardly requisite to point out that the doctrine of -spontaneous generation is due to the unsuspected presence<span class="pagenum" id="Page_433">433</span> -of germs, even after the most careful efforts to exclude -them, and in the case of many diseases, both of animals -and plants, germs which we have no means as yet of detecting -are doubtless the active cause. It has long been -a subject of dispute, again, whether the plants which spring -from newly turned land grow from seeds long buried in -that land, or from seeds brought by the wind. Argument -is unphilosophical when direct trial can readily be applied; -for by turning up some old ground, and covering a portion -of it with a glass case, the conveyance of seeds by the -wind can be entirely prevented, and if the same plants -appear within and without the case, it will become clear -that the seeds are in the earth. By gross oversight some -experimenters have thought before now that crops of rye -had sprung up where oats had been sown.</p> - - -<h3><i>Blind or Test Experiments.</i></h3> - -<p>Every conclusive experiment necessarily consists in the -comparison of results between two different combinations -of circumstances. To give a fair probability that A is the -cause of X, we must maintain invariable all surrounding -objects and conditions, and we must then show that where -A is X is, and where A is not X is not. This cannot really -be accomplished in a single trial. If, for instance, a -chemist places a certain suspected substance in Marsh’s -test apparatus, and finds that it gives a small deposit of -metallic arsenic, he cannot be sure that the arsenic really -proceeds from the suspected substance; the impurity of the -zinc or sulphuric acid may have been the cause of its -appearance. It is therefore the practice of chemists to -make what they call a <i>blind experiment</i>, that is to try -whether arsenic appears in the absence of the suspected -substance. The same precaution ought to be taken in all -important analytical operations. Indeed, it is not merely -a precaution, it is an essential part of any experiment. If -the blind trial be not made, the chemist merely assumes -that he knows what would happen. Whenever we assert -that because A and X are found together A is the cause of -X, we assume that if A were absent X would be absent. -But wherever it is possible, we ought not to take this -as a mere assumption, or even as a matter of inference.<span class="pagenum" id="Page_434">434</span> -Experience is ultimately the basis of all our inferences, -but if we can bring immediate experience to bear upon the -point in question we should not trust to anything more -remote and liable to error. When Faraday examined the -magnetic properties of the bearing apparatus, in the absence -of the substance to be experimented on, he really made a -blind experiment (p. <a href="#Page_431">431</a>).</p> - -<p>We ought, also, to test the accuracy of a method of experiment -whenever we can, by introducing known amounts -of the substance or force to be detected. A new analytical -process for the quantitative estimation of an element -should be tested by performing it upon a mixture compounded -so as to contain a known quantity of that element. -The accuracy of the gold assay process greatly depends -upon the precaution of assaying alloys of gold of exactly -known composition.<a id="FNanchor_347" href="#Footnote_347" class="fnanchor">347</a> Gabriel Plattes’ works give evidence -of much scientific spirit, and when discussing the supposed -merits of the divining rod for the discovery of subterranean -treasure, he sensibly suggests that the rod should be tried -in places where veins of metal are known to exist.<a id="FNanchor_348" href="#Footnote_348" class="fnanchor">348</a></p> - - -<h3><i>Negative Results of Experiment.</i></h3> - -<p>When we pay proper regard to the imperfection of all -measuring instruments and the possible minuteness of -effects, we shall see much reason for interpreting with -caution the negative results of experiments. We may fail -to discover the existence of an expected effect, not because -that effect is really non-existent, but because it is of a -magnitude inappreciable to our senses, or confounded with -other effects of much greater amount. As there is no -limit on <i>à priori</i> grounds to the smallness of a phenomenon, -we can never, by a single experiment, prove the -non-existence of a supposed effect. We are always at -liberty to assume that a certain amount of effect might -have been detected by greater delicacy of measurement. -We cannot safely affirm that the moon has no atmosphere -at all. We may doubtless show that the atmosphere, if -present, is less dense than the air in the so-called vacuum<span class="pagenum" id="Page_435">435</span> -of an air-pump, as did Du Sejour. It is equally impossible -to prove that gravity occupies <i>no time</i> in transmission. -Laplace indeed ascertained that the velocity of propagation -of the influence was at least fifty million times greater than -that of light;<a id="FNanchor_349" href="#Footnote_349" class="fnanchor">349</a> but it does not really follow that it is instantaneous; -and were there any means of detecting the -action of one star upon another exceedingly distant star, -we might possibly find an appreciable interval occupied in -the transmission of the gravitating impulse. Newton -could not demonstrate the absence of all resistance to -matter moving through empty space; but he ascertained by -an experiment with the pendulum (p. <a href="#Page_443">443</a>), that if such -resistance existed, it was in amount less than one five-thousandth -part of the external resistance of the air.<a id="FNanchor_350" href="#Footnote_350" class="fnanchor">350</a></p> - -<p>A curious instance of false negative inference is furnished -by experiments on light. Euler rejected the corpuscular -theory on the ground that particles of matter -moving with the immense velocity of light would possess -momentum, of which there was no evidence. Bennet had -attempted to detect the momentum of light by concentrating -the rays of the sun upon a delicately balanced body. -Observing no result, it was considered to be proved that -light had no momentum. Mr. Crookes, however, having -suspended thin vanes, blacked on one side, in a nearly -vacuous globe, found that they move under the influence -of light. It is now allowed that this effect can be explained -in accordance with the undulatory theory of light, -and the molecular theory of gases. It comes to this—that -Bennet failed to detect an effect which he might have -detected with a better method of experimenting; but if he -had found it, the phenomenon would have confirmed, not -the corpuscular theory of light, as was expected, but the -rival undulatory theory. The conclusion drawn from -Bennet’s experiment was falsely drawn, but it was nevertheless -true in matter.</p> - -<p>Many incidents in the history of science tend to show -that phenomena, which one generation has failed to discover, -may become accurately known to a succeeding -generation. The compressibility of water which the<span class="pagenum" id="Page_436">436</span> -Academicians of Florence could not detect, because at a -low pressure the effect was too small to perceive, and at a -high pressure the water oozed through their silver vessel,<a id="FNanchor_351" href="#Footnote_351" class="fnanchor">351</a> -has now become the subject of exact measurement and -precise calculation. Independently of Newton, Hooke -entertained very remarkable notions concerning the nature -of gravitation. In this and other subjects he showed, -indeed, a genius for experimental investigation which -would have placed him in the first rank in any other age -than that of Newton. He correctly conceived that the -force of gravity would decrease as we recede from the -centre of the earth, and he boldly attempted to prove it by -experiment. Having exactly counterpoised two weights -in the scales of a balance, or rather one weight against -another weight and a long piece of fine cord, he removed -his balance to the top of the dome of St. Paul’s, and tried -whether the balance remained in equilibrium after one -weight was allowed to hang down to a depth of 240 feet. -No difference could be perceived when the weights were at -the same and at different levels, but Hooke rightly held -that the failure arose from the insufficient elevation. He -says, “Yet I am apt to think some difference might be discovered -in greater heights.”<a id="FNanchor_352" href="#Footnote_352" class="fnanchor">352</a> The radius of the earth -being about 20,922,000 feet, we can now readily calculate -from the law of gravity that a height of 240 would not -make a greater difference than one part in 40,000 of the -weight. Such a difference would doubtless be inappreciable -in the balances of that day, though it could readily be detected -by balances now frequently constructed. Again, the -mutual gravitation of bodies at the earth’s surface is so -small that Newton appears to have made no attempt to -demonstrate its existence experimentally, merely remarking -that it was too small to fall under the observation of -our senses.<a id="FNanchor_353" href="#Footnote_353" class="fnanchor">353</a> It has since been successfully detected and -measured by Cavendish, Baily, and others.</p> - -<p>The smallness of the quantities which we can sometimes -observe is astonishing. A balance will weigh to one -millionth part of the load. Whitworth can measure to -the millionth part of an inch. A rise of temperature of<span class="pagenum" id="Page_437">437</span> -the 8800th part of a degree centigrade has been detected -by Dr. Joule. The spectroscope has revealed the presence -of the 10,000,000th part of a gram. It is said that the -eye can observe the colour produced in a drop of water by -the 50,000,000th part of a gram of fuschine, and about the -same quantity of cyanine. By the sense of smell we can -probably feel still smaller quantities of odorous matter.<a id="FNanchor_354" href="#Footnote_354" class="fnanchor">354</a> -We must nevertheless remember that quantitative effects -of far less amount than these must exist, and we should -state our negative results with corresponding caution. We -can only disprove the existence of a quantitative phenomenon -by showing deductively from the laws of nature, that -if present it would amount to a perceptible quantity. As -in the case of other negative arguments (p. <a href="#Page_414">414</a>), we must -demonstrate that the effect would appear, where it is by -experiment found not to appear.</p> - - -<h3><i>Limits of Experiment.</i></h3> - -<p>It will be obvious that there are many operations of -nature which we are quite incapable of imitating in our -experiments. Our object is to study the conditions under -which a certain effect is produced; but one of those conditions -may involve a great length of time. There are -instances on record of experiments extending over five or -ten years, and even over a large part of a lifetime; but -such intervals of time are almost nothing to the time -during which nature may have been at work. The contents -of a mineral vein in Cornwall may have been undergoing -gradual change for a hundred million years. All -metamorphic rocks have doubtless endured high temperature -and enormous, pressure for inconceivable periods of -time, so that chemical geology is generally beyond the -scope of experiment.</p> - -<p>Arguments have been brought against Darwin’s theory, -founded upon the absence of any clear instance of the -production of a new species. During an historical interval -of perhaps four thousand years, no animal, it is said, has -been so much domesticated as to become different in<span class="pagenum" id="Page_438">438</span> -species. It might as well be argued that no geological -changes are taking place, because no new mountain has -risen in Great Britain within the memory of man. Our -actual experience of geological changes is like a point in -the infinite progression of time. When we know that rain -water falling on limestone will carry away a minute -portion of the rock in solution, we do not hesitate to -multiply that quantity by millions, and infer that in -course of time a mountain may be dissolved away. We -have actual experience concerning the rise of land in some -parts of the globe and its fall in others to the extent of -some feet. Do we hesitate to infer what may thus be done -in course of geological ages? As Gabriel Plattes long ago -remarked, “The sea never resting, but perpetually winning -land in one place and losing in another, doth show what -may be done in length of time by a continual operation, -not subject unto ceasing or intermission.”<a id="FNanchor_355" href="#Footnote_355" class="fnanchor">355</a> The action of -physical circumstances upon the forms and characters of -animals by natural selection is subject to exactly the same -remarks. As regards animals living in a state of nature, -the change of circumstances which can be ascertained to -have occurred is so slight, that we could not expect to -observe any change in those animals whatever. Nature -has made no experiment at all for us within historical -times. Man, however, by taming and domesticating dogs, -horses, oxen, pigeons, &c., has made considerable change -in their circumstances, and we find considerable change -also in their forms and characters. Supposing the state of -domestication to continue unchanged, these new forms -would continue permanent so far as we know, and in this -sense they are permanent. Thus the arguments against -Darwin’s theory, founded on the non-observation of natural -changes within the historical period, are of the weakest -character, being purely negative.</p> - -<hr class="chap x-ebookmaker-drop"> - -<div class="chapter"> -<p><span class="pagenum" id="Page_439">439</span></p> -<h2 class="nobreak" id="CHAPTER_XX">CHAPTER XX. - -<span class="title">METHOD OF VARIATIONS.</span></h2> -</div> - -<p class="ti0">Experiments may be of two kinds, experiments of -simple fact, and experiments of quantity. In the first -class of experiments we combine certain conditions, and -wish to ascertain whether or not a certain effect of any -quantity exists. Hooke wished to ascertain whether or -not there was any difference in the force of gravity at the -top and bottom of St. Paul’s Cathedral. The chemist -continually performs analyses for the purpose of ascertaining -whether or not a given element exists in a particular mineral -or mixture; all such experiments and analyses are -qualitative rather than quantitative, because though the -result may be more or less, the particular amount of the -result is not the object of the inquiry.</p> - -<p>So soon, however, as a result is known to be discoverable, -the scientific man ought to proceed to the quantitative -inquiry, how great a result follows from a certain amount -of the conditions which are supposed to constitute the -cause? The possible numbers of experiments are now infinitely -great, for every variation in a quantitative condition -will usually produce a variation in the amount of the effect. -The method of variation which thus arises is no narrow or -special method, but it is the general application of experiment -to phenomena capable of continuous variation. As -Mr. Fowler has well remarked,<a id="FNanchor_356" href="#Footnote_356" class="fnanchor">356</a> the observation of variations -is really an integration of a supposed infinite number of -applications of the so-called method of difference, that is -of experiment in its perfect form.</p> -<p><span class="pagenum" id="Page_440">440</span></p> -<p>In induction we aim at establishing a general law, and -if we deal with quantities that law must really be expressed -more or less obviously in the form of an equation, or -equations. We treat as before of conditions, and of what -happens under those conditions. But the conditions will -now vary, not in quality, but quantity, and the effect will -also vary in quantity, so that the result of quantitative induction -is always to arrive at some mathematical expression -involving the quantity of each condition, and expressing -the quantity of the result. In other words, we wish to -know what function the effect is of its conditions. We -shall find that it is one thing to obtain the numerical -results, and quite another thing to detect the law obeyed -by those results, the latter being an operation of an inverse -and tentative character.</p> - - -<h3><i>The Variable and the Variant.</i></h3> - -<p>Almost every series of quantitative experiments is -directed to obtain the relation between the different -values of one quantity which is varied at will, and another -quantity which is caused thereby to vary. We -may conveniently distinguish these as respectively the -<i>variable</i> and the <i>variant</i>. When we are examining the -effect of heat in expanding bodies, heat, or one of its -dimensions, temperature, is the variable, length the -variant. If we compress a body to observe how much -it is thereby heated, pressure, or it may be the dimensions -of the body, forms the variable, heat the variant. In -the thermo-electric pile we make heat the variable and -measure electricity as the variant. That one of the two -measured quantities which is an antecedent condition of -the other will be the variable.</p> - -<p>It is always convenient to have the variable entirely -under our command. Experiments may indeed be made -with accuracy, provided we can exactly measure the variable -at the moment when the quantity of the effect is -determined. But if we have to trust to the action of -some capricious force, there may be great difficulty in -making exact measurements, and those results may not -be disposed over the whole range of quantity in a convenient -manner. It is one prime object of the experimenter,<span class="pagenum" id="Page_441">441</span> -therefore, to obtain a regular and governable -supply of the force which he is investigating. To determine -correctly the efficiency of windmills, when the -natural winds were constantly varying in force, would be -exceedingly difficult. Smeaton, therefore, in his experiments -on the subject, created a uniform wind of the -required force by moving his models against the air on the -extremity of a revolving arm.<a id="FNanchor_357" href="#Footnote_357" class="fnanchor">357</a> The velocity of the wind -could thus be rendered greater or less, it could be maintained -uniform for any length of time, and its amount -could be exactly ascertained. In determining the laws of -the chemical action of light it would be out of the question -to employ the rays of the sun, which vary in intensity with -the clearness of the atmosphere, and with every passing -cloud. One great difficulty in photometry and the investigation -of the chemical action of light consists in obtaining -a uniform and governable source of light rays.<a id="FNanchor_358" href="#Footnote_358" class="fnanchor">358</a></p> - -<p>Fizeau’s method of measuring the velocity of light -enabled him to appreciate the time occupied by light in -travelling through a distance of eight or nine thousand -metres. But the revolving mirror of Wheatstone subsequently -enabled Foucault and Fizeau to measure the -velocity in a space of four metres. In this latter method -there was the advantage that various media could be substituted -for air, and the temperature, density, and other -conditions of the experiment could be accurately governed -and measured.</p> - - -<h3><i>Measurement of the Variable.</i></h3> - -<p>There is little use in obtaining exact measurements of -an effect unless we can also exactly measure its conditions.</p> - -<p>It is absurd to measure the electrical resistance of a -piece of metal, its elasticity, tenacity, density, or other -physical qualities, if these vary, not only with the minute -impurities of the metal, but also with its physical condition. -If the same bar changes its properties by being<span class="pagenum" id="Page_442">442</span> -heated and cooled, and we cannot exactly define the state -in which it is at any moment, our care in measuring will -be wasted, because it can lead to no law. It is of little -use to determine very exactly the electric conductibility of -carbon, which as graphite or gas carbon conducts like a -metal, as diamond is almost a non-conductor, and in -several other forms possesses variable and intermediate -powers of conduction. It will be of use only for -immediate practical applications. Before measuring these -we ought to have something to measure of which the conditions -are capable of exact definition, and to which at a -future time we can recur. Similarly the accuracy of our -measurement need not much surpass the accuracy with -which we can define the conditions of the object treated.</p> - -<p>The speed of electricity in passing through a conductor -mainly depends upon the inductive capacity of the surrounding -substances, and, except for technical or special -purposes, there is little use in measuring velocities which -in some cases are one hundred times as great as in other -cases. But the maximum speed of electric conduction is -probably a constant quantity of great scientific importance, -and according to Prof. Clerk Maxwell’s determination in -1868 is 174,800 miles per second, or little less than that -of light. The true boiling point of water is a point on -which practical thermometry depends, and it is highly -important to determine that point in relation to the absolute -thermometric scale. But when water free from air -and impurity is heated there seems to be no definite limit -to the temperature it may reach, a temperature of 180° -Cent. having been actually observed. Such temperatures, -therefore, do not require accurate measurement. All -meteorological measurements depending on the accidental -condition of the sky are of far less importance than -physical measurements in which such accidental conditions -do not intervene. Many profound investigations -depend upon our knowledge of the radiant energy continually -poured upon the earth by the sun; but this must -be measured when the sky is perfectly clear, and the -absorption of the atmosphere at its minimum. The -slightest interference of cloud destroys the value of such -a measurement, except for meteorological purposes, which -are of vastly less generality and importance. It is seldom<span class="pagenum" id="Page_443">443</span> -useful, again, to measure the height of a snow-covered -mountain within a foot, when the thickness of the snow -alone may cause it to vary 25 feet or more, when in short -the height itself is indefinite to that extent.<a id="FNanchor_359" href="#Footnote_359" class="fnanchor">359</a></p> - - -<h3><i>Maintenance of Similar Conditions.</i></h3> - -<p>Our ultimate object in induction must be to obtain the -complete relation between the conditions and the effect, -but this relation will generally be so complex that we can -only attack it in detail. We must, as far as possible, -confine the variation to one condition at a time, and establish -a separate relation between each condition and the -effect. This is at any rate the first step in approximating -to the complete law, and it will be a subsequent question -how far the simultaneous variation of several conditions -modifies their separate actions. In many experiments, -indeed, it is only one condition which we wish to study, -and the others are interfering forces which we would avoid -if possible. One of the conditions of the motion of a pendulum -is the resistance of the air, or other medium in -which it swings; but when Newton was desirous of proving -the equal gravitation of all substances, he had no -interest in the air. His object was to observe a single -force only, and so it is in a great many other experiments. -Accordingly, one of the most important precautions in -investigation consists in maintaining all conditions constant -except that which is to be studied. As that admirable -experimental philosopher, Gilbert, expressed it,<a id="FNanchor_360" href="#Footnote_360" class="fnanchor">360</a> -“There is always need of similar preparation, of similar -figure, and of equal magnitude, for in dissimilar and unequal -circumstances the experiment is doubtful.”</p> - -<p>In Newton’s decisive experiment similar conditions were -provided for, with the simplicity which characterises the -highest art. The pendulums of which the oscillations were -compared consisted of equal boxes of wood, hanging by -equal threads, and filled with different substances, so that -the total weights should be equal and the centres of oscillation -at the same distance from the points of suspension.<span class="pagenum" id="Page_444">444</span> -Hence the resistance of the air became approximately a -matter of indifference; for the outward size and shape of -the pendulums being the same, the absolute force of resistance -would be the same, so long as the pendulums -vibrated with equal velocity; and the weights being equal -the resistance would diminish the velocity equally. Hence -if any inequality were observed in the vibrations of the two -pendulums, it must arise from the only circumstance which -was different, namely the chemical nature of the matter -within the boxes. No inequality being observed, the -chemical nature of substances can have no appreciable -influence upon the force of gravitation.<a id="FNanchor_361" href="#Footnote_361" class="fnanchor">361</a></p> - -<p>A beautiful experiment was devised by Dr. Joule for -the purpose of showing that the gain or loss of heat by a -gas is connected, not with the mere change of its volume -and density, but with the energy received or given out by -the gas. Two strong vessels, connected by a tube and stopcock, -were placed in water after the air had been exhausted -from one vessel and condensed in the other to the extent -of twenty atmospheres. The whole apparatus having -been brought to a uniform temperature by agitating the -water, and the temperature having been exactly observed, -the stopcock was opened, so that the air at once expanded -and filled the two vessels uniformly. The temperature of -the water being again noted was found to be almost unchanged. -The experiment was then repeated in an exactly -similar manner, except that the strong vessels were placed -in separate portions of the water. Now cold was produced -in the vessel from which the air rushed, and an almost -exactly equal quantity of heat appeared in that to which -it was conducted. Thus Dr. Joule clearly proved that -rarefaction produces as much heat as cold, and that only -when there is disappearance of mechanical energy will -there be production of heat.<a id="FNanchor_362" href="#Footnote_362" class="fnanchor">362</a> What we have to notice, -however, is not so much the result of the experiment, as -the simple manner in which a single change in the apparatus, -the separation of the portions of water surrounding -the air vessels, is made to give indications of the utmost -significance.</p> -<p><span class="pagenum" id="Page_445">445</span></p> - -<h3><i>Collective Experiments.</i></h3> - -<p>There is an interesting class of experiments which -enable us to observe a number of quantitative results in -one act. Generally speaking, each experiment yields us -but one number, and before we can approach the real -processes of reasoning we must laboriously repeat measurement -after measurement, until we can lay out a curve of -the variation of one quantity as depending on another. -We can sometimes abbreviate this labour, by making a -quantity vary in different parts of the same apparatus -through every required amount. In observing the height -to which water rises by the capillary attraction of a glass -vessel, we may take a series of glass tubes of different -bore, and measure the height through which it rises in each. -But if we take two glass plates, and place them vertically -in water, so as to be in contact at one vertical side, and -slightly separated at the other side, the interval between -the plates varies through every intermediate width, and -the water rises to a corresponding height, producing at its -upper surface a hyperbolic curve.</p> - -<p>The absorption of light in passing through a coloured -liquid may be beautifully shown by enclosing the liquid in -a wedge-shaped glass, so that we have at a single glance -an infinite variety of thicknesses in view. As Newton -himself remarked, a red liquid viewed in this manner is -found to have a pale yellow colour at the thinnest part, -and it passes through orange into red, which gradually -becomes of a deeper and darker tint.<a id="FNanchor_363" href="#Footnote_363" class="fnanchor">363</a> The effect may be -noticed in a conical wine-glass. The prismatic analysis of -light from such a wedge-shaped vessel discloses the reason, -by exhibiting the progressive absorption of different rays -of the spectrum as investigated by Dr. J. H. Gladstone.<a id="FNanchor_364" href="#Footnote_364" class="fnanchor">364</a></p> - -<p>A moving body may sometimes be made to mark out -its own course, like a shooting star which leaves a tail -behind it. Thus an inclined jet of water exhibits in the -clearest manner the parabolic path of a projectile. In -Wheatstone’s Kaleidophone the curves produced by the -combination of vibrations of different ratios are shown by<span class="pagenum" id="Page_446">446</span> -placing bright reflective buttons on the tops of wires of -various forms. The motions are performed so quickly that -the eye receives the impression of the path as a complete -whole, just as a burning stick whirled round produces a -continuous circle. The laws of electric induction are -beautifully shown when iron filings are brought under the -influence of a magnet, and fall into curves corresponding -to what Faraday called the Lines of Magnetic Force. -When Faraday tried to define what he meant by his lines -of force, he was obliged to refer to the filings. “By magnetic -curves,” he says,<a id="FNanchor_365" href="#Footnote_365" class="fnanchor">365</a> “I mean lines of magnetic forces -which would be depicted by iron filings.” Robison had -previously produced similar curves by the action of frictional -electricity, and from a mathematical investigation of -the forms of such curves we may infer that magnetic and -electric attractions obey the general law of emanation, -that of the inverse square of the distance. In the electric -brush we have a similar exhibition of the laws of electric -attraction.</p> - -<p>There are several branches of science in which collective -experiments have been used with great advantage. Lichtenberg’s -electric figures, produced by scattering electrified -powder on an electrified resin cake, so as to show the condition -of the latter, suggested to Chladni the notion of -discovering the state of vibration of plates by strewing -sand upon them. The sand collects at the points where the -motion is least, and we gain at a glance a comprehension -of the undulations of the plate. To this method of experiment -we owe the beautiful observations of Savart. The -exquisite coloured figures exhibited by plates of crystal, -when examined by polarised light, afford a more complicated -example of the same kind of investigation. They -led Brewster and Fresnel to an explanation of the properties -of the optic axes of crystals. The unequal conduction of -heat in crystalline substances has also been shown in a -similar manner, by spreading a thin layer of wax over the -plate of crystal, and applying heat to a single point. The -wax then melts in a circular or elliptic area according as -the rate of conduction is uniform or not. Nor should we -forget that Newton’s rings were an early and most important<span class="pagenum" id="Page_447">447</span> -instance of investigations of the same kind, showing -the effects of interference of light undulations of all -magnitudes at a single view. Herschel gave to all such -opportunities of observing directly the results of a general -law, the name of <i>Collective Instances</i>,<a id="FNanchor_366" href="#Footnote_366" class="fnanchor">366</a> and I propose to -adopt the name <i>Collective Experiments</i>.</p> - -<p>Such experiments will in many subjects only give the -first hint of the nature of the law in question, but will not -admit of any exact measurements. The parabolic form of -a jet of water may well have suggested to Galileo his views -concerning the path of a projectile; but it would not serve -now for the exact investigation of the laws of gravity. It -is unlikely that capillary attraction could be exactly -measured by the use of inclined plates of glass, and tubes -would probably be better for precise investigation. As a -general rule, these collective experiments would be most -useful for popular illustration. But when the curves are -of a precise and permanent character, as in the coloured -figures produced by crystalline plates, they may admit of -exact measurement. Newton’s rings and diffraction fringes -allow of very accurate measurements.</p> - -<p>Under collective experiments we may perhaps place -those in which we render visible the motions of gas or -liquid by diffusing some opaque substance in it. The -behaviour of a body of air may often be studied in a -beautiful way by the use of smoke, as in the production -of smoke rings and jets. In the case of liquids lycopodium -powder is sometimes employed. To detect the mixture of -currents or strata of liquid, I employed very dilute solutions -of common salt and silver nitrate, which produce a visible -cloud wherever they come into contact.<a id="FNanchor_367" href="#Footnote_367" class="fnanchor">367</a> Atmospheric -clouds often reveal to us the movements of great volumes -of air which would otherwise be quite unapparent.</p> - - -<h3><i>Periodic Variations.</i></h3> - -<p>A large class of investigations is concerned with Periodic -Variations. We may define a periodic phenomenon as one -which, with the uniform change of the variable, returns<span class="pagenum" id="Page_448">448</span> -time after time to the same value. If we strike a pendulum -it presently returns to the point from which we -disturbed it, and while time, the variable, progresses -uniformly, it goes on making excursions and returning, -until stopped by the dissipation of its energy. If one body -in space approaches by gravity towards another, they will -revolve round each other in elliptic orbits, and return for -an indefinite number of times to the same relative positions. -On the other hand a single body projected into empty -space, free from the action of any extraneous force, would -go on moving for ever in a straight line, according to the -first law of motion. In the latter case the variation is -called <i>secular</i>, because it proceeds during ages in a similar -manner, and suffers no περίοδος or going round. It may -be doubted whether there really is any motion in the -universe which is not periodic. Mr. Herbert Spencer long -since adopted the doctrine that all motion is ultimately -rhythmical,<a id="FNanchor_368" href="#Footnote_368" class="fnanchor">368</a> and abundance of evidence may be adduced -in favour of his view.</p> - -<p>The so-called secular acceleration of the moon’s motion -is certainly periodic, and as, so far as we can tell, no body -is beyond the attractive power of other bodies, rectilinear -motion becomes purely hypothetical, or at least infinitely -improbable. All the motions of all the stars must tend to -become periodic. Though certain disturbances in the planetary -system seem to be uniformly progressive, Laplace -is considered to have proved that they really have their -limits, so that after an immense time, all the planetary -bodies might return to the same places, and the stability of -the system be established. Such a theory of periodic stability -is really hypothetical, and does not take into account -phenomena resulting in the dissipation of energy, which -may be a really secular process. For our present purposes -we need not attempt to form an opinion on such questions. -Any change which does not present the appearance of a -periodic character will be empirically regarded as a secular -change, so that there will be plenty of non-periodic variations.</p> - -<p>The variations which we produce experimentally will -often be non-periodic. When we communicate heat to a<span class="pagenum" id="Page_449">449</span> -gas it increases in bulk or pressure, and as far as we can go -the higher the temperature the higher the pressure. Our -experiments are of course restricted in temperature both -above and below, but there is every reason to believe that -the bulk being the same, the pressure would never return -to the same point at any two different temperatures. We -may of course repeatedly raise and lower the temperature -at regular or irregular intervals entirely at our will, and -the pressure of the gas will vary in like manner and -exactly at the same intervals, but such an arbitrary series -of changes would not constitute Periodic Variation. It -would constitute a succession of distinct experiments, -which would place beyond reasonable doubt the connexion -of cause and effect.</p> - -<p>Whenever a phenomenon recurs at equal or nearly -equal intervals, there is, according to the theory of probability, -considerable evidence of connexion, because if the -recurrences were entirely casual it is unlikely that they -would happen at equal intervals. The fact that a brilliant -comet had appeared in the years 1301, 1378, 1456, 1531, -1607, and 1682 gave considerable presumption in favour -of the identity of the body, apart from similarity of the -orbit. There is nothing which so fascinates the attention -of men as the recurrence time after time of some unusual -event. Things and appearances which remain ever the -same, like mountains and valleys, fail to excite the curiosity -of a primitive people. It has been remarked by Laplace -that even in his day the rising of Venus in its brightest -phase never failed to excite surprise and interest. So -there is little doubt that the first germ of science arose -in the attention given by Eastern people to the changes -of the moon and the motions of the planets. Perhaps the -earliest astronomical discovery consisted in proving the -identity of the morning and evening stars, on the grounds -of their similarity of aspect and invariable alternation.<a id="FNanchor_369" href="#Footnote_369" class="fnanchor">369</a> -Periodical changes of a somewhat complicated kind must -have been understood by the Chaldeans, because they were -aware of the cycle of 6585 days or 19 years which brings -round the new and full moon upon the same days, hours, -and even minutes of the year. The earliest efforts of<span class="pagenum" id="Page_450">450</span> -scientific prophecy were founded upon this knowledge, -and if at present we cannot help wondering at the precise -anticipations of the nautical almanack, we may imagine -the wonder excited by such predictions in early times.</p> - - -<h3><i>Combined Periodic Changes.</i></h3> - -<p>We shall seldom find a body subject to a single periodic -variation, and free from other disturbances. We may expect -the periodic variation itself to undergo variation, -which may possibly be secular, but is more likely to -prove periodic; nor is there any limit to the complication -of periods beyond periods, or periods within periods, which -may ultimately be disclosed. In studying a phenomenon -of rhythmical character we have a succession of questions -to ask. Is the periodic variation uniform? If not, is the -change uniform? If not, is the change itself periodic? -Is that new period uniform, or subject to any other change, -or not? and so on <i>ad infinitum</i>.</p> - -<p>In some cases there may be many distinct causes of -periodic variations, and according to the principle of the -superposition of small effects, to be afterwards considered, -these periodic effects will be simply added together, or at -least approximately so, and the joint result may present a -very complicated subject of investigation. The tides of -the ocean consist of a series of superimposed undulations. -Not only are there the ordinary semi-diurnal tides caused -by sun and moon, but a series of minor tides, such as the -lunar diurnal, the solar diurnal, the lunar monthly, the -lunar fortnightly, the solar annual and solar semi-annual -are gradually being disentangled by the labours of Sir W. -Thomson, Professor Haughton and others.</p> - -<p>Variable stars present interesting periodic phenomena; -while some stars, δ Cephei for instance, are subject to very -regular variations, others, like Mira Ceti, are less constant -in the degrees of brilliancy which they attain or the -rapidity of the changes, possibly on account of some longer -periodic variation.<a id="FNanchor_370" href="#Footnote_370" class="fnanchor">370</a> The star β Lyræ presents a double -maximum and minimum in each of its periods of nearly 13 -days, and since the discovery of this variation the period<span class="pagenum" id="Page_451">451</span> -in a period has probably been on the increase. “At first -the variability was more rapid, then it became gradually -slower; and this decrease in the length of time reached -its limit between the years 1840 and 1844. During that -time its period was nearly invariable; at present it is again -decidedly on the decrease.”<a id="FNanchor_371" href="#Footnote_371" class="fnanchor">371</a> The tracing out of such -complicated variations presents an unlimited field for interesting -investigation. The number of such variable stars -already known is considerable, and there is no reason -to suppose that any appreciable fraction of the whole -number has yet been detected.</p> - - -<h3><i>Principle of Forced Vibrations.</i></h3> - -<p>Investigations of the connection of periodic causes and -effects rest upon a principle, which has been demonstrated -by Sir John Herschel for some special cases, and clearly -explained by him in several of his works.<a id="FNanchor_372" href="#Footnote_372" class="fnanchor">372</a> The principle -may be formally stated in the following manner: “If one -part of any system connected together either by material -ties, or by the mutual attractions of its members, be continually -maintained by any cause, whether inherent in -the constitution of the system or external to it, in a state -of regular periodic motion, that motion will be propagated -throughout the whole system, and will give rise, in every -member of it, and in every part of each member, to -periodic movements executed in equal periods, with that -to which they owe their origin, though not necessarily -synchronous with them in their maxima and minima.” -The meaning of the proposition is that the effect of a -periodic cause will be periodic, and will recur at intervals -equal to those of the cause. Accordingly when we find -two phenomena which do proceed, time after time, through -changes of the same period, there is much probability -that they are connected. In this manner, doubtless, Pliny -correctly inferred that the cause of the tides lies in the -sun and the moon, the intervals between successive high -tides being equal to the intervals between the moon’s<span class="pagenum" id="Page_452">452</span> -passage across the meridian. Kepler and Descartes too -admitted the connection previous to Newton’s demonstration -of its precise nature. When Bradley discovered the -apparent motion of the stars arising from the aberration -of light, he was soon able to attribute it to the earth’s -annual motion, because it went through its phases in a -year.</p> - -<p>The most beautiful instance of induction concerning -periodic changes which can be cited, is the discovery of -an eleven-year period in various meteorological phenomena. -It would be difficult to mention any two things -apparently more disconnected than the spots upon the -sun and auroras. As long ago as 1826, Schwabe commenced -a regular series of observations of the spots upon -the sun, which has been continued to the present time, -and he was able to show that at intervals of about -eleven years the spots increased much in size and number. -Hardly was this discovery made known, when Lamont -pointed out a nearly equal period of variation in the -declination of the magnetic needle. Magnetic storms or -sudden disturbances of the needle were next shown to -take place most frequently at the times when sun-spots -were prevalent, and as auroras are generally coincident -with magnetic storms, these phenomena were brought -into the cycle. It has since been shown by Professor -Piazzi Smyth and Mr. E. J. Stone, that the temperature -of the earth’s surface as indicated by sunken thermometers -gives some evidence of a like period. The existence -of a periodic cause having once been established, it is -quite to be expected, according to the principle of forced -vibrations, that its influence will be detected in all -meteorological phenomena.</p> - - -<h3><i>Integrated Variations.</i></h3> - -<p>In considering the various modes in which one effect -may depend upon another, we must set in a distinct -class those which arise from the accumulated effects of -a constantly acting cause. When water runs out of a -cistern, the velocity of motion depends, according to -Torricelli’s theorem, on the height of the surface of the -water above the vent; but the amount of water which<span class="pagenum" id="Page_453">453</span> -leaves the cistern in a given time depends upon the -aggregate result of that velocity, and is only to be -ascertained by the mathematical process of integration. -When one gravitating body falls towards another, the -force of gravity varies according to the inverse square -of the distance; to obtain the velocity produced we -must integrate or sum the effects of that law; and to -obtain the space passed over by the body in a given -time, we must integrate again.</p> - -<p>In periodic variations the same distinction must be -drawn. The heating power of the sun’s rays at any -place on the earth varies every day with the height -attained, and is greatest about noon; but the temperature -of the air will not be greatest at the same time. -This temperature is an integrated effect of the sun’s -heating power, and as long as the sun is able to give -more heat to the air than the air loses in other ways, -the temperature continues to rise, so that the maximum -is deferred until about 3 <span class="allsmcap">P.M.</span> Similarly the hottest day of -the year falls, on an average, about one month later than -the summer solstice, and all the seasons lag about a month -behind the motions of the sun. In the case of the tides, -too, the effect of the moon’s attractive power is never -greatest when the power is greatest; the effect always -lags more or less behind the cause. Yet the intervals -between successive tides are equal, in the absence of disturbance, -to the intervals between the passages of the -moon across the meridian. Thus the principle of forced -vibrations holds true.</p> - -<p>In periodic phenomena, however, curious results sometimes -follow from the integration of effects. If we strike -a pendulum, and then repeat the stroke time after time at -the same part of the vibration, all the strokes concur in -adding to the momentum, and we can thus increase the -extent and violence of the vibrations to any degree. We -can stop the pendulum again by strokes applied when it -is moving in the opposite direction, and the effects being -added together will soon bring it to rest. Now if we -alter the intervals of the strokes so that each two successive -strokes act in opposite manners they will neutralise -each other, and the energy expended will be turned into -heat or sound at the point of percussion. Similar effects<span class="pagenum" id="Page_454">454</span> -occur in all cases of rhythmical motion. If a musical note -is sounded in a room containing a piano, the string corresponding -to it will be thrown into vibration, because every -successive stroke of the air-waves upon the string finds -it in like position as regards the vibration, and thus adds -to its energy of motion. But the other strings being incapable -of vibrating with the same rapidity are struck at -various points of their vibrations, and one stroke will -soon be opposed by one contrary in effect. All phenomena -of <i>resonance</i> arise from this coincidence in time of -undulation. The air in a pipe closed at one end, and about -12 inches in length, is capable of vibrating 512 times in -a second. If, then, the note C is sounded in front of the -open end of the pipe, every successive vibration of the air -is treasured up as it were in the motion of the air. In -a pipe of different length the pulses of air would strike -each other, and the mechanical energy being transmuted -into heat would become no longer perceptible as sound.</p> - -<p>Accumulated vibrations sometimes become so intense -as to lead to unexpected results. A glass vessel if touched -with a violin bow at a suitable point may be fractured with -the violence of vibration. A suspension bridge may be -broken down if a company of soldiers walk across it in -steps the intervals of which agree with the vibrations of -the bridge itself. But if they break the step or march -in either quicker or slower pace, they may have no perceptible -effect upon the bridge. In fact if the impulses -communicated to any vibrating body are synchronous with -its vibrations, the energy of those vibrations will be unlimited, -and may fracture any body.</p> - -<p>Let us now consider what will happen if the strokes be -not exactly at the same intervals as the vibrations of the -body, but, say, a little slower. Then a succession of strokes -will meet the body in nearly but not quite the same position, -and their efforts will be accumulated. Afterwards the -strokes will begin to fall when the body is in the opposite -phase. Imagine that one pendulum moving from one extreme -point to another in a second, should be struck by -another pendulum which makes 61 beats in a minute; -then, if the pendulums commence together, they will at -the end of <span class="nowrap">30 <span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span> beats be moving in opposite directions. -Hence whatever energy was communicated in the first<span class="pagenum" id="Page_455">455</span> -half minute will be neutralised by the opposite effect of -that given in the second half. The effect of the strokes of -the second pendulum will therefore be alternately to increase -and decrease the vibrations of the first, so that a -new kind of vibration will be produced running through -its phases in 61 seconds. An effect of this kind was -actually observed by Ellicott, a member of the Royal -Society, in the case of two clocks.<a id="FNanchor_373" href="#Footnote_373" class="fnanchor">373</a> He found that -through the wood-work by which the clocks were connected -a slight impulse was transmitted, and each pendulum -alternately lost and gained momentum. Each -clock, in fact, tended to stop the other at regular intervals, -and in the intermediate times to be stopped by the other.</p> - -<p>Many disturbances in the planetary system depend -upon the same principle; for if one planet happens -always to pull another in the same direction in similar -parts of their orbits, the effects, however slight, will be -accumulated, and a disturbance of large ultimate amount -and of long period will be produced. The long inequality -in the motions of Jupiter and Saturn is thus due to the -fact that five times the mean motion of Saturn is very -nearly equal to twice the mean motion of Jupiter, causing -a coincidence in their relative positions and disturbing -powers. The rolling of ships depends mainly upon the -question whether the period of vibration of the ship -corresponds or not with the intervals at which the waves -strike her. Much which seems at first sight unaccountable -in the behaviour of vessels is thus explained, and the -loss of the <i>Captain</i> is a sad case in point.</p> -<hr class="chap x-ebookmaker-drop"> - -<div class="chapter"> -<p><span class="pagenum" id="Page_456">456</span></p> - -<h2 class="nobreak" id="CHAPTER_XXI">CHAPTER XXI.<br> - -<span class="title">THEORY OF APPROXIMATION.</span></h2> -</div> - -<p class="ti0">In order that we may gain a true understanding of the -kind, degree, and value of the knowledge which we acquire -by experimental investigation, it is requisite that -we should be fully conscious of its approximate character. -We must learn to distinguish between what we can know -and cannot know—between the questions which admit of -solution, and those which only seem to be solved. Many -persons may be misled by the expression <i>exact science</i>, -and may think that the knowledge acquired by scientific -methods admits of our reaching absolutely true laws, -exact to the last degree. There is even a prevailing -impression that when once mathematical formulæ have -been successfully applied to a branch of science, this portion -of knowledge assumes a new nature, and admits of -reasoning of a higher character than those sciences which -are still unmathematical.</p> - -<p>The very satisfactory degree of accuracy attained in the -science of astronomy gives a certain plausibility to erroneous -notions of this kind. Some persons no doubt consider -it to be <i>proved</i> that planets move in ellipses, in such -a manner that all Kepler’s laws hold exactly true; but -there is a double error in any such notions. In the first -place, Kepler’s laws are <i>not proved</i>, if by proof we mean -certain demonstration of their exact truth. In the next -place, even assuming Kepler’s laws to be exactly true in a -theoretical point of view, the planets never move according -to those laws. Even if we could observe the motions of a -planet, of a perfect globular form, free from all perturbing<span class="pagenum" id="Page_457">457</span> -or retarding forces, we could never prove that it moved -in a perfect ellipse. To prove the elliptical form we -should have to measure infinitely small angles, and infinitely -small fractions of a second; we should have to -perform impossibilities. All we can do is to show that -the motion of an unperturbed planet approaches <i>very -nearly</i> to the form of an ellipse, and more nearly the -more accurately our observations are made. But if we go -on to assert that the path <i>is</i> an ellipse we pass beyond -our data, and make an assumption which cannot be verified -by observation.</p> - -<p>But, secondly, as a matter of fact no planet does move -in a perfect ellipse, or manifest the truth of Kepler’s laws -exactly. The law of gravity prevents its own results -from being clearly exhibited, because the mutual perturbations -of the planets distort the elliptical paths. Those -laws, again, hold exactly true only of infinitely small -bodies, and when two great globes, like the sun and -Jupiter, attract each other, the law must be modified. -The periodic time is then shortened in the ratio of the -square root of the number expressing the sun’s mass, to -that of the sum of the numbers expressing the masses of -the sun and planet, as was shown by Newton.<a id="FNanchor_374" href="#Footnote_374" class="fnanchor">374</a> Even at -the present day discrepancies exist between the observed -dimensions of the planetary orbits and their theoretical -magnitudes, after making allowance for all disturbing -causes.<a id="FNanchor_375" href="#Footnote_375" class="fnanchor">375</a> Nothing is more certain in scientific method -than that approximate coincidence alone can be expected. -In the measurement of continuous quantity perfect correspondence -must be accidental, and should give rise to -suspicion rather than to satisfaction.</p> - -<p>One remarkable result of the approximate character of -our observations is that we could never prove the existence -of perfectly circular or parabolic movement, even if it -existed. The circle is a singular case of the ellipse, for -which the eccentricity is zero; it is infinitely improbable -that any planet, even if undisturbed by other bodies, -would have a circle for its orbit; but if the orbit were a -circle we could never prove the entire absence of eccentricity.<span class="pagenum" id="Page_458">458</span> -All that we could do would be to declare the -divergence from the circular form to be inappreciable. -Delambre was unable to detect the slightest ellipticity -in the orbit of Jupiter’s first satellite, but he could only -infer that the orbit was <i>nearly</i> circular. The parabola is -the singular limit between the ellipse and the hyperbola. -As there are elliptic and hyperbolic comets, so we might -conceive the existence of a parabolic comet. Indeed if an -undisturbed comet fell towards the sun from an infinite -distance it would move in a parabola; but we could never -prove that it so moved.</p> - - -<h3><i>Substitution of Simple Hypotheses.</i></h3> - -<p>In truth men never can solve problems fulfilling the -complex circumstances of nature. All laws and explanations -are in a certain sense hypothetical, and apply exactly -to nothing which we can know to exist. In place of the -actual objects which we see and feel, the mathematician -substitutes imaginary objects, only partially resembling -those represented, but so devised that the discrepancies -are not of an amount to alter seriously the character of -the solution. When we probe the matter to the bottom -physical astronomy is as hypothetical as Euclid’s elements. -There may exist in nature perfect straight lines, triangles, -circles, and other regular geometrical figures; to our -science it is a matter of indifference whether they do or -do not exist, because in any case they must be beyond -our powers of perception. If we submitted a perfect -circle to the most rigorous scrutiny, it is impossible that -we should discover whether it were perfect or not. -Nevertheless in geometry we argue concerning perfect -curves, and rectilinear figures, and the conclusions apply -to existing objects so far as we can assure ourselves that -they agree with the hypothetical conditions of our -reasoning. This is in reality all that we can do in the -most perfect of the sciences.</p> - -<p>Doubtless in astronomy we meet with the nearest approximation -to actual conditions. The law of gravity is -not a complex one in itself, and we believe it with much -probability to be exactly true; but we cannot calculate -out in any real case its accurate results. The law asserts<span class="pagenum" id="Page_459">459</span> -that every particle of matter in the universe attracts every -other particle, with a force depending on the masses of -the particles and their distances. We cannot know the -force acting on any particle unless we know the masses -and distances and positions of all other particles in the -universe. The physical astronomer has made a sweeping -assumption, namely, that all the millions of existing -systems exert no perturbing effects on our planetary -system, that is to say, no effects in the least appreciable. -The problem at once becomes hypothetical, because there -is little doubt that gravitation between our sun and planets -and other systems does exist. Even when they consider -the relations of our planetary bodies <i>inter se</i>, all their -processes are only approximate. In the first place they -assume that each of the planets is a perfect ellipsoid, -with a smooth surface and a homogeneous interior. That -this assumption is untrue every mountain and valley, every -sea, every mine affords conclusive evidence. If astronomers -are to make their calculations perfect, they must not only -take account of the Himalayas and the Andes, but must -calculate separately the attraction of every hill, nay, of -every ant-hill. So far are they from having considered -any local inequality of the surface, that they have not yet -decided upon the general form of the earth; it is still a -matter of speculation whether or not the earth is an ellipsoid -with three unequal axes. If, as is probable, the globe -is irregularly compressed in some directions, the calculations -of astronomers will have to be repeated and refined, -in order that they may approximate to the attractive -power of such a body. If we cannot accurately learn the -form of our own earth, how can we expect to ascertain -that of the moon, the sun, and other planets, in some of -which probably are irregularities of greater proportional -amount?</p> - -<p>In a further way the science of physical astronomy is -merely approximate and hypothetical. Given homogeneous -ellipsoids acting upon each other according to the law of -gravity, the best mathematicians have never and perhaps -never will determine exactly the resulting movements. -Even when three bodies simultaneously attract each other -the complication of effects is so great that only approximate -calculations can be made. Astronomers have not<span class="pagenum" id="Page_460">460</span> -even attempted the general problem of the simultaneous -attractions of four, five, six, or more bodies; they resolve -the general problem into so many different problems of -three bodies. The principle upon which the calculations -of physical astronomy proceed, is to neglect every quantity -which does not seem likely to lead to an effect appreciable -in observation, and the quantities rejected are far more -numerous and complex than the few larger terms which -are retained. All then is merely approximate.</p> - -<p>Concerning other branches of physical science the same -statements are even more evidently true. We speak and -calculate about inflexible bars, inextensible lines, heavy -points, homogeneous substances, uniform spheres, perfect -fluids and gases, and we deduce a great number of beautiful -theorems; but all is hypothetical. There is no such -thing as an inflexible bar, an inextensible line, nor any one -of the other perfect objects of mechanical science; they -are to be classed with those mythical existences, the -straight line, triangle, circle, &c., about which Euclid so -freely reasoned. Take the simplest operation considered -in statics—the use of a crowbar in raising a heavy stone, -and we shall find, as Thomson and Tait have pointed out, -that we neglect far more than we observe.<a id="FNanchor_376" href="#Footnote_376" class="fnanchor">376</a> If we suppose -the bar to be quite rigid, the fulcrum and stone perfectly -hard, and the points of contact real points, we may give -the true relation of the forces. But in reality the bar must -bend, and the extension and compression of different parts -involve us in difficulties. Even if the bar be homogeneous -in all its parts, there is no mathematical theory -capable of determining with accuracy all that goes on; if, -as is infinitely more probable, the bar is not homogeneous, -the complete solution will be immensely more complicated, -but hardly more hopeless. No sooner had we determined -the change of form according to simple mechanical principles, -than we should discover the interference of thermodynamic -principles. Compression produces heat and -extension cold, and thus the conditions of the problem are -modified throughout. In attempting a fourth approximation -we should have to allow for the conduction of heat -from one part of the bar to another. All these effects are<span class="pagenum" id="Page_461">461</span> -utterly inappreciable in a practical point of view, if the -bar be a good stout one; but in a theoretical point of -view they entirely prevent our saying that we have solved -a natural problem. The faculties of the human mind, -even when aided by the wonderful powers of abbreviation -conferred by analytical methods, are utterly unable to cope -with the complications of any real problem. And had -we exhausted all the known phenomena of a mechanical -problem, how can we tell that hidden phenomena, as yet -undetected, do not intervene in the commonest actions? -It is plain that no phenomenon comes within the sphere of -our senses unless it possesses a momentum capable of -irritating the appropriate nerves. There may then be -worlds of phenomena too slight to rise within the scope of -our consciousness.</p> - -<p>All the instruments with which we perform our measurements -are faulty. We assume that a plumb-line gives a -vertical line; but this is never true in an absolute sense, -owing to the attraction of mountains and other inequalities -in the surface of the earth. In an accurate trigonometrical -survey, the divergencies of the plumb-line must be approximately -determined and allowed for. We assume a -surface of mercury to be a perfect plane, but even in the -breadth of 5 inches there is a calculable divergence from a -true plane of about one ten-millionth part of an inch; and -this surface further diverges from true horizontality as the -plumb-line does from true verticality. That most perfect -instrument, the pendulum, is not theoretically perfect, -except for infinitely small arcs of vibration, and the -delicate experiments performed with the torsion balance -proceed on the assumption that the force of torsion of a -wire is proportional to the angle of torsion, which again is -only true for infinitely small angles.</p> - -<p>Such is the purely approximate character of all our -operations that it is not uncommon to find the theoretically -worse method giving truer results than the theoretically -perfect method. The common pendulum which is not -isochronous is better for practical purposes than the -cycloidal pendulum, which is isochronous in theory but -subject to mechanical difficulties. The spherical form is -not the correct form for a speculum or lense, but it differs -so slightly from the true form, and is so much more easily<span class="pagenum" id="Page_462">462</span> -produced mechanically, that it is generally best to rest -content with the spherical surface. Even in a six-feet -mirror the difference between the parabola and the sphere -is only about one ten-thousandth part of an inch, a thickness -which would be taken off in a few rubs of the polisher. -Watts’ ingenious parallel motion was intended to produce -rectilinear movement of the piston-rod. In reality the -motion was always curvilinear, but for his purposes a -certain part of the curve approximated sufficiently to a -straight line.</p> - - -<h3><i>Approximation to Exact Laws.</i></h3> - -<p>Though we can not prove numerical laws with perfect -accuracy, it would be a great mistake to suppose that -there is any inexactness in the laws of nature. We -may even discover a law which we believe to represent -the action of forces with perfect exactness. The mind -may seem to pass in advance of its data, and choose out -certain numerical results as absolutely true. We can -never really pass beyond our data, and so far as assumption -enters in, so far want of certainty will attach to our -conclusions; nevertheless we may sometimes rightly prefer -a probable assumption of a precise law to numerical results, -which are at the best only approximate. We must accordingly -draw a strong distinction between the laws of nature -which we believe to be accurately stated in our formulas, -and those to which our statements only make an approximation, -so that at a future time the law will be differently -stated.</p> - -<p>The law of gravitation is expressed in the form -F = <span class="nowrap"><span class="fraction2"><span class="fnum2">Mm</span><span class="bar">/</span><span class="fden2">D<sup>2</sup></span></span></span>, - meaning that gravity is proportional directly to -the product of the gravitating masses, and indirectly to the -square of their distance. The latent heat of steam is expressed -by the equation log F = <i>a</i> + <i>b</i>α<sup>t</sup> + <i>c</i>β<sup>t</sup>, in which are -five quantities <i>a</i>, <i>b</i>, <i>c</i>, α, β, to be determined by experiment. -There is every reason to believe that in the progress of -science the law of gravity will remain entirely unaltered, -and the only effect of further inquiry will be to render it a -more and more probable expression of the absolute truth. -The law of the latent heat of steam on the other hand, will<span class="pagenum" id="Page_463">463</span> -be modified by every new series of experiments, and it may -not improbably be shown that the assumed law can never -be made to agree exactly with the results of experiment.</p> - -<p>Philosophers have not always supposed that the law of -gravity was exactly true. Newton, though he had the -highest confidence in its truth, admitted that there were -motions in the planetary system which he could not -reconcile with the law. Euler and Clairaut who were, -with D’Alembert, the first to apply the full powers of -mathematical analysis to the theory of gravitation as explaining -the perturbations of the planets, did not think -the law sufficiently established to attribute all discrepancies -to the errors of calculation and observation. They did -not feel certain that the force of gravity exactly obeyed -the well-known rule. The law might involve other powers -of the distance. It might be expressed in the form</p> - -<div class="ml5em"> -F = . . . + <span class="nowrap"><span class="fraction2"><span class="fnum2"><i>a</i></span><span class="bar">/</span><span class="fden2">D</span></span></span> - + <span class="nowrap"><span class="fraction2"><span class="fnum2"><i>b</i></span><span class="bar">/</span><span class="fden2">D<sup>2</sup></span></span></span> - + <span class="nowrap"><span class="fraction2"><span class="fnum2"><i>c</i></span><span class="bar">/</span><span class="fden2">D<sup>3</sup></span></span></span> + . . . -</div> - -<p class="ti0">and the coefficients <i>a</i> and <i>c</i> might be so small that those -terms would become apparent only in very accurate -comparisons with fact. Attempts have been made to -account for difficulties, by attributing value to such -neglected terms. Gauss at one time thought the even -more fundamental principle of gravity, that the force -is dependent only on mass and distance, might not -be exactly true, and he undertook accurate pendulum -experiments to test this opinion. Only as repeated -doubts have time after time been resolved in favour of -the law of Newton, has it been assumed as precisely -correct. But this belief does not rest on experiment or -observation only. The calculations of physical astronomy, -however accurate, could never show that the other terms -of the above expression were absolutely devoid of value. -It could only be shown that they had such slight value -as never to become apparent.</p> - -<p>There are, however, other reasons why the law is probably -complete and true as commonly stated. Whatever -influence spreads from a point, and expands uniformly -through space, will doubtless vary inversely in intensity -as the square of the distance, because the area over which -it is spread increases as the square of the radius. This -part of the law of gravity may be considered as due to<span class="pagenum" id="Page_464">464</span> -the properties of space, and there is a perfect analogy -in this respect between gravity and all other <i>emanating</i> -forces, as was pointed out by Keill.<a id="FNanchor_377" href="#Footnote_377" class="fnanchor">377</a> Thus the undulations -of light, heat, and sound, and the attractions of electricity -and magnetism obey the very same law so far as we can -ascertain. If the molecules of a gas or the particles -of matter constituting odour were to start from a point -and spread uniformly, their distances would increase and -their density decrease according to the same principle.</p> - -<p>Other laws of nature stand in a similar position. Dalton’s -laws of definite combining proportions never have been, -and never can be, exactly proved; but chemists having -shown, to a considerable degree of approximation, that -the elements combine together as if each element had -atoms of an invariable mass, assume that this is exactly -true. They go even further. Prout pointed out in 1815 -that the equivalent weights of the elements appeared to -be simple numbers; and the researches of Dumas, Pelouze, -Marignac, Erdmann, Stas, and others have gradually rendered -it likely that the atomic weights of hydrogen, carbon, -oxygen, nitrogen, chlorine, and silver, are in the ratios of -the numbers 1, 12, 16, 14, 35·5, and 108. Chemists then -step beyond their data; they throw aside their actual -experimental numbers, and assume that the true ratios -are not those exactly indicated by any weighings, but the -simple ratios of these numbers. They boldly assume that -the discrepancies are due to experimental errors, and they -are justified by the fact that the more elaborate and skilful -the researches on the subject, the more nearly their assumption -is verified. Potassium is the only element whose -atomic weight has been determined with great care, but -which has not shown an approach to a simple ratio with -the other elements. This exception may be due to some -unsuspected cause of error.<a id="FNanchor_378" href="#Footnote_378" class="fnanchor">378</a> A similar assumption is -made in the law of definite combining volumes of gases, -and Brodie has clearly pointed out the line of argument -by which the chemist, observing that the discrepancies -between the law and fact are within the limits of experimental -error, assumes that they are due to error.<a id="FNanchor_379" href="#Footnote_379" class="fnanchor">379</a></p> - -<p><span class="pagenum" id="Page_465">465</span></p> - -<p>Faraday, in one of his researches, expressly makes an -assumption of the same kind. Having shown, with some -degree of experimental precision, that there exists a simple -proportion between quantities of electrical energy and the -quantities of chemical substances which it can decompose, -so that for every atom dissolved in the battery cell an -atom ought theoretically, that is without regard to dissipation -of some of the energy, to be decomposed in the -electrolytic cell, he does not stop at his numerical results. -“I have not hesitated,” he says,<a id="FNanchor_380" href="#Footnote_380" class="fnanchor">380</a> “to apply the more strict -results of chemical analysis to correct the numbers obtained -as electrolytic results. This, it is evident, may be done -in a great number of cases, without using too much liberty -towards the due severity of scientific research.”</p> - -<p>The law of the conservation of energy, one of the widest -of all physical generalisations, rests upon the same footing. -The most that we can do by experiment is to show that -the energy entering into any experimental combination is -almost equal to what comes out of it, and more nearly so -the more accurately we perform the measurements. Absolute -equality is always a matter of assumption. We -cannot even prove the indestructibility of matter; for -were an exceedingly minute fraction of existing matter to -vanish in any experiment, say one part in ten millions, -we could never detect the loss.</p> - - -<h3><i>Successive Approximations to Natural Conditions.</i></h3> - -<p>When we examine the history of scientific problems, we -find that one man or one generation is usually able to -make but a single step at a time. A problem is solved -for the first time by making some bold hypothetical -simplification, upon which the next investigator makes -hypothetical modifications approaching more nearly to -the truth. Errors are successively pointed out in previous -solutions, until at last there might seem little more to -be desired. Careful examination, however, will show that -a series of minor inaccuracies remain to be corrected and -explained, were our powers of reasoning sufficiently great, -and the purpose adequate in importance.</p> - -<p><span class="pagenum" id="Page_466">466</span></p> - -<p>Newton’s successful solution of the problem of the -planetary movements entirely depended at first upon a -great simplification. The law of gravity only applies -directly to two infinitely small particles, so that when we -deal with vast globes like the earth, Jupiter, and the -sun, we have an immense aggregate of separate attractions -to deal with, and the law of the aggregate need not coincide -with the law of the elementary particles. But Newton, -by a great effort of mathematical reasoning, was able to -show that two homogeneous spheres of matter act as if -the whole of their masses were concentrated at the centres; -in short, that such spheres are centrobaric bodies (p. <a href="#Page_364">364</a>). -He was then able with comparative ease to calculate the -motions of the planets on the hypothesis of their being -spheres, and to show that the results roughly agreed with -observation. Newton, indeed, was one of the few men -who could make two great steps at once. He did not -rest contented with the spherical hypothesis; having -reason to believe that the earth was really a spheroid -with a protuberance around the equator, he proceeded to -a second approximation, and proved that the attraction of -the protuberant matter upon the moon accounted for the -precession of the equinoxes, and led to various complicated -effects. But, (p. <a href="#Page_459">459</a>), even the spheroidal hypothesis is -far from the truth. It takes no account of the irregularities -of surface, the great protuberance of land in -Central Asia and South America, and the deficiency in -the bed of the Atlantic.</p> - -<p>To determine the law according to which a projectile, -such as a cannon ball, moves through the atmosphere is -a problem very imperfectly solved at the present day, but -in which many successive advances have been made. So -little was known concerning the subject three or four -centuries ago that a cannon ball was supposed to move -at first in a straight line, and after a time to be deflected -into a curve. Tartaglia ventured to maintain that the -path was curved throughout, as by the principle of continuity -it should be; but the ingenuity of Galileo was -required to prove this opinion, and to show that the curve -was approximately a parabola. It is only, however, under -forced hypotheses that we can assert the path of a projectile -to be truly a parabola: the path must be through a<span class="pagenum" id="Page_467">467</span> -perfect vacuum, where there is no resisting medium of any -kind; the force of gravity must be uniform and act in -parallel lines; or else the moving body must be either a -mere point, or a perfect centrobaric body, that is a body -possessing a definite centre of gravity. These conditions -cannot be really fulfilled in practice. The next great step -in the problem was made by Newton and Huyghens, the -latter of whom asserted that the atmosphere would offer a -resistance proportional to the velocity of the moving body, -and concluded that the path would have in consequence -a logarithmic character. Newton investigated in a general -manner the subject of resisting media, and came to the -conclusion that the resistance is more nearly proportional -to the square of the velocity. The subject then fell into -the hands of Daniel Bernoulli, who pointed out the enormous -resistance of the air in cases of rapid movement, -and calculated that a cannon ball, if fired vertically in a -vacuum, would rise eight times as high as in the atmosphere. -In recent times an immense amount both of -theoretical and experimental investigation has been spent -upon the subject, since it is one of importance in the art -of war. Successive approximations to the true law have -been made, but nothing like a complete and final solution -has been achieved or even hoped for.<a id="FNanchor_381" href="#Footnote_381" class="fnanchor">381</a></p> - -<p>It is quite to be expected that the earliest experimenters -in any branch of science will overlook errors which afterwards -become most apparent. The Arabian astronomers -determined the meridian by taking the middle point between -the places of the sun when at equal altitudes on -the same day. They overlooked the fact that the sun has -its own motion in the time between the observations. -Newton thought that the mutual disturbances of the -planets might be disregarded, excepting perhaps the effect -of the mutual attraction of the greater planets, Jupiter -and Saturn, near their conjunction.<a id="FNanchor_382" href="#Footnote_382" class="fnanchor">382</a> The expansion of -quicksilver was long used as the measure of temperature, -no clear idea being possessed of temperature apart from -some of its more obvious effects. Rumford, in the first -experiment leading to a determination of the mechanical<span class="pagenum" id="Page_468">468</span> -equivalent of heat, disregarded the heat absorbed by the -apparatus, otherwise he would, in Dr. Joule’s opinion, have -come nearly to the correct result.</p> - -<p>It is surprising to learn the number of causes of error -which enter into the simplest experiment, when we strive -to attain rigid accuracy. We cannot accurately perform -the simple experiment of compressing gas in a bent tube -by a column of mercury, in order to test the truth of -Boyle’s Law, without paying regard to—(1) the variations -of atmospheric pressure, which are communicated to the -gas through the mercury; (2) the compressibility of -mercury, which causes the column of mercury to vary -in density; (3) the temperature of the mercury throughout -the column; (4) the temperature of the gas, which is -with difficulty maintained invariable; (5) the expansion -of the glass tube containing the gas. Although Regnault -took all these circumstances into account in his examination -of the law,<a id="FNanchor_383" href="#Footnote_383" class="fnanchor">383</a> there is no reason to suppose that he -exhausted the sources of inaccuracy.</p> - -<p>The early investigations concerning the nature of waves -in elastic media proceeded upon the assumption that -waves of different lengths would travel with equal speed. -Newton’s theory of sound led him to this conclusion, and -observation (p. <a href="#Page_295">295</a>) had verified the inference. When -the undulatory theory came to be applied at the commencement -of this century to explain the phenomena of -light, a great difficulty was encountered. The angle at -which a ray of light is refracted in entering a denser -medium depends, according to that theory, on the velocity -with which the wave travels, so that if all waves -of light were to travel with equal velocity in the same -medium, the dispersion of mixed light by the prism and -the production of the spectrum could not take place. -Some most striking phenomena were thus in direct conflict -with the theory. Cauchy first pointed out the explanation, -namely, that all previous investigators had made -an arbitrary assumption for the sake of simplifying the -calculations. They had assumed that the particles of the -vibrating medium are so close together that the intervals -are inconsiderable compared with the length of the wave.<span class="pagenum" id="Page_469">469</span> -This hypothesis happened to be approximately true in -the case of air, so that no error was discovered in experiments -on sound. Had it not been so, the earlier -analysts would probably have failed to give any solution, -and the progress of the subject might have been retarded. -Cauchy was able to make a new approximation under -the more difficult supposition, that the particles of the -vibrating medium are situated at considerable distances, -and act and react upon the neighbouring particles by -attractive and repulsive forces. To calculate the rate of -propagation of disturbance in such a medium is a work -of excessive difficulty. The complete solution of the -problem appears indeed to be beyond human power, so -that we must be content, as in the case of the planetary -motions, to look forward to successive approximations. -All that Cauchy could do was to show that certain quantities, -neglected in previous theories, became of considerable -amount under the new conditions of the problem, -so that there will exist a relation between the length of -the wave, and the velocity at which it travels. To remove, -then, the difficulties in the way of the undulatory -theory of light, a new approach to probable conditions -was needed.<a id="FNanchor_384" href="#Footnote_384" class="fnanchor">384</a></p> - -<p>In a similar manner Fourier’s theory of the conduction -and radiation of heat was based upon the hypothesis that -the quantity of heat passing along any line is simply proportional -to the rate of change of temperature. But it -has since been shown by Forbes that the conductivity of a -body diminishes as its temperature increases. All the -details of Fourier’s solution therefore require modification, -and the results are in the meantime to be regarded as -only approximately true.<a id="FNanchor_385" href="#Footnote_385" class="fnanchor">385</a></p> - -<p>We ought to distinguish between those problems which -are physically and those which are merely mathematically -incomplete. In the latter case the physical law is correctly -seized, but the mathematician neglects, or is more -often unable to follow out the law in all its results. The -law of gravitation and the principles of harmonic or undulatory -movement, even supposing the data to be correct,<span class="pagenum" id="Page_470">470</span> -can never be followed into all their ultimate results. -Young explained the production of Newton’s rings by -supposing that the rays reflected from the upper and -lower surfaces of a thin film of a certain thickness were in -opposite phases, and thus neutralised each other. It was -pointed out, however, that as the light reflected from the -nearer surface must be undoubtedly a little brighter than -that from the further surface, the two rays ought not to -neutralise each other so completely as they are observed -to do. It was finally shown by Poisson that the discrepancy -arose only from incomplete solution of the -problem; for the light which has once got into the film -must be to a certain extent reflected backwards and -forwards <i>ad infinitum</i>; and if we follow out this course of -the light by perfect mathematical analysis, absolute darkness -may be shown to result from the interference of -the rays.<a id="FNanchor_386" href="#Footnote_386" class="fnanchor">386</a> In this case the natural laws concerned, those -of reflection and refraction, are accurately known, and -the only difficulty consists in developing their full -consequences.</p> - - -<h3><i>Discovery of Hypothetically Simple Laws.</i></h3> - -<p>In some branches of science we meet with natural laws -of a simple character which are in a certain point of view -exactly true and yet can never be manifested as exactly -true in natural phenomena. Such, for instance, are the -laws concerning what is called a <i>perfect gas</i>. The gaseous -state of matter is that in which the properties of matter -are exhibited in the simplest manner. There is much -advantage accordingly in approaching the question of -molecular mechanics from this side. But when we ask -the question—What is a gas? the answer must be a -hypothetical one. Finding that gases <i>nearly</i> obey the -law of Boyle and Mariotte; that they <i>nearly</i> expand by -heat at the uniform rate of one part in 272·9 of their -volume at 0° for each degree centigrade; and that they -<i>more nearly</i> fulfil these conditions the more distant the -point of temperature at which we examine them from -the liquefying point, we pass by the principle of continuity<span class="pagenum" id="Page_471">471</span> -to the conception of a perfect gas. Such a gas -would probably consist of atoms of matter at so great a -distance from each other as to exert no attractive forces -upon each other; but for this condition to be fulfilled the -distances must be infinite, so that an absolutely perfect -gas cannot exist. But the perfect gas is not merely a -limit to which we may approach, it is a limit passed by -at least one real gas. It has been shown by Despretz, -Pouillet, Dulong, Arago, and finally Regnault, that all -gases diverge from the Boylean law, and in nearly all -cases the density of the gas increases in a somewhat greater -ratio than the pressure, indicating a tendency on the -part of the molecules to approximate of their own accord. -In the more condensable gases such as sulphurous acid, -ammonia, and cyanogen, this tendency is strongly apparent -near the liquefying point. Hydrogen, on the contrary, -diverges from the law of a perfect gas in the opposite -direction, that is, the density increases less than in the -ratio of the pressure.<a id="FNanchor_387" href="#Footnote_387" class="fnanchor">387</a> This is a singular exception, the -bearing of which I am unable to comprehend.</p> - -<p>All gases diverge again from the law of uniform expansion -by heat, but the divergence is less as the gas in -question is less condensable, or examined at a temperature -more removed from its liquefying point. Thus the perfect -gas must have an infinitely high temperature. According -to Dalton’s law each gas in a mixture retains its own -properties unaffected by the presence of any other gas.<a id="FNanchor_388" href="#Footnote_388" class="fnanchor">388</a> -This law is probably true only by approximation, but it -is obvious that it would be true of the perfect gas with -infinitely distant particles.<a id="FNanchor_389" href="#Footnote_389" class="fnanchor">389</a></p> - - -<h3><i>Mathematical Principles of Approximation.</i></h3> - -<p>The approximate character of physical science will be -rendered more plain if we consider it from a mathematical -point of view. Throughout quantitative investigations we -deal with the relation of one quantity to other quantities,<span class="pagenum" id="Page_472">472</span> -of which it is a function; but the subject is sufficiently -complicated if we view one quantity as a function of -one other. Now, as a general rule, a function can be -developed or expressed as the sum of quantities, the -values of which depend upon the successive powers of the -variable quantity. If <i>y</i> be a function of <i>x</i> then we may -say that</p> - -<div class="ml5em"> -<i>y</i> = A + B<i>x</i> + C<i>x</i><sup>2</sup> + D<i>x</i><sup>3</sup> + E<i>x</i><sup>4</sup> . . . -</div> - -<p class="ti0">In this equation, A, B, C, D, &c., are fixed quantities, of -different values in different cases. The terms may be -infinite in number or after a time may cease to have any -value. Any of the coefficients A, B, C, &c., may be -zero or negative; but whatever they be they are fixed. -The quantity <i>x</i> on the other hand may be made what we -like, being variable. Suppose, in the first place, that <i>x</i> and -<i>y</i> are both lengths. Let us assume that <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">10,000</span></span></span> - part of an -inch is the least that we can take note of. Then when <i>x</i> -is one hundredth of an inch, we have <i>x</i><sup>2</sup> = - <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">10,000</span></span></span>, - and -if C be less than unity, the term C<i>x</i><sup>2</sup> will be inappreciable, -being less than we can measure. Unless any of the -quantities D, E, &c., should happen to be very great, it -is evident that all the succeeding terms will also be inappreciable, -because the powers of <i>x</i> become rapidly -smaller in geometrical ratio. Thus when <i>x</i> is made small -enough the quantity <i>y</i> seems to obey the equation</p> - -<div class="ml5em"> -<i>y</i> = A + B<i>x</i>. -</div> - -<p>If <i>x</i> should be still less, if it should become as small, -for instance, as <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">1,000,000</span></span></span> of an inch, and B should not -be very great, then <i>y</i> would appear to be the fixed -quantity A, and would not seem to vary with <i>x</i> at all. -On the other hand, were x to grow greater, say equal to -<span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">10</span></span></span> - inch, and C not be very small, the term C<i>x</i><sup>2</sup> - would -become appreciable, and the law would now be more -complicated.</p> - -<p>We can invert the mode of viewing this question, and -suppose that while the quantity <i>y</i> undergoes variations -depending on many powers of <i>x</i>, our power of detecting -the changes of value is more or less acute. While -our powers of observation remain very rude we may be -unable to detect any change in the quantity at all, that -is to say, B<i>x</i> may always be too small to come within<span class="pagenum" id="Page_473">473</span> -our notice, just as in former days the fixed stars were so -called because they remained at apparently fixed distances -from each other. With the use of telescopes and micrometers -we become able to detect the existence of some -motion, so that the distance of one star from another may -be expressed by A + B<i>x</i>, the term including <i>x</i><sup>2</sup> being -still inappreciable. Under these circumstances the star -will seem to move uniformly, or in simple proportion to -the time <i>x</i>. With much improved means of measurement -it will probably be found that this uniformity of motion -is only apparent, and that there exists some acceleration -or retardation. More careful investigation will show the -law to be more and more complicated than was previously -supposed.</p> - -<p>There is yet another way of explaining the apparent -results of a complicated law. If we take any curve and -regard a portion of it free from any kind of discontinuity, -we may represent the character of such portion by an -equation of the form</p> - -<div class="ml5em"> -<i>y</i> = A + B<i>x</i> + C<i>x</i><sup>2</sup> + D<i>x</i><sup>3</sup> + . . . -</div> - -<p class="ti0">Restrict the attention to a very small portion of the curve, -and the eye will be unable to distinguish its difference -from a straight line, which amounts to saying that in the -portion examined the term C<i>x</i><sup>2</sup> has no value appreciable -by the eye. Take a larger portion of the curve and it will -be apparent that it possesses curvature, but it will be -possible to draw a parabola or ellipse so that the curve -shall apparently coincide with a portion of that parabola -or ellipse. In the same way if we take larger and larger -arcs of the curve it will assume the character successively -of a curve of the third, fourth, and perhaps higher degrees; -that is to say, it corresponds to equations involving the -third, fourth, and higher powers of the variable quantity.</p> - -<p>We have arrived then at the conclusion that every phenomenon, -when its amount can only be rudely measured, -will either be of fixed amount, or will seem to vary uniformly -like the distance between two inclined straight -lines. More exact measurement may show the error of -this first assumption, and the variation will then appear -to be like that of the distance between a straight line -and a parabola or ellipse. We may afterwards find that -a curve of the third or higher degrees is really required<span class="pagenum" id="Page_474">474</span> -to represent the variation. I propose to call the variation -of a quantity <i>linear</i>, <i>elliptic</i>, <i>cubic</i>, <i>quartic</i>, <i>quintic</i>, &c., -according as it is discovered to involve the first, second, -third, fourth, fifth, or higher powers of the variable. It is -a general rule in quantitative investigation that we commence -by discovering linear, and afterwards proceed to -elliptic or more complicated laws of variation. The approximate -curves which we employ are all, according to -De Morgan’s use of the name, parabolas of some order -or other; and since the common parabola of the second -order is approximately the same as a very elongated -ellipse, and is in fact an infinitely elongated ellipse, -it is convenient and proper to call variation of the -second order <i>elliptic</i>. It might also be called <i>quadric</i> -variation.</p> - -<p>As regards many important phenomena we are yet only -in the first stage of approximation. We know that the -sun and many so-called fixed stars, especially 61 Cygni, -have a proper motion through space, and the direction of -this motion at the present time is known with some degree -of accuracy. But it is hardly consistent with the theory -of gravity that the path of any body should really be a -straight line. Hence, we must regard a rectilinear path -as only a provisional description of the motion, and look -forward to the time when its curvature will be detected, -though centuries perhaps must first elapse.</p> - -<p>We are accustomed to assume that on the surface of the -earth the force of gravity is uniform, because the variation -is of so slight an amount that we are scarcely able to -detect it. But supposing we could measure the variation, -we should find it simply proportional to the height. -Taking the earth’s radius to be unity, let <i>h</i> be the height -at which we measure the force of gravity. Then by the -well-known law of the inverse square, that force will be -proportional to</p> - -<div class="ml5em"> -<span class="nowrap"><span class="fraction2"><span class="fnum2"><i>g</i></span><span class="bar">/</span><span class="fden2">(1 + <i>h</i>)<sup>2</sup></span></span></span>, or to <i>g</i>(1 - 2<i>h</i> + 3<i>h</i><sup>2</sup> - 4<i>h</i><sup>3</sup> + . . .). -</div> - -<p class="ti0">But at all heights to which we can attain <i>h</i> will be -so small a fraction of the earth’s radius that 3<i>h</i><sup>2</sup> will -be inappreciable, and the force of gravity will seem -to follow the law of linear variation, being proportional -to 1 - 2<i>h</i>.</p> - -<p><span class="pagenum" id="Page_475">475</span></p> - -<p>When the circumstances of an experiment are much -altered, different powers of the variable may become prominent. -The resistance of a liquid to a body moving -through it may be approximately expressed as the sum -of two terms respectively involving the first and second -powers of the velocity. At very low velocities the first -power is of most importance, and the resistance, as Professor -Stokes has shown, is nearly in simple proportion to -the velocity. When the motion is rapid the resistance -increases in a still greater degree, and is more nearly proportional -to the square of the velocity.</p> - - -<h3><i>Approximate Independence of Small Effects.</i></h3> - -<p>One result of the theory of approximation possesses such -importance in physical science, and is so often applied, -that we may consider it separately. The investigation of -causes and effects is immensely simplified when we may -consider each cause as producing its own effect invariably, -whether other causes are acting or not. Thus, if the body -P produces <i>x</i>, and Q produces <i>y</i>, the question is whether P -and Q acting together will produce the sum of the separate -effects, <i>x</i> + <i>y</i>. It is under this supposition that we treated -the methods of eliminating error (Chap. XV.), and errors of -a less amount would still remain if the supposition was a -forced one. There are probably some parts of science in -which the supposition of independence of effects holds -rigidly true. The mutual gravity of two bodies is entirely -unaffected by the presence of other gravitating bodies. -People do not usually consider that this important principle -is involved in such a simple thing as putting two -pound weights in the scale of a balance. How do we -know that two pounds together will weigh twice as much -as one? Do we know it to be exactly so? Like other -results founded on induction we cannot prove it absolutely, -but all the calculations of physical astronomy proceed -upon the assumption, so that we may consider it proved -to a very high degree of approximation. Had not this -been true, the calculations of physical astronomy would -have been infinitely more complex than they actually are, -and the progress of knowledge would have been much -slower.</p> - -<p><span class="pagenum" id="Page_476">476</span></p> - -<p>It is a general principle of scientific method that if -effects be of small amount, comparatively to our means of -observation, all joint effects will be of a higher order of -smallness, and may therefore be rejected in a first approximation. -This principle was employed by Daniel -Bernoulli in the theory of sound, under the title of <i>The -Principle of the Coexistence of Small Vibrations</i>. He -showed that if a string is affected by two kinds of -vibrations, we may consider each to be going on as -if the other did not exist. We cannot perceive that -the sounding of one musical instrument prevents or -even modifies the sound of another, so that all sounds -would seem to travel through the air, and act upon -the ear in independence of each other. A similar -assumption is made in the theory of tides, which are -great waves. One wave is produced by the attraction -of the moon, and another by the attraction of the -sun, and the question arises, whether when these waves -coincide, as at the time of spring tides, the joint wave -will be simply the sum of the separate waves. On the -principle of Bernoulli this will be so, because the tides -on the ocean are very small compared with the depth of -the ocean.</p> - -<p>The principle of Bernoulli, however, is only approximately -true. A wave never is exactly the same when -another wave is interfering with it, but the less the displacement -of particles due to each wave, the less in a still -higher degree is the effect of one wave upon the other. -In recent years Helmholtz was led to suspect that some -of the phenomena of sound might after all be due to -resultant effects overlooked by the assumption of previous -physicists. He investigated the secondary waves which -would arise from the interference of considerable disturbances, -and was able to show that certain summation of -resultant tones ought to be heard, and experiments subsequently -devised for the purpose showed that they might -be heard.</p> - -<figure class="figright illowp88" id="p477" style="max-width: 12.125em;"> - <img class="w100" src="images/p477.jpg" alt=""> -</figure> - -<p>Throughout the mechanical sciences the <i>Principle of the -Superposition of Small Motions</i> is of fundamental importance,<a id="FNanchor_390" href="#Footnote_390" class="fnanchor">390</a> -and it may be thus explained. Suppose<span class="pagenum" id="Page_477">477</span> -that two forces, acting from the points B and C, are -simultaneously moving a body A. Let the force acting -from B be such that in one second it would move A -to <i>p</i>, and similarly let the second force, acting alone, -move A to <i>r</i>. The question -arises, then, whether their joint -action will urge A to <i>q</i> along -the diagonal of the parallelogram. -May we say that A will -move the distance A<i>p</i> in the -direction AB, and A<i>r</i> in the -direction AC, or, what is the -same thing, along the parallel -line <i>pq</i>? In strictness we cannot say so; for when A has -moved towards <i>p</i>, the force from C will no longer act along -the line AC, and similarly the motion of A towards <i>r</i> will -modify the action of the force from B. This interference -of one force with the line of action of the other will -evidently be greater the larger is the extent of motion -considered; on the other hand, as we reduce the parallelogram -A<i>pqr</i>, compared with the distances AB and AC, -the less will be the interference of the forces. Accordingly -mathematicians avoid all error by considering the -motions as infinitely small, so that the interference becomes -of a still higher order of infinite smallness, and -may be entirely neglected. By the resources of the differential -calculus it is possible to calculate the motion of the -particle A, as if it went through an infinite number of -infinitely small diagonals of parallelograms. The great -discoveries of Newton really arose from applying this -method of calculation to the movements of the moon -round the earth, which, while constantly tending to move -onward in a straight line, is also deflected towards the -earth by gravity, and moves through an elliptic curve, -composed as it were of the infinitely small diagonals of -infinitely numerous parallelograms. The mathematician, -in his investigation of a curve, always treats it as made -up of a great number of straight lines, and it may be -doubted whether he could treat it in any other manner. -There is no error in the final results, because having obtained -the formulæ flowing from this supposition, each -straight line is then regarded as becoming infinitely small,<span class="pagenum" id="Page_478">478</span> -and the polygonal line becomes undistinguishable from a -perfect curve.<a id="FNanchor_391" href="#Footnote_391" class="fnanchor">391</a></p> - -<p>In abstract mathematical theorems the approximation -to absolute truth is perfect, because we can treat of infinitesimals. -In physical science, on the contrary, we -treat of the least quantities which are perceptible. Nevertheless, -while carefully distinguishing between these two -different cases, we may fearlessly apply to both the principle -of the superposition of small effects. In physical -science we have only to take care that the effects really -are so small that any joint effect will be unquestionably -imperceptible. Suppose, for instance, that there is some -cause which alters the dimensions of a body in the ratio -of 1 to 1 + α, and another cause which produces an alteration -in the ratio of 1 to 1 + β. If they both act at once -the change will be in the ratio of 1 to (1 + α)(1 + β), -or as 1 to 1 + α + β + αβ. But if α and β be both very -small fractions of the total dimensions, αβ will be yet far -smaller and may be disregarded; the ratio of change is -then approximately that of 1 to 1 + α + β, or the joint -effect is the sum of the separate effects. Thus if a body -were subjected to three strains, at right angles to each -other, the total change in the volume of the body would -be approximately equal to the sum of the changes produced -by the separate strains, provided that these are very -small. In like manner not only is the expansion of every -solid and liquid substance by heat approximately proportional -to the change of temperature, when this change is -very small in amount, but the cubic expansion may also -be considered as being three times as great as the linear -expansion. For if the increase of temperature expands -a bar of metal in the ratio of 1 to 1 + α, and the expansion -be equal in all directions, then a cube of the same metal -would expand as 1 to (1 + α)<sup>3</sup>, or as 1 to 1 + 3α + 3α<sup>2</sup> + α<sup>3</sup>. -When α is a very small quantity the third term 3α<sup>2</sup> will -be imperceptible, and still more so the fourth term α<sup>3</sup>. -The coefficients of expansion of solids are in fact so -small, and so imperfectly determined, that physicists -seldom take into account their second and higher powers.</p> - -<p><span class="pagenum" id="Page_479">479</span></p> - -<p>It is a result of these principles that all small errors may -be assumed to vary in simple proportion to their causes—a -new reason why, in eliminating errors, we should first of -all make them as small as possible. Let us suppose that -there is a right-angled triangle of which the two sides -containing the right angle are really of the lengths 3 and -4, so that the hypothenuse is √<span class="o">3<sup>2</sup> + 4<sup>2</sup></span> or 5. Now, if in -two measurements of the first side we commit slight -errors, making it successively 4·001 and 4·002, then calculation -will give the lengths of the hypothenuse as almost -exactly 5·0008 and 5·0016, so that the error in the -hypothenuse will seem to vary in simple proportion to -that of the side, although it does not really do so with -perfect exactness. The logarithm of a number does not -vary in proportion to that number—nevertheless we find -the difference between the logarithms of the numbers -100000 and 100001 to be almost exactly equal to that -between the numbers 100001 and 100002. It is thus a -general rule that very small differences between successive -values of a function are approximately proportional to -the small differences of the variable quantity.</p> - -<p>On these principles it is easy to draw up a series of -rules such as those given by Kohlrausch<a id="FNanchor_392" href="#Footnote_392" class="fnanchor">392</a> for performing -calculations in an abbreviated form when the variable -quantity is very small compared with unity. Thus for -1 ÷ (1 + α) we may substitute 1 – α; for 1 ÷ (1 – α) -we may put 1 + α; 1 ÷ √<span class="o">1 + α</span> -becomes 1 – <span class="nowrap"><span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span>α, -and so forth.</p> - - -<h3><i>Four Meanings of Equality.</i></h3> - -<p>Although it might seem that there are few terms more -free from ambiguity than the term <i>equal</i>, yet scientific -men do employ it with at least four meanings, which it -is desirable to distinguish. These meanings I may describe -as</p> - -<div class="ml5em"> -(1) Absolute Equality.<br> -(2) Sub-equality.<br> -(3) Apparent Equality.<br> -(4) Probable Equality. -</div> - -<p><span class="pagenum" id="Page_480">480</span></p> - -<p>By <i>absolute equality</i> we signify that which is complete -and perfect to the last degree; but it is obvious that we -can only know such equality in a theoretical or hypothetical -manner. The areas of two triangles standing upon the -same base and between the same parallels are absolutely -equal. Hippocrates beautifully proved that the area of a -lunula or figure contained between two segments of circles -was absolutely equal to that of a certain right-angled -triangle. As a general rule all geometrical and other -elementary mathematical theorems involve absolute -equality.</p> - -<p>De Morgan proposed to describe as <i>sub-equal</i> those -quantities which are equal within an infinitely small -quantity, so that <i>x</i> is sub-equal to <i>x</i> + <i>dx</i>. The differential -calculus may be said to arise out of the neglect -of infinitely small quantities, and in mathematical science -other subtle distinctions may have to be drawn between -kinds of equality, as De Morgan has shown in a remarkable -memoir “On Infinity; and on the sign of Equality.”<a id="FNanchor_393" href="#Footnote_393" class="fnanchor">393</a></p> - -<p><i>Apparent equality</i> is that with which physical science -deals. Those magnitudes are apparently equal which differ -only by an imperceptible quantity. To the carpenter -anything less than the hundredth part of an inch is non-existent; -there are few arts or artists to which the hundred-thousandth -of an inch is of any account. Since all -coincidence between physical magnitudes is judged by one -or other sense, we must be restricted to a knowledge of -apparent equality.</p> - -<p>In reality even apparent equality is rarely to be expected. -More commonly experiments will give only -<i>probable equality</i>, that is results will come so near to each -other that the difference may be ascribed to unimportant -disturbing causes. Physicists often assume quantities to -be equal provided that they fall within the limits of -probable error of the processes employed. We cannot -expect observations to agree with theory more closely -than they agree with each other, as Newton remarked of -his investigations concerning Halley’s Comet.</p> - -<p><span class="pagenum" id="Page_481">481</span></p> - - -<h3><i>Arithmetic of Approximate Quantities.</i></h3> - -<p>Considering that almost all the quantities which we -treat in physical and social science are approximate only, -it seems desirable that attention should be paid in the -teaching of arithmetic to the correct interpretation and -treatment of approximate numerical statements. We seem -to need notation for expressing the approximateness or -exactness of decimal numbers. The fraction ·025 may -mean either precisely one 40th part, or it may mean -anything between ·0245 and ·0255. I propose that when -a decimal fraction is completely and exactly given, a -<i>small cipher</i> or circle should be added to indicate that -there is nothing more to come, as in ·025◦. When the -first figure of the decimals rejected is 5 or more, the first -figure retained should be raised by a unit, according to a -rule approved by De Morgan, and now generally recognised. -To indicate that the fraction thus retained is more -than the truth, a point has been placed over the last figure -in some tables of logarithms; but a similar point is used -to denote the period of a repeating decimal, and I should -therefore propose to employ a colon <i>after</i> the figure; thus -·025: would mean that the true quantity lies between -·0245° and ·025° inclusive of the lower but not the higher -limit. When the fraction is less than the truth, two dots -might be placed horizontally as in 025.. which would -mean anything between ·025° and ·0255° not inclusive.</p> - -<p>When approximate numbers are added, subtracted, multiplied, -or divided, it becomes a matter of some complexity -to determine the degree of accuracy of the result. There -are few persons who could assert off-hand that the sum -of the approximate numbers 34·70, 52·693, 80·1, is 167·5 -<i>within less than</i> ·07. Mr. Sandeman has traced out the -rules of approximate arithmetic in a very thorough manner, -and his directions are worthy of careful attention.<a id="FNanchor_394" href="#Footnote_394" class="fnanchor">394</a> The -third part of Sonnenschein and Nesbitt’s excellent book -on arithmetic<a id="FNanchor_395" href="#Footnote_395" class="fnanchor">395</a> describes fully all kinds of approximate -calculations, and shows both how to avoid needless labour<span class="pagenum" id="Page_482">482</span> -and how to take proper account of inaccuracy in operating -with approximate decimal fractions. A simple investigation -of the subject is to be found in Sonnet’s <i>Algèbre -Elémentaire</i> (Paris, 1848) chap. xiv., “Des Approximations -Absolues et Relatives.” There is also an American work -on the subject.<a id="FNanchor_396" href="#Footnote_396" class="fnanchor">396</a></p> - -<p>Although the accuracy of measurement has so much -advanced since the time of Leslie, it is not superfluous to -repeat his protest against the unfairness of affecting by a -display of decimal fractions a greater degree of accuracy -than the nature of the case requires and admits.<a id="FNanchor_397" href="#Footnote_397" class="fnanchor">397</a> I have -known a scientific man to register the barometer to a -second of time when the nearest quarter of an hour would -have been amply sufficient. Chemists often publish results -of analysis to the ten-thousandth or even the millionth -part of the whole, when in all probability the processes -employed cannot be depended on beyond the hundredth -part. It is seldom desirable to give more than one place -of figures of uncertain amount; but it must be allowed -that a nice perception of the degree of accuracy possible -and desirable is requisite to save misapprehension and -needless computation on the one hand, and to secure all -attainable exactness on the other hand.</p> - -<hr class="chap x-ebookmaker-drop"> - -<div class="chapter"> -<p><span class="pagenum" id="Page_483">483</span></p> -<h2 class="nobreak" id="CHAPTER_XXII">CHAPTER XXII.<br> - -<span class="title">QUANTITATIVE INDUCTION.</span></h2> -</div> - -<p class="ti0">We have not yet formally considered any processes -of reasoning which have for their object to disclose laws -of nature expressed in quantitative equations. We have -been inquiring into the modes by which a phenomenon -may be measured, and, if it be a composite phenomenon, -may be resolved, by the aid of several measurements, into -its component parts. We have also considered the precautions -to be taken in the performance of observations -and experiments in order that we may know what phenomena -we really do measure, but we must remember that, -no number of facts and observations can by themselves -constitute science. Numerical facts, like other facts, are -but the raw materials of knowledge, upon which our -reasoning faculties must be exerted in order to draw -forth the principles of nature. It is by an inverse process -of reasoning that we can alone discover the mathematical -laws to which varying quantities conform. By well-conducted -experiments we gain a series of values of a -variable, and a corresponding series of values of a variant, -and we now want to know what mathematical function -the variant is as regards the variable. In the usual progress -of a science three questions will have to be answered -as regards every important quantitative phenomenon:—</p> - -<p class="ti1">(1) Is there any constant relation between a variable -and a variant?</p> - -<p class="ti1">(2) What is the empirical formula expressing this relation?</p> - -<p class="ti1">(3) What is the rational formula expressing the law of -nature involved?</p> - -<p><span class="pagenum" id="Page_484">484</span></p> - - -<h3><i>Probable Connection of Varying Quantities.</i></h3> - -<p>We find it stated by Mill,<a id="FNanchor_398" href="#Footnote_398" class="fnanchor">398</a> that “Whatever phenomenon -varies in any manner whenever another phenomenon -varies in some particular manner, is either a cause -or an effect of that phenomenon, or is connected with it -through some fact of causation.” This assertion may be -considered true when it is interpreted with sufficient -caution; but it might otherwise lead us into error. There -is nothing whatever in the nature of things to prevent the -existence of two variations which should apparently follow -the same law, and yet have no connection with each other. -One binary star might be going through a revolution -which, so far as we could tell, was of equal period with -that of another binary star, and according to the above -rule the motion of one would be the cause of the motion -of the other, which would not be really the case. Two -astronomical clocks might conceivably be made so nearly -perfect that, for several years, no difference could be detected, -and we might then infer that the motion of one -clock was the cause or effect of the motion of the other. -This matter requires careful discrimination. We must -bear in mind that the continuous quantities of space, -time, force, &c., which we measure, are made up of an -infinite number of infinitely small units. We may then -meet with two variable phenomena which follow laws -so nearly the same, that in no part of the variations open -to our observation can any discrepancy be discovered. -I grant that if two clocks could be shown to have kept -<i>exactly</i> the same time during any finite interval, the probability -would become infinitely high that there was a -connection between their motions. But we can never -absolutely prove such coincidences to exist. Allow that -we may observe a difference of one-tenth of a second in -their time, yet it is possible that they were independently -regulated so as to go together within less than that -quantity of time. In short, it would require either an infinitely -long time of observation, or infinitely acute powers -of measuring discrepancy, to decide positively whether -two clocks were or were not in relation with each other.</p> -<p><span class="pagenum" id="Page_485">485</span></p> -<p>A similar question actually occurs in the case of the -moon’s motion. We have no record that any other portion -of the moon was ever visible to men than such as we -now see. This fact sufficiently proves that within the -historical period the rotation of the moon on its own axis -has coincided with its revolutions round the earth. Does -this coincidence prove a relation of cause and effect to -exist? The answer must be in the negative, because -there might have been so slight a discrepancy between -the motions that there has not yet been time to produce -any appreciable effect. There may nevertheless be a high -probability of connection.</p> - -<p>The whole question of the relation of quantities thus -resolves itself into one of probability. When we can -only rudely measure a quantitative result, we can assign -but slight importance to any correspondence. Because -the brightness of two stars seems to vary in the same -manner, there is no considerable probability that they have -any relation with each other. Could it be shown that -their periods of variation were the same to infinitely -small quantities it would be certain, that is infinitely probable, -that they were connected, however unlikely this -might be on other grounds. The general mode of estimating -such probabilities is identical with that applied -to other inductive problems. That any two periods of -variation should by chance become <i>absolutely equal</i> is infinitely -improbable; hence if, in the case of the moon or -other moving bodies, we could prove absolute coincidence -we should have certainty of connection.<a id="FNanchor_399" href="#Footnote_399" class="fnanchor">399</a> With approximate -measurements, which alone are within our power, we must -hope for approximate certainty at the most.</p> - -<p>The principles of inference and probability, according -to which we treat causes and effects varying in amount, -are exactly the same as those by which we treated simple -experiments. Continuous quantity, however, affords us -an infinitely more extensive sphere of observation, because -every different amount of cause, however little different, -ought to be followed by a different amount of effect. -If we can measure temperature to the one-hundredth part -of a degree centigrade, then between 0° and 100° we have<span class="pagenum" id="Page_486">486</span> -10,000 possible trials. If the precision of our measurements -is increased, so that the one-thousandth part of a -degree can be appreciated, our trials may be increased -tenfold. The probability of connection will be proportional -to the accuracy of our measurements.</p> - -<p>When we can vary the quantity of a cause at will it -is easy to discover whether a certain effect is due to that -cause or not. We can then make as many irregular -changes as we like, and it is quite incredible that the -supposed effect should by chance go through exactly the -corresponding series of changes except by dependence. -If we have a bell ringing <i>in vacuo</i>, the sound increases as -we let in the air, and it decreases again as we exhaust the -air. Tyndall’s singing flames evidently obeyed the directions -of his own voice; and Faraday when he discovered -the relation of magnetism and light found that, by making -or breaking or reversing the current of the electro-magnet, -he had complete command over a ray of light, proving -beyond all reasonable doubt the dependence of cause and -effect. In such cases it is the perfect coincidence in time -between the change in the effect and that in the cause -which raises a high improbability of casual coincidence.</p> - -<p>It is by a simple case of variation that we infer the -existence of a material connection between two bodies -moving with exactly equal velocity, such as the locomotive -engine and the train which follows it. Elaborate observations -were requisite before astronomers could all be -convinced that the red hydrogen flames seen during solar -eclipses belonged to the sun, and not to the moon’s atmosphere -as Flamsteed assumed. As early as 1706, Stannyan -noticed a blood-red streak in an eclipse which he witnessed -at Berne, and he asserted that it belonged to the sun; -but his opinion was not finally established until photographs -of the eclipse in 1860, taken by Mr. De la Rue, -showed that the moon’s dark body gradually covered the -red prominences on one side, and uncovered those on the -other; in short, that these prominences moved precisely as -the sun moved, and not as the moon moved.</p> - -<p>Even when we have no means of accurately measuring -the variable quantities we may yet be convinced of their -connection, if one always varies perceptibly at the same -time as the other. Fatigue increases with exertion;<span class="pagenum" id="Page_487">487</span> -hunger with abstinence from food; desire and degree of -utility decrease with the quantity of commodity consumed. -We know that the sun’s heating power depends -upon his height of the sky; that the temperature of the -air falls in ascending a mountain; that the earth’s crust -is found to be perceptibly warmer as we sink mines into -it; we infer the direction in which a sound comes from -the change of loudness as we approach or recede. The -facility with which we can time after time observe the -increase or decrease of one quantity with another sufficiently -shows the connection, although we may be unable -to assign any precise law of relation. The probability -in such cases depends upon frequent coincidence in time.</p> - - -<h3><i>Empirical Mathematical Laws.</i></h3> - -<p>It is important to acquire a clear comprehension of the -part which is played in scientific investigation by empirical -formulæ and laws. If we have a table containing -certain values of a variable and the corresponding values -of the variant, there are mathematical processes by which -we can infallibly discover a mathematical formula yielding -numbers in more or less exact agreement with the -table. We may generally assume that the quantities will -approximately conform to a law of the form</p> - -<div class="ml5em"> -<i>y</i> = A + B<i>x</i> + C<i>x</i><sup>2</sup>, -</div> - -<p class="ti0">in which <i>x</i> is the variable and <i>y</i> the variant. We can -then select from the table three values of <i>y</i>, and the corresponding -values of <i>x</i>; inserting them in the equation, -we obtain three equations by the solution of which we -gain the values of A, B, and C. It will be found as a -general rule that the formula thus obtained yields the -other numbers of the table to a considerable degree of -approximation.</p> - -<p>In many cases even the second power of the variable -will be unnecessary; Regnault found that the results -of his elaborate inquiry into the latent heat of steam at -different pressures were represented with sufficient accuracy -by the empirical formula</p> - -<div class="ml5em"> -λ = 606·5 + 0·305 <i>t</i>,<br> -</div> - -<p class="ti0">in which λ is the total heat of the steam, and <i>t</i> - the temperature.<span class="pagenum" id="Page_488">488</span><a id="FNanchor_400" href="#Footnote_400" class="fnanchor">400</a> -In other cases it may be requisite to include -the third power of the variable. Thus physicists assume -the law of the dilatation of liquids to be of the form</p> - -<div class="ml5em"> -δ<sub>t</sub> = <i>at</i> + <i>bt</i><sup>2</sup> + <i>ct</i><sup>3</sup>, -</div> - -<p class="ti0">and they calculate from results of observation the values -of the three constants <i>a</i>, <i>b</i>, <i>c</i>, which are usually small -quantities not exceeding one-hundredth part of a unit, -but requiring to be determined with great accuracy.<a id="FNanchor_401" href="#Footnote_401" class="fnanchor">401</a> -Theoretically speaking, this process of empirical representation -might be applied with any degree of accuracy; -we might include still higher powers in the formula, and -with sufficient labour obtain the values of the constants, -by using an equal number of experimental results. The -method of least squares may also be employed to obtain -the most probable values of the constants.</p> - -<p>In a similar manner all periodic variations may be represented -with any required degree of accuracy by formulæ -involving the sines and cosines of angles and their multiples. -The form of any tidal or other wave may thus be -expressed, as Sir G. B. Airy has explained.<a id="FNanchor_402" href="#Footnote_402" class="fnanchor">402</a> Almost all -the phenomena registered by meteorologists are periodic -in character, and when freed from disturbing causes may -be embodied in empirical formulæ. Bessel has given a -rule by which from any regular series of observations we -may, on the principle of the method of least squares, -calculate out with a moderate amount of labour a formula -expressing the variation of the quantity observed, in the -most probable manner. In meteorology three or four -terms are usually sufficient for representing any periodic -phenomenon, but the calculation might be carried to any -higher degree of accuracy. As the details of the process -have been described by Herschel in his treatise on -Meteorology,<a id="FNanchor_403" href="#Footnote_403" class="fnanchor">403</a> I need not further enter into them.</p> - -<p>The reader might be tempted to think that in these -processes of calculation we have an infallible method of -discovering inductive laws, and that my previous statements -(Chap. VII.) as to the purely tentative and inverse -character of the inductive process are negatived. Were<span class="pagenum" id="Page_489">489</span> -there indeed any general method of inferring laws from -facts it would overturn my statement, but it must be -carefully observed that these empirical formulæ do not -coincide with natural laws. They are only approximations -to the results of natural laws founded upon the general -principles of approximation. It has already been pointed -out that however complicated be the nature of a curve, -we may examine so small a portion of it, or we may examine -it with such rude means of measurement, that its -divergence from an elliptic curve will not be apparent. -As a still ruder approximation a portion of a straight line -will always serve our purpose; but if we need higher precision -a curve of the third or fourth degree will almost -certainly be sufficient. Now empirical formulæ really represent -these approximate curves, but they give us no -information as to the precise nature of the curve itself to -which we are approximating. We do not learn what function -the variant is of the variable, but we obtain another -function which, within the bounds of observation, gives -nearly the same values.</p> - - -<h3><i>Discovery of Rational Formulæ.</i></h3> - -<p>Let us now proceed to consider the modes in which -from numerical results we can establish the actual relation -between the quantity of the cause and that of the effect. -What we want is a <i>rational</i> formula or function, which -will exhibit the <i>reason</i> or exact nature and origin of the -law in question. There is no word more frequently used -by mathematicians than the word <i>function</i>, and yet it -is difficult to define its meaning with perfect accuracy. -Originally it meant performance or execution, being equivalent -to the Greek λειτουργία or τέλεσμα. Mathematicians -at first used it to mean <i>any power of a quantity</i>, but -afterwards generalised it so as to include “any quantity -formed in any manner whatsoever from another quantity.”<a id="FNanchor_404" href="#Footnote_404" class="fnanchor">404</a> -Any quantity, then, which depends upon and varies with -another quantity may be called a function of it, and -either may be considered a function of the other.</p> - -<p>Given the quantities, we want the function of which<span class="pagenum" id="Page_490">490</span> -they are the values. Simple inspection of the numbers -cannot as a general rule disclose the function. In an -earlier chapter (p. <a href="#Page_124">124</a>) I put before the reader certain -numbers, and requested him to point out the law which -they obey, and the same question will have to be asked -in every case of quantitative induction. There are perhaps -three methods, more or less distinct, by which we -may hope to obtain an answer:</p> - -<p class="ti1">(1) By purely haphazard trial.</p> - -<p class="ti1">(2) By noting the general character of the variation of -the quantities, and trying by preference functions which -give a similar form of variation.</p> - -<p class="ti1">(3) By deducing from previous knowledge the form of -the function which is most likely to suit.</p> - -<p>Having numerical results we are always at liberty -to invent any kind of mathematical formula we like, and -then try whether, by the suitable selection of values for -the unknown constant quantities, we can make it give the -required results. If ever we fall upon a formula which -does so, to a fair degree of approximation, there is a presumption -in favour of its being the true function, although -there is no certainty whatever in the matter. In this way -I discovered a simple mathematical law which closely -agreed with the results of my experiments on muscular -exertion. This law was afterwards shown by Professor -Haughton to be the true rational law according to his -theory of muscular action.<a id="FNanchor_405" href="#Footnote_405" class="fnanchor">405</a></p> - -<p>But the chance of succeeding in this manner is small. -The number of possible functions is infinite, and even the -number of comparatively simple functions is so large -that the probability of falling upon the correct one by -mere chance is very slight. Even when we obtain the -law it is by a deductive process, not by showing that the -numbers give the law, but that the law gives the numbers.</p> - -<p>In the second way, we may, by a survey of the -numbers, gain a general notion of the kind of law they -are likely to obey, and we may be much assisted in this<span class="pagenum" id="Page_491">491</span> -process by drawing them out in the form of a curve. We -can in this way ascertain with some probability whether -the curve is likely to return into itself, or whether it has -infinite branches; whether such branches are asymptotic, -that is, approach infinitely towards straight lines; whether -it is logarithmic in character, or trigonometric. This -indeed we can only do if we remember the results of previous -investigations. The process is still inversely deductive, -and consists in noting what laws give particular curves, -and then inferring inversely that such curves belong to -such laws. If we can in this way discover the class of -functions to which the required law belongs, our chances -of success are much increased, because our haphazard -trials are now reduced within a narrower sphere. But, -unless we have almost the whole curve before us, the -identification of its character must be a matter of great -uncertainty; and if, as in most physical investigations, -we have a mere fragment of the curve, the assistance -given would be quite illusory. Curves of almost any -character can be made to approximate to each other for -a limited extent, so that it is only by a kind of <i>divination</i> -that we fall upon the actual function, unless we have -theoretical knowledge of the kind of function applicable -to the case.</p> - -<p>When we have once obtained what we believe to be the -correct form of function, the remainder of the work is -mere mathematical computation to be performed infallibly -according to fixed rules,<a id="FNanchor_406" href="#Footnote_406" class="fnanchor">406</a> which include those employed -in the determination of empirical formulæ (p. <a href="#Page_487">487</a>). The -function will involve two or three or more unknown -constants, the values of which we need to determine by -our experimental results. Selecting some of our results -widely apart and nearly equidistant, we form by means -of them as many equations as there are constant quantities -to be determined. The solution of these equations will -then give us the constants required, and having now the -actual function we can try whether it gives with sufficient -accuracy the remainder of our experimental results. If -not, we must either make a new selection of results to -give a new set of equations, and thus obtain a new set of -values for the constants, or we must acknowledge that our<span class="pagenum" id="Page_492">492</span> -form of function has been wrongly chosen. If it appears -that the form of function has been correctly ascertained, -we may regard the constants as only approximately accurate -and may proceed by the Method of Least Squares (p. <a href="#Page_393">393</a>) -to determine the most probable values as given by the -whole of the experimental results.</p> - -<p>In most cases we shall find ourselves obliged to fall -back upon the third mode, that is, anticipation of the -form of the law to be expected on the ground of previous -knowledge. Theory and analogical reasoning must be our -guides. The general nature of the phenomenon will often -indicate the kind of law to be looked for. If one form of -energy or one kind of substance is being converted into -another, we may expect the law of direct simple proportion. -In one distinct class of cases the effect already produced -influences the amount of the ensuing effect, as for instance -in the cooling of a heated body, when the law will be of -an exponential form. When the direction of a force influences -its action, trigonometrical functions enter. Any -influence which spreads freely through tridimensional -space will be subject to the law of the inverse square -of the distance. From such considerations we may sometimes -arrive deductively and analogically at the general -nature of the mathematical law required.</p> - - -<h3><i>The Graphical Method.</i></h3> - -<p>In endeavouring to discover the mathematical law -obeyed by experimental results it is often desirable to -call in the aid of space-representations. Every equation -involving two variable quantities corresponds to some kind -of plane curve, and every plane curve may be represented -symbolically in an equation containing two unknown -quantities. Now in an experimental research we obtain -a number of values of the variant corresponding to an -equal number of values of the variable; but all the -numbers are affected by more or less error, and the values -of the variable will often be irregularly disposed. Even -if the numbers were absolutely correct and disposed at -regular intervals, there is, as we have seen, no direct mode -of discovering the law, but the difficulty of discovery is much -increased by the uncertainty and irregularity of the results.</p> - -<p><span class="pagenum" id="Page_493">493</span></p> - -<p>Under such circumstances, the best mode of proceeding -is to prepare a paper divided into equal rectangular spaces, -a convenient size for the spaces being one-tenth of an -inch square. The values of the variable being marked -off on the lowest horizontal line, a point is marked for -each corresponding value of the variant perpendicularly -above that of the variable, and at such a height as corresponds -to the value of the variant.</p> - -<p>The exact scale of the drawing is not of much importance, -but it may require to be adjusted according to -circumstances, and different values must often be attributed -to the upright and horizontal divisions, so as to -make the variations conspicuous but not excessive. If -a curved line be drawn through all the points or ends -of the ordinates, it will probably exhibit irregular inflections, -owing to the errors which affect the numbers. But, -when the results are numerous, it becomes apparent which -results are more divergent than others, and guided by a -so-called <i>sense of continuity</i>, it is possible to trace a line -among the points which will approximate to the true law -more nearly than the points themselves. The accompanying -figure sufficiently explains itself.</p> - -<figure class="figcenter illowp100" id="p493" style="max-width: 26.875em;"> - <img class="w100" src="images/p493.jpg" alt=""> -</figure> - -<p>Perkins employed this graphical method with much -care in exhibiting the results of his experiments on the -compression of water.<a id="FNanchor_407" href="#Footnote_407" class="fnanchor">407</a> The numerical results were marked<span class="pagenum" id="Page_494">494</span> -upon a sheet of paper very exactly ruled at intervals of -one-tenth of an inch, and the original marks were left -in order that the reader might judge of the correctness of -the curve drawn, or choose another for himself. Regnault -carried the method to perfection by laying off the points -with a screw dividing engine;<a id="FNanchor_408" href="#Footnote_408" class="fnanchor">408</a> and he then formed a -table of results by drawing a continuous curve, and -measuring its height for equidistant values of the variable. -Not only does a curve drawn in this manner enable us to -infer numerical results more free from accidental errors -than any of the numbers obtained directly from experiment, -but the form of the curve sometimes indicates the class of -functions to which our results belong.</p> - -<p>Engraved sheets of paper prepared for the drawing of -curves may be obtained from Mr. Stanford at Charing -Cross, Messrs. W. and A. K. Johnston, of London and -Edinburgh, Waterlow and Sons, Letts and Co., and probably -other publishers. When we do not require great accuracy, -paper ruled by the common machine-ruler into equal -squares of about one-fifth or one-sixth of an inch square -will serve well enough. I have met with engineers’ and -surveyors’ memorandum books ruled with one-twelfth inch -squares. When a number of curves have to be drawn, I -have found it best to rule a good sheet of drawing paper -with lines carefully adjusted at the most convenient -distances, and then to prick the points of the curve -through it upon another sheet fixed underneath. In this -way we obtain an accurate curve upon a blank sheet, -and need only introduce such division lines as are requisite -to the understanding of the curve.</p> - -<p>In some cases our numerical results will correspond, -not to the height of single ordinates, but to the area of -the curve between two ordinates, or the average height of -ordinates between certain limits. If we measure, for instance, -the quantities of heat absorbed by water when -raised in temperature from 0° to 5°, from 5° to 10°, and so -on, these quantities will really be represented by <i>areas</i> of -the curve denoting the specific heat of water; and since -the specific heat varies continuously between every two -points of temperature, we shall not get the correct curve<span class="pagenum" id="Page_495">495</span> -by simply laying off the quantities of heat at the mean temperatures, -namely <span class="nowrap">2 <span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span>°, - and <span class="nowrap">7 <span class="fraction"><span class="fnum">1</span><span class="bar">/</span><span class="fden">2</span></span></span>°, and so on. Lord Rayleigh -has shown that if we have drawn such an incorrect curve, -we can with little trouble correct it by a simple geometrical -process, and obtain to a close approximation the -true ordinates instead of those denoting areas.<a id="FNanchor_409" href="#Footnote_409" class="fnanchor">409</a></p> - - -<h3><i>Interpolation and Extrapolation.</i></h3> - -<p>When we have by experiment obtained two or more -numerical results, and endeavour, without further experiment, -to calculate intermediate results, we are said to -<i>interpolate</i>. If we wish to assign by reasoning results -lying beyond the limits of experiment, we may be said, -using an expression of Sir George Airy, to <i>extrapolate</i>. -These two operations are the same in principle, but differ -in practicability. It is a matter of great scientific importance -to apprehend precisely how far we can practise -interpolation or extrapolation, and on what grounds we -proceed.</p> - -<p>In the first place, if the interpolation is to be more than -empirical, we must have not only the experimental results, -but the laws which they obey—we must in fact go through -the complete process of scientific investigation. Having -discovered the laws of nature applying to the case, and -verified them by showing that they agree with the experiments -in question, we are then in a position to anticipate -the results of similar experiments. Our knowledge even -now is not certain, because we cannot completely prove -the truth of any assumed law, and we cannot possibly -exhaust all the circumstances which may affect the result. -At the best then our interpolations will partake of the -want of certainty and precision attaching to all our knowledge -of nature. Yet, having the supposed laws, our results -will be as sure and accurate as any we can attain to. But -such a complete procedure is more than we commonly -mean by interpolation, which usually denotes some method -of estimating in a merely approximate manner the results<span class="pagenum" id="Page_496">496</span> -which might have been expected independently of a theoretical -investigation.</p> - -<p>Regarded in this light, interpolation is in reality an indeterminate -problem. From given values of a function it is -impossible to determine that function; for we can invent -an infinite number of functions which will give those -values if we are not restricted by any conditions, just as -through a given series of points we can draw an infinite -number of curves, if we may diverge between or beyond -the points into bends and cusps as we think fit.<a id="FNanchor_410" href="#Footnote_410" class="fnanchor">410</a> In interpolation -we must in fact be guided more or less by <i>à priori</i> -considerations; we must know, for instance, whether or not -periodical fluctuations are to be expected. Supposing that -the phenomenon is non-periodic, we proceed to assume that -the function can be expressed in a limited series of the -powers of the variable. The number of powers which can -be included depends upon the number of experimental -results available, and must be at least one less than this -number. By processes of calculation, which have been -already alluded to in the section on empirical formulæ, we -then calculate the coefficients of the powers, and obtain an -empirical formula which will give the required intermediate -results. In reality, then, we return to the methods treated -under the head of approximation and empirical formulæ; -and interpolation, as commonly understood, consists in -assuming that a curve of simple character is to pass through -certain determined points. If we have, for instance, two -experimental results, and only two, we assume that the -curve is a straight line; for the parabolas which can be -passed through two points are infinitely various in magnitude, -and quite indeterminate. One straight line alone -can pass through two points, and it will have an equation -of the form, <i>y</i> = <i>mx</i> + <i>n</i>, the constant quantities of which -can be determined from two results. Thus, if the two -values for <i>x</i>, 7 and 11, give the values for <i>y</i>, 35 and 53, -the solution of two equations gives <i>y</i> = 4·5 × <i>x</i> + 3·5 -as the equation, and for any other value of <i>x</i>, for instance -10, we get a value of <i>y</i>, that is 48·5. When we take -a mean value of <i>x</i>, namely 9, this process yields a simple -mean result, namely 44. Three experimental results<span class="pagenum" id="Page_497">497</span> -being given, we assume that they fall upon a portion of a -parabola and algebraic calculation gives the position of -any intermediate point upon the parabola. Concerning -the process of interpolation as practised in the science -of meteorology the reader will find some directions in the -French edition of Kaëmtz’s Meteorology.<a id="FNanchor_411" href="#Footnote_411" class="fnanchor">411</a></p> - -<p>When we have, either by direct experiment or by -the use of a curve, a series of values of the variant for -equidistant values of the variable, it is instructive to take -the differences between each value of the variant and the -next, and then the differences between those differences, -and so on. If any series of differences approaches closely -to zero it is an indication that the numbers may be -correctly represented by a finite empirical formula; if -the <i>n</i>th differences are zero, then the formula will contain -only the first <i>n</i> - 1 powers of the variable. Indeed we -may sometimes obtain by the calculus of differences a -correct empirical formula; for if <i>p</i> be the first term of -the series of values, and Δ<i>p</i>, Δ<sup>2</sup><i>p</i>, Δ<sup>3</sup><i>p</i>, be the first number -in each column of differences, then the <i>m</i>th term of -the series of values will be</p> - -<div class="ml5em"> -<i>p</i> + <i>m</i>Δ<i>p</i> + <i>m</i> <span class="nowrap"><span class="fraction2"><span class="fnum2"><i>m</i> – 1</span><span class="bar">/</span><span class="fden2">2</span></span></span> Δ<sup>2</sup><i>p</i> - + <i>m</i> <span class="nowrap"><span class="fraction2"><span class="fnum2"><i>m</i> – 1</span><span class="bar">/</span><span class="fden2">2</span></span></span> <span class="nowrap"><span class="fraction2"><span class="fnum2"><i>m</i> – 2</span><span class="bar">/</span><span class="fden2">3</span></span></span> Δ<sup>3</sup><i>p</i> + &c. -</div> - -<p>A closely equivalent but more practicable formula for -interpolation by differences, as devised by Lagrange, will -be found in Thomson and Tait’s <i>Elements of Natural -Philosophy</i>, p. 115.</p> - -<p>If no column of differences shows any tendency to -become zero throughout, it is an indication that the law -is of a more complicated, for instance of an exponential -character, so that it requires different treatment. Dr. J. -Hopkinson has suggested a method of arithmetical interpolation,<a id="FNanchor_412" href="#Footnote_412" class="fnanchor">412</a> -which is intended to avoid much that is -arbitrary in the graphical method. His process will yield -the same results in all hands.</p> - -<p>So far as we can infer the results likely to be obtained -by variations beyond the limits of experiment, we must<span class="pagenum" id="Page_498">498</span> -proceed upon the same principles. If possible we must -detect the exact laws in action, and then trust to them as -a guide when we have no experience. If not, an empirical -formula of the same character as those employed in interpolation -is our only resource. But to extend our inference -far beyond the limits of experience is exceedingly unsafe. -Our knowledge is at the best only approximate, and -takes no account of small tendencies. Now it usually -happens that tendencies small within our limits of observation -become perceptible or great under extreme -circumstances. When the variable in our empirical -formula is small, we are justified in overlooking the higher -powers, and taking only two or three lower powers. But -as the variable increases, the higher powers gain in importance, -and in time yield the principal part of the value of -the function.</p> - -<p>This is no mere theoretical inference. Excepting the -few primary laws of nature, such as the law of gravity, -of the conservation of energy, &c., there is hardly any -natural law which we can trust in circumstances widely -different from those with which we are practically acquainted. -From the expansion or contraction, fusion or -vaporisation of substances by heat at the surface of the -earth, we can form a most imperfect notion of what would -happen near the centre of the earth, where the pressure -almost infinitely exceeds anything possible in our experiments. -The physics of the earth give us a feeble, and probably -a misleading, notion of a body like the sun, in -which an inconceivably high temperature is united with an -inconceivably high pressure. If there are in the realms of -space nebulæ consisting of incandescent and unoxidised -vapours of metals and other elements, so highly heated -perhaps that chemical composition is out of the question, -we are hardly able to treat them as subjects of scientific -inference. Hence arises the great importance of experiments -in which we investigate the properties of substances -under extreme circumstances of cold or heat, density or -rarity, intense electric excitation, &c. This insecurity -in extending our inferences arises from the approximate -character of our measurements. Had we the power of -appreciating infinitely small quantities, we should by -the principle of continuity discover some trace of every<span class="pagenum" id="Page_499">499</span> -change which a substance could undergo under unattainable -circumstances. By observing, for instance, the tension -of aqueous vapour between 0° and 100° C., we ought -theoretically to be able to infer its tension at every other -temperature; but this is out of the question practically -because we cannot really ascertain the law precisely between -those temperatures.</p> - -<p>Many instances might be given to show that laws -which appear to represent correctly the results of experiments -within certain limits altogether fail beyond those -limits. The experiments of Roscoe and Dittmar, on the -absorption of gases in water<a id="FNanchor_413" href="#Footnote_413" class="fnanchor">413</a> afford interesting illustrations, -especially in the case of hydrochloric acid, the quantity of -which dissolved in water under different pressures follows -very closely a linear law of variation, from which however -it diverges widely at low pressures.<a id="FNanchor_414" href="#Footnote_414" class="fnanchor">414</a> Herschel, having -deduced from observations of the double star γ Virginis -an elliptic orbit for the motion of one component round -the centre of gravity of both, found that for a time -the motion of the star agreed very well with this orbit. -Nevertheless divergence began to appear and after a time -became so great that an entirely new orbit, of more than -double the dimensions of the old one, had ultimately to be -adopted.<a id="FNanchor_415" href="#Footnote_415" class="fnanchor">415</a></p> - - -<h3><i>Illustrations of Empirical Quantitative Laws.</i></h3> - -<p>Although our object in quantitative inquiry is to discover -the exact or rational formulæ, expressing the laws which -apply to the subject, it is instructive to observe in how -many important branches of science, no precise laws have -yet been detected. The tension of aqueous vapour at -different temperatures has been determined by a succession -of eminent experimentalists—Dalton, Kaëmtz, Dulong, -Arago, Magnus, and Regnault—and by the last mentioned -the measurements were conducted with extraordinary care.<span class="pagenum" id="Page_500">500</span> -Yet no incontestable general law has been established. -Several functions have been proposed to express the -elastic force of the vapour as depending on the temperature. -The first form is that of Young, namely -F = (<i>a</i> + <i>b t</i>)<sup>m</sup>, in which <i>a</i>, <i>b</i>, and <i>m</i> are unknown quantities -to be determined by observation. Roche proposed, -on theoretical grounds, a complicated formula of an exponential -form, and a third form of function is that of -Biot,<a id="FNanchor_416" href="#Footnote_416" class="fnanchor">416</a> as follows—log F = <i>a</i> + <i>b</i>α<sup>t</sup> + <i>c</i>β<sup>t</sup>. I mention -these formulæ, because they well illustrate the feeble -powers of empirical inquiry. None of the formulæ can be -made to correspond closely with experimental results, and -the two last forms correspond almost equally well. There is -very little probability that the real law has been reached, -and it is unlikely that it will be discovered except by -deduction from mechanical theory.</p> - -<p>Much ingenious labour has been spent upon the discovery -of some general law of atmospheric refraction. -Tycho Brahe and Kepler commenced the inquiry: Cassini -first formed a table of refractions, calculated on theoretical -grounds: Newton entered into some profound investigations -upon the subject: Brooke Taylor, Bouguer, Simpson, -Bradley, Mayer, and Kramp successively attacked the -question, which is of the highest practical importance -as regards the correction of astronomical observations. -Laplace next laboured on the subject without exhausting -it, and Brinkley and Ivory have also treated it. The true -law is yet undiscovered. A closely connected problem, -that regarding the relation between the pressure and -elevation in different strata of the atmosphere, has received -the attention of a long succession of physicists and was -most carefully investigated by Laplace. Yet no invariable -and general law has been detected. The same may be -said concerning the law of human mortality; abundant -statistics on this subject are available, and many hypotheses -more or less satisfactory have been put forward as to the -form of the curve of mortality, but it seems to be impossible -to discover more than an approximate law.</p> - -<p>It may perhaps be urged that in such subjects no single -invariable law can be expected. The atmosphere may be<span class="pagenum" id="Page_501">501</span> -divided into several variable strata which by their unconnected -changes frustrate the exact calculations of astronomers. -Human life may be subject at different ages to -a succession of different influences incapable of reduction -under any one law. The results observed may in fact be -aggregates of an immense number of separate results each -governed by its own separate laws, so that the subjects -may be complicated beyond the possibility of complete -resolution by empirical methods. This is certainly true -of the mathematical functions which must some time or -other be introduced into the science of political economy.</p> - - -<h3><i>Simple Proportional Variation.</i></h3> - -<p>When we first treat numerical results in any novel kind -of investigation, our impression will probably be that one -quantity varies in <i>simple proportion</i> to another, so as to -obey the law <i>y</i> = <i>mx</i> + <i>n</i>. We must learn to distinguish -carefully between the cases where this proportionality is -really, and where it is only apparently true. In considering -the principles of approximation we found that a -small portion of any curve will appear to be a straight line. -When our modes of measurement are comparatively rude, -we must expect to be unable to detect the curvature. -Kepler made meritorious attempts to discover the law of -refraction, and he approximated to it when he observed -that the angles of incidence and refraction <i>if small</i> bear -a constant ratio to each other. Angles when small are -nearly as their sines, so that he reached an approximate -result of the true law. Cardan assumed, probably as a -mere guess, that the force required to sustain a body on -an inclined plane was simply proportional to the angle of -elevation of the plane. This is approximately the case -when the angle is small, but in reality the law is much -more complicated, the power required being proportional -to the sine of the angle. The early thermometer-makers -were unaware whether the expansion of mercury was -proportional or not to the heat communicated to it, and -it is only in the present century that we have learnt it -to be not so. We now know that even gases obey the -law of uniform expansion by heat only in an approximate<span class="pagenum" id="Page_502">502</span> -manner. Until reason to the contrary is shown, we should -do well to look upon every law of simple proportion as -only provisionally true.</p> - -<p>Nevertheless many important laws of nature are in the -form of simple proportions. Wherever a cause acts in -independence of its previous effects, we may expect this -relation. An accelerating force acts equally upon a -moving and a motionless body. Hence the velocity -produced is in simple proportion to the force, and to the -duration of its uniform action. As gravitating bodies -never interfere with each other’s gravity, this force is in -direct simple proportion to the mass of each of the attracting -bodies, the mass being measured by, or proportional -to inertia. Similarly, in all cases of “direct unimpeded -action,” as Herschel has remarked,<a id="FNanchor_417" href="#Footnote_417" class="fnanchor">417</a> we may expect simple -proportion to manifest itself. In such cases the equation -expressing the relation may have the simple form <i>y</i> = <i>mx</i>.</p> - -<p>A similar relation holds true when there is conversion -of one substance or form of energy into another. The -quantity of a compound is equal to the quantity of the -elements which combine. The heat produced in friction -is exactly proportional to the mechanical energy absorbed. -It was experimentally proved by Faraday that “the chemical -power of the current of electricity is in direct proportion -to the quantity of electricity which passes.” When -an electric current is produced, the quantity of electric -energy is simply proportional to the weight of metal -dissolved. If electricity is turned into heat, there is -again simple proportion. Wherever, in fact, one thing -is but another thing with a new aspect, we may expect -to find the law of simple proportion. But it is only in -the most elementary cases that this simple relation will -hold true. Simple conditions do not, generally speaking, -produce simple results. The planets move in approximate -circles round the sun, but the apparent motions, as seen -from the earth, are very various. All those motions, again, -are summed up in the law of gravity, of no great complexity; -yet men never have been, and never will be, able -to exhaust the complications of action and reaction arising -from that law, even among a small number of planets.<span class="pagenum" id="Page_503">503</span> -We should be on our guard against a tendency to assume -that the connection of cause and effect is one of direct -proportion. Bacon reminds us of the woman in Æsop’s -fable, who expected that her hen, with a double measure -of barley, would lay two eggs a day instead of one, whereas -it grew fat, and ceased to lay any eggs at all. It is a -wise maxim that the half is often better than the whole.</p> -<hr class="chap x-ebookmaker-drop"> - -<div class="chapter"> -<p><span class="pagenum" id="Page_504">504</span></p> - -<h2 class="nobreak" id="CHAPTER_XXIII">CHAPTER XXIII.<br> - -<span class="title">THE USE OF HYPOTHESIS.</span></h2> -</div> - -<p class="ti0">If the views upheld in this work be correct, all inductive -investigation consists in the marriage of hypothesis and -experiment. When facts are in our possession, we frame -an hypothesis to explain their relations, and by the success -of this explanation is the value of the hypothesis to be -judged. In the invention and treatment of such hypotheses, -we must avail ourselves of the whole body of science -already accumulated, and when once we have obtained a -probable hypothesis, we must not rest until we have verified -it by comparison with new facts. We must endeavour by -deductive reasoning to anticipate such phenomena, especially -those of a singular and exceptional nature, as would -happen if the hypothesis be true. Out of the infinite -number of experiments which are possible, theory must -lead us to select those critical ones which are suitable for -confirming or negativing our anticipations.</p> - -<p>This work of inductive investigation cannot be guided -by any system of precise and infallible rules, like those of -deductive reasoning. There is, in fact, nothing to which -we can apply rules of method, because the laws of nature -must be in our possession before we can treat them. If -there were any rule of inductive method, it would direct -us to make an exhaustive arrangement of facts in all -possible orders. Given the specimens in a museum, we -might arrive at the best classification by going systematically -through all possible classifications, and, were we endowed -with infinite time and patience, this would be an effective -method. It is the method by which the first simple steps<span class="pagenum" id="Page_505">505</span> -are taken in an incipient branch of science. Before the dignified -name of science is applicable, some coincidences will -force themselves upon the attention. Before there was a -science of meteorology observant persons learned to associate -clearness of the atmosphere with coming rain, and a -colourless sunset with fine weather. Knowledge of this -kind is called <i>empirical</i>, as seeming to come directly from -experience; and there is a considerable portion of knowledge -which bears this character.</p> - -<p>We may be obliged to trust to the casual detection -of coincidences in those branches of knowledge where -we are deprived of the aid of any guiding notions; but -a little reflection will show the utter insufficiency of -haphazard experiment, when applied to investigations of -a complicated nature. At the best, it will be the simple -identity, or partial identity, of classes, as illustrated -in pages <a href="#Page_127">127</a> or <a href="#Page_134">134</a>, which can be thus detected. It was -pointed out that, even when a law of nature involves only -two circumstances, and there are one hundred distinct circumstances -which may possibly be connected, there will -be no less than 4,950 pairs of circumstances between -which coincidence may exist. When a law involves three -or more circumstances, the possible number of relations -becomes vastly greater. When considering the subject -of combinations and permutations, it became apparent -that we could never cope with the possible variety of -nature. An exhaustive examination of the possible metallic -alloys, or chemical compounds, was found to be out -of the question (p. <a href="#Page_191">191</a>).</p> - -<p>It is on such considerations that we can explain the -very small additions made to our knowledge by the alchemists. -Many of them were men of the greatest acuteness, -and their indefatigable labours were pursued through -many centuries. A few things were discovered by them, -but a true insight into nature, now enables chemists to -discover more useful facts in a year than were yielded by -the alchemists during many centuries. There can be no -doubt that Newton was an alchemist, and that he often -laboured night and day at alchemical experiments. But -in trying to discover the secret by which gross metals -might be rendered noble, his lofty powers of deductive -investigation were wholly useless. Deprived of all<span class="pagenum" id="Page_506">506</span> -guiding clues, his experiments were like those of all the -alchemists, purely tentative and haphazard. While his -hypothetical and deductive investigations have given us -the true system of the Universe, and opened the way in -almost all the great branches of natural philosophy, the -whole results of his tentative experiments are comprehended -in a few happy guesses, given in his celebrated -“Queries.”</p> - -<p>Even when we are engaged in apparently passive -observation of a phenomenon, which we cannot modify -experimentally, it is advantageous that our attention -should be guided by theoretical anticipations. A phenomenon -which seems simple is, in all probability, really -complex, and unless the mind is actively engaged in -looking for particular details, it is likely that the critical -circumstances will be passed over. Bessel regretted that -no distinct theory of the constitution of comets had -guided his observations of Halley’s comet;<a id="FNanchor_418" href="#Footnote_418" class="fnanchor">418</a> in attempting -to verify or refute a hypothesis, not only would there be -a chance of establishing a true theory, but if confuted, -the confutation would involve a store of useful observations.</p> - -<p>It would be an interesting work, but one which I cannot -undertake, to trace out the gradual reaction which has -taken place in recent times against the purely empirical -or Baconian theory of induction. Francis Bacon, seeing -the futility of the scholastic logic, which had long been -predominant, asserted that the accumulation of facts and -the orderly abstraction of axioms, or general laws from -them, constituted the true method of induction. Even -Bacon was not wholly unaware of the value of hypothetical -anticipation. In one or two places he incidentally -acknowledges it, as when he remarks that the subtlety of -nature surpasses that of reason, adding that “axioms abstracted -from particular facts in a careful and orderly -manner, readily suggest and mark out new particulars.”</p> - -<p>Nevertheless Bacon’s method, as far as we can gather -the meaning of the main portions of his writings, would -correspond to the process of empirically collecting facts<span class="pagenum" id="Page_507">507</span> -and exhaustively classifying them, to which I alluded. -The value of this method may be estimated historically -by the fact that it has not been followed by any of -the great masters of science. Whether we look to Galileo, -who preceded Bacon, to Gilbert, his contemporary, or -to Newton and Descartes, Leibnitz and Huyghens, his -successors, we find that discovery was achieved by the -opposite method to that advocated by Bacon. Throughout -Newton’s works, as I shall show, we find deductive -reasoning wholly predominant, and experiments are employed, -as they should be, to confirm or refute hypothetical -anticipations of nature. In my “Elementary Lessons -in Logic” (p. 258), I stated my belief that there was no -kind of reference to Bacon in Newton’s works. I have -since found that Newton does once or twice employ the -expression <i>experimentum crucis</i> in his “Opticks,” but this -is the only expression, so far as I am aware, which could -indicate on the part of Newton direct or indirect acquaintance -with Bacon’s writings.<a id="FNanchor_419" href="#Footnote_419" class="fnanchor">419</a></p> - -<p>Other great physicists of the same age were equally -prone to the use of hypotheses rather than the blind -accumulation of facts in the Baconian manner. Hooke -emphatically asserts in his posthumous work on Philosophical -Method, that the first requisite of the Natural -Philosopher is readiness at guessing the solution of phenomena -and making queries. “He ought to be very well -skilled in those several kinds of philosophy already -known, to understand their several hypotheses, suppositions, -collections, observations, &c., their various ways -of ratiocinations and proceedings, the several failings and -defects, both in their way of raising and in their way of -managing their several theories: for by this means the -mind will be somewhat more ready at guessing at the -solution of many phenomena almost at first sight, and -thereby be much more prompt at making queries, and at -tracing the subtlety of Nature, and in discovering and -searching into the true reason of things.”</p> - -<p>We find Horrocks, again, than whom no one was more<span class="pagenum" id="Page_508">508</span> -filled with the scientific spirit, telling us how he tried -theory after theory in order to discover one which was in -accordance with the motions of Mars.<a id="FNanchor_420" href="#Footnote_420" class="fnanchor">420</a> Huyghens, who -possessed one of the most perfect philosophical intellects, -followed the deductive process combined with continual -appeal to experiment, with a skill closely analogous to -that of Newton. As to Descartes and Leibnitz, they fell -into excess in the use of hypothesis, since they sometimes -adopted hypothetical reasoning to the exclusion of experimental -verification. Throughout the eighteenth century -science was supposed to be advancing by the pursuance -of the Baconian method, but in reality hypothetical -investigation was the main instrument of progress. It is -only in the present century that physicists began to recognise -this truth. So much opprobrium had been attached -by Bacon to the use of hypotheses, that we find Young -speaking of them in an apologetic tone. “The practice of -advancing general principles and applying them to particular -instances is so far from being fatal to truth in all -sciences, that when those principles are advanced on sufficient -grounds, it constitutes the essence of true philosophy;”<a id="FNanchor_421" href="#Footnote_421" class="fnanchor">421</a> -and he quotes cases in which Davy trusted -to his theories rather than his experiments.</p> - -<p>Herschel, who was both a practical physicist and an -abstract logician, entertained the deepest respect for -Bacon, and made the “Novum Organum” as far as -possible the basis of his own admirable <i>Discourse on -the Study of Natural Philosophy</i>. Yet we find him in -Chapter VII. recognising the part which the formation -and verification of theories takes in the higher and more -general investigations of physical science. J. S. Mill -carried on the reaction by describing the Deductive -Method in which ratiocination, that is deductive reasoning, -is employed for the discovery of new opportunities -of testing and verifying an hypothesis. Nevertheless -throughout the other parts of his system he -inveighed against the value of the deductive process, -and even asserted that empirical inference from particulars -to particulars is the true type of reasoning.<span class="pagenum" id="Page_509">509</span> -The irony of fate will probably decide that the most -original and valuable part of Mill’s System of Logic is -irreconcilable with those views of the syllogism and of -the nature of inference which occupy the main part of -the treatise, and are said to have effected a revolution -in logical science. Mill would have been saved from -much confusion of thought had he not failed to observe -that the inverse use of deduction constitutes induction. -In later years Professor Huxley has strongly insisted -upon the value of hypothesis. When he advocates the -use of “working hypotheses” he means no doubt that -any hypothesis is better that none, and that we cannot -avoid being guided in our observations by some hypothesis -or other. Professor Tyndall’s views as to the -use of the Imagination in the pursuit of Science put the -same truth in another light.</p> - -<p>It ought to be pointed out that Neil in his <i>Art of -Reasoning</i>, a popular but able exposition of the principles -of Logic, published in 1853, fully recognises in Chapter -XI. the value and position of hypothesis in the discovery -of truth. He endeavours to show, too (p. 109), that -Francis Bacon did not object to the use of hypothesis.</p> - -<p>The true course of inductive procedure is that which -has yielded all the more lofty results of science. It -consists in <i>Anticipating Nature</i>, in the sense of forming -hypotheses as to the laws which are probably in operation; -and then observing whether the combinations of -phenomena are such as would follow from the laws -supposed. The investigator begins with facts and ends -with them. He uses facts to suggest probable hypotheses; -deducing other facts which would happen if a particular -hypothesis is true, he proceeds to test the truth -of his notion by fresh observations. If any result prove -different from what he expects, it leads him to modify -or to abandon his hypothesis; but every new fact may -give some new suggestion as to the laws in action. -Even if the result in any case agrees with his anticipations, -he does not regard it as finally confirmatory of his -theory, but proceeds to test the truth of the theory by new -deductions and new trials.</p> - -<p>In such a process the investigator is assisted by the -whole body of science previously accumulated. He may<span class="pagenum" id="Page_510">510</span> -employ analogy, as I shall point out, to guide him in the -choice of hypotheses. The manifold connections between -one science and another give him clues to the kind of laws -to be expected, and out of the infinite number of possible -hypotheses he selects those which are, as far as can be -foreseen at the moment, most probable. Each experiment, -therefore, which he performs is that most likely to throw -light upon his subject, and even if it frustrate his first -views, it tends to put him in possession of the correct -clue.</p> - - -<h3><i>Requisites of a good Hypothesis.</i></h3> - -<p>There is little difficulty in pointing out to what condition -an hypothesis must conform in order to be accepted -as probable and valid. That condition, as I conceive, is -the single one of enabling us to infer the existence of -phenomena which occur in our experience. <i>Agreement -with fact is the sole and sufficient test of a true hypothesis.</i></p> - -<p>Hobbes has named two conditions which he considers -requisite in an hypothesis, namely (1) That it should be -conceivable and not absurd; (2) That it should allow of -phenomena being necessarily inferred. Boyle, in noticing -Hobbes’ views, proposed to add a third condition, to the -effect that the hypothesis should not be inconsistent with -any other truth on phenomenon of nature.<a id="FNanchor_422" href="#Footnote_422" class="fnanchor">422</a> I think that -of these three conditions, the first cannot be accepted, -unless by <i>inconceivable</i> and <i>absurd</i> we mean self-contradictory -or inconsistent with the laws of thought and -nature. I shall have to point out that some satisfactory -theories involve suppositions which are wholly <i>inconceivable</i> -in a certain sense of the word, because the mind cannot -sufficiently extend its ideas to frame a notion of the -actions supposed to take place. That the force of gravity -should act instantaneously between the most distant parts -of the planetary system, or that a ray of violet light -should consist of about 700 billions of vibrations in a -second, are statements of an inconceivable and absurd -character in one sense; but they are so far from being -opposed to fact that we cannot on any other suppositions -account for phenomena observed. But if an hypothesis -involve self-contradiction, or is inconsistent with known<span class="pagenum" id="Page_511">511</span> -laws of nature, it is self-condemned. We cannot even -apply deductive reasoning to a self-contradictory notion; -and being opposed to the most general and certain laws -known to us, the primary laws of thought, it thereby conspicuously -fails to agree with facts. Since nature, again, -is never self-contradictory, we cannot at the same time -accept two theories which lead to contradictory results. -If the one agrees with nature, the other cannot. Hence if -there be a law which we believe with high probability to -be verified by observation, we must not frame an hypothesis -in conflict with it, otherwise the hypothesis will necessarily -be in disagreement with observation. Since no law or -hypothesis is proved, indeed, with absolute certainty, there -is always a chance, however slight, that the new hypothesis -may displace the old one; but the greater the probability -which we assign to that old hypothesis, the greater -must be the evidence required in favour of the new and -conflicting one.</p> - -<p>I assert, then, that there is but one test of a good -hypothesis, namely, <i>its conformity with observed facts</i>; but -this condition may be said to involve three constituent -conditions, nearly equivalent to those suggested by Hobbes -and Boyle, namely:—</p> - -<p class="ti1">(1) That it allow of the application of deductive reasoning -and the inference of consequences capable of comparison -with the results of observation.</p> - -<p class="ti1">(2) That it do not conflict with any laws of nature, or -of mind, which we hold to be true.</p> - -<p class="ti1">(3) That the consequences inferred do agree with facts -of observation.</p> - - -<h3><i>Possibility of Deductive Reasoning.</i></h3> - -<p>As the truth of an hypothesis is to be proved by its -conformity with fact, the first condition is that we be able -to apply methods of deductive reasoning, and learn what -should happen according to such an hypothesis. Even -if we could imagine an object acting according to laws -hitherto wholly unknown it would be useless to do so, -because we could never decide whether it existed or not. -We can only infer what would happen under supposed -conditions by applying the knowledge of nature we possess<span class="pagenum" id="Page_512">512</span> -to those conditions. Hence, as Boscovich truly said, we -are to understand by hypotheses “not fictions altogether -arbitrary, but suppositions conformable to experience or -analogy.” It follows that every hypothesis worthy of -consideration must suggest some likeness, analogy, or -common law, acting in two or more things. If, in order -to explain certain facts, <i>a</i>, <i>a′</i>, <i>a″</i>, &c., we invent a cause A, -then we must in some degree appeal to experience as to -the mode in which A will act. As the laws of nature are -not known to the mind intuitively, we must point out -some other cause, B, which supplies the requisite notions, -and all we do is to invent a fourth term to an analogy. -As B is to its effects <i>b</i>, <i>b′</i>, <i>b″</i>, &c., so is A to its effects <i>a</i>, -<i>a′</i>, <i>a″</i>, &c. When we attempt to explain the passage of -light and heat radiations through space unoccupied by -matter, we imagine the existence of the so-called <i>ether</i>. -But if this ether were wholly different from anything -else known to us, we should in vain try to reason about it. -We must apply to it at least the laws of motion, that is -we must so far liken it to matter. And as, when applying -those laws to the elastic medium air, we are able to infer -the phenomena of sound, so by arguing in a similar manner -concerning ether we are able to infer the existence of light -phenomena corresponding to what do occur. All that we -do is to take an elastic substance, increase its elasticity -immensely, and denude it of gravity and some other -properties of matter, but we must retain sufficient likeness -to matter to allow of deductive calculations.</p> - -<p>The force of gravity is in some respects an incomprehensible -existence, but in other respects entirely conformable -to experience. We observe that the force is -proportional to mass, and that it acts in entire independence -of other matter which may be present or intervening. -The law of the decrease of intensity, as the square of the -distance increases, is observed to hold true of light, sound, -and other influences emanating from a point, and spreading -uniformly through space. The law is doubtless connected -with the properties of space, and is so far in agreement -with our necessary ideas.</p> - -<p>It may be said, however, that no hypothesis can be so -much as framed in the mind unless it be more or less -conformable to experience. As the material of our ideas<span class="pagenum" id="Page_513">513</span> -is derived from sensation we cannot figure to ourselves -any agent, but as endowed with some of the properties of -matter. All that the mind can do in the creation of new -existences is to alter combinations, or the intensity of -sensuous properties. The phenomenon of motion is -familiar to sight and touch, and different degrees of rapidity -are also familiar; we can pass beyond the limits of sense, -and imagine the existence of rapid motion, such as our -senses could not observe. We know what is elasticity, -and we can therefore in a way figure to ourselves elasticity -a thousand or a million times greater than any which is -sensuously known to us. The waves of the ocean are many -times higher than our own bodies; other waves, are many -times less; continue the proportion, and we ultimately -arrive at waves as small as those of light. Thus it is that -the powers of mind enable us from a sensuous basis to -reason concerning agents and phenomena different in an -unlimited degree. If no hypothesis then can be absolutely -opposed to sense, accordance with experience must always -be a question of degree.</p> - -<p>In order that an hypothesis may allow of satisfactory -comparison with experience, it must possess definiteness -and in many cases mathematical exactness allowing of -the precise calculation of results. We must be able to -ascertain whether it does or does not agree with facts. -The theory of vortices is an instance to the contrary, for -it did not present any mode of calculating the exact -relations between the distances and periods of the planets -and satellites; it could not, therefore, undergo that rigorous -testing to which Newton scrupulously submitted his theory -of gravity before its promulgation. Vagueness and incapability -of precise proof or disproof often enable a false -theory to live; but with those who love truth, vagueness -should excite suspicion. The upholders of the ancient -doctrine of Nature’s abhorrence of a vacuum, had been -unable to anticipate the important fact that water would -not rise more than 33 feet in a common suction pump. -Nor when the fact was pointed out could they explain it, -except by introducing a special alteration of the theory to -the effect that Nature’s abhorrence of a vacuum was -limited to 33 feet.</p> - -<p><span class="pagenum" id="Page_514">514</span></p> - - -<h3><i>Consistency with the Laws of Nature.</i></h3> - -<p>In the second place an hypothesis must not be contradictory -to what we believe to be true concerning Nature. -It must not involve self-inconsistency which is opposed to -the highest and simplest laws, namely, those of Logic. -Neither ought it to be irreconcilable with the simple -laws of motion, of gravity, of the conservation of energy, -nor any parts of physical science which we consider to be -established beyond reasonable doubt. Not that we are -absolutely forbidden to entertain such an hypothesis, but -if we do so we must be prepared to disprove some of the -best demonstrated truths in the possession of mankind. -The fact that conflict exists means that the consequences -of the theory are not verified if previous discoveries are -correct, and we must therefore show that previous discoveries -are incorrect before we can verify our theory.</p> - -<p>An hypothesis will be exceedingly improbable, not to -say absurd, if it supposes a substance to act in a manner -unknown in other cases; for it then fails to be verified in -our knowledge of that substance. Several physicists, -especially Euler and Grove, have supposed that we might -dispense with an ethereal basis of light, and infer from -the interstellar passage of rays that there was a kind of -rare gas occupying space. But if so, that gas must be -excessively rare, as we may infer from the apparent -absence of an atmosphere around the moon, and from -other facts known to us concerning gases and the atmosphere; -yet it must possess an elastic force at least a -billion times as great as atmospheric air at the earth’s -surface, in order to account for the extreme rapidity of -light rays. Such an hypothesis then is inconsistent with -our knowledge concerning gases.</p> - -<p>Provided that there be no clear and absolute conflict -with known laws of nature, there is no hypothesis so -improbable or apparently inconceivable that it may not -be rendered probable, or even approximately certain, by -a sufficient number of concordances. In fact the two best -founded and most successful theories in physical science -involve the most absurd suppositions. Gravity is a force -which appears to act between bodies through vacuous<span class="pagenum" id="Page_515">515</span> -space; it is in positive contradiction to the old dictum -that nothing can act but through some medium. It is -even more puzzling that the force acts in perfect indifference -to intervening obstacles. Light in spite of its -extreme velocity shows much respect to matter, for it is -almost instantaneously stopped by opaque substances, and -to a considerable extent absorbed and deflected by transparent -ones. But to gravity all media are, as it were, -absolutely transparent, nay non-existent; and two particles -at opposite points of the earth affect each other exactly as -if the globe were not between. The action is, so far as -we can observe, instantaneous, so that every particle of the -universe is at every moment in separate cognisance, as it -were, of the relative position of every other particle throughout -the universe at that same moment of time. Compared -with such incomprehensible conditions, the theory of -vortices deals with commonplace realities. Newton’s -celebrated saying <i>hypotheses non fingo</i>, bears the appearance -of irony; and it was not without apparent grounds that -Leibnitz and the continental philosophers charged Newton -with re-introducing occult powers and qualities.</p> - -<p>The undulatory theory of light presents almost equal -difficulties of conception. We are asked by physical -philosophers to give up our prepossessions, and to believe -that interstellar space which seems empty is not empty at -all, but filled with <i>something</i> immensely more solid and -elastic than steel. As Young himself remarked,<a id="FNanchor_423" href="#Footnote_423" class="fnanchor">423</a> “the -luminiferous ether, pervading all space, and penetrating -almost all substances, is not only highly elastic, but -absolutely solid!!!” Herschel calculated the force which -may be supposed, according to the undulatory theory of -light, to be constantly exerted at each point in space, and -finds it to be 1,148,000,000,000 times the elastic force of -ordinary air at the earth’s surface, so that the pressure -of ether per square inch must be about seventeen billions -of pounds.<a id="FNanchor_424" href="#Footnote_424" class="fnanchor">424</a> Yet we live and move without appreciable -resistance through this medium, immensely harder and -more elastic than adamant. All our ordinary notions -must be laid aside in contemplating such an hypothesis;<span class="pagenum" id="Page_516">516</span> -yet it is no more than the observed phenomena of light -and heat force us to accept. We cannot deny even the -strange suggestion of Young, that there may be independent -worlds, some possibly existing in different parts of space, -but others perhaps pervading each other unseen and -unknown in the same space.<a id="FNanchor_425" href="#Footnote_425" class="fnanchor">425</a> For if we are bound to -admit the conception of this adamantine firmament, it is -equally easy to admit a plurality of such. We see, then, -that mere difficulties of conception must not discredit a -theory which otherwise agrees with facts, and we must -only reject hypotheses which are inconceivable in the -sense of breaking distinctly the primary laws of thought -and nature.</p> - - -<h3><i>Conformity with Facts.</i></h3> - -<p>Before we accept a new hypothesis it must be shown -to agree not only with the previously known laws of nature, -but also with the particular facts which it is framed -to explain. Assuming that these facts are properly -established, it must agree with all of them. A single -absolute conflict between fact and hypothesis, is fatal to -the hypothesis; <i>falsa in uno, falsa in omnibus</i>.</p> - -<p>Seldom, indeed, shall we have a theory free from -difficulties and apparent inconsistency with facts. Though -one real inconsistency would overturn the most plausible -theory, yet there is usually some probability that the fact -may be misinterpreted, or that some supposed law of -nature, on which we are relying, may not be true. It may -be expected, moreover, that a good hypothesis, besides -agreeing with facts already noticed, will furnish us with -distinct credentials by enabling us to anticipate deductively -series of facts which are not already connected and -accounted for by any equally probable hypothesis. We -cannot lay down any precise rule as to the number of -accordances which can establish the truth of an hypothesis, -because the accordances will vary much in value. While, -on the one hand, no finite number of accordances will -give entire certainty, the probability of the hypothesis -will increase very rapidly with the number of accordances.<span class="pagenum" id="Page_517">517</span> -Almost every problem in science thus takes the form of -a balance of probabilities. It is only when difficulty -after difficulty has been successfully explained away, and -decisive <i>experimenta crucis</i> have, time after time, resulted -in favour of our theory, that we can venture to assert the -falsity of all objections.</p> - -<p>The sole real test of an hypothesis is its accordance -with fact. Descartes’ celebrated system of vortices is -exploded, not because it was intrinsically absurd and -inconceivable, but because it could not give results in -accordance with the actual motions of the heavenly bodies. -The difficulties of conception involved in the apparatus -of vortices, are child’s play compared with those of gravitation -and the undulatory theory already described. -Vortices are on the whole plausible suppositions; for -planets and satellites bear at first sight much resemblance -to objects carried round in whirlpools, an analogy which -doubtless suggested the theory. The failure was in the -first and third requisites; for, as already remarked, the -theory did not allow of precise calculation of planetary -motions, and was thus incapable of rigorous verification. -But so far as we can institute a comparison, facts are entirely -against the vortices. Newton did not ridicule the -theory as absurd, but showed<a id="FNanchor_426" href="#Footnote_426" class="fnanchor">426</a> that it was “pressed with -many difficulties.” He carefully pointed out that the -Cartesian theory was inconsistent with the laws of Kepler, -and would represent the planets as moving more rapidly -at their aphelia than at their perihelia.<a id="FNanchor_427" href="#Footnote_427" class="fnanchor">427</a> The rotatory -motion of the sun and planets on their own axes is in -striking conflict with the revolutions of the satellites -carried round them; and comets, the most flimsy of bodies, -calmly pursue their courses in elliptic paths, irrespective -of the vortices which they pass through. We may now -also point to the interlacing orbits of the minor planets -as a new and insuperable difficulty in the way of the -Cartesian ideas.</p> - -<p>Newton, though he established the best of theories, was -also capable of proposing one of the worst; and if we -want an instance of a theory decisively contradicted by<span class="pagenum" id="Page_518">518</span> -facts, we have only to turn to his views concerning the -origin of natural colours. Having analysed, with incomparable -skill, the origin of the colours of thin plates, he -suggests that the colours of all bodies are determined -in like manner by the size of their ultimate particles. -A thin plate of a definite thickness will reflect a definite -colour; hence, if broken up into fragments it will -form a powder of the same colour. But, if this be a -sufficient explanation of coloured substances, then every -coloured fluid ought to reflect the complementary colour of -that which it transmits. Colourless transparency arises, -according to Newton, from particles being too minute to -reflect light; but if so, every black substance should be -transparent. Newton himself so acutely felt this last difficulty -as to suggest that true blackness is due to some -internal refraction of the rays to and fro, and an ultimate -stifling of them, which he did not attempt to explain -further. Unless some other process comes into operation, -neither refraction nor reflection, however often repeated, -will destroy the energy of light. The theory therefore -gives no account, as Brewster shows, of 24 parts out of -25 of the light which falls upon a black coal, and the remaining -part which is reflected from the lustrous surface -is equally inconsistent with the theory, because fine coal-dust -is almost entirely devoid of reflective power.<a id="FNanchor_428" href="#Footnote_428" class="fnanchor">428</a> It is -now generally believed that the colours of natural bodies -are due to the unequal absorption of rays of light of different -refrangibility.</p> - - -<h3><i>Experimentum Crucis.</i></h3> - -<p>As we deduce more and more conclusions from a theory, -and find them verified by trial, the probability of the -theory increases in a rapid manner; but we never escape -the risk of error altogether. Absolute certainty is beyond -the powers of inductive investigation, and the most -plausible supposition may ultimately be proved false. -Such is the groundwork of similarity in nature, that -two very different conditions may often give closely -similar results. We sometimes find ourselves therefore<span class="pagenum" id="Page_519">519</span> -in possession of two or more hypotheses which both agree -with so many experimental facts as to have great appearance -of truth. Under such circumstances we have need -of some new experiment, which shall give results agreeing -with one hypothesis but not with the other.</p> - -<p>Any such experiment which decides between two rival -theories may be called an <i>Experimentum Crucis</i>, an -Experiment of the Finger Post. Whenever the mind -stands, as it were, at cross-roads and knows not which -way to select, it needs some decisive guide, and Bacon -therefore assigned great importance and authority to instances -which serve in this capacity. The name given by -Bacon has become familiar; it is almost the only one of -Bacon’s figurative expressions which has passed into common -use. Even Newton, as I have mentioned (p. <a href="#Page_507">507</a>), -used the name.</p> - -<p>I do not think, indeed, that the common use of the -word at all agrees with that intended by Bacon. Herschel -says that “we make an experiment of the crucial -kind when we form combinations, and put in action -causes from which some particular one shall be deliberately -excluded, and some other purposely admitted.”<a id="FNanchor_429" href="#Footnote_429" class="fnanchor">429</a> This, -however, seems to be the description of any special experiment -not made at haphazard. Pascal’s experiment -of causing a barometer to be carried to the top of -the Puy-de-Dôme has often been considered as a perfect -<i>experimentum crucis</i>, if not the first distinct one on -record;<a id="FNanchor_430" href="#Footnote_430" class="fnanchor">430</a> but if so, we must dignify the doctrine of -Nature’s abhorrence of a vacuum with the position of a -rival theory. A crucial experiment must not simply -confirm one theory, but must negative another; it must -decide a mind which is in equilibrium, as Bacon says,<a id="FNanchor_431" href="#Footnote_431" class="fnanchor">431</a> -between two equally plausible views. “When in search -of any nature, the understanding comes to an equilibrium, -as it were, or stands suspended as to which of two or -more natures the cause of nature inquired after should -be attributed or assigned, by reason of the frequent and -common occurrence of several natures, then these Crucial -Instances show the true and inviolable association of one<span class="pagenum" id="Page_520">520</span> -of these natures to the nature sought, and the uncertain -and separable alliance of the other, whereby the question -is decided, the former nature admitted for the cause, -and the other rejected. These instances, therefore, afford -great light, and have a kind of overruling authority, so -that the course of interpretation will sometimes terminate -in them, or be finished by them.”</p> - -<p>The long-continued strife between the Corpuscular and -Undulatory theories of light forms the best possible illustration -of an Experimentum Crucis. It is remarkable in -how plausible a manner both these theories agreed with -the ordinary laws of geometrical optics, relating to reflection -and refraction. According to the first law of motion -a moving particle proceeds in a perfectly straight line, -when undisturbed by extraneous forces. If the particle -being perfectly elastic, strike a perfectly elastic plane, it -will bound off in such a path that the angles of incidence -and reflection will be equal. Now a ray of light proceeds -in a straight line, or appears to do so, until it meets a reflecting -body, when its path is altered in a manner exactly -similar to that of the elastic particle. Here is a remarkable -correspondence which probably suggested to Newton’s -mind the hypothesis that light consists of minute elastic -particles moving with excessive rapidity in straight lines. -The correspondence was found to extend also to the law -of simple refraction; for if particles of light be supposed -capable of attracting matter, and being attracted by it at -insensibly small distances, then a ray of light, falling on -the surface of a transparent medium, will suffer an increase -in its velocity perpendicular to the surface, and the law -of sines is the consequence. This remarkable explanation -of the law of refraction had doubtless a very strong -effect in leading Newton to entertain the corpuscular -theory, and he appears to have thought that the analogy -between the propagation of rays of light and the motion -of bodies was perfectly exact, whatever might be the -actual nature of light.<a id="FNanchor_432" href="#Footnote_432" class="fnanchor">432</a> It is highly remarkable, again, -that Newton was able to give by his corpuscular theory, -a plausible explanation of the inflection of light as discovered<span class="pagenum" id="Page_521">521</span> -by Grimaldi. The theory would indeed have been -a very probable one could Newton’s own law of gravity -have applied; but this was out of the question, because the -particles of light, in order that they may move in straight -lines, must be devoid of any influence upon each other.</p> - -<p>The Huyghenian or Undulatory theory of light was also -able to explain the same phenomena, but with one remarkable -difference. If the undulatory theory be true, -light must move more slowly in a dense refracting medium -than in a rarer one; but the Newtonian theory assumed -that the attraction of the dense medium caused the particles -of light to move more rapidly than in the rare -medium. On this point, then, there was complete discrepancy -between the theories, and observation was required -to show which theory was to be preferred. Now by -simply cutting a uniform plate of glass into two pieces, -and slightly inclining one piece so as to increase the -length of the path of a ray passing through it, experimenters -were able to show that light does move more -slowly in glass than in air.<a id="FNanchor_433" href="#Footnote_433" class="fnanchor">433</a> More recently Fizeau and -Foucault independently measured the velocity of light in -air and in water, and found that the velocity is greater in -air.<a id="FNanchor_434" href="#Footnote_434" class="fnanchor">434</a></p> - -<p>There are a number of other points at which experience -decides against Newton, and in favour of Huyghens -and Young. Laplace pointed out that the attraction supposed -to exist between matter and the corpuscular particles -of light would cause the velocity of light to vary -with the size of the emitting body, so that if a star were -250 times as great in diameter as our sun, its attraction -would prevent the emanation of light altogether.<a id="FNanchor_435" href="#Footnote_435" class="fnanchor">435</a> But -experience shows that the velocity of light is uniform, -and independent of the magnitude of the emitting body, as -it should be according to the undulatory theory. Lastly, -Newton’s explanation of diffraction or inflection fringes -of colours was only <i>plausible</i>, and not true; for Fresnel -ascertained that the dimensions of the fringes are not what -they would be according to Newton’s theory.</p> - -<p>Although the Science of Light presents us with the<span class="pagenum" id="Page_522">522</span> -most beautiful examples of crucial experiments and observations, -instances are not wanting in other branches of -science. Copernicus asserted, in opposition to the ancient -Ptolemaic theory, that the earth moved round the sun, and -he predicted that if ever the sense of sight could be -rendered sufficiently acute and powerful, we should see -phases in Mercury and Venus. Galileo with his telescope -was able, in 1610 to verify the prediction as regards Venus, -and subsequent observations of Mercury led to a like conclusion. -The discovery of the aberration of light added a -new proof, still further strengthened by the more recent -determination of the parallax of fixed stars. Hooke proposed -to prove the existence of the earth’s diurnal motion -by observing the deviation of a falling body, an experiment -successfully accomplished by Benzenberg; and -Foucault’s pendulum has since furnished an additional -indication of the same motion, which is indeed also -apparent in the trade winds. All these are crucial facts in -favour of the Copernican theory.</p> - - -<h3><i>Descriptive Hypotheses.</i></h3> - -<p>There are hypotheses which we may call <i>descriptive -hypotheses</i>, and which serve for little else than to furnish -convenient names. When a phenomenon is of an unusual -kind, we cannot even speak of it without using some -analogy. Every word implies some resemblance between -the thing to which it is applied, and some other thing, -which fixes the meaning of the word. If we are to speak -of what constitutes electricity, we must search for the -nearest analogy, and as electricity is characterised by the -rapidity and facility of its movements, the notion of a fluid -of a very subtle character presents itself as appropriate. -There is the single-fluid and the double-fluid theory of -electricity, and a great deal of discussion has been uselessly -spent upon them. The fact is, that if these theories be -understood as more than convenient modes of describing -the phenomena, they are altogether invalid. The analogy -extends only to the rapidity of motion, or rather the fact -that a phenomenon occurs successively at different points -of the body. The so-called electric fluid adds nothing to -the weight of the conductor, and to suppose that it really<span class="pagenum" id="Page_523">523</span> -consists of particles of matter is even more absurd than to -reinstate the corpuscular theory of light. A far closer -analogy exists between electricity and light undulations, -which are about equally rapid in propagation. We shall -probably continue for a long time to talk of the <i>electric -fluid</i>, but there can be no doubt that this expression -represents merely a phase of molecular motion, a wave of -disturbance. The invalidity of these fluid theories is -shown moreover in the fact that they have not led to the -invention of a single new experiment.</p> - -<p>Among these merely descriptive hypotheses I should -place Newton’s theory of Fits of Easy Reflection and -Refraction. That theory did not do more than describe -what took place. It involved no analogy to other phenomena -of nature, for Newton could not point to any other -substance which went through these extraordinary fits. -We now know that the true analogy would have been -waves of sound, of which Newton had acquired in other -respects so complete a comprehension. But though the -notion of interference of waves had distinctly occurred to -Hooke, Newton failed to see how the periodic phenomena -of light could be connected with the periodic character of -waves. His hypothesis fell because it was out of analogy -with everything else in nature, and it therefore did not -allow him, as in other cases, to descend by mathematical -deduction to consequences which could be verified or -refuted.</p> - -<p>We are at freedom to imagine the existence of a new -agent, and to give it an appropriate name, provided there -are phenomena incapable of explanation from known -causes. We may speak of <i>vital force</i> as occasioning life, -provided that we do not take it to be more than a name -for an undefined something giving rise to inexplicable -facts, just as the French chemists called Iodine the Substance -X, so long as they were unaware of its real character -and place in chemistry.<a id="FNanchor_436" href="#Footnote_436" class="fnanchor">436</a> Encke was quite justified -in speaking of the <i>resisting medium</i> in space so long as the -retardation of his comet could not be otherwise accounted -for. But such hypotheses will do much harm whenever -they divert us from attempts to reconcile the facts with<span class="pagenum" id="Page_524">524</span> -known laws, or when they lead us to mix up discrete things. -Because we speak of vital force we must not assume that it -is a really existing physical force like electricity; we do not -know what it is. We have no right to confuse Encke’s -supposed resisting medium with the basis of light without -distinct evidence of identity. The name protoplasm, now -so familiarly used by physiologists, is doubtless legitimate -so long as we do not mix up different substances under it, -or imagine that the name gives us any knowledge of the -obscure origin of life. To name a substance protoplasm -no more explains the infinite variety of forms of life which -spring out of the substance, than does the <i>vital force</i> which -may be supposed to reside in the protoplasm. Both expressions -are mere names for an inexplicable series of -causes which out of apparently similar conditions produce -the most diverse results.</p> - -<p>Hardly to be distinguished from descriptive hypotheses -are certain imaginary objects which we frame for the -ready comprehension of a subject. The mathematician, -in treating abstract questions of probability, finds it convenient -to represent the conditions by a concrete hypothesis -in the shape of a ballot-box. Poisson proved the -principle of the inverse method of probabilities by imagining -a number of ballot-boxes to have their contents -mixed in one great ballot-box (p. <a href="#Page_244">244</a>). Many such -devices are used by mathematicians. The Ptolemaic -theory of <i>cycles</i> and <i>epi-cycles</i> was no grotesque and useless -work of the imagination, but a perfectly valid mode -of analysing the motions of the heavenly bodies; in reality -it is used by mathematicians at the present day. Newton -employed the pendulum as a means of representing the -nature of an undulation. Centres of gravity, oscillation, -&c., poles of the magnet, lines of force, are other imaginary -existences employed to assist our thoughts (p. <a href="#Page_364">364</a>). Such -devices may be called <i>Representative Hypotheses</i>, and they -are only permissible so far as they embody analogies. -Their further consideration belongs either to the subject -of Analogy, or to that of language and representation, -founded upon analogy.</p> -<hr class="chap x-ebookmaker-drop"> - -<div class="chapter"> -<p><span class="pagenum" id="Page_525">525</span></p> - -<h2 class="nobreak" id="CHAPTER_XXIV">CHAPTER XXIV.<br> - -<span class="title">EMPIRICAL KNOWLEDGE, EXPLANATION, AND PREDICTION.</span></h2> -</div> - -<p class="ti0">Inductive investigation, as we have seen, consists in the -union of hypothesis and experiment, deductive reasoning -being the link by which experimental results are made to -confirm or confute the hypothesis. Now when we consider -this relation between hypothesis and experiment it is -obvious that we may classify our knowledge under four -heads.</p> - -<p>(1) We may be acquainted with facts which have not -yet been brought into accordance with any hypothesis. -Such facts constitute what is called <i>Empirical Knowledge</i>.</p> - -<p>(2) Another extensive portion of our knowledge consists -of facts which having been first observed empirically, -have afterwards been brought into accordance with other -facts by an hypothesis concerning the general laws applying -to them. This portion of our knowledge may be said -to be <i>explained</i>, <i>reasoned</i>, or <i>generalised</i>.</p> - -<p>(3) In the third place comes the collection of facts, minor -in number, but most important as regards their scientific -interest, which have been anticipated by theory and afterwards -verified by experiment.</p> - -<p>(4) Lastly, there exists knowledge which is accepted -solely on the ground of theory, and is incapable of experimental -confirmation, at least with the instrumental means -in our possession.</p> - -<p>It is a work of much interest to compare and illustrate -the relative extent and value of these four groups of knowledge. -We shall observe that as a general rule a great -branch of science originates in facts observed accidentally,<span class="pagenum" id="Page_526">526</span> -or without distinct consciousness of what is to be expected. -As a science progresses, its power of foresight rapidly -increases, until the mathematician in his library acquires -the power of anticipating nature, and predicting what will -happen in circumstances which the eye of man has never -examined.</p> - - -<h3><i>Empirical Knowledge.</i></h3> - -<p>By empirical knowledge we mean such as is derived -directly from the examination of detached facts, and rests -entirely on those facts, without corroboration from other -branches of knowledge. It is contrasted with generalised -and theoretical knowledge, which embraces many series of -facts under a few comprehensive principles, so that each -series serves to throw light upon each other series of facts. -Just as, in the map of a half-explored country, we see -detached bits of rivers, isolated mountains, and undefined -plains, not connected into any complete plan, so a new -branch of knowledge consists of groups of facts, each group -standing apart, so as not to allow us to reason from one to -another.</p> - -<p>Before the time of Descartes, and Newton, and Huyghens, -there was much empirical knowledge of the phenomena of -light. The rainbow had always struck the attention of -the most careless observers, and there was no difficulty -in perceiving that its conditions of occurrence consisted in -rays of the sun shining upon falling drops of rain. It was -impossible to overlook the resemblance of the ordinary -rainbow to the comparatively rare lunar rainbow, to the -bow which appears upon the spray of a waterfall, or even -upon beads of dew suspended on grass and spiders’ webs. -In all these cases the uniform conditions are rays of light -and round drops of water. Roger Bacon had noticed these -conditions, as well as the analogy of the rainbow colours -to those produced by crystals.<a id="FNanchor_437" href="#Footnote_437" class="fnanchor">437</a> But the knowledge was -empirical until Descartes and Newton showed how the -phenomena were connected with facts concerning the -refraction of light.</p> - -<p>There can be no better instance of an empirical truth<span class="pagenum" id="Page_527">527</span> -than that detected by Newton concerning the high refractive -powers of combustible substances. Newton’s -chemical notions were almost as vague as those prevalent -in his day, but he observed that certain “fat, sulphureous, -unctuous bodies,” as he calls them, such as camphor, oils -spirit of turpentine, amber, &c., have refractive powers -two or three times greater than might be anticipated from -their densities.<a id="FNanchor_438" href="#Footnote_438" class="fnanchor">438</a> The enormous refractive index of diamond, -led him with great sagacity to regard this substance as -of the same unctuous or inflammable nature, so that he -may be regarded as predicting the combustibility of the -diamond, afterwards demonstrated by the Florentine -Academicians in 1694. Brewster having entered into a -long investigation of the refractive powers of different -substances, confirmed Newton’s assertions, and found that -the three elementary combustible substances, diamond, -phosphorus, and sulphur, have, in comparison with their -densities, by far the highest known refractive indices,<a id="FNanchor_439" href="#Footnote_439" class="fnanchor">439</a> and -there are only a few substances, such as chromate of lead -or glass of antimony, which exceed them in absolute power -of refraction. The oils and hydrocarbons generally possess -excessive indices. But all this knowledge remains to the -present day purely empirical, no connection having been -pointed out between this coincidence of inflammability and -high refractive power, with other laws of chemistry or optics. -It is worth notice, as pointed out by Brewster, that if -Newton had argued concerning two minerals, Greenockite -and Octahedrite, as he did concerning diamond, his predictions -would have proved false, showing sufficiently that -he did not make any sure induction on the subject. In -the present day, the relation of the refractive index to the -density and atomic weight of a substance is becoming a -matter of theory; yet there remain specific differences of -refracting power known only on empirical grounds, and it -is curious that in hydrogen an abnormally high refractive -power has been found to be joined to inflammability.</p> - -<p>The science of chemistry, however much its theory may -have progressed, still presents us with a vast body of empirical -knowledge. Not only is it as yet hopeless to attempt<span class="pagenum" id="Page_528">528</span> -to account for the particular group of qualities belonging to -each element, but there are multitudes of particular facts -of which no further account can be given. Why should -the sulphides of many metals be intensely black? Why -should a slight amount of phosphoric acid have so great -a power of interference with the crystallisation of vanadic -acid?<a id="FNanchor_440" href="#Footnote_440" class="fnanchor">440</a> Why should the compound silicates of alkalies and -alkaline metals be transparent? Why should gold be so -highly ductile, and gold and silver the only two sensibly -translucent metals? Why should sulphur be capable of -so many peculiar changes into allotropic modifications?</p> - -<p>There are whole branches of chemical knowledge which -are mere collections of disconnected facts. The properties -of alloys are often remarkable; but no laws have yet been -detected, and the laws of combining proportions seem to have -no clear application.<a id="FNanchor_441" href="#Footnote_441" class="fnanchor">441</a> Not the slightest explanation can -be given of the wonderful variations of the qualities of iron, -according as it contains more or less carbon and silicon, nay, -even the facts of the case are often involved in uncertainty. -Why, again, should the properties of steel be remarkably -affected by the presence of a little tungsten or manganese? -All that was determined by Matthiessen concerning the -conducting powers of copper, was of a purely empirical -character.<a id="FNanchor_442" href="#Footnote_442" class="fnanchor">442</a> Many animal substances cannot be shown to -obey the laws of combining proportions. Thus for the most -part chemistry is yet an empirical science occupied with -the registration of immense numbers of disconnected facts, -which may at some future time become the basis of a -greatly extended theory.</p> - -<p>We must not indeed suppose that any science will ever -entirely cease to be empirical. Multitudes of phenomena -have been explained by the undulatory theory of light; -but there yet remain many facts to be treated. The -natural colours of bodies and the rays given off by them -when heated, are unexplained, and yield few empirical -coincidences. The theory of electricity is partially understood, -but the conditions of the production of frictional -electricity defy explanation, although they have been<span class="pagenum" id="Page_529">529</span> -studied for two centuries. I shall subsequently point out -that even the establishment of a wide and true law of -nature is but the starting-point for the discovery of exceptions -and divergences giving a new scope to empirical -discovery.</p> - -<p>There is probably no science, I have said, which is -entirely free from empirical and unexplained facts. Logic -approaches most nearly to this position, as it is merely a -deductive development of the laws of thought and the -principle of substitution. Yet some of the facts established -in the investigation of the inverse logical problem may be -considered empirical. That a proposition of the form -A = BC ꖌ <i>b c</i> possesses the least number of distinct logical -variations, and the greatest number of logical equivalents -of the same form among propositions involving three -classes (p. <a href="#Page_141">141</a>), is a case in point. So also is the fact -discovered by Professor Clifford that in regard to statements -involving four classes, there is only one example of two -dissimilar statements having the same distances (p. <a href="#Page_144">144</a>). -Mathematical science often yields empirical truths. Why, -for instance, should the value of π, when expressed to a great -number of figures, contain the digit 7 much less frequently -than any other digit?<a id="FNanchor_443" href="#Footnote_443" class="fnanchor">443</a> Even geometry may allow of -empirical truths, when the matter does not involve -quantities of space, but numerical results and the positive -or negative character of quantities, as in De Morgan’s -theorem concerning negative areas.</p> - - -<h3><i>Accidental Discovery.</i></h3> - -<p>There are not a few cases where almost pure accident -has determined the moment when a new branch of knowledge -was to be created. The laws of the structure of crystals -were not discovered until Haüy happened to drop a -beautiful crystal of calc-spar upon a stone pavement. His -momentary regret at destroying a choice specimen was -quickly removed when, in attempting to join the fragments -together, he observed regular geometrical faces, which did -not correspond with the external facets of the crystals. A -great many more crystals were soon broken intentionally,<span class="pagenum" id="Page_530">530</span> -to observe the planes of cleavage, and the discovery of the -internal structure of crystalline substances was the result. -Here we see how much more was due to the reasoning -power of the philosopher, than to an accident which must -often have happened to other persons.</p> - -<p>In a similar manner, a fortuitous occurrence led Malus -to discover the polarisation of light by reflection. The -phenomena of double refraction had been long known, and -when engaged in Paris in 1808, in investigating the character -of light thus polarised, Malus chanced to look -through a double refracting prism at the light of the setting -sun, reflected from the windows of the Luxembourg Palace. -In turning the prism round, he was surprised to find that -the ordinary image disappeared at two opposite positions -of the prism. He remarked that the reflected light behaved -like light which had been polarised by passing through -another prism. He was induced to test the character of -light reflected under other circumstances, and it was -eventually proved that polarisation is invariably connected -with reflection. Some of the general laws of optics, -previously unsuspected, were thus discovered by pure -accident. In the history of electricity, accident has had a -large part. For centuries some of the more common -effects of magnetism and of frictional electricity had presented -themselves as unaccountable deviations from the -ordinary course of Nature. Accident must have first -directed attention to such phenomena, but how few of -those who witnessed them had any conception of the all-pervading -character of the power manifested. The very -existence of galvanism, or electricity of low tension, was -unsuspected until Galvani accidentally touched the leg of -a frog with pieces of metal. The decomposition of water -by voltaic electricity also was accidentally discovered by -Nicholson in 1801, and Davy speaks of this discovery as -the foundation of all that had since been done in electro-chemical -science.</p> - -<p>It is otherwise with the discovery of electro-magnetism. -Oersted, in common with many others, had suspected the -existence of some relation between the magnet and -electricity, and he appears to have tried to detect its exact -nature. Once, as we are told by Hansteen, he had employed -a strong galvanic battery during a lecture, and at<span class="pagenum" id="Page_531">531</span> -the close it occurred to him to try the effect of placing -the conducting wire parallel to a magnetic needle, instead -of at right angles, as he had previously done. The needle -immediately moved and took up a position nearly at right -angles to the wire; he inverted the direction of the -current, and the needle deviated in a contrary direction. -The great discovery was made, and if by accident, it was -such an accident as happens, as Lagrange remarked of -Newton, only to those who deserve it.<a id="FNanchor_444" href="#Footnote_444" class="fnanchor">444</a> There was, -in fact, nothing accidental, except that, as in all totally -new discoveries, Oersted did not know what to look for. -He could not infer from previous knowledge the nature -of the relation, and it was only repeated trial in different -modes which could lead him to the right combination. -High and happy powers of inference, and not accident, -subsequently led Faraday to reverse the process, and to -show that the motion of the magnet would occasion an -electric current in the wire.</p> - -<p>Sufficient investigation would probably show that almost -every branch of art and science had an accidental beginning. -In historical times almost every important new -instrument as the telescope, the microscope, or the compass, -was probably suggested by some accidental occurrence. -In pre-historic times the germs of the arts must have -arisen still more exclusively in the same way. Cultivation -of plants probably arose, in Mr. Darwin’s opinion, -from some such accident as the seeds of a fruit falling upon -a heap of refuse, and producing an unusually fine variety. -Even the use of fire must, some time or other, have been -discovered in an accidental manner.</p> - -<p>With the progress of a branch of science, the element -of chance becomes much reduced. Not only are laws -discovered which enable results to be predicted, as we -shall see, but the systematic examination of phenomena -and substances often leads to discoveries which can in no -sense be said to be accidental. It has been asserted that -the anæsthetic properties of chloroform were disclosed by a -little dog smelling at a saucerful of the liquid in a chemist’s -shop in Linlithgow, the singular effects upon the dog being -reported to Simpson, who turned the incident to good<span class="pagenum" id="Page_532">532</span> -account. This story, however, has been shown to be a -fabrication, the fact being that Simpson had for many -years been endeavouring to discover a better anæsthetic -than those previously employed, and that he tested the -properties of chloroform, among other substances, at the -suggestion of Waldie, a Liverpool chemist. The valuable -powers of chloral hydrate have since been discovered in -a like manner, and systematic inquiries are continually -being made into the therapeutic or economic values of -new chemical compounds.</p> - -<p>If we must attempt to draw a conclusion concerning -the part which chance plays in scientific discovery, it -must be allowed that it more or less affects the success of -all inductive investigation, but becomes less important -with the progress of science. Accident may bring a new -and valuable combination to the notice of some person who -had never expressly searched for a discovery of the kind, -and the probabilities are certainly in favour of a discovery -being occasionally made in this manner. But the greater -the tact and industry with which a physicist applies himself -to the study of nature, the greater is the probability -that he will meet with fortunate accidents, and will turn -them to good account. Thus it comes to pass that, in the -refined investigations of the present day, genius united to -extensive knowledge, cultivated powers, and indomitable -industry, constitute the characteristics of the successful -discoverer.</p> - - -<h3><i>Empirical Observations subsequently Explained.</i></h3> - -<p>The second great portion of scientific knowledge consists -of facts which have been first learnt in a purely empirical -manner, but have afterwards been shown to follow from -some law of nature, that is, from some highly probable -hypothesis. Facts are said to be explained when they are -thus brought into harmony with other facts, or bodies of -general knowledge. There are few words more familiarly -used in scientific phraseology than this word <i>explanation</i>, -and it is necessary to decide exactly what we mean by it, -since the question touches the deepest points concerning -the nature of science. Like most terms referring to mental -actions, the verbs <i>to explain</i>, or <i>to explicate</i>, involve<span class="pagenum" id="Page_533">533</span> -material similes. The action is <i>ex plicis plana reddere</i>, -to take out the folds, and render a thing plain or even. -Explanation thus renders a thing clearly comprehensible -in all its points, so that there is nothing left outstanding -or obscure.</p> - -<p>Every act of explanation consists in pointing out a -resemblance between facts, or in showing that similarity -exists between apparently diverse phenomena. This similarity -may be of any extent and depth; it may be a -general law of nature, which harmonises the motions of -all the heavenly bodies by showing that there is a similar -force which governs all those motions, or the explanation -may involve nothing more than a single identity, as when -we explain the appearance of shooting stars by showing -that they are identical with portions of a comet. Wherever -we detect resemblance, there is a more or less explanation. -The mind is disquieted when it meets a novel phenomenon, -one which is <i>sui generis</i>; it seeks at once for -parallels which may be found in the memory of past -sensations. The so-called sulphurous smell which attends -a stroke of lightning often excited attention, and it was -not explained until the exact similarity of the smell -to that of ozone was pointed out. The marks upon a -flagstone are explained when they are shown to correspond -with the feet of an extinct animal, whose bones are elsewhere -found. Explanation, in fact, generally commences -by the discovery of some simple resemblance; the theory -of the rainbow began as soon as Antonio de Dominis -pointed out the resemblance between its colours and those -presented by a ray of sunlight passing through a glass -globe full of water.</p> - -<p>The nature and limits of explanation can only be fully -considered, after we have entered upon the subjects of -generalisation and analogy. It must suffice to remark, in -this place, that the most important process of explanation -consists in showing that an observed fact is one case of a -general law or tendency. Iron is always found combined -with sulphur, when it is in contact with coal, whereas in -other parts of the carboniferous strata it always occurs as -a carbonate. We explain this empirical fact as being due -to the reducing power of carbon and hydrogen, which prevents -the iron from combining with oxygen, and leaves it<span class="pagenum" id="Page_534">534</span> -open to the affinity of sulphur. The uniform strength and -direction of the trade-winds were long familiar to mariners, -before they were explained by Halley on hydrostatical -principles. The winds were found to arise from the action -of gravity, which causes a heavier body to displace a lighter -one, while the direction from east to west was explained -as a result of the earth’s rotation. Whatever body in the -northern hemisphere changes its latitude, whether it be a -bird, or a railway train, or a body of air, must tend towards -the right hand. Dove’s law of the winds is that the winds -tend to veer in the northern hemisphere in the direction -N.E.S.W., and in the southern hemisphere in the direction -N.W.S.E. This tendency was shown by him to be the -necessary effect of the same conditions which apply to the -trade winds. Whenever, then, any fact is connected by -resemblance, law, theory, or hypothesis, with other facts, it -is explained.</p> - -<p>Although the great mass of recorded facts must be -empirical, and awaiting explanation, such knowledge is of -minor value, because it does not admit of safe and extensive -inference. Each recorded result informs us exactly what -will be experienced again in the same circumstances, -but has no bearing upon what will happen in other circumstances.</p> - - -<h3><i>Overlooked Results of Theory.</i></h3> - -<p>We must by no means suppose that, when a scientific -truth is in our possession, all its consequences will be -foreseen. Deduction is certain and infallible, in the sense -that each step in deductive reasoning will lead us to some -result, as certain as the law itself. But it does not follow -that deduction will lead the reasoner to every result of a law -or combination of laws. Whatever road a traveller takes, -he is sure to arrive somewhere, but unless he proceeds in -a systematic manner, it is unlikely that he will reach -every place to which a network of roads will conduct him.</p> - -<p>In like manner there are many phenomena which were -virtually within the reach of philosophers by inference from -their previous knowledge, but were never discovered until -accident or systematic empirical observation disclosed their -existence.</p> - -<p><span class="pagenum" id="Page_535">535</span></p> - -<p>That light travels with a uniform high velocity was -proved by Roemer from observations of the eclipses of -Jupiter’s satellites. Corrections were thenceforward made -in all astronomical observations requiring it, for the -difference of absolute time at which an event happened, -and that at which it would be seen on the earth. But -no person happened to remark that the motion of light -compounded with that of the earth in its orbit would -occasion a small apparent displacement of the greater -part of the heavenly bodies. Fifty years elapsed before -Bradley empirically discovered this effect, called by him -aberration, when reducing his observations of the fixed -stars.</p> - -<p>When once the relation between an electric current and -a magnet had been detected by Oersted and Faraday, it -ought to have been possible for them to foresee the diverse -results which must ensue in different circumstances. If, -for instance, a plate of copper were placed beneath an -oscillating magnetic needle, it should have been seen that -the needle would induce currents in the copper, but as -this could not take place without a certain reaction against -the needle, it ought to have been seen that the needle -would come to rest more rapidly than in the absence of the -copper. This peculiar effect was accidentally discovered -by Gambey in 1824. Arago acutely inferred from -Gambey’s experiment that if the copper were set in -rotation while the needle was stationary the motion -would gradually be communicated to the needle. The -phenomenon nevertheless puzzled the whole scientific -world, and it required the deductive genius of Faraday -to show that it was a result of the principles of electro-magnetism.<a id="FNanchor_445" href="#Footnote_445" class="fnanchor">445</a></p> - -<p>Many other curious facts might be mentioned which -when once noticed were explained as the effects of well-known -laws. It was accidentally discovered that the -navigation of canals of small depth could be facilitated -by increasing the speed of the boats, the resistance being -actually reduced by this increase of speed, which enables -the boat to ride as it were upon its own forced wave. -Now mathematical theory might have predicted this<span class="pagenum" id="Page_536">536</span> -result had the right application of the formulæ occurred -to any one.<a id="FNanchor_446" href="#Footnote_446" class="fnanchor">446</a> Giffard’s injector for supplying steam boilers -with water by the force of their own steam, was, I -believe, accidentally discovered, but no new principles of -mechanics are involved in it, so that it might have been -theoretically invented. The same may be said of the -curious experiment in which a stream of air or steam -issuing from a pipe is made to hold a free disc upon the -end of the pipe and thus obstruct its own outlet. The -possession then of a true theory does not by any means -imply the foreseeing of all the results. The effects of even -a few simple laws may be manifold, and some of the -most curious and useful effects may remain undetected -until accidental observation brings them to our notice. -.</p> - -<h3><i>Predicted Discoveries.</i></h3> - -<p>The most interesting of the four classes of facts specified -in p. <a href="#Page_525">525</a>, is probably the third, containing those the -occurrence of which has been first predicted by theory and -then verified by observation. There is no more convincing -proof of the soundness of knowledge than that it confers -the gift of foresight. Auguste Comte said that “Prevision -is the test of true theory;” I should say that it is <i>one test</i> -of true theory, and that which is most likely to strike -the public attention. Coincidence with fact is the test of -true theory, but when the result of theory is announced -before-hand, there can be no doubt as to the unprejudiced -spirit in which the theorist interprets the results of his -own theory.</p> - -<p>The earliest instance of scientific prophecy is naturally -furnished by the science of Astronomy, which was the -earliest in development. Herodotus<a id="FNanchor_447" href="#Footnote_447" class="fnanchor">447</a> narrates that, in -the midst of a battle between the Medes and Lydians, the -day was suddenly turned into night, and the event had -been foretold by Thales, the Father of Philosophy. A -cessation of the combat and peace confirmed by marriages -were the consequences of this happy scientific effort. -Much controversy has taken place concerning the date of<span class="pagenum" id="Page_537">537</span> -this occurrence, Baily assigning the year 610 <span class="allsmcap">B.C</span>., but -Airy has calculated that the exact day was the 28th of -May, 584 <span class="allsmcap">B.C.</span> There can be no doubt that this and other -predictions of eclipses attributed to ancient philosophers -were due to a knowledge of the Metonic Cycle, a period of -6,585 days, or 223 lunar months, or about 19 years, after -which a nearly perfect recurrence of the phases and -eclipses of the moon takes place; but if so, Thales must -have had access to long series of astronomical records of -the Egyptians or the Chaldeans. There is a well-known -story as to the happy use which Columbus made of the -power of predicting eclipses in overawing the islanders of -Jamaica who refused him necessary supplies of food for his -fleet. He threatened to deprive them of the moon’s light. -“His threat was treated at first with indifference, but -when the eclipse actually commenced, the barbarians vied -with each other in the production of the necessary supplies -for the Spanish fleet.”</p> - -<p>Exactly the same kind of awe which the ancients experienced -at the prediction of eclipses, has been felt in -modern times concerning the return of comets. Seneca -asserted in distinct terms that comets would be found to -revolve in periodic orbits and return to sight. The ancient -Chaldeans and the Pythagoreans are also said to have -entertained a like opinion. But it was not until the age -of Newton and Halley that it became possible to calculate -the path of a comet in future years. A great comet -appeared in 1682, a few years before the first publication of -the <i>Principia</i>, and Halley showed that its orbit corresponded -with that of remarkable comets recorded to have appeared -in the years 1531 and 1607. The intervals of time were -not quite equal, but Halley conceived the bold idea that -this difference might be due to the disturbing power of -Jupiter, near which the comet had passed in the interval -1607–1682. He predicted that the comet would return -about the end of 1758 or the beginning of 1759, and -though Halley did not live to enjoy the sight, it was -actually detected on the night of Christmas-day, 1758. -A second return of the comet was witnessed in 1835 -nearly at the anticipated time.</p> - -<p>In recent times the discovery of Neptune has been the -most remarkable instance of prevision in astronomical<span class="pagenum" id="Page_538">538</span> -science. A full account of this discovery may be found in -several works, as for instance Herschel’s <i>Outlines of -Astronomy</i>, and <i>Grant’s History of Physical Astronomy</i>, -Chapters XII and XIII.</p> - - -<h3><i>Predictions in the Science of Light.</i></h3> - -<p>Next after astronomy the science of physical optics has -furnished the most beautiful instances of the prophetic -power of correct theory. These cases are the more striking -because they proceed from the profound application of -mathematical analysis and show an insight into the mysterious -workings of matter which is surprising to all, but -especially to those who are unable to comprehend the -methods of research employed. By its power of prevision -the truth of the undulatory theory of light has been conspicuously -proved, and the contrast in this respect between -the undulatory and Corpuscular theories is remarkable. -Even Newton could get no aid from his corpuscular theory -in the invention of new experiments, and to his followers -who embraced that theory we owe little or nothing in the -science of light. Laplace did not derive from the theory a -single discovery. As Fresnel remarks:<a id="FNanchor_448" href="#Footnote_448" class="fnanchor">448</a></p> - -<p>“The assistance to be derived from a good theory is not -to be confined to the calculation of the forces when the -laws of the phenomena are known. There are certain -laws so complicated and so singular, that observation alone, -aided by analogy, could never lead to their discovery. To -divine these enigmas we must be guided by theoretical -ideas founded on a <i>true</i> hypothesis. The theory of luminous -vibrations presents this character, and these precious -advantages; for to it we owe the discovery of optical laws -the most complicated and most difficult to divine.”</p> - -<p>Physicists who embraced the corpuscular theory had -nothing but their own quickness of observation to rely -upon. Fresnel having once seized the conditions of the -true undulatory theory, as previously stated by Young, was -enabled by the mere manipulation of his mathematical -symbols to foresee many of the complicated phenomena of -light. Who could possibly suppose, that by stopping a<span class="pagenum" id="Page_539">539</span> -portion of the rays passing through a circular aperture, -the illumination of a point upon a screen behind the aperture -might be many times multiplied. Yet this paradoxical -effect was predicted by Fresnel, and verified both by himself, -and in a careful repetition of the experiment, by Billet. -Few persons are aware that in the middle of the shadow -of an opaque circular disc is a point of light sensibly as -bright as if no disc had been interposed. This startling -fact was deduced from Fresnel’s theory by Poisson, and -was then verified experimentally by Arago. Airy, again, -was led by pure theory to predict that Newton’s rings -would present a modified appearance if produced between -a lens of glass and a plate of metal. This effect happened -to have been observed fifteen years before by Arago, unknown -to Airy. Another prediction of Airy, that there -would be a further modification of the rings when made -between two substances of very different refractive indices, -was verified by subsequent trial with a diamond. A -reversal of the rings takes place when the space intervening -between the plates is filled with a substance of intermediate -refractive power, another phenomenon predicted by theory -and verified by experiment. There is hardly a limit to the -number of other complicated effects of the interference of -rays of light under different circumstances which might be -deduced from the mathematical expressions, if it were -worth while, or which, being previously observed, can be -explained. An interesting case was observed by Herschel -and explained by Airy.<a id="FNanchor_449" href="#Footnote_449" class="fnanchor">449</a></p> - -<p>By a somewhat different effort of scientific foresight, -Fresnel discovered that any solid transparent medium -might be endowed with the power of double refraction by -mere compression. As he attributed the double refracting -power of crystals to unequal elasticity in different directions, -he inferred that unequal elasticity, if artificially -produced, would give similar phenomena. With a powerful -screw and a piece of glass, he then produced not only -the colours due to double refraction, but the actual duplication -of images. Thus, by a great scientific generalisation, -are the remarkable properties of Iceland spar shown to -belong to all transparent substances under certain conditions.<a id="FNanchor_450" href="#Footnote_450" class="fnanchor">450</a></p> -<p><span class="pagenum" id="Page_540">540</span></p> -<p>All other predictions in optical science are, however, -thrown into the shade by the theoretical discovery of -conical refraction by the late Sir W. R. Hamilton, of -Dublin. In investigating the passage of light through -certain crystals, Hamilton found that Fresnel had slightly -misinterpreted his own formulæ, and that, when rightly -understood, they indicated a phenomenon of a kind never -witnessed. A small ray of light sent into a crystal of -arragonite in a particular direction, becomes spread out -into an infinite number of rays, which form a hollow -cone within the crystal, and a hollow cylinder when -emerging from the opposite side. In another case, a -different, but equally strange, effect is produced, a ray of -light being spread out into a hollow cone at the point -where it quits the crystal. These phenomena are peculiarly -interesting, because cones and cylinders of light are -not produced in any other cases. They are opposed to all -analogy, and constitute singular exceptions, of a kind which -we shall afterwards consider more fully. Their strangeness -rendered them peculiarly fitted to test the truth of the -theory by which they were discovered; and when Professor -Lloyd, at Hamilton’s request, succeeded, after considerable -difficulty, in witnessing the new appearances, no further -doubt could remain of the validity of the wave theory -which we owe to Huyghens, Young, and Fresnel.<a id="FNanchor_451" href="#Footnote_451" class="fnanchor">451</a></p> - - -<h3><i>Predictions from the Theory of Undulations.</i></h3> - -<p>It is curious that the undulations of light, although inconceivably -rapid and small, admit of more accurate measurement -than waves of any other kind. But so far as we -can carry out exact experiments on other kinds of waves, -we find the phenomena of interference repeated, and -analogy gives considerable power of prediction. Herschel -was perhaps the first to suggest that two sounds might be -made to destroy each other by interference.<a id="FNanchor_452" href="#Footnote_452" class="fnanchor">452</a> For if one-half -of a wave travelling through a tube could be separated,<span class="pagenum" id="Page_541">541</span> -and conducted by a longer passage, so as, on rejoining -the other half, to be one-quarter of a vibration behind-hand, -the two portions would exactly neutralise each -other. This experiment has been performed with success. -The interference arising between the waves from the two -prongs of a tuning-fork was also predicted by theory, and -proved to exist by Weber; indeed it may be observed by -merely holding a vibrating fork close to the ear and turning -it round.<a id="FNanchor_453" href="#Footnote_453" class="fnanchor">453</a></p> - -<p>It is a result of the theory of sound that, if we move -rapidly towards a sounding body, or if it move rapidly -towards us, the pitch of the sound will be a little more -acute; and, <i>vice versâ</i>, when the relative motion is in the -opposite direction, the pitch will be more grave. This arises -from the less or greater intervals of time elapsing between -the successive strokes of waves upon the auditory nerve, -according as the ear moves towards or from the source -of sound relatively speaking. This effect was predicted -by theory, and afterwards verified by the experiments of -Buys Ballot, on Dutch railways, and of Scott Russell, in -England. Whenever one railway train passes another, -on the locomotive of which the whistle is being sounded, -the drop in the acuteness of the sound may be noticed at -the moment of passing. This change gives the sound a -peculiar howling character, which many persons must have -noticed. I have calculated that with two trains travelling -thirty miles an hour, the effect would amount to rather -more than half a tone, and with some express trains it -would amount to a tone. A corresponding effect is produced -in the case of light undulations, when the eye and -the luminous body approach or recede from each other. It -is shown by a slight change in the refrangibility of the -rays of light, and a consequent change in the place of the -lines of the spectrum, which has been made to give important -and unexpected information concerning the relative -approach or recession of stars.</p> - -<p>Tides are vast waves, and were the earth’s surface entirely -covered by an ocean of uniform depth, they would -admit of exact theoretical investigation. The irregular -form of the seas introduces unknown quantities and complexities<span class="pagenum" id="Page_542">542</span> -with which theory cannot cope. Nevertheless, -Whewell, observing that the tides of the German Ocean -consist of interfering waves, which arrive partly round the -North of Scotland and partly through the British Channel, -was enabled to predict that at a point about midway between -Brill on the coast of Holland, and Lowestoft no tides -would be found to exist. At that point the two waves -would be of the same amount, but in opposite phases, so -as to neutralise each other. This prediction was verified -by a surveying vessel of the British navy.<a id="FNanchor_454" href="#Footnote_454" class="fnanchor">454</a></p> - - -<h3><i>Prediction in other Sciences.</i></h3> - -<p>Generations, or even centuries, may elapse before mankind -are in possession of a mathematical theory of the constitution -of matter as complete as the theory of gravitation. -Nevertheless, mathematical physicists have in recent years -acquired a hold of some of the relations of the physical -forces, and the proof is found in anticipations of curious -phenomena which had never been observed. Professor -James Thomson deduced from Carnot’s theory of heat that -the application of pressure would lower the melting-point -of ice. He even ventured to assign the amount of this -effect, and his statement was afterwards verified by Sir W. -Thomson.<a id="FNanchor_455" href="#Footnote_455" class="fnanchor">455</a> “In this very remarkable speculation, an entirely -novel physical phenomenon was <i>predicted</i>, in anticipation -of any direct experiments on the subject; and -the actual observation of the phenomenon was pointed out -as a highly interesting object for experimental research.” -Just as liquids which expand in solidifying will have the -temperature of solidification lowered by pressure, so liquids -which contract in solidifying will exhibit the reverse effect. -They will be assisted in solidifying, as it were, by pressure, -so as to become solid at a higher temperature, as the pressure -is greater. This latter result was verified by Bunsen -and Hopkins, in the case of paraffin, spermaceti, wax, and -stearin. The effect upon water has more recently been -carried to such an extent by Mousson, that under the vast<span class="pagenum" id="Page_543">543</span> -pressure of 1300 atmospheres, water did not freeze until -cooled down to -18°C. Another remarkable prediction -of Professor Thomson was to the effect that, if a metallic -spring be weakened by a rise of temperature, work done -against the spring in bending it will cause a cooling effect. -Although the effect to be expected in a certain apparatus -was only about four-thousandths of a degree Centigrade, -Dr. Joule<a id="FNanchor_456" href="#Footnote_456" class="fnanchor">456</a> succeeded in measuring it to the extent of three-thousandths -of a degree, such is the delicacy of modern -heat measurements. I cannot refrain from quoting Dr. Joule’s -reflections upon this fact. “Thus even in the above delicate -case,” he says, “is the formula of Professor Thomson -completely verified. The mathematical investigation of the -thermo-elastic qualities of metals has enabled my illustrious -friend to predict with certainty a whole class of highly interesting -phenomena. To him especially do we owe the -important advance which has been recently made to a new -era in the history of science, when the famous philosophical -system of Bacon will be to a great extent superseded, -and when, instead of arriving at discovery by induction -from experiment, we shall obtain our largest accessions of -new facts by reasoning deductively from fundamental -principles.”</p> - -<p>The theory of electricity is a necessary part of the -general theory of matter, and is rapidly acquiring the -power of prevision. As soon as Wheatstone had proved -experimentally that the conduction of electricity occupies -time, Faraday remarked in 1838, with wonderful sagacity, -that if the conducting wires were connected with the -coatings of a large Leyden jar, the rapidity of conduction -would be lessened. This prediction remained unverified -for sixteen years, until the submarine cable was laid beneath -the Channel. A considerable retardation of the -electric spark was then detected, and Faraday at once -pointed out that the wire surrounded by water resembles -a Leyden jar on a large scale, so that each message sent -through the cable verified his remark of 1838.<a id="FNanchor_457" href="#Footnote_457" class="fnanchor">457</a></p> - -<p>The joint relations of heat and electricity to the metals -constitute a new science of thermo-electricity by which<span class="pagenum" id="Page_544">544</span> -Sir W. Thomson was enabled to anticipate the following -curious effect, namely, that an electric current passing in -an iron bar from a hot to a cold part produces a cooling -effect, but in a copper bar the effect is exactly opposite in -character, that is, the bar becomes heated.<a id="FNanchor_458" href="#Footnote_458" class="fnanchor">458</a> The action -of crystals with regard to heat and electricity was partly -foreseen on the grounds of theory by Poisson.</p> - -<p>Chemistry, although to a great extent an empirical -science, has not been without prophetic triumphs. The -existence of the metals potassium and sodium was foreseen -by Lavoisier, and their elimination by Davy was one -of the chief <i>experimenta crucis</i> which established Lavoisier’s -system. The existence of many other metals -which eye had never seen was a natural inference, and -theory has not been at fault. In the above cases the -compounds of the metal were well known, and it was -the result of decomposition that was foretold. The discovery -in 1876 of the metal gallium is peculiarly interesting -because the existence of this metal, previously -wholly unknown, had been inferred from theoretical considerations -by M. Mendelief, and some of its properties -had been correctly predicted. No sooner, too, had a -theory of organic compounds been conceived by Professor -A. W. Williamson than he foretold the formation of a -complex substance consisting of water in which both -atoms of hydrogen are replaced by atoms of acetyle. -This substance, known as the acetic anhydride, was afterwards -produced by Gerhardt. In the subsequent progress -of organic chemistry occurrences of this kind have become -common. The theoretical chemist by the classification of -his specimens and the manipulation of his formulæ can -plan out whole series of unknown oils, acids, and alcohols, -just as a designer might draw out a multitude of patterns. -Professor Cayley has even calculated for certain cases the -possible numbers of chemical compounds.<a id="FNanchor_459" href="#Footnote_459" class="fnanchor">459</a> The formation -of many such substances is a matter of course; but there -is an interesting prediction given by Hofmann, concerning -the possible existence of new compounds of sulphur and<span class="pagenum" id="Page_545">545</span> -selenium, and even oxides of ammonium, which it remains -for chemists to verify.<a id="FNanchor_460" href="#Footnote_460" class="fnanchor">460</a></p> - - -<h3><i>Prediction by Inversion of Cause and Effect.</i></h3> - -<p>There is one process of experiment which has so often led -to important discoveries as to deserve separate illustration—I -mean the inversion of Cause and Effect. Thus if -A and B in one experiment produce C as a consequent, -then antecedents of the nature of B and C may usually be -made to produce a consequent of the nature of A inverted -in direction. When we apply heat to a gas it tends to -expand; hence if we allow the gas to expand by its own -elastic force, cold is the result; that is, B (air) and C -(expansion) produce the negative of A (heat). Again, B -(air) and compression, the negative of C, produce A (heat). -Similar results may be expected in a multitude of cases. -It is a familiar law that heat expands iron. What may be -expected, then, if instead of increasing the length of an -iron bar by heat we use mechanical force and stretch the -bar? Having the bar and the former consequent, expansion, -we should expect the negative of the former antecedent, -namely cold. The truth of this inference was proved -by Dr. Joule, who investigated the amount of the effect -with his usual skill.<a id="FNanchor_461" href="#Footnote_461" class="fnanchor">461</a></p> - -<p>This inversion of cause and effect in the case of heat -may be itself inverted in a highly curious manner. It -happens that there are a few substances which are unexplained -exceptions to the general law of expansion by heat. -India-rubber especially is remarkable for <i>contracting</i> when -heated. Since, then, iron and india-rubber are oppositely -related to heat, we may expect that as distension of the -iron produced cold, distension of the india-rubber will -produce heat. This is actually found to be the case, and -anyone may detect the effect by suddenly stretching an -india-rubber band while the middle part is in the mouth. -When being stretched it grows slightly warm, and when -relaxed cold.</p> - -<p>The reader will see that some of the scientific predictions -mentioned in preceding sections were due to the principle<span class="pagenum" id="Page_546">546</span> -of inversion; for instance, Thomson’s speculations on the -relation between pressure and the melting-point. But -many other illustrations could be adduced. The usual -agent by which we melt a substance is heat; but if we can -melt a substance without heat, then we may expect the -negative of heat as an effect. This is the foundation of all -freezing mixtures. The affinity of salt for water causes it -to melt ice, and we may thus reduce the temperature to -Fahrenheit’s zero. Calcium chloride has so much higher -an attraction for water that a temperature of -45° C. may -be attained by its use. Even the solution of a certain -alloy of lead, tin, and bismuth in mercury, may be made -to reduce the temperature through 27° C. All the other -modes of producing cold are inversions of more familiar -uses of heat. Carré’s freezing machine is an inverted -distilling apparatus, the distillation being occasioned by -chemical affinity instead of heat. Another kind of freezing -machine is the exact inverse of the steam-engine.</p> - -<p>A very paradoxical effect is due to another inversion. -It is hard to believe that a current of steam at 100° C. can -raise a body of liquid to a higher temperature than the -steam itself possesses. But Mr. Spence has pointed out -that if the boiling-point of a saline solution be above 100°, -it will continue, on account of its affinity for water, to condense -steam when above 100° in temperature. It will condense -the steam until heated to the point at which the tension -of its vapour is equal to that of the atmosphere, that -is, its own boiling-point.<a id="FNanchor_462" href="#Footnote_462" class="fnanchor">462</a> Again, since heat melts ice, we -might expect to produce heat by the inverse change from -water into ice. This is accomplished in the phenomenon -of suspended freezing. Water may be cooled in a clean -glass vessel many degrees below the freezing-point, and -yet retained in the liquid condition. But if disturbed, and -especially if brought into contact with a small particle of -ice, it instantly solidifies and rises in temperature to 0° C. -The effect is still better displayed in the lecture-room -experiment of the suspended crystallisation of a solution -of sodium sulphate, in which a sudden rise of temperature -of 15° or 20° C. is often manifested.</p> - -<p>The science of electricity is full of most interesting cases<span class="pagenum" id="Page_547">547</span> -of inversion. As Professor Tyndall has remarked, Faraday -had a profound belief in the reciprocal relations of the -physical forces. The great starting-point of his researches, -the discovery of electro-magnetism, was clearly an inversion. -Oersted and Ampère had proved that with an electric current -and a magnet in a particular position as antecedents, -motion is the consequent. If then a magnet, a wire and -motion be the antecedents, an <i>opposite</i> electric current will -be the consequent. It would be an endless task to trace -out the results of this fertile relationship. Another part of -Faraday’s researches was occupied in ascertaining the direct -and inverse relations of magnetic and diamagnetic, amorphous -and crystalline substances in various circumstances. -In all other relations of electricity the principle of inversion -holds. The voltameter or the electro-plating cell is -the inverse of the galvanic battery. As heat applied to a -junction of antimony and bismuth bars produces electricity, -it follows that an electric current passed through such -a junction will produce cold. But it is now sufficiently apparent -that inversion of cause and effect is a most fertile -means of discovery and prediction.</p> - - -<h3><i>Facts known only by Theory.</i></h3> - -<p>Of the four classes of facts enumerated in p. <a href="#Page_525">525</a> the -last remains unconsidered. It includes the unverified predictions -of science. Scientific prophecy arrests the attention -of the world when it refers to such striking events as -an eclipse, the appearance of a great comet, or any phenomenon -which people can verify with their own eyes. But -it is surely a matter for greater wonder that a physicist -describes and measures phenomena which eye cannot see, -nor sense of any kind detect. In most cases this arises -from the effect being too small in amount to affect our -organs of sense, or come within the powers of our instruments -as at present constructed. But there is a class of -yet more remarkable cases, in which a phenomenon cannot -possibly be observed, and yet we can say what it would be -if it were observed.</p> - -<p>In astronomy, systematic aberration is an effect of the -sun’s proper motion almost certainly known to exist, but -which we have no hope of detecting by observation in the<span class="pagenum" id="Page_548">548</span> -present age of the world. As the earth’s motion round the -sun combined with the motion of light causes the stars to -deviate apparently from their true positions to the extent -of about 18″ at the most, so the motion of the whole planetary -system through space must occasion a similar displacement -of at most 5″. The ordinary aberration can be readily -detected with modern astronomical instruments, because it -goes through a yearly change in direction or amount; but -systematic aberration is constant so long as the planetary -system moves uniformly in a sensibly straight line. Only -then in the course of ages, when the curvature of the sun’s -path becomes apparent, can we hope to verify the existence -of this kind of aberration. A curious effect must also be -produced by the sun’s proper motion upon the apparent -periods of revolution of the binary stars.</p> - -<p>To my mind, some of the most interesting truths in the -whole range of science are those which have not been, and -in many cases probably never can be, verified by trial. -Thus the chemist assigns, with a very high degree of probability, -the vapour densities of such elements as carbon -and silicon, which have never been observed separately in -a state of vapour. The chemist is also familiar with the -vapour densities of elements at temperatures at which the -elements in question never have been, and probably never -can be, submitted to experiment in the form of vapour.</p> - -<p>Joule and others have calculated the actual velocity of -the molecules of a gas, and even the number of collisions -which must take place per second during their constant -circulation. Physicists have not yet given us the exact -magnitudes of the particles of matter, but they have ascertained -by several methods the limits within which their -magnitudes must lie. Such scientific results must be for -ever beyond the power of verification by the senses. I -have elsewhere had occasion to remark that waves of light, -the intimate processes of electrical changes, the properties -of the ether which is the base of all phenomena, are necessarily -determined in a hypothetical, but not therefore a -less certain manner.</p> - -<p>Though only two of the metals, gold and silver, have -ever been observed to be transparent, we know on the -grounds of theory that they are all more or less so; we -can even estimate by theory their refractive indices, and<span class="pagenum" id="Page_549">549</span> -prove that they are exceedingly high. The phenomena -of elliptic polarisation, and perhaps also those of internal -radiation,<a id="FNanchor_463" href="#Footnote_463" class="fnanchor">463</a> depend upon the refractive index, and thus, even -when we cannot observe any refracted rays, we can indirectly -learn how they would be refracted.</p> - -<p>In many cases large quantities of electricity must be -produced, which we cannot observe because it is instantly -discharged. In the common electric machine the cylinder -and rubber are made of non-conductors, so that we can -separate and accumulate the electricity. But a little damp, -by serving as a conductor, prevents this separation from -enduring any sensible time. Hence there is no doubt that -when we rub two good conductors against each other, for -instance two pieces of metals, much electricity is produced, -but instantaneously converted into some other form of -energy. Joule believes that all the heat of friction is -transmuted electricity.</p> - -<p>As regards phenomena of insensible amount, nature is -absolutely full of them. We must regard those changes -which we can observe as the comparatively rare aggregates -of minuter changes. On a little reflection we must allow -that no object known to us remains for two instants of -exactly the same temperature. If so, the dimensions of -objects must be in a perpetual state of variation. The -minor planetary and lunar perturbations are infinitely -numerous, but usually too small to be detected by observation, -although their amounts may be assigned by theory. -There is every reason to believe that chemical and electric -actions of small amount are constantly in progress. The -hardest substances, if reduced to extremely small particles, -and diffused in pure water, manifest oscillatory movements -which must be due to chemical and electric changes, so -slight that they go on for years without affecting appreciably -the weight of the particles.<a id="FNanchor_464" href="#Footnote_464" class="fnanchor">464</a> The earth’s magnetism must -more or less affect every object which we handle. As -Tyndall remarks, “An upright iron stone influenced by the -earth’s magnetism becomes a magnet, with its bottom a -north and its top a south pole. Doubtless, though in an -immensely feebler degree, every erect marble statue is a<span class="pagenum" id="Page_550">550</span> -true diamagnet, with its head a north pole and its feet a -south pole. The same is certainly true of man as he stands -upon the earth’s surface, for all the tissues of the human -body are diamagnetic.”<a id="FNanchor_465" href="#Footnote_465" class="fnanchor">465</a> The sun’s light produces a very -quick and perceptible effect upon the photographic plate; -in all probability it has a less effect upon a great variety -of substances. We may regard every phenomenon as an -exaggerated and conspicuous case of a process which is, in -infinitely numerous cases, beyond the means of observation.</p> - - -<hr class="chap x-ebookmaker-drop"> - -<div class="chapter"> -<p><span class="pagenum" id="Page_551">551</span></p> -<h2 class="nobreak" id="CHAPTER_XXV">CHAPTER XXV. - -<span class="title">ACCORDANCE OF QUANTITATIVE THEORIES.</span></h2> -</div> - -<p class="ti0">In the preceding chapter we found that facts may be -classed under four heads as regards their connection with -theory, and our powers of explanation or prediction. The -facts hitherto considered were generally of a qualitative -rather than a quantitative nature; but when we look -exclusively to the quantity of a phenomenon, and the -various modes in which we may determine its amount, -nearly the same system of classification will hold good. -There will, however, be five possible cases:—</p> - -<p>(1) We may directly and empirically measure a phenomenon, -without being able to explain why it should -have any particular quantity, or to connect it by theory -with other quantities.</p> - -<p>(2) In a considerable number of cases we can theoretically -predict the existence of a phenomenon, but are -unable to assign its amount, except by direct measurement, -or to explain the amount theoretically when thus -ascertained.</p> - -<p>(3) We may measure a quantity, and afterwards explain -it as related to other quantities, or as governed by known -quantitative laws.</p> - -<p>(4) We may predict the quantity of an effect on theoretical -grounds, and afterwards confirm the prediction by -direct measurement.</p> - -<p>(5) We may indirectly determine the quantity of an -effect without being able to verity it by experiment.</p> - -<p>These classes of quantitative facts might be illustrated -by an immense number of interesting points in the history<span class="pagenum" id="Page_552">552</span> -of physical science. Only a few instances of each class -can be given here.</p> - - -<h3><i>Empirical Measurements.</i></h3> - -<p>Under the first head of purely empirical measurements, -which have not been brought under any theoretical system, -may be placed the great bulk of quantitative facts recorded -by scientific observers. The tables of numerical results -which abound in books on chemistry and physics, the huge -quartos containing the observations of public observatories, -the multitudinous tables of meteorological observations, -which are continually being published, the more abstruse -results concerning terrestrial magnetism—such results of -measurement, for the most part, remain empirical, either -because theory is defective, or the labour of calculation -and comparison is too formidable. In the Greenwich -Observatory, indeed, the salutary practice has been maintained -by the present Astronomer Royal, of always reducing -the observations, and comparing them with the theories -of the several bodies. The divergences from theory thus -afford material for the discovery of errors or of new phenomena; -in short, the observations have been turned to -the use for which they were intended. But it is to be -feared that other establishments are too often engaged in -merely recording numbers of which no real use is made, -because the labour of reduction and comparison with -theory is too great for private inquirers to undertake. In -meteorology, especially, great waste of labour and money -is taking place, only a small fraction of the results recorded -being ever used for the advancement of the science. For -one meteorologist like Quetelet, Dove, or Baxendell, who -devotes himself to the truly useful labour of reducing other -people’s observations, there are hundreds who labour under -the delusion that they are advancing science by loading -our book-shelves with numerical tables. It is to be feared, -in like manner, that almost the whole bulk of statistical -numbers, whether commercial, vital, or moral, is of little -scientific value. Purely empirical measurements may -have a direct practical value, as when tables of the specific -gravity, or strength of materials, assist the engineer; the -specific gravities of mixtures of water with acids, alcohols,<span class="pagenum" id="Page_553">553</span> -salts, &c., are useful in chemical manufactories, custom-house -gauging, &c.; observations of rainfall are requisite -for questions of water supply; the refractive index of -various kinds of glass must be known in making achromatic -lenses; but in all such cases the use made of the -measurements is not scientific but practical. It may be -asserted, that no number which remains isolated, and -uncompared by theory with other numbers, is of scientific -value. Having tried the tensile strength of a piece of iron -in a particular condition, we know what will be the strength -of the same kind of iron in a similar condition, provided -we can ever meet with that exact kind of iron again; but -we cannot argue from piece to piece, nor lay down any laws -exactly connecting the strength of iron with the quantity -of its impurities.</p> - - -<h3><i>Quantities indicated by Theory, but Empirically Measured.</i></h3> - -<p>In many cases we are able to foresee the existence of -a quantitative effect, on the ground of general principles, -but are unable, either from the want of numerical data, -or from the entire absence of any mathematical theory, to -assign the amount of such effect. We then have recourse -to direct experiment to determine its amount. Whether -we argued from the oceanic tides by analogy, or deductively -from the theory of gravitation, there could be no -doubt that atmospheric tides of some amount must occur -in the atmosphere. Theory, however, even in the hands -of Laplace, was not able to overcome the complicated -mechanical conditions of the atmosphere, and predict the -amounts of such tides; and, on the other hand, these -amounts were so small, and were so masked by far larger -undulations arising from the heating power of the sun, -and from other meteorological disturbances, that they -would probably have never been discovered by purely -empirical observations. Theory having, however, indicated -their existence and their periods, it was easy to -make series of barometrical observations in places selected -so as to be as free as possible from casual fluctuations, and -then, by the suitable application of the method of means, to -detect the small effects in question. The principal lunar<span class="pagenum" id="Page_554">554</span> -atmospheric tide was thus proved to amount to between -·003 and ·004 inch.<a id="FNanchor_466" href="#Footnote_466" class="fnanchor">466</a></p> - -<p>Theory yields the greatest possible assistance in applying -the method of means. For if we have a great number of -empirical measurements, each representing the joint effect -of a number of causes, our object will be to take the mean -of all those in which the effect to be measured is present, -and compare it with the mean of the remainder in which -the effect is absent, or acts in the opposite direction. The -difference will then represent the amount of the effect, or -double the amount respectively. Thus, in the case of the -atmospheric tides, we take the mean of all the observations -when the moon was on the meridian, and compare it with -the mean of all observations when she was on the horizon. -In this case we trust to chance that all other effects will -lie about as often in one direction as the other, and will -neutralise themselves in the drawing of each mean. It is -a great advantage, however, to be able to decide by theory -when each principal disturbing effect is present or absent; -for the means may then be drawn so as to separate each -such effect, leaving only minor and casual divergences to -the law of error. Thus, if there be three principal effects, -and we draw means giving respectively the sum of all -three, the sum of the first two, and the sum of the last -two, then we gain three simple equations, by the solution -of which each quantity is determined.</p> - - -<h3><i>Explained Results of Measurement.</i></h3> - -<p>The second class of measured phenomena contains those -which, after being determined in a direct and purely empirical -application of measuring instruments, are afterwards -shown to agree with some hypothetical explanation. Such -results are turned to their proper use, and several advantages -may arise from the comparison. The correspondence -with theory will seldom or never be precise; and, even if -it be so, the coincidence must be regarded as accidental.</p> - -<p>If the divergences between theory and experiment be -comparatively small, and variable in amount and direction, -they may often be safely attributed to inconsiderable<span class="pagenum" id="Page_555">555</span> -sources of error in the experimental processes. The strict -method of procedure is to calculate the probable error of -the mean of the observed results (p. <a href="#Page_387">387</a>), and then observe -whether the theoretical result falls within the limits of -probable error. If it does, and if the experimental results -agree as well with theory as they agree with each other, -then the probability of the theory is much increased, and -we may employ the theory with more confidence in the -anticipation of further results. The probable error, it -should be remembered, gives a measure only of the effects -of incidental and variable sources of error, but in no degree -indicates the amount of fixed causes of error. Thus, if the -mean results of two modes of determining a quantity are -so far apart that the limits of probable error do not overlap, -we may infer the existence of some overlooked source of -fixed error in one or both modes. We will further consider -in a subsequent section the discordance of measurements.</p> - - -<h3><i>Quantities determined by Theory and verified by -Measurement.</i></h3> - -<p>One of the most satisfactory tests of a theory consists in -its application not only to predict the nature of a phenomenon, -and the circumstances in which it may be observed, -but also to assign the precise quantity of the phenomenon. -If we can subsequently apply accurate instruments and -measure the amount of the phenomenon witnessed, we have -an excellent opportunity of verifying or negativing the -theory. It was in this manner that Newton first attempted -to verify his theory of gravitation. He knew approximately -the velocity produced in falling bodies at the earth’s surface, -and if the law of the inverse square of the distance held -true, and the reputed distance of the moon was correct, he -could infer that the moon ought to fall towards the earth at -the rate of fifteen feet in one minute. Now, the actual -divergence of the moon from the tangent of its orbit appeared -to amount only to thirteen feet in one minute, and -there was a discrepancy of two feet in fifteen, which caused -Newton to lay “aside at that time any further thoughts of -this matter.” Many years afterwards, probably fifteen or -sixteen years, Newton obtained more precise data from<span class="pagenum" id="Page_556">556</span> -which he could calculate the size of the moon’s orbit, and -he then found the discrepancy to be inconsiderable.</p> - -<p>His theory of gravitation was thus verified as far as the -moon was concerned; but this was to him only the beginning -of a long course of deductive calculations, each ending -in a verification. If the earth and moon attract each other, -and also the sun and the earth, there is reason to expect -that the sun and moon should attract each other. Newton -followed out the consequences of this inference, and showed -that the moon would not move as if attracted by the earth -only, but sometimes faster and sometimes slower. Comparison -with Flamsteed’s observations of the moon showed -that such was the case. Newton argued again, that as the -waters of the ocean are not rigidly attached to the earth, -they might attract the moon, and be attracted in return, -independently of the rest of the earth. Certain daily -motions resembling the tides would then be caused, and -there were the tides to verify the reasoning. It was the -extraordinary power with which Newton traced out geometrically -the consequences of his theory, and submitted them -to repeated comparison with experience, which constitutes -his pre-eminence over all physicists.</p> - - -<h3><i>Quantities determined by Theory and not verified.</i></h3> - -<p>It will continually happen that we are able, from certain -measured phenomena and a correct theory, to determine -the amount of some other phenomenon which we may -either be unable to measure at all, or to measure with an -accuracy corresponding to that required to verify the prediction. -Thus Laplace having worked out a theory of the -motions of Jupiter’s satellites on the hypothesis of gravitation, -found that these motions were greatly affected by -the spheroidal form of Jupiter. The motions of the -satellites can be observed with great accuracy owing to -their frequent eclipses and transits, and from these motions -he was able to argue inversely, and assign the ellipticity -of the planet. The ratio of the polar and equatorial axes -thus determined was very nearly that of 13 to 14; and it -agrees well with such direct micrometrical measurements -of the planet as have been made; but Laplace believed that -the theory gave a more accurate result than direct observation<span class="pagenum" id="Page_557">557</span> -could yield, so that the theory could hardly be said -to admit of direct verification.</p> - -<p>The specific heat of air was believed on the grounds of -direct experiment to amount to 0·2669, the specific heat of -water being taken as unity; but the methods of experiment -were open to considerable causes of error. Rankine -showed in 1850 that it was possible to calculate from the -mechanical equivalent of heat and other thermodynamic -data, what this number should be, and he found it to be -0·2378. This determination was at the time accepted as -the most satisfactory result, although not verified; subsequently -in 1853 Regnault obtained by direct experiment -the number 0·2377, proving that the prediction had been -well grounded.</p> - -<p>It is readily seen that in quantitative questions verification -is a matter of degree and probability. A less -accurate method of measurement cannot verify the results -of a more accurate method, so that if we arrive at a -determination of the same physical quantity in several -distinct modes it is often a delicate matter to decide which -result is most reliable, and should be used for the indirect -determination of other quantities. For instance, Joule’s -and Thomson’s ingenious experiments upon the thermal -phenomena of fluids in motion<a id="FNanchor_467" href="#Footnote_467" class="fnanchor">467</a> involved, as one physical -constant, the mechanical equivalent of heat; if requisite, -then, they might have been used to determine that important -constant. But if more direct methods of experiment -give the mechanical equivalent of heat with superior -accuracy, then the experiments on fluids will be turned to -a better use in determining various quantities relating to -the theory of fluids. We will further consider questions -of this kind in succeeding sections.</p> - -<p>There are of course many quantities assigned on theoretical -grounds which we are quite unable to verify with -corresponding accuracy. The thickness of a film of gold -leaf, the average depths of the oceans, the velocity of a -star’s approach to or regression from the earth as inferred -from spectroscopic data (pp. <a href="#Page_296">296</a>–99), are cases in point; -but many others might be quoted where direct verification -seems impossible. Newton and subsequent physicists<span class="pagenum" id="Page_558">558</span> -have measured light undulations, and by several methods -we learn the velocity with which light travels. Since an -undulation of the middle green is about five ten-millionths -of a metre in length, and travels at the rate of nearly -300,000,000 of metres per second, it follows that about -600,000,000,000,000 undulations must strike in one -second the retina of an eye which perceives such light. -But how are we to verify such an astounding calculation -by directly counting pulses which recur six hundred -billions of times in a second?</p> - - -<h3><i>Discordance of Theory and Experiment.</i></h3> - -<p>When a distinct want of accordance is found to exist -between the results of theory and direct measurement, -interesting questions arise as to the mode in which we can -account for this discordance. The ultimate explanation -of the discrepancy may be accomplished in at least four -ways as follows:—</p> - -<p>(1) The direct measurement may be erroneous owing to -various sources of casual error.</p> - -<p>(2) The theory may be correct as far as regards the -general form of the supposed laws, but some of the constant -numbers or other quantitative data employed in the -theoretical calculations may be inaccurate.</p> - -<p>(3) The theory may be false, in the sense that the forms -of the mathematical equations assumed to express the laws -of nature are incorrect.</p> - -<p>(4) The theory and the involved quantities may be -approximately accurate, but some regular unknown cause -may have interfered, so that the divergence may be regarded -as a <i>residual effect</i> representing possibly a new and -interesting phenomenon.</p> - -<p>No precise rules can be laid down as to the best mode -of proceeding to explain the divergence, and the experimentalist -will have to depend upon his own insight and -knowledge; but the following recommendations may be -made.</p> - -<p>If the experimental measurements are not numerous, -repeat them and take a more extensive mean result, the probable -accuracy of which, as regards casual errors, will increase -as the square root of the number of experiments. Supposing<span class="pagenum" id="Page_559">559</span> -that no considerable modification of the result is thus -effected, we may suspect the existence of more deep-seated -sources of error in our method of measurement. The next -resource will be to change the size and form of the apparatus -employed, and to introduce various modifications in -the materials employed or the course of procedure, in the -hope (p. <a href="#Page_396">396</a>) that some cause of constant error may thus -be removed. If the inconsistency with theory still remains -unreduced we may attempt to invent some widely different -mode of arriving at the same physical quantity, so that we -may be almost sure that the same cause of error will not -affect both the new and old results. In some cases it is -possible to find five or six essentially different modes of -arriving at the same determination.</p> - -<p>Supposing that the discrepancy still exists we may -begin to suspect that our direct measurements are correct, -and that the data employed in the theoretical calculations -are inaccurate. We must now review the grounds on -which these data depend, consisting as they must ultimately -do of direct measurements. A comparison of the -recorded data will show the degree of probability attaching -to the mean result employed; and if there is any ground -for imagining the existence of error, we should repeat the -observations, and vary the forms of experiment just as in -the case of the previous direct measurements. The continued -existence of the discrepancy must show that we -have not attained to a complete acquaintance with the -theory of the causes in action, but two different cases still -remain. We may have misunderstood the action of those -causes which we know to exist, or we may have overlooked -the existence of one or more other causes. In the first -case our hypothesis appears to be wrongly chosen and -inapplicable; but whether we are to reject it will depend -upon whether we can form another hypothesis which -yields a more accurate accordance. The probability of an -hypothesis, it will be remembered (p. <a href="#Page_243">243</a>), is to be judged, -in the absence of <i>à priori</i> grounds of judgment, by the -probability that if the supposed causes exist the observed -result follows; but as there is now little probability of -reconciling the original hypothesis with our direct measurements -the field is open for new hypotheses, and any one -which gives a closer accordance with measurement will so<span class="pagenum" id="Page_560">560</span> -far have better claims to attention. Of course we must -never estimate the probability of an hypothesis merely by -its accordance with a few results only. Its general analogy -and accordance with other known laws of nature, and the -fact that it does not conflict with other probable theories, -must be taken into account, as we shall see in the next -book. The requisite condition of a good hypothesis, that -it must admit of the deduction of facts verified in observation, -must be interpreted in the widest manner, as including -all ways in which there may be accordance or discordance. -All our attempts at reconciliation having failed, the only -conclusion we can come to is that some unknown cause of -a new character exists. If the measurements be accurate -and the theory probable, then there remains a <i>residual phenomenon</i>, -which, being devoid of theoretical explanation, -must be set down as a new empirical fact worthy of further -investigation. Outstanding residual discrepancies have -often been found to involve new discoveries of the greatest -importance.</p> - - -<h3><i>Accordance of Measurements of Astronomical Distances.</i></h3> - -<p>One of the most instructive instances which we can -meet, of the manner in which different measurements confirm -or check each other, is furnished by the determination -of the velocity of light, and the dimensions of the planetary -system. Roemer first discovered that light requires time -to travel, by observing that the eclipses of Jupiter’s satellites, -although they occur at fixed moments of absolute time, are -visible at different moments in different parts of the earth’s -orbit, according to the distance between the earth and -Jupiter. The time occupied by light in traversing the -mean semi-diameter of the earth’s orbit is found to be -about eight minutes. The mean distance of the sun and -earth was long assumed by astronomers as being about -95,274,000 miles, this result being deduced by Bessel from -the observations of the transit of Venus, which occurred in -1769, and which were found to give the solar parallax, or -which is the same thing, the apparent angular magnitude -of the earth seen from the sun, as equal to 8″·578. -Dividing the mean distance of the sun and earth by the<span class="pagenum" id="Page_561">561</span> -number of seconds in 8<sup>m</sup>. 13<sup>s</sup>.3 we find the velocity of light -to be about 192,000 miles per second.</p> - -<p>Nearly the same result was obtained in what seems a -different manner. The aberration of light is the apparent -change in the direction of a ray of light owing to the composition -of its motion with that of the earth’s motion -round the sun. If we know the amount of aberration and -the mean velocity of the earth, we can estimate that of -light, which is thus found to be 191,100 miles per second. -Now this determination depends upon a new physical -quantity, that of aberration, which is ascertained by direct -observation of the stars, so that the close accordance of the -estimates of the velocity of light as thus arrived at by different -methods might seem to leave little room for doubt, -the difference being less than one per cent.</p> - -<p>Nevertheless, experimentalists were not satisfied until -they had succeeded in measuring the velocity of light by -direct experiments performed upon the earth’s surface. -Fizeau, by a rapidly revolving toothed wheel, estimated the -velocity at 195,920 miles per second. As this result differed -by about one part in sixty from estimates previously -accepted, there was thought to be room for further investigation. -The revolving mirror, used by Wheatstone in -measuring the velocity of electricity, was now applied in a -more refined manner by Fizeau and by Foucault to determine -the velocity of light. The latter physicist came to -the startling conclusion that the velocity was not really -more than 185,172 miles per second. No repetition of the -experiment would shake this result, and there was accordingly -a discrepancy between the astronomical and the experimental -results of about 7,000 miles per second. The -latest experiments, those of M. Cornu, only slightly raise -the estimate, giving 186,660 miles per second. A little -consideration shows that both the astronomical determinations -involve the magnitude of the earth’s orbit as one -datum, because our estimate of the earth’s velocity in its -orbit depends upon our estimate of the sun’s mean distance. -Accordingly as regards this quantity the two astronomical -results count only for one. Though the transit of Venus -had been considered to give the best data for the calculation -of the sun’s parallax, yet astronomers had not neglected -less favourable opportunities. Hansen, calculating from<span class="pagenum" id="Page_562">562</span> -certain inequalities in the moon’s motion, had estimated -it at 8″·916; Winneke, from observations of Mars, at -8″·964; Leverrier, from the motions of Mars, Venus, and -the moon, at 8″·950. These independent results agree -much better with each other than with that of Bessel -(8″·578) previously received, or that of Encke (8″·58) -deduced from the transits of Venus in 1761 and 1769, and -though each separately might be worthy of less credit, yet -their close accordance renders their mean result (8″·943) -comparable in probability with that of Bessel. It was -further found that if Foucault’s value for the velocity of -light were assumed to be correct, and the sun’s distance -were inversely calculated from that, the sun’s parallax -would be 8″·960, which closely agreed with the above -mean result. This further correspondence of independent -results threw the balance of probability strongly against -the results of the transit of Venus, and rendered it desirable -to reconsider the observations made on that occasion. -Mr. E. J. Stone, having re-discussed those observations,<a id="FNanchor_468" href="#Footnote_468" class="fnanchor">468</a> -found that grave oversights had been made in the calculations, -which being corrected would alter the estimate of -parallax to 8″·91, a quantity in such comparatively close -accordance with the other results that astronomers did not -hesitate at once to reduce their estimate of the sun’s mean -distance from 95,274,000 to 91,771,000, miles, although -this alteration involved a corresponding correction in the -assumed magnitudes and distances of most of the heavenly -bodies. The solar parallax is now (1875) believed to be -about 8″·878, the number deduced from Cornu’s experiments -on the velocity of light. This result agrees very -closely with 8″·879, the estimate obtained from new observations -on the transit of Venus, by the French observers, -and with 8″·873, the result of Galle’s observations of the -planet Flora. When all the observations of the late transit -of Venus are fully discussed the sun’s distance will probably -be known to less than one part in a thousand, if not one -part in ten thousand.<a id="FNanchor_469" href="#Footnote_469" class="fnanchor">469</a></p> - -<p><span class="pagenum" id="Page_563">563</span></p> - -<p>In this question the theoretical relations between the -velocity of light, the constant of aberration, the sun’s parallax, -and the sun’s mean distance, are of the simplest -character, and can hardly be open to any doubt, so that -the only doubt was as to which result of observation was -the most reliable. Eventually the chief discrepancy was -found to arise from misapprehension in the reduction -of observations, but we have a satisfactory example of the -value of different methods of estimation in leading to the -detection of a serious error. Is it not surprising that -Foucault by measuring the velocity of light when passing -through the space of a few yards, should lead the way -to a change in our estimates of the magnitudes of the -whole universe?</p> - - -<h3><i>Selection of the best Mode of Measurement.</i></h3> - -<p>When we once obtain command over a question of -physical science by comprehending the theory of the subject, -we often have a wide choice opened to us as regards -the methods of measurement, which may thenceforth be -made to give the most accurate results. If we can measure -one fundamental quantity very precisely we may be able -by theory to determine accurately many other quantitative -results. Thus, if we determine satisfactorily the atomic -weights of certain elements, we do not need to determine -with equal accuracy the composition and atomic weights of -their several compounds. Having learnt the relative -atomic weights of oxygen and sulphur, we can calculate the -composition by weight of the several oxides of sulphur. -Chemists accordingly select with the greatest care that -compound of two elements which seems to allow of the -most accurate analysis, so as to give the ratio of their -atomic weights. It is obvious that we only need the ratio -of the atomic weight of each element to that of some common -element, in order to calculate, that of each to each. -Moreover the atomic weight stands in simple relation to -other quantitative facts. The weights of equal volumes of -elementary gases at equal temperature and pressure have<span class="pagenum" id="Page_564">564</span> -the same ratios as the atomic weights; now, as nitrogen -under such circumstances weighs 14·06 times as much as -hydrogen, we may infer that the atomic weight of nitrogen -is about 14·06, or more probably 14·00, that of hydrogen -being unity. There is much evidence, again, that the -specific heats of elements are inversely as their atomic -weights, so that these two classes of quantitative data -throw light mutually upon each other. In fact the atomic -weight, the atomic volume, and the atomic heat of an -element, are quantities so closely connected that the determination -of one will lead to that of the others. The -chemist has to solve a complicated problem in deciding in -the case of each of 60 or 70 elements which mode of determination -is most accurate. Modern chemistry presents us -with an almost infinitely extensive web of numerical ratios -developed out of a few fundamental ratios.</p> - -<p>In hygrometry we have a choice among at least four -modes of measuring the quantity of aqueous vapour contained -in a given bulk of air. We can extract the vapour -by absorption in sulphuric acid, and directly weigh its -amount; we can place the air in a barometer tube and -observe how much the absorption of the vapour alters -the elastic force of the air; we can observe the dew-point -of the air, that is the temperature at which the vapour -becomes saturated; or, lastly, we can insert a dry and wet -bulb thermometer and observe the temperature of an -evaporating surface. The results of each mode can be connected -by theory with those of the other modes, and we -can select for each experiment that mode which is most -accurate or most convenient. The chemical method of -direct measurement is capable of the greatest accuracy, but -is troublesome; the dry and wet bulb thermometer is -sufficiently exact for meteorological purposes and is most -easy to use.</p> - - -<h3><i>Agreement of Distinct Modes of Measurement.</i></h3> - -<p>Many illustrations might be given of the accordance -which has been found to exist in some cases between the -results of entirely different methods of arriving at the -measurement of a physical quantity. While such accordance -must, in the absence of information to the contrary,<span class="pagenum" id="Page_565">565</span> -be regarded as the best possible proof of the approximate -correctness of the mean result, yet instances have occurred -to show that we can never take too much trouble in confirming -results of great importance. When three or even -more distinct methods have given nearly coincident numbers, -a new method has sometimes disclosed a discrepancy -which it is yet impossible to explain.</p> - -<p>The ellipticity of the earth is known with considerable -approach to certainty and accuracy, for it has been estimated -in three independent ways. The most direct mode -is to measure long arcs extending north and south upon -the earth’s surface, by means of trigonometrical surveys, -and then to compare the lengths of these arcs with their -curvature as determined by observations of the altitude -of certain stars at the terminal points. The most probable -ellipticity of the earth deduced from all measurements of -this kind was estimated by Bessel at <span class="nowrap"><span class="fraction2"><span class="fnum2">1</span><span class="bar">/</span><span class="fden2">300</span></span></span>, though subsequent -measurements might lead to a slightly different -estimate. The divergence from a globular form causes a -small variation in the force of gravity at different parts of -the earth’s surface, so that exact pendulum observations -give the data for an independent estimate of the ellipticity, -which is thus found to be <span class="nowrap"><span class="fraction2"><span class="fnum2">1</span><span class="bar">/</span><span class="fden2">320</span></span></span>. In the third place the -spheroidal protuberance about the earth’s equator leads to -a certain inequality in the moon’s motion, as shown by -Laplace; and from the amount of that inequality, as given -by observations, Laplace was enabled to calculate back to -the amount of its cause. He thus inferred that the ellipticity -is <span class="nowrap"><span class="fraction2"><span class="fnum2">1</span><span class="bar">/</span><span class="fden2">305</span></span></span>, which lies between the two numbers previously -given, and was considered by him the most satisfactory -determination. In this case the accordance is undisturbed -by subsequent results, so that we are obliged to accept -Laplace’s result as a highly probable one.</p> - -<p>The mean density of the earth is a constant of high -importance, because it is necessary for the determination -of the masses of all the other heavenly bodies. Astronomers -and physicists accordingly have bestowed a great -deal of labour upon the exact estimation of this constant. -The method of procedure consists in comparing the<span class="pagenum" id="Page_566">566</span> -gravitation of the globe with that of some body of matter of -which the mass is known in terms of the assumed unit of -mass. This body of matter, serving as an intermediate -term of comparison, may be variously chosen; it may -consist of a mountain, or a portion of the earth’s crust, or -a heavy ball of metal. The method of experiment varies -so much according as we select one body or the other, that -we may be said to have three independent modes of arriving -at the desired result.</p> - -<p>The mutual gravitation of two balls is so exceedingly -small compared with their gravitation towards the immense -mass of the earth, that it is usually quite imperceptible, -and although asserted by Newton to exist, on the ground -of theory, was never observed until the end of the 18th -century. Michell attached two small balls to the extremities -of a delicately suspended torsion balance, and then -bringing heavy balls of lead alternately to either side of -these small balls was able to detect a slight deflection of -the torsion balance. He thus furnished a new verification -of the theory of gravitation. Cavendish carried out the -experiment with more care, and estimated the gravitation -of the balls by treating the torsion balance as a pendulum; -then taking into account the respective distances of the -balls from each other and from the centre of the earth, -he was able to assign 5·48 (or as re-computed by Baily, -5·448) as the probable mean density of the earth. Newton’s -sagacious guess to the effect that the density of the -earth was between five and six times that of water, was -thus remarkably confirmed. The same kind of experiment -repeated by Reich gave 5·438. Baily having again performed -the experiment with every possible refinement -obtained a slightly higher number, 5·660.</p> - -<p>A different method of procedure consisted in ascertaining -the effect of a mountain mass in deflecting the plumb-line; -for, assuming that we can determine the dimensions and -mean density of the mountain, the plumb-line enables us -to compare its mass with that of the whole earth. The mountain -Schehallien was selected for the experiment, and observations -and calculations performed by Maskelyne, Hutton, -and Playfair, gave as the most probable result 4·713. The -difference from the experimental results already mentioned -is considerable and is important, because the instrumental<span class="pagenum" id="Page_567">567</span> -operations are of an entirely different character from those -of Cavendish and Baily’s experiments. Sir Henry James’ -similar determination from the attraction of Arthur’s Seat -gave 5·14.</p> - -<p>A third distinct method consists in determining the force -of gravity at points elevated above the surface of the earth -on mountain ranges, or sunk below it in mines. Carlini -experimented with a pendulum at the hospice of Mont -Cenis, 6,375 feet above the sea, and by comparing the -attractive forces of the earth and the Alps, found the -density to be still smaller, namely, 4·39, or as corrected -by Giulio, 4·950. Lastly, the Astronomer Royal has on -two occasions adopted the opposite method of observing -a pendulum at the bottom of a deep mine, so as to compare -the density of the strata penetrated with the density -of the whole earth. On the second occasion he carried his -method into effect at the Harton Colliery, 1,260 feet deep; -all that could be done by skill in measurement and careful -consideration of all the causes of error, was accomplished -in this elaborate series of observations<a id="FNanchor_470" href="#Footnote_470" class="fnanchor">470</a> (p. 291). No doubt -Sir George Airy was much perplexed when he found that -his new result considerably exceeded that obtained by any -other method, being no less than 6·566, or 6·623 as finally -corrected. In this case we learn an impressive lesson -concerning the value of repeated determinations by distinct -methods in disabusing our minds of the reliance which we -are only too apt to place in results which show a certain -degree of coincidence.</p> - -<p>In 1844 Herschel remarked in his memoir of Francis -Baily,<a id="FNanchor_471" href="#Footnote_471" class="fnanchor">471</a> “that the mean specific gravity of this our planet is, -in all human probability, quite as well determined as that -of an ordinary hand-specimen in a mineralogical cabinet,—a -marvellous result, which should teach us to despair of -nothing which lies within the compass of number, weight -and measure.” But at the same time he pointed out that -Baily’s final result, of which the probable error was only -0·0032, was the highest of all determinations then known, -and Airy’s investigation has since given a much higher -result, quite beyond the limits of probable error of any of<span class="pagenum" id="Page_568">568</span> -the previous experiments. If we treat all determinations -yet made as of equal weight, the simple mean is about -5·45, the mean error nearly 0·5, and the probable error -almost 0·2, so that it is as likely as not that the truth lies -between 5·65 and 5·25 on this view of the matter. But it -is remarkable that the two most recent and careful series -of observations by Baily and Airy,<a id="FNanchor_472" href="#Footnote_472" class="fnanchor">472</a> lie beyond these limits, -and as with the increase of care the estimate rises, it seems -requisite to reject the earlier results, and look upon the -question as still requiring further investigation. Physicists -often take <span class="nowrap">5 <span class="fraction"><span class="fnum">2</span><span class="bar">/</span><span class="fden">3</span></span></span> or 5·67 as the best guess at the truth, but it -is evident that new experiments are much required. I -cannot help thinking that a portion of the great sums of -money which many governments and private individuals -spent upon the transit of Venus expeditions in 1874, and -which they will probably spend again in 1882 (p. <a href="#Page_562">562</a>), -would be better appropriated to new determinations of -the earth’s density. It seems desirable to repeat Baily’s -experiment in a vacuous case, and with the greater mechanical -refinements which the progress of the last forty -years places at the disposal of the experimentalist. It -would be desirable, also, to renew the pendulum experiments -of Airy in some other deep mine. It might even -be well to repeat upon some suitable mountain the observations -performed at Schehallien. All these operations -might be carried out for the cost of one of the superfluous -transit expeditions.</p> - -<p>Since the establishment of the dynamical theory of heat -it has become a matter of the greatest importance to -determine with accuracy the mechanical equivalent of -heat, or the quantity of energy which must be given, or -received, in a definite change of temperature effected in a -definite quantity of a standard substance, such as water. -No less than seven almost entirely distinct modes of -determining this constant have been tried. Dr. Joule first -ascertained by the friction of water that to raise the temperature -of one kilogram of water through one degree -centigrade, we must employ energy sufficient to raise 424 -kilograms through the height of one metre against the -force of gravity at the earth’s surface. Joule, Mayer,<span class="pagenum" id="Page_569">569</span> -Clausius,<a id="FNanchor_473" href="#Footnote_473" class="fnanchor">473</a> Favre and other experimentalists have made -determinations by less direct methods. Experiments on -the mechanical properties of gases give 426 kilogrammetres -as the constant; the work done by a steam-engine -gives 413; from the heat evolved in electrical experiments -several determinations have been obtained; thus from -induced electric currents we get 452; from the electro-magnetic -engine 443; from the circuit of a battery 420; -and, from an electric current, the lowest result of all, -namely, 400.<a id="FNanchor_474" href="#Footnote_474" class="fnanchor">474</a></p> - -<p>Considering the diverse and in many cases difficult -methods of observation, these results exhibit satisfactory -accordance, and their mean (423·9) comes very close to -the number derived by Dr. Joule from the apparently -most accurate method. The constant generally assumed -as the most probable result is 423·55 kilogrammetres.</p> - - -<h3><i>Residual Phenomena.</i></h3> - -<p>Even when the experimental data employed in the -verification of a theory are sufficiently accurate, and the -theory itself is sound, there may exist discrepancies -demanding further investigation. Herschel pointed out -the importance of such outstanding quantities, and called -them <i>residual phenomena</i>.<a id="FNanchor_475" href="#Footnote_475" class="fnanchor">475</a> Now if the observations and -the theory be really correct, such discrepancies must be -due to the incompleteness of our knowledge of the causes -in action, and the ultimate explanation must consist in -showing that there is in action, either</p> - -<p>(1) Some agent of known nature whose presence was -not suspected;</p> - -<p>Or (2) Some new agent of unknown nature.</p> - -<p>In the first case we can hardly be said to make a new -discovery, for our ultimate success consists merely in -reconciling the theory with known facts when our investigation -is more comprehensive. But in the second -case we meet with a totally new fact, which may lead us<span class="pagenum" id="Page_570">570</span> -to realms of new discovery. Take the instance adduced by -Herschel. The theory of Newton and Halley concerning -comets was that they were gravitating bodies revolving -round the sun in elliptic orbits, and the return of Halley’s -Comet, in 1758, verified this theory. But, when accurate -observations of Encke’s Comet came to be made, the verification -was not found to be exact. Encke’s Comet returned -each time a little sooner than it ought to do, the period -regularly decreasing from 1212·79 days, between 1786 and -1789, to 1210·44 between 1855 and 1858; and the hypothesis -has been started that there is a resisting medium -filling the space through which the comet passes. This -hypothesis is a <i>deus ex machinâ</i> for explaining this solitary -phenomenon, and cannot possess much probability unless -it can be shown that other phenomena are deducible from it. -Many persons have identified this medium with that through -which light undulations pass, but I am not aware that -there is anything in the undulatory theory of light to show -that the medium would offer resistance to a moving body. -If Professor Balfour Stewart can prove that a rotating disc -would experience resistance in a vacuous receiver, here is -an experimental fact which distinctly supports the hypothesis. -But in the mean time it is open to question -whether other known agents, for instance electricity, may -not be brought in, and I have tried to show that if, as is -believed, the tail of a comet is an electrical phenomenon, -it is a necessary result of the conservation of energy -that the comet shall exhibit a loss of energy manifested -in a diminution of its mean distance from the sun -and its period of revolution.<a id="FNanchor_476" href="#Footnote_476" class="fnanchor">476</a> It should be added that if<span class="pagenum" id="Page_571">571</span> -Professor Tait’s theory be correct, as seems very probable, -and comets consist of swarms of small meteors, there is no -difficulty in accounting for the retardation. It has long -been known that a collection of small bodies travelling -together in an orbit round a central body will tend to fall -towards it. In either case, then, this residual phenomenon -seems likely to be reconciled with known laws of nature.</p> - -<p>In other cases residual phenomena have involved important -inferences not recognised at the time. Newton -showed how the velocity of sound in the atmosphere -could be calculated by a theory of pulses or undulations -from the observed tension and density of the air. He -inferred that the velocity in the ordinary state of the -atmosphere at the earth’s surface would be 968 feet per -second, and rude experiments made by him in the cloisters -of Trinity College seemed to show that this was not far -from the truth. Subsequently it was ascertained by other -experimentalists that the velocity of sound was more -nearly 1,142 feet, and the discrepancy being one-sixth -part of the whole was far too much to attribute to casual -errors in the numerical data. Newton attempted to -explain away this discrepancy by hypotheses as to the -reactions of the molecules of air, but without success.</p> - -<p>New investigations having been made from time to time -concerning the velocity of sound, both as observed experimentally -and as calculated from theory, it was found that -each of Newton’s results was inaccurate, the theoretical -velocity being 916 feet per second, and the real velocity -about 1,090 feet. The discrepancy, nevertheless, remained -as serious as ever, and it was not until the year 1816 that -Laplace showed it to be due to the heat developed by the -sudden compression of the air in the passage of the wave, -this heat having the effect of increasing the elasticity of -the air and accelerating the impulse. It is now perceived<span class="pagenum" id="Page_572">572</span> -that this discrepancy really involves the doctrine of the -equivalence of heat and energy, and it was applied by -Mayer, at least by implication, to give an estimate of the -mechanical equivalent of heat. The estimate thus derived -agrees satisfactorily with direct determinations by Dr. -Joule and other physicists, so that the explanation of the -residual phenomenon which exercised Newton’s ingenuity -is now complete, and forms an important part of the new -science of thermodynamics.</p> - -<p>As Herschel observed, almost all great astronomical discoveries -have been disclosed in the form of residual differences. -It is the practice at well-conducted observatories -to compare the positions of the heavenly bodies as actually -observed with what might have been expected theoretically. -This practice was introduced by Halley when Astronomer -Royal, and his reduction of the lunar observations gave a -series of residual errors from 1722 to 1739, by the examination -of which the lunar theory was improved. Most of -the greater astronomical variations arising from nutation, -aberration, planetary perturbation were discovered in the -same manner. The precession of the equinox was perhaps -the earliest residual difference observed; the systematic -divergence of Uranus from its calculated places was one of -the latest, and was the clue to the remarkable discovery -of Neptune. We may also class under residual phenomena -all the so-called <i>proper motions</i> of the stars. A complete -star catalogue, such as that of the British Association, gives -a greater or less amount of proper motion for almost every -star, consisting in the apparent difference of position of the -star as derived from the earliest and latest good observations. -But these apparent motions are often due, as -explained by Baily,<a id="FNanchor_477" href="#Footnote_477" class="fnanchor">477</a> the author of the catalogue, to errors -of observation and reduction. In many cases the best -astronomical authorities have differed as to the very direction -of the supposed proper motion of stars, and as regards -the amount of the motion, for instance of α Polaris, the -most different estimates have been formed. Residual -quantities will often be so small that their very existence -is doubtful. Only the gradual progress of theory and of -measurement will show clearly whether a discrepancy is to<span class="pagenum" id="Page_573">573</span> -be referred to casual errors of observation or to some new -phenomenon. But nothing is more requisite for the progress -of science than the careful recording and investigation -of such discrepancies. In no part of physical science can -we be free from exceptions and outstanding facts, of which -our present knowledge can give no account. It is among -such anomalies that we must look for the clues to new -realms of facts worthy of discovery. They are like the -floating waifs which led Columbus to suspect the existence -of the new world.</p> - -<hr class="chap x-ebookmaker-drop"> - -<div class="chapter"> -<p><span class="pagenum" id="Page_574">574</span></p> -<h2 class="nobreak" id="CHAPTER_XXVI">CHAPTER XXVI.<br> - -<span class="title">CHARACTER OF THE EXPERIMENTALIST.</span></h2> -</div> - -<p class="ti0">In the present age there seems to be a tendency to believe -that the importance of individual genius is less than -it was—</p> - -<p class="ml5em fs90"> -“The individual withers, and the world is more and more.”<br> -</p> - -<p>Society, it is supposed, has now assumed so highly developed -a form, that what was accomplished in past times by -the solitary exertions of a great intellect, may now be -worked out by the united labours of an army of investigators. -Just as the well-organised power of a modern army -supersedes the single-handed bravery of the mediæval -knights, so we are to believe that the combination of intellectual -labour has superseded the genius of an Archimedes, -a Newton, or a Laplace. So-called original research -is now regarded as a profession, adopted by hundreds of -men, and communicated by a system of training. All that -we need to secure additions to our knowledge of nature is -the erection of great laboratories, museums, and observatories, -and the offering of pecuniary rewards to those who -can invent new chemical compounds, detect new species, or -discover new comets. Doubtless this is not the real meaning -of the eminent men who are now urging upon Government -the endowment of physical research. They can only -mean that the greater the pecuniary and material assistance -given to men of science, the greater the result which the -available genius of the country may be expected to -produce. Money and opportunities of study can no more -produce genius than sunshine and moisture can generate<span class="pagenum" id="Page_575">575</span> -living beings; the inexplicable germ is wanting in both -cases. But as, when the germ is present, the plant will grow -more or less vigorously according to the circumstances in -which it is placed, so it may be allowed that pecuniary assistance -may favour development of intellect. Public opinion -however is not discriminating, and is likely to interpret -the agitation for the endowment of science as meaning -that science can be had for money.</p> - -<p>All such notions are erroneous. In no branch of human -affairs, neither in politics, war, literature, industry, nor -science, is the influence of genius less considerable than it -was. It is possible that the extension and organisation of -scientific study, assisted by the printing-press and the -accelerated means of communication, has increased the -rapidity with which new discoveries are made known, and -their details worked out by many heads and hands. A -Darwin now no sooner propounds original ideas concerning -the evolution of living creatures, than those ideas are discussed -and illustrated, and applied by naturalists in every -part of the world. In former days his discoveries would -have been hidden for decades of years in scarce manuscripts, -and generations would have passed away before his -theory had enjoyed the same amount of criticism and corroboration -as it has already received. The result is that -the genius of Darwin is more valuable, not less valuable, -than it would formerly have been. The advance of military -science and the organisation of enormous armies has -not decreased the value of a skilful general; on the contrary, -the rank and file are still more in need than they -used to be of the guiding power of a far-seeing intellect. -The swift destruction of the French military power was -not due alone to the perfection of the German army, nor to -the genius of Moltke; it was due to the combination of a -well-disciplined multitude with a leader of the highest -powers. So in every branch of human affairs the influence -of the individual is not withering, but is growing -with the extent of the material resources which are at -his command.</p> - -<p>Turning to our own subject, it is a work of undiminished -interest to reflect upon those qualities of mind which lead -to great advances in natural knowledge. Nothing, indeed, -is less amenable than genius to scientific analysis and<span class="pagenum" id="Page_576">576</span> -explanation. Even definition is out of the question. Buffon -said that “genius is patience,” and certainly patience is one -of its most requisite components. But no one can suppose -that patient labour alone will invariably lead to those conspicuous -results which we attribute to genius. In every -branch of science, literature, art, or industry, there are -thousands of men and women who work with unceasing -patience, and thereby ensure moderate success; but it -would be absurd to suppose that equal amounts of intellectual -labour yield equal results. A Newton may modestly -attribute his discoveries to industry and patient thought, -and there is reason to believe that genius is unconscious -and unable to account for its own peculiar powers. As -genius is essentially creative, and consists in divergence -from the ordinary grooves of thought and action, it must -necessarily be a phenomenon beyond the domain of the -laws of nature. Nevertheless, it is always an interesting -and instructive work to trace out, as far as possible, the -characteristics of mind by which great discoveries have -been achieved, and we shall find in the analysis much to -illustrate the principles of scientific method.</p> - - -<h3><i>Error of the Baconian Method.</i></h3> - -<p>Hundreds of investigators may be constantly engaged in -experimental inquiry; they may compile numberless note-books -full of scientific facts, and endless tables of numerical -results; but, if the views of induction here maintained be -true, they can never by such work alone rise to new and -great discoveries. By a system of research they may work -out deductively the details of a previous discovery, but to -arrive at a new principle of nature is another matter. -Francis Bacon spread abroad the notion that to advance -science we must begin by accumulating facts, and then -draw from them, by a process of digestion, successive laws -of higher and higher generality. In protesting against the -false method of the scholastic logicians, he exaggerated a -partially true philosophy, until it became as false as that -which preceded it. His notion of scientific method was a -kind of scientific bookkeeping. Facts were to be indiscriminately -gathered from every source, and posted in a -ledger, from which would emerge in time a balance of<span class="pagenum" id="Page_577">577</span> -truth. It is difficult to imagine a less likely way of arriving -at great discoveries. The greater the array of facts, -the less is the probability that they will by any routine -system of classification disclose the laws of nature they -embody. Exhaustive classification in all possible orders is -out of the question, because the possible orders are practically -infinite in number.</p> - -<p>It is before the glance of the philosophic mind that -facts must display their meaning, and fall into logical order. -The natural philosopher must therefore have, in the first -place, a mind of impressionable character, which is affected -by the slightest exceptional phenomenon. His associating -and identifying powers must be great, that is, a strange fact -must suggest to his mind whatever of like nature has previously -come within his experience. His imagination must -be active, and bring before his mind multitudes of relations -in which the unexplained facts may possibly stand with -regard to each other, or to more common facts. Sure and -vigorous powers of deductive reasoning must then come into -play, and enable him to infer what will happen under each -supposed condition. Lastly, and above all, there must be -the love of certainty leading him diligently and with perfect -candour, to compare his speculations with the test of -fact and experiment.</p> - - -<h3><i>Freedom of Theorising.</i></h3> - -<p>It would be an error to suppose that the great discoverer -seizes at once upon the truth, or has any unerring method -of divining it. In all probability the errors of the great -mind exceed in number those of the less vigorous one. -Fertility of imagination and abundance of guesses at truth -are among the first requisites of discovery; but the erroneous -guesses must be many times as numerous as those which -prove well founded. The weakest analogies, the most -whimsical notions, the most apparently absurd theories, -may pass through the teeming brain, and no record remain -of more than the hundredth part. There is nothing really -absurd except that which proves contrary to logic and experience. -The truest theories involve suppositions which -are inconceivable, and no limit can really be placed to the -freedom of hypothesis.</p> - -<p><span class="pagenum" id="Page_578">578</span></p> - -<p>Kepler is an extraordinary instance to this effect. No -minor laws of nature are more firmly established than those -which he detected concerning the orbits and motions of -planetary masses, and on these empirical laws the theory -of gravitation was founded. Did we not learn from his -own writings the multitude of errors into which he fell, we -might have imagined that he had some special faculty of -seizing on the truth. But, as is well known, he was full of -chimerical notions; his favourite and long-studied theory -was founded on a fanciful analogy between the planetary -orbits and the regular solids. His celebrated laws were the -outcome of a lifetime of speculation, for the most part vain -and groundless. We know this because he had a curious -pleasure in dwelling upon erroneous and futile trains of -reasoning, which most persons consign to oblivion. But -Kepler’s name was destined to be immortal, on account of -the patience with which he submitted his hypotheses to -comparison with observation, the candour with which he -acknowledged failure after failure, and the perseverance -and ingenuity with which he renewed his attack upon the -riddles of nature.</p> - -<p>Next after Kepler perhaps Faraday is the physical philosopher -who has given us the best insight into the progress -of discovery, by recording erroneous as well as successful -speculations. The recorded notions, indeed, are probably -but a tithe of the fancies which arose in his active brain. -As Faraday himself said—“The world little knows how -many of the thoughts and theories which have passed -through the mind of a scientific investigator, have been -crushed in silence and secrecy by his own severe criticism -and adverse examination; that in the most successful instances -not a tenth of the suggestions, the hopes, the wishes, -the preliminary conclusions have been realised.”</p> - -<p>Nevertheless, in Faraday’s researches, published in the -<i>Philosophical Transactions</i>, in minor papers, in manuscript -note-books, or in other materials, made known in his interesting -life by Dr. Bence Jones, we find invaluable lessons -for the experimentalist. These writings are full of speculations -which we must not judge by the light of subsequent -discovery. It may perhaps be said that Faraday committed -to the printing press crude ideas which a friend -would have counselled him to keep back. There was<span class="pagenum" id="Page_579">579</span> -occasionally even a wildness and vagueness in his notions, -which in a less careful experimentalist would have been -fatal to the attainment of truth. This is especially apparent -in a curious paper concerning Ray-vibrations; but fortunately -Faraday was aware of the shadowy character of his -speculations, and expressed the feeling in words which -must be quoted. “I think it likely,” he says,<a id="FNanchor_478" href="#Footnote_478" class="fnanchor">478</a> “that I -have made many mistakes in the preceding pages, for -even to myself my ideas on this point appear only as the -shadow of a speculation, or as one of those impressions -upon the mind, which are allowable for a time as guides to -thought and research. He who labours in experimental -inquiries knows how numerous these are, and how often -their apparent fitness and beauty vanish before the progress -and development of real natural truth.” If, then, the experimentalist -has no royal road to the discovery of the -truth, it is an interesting matter to consider by what logical -procedure he attains the truth.</p> - -<p>If I have taken a correct view of logical method, there -is really no such thing as a distinct process of induction. -The probability is infinitely small that a collection of -complicated facts will fall into an arrangement capable -of exhibiting directly the laws obeyed by them. The -mathematician might as well expect to integrate his -functions by a ballot-box, as the experimentalist to draw -deep truths from haphazard trials. All induction is but -the inverse application of deduction, and it is by the -inexplicable action of a gifted mind that a multitude of -heterogeneous facts are ranged in luminous order as the -results of some uniformly acting law. So different, indeed, -are the qualities of mind required in different branches of -science, that it would be absurd to attempt to give an -exhaustive description of the character of mind which -leads to discovery. The labours of Newton could not -have been accomplished except by a mind of the utmost -mathematical genius; Faraday, on the other hand, has -made the most extensive additions to human knowledge -without passing beyond common arithmetic. I do not -remember meeting in Faraday’s writings with a single<span class="pagenum" id="Page_580">580</span> -algebraic formula or mathematical problem of any complexity. -Professor Clerk Maxwell, indeed, in the preface -to his new <i>Treatise on Electricity</i>, has strongly recommended -the reading of Faraday’s researches by all students of -science, and has given his opinion that though Faraday -seldom or never employed mathematical formulæ, his -methods and conceptions were not the less mathematical -in their nature.<a id="FNanchor_479" href="#Footnote_479" class="fnanchor">479</a> I have myself protested against the -prevailing confusion between a mathematical and an exact -science,<a id="FNanchor_480" href="#Footnote_480" class="fnanchor">480</a> yet I certainly think that Faraday’s experiments -were for the most part qualitative, and that his mathematical -ideas were of a rudimentary character. It is true -that he could not possibly investigate such a subject as -magne-crystallic action without involving himself in -geometrical relations of some complexity. Nevertheless -I think that he was deficient in mathematical deductive -power, that power which is so highly developed by -the modern system of mathematical training at Cambridge.</p> - -<p>Faraday was acquainted with the forms of his celebrated -lines of force, but I am not aware that he ever entered -into the algebraic nature of those curves, and I feel sure -that he could not have explained their forms as depending -on the resultant attractions of all the magnetic particles. -There are even occasional indications that he did not -understand some of the simpler mathematical doctrines of -modern physical science. Although he so clearly foresaw -the correlation of the physical forces, and laboured so hard -with his own hands to connect gravity with other forces, -it is doubtful whether he understood the doctrine of the -conservation of energy as applied to gravitation. Faraday -was probably equal to Newton in experimental skill, and -in that peculiar kind of deductive power which leads to -the invention of simple qualitative experiments; but it -must be allowed that he exhibited little of that mathematical -power which enabled Newton to follow out intuitively -the quantitative results of a complicated problem -with such wonderful facility. Two instances, Newton and -Faraday, are sufficient to show that minds of widely<span class="pagenum" id="Page_581">581</span> -different conformation will meet with suitable regions of -research. Nevertheless, there are certain traits which we -may discover in all the highest scientific minds.</p> - - -<h3><i>The Newtonian Method, the True Organum.</i></h3> - -<p>Laplace was of opinion that the <i>Principia</i> and the -<i>Opticks</i> of Newton furnished the best models then available -of the delicate art of experimental and theoretical -investigation. In these, as he says, we meet with the -most happy illustrations of the way in which, from a -series of inductions, we may rise to the causes of phenomena, -and thence descend again to all the resulting -details.</p> - -<p>The popular notion concerning Newton’s discoveries is -that in early life, when driven into the country by the -Great Plague, a falling apple accidentally suggested to -him the existence of gravitation, and that, availing himself -of this hint, he was led to the discovery of the law of -gravitation, the explanation of which constitutes the -<i>Principia</i>. It is difficult to imagine a more ludicrous and -inadequate picture of Newton’s labours. No originality, -or at least priority, was claimed by Newton as regards the -discovery of the law of the inverse square, so closely -associated with his name. In a well-known Scholium<a id="FNanchor_481" href="#Footnote_481" class="fnanchor">481</a> -he acknowledges that Sir Christopher Wren, Hooke, and -Halley, had severally observed the accordance of Kepler’s -third law of motion with the principle of the inverse -square.</p> - -<p>Newton’s work was really that of developing the -methods of deductive reasoning and experimental verification, -by which alone great hypotheses can be brought to -the touchstone of fact. Archimedes was the greatest of -ancient philosophers, for he showed how mathematical -theory could be wedded to physical experiments; and -his works are the first true Organum. Newton is the -modern Archimedes, and the <i>Principia</i> forms the true -Novum Organum of scientific method. The laws which -he established are great, but his example of the manner of -establishing them is greater still. Excepting perhaps<span class="pagenum" id="Page_582">582</span> -chemistry and electricity, there is hardly a progressive -branch of physical and mathematical science, which has -not been developed from the germs of true scientific procedure -which he disclosed in the <i>Principia</i> or the <i>Opticks</i>. -Overcome by the success of his theory of universal gravitation, -we are apt to forget that in his theory of sound he -originated the mathematical investigation of waves and -the mutual action of particles; that in his corpuscular -theory of light, however mistaken, he first ventured to -apply mathematical calculation to molecular attractions -and repulsions; that in his prismatic experiments he -showed how far experimental verification could be pushed; -that in his examination of the coloured rings named after -him, he accomplished the most remarkable instance of -minute measurement yet known, a mere practical application -of which by Fizeau was recently deemed worthy -of a medal by the Royal Society. We only learn by degrees -how complete was his scientific insight; a few words in his -third law of motion display his acquaintance with the -fundamental principles of modern thermodynamics and -the conservation of energy, while manuscripts long overlooked -prove that in his inquiries concerning atmospheric -refraction he had overcome the main difficulties of applying -theory to one of the most complex of physical -problems.</p> - -<p>After all, it is only by examining the way in which he -effected discoveries, that we can rightly appreciate his -greatness. The <i>Principia</i> treats not of gravity so much -as of forces in general, and the methods of reasoning -about them. He investigates not one hypothesis only, -but mechanical hypotheses in general. Nothing so much -strikes the reader of the work as the exhaustiveness of his -treatment, and the unbounded power of his insight. If he -treats of central forces, it is not one law of force which he -discusses, but many, or almost all imaginable laws, the -results of each of which he sketches out in a few pregnant -words. If his subject is a resisting medium, it is not air -or water alone, but resisting media in general. We have -a good example of his method in the scholium to the -twenty-second proposition of the second book, in which he -runs rapidly over many suppositions as to the laws of the -compressing forces which might conceivably act in an<span class="pagenum" id="Page_583">583</span> -atmosphere of gas, a consequence being drawn from each -case, and that one hypothesis ultimately selected which -yields results agreeing with experiments upon the pressure -and density of the terrestrial atmosphere.</p> - -<p>Newton said that he did not frame hypotheses, but, in -reality, the greater part of the <i>Principia</i> is purely hypothetical, -endless varieties of causes and laws being -imagined which have no counterpart in nature. The -most grotesque hypotheses of Kepler or Descartes were -not more imaginary. But Newton’s comprehension of -logical method was perfect; no hypothesis was entertained -unless it was definite in conditions, and admitted of unquestionable -deductive reasoning; and the value of each -hypothesis was entirely decided by the comparison of its -consequences with facts. I do not entertain a doubt that -the general course of his procedure is identical with that -view of the nature of induction, as the inverse application -of deduction, which I advocate throughout this book. -Francis Bacon held that science should be founded on -experience, but he mistook the true mode of using experience, -and, in attempting to apply his method, ludicrously -failed. Newton did not less found his method on experience, -but he seized the true method of treating it, and -applied it with a power and success never since equalled. -It is a great mistake to say that modern science is the -result of the Baconian philosophy; it is the Newtonian -philosophy and the Newtonian method which have led to -all the great triumphs of physical science, and I repeat -that the <i>Principia</i> forms the true “Novum Organum.”</p> - -<p>In bringing his theories to a decisive experimental verification, -Newton showed, as a general rule, exquisite skill -and ingenuity. In his hands a few simple pieces of apparatus -were made to give results involving an unsuspected -depth of meaning. His most beautiful experimental inquiry -was that by which he proved the differing refrangibility -of rays of light. To suppose that he originally discovered -the power of a prism to break up a beam of white -light would be a mistake, for he speaks of procuring a -glass prism to try the “celebrated phenomena of colours.” -But we certainly owe to him the theory that white light is -a mixture of rays differing in refrangibility, and that lights -which differ in colour, differ also in refrangibility. Other<span class="pagenum" id="Page_584">584</span> -persons might have conceived this theory; in fact, any -person regarding refraction as a quantitative effect must -see that different parts of the spectrum have suffered -different amounts of refraction. But the power of Newton -is shown in the tenacity with which he followed his theory -into every consequence, and tested each result by a simple -but conclusive experiment. He first shows that different -coloured spots are displaced by different amounts when -viewed through a prism, and that their images come to a -focus at different distances from the lense, as they should -do, if the refrangibility differed. After excluding by many -experiments a variety of indifferent circumstances, he fixes -his attention upon the question whether the rays are -merely shattered, disturbed, and spread out in a chance -manner, as Grimaldi supposed, or whether there is a constant -relation between the colour and the refrangibility.</p> - -<p>If Grimaldi was right, it might be expected that a part -of the spectrum taken separately, and subjected to a second -refraction, would suffer a new breaking up, and produce -some new spectrum. Newton inferred from his own theory -that a particular ray of the spectrum would have a constant -refrangibility, so that a second prism would merely -bend it more or less, but not further disperse it in any considerable -degree. By simply cutting off most of the rays of -the spectrum by a screen, and allowing the remaining -narrow ray to fall on a second prism, he proved the truth -of this conclusion; and then slowly turning the first prism, -so as to vary the colour of the ray falling on the second one, -he found that the spot of light formed by the twice-refracted -ray travelled up and down, a palpable proof that the amount -of refrangibility varies with the colour. For his further -satisfaction, he sometimes refracted the light a third or -fourth time, and he found that it might be refracted upwards -or downwards or sideways, and yet for each colour -there was a definite amount of refraction through each -prism. He completed the proof by showing that the separated -rays may again be gathered together into white light -by an inverted prism, so that no number of refractions -alters the character of the light. The conclusion thus obtained -serves to explain the confusion arising in the use of -a common lense; he shows that with homogeneous light -there is one distinct focus, with mixed light an infinite<span class="pagenum" id="Page_585">585</span> -number of foci, which prevent a clear view from being obtained -at any point.</p> - -<p>What astonishes the reader of the <i>Opticks</i> is the persistence -with which Newton follows out the consequences -of a preconceived theory, and tests the one notion by a -wonderful variety of simple comparisons with fact. The -ease with which he invents new combinations, and foresees -the results, subsequently verified, produces an insuperable -conviction in the reader that he has possession of the -truth. And it is certainly the theory which leads him to -the experiments, most of which could hardly be devised by -accident. Newton actually remarks that it was by mathematically -determining all kinds of phenomena of colours -which could be produced by refraction that he had “invented” -almost all the experiments in the book, and he -promises that others who shall “argue truly,” and try the -experiments with care, will not be disappointed in the -results.<a id="FNanchor_482" href="#Footnote_482" class="fnanchor">482</a></p> - -<p>The philosophic method of Huyghens was the same as -that of Newton, and Huyghens’ investigation of double -refraction furnishes almost equally beautiful instances of -theory guiding experiment. So far as we know double refraction -was first discovered by accident, and was described -by Erasmus Bartholinus in 1669. The phenomenon then -appeared to be entirely exceptional, and the laws governing -the two paths of the refracted rays were so unapparent -and complicated, that Newton altogether misunderstood the -phenomenon, and it was only at the latter end of the last -century that scientific men began to comprehend its laws.</p> - -<p>Nevertheless, Huyghens had, with rare genius, arrived -at the true theory as early as 1678. He regarded light -as an undulatory motion of some medium, and in his -<i>Traité de la Lumière</i> he pointed out that, in ordinary -refraction, the velocity of propagation of the wave is -equal in all directions, so that the front of an advancing -wave is spherical, and reaches equal distances in equal -times. But in crystals, as he supposed, the medium would -be of unequal elasticity in different directions, so that a -disturbance would reach unequal distances in equal times, -and the wave produced would have a spheroidal form.<span class="pagenum" id="Page_586">586</span> -Huyghens was not satisfied with an unverified theory. -He calculated what might be expected to happen when a -crystal of calc-spar was cut in various directions, and he -says: “I have examined in detail the properties of the -extraordinary refraction of this crystal, to see if each -phenomenon which is deduced from theory would agree -with what is really observed. And this being so, it is -no slight proof of the truth of our suppositions and principles; -but what I am going to add here confirms them -still more wonderfully; that is, the different modes of -cutting this crystal, in which the surfaces produced give -rise to refraction exactly such as they ought to be, and as -I had foreseen them, according to the preceding theory.”</p> - -<p>Newton’s mistaken corpuscular theory of light caused -the theories and experiments of Huyghens to be disregarded -for more than a century; but it is not easy to imagine a -more beautiful or successful application of the true method -of inductive investigation, theory guiding experiment, and -yet wholly relying on experiment for confirmation.</p> - - -<h3><i>Candour and Courage of the Philosophic Mind.</i></h3> - -<p>Perfect readiness to reject a theory inconsistent with -fact is a primary requisite of the philosophic mind. But it -would be a mistake to suppose that this candour has anything -akin to fickleness; on the contrary, readiness to reject -a false theory may be combined with a peculiar pertinacity -and courage in maintaining an hypothesis as long as its -falsity is not actually apparent. There must, indeed, be no -prejudice or bias distorting the mind, and causing it to pass -over the unwelcome results of experiment. There must be -that scrupulous honesty and flexibility of mind, which -assigns adequate value to all evidence; indeed, the more a -man loves his theory, the more scrupulous should be his -attention to its faults. It is common in life to meet -with some theorist, who, by long cogitation over a single -theory, has allowed it to mould his mind, and render him -incapable of receiving anything but as a contribution to the -truth of his one theory. A narrow and intense course of -thought may sometimes lead to great results, but the adoption -of a wrong theory at the outset is in such a mind irretrievable. -The man of one idea has but a single chance of<span class="pagenum" id="Page_587">587</span> -truth. The fertile discoverer, on the contrary, chooses -between many theories, and is never wedded to any one, -unless impartial and repeated comparison has convinced -him of its validity. He does not choose and then compare; -but he compares time after time, and then chooses.</p> - -<p>Having once deliberately chosen, the philosopher may -rightly entertain his theory with the strongest fidelity. -He will neglect no objection; for he may chance at any -time to meet a fatal one; but he will bear in mind the inconsiderable -powers of the human mind compared with -the tasks it has to undertake. He will see that no theory -can at first be reconciled with all objections, because there -may be many interfering causes, and the very consequences -of the theory may have a complexity which prolonged -investigation by successive generations of men may not -exhaust. If, then, a theory exhibit a number of striking -coincidences with fact, it must not be thrown aside until at -least one <i>conclusive discordance</i> is proved, regard being had -to possible error in establishing that discordance. In -science and philosophy something must be risked. He -who quails at the least difficulty will never establish a new -truth, and it was not unphilosophic in Leslie to remark -concerning his own inquiries into the nature of heat—</p> - -<p>“In the course of investigation, I have found myself -compelled to relinquish some preconceived notions; but -I have not abandoned them hastily, nor, till after a -warm and obstinate defence, I was driven from every -post.”<a id="FNanchor_483" href="#Footnote_483" class="fnanchor">483</a></p> - -<p>Faraday’s life, again, furnishes most interesting illustrations -of this tenacity of the philosophic mind. Though so -candid in rejecting some theories, there were others to -which he clung through everything. One of his favourite -notions resulted in a brilliant discovery; another remains -in doubt to the present day.</p> - - -<h3><i>The Philosophic Character of Faraday.</i></h3> - -<p>In Faraday’s researches concerning the connection of -magnetism and light, we find an excellent instance of the -pertinacity with which a favourite theory may be pursued,<span class="pagenum" id="Page_588">588</span> -so long as the results of experiment do not clearly negative -the notions entertained. In purely quantitative questions, -as we have seen, the absence of apparent effect can seldom -be regarded as proving the absence of all effect. Now -Faraday was convinced that some mutual relation must -exist between magnetism and light. As early as 1822, he -attempted to produce an effect upon a ray of polarised light, -by passing it through water placed between the poles of a -voltaic battery; but he was obliged to record that not the -slightest effect was observable. During many years the -subject, we are told,<a id="FNanchor_484" href="#Footnote_484" class="fnanchor">484</a> rose again and again to his mind, -and no failure could make him relinquish his search after -this unknown relation. It was in the year 1845 that he -gained the first success; on August 30th he began to -work with common electricity, vainly trying glass, quartz, -Iceland spar, &c. Several days of labour gave no result; -yet he did not desist. Heavy glass, a transparent medium -of great refractive powers, composed of borate of lead, was -now tried, being placed between the poles of a powerful -electro-magnet while a ray of polarised light was transmitted -through it. When the poles of the electro-magnet -were arranged in certain positions with regard to the -substance under trial, no effects were apparent; but at -last Faraday happened fortunately to place a piece of -heavy glass so that contrary magnetic poles were on the -same side, and now an effect was witnessed. The glass -was found to have the power of twisting the plane of -polarisation of the ray of light.</p> - -<p>All Faraday’s recorded thoughts upon this great experiment -are replete with curious interest. He attributes his -success to the opinion, almost amounting to a conviction, -that the various forms, under which the forces of matter -are made manifest, have one common origin, and are so -directly related and mutually dependent that they are -convertible. “This strong persuasion,” he says,<a id="FNanchor_485" href="#Footnote_485" class="fnanchor">485</a> “extended -to the powers of light, and led to many exertions having -for their object the discovery of the direct relation of light -and electricity. These ineffectual exertions could not -remove my strong persuasion, and I have at last succeeded.”<span class="pagenum" id="Page_589">589</span> -He describes the phenomenon in somewhat figurative -language as <i>the magnetisation of a ray of light</i>, -and also as <i>the illumination of a magnetic curve or line -of force</i>. He has no sooner got the effect in one case, -than he proceeds, with his characteristic comprehensiveness -of research, to test the existence of a like phenomenon -in all the substances available. He finds that not only -heavy glass, but solids and liquids, acids and alkalis, -oils, water, alcohol, ether, all possess this power; but he -was not able to detect its existence in any gaseous substance. -His thoughts cannot be restrained from running -into curious speculations as to the possible results of the -power in certain cases. “What effect,” he says, “does this -force have in the earth where the magnetic curves of the -earth traverse its substance? Also what effect in a magnet?” -And then he falls upon the strange notion that -perhaps this force tends to make iron and oxide of iron -transparent, a phenomenon never observed. We can meet -with nothing more instructive as to the course of mind by -which great discoveries are made, than these records of -Faraday’s patient labours, and his varied success and -failure. Nor are his unsuccessful experiments upon the -relation of gravity and electricity less interesting, or less -worthy of study.</p> - -<p>Throughout a large part of his life, Faraday was possessed -by the idea that gravity cannot be unconnected -with the other forces of nature. On March 19th, 1849, -he wrote in his laboratory book,—“Gravity. Surely this -force must be capable of an experimental relation to electricity, -magnetism, and the other forces, so as to bind it -up with them in reciprocal action and equivalent effect?”<a id="FNanchor_486" href="#Footnote_486" class="fnanchor">486</a> -He filled twenty paragraphs or more with reflections and -suggestions, as to the mode of treating the subject by experiment. -He anticipated that the mutual approach of -two bodies would develop electricity in them, or that a -body falling through a conducting helix would excite a -current changing in direction as the motion was reversed. -“<i>All this is a dream</i>,” he remarks; “still examine it by a -few experiments. Nothing is too wonderful to be true, if<span class="pagenum" id="Page_590">590</span> -it be consistent with the laws of nature; and in such -things as these, experiment is the best test of such consistency.”</p> - -<p>He executed many difficult and tedious experiments, -which are described in the 24th Series of Experimental -Researches. The result was <i>nil</i>, and yet he concludes: -“Here end my trials for the present. The results are -negative; they do not shake my strong feeling of the -existence of a relation between gravity and electricity, -though they give no proof that such a relation exists.”</p> - -<p>He returned to the work when he was ten years older, -and in 1858–9 recorded many remarkable reflections and -experiments. He was much struck by the fact that electricity -is essentially a <i>dual force</i>, and it had always been -a conviction of Faraday that no body could be electrified -positively without some other body becoming electrified -negatively; some of his researches had been simple developments -of this relation. But observing that between -two mutually gravitating bodies there was no apparent -circumstance to determine which should be positive and -which negative, he does not hesitate to call in question an -old opinion. “The evolution of <i>one</i> electricity would be a -new and very remarkable thing. The idea throws a doubt -on the whole; but still try, for who knows what is possible -in dealing with gravity?” We cannot but notice the -candour with which he thus acknowledges in his laboratory -book the doubtfulness of the whole thing, and is yet prepared -as a forlorn hope to frame experiments in opposition -to all his previous experience of the course of nature. For -a time his thoughts flow on as if the strange detection were -already made, and he had only to trace out its consequences -throughout the universe. “Let us encourage -ourselves by a little more imagination prior to experiment,” -he says; and then he reflects upon the infinity of actions -in nature, in which the mutual relations of electricity and -gravity would come into play; he pictures to himself the -planets and the comets charging themselves as they approach -the sun; cascades, rain, rising vapour, circulating -currents of the atmosphere, the fumes of a volcano, the -smoke in a chimney become so many electrical machines. -A multitude of events and changes in the atmosphere -seem to be at once elucidated by such actions; for a<span class="pagenum" id="Page_591">591</span> -moment his reveries have the vividness of fact. “I think -we have been dull and blind not to have suspected some -such results,” and he sums up rapidly the consequences of -his great but imaginary theory; an entirely new mode of -exciting heat or electricity, an entirely new relation of the -natural forces, an analysis of gravitation, and a justification -of the conservation of force.</p> - -<p>Such were Faraday’s fondest dreams of what might be, -and to many a philosopher they would have been sufficient -basis for the writing of a great book. But Faraday’s -imagination was within his full control; as he himself -says, “Let the imagination go, guarding it by judgment -and principle, and holding it in and directing it by experiment.” -His dreams soon took a very practical form, and -for many days he laboured with ceaseless energy, on the -staircase of the Royal Institution, in the clock tower of the -Houses of Parliament, or at the top of the Shot Tower in -Southwark, raising and lowering heavy weights, and combining -electrical helices and wires in every conceivable -way. His skill and long experience in experiment were -severely taxed to eliminate the effects of the earth’s magnetism, -and time after time he saved himself from accepting -mistaken indications, which to another man might have -seemed conclusive verifications of his theory. When all -was done there remained absolutely no results. “The -experiments,” he says, “were well made, but the results -are negative;” and yet, he adds, “I cannot accept them as -conclusive.” In this position the question remains to the -present day; it may be that the effect was too slight to be -detected, or it may be that the arrangements adopted were -not suited to develop the particular relation which exists, -just as Oersted could not detect electro-magnetism, so long -as his wire was perpendicular to the plane of motion of his -needle. But these are not matters which concern us -further here. We have only to notice the profound conviction -in the unity of natural laws, the active powers of -inference and imagination, the unbounded licence of theorising, -combined above all with the utmost diligence in -experimental verification which this remarkable research -exhibits.</p> - -<p><span class="pagenum" id="Page_592">592</span></p> - - -<h3><i>Reservation of Judgment.</i></h3> - -<p>There is yet another characteristic needed in the -philosophic mind; it is that of suspending judgment -when the data are insufficient. Many people will express -a confident opinion on almost any question which is put -before them, but they thereby manifest not strength, but -narrowness of mind. To see all sides of a complicated -subject, and to weigh all the different facts and probabilities -correctly, require no ordinary powers of comprehension. -Hence it is most frequently the philosophic mind which is -in doubt, and the ignorant mind which is ready with a -positive decision. Faraday has himself said, in a very -interesting lecture:<a id="FNanchor_487" href="#Footnote_487" class="fnanchor">487</a> “Occasionally and frequently the -exercise of the judgment ought to end in <i>absolute reservation</i>. -It may be very distasteful, and great fatigue, to -suspend a conclusion; but as we are not infallible, so we -ought to be cautious; we shall eventually find our advantage, -for the man who rests in his position is not so far -from right as he who, proceeding in a wrong direction, is -ever increasing his distance.”</p> - -<p>Arago presented a conspicuous example of this high -quality of mind, as Faraday remarks; for when he made -known his curious discovery of the relation of a magnetic -needle to a revolving copper plate, a number of supposed -men of science in different countries gave immediate and -confident explanations of it, which were all wrong. But -Arago, who had both discovered the phenomenon and -personally investigated its conditions, declined to put -forward publicly any theory at all.</p> - -<p>At the same time we must not suppose that the truly -philosophic mind can tolerate a state of doubt, while a -chance of decision remains open. In science nothing like -compromise is possible, and truth must be one. Hence, -doubt is the confession of ignorance, and involves a painful -feeling of incapacity. But doubt lies between error and -truth, so that if we choose wrongly we are further away -than ever from our goal.</p> - -<p>Summing up, then, it would seem as if the mind of -the great discoverer must combine contradictory attributes.<span class="pagenum" id="Page_593">593</span> -He must be fertile in theories and hypotheses, and yet full -of facts and precise results of experience. He must entertain -the feeblest analogies, and the merest guesses at -truth, and yet he must hold them as worthless till they -are verified in experiment. When there are any grounds -of probability he must hold tenaciously to an old opinion, -and yet he must be prepared at any moment to relinquish -it when a clearly contradictory fact is encountered. “The -philosopher,” says Faraday,<a id="FNanchor_488" href="#Footnote_488" class="fnanchor">488</a> “should be a man willing to -listen to every suggestion, but determined to judge for -himself. He should not be biased by appearances; have -no favourite hypothesis; be of no school; and in doctrine -have no master. He should not be a respecter of persons, -but of things. Truth should be his primary object. If to -these qualities be added industry, he may indeed hope to -walk within the veil of the temple of nature.”</p> - -<hr class="chap x-ebookmaker-drop"> - -<div class="chapter"> -<p><span class="pagenum" id="Page_594">594</span></p> -<p class="nobreak ph2 ti0" id="BOOK_V">BOOK V.<br> - -<span class="title">GENERALISATION, ANALOGY, AND CLASSIFICATION.</span></p> -</div> - -<hr class="r30"> - -<div class="chapter"> -<h2 class="nobreak" id="CHAPTER_XXVII">CHAPTER XXVII.<br> - -<span class="title">GENERALISATION.</span></h2> -</div> - -<p class="ti0">I have endeavoured to show in preceding chapters that -all inductive reasoning is an inverse application of deductive -reasoning, and consists in demonstrating that the -consequences of certain assumed laws agree with facts of -nature gathered by active or passive observation. The -fundamental process of reasoning, as stated in the outset, -consists in inferring of a thing what we know of similar -objects, and it is on this principle that the whole of deductive -reasoning, whether simply logical or mathematico-logical, -is founded. All inductive reasoning must be -founded on the same principle. It might seem that by a -plain use of this principle we could avoid the complicated -processes of induction and deduction, and argue directly -from one particular case to another, as Mill proposed. If -the Earth, Venus, Mars, Jupiter, and other planets move -in elliptic orbits, cannot we dispense with elaborate precautions, -and assert that Neptune, Ceres, and the last -discovered planet must do so likewise? Do we not know -that Mr. Gladstone must die, because he is like other<span class="pagenum" id="Page_595">595</span> -men? May we not argue that because some men die -therefore he must? Is it requisite to ascend by induction -to the general proposition “all men must die,” and then -descend by deduction from that general proposition to the -case of Mr. Gladstone? My answer undoubtedly is that -we must ascend to general propositions. The fundamental -principle of the substitution of similars gives us no warrant -in affirming of Mr. Gladstone what we know of other men, -because we cannot be sure that Mr. Gladstone is exactly -similar to other men. Until his death we cannot be perfectly -sure that he possesses all the attributes of other -men; it is a question of probability, and I have endeavoured -to explain the mode in which the theory of probability is -applied to calculate the probability that from a series of -similar events we may infer the recurrence of like events -under identical circumstances. There is then no such -process as that of inferring from particulars to particulars. -A careful analysis of the conditions under which such an -inference appears to be made, shows that the process is -really a general one, and that what is inferred of a particular -case might be inferred of all similar cases. All -reasoning is essentially general, and all science implies -generalisation. In the very birth-time of philosophy this -was held to be so: “Nulla scientia est de individuis, sed -de solis universalibus,” was the doctrine of Plato, delivered -by Porphyry. And Aristotle<a id="FNanchor_489" href="#Footnote_489" class="fnanchor">489</a> held a like -opinion—Οὐδεμία δὲ τέχνη σκοπεȋ τὸ καθ’ ἕκαστον ... τὸ δὲ καθ’ -ἕκαστον ἄπειρον καὶ οὐκ ἐπιστητόν. “No art treats of -particular cases; for particulars are infinite and cannot be -known.” No one who holds the doctrine that reasoning -may be from particulars to particulars, can be supposed -to have the most rudimentary notion of what constitutes -reasoning and scíence.</p> - -<p>At the same time there can be no doubt that practically -what we find to be true of many similar objects will -probably be true of the next similar object. This is the -result to which an analysis of the Inverse Method of -Probabilities leads us, and, in the absence of precise data -from which we may calculate probabilities, we are usually -obliged to make a rough assumption that similars in some<span class="pagenum" id="Page_596">596</span> -respects are similars in other respects. Thus it comes to -pass that a large part of the reasoning processes in which -scientific men are engaged, consists in detecting similarities -between objects, and then rudely assuming that the like -similarities will be detected in other cases.</p> - - -<h3><i>Distinction of Generalisation and Analogy.</i></h3> - -<p>There is no distinction but that of degree between what -is known as reasoning by <i>generalisation</i> and reasoning by -<i>analogy</i>. In both cases from certain observed resemblances -we infer, with more or less probability, the existence of -other resemblances. In generalisation the resemblances -have great extension and usually little intension, whereas -in analogy we rely upon the great intension, the extension -being of small amount (p. <a href="#Page_26">26</a>). If we find that the -qualities A and B are associated together in a great -many instances, and have never been found separate, it is -highly probable that on the next occasion when we meet -with A, B will also be present, and <i>vice versâ</i>. Thus -wherever we meet with an object possessing gravity, it is -found to possess inertia also, nor have we met with any -material objects possessing inertia without discovering that -they also possess gravity. The probability has therefore -become very great, as indicated by the rules founded on -the Inverse Method of Probabilities (p. <a href="#Page_257">257</a>), that whenever -in the future we meet an object possessing either of the -properties of gravity and inertia, it will be found on -examination to possess the other of these properties. -This is a clear instance of the employment of generalisation.</p> - -<p>In analogy, on the other hand, we reason from likeness -in many points to likeness in other points. The qualities -or points of resemblance are now numerous, not the -objects. At the poles of Mars are two white spots which -resemble in many respects the white regions of ice and -snow at the poles of the earth. There probably exist no -other similar objects with which to compare these, yet the -exactness of the resemblance enables us to infer, with high -probability, that the spots on Mars consist of ice and snow. -In short, many points of resemblance imply many more. -From the appearance and behaviour of those white spots -we infer that they have all the chemical and physical<span class="pagenum" id="Page_597">597</span> -properties of frozen water. The inference is of course only -probable, and based upon the improbability that aggregates -of many qualities should be formed in a like manner in -two or more cases, without being due to some uniform -condition or cause.</p> - -<p>In reasoning by analogy, then, we observe that two -objects ABCDE . . . . and A′B′C′D′E′ . . . . have -many like qualities, as indicated by the identity of the -letters, and we infer that, since the first has another -quality, X, we shall discover this quality in the second case -by sufficiently close examination. As Laplace says,—“Analogy -is founded on the probability that similar things -have causes of the same kind, and produce the same effects. -The more perfect this similarity, the greater is this probability.”<a id="FNanchor_490" href="#Footnote_490" class="fnanchor">490</a> -The nature of analogical inference is aptly -described in the work on Logic attributed to Kant, where -the rule of ordinary induction is stated in the words, “<i>Eines -in vielen, also in allen</i>,” one quality in many things, therefore -in all; and the rule of analogy is “<i>Vieles in einem, also -auch das übrige in demselben</i>,”<a id="FNanchor_491" href="#Footnote_491" class="fnanchor">491</a> many (qualities) in one, -therefore also the remainder in the same. It is evident -that there may be intermediate cases in which, from the -identity of a moderate number of objects in several properties, -we may infer to other objects. Probability must -rest either upon the number of instances or the depth of -resemblance, or upon the occurrence of both in sufficient -degrees. What there is wanting in extension must be -made up by intension, and <i>vice versâ</i>.</p> - - -<h3><i>Two Meanings of Generalisation.</i></h3> - -<p>The term generalisation, as commonly used, includes two -processes which are of different character, but are often -closely associated together. In the first place, we generalise -when we recognise even in two objects a common nature. -We cannot detect the slightest similarity without opening -the way to inference from one case to the other. If we -compare a cubical crystal with a regular octahedron, there -is little apparent similarity; but, as soon as we perceive<span class="pagenum" id="Page_598">598</span> -that either can be produced by the symmetrical modification -of the other, we discover a groundwork of similarity in the -crystals, which enables us to infer many things of one, -because they are true of the other. Our knowledge of -ozone took its rise from the time when the similarity of -smell, attending electric sparks, strokes of lightning, and -the slow combustion of phosphorus, was noticed by -Schönbein. There was a time when the rainbow was an -inexplicable phenomenon—a portent, like a comet, and a -cause of superstitious hopes and fears. But we find the -true spirit of science in Roger Bacon, who desires us to -consider the objects which present the same colours as the -rainbow; he mentions hexagonal crystals from Ireland and -India, but he bids us not suppose that the hexagonal form -is essential, for similar colours may be detected in many -transparent stones. Drops of water scattered by the oar -in the sun, the spray from a water-wheel, the dewdrops -lying on the grass in the summer morning, all display a -similar phenomenon. No sooner have we grouped together -these apparently diverse instances, than we have begun to -generalise, and have acquired a power of applying to one -instance what we can detect of others. Even when we do -not apply the knowledge gained to new objects, our comprehension -of those already observed is greatly strengthened -and deepened by learning to view them as particular cases -of a more general property.</p> - -<p>A second process, to which the name of generalisation -is often given, consists in passing from a fact or partial law -to a multitude of unexamined cases, which we believe to -be subject to the same conditions. Instead of merely -recognising similarity as it is brought before us, we predict -its existence before our senses can detect it, so that -generalisation of this kind endows us with a prophetic -power of more or less probability. Having observed that -many substances assume, like water and mercury, the three -states of solid, liquid, and gas, and having assured ourselves -by frequent trial that the greater the means we possess of -heating and cooling, the more substances we can vaporise -and freeze, we pass confidently in advance of fact, and -assume that all substances are capable of these three forms. -Such a generalisation was accepted by Lavoisier and -Laplace before many of the corroborative facts now in our<span class="pagenum" id="Page_599">599</span> -possession were known. The reduction of a single comet -beneath the sway of gravity was considered sufficient -indication that all comets obey the same power. Few -persons doubted that the law of gravity extended over the -whole heavens; certainly the fact that a few stars out of -many millions manifest the action of gravity, is now held -to be sufficient evidence of its general extension over the -visible universe.</p> - - -<h3><i>Value of Generalisation.</i></h3> - -<p>It might seem that if we know particular facts, there can -be little use in connecting them together by a general law. -The particulars must be more full of useful information -than an abstract general statement. If we know, for -instance, the properties of an ellipse, a circle, a parabola, -and hyperbola, what is the use of learning all these properties -over again in the general theory of curves of the -second degree? If we understand the phenomena of sound -and light and water-waves separately, what is the need of -erecting a general theory of waves, which, after all, is -inapplicable to practice until resolved again into particular -cases? But, in reality, we never do obtain an adequate -knowledge of particulars until we regard them as cases of -the general. Not only is there a singular delight in discovering -the many in the one, and the one in the many, -but there is a constant interchange of light and knowledge. -Properties which are unapparent in the hyperbola may be -readily observed in the ellipse. Most of the complex -relations which old geometers discovered in the circle will -be reproduced <i>mutatis mutandis</i> in the other conic sections. -The undulatory theory of light might have been unknown -at the present day, had not the theory of sound supplied -hints by analogy. The study of light has made known -many phenomena of interference and polarisation, the -existence of which had hardly been suspected in the case -of sound, but which may now be sought out, and perhaps -found to possess unexpected interest. The careful study -of water-waves shows how waves alter in form and velocity -with varying depth of water. Analogous changes may -some time be detected in sound waves. Thus there is -mutual interchange of aid.</p> - -<p><span class="pagenum" id="Page_600">600</span></p> - -<p>“Every study of a generalisation or extension,” De -Morgan has well said,<a id="FNanchor_492" href="#Footnote_492" class="fnanchor">492</a> “gives additional power over the -particular form by which the generalisation is suggested. -Nobody who has ever returned to quadratic equations -after the study of equations of all degrees, or who has -done the like, will deny my assertion that οὐ βλέπει -βλέπων may be predicated of any one who studies a branch -or a case, without afterwards making it part of a larger -whole. Accordingly it is always worth while to generalise, -were it only to give power over the <i>particular</i>. This -principle, of daily familiarity to the mathematician, is -almost unknown to the logician.”</p> - - -<h3><i>Comparative Generality of Properties.</i></h3> - -<p>Much of the value of science depends upon the knowledge -which we gradually acquire of the different degrees -of generality of properties and phenomena of various kinds. -The use of science consists in enabling us to act with -confidence, because we can foresee the result. Now this -foresight must rest upon the knowledge of the powers -which will come into play. That knowledge, indeed, can -never be certain, because it rests upon imperfect induction, -and the most confident beliefs and predictions of the -physicist may be falsified. Nevertheless, if we always -estimate the probability of each belief according to the -due teaching of the data, and bear in mind that probability -when forming our anticipations, we shall ensure the minimum -of disappointment. Even when he cannot exactly -apply the theory of probabilities, the physicist may acquire -the habit of making judgments in general agreement with -its principles and results.</p> - -<p>Such is the constitution of nature, that the physicist -learns to distinguish those properties which have wide -and uniform extension, from those which vary between -case and case. Not only are certain laws distinctly laid -down, with their extension carefully defined, but a scientific -training gives a kind of tact in judging how far other -laws are likely to apply under any particular circumstances. -We learn by degrees that crystals exhibit phenomena depending<span class="pagenum" id="Page_601">601</span> -upon the directions of the axes of elasticity, which -we must not expect in uniform solids. Liquids, compared -even with non-crystalline solids, exhibit laws of far less -complexity and variety; and gases assume, in many -respects, an aspect of nearly complete uniformity. To -trace out the branches of science in which varying degrees -of generality prevail, would be an inquiry of great interest -and importance; but want of space, if there were no other -reason, would forbid me to attempt it, except in a very -slight manner.</p> - -<p>Gases, so far as they are really gaseous, not only have -exactly the same properties in all directions of space, but -one gas exactly resembles other gases in many qualities. -All gases expand by heat, according to the same law, and -by nearly the same amount; the specific heats of equivalent -weights are equal, and the densities are exactly proportional -to the atomic weights. All such gases obey the -general law, that the volume multiplied by the pressure, -and divided by the absolute temperature, is constant or -nearly so. The laws of diffusion and transpiration are the -same in all cases, and, generally speaking, all physical -laws, as distinguished from chemical laws, apply equally -to all gases. Even when gases differ in chemical or physical -properties, the differences are minor in degree. Thus -the differences of viscosity are far less marked than in the -liquid and solid states. Nearly all gases, again, are colourless, -the exceptions being chlorine, the vapours of iodine, -bromine, and a few other substances.</p> - -<p>Only in one single point, so far as I am aware, do gases -present distinguishing marks unknown or nearly so, in the -solid and liquid states. I mean as regards the light given -off when incandescent. Each gas when sufficiently heated, -yields its own peculiar series of rays, arising from the free -vibrations of the constituent parts of the molecules. Hence -the possibility of distinguishing gases by the spectroscope. -But the molecules of solids and liquids appear to be continually -in conflict with each other, so that only a confused -<i>noise</i> of atoms is produced, instead of a definite series of -luminous chords. At the same temperature, accordingly, -all solids and liquids give off nearly the same rays when -strongly heated, and we have in this case an exception to -the greater generality of properties in gases.</p> - -<p><span class="pagenum" id="Page_602">602</span></p> - -<p>Liquids are in many ways intermediate in character -between gases and solids. While incapable of possessing -different elasticity in different directions, and thus denuded -of the rich geometrical complexity of solids, they retain the -variety of density, colour degrees of transparency, great -diversity in surface tension, viscosity, coefficients of expansion, -compressibility, and many other properties which we -observe in solids, but not for the most part in gases. -Though our knowledge of the physical properties of liquids -is much wanting in generality at present, there is ground -to hope that by degrees laws connecting and explaining the -variations may be traced out.</p> - -<p>Solids are in every way contrasted to gases. Each solid -substance has its own peculiar degree of density, hardness, -compressibility, transparency, tenacity, elasticity, power -of conducting heat and electricity, magnetic properties, -capability of producing frictional electricity, and so forth. -Even different specimens of the same kind of substance will -differ widely, according to the accidental treatment received. -And not only has each substance its own specific properties, -but, when crystallised, its properties vary in each direction -with regard to the axes of crystallisation. The velocity of -radiation, the rate of conduction of heat, the coefficients of -expansibility and compressibility, the thermo-electric properties, -all vary in different crystallographic directions.</p> - -<p>It is probable that many apparent differences between -liquids, and even between solids, will be explained when -we learn to regard them under exactly corresponding -circumstances. The extreme generality of the properties -of gases is in reality only true at an infinitely high temperature, -when they are all equally remote from their condensing -points. Now, it is found that if we compare -liquids—for instance, different kinds of alcohols—not -at equal temperatures, but at points equally distant -from their respective boiling points, the laws and coefficients -of expansion are nearly equal. The vapour-tensions -of liquids also are more nearly equal, when compared -at corresponding points, and the boiling-points -appear in many cases to be simply related to the chemical -composition. No doubt the progress of investigation will -enable us to discover generality, where at present we only -see variety and puzzling complexity.</p> - -<p><span class="pagenum" id="Page_603">603</span></p> - -<p>In some cases substances exhibit the same physical properties -in the liquid as in the solid state. Lead has a high -refractive power, whether in solution, or in solid salts, -crystallised or vitreous. The magnetic power of iron is -conspicuous, whatever be its chemical condition; indeed, -the magnetic properties of substances, though varying -with temperature, seem not to be greatly affected by other -physical changes. Colour, absorptive power for heat or -light rays, and a few other properties are also often the -same in liquids and gases. Iodine and bromine possess a -deep colour whenever they are chemically uncombined. -Nevertheless, we can seldom argue safely from the properties -of a substance in one condition to those in another -condition. Ice is an insulator, water a conductor of -electricity, and the same contrast exists in most other -substances. The conducting power of a liquid for electricity -increases with the temperature, while that of a solid -decreases. By degrees we may learn to distinguish -between those properties of matter which depend upon the -intimate construction of the chemical molecule, and those -which depend upon the contact, conflict, mutual attraction, -or other relations of distinct molecules. The properties -of a substance with respect to light seem generally to -depend upon the molecule; thus, the power of certain -substances to cause the plane of polarisation of a ray of -light to rotate, is exactly the same whatever be its degree -of density, or the diluteness of the solution in which it is -contained. Taken as a whole, the physical properties of -substances and their quantitative laws, present a problem -of infinite complexity, and centuries must elapse before any -moderately complete generalisations on the subject become -possible.</p> - - -<h3><i>Uniform Properties of all Matter.</i></h3> - -<p>Some laws are held to be true of all matter in the -universe absolutely, without exception, no instance to the -contrary having ever been noticed. This is the case with -the laws of motion, as laid down by Galileo and Newton. -It is also conspicuously true of the law of universal gravitation. -The rise of modern physical science may perhaps -be considered as beginning at the time when Galileo<span class="pagenum" id="Page_604">604</span> -showed, in opposition to the Aristotelians, that matter is -equally affected by gravity, irrespective of its form, -magnitude, or texture. All objects fall with equal rapidity, -when disturbing causes, such as the resistance of the air, -are removed or allowed for. That which was rudely -demonstrated by Galileo from the leaning tower of Pisa, -was proved by Newton to a high degree of approximation, -in an experiment which has been mentioned (p. <a href="#Page_443">443</a>).</p> - -<p>Newton formed two pendulums, as nearly as possible the -same in outward shape and size by taking two equal round -wooden boxes, and suspending them by equal threads, -eleven feet long. The pendulums were therefore equally -subject to the resistance of the air. He filled one box -with wood, and in the centre of oscillation of the other he -placed an equal weight of gold. The pendulums were then -equal in weight as well as in size; and, on setting them -simultaneously in motion, Newton found that they vibrated -for a length of time with equal vibrations. He tried the -same experiment with silver, lead, glass, sand, common -salt, water, and wheat, in place of the gold, and ascertained -that the motion of his pendulum was exactly the same -whatever was the kind of matter inside.<a id="FNanchor_493" href="#Footnote_493" class="fnanchor">493</a> He considered -that a difference of a thousandth part would have been -apparent. The reader must observe that the pendulums -were made of equal weight only in order that they might -suffer equal retardation from the air. The meaning of the -experiment is that all substances manifest exactly equal -acceleration from the force of gravity, and that therefore the -inertia or resistance of matter to force, which is the only -independent measure of mass known to us, is always -proportional to gravity.</p> - -<p>These experiments of Newton were considered conclusive -up to very recent times, when certain discordances -between the theory and observations of the movements -of planets led Nicolai, in 1826, to suggest that the equal -gravitation of different kinds of matter might not be -absolutely exact. It is perfectly philosophical thus to -call in question, from time to time, some of the best -accepted laws. On this occasion Bessel carefully repeated -the experiments of Newton with pendulums composed of<span class="pagenum" id="Page_605">605</span> -ivory, glass, marble, quartz, meteoric stones, &c., but was -unable to detect the least difference. This conclusion is -also confirmed by the ultimate agreement of all the calculations -of physical astronomy based upon it. Whether -the mass of Jupiter be calculated from the motion of its -own satellites, from the effect upon the small planets, -Vesta, Juno, &c., or from the perturbation of Encke’s -Comet, the results are closely accordant, showing that -precisely the same law of gravity applies to the most -different bodies which we can observe. The gravity of -a body, again, appears to be entirely independent of its -other physical conditions, being totally unaffected by -any alteration in the temperature, density, electric or -magnetic condition, or other physical properties of the -substance.</p> - -<p>One paradoxical result of the law of equal gravitation -is the theorem of Torricelli, to the effect that all liquids -of whatever density fall or flow with equal rapidity. If -there be two equal cisterns respectively filled with mercury -and water, the mercury, though thirteen times as -heavy, would flow from an aperture neither more rapidly -nor more slowly than the water, and the same would be -true of ether, alcohol, and other liquids, allowance being -made, however, for the resistance of the air, and the -differing viscosities of the liquids.</p> - -<p>In its exact equality and its perfect independence of -all circumstances, except mass and distance, the force of -gravity stands apart from all the other forces and phenomena -of nature, and has not yet been brought into any -relation with them except through the general principle -of the conservation of energy. Magnetic attraction, as -remarked by Newton, follows very different laws, depending -upon the chemical quality and molecular structure -of each particular substance.</p> - -<p>We must remember that in saying “all matter gravitates,” -we exclude from the term matter the basis of light-undulations, -which is immensely more extensive in amount, -and obeys in many respects the laws of mechanics. This -adamantine substance appears, so far as can be ascertained, -to be perfectly uniform in its properties when existing in -space unoccupied by matter. Light and heat are conveyed -by it with equal velocity in all directions, and in all parts<span class="pagenum" id="Page_606">606</span> -of space so far as observation informs us. But the presence -of gravitating matter modifies the density and mechanical -properties of the so-called ether in a way which is yet -quite unexplained.<a id="FNanchor_494" href="#Footnote_494" class="fnanchor">494</a></p> - -<p>Leaving gravity, it is somewhat difficult to discover -other laws which are equally true of all matter. Boerhaave -was considered to have established that all bodies -expand by heat; but not only is the expansion very different -in different substances, but we now know positive -exceptions. Many liquids and a few solids contract by -heat at certain temperatures. There are indeed other -relations of heat to matter which seem to be universal -and uniform; all substances begin to give off rays of light -at the same temperature, according to the law of Draper; -and gases will not be an exception if sufficiently condensed, -as in the experiments of Frankland. Grove considers it -to be universally true that all bodies in combining produce -heat; with the doubtful exception of sulphur and selenium, -all solids in becoming liquids, and all liquids in becoming -gases, absorb heat; but the quantities of heat absorbed -vary with the chemical qualities of the matter. Carnot’s -Thermodynamic Law is held to be exactly true of all matter -without distinction; it expresses the fact that the amount -of mechanical energy which might be theoretically obtained -from a certain amount of heat energy depends only upon -the change of the temperatures, so that whether an engine -be worked by water, air, alcohol, ammonia, or any other -substance, the result would theoretically be the same, if -the boiler and condenser were maintained at similar -temperatures.</p> - - -<h3><i>Variable Properties of Matter.</i></h3> - -<p>I have enumerated some of the few properties of matter, -which are manifested in exactly the same manner by all -substances, whatever be their differences of chemical or -physical constitution. But by far the greater number of<span class="pagenum" id="Page_607">607</span> -qualities vary in degree; substances are more or less -dense, more or less transparent, more or less compressible, -more or less magnetic, and so on. One common result of -the progress of science is to show that qualities supposed -to be entirely absent from many substances are present -only in so low a degree of intensity that the means of -detection were insufficient. Newton believed that most -bodies were quite unaffected by the magnet; Faraday and -Tyndall have rendered it very doubtful whether any substance -whatever is wholly devoid of magnetism, including -under that term diamagnetism. We are rapidly learning -to believe that there are no substances absolutely opaque, -or non-conducting, non-electric, non-elastic, non-viscous, -non-compressible, insoluble, infusible, or non-volatile. All -tends to become a matter of degree, or sometimes of direction. -There may be some substances oppositely affected -to others, as ferro-magnetic substances are oppositely -affected to diamagnetics, or as substances which contract -by heat are opposed to those which expand; but the -tendency is certainly for every affection of one kind of -matter to be represented by something similar in other -kinds. On this account one of Newton’s rules of philosophising -seems to lose all validity; he said, “Those -qualities of bodies which are not capable of being -heightened, and remitted, and which are found in all -bodies on which experiment can be made, must be considered -as universal qualities of all bodies.” As far as I -can see, the contrary is more probable, namely, that -qualities variable in degree will be found in every substance -in a greater or less degree.</p> - -<p>It is remarkable that Newton whose method of investigation -was logically perfect, seemed incapable of generalising -and describing his own procedure. His celebrated -“Rules of Reasoning in Philosophy,” described at the -commencement of the third book of the <i>Principia</i>, are -of questionable truth, and still more questionable value.</p> - - -<h3><i>Extreme Instances of Properties.</i></h3> - -<p>Although substances usually differ only in degree, great -interest may attach to particular substances which manifest -a property in a conspicuous and intense manner. Every<span class="pagenum" id="Page_608">608</span> -branch of physical science has usually been developed from -the attention forcibly drawn to some singular substance. -Just as the loadstone disclosed magnetism and amber -frictional electricity, so did Iceland spar show the existence -of double refraction, and sulphate of quinine the phenomenon -of fluorescence. When one such startling instance -has drawn the attention of the scientific world, numerous -less remarkable cases of the phenomenon will be detected, -and it will probably prove that the property in question is -actually universal to all matter. Nevertheless, the extreme -instances retain their interest, partly in a historical point -of view, partly because they furnish the most convenient -substances for experiment.</p> - -<p>Francis Bacon was fully aware of the value of such -examples, which he called <i>Ostensive Instances</i> or Light-giving, -Free and Predominant Instances. “They are those,” -he says,<a id="FNanchor_495" href="#Footnote_495" class="fnanchor">495</a> “which show the nature under investigation -naked, in an exalted condition, or in the highest degree -of power; freed from impediments, or at least by its -strength predominating over and suppressing them.” He -mentions quicksilver as an ostensive instance of weight or -density, thinking it not much less dense than gold, and -more remarkable than gold as joining density to liquidity. -The magnet is mentioned as an ostensive instance of -attraction. It would not be easy to distinguish clearly -between these ostensive instances and those which he calls -<i>Instantiae Monodicae</i>, or <i>Irregulares</i>, or <i>Heteroclitae</i>, under -which he places whatever is extravagant in its properties -or magnitude, or exhibits least similarity to other things, -such as the sun and moon among the heavenly bodies, the -elephant among animals, the letter <i>s</i> among letters, or the -magnet among stones.<a id="FNanchor_496" href="#Footnote_496" class="fnanchor">496</a></p> - -<p>In optical science great use has been made of the high -dispersive power of the transparent compounds of lead, -that is, the power of giving a long spectrum (p. <a href="#Page_432">432</a>). -Dollond, having noticed this peculiar dispersive power in -lenses made of flint glass, employed them to produce an -achromatic arrangement. The element strontium presents -a contrast to lead in this respect, being characterised by a -remarkably low dispersive power; but I am not aware -that this property has yet been turned to account.</p> - -<p><span class="pagenum" id="Page_609">609</span></p> - -<p>Compounds of lead have both a high dispersive and -a high refractive index, and in the latter respect they -proved very useful to Faraday. Having spent much -labour in preparing various kinds of optical glass, Faraday -happened to form a compound of lead, silica, and -boracic acid, now known as <i>heavy glass</i>, which possessed -an intensely high refracting power. Many years afterwards -in attempting to discover the action of magnetism -upon light he failed to detect any effect, as has been -already mentioned, (p. <a href="#Page_588">588</a>), until he happened to test a -piece of the heavy glass. The peculiar refractive power of -this medium caused the magnetic strain to be apparent, -and the rotation of the plane of polarisation was discovered.</p> - -<p>In almost every part of physical science there is some -substance of powers pre-eminent for the special purpose to -which it is put. Rock-salt is invaluable for its extreme -diathermancy or transparency to the least refrangible rays -of the spectrum. Quartz is equally valuable for its transparency, -as regards the ultra-violet or most refrangible rays. -Diamond is the most highly refracting substance which is -at the same time transparent; were it more abundant and -easily worked it would be of great optical importance. -Cinnabar is distinguished by possessing a power of rotating -the plane of polarisation of light, from 15 to 17 times as -much as quartz. In electric experiments copper is employed -for its high conducting powers and exceedingly low -magnetic properties; iron is of course indispensable for its -enormous magnetic powers; while bismuth holds a like -place as regards its diamagnetic powers, and was of much -importance in Tyndall’s decisive researches upon the polar -character of the diamagnetic force.<a id="FNanchor_497" href="#Footnote_497" class="fnanchor">497</a> In regard to -magne-crystallic action the mineral cyanite is highly -remarkable, being so powerfully affected by the earth’s -magnetism, that, when delicately suspended, it assumes a -constant position with regard to the magnetic meridian, -and may almost be used like the compass needle. Sodium -is distinguished by its unique light-giving powers, which -are so extraordinary that probably one half of the whole -number of stars in the heavens have a yellow tinge in -consequence.</p> - -<p><span class="pagenum" id="Page_610">610</span></p> - -<p>It is remarkable that water, though the most common -of all fluids, is distinguished in almost every respect by -extreme qualities. Of all known substances water has the -highest specific heat, being thus peculiarly fitted for the -purpose of warming and cooling, to which it is often put. -It rises by capillary attraction to a height more than twice -that of any other liquid. In the state of ice it is nearly -twice as dilatable by heat as any other known solid -substance.<a id="FNanchor_498" href="#Footnote_498" class="fnanchor">498</a> In proportion to its density it has a far -higher surface tension than any other substance, being -surpassed in absolute tension only by mercury; and it -would not be difficult to extend considerably the list of its -remarkable and useful properties.</p> - -<p>Under extreme instances we may include cases of remarkably -low powers or qualities. Such cases seem to -correspond to what Bacon calls <i>Clandestine Instances</i>, which -exhibit a given nature in the least intensity, and as it -were in a rudimentary state.<a id="FNanchor_499" href="#Footnote_499" class="fnanchor">499</a> They may often be important, -he thinks, as allowing the detection of the cause -of the property by difference. I may add that in some -cases they may be of use in experiments. Thus hydrogen -is the least dense of all known substances, and has the least -atomic weight. Liquefied nitrous oxide has the lowest -refractive index of all known fluids.<a id="FNanchor_500" href="#Footnote_500" class="fnanchor">500</a> The compounds of -strontium have the lowest dispersive power. It is obvious -that a property of very low degree may prove as curious -and valuable a phenomenon as a property of very high -degree.</p> - - -<h3><i>The Detection of Continuity.</i></h3> - -<p>We should bear in mind that phenomena which are in -reality of a closely similar or even identical nature, may -present to the senses very different appearances. Without -a careful analysis of the changes which take place, we may -often be in danger of widely separating facts and processes, -which are actually instances of the same law. Extreme -difference of degree or magnitude is a frequent cause of<span class="pagenum" id="Page_611">611</span> -error. It is truly difficult at the first moment to recognise -any similarity between the gradual rusting of a piece of -iron, and the rapid combustion of a heap of straw. Yet -Lavoisier’s chemical theory was founded upon the similarity -of the oxydising process in one case and the other. We -have only to divide the iron into excessively small particles -to discover that it is really the more combustible of the -two, and that it actually takes fire spontaneously and burns -like tinder. It is the excessive slowness of the process in -the case of a massive piece of iron which disguises its real -character.</p> - -<p>If Xenophon reports truly, Socrates was misled by not -making sufficient allowance for extreme differences of degree -and quantity. Anaxagoras held that the sun is a fire, -but Socrates rejected this opinion, on the ground that we -can look at a fire, but not at the sun, and that plants grow -by sunshine while they are killed by fire. He also pointed -out that a stone heated in a fire is not luminous, and soon -cools, whereas the sun ever remains equally luminous and -hot.<a id="FNanchor_501" href="#Footnote_501" class="fnanchor">501</a> All such mistakes evidently arise from not perceiving -that difference of quantity may be so extreme as to -assume the appearance of difference of quality. It is the -least creditable thing we know of Socrates, that after pointing -out these supposed mistakes of earlier philosophers, he -advised his followers not to study astronomy.</p> - -<p>Masses of matter of very different size may be expected -to exhibit apparent differences of conduct, arising from the -various intensity of the forces brought into play. Many -persons have thought it requisite to imagine occult forces -producing the suspension of the clouds, and there have even -been absurd theories representing cloud particles as minute -water-balloons buoyed up by the warm air within them. -But we have only to take proper account of the enormous -comparative resistance which the air opposes to the fall of -minute particles, to see that all cloud particles are probably -constantly falling through the air, but so slowly that there -is no apparent effect. Mineral matter again is always regarded -as inert and incapable of spontaneous movement. -We are struck by astonishment on observing in a powerful -microscope, that every kind of solid matter suspended in<span class="pagenum" id="Page_612">612</span> -extremely minute particles in pure water, acquires an -oscillatory movement, often so marked as to resemble dancing -or skipping. I conceive that this movement is due to -the comparatively vast intensity of chemical action when -exerted upon minute particles, the effect being 5,000 or -10,000 greater in proportion to the mass than in fragments -of an inch diameter (p. <a href="#Page_406">406</a>).</p> - -<p>Much that was formerly obscure in the science of electricity -arose from the extreme differences of intensity and -quantity in which this form of energy manifests itself. -Between the brilliant explosive discharge of a thunder-cloud -and the gentle continuous current produced by two pieces -of metal and some dilute acid, there is no apparent analogy -whatever. It was therefore a work of great importance -when Faraday demonstrated the identity of the forces in -action, showing that common frictional electricity would -decompose water like that from the voltaic battery. The -relation of the phenomena became plain when he succeeded -in showing that it would require 800,000 discharges of his -large Leyden battery to decompose one single grain of -water. Lightning was now seen to be electricity of excessively -high tension, but extremely small quantity, the -difference being somewhat analogous to that between the -force of one million gallons of water falling through one -foot, and one gallon of water falling through one million -feet. Faraday estimated that one grain of water acting on -four grains of zinc, would yield electricity enough for a -great thunderstorm.</p> - -<p>It was long believed that electrical conductors and insulators -belonged to two opposed classes of substances. -Between the inconceivable rapidity with which the current -passes through pure copper wire, and the apparently complete -manner in which it is stopped by a thin partition of -gutta-percha or gum-lac, there seemed to be no resemblance. -Faraday again laboured successfully to show that -these were but the extreme cases of a chain of substances -varying in all degrees in their powers of conduction. Even -the best conductors, such as pure copper or silver, offer -resistance to the electric current. The other metals have -considerably higher powers of resistance, and we pass -gradually down through oxides and sulphides. The best -insulators, on the other hand, allow of an atomic induction<span class="pagenum" id="Page_613">613</span> -which is the necessary antecedent of conduction. Hence -Faraday inferred that whether we can measure the effect or -not, all substances discharge electricity more or less.<a id="FNanchor_502" href="#Footnote_502" class="fnanchor">502</a> One -consequence of this doctrine must be, that every discharge -of electricity produces an induced current. In the case of -the common galvanic current we can readily detect the induced -current in any parallel wire or other neighbouring -conductor, and can separate the opposite currents which -arise at the moments when the original current begins and -ends. But a discharge of high tension electricity like -lightning, though it certainly occupies time and has a -beginning and an end, yet lasts so minute a fraction of a -second, that it would be hopeless to attempt to detect and -separate the two opposite induced currents, which are -nearly simultaneous and exactly neutralise each other. -Thus an apparent failure of analogy is explained away, and -we are furnished with another instance of a phenomenon -incapable of observation and yet theoretically known to -exist.<a id="FNanchor_503" href="#Footnote_503" class="fnanchor">503</a></p> - -<p>Perhaps the most extraordinary case of the detection of -unsuspected continuity is found in the discovery of Cagniard -de la Tour and Professor Andrews, that the liquid -and gaseous conditions of matter are only remote points in -a continuous course of change. Nothing is at first sight -more apparently distinct than the physical condition of -water and aqueous vapour. At the boiling-point there is -an entire breach of continuity, and the gas produced is subject -to laws incomparably more simple than the liquid from -which it arose. But Cagniard de la Tour showed that if -we maintain a liquid under sufficient pressure its boiling -point may be indefinitely raised, and yet the liquid will -ultimately assume the gaseous condition with but a small -increase of volume. Professor Andrews, recently following -out this course of inquiry, has shown that liquid carbonic -acid may, at a particular temperature (30°·92 C.), and -under the pressure of 74 atmospheres, be at the same time -in a state indistinguishable from that of liquid and gas. -At higher pressures carbonic acid may be made to pass -from a palpably liquid state to a truly gaseous state without<span class="pagenum" id="Page_614">614</span> -any abrupt change whatever. As the pressure is greater -the abruptness of the change from liquid to gas gradually -decreases, and finally vanishes. Similar phenomena or an -approximation to them have been observed in other liquids, -and there is little doubt that we may make a wide generalisation, -and assert that, under adequate pressure, every -liquid might be made to pass into a gas without breach of -continuity.<a id="FNanchor_504" href="#Footnote_504" class="fnanchor">504</a> The liquid state, moreover, is considered by -Professor Andrews to be but an intermediate step between -the solid and gaseous conditions. There are various indications -that the process of melting is not perfectly abrupt; -and could experiments be made under adequate pressures, -it is believed that every solid could be made to pass by insensible -degrees into the state of liquid, and subsequently -into that of gas.</p> - -<p>These discoveries appear to open the way to most important -and fundamental generalisations, but it is probable -that in many other cases phenomena now regarded as discrete -may be shown to be different degrees of the same -process. Graham was of opinion that chemical affinity -differs but in degree from the ordinary attraction which -holds different particles of a body together. He found that -sulphuric acid continued to evolve heat when mixed even -with the fiftieth equivalent of water, so that there seemed -to be no distinct limit to chemical affinity. He concludes, -“There is reason to believe that chemical affinity passes -in its lowest degree into the attraction of aggregation.”<a id="FNanchor_505" href="#Footnote_505" class="fnanchor">505</a></p> - -<p>The atomic theory is well established, but its limits are -not marked out. As Grove points out, we may by -selecting sufficiently high multipliers express any combination -or mixture of elements in terms of their equivalent -weights.<a id="FNanchor_506" href="#Footnote_506" class="fnanchor">506</a> Sir W. Thomson has suggested that the power -which vegetable fibre, oatmeal, and other substances possess -of attracting and condensing aqueous vapour is probably -continuous, or, in fact, identical with capillary attraction, -which is capable of interfering with the pressure of aqueous -vapour and aiding its condensation.<a id="FNanchor_507" href="#Footnote_507" class="fnanchor">507</a> There are many cases -of so-called catalytic or surface action, such as the extraordinary<span class="pagenum" id="Page_615">615</span> -power of animal charcoal for attracting organic -matter, or of spongy platinum for condensing hydrogen, -which can only be considered as exalted cases of a more -general power of attraction. The number of substances -which are decomposed by light in a striking manner is very -limited; but many other substances, such as vegetable -colours, are affected by long exposure; on the principle of -continuity we might expect to find that all kinds of matter -are more or less susceptible of change by the incidence of -light rays.<a id="FNanchor_508" href="#Footnote_508" class="fnanchor">508</a> It is the opinion of Grove that wherever an -electric current passes there is a tendency to decomposition, -a strain on the molecules, which when sufficiently intense -leads to disruption. Even a metallic conducting wire may -be regarded as tending to decomposition. Davy was probably -correct in describing electricity as chemical affinity -acting on masses, or rather, as Grove suggests, creating a -disturbance through a chain of particles.<a id="FNanchor_509" href="#Footnote_509" class="fnanchor">509</a> Laplace went so -far as to suggest that all chemical phenomena may be results -of the Newtonian law of attraction, applied to atoms of -various mass and position; but the time is probably far -distant when the progress of molecular philosophy and of -mathematical methods will enable such a generalisation to -be verified or refuted.</p> - - -<h3><i>The Law of Continuity.</i></h3> - -<p>Under the title of the Law of Continuity we may place -many applications of the general principle of reasoning, -that what is true of one case will be true of similar cases, -and probably true of what are probably similar. Whenever -we find that a law or similarity is rigorously fulfilled -up to a certain point in time or space, we expect with a -high degree of probability that it will continue to be -fulfilled at least a little further. If we see part only of a -circle, we naturally expect that the circular form will be -continued in the part hidden from us. If a body has moved -uniformly over a certain space, we expect that it will -continue to move uniformly. The ground of such inferences -is doubtless identical with that of other inductive inferences.<span class="pagenum" id="Page_616">616</span> -In continuous motion every infinitely small space passed -over constitutes a separate constituent fact, and had we -perfect powers of observation the smallest finite motion -would include an infinity of information, which, by the -principles of the inverse method of probabilities, would -enable us to infer with certainty to the next infinitely -small portion of the path. But when we attempt to infer -from one finite portion of a path to another finite portion, -inference will be only more or less probable, according to -the comparative lengths of the portions and the accuracy -of observation; the longer our experience is, the more -probable our inference will be; the greater the length of -time or space over which the inference extends, the less -probable.</p> - -<p>This principle of continuity presents itself in nature in -a great variety of forms and cases. It is familiarly expressed -in the dictum <i>Natura non agit per saltum</i>. As -Graham expressed the maxim, there are in nature no abrupt -transitions, and the distinctions of class are never absolute.<a id="FNanchor_510" href="#Footnote_510" class="fnanchor">510</a> -There is always some notice—some forewarning of every -phenomenon, and every change begins by insensible -degrees, could we observe it with perfect accuracy. The -cannon ball, indeed, is forced from the cannon in an -inappreciable portion of time; the trigger is pulled, the fuze -fired, the powder inflamed, the ball expelled, all simultaneously -to our senses. But there is no doubt that time -is occupied by every part of the process, and that the ball -begins to move at first with infinite slowness. Captain -Noble is able to measure by his chronoscope the progress -of the shot in a 300-pounder gun, and finds that the whole -motion within the barrel takes place in something less than -one 200th part of a second. It is certain that no finite -force can produce motion, except in a finite space of time. -The amount of momentum communicated to a body is -proportional to the accelerating force multiplied by the time -during which it acts uniformly. Thus a slight force produces -a great velocity only by long-continued action. In -a powerful shock, like that of a railway collision, the stroke -of a hammer on an anvil, or the discharge of a gun, the<span class="pagenum" id="Page_617">617</span> -time is very short, and therefore the accelerating forces -brought into play are exceedingly great, but never infinite. -In the case of a large gun the powder in exploding is said -to exert for a moment a force equivalent to at least 2,800,000 -horses.</p> - -<p>Our belief in some of the fundamental laws of nature -rests upon the principle of continuity. Galileo is held to -be the first philosopher who consciously employed this -principle in his arguments concerning the nature of motion, -and it is certain that we can never by mere experience -assure ourselves of the truth even of the first law of motion. -<i>A material particle</i>, we are told, <i>when not acted on by -extraneous forces will continue in the same state of rest or -motion.</i> This may be true, but as we can find no body -which is free from the action of extraneous causes, how are -we to prove it? Only by observing that the less the -amount of those forces the more nearly is the law found to -be true. A ball rolled along rough ground is soon stopped; -along a smooth pavement it continues longer in movement. -A delicately suspended pendulum is almost free from -friction against its supports, but it is gradually stopped by -the resistance of the air; place it in the vacuous receiver of -an air-pump and we find the motion much prolonged. A -large planet like Jupiter experiences almost infinitely less -friction, in comparison to its vast momentum, than we can -produce experimentally, and we find in such a case that -there is not the least evidence of the falsity of the law. -Experience, then, informs us that we may approximate -indefinitely to a uniform motion by sufficiently decreasing -the disturbing forces. It is an act of inference which -enables us to travel on beyond experience, and assert that, -in the total absence of any extraneous force, motion would -be absolutely uniform. The state of rest, again, is a -limiting case in which motion is infinitely small or zero, -to which we may attain, on the principle of continuity, by -successively considering cases of slower and slower motion. -There are many classes of phenomena, in which, by -gradually passing from the apparent to the obscure, we can -assure ourselves of the nature of phenomena which would -otherwise be a matter of great doubt. Thus we can sufficiently -prove in the manner of Galileo, that a musical -sound consists of rapid uniform pulses, by causing strokes<span class="pagenum" id="Page_618">618</span> -to be made at intervals which we gradually diminish until -the separate strokes coalesce into a uniform hum or note. -With great advantage we approach, as Tyndall says, the -sonorous through the grossly mechanical. In listening to -a great organ we cannot fail to perceive that the longest -pipes, or their partial tones, produce a tremor and fluttering -of the building. At the other extremity of the scale, there -is no fixed limit to the acuteness of sounds which we can -hear; some individuals can hear sounds too shrill for other -ears, and as there is nothing in the nature of the atmosphere -to prevent the existence of undulations far more rapid than -any of which we are conscious, we may infer, by the principle -of continuity, that such undulations probably exist.</p> - -<p>There are many habitual actions which we perform we -know not how. So rapidly are acts of minds accomplished -that analysis seems impossible. We can only investigate -them when in process of formation, observing that the best -formed habit is slowly and continuously acquired, and it is -in the early stages that we can perceive the rationale of -the process.</p> - -<p>Let it be observed that this principle of continuity must -be held of much weight only in exact physical laws, those -which doubtless repose ultimately upon the simple laws of -motion. If we fearlessly apply the principle to all kinds -of phenomena, we may often be right in our inferences, but -also often wrong. Thus, before the development of spectrum -analysis, astronomers had observed that the more they -increased the powers of their telescopes the more nebulæ -they could resolve into distinct stars. This result had -been so often found true that they almost irresistibly -assumed that all nebulæ would be ultimately resolved by -telescopes of sufficient power; yet Huggins has in recent -years proved by the spectroscope, that certain nebulæ are -actually gaseous, and in a truly nebulous state.</p> - -<p>The principle of continuity must have been continually -employed in the inquiries of Galileo, Newton, and other -experimental philosophers, but it appears to have been -distinctly formulated for the first time by Leibnitz. He at -least claims to have first spoken of “the law of continuity” -in a letter to Bayle, printed in the <i>Nouvelles de la République -des Lettres</i>, an extract from which is given in -Erdmann’s edition of Leibnitz’s works, p. 104, under the<span class="pagenum" id="Page_619">619</span> -title “Sur un Principe Général utile à l’explication des -Lois de la Nature.”<a id="FNanchor_511" href="#Footnote_511" class="fnanchor">511</a> It has indeed been asserted that the -doctrine of the <i>latens processus</i> of Francis Bacon involves -the principle of continuity,<a id="FNanchor_512" href="#Footnote_512" class="fnanchor">512</a> but I think that this doctrine, -like that of the <i>natures</i> of substances, is merely a vague -statement of the principle of causation.</p> - - -<h3><i>Failure of the Law of Continuity.</i></h3> - -<p>There are certain cautions which must be given as to the -application of the principle of continuity. In the first -place, where this principle really holds true, it may seem to -fail owing to our imperfect means of observation. Though -a physical law may not admit of perfectly abrupt change, -there is no limit to the approach which it may make to -abruptness. When we warm a piece of very cold ice, the -absorption of heat, the temperature, and the dilatation of -the ice vary according to apparently simple laws until we -come to the zero of the Centigrade scale. Everything is -then changed; an enormous absorption of heat takes place -without any rise of temperature, and the volume of the ice -decreases as it changes into water. Unless carefully investigated, -this change appears to be perfectly abrupt; but -accurate observation seems to show that there is a certain -forewarning; the ice does not turn into water all at once, -but through a small fraction of a degree the change is -gradual. All the phenomena concerned, if measured very -exactly, would be represented not by angular lines, but -continuous curves, undergoing rapid flexures; and we may -probably assert with safety that between whatever points -of temperature we examine ice, there would be found some -indication, though almost infinitesimally small, of the -apparently abrupt change which was to occur at a higher -temperature. It might also be pointed out that the important -and apparently simple physical laws, such as those -of Boyle and Mariotte, Dalton and Gay-Lussac, &c., are -only approximately true, and the divergences from the -simple laws are forewarnings of abrupt changes, which -would otherwise break the law of continuity.</p> - -<p><span class="pagenum" id="Page_620">620</span></p> - -<p>Secondly, it must be remembered that mathematical laws -of some complexity will probably present singular cases or -negative results, which may bear the appearance of discontinuity, -as when the law of retraction suddenly yields us -with perfect abruptness the phenomenon of total internal -reflection. In the undulatory theory, however, there is -no real change of law between refraction and reflection. -Faraday in the earlier part of his career found so many -substances possessing magnetic power, that he ventured on -a great generalisation, and asserted that all bodies shared -in the magnetic property of iron. His mistake, as he -afterwards discovered, consisted in overlooking the fact -that though magnetic in a certain sense, some substances -have negative magnetism, and are repelled instead of being -attracted by the magnet.</p> - -<p>Thirdly, where we might expect to find a uniform -mathematical law prevailing, the law may undergo abrupt -change at singular points, and actual discontinuity may -arise. We may sometimes be in danger of treating under -one law phenomena which really belong to different laws. -For instance, a spherical shell of uniform matter attracts -an external particle of matter with a force varying inversely -as the square of the distance from the centre of the sphere. -But this law only holds true so long as the particle is -external to the shell. Within the shell the law is wholly -different, and the aggregate gravity of the sphere becomes -zero, the force in every direction being neutralised by -an exactly equal opposite force. If an infinitely small -particle be in the superficies of a sphere, the law is again -different, and the attractive power of the shell is half what -it would be with regard to particles infinitely close to the -surface of the shell. Thus in approaching the centre of a -shell from a distance, the force of gravity shows double -discontinuity in passing through the shell.<a id="FNanchor_513" href="#Footnote_513" class="fnanchor">513</a></p> - -<p>It may admit of question, too, whether discontinuity is -really unknown in nature. We perpetually do meet with -events which are real breaks upon the previous law, though -the discontinuity may be a sign that some independent -cause has come into operation. If the ordinary course of<span class="pagenum" id="Page_621">621</span> -the tides is interrupted by an enormous irregular wave, we -attribute it to an earthquake, or some gigantic natural disturbance. -If a meteoric stone falls upon a person and kills -him, it is clearly a discontinuity in his life, of which he -could have had no anticipation. A sudden sound may pass -through the air neither preceded nor followed by any continuous -effect. Although, then, we may regard the Law of -Continuity as a principle of nature holding rigorously true -in many of the relations of natural forces, it seems to be a -matter of difficulty to assign the limits within which the -law is verified. Much caution is required in its application.</p> - - -<h3><i>Negative Arguments on the Principle of Continuity.</i></h3> - -<p>Upon the principle of continuity we may sometimes -found arguments of great force which prove an hypothesis -to be impossible, because it would involve a continual repetition -of a process <i>ad infinitum</i>, or else a purely arbitrary -breach at some point. Bonnet’s famous theory of reproduction -represented every living creature as containing germs -which were perfect representatives of the next generation, -so that on the same principle they necessarily included -germs of the next generation, and so on indefinitely. The -theory was sufficiently refuted when once clearly stated, -as in the following poem called the Universe,<a id="FNanchor_514" href="#Footnote_514" class="fnanchor">514</a> by Henry -Baker:—</p> - -<div class="poetry-container"> -<div class="poetry"> - <div class="stanza"> - <div class="verse indent0">“Each seed includes a plant: that plant, again,</div> - <div class="verse indent1">Has other seeds, which other plants contain:</div> - <div class="verse indent1">Those other plants have all their seeds, and those</div> - <div class="verse indent1">More plants again, successively inclose.</div> - </div> - <div class="stanza"> - <div class="verse indent0">“Thus, ev’ry single berry that we find,</div> - <div class="verse indent1">Has, really, in itself whole forests of its kind,</div> - <div class="verse indent1">Empire and wealth one acorn may dispense,</div> - <div class="verse indent1">By fleets to sail a thousand ages hence.”</div> - </div> -</div> -</div> - -<p>The general principle of inference, that what we know -of one case must be true of similar cases, so far as they -are similar, prevents our asserting anything which we cannot -apply time after time under the same circumstances.<span class="pagenum" id="Page_622">622</span> -On this principle Stevinus beautifully demonstrated that -weights resting on two inclined planes and balancing each -other must be proportional to the lengths of the planes between -their apex and a horizontal plane. He imagined a -uniform endless chain to be hung over the planes, and to -hang below in a symmetrical festoon. If the chain were -ever to move by gravity, there would be the same reason -for its moving on for ever, and thus producing a perpetual -motion. As this is absurd, the portions of the chain -lying on the planes, and equal in length to the planes, -must balance each other. On similar grounds we may -disprove the existence of any <i>self-moving machine</i>; for if -it could once alter its own state of motion or rest, in however -small a degree, there is no reason why it should not -do the like time after time <i>ad infinitum</i>. Newton’s proof -of his third law of motion, in the case of gravity, is of -this character. For he remarks that if two gravitating -bodies do not exert exactly equal forces in opposite directions, -the one exerting the strongest pull will carry both -away, and the two bodies will move off into space together -with velocity increasing <i>ad infinitum</i>. But though the -argument might seem sufficiently convincing, Newton in his -characteristic way made an experiment with a loadstone -and iron floated upon the surface of water.<a id="FNanchor_515" href="#Footnote_515" class="fnanchor">515</a> In recent -years the very foundation of the principle of conservation -of energy has been placed on the assumption that it is -impossible by any combination of natural bodies to produce -force continually from nothing.<a id="FNanchor_516" href="#Footnote_516" class="fnanchor">516</a> The principle admits -of application in various subtle forms.</p> - -<p>Lucretius attempted to prove, by a most ingenious argument -of this kind, that matter must be indestructible. -For if a finite quantity, however small, were to fall out -of existence in any finite time, an equal quantity might -be supposed to lapse in every equal interval of time, so -that in the infinity of past time the universe must have -ceased to exist.<a id="FNanchor_517" href="#Footnote_517" class="fnanchor">517</a> But the argument, however ingenious, -seems to fail at several points. If past time be infinite, -why may not matter have been created infinite also? It -would be most reasonable, again, to suppose the matter<span class="pagenum" id="Page_623">623</span> -destroyed in any time to be proportional to the matter -then remaining, and not to the original quantity; under -this hypothesis even a finite quantity of original matter -could never wholly disappear from the universe. For like -reasons we cannot hold that the doctrine of the conservation -of energy is really proved, or can ever be proved to -be absolutely true, however probable it may be regarded.</p> - - -<h3><i>Tendency to Hasty Generalisation.</i></h3> - -<p>In spite of all the powers and advantages of generalisation, -men require no incitement to generalise; they are -too apt to draw hasty and ill-considered inferences. As -Francis Bacon said, our intellects want not wings, but -rather weights of lead to moderate their course.<a id="FNanchor_518" href="#Footnote_518" class="fnanchor">518</a> The -process is inevitable to the human mind; it begins with -childhood and lasts through the second childhood. The -child that has once been hurt fears the like result on all -similar occasions, and can with difficulty be made to distinguish -between case and case. It is caution and discrimination -in the adoption of conclusions that we have -chiefly to learn, and the whole experience of life is one -continued lesson to this effect. Baden Powell has excellently -described this strong natural propensity to hasty -inference, and the fondness of the human mind for tracing -resemblances real or fanciful. “Our first inductions,” he -says,<a id="FNanchor_519" href="#Footnote_519" class="fnanchor">519</a> “are always imperfect and inconclusive; we advance -towards real evidence by successive approximations; and -accordingly we find false generalisation the besetting error -of most first attempts at scientific research. The faculty -to generalise accurately and philosophically requires large -caution and long training, and is not fully attained, especially -in reference to more general views, even by some -who may properly claim the title of very accurate scientific -observers in a more limited field. It is an intellectual -habit which acquires immense and accumulating force -from the contemplation of wider analogies.”</p> - -<p>Hasty and superficial generalisations have always been -the bane of science, and there would be no difficulty in<span class="pagenum" id="Page_624">624</span> -finding endless illustrations. Between things which are -the same in number there is a certain resemblance, namely -in number; but in the infancy of science men could not be -persuaded that there was not a deeper resemblance implied -in that of number. Pythagoras was not the inventor -of a mystical science of number. In the ancient Oriental -religions the seven metals were connected with the seven -planets, and in the seven days of the week we still have, -and probably always shall have, a relic of the septiform -system ascribed by Dio Cassius to the ancient Egyptians. -The disciples of Pythagoras carried the doctrine of the -number seven into great detail. Seven days are mentioned -in Genesis; infants acquire their teeth at the end -of seven months; they change them at the end of seven -years; seven feet was the limit of man’s height; every -seventh year was a climacteric or critical year, at which a -change of disposition took place. Then again there were -the seven sages of Greece, the seven wonders of the world, -the seven rites of the Grecian games, the seven gates of -Thebes, and the seven generals destined to conquer that -city.</p> - -<p>In natural science there were not only the seven -planets, and the seven metals, but also the seven primitive -colours, and the seven tones of music. So deep a -hold did this doctrine take that we still have its results -in many customs, not only in the seven days of the week, -but the seven years’ apprenticeship, puberty at fourteen -years, the second climacteric, and legal majority at twenty-one -years, the third climacteric. The idea was reproduced -in the seven sacraments of the Roman Catholic Church, -and the seven year periods of Comte’s grotesque system -of domestic worship. Even in scientific matters the loftiest -intellects have occasionally yielded, as when Newton was -misled by the analogy between the seven tones of music -and the seven colours of his spectrum. Other numerical -analogies, though rejected by Galileo, held Kepler in thraldom; -no small part of Kepler’s labours during seventeen -years was spent upon numerical and geometrical analogies -of the most baseless character; and he gravely held that -there could not be more than six planets, because there -were not more than five regular solids. Even the genius -of Huyghens did not prevent him from inferring that but<span class="pagenum" id="Page_625">625</span> -one satellite could belong to Saturn, because, with those of -Jupiter and the Earth, it completed the perfect number of -six. A whole series of other superstitions and fallacies -attach to the numbers six and nine.</p> - -<p>It is by false generalisation, again, that the laws of -nature have been supposed to possess that perfection which -we attribute to simple forms and relations. The heavenly -bodies, it was held, must move in circles, for the circle was -the perfect figure. Newton seemed to adopt the questionable -axiom that nature always proceeds in the simplest -way; in stating his first rule of philosophising, he adds:<a id="FNanchor_520" href="#Footnote_520" class="fnanchor">520</a> -“To this purpose the philosophers say, that nature does -nothing in vain, when less will serve; for nature is pleased -with simplicity, and affects not the pomp of superfluous -causes.” Keill lays down<a id="FNanchor_521" href="#Footnote_521" class="fnanchor">521</a> as an axiom that “The causes -of natural things are such, as are the most simple, and are -sufficient to explain the phenomena: for nature always -proceeds in the simplest and most expeditious method; -because by this manner of operating the Divine Wisdom -displays itself the more.” If this axiom had any clear -grounds of truth, it would not apply to proximate laws; -for even when the ultimate law is simple the results may -be infinitely diverse, as in the various elliptic, hyperbolic, -parabolic, or circular orbits of the heavenly bodies. Simplicity -is naturally agreeable to a mind of limited powers, -but to an infinite mind all things are simple.</p> - -<p>Every great advance in science consists in a great generalisation, -pointing out deep and subtle resemblances. -The Copernican system was a generalisation, in that it -classed the earth among the planets; it was, as Bishop -Wilkins expressed it, “the discovery of a new planet,” but -it was opposed by a more shallow generalisation. Those -who argued from the condition of things upon the earth’s -surface, thought that every object must be attached to -and rest upon something else. Shall the earth, they said, -alone be free? Accustomed to certain special results of -gravity they could not conceive its action under widely -different circumstances.<a id="FNanchor_522" href="#Footnote_522" class="fnanchor">522</a> No hasty thinker could seize -the deep analogy pointed out by Horrocks between a pendulum<span class="pagenum" id="Page_626">626</span> -and a planet, true in substance though mistaken in -some details. All the advances of modern science rise -from the conception of Galileo, that in the heavenly -bodies, however apparently different their condition, we -shall ultimately recognise the same fundamental principles -of mechanical science which are true on earth.</p> - -<p>Generalisation is the great prerogative of the intellect, -but it is a power only to be exercised safely with much -caution and after long training. Every mind must generalise, -but there are the widest differences in the depth of -the resemblances discovered and the care with which the -discovery is verified. There seems to be an innate power -of insight which a few men have possessed pre-eminently, -and which enabled them, with no exemption indeed from -labour or temporary error, to discover the one in the -many. Minds of excessive acuteness may exist, which -have yet only the powers of minute discrimination, and of -storing up, in the treasure-house of memory, vast accumulations -of words and incidents. But the power of discovery -belongs to a more restricted class of minds. Laplace -said that, of all inventors who had contributed the -most to the advancement of human knowledge, Newton -and Lagrange appeared to possess in the highest degree -the happy tact of distinguishing general principles among -a multitude of objects enveloping them, and this tact -he conceived to be the true characteristic of scientific -genius.<a id="FNanchor_523" href="#Footnote_523" class="fnanchor">523</a></p> - -<hr class="chap x-ebookmaker-drop"> - -<div class="chapter"> -<p><span class="pagenum" id="Page_627">627</span></p> -<h2 class="nobreak" id="CHAPTER_XXVIII">CHAPTER XXVIII.<br> - -<span class="title">ANALOGY.</span></h2> -</div> - -<p class="ti0">As we have seen in the previous chapter, generalisation -passes insensibly into reasoning by analogy, and the difference -is one of degree. We are said to generalise when we -view many objects as agreeing in a few properties, so that -the resemblance is extensive rather than deep. When we -have only a few objects of thought, but are able to discover -many points of resemblance, we argue by analogy that the -correspondence will be even deeper than appears. It -may not be true that the words are always used in such -distinct senses, and there is great vagueness in the employment -of these and many logical terms; but if any clear -discrimination can be drawn between generalisation and -analogy, it is as indicated above.</p> - -<p>It has been said, indeed, that analogy denotes not a -resemblance between things, but between the relations of -things. A pilot is a very different man from a prime -minister, but he bears the same relation to a ship that the -minister does to the state, so that we may analogically -describe the prime minister as the pilot of the state. A -man differs still more from a horse, nevertheless four men -bear to three men the same relation as four horses bear to -three horses. There is a real analogy between the tones of -the Monochord, the Sages of Greece, and the Gates of -Thebes, but it does not extend beyond the fact that they -were all seven in number. Between the most discrete -notions, as, for instance, those of time and space, analogy -may exist, arising from the fact that the mathematical -conditions of the lapse of time and of motion along a line<span class="pagenum" id="Page_628">628</span> -are similar. There is no identity of nature between a word -and the thing it signifies; the substance <i>iron</i> is a heavy -solid, the word <i>iron</i> is either a momentary disturbance of -the air, or a film of black pigment on white paper; but -there is analogy between words and their significates. -The substance iron is to the substance iron-carbonate, as -the name iron is to the name iron-carbonate, when these -names are used according to their scientific definitions. -The whole structure of language and the whole utility of -signs, marks, symbols, pictures, and representations of -various kinds, rest upon analogy. I may hope perhaps -to enter more fully upon this important subject at some -future time, and to attempt to show how the invention of -signs enables us to express, guide, and register our thoughts. -It will be sufficient to observe here that the use of words -constantly involves analogies of a subtle kind; we should -often be at a loss how to describe a notion, were we not -at liberty to employ in a metaphorical sense the name of -anything sufficiently resembling it. There would be no -expression for the sweetness of a melody, or the brilliancy -of an harangue, unless it were furnished by the taste of -honey and the brightness of a torch.</p> - -<p>A cursory examination of the way in which we popularly -use the word analogy, shows that it includes all -degrees of resemblance or similarity. The analogy may -consist only in similarity of number or ratio, or in like relations -of time and space. It may also consist in simple -resemblance between physical properties. We should not -be using the word inconsistently with custom, if we said -that there was an analogy between iron, nickel, and -cobalt, manifested in the strength of their magnetic -powers. There is a still more perfect analogy between -iodine and chlorine; not that every property of iodine is -identical with the corresponding property of chlorine; -for then they would be one and the same kind of substance, -and not two substances; but every property of -iodine resembles in all but degree some property of chlorine. -For almost every substance in which iodine forms -a component, a corresponding substance may be discovered -containing chlorine, so that we may confidently -infer from the compounds of the one to the compounds -of the other substance. Potassium iodide crystallises in<span class="pagenum" id="Page_629">629</span> -cubes; therefore it is to be expected that potassium chloride -will also crystallise in cubes. The science of chemistry -as now developed rests almost entirely upon a careful -and extensive comparison of the properties of substances, -bringing deep-lying analogies to light. When any new -substance is encountered, the chemist is guided in his -treatment of it by the analogies which it seems to present -with previously known substances.</p> - -<p>In this chapter I cannot hope to illustrate the all-pervading -influence of analogy in human thought and -science. All science, it has been said, at the outset, arises -from the discovery of identity, and analogy is but one -name by which we denote the deeper-lying cases of resemblance. -I shall only try to point out at present how -analogy between apparently diverse classes of phenomena -often serves as a guide in discovery. We thus commonly -gain the first insight into the nature of an apparently -unique object, and thus, in the progress of a science, we -often discover that we are treating over again, in a new -form, phenomena which were well known to us in another -form.</p> - - -<h3><i>Analogy as a Guide in Discovery.</i></h3> - -<p>There can be no doubt that discovery is most frequently -accomplished by following up hints received from analogy, -as Jeremy Bentham remarked.<a id="FNanchor_524" href="#Footnote_524" class="fnanchor">524</a> Whenever a phenomenon -is perceived, the first impulse of the mind is to connect it -with the most nearly similar phenomenon. If we could -ever meet a thing wholly <i>sui generis</i>, presenting no -analogy to anything else, we should be incapable of -investigating its nature, except by purely haphazard -trial. The probability of success by such a process is -so slight, that it is preferable to follow up the faintest -clue. As I have pointed out already (p. <a href="#Page_418">418</a>), the possible -experiments are almost infinite in number, and very -numerous also are the hypotheses upon which we may -proceed. Now it is self-evident that, however slightly -superior the probability of success by one course of procedure -may be over another, the most probable one should -always be adopted first.</p> -<p><span class="pagenum" id="Page_630">630</span></p> -<p>The chemist having discovered what he believes to be a -new element, will have before him an infinite variety of -modes of treating and investigating it. If in any of its -qualities the substance displays a resemblance to an alkaline -metal, for instance, he will naturally proceed to try whether -it possesses other properties of the alkaline metals. Even -the simplest phenomenon presents so many points for -notice that we have a choice from among many hypotheses.</p> - -<p>It would be difficult to find a more instructive instance -of the way in which the mind is guided by analogy than -in the description by Sir John Herschel of the course of -thought by which he was led to anticipate in theory one -of Faraday’s greatest discoveries. Herschel noticed that -a screw-like form, technically called helicoidal dissymmetry, -was observed in three cases, namely, in electrical helices, -plagihedral quartz crystals, and the rotation of the plane -of polarisation of light. As he said,<a id="FNanchor_525" href="#Footnote_525" class="fnanchor">525</a> “I reasoned thus: -Here are three phenomena agreeing in a <i>very strange -peculiarity</i>. Probably, this peculiarity is a connecting -link, physically speaking, among them. Now, in the case -of the crystals and the light, this probability has been -turned into certainty by my own experiments. Therefore, -induction led me to conclude that a similar connection -exists, and must turn up, somehow or other, between the -electric current and polarised light, and that the plane of -polarisation would be deflected by magneto-electricity.” -By this course of analogical thought Herschel had actually -been led to anticipate Faraday’s great discovery of the -influence of magnetic strain upon polarised light. He had -tried in 1822–25 to discover the influence of electricity on -light, by sending a ray of polarised light through a helix, -or near a long wire conveying an electric current. Such a -course of inquiry, followed up with the persistency of -Faraday, and with his experimental resources, would -doubtless have effected the discovery. Herschel also -suggests that the plagihedral form of quartz crystals must -be due to a screw-like strain during crystallisation; but -the notion remains unverified by experiment.</p> -<p><span class="pagenum" id="Page_631">631</span></p> - -<h3><i>Analogy in the Mathematical Sciences.</i></h3> - -<p>Whoever wishes to acquire a deep acquaintance with -Nature must observe that there are analogies which connect -whole branches of science in a parallel manner, -and enable us to infer of one class of phenomena what -we know of another. It has thus happened on several -occasions that the discovery of an unsuspected analogy -between two branches of knowledge has been the starting-point -for a rapid course of discovery. The truths readily -observed in the one may be of a different character from -those which present themselves in the other. The analogy, -once pointed out, leads us to discover regions of one -science yet undeveloped, to which the key is furnished by -the corresponding truths in the other science. An interchange -of aid most wonderful in its results may thus -take place, and at the same time the mind rises to a higher -generalisation, and a more comprehensive view of nature.</p> - -<p>No two sciences might seem at first sight more different -in their subject matter than geometry and algebra. The -first deals with circles, squares, parallelograms, and other -forms in space; the latter with mere symbols of number. -Prior to the time of Descartes, the sciences were developed -slowly and painfully in almost entire independence of each -other. The Greek philosophers indeed could not avoid -noticing occasional analogies, as when Plato in the Thæetetus -describes a square number as <i>equally equal</i>, and a -number produced by multiplying two unequal factors -as <i>oblong</i>. Euclid, in the 7th and 8th books of his Elements, -continually uses expressions displaying a consciousness -of the same analogies, as when he calls a number -of two factors a <i>plane number</i>, ἐπίπεδος ἀριθμός, and -distinguishes a square number of which the two factors are -equal as an equal-sided and plane number, ἰσόπλευρος -καὶ ἐπίπεδος ἀριθμός. He also calls the root of a cubic -number its side, πλευρά. In the Diophantine algebra -many problems of a geometrical character were solved by -algebraic or numerical processes; but there was no general -system, so that the solutions were of an isolated character. -In general the ancients were far more advanced in geometric -than symbolic methods; thus Euclid in his 4th book gives<span class="pagenum" id="Page_632">632</span> -the means of dividing a circle by purely geometric means -into 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30 parts, but he -was totally unacquainted with the theory of the roots of -unity exactly corresponding to this division of the circle.</p> - -<p>During the middle ages, on the contrary, algebra advanced -beyond geometry, and modes of solving equations -were gradually discovered by those who had no notion that -at every step they were implicitly solving geometric problems. -It is true that Regiomontanus, Tartaglia, Bombelli, -and possibly other early algebraists, solved isolated geometrical -problems by the aid of algebra, but particular -numbers were always used, and no consciousness of a -general method was displayed. Vieta in some degree -anticipated the final discovery, and occasionally represented -the roots of an equation geometrically, but it was -reserved for Descartes to show, in the most general manner, -that every equation may be represented by a curve or -figure in space, and that every bend, point, cusp, or other -peculiarity in the curve indicates some peculiarity in the -equation. It is impossible to describe in any adequate -manner the importance of this discovery. The advantage -was two-fold: algebra aided geometry, and geometry gave -reciprocal aid to algebra. Curves such as the well-known -sections of the cone were found to correspond to quadratic -equations; and it was impossible to manipulate the equations -without discovering properties of those all-important -curves. The way was thus opened for the algebraic -treatment of motions and forces, without which Newton’s -<i>Principia</i> could never have been worked out. Newton -indeed was possessed by a strong infatuation in favour of -the ancient geometrical methods; but it is well known -that he employed symbolic methods to discover his theorems, -and he now and then, by some accidental use of -algebraic expression, confessed its greater power and -generality.</p> - -<p>Geometry, on the other hand, gave great assistance to -algebra, by affording concrete representations of relations -which would otherwise be too abstract for easy comprehension. -A curve of no great complexity may give the -whole history of the variations of value of a troublesome -mathematical expression. As soon as we know, too, that -every regular geometrical curve represents some algebraic<span class="pagenum" id="Page_633">633</span> -equation, we are presented by observation of mechanical -movements with abundant suggestions towards the discovery -of mathematical problems. Every particle of a -carriage-wheel when moving on a level road is constantly -describing a cycloidal curve, the curious properties of -which exercised the ingenuity of all the most skilful -mathematicians of the seventeenth century, and led to -important advancements in algebraic power. It may be -held that the discovery of the Differential Calculus was -mainly due to geometrical analogy, because mathematicians, -in attempting to treat algebraically the tangent of a curve, -were obliged to entertain the notion of infinitely small -quantities.<a id="FNanchor_526" href="#Footnote_526" class="fnanchor">526</a> There can be no doubt that Newton’s -fluxional, that is, geometrical mode of stating the differential -calculus, however much it subsequently retarded -its progress in England, facilitated its apprehension at first, -and I should think it almost certain that Newton discovered -the principles of the calculus geometrically.</p> - -<p>We may accordingly look upon this discovery of -analogy, this happy alliance, as Bossut calls it,<a id="FNanchor_527" href="#Footnote_527" class="fnanchor">527</a> between -geometry and algebra, as the chief source of discoveries -which have been made for three centuries past in mathematical -methods. This is certainly the opinion of Lagrange, -who says, “So long as algebra and geometry have -been separate, their progress was slow, and their employment -limited; but since these two sciences have been -united, they have lent each other mutual strength, and -have marched together with a rapid step towards perfection.”</p> - -<p>The advancement of mechanical science has also been -greatly aided by analogy. An abstract and intangible -existence like force demands much power of conception, -but it has a perfect concrete representative in a line, the -end of which may denote the point of application, and the -direction the line of action of the force, while the length -can be made arbitrarily to denote the amount of the force. -Nor does the analogy end here; for the moment of the -force about any point, or its product into the perpendicular -distance of its line of action from the point, is<span class="pagenum" id="Page_634">634</span> -found to be represented by an area, namely twice the area -of the triangle contained between the point and the ends -of the line representing the force. Of late years a great -generalisation has been effected; the Double Algebra of De -Morgan is true not only of space relations, but of forces, so -that the triangle of forces is reduced to a case of pure -geometrical addition. Nay, the triangle of lines, the triangle -of velocities, the triangle of forces, the triangle of -couples, and perhaps other cognate theorems, are reduced -by analogy to one simple theorem, which amounts to this, -that there are two ways of getting from one angular point -of a triangle to another, which ways, though different in -length, are identical in their final results.<a id="FNanchor_528" href="#Footnote_528" class="fnanchor">528</a> In the system -of quaternions of the late Sir W. R. Hamilton, these -analogies are embodied and carried out in the most -general manner, so that whatever problem involves the -threefold dimensions of space, or relations analogous to -those of space, is treated by a symbolic method of the -most comprehensive simplicity.</p> - -<p>It ought to be added that to the discovery of analogy -between the forms of mathematical and logical expressions, -we owe the greatest advance in logical science. Boole -based his extension of logical processes upon the notion -that logic is an algebra of two quantities 0 and 1. His -profound genius for symbolic investigation led him to perceive -by analogy that there must exist a general system of -logical deduction, of which the old logicians had seized -only a few fragments. Mistaken as he was in placing -algebra as a higher science than logic, no one can deny that -the development of the more complex and dependent -science had advanced far beyond that of the simpler science, -and that Boole, in drawing attention to the connection, -made one of the most important discoveries in the history -of science. As Descartes had wedded algebra and geometry,<span class="pagenum" id="Page_635">635</span> -so did Boole accomplish the marriage of logic and -algebra.</p> - - -<h3><i>Analogy in the Theory of Undulations.</i></h3> - -<p>There is no class of phenomena which more thoroughly -illustrates alike the power and weakness of analogy than -the waves which agitate every kind of medium. All waves, -whatsoever be the matter through which they pass, obey -the principles of rhythmical or harmonic motion, and the -subject therefore presents a fine field for mathematical -generalisation. Each kind of medium may allow of waves -peculiar in their conditions, so that it is a beautiful exercise -in analogical reasoning to decide how, in making inferences -from one kind of medium to another, we must make allowance -for difference of circumstances. The waves of the -ocean are large and visible, and there are the yet greater -tidal waves which extend around the globe. From such -palpable cases of rhythmical movement we pass to waves -of sound, varying in length from about 32 feet to a small -fraction of an inch. We have but to imagine, if we can, -the fortieth octave of the middle C of a piano, and we -reach the undulations of yellow light, the ultra-violet being -about the forty-first octave. Thus we pass from the -palpable and evident to that which is obscure, if not incomprehensible. -Yet the same phenomena of reflection, -interference, and refraction, which we find in some kinds of -waves, may be expected to occur, <i>mutatis mutandis</i>, in -other kinds.</p> - -<p>From the great to the small, from the evident to the -obscure, is not only the natural order of inference, but it is -the historical order of discovery. The physical science of -the Greek philosophers must have remained incomplete, -and their theories groundless, because they did not understand -the nature of undulations. Their systems were based -upon the notion of movement of translation from place to -place. Modern science tends to the opposite notion that -all motion is alternating or rhythmical, energy flowing onwards -but matter remaining comparatively fixed in position. -Diogenes Laertius indeed correctly compared the propagation -of sound with the spreading of waves on the surface -of water when disturbed by a stone, and Vitruvius displayed<span class="pagenum" id="Page_636">636</span> -a more complete comprehension of the same analogy. -It remained for Newton to create the theory of undulatory -motion in showing by mathematical deductive -reasoning that the particles of an elastic fluid by vibrating -backwards and forwards, might carry a pulse or wave moving -from the source of disturbance, while the disturbed particles -return to their place of rest. He was even able to make a -first approximation by theoretical calculation to the velocity -of sound-waves in the atmosphere. His theory of sound -formed a hardly less important epoch in science than his far -more celebrated theory of gravitation. It opened the way to -all the subsequent applications of mechanical principles to -the insensible motion of molecules. He seems to have been, -too, upon the brink of another application of the same -principles which would have advanced science by a century -of progress, and made him the undisputed founder of all the -theories of matter. He expressed opinions at various times -that light might be due to undulatory movements of a -medium occupying space, and in one intensely interesting -sentence remarks<a id="FNanchor_529" href="#Footnote_529" class="fnanchor">529</a> that colours are probably vibrations of -different lengths, “much after the manner that, in the sense -of hearing, nature makes use of aërial vibrations of several -bignesses to generate sounds of divers tones, for the analogy -of nature is to be observed.” He correctly foresaw that -red and yellow light would consist of the longer undulations, -and blue and violet of the shorter, while white light would -be composed of an indiscriminate mixture of waves of -various lengths. Newton almost overcame the strongest -apparent difficulty of the undulatory theory of light, -namely, the propagation of light in straight lines. For he -observed that though waves of sound bend round an obstacle -to some extent, they do not do so in the same degree -as water-waves.<a id="FNanchor_530" href="#Footnote_530" class="fnanchor">530</a> He had but to extend the analogy -proportionally to light-waves, and not only would the -difficulty have vanished, but the true theory of diffraction -would have been open to him. Unfortunately he had a -preconceived theory that rays of light are bent from and -not towards the shadow of a body, a theory which for once -he did not sufficiently compare with observation to detect<span class="pagenum" id="Page_637">637</span> -its falsity. I am not aware, too, that Newton has, in any -of his works, displayed an understanding of the phenomena -of interference without which his notion of waves must -have been imperfect.</p> - -<p>While the general principles of undulatory motion will -be the same in whatever medium the motion takes place, -the circumstances may be excessively different. Between -light travelling 186,000 miles per second and sound -travelling in air only about 1,100 feet in the same time, or -almost 900,000 times as slowly, we cannot expect a close -outward resemblance. There are great differences, too, in -the character of the vibrations. Gases scarcely admit of -transverse vibration, so that sound travelling in air is a -longitudinal wave, the particles of air moving backwards -and forwards in the same line in which the wave moves onwards. -Light, on the other hand, appears to consist entirely -in the movement of points of force transversely to the direction -of propagation of the ray. The light-wave is partially -analogous to the bending of a rod or of a stretched cord -agitated at one end. Now this bending motion may take -place in any one of an infinite number of planes, and waves -of which the planes are perpendicular to each other cannot -interfere any more than two perpendicular forces can -interfere. The complicated phenomena of polarised light -arise out of this transverse character of the luminous wave, -and we must not expect to meet analogous phenomena in -atmospheric sound-waves. It is conceivable that in solids -we might produce transverse sound undulations, in which -phenomena of polarisation might be reproduced. But it -would appear that even between transverse sound and light-waves -the analogy holds true rather of the principles of -harmonic motion than the circumstances of the vibrating -medium; from experiment and theory it is inferred that the -plane of polarisation in plane polarised light is perpendicular -to instead of being coincident with the direction of -vibration, as it would be in the case of transverse sound -undulations. If so the laws of elastic forces are essentially -different in application to the luminiferous ether and to -ordinary solid bodies.<a id="FNanchor_531" href="#Footnote_531" class="fnanchor">531</a></p> -<p><span class="pagenum" id="Page_638">638</span></p> - -<h3><i>Analogy in Astronomy.</i></h3> - -<p>We shall be much assisted in gaining a true appreciation -of the value of analogy in its feebler degrees, by considering -how much it has contributed to the progress of -astronomical science. Our point of observation is so fixed -with regard to the universe, and our means of examining -distant bodies are so restricted, that we are necessarily -guided by limited and apparently feeble resemblances. In -many cases the result has been confirmed by subsequent -direct evidence of the most forcible character.</p> - -<p>While the scientific world was divided in opinion -between the Copernican and Ptolemaic systems, it was -analogy which furnished the most satisfactory argument. -Galileo discovered, by the use of his new telescope, the -four small satellites which circulate round Jupiter, and -make a miniature planetary world. These four Medicean -Stars, as they were called, were plainly seen to revolve -round Jupiter in various periods, but approximately in -one plane, and astronomers irresistibly inferred that what -might happen on the smaller scale might also be found true -of the greater planetary system. This discovery gave “the -holding turn,” as Herschel expressed it, to the opinions of -mankind. Even Francis Bacon, who, little to the credit of -his scientific sagacity, had previously opposed the Copernican -views, now became convinced, saying “We affirm the -solisequium of Venus and Mercury; since it has been found -by Galileo that Jupiter also has attendants.” Nor did -Huyghens think it superfluous to adopt the analogy as a -valid argument.<a id="FNanchor_532" href="#Footnote_532" class="fnanchor">532</a> Even in an advanced stage of physical -astronomy, the Jovian system has not lost its analogical -interest; for the mutual perturbations of the four satellites -pass through all their phases within a few centuries, and -thus enable us to verify in a miniature case the principles -of stability, which Laplace established for the great planetary -system. Oscillations or disturbances which in the -motions of the planets appear to be secular, because their -periods extend over millions of years, can be watched, in -the case of Jupiter’s satellites, through complete revolutions -within the historical period of astronomy.<a id="FNanchor_533" href="#Footnote_533" class="fnanchor">533</a></p> - -<p><span class="pagenum" id="Page_639">639</span></p> - -<p>In obtaining a knowledge of the stellar universe we -must sometimes depend upon precarious analogies. We -still hold upon this ground the opinion, entertained by -Bruno as long ago as 1591, that the stars may be suns -attended by planets like our earth. This is the most -probable first assumption, and it is supported by spectrum -observations, which show the similarity of light derived -from many stars with that of the sun. But at the same -time we learn by the prism that there are nebulæ and stars -in conditions widely different from anything known in our -system. In the course of time the analogy may perhaps -be restored to comparative completeness by the discovery -of suns in various stages of nebulous condensation. The -history of the evolution of our own world may be traced -back in bodies less developed, or traced forwards in systems -more advanced towards the dissipation of energy, and the -extinction of life. As in a great workshop, we may perhaps -see the material work of Creation as it has progressed -through thousands of millions of years.</p> - -<p>In speculations concerning the physical condition of -the planets and their satellites, we depend upon analogies -of a weak character. We may be said to know that the -moon has mountains and valleys, plains and ridges, volcanoes -and streams of lava, and, in spite of the absence of -air and water, the rocky surface of the moon presents so -many familiar appearances that we do not hesitate to -compare them with the features of our globe. We infer -with high probability that Mars has polar snow and an -atmosphere absorbing blue rays like our own; Jupiter -undoubtedly possesses a cloudy atmosphere, possibly not -unlike a magnified copy of that surrounding the earth, but -our tendency to adopt analogies receives a salutary correction -in the recently discovered fact that the atmosphere of -Uranus contains hydrogen.</p> - -<p>Philosophers have not stopped at these comparatively -safe inferences, but have speculated on the existence of -living creatures in other planets. Huyghens remarked -that as we infer by analogy from the dissected body of a -dog to that of a pig and ox or other animal of the same -general form, and as we expect to find the same viscera, -the heart, stomach, lungs, intestines, &c., in corresponding -positions, so when we notice the similarity of the planets<span class="pagenum" id="Page_640">640</span> -in many respects, we must expect to find them alike in -other respects.<a id="FNanchor_534" href="#Footnote_534" class="fnanchor">534</a> He even enters into an inquiry whether -the inhabitants of other planets would possess reason and -knowledge of the same sort as ours, concluding in the -affirmative. Although the power of intellect might be -different, he considers that they would have the same -geometry if they had any at all, and that what is true -with us would be true with them.<a id="FNanchor_535" href="#Footnote_535" class="fnanchor">535</a> As regards the sun, -he wisely observes that every conjecture fails. Laplace -entertained a strong belief in the existence of inhabitants -on other planets. The benign influence of the sun gives -birth to animals and plants upon the surface of the earth, -and analogy induces us to believe that his rays would tend -to have a similar effect elsewhere. It is not probable that -matter which is here so fruitful of life would be sterile -upon so great a globe as Jupiter, which, like the earth, has -its days and nights and years, and changes which indicate -active forces. Man indeed is formed for the temperature -and atmosphere in which he lives, and, so far as appears, -could not live upon the other planets. But there might -be an infinity of organisations relative to the diverse -constitutions of the bodies of the universe. The most -active imagination cannot form any idea of such various -creatures, but their existence is not unlikely.<a id="FNanchor_536" href="#Footnote_536" class="fnanchor">536</a></p> - -<p>We now know that many metals and other elements -never found in organic structures are yet capable of forming -compounds with substances of vegetable or animal -origin. It is therefore just possible that at different temperatures -creatures formed of different yet analogous compounds -might exist, but it would seem indispensable that -carbon should form the basis of organic structures. We -have no analogies to lead us to suppose that in the absence -of that complex element life can exist. Could we find -globes surrounded by atmospheres resembling our own in -temperature and composition, we should be almost forced -to believe them inhabited, but the probability of any analogical -argument decreases rapidly as the condition of a -globe diverges from that of our own. The Cardinal -Nicholas de Cusa held long ago that the moon was<span class="pagenum" id="Page_641">641</span> -inhabited, but the absence of any appreciable atmosphere -renders the existence of inhabitants highly improbable. -Speculations resting upon weak analogies hardly belong -to the scope of true science, and can only be tolerated as -an antidote to the far worse dogmas which assert that the -thousand million of persons on earth, or rather a small -fraction of them, are the sole objects of care of the Power -which designed this limitless Universe.</p> - - -<h3><i>Failures of Analogy.</i></h3> - -<p>So constant is the aid which we derive from the use of -analogy in all attempts at discovery or explanation, that it -is most important to observe in what cases it may lead us -into difficulties. That which we expect by analogy to -exist</p> - -<p>(1) May be found to exist;</p> - -<p>(2) May seem not to exist, but nevertheless may really -exist;</p> - -<p>(3) May actually be non-existent.</p> - -<p>In the second case the failure is only apparent, and -arises from our obtuseness of perception, the smallness of -the phenomenon to be noticed, or the disguised character -in which it appears. I have already pointed out that the -analogy of sound and light seems to fail because light does -not apparently bend round a corner, the fact being that -it does so bend in the phenomena of diffraction, which -present the effect, however, in such an unexpected and -minute form, that even Newton was misled, and turned -from the correct hypothesis of undulations which he had -partially entertained.</p> - -<p>In the third class of cases analogy fails us altogether, -and we expect that to exist which really does not exist. -Thus we fail to discover the phenomena of polarisation in -sound travelling through the atmosphere, since air is not -capable of any appreciable transverse undulations. These -failures of analogy are of peculiar interest, because they -make the mind aware of its superior powers. There have -been many philosophers who said that we can conceive -nothing in the intellect which we have not previously -received through the senses. This is true in the sense -that we cannot <i>image</i> them to the mind in the concrete<span class="pagenum" id="Page_642">642</span> -form of a shape or a colour; but we can speak of them and -reason concerning them; in short, we often know them -in everything but a sensuous manner. Accurate investigation -shows that all material substances retard the -motion of bodies through them by subtracting energy -by impact. By the law of continuity we can frame the -notion of a vacuous space in which there is no resistance -whatever, nor need we stop there; for we have only to -proceed by analogy to the case where a medium should -accelerate the motion of bodies passing through it, somewhat -in the mode which Aristotelians attributed falsely -to the air. Thus we can frame the notion of <i>negative -density</i>, and Newton could reason exactly concerning it, -although no such thing exists.<a id="FNanchor_537" href="#Footnote_537" class="fnanchor">537</a></p> - -<p>In every direction of thought we may meet ultimately -with similar failures of analogy. A moving point generates -a line, a moving line generates a surface, a moving -surface generates a solid, but what does a moving solid -generate? When we compare a polyhedron, or many-sided -solid, with a polygon, or plane figure of many sides, -the volume of the first is analogous to the area of the -second; the face of the solid answers to the side of the -polygon; the edge of the solid to the point of the figure; -but the corner, or junction of edges in the polyhedron, -is left wholly unrepresented in the plane of the polygon. -Even if we attempted to draw the analogies in some -other manner, we should still find a geometrical notion -embodied in the solid which has no representative in the -figure of two dimensions.<a id="FNanchor_538" href="#Footnote_538" class="fnanchor">538</a></p> - -<p>Faraday was able to frame some notion of matter in a -fourth condition, which should be to gas what gas is to -liquid.<a id="FNanchor_539" href="#Footnote_539" class="fnanchor">539</a> Such substance, he thought, would not fall far -short of <i>radiant matter</i>, by which apparently he meant -the supposed caloric or matter assumed to constitute heat, -according to the corpuscular theory. Even if we could -frame the notion, matter in such a state cannot be known -to exist, and recent discoveries concerning the continuity<span class="pagenum" id="Page_643">643</span> -of the solid, liquid, and gaseous states remove the basis -of the speculation.</p> - -<p>From these and many other instances which might be -adduced, we learn that analogical reasoning leads us to -the conception of many things which, so far as we can -ascertain, do not exist. In this way great perplexities -have arisen in the use of language and mathematical -symbols. All language depends upon analogy; for we -join and arrange words so that they may represent the -corresponding junctions or arrangements of things and -their equalities. But in the use of language we are -obviously capable of forming many combinations of words -to which no corresponding meaning apparently exists. -The same difficulty arises in the use of mathematical -signs, and mathematicians have needlessly puzzled themselves -about the square root of a negative quantity, which -is, in many applications of algebraic calculation, simply a -sign without any analogous meaning, there being a failure -of analogy.</p> -<hr class="chap x-ebookmaker-drop"> - -<div class="chapter"> -<p><span class="pagenum" id="Page_644">644</span></p> - -<h2 class="nobreak" id="CHAPTER_XXIX">CHAPTER XXIX.<br> - -<span class="title">EXCEPTIONAL PHENOMENA.</span></h2> -</div> - -<p class="ti0">If science consists in the detection of identity and the -recognition of uniformity existing in many objects, it -follows that the progress of science depends upon the study -of exceptional phenomena. Such new phenomena are the -raw material upon which we exert our faculties of observation -and reasoning, in order to reduce the new facts -beneath the sway of the laws of nature, either those laws -already well known, or those to be discovered. Not only -are strange and inexplicable facts those which are on the -whole most likely to lead us to some novel and important -discovery, but they are also best fitted to arouse our -attention. So long as events happen in accordance with -our anticipations, and the routine of every-day observation -is unvaried, there is nothing to impress upon the mind the -smallness of its knowledge, and the depth of mystery, which -may be hidden in the commonest sights and objects. In -early times the myriads of stars which remained in apparently -fixed relative positions upon the heavenly sphere, -received less notice from astronomers than those few -planets whose wandering and inexplicable motions formed -a riddle. Hipparchus was induced to prepare the first -catalogue of stars, because a single new star had been -added to those nightly visible; and in the middle ages two -brilliant but temporary stars caused more popular interest -in astronomy than any other events, and to one of them we -owe all the observations of Tycho Brahe, the mediæval -Hipparchus.</p> - -<p>In other sciences, as well as in that of the heavens,<span class="pagenum" id="Page_645">645</span> -exceptional events are commonly the points from which -we start to explore new regions of knowledge. It has been -beautifully said that Wonder is the daughter of Ignorance, -but the mother of Invention; and though the most familiar -and slight events, if fully examined, will afford endless food -for wonder and for wisdom, yet it is the few peculiar and -unlooked-for events which most often lead to a course of -discovery. It is true, indeed, that it requires much -philosophy to observe things which are too near to us.</p> - -<p>The high scientific importance attaching, then, to exceptions, -renders it desirable that we should carefully -consider the various modes in which an exception may be -disposed of; while some new facts will be found to confirm -the very laws to which they seem at first sight clearly -opposed, others will cause us to limit the generality of our -previous statements. In some cases the exception may be -proved to be no exception; occasionally it will prove fatal -to our previous most confident speculations; and there are -some new phenomena which, without really destroying any -of our former theories, open to us wholly new fields of scientific -investigation. The study of this subject is especially -interesting and important, because, as I have before said -(p. <a href="#Page_587">587</a>), no important theory can be built up complete -and perfect all at once. When unexplained phenomena -present themselves as objections to the theory, it will often -demand the utmost judgment and sagacity to assign to -them their proper place and force. The acceptance or -rejection of a theory will depend upon discriminating the -one insuperable contradictory fact from many, which, -however singular and inexplicable at first sight, may -afterwards be shown to be results of different causes, or -possibly the most striking results of the very law with -which they stand in apparent conflict.</p> - -<p>I can enumerate at least eight classes or kinds of exceptional -phenomena, to one or other of which any -supposed exception to the known laws of nature can -usually be referred; they may be briefly described as -below, and will be sufficiently illustrated in the succeeding -sections.</p> - -<p>(1) Imaginary, or false exceptions, that is, facts, objects, -or events which are not really what they are supposed -to be.</p> - -<p><span class="pagenum" id="Page_646">646</span></p> - -<p>(2) Apparent, but congruent exceptions, which, though -apparently in conflict with a law of nature, are really in -agreement with it.</p> - -<p>(3) Singular exceptions, which really agree with a law -of nature, but exhibit remarkable and unique results of it.</p> - -<p>(4) Divergent exceptions, which really proceed from the -ordinary action of known processes of nature, but which -are excessive in amount or monstrous in character.</p> - -<p>(5) Accidental exceptions, arising from the interference -of some entirely distinct but known law of nature.</p> - -<p>(6) Novel and unexplained exceptions, which lead to -the discovery of a new series of laws and phenomena, -modifying or disguising the effects of previously known -laws, without being inconsistent with them.</p> - -<p>(7) Limiting exceptions showing the falsity of a supposed -law in some cases to which it had been extended, -but not affecting its truth in other cases.</p> - -<p>(8) Contradictory or real exceptions which lead us to -the conclusion that a supposed hypothesis or theory is in -opposition to the phenomena of nature, and must therefore -be abandoned.</p> - -<p>It ought to be clearly understood that in no case is a -law of nature really thwarted or prevented from being -fulfilled. The effects of a law may be disguised and -hidden from our view in some instances: in others the -law itself may be rendered inapplicable altogether; but -if a law is applicable it must be carried out. Every -law of nature must therefore be stated with the utmost -generality of all the instances really coming under it. -Babbage proposed to distinguish between <i>universal principles</i>, -which do not admit of a single exception, such -as that every number ending in 5 is divisible by five, -and <i>general principles</i> which are more frequently obeyed -than violated, as that “men will be governed by what -they believe to be their interest.”<a id="FNanchor_540" href="#Footnote_540" class="fnanchor">540</a> But in a scientific -point of view general principles must be universal as -regards some distinct class of objects, or they are not -principles at all. If a law to which exceptions exist is -stated without allusion to those exceptions, the statement -is erroneous. I have no right to say that “All liquids<span class="pagenum" id="Page_647">647</span> -expand by heat,” if I know that water below 4° C. does -not; I ought to say, “All liquids, except water below 4° C., -expand by heat;” and every new exception discovered will -falsify the statement until inserted in it. To speak of -some laws as being <i>generally</i> true, meaning not universally -but in the majority of cases, is a hurtful abuse of the word, -but is quite usual. <i>General</i> should mean that which is -true of a whole <i>genus</i> or class, and every true statement -must be true of some assigned or assignable class.</p> - - -<h3><i>Imaginary or False Exceptions.</i></h3> - -<p>When a supposed exception to a law of nature is brought -to our notice, the first inquiry ought properly to be—Is -there any breach of the law at all? It may be that the -supposed exceptional fact is not a fact at all, but a mere -figment of the imagination. When King Charles requested -the Royal Society to investigate the curious fact that a live -fish put into a bucket of water does not increase the weight -of the bucket and its contents, the Royal Society wisely -commenced their deliberations by inquiring whether the -fact was so or not. Every statement, however false, must -have some cause or prior condition, and the real question -for the Royal Society to investigate was, how the King -came to think that the fact was so. Mental conditions, as -we have seen, enter into all acts of observation, and are -often a worthy subject of inquiry. But there are many -instances in the history of science, in which trouble and -error have been caused by false assertions carelessly made, -and carelessly accepted without verification.</p> - -<p>The reception of the Copernican theory was much -impeded by the objection, that if the earth were moving, a -stone dropped from the top of a high tower should be left -behind, and should appear to move towards the west, just -as a stone dropped from the mast-head of a moving ship -would fall behind, owing to the motion of the ship. The -Copernicans attempted to meet this grave objection in every -way but the true one, namely, showing by trial that the -asserted facts are not correct. In the first place, if a stone -had been dropped with suitable precautions from the mast-head -of a moving ship, it would have fallen close to the foot -of the mast, because, by the first law of motion, it would<span class="pagenum" id="Page_648">648</span> -remain in the same state of horizontal motion communicated -to it by the mast. As the anti-Copernicans had -assumed the contrary result as certain to ensue, their -argument would of course have fallen through. Had the -Copernicans next proceeded to test with great care the other -assertion involved, they would have become still better -convinced of the truth of their own theory. A stone -dropped from the top of a high tower, or into a deep well, -would certainly not have been deflected from the vertical -direction in the considerable degree required to agree with -the supposed consequences of the Copernican views; but, -with very accurate observation, they might have discovered, -as Benzenberg subsequently did, a very small deflection -towards the east, showing that the eastward velocity is -greater at the top than the bottom. Had the Copernicans -then been able to detect and interpret the meaning of the -small divergence thus arising, they would have found in it -corroboration of their own views.</p> - -<p>Multitudes of cases might be cited in which laws of -nature seem to be evidently broken, but in which the -apparent breach arises from a misapprehension of the case. -It is a general law, absolutely true of all crystals yet submitted -to examination, that no crystal has a re-entrant -angle, that is an angle which towards the axis of the crystal -is greater than two right angles. Wherever the faces of a -crystal meet they produce a projecting edge, and wherever -edges meet they produce a corner. Many crystals, however, -when carelessly examined, present exceptions to this law, -but closer observation always shows that the apparently -re-entrant angle really arises from the oblique union of two -distinct crystals. Other crystals seem to possess faces -contradicting all the principles of crystallography; but -careful examination shows that the supposed faces are not -true faces, but surfaces produced by the orderly junction -of an immense number of distinct thin crystalline plates, -each plate being in fact a separate crystal, in which the -laws of crystallography are strictly observed. The roughness -of the supposed face, the striæ detected by the -microscope, or inference by continuity from other specimens -where the true faces of the plates are clearly seen, prove the -mistaken character of the supposed exceptions. Again, four -of the faces of a regular octahedron may become so enlarged<span class="pagenum" id="Page_649">649</span> -in the crystallisation of iron pyrites and some other substances, -that the other four faces become imperceptible and -a regular tetrahedron appears to be produced, contrary to -the laws of crystallographic symmetry. Many other crystalline -forms are similarly modified, so as to produce a -series of what are called <i>hemihedral</i> forms.</p> - -<p>In tracing out the isomorphic relations of the elements, -great perplexity has often been caused by mistaking one -substance for another. It was pointed out that though -arsenic was supposed to be isomorphous with phosphorus, -the arseniate of soda crystallised in a form distinct from -that of the corresponding phosphate. Some chemists held -this to be a fatal objection to the doctrine of isomorphism; -but it was afterwards pointed out by Clarke, that the -arseniate and phosphate in question were not corresponding -compounds, as they differed in regard to the water -of crystallisation.<a id="FNanchor_541" href="#Footnote_541" class="fnanchor">541</a> Vanadium again appeared to be an -exception to the laws of isomorphism, until it was proved -by Professor Roscoe, that what Berzelius supposed to be -metallic vanadium was really an oxide of vanadium.<a id="FNanchor_542" href="#Footnote_542" class="fnanchor">542</a></p> - - -<h3><i>Apparent but Congruent Exceptions.</i></h3> - -<p>Not unfrequently a law of nature will present results -in certain circumstances which appear to be entirely in -conflict with the law itself. Not only may the action of -the law be much complicated and disguised, but it may -in various ways be reversed or inverted, so that careless -observers are misled. Ancient philosophers generally -believed that while some bodies were heavy by nature, -others, such as flame, smoke, bubbles, clouds, &c., were -essentially light, or possessed a tendency to move upwards. -So acute an inquirer as Aristotle failed to perceive the -true nature of buoyancy, and the doctrine of intrinsic -lightness, expounded in his works, became the accepted -view for many centuries. It is true that Lucretius was -aware why flame tends to rise, holding that—</p> - -<div class="poetry-container"> -<div class="content"> -“The flame has weight, though highly rare,<br> - Nor mounts but when compelled by heavier air.” -</div> -</div> - -<p><span class="pagenum" id="Page_650">650</span></p> - -<p class="ti0">Archimedes also was so perfectly acquainted with the -buoyancy of bodies immersed in water, that he could not -fail to perceive the existence of a parallel effect in air. -Yet throughout the early middle ages the light of true -science could not contend with the glare of the Peripatetic -doctrine. The genius of Galileo and Newton was required -to convince people of the simple truth that all matter -is heavy, but that the gravity of one substance may be -overborne by that of another, as one scale of a balance is -carried up by the preponderating weight in the opposite -scale. It is curious to find Newton gravely explaining -the difference of absolute and relative gravity, as if it -were a new discovery proceeding from his theory.<a id="FNanchor_543" href="#Footnote_543" class="fnanchor">543</a> More -than a century elapsed before other apparent exceptions -to the Newtonian philosophy were explained away.</p> - -<p>Newton himself allowed that the motion of the apsides -of the moon’s orbit appeared to be irreconcilable with the -law of gravity, and it remained for Clairaut to remove the -difficulty by more complete mathematical analysis. There -must always remain, in the motions of the heavenly bodies, -discrepancies of some amount between theory and observation; -but such discrepancies have so often yielded in past -times to prolonged investigation that physicists now regard -them as merely apparent exceptions, which will afterwards -be found to agree with the law of gravity.</p> - -<p>The most beautiful instance of an apparent exception, is -found in the total reflection of light, which occurs when a -beam of light within a medium falls very obliquely upon -the boundary separating it from a rarer medium. The -general law is that when a ray strikes the limit between two -media of different refractive indices, part of the light is -reflected and part is refracted; but when the obliquity of -the ray within the denser medium passes beyond a certain -point, there is a sudden apparent breach of continuity, and -the whole of the light is reflected. A clear reason can be -given for this exceptional conduct of the light. According -to the law of refraction, the sine of the angle of incidence -bears a fixed ratio to the sine of the angle of refraction, so -that the greater of the two angles, which is always that in -the less dense medium, may increase up to a right angle;<span class="pagenum" id="Page_651">651</span> -but when the media differ in refractive power, the less -angle cannot become a right angle, as this would require -the sine of an angle to be greater than the radius. It might -seem that this is an exception of the kind described below -as a limiting exception, by which a law is shown to be inapplicable -beyond certain limits; but in the explanation -of the exception according to the undulatory theory, we -find that there is really no breach of the general law. -When an undulation strikes a point in a bounding surface, -spherical waves are produced and spread from the point. -The refracted ray is the resultant of an infinite number of -such spherical waves, and the bending of the ray at the -common surface of two media depends upon the comparative -velocities of propagation of the undulations in those -media. But if a ray falls very obliquely upon the surface -of a rarer medium, the waves proceeding from successive -points of the surface spread so rapidly as never to intersect, -and no resultant wave will then be produced. We thus -perceive that from similar mathematical conditions arise -distinct apparent effects.</p> - -<p>There occur from time to time failures in our best -grounded predictions. A comet, of which the orbit has been -well determined, may fail, like Lexell’s Comet, to appear at -the appointed time and place in the heavens. In the -present day we should not allow such an exception to our -successful predictions to weigh against our belief in the -theory of gravitation, but should assume that some unknown -body had through the action of gravitation deflected the -comet. As Clairaut remarked, in publishing his calculations -concerning the expected reappearance of Halley’s Comet, a -body which passes into regions so remote, and which is -hidden from our view during such long periods, might be -exposed to the influence of forces totally unknown to us, -such as the attraction of other comets, or of planets too far -removed from the sun to be ever perceived by us. In the -case of Lexell’s Comet it was afterwards shown, curiously -enough, that its appearance was not one of a regular series -of periodical returns within the sphere of our vision, but a -single exceptional visit never to be repeated, and probably -due to the perturbing powers of Jupiter. This solitary -visit became a strong confirmation of the law of gravity -with which it seemed to be in conflict.</p> - -<p><span class="pagenum" id="Page_652">652</span></p> - - -<h3><i>Singular Exceptions.</i></h3> - -<p>Among the most interesting of apparent exceptions are -those which I call <i>singular exceptions</i>, because they are -more or less analogous to the singular cases or solutions -which occur in mathematical science. A general mathematical -law embraces an infinite multitude of cases which -perfectly agree with each other in a certain respect. It may -nevertheless happen that a single case, while really obeying -the general law, stands out as apparently different from all -the rest. The rotation of the earth upon its axis gives to -all the stars an apparent motion of rotation from east to -west; but while countless thousands obey the rule, the Pole -Star alone seems to break it. Exact observations indeed -show that it also revolves in a small circle, but a star -might happen for a short time to exist so close to the pole -that no appreciable change of place would be caused by the -earth’s rotation. It would then constitute a perfect singular -exception; while really obeying the law, it would break the -terms in which it is usually stated. In the same way the -poles of every revolving body are singular points.</p> - -<p>Whenever the laws of nature are reduced to a mathematical -form we may expect to meet with singular cases, -and, as all the physical sciences will meet in the mathematical -principles of mechanics, there is no part of nature -where we may not encounter them. In mechanical -science the motion of rotation may be considered an exception -to the motion of translation. It is a general law -that any number of parallel forces, whether acting in the -same or opposite directions, will have a resultant which -may be substituted for them with like effect. This resultant -will be equal to the algebraic sum of the forces, or -the difference of those acting in one direction and the -other; it will pass through a point which is determined by -a simple formula, and which may be described as the mean -point of all the points of application of the parallel forces -(p. <a href="#Page_364">364</a>). Thus we readily determine the resultant of -parallel forces except in one peculiar case, namely, when -two forces are equal and opposite but not in the same -straight line. Being equal and opposite the amount of the -resultant is nothing, yet, as the forces are not in the same<span class="pagenum" id="Page_653">653</span> -straight line, they do not balance each other. Examining -the formula for the point of application of the resultant, we -find that it gives an infinitely great magnitude, so that the -resultant is nothing at all, and acts at an infinite distance, -which is practically the same as to say that there is no -resultant. Two such forces constitute what is known in -mechanical science as a <i>couple</i>, which occasions rotatory -instead of rectilinear motion, and can only be neutralised -by an equal and opposite couple of forces.</p> - -<p>The best instances of singular exceptions are furnished -by the science of optics. It is a general law that in passing -through transparent media the plane of vibration of polarised -light remains unchanged. But in certain liquids, -some peculiar crystals of quartz, and transparent solid -media subjected to a magnetic strain, as in Faraday’s experiment -(pp. <a href="#Page_588">588</a>, <a href="#Page_630">630</a>), the plane of polarisation is rotated -in a screw-like manner. This effect is so entirely <i>sui -generis</i>, so unlike any other phenomena in nature, as to -appear truly exceptional; yet mathematical analysis shows -it to be only a single case of much more general laws. As -stated by Thomson and Tait,<a id="FNanchor_544" href="#Footnote_544" class="fnanchor">544</a> it arises from the composition -of two uniform circular motions. If while a point -is moving round a circle, the centre of that circle move -upon another circle, a great variety of curious curves will -be produced according as we vary the dimensions of the -circles, the rapidity or the direction of the motions. When -the two circles are exactly equal, the rapidities nearly so, -and the directions opposite, the point will be found to -move gradually round the centre of the stationary circle, -and describe a curious star-like figure connected with the -molecular motions out of which the rotational power of the -media rises. Among other singular exceptions in optics -may be placed the conical refraction of light, already -noticed (p. <a href="#Page_540">540</a>), arising from the peculiar form assumed -by a wave of light when passing through certain double-refracting -crystals. The laws obeyed by the wave are -exactly the same as in other cases, yet the results are -entirely <i>sui generis</i>. So far are such cases from contradicting -the law of ordinary cases, that they afford the best -opportunities for verification.</p> -<p><span class="pagenum" id="Page_654">654</span></p> -<p>In astronomy singular exceptions might occur, and in an -approximate manner they do occur. We may point to the -rings of Saturn as objects which, though undoubtedly obeying -the law of gravity, are yet unique, as far as our observation -of the universe has gone. They agree, indeed, with -the other bodies of the planetary system in the stability of -their movements, which never diverge far from the mean -position. There seems to be little doubt that these rings -are composed of swarms of small meteoric stones; formerly -they were thought to be solid continuous rings, and mathematicians -proved that if so constituted an entirely exceptional -event might have happened under certain circumstances. -Had the rings been exactly uniform all round, and with a -centre of gravity coinciding for a moment with that of -Saturn, a singular case of unstable equilibrium would have -arisen, necessarily resulting in the sudden collapse of the -rings, and the fall of their debris upon the surface of the -planet. Thus in one single case the theory of gravity would -give a result wholly unlike anything else known in the -mechanism of the heavens.</p> - -<p>It is possible that we might meet with singular exceptions -in crystallography. If a crystal of the second or dimetric -system, in which the third axis is usually unequal to either -of the other two, happened to have the three axes equal, it -might be mistaken for a crystal of the cubic system, but -would exhibit different faces and dissimilar properties. -There is, again, a possible class of diclinic crystals in which -two axes are at right angles and the third axis inclined to -the other two. This class is chiefly remarkable for its -non-existence, since no crystals have yet been proved to have -such axes. It seems likely that the class would constitute -only a singular case of the more general triclinic system, in -which all three axes are inclined to each other at various -angles. Now if the diclinic form were merely accidental, -and not produced by any general law of molecular constitution, -its actual occurrence would be infinitely improbable, -just as it is infinitely improbable that any star should indicate -the North Pole with perfect exactness.</p> - -<p>In the curves denoting the relation between the temperature -and pressure of water there is, as shown by -Professor J. Thomson, one very remarkable point entirely -unique, at which alone water can remain in the three<span class="pagenum" id="Page_655">655</span> -conditions of gas, liquid, and solid in the same vessel. It is -the triple point at which three lines meet, namely (1) the -steam line, which shows at what temperatures and pressures -water is just upon the point of becoming gaseous; (2) the -ice line, showing when ice is just about to melt; and (3) the -hoar-frost line, which similarly indicates the pressures and -temperatures at which ice is capable of passing directly -into the state of gaseous vapour.<a id="FNanchor_545" href="#Footnote_545" class="fnanchor">545</a></p> - - -<h3><i>Divergent Exceptions.</i></h3> - -<p>Closely analogous to singular exceptions are those divergent -exceptions, in which a phenomenon manifests itself in -unusual magnitude or character, without becoming subject -to peculiar laws. Thus in throwing ten coins, it happened -in four cases out of 2,048 throws, that all the coins fell with -heads uppermost (p. <a href="#Page_208">208</a>); these would usually be regarded -as very singular events, and, according to the theory of -probabilities, they would be rare; yet they proceed only -from an unusual conjunction of accidental events, and from -no really exceptional causes. In all classes of natural -phenomena we may expect to meet with similar divergencies -from the average, sometimes due merely to the principles -of probability, sometimes to deeper reasons. Among every -large collection of persons, we shall probably find some -persons who are remarkably large or remarkably small, -giants or dwarfs, whether in bodily or mental conformation. -Such cases appear to be not mere <i>lusus naturæ</i>, since they -occur with a frequency closely accordant with the law of -error or divergence from an average, as shown by Quetelet -and Mr. Galton.<a id="FNanchor_546" href="#Footnote_546" class="fnanchor">546</a> The rise of genius, and the occurrence of -extraordinary musical or mathematical faculties, are attributed -by Mr. Galton to the same principle of divergence.</p> - -<p>When several distinct forces happen to concur together, -we may have surprising or alarming results. Great storms, -floods, droughts, and other extreme deviations from the -average condition of the atmosphere thus arise. They -must be expected to happen from time to time, and will -yet be very infrequent compared with minor disturbances.<span class="pagenum" id="Page_656">656</span> -They are not anomalous but only extreme events, analogous -to extreme runs of luck. There seems, indeed, to be a -fallacious impression in the minds of many persons, that -the theory of probabilities necessitates uniformity in the -happening of events, so that in the same space of time there -will always be nearly the same number of railway accidents -and murders. Buckle has superficially remarked upon the -constancy of such events as ascertained by Quetelet, and -some of his readers acquire the false notion that there is a -mysterious inexorable law producing uniformity in human -affairs. But nothing can be more opposed to the teachings -of the theory of probability, which always contemplates the -occurrence of unusual runs of luck. That theory shows -the great improbability that the number of railway accidents -per month should be always equal, or nearly so. The -public attention is strongly attracted to any unusual conjunction -of events, and there is a fallacious tendency to -suppose that such conjunction must be due to a peculiar -new cause coming into operation. Unless it can be clearly -shown that such unusual conjunctions occur more frequently -than they should do according to the theory of probabilities, -we should regard them as merely divergent exceptions.</p> - -<p>Eclipses and remarkable conjunctions of the heavenly -bodies may also be regarded as results of ordinary laws -which nevertheless appear to break the regular course of -nature, and never fail to excite surprise. Such events vary -greatly in frequency. One or other of the satellites of -Jupiter is eclipsed almost every day, but the simultaneous -eclipse of three satellites can only take place, according to -the calculations of Wargentin, after the lapse of 1,317,900 -years. The relations of the four satellites are so remarkable, -that it is actually impossible, according to the theory of -gravity, that they should all suffer eclipse simultaneously. -But it may happen that while some of the satellites are -really eclipsed by entering Jupiter’s shadow, the others are -either occulted or rendered invisible by passing over his -disk. Thus on four occasions, in 1681, 1802, 1826, and -1843, Jupiter has been witnessed in the singular condition -of being apparently deprived of satellites. A close conjunction -of two planets always excites admiration, though -such conjunctions must occur at intervals in the ordinary -course of their motions. We cannot wonder that when<span class="pagenum" id="Page_657">657</span> -three or four planets approach each other closely, the event -is long remembered. A most remarkable conjunction of -Mars, Jupiter, Saturn, and Mercury, which took place in -the year 2446 <span class="allsmcap">B.C.</span>, was adopted by the Chinese Emperor, -Chuen Hio, as a new epoch for the chronology of his -Empire, though there is some doubt whether the conjunction -was really observed, or was calculated from the supposed -laws of motion of the planets. It is certain that on the -11th November, 1524, the planets Venus, Jupiter, Mars, -and Saturn were seen very close together, while Mercury -was only distant by about 16° or thirty apparent diameters -of the sun, this conjunction being probably the most remarkable -which has occurred in historical times.</p> - -<p>Among the perturbations of the planets we find divergent -exceptions arising from the peculiar accumulation of effects, -as in the case of the long inequality of Jupiter and Saturn -(p. <a href="#Page_455">455</a>). Leverrier has shown that there is one place between -the orbits of Mercury and Venus, and another between those -of Mars and Jupiter, in either of which, if a small planet -happened to exist, it would suffer comparatively immense -disturbance in the elements of its orbit. Now between -Mars and Jupiter there do occur the minor planets, the -orbits of which are in many cases exceptionally divergent.<a id="FNanchor_547" href="#Footnote_547" class="fnanchor">547</a></p> - -<p>Under divergent exceptions we might place all or nearly -all the instances of substances possessing physical properties -in a very high or low degree, which were described -in the chapter on Generalisation (p. <a href="#Page_607">607</a>). Quicksilver is -divergent among metals as regards its melting point, and -potassium and sodium as regards their specific gravities. -Monstrous productions and variations, whether in the animal -or vegetable kingdoms, should probably be assigned to this -class of exceptions.</p> - -<p>It is worthy of notice that even in such a subject as -formal logic, divergent exceptions seem to occur, not of -course due to chance, but exhibiting in an unusual degree -a phenomenon which is more or less manifested in all -other cases. I pointed out in p. <a href="#Page_141">141</a> that propositions of -the general type A = BC ꖌ <i>bc</i> are capable of expression -in six equivalent logical forms, so that they manifest in a -higher degree than any other proposition yet discovered -the phenomenon of logical equivalence.</p> - -<p><span class="pagenum" id="Page_658">658</span></p> - - -<h3><i>Accidental Exceptions.</i></h3> - -<p>The third and largest class of exceptions contains those -which arise from the casual interference of extraneous -causes. A law may be in operation, and, if so, must be -perfectly fulfilled; but, while we conceive that we are -examining its results, we may have before us the effects -of a different cause, possessing no connexion with the -subject of our inquiry. The law is not really broken, but -at the same time the supposed exception is not illusory. -It may be a phenomenon which cannot occur but under -the condition of the law in question, yet there has been -such interference that there is an apparent failure of -science. There is, for instance, no subject in which more -rigorous and invariable laws have been established than in -crystallography. As a general rule, each chemical substance -possesses its own definite form, by which it can be -infallibly recognised; but the mineralogist has to be on his -guard against what are called <i>pseudomorphic</i> crystals. In -some circumstances a substance, having assumed its proper -crystalline form, may afterwards undergo chemical change; -a new ingredient may be added, a former one removed, or -one element may be substituted for another. In calcium -carbonate the carbonic acid is sometimes replaced by -sulphuric acid, so that we find gypsum in the form of -calcite; other cases are known where the change is inverted -and calcite is found in the form of gypsum. Mica, talc, -steatite, hematite, are other minerals subject to these curious -transmutations. Sometimes a crystal embedded in a matrix -is entirely dissolved away, and a new mineral is subsequently -deposited in the cavity as in a mould. Quartz is -thus found cast in many forms wholly unnatural to it. A -still more perplexing case sometimes occurs. Calcium -carbonate is capable of assuming two distinct forms of -crystallisation, in which it bears respectively the names of -calcite and arragonite. Now arragonite, while retaining its -outward form unchanged, may undergo an internal molecular -change into calcite, as indicated by the altered -cleavage. Thus we may come across crystals apparently -of arragonite, which seem to break all the laws of crystallography, -by possessing the cleavage of a different system of -crystallisation.</p> - -<p><span class="pagenum" id="Page_659">659</span></p> - -<p>Some of the most invariable laws of nature are disguised -by interference of unlooked-for causes. While the barometer -was yet a new and curious subject of investigation, -its theory, as stated by Torricelli and Pascal, seemed to be -contradicted by the fact that in a well-constructed instrument -the mercury would often stand far above 31 inches -in height. Boyle showed<a id="FNanchor_548" href="#Footnote_548" class="fnanchor">548</a> that mercury could be made -to stand as high as 75 inches in a perfectly cleansed tube, -or about two and a half times as high as could be due to -the pressure of the atmosphere. Many theories about -the pressure of imaginary fluids were in consequence put -forth,<a id="FNanchor_549" href="#Footnote_549" class="fnanchor">549</a> and the subject was involved in much confusion -until the adhesive or cohesive force between glass and -mercury, when brought into perfect contact, was pointed -out as the real interfering cause. It seems to me, however, -that the phenomenon is not thoroughly understood -as yet.</p> - -<p>Gay-Lussac observed that the temperature of boiling -water was very different in some kinds of vessels from -what it was in others. It is only when in contact with -metallic surfaces or sharply broken edges that the temperature -is fixed at 100° C. The suspended freezing of -liquids is another case where the action of a law of nature -appears to be interrupted. Spheroidal ebullition was at -first sight a most anomalous phenomenon; it was almost -incredible that water should not boil in a red-hot vessel, or -that ice could actually be produced in a red-hot crucible. -These paradoxical results are now fully explained as due to -the interposition of a non-conducting film of vapour between -the globule of liquid and the sides of the vessel. The feats -of conjurors who handle liquid metals are accounted for in -the same manner. At one time the <i>passive state</i> of steel -was regarded as entirely anomalous. It may be assumed -as a general law that when pieces of electro-negative and -electro-positive metal are placed in nitric acid, and made to -touch each other, the electro-negative metal will undergo -rapid solution. But when iron is the electro-negative and -platinum the electro-positive, the solution of the iron -entirely and abruptly ceases. Faraday ingeniously proved<span class="pagenum" id="Page_660">660</span> -that this effect is due to a thin film of oxide of iron, which -forms upon the surface of the iron and protects it.<a id="FNanchor_550" href="#Footnote_550" class="fnanchor">550</a></p> - -<p>The law of gravity is so simple, and disconnected from -the other laws of nature, that it never suffers any disturbance, -and is in no way disguised, but by the complication -of its own effects. It is otherwise with those secondary -laws of the planetary system which have only an empirical -basis. The fact that all the long known planets -and satellites have a similar motion from west to east is -not necessitated by any principles of mechanics, but -points to some common condition existing in the nebulous -mass from which our system has been evolved. The -retrograde motions of the satellites of Uranus constituted -a distinct breach in this law of uniform direction, which -became all the more interesting when the single satellite of -Neptune was also found to be retrograde. It now became -probable, as Baden Powell well observed, that the anomaly -would cease to be singular, and become a case of another -law, pointing to some general interference which has taken -place on the bounds of the planetary system. Not only -have the satellites suffered from this perturbance, but -Uranus is also anomalous in having an axis of rotation -lying nearly in the ecliptic; and Neptune constitutes a -partial exception to the empirical law of Bode concerning -the distances of the planets, which circumstance may -possibly be due to the same disturbance.</p> - -<p>Geology is a science in which accidental exceptions are -likely to occur. Only when we find strata in their original -relative positions can we surely infer that the order of -succession is the order of time. But it not uncommonly -happens that strata are inverted by the bending and -doubling action of extreme pressure. Landslips may carry -one body of rock into proximity with an unrelated series, -and produce results apparently inexplicable.<a id="FNanchor_551" href="#Footnote_551" class="fnanchor">551</a> Floods, -streams, icebergs, and other casual agents, may lodge -remains in places where they would be wholly unexpected. -Though such interfering causes have been sometimes -wrongly supposed to explain important discoveries, the -geologist must bear the possibility of interference in mind.<span class="pagenum" id="Page_661">661</span> -Scarcely more than a century ago it was held that fossils -were accidental productions of nature, mere forms into -which minerals had been shaped by no peculiar cause. -Voltaire appears not to have accepted such an explanation; -but fearing that the occurrence of fossil fishes on the Alps -would support the Mosaic account of the deluge, he did -not hesitate to attribute them to the remains of fishes -accidentally brought there by pilgrims. In archæological -investigations the greatest caution is requisite in allowing -for secondary burials in ancient tombs and tumuli, for -imitations, forgeries, casual coincidences, disturbance by -subsequent races or by other archæologists. In common -life extraordinary events will happen from time to time, -as when a shepherdess in France was astonished at an iron -chain falling out of the sky close to her, the fact being that -Gay-Lussac had thrown it out of his balloon, which was -passing over her head at the time.</p> - - -<h3><i>Novel and Unexplained Exceptions.</i></h3> - -<p>When a law of nature appears to fail because some other -law has interfered with its action, two cases may present -themselves;—the interfering law may be a known one, or -it may have been previously undetected. In the first case, -which we have sufficiently considered in the preceding -section, we have nothing to do but calculate as exactly as -possible the amount of interference, and make allowance -for it; the apparent failure of the law under examination -should then disappear. But in the second case the results -may be much more important. A phenomenon which -cannot be explained by any known laws may indicate the -interference of undiscovered natural forces. The ancients -could not help perceiving that the general tendency of -bodies downwards failed in the case of the loadstone, nor -would the doctrine of essential lightness explain the exception, -since the substance drawn upwards by the loadstone -is a heavy metal. We now see that there was no breach in -the perfect generality of the law of gravity, but that a new -form of energy manifested itself in the loadstone for the first -time.</p> - -<p>Other sciences show us that laws of nature, rigorously -true and exact, may be developed by those who are<span class="pagenum" id="Page_662">662</span> -ignorant of more complex phenomena involved in their -application. Newton’s comprehension of geometrical optics -was sufficient to explain all the ordinary refractions and -reflections of light. The simple laws of the bending of -rays apply to all rays, whatever the character of the -undulations composing them. Newton suspected the -existence of other classes of phenomena when he spoke of -rays as <i>having sides</i>; but it remained for later experimentalists -to show that light is a transverse undulation, -like the bending of a rod or cord.</p> - -<p>Dalton’s atomic theory is doubtless true of all chemical -compounds, and the essence of it is that the same compound -will always be found to contain the same elements -in the same definite proportions. Pure calcium carbonate -contains 48 parts by weight of oxygen to 40 of calcium -and 12 of carbon. But when careful analyses were made -of a great many minerals, this law appeared to fail. What -was unquestionably the same mineral, judging by its -crystalline form and physical properties, would give varying -proportions of its components, and would sometimes contain -unusual elements which yet could not be set down as -mere impurities. Dolomite, for instance, is a compound of -the carbonates of magnesia and lime, but specimens from -different places do not exhibit any fixed ratio between the -lime and magnesia. Such facts could be reconciled with -the laws of Dalton only by supposing the interference of a -new law, that of Isomorphism.</p> - -<p>It is now established that certain elements are related to -each other, so that they can, as it were, step into each other’s -places without apparently altering the shapes of the crystals -which they constitute. The carbonates of iron, calcium, -and magnesium, are nearly identical in their crystalline -forms, hence they may crystallise together in harmony, -producing mixed minerals of considerable complexity, -which nevertheless perfectly verify the laws of equivalent -proportions. This principle of isomorphism once established, -not only explains what was formerly a stumbling-block, -but gives valuable aid to chemists in deciding upon -the constitution of new salts, since compounds of isomorphous -elements which have identical crystalline forms -must possess corresponding chemical formulæ.</p> - -<p>We may expect that from time to time extraordinary<span class="pagenum" id="Page_663">663</span> -phenomena will be discovered, and will lead to new views -of nature. The recent observation, for instance, that the -resistance of a bar of selenium to a current of electricity is -affected in an extraordinary degree by rays of light falling -upon the selenium, points to a new relation between light -and electricity. The allotropic changes which sulphur, -selenium, and phosphorus undergo by an alteration in the -amount of latent heat which they contain, will probably -lead at some future time to important inferences concerning -the molecular constitution of solids and liquids. The -curious substance ozone has perplexed many chemists, and -Andrews and Tait thought that it afforded evidence of the -decomposition of oxygen by the electric discharge. The -researches of Sir B. C. Brodie negative this notion, and afford -evidence of the real constitution of the substance,<a id="FNanchor_552" href="#Footnote_552" class="fnanchor">552</a> which -still, however, remains exceptional in its properties and -relations, and affords a hope of important discoveries in -chemical theory.</p> - - -<h3><i>Limiting Exceptions.</i></h3> - -<p>We pass to cases where exceptional phenomena are -actually irreconcilable with a law of nature previously -regarded as true. Error must now be allowed to have been -committed, but the error may be more or less extensive. -It may happen that a law holding rigorously true of the -facts actually under notice had been extended by generalisation -to other series of facts then unexamined. Subsequent -investigation may show the falsity of this generalisation, -and the result must be to limit the law for the future to -those objects of which it is really true. The contradiction -to our previous opinions is partial and not total.</p> - -<p>Newton laid down as a result of experiment that every -ray of homogeneous light has a definite refrangibility, which -it preserves throughout its course until extinguished. This -is one case of the general principle of undulatory movement, -which Herschel stated under the title “Principle of Forced -Vibrations” (p. <a href="#Page_451">451</a>), and asserted to be absolutely without -exception. But Herschel himself described in the <i>Philosophical -Transactions</i> for 1845 a curious appearance in a<span class="pagenum" id="Page_664">664</span> -solution of quinine; as viewed by transmitted light the -solution appeared colourless, but in certain aspects it exhibited -a beautiful celestial blue tint. Curiously enough the -colour is seen only in the first portion of liquid which the -light enters. Similar phenomena in fluor-spar had been -described by Brewster in 1838. Professor Stokes, having -minutely investigated the phenomena, discovered that they -were more or less present in almost all vegetable infusions, -and in a number of mineral substances. He came to the -conclusion that this phenomenon, called by him Fluorescence, -could only be explained by an alteration in the -refrangibility of the rays of light; he asserts that light-rays -of very short length of vibration in falling upon certain -atoms excite undulations of greater length, in opposition to -the principle of forced vibrations. No complete explanation -of the mode of change is yet possible, because it depends -upon the intimate constitution of the atoms of the substances -concerned; but Professor Stokes believes that the -principle of forced vibrations is true only so long as the -excursions of an atom are very small compared with the -magnitude of the complex molecules.<a id="FNanchor_553" href="#Footnote_553" class="fnanchor">553</a></p> - -<p>It is well known that in Calorescence the refrangibility -of rays is increased and the wave-length diminished. Rays -of obscure heat and low refrangibility may be concentrated -so as to heat a solid substance, and make it give out rays -belonging to any part of the spectrum, and it seems probable -that this effect arises from the impact of distinct but -conflicting atoms. Nor is it in light only that we discover -limiting exceptions to the law of forced vibrations; for if -we notice gentle waves lapping upon the stones at the edge -of a lake we shall see that each larger wave in breaking -upon a stone gives rise to a series of smaller waves. Thus -there is constantly in progress a degradation in the magnitude -of water-waves. The principle of forced vibrations -seems then to be too generally stated by Herschel, but it -must be a difficult question of mechanical theory to discriminate -the circumstances in which it does and does not -hold true.</p> - -<p>We sometimes foresee the possible existence of exceptions -yet unknown by experience, and limit the statement of our -discoveries accordingly. Extensive inquiries have shown<span class="pagenum" id="Page_665">665</span> -that all substances yet examined fall into one of two classes; -they are all either ferro-magnetic, that is, magnetic in the -same way as iron, or they are diamagnetic like bismuth. -But it does not follow that every substance must be ferro-magnetic -or diamagnetic. The magnetic properties are -shown by Sir W. Thomson<a id="FNanchor_554" href="#Footnote_554" class="fnanchor">554</a> to depend upon the specific -inductive capacities of the substance in three rectangular -directions. If these inductive capacities are all positive, we -have a ferro-magnetic substance; if negative, a diamagnetic -substance; but if the specific inductive capacity were -positive in one direction and negative in the others, we -should have an exception to previous experience, and -could not place the substance under either of the present -recognised classes.</p> - -<p>So many gases have been reduced to the liquid state, and -so many solids fused, that scientific men rather hastily -adopted the generalisation that all substances could exist -in all three states. A certain number of gases, such as -oxygen, hydrogen, and nitrogen, have resisted all efforts to -liquefy them, and it now seems probable from the experiments -of Dr. Andrews that they are limiting exceptions. -He finds that above 31° C. carbonic acid cannot be liquefied -by any pressure he could apply, whereas below this temperature -liquefaction is always possible. By analogy it -becomes probable that even hydrogen might be liquefied if -cooled to a very low temperature. We must modify our -previous views, and either assert that <i>below a certain critical -temperature</i> every gas may be liquefied, or else we must -assume that a highly condensed gas is, when above the -critical temperature, undistinguishable from a liquid. At -the same time we have an explanation of a remarkable -exception presented by liquid carbonic acid to the general -rule that gases expand more by heat than liquids. Liquid -carbonic acid was found by Thilorier in 1835 to expand -more than four times as much as air; but by the light of -Andrews’ experiments we learn to regard the liquid as -rather a highly condensed gas than an ordinary liquid, and -it is actually possible to reduce the gas to the apparently -liquid condition without any abrupt condensation.<a id="FNanchor_555" href="#Footnote_555" class="fnanchor">555</a></p> -<p><span class="pagenum" id="Page_666">666</span></p> -<p>Limiting exceptions occur most frequently in the natural -sciences of Botany, Zoology, Geology, &c., the laws of which -are empirical. In innumerable instances the confident -belief of one generation has been falsified by the wider -observation of a succeeding one. Aristotle confidently -held that all swans are white,<a id="FNanchor_556" href="#Footnote_556" class="fnanchor">556</a> and the proposition seemed -true until not a hundred years ago black swans were discovered -in Western Australia. In zoology and physiology -we may expect a fundamental identity to exist in the vital -processes, but continual discoveries show that there is no -limit to the apparently anomalous expedients by which -life is reproduced. Alternate generation, fertilisation for -several successive generations, hermaphroditism, are opposed -to all we should expect from induction founded -upon the higher animals. But such phenomena are only -limiting exceptions showing that what is true of one -class is not true of another. In certain of the cephalopoda -we meet the extraordinary fact that an arm of the -male is cast off and lives independently until it encounters -the female.</p> - - -<h3><i>Real Exceptions to Supposed Laws.</i></h3> - -<p>The exceptions which we have lastly to consider are -the most important of all, since they lead to the entire -rejection of a law or theory before accepted. No law of -nature can fail; there are no such things as real exceptions -to real laws. Where contradiction exists it must be -in the mind of the experimentalist. Either the law is -imaginary or the phenomena which conflict with it; if, -then, by our senses we satisfy ourselves of the actual -occurrence of the phenomena, the law must be rejected -as illusory. The followers of Aristotle held that nature -abhors a vacuum, and thus accounted for the rise of water -in a pump. When Torricelli pointed out the visible fact -that water would not rise more than 33 feet in a pump, -nor mercury more than about 30 inches in a glass tube, -they attempted to represent these facts as limiting exceptions, -saying that nature abhorred a vacuum to a certain -extent and no further. But the Academicians del Cimento<span class="pagenum" id="Page_667">667</span> -completed their discomfiture by showing that if we remove -the pressure of the surrounding air, and in proportion as -we remove it, nature’s feelings of abhorrence decrease and -finally disappear altogether. Even Aristotelian doctrines -could not stand such direct contradiction.</p> - -<p>Lavoisier’s ideas concerning the constitution of acids -received complete refutation. He named oxygen the <i>acid -generator</i>, because he believed that all acids were compounds -of oxygen, a generalisation based on insufficient -data. Berthollet, as early as 1789, proved by analysis that -hydrogen sulphide and prussic acid, both clearly acting -the part of acids, were devoid of oxygen; the former might -perhaps have been interpreted as a limiting exception, but -when so powerful an acid as hydrogen chloride (muriatic -acid) was found to contain no oxygen the theory had to be -relinquished. Berzelius’ theory of the dual formation of -chemical compounds met a similar fate.</p> - -<p>It is obvious that all conclusive <i>experimenta crucis</i> constitute -real exceptions to the supposed laws of the theory -which is overthrown. Newton’s corpuscular theory of light -was not rejected on account of its absurdity or inconceivability, -for in these respects it is, as we have seen, far -superior to the undulatory theory. It was rejected because -certain small fringes of colour did not appear in the exact -place and of the exact size in which calculation showed -that they ought to appear according to the theory (pp. <a href="#Page_516">516</a>–521). -One single fact clearly irreconcilable with a theory -involves its rejection. In the greater number of cases, -what appears to be a fatal exception may be afterwards -explained away as a singular or disguised result of the -laws with which it seems to conflict, or as due to the interference -of extraneous causes; but if we fail thus to reduce -the fact to congruity, it remains more powerful than any -theories or any dogmas.</p> - -<p>Of late years not a few of the favourite doctrines of -geologists have been rudely destroyed. It was the general -belief that human remains were to be found only in those -deposits which are actually in progress at the present day, -so that the creation of man appeared to have taken place -in this geological age. The discovery of a single worked -flint in older strata and in connexion with the remains of -extinct mammals was sufficient to explode such a doctrine.<span class="pagenum" id="Page_668">668</span> -Similarly, the opinions of geologists have been altered by -the discovery of the Eozoön in the Laurentian rocks of -Canada; it was previously held that no remains of life -occurred in any older strata than those of the Cambrian -system. As the examination of the strata of the globe -becomes more complete, our views of the origin and succession -of life upon the globe must undergo many changes.</p> - - -<h3><i>Unclassed Exceptions.</i></h3> - -<p>At every period of scientific progress there will exist a -multitude of unexplained phenomena which we know not -how to regard. They are the outstanding facts upon -which the labours of investigators must be exerted,—the -ore from which the gold of future discovery is to be extracted. -It might be thought that, as our knowledge of -the laws of nature increases, the number of such exceptions -should decrease; but, on the contrary, the more we know -the more there is yet to explain. This arises from several -reasons; in the first place, the principal laws and forces in -nature are numerous, so that he who bears in mind the -wonderfully large numbers developed in the doctrine of -combinations, will anticipate the existence of immensely -numerous relations of one law to another. When we are -once in possession of a law, we are potentially in possession -of all its consequences; but it does not follow that the -mind of man, so limited in its powers and capacities, can -actually work them all out in detail. Just as the aberration -of light was discovered empirically, though it should -have been foreseen, so there are multitudes of unexplained -facts, the connexion of which with laws of nature already -known to us, we should perceive, were we not hindered by -the imperfection of our deductive powers. But, in the -second place, as will be more fully pointed out, it is not to -be supposed that we have approximated to an exhaustive -knowledge of nature’s powers. The most familiar facts -may teem with indications of forces, now secrets hidden -from us, because we have not mind-directed eyes to -discriminate them. The progress of science will consist -in the discovery from time to time of new exceptional -phenomena, and their assignment by degrees to one or -other of the heads already described. When a new fact<span class="pagenum" id="Page_669">669</span> -proves to be merely a false, apparent, singular, divergent, -or accidental exception, we gain a more minute and accurate -acquaintance with the effects of laws already known -to exist. We have indeed no addition to what was implicitly -in our possession, but there is much difference -between knowing the laws of nature and perceiving all -their complicated effects. Should a new fact prove to be a -limiting or real exception, we have to alter, in part or in -whole, our views of nature, and are saved from errors into -which we had fallen. Lastly, the new fact may come -under the sixth class, and may eventually prove to be a -novel phenomenon, indicating the existence of new laws -and forces, complicating but not otherwise interfering with -the effects of laws and forces previously known.</p> - -<p>The best instance which I can find of an unresolved -exceptional phenomenon, consists in the anomalous vapour-densities -of phosphorus, arsenic, mercury, and cadmium. -It is one of the most important laws of chemistry, discovered -by Gay-Lussac, that equal volumes of gases exactly -correspond to equivalent weights of the substances. Nevertheless -phosphorus and arsenic give vapours exactly twice -as dense as they should do by analogy, and mercury and -cadmium diverge in the other direction, giving vapours -half as dense as we should expect. We cannot treat these -anomalies as limiting exceptions, and say that the law -holds true of substances generally but not of these; for -the properties of gases (p. <a href="#Page_601">601</a>), usually admit of the -widest generalisations. Besides, the preciseness of the -ratio of divergence points to the real observance of the law -in a modified manner. We might endeavour to reduce the -exceptions by doubling the atomic weights of phosphorus -and arsenic, and halving those of mercury and cadmium. -But this step has been maturely considered by chemists, -and is found to conflict with all the other analogies of the -substances and with the principle of isomorphism. One -of the most probable explanations is, that phosphorus and -arsenic produce vapour in an allotropic condition, which -might perhaps by intense heat be resolved into a simpler -gas of half the density; but facts are wanting to support -this hypothesis, and it cannot be applied to the other two -exceptions without supposing that gases and vapours -generally are capable of resolution into something simpler.<span class="pagenum" id="Page_670">670</span> -In short, chemists can at present make nothing of these -anomalies. As Hofmann says, “Their philosophical interpretation -belongs to the future.... They may turn out to -be typical facts, round which many others of the like kind -may come hereafter to be grouped; and they may prove to -be allied with special properties, or dependent on particular -conditions as yet unsuspected.”<a id="FNanchor_557" href="#Footnote_557" class="fnanchor">557</a></p> - -<p>It would be easy to point out a great number of other -unexplained anomalies. Physicists assert, as an absolutely -universal law, that in liquefaction heat is absorbed;<a id="FNanchor_558" href="#Footnote_558" class="fnanchor">558</a> -yet sulphur is at least an apparent exception. The two -substances, sulphur and selenium, are, in fact, very anomalous -in their relations to heat. Sulphur may be said -to have two melting points, for, though liquid like water -at 120° C., it becomes quite thick and tenacious between -221° and 249°, and melts again at a higher temperature. -Both sulphur and selenium may be thrown into several -curious states, which chemists conveniently dispose of by -calling them <i>allotropic</i>, a term freely used when they are -puzzled to know what has happened. The chemical and -physical history of iron, again, is full of anomalies; not -only does it undergo inexplicable changes of hardness and -texture in its alloys with carbon and other elements, but -it is almost the only substance which conveys sound with -greater velocity at a higher than at a lower temperature, -the velocity increasing from 20° to 100° C., and then decreasing. -Silver also is anomalous in regard to sound. -These are instances of inexplicable exceptions, the bearing -of which must be ascertained in the future progress of -science.</p> - -<p>When the discovery of new and peculiar phenomena -conflicting with our theories of the constitution of nature -is reported to us, it becomes no easy task to steer a philosophically -correct course between credulity and scepticism. -We are not to assume, on the one hand, that there is any -limit to the wonders which nature can present to us. -Nothing except the contradictory is really impossible, and -many things which we now regard as common-place were -considered as little short of the miraculous when first<span class="pagenum" id="Page_671">671</span> -perceived. The electric telegraph was a visionary dream -among mediæval physicists;<a id="FNanchor_559" href="#Footnote_559" class="fnanchor">559</a> it has hardly yet ceased to -excite our wonder; to our descendants centuries hence -it will probably appear inferior in ingenuity to some -inventions which they will possess. Now every strange -phenomenon may be a secret spring which, if rightly -touched, will open the door to new chambers in the palace -of nature. To refuse to believe in the occurrence of anything -strange would be to neglect the most precious chances -of discovery. We may say with Hooke, that “the believing -strange things possible may perhaps be an occasion of taking -notice of such things as another would pass by without -regard as useless.” We are not, therefore, to shut our ears -even to such apparently absurd stories as those concerning -second-sight, clairvoyance, animal magnetism, ode force, -table-turning, or any of the popular delusions which from -time to time are current. The facts recorded concerning -these matters are facts in some sense or other, and they -demand explanation, either as new natural phenomena, or -as the results of credulity and imposture. Most of the -supposed phenomena referred to have been, or by careful -investigation would doubtless be, referred to the latter -head, and the absence of scientific ability in many of -those who describe them is sufficient to cast a doubt upon -their value.</p> - -<p>It is to be remembered that according to the principle -of the inverse method of probability, the probability -of any hypothetical explanation is affected by the probability -of each other possible explanation. If no other -reasonable explanation could be suggested, we should be -forced to look upon spiritualist manifestations as indicating -mysterious causes. But as soon as it is shown that fraud -has been committed in several important cases, and that in -other cases persons in a credulous and excited state of mind -have deceived themselves, the probability becomes very considerable -that similar explanations may apply to most like -manifestations. The performances of conjurors sufficiently -prove that it requires no very great skill to perform tricks -the <i>modus operandi</i> of which shall entirely escape the<span class="pagenum" id="Page_672">672</span> -notice of spectators. It is on these grounds of probability -that we should reject the so-called spiritualist -stories, and not simply because they are strange.</p> - -<p>Certainly in the obscure phenomena of mind, those -relating to memory, dreams, somnambulism, and other -peculiar states of the nervous system, there are many -inexplicable and almost incredible facts, and it is equally -unphilosophical to believe or to disbelieve without clear -evidence. There are many facts, too, concerning the -instincts of animals, and the mode in which they find -their way from place to place, which are at present quite -inexplicable. No doubt there are many strange things -not dreamt of in our philosophy, but this is no reason -why we should believe in every strange thing which is -reported to have happened.</p> -<hr class="chap x-ebookmaker-drop"> - -<div class="chapter"> -<p><span class="pagenum" id="Page_673">673</span></p> - -<h2 class="nobreak" id="CHAPTER_XXX">CHAPTER XXX.<br> - -<span class="title">CLASSIFICATION.</span></h2> -</div> - -<p class="ti0">The extensive subject of Classification has been deferred -to a late part of this treatise, because it involves questions -of difficulty, and did not seem naturally to fall into an -earlier place. But it must not be supposed that, in now -formally taking up the subject, we are for the first time -entertaining the notion of classification. All logical inference -involves classification, which is indeed the necessary -accompaniment of the action of judgment. It is impossible -to detect similarity between objects without thereby joining -them together in thought, and forming an incipient class. -Nor can we bestow a common name upon objects without -implying the existence of a class. Every common name is -the name of a class, and every name of a class is a common -name. It is evident also that to speak of a general notion -or concept is but another way of speaking of a class. Usage -leads us to employ the word classification in some cases -and not in others. We are said to form the <i>general notion</i> -parallelogram when we regard an infinite number of possible -four-sided rectilinear figures as resembling each other in -the common property of possessing parallel sides. We -should be said to form a <i>class</i>, Trilobite, when we place -together in a museum a number of specimens resembling -each other in certain defined characters. But the logical -nature of the operation is the same in both cases. We -form a <i>class</i> of figures called parallelograms and we form -a <i>general notion</i> of trilobites.</p> - -<p>Science, it was said at the outset, is the detection of -identify, and classification is the placing together, either in<span class="pagenum" id="Page_674">674</span> -thought or in actual proximity of space, those objects between -which identity has been detected. Accordingly, the -value of classification is co-extensive with the value of -science and general reasoning. Whenever we form a class -we reduce multiplicity to unity, and detect, as Plato said, -the one in the many. The result of such classification is -to yield generalised knowledge, as distinguished from the -direct and sensuous knowledge of particular facts. Of -every class, so far as it is correctly formed, the principle -of substitution is true, and whatever we know of one object -in a class we know of the other objects, so far as identity -has been detected between them. The facilitation and -abbreviation of mental labour is at the bottom of all mental -progress. The reasoning faculties of Newton were not -different in nature from those of a ploughman; the difference -lay in the extent to which they were exerted, and -the number of facts which could be treated. Every thinking -being generalises more or less, but it is the depth and -extent of his generalisations which distinguish the philosopher. -Now it is the exertion of the classifying and -generalising powers which enables the intellect of man to -cope in some degree with the infinite number of natural -phenomena. In the chapters upon combinations and -permutations it was made evident, that from a few elementary -differences immense numbers of combinations can be -produced. The process of classification enables us to resolve -these combinations, and refer each one to its place according -to one or other of the elementary circumstances out of which -it was produced. We restore nature to the simple conditions -out of which its endless variety was developed. As -Professor Bowen has said,<a id="FNanchor_560" href="#Footnote_560" class="fnanchor">560</a> “The first necessity which is -imposed upon us by the constitution of the mind itself, is -to break up the infinite wealth of Nature into groups and -classes of things, with reference to their resemblances and -affinities, and thus to enlarge the grasp of our mental -faculties, even at the expense of sacrificing the minuteness -of information which can be acquired only by studying -objects in detail. The first efforts in the pursuit of knowledge, -then, must be directed to the business of classification.<span class="pagenum" id="Page_675">675</span> -Perhaps it will be found in the sequel, that classification -is not only the beginning, but the culmination and the end, -of human knowledge.”</p> - - -<h3><i>Classification Involving Induction.</i></h3> - -<p>The purpose of classification is the detection of the laws -of nature. However much the process may in some cases -be disguised, classification is not really distinct from the -process of perfect induction, whereby we endeavour to -ascertain the connexions existing between properties of the -objects under treatment. There can be no use in placing -an object in a class unless something more than the fact -of being in the class is implied. If we arbitrarily formed -a class of metals and placed therein a selection from the -list of known metals made by ballot, we should have no -reason to expect that the metals in question would resemble -each other in any points except that they are metals, and -have been selected by the ballot. But when chemists -select from the list the five metals, potassium, sodium, -cæsium, rubidium, and lithium and call them the Alkaline -metals, a great deal is implied in this classification. On -comparing the qualities of these substances they are all -found to combine very energetically with oxygen, to decompose -water at all temperatures, and to form strongly basic -oxides, which are highly soluble in water, yielding powerfully -caustic and alkaline hydrates from which water cannot -be expelled by heat. Their carbonates are also soluble in -water, and each metal forms only one chloride. It may also -be expected that each salt of one of the metals will correspond -to a salt of each other metal, there being a general analogy -between the compounds of these metals and their properties.</p> - -<p>Now in forming this class of alkaline metals, we have -done more than merely select a convenient order of -statement. We have arrived at a discovery of certain -empirical laws of nature, the probability being very considerable -that a metal which exhibits some of the properties -of alkaline metals will also possess the others. If we -discovered another metal whose carbonate was soluble in -water, and which energetically combined with water at all -temperatures, producing a strongly basic oxide, we should -infer that it would form only a single chloride, and that<span class="pagenum" id="Page_676">676</span> -generally speaking, it would enter into a series of compounds -corresponding to the salts of the other alkaline -metals. The formation of this class of alkaline metals -then, is no mere matter of convenience; it is an important -and successful act of inductive discovery, enabling us to -register many undoubted propositions as results of perfect -induction, and to make a great number of inferences -depending upon the principles of imperfect induction.</p> - -<p>An excellent instance as to what classification can do, is -found in Mr. Lockyer’s researches on the sun.<a id="FNanchor_561" href="#Footnote_561" class="fnanchor">561</a> Wanting -some guide as to what more elements to look for in the -sun’s photosphere, he prepared a classification of the elements -according as they had or had not been traced in -the sun, together with a detailed statement of the chief -chemical characters of each element. He was then able -to observe that the elements found in the sun were for the -most part those forming stable compounds with oxygen. -He then inferred that other elements forming stable -oxides would probably exist in the sun, and he was -rewarded by the discovery of five such metals. Here -we have empirical and tentative classification leading to -the detection of the correlation between existence in the -sun, and the power of forming stable oxides and then -leading by imperfect induction to the discovery of more -coincidences between these properties.</p> - -<p>Professor Huxley has defined the process of classification -in the following terms.<a id="FNanchor_562" href="#Footnote_562" class="fnanchor">562</a> “By the classification of any -series of objects, is meant the actual or ideal arrangement -together of those which are like and the separation of -those which are unlike; the purpose of this arrangement -being to facilitate the operations of the mind in clearly -conceiving and retaining in the memory the characters of -the objects in question.”</p> - -<p>This statement is doubtless correct, so far as it goes, but it -does not include all that Professor Huxley himself implicitly -treats under classification. He is fully aware that deep -correlations, or in other terms deep uniformities or laws of -nature, will be disclosed by any well chosen and profound -system of classification. I should therefore propose to<span class="pagenum" id="Page_677">677</span> -modify the above statement, as follows:—“By the classification -of any series of objects, is meant the actual or ideal -arrangement together of those which are like and the separation -of those which are unlike, the purpose of this arrangement -being, primarily, to disclose the correlations or laws of -union of properties and circumstances, and, secondarily, to -facilitate the operations of the mind in clearly conceiving -and retaining in the memory the characters of the objects -in question.”</p> - - -<h3><i>Multiplicity of Modes of Classification.</i></h3> - -<p>In approaching the question how any given group -of objects may be best classified, let it be remarked that -there must generally be an unlimited number of modes -of classifying a group of objects. Misled, as we shall see, -by the problem of classification in the natural sciences, -philosophers seem to think that in each subject there -must be one essentially natural system of classification -which is to be selected, to the exclusion of all others. -This erroneous notion probably arises also in part from the -limited powers of thought and the inconvenient mechanical -conditions under which we labour. If we arrange the -books in a library catalogue, we must arrange them in -some one order; if we compose a treatise on mineralogy, -the minerals must be successively described in some one -arrangement; if we treat such simple things as geometrical -figures, they must be taken in some fixed order. We shall -naturally select that arrangement which appears to be most -convenient and instructive for our principal purpose. But -it does not follow that this method of arrangement possesses -any exclusive excellence, and there will be usually many -other possible arrangements, each valuable in its own way. -A perfect intellect would not confine itself to one order of -thought, but would simultaneously regard a group of -objects as classified in all the ways of which they are -capable. Thus the elements may be classified according -to their atomicity into the groups of monads, dyads, triads, -tetrads, pentads, and hexads, and this is probably the most -instructive classification; but it does not prevent us from -also classifying them according as they are metallic or non-metallic, -solid, liquid or gaseous at ordinary temperatures,<span class="pagenum" id="Page_678">678</span> -useful or useless, abundant or scarce, ferro-magnetic or -diamagnetic, and so on.</p> - -<p>Mineralogists have spent a great deal of labour in trying -to discover the supposed natural system of classification for -minerals. They have constantly encountered the difficulty -that the chemical composition does not run together with -the crystallographic form, and the various physical properties -of the mineral. Substances identical in the forms -of their crystals, especially those belonging to the first or -cubical system of crystals, are often found to have no -resemblance in chemical composition. The same substance, -again, is occasionally found crystallised in two -essentially different crystallographic forms; calcium carbonate, -for instance, appearing as calc-spar and arragonite. -The simple truth is that if we are unable to discover any -correspondence, or, as we may call it, any <i>correlation</i> between -the properties of minerals, we cannot make any one arrangement -which will enable us to treat all these properties in a -single system of classification. We must classify minerals -in as many different ways as there are different groups of -unrelated properties of sufficient importance. Even if, for -the purpose of describing minerals successively in a treatise, -we select one chief system, that, for instance, having regard -to chemical composition, we ought mentally to regard the -minerals as classified in all other useful modes.</p> - -<p>Exactly the same may be said of the classification of -plants. An immense number of different modes of classifying -plants have been proposed at one time or other, an -exhaustive account of which will be found in the article on -classification in Rees’s “Cyclopædia,” or in the introduction -to Lindley’s “Vegetable Kingdom.” There have been -the Fructists, such as Cæsalpinus, Morison, Hermann, -Boerhaave or Gaertner, who arranged plants according to -the form of the fruit. The Corollists, Rivinus, Ludwig, -and Tournefort, paid attention chiefly to the number and -arrangement of the parts of the corolla. Magnol selected -the calyx as the critical part, while Sauvage arranged plants -according to their leaves; nor are these instances more than -a small selection from the actual variety of modes of classification -which have been tried. Of such attempts it may -be said that every system will probably yield some information -concerning the relations of plants, and it is only<span class="pagenum" id="Page_679">679</span> -after trying many modes that it is possible to approximate -to the best.</p> - - -<h3><i>Natural and Artificial Systems of Classification.</i></h3> - -<p>It has been usual to distinguish systems of classification -as natural and artificial, those being called natural -which seemed to express the order of existing things as -determined by nature. Artificial methods of classification, -on the other hand, included those formed for the mere -convenience of men in remembering or treating natural -objects.</p> - -<p>The difference, as it is commonly regarded, has been well -described by Ampére,<a id="FNanchor_563" href="#Footnote_563" class="fnanchor">563</a> as follows: “We can distinguish -two kinds of classifications, the natural and the artificial. -In the latter kind, some characters, arbitrarily chosen, -serve to determine the place of each object; we abstract -all other characters, and the objects are thus found to be -brought near to or to be separated from each other, often -in the most bizarre manner. In natural systems of classification, -on the contrary, we employ concurrently all the -characters essential to the objects with which we are -occupied, discussing the importance of each of them; and -the results of this labour are not adopted unless the -objects which present the closest analogy are brought -most near together, and the groups of the several orders -which are formed from them are also approximated in proportion -as they offer more similar characters. In this way -it arises that there is always a kind of connexion, more or -less marked, between each group and the group which -follows it.”</p> - -<p>There is much, however, that is vague and logically -false in this and other definitions which have been proposed -by naturalists to express their notion of a natural -system. We are not informed how the <i>importance</i> of a -resemblance is to be determined, nor what is the measure -of the <i>closeness</i> of analogy. Until all the words employed -in a definition are made clear in meaning, the definition -itself is worse than useless. Now if the views concerning -classification here upheld are true, there can be no sharp<span class="pagenum" id="Page_680">680</span> -and precise distinction between natural and artificial -systems. All arrangements which serve any purpose at -all must be more or less natural, because, if closely enough -scrutinised, they will involve more resemblances than -those whereby the class was defined.</p> - -<p>It is true that in the biological sciences there would be -one arrangement of plants or animals which would be -conspicuously instructive, and in a certain sense natural, -if it could be attained, and it is that after which naturalists -have been in reality striving for nearly two centuries, -namely, that <i>arrangement which would display the genealogical -descent of every form from the original life germ</i>. -Those morphological resemblances upon which the classification -of living beings is almost always based are inherited -resemblances, and it is evident that descendants -will usually resemble their parents and each other in a -great many points.</p> - -<p>I have said that a natural is distinguished from an -arbitrary or artificial system only in degree. It will be -found almost impossible to arrange objects according to -any circumstance without finding that some correlation of -other circumstances is thus made apparent. No arrangement -could seem more arbitrary than the common alphabetical -arrangement according to the initial letter of the name. -But we cannot scrutinise a list of names of persons without -noticing a predominance of Evans’s and Jones’s, under the -letters E and J, and of names beginning with Mac under -the letter M. The predominance is so great that we could -not attribute it to chance, and inquiry would of course -show that it arose from important facts concerning the -nationality of the persons. It would appear that the -Evans’s and Jones’s were of Welsh descent, and those -whose names bear the prefix Mac of Keltic descent. -With the nationality would be more or less strictly -correlated many peculiarities of physical constitution, -language, habits, or mental character. In other cases I -have been interested in noticing the empirical inferences -which are displayed in the most arbitrary arrangements. -If a large register of the names of ships be examined it -will often be found that a number of ships bearing the same -name were built about the same time, a correlation due to -the occurrence of some striking incident shortly previous<span class="pagenum" id="Page_681">681</span> -to the building of the ships. The age of ships or other -structures is usually correlated with their general form, -nature of materials, &c., so that ships of the same name will -often resemble each other in many points.</p> - -<p>It is impossible to examine the details of some of the -so-called artificial systems of classification of plants, -without finding that many of the classes are natural in -character. Thus in Tournefort’s arrangement, depending -almost entirely on the formation of the corolla, we find -the natural orders of the Labiatæ, Cruciferæ, Rosaceæ, -Umbelliferæ, Liliaceæ, and Papilionaceæ, recognised in his -4th, 5th, 6th, 7th, 9th, and 10th classes. Many of the -classes in Linnæus’ celebrated sexual system also approximate -to natural classes.</p> - - -<h3><i>Correlation of Properties.</i></h3> - -<p>Habits and usages of language are apt to lead us into -the error of imagining that when we employ different -words we always mean different things. In introducing the -subject of classification nominally I was careful to draw -the reader’s attention to the fact that all reasoning and all -operations of scientific method really involve classification, -though we are accustomed to use the name in some cases -and not in others. The name <i>correlation</i> requires to be -used with the same qualification. Things are correlated -(<i>con</i>, <i>relata</i>) when they are so related or bound to each -other that <i>where one is the other is, and where one is not the -other is not</i>. Throughout this work we have then been -dealing with correlations. In geometry the occurrence -of three equal angles in a triangle is correlated with the -existence of three equal sides; in physics gravity is correlated -with inertia; in botany exogenous growth is correlated -with the possession of two cotyledons, or the production -of flowers with that of spiral vessels. Wherever a proposition -of the form A = B is true there correlation exists. -But it is in the classificatory sciences especially that -the word correlation has been employed.</p> - -<p>We find it stated that in the class Mammalia the -possession of two occipital condyles, with a well-ossified -basi-occipital, is correlated with the possession of mandibles, -each ramus of which is composed of a single piece<span class="pagenum" id="Page_682">682</span> -of bone, articulated with the squamosal element of the -skull, and also with the possession of mammæ and non-nucleated -red blood-corpuscles. Professor Huxley remarks<a id="FNanchor_564" href="#Footnote_564" class="fnanchor">564</a> -that this statement of the character of the class mammalia -is something more than an arbitrary definition; it is a -statement of a law of correlation or co-existence of animal -structures, from which most important conclusions are -deducible. It involves a generalisation to the effect that -in nature the structures mentioned are always found -associated together. This amounts to saying that the -formation of the class mammalia involves an act of inductive -discovery, and results in the establishment of certain -empirical laws of nature. Professor Huxley has excellently -expressed the mode in which discoveries of this kind enable -naturalists to make deductions or predictions with considerable -confidence, but he has also pointed out that such -inferences are likely from time to time to prove mistaken. -I will quote his own words:</p> - -<p>“If a fragmentary fossil be discovered, consisting of no -more than a ramus of a mandible, and that part of the -skull with which it articulated, a knowledge of this law -may enable the palæontologist to affirm, with great confidence, -that the animal of which it formed a part -suckled its young, and had non-nucleated red blood-corpuscles; -and to predict that should the back part of that -skull be discovered, it will exhibit two occipital condyles -and a well-ossified basi-occipital bone.</p> - -<p>“Deductions of this kind, such as that made by Cuvier -in the famous case of the fossil opossum of Montmartre, -have often been verified, and are well calculated to impress -the vulgar imagination; so that they have taken -rank as the triumphs of the anatomist. But it should -carefully be borne in mind, that, like all merely empirical -laws, which rest upon a comparatively narrow observational -basis, the reasoning from them may at any time -break down. If Cuvier, for example, had had to do with a -fossil Thylacinus instead of a fossil Opossum, he would -not have found the marsupial bones, though the inflected -angle of the jaw would have been obvious enough. And<span class="pagenum" id="Page_683">683</span> -so, though, practically, any one who met with a characteristically -mammalian jaw would be justified in expecting -to find the characteristically mammalian occiput associated -with it; yet, he would be a bold man indeed, who -should strictly assert the belief which is implied in this -expectation, viz., that at no period of the world’s history -did animals exist which combined a mammalian occiput -with a reptilian jaw, or <i>vice versâ</i>.”</p> - -<p>One of the most distinct and remarkable instances of -correlation in the animal world is that which occurs in -ruminating animals, and which could not be better stated -than in the following extract from the classical work of -Cuvier:<a id="FNanchor_565" href="#Footnote_565" class="fnanchor">565</a></p> - -<p>“I doubt if any one would have divined, if untaught -by observation, that all ruminants have the foot cleft, -and that they alone have it. I doubt if any one would -have divined that there are frontal horns only in this -class: that those among them which have sharp canines -for the most part lack horns.</p> - -<p>“However, since these relations are constant, they must -have some sufficient cause; but since we are ignorant of -it, we must make good the defect of the theory by means -of observation: it enables us to establish empirical laws -which become almost as certain as rational laws when -they rest on sufficiently repeated observations; so that -now whoso sees merely the print of a cleft foot may conclude -that the animal which left this impression ruminated, -and this conclusion is as certain as any other in -physics or morals. This footprint alone then, yields, to -him who observes it, the form of the teeth, the form of -the jaws, the form of the vertebræ, the form of all the -bones of the legs, of the thighs, of the shoulders, and of -the pelvis of the animal which has passed by: it is a -surer mark than all those of Zadig.”</p> - -<p>We meet with a good instance of the purely empirical -correlation of circumstances when we classify the planets -according to their densities and periods of axial rotation.<a id="FNanchor_566" href="#Footnote_566" class="fnanchor">566</a> -If we examine a table specifying the usual astronomical -elements of the solar system, we find that four planets<span class="pagenum" id="Page_684">684</span> -resemble each other very closely in the period of axial -rotation, and the same four planets are all found to have -high densities, thus:—</p> - -<table class="ml5em mtb05em"> -<tr class="fs75"> -<td class="tac pall"><div>Name of<br>Planet.</div></td> -<td class="tac pall" colspan="4"><div> Period of Axial<br>Rotation.</div></td> -<td class="tac pall"><div> Density.</div></td> -</tr> -<tr class="fs90"> -<td class="tal">Mercury</td> -<td class="tar pl2"><div>24</div></td> -<td class="tac"><div> hours</div></td> -<td class="tar"><div>5</div></td> -<td class="tac"><div> minutes</div></td> -<td class="tac pl2"><div>7·94</div></td> -</tr> -<tr class="fs90"> -<td class="tal">Venus</td> -<td class="tar pl2"><div>23</div></td> -<td class="tac"><div>"</div></td> -<td class="tar"><div>21</div></td> -<td class="tac"><div>"</div></td> -<td class="tac pl2"><div>5·33</div></td> -</tr> -<tr class="fs90"> -<td class="tal">Earth</td> -<td class="tar pl2"><div>23</div></td> -<td class="tac"><div>"</div></td> -<td class="tar"><div>56</div></td> -<td class="tac"><div>"</div></td> -<td class="tac pl2"><div>5·67</div></td> -</tr> -<tr class="fs90"> -<td class="tal">Mars</td> -<td class="tar pl2"><div>24</div></td> -<td class="tac"><div>"</div></td> -<td class="tar"><div>37</div></td> -<td class="tac"><div>"</div></td> -<td class="tac pl2"><div>5·84</div></td> -</tr> -</table> - -<p>A similar table for the other larger planets, is as -follows:—</p> - -<table class="ml5em mtb05em"> -<tr class="fs90"> -<td class="tal">Jupiter</td> -<td class="tar pl2">9</td> -<td class="tac"> hours</td> -<td class="tar">55</td> -<td class="tac"> minutes</td> -<td class="tac pl2"> 1·36</td> -</tr> -<tr class="fs90"> -<td class="tal">Saturn</td> -<td class="tar pl2">10</td> -<td class="tac">"</td> -<td class="tar">29</td> -<td class="tac">"</td> -<td class="tac pl2"> ·74</td> -</tr> -<tr class="fs90"> -<td class="tal">Uranus</td> -<td class="tar pl2">9</td> -<td class="tac">"</td> -<td class="tar">30</td> -<td class="tac">"</td> -<td class="tac pl2"> ·97</td> -</tr> -<tr class="fs90"> -<td class="tal">Neptune</td> -<td class="tar">—</td> -<td class="tac">"</td> -<td class="tar">—</td> -<td class="tac"></td> -<td class="tac pl2"> 1·02</td> -</tr> -</table> - -<p>It will be observed that in neither group is the equality -of the rotational period or the density more than rudely -approximate; nevertheless the difference of the numbers in -the first and second group is so very well marked, the -periods of the first being at least double and the densities -four or five times those of the second, that the coincidence -cannot be attributed to accident. The reader will also -notice that the first group consists of the planets nearest -to the sun; that with the exception of the earth none of -them possess satellites; and that they are all comparatively -small. The second group are furthest from the sun, and -all of them possess several satellites, and are comparatively -great. Therefore, with but slight exceptions, the following -correlations hold true:—</p> - -<table class="ml5em fs90 mtb05em"> -<tr> -<td class="tal pr2">Interior planets.</td> -<td class="tal pr2">Long period.</td> -<td class="tal pr2">Small size.</td> -<td class="tal pr2">High Density.</td> -<td class="tal">No satellites.</td> -</tr> -<tr> -<td class="tal">Exterior "</td> -<td class="tal">Short "</td> -<td class="tal">Great "</td> -<td class="tal">Low "</td> -<td class="tal">Many "</td> -</tr> -</table> - -<p>These coincidences point with much probability to a -difference in the origin of the two groups, but no further -explanation of the matter is yet possible.</p> - -<p>The classification of comets according to their periods -by Mr. Hind and Mr. A. S. Davies, tends to establish the -conclusion that distinct groups of comets have been -brought into the solar system by the attractive powers of -Jupiter, Uranus, or other planets.<a id="FNanchor_567" href="#Footnote_567" class="fnanchor">567</a> The classification of -nebulæ as commenced by the two Herschels, and continued<span class="pagenum" id="Page_685">685</span> -by Lord Rosse, Mr. Huggins, and others, will probably lead -at some future time to the discovery of important empirical -laws concerning the constitution of the universe. The -minute examination and classification of meteorites, as -carried on by Mr. Sorby and others, seems likely to afford -us an insight into the formation of the heavenly bodies.</p> - -<p>We should never fail to remember the slightest and most -inexplicable correlations, for they may prove of importance -in the future. Discoveries begin when we are least expecting -them. It is a significant fact, for instance, that -the greater number of variable stars are of a reddish -colour. Not all variable stars are red, nor all red stars -variable; but considering that only a small fraction of the -observed stars are known to be variable, and only a small -fraction are red, the number which fall into both classes is -too great to be accidental.<a id="FNanchor_568" href="#Footnote_568" class="fnanchor">568</a> It is also remarkable that the -greater number of stars possessing great proper motion are -double stars, the star 61 Cygni being especially noticeable -in this respect.<a id="FNanchor_569" href="#Footnote_569" class="fnanchor">569</a> The correlation in these cases is not -without exception, but the preponderance is so great as -to point to some natural connexion, the exact nature of -which must be a matter for future investigation. Herschel -remarked that the two double stars 61 Cygni and α Centauri -of which the orbits were well ascertained, evidently belonged -to the same family or genus.<a id="FNanchor_570" href="#Footnote_570" class="fnanchor">570</a></p> - - -<h3><i>Classification in Crystallography.</i></h3> - -<p>Perhaps the most perfect and instructive instance of -classification which we can find is furnished by the science -of crystallography (p. <a href="#Page_133">133</a>). The system of arrangement -now generally adopted is conspicuously natural, and is even -mathematically perfect. A crystal consists in every part -of similar molecules similarly related to the adjoining -molecules, and connected with them by forces the nature -of which we can only learn by their apparent effects. But -these forces are exerted in space of three dimensions, so -that there is a limited number of suppositions which can -be entertained as to the relations of these forces. In one<span class="pagenum" id="Page_686">686</span> -case each molecule will be similarly related to all those -which are next to it; in a second case, it will be similarly -related to those in a certain plane, but differently related -to those not in that plane. In the simpler cases the arrangement -of molecules is rectangular; in the remaining cases -oblique either in one or two planes.</p> - -<p>In order to simplify the explanation and conception of -the complicated phenomena which crystals exhibit, an -hypothesis has been invented which is an excellent instance -of the Descriptive Hypotheses before mentioned (p. <a href="#Page_522">522</a>). -Crystallographers imagine that there are within each -crystal certain axes, or lines of direction, by the comparative -length and the mutual inclination of which the nature of -the crystal is determined. In one class of crystals there -are three such axes lying in one plane, and a fourth perpendicular -to that plane; but in all the other classes there are -imagined to be only three axes. Now these axes can be -varied in three ways as regards length: they may be (1) all -equal, or (2) two equal and one unequal, or (3) all unequal. -They may also be varied in four ways as regards direction: -(1) they may be all at right angles to each other; (2) two -axes may be oblique to each other and at right angles to -the third; (3) two axes may be at right angles to each other -and the third oblique to both; (4) the three axes may be -all oblique. Now, if all the variations as regards length -were combined with those regarding direction, it would -seem to be possible to have twelve classes of crystals in all, -the enumeration being then logically and geometrically -complete. But as a matter of empirical observation, many -of these classes are not found to occur, oblique axes being -seldom or never equal. There remain seven recognised -classes of crystals, but even of these one class is not positively -known to be represented in nature.</p> - -<p>The first class of crystals is defined by possessing three -equal rectangular axes, and equal elasticity in all directions. -The primary or simple form of the crystals is the cube, but -by the removal of the corners of the cube by planes variously -inclined to the axes, we have the regular octohedron, -the dodecahedron, and various combinations of these forms. -Now it is a law of this class of crystals that as each axis is -exactly like each other axis, every modification of any -corner of a crystal must be repeated symmetrically with<span class="pagenum" id="Page_687">687</span> -regard to the other axes; thus the forms produced are -symmetrical or regular, and the class is called the <i>Regular -System</i> of crystals. It includes a great variety of substances, -some of them being elements, such as carbon in the form -of diamond, others more or less complex compounds, such -as rock-salt, potassium iodide and bromide, the several -kinds of alum, fluor-spar, iron bisulphide, garnet, spinelle, -&c. No correlation then is apparent between the form of -crystallisation and the chemical composition. But what -we have to notice is that the physical properties of the -crystallised substances with regard to light, heat, electricity, -&c., are closely similar. Light and heat undulations, wherever -they enter a crystal of the regular system, spread with -equal rapidity in all directions, just as they would in a uniform -fluid. Crystals of the regular system accordingly do -not in any case exhibit the phenomena of double refraction, -unless by mechanical compression we alter the conditions -of elasticity. These crystals, again, expand equally in all -directions when heated, and if we could cut a sufficiently -large plate from a cubical crystal, and examine the sound -vibrations of which it is capable, we should find that they -indicated an equal elasticity in every direction. Thus we -see that a great number of important properties are correlated -with that of crystallisation in the regular system, and -as soon as we know that the primary form of a substance -is the cube, we are able to infer with approximate certainty -that it possesses all these properties. The class of regular -crystals is then an eminently natural class, one disclosing -many general laws connecting together the physical and -mechanical properties of the substances classified.</p> - -<p>In the second class of crystals, called the dimetric, square -prismatic, or pyramidal system, there are also three axes at -right angles to each other; two of the axes are equal, but -the third or principal axis is unequal, being either greater -or less than either of the other two. In such crystals -accordingly the elasticity and other properties are alike -in all directions perpendicular to the principal axis, but -vary in all other directions. If a point within a crystal of -this system be heated, the heat spreads with equal rapidity -in planes perpendicular to the principal axis, but more or -less rapidly in the direction of this axis, so that the isothermal -surface is an ellipsoid of revolution round that axis.</p> - -<p><span class="pagenum" id="Page_688">688</span></p> - -<p>Nearly the same statement may be made concerning the -third or hexagonal or rhombohedral system of crystals, in -which there are three axes lying in one plane and meeting -at angles of 60°, while the fourth axis is perpendicular to -the other three. The hexagonal prism and rhombohedron -are the commonest forms assumed by crystals of this system, -and in ice, quartz, and calc-spar, we have abundance of -beautiful specimens of the various shapes produced by the -modification of the primitive form. Calc-spar alone is said -to crystallise in at least 700 varieties of form. Now of all -the crystals belonging both to this and the dimetric class, -we know that a ray of light passing in the direction of the -principal axis will be refracted singly as in a crystal of -the regular system; but in every other direction the light -will suffer double refraction being separated into two rays, -one of which obeys the ordinary law of refraction, but the -other a much more complicated law. The other physical -properties vary in an analogous manner. Thus calc-spar -expands by heat in the direction of the principal axis, but -contracts a little in directions perpendicular to it. So -closely are the physical properties correlated that Mitscherlich, -having observed the law of expansion in calc-spar, -was enabled to predict that the double refracting -power of the substance would be decreased by a rise of -temperature, as was proved by experiment to be the -case.</p> - -<p>In the fourth system, called the trimetric, rhombic, or -right prismatic system, there are three axes, at right angles, -but all unequal in length. It may be asserted in general -terms that the mechanical properties vary in such crystals -in every direction, and heat spreads so that the isothermal -surface is an ellipsoid with three unequal axes.</p> - -<p>In the remaining three classes, called the monoclinic, -diclinic, and triclinic, the axes are more or less oblique, -and at the same time unequal. The complication of -phenomena is therefore greatly increased, and it need only -be stated that there are always two directions in which a -ray is singly refracted, but that in all other directions -double refraction takes place. The conduction of heat is -unequal in all directions, the isothermal surface being an -ellipsoid of three unequal axes. The relations of such -crystals to other phenomena are often very complicated,<span class="pagenum" id="Page_689">689</span> -and hardly yet reduced to law. Some crystals, called -pyro-electric, manifest vitreous electricity at some points -of their surface, and resinous electricity at other points -when rising in temperature, the character of the electricity -being changed when the temperature sinks again. This -production of electricity is believed to be connected with -the hemihedral character of the crystals exhibiting it. -The crystalline structure of a substance again influences -its magnetic behaviour, the general law being that the -direction in which the molecules of a crystal are most -approximated tends to place itself axially or equatorially -between the poles of a magnet, respectively as the body is -magnetic or diamagnetic. Further questions arise if we -apply pressure to crystals. Thus doubly refracting crystals -with one principal axis acquire two axes when the pressure -is perpendicular in direction to the principal axis.</p> - -<p>All the phenomena peculiar to crystalline bodies are -thus closely correlated with the formation of the crystal, or -will almost certainly be found to be so as investigation -proceeds. It is upon empirical observation indeed that -the laws of connexion are in the first place founded, but -the simple hypothesis that the elasticity and approximation -of the particles vary in the directions of the crystalline -axes allows of the application of deductive reasoning. -The whole of the phenomena are gradually being proved -to be consistent with this hypothesis, so that we have in -this subject of crystallography a beautiful instance of -successful classification, connected with a nearly perfect -physical hypothesis. Moreover this hypothesis was verified -experimentally as regards the mechanical vibrations of -sound by Savart, who found that the vibrations in a plate -of biaxial crystal indicated the existence of varying -elasticity in varying directions.</p> - - -<h3><i>Classification an Inverse and Tentative Operation.</i></h3> - -<p>If attempts at so-called natural classification are really -attempts at perfect induction, it follows that they are -subject to the remarks which were made upon the inverse -character of the inductive process, and upon the difficulty -of every inverse operation (pp. <a href="#Page_11">11</a>, <a href="#Page_12">12</a>, <a href="#Page_122">122</a>, &c.). There -will be no royal road to the discovery of the best system,<span class="pagenum" id="Page_690">690</span> -and it will even be impossible to lay down rules of procedure -to assist those who are in search of a good arrangement. -The only logical rule would be as follows:—Having -given certain objects, group them in every way in which -they can be grouped, and then observe in which method -of grouping the correlation of properties is most conspicuously -manifested. But this method of exhaustive -classification will in almost every case be impracticable, -owing to the immensely great number of modes in which -a comparatively small number of objects may be grouped -together. About sixty-three elements have been classified -by chemists in six principal groups as monad, dyad, triad, -&c., elements, the numbers in the classes varying from three -to twenty elements. Now if we were to calculate the -whole number of ways in which sixty-three objects can be -arranged in six groups, we should find the number to be so -great that the life of the longest lived man would be wholly -inadequate to enable him to go through these possible -groupings. The rule of exhaustive arrangement, then, is -absolutely impracticable. It follows that mere haphazard -trial cannot as a general rule give any useful result. If -we were to write the names of the elements in succession -upon sixty-three cards, throw them into a ballot-box, and -draw them out haphazard in six handfuls time after time, -the probability is excessively small that we should take -them out in a specified order, that for instance at present -adopted by chemists.</p> - -<p>The usual mode in which an investigator proceeds to -form a classification of a new group of objects seems to -consist in tentatively arranging them according to their -most obvious similarities. Any two objects which present -a close resemblance to each other will be joined and formed -into the rudiment of a class, the definition of which will -at first include all the apparent points of resemblance. -Other objects as they come to our notice will be gradually -assigned to those groups with which they present the -greatest number of points of resemblance, and the definition -of a class will often have to be altered in order to -admit them. The early chemists could hardly avoid -classing together the common metals, gold, silver, copper, -lead, and iron, which present such conspicuous points of -similarity as regards density, metallic lustre, malleability,<span class="pagenum" id="Page_691">691</span> -&c. With the progress of discovery, however, difficulties -began to present themselves in such a grouping. Antimony, -bismuth, and arsenic are distinctly metallic as -regards lustre, density, and some chemical properties, but -are wanting in malleability. The recently discovered -tellurium presents greater difficulties, for it has many of -the physical properties of metal, and yet all its chemical -properties are analogous to those of sulphur and selenium, -which have never been regarded as metals. Great chemical -differences again are discovered by degrees between the five -metals mentioned; and the class, if it is to have any chemical -validity, must be made to include other elements, -having none of the original properties on which the class -was founded. Hydrogen is a transparent colourless gas, -and the least dense of all substances; yet in its chemical -analogies it is a metal, as suggested by Faraday<a id="FNanchor_571" href="#Footnote_571" class="fnanchor">571</a> in 1838, -and almost proved by Graham;<a id="FNanchor_572" href="#Footnote_572" class="fnanchor">572</a> it must be placed in -the same class as silver. In this way it comes to pass that -almost every classification which is proposed in the early -stages of a science will be found to break down as the -deeper similarities of the objects come to be detected. The -most obvious points of difference will have to be neglected. -Chlorine is a gas, bromine a liquid, and iodine a solid, and -at first sight these might have seemed formidable circumstances -to overlook; but in chemical analogy the substances -are closely united. The progress of organic chemistry, -again, has yielded wholly new ideas of the similarities of -compounds. Who, for instance, would recognise without -extensive research a close similarity between glycerine and -alcohol, or between fatty substances and ether? The class -of paraffins contains three substances gaseous at ordinary -temperatures, several liquids, and some crystalline solids. -It required much insight to detect the analogy which exists -between such apparently different substances.</p> - -<p>The science of chemistry now depends to a great extent -on a correct classification of the elements, as will be learnt -by consulting the able article on Classification by Professor -G. C. Foster in Watts’ <i>Dictionary of Chemistry</i>. -But the present system of chemical classification was not<span class="pagenum" id="Page_692">692</span> -reached until at least three previous false systems had -been long entertained. And though there is much reason -to believe that the present mode of classification according -to atomicity is substantially correct, errors may yet be -discovered in the details of the grouping.</p> - - -<h3><i>Symbolic Statement of the Theory of Classification.</i></h3> - -<p>The theory of classification can be explained in the most -complete and general manner, by reverting for a time to -the use of the Logical Alphabet, which was found to be of -supreme importance in Formal Logic. That form expresses -the necessary classification of all objects and ideas as depending -on the laws of thought, and there is no point concerning -the purpose and methods of classification which may not be -stated precisely by the use of letter combinations, the only -inconvenience being the abstract form in which the subject -is thus represented.</p> - -<p>If we pay regard only to three qualities in which things -may resemble each other, namely, the qualities A, B, C, -there are according to the laws of thought eight possible -classes of objects, shown in the fourth column of the -Logical Alphabet (p. <a href="#Page_94">94</a>). If there exist objects belonging -to all these eight classes, it follows that the qualities A, B, -C, are subject to no conditions except the primary laws of -thought and things (p. <a href="#Page_5">5</a>). There is then no special law of -nature to discover, and, if we arrange the objects in any -one order rather than another, it must be for the purpose of -showing that the combinations are logically complete.</p> - -<p>Suppose, however, that there are but four kinds of objects -possessing the qualities A, B, C, and that these kinds are -represented by the combinations ABC, A<i>b</i>C, <i>a</i>B<i>c</i>, <i>abc</i>. -The order of arrangement will now be of importance; for if -we place them in the order</p> - -<table class="ml5em"> -<tr> -<td class="tar vab" rowspan="2"><div><img src="images/31x8bl.png" width="8" height="31" alt="" ></div></td> -<td class="tal pr3">ABC</td> -<td class="tar vab" rowspan="2"><div><img src="images/31x8bl.png" width="8" height="31" alt="" ></div></td> -<td class="tal">A<i>b</i>C</td> -</tr> -<tr> -<td class="tal"><i>a</i>B<i>c</i></td> -<td class="tal"><i>abc</i></td> -</tr> -</table> - -<p class="ti0">placing the B’s first and those which are <i>b</i>’s last, we shall -perhaps overlook the law of correlation of properties involved. -But if we arrange the combinations as follows</p> - -<table class="ml5em"> -<tr> -<td class="tar vab" rowspan="2"><div><img src="images/31x8bl.png" width="8" height="31" alt="" ></div></td> -<td class="tal pr3">ABC</td> -<td class="tar vab" rowspan="2"><div><img src="images/31x8bl.png" width="8" height="31" alt="" ></div></td> -<td class="tal"><i>a</i>B<i>c</i></td> -</tr> -<tr> -<td class="tal">A<i>b</i>C</td> -<td class="tal"><i>abc</i></td> -</tr> -</table> - -<p class="ti0">it becomes apparent at once that where A is, and only -where A is, the property C is to be found, B being<span class="pagenum" id="Page_693">693</span> -indifferently present and absent. The second arrangement -then would be called a natural one, as rendering manifest -the conditions under which the combinations exist.</p> - -<p>As a further instance, let us suppose that eight objects -are presented to us for classification, which exhibit combinations -of the five properties, A, B, C, D, E, in the following -manner:—</p> - -<table class="ml5em"> -<tr> -<td class="tal pr3">ABC<i>d</i>E</td> -<td class="tal"><i>a</i>BC<i>d</i>E</td> -</tr> -<tr> -<td class="tal">AB<i>cde</i></td> -<td class="tal"><i>a</i>B<i>cde</i></td> -</tr> -<tr> -<td class="tal pr3">A<i>b</i>CDE</td> -<td class="tal"><i>ab</i>CDE</td> -</tr> -<tr> -<td class="tal">A<i>bc</i>D<i>e</i></td> -<td class="tal"><i>abc</i>D<i>e</i></td> -</tr> -</table> - -<p class="ti0">They are now classified, so that those containing A stand -first, and those devoid of A second, but no other property -seems to be correlated with A. Let us alter this arrangement -and group the combinations thus:—</p> - -<table class="ml5em"> -<tr> -<td class="tal pr3">ABC<i>d</i>E</td> -<td class="tal">A<i>b</i>CDE</td> -</tr> -<tr> -<td class="tal">AB<i>cde</i></td> -<td class="tal">A<i>bc</i>D<i>e</i></td> -</tr> -<tr> -<td class="tal pr3"><i>a</i>BC<i>d</i>E</td> -<td class="tal"><i>ab</i>CDE</td> -</tr> -<tr> -<td class="tal"><i>a</i>B<i>cde</i></td> -<td class="tal"><i>abc</i>D<i>e</i></td> -</tr> -</table> - -<p class="ti0">It requires little examination to discover that in the first -group B is always present and D absent, whereas in the -second group, B is always absent and D present. This is -the result which follows from a law of the form B = d -(p. <a href="#Page_136">136</a>), so that in this mode of arrangement we readily -discover correlation between two letters. Altering the -groups again as follows:—</p> - -<table class="ml5em"> -<tr> -<td class="tal pr3">ABC<i>d</i>E</td> -<td class="tal">AB<i>cde</i></td> -</tr> -<tr> -<td class="tal"><i>a</i>BC<i>d</i>E</td> -<td class="tal"><i>a</i>B<i>cde</i></td> -</tr> -<tr> -<td class="tal pr3">A<i>b</i>CDE</td> -<td class="tal">A<i>bc</i>D<i>e</i></td> -</tr> -<tr> -<td class="tal"><i>ab</i>CDE</td> -<td class="tal"><i>abc</i>D<i>e</i>,</td> -</tr> -</table> - -<p class="ti0">we discover another evident correlation between C and E. -Between A and the other letters, or between the two pairs -of letters B, D and C, E, there is no logical connexion.</p> - -<p>This example may seem tedious, but it will be found -instructive in this way. We are classifying only eight -objects or combinations, in each of which only five qualities -are considered. There are only two laws of correlation -between four of those five qualities, and those laws are -of the simplest logical character. Yet the reader would -hardly discover what those laws are, and confidently assign -them by rapid contemplation of the combinations, as given -in the first group. Several tentative classifications must<span class="pagenum" id="Page_694">694</span> -probably be made before we can resolve the question. Let -us now suppose that instead of eight objects and five -qualities, we have, say, five hundred objects and fifty -qualities. If we were to attempt the same method of -exhaustive grouping which we before employed, we should -have to arrange the five hundred objects in fifty different -ways, before we could be sure that we had discovered -even the simpler laws of correlation. But even the successive -grouping of all those possessing each of the fifty -properties would not necessarily give us all the laws. -There might exist complicated relations between several -properties simultaneously, for the detection of which no -rule of procedure whatever can be given.</p> - - -<h3><i>Bifurcate Classification.</i></h3> - -<p>Every system of classification ought to be formed on -the principles of the Logical Alphabet. Each superior -class should be divided into two inferior classes, distinguished -by the possession and non-possession of a single -specified difference. Each of these minor classes, again, is -divisible by any other quality whatever which can be -suggested, and thus every classification logically consists -of an infinitely extended series of subaltern genera and -species. The classifications which we form are in reality -very small fragments of those which would correctly and -fully represent the relations of existing things. But if we -take more than four or five qualities into account, the -number of subdivisions grows impracticably large. Our -finite minds are unable to treat any complex group exhaustively, -and we are obliged to simplify and generalise -scientific problems, often at the risk of overlooking -particular conditions and exceptions.</p> - -<p>Every system of classes displayed in the manner of the -Logical Alphabet may be called <i>bifurcate</i>, because every -class branches out at each step into two minor classes, -existent or imaginary. It would be a great mistake to -regard this arrangement as in any way a peculiar or -special method; it is not only a natural and important -one, but it is the inevitable and only system which is -logically perfect, according to the fundamental laws of -thought. All other arrangements of classes correspond to -the bifurcate arrangement, with the implication that some<span class="pagenum" id="Page_695">695</span> -of the minor classes are not represented among existing -things. If we take the genus A and divide it into the -species AB and AC, we imply two propositions, namely -that in the class A, the properties of B and C never occur -together, and that they are never both absent; these -propositions are logically equivalent to one, namely -AB = A<i>c</i>. Our classification is then identical with the -following bifurcate one:—</p> - -<table class="ml5em fs85"> -<tr> -<td class="tac" colspan="8">A</td> -</tr> -<tr class="fs50"> -<td class="tac" colspan="3"> </td> -<td class="tac br"></td> -<td class="tac bl"></td> -<td class="tac" colspan="3"></td> -</tr> -<tr class="fs50"> -<td class="tac"> </td> -<td class="tac br"></td> -<td class="tac btl"></td> -<td class="tac bt"></td> -<td class="tac bt"></td> -<td class="tac btr"></td> -<td class="tac bl"></td> -<td class="tac"> </td> -</tr> -<tr> -<td class="tac"></td> -<td class="tac pl2 pr1" colspan="2"><div>AB</div></td> -<td class="tac" colspan="2"></td> -<td class="tac pl1 pr2" colspan="2"><div>A<i>b</i></div></td> -<td class="tac"></td> -</tr> -<tr class="fs50"> -<td class="tac"> </td> -<td class="tac br"> </td> -<td class="tac bl"> </td> -<td class="tac" colspan="2"> </td> -<td class="tac br"> </td> -<td class="tac bl"> </td> -<td class="tac"> </td> -</tr> -<tr class="fs50"> -<td class="tac br"> </td> -<td class="tac btl"> </td> -<td class="tac btr"> </td> -<td class="tac bl"> </td> -<td class="tac br"> </td> -<td class="tac btl"> </td> -<td class="tac btr"> </td> -<td class="tac bl"> </td> -</tr> -<tr> -<td class="tac prl15" colspan="2"><div>ABC = 0</div></td> -<td class="tac prl15" colspan="2"><div>AB<i>c</i></div></td> -<td class="tac prl15" colspan="2"><div>A<i>b</i>C</div></td> -<td class="tac prl15" colspan="2"><div>A<i>bc</i> = 0</div></td> -</tr> -</table> - -<p>If, again, we divide the genus A into three species, AB, -AC, AD, we are either logically in error, or else we must -be understood to imply that, as regards the other letters, -there exist only three combinations containing A, namely -AB<i>cd</i>, A<i>b</i>C<i>d</i>, and A<i>bc</i>D.</p> - -<p>The logical necessity of bifurcate classification has been -clearly and correctly stated in the <i>Outline of a New System -of Logic</i> by George Bentham, the eminent botanist, a work -of which the logical value has been quite overlooked until -lately. Mr. Bentham points out, in p. 113, that every -classification must be essentially bifurcate, and takes, as -an example, the division of vertebrate animals into four -sub-classes, as follows:—</p> - -<table class="ml5em"> -<tr> -<td>Mammifera—</td><td>endowed with mammæ and lungs.</td> -</tr> -<tr> -<td>Birds</td> <td>without mammæ but with lungs and wings.</td> -</tr> -<tr> -<td>Fish</td> <td>deprived of lungs.</td> -</tr> -<tr> -<td>Reptiles</td> <td>deprived of mammæ and wings but with lungs.</td> -</tr> -</table> - -<p>We have, then, as Mr. Bentham says, three bifid divisions, -thus represented:—</p> - -<table class="ml5em fs85"> -<tr> -<td class="tac" colspan="8"><div>Vertebrata</div></td> -</tr> -<tr class="fs50"> -<td class="tac" colspan="3"> </td> -<td class="tac br"> </td> -<td class="tac bl"> </td> -<td class="tac" colspan="3"> </td> -</tr> -<tr class="fs50"> -<td class="tac"> </td> -<td class="tac br"> </td> -<td class="tac btl"> </td> -<td class="tac bt"> </td> -<td class="tac bt"> </td> -<td class="tac bt"> </td> -<td class="tac btr"> </td> -<td class="tal bl"> </td> -</tr> -<tr> -<td class="tac"> </td> -<td class="tac vat" colspan="2"><div>Endowed with<br>lungs</div></td> -<td class="tac" colspan="3"> </td> -<td class="tac" colspan="2"><div>deprived of lungs<br>= Fish.</div></td> -</tr> -<tr class="fs50"> -<td class="tac"> </td> -<td class="tac br"> </td> -<td class="tac bl"> </td> -<td class="tac" colspan="5"> </td> -</tr> -<tr class="fs50"> -<td class="tac br"> </td> -<td class="tac btl"> </td> -<td class="tac bt"> </td> -<td class="tac bt"> </td> -<td class="tac btr"> </td> -<td class="tac bl"> </td> -<td class="tac" colspan="2"> </td> -</tr> -<tr> -<td class="tac" colspan="2" rowspan="2"><div>Endowed with<br>mammæ<br>= Mammifera.</div></td> -<td class="tac" colspan="2"> </td> -<td class="tac vat" colspan="2"><div>deprived of<br>mammæ</div></td> -<td class="tac" colspan="2"> </td> -</tr> -<tr class="fs50"> -<td class="tac" colspan="2"> </td> -<td class="tac br"> </td> -<td class="tac bl"> </td> -<td class="tac" colspan="2"> </td> -</tr> -<tr class="fs50"> -<td class="tac" colspan="2"> </td> -<td class="tac br"> </td> -<td class="tac btl"> </td> -<td class="tac bt"> </td> -<td class="tac bt"> </td> -<td class="tac btr"> </td> -<td class="tal bl"> </td> -</tr> -<tr> -<td class="tac" colspan="2"> </td> -<td class="tac" colspan="2"><div>with wings<br>= Birds.</div></td> -<td class="tac" colspan="2"> </td> -<td class="tac" colspan="2"><div>without wings<br>= Reptiles.</div></td> -</tr> -</table> - -<p><span class="pagenum" id="Page_696">696</span></p> - -<p>It is quite evident that according to the laws of thought -even this arrangement is incomplete. The sub-class mammifera -must either have wings or be deprived of them; we -must either subdivide this class, or assume that none of -the mammifera have wings, which is, as a matter of fact, the -case, the wings of bats not being true wings in the meaning -of the term as applied to birds. Fish, again, ought to be -considered with regard to the possession of mammæ and -wings; and in leaving them undivided we really imply that -they never have mammæ nor wings, the wings of the flying-fish, -again, being no exception. If we resort to the use of -our letters and define them as follows—</p> - -<div class="ml5em">A = vertebrata,<br> -B = having lungs,<br> -C = having mammæ,<br> -D = having wings, -</div> - -<p class="ti0">then there are four existent classes of vertebrata which -appear to be thus described—</p> - -<div class="ml5em"> -ABC AB<i>c</i>D AB<i>cd</i> A<i>b</i>. -</div> - -<p class="ti0">But in reality the combinations are implied to be</p> - -<div class="ml5em"> -ABC<i>d</i> = Mammifera,<br> -AB<i>c</i>D = Birds,<br> -AB<i>cd</i> = Reptiles,<br> -A<i>bcd</i> = Fish, -</div> - -<p class="ti0">and we imply at the same time that the other four conceivable -combinations containing B, C, or D, namely ABCD, -A<i>b</i>CD, A<i>b</i>C<i>d</i>, and A<i>bc</i>D, do not exist in nature.</p> - -<p>Mr. Bentham points out<a id="FNanchor_573" href="#Footnote_573" class="fnanchor">573</a> that it is really this method of -classification which was employed by Lamarck and De Candolle -in their so-called analytical arrangement of the French -Flora. He gives as an example a table of the principal -classes of De Candolle’s system, as also a bifurcate arrangement -of animals after the method proposed by Duméril in -his <i>Zoologie Analytique</i>, this naturalist being distinguished -by his clear perception of the logical importance of the -method. A bifurcate classification of the animal kingdom -may also be found in Professor Reay Greene’s <i>Manual of -the Cœlenterata</i>, p. 18.</p> - -<p>The bifurcate form of classification seems to be needless -when the quality according to which we classify any group<span class="pagenum" id="Page_697">697</span> -of things admits of numerical discrimination. It would -seem absurd to arrange things according as they have one -degree of the quality or not one degree, two degrees or not -two degrees, and so on. The elements are classified according -as the atom of each saturates one, two, three, or more -atoms of a monad element, such as chlorine, and they are -called accordingly monad, dyad, triad, tetrad elements, and -so on. It would be useless to apply the bifid arrangement, -thus:—</p> - -<table class="ml5em fs85 mtb05em"> -<tr> -<td class="tal pl3" colspan="12"><div>Element</div></td> -</tr> -<tr class="fs50"> -<td class="tac" colspan="2"> </td> -<td class="tac br"> </td> -<td class="tac bl"> </td> -<td class="tac" colspan="8"> </td> -</tr> -<tr class="fs50"> -<td class="tac br"> </td> -<td class="tac btl"> </td> -<td class="tac bt"> </td> -<td class="tac bt"> </td> -<td class="tac btr"> </td> -<td class="tac bl"> </td> -<td class="tac" colspan="6"> </td> -</tr> -<tr> -<td class="tac" colspan="2"><div>Monad</div></td> -<td class="tac" colspan="2"> </td> -<td class="tac" colspan="2">not-Monad</td> -<td class="tac" colspan="6"> </td> -</tr> -<tr class="fs50"> -<td class="tac" colspan="4"> </td> -<td class="tac br"> </td> -<td class="tac bl"> </td> -<td class="tac" colspan="6"> </td> -</tr> -<tr class="fs50"> -<td class="tac" colspan="2"> </td> -<td class="tac br"> </td> -<td class="tac btl"> </td> -<td class="tac bt"> </td> -<td class="tac bt"> </td> -<td class="tac btr"> </td> -<td class="tac bl"> </td> -<td class="tac" colspan="4"> </td> -</tr> -<tr> -<td class="tac" colspan="2"> </td> -<td class="tac" colspan="2"><div>Dyad</div></td> -<td class="tac" colspan="2"> </td> -<td class="tac" colspan="2">not-Dyad</td> -<td class="tac" colspan="4"> </td> -</tr> -<tr class="fs50"> -<td class="tac" colspan="6"> </td> -<td class="tac br"> </td> -<td class="tac bl"> </td> -<td class="tac" colspan="4"> </td> -</tr> -<tr class="fs50"> -<td class="tac" colspan="5"> </td> -<td class="tac btl"> </td> -<td class="tac bt"> </td> -<td class="tac bt"> </td> -<td class="tac btr"> </td> -<td class="tac" colspan="3"> </td> -</tr> -<tr> -<td class="tac" colspan="4"> </td> -<td class="tac" colspan="2"><div>Triad</div></td> -<td class="tac" colspan="2"> </td> -<td class="tac" colspan="2">not-Triad</td> -<td class="tac" colspan="2"> </td> -</tr> -<tr class="fs50"> -<td class="tac" colspan="8"> </td> -<td class="tac br"> </td> -<td class="tac bl"> </td> -<td class="tac" colspan="2"> </td> -</tr> -<tr class="fs50"> -<td class="tac" colspan="6"> </td> -<td class="tac br"> </td> -<td class="tac btl"> </td> -<td class="tac bt"> </td> -<td class="tac bt"> </td> -<td class="tac btr"> </td> -<td class="tac bl"> </td> -</tr> -<tr> -<td class="tac" colspan="6"> </td> -<td class="tac" colspan="2"><div>Tetrad</div></td> -<td class="tac" colspan="2"> </td> -<td class="tac" colspan="2"><div>not-Tetrad.</div></td> -</tr> -</table> - -<p class="ti0">The reason of this is that, by the nature of number (p. <a href="#Page_157">157</a>) -every number is logically discriminated from every other -number. There can thus be no logical confusion in a numerical -arrangement, and the series of numbers indefinitely -extended is also exhaustive. Every thing admitting of a -quality expressible in numbers must find its place somewhere -in the series of numbers. The chords in music -correspond to the simpler numerical ratios and must admit -of complete exhaustive classification in respect to the -complexity of the ratios forming them. Plane rectilinear -figures may be classified according to the numbers of their -sides, as triangles, quadrilateral figures, pentagons, hexagons, -heptagons, &c. The bifurcate arrangement is not false when -applied to such series of objects; it is even necessarily -involved in the arrangement which we do apply, so that -its formal statement is needless and tedious. The same -may be said of the division of portions of space. Reid -and Kames endeavoured to cast ridicule on the bifurcate -arrangement<a id="FNanchor_574" href="#Footnote_574" class="fnanchor">574</a> by proposing to classify the parts of England -into Middlesex and what is not Middlesex, dividing the -latter again into Kent and what is not Kent, Sussex and<span class="pagenum" id="Page_698">698</span> -what is not Sussex; and so on. This is so far, however, -from being an absurd proceeding that it is requisite to -assure us that we have made an exhaustive enumeration of -the parts of England.</p> - - -<h3><i>The Five Predicables.</i></h3> - -<p>As a rule it is highly desirable to consign to oblivion the -ancient logical names and expressions, which have infested -the science for many centuries past. If logic is ever to be -a useful and progressive science, logicians must distinguish -between logic and the history of logic. As in the case of -any other science it may be desirable to examine the course -of thought by which logic has, before or since the time of -Aristotle, been brought to its present state; the history of a -science is always instructive as giving instances of the -mode in which discoveries take place. But at the same -time we ought carefully to disencumber the statement of -the science itself of all names and other vestiges of antiquity -which are not actually useful at the present day.</p> - -<p>Among the ancient expressions which may well be -excepted from such considerations and retained in use, are -the “Five Words” or “Five Predicables” which were -described by Porphyry in his introduction to Aristotle’s -Organum. Two of them, <i>Genus</i> and <i>Species</i>, are the most -venerable names in philosophy, having probably been first -employed in their present logical meanings by Socrates. -In the present day it requires some mental effort, as -remarked by Grote, to see anything important in the -invention of notions now so familiar as those of Genus and -Species. But in reality the introduction of such terms -showed the rise of the first germs of logic and scientific -method; it showed that men were beginning to analyse -their processes of thought.</p> - -<p>The Five Predicables are Genus, Species, Difference, -Property, and Accident, or in the original Greek, γένος, -εἶδος, διαφορά, ἴδιον, συμβεβηκός. Of these, Genus may -be taken to mean any class of objects which is regarded as -broken up into two minor classes, which form Species of it. -The genus is defined by a certain number of qualities or -circumstances which belong to all objects included in the -class, and which are sufficient to mark out these objects<span class="pagenum" id="Page_699">699</span> -from all others which we do not intend to include. Interpreted -as regards intension, then, the genus is a group of -qualities; interpreted as regards extension, it is a group of -objects possessing those qualities. If another quality be -taken into account which is possessed by some of the -objects and not by the others, this quality becomes a -difference which divides the genus into two species. We -may interpret the species either in intension or extension; -in the former respect it is more than the genus as containing -one more quality, the difference: in the latter respect it is -less than the genus as containing only a portion of the group -constituting the genus. We may say, then, with Aristotle, -that in one sense the genus is in the species, namely in -intension, and in another sense the species is in the genus, -namely in extension. The difference, it is evident, can be -interpreted in intension only.</p> - -<p>A Property is a quality which belongs to the whole of -a class, but does not enter into the definition of that class. -A generic property belongs to every individual object -contained in the genus. It is a property of the genus -parallelogram that the opposite angles are equal. If we -regard a rectangle as a species of parallelogram, the -difference being that <i>one</i> angle is a right angle, it follows -as a specific property that all the angles are right angles. -Though a property in the strict logical sense must belong -to each of the objects included in the class of which it is a -property, it may or may not belong to other objects. The -property of having the opposite angles equal may belong -to many figures besides parallelograms, for instance, -regular hexagons. It is a property of the circle that all -triangles constructed upon the diameter with the apex -upon the circumference are right-angled triangles, and -<i>vice versâ</i>, all curves of which this is true must be circles. -A property which thus belongs to the whole of a class and -only to that class, corresponds to the ἴδιον of Aristotle and -Porphyry; we might conveniently call it <i>a peculiar property</i>. -Every such property enables us to make a statement in the -form of a simple identity (p. <a href="#Page_37">37</a>). Thus we know it to be -a peculiar property of the circle that for a given length of -perimeter it encloses a greater area than any other possible -curve; hence we may say—</p> - -<div class="ml3em"> -Curve of equal curvature = curve of greatest area. -</div> - -<p><span class="pagenum" id="Page_700">700</span></p> - -<p>It is a peculiar property of equilateral triangles that they -are equiangular, and <i>vice versâ</i>, it is a peculiar property of -equiangular triangles that they are equilateral. It is a -property of crystals of the regular system that they are -devoid of the power of double refraction, but this is not a -property peculiar to them, because liquids and gases are -devoid of the same property.</p> - -<p>An Accident, the fifth and last of the Predicables, is any -quality which may or may not belong to certain objects, -and which has no connexion with the classification adopted. -The particular size of a crystal does not in the slightest -degree affect the form of the crystal, nor does the manner -in which it is grouped with other crystals; these, then, are -accidents as regards a crystallographic classification. With -respect to the chemical composition of a substance, again, -it is an accident whether the substance be crystallised or -not, or whether it be organised or not. As regards botanical -classification the absolute size of a plant is an accident. -Thus we see that a logical accident is any quality or circumstance -which is not known to be correlated with those -qualities or circumstances forming the definition of the -species.</p> - -<p>The meanings of the Predicables can be clearly explained -by our symbols. Let A be any definite group of qualities -and B another quality or group of qualities; then A will -constitute a genus, and AB, A<i>b</i> will be species of it, B -being the difference. Let C, D and E be other qualities -or groups of qualities, and on examining the combinations -in which A, B, C, D, E occur let them be as follows:—</p> - -<table class="ml5em"> -<tr> -<td class="tal pr3">ABCDE</td> -<td class="tal">A<i>b</i>C<i>d</i>E</td> -</tr> -<tr> -<td class="tal">ABCD<i>e</i></td> -<td class="tal">A<i>b</i>C<i>de</i>.</td> -</tr> -</table> - -<p class="ti0">Here we see that wherever A is we also find C, so that -C is a generic property; D occurs always with B, so that it -constitutes a specific property, while E is indifferently -present and absent, so as not to be related to any other -letter; it represents, therefore, an accident. It will now be -seen that the Logical Alphabet represents an interminable -series of subordinate genera and species; it is but a concise -symbolic statement of what was involved in the ancient -doctrine of the Predicables.</p> - -<p><span class="pagenum" id="Page_701">701</span></p> - - -<h3><i>Summum Genus and Infima Species.</i></h3> - -<p>As a genus means any class whatever which is regarded -as composed of minor classes or species, it follows that the -same class will be a genus in one point of view and a -species in another. Metal is a genus as regards alkaline -metal, a species as regards element, and any extensive -system of classes consists of a series of subordinate, or as -they are technically called, <i>subaltern</i> genera and species. -The question, however, arises, whether such a chain of -classes has a definite termination at either end. The -doctrine of the old logicians was to the effect that it terminated -upwards in a <i>genus generalissimum</i> or <i>summum genus</i>, -which was not a species of any wider class. Some very -general notion, such as substance, object, or thing, was -supposed to be so comprehensive as to include all thinkable -objects, and for all practical purposes this might be so. -But as I have already explained (p. <a href="#Page_74">74</a>), we cannot really -think of any object or class without thereby separating it -from what is not that object or class. All thinking is -relative, and implies discrimination, so that every class -and every logical notion must have its negative. If so, -there is no such thing as a <i>summum genus</i>; for we cannot -frame the requisite notion of a class forming it without -implying the existence of another class discriminated from -it; add this new negative class to the supposed <i>summum -genus</i>, and we form a still higher genus, which is absurd.</p> - -<p>Although there is no absolute summum genus, nevertheless -relatively to any branch of knowledge or any particular -argument, there is always some class or notion which bounds -our horizon as it were. The chemist restricts his view to -material substances and the forces manifested in them; -the mathematician extends his view so as to comprehend -all notions capable of numerical discrimination. The biologist, -on the other hand, has a narrower sphere containing -only organised bodies, and of these the botanist and the -zoologist take parts. In other subjects there may be a -still narrower summum genus, as when the lawyer regards -only reasoning beings of his own country together with -their property.</p> - -<p>In the description of the Logical Alphabet it was pointed -out (p. <a href="#Page_93">93</a>) that every series of combinations is really the<span class="pagenum" id="Page_702">702</span> -development of a single class, denoted by X, which letter -was accordingly placed in the first column of the table on -p. <a href="#Page_94">94</a>. This is the formal acknowledgment of the principle -clearly stated by De Morgan, that all reasoning proceeds -within an assumed summum genus. But at the same time -the fact that X as a logical term must have its negative -<i>x</i>, shows that it cannot be an absolute summum genus.</p> - -<p>There arises, again, the question whether there be any -such thing as an <i>infima species</i>, which cannot be divided -into minor species. The ancient logicians were of opinion -that there always was some assignable class which could -only be divided into individuals, but this doctrine appears -to be theoretically incorrect, as Mr. George Bentham -long ago stated.<a id="FNanchor_575" href="#Footnote_575" class="fnanchor">575</a> We may put an arbitrary limit to the -subdivision of our classes at any point convenient to our -purpose. The crystallographer would not generally treat -as different species crystalline forms which differ only -in the degree of development of the faces. The naturalist -overlooks innumerable slight differences between animals -which he refers to the same species. But in a strictly -logical point of view classification might be carried on as -long as there is a difference, however minute, between -two objects, and we might thus go on until we arrive at -individual objects which are numerically distinct in the -logical sense attributed to that expression in the chapter -upon Number. Either, then, we must call the individual -the <i>infima species</i> or allow that there is no such thing at all.</p> - - -<h3><i>The Tree of Porphyry.</i></h3> - -<p>Both Aristotle and Plato were acquainted with the value -of bifurcate classification, which they occasionally employed -in an explicit manner. It is impossible too that Aristotle -should state the laws of thought, and employ the predicables -without implicitly recognising the logical necessity of that -method. It is, however, in Porphyry’s remarkable and in -many respects excellent <i>Introduction to the Categories of -Aristotle</i> that we find the most distinct account of it. -Porphyry not only fully and accurately describes the -Predicables, but incidentally introduces an example for<span class="pagenum" id="Page_703">703</span> -illustrating those predicables, which constitutes a good -specimen of bifurcate classification. Translating his words<a id="FNanchor_576" href="#Footnote_576" class="fnanchor">576</a> -freely we may say that he takes Substance as the genus to -be divided, under which are successively placed as Species—Body, -Animated Body, Animal, Rational Animal, and Man. -Under Man, again, come Socrates, Plato, and other particular -men. Now of these notions Substance is the genus -generalissimum, and is a genus only, not a species. Man, -on the other hand, is the species specialissima (infima -species), and is a species only, not a genus. Body is a -species of substance, but a genus of animated body, which, -again, is a species of body but a genus of animal. -Animal is a species of animated body, but a genus of -rational animal, which, again, is a species of animal, but a -genus of man. Finally, man is a species of rational animal, -but is a species merely and not a genus, being divisible -only into particular men.</p> - -<p>Porphyry proceeds at some length to employ his example -in further illustration of the predicables. We do not -find in Porphyry’s own work any scheme or diagram -exhibiting this curious specimen of classification, but some -of the earlier commentators and epitome writers drew what -has long been called the Tree of Porphyry. This diagram, -which may be found in most elementary works on Logic,<a id="FNanchor_577" href="#Footnote_577" class="fnanchor">577</a> -is also called the Ramean Tree, because Ramus insisted -much upon the value of Dichotomy. With the exception -of Jeremy Bentham<a id="FNanchor_578" href="#Footnote_578" class="fnanchor">578</a> and George Bentham, hardly any -modern logicians have shown an appreciation of the value -of bifurcate classification. The latter author has treated -the subject, both in his <i>Outline of a New System of Logic</i> -(pp. 105–118), and in his earlier work entitled <i>Essai sur la -Nomenclature et la Classification des Principales Branches -d’Art-et-Science</i> (Paris, 1823), which consists of a free -translation or improved version of his uncle’s Essay on -Classification in the <i>Chrestomathia</i>. Some interest attaches -to the history of the Tree of Porphyry and Ramus, because it -is the prototype of the Logical Alphabet which lies at the -basis of logical method. Jeremy Bentham speaks truly<span class="pagenum" id="Page_704">704</span> -of “the matchless beauty of the Ramean Tree.” After -fully showing its logical value as an exhaustive method of -classification, and refuting the objections of Reid and -Kames, on a wrong ground, as I think, he proceeds to -inquire to what length it may be carried. He correctly -points out two objections to the extensive use of bifid -arrangements, (1) that they soon become impracticably -extensive and unwieldy, and (2) that they are uneconomical. -In his day the recorded number of different species -of plants was 40,000, and he leaves the reader to estimate -the immense number of branches and the enormous area of -a bifurcate table which should exhibit all these species in -one scheme. He also points out the apparent loss of -labour in making any large bifurcate classification; but -this he considers to be fully recompensed by the logical -value of the result, and the logical training acquired in its -execution. Jeremy Bentham, then, fully recognises the -value of the Logical Alphabet under another name, though -he apprehends also the limit to its use placed by the -finiteness of our mental and manual powers.</p> - - -<h3><i>Does Abstraction imply Generalisation?</i></h3> - -<p>Before we can acquire a sound comprehension of the -subject of classification we must answer the very difficult -question whether logical abstraction does or does not imply -generalisation. It comes to exactly the same thing if we -ask whether a species may be coextensive with its genus, -or whether, on the other hand, the genus must contain -more than the species. To abstract logically is (p. <a href="#Page_27">27</a>), -to overlook or withdraw our notice from some point of -difference. Whenever we form a class we abstract, for the -time being, the differences of the objects so united in respect -of some common quality. If we class together a great -number of objects as dwelling-houses, we overlook the fact -that some dwelling-houses are constructed of stone, others -of brick, wood, iron, &c. Often at least the abstraction of a -circumstance increases the number of objects included -under a class according to the law of the inverse relation -of the quantities of extension and intension (p. <a href="#Page_26">26</a>). -Dwelling-house is a wider term than brick-dwelling-house. -House is more general than dwelling-house. But the<span class="pagenum" id="Page_705">705</span> -question before us is, whether abstraction <i>always</i> increases -the number of objects included in a class, which amounts to -asking whether the law of the inverse relation of logical -quantities is <i>always</i> true. The interest of the question -partly arises from the fact, that so high a philosophical -authority as Mr. Herbert Spencer has denied that generalisation -is implied in abstraction,<a id="FNanchor_579" href="#Footnote_579" class="fnanchor">579</a> making this doctrine -the ground for rejecting previous methods of classifying -the sciences, and for forming an ingenious but peculiar -method of his own. The question is also a fundamental -one of the highest logical importance, and involves subtle -difficulties which have made me long hesitate in forming -a decisive opinion.</p> - -<p>Let us attempt to answer the question by examination of -a few examples. Compare the two classes <i>gun</i> and <i>iron -gun</i>. It is certain that there are many guns which are not -made of iron, so that abstraction of the circumstance “made -of iron” increases the extent of the notion. Next compare -<i>gun</i> and <i>metallic gun</i>. All guns made at the present day -consist of metal, so that the two notions seem to be coextensive; -but guns were at first made of pieces of wood -bound together like a tub, and as the logical term gun -takes no account of time, it must include all guns that -have ever existed. Here again extension increases as intension -decreases. Compare once more “steam-locomotive -engine” and “locomotive engine.” In the present day, as -far as I am aware, all locomotives are worked by steam, so -that the omission of that qualification might seem not to -widen the term; but it is quite possible that in some future -age a different motive power may be used in locomotives; -and as there is no limitation of time in the use of logical -terms, we must certainly assume that there is a class of -locomotives not worked by steam, as well as a class that is -worked by steam. When the natural class of Euphorbiaceæ -was originally formed, all the plants known to belong to it -were devoid of corollas; it would have seemed therefore -that the two classes “Euphorbiaceæ,” and “Euphorbiaceæ -devoid of Corollas,” were of equal extent. Subsequently a -number of plants plainly belonging to the same class were -found in tropical countries, and they possessed bright<span class="pagenum" id="Page_706">706</span> -coloured corollas. Naturalists believe with the utmost confidence -that “Ruminants” and “Ruminants with cleft feet” -are identical terms, because no ruminant has yet been discovered -without cleft feet. But we can see no impossibility -in the conjunction of rumination with uncleft feet, and it -would be too great an assumption to say that we are -certain that an example of it will never be met with. -Instances can be quoted, without end, of objects being -ultimately discovered combining properties which had never -before been seen together. In the animal kingdom the -Black Swan, the Ornithorhynchus Paradoxus, and more -recently the singular fish called Ceratodus Forsteri, all -discovered in Australia, have united characters never -previously known to coexist. At the present time deep-sea -dredging is bringing to light many animals of an unprecedented -nature. Singular exceptional discoveries may -certainly occur in other branches of science. When Davy -first discovered metallic potassium, it was a well established -empirical law that all metallic substances possessed a high -specific gravity, the least dense of the metals then known -being zinc, of which the specific gravity is 7·1. Yet to -the surprise of chemists, potassium was found to be an -undoubted metal of less density than water, its specific -gravity being 0·865.</p> - -<p>It is hardly requisite to prove by further examples that -our knowledge of nature is incomplete, so that we cannot -safely assume the non-existence of new combinations. -Logically speaking, we ought to leave a place open for -animals which ruminate but are without cleft feet, and -for every possible intermediate form of animal, plant, or -mineral. A purely logical classification must take account -not only of what certainly does exist, but of what may in -after ages be found to exist.</p> - -<p>I will go a step further, and say that we must have -places in our scientific classifications for purely imaginary -existences. A large proportion of the mathematical functions -which are conceivable have no application to the circumstances -of this world. Physicists certainly do investigate -the nature and consequences of forces which nowhere -exist. Newton’s <i>Principia</i> is full of such investigations. -In one chapter of his <i>Mécanique Céleste</i> Laplace indulges -in a remarkable speculation as to what the laws of motion<span class="pagenum" id="Page_707">707</span> -would have been if momentum, instead of varying simply -as the velocity, had been a more complicated function of -it. I have already mentioned (p. <a href="#Page_223">223</a>) that Airy contemplated -the existence of a world in which the laws of force -should be such that a perpetual motion would be possible, -and the Law of Conservation of Energy would not hold -true.</p> - -<p>Thought is not bound down to the limits of what is -materially existent, but is circumscribed only by those -Fundamental Laws of Identity, Contradiction and Duality, -which were laid down at the outset. This is the point at -which I should differ from Mr. Spencer. He appears to -suppose that a classification is complete if it has a place -for every existing object, and this may perhaps seem to be -practically sufficient; but it is subject to two profound -objections. Firstly, we do not know all that exists, and -therefore in limiting our classes we are erroneously omitting -multitudes of objects of unknown form and nature which -may exist either on this earth or in other parts of space. -Secondly, as I have explained, the powers of thought are -not limited by material existences, and we may, or, for some -purposes, must imagine objects which probably do not -exist, and if we imagine them we ought to find places for -them in the classifications of science.</p> - -<p>The chief difficulty of this subject, however, consists in -the fact that mathematical or other certain laws may -entirely forbid the existence of some combinations. The -circle may be defined as a plane curve of equal curvature, -and it is a property of the circle that it contains the greatest -area within the least possible perimeter. May we then -contemplate mentally a circle not a figure of greatest possible -area? Or, to take a still simpler example, a parallelogram -possesses the property of having the opposite angles -equal. May we then mentally divide parallelograms into -two classes according as they do or do not have their opposite -angles equal? It might seem absurd to do so, because -we know that one of the two species of parallelogram -would be non-existent. But, then, unless the student had -previously contemplated the existence of both species as -possible, what is the meaning of the thirty-fourth proposition -of Euclid’s first book? We cannot deny or disprove -the existence of a certain combination without thereby in<span class="pagenum" id="Page_708">708</span> -a certain way recognising that combination as an object of -thought.</p> - -<p>The conclusion at which I arrive is in opposition to -that of Mr. Spencer. I think that whenever we abstract -a quality or circumstance we do generalise or widen the -notion from which we abstract. Whatever the terms A, -B, and C may be, I hold that in strict logic AB is mentally -a wider term than ABC, because AB includes the two -species ABC and AB<i>c</i>. The term A is wider still, for it -includes the four species ABC, AB<i>c</i>, A<i>b</i>C, A<i>bc</i>. The Logical -Alphabet, in short, is the only limit of the classes of -objects which we must contemplate in a purely logical -point of view. Whatever notions be brought before us, -we must mentally combine them in all the ways sanctioned -by the laws of thought and exhibited in the Logical -Alphabet, and it is a matter for after consideration to -determine how many of these combinations exist in outward -nature, or how many are actually forbidden by the -conditions of space. A classification is essentially a -mental, not a material thing.</p> - - -<h3><i>Discovery of Marks or Characteristics.</i></h3> - -<p>Although the chief purpose of classification is to disclose -the deepest and most general resemblances of the objects -classified, yet the practical value of a system will depend -partly upon the ease with which we can refer an object to -its proper class, and thus infer concerning it all that is -known generally of that class. This operation of discovering -to which class of a system a certain specimen or case belongs, -is generally called <i>Diagnosis</i>, a technical term familiarly -used by physicians, who constantly require to diagnose or -determine the nature of the disease from which a patient is -suffering. Now every class is defined by certain specified -qualities or circumstances, the whole of which are present -in every object contained in the class, and <i>not all present</i> in -any object excluded from it. These defining circumstances -ought to consist of the deepest and most important circumstances, -by which we vaguely mean those probably forming -the conditions with which the minor circumstances are -correlated. But it will often happen that the so-called -important points of an object are not those which can<span class="pagenum" id="Page_709">709</span> -most readily be observed. Thus the two great classes of -phanerogamous plants are defined respectively by the -possession of two cotyledons or seed-leaves, and one cotyledon. -But when a plant comes to our notice and we -want to refer it to the right class, it will often happen -that we have no seed at all to examine, in order to discover -whether there be one seed-leaf or two in the germ. -Even if we have a seed it will often be small, and a careful -dissection under the microscope will be requisite to ascertain -the number of cotyledons. Occasionally the examination -of the germ would mislead us, for the cotyledons may -be obsolete, as in Cuscuta, or united together, as in Clintonia. -Botanists therefore seldom actually refer to the -seed for such information. Certain other characters of a -plant are correlated with the number of seed-leaves; thus -monocotyledonous plants almost always possess leaves with -parallel veins like those of grass, while dicotyledonous -plants have leaves with reticulated veins like those of an -oak leaf. In monocotyledonous plants, too, the parts of the -flower are most often three or some multiple of three in -number, while in dicotyledonous plants the numbers four -and five and their multiples prevail. Botanists, therefore, -by a glance at the leaves and flowers can almost certainly -refer a plant to its right class, and can infer not only the -number of cotyledons which would be found in the seed or -young plant, but also the structure of the stem and other -general characters.</p> - -<p>Any conspicuous and easily discriminated property -which we thus select for the purpose of deciding to which -class an object belongs, may be called a <i>characteristic</i>. The -logical conditions of a good characteristic mark are very -simple, namely, that it should be possessed by all objects -entering into a certain class, and by none others. Every -characteristic should enable us to assert a simple identity; -if A is a characteristic, and B, viewed intensively, the class -of objects of which it is the mark, then A = B ought to be -true. The characteristic may consist either of a single -quality or circumstance, or of a group of such, provided -that they all be constant and easily detected. Thus in the -classification of mammals the teeth are of the greatest -assistance, not because a slight variation in the number -and form of the teeth is of importance in the general<span class="pagenum" id="Page_710">710</span> -economy of the animal, but because such variations are -proved by empirical observation to coincide with most important -differences in the general affinities. It is found -that the minor classes and genera of mammals can be -discriminated accurately by their teeth, especially by the -foremost molars and the hindmost pre-molars. Some teeth, -indeed, are occasionally missing, so that zoologists prefer to -trust to those characteristic teeth which are most constant,<a id="FNanchor_580" href="#Footnote_580" class="fnanchor">580</a> -and to infer from them not only the arrangement of the -other teeth, but the whole conformation of the animal.</p> - -<p>It is a very difficult matter to mark out a boundary-line -between the animal and vegetable kingdoms, and it may -even be doubted whether a rigorous boundary can be established. -The most fundamental and important difference of -a vegetable as compared with an animal substance probably -consists in the absence of nitrogen from the constituent -membranes. Supposing this to be the case, the difficulty -arises that in examining minute organisms we cannot ascertain -directly whether they contain nitrogen or not. Some -minor but easily detected circumstance is therefore needed -to discriminate between animals and vegetables, and this is -furnished to some extent by the fact that the production -of starch granules is restricted to the vegetable kingdom. -Thus the Desmidiaceæ may be safely assigned to the vegetable -kingdom, because they contain starch. But we -must not employ this characteristic negatively; the Diatomaceæ -are probably vegetables, though they do not produce -starch.</p> - - -<h3><i>Diagnostic Systems of Classification.</i></h3> - -<p>We have seen that diagnosis is the process of discovering -the place in any system of classes, to which an object -has been referred by some previous investigation, the -object being to avail ourselves of the information relating -to such an object which has been accumulated and recorded. -It is obvious that this is a matter of great importance, -for, unless we can recognise, from time to time, -objects or substances which have been investigated, recorded -discoveries would lose their value. Even a single investigator<span class="pagenum" id="Page_711">711</span> -must have means of recording and systematising his -observations of any large groups of objects like the vegetable -and animal kingdoms.</p> - -<p>Now whenever a class has been properly formed, a -definition must have been laid down, stating the qualities -and circumstances possessed by all the objects which are -intended to be included in the class, and not possessed -<i>completely</i> by any other objects. Diagnosis, therefore, -consists in comparing the qualities of a certain object -with the definitions of a series of classes; the absence -in the object of any one quality stated in the definition -excludes it from the class thus defined; whereas, if we -find every point of a definition exactly fulfilled in the -specimen, we may at once assign it to the class in -question. It is of course by no means certain that everything -which has been affirmed of a class is true of all -objects afterwards referred to the class; for this would -be a case of imperfect inference, which is never more -than matter of probability. A definition can only make -known a finite number of the qualities of an object, and -it always remains possible that objects agreeing in those -assigned qualities will differ in others. <i>An individual -cannot be defined</i>, and can only be made known by the -exhibition of the individual itself, or by a material specimen -exactly representing it. But this and other questions -relating to definition must be treated when I am able to -take up the subject of language in another work.</p> - -<p>Diagnostic systems of classification should, as a general -rule, be arranged on the bifurcate method explicitly. Any -quality may be chosen which divides the whole group of -objects into two distinct parts, and each part may be sub-divided -successively by any prominent and well-marked -circumstance which is present in a large part of the genus -and not in the other. To refer an object to its proper -place in such an arrangement we have only to note whether -it does or does not possess the successive critical differentiæ. -Dana devised a classification of this kind<a id="FNanchor_581" href="#Footnote_581" class="fnanchor">581</a> by which to refer -a crystal to its place in the series of six or seven classes -already described. If a crystal has all its edges modified -alike or the angles replaced by three or six similar planes,<span class="pagenum" id="Page_712">712</span> -it belongs to the monometric system; if not, we observe -whether the number of similar planes at the extremity of -the crystal is three or some multiple of three, in which -case it is a crystal of the hexagonal system; and so we -proceed with further successive discriminations. To ascertain -the name of a mineral by examination with the blow-pipe, -an arrangement more or less evidently on the bifurcate -plan, has been laid down by Von Kobell.<a id="FNanchor_582" href="#Footnote_582" class="fnanchor">582</a> Minerals -are divided according as they possess or do not possess -metallic lustre; as they are fusible or not fusible, according -as they do or do not on charcoal give a metallic bead, -and so on.</p> - -<p>Perhaps the best example to be found of an arrangement -devised simply for the purpose of diagnosis, is -Mr. George Bentham’s <i>Analytical Key to the Natural -Orders and Anomalous Genera of the British Flora</i>, given -in his <i>Handbook of the British Flora</i>.<a id="FNanchor_583" href="#Footnote_583" class="fnanchor">583</a> In this scheme, -the great composite family of plants, together with the -closely approximate genus Jasione, are first separated -from all other flowering plants by the compound character -of their flowers. The remaining plants are sub-divided -according as the perianth is double or single. Since no -plants are yet known in which the perianth can be said -to have three or more distinct rings, this division becomes -practically the same as one into double and not-double. -Flowers with a double perianth are next discriminated -according as the corolla does or does not consist of one -piece; according as the ovary is free or not free; as it is -simple or not simple; as the corolla is regular or irregular; -and so on. On looking over this arrangement, it will -be found that numerical discriminations often occur, the -numbers of petals, stamens, capsules, or other parts being -the criteria, in which cases, as already explained (p. <a href="#Page_697">697</a>), -the actual exhibition of the bifid division would be tedious.</p> - -<p>Linnæus appears to have been perfectly acquainted -with the nature and uses of diagnostic classification, which -he describes under the name of Synopsis, saying:<a id="FNanchor_584" href="#Footnote_584" class="fnanchor">584</a>—“Synopsis<span class="pagenum" id="Page_713">713</span> -tradit Divisiones arbitrarias, longiores aut breviores, -plures aut pauciores: a Botanicis in genere non -agnoscenda. Synopsis est dichotomia arbitraria, quæ -instar viæ ad Botanicem ducit. Limites autem non determinat.”</p> - -<p>The rules and tables drawn out by chemists to facilitate -the discovery of the nature of a substance in qualitative -analysis are usually arranged on the bifurcate method, -and form excellent examples of diagnostic classification, -the qualities of the substances produced in testing being -in most cases merely characteristic properties of little importance -in other respects. The chemist does not detect -potassium by reducing it to the state of metallic potassium, -and then observing whether it has all the principal -qualities belonging to potassium. He selects from among -the whole number of compounds of potassium that salt, -namely the compound of platinum tetra-chloride, and -potassium chloride, which has the most distinctive appearance, -as it is comparatively insoluble and produces -a peculiar yellow and highly crystalline precipitate. Accordingly, -potassium is present whenever this precipitate -can be produced by adding platinum chloride to a solution. -The fine purple or violet colour which potassium -salts communicate to the blowpipe flame, had long been -used as a characteristic mark. Some other elements were -readily detected by the colouring of the blowpipe flame, -barium giving a pale yellowish green, and salts of strontium -a bright red. By the use of the spectroscope the -coloured light given off by an incandescent vapour is made -to give perfectly characteristic marks of the elements contained -in the vapour.</p> - -<p>Diagnosis seems to be identical with the process termed -by the ancient logicians <i>abscissio infiniti</i>, the cutting off -of the infinite or negative part of a genus when we discover -by observation that an object possesses a particular -difference. At every step in a bifurcate division, some -objects possessing the difference will fall into the affirmative -part or species; all the remaining objects in the world -fall into the negative part, which will be infinite in extent. -Diagnosis consists in the successive rejection from further -notice of those infinite classes with which the specimen in -question does not agree.</p> - -<p><span class="pagenum" id="Page_714">714</span></p> - - -<h3><i>Index Classifications.</i></h3> - -<p>Under classification we may include all arrangements of -objects or names, which we make for saving labour in the -discovery of an object. Even alphabetical indices are real -classifications. No such arrangement can be of use unless -it involves some correlation of circumstances, so that -knowing one thing we learn another. If we merely -arrange letters in the pigeon-holes of a secretaire we -establish a correlation, for all letters in the first hole will -be written by persons, for instance, whose names begin -with A, and so on. Knowing then the initial letter of -the writer’s name, we know also the place of the letter, and -the labour of search is thus reduced to one twenty-sixth -part of what it would be without arrangement.</p> - -<p>Now the purpose of a catalogue is to discover the place -in which an object is to be found; but the art of cataloguing -involves logical considerations of some importance. We -want to establish a correlation between the place of an -object and some circumstance about the object which -shall enable us readily to refer to it; this circumstance -therefore should be that which will most readily dwell in -the memory of the searcher. A piece of poetry will be -best remembered by the first line of the piece, and the -name of the author will be the next most definite circumstance; -a catalogue of poetry should therefore be arranged -alphabetically according to the first word of the piece, or -the name of the author, or, still better, in both ways. It -would be impossible to arrange poems according to their -subjects, so vague and mixed are these found to be when -the attempt is made.</p> - -<p>It is a matter of considerable literary importance to -decide upon the best mode of cataloguing books, so that -any required book in a library shall be most readily -found. Books may be classified in a great number of -ways, according to subject, language, date, or place of -publication, size, the initial words of the text or title-page, -or colophon, the author’s name, the publisher’s name, the -printer’s name, the character of the type, and so on. Every -one of these modes of arrangement may be useful, for we -may happen to remember one circumstance about a book<span class="pagenum" id="Page_715">715</span> -when we have forgotten all others; but as we cannot usually -go to the expense of forming more than two or three -indices, we must select those circumstances which will -lead to the discovery of a book most frequently. Many -of the criteria mentioned are evidently inapplicable.</p> - -<p>The language in which a book is written is definite -enough, provided that the whole book is written in the -same language; but it is obvious that language gives no -means for the subdivision and arrangement of the literature -of any one people. Classification by subjects would be an -exceedingly useful method if it were practicable, but experience -shows it to be a logical absurdity. It is a very -difficult matter to classify the sciences, so complicated -are the relations between them. But with books the -complication is vastly greater, since the same book -may treat of different sciences, or it may discuss a -problem involving many branches of knowledge. A -good account of the steam-engine will be antiquarian, so -far as it traces out the earliest efforts at discovery; purely -scientific, as regards the principles of thermodynamics involved; -technical, as regards the mechanical means of applying -those principles; economical, as regards the industrial -results of the invention; biographical, as regards the lives -of the inventors. A history of Westminster Abbey might -belong either to the history of architecture, the history of -the Church, or the history of England. If we abandon the -attempt to carry out an arrangement according to the -natural classification of the sciences, and form comprehensive -practical groups, we shall be continually perplexed by -the occurrence of intermediate cases, and opinions will -differ <i>ad infinitum</i> as to the details. If, to avoid the difficulty -about Westminster Abbey, we form a class of books -devoted to the History of Buildings, the question will then -arise whether Stonehenge is a building, and if so, whether -cromlechs, mounds, and monoliths are so. We shall be -uncertain whether to include lighthouses, monuments, -bridges, &c. In regard to literary works, rigorous classification -is still less possible. The same work may partake -of the nature of poetry, biography, history, philosophy, or -if we form a comprehensive class of Belles-lettres, nobody -can say exactly what does or does not come under the -term.</p> - -<p><span class="pagenum" id="Page_716">716</span></p> - -<p>My own experience entirely bears out the opinion of De -Morgan, that classification according to the name of the -author is the only one practicable in a large library, and -this method has been admirably carried out in the great -catalogue of the British Museum. The name of the author -is the most precise circumstance concerning a book, which -usually dwells in the memory. It is a better characteristic -of the book than anything else. In an alphabetical -arrangement we have an exhaustive classification, including -a place for every name. The following remarks<a id="FNanchor_585" href="#Footnote_585" class="fnanchor">585</a> -of De Morgan seem therefore to be entirely correct. -“From much, almost daily use, of catalogues for many -years, I am perfectly satisfied that a classed catalogue is -more difficult to use than to make. It is one man’s theory -of the subdivision of knowledge, and the chances are -against its suiting any other man. Even if all doubtful -works were entered under several different heads, the -frontier of the dubious region would itself be a mere matter -of doubt. I never turn from a classed catalogue to an -alphabetical one without a feeling of relief and security. -With the latter I can always, by taking proper pains, make -a library yield its utmost; with the former I can never -be satisfied that I have taken proper pains, until I have -made it, in fact, as many different catalogues as there are -different headings, with separate trouble for each. Those -to whom bibliographical research is familiar, know that -they have much more frequently to hunt an author than -a subject: they know also that in searching for a subject, -it is never safe to take another person’s view, however -good, of the limits of that subject with reference to their -own particular purposes.”</p> - -<p>It is often desirable, however, that a name catalogue -should be accompanied by a subordinate subject catalogue, -but in this case no attempt should be made to devise a -theoretically complete classification. Every principal -subject treated in a book should be entered separately in -an alphabetical list, under the name most likely to occur<span class="pagenum" id="Page_717">717</span> -to the searcher, or under several names. This method was -partially carried out in Watts’ <i>Bibliotheca Britannica</i>, but -it was excellently applied in the admirable subject index -to the <i>British Catalogue of Books</i>, and equally well in the -<i>Catalogue of the Manchester Free Library</i> at Campfield, -drawn up under the direction of Mr. Crestadoro, this -latter being the most perfect model of a printed catalogue -with which I am acquainted. The Catalogue of the -London Library is also in the right form, and has a useful -index of subjects, though it is too much condensed and -abbreviated. The public catalogue of the British Museum -is arranged as far as possible according to the alphabetical -order of the authors’ names, but in writing the titles for -this catalogue several copies are simultaneously produced -by a manifold writer, so that a catalogue according to the -order of the books on the shelves, and another according -to the first words of the title-page, are created by a mere -rearrangement of the spare copies. In the <i>English Cyclopædia</i> -it is suggested that twenty copies of the book titles -might readily have been utilised in forming additional -catalogues, arranged according to the place of publication, -the language of the book, the general nature of the subject, -and so forth.<a id="FNanchor_586" href="#Footnote_586" class="fnanchor">586</a> An excellent suggestion has also been made -to the effect that each book when published should have a -fly-leaf containing half a dozen printed copies of the title, -drawn up in a form suitable for insertion in catalogues. -Every owner of a library could then easily make accurate -printed catalogues to suit his own purposes, by merely -cutting out these titles and pasting them in books in any -desirable order.</p> - -<p>It will hardly be a digression to point out the enormous -saving of labour, or, what comes to the same thing, the -enormous increase in our available knowledge, both literary -and scientific, which arises from the formation of extensive -indices. The “State Papers,” containing the whole history -of the nation, were practically sealed to literary inquirers -until the Government undertook the task of calendaring -and indexing them. The British Museum Catalogue is -another national work, of which the importance in -advancing knowledge cannot be overrated. The Royal<span class="pagenum" id="Page_718">718</span> -Society is doing great service in publishing a complete -catalogue of memoirs upon physical science. The time -will perhaps come when our views upon this subject will -be extended, and either Government or some public society -will undertake the systematic cataloguing and indexing of -masses of historical and scientific information which are -now almost closed against inquiry.</p> - - -<h3><i>Classification in the Biological Sciences.</i></h3> - -<p>The great generalisations established in the works of -Herbert Spencer and Charles Darwin have thrown much -light upon other sciences, and have removed several -difficulties out of the way of the logician. The subject of -classification has long been studied in almost exclusive -reference to the arrangement of animals and plants. -Systematic botany and zoology have been commonly -known as the Classificatory Sciences, and scientific men -seemed to suppose that the methods of arrangement, -which were suitable for living creatures, must be the best -for all other classes of objects. Several mineralogists, -especially Mohs, have attempted to arrange minerals in -genera and species, just as if they had been animals -capable of reproducing their kind with variations. This -confusion of ideas between the relationship of living forms -and the logical relationship of things in general prevailed -from the earliest times, as manifested in the etymology of -words. We familiarly speak of a <i>kind</i> of things meaning -a class of things, and the kind consists of those things -which are <i>akin</i>, or come of the same race. When Socrates -and his followers wanted a name for a class regarded in a -philosophical light, they adopted the analogy in question, -and called it a γένος, or race, the root γεν- being connected -with the notion of generation.</p> - -<p>So long as species of plants and animals were believed -to proceed from distinct acts of Creation, there was no -apparent reason why methods of classification suitable to -them should not be treated as a guide to the classification -of other objects generally. But when once we regard -these resemblances as hereditary in their origin, we see -that the sciences of systematic botany and zoology have -a special character of their own. There is no reason to<span class="pagenum" id="Page_719">719</span> -suppose that the same kind of natural classification which -is best in biology will apply also in mineralogy, in -chemistry, or in astronomy. The logical principles which -underlie all classification are of course the same in natural -history as in the sciences of lifeless matter, but the special -resemblances which arise from the relation of parent and -offspring will not be found to prevail between different -kinds of crystals or mineral bodies.</p> - -<p>The genealogical view of the relations of animals and -plants leads us to discard all notions of a regular progression -of living forms, or any theory as to their symmetrical -relations. It was at one time a question whether the -ultimate scheme of natural classification would lead to -arrangement in a simple line, or a circle, or a combination -of circles. Macleay’s once celebrated system was a circular -one, and each class-circle was composed of five order-circles, -each of which was composed again of five tribe-circles, -and so on, the subdivision being at each step into -five minor circles. Macleay held that in the animal -kingdom there are five sub-kingdoms—the Vertebrata, -Annulosa, Radiata, Acrita, and Mollusca. Each of these -was again divided into five—the Vertebrata, consisting of -Mammalia, Reptilia, Pisces, Amphibia, and Aves.<a id="FNanchor_587" href="#Footnote_587" class="fnanchor">587</a> It is -evident that in such a symmetrical system the animals -were made to suit themselves to the classes instead of the -classes being suited to the animals.</p> - -<p>We now perceive that the ultimate system will have the -form of an immensely extended genealogical tree, which -will be capable of representation by lines on a plane -surface of sufficient extent. Strictly speaking, this genealogical -tree ought to represent the descent of each individual -living form now existing or which has existed. It -should be as personal and minute in its detail of relations, -as the Stemma of the Kings of England. We must not -assume that any two forms are exactly alike, and in any -case they are numerically distinct. Every parent then -must be represented at the apex of a series of divergent -lines, representing the generation of so many children. Any -complete system of classification must regard individuals -as the infimæ species. But as in the lower races of animals<span class="pagenum" id="Page_720">720</span> -and plants the differences between individuals are slight -and apparently unimportant, while the numbers of such -individuals are immensely great, beyond all possibility of -separate treatment, scientific men have always stopped at -some convenient but arbitrary point, and have assumed -that forms so closely resembling each other as to present -no constant difference were all of one kind. They have, -in short, fixed their attention entirely upon the main -features of family difference. In the genealogical tree -which they have been unconsciously aiming to construct, -diverging lines meant races diverging in character, and -the purpose of all efforts at so-called natural classification -was to trace out the descents between existing groups of -plants or animals.</p> - -<p>Now it is evident that hereditary descent may have in -different cases produced very different results as regards -the problem of classification. In some cases the differentiation -of characters may have been very frequent, and -specimens of all the characters produced may have -been transmitted to the present time. A living form -will then have, as it were, an almost infinite number of -cousins of various degrees, and there will be an immense -number of forms finely graduated in their resemblances. -Exact and distinct classification will then be almost -impossible, and the wisest course will be not to attempt -arbitrarily to distinguish forms closely related in nature, -but to allow that there exist transitional forms of every -degree, to mark out if possible the extreme limits of the -family relationship, and perhaps to select the most -generalised form, or that which presents the greatest -number of close resemblances to others of the family, as -the <i>type</i> of the whole.</p> - -<p>Mr. Darwin, in his most interesting work upon Orchids, -points out that the tribe of Malaxeæ are distinguished from -Epidendreæ by the absence of a caudicle to the pollinia; -but as some of the Malaxeæ have a minute caudicle, the -division really breaks down in the most essential point. -“This is a misfortune,” he remarks,<a id="FNanchor_588" href="#Footnote_588" class="fnanchor">588</a> “which every naturalist -encounters in attempting to classify a largely -developed or so-called natural group, in which, relatively<span class="pagenum" id="Page_721">721</span> -to other groups, there has been little extinction. In order -that the naturalist may be enabled to give precise and -clear definitions of his divisions, whole ranks of intermediate -or gradational forms must have been utterly swept -away: if here and there a member of the intermediate -ranks has escaped annihilation, it puts an effectual bar to -any absolutely distinct definition.”</p> - -<p>In other cases a particular plant or animal may perhaps -have transmitted its form from generation to generation -almost unchanged, or, what comes to the same result, those -forms which diverged in character from the parent stock -may have proved unsuitable to their circumstances, and -perished. We shall then find a particular form standing -apart from all others, and marked by many distinct -characters. Occasionally we may meet with specimens of -a race which was formerly far more common but is now -undergoing extinction, and is nearly the last of its kind. -Thus we explain the occurrence of exceptional forms such -as are found in the Amphioxus. The Equisetaceæ perplex -botanists by their want of affinity to other orders of Acrogenous -plants. This doubtless indicates that their genealogical -connection with other plants must be sought for in -the most distant ages of geological development.</p> - -<p>Constancy of character, as Mr. Darwin has said,<a id="FNanchor_589" href="#Footnote_589" class="fnanchor">589</a> is -what is chiefly valued and sought after by naturalists; -that is to say, naturalists wish to find some distinct family -mark, or group of characters, by which they may clearly -recognise the relationship of descent between a large -group of living forms. It is accordingly a great relief to -the mind of the naturalist when he comes upon a definitely -marked group, such as the Diatomaceæ, which are -clearly separated from their nearest neighbours the Desmidiaceæ -by their siliceous framework and the absence of -chlorophyll. But we must no longer think that because -we fail in detecting constancy of character the fault is -in our classificatory sciences. Where gradation of character -really exists, we must devote ourselves to defining and -registering the degrees and limits of that gradation. The -ultimate natural arrangement will often be devoid of strong -lines of demarcation.</p> -<p><span class="pagenum" id="Page_722">722</span></p> -<p>Let naturalists, too, form their systems of natural -classification with all care they can, yet it will certainly -happen from time to time that new and exceptional forms -of animals or vegetables will be discovered and will -require the modification of the system. A natural system -is directed, as we have seen, to the discovery of empirical -laws of correlation, but these laws being purely empirical -will frequently be falsified by more extensive investigation. -From time to time the notions of naturalists have -been greatly widened, especially in the case of Australian -animals and plants, by the discovery of unexpected combinations -of organs, and such events must often happen -in the future. If indeed the time shall come when all -the forms of plants are discovered and accurately described, -the science of Systematic Botany will then be -placed in a new and more favourable position, as remarked -by Alphonse Decandolle.<a id="FNanchor_590" href="#Footnote_590" class="fnanchor">590</a></p> - -<p>It ought to be remembered that though the genealogical -classification of plants or animals is doubtless the most instructive -of all, it is not necessarily the best for all purposes. -There may be correlations of properties important for -medicinal, or other practical purposes, which do not correspond -to the correlations of descent. We must regard -the bamboo as a tree rather than a grass, although it is -botanically a grass. For legal purposes we may continue -with advantage to treat the whale, seal, and other cetaceæ, -as fish. We must also class plants according as they -belong to arctic, alpine, temperate, sub-tropical or tropical -regions. There are causes of likeness apart from hereditary -relationship, and <i>we must not attribute exclusive excellence -to any one method of classification</i>.</p> - - -<h3><i>Classification by Types.</i></h3> - -<p>Perplexed by the difficulties arising in natural history -from the discovery of intermediate forms, naturalists have -resorted to what they call classification by types. Instead -of forming one distinct class defined by the invariable -possession of certain assigned properties, and rigidly including -or excluding objects according as they do or do not<span class="pagenum" id="Page_723">723</span> -possess all these properties, naturalists select a typical -specimen, and they group around it all other specimens -which resemble this type more than any other selected -type. “The type of each genus,” we are told,<a id="FNanchor_591" href="#Footnote_591" class="fnanchor">591</a> “should be -that species in which the characters of its group are -best exhibited and most evenly balanced.” It would -usually consist of those descendants of a form which had -undergone little alteration, while other descendants had -suffered slight differentiation in various directions.</p> - -<p>It would be a great mistake to suppose that this classification -by types is a logically distinct method. It is -either not a real method of classification at all, or it is -merely an abbreviated mode of representing a complicated -system of arrangement. A class must be defined by the -invariable presence of certain common properties. If, -then, we include an individual in which one of these -properties does not appear, we either fall into logical contradiction, -or else we form a new class with a new -definition. Even a single exception constitutes a new -class by itself, and by calling it an exception we merely -imply that this new class closely resembles that from -which it diverges in one or two points only. Thus in the -definition of the natural order of Rosaceæ, we find that -the seeds are one or two in each carpel, but that in the -genus Spiræa there are three or four; this must mean -either that the number of seeds is not a part of the fixed -definition of the class, or else that Spiræa does not belong -to that class, though it may closely approximate to it. -Naturalists continually find themselves between two horns -of a dilemma; if they restrict the number of marks -specified in a definition so that every form intended to -come within the class shall possess all those marks, it will -then be usually found to include too many forms; if the -definition be made more particular, the result is to produce -so-called anomalous genera, which, while they are held to -belong to the class, do not in all respects conform to its -definition. The practice has hence arisen of allowing considerable -latitude in the definition of natural orders. The -family of Cruciferæ, for instance, forms an exceedingly well-marked -natural order, and among its characters we find it<span class="pagenum" id="Page_724">724</span> -specified that the fruit is a pod, divided into two cells by -a thin partition, from which the valves generally separate -at maturity; but we are also informed that, in a few genera, -the pod is one-celled, or indehiscent, or separates transversely -into several joints.<a id="FNanchor_592" href="#Footnote_592" class="fnanchor">592</a> Now this must either mean -that the formation of the pod is not an essential point in -the definition of the family, or that there are several closely -associated families.</p> - -<p>The same holds true of typical classification. The type -itself is an individual, not a class, and no other object can -be exactly like the type. But as soon as we abstract the -individual peculiarities of the type and thus specify a -finite number of qualities in which other objects may -resemble the type, we immediately constitute a class. If -some objects resemble the type in some points, and others -in other points, then each definite collection of points of -resemblance constitutes intensively a separate class. The -very notion of classification by types is in fact erroneous -in a logical point of view. The naturalist is constantly -occupied in endeavouring to mark out definite groups -of living forms, where the forms themselves do not in -many cases admit of such rigorous lines of demarcation. -A certain laxity of logical method is thus apt to creep in, -the only remedy for which will be the frank recognition of -the fact, that, according to the theory of hereditary descent, -gradation of characters is probably the rule, and precise -demarcation between groups the exception.</p> - - -<h3><i>Natural Genera and Species.</i></h3> - -<p>One important result of the establishment of the theory -of evolution is to explode all notions about natural groups -constituting separate creations. Naturalists long held that -every plant belongs to some species, marked out by invariable -characters, which do not change by difference of -soil, climate, cross-breeding, or other circumstances. They -were unable to deny the existence of such things as sub-species, -varieties, and hybrids, so that a species of plants -was often subdivided and classified within itself. But -then the differences upon which this sub-classification<span class="pagenum" id="Page_725">725</span> -depended were supposed to be variable, and thus distinguished -from the invariable characters imposed upon the -whole species at its creation. Similarly a natural genus -was a group of species, and was marked out from other -genera by eternal differences of still greater importance.</p> - -<p>We now, however, perceive that the existence of any -such groups as genera and species is an arbitrary creation -of the naturalist’s mind. All resemblances of plants are -natural so far as they express hereditary affinities; but this -applies as well to the variations within the species as to -the species itself, or to the larger groups. All is a matter -of degree. The deeper differences between plants have -been produced by the differentiating action of circumstances -during millions of years, so that it would naturally -require millions of years to undo this result, and prove -experimentally that the forms can be approximated again. -Sub-species may sometimes have arisen within historical -times, and varieties approaching to sub-species may often -be produced by the horticulturist in a few years. Such -varieties can easily be brought back to their original forms, -or, if placed in the original circumstances, will themselves -revert to those forms; but according to Darwin’s views -all forms are capable of unlimited change, and it might -possibly be, unlimited reversion if suitable circumstances -and sufficient time be granted.</p> - -<p>Many fruitless attempts have been made to establish a -rigorous criterion of specific and generic difference, so that -these classes might have a definite value and rank in all -branches of biology. Linnæus adopted the view that the -species was to be defined as a distinct creation, saying,<a id="FNanchor_593" href="#Footnote_593" class="fnanchor">593</a> -“Species tot numeramus, quot diversæ formæ in principio -sunt creatæ;” or again, “Species tot sunt, quot diversas -formas ab initio produxit Infinitum Ens; quæ formæ, -secundum generationis inditas leges, produxere plures, at -sibi semper similes.” Of genera he also says,<a id="FNanchor_594" href="#Footnote_594" class="fnanchor">594</a> “Genus -omne est naturale, in primordio tale creatum.” It was a -common doctrine added to and essential to that of distinct -creation that these species could not produce intermediate -and variable forms, so that we find Linnæus obliged by the -ascertained existence of hybrids to take a different view<span class="pagenum" id="Page_726">726</span> -in another work; he says,<a id="FNanchor_595" href="#Footnote_595" class="fnanchor">595</a> “Novas species immo et genera -ex copula diversarum specierum in regno vegetabilium oriri -primo intuitu paradoxum videtur; interim observationes sic -fieri non ita dissuadent.” Even supposing in the present -day that we could assent to the notion of a certain number -of distinct creational acts, this notion would not help us in -the theory of classification. Naturalists have never pointed -out any method of deciding what are the results of distinct -creations, and what are not. As Darwin says,<a id="FNanchor_596" href="#Footnote_596" class="fnanchor">596</a> “the definition -must not include an element which cannot possibly -be ascertained, such as an act of creation.” It is, in fact, -by investigation of forms and classification that we should -ascertain what were distinct creations and what were not; -this information would be a result and not a means of -classification.</p> - -<p>Agassiz seemed to consider that he had discovered an important -principle, to the effect that general plan or structure -is the true ground for the discrimination of the great classes -of animals, which may be called branches of the animal -kingdom.<a id="FNanchor_597" href="#Footnote_597" class="fnanchor">597</a> He also thought that genera are definite and -natural groups. “Genera,” he says,<a id="FNanchor_598" href="#Footnote_598" class="fnanchor">598</a> “are most closely -allied groups of animals, differing neither in form, nor in -complication of structure, but simply in the ultimate structural -peculiarities of some of their parts; and this is, I believe, -the best definition which can be given of genera.” -But it is surely apparent that there are endless degrees both -of structural peculiarity and of complication of structure. -It is impossible to define the amount of structural peculiarity -which constitutes the genus as distinguished from -the species.</p> - -<p>The form which any classification of plants or animals -tends to take is that of an unlimited series of subaltern -classes. Originally botanists confined themselves for the -most part to a small number of such classes. Linnæus -adopted Class, Order, Genus, Species, and Variety, and even -seemed to think that there was something essentially natural -in a five-fold arrangement of groups.<a id="FNanchor_599" href="#Footnote_599" class="fnanchor">599</a></p> -<p><span class="pagenum" id="Page_727">727</span></p> -<p>With the progress of botany intermediate and additional -groups have gradually been introduced. According to the -Laws of Botanical Nomenclature adopted by the International -Botanical Congress, held at Paris<a id="FNanchor_600" href="#Footnote_600" class="fnanchor">600</a> in August -1867, no less than twenty-one names of classes are recognised—namely, -Kingdom, Division, Sub-division, Class, -Sub-class, Cohort, Sub-cohort, Order, Sub-order, Tribe, Sub-tribe, -Genus, Sub-genus, Section, Sub-section, Species, Sub-species, -Variety, Sub-variety, Variation, Sub-variation. It -is allowed by the authors of this scheme, that the rank or -degree of importance to be attributed to any of these divisions -may vary in a certain degree according to individual -opinion. The only point on which botanists are not allowed -discretion is as to the order of the successive sub-divisions; -any inversion of the arrangement, such as division of a -genus into tribes, or of a tribe into orders, is quite inadmissible. -There is no reason to suppose that even the -above list is complete and inextensible. The Botanical -Congress itself recognised the distinction between variations -according as they are Seedlings, Half-breeds, or <i>Lusus -Naturæ</i>. The complication of the inferior classes is increased -again by the existence of <i>hybrids</i>, arising from the -fertilisation of one species by another deemed a distinct -species, nor can we place any limit to the minuteness of -discrimination of degrees of breeding short of an actual -pedigree of individuals.</p> - -<p>It will be evident to the reader that in the remarks -upon classification as applied to the Natural Sciences, -given in this and the preceding sections, I have not in the -least attempted to treat the subject in a manner adequate -to its extent and importance. A volume would be insufficient -for tracing out the principles of scientific method -specially applicable to these branches of science. What -more I may be able to say upon the subject will be better -said, if ever, when I am able to take up the closely-connected -subjects of Scientific Nomenclature, Terminology, -and Descriptive Representation. In the meantime, I have -wished to show, in a negative point of view, that natural -classification in the animal and vegetable kingdoms is -a special problem, and that the particular methods and<span class="pagenum" id="Page_728">728</span> -difficulties to which it gives rise are not those common -to all cases of classification, as so many physicists have -supposed. Genealogical resemblances are only a special -case of resemblances in general.</p> - - -<h3><i>Unique or Exceptional Objects.</i></h3> - -<p>In framing a system of classification in almost any -branch of science, we must expect to meet with unique -or peculiar objects, which stand alone, having comparatively -few analogies with other objects. They may also be said -to be <i>sui generis</i>, each unique object forming, as it were, a -genus by itself; or they are called <i>nondescript</i>, because from -thus standing apart it is difficult to find terms in which to -describe their properties. The rings of Saturn, for instance, -form a unique object among the celestial bodies. We -have indeed considered this and many other instances of -unique objects in the preceding chapter on Exceptional -Phenomena. Apparent, Singular, and Divergent Exceptions -especially, are analogous to unique objects.</p> - -<p>In the classification of the elements, Carbon stands -apart as a substance entirely unique in its powers of -producing compounds. It is considered to be a quadrivalent -element, and it obeys all the ordinary laws of -chemical combination. Yet it manifests powers of affinity -in such an exalted degree that the substances in which it -appears are more numerous than all the other compounds -known to chemists. Almost the whole of the substances -which have been called organic contain carbon, and are -probably held together by the carbon atoms, so that many -chemists are now inclined to abandon the name Organic -Chemistry, and substitute the name Chemistry of the -Carbon Compounds. It used to be believed that the -production of organic compounds could be effected only -by the action of vital force, or of some inexplicable cause -involved in the phenomena of life; but it is now found -that chemists are able to commence with the elementary -materials, pure carbon, hydrogen, and oxygen, and by -strictly chemical operations to combine these so as to form -complicated organic compounds. So many substances have -already been formed that we might be inclined to generalise -and infer that all organic compounds might ultimately<span class="pagenum" id="Page_729">729</span> -be produced without the agency of living beings. Thus -the distinction between the organic and the inorganic -kingdoms seems to be breaking down, but our wonder at -the peculiar powers of carbon must increase at the same -time.</p> - -<p>In considering generalisation, the law of continuity was -applied chiefly to physical properties capable of mathematical -treatment. But in the classificatory sciences, also, -the same important principle is often beautifully exemplified. -Many objects or events seem to be entirely -exceptional and abnormal, and in regard to degree or -magnitude they may be so termed; but it is often easy to -show that they are connected by intermediate links with -ordinary cases. In the organic kingdoms there is a common -groundwork of similarity running through all classes, -but particular actions and processes present themselves -conspicuously in particular families and classes. Tenacity -of life is most marked in the Rotifera, and some other -kinds of microscopic organisms, which can be dried and -boiled without loss of life. Reptiles are distinguished -by torpidity, and the length of time they can live without -food. Birds, on the contrary, exhibit ceaseless activity and -high muscular power. The ant is as conspicuous for -intelligence and size of brain among insects as the quadrumana -and man among vertebrata. Among plants the -Leguminosæ are distinguished by a tendency to sleep, -folding their leaves at the approach of night. In the -genus Mimosa, especially the Mimosa pudica, commonly -called the sensitive plant, the same tendency is magnified -into an extreme irritability, almost resembling voluntary -motion. More or less of the same irritability probably -belongs to vegetable forms of every kind, but it is of -course to be investigated with special ease in such an -extreme case. In the Gymnotus and Torpedo, we find that -organic structures can act like galvanic batteries. Are we -to suppose that such animals are entirely anomalous exceptions; -or may we not justly expect to find less intense -manifestations of electric action in all animals?</p> - -<p>Some extraordinary differences between the modes of reproduction -of animals have been shown to be far less than -was at first sight apparent. The lower animals seem to -differ entirely from the higher ones in the power of reproducing<span class="pagenum" id="Page_730">730</span> -lost limbs. A kind of crab has the habit of casting -portions of its claws when much frightened, but they soon -grow again. There are multitudes of smaller animals -which, like the Hydra, may be cut in two and yet live and -develop into new complete individuals. No mammalian -animal can reproduce a limb, and in appearance there is no -analogy. But it was suggested by Blumenbach that the -healing of a wound in the higher animals really represents -in a lower degree the power of reproducing a limb. That -this is true may be shown by adducing a multitude of intermediate -cases, each adjoining pair of which are clearly -analogous, so that we pass gradually from one extreme to -the other. Darwin holds, moreover, that any such restoration -of parts is closely connected with that perpetual -replacement of the particles which causes every organised -body to be after a time entirely new as regards its constituent -substance. In short, we approach to a great -generalisation under which all the phenomena of growth, -restoration, and maintenance of organs are effects of one -and the same power.<a id="FNanchor_601" href="#Footnote_601" class="fnanchor">601</a> It is perhaps still more surprising -to find that the complicated process of reproduction -in the higher animals may be gradually traced down -to a simpler and simpler form, which at last becomes undistinguishable -from the budding out of one plant from the -stem of another. By a great generalisation we may regard -all the modes of reproduction of organic life as alike in their -nature, and varying only in complexity of development.<a id="FNanchor_602" href="#Footnote_602" class="fnanchor">602</a></p> - - -<h3><i>Limits of Classification.</i></h3> - -<p>Science can extend only so far as the power of accurate -classification extends. If we cannot detect resemblances, -and assign their exact character and amount, we cannot -have that generalised knowledge which constitutes science; -we cannot infer from case to case. Classification is the -opposite process to discrimination. If we feel that two -tastes differ, the tastes of two kinds of wine for instance, -the mere fact of difference existing prevents inference. -The detection of the difference saves us, indeed, from false<span class="pagenum" id="Page_731">731</span> -inference, because so far as difference exists, inference is -impossible. But classification consists in detecting resemblances -of all degrees of generality, and ascertaining -exactly how far such resemblances extend, while assigning -precisely the points at which difference begins. It enables -us, then, to generalise, and make inferences where it is -possible, and it saves us at the same time from going too -far. A full classification constitutes a complete record of -all our knowledge of the objects or events classified, and -the limits of exact knowledge are identical with the limits -of classification.</p> - -<p>It must by no means be supposed that every group -of natural objects will be found capable of rigorous -classification. There may be substances which vary by -insensible degrees, consisting, for instance, in varying -mixtures of simpler substances. Granite is a mixture -of quartz, felspar, and mica, but there are hardly two -specimens in which the proportions of these three constituents -are alike, and it would be impossible to lay -down definitions of distinct species of granite without -finding an infinite variety of intermediate species. The -only true classification of granites, then, would be founded -on the proportions of the constituents present, and a -chemical or microscopic analysis would be requisite, in -order that we might assign a specimen to its true position -in the series. Granites vary, again, by insensible degrees, -as regards the magnitude of the crystals of felspar and -mica. Precisely similar remarks might be made concerning -the classification of other plutonic rocks, such as -syenite, basalt, pumice-stone, lava.</p> - -<p>The nature of a ray of homogeneous light is strictly -defined, either by its place in the spectrum or by the corresponding -wave-length, but a ray of mixed light admits -of no simple classification; any of the infinitely numerous -rays of the continuous spectrum may be present or absent, -or present in various intensities, so that we can only class -and define a mixed colour by defining the intensity and -wave-length of each ray of homogeneous light which is -present in it. Complete spectroscopic analysis and the -determination of the intensity of every part of the spectrum -yielded by a mixed ray is requisite for its accurate -classification. Nearly the same may be said of complex<span class="pagenum" id="Page_732">732</span> -sounds. A simple sound undulation, if we could meet -with such a sound, would admit of precise and exhaustive -classification as regards pitch, the length of wave, or the -number of waves reaching the ear per second being a sufficient -criterion. But almost all ordinary sounds, even -those of musical instruments, consist of complex aggregates -of undulations of different pitches, and in order to classify -the sound we should have to measure the intensities of -each of the constituent sounds, a work which has been -partially accomplished by Helmholtz, as regards the vowel -sounds. The different tones of voice distinctive of different -individuals must also be due to the intermixture of minute -waves of various pitch, which are yet quite beyond the -range of experimental investigation. We cannot, then, at -present attempt to classify the different kinds or <i>timbres</i> of -sound.</p> - -<p>The difficulties of classification are still greater when a -varying phenomenon cannot be shown to be a mixture of -simpler phenomena. If we attempt to classify tastes, we -may rudely group them according as they are sweet, bitter, -saline, alkaline, acid, astringent or fiery; but it is evident -that these groups are bounded by no sharp lines of definition. -Tastes of mixed or intermediate character may exist -almost <i>ad infinitum</i>, and what is still more troublesome, -the tastes clearly united within one class may differ more -or less from each other, without our being able to arrange -them in subordinate genera and species. The same remarks -may be made concerning the classification of odours, which -may be roughly grouped according to the arrangement of -Linnæus as, aromatic, fragrant, ambrosiac, alliaceous, fetid, -virulent, nauseous. Within each of these vague classes, -however, there would be infinite shades of variety, and -each class would graduate into other classes. The odours -which can be discriminated by an acute nose are infinite; -every rock, stone, plant, or animal has some slight smell, -and it is well known that dogs, or even blind men, can -discriminate persons by a slight distinctive odour which -usually passes unnoticed.</p> - -<p>Similar remarks may be made concerning the feelings -of the human mind, called emotions. We know what is -anger, grief, fear, hatred, love; and many systems for -classifying these feelings have been proposed. They may<span class="pagenum" id="Page_733">733</span> -be roughly distinguished according as they are pleasurable -or painful, prospective or retrospective, selfish or sympathetic, -active or passive, and possibly in many other ways; -but each mode of arrangement will be indefinite and unsatisfactory -when followed into details. As a general rule, -the emotional state of the mind at any moment will be -neither pure anger nor pure fear, nor any one pure feeling, -but an indefinite and complex aggregate of feelings. It -may be that the state of mind is really a sum of several -distinct modes of agitation, just as a mixed colour is the -sum of the several rays of the spectrum. In this case -there may be more hope of some method of analysis being -successfully applied at a future time. But it may be -found that states of mind really graduate into each other -so that rigorous classification would be hopeless.</p> - -<p>A little reflection will show that there are whole worlds -of existences which in like manner are incapable of logical -analysis and classification. One friend may be able to -single out and identify another friend by his countenance -among a million other countenances. Faces are capable of -infinite discrimination, but who shall classify and define -them, or say by what particular shades of feature he does -judge? There are of course certain distinct types of face, -but each type is connected with each other type by infinite -intermediate specimens. We may classify melodies -according to the major or minor key, the character of the -time, and some other distinct points; but every melody -has, independently of such circumstances, its own distinctive -character and effect upon the mind. We can detect differences -between the styles of literary, musical, or artistic -compositions. We can even in some cases assign a picture -to its painter, or a symphony to its composer, by a subtle -feeling of resemblances or differences which may be felt, -but cannot be described.</p> - -<p>Finally, it is apparent that in human character there is -unfathomable and inexhaustible diversity. Every mind is -more or less like every other mind; there is always a basis -of similarity, but there is a superstructure of feelings, -impulses, and motives which is distinctive for each person. -We can sometimes predict the general character of the -feelings and actions which will be produced by a given -external event in an individual well known to us; but<span class="pagenum" id="Page_734">734</span> -we also know that we are often inexplicably at fault in -our inferences. No one can safely generalise upon the -subtle variations of temper and emotion which may arise -even in a person of ordinary character. As human knowledge -and civilisation progress, these characteristic differences -tend to develop and multiply themselves, rather than -decrease. Character grows more many-sided. Two well -educated Englishmen are far better distinguished from -each other than two common labourers, and these are -better distinguished than two Australian aborigines. The -complexities of existing phenomena probably develop themselves -more rapidly than scientific method can overtake -them. In spite of all the boasted powers of science, we -cannot really apply scientific method to our own minds -and characters, which are more important to us than all -the stars and nebulæ.</p> -<hr class="chap x-ebookmaker-drop"> - -<div class="chapter"> -<p><span class="pagenum" id="Page_735">735</span></p> - -<p class="nobreak ph2 ti0" id="BOOK_VI">BOOK VI.</p> -</div> - - -<hr class="r30"> - -<div class="chapter"> -<h2 class="nobreak" id="CHAPTER_XXXI">CHAPTER XXXI.<br> - -<span class="title">REFLECTIONS ON THE RESULTS AND LIMITS OF -SCIENTIFIC METHOD.</span></h2> -</div> - -<p class="ti0">Before concluding a work on the Principles of Science, -it will not be inappropriate to add some remarks upon -the limits and ultimate bearings of the knowledge which -we may acquire by the employment of scientific method. -All science consists, it has several times been stated, in the -detection of identities in the action of natural agents. The -purpose of inductive inquiry is to ascertain the apparent -existence of necessary connection between causes and -effects, expressed in the form of natural laws. Now so far -as we thus learn the invariable course of nature, the future -becomes the necessary sequel of the present, and we are -brought beneath the sway of powers with which nothing -can interfere.</p> - -<p>By degrees it is found, too, that the chemistry of -organised substances is not entirely separated from, but is -continuous with, that of earth and stones. Life seems to -be nothing but a special form of energy which is manifested -in heat and electricity and mechanical force. The -time may come, it almost seems, when the tender mechanism -of the brain will be traced out, and every thought -reduced to the expenditure of a determinate weight of<span class="pagenum" id="Page_736">736</span> -nitrogen and phosphorus. No apparent limit exists to the -success of scientific method in weighing and measuring, -and reducing beneath the sway of law, the phenomena both -of matter and of mind. And if mental phenomena be thus -capable of treatment by the balance and the micrometer, -can we any longer hold that mind is distinct from matter? -Must not the same inexorable reign of law which is -apparent in the motions of brute matter be extended to the -subtle feelings of the human heart? Are not plants and -animals, and ultimately man himself, merely crystals, as it -were, of a complicated form? If so, our boasted free will -becomes a delusion, moral responsibility a fiction, spirit a -mere name for the more curious manifestations of material -energy. All that happens, whether right or wrong, pleasurable -or painful, is but the outcome of the necessary -relations of time and space and force.</p> - -<p>Materialism seems, then, to be the coming religion, and -resignation to the nonentity of human will the only duty. -Such may not generally be the reflections of men of -science, but I believe that we may thus describe the -secret feelings of fear which the constant advance of -scientific investigation excites in the minds of many. Is -science, then, essentially atheistic and materialistic in its -tendency? Does the uniform action of material causes, -which we learn with an ever-increasing approximation to -certainty, preclude the hypothesis of a benevolent Creator, -who has not only designed the existing universe, but who -still retains the power to alter its course from time -to time?</p> - -<p>To enter upon actual theological discussions would be -evidently beyond the scope of this work. It is with the -scientific method common to all the sciences, and not with -any of the separate sciences, that we are concerned. -Theology therefore would be at least as much beyond -my scope as chemistry or geology. But I believe that -grave misapprehensions exist as regards the very nature -of scientific method. There are scientific men who assert -that the interposition of Providence is impossible, and -prayer an absurdity, because the laws of nature are inductively -proved to be invariable. Inferences are drawn -not so much from particular sciences as from the logical -nature of science itself, to negative the impulses and<span class="pagenum" id="Page_737">737</span> -hopes of men. Now I may state that my own studies in -logic lead me to call in question such negative inferences. -Laws of nature are uniformities observed to exist in the action -of certain material agents, but it is logically impossible -to show that all other agents must behave as these do. -The too exclusive study of particular branches of physical -science seems to generate an over-confident and dogmatic -spirit. Rejoicing in the success with which a few groups -of facts are brought beneath the apparent sway of laws, the -investigator hastily assumes that he is close upon the ultimate -springs of being. A particle of gelatinous matter is -found to obey the ordinary laws of chemistry; yet it moves -and lives. The world is therefore asked to believe that -chemistry can resolve the mysteries of existence.</p> - - -<h3><i>The Meaning of Natural Law.</i></h3> - -<p>Pindar speaks of Law as the Ruler of the Mortals and -the Immortals, and it seems to be commonly supposed -that the so-called Laws of Nature, in like manner, rule -man and his Creator. The course of nature is regarded -as being determined by invariable principles of mechanics -which have acted since the world began, and will act for -evermore. Even if the origin of all things is attributed -to an intelligent creative mind, that Being is regarded as -having yielded up arbitrary power, and as being subject like -a human legislator to the laws which he has himself -enacted. Such notions I should describe as superficial and -erroneous, being derived, as I think, from false views of -the nature of scientific inference, and the degree of certainty -of the knowledge which we acquire by inductive investigation.</p> - -<p>A law of nature, as I regard the meaning of the -expression, is not a uniformity which must be obeyed by -all objects, but merely a uniformity which is as a matter of -fact obeyed by those objects which have come beneath -our observation. There is nothing whatever incompatible -with logic in the discovery of objects which should -prove exceptions to any law of nature. Perhaps the best -established law is that which asserts an invariable correlation -to exist between gravity and inertia, so that all -gravitating bodies are found to possess inertia, and all<span class="pagenum" id="Page_738">738</span> -bodies possessing inertia are found to gravitate. But it -would be no reproach to our scientific method, if something -were ultimately discovered to possess gravity without -inertia. Strictly defined and correctly interpreted, the law -itself would acknowledge the possibility; for with the -statement of every law we ought properly to join an estimate -of the number of instances in which it has been -observed to hold true, and the probability thence calculated, -that it will hold true in the next case. Now, as we -found (p. <a href="#Page_259">259</a>), no finite number of instances can warrant -us in expecting with certainty that the next instance will -be of like nature; in the formulas yielded by the inverse -method of probabilities a unit always appears to represent -the probability that our inference will be mistaken. I -demur to the assumption that there is any necessary truth -even in such fundamental laws of nature as the Indestructibility -of Matter, the Conservation of Energy, or the Laws -of Motion. Certain it is that men of science have recognised -the conceivability of other laws, and even investigated -their mathematical consequences. Airy investigated the -mathematical conditions of a perpetual motion (p. <a href="#Page_223">223</a>), -and Laplace and Newton discussed imaginary laws of forces -inconsistent with those observed to operate in the universe -(pp. <a href="#Page_642">642</a>, <a href="#Page_706">706</a>).</p> - -<p>The laws of nature, as I venture to regard them, are -simply general propositions concerning the correlation of -properties which have been observed to hold true of -bodies hitherto observed. On the assumption that our -experience is of adequate extent, and that no arbitrary -interference takes place, we are then able to assign the -probability, always less than certainty, that the next -object of the same apparent nature will conform to the -same laws.</p> - - -<h3><i>Infiniteness of the Universe.</i></h3> - -<p>We may safely accept as a satisfactory scientific hypothesis -the doctrine so grandly put forth by Laplace, who -asserted that a perfect knowledge of the universe, as it -existed at any given moment, would give a perfect knowledge -of what was to happen thenceforth and for ever -after. Scientific inference is impossible, unless we may<span class="pagenum" id="Page_739">739</span> -regard the present as the outcome of what is past, and the -cause of what is to come. To the view of perfect intelligence -nothing is uncertain. The astronomer can calculate -the positions of the heavenly bodies when thousands of -generations of men shall have passed away, and in this fact -we have some illustration, as Laplace remarks, of the power -which scientific prescience may attain. Doubtless, too, all -efforts in the investigation of nature tend to bring us nearer -to the possession of that ideally perfect power of intelligence. -Nevertheless, as Laplace with profound wisdom -adds,<a id="FNanchor_603" href="#Footnote_603" class="fnanchor">603</a> we must ever remain at an infinite distance from the -goal of our aspirations.</p> - -<p>Let us assume, for a time at least, as a highly probable -hypothesis, that whatever is to happen must be the outcome -of what is; there then arises the question, What is? -Now our knowledge of what exists must ever remain imperfect -and fallible in two respects. Firstly, we do not -know all the matter that has been created, nor the exact -manner in which it has been distributed through space. -Secondly, assuming that we had that knowledge, we -should still be wanting in a perfect knowledge of the -way in which the particles of matter will act upon each -other. The power of scientific prediction extends at the -most to the limits of the data employed. Every conclusion -is purely hypothetical and conditional upon the -non-interference of agencies previously undetected. The -law of gravity asserts that every body tends to approach -towards every other body, with a certain determinate -force; but, even supposing the law to hold true, it does -not assert that the body <i>will</i> approach. No single law -of nature can warrant us in making an absolute prediction. -We must know all the laws of nature and all the -existing agents acting according to those laws before we -can say what will happen. To assume, then, that scientific -method can take everything within its cold embrace of -uniformity, is to imply that the Creator cannot outstrip -the intelligence of his creatures, and that the existing -Universe is not infinite in extent and complexity, an assumption -for which I see no logical basis whatever.</p> - -<p><span class="pagenum" id="Page_740">740</span></p> - - -<h3><i>The Indeterminate Problem of Creation.</i></h3> - -<p>A second and very serious misapprehension concerning -the import of a law of nature may now be pointed -out. It is not uncommonly supposed that a law determines -the character of the results which shall take place, -as, for instance, that the law of gravity determines what -force of gravity shall act upon a given particle. Surely -a little reflection must render it plain that a law by itself -determines nothing. It is <i>law plus agents obeying law -which has results</i>, and it is no function of law to govern or -define the number and place of its own agents. Whether -a particle of matter shall gravitate, depends not only upon -the law of Newton, but also upon the distribution of surrounding -particles. The theory of gravitation may perhaps -be true throughout all time and in all parts of space, and -the Creator may never find occasion to create those possible -exceptions to it which I have asserted to be conceivable. -Let this be as it may; our science cannot certainly determine -the question. Certain it is, that the law of gravity -does not alone determine the forces which may be brought -to bear at any point of space. The force of gravitation acting -upon any particle depends upon the mass, distance, and -relative position of all the other particles of matter within -the bounds of space at the instant in question. Even -assuming that all matter when once distributed through -space at the Creation was thenceforth to act in an invariable -manner without subsequent interference, yet the -actual configuration of matter at any moment, and the -consequent results of the law of gravitation, must have -been entirely a matter of free choice.</p> - -<p>Chalmers has most distinctly pointed out that the -existing <i>collocations</i> of the material world are as important -as the laws which the objects obey. He remarks that a -certain class of writers entirely overlook the distinction, -and forget that mere laws without collocations would -have afforded no security against a turbid and disorderly -chaos.<a id="FNanchor_604" href="#Footnote_604" class="fnanchor">604</a> Mill has recognised<a id="FNanchor_605" href="#Footnote_605" class="fnanchor">605</a> the truth of Chalmers’ -statement, without drawing the proper inferences from<span class="pagenum" id="Page_741">741</span> -it. He says<a id="FNanchor_606" href="#Footnote_606" class="fnanchor">606</a> of the distribution of matter through space, -“We can discover nothing regular in the distribution itself; -we can reduce it to no uniformity, to no law.” More lately -the Duke of Argyll in his well-known work on the <i>Reign -of Law</i> has drawn attention to the profound distinction -between laws and collocations of causes.</p> - -<p>The original conformation of the material universe, as -far as we can tell, was free from all restriction. There -was unlimited space in which to frame it, and an unlimited -number of material particles, each of which could be placed -in any one of an infinite number of different positions. It -should be added, that each particle might be endowed -with any one of an infinite number of quantities of <i>vis -viva</i> acting in any one of an infinite number of different -directions. The problem of Creation was, then, what a -mathematician would call <i>an indeterminate problem</i>, and it -was indeterminate in a great number of ways. Infinitely -numerous and various universes might then have been -fashioned by the various distribution of the original -nebulous matter, although all the particles of matter -should obey the law of gravity.</p> - -<p>Lucretius tells us how in the original rain of atoms -some of these little bodies diverged from the rectilinear -direction, and coming into contact with other atoms gave -rise to the various combinations of substances which exist. -He omitted to tell us whence the atoms came, or by what -force some of them were caused to diverge; but surely -these omissions involve the whole question. I accept the -Lucretian conception of creation when properly supplemented. -Every atom which existed in any point of space -must have existed there previously, or must have been -created there by a previously existing Power. When -placed there it must have had a definite mass and a -definite energy. Now, as before remarked, an unlimited -number of atoms can be placed in unlimited space in an -unlimited number of modes of distribution. Out of infinitely -infinite choices which were open to the Creator, -that one choice must have been made which has yielded -the Universe as it now exists.</p> - -<p>It would be a mistake, indeed, to suppose that the law<span class="pagenum" id="Page_742">742</span> -of gravity, when it holds true, is no restriction on the -distribution of force. That law is a geometrical law, and -it would in many cases be mathematically impossible, as -far as we can see, that the force of gravity acting on one -particle should be small while that on a neighbouring -particle is great. We cannot conceive that even Omnipotent -Power should make the angles of a triangle greater -than two right angles. The primary laws of thought and -the fundamental notions of the mathematical sciences do -not seem to admit of error or alteration. Into the metaphysical -origin and meaning of the apparent necessity -attaching to such laws I have not attempted to inquire in -this work, and it is not requisite for my present purpose. -If the law of gravity were the only law of nature and the -Creator had chosen to render all matter obedient to that -law, there would doubtless be restrictions upon the effects -derivable from any one distribution of matter.</p> - - -<h3><i>Hierarchy of Natural Laws.</i></h3> - -<p>A further consideration presents itself. A natural law -like that of gravity expresses a certain uniformity in the -action of agents submitted to it, and this produces, as we -have seen, certain geometrical restrictions upon the effects -which those agents may produce. But there are other -forces and laws besides gravity. One force may override -another, and two laws may each be obeyed and may each -disguise the action of the other. In the intimate constitution -of matter there may be hidden springs which, while -acting in accordance with their own fixed laws, may lead -to sudden and unexpected changes. So at least it has -been found from time to time in the past, and so there -is every reason to believe it will be found in the future. -To the ancients it seemed incredible that one lifeless stone -could make another leap towards it. A piece of iron -while it obeys the magnetic force of the loadstone does -not the less obey the law of gravity. A plant gravitates -downwards as regards every constituent cell or fibre, and -yet it persists in growing upwards. Life is altogether an -exception to the simpler phenomena of mineral substances, -not in the sense of disproving those laws, but in superadding -forces of new and inexplicable character. Doubtless no<span class="pagenum" id="Page_743">743</span> -law of chemistry is broken by the action of the nervous -cells, and no law of physics by the pulses of the nervous -fibres, but something requires to be added to our sciences -in order that we may explain these subtle phenomena.</p> - -<p>Now there is absolutely nothing in science or in scientific -method to warrant us in assigning a limit to this -hierarchy of laws. When in many undoubted cases we -find law overriding law, and at certain points in our -experience producing unexpected results, we cannot -venture to affirm that we have exhausted the strange -phenomena which may have been provided for in the -original constitution of matter. The Universe might have -been so designed that it should go for long intervals -through the same round of unvaried existence, and yet -that events of exceptional character should be produced -from time to time. Babbage showed in that most profound -and eloquent work, <i>The Ninth Bridgewater Treatise</i>, that it -was theoretically possible for human artists to design a -machine, consisting of metallic wheels and levers, which -should work invariably according to a simple law of action -during any finite number of steps, and yet at a fixed -moment, however distant, should manifest a single breach -of law. Such an engine might go on counting, for instance, -the natural numbers until they would reach a number -requiring for its expression a hundred million digits. “If -every letter in the volume now before the reader’s eyes,” -says Babbage,<a id="FNanchor_607" href="#Footnote_607" class="fnanchor">607</a> “were changed into a figure, and if all the -figures contained in a thousand such volumes were arranged -in order, the whole together would yet fall far short of the -vast induction the observer would have had in favour of -the truth of the law of natural numbers.... Yet shall -the engine, true to the prediction of its director, after the -lapse of myriads of ages, fulfil its task, and give that one, -the first and only exception to that time-sanctioned law. -What would have been the chances against the appearance -of the excepted case, immediately prior to its occurrence?”</p> - -<p>As Babbage further showed,<a id="FNanchor_608" href="#Footnote_608" class="fnanchor">608</a> a calculating engine, after -proceeding through any required number of motions -according to a first law, may be made suddenly to suffer -a change, so that it shall then commence to calculate<span class="pagenum" id="Page_744">744</span> -according to a wholly new law. After giving the natural -numbers for a finite time, it might suddenly begin to give -triangular, or square, or cube numbers, and these changes -might be conceived theoretically as occurring time after -time. Now if such occurrences can be designed and foreseen -by a human artist, it is surely within the capacity of -the Divine Artist to provide for analogous changes of law -in the mechanism of the atom, or the construction of the -heavens.</p> - -<p>Physical science, so far as its highest speculations can -be trusted, gives some indication of a change of law in -the past history of the Universe. According to Sir W. -Thomson’s deductions from Fourier’s <i>Theory of Heat</i>, we -can trace down the dissipation of heat by conduction and -radiation to an infinitely distant time when all things will -be uniformly cold. But we cannot similarly trace the -heat-history of the Universe to an infinite distance in the -past. For a certain negative value of the time the formulæ -give impossible values, indicating that there was some -initial distribution of heat which could not have resulted, -according to known laws of nature,<a id="FNanchor_609" href="#Footnote_609" class="fnanchor">609</a> from any previous -distribution.<a id="FNanchor_610" href="#Footnote_610" class="fnanchor">610</a> There are other cases in which a consideration -of the dissipation of energy leads to the conception of -a limit to the antiquity of the present order of things.<a id="FNanchor_611" href="#Footnote_611" class="fnanchor">611</a> -Human science, of course, is fallible, and some oversight -or erroneous simplification in these theoretical calculations -may afterwards be discovered; but as the present state of -scientific knowledge is the only ground on which erroneous -inferences from the uniformity of nature and the supposed -reign of law are founded, I am right in appealing to the -present state of science in opposition to these inferences. -Now the theory of heat places us in the dilemma either of<span class="pagenum" id="Page_745">745</span> -believing in Creation at an assignable date in the past, or -else of supposing that some inexplicable change in the -working of natural laws then took place. Physical science -gives no countenance to the notion of infinite duration of -matter in one continuous course of existence. And if in -time past there has been a discontinuity of law, why may -there not be a similar event awaiting the world in the -future? Infinite ingenuity could have implanted some -agency in matter so that it might never yet have made -its tremendous powers manifest. We have a very good -theory of the conservation of energy, but the foremost -physicists do not deny that there may possibly be forms of -energy, neither kinetic nor potential, and therefore of unknown -nature.<a id="FNanchor_612" href="#Footnote_612" class="fnanchor">612</a></p> - -<p>We can imagine reasoning creatures dwelling in a world -where the atmosphere was a mixture of oxygen and inflammable -gas like the fire-damp of coal-mines. If devoid -of fire, they might have lived through long ages unconscious -of the tremendous forces which a single spark would call -into play. In the twinkling of an eye new laws might come -into action, and the poor reasoning creatures, so confident -about their knowledge of the reign of law in their world, -would have no time to speculate upon the overthrow of all -their theories. Can we with our finite knowledge be sure -that such an overthrow of our theories is impossible?</p> - - -<h3><i>The Ambiguous Expression, “Uniformity of Nature.”</i></h3> - -<p>I have asserted that serious misconception arises from -an erroneous interpretation of the expression Uniformity of -Nature. Every law of nature is the statement of a certain -uniformity observed to exist among phenomena, and since -the laws of nature are invariably obeyed, it seems to follow -that the course of nature itself is uniform, so that we can -safely judge of the future by the present. This inference -is supported by some of the results of physical astronomy. -Laplace proved that the planetary system is stable, so that -no perturbation which planet produces upon planet can -become so great as to cause disruption and permanent -alteration of the planetary orbits. A full comprehension<span class="pagenum" id="Page_746">746</span> -of the law of gravity shows that all such disturbances are -essentially periodic, so that after the lapse of millions of -years the planets will return to the same relative positions, -and a new cycle of disturbances will then commence.</p> - -<p>As other branches of science progress, we seem to gain -assurance that no great alteration of the world’s condition -is to be expected. Conflict with a comet has long been the -cause of fear, but now it is credibly asserted that we have -passed through a comet’s tail without the fact being known -at the time, or manifested by any more serious a phenomenon -than a slight luminosity of the sky. More recently still -the earth is said to have touched the comet Biela, and the -only result was a beautiful and perfectly harmless display -of meteors. A decrease in the heating power of the sun -seems to be the next most probable circumstance from -which we might fear the extinction of life on the earth. -But calculations founded on reasonable physical data show -that no appreciable change can be going on, and experimental -data to indicate a change are wholly wanting. -Geological investigations show indeed that there have been -extensive variations of climate in past times; vast glaciers -and icebergs have swept over the temperate regions at one -time, and tropical vegetation has flourished near the poles -at another time. But here again the vicissitudes of climate -assume a periodic character, so that the stability of the -earth’s condition does not seem to be threatened.</p> - -<p>All these statements may be reasonable, but they do not -establish the Uniformity of Nature in the sense that extensive -alterations or sudden catastrophes are impossible. In -the first place, Laplace’s theory of the stability of the -planetary system is of an abstract character, as paying -regard to nothing but the mutual gravitation of the -planetary bodies and the sun. It overlooks several -physical causes of change and decay in the system which -were not so well known in his day as at present, and it also -presupposes the absence of any interruption of the course -of things by conflict with foreign astronomical bodies.</p> - -<p>It is now acknowledged by astronomers that there are at -least two ways in which the <i>vis viva</i> of the planets and -satellites may suffer loss. The friction of the tides upon -the earth produces a small quantity of heat which is -radiated into space, and this loss of energy must result in a<span class="pagenum" id="Page_747">747</span> -decrease of the rotational velocity, so that ultimately the -terrestrial day will become identical with the year, just as -the periods of revolution of the moon upon its axis and -around the earth have already become equal. Secondly, -there can be little doubt that certain manifestations of -electricity upon the earth’s surface depend upon the -relative motions of the planets and the sun, which give rise -to periods of increased intensity. Such electrical phenomena -must result in the production and dissipation of heat, -the energy of which must be drawn, partially at least, from -the moving bodies. This effect is probably identical (p. <a href="#Page_570">570</a>) -with the loss of energy of comets attributed to the so-called -resisting medium. But whatever be the theoretical explanation -of these phenomena, it is almost certain that there -exists a tendency to the dissipation of the energy of the -planetary system, which will, in the indefinite course of -time, result in the fall of the planets into the sun.</p> - -<p>It is hardly probable, however, that the planetary system -will be left undisturbed throughout the enormous interval -of time required for the dissipation of its energy in this way. -Conflict with other bodies is so far from being improbable, -that it becomes approximately certain when we take very -long intervals of time into account. As regards cometary -conflicts, I am by no means satisfied with the negative -conclusions drawn from the remarkable display on the -evening of the 27th of November, 1872. We may often -have passed through the tail of a comet, the light of which -is probably an electrical manifestation no more substantial -than the aurora borealis. Every remarkable shower of -shooting stars may also be considered as proceeding from a -cometary body, so that we may be said to have passed -through the thinner parts of innumerable comets. But the -earth has probably never passed, in times of which we have -any record, through the nucleus of a comet, which consists -perhaps of a dense swarm of small meteorites. We can -only speculate upon the effects which might be produced -by such a conflict, but it would probably be a much more -serious event than any yet registered in history. The -probability of its occurrence, too, cannot be assigned; for -though the probability of conflict with any one cometary -nucleus is almost infinitesimal, yet the number of comets -is immensely great (p. <a href="#Page_408">408</a>).</p> - -<p><span class="pagenum" id="Page_748">748</span></p> - -<p>It is far from impossible, again, that the planetary -system may be invaded by bodies of greater mass than -comets. The sun seems to be placed in so extensive a -portion of empty space that its own proper motion would -not bring it to the nearest known star (α Centauri) in less -than 139,200 years. But in order to be sure that this -interval of undisturbed life is granted to our globe, we -must prove that there are no stars moving so as to meet -us, and no dark bodies of considerable size flying through -intervening space unknown to us. The intrusion of comets -into our system, and the fact that many of them have -hyperbolic paths, is sufficient to show that the surrounding -parts of space are occupied by multitudes of dark -bodies of some size. It is quite probable that small suns -may have cooled sufficiently to become non-luminous; -for even if we discredit the theory that the variation of -brightness of periodic stars is due to the revolution of -dark companion stars, yet there is in our own globe -an unquestionable example of a smaller body which has -cooled below the luminous point.</p> - -<p>Altogether, then, it is a mere assumption that the -uniformity of nature involves the unaltered existence of -our own globe. There is no kind of catastrophe which -is too great or too sudden to be theoretically consistent -with the reign of law. For all that our science can tell, -human history may be closed in the next instant of time. -The world may be dashed to pieces against a wandering -star; it may be involved in a nebulous atmosphere of -hydrogen to be exploded a second afterwards; it may be -scorched up or dissipated into vapour by some great -explosion in the sun; there might even be within the -globe itself some secret cause of disruption, which only -needs time for its manifestation.</p> - -<p>There are some indications, as already noticed (p. <a href="#Page_660">660</a>), -that violent disturbances have actually occurred in the -history of the solar system. Olbers sought for the minor -planets on the supposition that they were fragments of an -exploded planet, and he was rewarded with the discovery -of some of them. The retrograde motion of the satellites -of the more distant planets, the abnormal position of the -poles of Uranus and the excessive distance of Neptune, are -other indications of some violent event, of which we have<span class="pagenum" id="Page_749">749</span> -no other evidence. I adduce all these facts and arguments, -not to show that there is any considerable probability, as -far as we can judge, of interruption within the scope of -human history, but to prove that the Uniformity of Nature -is theoretically consistent with the most unexpected events -of which we can form a conception.</p> - - -<h3><i>Possible States of the Universe.</i></h3> - -<p>When we give the rein to scientific imagination, it -becomes apparent that conflict of body with body must -not be regarded as the rare exception, but as the general -rule and the inevitable fate of each star system. So far -as we can trace out the results of the law of gravitation, -and of the dissipation of energy, the universe must be regarded -as undergoing gradual condensation into a single -cold solid body of gigantic dimensions. Those who so -frequently use the expression Uniformity of Nature seem -to forget that the Universe might exist consistently with -the laws of nature in the most diverse conditions. It -might consist, on the one hand, of a glowing nebulous -mass of gaseous substances. The heat might be so intense -that all elements, even carbon and silicon, would be -in the state of gas, and all atoms, of whatever nature, -would be flying about in chemical independence, diffusing -themselves almost uniformly in the neighbouring parts -of space. There would then be no life, unless we can -apply that name to the passage through each part of -space of similar average trains of atoms, the particular -succession of atoms being governed only by the theory -of probability, and the law of divergence from a mean -exhibited in the Arithmetical Triangle. Such a universe -would correspond partially to the Lucretian rain of atoms, -and to that nebular hypothesis out of which Laplace -proposed philosophically to explain the evolution of the -planetary system.</p> - -<p>According to another extreme supposition, the intense -heat-energy of this nebulous mass might be radiated away -into the unknown regions of outer space. The attraction -of gravity would exert itself between each two particles, -and the energy of motion thence arising would, by incessant -conflicts, be resolved into heat and dissipated.<span class="pagenum" id="Page_750">750</span> -Inconceivable ages might be required for the completion of -this process, but the dissipation of energy thus proceeding -could end only in the production of a cold and motionless -universe. The relation of cause and effect, as we see it -manifested in life and growth, would degenerate into the constant -existence of every particle in a fixed position relative -to every other particle. Logical and geometrical resemblances -would still exist between atoms, and between -groups of atoms crystallised in their appropriate forms -for evermore. But time, the great variable, would bring -no variation, and as to human hopes and troubles, they -would have gone to eternal rest.</p> - -<p>Science is not really adequate to proving that such is -the inevitable fate of the universe, for we can seldom trust -our best-established theories far from their data. Nevertheless, -the most probable speculations which we can -form as to the history, especially of our own planetary -system, is that it originated in a heated revolving nebulous -mass of gas, and is in a state of excessively slow progress -towards the cold and stony condition. Other speculative -hypotheses might doubtless be entertained. Every hypothesis -is pressed by difficulties. If the whole universe be -cooling, whither does the heat go? If we are to get rid -of it entirely, outer space must be infinite in extent, so -that it shall never be stopped and reflected back. But not -to speak of metaphysical difficulties, if the medium of heat -undulations be infinite in extent, why should not the -material bodies placed in it be infinite also in number and -aggregate mass? It is apparent that we are venturing into -speculations which surpass our powers of scientific inference. -But then I am arguing negatively; I wish to show that -those who speak of the uniformity of nature, and the reign -of law, misinterpret the meaning involved in those expressions. -Law is not inconsistent with extreme diversity, -and, so far as we can read the history of this planetary -system, it did probably originate in heated nebulous matter, -and man’s history forms but a brief span in its progress -towards the cold and stony condition. It is by doubtful -and speculative hypotheses alone that we can avoid -such a conclusion, and I depart least from undoubted -facts and well-established laws when I assert that, whatever -uniformities may underlie the phenomena of nature,<span class="pagenum" id="Page_751">751</span> -constant variety and ever-progressing change is the real -outcome.</p> - - -<h3><i>Speculations on the Reconcentration of Energy.</i></h3> - -<p>There are unequivocal indications, as I have said, that -the material universe, as we at present see it, is progressing -from some act of creation, or some discontinuity of existence -of which the date may be approximately fixed by -scientific inference. It is progressing towards a state in -which the available energy of matter will be dissipated -through infinite surrounding space, and all matter will -become cold and lifeless. This constitutes, as it were, the -historical period of physical science, that over which our -scientific foresight may more or less extend. But in this, -as in other cases, we have no right to interpret our experience -negatively, so as to infer that because the present -state of things began at a particular time, there was no -previous existence. It may be that the present period of -material existence is but one of an indefinite series of like -periods. All that we can see, and feel, and infer, and -reason about may be, as it were, but a part of one single -pulsation in the existence of the universe.</p> - -<p>After Sir W. Thomson had pointed out the preponderating -tendency which now seems to exist towards the -conversion of all energy into heat-energy, and its equal -diffusion by radiation throughout space, the late Professor -Rankine put forth a remarkable speculation.<a id="FNanchor_613" href="#Footnote_613" class="fnanchor">613</a> He suggested -that the ethereal, or, as I have called it, the <i>adamantine</i> -medium in which all the stars exist, and all radiation -takes place, may have bounds, beyond which only empty -space exists. All heat undulations reaching this boundary -will be totally reflected, according to the theory of undulations, -and will be reconcentrated into foci situated in -various parts of the medium. Whenever a cold and -extinct star happens to pass through one of these foci, it -will be instantly ignited and resolved by intense heat into -its constituent elements. Discontinuity will occur in the -history of that portion of matter, and the star will begin -its history afresh with a renewed store of energy.</p> -<p><span class="pagenum" id="Page_752">752</span></p> -<p>This is doubtless a mere speculation, practically incapable -of verification by observation, and almost free -from restrictions afforded by present knowledge. We -might attribute various shapes to the adamantine medium, -and the consequences would be various. But there is this -value in such speculations, that they draw attention to the -finiteness of our knowledge. We cannot deny the possible -truth of such an hypothesis, nor can we place a limit to -the scientific imagination in the framing of other like -hypotheses. It is impossible, indeed, to follow out our -scientific inferences without falling into speculation. If -heat be radiated into outward space, it must either proceed -<i>ad infinitum</i>, or it must be stopped somewhere. In the -latter case we fall upon Rankine’s hypothesis. But if the -material universe consist of a finite collection of heated -matter situated in a finite portion of an infinite adamantine -medium, then either this universe must have existed for a -finite time, or else it must have cooled down during the -infinity of past time indefinitely near to the absolute zero -of temperature. I objected to Lucretius’ argument against -the destructibility of matter, that we have no knowledge -whatever of the laws according to which it would undergo -destruction. But we do know the laws according to which -the dissipation of heat appears to proceed, and the conclusion -inevitably is that a finite heated material body -placed in a perfectly cold infinitely extended medium -would in an infinite time sink to zero of temperature. -Now our own world is not yet cooled down near to zero, -so that physical science seems to place us in the dilemma -of admitting either the finiteness of past duration of the -world, or else the finiteness of the portion of medium in -which we exist. In either case we become involved in -metaphysical and mechanical difficulties surpassing our -mental powers.</p> - - -<h3><i>The Divergent Scope for New Discovery.</i></h3> - -<p>In the writings of some recent philosophers, especially -of Auguste Comte, and in some degree John Stuart Mill, -there is an erroneous and hurtful tendency to represent -our knowledge as assuming an approximately complete -character. At least these and many other writers fail to<span class="pagenum" id="Page_753">753</span> -impress upon their readers a truth which cannot be too -constantly borne in mind, namely, that the utmost successes -which our scientific method can accomplish will not enable -us to comprehend more than an infinitesimal fraction of -what there doubtless is to comprehend.<a id="FNanchor_614" href="#Footnote_614" class="fnanchor">614</a> Professor Tyndall -seems to me open to the same charge in a less degree. He -remarks<a id="FNanchor_615" href="#Footnote_615" class="fnanchor">615</a> that we can probably never bring natural phenomena -completely under mathematical laws, because the -approach of our sciences towards completeness may be -asymptotic, so that however far we may go, there may -still remain some facts not subject to scientific explanation. -He thus likens the supply of novel phenomena to a convergent -series, the earlier and larger terms of which have -been successfully disposed of, so that comparatively minor -groups of phenomena alone remain for future investigators -to occupy themselves upon.</p> - -<p>On the contrary, as it appears to me, the supply of new -and unexplained facts is divergent in extent, so that the -more we have explained, the more there is to explain. -The further we advance in any generalisation, the more -numerous and intricate are the exceptional cases still -demanding further treatment. The experiments of Boyle, -Mariotte, Dalton, Gay-Lussac, and others, upon the physical -properties of gases, might seem to have exhausted that -subject by showing that all gases obey the same laws -as regards temperature, pressure, and volume. But in -reality these laws are only approximately true, and the -divergences afford a wide and quite unexhausted field for -further generalisation. The recent discoveries of Professor -Andrews have summed up some of these exceptional facts -under a wider generalisation, but in reality they have -opened to us vast new regions of interesting inquiry, and -they leave wholly untouched the question why one gas -behaves differently from another.</p> - -<p><span class="pagenum" id="Page_754">754</span></p> - -<p>The science of crystallography is that perhaps in which -the most precise and general laws have been detected, but -it would be untrue to assert that it has lessened the area of -future discovery. We can show that each one of the seven -or eight hundred forms of calcite is derivable by geometrical -modifications from an hexagonal prism; but who has -attempted to explain the molecular forces producing these -modifications, or the chemical conditions in which they arise? -The law of isomorphism is an important generalisation, for -it establishes a general resemblance between the forms of -crystallisation of natural classes of elements. But if we -examine a little more closely we find that these forms are -only approximately alike, and the divergence peculiar to -each substance is an unexplained exception.</p> - -<p>By many similar illustrations it might readily be shown -that in whatever direction we extend our investigations -and successfully harmonise a few facts, the result is only -to raise up a host of other unexplained facts. Can any -scientific man venture to state that there is less opening -now for new discoveries than there was three centuries ago? -Is it not rather true that we have but to open a scientific -book and read a page or two, and we shall come to some -recorded phenomenon of which no explanation can yet -be given? In every such fact there is a possible opening -for new discoveries, and it can only be the fault of the -investigator’s mind if he can look around him and find -no scope for the exercise of his faculties.</p> - - -<h3><i>Infinite Incompleteness of the Mathematical Sciences.</i></h3> - -<p>There is one privilege which a certain amount of knowledge -should confer; it is that of becoming aware of the -weakness of our powers compared with the tasks which -they might undertake if stronger. To the poor savage who -cannot count twenty the arithmetical accomplishments of -the schoolboy are miraculously great. The schoolboy cannot -comprehend the vastly greater powers of the student, who -has acquired facility in algebraic processes. The student -can but look with feelings of surprise and reverence at the -powers of a Newton or a Laplace. But the question at -once suggests itself, Do the powers of the highest human -intellect bear a finite ratio to the things which are to be<span class="pagenum" id="Page_755">755</span> -understood and calculated? How many further steps must -we take in the rise of mental ability and the extension of -mathematical methods before we begin to exhaust the -knowable?</p> - -<p>I am inclined to find fault with mathematical writers -because they often exult in what they can accomplish, and -omit to point out that what they do is but an infinitely -small part of what might be done. They exhibit a general -inclination, with few exceptions, not to do so much as -mention the existence of problems of an impracticable -character. This may be excusable as far as the immediate -practical result of their researches is in question, but the -custom has the effect of misleading the general public into -the fallacious notion that mathematics is a <i>perfect</i> science, -which accomplishes what it undertakes in a complete -manner. On the contrary, it may be said that if a mathematical -problem were selected by chance out of the whole -number which might be proposed, the probability is infinitely -slight that a human mathematician could solve it. -Just as the numbers we can count are nothing compared with -the numbers which might exist, so the accomplishments -of a Laplace or a Lagrange are, as it were, the little corner -of the multiplication-table, which has really an infinite -extent.</p> - -<p>I have pointed out that the rude character of our observations -prevents us from being aware of the greater -number of effects and actions in nature. It must be added -that, if we perceive them, we should usually be incapable -of including them in our theories from want of mathematical -power. Some persons may be surprised that -though nearly two centuries have elapsed since the time -of Newton’s discoveries, we have yet no general theory of -molecular action. Some approximations have been made -towards such a theory. Joule and Clausius have measured -the velocity of gaseous atoms, or even determined the -average distance between the collisions of atom and atom. -Thomson has approximated to the number of atoms in a -given bulk of substance. Rankine has formed some reasonable -hypotheses as to the actual constitution of atoms. -It would be a mistake to suppose that these ingenious -results of theory and experiment form any appreciable -approach to a complete solution of molecular motions.<span class="pagenum" id="Page_756">756</span> -There is every reason to believe, judging from the spectra -of the elements, their atomic weights and other data, that -chemical atoms are very complicated structures. An atom -of pure iron is probably a far more complicated system -than that of the planets and their satellites. A compound -atom may perhaps be compared with a stellar system, each -star a minor system in itself. The smallest particle of -solid substance will consist of a great number of such stellar -systems united in regular order, each bounded by the other, -communicating with it in some manner yet wholly incomprehensible. -What are our mathematical powers in comparison -with this problem?</p> - -<p>After two centuries of continuous labour, the most gifted -men have succeeded in calculating the mutual effects of -three bodies each upon the other, under the simple -hypothesis of the law of gravity. Concerning these calculations -we must further remember that they are purely -approximate, and that the methods would not apply where -four or more bodies are acting, and all produce considerable -effects upon each other. There is reason to believe that -each constituent of a chemical atom goes through an orbit -in the millionth part of the twinkling of an eye. In each -revolution it is successively or simultaneously under the -influence of many other constituents, or possibly comes into -collision with them. It is no exaggeration to say that -mathematicians have the least notion of the way in which -they could successfully attack so difficult a problem of -forces and motions. As Herschel has remarked,<a id="FNanchor_616" href="#Footnote_616" class="fnanchor">616</a> each of -these particles is for ever solving differential equations, -which, if written out in full, might belt the earth.</p> - -<p>Some of the most extensive calculations ever made -were those required for the reduction of the measurements -executed in the course of the Trigonometrical Survey of -Great Britain. The calculations arising out of the principal -triangulation occupied twenty calculators during three or -four years, in the course of which the computers had to -solve simultaneous equations involving seventy-seven -unknown quantities. The reduction of the levellings -required the solution of a system of ninety-one equations. -But these vast calculations present no approach whatever to<span class="pagenum" id="Page_757">757</span> -what would be requisite for the complete treatment of any -one physical problem. The motion of glaciers is supposed -to be moderately well understood in the present day. A -glacier is a viscid, slowly yielding mass, neither absolutely -solid nor absolutely rigid, but it is expressly remarked by -Forbes,<a id="FNanchor_617" href="#Footnote_617" class="fnanchor">617</a> that not even an approximate solution of the -mathematical conditions of such a moving mass can yet be -possible. “Every one knows,” he says, “that such problems -are beyond the compass of exact mathematics;” but though -mathematicians may know this, they do not often enough -impress that knowledge on other people.</p> - -<p>The problems which are solved in our mathematical -books consist of a small selection of those which happen -from peculiar conditions to be solvable. But the very -simplest problem in appearance will often give rise to -impracticable calculations. Mr. Todhunter<a id="FNanchor_618" href="#Footnote_618" class="fnanchor">618</a> seems to blame -Condorcet, because in one of his memoirs he mentions a -problem to solve which would require a great and impracticable -number of successive integrations. Now, if our -mathematical sciences are to cope with the problems which -await solution, we must be prepared to effect an unlimited -number of successive integrations; yet at present, and -almost beyond doubt for ever, the probability that an -integration taken haphazard will come within our powers -is exceedingly small.</p> - -<p>In some passages of that remarkable work, the <i>Ninth -Bridgewater Treatise</i> (pp. 113–115), Babbage has pointed -out that if we had power to follow and detect the minutest -effects of any disturbance, each particle of existing matter -would furnish a register of all that has happened. “The -track of every canoe—of every vessel that has yet disturbed -the surface of the ocean, whether impelled by manual force -or elemental power, remains for ever registered in the future -movement of all succeeding particles which may occupy its -place. The furrow which it left is, indeed, instantly filled -up by the closing waters; but they draw after them other -and larger portions of the surrounding element, and these -again, once moved, communicate motion to others in endless -succession.” We may even say that “The air itself is one -vast library, on whose pages are for ever written all that<span class="pagenum" id="Page_758">758</span> -man has ever said or even whispered. There, in their -mutable but unerring characters, mixed with the earliest -as well as the latest sighs of mortality, stand for ever -recorded, vows unredeemed, promises unfulfilled, perpetuating -in the united movements of each particle the -testimony of man’s changeful will.”</p> - -<p>When we read reflections such as these, we may congratulate -ourselves that we have been endowed with minds -which, rightly employed, can form some estimate of their -incapacity to trace out and account for all that proceeds -in the simpler actions of material nature. It ought to be -added that, wonderful as is the extent of physical phenomena -open to our investigation, intellectual phenomena are -yet vastly more extensive. Of this I might present one -satisfactory proof were space available by pointing out that -the mathematical functions employed in the calculations -of physical science form an infinitely small fraction of the -functions which might be invented. Common trigonometry -consists of a great series of useful formulæ, all of which arise -out of the relation of the sine and cosine expressed in one -equation, sin <sup>2</sup><i>x</i> + cos <sup>2</sup><i>x</i> = 1. But this is not the only -trigonometry which may exist; mathematicians also recognise -hyperbolic trigonometry, of which the fundamental -equation is cos <sup>2</sup><i>x</i> - sin <sup>2</sup><i>x</i> = 1. De Morgan has pointed -out that the symbols of ordinary algebra form but three -of an interminable series of conceivable systems.<a id="FNanchor_619" href="#Footnote_619" class="fnanchor">619</a> As the -logarithmic operation is to addition or addition to multiplication, -so is the latter to a higher operation, and so on -without limit.</p> - -<p>We may rely upon it that immense, and to us inconceivable, -advances will be made by the human intellect, in -the absence of any catastrophe to the species or the globe. -Within historical periods we can trace the rise of mathematical -science from its simplest germs. We can prove -our descent from ancestors who counted only on their -fingers. How infinitely is a Newton or a Laplace above -those simple savages. Pythagoras is said to have sacrificed -a hecatomb when he discovered the forty-seventh proposition -of Euclid, and the occasion was worthy of the sacrifice. -Archimedes was beside himself when he first perceived<span class="pagenum" id="Page_759">759</span> -his beautiful mode of determining specific gravities. Yet -these great discoveries are the commonplaces of our school -books. Step by step we can trace upwards the acquirement -of new mental powers. What could be more wonderful -than Napier’s discovery of logarithms, a new mode of -calculation which has multiplied perhaps a hundredfold -the working powers of every computer, and has rendered -easy calculations which were before impracticable? Since -the time of Newton and Leibnitz worlds of problems have -been solved which before were hardly conceived as matters -of inquiry. In our own day extended methods of mathematical -reasoning, such as the system of quaternions, have -been brought into existence. What intelligent man will -doubt that the recondite speculations of a Cayley, a Sylvester, -or a Clifford may lead to some new development of -new mathematical power, at the simplicity of which a -future age will wonder, and yet wonder more that to us they -were so dark and difficult. May we not repeat the words -of Seneca: “Veniet tempus, quo ista quæ nunc latent, in -lucem dies extrahat, et longioris ævi diligentia: ad inquisitionem -tantorum ætas una non sufficit. Veniet tempus, -quo posteri nostri tam aperta nos nescisse mirentur.”</p> - - -<h3><i>The Reign of Law in Mental and Social Phenomena.</i></h3> - -<p>After we pass from the so-called physical sciences to -those which attempt to investigate mental and social -phenomena, the same general conclusions will hold true. -No one will be found to deny that there are certain uniformities -of thinking and acting which can be detected -in reasoning beings, and so far as we detect such laws -we successfully apply scientific method. But those who -attempt to establish social or moral sciences soon become -aware that they are dealing with subjects of enormous -perplexity. Take as an instance the science of political -economy. If a science at all, it must be a mathematical -science, because it deals with quantities of commodities. -But as soon as we attempt to draw out the equations -expressing the laws of demand and supply, we discover -that they have a complexity entirely surpassing our powers -of mathematical treatment. We may lay down the general -form of the equations, expressing the demand and supply<span class="pagenum" id="Page_760">760</span> -for two or three commodities among two or three trading -bodies, but all the functions involved are so complicated in -character that there is not much fear of scientific method -making rapid progress in this direction. If such be the -prospects of a comparatively formal science, like political -economy, what shall we say of moral science? Any -complete theory of morals must deal with quantities of -pleasure and pain, as Bentham pointed out, and must sum -up the general tendency of each kind of action upon the -good of the community. If we are to apply scientific -method to morals, we must have a calculus of moral effects, -a kind of physical astronomy investigating the mutual perturbations -of individuals. But as astronomers have not -yet fully solved the problem of three gravitating bodies, -when shall we have a solution of the problem of three -moral bodies?</p> - -<p>The sciences of political economy and morality are comparatively -abstract and general, treating mankind from -simple points of view, and attempting to detect general -principles of action. They are to social phenomena what -the abstract sciences of chemistry, heat, and electricity -are to the concrete science of meteorology. Before we can -investigate the actions of any aggregate of men, we must -have fairly mastered all the more abstract sciences applying -to them, somewhat in the way that we have acquired a -fair comprehension of the simpler truths of chemistry and -physics. But all our physical sciences do not enable us to -predict the weather two days hence with any great probability, -and the general problem of meteorology is almost -unattempted as yet. What shall we say then of the general -problem of social science, which shall enable us to predict -the course of events in a nation?</p> - -<p>Several writers have proposed to lay the foundations of -the science of history. Buckle undertook to write the -<i>History of Civilisation in England</i>, and to show how the -character of a nation could be explained by the nature of -the climate and the fertility of the soil. He omitted to -explain the contrast between the ancient Greek nation and -the present one; there must have been an extraordinary -revolution in the climate or the soil. Auguste Comte -detected the simple laws of the course of development -through which nations pass. There are always three<span class="pagenum" id="Page_761">761</span> -phases of intellectual condition,—the theological, the -metaphysical, and the positive; applying this general -law of progress to concrete cases, Comte was enabled -to predict that in the hierarchy of European nations, -Spain would necessarily hold the highest place. Such -are the parodies of science offered to us by the <i>positive</i> -philosophers.</p> - -<p>A science of history in the true sense of the term is -an absurd notion. A nation is not a mere sum of individuals -whom we can treat by the method of averages; -it is an organic whole, held together by ties of infinite -complexity. Each individual acts and re-acts upon his -smaller or greater circle of friends, and those who acquire -a public position exert an influence on much larger sections -of the nation. There will always be a few great leaders -of exceptional genius or opportunities, the unaccountable -phases of whose opinions and inclinations sway the whole -body. From time to time arise critical situations, battles, -delicate negotiations, internal disturbances, in which the -slightest incidents may change the course of history. A -rainy day may hinder a forced march, and change the course -of a campaign; a few injudicious words in a despatch may -irritate the national pride; the accidental discharge of a -gun may precipitate a collision the effects of which will -last for centuries. It is said that the history of Europe -depended at one moment upon the question whether the -look-out man upon Nelson’s vessel would or would not -descry a ship of Napoleon’s expedition to Egypt which was -passing not far off. In human affairs, then, the smallest -causes may produce the greatest effects, and the real application -of scientific method is out of the question.</p> - - -<h3><i>The Theory of Evolution.</i></h3> - -<p>Profound philosophers have lately generalised concerning -the production of living forms and the mental and moral -phenomena regarded as their highest development. Herbert -Spencer’s theory of evolution purports to explain the origin -of all specific differences, so that not even the rise of a -Homer or a Beethoven would escape from his broad theories. -The homogeneous is unstable and must differentiate -itself, says Spencer, and hence comes the variety of human<span class="pagenum" id="Page_762">762</span> -institutions and characters. In order that a living form -shall continue to exist and propagate its kind, says Darwin, -it must be suitable to its circumstances, and the most -suitable forms will prevail over and extirpate those which -are less suitable. From these fruitful ideas are developed -theories of evolution and natural selection which go far -towards accounting for the existence of immense numbers -of living creatures—plants, and animals. Apparent adaptations -of organs to useful purposes, which Paley regarded -as distinct products of creative intelligence, are now seen -to follow as natural effects of a constantly acting tendency. -Even man, according to these theories, is no distinct creation, -but rather an extreme case of brain development. -His nearest cousins are the apes, and his pedigree extends -backwards until it joins that of the lowliest zoophytes.</p> - -<p>The theories of Darwin and Spencer are doubtless not -demonstrated; they are to some extent hypothetical, just -as all the theories of physical science are to some extent -hypothetical, and open to doubt. Judging from the -immense numbers of diverse facts which they harmonise -and explain, I venture to look upon the theories of evolution -and natural selection in their main features as two of -the most probable hypotheses ever proposed. I question -whether any scientific works which have appeared since the -<i>Principia</i> of Newton are comparable in importance with -those of Darwin and Spencer, revolutionising as they do all -our views of the origin of bodily, mental, moral, and social -phenomena.</p> - -<p>Granting all this, I cannot for a moment admit that the -theory of evolution will destroy theology. That theory -embraces several laws or uniformities which are observed -to be true in the production of living forms; but these laws -do not determine the size and figure of living creatures, any -more than the law of gravitation determines the magnitudes -and distances of the planets. Suppose that Darwin is -correct in saying that man is descended from the Ascidians: -yet the precise form of the human body must have been -influenced by an infinite train of circumstances affecting -the reproduction, growth, and health of the whole chain of -intermediate beings. No doubt, the circumstances being -what they were, man could not be otherwise than he is, and -if in any other part of the universe an exactly similar earth,<span class="pagenum" id="Page_763">763</span> -furnished with exactly similar germs of life, existed, a -race must have grown up there exactly similar to the -human race.</p> - -<p>By a different distribution of atoms in the primeval world -a different series of living forms on this earth would have -been produced. From the same causes acting according to -the same laws, the same results will follow; but from -different causes acting according to the same laws, different -results will follow. So far as we can see, then, infinitely -diverse living creatures might have been created consistently -with the theory of evolution, and the precise -reason why we have a backbone, two hands with opposable -thumbs, an erect stature, a complex brain, about 223 bones, -and many other peculiarities, is only to be found in the -original act of creation. I do not, any less than Paley, -believe that the eye of man manifests design. I believe -that the eye was gradually developed, and we can in fact -trace its gradual development from the first germ of a nerve -affected by light-rays in some simple zoophyte. In proportion -as the eye became a more accurate instrument of -vision, it enabled its possessor the better to escape destruction, -but the ultimate result must have been contained in -the aggregate of the causes, and these causes, as far as we -can see, were subject to the arbitrary choice of the Creator.</p> - -<p>Although Agassiz was clearly wrong in holding that -every species of living creature appeared on earth by the -immediate intervention of the Creator, which would amount -to saying that no laws of connection between forms are -discoverable, yet he seems to be right in asserting that -living forms are distinct from those produced by purely -physical causes. “The products of what are commonly -called physical agents,” he says,<a id="FNanchor_620" href="#Footnote_620" class="fnanchor">620</a> “are everywhere the -same (<i>i.e.</i> upon the whole surface of the earth), and have -always been the same (<i>i.e.</i> during all geological periods); -while organised beings are everywhere different and have -differed in all ages. Between two such series of phenomena -there can be no causal or genetic connection.” Living forms -as we now regard them are essentially variable, but from -constant mechanical causes constant effects would ensue. -If vegetable cells are formed on geometrical principles<span class="pagenum" id="Page_764">764</span> -being first spherical, and then by mutual compression -dodecahedral, then all cells should have similar forms. In -the Foraminifera and some other lowly organisms, we seem -to observe the production of complex forms on geometrical -principles. But from similar causes acting according to -similar laws only similar results could be produced. If -the original life germ of each creature is a simple particle -of protoplasm, unendowed with any distinctive forces, then -the whole of the complex phenomena of animal and vegetable -life are effects without causes. Protoplasm may be -chemically the same substance, and the germ-cell of a man -and of a fish may be apparently the same, so far as the -microscope can decide; but if certain cells produce men, -and others as uniformly produce a species of fish, there -must be a hidden constitution determining the extremely -different results. If this were not so, the generation of -every living creature from the uniform germ would have -to be regarded as a distinct act of creation.</p> - -<p>Theologians have dreaded the establishment of the -theories of Darwin and Huxley and Spencer, as if they -thought that those theories could explain everything upon -the purest mechanical and material principles, and exclude -all notions of design. They do not see that those theories -have opened up more questions than they have closed. -The doctrine of evolution gives a complete explanation of -no single living form. While showing the general principles -which prevail in the variation of living creatures, it -only points out the infinite complexity of the causes and -circumstances which have led to the present state of things. -Any one of Mr. Darwin’s books, admirable though they all -are, consists but in the setting forth of a multitude of -indeterminate problems. He proves in the most beautiful -manner that each flower of an orchid is adapted to some -insect which frequents and fertilises it, and these adaptations -are but a few cases of those immensely numerous ones -which have occurred in the lives of plants and animals. -But why orchids should have been formed so differently -from other plants, why anything, indeed, should be as it is, -rather than in some of the other infinitely numerous possible -modes of existence, he can never show. The origin of everything -that exists is wrapped up in the past history of the -universe. At some one or more points in past time there<span class="pagenum" id="Page_765">765</span> -must have been arbitrary determinations which led to the -production of things as they are.</p> - - -<h3><i>Possibility of Divine Interference.</i></h3> - -<p>I will now draw the reader’s attention to pages 149 to 152. -I there pointed out that all inductive inference involves -the assumption that our knowledge of what exists is complete, -and that the conditions of things remain unaltered -between the time of our experience and the time to which -our inferences refer. Recurring to the illustration of a -ballot-box, employed in the chapter on the inverse method -of probabilities, we assume when predicting the probable -nature of the next drawing, firstly, that our previous -drawings have been sufficiently numerous to give us -knowledge of the contents of the box; and, secondly, that -no interference with the ballot-box takes place between -the previous and the next drawings. The results yielded -by the theory of probability are quite plain. No finite -number of casual drawings can give us sure knowledge of -the contents of the box, so that, even in the absence of all -disturbance, our inferences are merely the best which can -be made, and do not approach to infallibility. If, however, -interference be possible, even the theory of probability -ceases to be applicable, for, the amount and nature of that -interference being arbitrary and unknown, there ceases to -be any connection between premises and conclusion. Many -years of reflection have not enabled me to see the way of -avoiding this hiatus in scientific certainty. The conclusions -of scientific inference appear to be always of a hypothetical -and provisional nature. Given certain experience, the -theory of probability yields us the true interpretation of -that experience and is the surest guide open to us. But -the best calculated results which it can give are never -absolute probabilities; they are purely relative to the extent -of our information. It seems to be impossible for us to -judge how far our experience gives us adequate information -of the universe as a whole, and of all the forces and phenomena -which can have place therein.</p> - -<p>I feel that I cannot in the space remaining at my command -in the present volume, sufficiently follow out the -lines of thought suggested, or define with precision my<span class="pagenum" id="Page_766">766</span> -own conclusions. This chapter contains merely <i>Reflections</i> -upon subjects of so weighty a character that I should -myself wish for many years—nay for more than a lifetime -of further reflection. My purpose, as I have repeatedly -said, is the purely negative one of showing that atheism -and materialism are no necessary results of scientific -method. From the preceding reviews of the value of our -scientific knowledge, I draw one distinct conclusion, that -we cannot disprove the possibility of Divine interference -in the course of nature. Such interference might arise, so -far as our knowledge extends, in two ways. It might -consist in the disclosure of the existence of some agent or -spring of energy previously unknown, but which effects a -given purpose at a given moment. Like the pre-arranged -change of law in Babbage’s imaginary calculating machine, -there may exist pre-arranged surprises in the order of -nature, as it presents itself to us. Secondly, the same -Power, which created material nature, might, so far as -I can see, create additions to it, or annihilate portions -which do exist. Such events are in a certain sense inconceivable -to us; yet they are no more inconceivable than -the existence of the world as it is. The indestructibility of -matter, and the conservation of energy, are very probable -scientific hypotheses, which accord satisfactorily with experiments -of scientific men during a few years past, but it -would be gross misconception of scientific inference to -suppose that they are certain in the sense that a proposition -in geometry is certain. Philosophers no doubt hold -that <i>de nihilo nihil fit</i>, that is to say, their senses give them -no means of imagining to the mind how creation can take -place. But we are on the horns of a trilemma; we must -either deny that anything exists, or we must allow that it -was created out of nothing at some moment of past time, -or that it existed from eternity. The first alternative is -absurd; the other two seem to me equally conceivable.</p> - - -<h3><i>Conclusion.</i></h3> - -<p>It may seem that there is one point where our speculations -must end, namely where contradiction begins. The -laws of Identity and Difference and Duality were the<span class="pagenum" id="Page_767">767</span> -foundations from which we started, and they are, so far as -I can see, the foundations which we can never quit without -tottering. Scientific Method must begin and end with the -laws of thought, but it does not follow that it will save us -from encountering inexplicable, and at least apparently -contradictory results. The nature of continuous quantity -leads us into extreme difficulties. Any finite space is -composed of an infinite number of infinitely small spaces, -each of which, again, is composed of an infinite number of -spaces of a second order of smallness; these spaces of the -second order are composed, again, of infinitely small -spaces of the third order. Even these spaces of the third -order are not absolute geometrical points answering to -Euclid’s definition of a point, as position without magnitude. -Go on as far as we will, in the subdivision of -continuous quantity, yet we never get down to the absolute -point. Thus scientific method leads us to the -inevitable conception of an infinite series of successive -orders of infinitely small quantities. If so, there is nothing -impossible in the existence of a myriad universes within -the compass of a needle’s point, each with its stellar systems, -and its suns and planets, in number and variety -unlimited. Science does nothing to reduce the number -of strange things that we may believe. When fairly -pursued it makes absurd drafts upon our powers of comprehension -and belief.</p> - -<p>Some of the most precise and beautiful theorems in -mathematical science seem to me to involve apparent contradiction. -Can we imagine that a point moving along a -perfectly straight line towards the west would ever get -round to the east and come back again, having performed, -as it were, a circuit through infinite space, yet without -ever diverging from a perfectly straight direction? Yet -this is what happens to the intersecting point of two -straight lines in the same plane, when one line revolves. -The same paradox is exhibited in the hyperbola regarded -as an infinite ellipse, one extremity of which has passed to -an infinite distance and come back in the opposite direction. -A varying quantity may change its sign by passing either -through zero or through infinity. In the latter case there -must be one intermediate value of the variable for which -the variant is indifferently negative infinity and positive<span class="pagenum" id="Page_768">768</span> -infinity. Professor Clifford tells me that he has found a -mathematical function which approaches infinity as the -variable approaches a certain limit; yet at the limit the -function is finite! Mathematicians may shirk difficulties, -but they cannot make such results of mathematical principles -appear otherwise than contradictory to our common -notions of space.</p> - -<p>The hypothesis that there is a Creator at once all-powerful -and all-benevolent is pressed, as it must seem to every -candid investigator, with difficulties verging closely upon -logical contradiction. The existence of the smallest amount -of pain and evil would seem to show that He is either not -perfectly benevolent, or not all-powerful. No one can -have lived long without experiencing sorrowful events -of which the significance is inexplicable. But if we -cannot succeed in avoiding contradiction in our notions of -elementary geometry, can we expect that the ultimate -purposes of existence shall present themselves to us with -perfect clearness? I can see nothing to forbid the notion -that in a higher state of intelligence much that is now -obscure may become clear. We perpetually find ourselves -in the position of finite minds attempting infinite problems, -and can we be sure that where we see contradiction, an -infinite intelligence might not discover perfect logical -harmony?</p> - -<p>From science, modestly pursued, with a due consciousness -of the extreme finitude of our intellectual powers, -there can arise only nobler and wider notions of the purpose -of Creation. Our philosophy will be an affirmative -one, not the false and negative dogmas of Auguste Comte, -which have usurped the name, and misrepresented the -tendencies of a true <i>positive philosophy</i>. True science will -not deny the existence of things because they cannot be -weighed and measured. It will rather lead us to believe -that the wonders and subtleties of possible existence surpass -all that our mental powers allow us clearly to perceive. -The study of logical and mathematical forms has convinced -me that even space itself is no requisite condition of conceivable -existence. Everything, we are told by materialists, -must be here or there, nearer or further, before or after. I -deny this, and point to logical relations as my proof.</p> - -<p>There formerly seemed to me to be something mysterious<span class="pagenum" id="Page_769">769</span> -in the denominators of the binomial expansion (p. <a href="#Page_190">190</a>), -which are reproduced in the natural constant ε, or</p> - -<div class="center"> -1 + <span class="nowrap"><span class="fraction2"><span class="fnum2">1</span><span class="bar">/</span><span class="fden2">1</span></span></span> + <span class="nowrap"><span class="fraction2"><span class="fnum2">1</span><span class="bar">/</span><span class="fden2">1 . 2</span></span></span> + <span class="nowrap"><span class="fraction2"><span class="fnum2">1</span><span class="bar">/</span><span class="fden2">1 . 2 . 3</span></span></span> + . . . -</div> - -<p class="ti0">and in many results of mathematical analysis. I now -perceive, as already explained (pp. <a href="#Page_33">33</a>, <a href="#Page_160">160</a>, <a href="#Page_383">383</a>), that they -arise out of the fact that the relations of space do not apply -to the logical conditions governing the numbers of combinations -as contrasted to those of permutations. So far -am I from accepting Kant’s doctrine that space is a -necessary form of thought, that I regard it as an accident, -and an impediment to pure logical reasoning. Material -existences must exist in space, no doubt, but intellectual -existences may be neither in space nor out of space; they -may have no relation to space at all, just as space itself -has no relation to time. For all that I can see, then, there -may be intellectual existences to which both time and -space are nullities.</p> - -<p>Now among the most unquestionable rules of scientific -method is that first law that <i>whatever phenomenon is, is</i>. -We must ignore no existence whatever; we may variously -interpret or explain its meaning and origin, but, if a phenomenon -does exist, it demands some kind of explanation. -If then there is to be competition for scientific recognition, -the world without us must yield to the undoubted -existence of the spirit within. Our own hopes and wishes -and determinations are the most undoubted phenomena -within the sphere of consciousness. If men do act, feel, -and live as if they were not merely the brief products of a -casual conjunction of atoms, but the instruments of a far-reaching -purpose, are we to record all other phenomena -and pass over these? We investigate the instincts of the -ant and the bee and the beaver, and discover that they are -led by an inscrutable agency to work towards a distant -purpose. Let us be faithful to our scientific method, and -investigate also those instincts of the human mind by -which man is led to work as if the approval of a Higher -Being were the aim of life.</p> -<hr class="chap x-ebookmaker-drop"> - -<div class="chapter"> -<p><span class="pagenum" id="Page_771">771</span></p> - -<h2 class="nobreak" id="INDEX">INDEX.</h2> -</div> - -<div class="center"> -<p class="hide"><a id="alpha-table"></a>alpha-table</p> - -<table id="alpha"> -<tr class="center"> - <td class="tac"><div><a href="#IX_A">A</a></div></td> - <td class="tac"><div><a href="#IX_B">B</a></div></td> - <td class="tac"><div><a href="#IX_C">C</a></div></td> - <td class="tac"><div><a href="#IX_D">D</a></div></td> - <td class="tac"><div><a href="#IX_E">E</a></div></td> - <td class="tac"><div><a href="#IX_F">F</a></div></td> - <td class="tac"><div><a href="#IX_G">G</a></div></td> - <td class="tac"><div><a href="#IX_H">H</a></div></td> - <td class="tac"><div><a href="#IX_I">I</a></div></td> - <td class="tac"><div><a href="#IX_J">J</a></div></td> - <td class="tac"><div><a href="#IX_K">K</a></div></td> - <td class="tac"><div><a href="#IX_L">L</a></div></td> - <td class="tac"><div><a href="#IX_M">M</a></div></td> -</tr> -<tr class="center"> - <td class="tac"><div><a href="#IX_N">N</a></div></td> - <td class="tac"><div><a href="#IX_O">O</a></div></td> - <td class="tac"><div><a href="#IX_P">P</a></div></td> - <td class="tac"><div><a href="#IX_Q">Q</a></div></td> - <td class="tac"><div><a href="#IX_R">R</a></div></td> - <td class="tac"><div><a href="#IX_S">S</a></div></td> - <td class="tac"><div><a href="#IX_T">T</a></div></td> - <td class="tac"><div><a href="#IX_U">U</a></div></td> - <td class="tac"><div><a href="#IX_V">V</a></div></td> - <td class="tac"><div><a href="#IX_W">W</a></div></td> - <td class="tac"><div><a href="#IX_X">X</a></div></td> - <td class="tac"><div><a href="#IX_Y">Y</a></div></td> - <td class="tac"><div><a href="#IX_Z">Z</a></div></td> -</tr> -</table> -</div> - -<ul class="index"> -<li class="abet"><span class="alpha"><a id="IX_A"></a><a href="#alpha-table">A</a></span></li> -<li class="ifrst">Abacus, logical, <a href="#Page_104">104</a>;</li> -<li class="isub1">arithmetical, <a href="#Page_107">107</a>;</li> -<li class="isub1">Panchrestus, <a href="#Page_182">182</a>.</li> - -<li class="indx">Aberration of light, <a href="#Page_561">561</a>;</li> -<li class="isub1">systematic, <a href="#Page_547">547</a>.</li> - -<li class="indx">Abscissio infiniti, <a href="#Page_79">79</a>, <a href="#Page_713">713</a>.</li> - -<li class="indx">Abstract terms, <a href="#Page_27">27</a>;</li> -<li class="isub1">number, <a href="#Page_159">159</a>.</li> - -<li class="indx">Abstraction, <a href="#Page_704">704</a>;</li> -<li class="isub1">logical, <a href="#Page_25">25</a>;</li> -<li class="isub1">numerical, <a href="#Page_158">158</a>;</li> -<li class="isub1">of indifferent circumstances, <a href="#Page_97">97</a>.</li> - -<li class="indx">Accademia del Cimento, <a href="#Page_427">427</a>, <a href="#Page_432">432</a>, <a href="#Page_436">436</a>, <a href="#Page_527">527</a>.</li> - -<li class="indx">Accident, logical, <a href="#Page_700">700</a>.</li> - -<li class="indx">Accidental discovery, <a href="#Page_529">529</a>.</li> - -<li class="indx">Achromatic lenses, <a href="#Page_432">432</a>.</li> - -<li class="indx">Actinometer, <a href="#Page_337">337</a>.</li> - -<li class="indx">Adamantine medium, <a href="#Page_605">605</a>, <a href="#Page_751">751</a>.</li> - -<li class="indx">Adjectives, <a href="#Page_14">14</a>, <a href="#Page_30">30</a>, <a href="#Page_31">31</a>, <a href="#Page_35">35</a>;</li> -<li class="isub1">indeterminate, <a href="#Page_41">41</a>.</li> - -<li class="indx">Adrain, of New Brunswick, <a href="#Page_375">375</a>.</li> - -<li class="indx">Affirmation, <a href="#Page_44">44</a>.</li> - -<li class="indx">Agassiz, on genera, <a href="#Page_726">726</a>;</li> -<li class="isub1">on creation of species, <a href="#Page_763">763</a>.</li> - -<li class="indx">Agreement, <a href="#Page_44">44</a>.</li> - -<li class="indx">Airy, Sir George Biddell, on perpetual motion, <a href="#Page_223">223</a>;</li> -<li class="isub1">new property of sphere, <a href="#Page_232">232</a>;</li> -<li class="isub1">pendulum experiments, <a href="#Page_291">291</a>, <a href="#Page_304">304</a>, <a href="#Page_348">348</a>, <a href="#Page_567">567</a>;</li> -<li class="isub1">standard clock, <a href="#Page_353">353</a>;</li> -<li class="isub1">book on <i>Errors of Observation</i>, <a href="#Page_395">395</a>;</li> -<li class="isub1">tides, <a href="#Page_488">488</a>;</li> -<li class="isub1">extra-polation, <a href="#Page_495">495</a>;</li> -<li class="isub1">Thales’ eclipse, <a href="#Page_537">537</a>;</li> -<li class="isub1">interference of light, <a href="#Page_539">539</a>;</li> -<li class="isub1">density of earth, <a href="#Page_291">291</a>.</li> - -<li class="indx">Alchemists, <a href="#Page_505">505</a>;</li> -<li class="isub1">how misled, <a href="#Page_428">428</a>.</li> - -<li class="indx">Algebra, <a href="#Page_123">123</a>, <a href="#Page_155">155</a>, <a href="#Page_164">164</a>;</li> -<li class="isub1">Diophantine, <a href="#Page_631">631</a>.</li> - -<li class="indx">Algebraic, equations, <a href="#Page_123">123</a>;</li> -<li class="isub1">geometry, <a href="#Page_633">633</a>.</li> - -<li class="indx">Allotropic state, <a href="#Page_663">663</a>, <a href="#Page_670">670</a>.</li> - -<li class="indx">Alloys, possible number, <a href="#Page_191">191</a>;</li> -<li class="isub1">properties, <a href="#Page_528">528</a>.</li> - -<li class="indx">Alphabet, the Logical, <a href="#Page_93">93</a>, <a href="#Page_104">104</a>, <a href="#Page_125">125</a>;</li> -<li class="isub1">Morse, <a href="#Page_193">193</a>.</li> - -<li class="indx">Alphabet, permutations of letters of the, <a href="#Page_174">174</a>, <a href="#Page_179">179</a>.</li> - -<li class="indx">Alphabetic indexes, <a href="#Page_714">714</a>.</li> - -<li class="indx">Alternative relations, <a href="#Page_67">67</a>;</li> -<li class="isub1">exclusive and unexclusive, <a href="#Page_205">205</a>.</li> - -<li class="indx">Ampère, electricity, <a href="#Page_547">547</a>;</li> -<li class="isub1">classification, <a href="#Page_679">679</a>.</li> - -<li class="indx">Anagrams, <a href="#Page_128">128</a>.</li> - -<li class="indx">Analogy, <a href="#Page_627">627</a>;</li> -<li class="isub1">of logical and numerical terms, <a href="#Page_160">160</a>;</li> -<li class="isub1">and generalisation, <a href="#Page_596">596</a>;</li> -<li class="isub1">in mathematical sciences, <a href="#Page_631">631</a>;</li> -<li class="isub1">in theory of undulations, <a href="#Page_635">635</a>;</li> -<li class="isub1">in astronomy, <a href="#Page_638">638</a>;</li> -<li class="isub1">failure of, <a href="#Page_641">641</a>.</li> - -<li class="indx">Analysis, logical, <a href="#Page_122">122</a>.</li> - -<li class="indx">Andrews, Prof. Thomas, experiments on gaseous state, <a href="#Page_71">71</a>, <a href="#Page_613">613</a>, <a href="#Page_665">665</a>, <a href="#Page_753">753</a>.</li> - -<li class="indx">Angström, on spectrum, <a href="#Page_424">424</a>.</li> - -<li class="indx">Angular magnitude, <a href="#Page_305">305</a>, <a href="#Page_306">306</a>, <a href="#Page_326">326</a>.</li> - -<li class="indx">Antecedent defined, <a href="#Page_225">225</a>.</li> - -<li class="indx">Anticipation of Nature, <a href="#Page_509">509</a>.</li> - -<li class="indx">Anticipations, of Principle of Substitution, <a href="#Page_21">21</a>;</li> -<li class="isub1">of electric telegraph, <a href="#Page_671">671</a>.</li> - -<li class="indx">Apparent, equality, <a href="#Page_275">275</a>;</li> -<li class="isub1">sequence of events, <a href="#Page_409">409</a>.</li> - -<li class="indx">Approximation, theory of, <a href="#Page_456">456</a>;</li> -<li class="isub1">to exact laws, <a href="#Page_462">462</a>;</li> -<li class="isub1">mathematical principles of, <a href="#Page_471">471</a>;</li> -<li class="isub1">arithmetic of, <a href="#Page_481">481</a>.</li> - -<li class="indx">Aqueous vapour, <a href="#Page_500">500</a>.</li> - -<li class="indx">Aquinas, on disjunctive propositions, <a href="#Page_69">69</a>.<span class="pagenum" id="Page_772">772</span></li> - -<li class="indx">Arago, photometer, <a href="#Page_288">288</a>;</li> -<li class="isub1">rotating disc, <a href="#Page_535">535</a>;</li> -<li class="isub1">his philosophic character, <a href="#Page_592">592</a>.</li> - -<li class="indx">Archimedes, <i>De Arenæ Numero</i>, <a href="#Page_195">195</a>;</li> -<li class="isub1">centre of gravity, <a href="#Page_363">363</a>.</li> - -<li class="indx">Arcual unit, <a href="#Page_306">306</a>, <a href="#Page_330">330</a>.</li> - -<li class="indx">Argyll, Duke of, <a href="#Page_741">741</a>.</li> - -<li class="indx">Aristarchus on sun’s and moon’s distances, <a href="#Page_294">294</a>.</li> - -<li class="indx">Aristotelian doctrines, <a href="#Page_666">666</a>.</li> - -<li class="indx">Aristotle, dictum, <a href="#Page_21">21</a>;</li> -<li class="isub1">singular terms, <a href="#Page_39">39</a>;</li> -<li class="isub1">overlooked simple identities, <a href="#Page_40">40</a>;</li> -<li class="isub1">order of premises, <a href="#Page_114">114</a>;</li> -<li class="isub1">logical error, <a href="#Page_117">117</a>;</li> -<li class="isub1">definition of time, <a href="#Page_307">307</a>;</li> -<li class="isub1">on science, <a href="#Page_595">595</a>;</li> -<li class="isub1">on white swans, <a href="#Page_666">666</a>.</li> - -<li class="indx">Arithmetic, reasoning in, <a href="#Page_167">167</a>;</li> -<li class="isub1">of approximate quantities, <a href="#Page_481">481</a>.</li> - -<li class="indx">Arithmetical triangle, <a href="#Page_93">93</a>, <a href="#Page_143">143</a>, <a href="#Page_182">182</a>, <a href="#Page_202">202</a>, <a href="#Page_378">378</a>, - <a href="#Page_383">383</a>;</li> -<li class="isub1">diagram of, <a href="#Page_184">184</a>;</li> -<li class="isub1">connection with Logical Alphabet, <a href="#Page_189">189</a>;</li> -<li class="isub1">in probability, <a href="#Page_208">208</a>.</li> - -<li class="indx">Asteroids, discovery of, <a href="#Page_412">412</a>, <a href="#Page_748">748</a>.</li> - -<li class="indx">Astronomy, physical, <a href="#Page_459">459</a>.</li> - -<li class="indx">Atmospheric tides, <a href="#Page_553">553</a>.</li> - -<li class="indx">Atomic theory, <a href="#Page_662">662</a>.</li> - -<li class="indx">Atomic weights, <a href="#Page_563">563</a>.</li> - -<li class="indx">Atoms, size of, <a href="#Page_195">195</a>;</li> -<li class="isub1">impossibility of observing, <a href="#Page_406">406</a>.</li> - -<li class="indx">Augustin on time, <a href="#Page_307">307</a>.</li> - -<li class="indx">Average, <a href="#Page_359">359</a>, <a href="#Page_360">360</a>;</li> -<li class="isub1">divergence from, <a href="#Page_188">188</a>;</li> -<li class="isub1">etymology of, <a href="#Page_363">363</a>.</li> - -<li class="indx">Axes of crystals, <a href="#Page_686">686</a>.</li> - -<li class="indx">Axioms of algebra, <a href="#Page_164">164</a>.</li> - - -<li class="abet"><span class="alpha"><a id="IX_B"></a><a href="#alpha-table">B</a></span></li> -<li class="ifrst">Babbage, Charles, calculating machine, <a href="#Page_107">107</a>, <a href="#Page_231">231</a>, <a href="#Page_743">743</a>;</li> -<li class="isub1">lighthouse signals, <a href="#Page_194">194</a>;</li> -<li class="isub1">natural constants, <a href="#Page_329">329</a>;</li> -<li class="isub1">Mosaic history, <a href="#Page_412">412</a>;</li> -<li class="isub1">universal and general truths, <a href="#Page_646">646</a>;</li> -<li class="isub1">change of law, <a href="#Page_230">230</a>;</li> -<li class="isub1">persistence of effects, <a href="#Page_757">757</a>.</li> - -<li class="indx">Bacon, Francis Lord, <i>Novum Organum</i>, <a href="#Page_107">107</a>;</li> -<li class="isub1">on induction, <a href="#Page_121">121</a>;</li> -<li class="isub1">biliteral cipher, <a href="#Page_193">193</a>;</li> -<li class="isub1">First Aphorism, <a href="#Page_219">219</a>;</li> -<li class="isub1">on causes, <a href="#Page_221">221</a>;</li> -<li class="isub1">Copernican system, <a href="#Page_249">249</a>, <a href="#Page_638">638</a>;</li> -<li class="isub1">deficient powers of senses, <a href="#Page_278">278</a>;</li> -<li class="isub1">observation, <a href="#Page_402">402</a>;</li> -<li class="isub1">Natural History, <a href="#Page_403">403</a>;</li> -<li class="isub1">use of hypothesis, <a href="#Page_506">506</a>;</li> -<li class="isub1">his method, <a href="#Page_507">507</a>;</li> -<li class="isub1"><i>experimentum crucis</i>, <a href="#Page_519">519</a>;</li> -<li class="isub1">error of his method, <a href="#Page_576">576</a>;</li> -<li class="isub1">ostensive, clandestine instances, &c., <a href="#Page_608">608</a>, <a href="#Page_610">610</a>;</li> -<li class="isub1"><i>latens precessus</i>, <a href="#Page_619">619</a>.</li> - -<li class="indx">Bacon, Roger, on the rainbow, <a href="#Page_526">526</a>, <a href="#Page_598">598</a>.</li> - -<li class="indx">Baily, Francis, <a href="#Page_272">272</a>;</li> -<li class="isub1">density of earth, <a href="#Page_342">342</a>, <a href="#Page_566">566</a>;</li> -<li class="isub1">experiments with torsion balance, <a href="#Page_370">370</a>, <a href="#Page_397">397</a>, <a href="#Page_432">432</a>, <a href="#Page_567">567–8</a>;</li> -<li class="isub1">motions of stars, <a href="#Page_572">572</a>.</li> - -<li class="indx">Bain, Alexander, on powers of mind, <a href="#Page_4">4</a>;</li> -<li class="isub1">Mill’s reform of logic, <a href="#Page_227">227</a>.</li> - -<li class="indx">Baker’s poem, <i>The Universe</i>, <a href="#Page_621">621</a>.</li> - -<li class="indx">Balance, use of the chemical, <a href="#Page_292">292</a>, <a href="#Page_351">351</a>, <a href="#Page_354">354</a>, <a href="#Page_369">369</a>;</li> -<li class="isub1">delicacy of, <a href="#Page_304">304</a>;</li> -<li class="isub1">vibrations of, <a href="#Page_369">369</a>.</li> - -<li class="indx">Ballot, Buys, experiment on sound, <a href="#Page_541">541</a>.</li> - -<li class="indx">Ballot-box, simile of, <a href="#Page_150">150</a>, <a href="#Page_251">251–6</a>, <a href="#Page_765">765</a>.</li> - -<li class="indx">Barbara, <a href="#Page_55">55</a>, <a href="#Page_57">57</a>, <a href="#Page_88">88</a>, <a href="#Page_105">105</a>, <a href="#Page_141">141</a>.</li> - -<li class="indx">Baroko, <a href="#Page_85">85</a>.</li> - -<li class="indx">Barometer, <a href="#Page_659">659</a>;</li> -<li class="isub1">Gay Lussac’s standard, <a href="#Page_346">346</a>;</li> -<li class="isub1">variations, <a href="#Page_337">337</a>, <a href="#Page_346">346</a>, <a href="#Page_349">349</a>.</li> - -<li class="indx">Bartholinus on double refraction, <a href="#Page_585">585</a>.</li> - -<li class="indx">Base-line, measurement of, <a href="#Page_304">304</a>.</li> - -<li class="indx">Bauhusius, verses of, <a href="#Page_175">175</a>.</li> - -<li class="indx">Baxendell, Joseph, <a href="#Page_552">552</a>.</li> - -<li class="indx">Beneke, on substitution, <a href="#Page_21">21</a>.</li> - -<li class="indx">Bennet, momentum of light, <a href="#Page_435">435</a>.</li> - -<li class="indx">Bentham, George, <a href="#Page_15">15</a>;</li> -<li class="isub1">bifurcate classification, <a href="#Page_695">695</a>;</li> -<li class="isub1">infima species, <a href="#Page_702">702</a>;</li> -<li class="isub1">works on classification, <a href="#Page_703">703</a>;</li> -<li class="isub1">analytical key to flora, <a href="#Page_712">712</a>.</li> - -<li class="indx">Bentham, Jeremy, on analogy, <a href="#Page_629">629</a>;</li> -<li class="isub1">bifurcate classification, <a href="#Page_703">703</a>.</li> - -<li class="indx">Benzenberg’s experiment, <a href="#Page_388">388</a>.</li> - -<li class="indx">Bernoulli, Daniel, planetary orbits, <a href="#Page_250">250</a>;</li> -<li class="isub1">resisting media and projectiles, <a href="#Page_467">467</a>;</li> -<li class="isub1">vibrations, <a href="#Page_476">476</a>.</li> - -<li class="indx">Bernoulli, James, <a href="#Page_154">154</a>;</li> -<li class="isub1">numbers of, <a href="#Page_124">124</a>;</li> -<li class="isub1">Protean verses, <a href="#Page_175">175</a>;</li> -<li class="isub1"><i>De Arte Conjectandi</i> quoted, <a href="#Page_176">176</a>, <a href="#Page_183">183</a>;</li> -<li class="isub1">on figurate numbers, <a href="#Page_183">183</a>;</li> -<li class="isub1">theorem of, <a href="#Page_209">209</a>;</li> -<li class="isub1">false solution in probability, <a href="#Page_213">213</a>;</li> -<li class="isub1">solution of inverse problem, <a href="#Page_261">261</a>.</li> - -<li class="indx">Bessel, F. W., <a href="#Page_375">375</a>;</li> -<li class="isub1">law of error, <a href="#Page_384">384</a>;</li> -<li class="isub1">formula for periodic variations, <a href="#Page_488">488</a>;</li> -<li class="isub1">use of hypothesis, <a href="#Page_506">506</a>;</li> -<li class="isub1">solar parallax, <a href="#Page_560">560–2</a>;</li> -<li class="isub1">ellipticity of earth, <a href="#Page_565">565</a>;</li> -<li class="isub1">pendulum experiments, <a href="#Page_604">604</a>.</li> - -<li class="indx">Bias, <a href="#Page_393">393</a>, <a href="#Page_402">402</a>.</li> - -<li class="indx">Biela’s comet, <a href="#Page_746">746</a>.</li> - -<li class="indx">Bifurcate classification, <a href="#Page_694">694</a>.<span class="pagenum" id="Page_773">773</span></li> - -<li class="indx">Binomial theorem, <a href="#Page_190">190</a>;</li> -<li class="isub1">discovery of, <a href="#Page_231">231</a>.</li> - -<li class="indx">Biot, on tension of vapour, <a href="#Page_500">500</a>.</li> - -<li class="indx">Blind experiments, <a href="#Page_433">433</a>.</li> - -<li class="indx">Bode’s law, <a href="#Page_147">147</a>, <a href="#Page_257">257</a>, <a href="#Page_660">660</a>.</li> - -<li class="indx">Boethius, quoted, <a href="#Page_33">33</a>;</li> -<li class="isub1">on kinds of mean, <a href="#Page_360">360</a>.</li> - -<li class="indx">Boiling point, <a href="#Page_442">442</a>, <a href="#Page_659">659</a>.</li> - -<li class="indx">Bonnet’s theory of reproduction, <a href="#Page_621">621</a>.</li> - -<li class="indx">Boole, George, on sign of equality, <a href="#Page_15">15</a>;</li> -<li class="isub1">his calculus of logic, <a href="#Page_23">23</a>, <a href="#Page_113">113</a>, <a href="#Page_634">634</a>;</li> -<li class="isub1">on logical terms, <a href="#Page_33">33</a>;</li> -<li class="isub1">law of commutativeness, <a href="#Page_35">35</a>;</li> -<li class="isub1">use of <i>some</i>, <a href="#Page_41">41–2</a>;</li> -<li class="isub1">disjunctive propositions, <a href="#Page_70">70</a>;</li> -<li class="isub1">Venn on his method, <a href="#Page_90">90</a>;</li> -<li class="isub1"><i>Laws of Thought</i>, <a href="#Page_155">155</a>;</li> -<li class="isub1">statistical conditions, <a href="#Page_168">168</a>;</li> -<li class="isub1">propositions numerically definite, <a href="#Page_172">172</a>;</li> -<li class="isub1">on probability, <a href="#Page_199">199</a>;</li> -<li class="isub1">general method in probabilities, <a href="#Page_206">206</a>;</li> -<li class="isub1">Laplace’s solution of inverse problem, <a href="#Page_256">256</a>;</li> -<li class="isub1">law of error, <a href="#Page_377">377</a>.</li> - -<li class="indx">Borda, his repeating circle, <a href="#Page_290">290</a>.</li> - -<li class="indx">Boscovich’s hypothesis, <a href="#Page_512">512</a>.</li> - -<li class="indx">Botany, <a href="#Page_666">666</a>, <a href="#Page_678">678</a>, <a href="#Page_681">681</a>;</li> -<li class="isub1">modes of classification, <a href="#Page_678">678</a>;</li> -<li class="isub1">systematic, <a href="#Page_722">722</a>;</li> -<li class="isub1">nomenclature of, <a href="#Page_727">727</a>.</li> - -<li class="indx">Bowen, Prof. Francis, on inference, <a href="#Page_118">118</a>;</li> -<li class="isub1">classification, <a href="#Page_674">674</a>.</li> - -<li class="indx">Boyle’s, Robert, law of gaseous pressure, <a href="#Page_468">468</a>, <a href="#Page_470">470</a>, <a href="#Page_619">619</a>;</li> -<li class="isub1">on hypothesis, <a href="#Page_510">510</a>;</li> -<li class="isub1">barometer, <a href="#Page_659">659</a>.</li> - -<li class="indx">Bradley, his observations, <a href="#Page_384">384</a>;</li> -<li class="isub1">accuracy of, <a href="#Page_271">271</a>;</li> -<li class="isub1">aberration of light, <a href="#Page_535">535</a>.</li> - -<li class="indx">Bravais, on law of error, <a href="#Page_375">375</a>.</li> - -<li class="indx">Brewer, W. H., <a href="#Page_142">142</a>.</li> - -<li class="indx">Brewster, Sir David, iridescent colours, <a href="#Page_419">419</a>;</li> -<li class="isub1">spectrum, <a href="#Page_429">429</a>;</li> -<li class="isub1">Newton’s theory of colours, <a href="#Page_518">518</a>;</li> -<li class="isub1">refractive indices, <a href="#Page_10">10</a>, <a href="#Page_527">527</a>;</li> -<li class="isub1">optic axes, <a href="#Page_446">446</a>.</li> - -<li class="indx">British Museum, catalogue of, <a href="#Page_717">717</a>.</li> - -<li class="indx">Brodie, Sir B. C., on errors of experiment, <a href="#Page_388">388</a>, <a href="#Page_464">464</a>;</li> -<li class="isub1">ozone, <a href="#Page_663">663</a>.</li> - -<li class="indx">Brown, Thomas, on cause, <a href="#Page_224">224</a>.</li> - -<li class="indx">Buckle, Thomas, on constancy of average, <a href="#Page_656">656</a>;</li> -<li class="isub1">science of history, <a href="#Page_760">760</a>.</li> - -<li class="indx">Buffon, on probability, <a href="#Page_215">215</a>;</li> -<li class="isub1">definition of genius, <a href="#Page_576">576</a>.</li> - -<li class="indx">Bunsen, Robert, spectrum, <a href="#Page_244">244</a>;</li> -<li class="isub1">photometrical researches, <a href="#Page_273">273</a>, <a href="#Page_324">324</a>, <a href="#Page_441">441</a>;</li> -<li class="isub1">calorimeter, <a href="#Page_343">343</a>.</li> - -<li class="indx">Butler, Bishop, on probability, <a href="#Page_197">197</a>.</li> - - -<li class="abet"><span class="alpha"><a id="IX_C"></a><a href="#alpha-table">C</a></span></li> -<li class="ifrst">Calorescence, <a href="#Page_664">664</a>.</li> - -<li class="indx">Camestres, <a href="#Page_84">84</a>.</li> - -<li class="indx">Canton, on compressibility of water, <a href="#Page_338">338</a>.</li> - -<li class="indx">Carbon, <a href="#Page_640">640</a>, <a href="#Page_728">728</a>;</li> -<li class="isub1">conductibility of, <a href="#Page_442">442</a>.</li> - -<li class="indx">Cardan, on inclined plane, <a href="#Page_501">501</a>.</li> - -<li class="indx">Cards, combinations of, <a href="#Page_190">190</a>.</li> - -<li class="indx">Carlini, pendulum experiments, <a href="#Page_567">567</a>.</li> - -<li class="indx">Carnot’s law, <a href="#Page_606">606</a>.</li> - -<li class="indx">Carpenter, Dr. W. B., <a href="#Page_412">412</a>.</li> - -<li class="indx">Catalogues, art of making, <a href="#Page_714">714</a>.</li> - -<li class="indx">Cauchy, undulatory theory, <a href="#Page_468">468</a>.</li> - -<li class="indx">Cause, <a href="#Page_220">220</a>;</li> -<li class="isub1">definition of, <a href="#Page_224">224</a>.</li> - -<li class="indx">Cavendish’s experiment, <a href="#Page_272">272</a>, <a href="#Page_566">566</a>.</li> - -<li class="indx">Cayley, Professor, <a href="#Page_145">145</a>;</li> -<li class="isub1">on mathematical tables, <a href="#Page_331">331</a>;</li> -<li class="isub1">numbers of chemical compounds, <a href="#Page_544">544</a>.</li> - -<li class="indx">Celarent, <a href="#Page_55">55</a>.</li> - -<li class="indx">Centre of gravity, <a href="#Page_363">363</a>, <a href="#Page_524">524</a>;</li> -<li class="isub1">of oscillation, gyration, &c., <a href="#Page_364">364</a>.</li> - -<li class="indx">Centrobaric bodies, <a href="#Page_364">364</a>.</li> - -<li class="indx">Certainty, <a href="#Page_235">235</a>, <a href="#Page_266">266</a>.</li> - -<li class="indx">Cesare, <a href="#Page_85">85</a>.</li> - -<li class="indx">Chalmers, on collocations, <a href="#Page_740">740</a>.</li> - -<li class="indx">Chance, <a href="#Page_198">198</a>.</li> - -<li class="indx">Character, human, <a href="#Page_733">733</a>.</li> - -<li class="indx">Characteristics, <a href="#Page_708">708</a>.</li> - -<li class="indx">Chauvenet, Professor W., on treatment of observations, <a href="#Page_391">391</a>.</li> - -<li class="indx">Chemical affinity, <a href="#Page_614">614</a>;</li> -<li class="isub1">analysis, <a href="#Page_713">713</a>.</li> - -<li class="indx">Chladni, <a href="#Page_446">446</a>.</li> - -<li class="indx">Chloroform, discovery of, <a href="#Page_531">531</a>.</li> - -<li class="indx">Chronoscope, <a href="#Page_616">616</a>.</li> - -<li class="indx">Cipher, <a href="#Page_32">32</a>;</li> -<li class="isub1">Bacon’s, <a href="#Page_193">193</a>.</li> - -<li class="indx">Circle, circumference of, <a href="#Page_389">389</a>.</li> - -<li class="indx">Circumstances, indifferent, <a href="#Page_419">419</a>.</li> - -<li class="indx">Circumstantial evidence, <a href="#Page_264">264</a>.</li> - -<li class="indx">Clairaut, <a href="#Page_650">650</a>, <a href="#Page_651">651</a>;</li> -<li class="isub1">on gravity, <a href="#Page_463">463</a>.</li> - -<li class="indx">Classes, <a href="#Page_25">25</a>;</li> -<li class="isub1">problem of common part of three, <a href="#Page_170">170</a>.</li> - -<li class="indx">Classification, <a href="#Page_673">673</a>;</li> -<li class="isub1">involving induction, <a href="#Page_675">675</a>;</li> -<li class="isub1">multiplicity of modes, <a href="#Page_677">677</a>;</li> -<li class="isub1">natural and artificial systems, <a href="#Page_679">679</a>;</li> -<li class="isub1">in crystallography, <a href="#Page_685">685</a>;</li> -<li class="isub1">symbolic statement of, <a href="#Page_692">692</a>;</li> -<li class="isub1">bifurcate, <a href="#Page_694">694</a>;</li> -<li class="isub1">an inverse and tentative operation, <a href="#Page_689">689</a>;</li> -<li class="isub1">diagnostic, <a href="#Page_710">710</a>;</li> -<li class="isub1">by indexes, <a href="#Page_714">714</a>;</li> -<li class="isub1">of books, <a href="#Page_715">715</a>;</li> -<li class="isub1">in biological sciences, <a href="#Page_718">718</a>;</li> -<li class="isub1">genealogical, <a href="#Page_719">719</a>;</li> -<li class="isub1">by types, <a href="#Page_722">722</a>;</li> -<li class="isub1">limits of, <a href="#Page_730">730</a>.</li> - -<li class="indx">Clifford, Professor, on types of<span class="pagenum" id="Page_774">774</span> compound statements, <a href="#Page_143">143</a>, <a href="#Page_529">529</a>;</li> -<li class="isub1">first and last catastrophe, <a href="#Page_744">744</a>;</li> -<li class="isub1">mathematical function, <a href="#Page_768">768</a>.</li> - -<li class="indx">Clocks, astronomical, <a href="#Page_340">340</a>, <a href="#Page_353">353</a>.</li> - -<li class="indx">Clouds, <a href="#Page_447">447</a>;</li> -<li class="isub1">cirrous, <a href="#Page_411">411</a>.</li> - -<li class="indx">Coincidences, <a href="#Page_128">128</a>;</li> -<li class="isub1">fortuitous, <a href="#Page_261">261</a>;</li> -<li class="isub1">measurement by, <a href="#Page_292">292</a>;</li> -<li class="isub1">method of, <a href="#Page_291">291</a>.</li> - -<li class="indx">Collective terms, <a href="#Page_29">29</a>, <a href="#Page_39">39</a>.</li> - -<li class="indx">Collocations of matter, <a href="#Page_740">740</a>.</li> - -<li class="indx">Colours, iridescent, <a href="#Page_419">419</a>;</li> -<li class="isub1">natural, <a href="#Page_518">518</a>;</li> -<li class="isub1">perception of, <a href="#Page_437">437</a>;</li> -<li class="isub1">of spectrum, <a href="#Page_584">584</a>.</li> - -<li class="indx">Combinations, <a href="#Page_135">135</a>, <a href="#Page_142">142</a>;</li> -<li class="isub1">doctrine of, <a href="#Page_173">173</a>;</li> -<li class="isub1">of letters of alphabet, <a href="#Page_174">174</a>;</li> -<li class="isub1">calculations of, <a href="#Page_180">180</a>;</li> -<li class="isub1">higher orders of, <a href="#Page_194">194</a>.</li> - -<li class="indx">Combinatorial analysis, <a href="#Page_176">176</a>.</li> - -<li class="indx">Comets, <a href="#Page_449">449</a>;</li> -<li class="isub1">number of, <a href="#Page_408">408</a>;</li> -<li class="isub1">hyperbolic, <a href="#Page_407">407</a>;</li> -<li class="isub1">classification of, <a href="#Page_684">684</a>;</li> -<li class="isub1">conflict with, <a href="#Page_746">746–7</a>;</li> -<li class="isub1">Halley’s comet, <a href="#Page_537">537</a>;</li> -<li class="isub1">Lexell’s comet, <a href="#Page_651">651</a>.</li> - -<li class="indx">Commutativeness, law of, <a href="#Page_35">35</a>, <a href="#Page_72">72</a>, <a href="#Page_177">177</a>.</li> - -<li class="indx">Comparative use of instruments, <a href="#Page_299">299</a>.</li> - -<li class="indx">Compass, variations of, <a href="#Page_281">281</a>.</li> - -<li class="indx">Complementary statements, <a href="#Page_144">144</a>.</li> - -<li class="indx">Compossible alternatives, <a href="#Page_69">69</a>.</li> - -<li class="indx">Compound statements, <a href="#Page_144">144</a>;</li> -<li class="isub1">events, <a href="#Page_204">204</a>.</li> - -<li class="indx">Compounds, chemical, <a href="#Page_192">192</a>.</li> - -<li class="indx">Comte, Auguste, on probability, <a href="#Page_200">200</a>, <a href="#Page_214">214</a>;</li> -<li class="isub1">on prevision, <a href="#Page_536">536</a>;</li> -<li class="isub1">his positive philosophy, <a href="#Page_752">752</a>, <a href="#Page_760">760</a>, <a href="#Page_768">768</a>.</li> - -<li class="indx">Concrete number, <a href="#Page_159">159</a>.</li> - -<li class="indx">Conditions, of logical symbols, <a href="#Page_32">32</a>;</li> -<li class="isub1">removal of usual, <a href="#Page_426">426</a>;</li> -<li class="isub1">interference of unsuspected, <a href="#Page_428">428</a>;</li> -<li class="isub1">maintenance of similar, <a href="#Page_443">443</a>;</li> -<li class="isub1">approximation to natural, <a href="#Page_465">465</a>.</li> - -<li class="indx">Condorcet, <a href="#Page_2">2</a>;</li> -<li class="isub1">his problem, <a href="#Page_253">253</a>.</li> - -<li class="indx">Confusion of elements, <a href="#Page_237">237</a>.</li> - -<li class="indx">Conical refraction, <a href="#Page_653">653</a>.</li> - -<li class="indx">Conjunction of planets, <a href="#Page_293">293</a>, <a href="#Page_657">657</a>.</li> - -<li class="indx">Consequent, definition of, <a href="#Page_225">225</a>.</li> - -<li class="indx">Conservation of energy, <a href="#Page_738">738</a>.</li> - -<li class="indx">Constant numbers of nature, <a href="#Page_328">328</a>;</li> -<li class="isub1">mathematical, <a href="#Page_330">330</a>;</li> -<li class="isub1">physical, <a href="#Page_331">331</a>;</li> -<li class="isub1">astronomical, <a href="#Page_332">332</a>;</li> -<li class="isub1">terrestrial, <a href="#Page_333">333</a>;</li> -<li class="isub1">organic, <a href="#Page_333">333</a>;</li> -<li class="isub1">social, <a href="#Page_334">334</a>.</li> - -<li class="indx">Continuity, law of, <a href="#Page_615">615</a>, <a href="#Page_729">729</a>;</li> -<li class="isub1">sense of, <a href="#Page_493">493</a>;</li> -<li class="isub1">detection of, <a href="#Page_610">610</a>;</li> -<li class="isub1">failure of, <a href="#Page_619">619</a>.</li> - -<li class="indx">Continuous quantity, <a href="#Page_274">274</a>, <a href="#Page_485">485</a>.</li> - -<li class="indx">Contradiction, law of, <a href="#Page_31">31</a>, <a href="#Page_74">74</a>.</li> - -<li class="indx">Contrapositive, proposition, <a href="#Page_84">84</a>, <a href="#Page_136">136</a>;</li> -<li class="isub1">conversion, <a href="#Page_83">83</a>.</li> - -<li class="indx">Conversion of propositions, <a href="#Page_46">46</a>, <a href="#Page_118">118</a>.</li> - -<li class="indx">Copernican theory, <a href="#Page_522">522</a>, <a href="#Page_625">625</a>, <a href="#Page_638">638</a>, <a href="#Page_647">647</a>.</li> - -<li class="indx">Copula, <a href="#Page_16">16</a>.</li> - -<li class="indx">Cornu, velocity of light, <a href="#Page_561">561</a>.</li> - -<li class="indx">Corpuscular theory, <a href="#Page_520">520</a>, <a href="#Page_538">538</a>, <a href="#Page_667">667</a>.</li> - -<li class="indx">Correction, method of, <a href="#Page_346">346</a>.</li> - -<li class="indx">Correlation, <a href="#Page_678">678</a>, <a href="#Page_681">681</a>.</li> - -<li class="indx">Cotes, Roger, use of mean, <a href="#Page_359">359</a>;</li> -<li class="isub1">method of least squares, <a href="#Page_377">377</a>.</li> - -<li class="indx">Coulomb, <a href="#Page_272">272</a>.</li> - -<li class="indx">Couple, mechanical, <a href="#Page_653">653</a>.</li> - -<li class="indx">Creation, problem of, <a href="#Page_740">740</a>.</li> - -<li class="indx">Crookes’ radiometer, <a href="#Page_435">435</a>.</li> - -<li class="indx">Cross divisions, <a href="#Page_144">144</a>.</li> - -<li class="indx">Crystallography, <a href="#Page_648">648</a>, <a href="#Page_654">654</a>, <a href="#Page_658">658</a>, <a href="#Page_678">678</a>, <a href="#Page_754">754</a>;</li> -<li class="isub1">systems of, <a href="#Page_133">133</a>;</li> -<li class="isub1">classification in, <a href="#Page_685">685</a>.</li> - -<li class="indx">Crystals, <a href="#Page_602">602</a>;</li> -<li class="isub1">Dana’s classification of, <a href="#Page_711">711</a>;</li> -<li class="isub1">pseudomorphic, <a href="#Page_658">658</a>.</li> - -<li class="indx">Curves, use of, <a href="#Page_392">392</a>, <a href="#Page_491">491</a>, <a href="#Page_496">496</a>;</li> -<li class="isub1">of various degrees, <a href="#Page_473">473</a>.</li> - -<li class="indx">Cuvier, on experiment, <a href="#Page_423">423</a>;</li> -<li class="isub1">on inferences, <a href="#Page_682">682</a>.</li> - -<li class="indx">Cyanite, <a href="#Page_609">609</a>.</li> - -<li class="indx">Cycloid, <a href="#Page_633">633</a>.</li> - -<li class="indx">Cycloidal pendulum, <a href="#Page_461">461</a>.</li> - -<li class="indx">Cypher, <a href="#Page_124">124</a>.</li> - - -<li class="abet"><span class="alpha"><a id="IX_D"></a><a href="#alpha-table">D</a></span></li> -<li class="ifrst">D’Alembert, blunders in probability, <a href="#Page_213">213</a>, <a href="#Page_214">214</a>;</li> -<li class="isub1">on gravity, <a href="#Page_463">463</a>.</li> - -<li class="indx">Dalton, laws of, <a href="#Page_464">464</a>, <a href="#Page_471">471</a>;</li> -<li class="isub1">atomic theory, <a href="#Page_662">662</a>.</li> - -<li class="indx">Darapti, <a href="#Page_59">59</a>.</li> - -<li class="indx">Darii, <a href="#Page_56">56</a>.</li> - -<li class="indx">Darwin, Charles, his works, <a href="#Page_131">131</a>;</li> -<li class="isub1">negative results of observation, <a href="#Page_413">413</a>;</li> -<li class="isub1">arguments against his theory, <a href="#Page_437">437</a>;</li> -<li class="isub1">cultivated plants, <a href="#Page_531">531</a>;</li> -<li class="isub1">his influence, <a href="#Page_575">575</a>;</li> -<li class="isub1">classification, <a href="#Page_718">718</a>;</li> -<li class="isub1">constancy of character in classification, <a href="#Page_720">720–1</a>;</li> -<li class="isub1">on definition, <a href="#Page_726">726</a>;</li> -<li class="isub1">restoration of limbs, <a href="#Page_730">730</a>;</li> -<li class="isub1">tendency of his theory, <a href="#Page_762">762</a>, <a href="#Page_764">764</a>.</li> - -<li class="indx">Davy, Sir H., on new instruments, <a href="#Page_270">270</a>;</li> -<li class="isub1">nature of heat, <a href="#Page_343">343</a>, <a href="#Page_417">417</a>;</li> -<li class="isub1">detection of salt in electrolysis, <a href="#Page_428">428</a>.</li> - -<li class="indx">Day, sidereal, <a href="#Page_310">310</a>;</li> -<li class="isub1">length of, <a href="#Page_289">289</a>.</li> - -<li class="indx">Decandolle, on classification, <a href="#Page_696">696</a>.<span class="pagenum" id="Page_775">775</span></li> - -<li class="indx">Decyphering, <a href="#Page_124">124</a>.</li> - -<li class="indx">Deduction, <a href="#Page_11">11</a>, <a href="#Page_49">49</a>.</li> - -<li class="indx">Deductive reasoning, <a href="#Page_534">534</a>;</li> -<li class="isub1">miscellaneous forms of, <a href="#Page_60">60</a>;</li> -<li class="isub1">probable, <a href="#Page_209">209</a>.</li> - -<li class="indx">Definition, <a href="#Page_39">39</a>, <a href="#Page_62">62</a>, <a href="#Page_711">711</a>, <a href="#Page_723">723</a>;</li> -<li class="isub1">purpose of, <a href="#Page_54">54</a>;</li> -<li class="isub1">of cause and power, <a href="#Page_224">224</a>.</li> - -<li class="indx">De Morgan, Augustus, negative terms, <a href="#Page_14">14</a>;</li> -<li class="isub1">Aristotle’s logic, <a href="#Page_18">18</a>;</li> -<li class="isub1">relatives, <a href="#Page_23">23</a>;</li> -<li class="isub1">logical universe, <a href="#Page_43">43</a>;</li> -<li class="isub1">complex propositions, <a href="#Page_75">75</a>;</li> -<li class="isub1">contraposition, <a href="#Page_83">83</a>;</li> -<li class="isub1">formal logic quoted, <a href="#Page_101">101</a>;</li> -<li class="isub1">error of his system, <a href="#Page_117">117</a>;</li> -<li class="isub1">anagram of his name, <a href="#Page_128">128</a>;</li> -<li class="isub1">numerically definite reasoning, <a href="#Page_168">168–172</a>;</li> -<li class="isub1">probability, <a href="#Page_198">198</a>;</li> -<li class="isub1">belief, <a href="#Page_199">199</a>;</li> -<li class="isub1">experiments in probability, <a href="#Page_207">207</a>;</li> -<li class="isub1">probable deductive arguments, <a href="#Page_209">209–210</a>;</li> -<li class="isub1">trisection of angle, <a href="#Page_233">233</a>;</li> -<li class="isub1">probability of inference, <a href="#Page_259">259</a>;</li> -<li class="isub1">arcual unit, <a href="#Page_306">306</a>;</li> -<li class="isub1">mathematical tables, <a href="#Page_331">331</a>;</li> -<li class="isub1">personal error, <a href="#Page_348">348</a>;</li> -<li class="isub1">average, <a href="#Page_363">363</a>;</li> -<li class="isub1">his works on probability, <a href="#Page_394">394–395</a>;</li> -<li class="isub1">apparent sequence, <a href="#Page_409">409</a>;</li> -<li class="isub1">sub-equality, <a href="#Page_480">480</a>;</li> -<li class="isub1">rule of approximation, <a href="#Page_481">481</a>;</li> -<li class="isub1">negative areas, <a href="#Page_529">529</a>;</li> -<li class="isub1">generalisation, <a href="#Page_600">600</a>;</li> -<li class="isub1">double algebra, <a href="#Page_634">634</a>;</li> -<li class="isub1">bibliography, <a href="#Page_716">716</a>;</li> -<li class="isub1">catalogues, <a href="#Page_716">716</a>;</li> -<li class="isub1">extensions of algebra, <a href="#Page_758">758</a>.</li> - -<li class="indx">Density, unit of, <a href="#Page_316">316</a>;</li> -<li class="isub1">of earth, <a href="#Page_387">387</a>;</li> -<li class="isub1">negative, <a href="#Page_642">642</a>.</li> - -<li class="indx">Descartes, vortices, <a href="#Page_517">517</a>;</li> -<li class="isub1">geometry, <a href="#Page_632">632</a>.</li> - -<li class="indx">Description, <a href="#Page_62">62</a>.</li> - -<li class="indx">Design, <a href="#Page_762">762–763</a>.</li> - -<li class="indx">Determinants, inference by, <a href="#Page_50">50</a>.</li> - -<li class="indx">Development, logical, <a href="#Page_89">89</a>, <a href="#Page_97">97</a>.</li> - -<li class="indx">Diagnosis, <a href="#Page_708">708</a>.</li> - -<li class="indx">Dichotomy, <a href="#Page_703">703</a>.</li> - -<li class="indx">Difference, <a href="#Page_44">44</a>;</li> -<li class="isub1">law of, <a href="#Page_5">5</a>;</li> -<li class="isub1">sign of, <a href="#Page_17">17</a>;</li> -<li class="isub1">representation of, <a href="#Page_45">45</a>;</li> -<li class="isub1">inference with, <a href="#Page_52">52</a>, <a href="#Page_166">166</a>;</li> -<li class="isub1">form of, <a href="#Page_158">158</a>.</li> - -<li class="indx">Differences of numbers, <a href="#Page_185">185</a>.</li> - -<li class="indx">Differential calculus, <a href="#Page_477">477</a>.</li> - -<li class="indx">Differential thermometer, <a href="#Page_345">345</a>.</li> - -<li class="indx">Diffraction of light, <a href="#Page_420">420</a>.</li> - -<li class="indx">Dimensions, theory of, <a href="#Page_325">325</a>.</li> - -<li class="indx">Dip-needle, observation of, <a href="#Page_355">355</a>.</li> - -<li class="indx">Direct deduction, <a href="#Page_49">49</a>.</li> - -<li class="indx">Direction of motion, <a href="#Page_47">47</a>.</li> - -<li class="indx">Discontinuity, <a href="#Page_620">620</a>.</li> - -<li class="indx">Discordance, of theory and experiment, <a href="#Page_558">558</a>;</li> -<li class="isub1">of theories, <a href="#Page_587">587</a>.</li> - -<li class="indx">Discoveries, accidental, <a href="#Page_529">529</a>;</li> -<li class="isub1">predicted, <a href="#Page_536">536</a>;</li> -<li class="isub1">scope for, <a href="#Page_752">752</a>.</li> - -<li class="indx">Discrimination, <a href="#Page_24">24</a>;</li> -<li class="isub1">power of, <a href="#Page_4">4</a>.</li> - -<li class="indx">Disjunctive, terms, <a href="#Page_66">66</a>;</li> -<li class="isub1">conjunction, <a href="#Page_67">67</a>;</li> -<li class="isub1">propositions, <a href="#Page_66">66</a>;</li> -<li class="isub1">syllogism, <a href="#Page_77">77</a>;</li> -<li class="isub1">argument, <a href="#Page_106">106</a>.</li> - -<li class="indx">Dissipation of energy, <a href="#Page_310">310</a>.</li> - -<li class="indx">Distance of statements, <a href="#Page_144">144</a>.</li> - -<li class="indx">Divergence from average, <a href="#Page_188">188</a>.</li> - -<li class="indx">Diversity, <a href="#Page_156">156</a>.</li> - -<li class="indx">Divine interference, <a href="#Page_765">765</a>.</li> - -<li class="indx">Dollond, achromatic lenses, <a href="#Page_608">608</a>.</li> - -<li class="indx">Donkin, Professor, <a href="#Page_375">375</a>;</li> -<li class="isub1">on probability, <a href="#Page_199">199</a>, <a href="#Page_216">216</a>;</li> -<li class="isub1">principle of inverse method, <a href="#Page_244">244</a>.</li> - -<li class="indx">Double refraction, <a href="#Page_426">426</a>.</li> - -<li class="indx">Dove’s law of winds, <a href="#Page_534">534</a>.</li> - -<li class="indx">Draper’s law, <a href="#Page_606">606</a>.</li> - -<li class="indx">Drobitsch, <a href="#Page_15">15</a>.</li> - -<li class="indx">Duality, <a href="#Page_73">73</a>, <a href="#Page_81">81</a>;</li> -<li class="isub1">law of, <a href="#Page_5">5</a>, <a href="#Page_45">45</a>, <a href="#Page_92">92</a>, <a href="#Page_97">97</a>.</li> - -<li class="indx">Dulong and Petit, <a href="#Page_341">341</a>, <a href="#Page_471">471</a>.</li> - -<li class="indx">Duration, <a href="#Page_308">308</a>.</li> - - -<li class="abet"><span class="alpha"><a id="IX_E"></a><a href="#alpha-table">E</a></span></li> -<li class="ifrst">ε, <a href="#Page_330">330</a>, <a href="#Page_769">769</a>.</li> - -<li class="indx">Earth, density of, <a href="#Page_387">387</a>;</li> -<li class="isub1">ellipticity, <a href="#Page_565">565</a>.</li> - -<li class="indx">Eclipses, <a href="#Page_656">656</a>;</li> -<li class="isub1">Egyptian records of, <a href="#Page_246">246</a>;</li> -<li class="isub1">of Jupiter’s satellites, <a href="#Page_294">294</a>, <a href="#Page_372">372</a>;</li> -<li class="isub1">solar, <a href="#Page_486">486</a>.</li> - -<li class="indx">Electric, sense, <a href="#Page_405">405</a>;</li> -<li class="isub1">acid, <a href="#Page_428">428</a>;</li> -<li class="isub1">fluid, <a href="#Page_523">523</a>.</li> - -<li class="indx">Electric telegraph, anticipations of, <a href="#Page_671">671</a>.</li> - -<li class="indx">Electricity, theories of, <a href="#Page_522">522</a>;</li> -<li class="isub1">duality of, <a href="#Page_590">590</a>.</li> - -<li class="indx">Electrolysis, <a href="#Page_428">428</a>, <a href="#Page_530">530</a>.</li> - -<li class="indx">Electro-magnet, use of, <a href="#Page_423">423</a>.</li> - -<li class="indx">Elements, confusion of, <a href="#Page_237">237</a>;</li> -<li class="isub1">definition, <a href="#Page_427">427</a>;</li> -<li class="isub1">classification, <a href="#Page_676">676</a>, <a href="#Page_677">677</a>, <a href="#Page_690">690</a>.</li> - -<li class="indx">Elimination, <a href="#Page_58">58</a>.</li> - -<li class="indx">Ellicott, observation on clocks, <a href="#Page_455">455</a>.</li> - -<li class="indx">Ellipsis, <a href="#Page_41">41</a>;</li> -<li class="isub1">of terms, <a href="#Page_57">57</a>.</li> - -<li class="indx">Elliptic variation, <a href="#Page_474">474</a>.</li> - -<li class="indx">Ellipticity of earth, <a href="#Page_565">565</a>.</li> - -<li class="indx">Ellis, A. J., contributions to formal logic, <a href="#Page_172">172</a>.</li> - -<li class="indx">Ellie, Leslie, <a href="#Page_23">23</a>, <a href="#Page_375">375</a>.</li> - -<li class="indx">Ellis, W., on moon’s influence, <a href="#Page_410">410</a>.</li> - -<li class="indx">Emanation, law of, <a href="#Page_463">463</a>.</li> - -<li class="indx">Emotions, <a href="#Page_732">732</a>.</li> - -<li class="indx">Empirical, knowledge, <a href="#Page_505">505</a>, <a href="#Page_525">525–526</a>;</li> -<li class="isub1">measurement, <a href="#Page_552">552</a>.</li> - -<li class="indx">Encke, on mean, <a href="#Page_386">386</a>, <a href="#Page_389">389</a>;</li> -<li class="isub1">his comet, <a href="#Page_570">570</a>, <a href="#Page_605">605</a>;</li> -<li class="isub1">on resisting medium, <a href="#Page_523">523</a>;<span class="pagenum" id="Page_776">776</span></li> -<li class="isub1">solar parallax, <a href="#Page_562">562</a>.</li> - -<li class="indx">Energy, unit of, <a href="#Page_322">322</a>;</li> -<li class="isub1">conservation of, <a href="#Page_465">465</a>;</li> -<li class="isub1">reconcentration of, <a href="#Page_751">751</a>.</li> - -<li class="indx">English language, words in, <a href="#Page_175">175</a>.</li> - -<li class="indx">Eözoon canadense, <a href="#Page_412">412</a>, <a href="#Page_668">668</a>.</li> - -<li class="indx">Equality, sign of, <a href="#Page_14">14</a>;</li> -<li class="isub1">axiom, <a href="#Page_163">163</a>;</li> -<li class="isub1">four meanings of, <a href="#Page_479">479</a>.</li> - -<li class="indx">Equations, <a href="#Page_46">46</a>, <a href="#Page_53">53</a>, <a href="#Page_160">160</a>;</li> -<li class="isub1">solution of, <a href="#Page_123">123</a>.</li> - -<li class="indx">Equilibrium, unstable, <a href="#Page_276">276</a>, <a href="#Page_654">654</a>.</li> - -<li class="indx">Equisetaceæ, <a href="#Page_721">721</a>.</li> - -<li class="indx">Equivalence of propositions, <a href="#Page_115">115</a>, <a href="#Page_120">120</a>, <a href="#Page_132">132</a>;</li> -<li class="isub1">remarkable case of, <a href="#Page_529">529</a>, <a href="#Page_657">657</a>.</li> - -<li class="indx">Eratosthenes, sieve of, <a href="#Page_82">82</a>, <a href="#Page_123">123</a>, <a href="#Page_139">139</a>;</li> -<li class="isub1">measurement of degree, <a href="#Page_293">293</a>.</li> - -<li class="indx">Error, function, <a href="#Page_330">330</a>, <a href="#Page_376">376</a>, <a href="#Page_381">381</a>;</li> -<li class="isub1">elimination of, <a href="#Page_339">339</a>, <a href="#Page_353">353</a>;</li> -<li class="isub1">personal, <a href="#Page_347">347</a>;</li> -<li class="isub1">law of, <a href="#Page_374">374</a>;</li> -<li class="isub1">origin of law, <a href="#Page_383">383</a>;</li> -<li class="isub1">verification of law, <a href="#Page_383">383</a>;</li> -<li class="isub1">probable, <a href="#Page_386">386</a>;</li> -<li class="isub1">mean, <a href="#Page_387">387</a>;</li> -<li class="isub1">constant, <a href="#Page_396">396</a>;</li> -<li class="isub1">variation of small errors, <a href="#Page_479">479</a>.</li> - -<li class="indx">Ether, luminiferous, <a href="#Page_512">512</a>, <a href="#Page_514">514</a>, <a href="#Page_605">605</a>.</li> - -<li class="indx">Euclid, axioms, <a href="#Page_51">51</a>, <a href="#Page_163">163</a>;</li> -<li class="isub1">indirect proof, <a href="#Page_84">84</a>;</li> -<li class="isub1">10th book, 117th proposition, <a href="#Page_275">275</a>;</li> -<li class="isub1">on analogy, <a href="#Page_631">631</a>.</li> - -<li class="indx">Euler, on certainty of inference, <a href="#Page_238">238</a>;</li> -<li class="isub1">corpuscular theory, <a href="#Page_435">435</a>;</li> -<li class="isub1">gravity, <a href="#Page_463">463</a>;</li> -<li class="isub1">on ether, <a href="#Page_514">514</a>.</li> - -<li class="indx">Everett, Professor, unit of angle, <a href="#Page_306">306</a>;</li> -<li class="isub1">metric system, <a href="#Page_328">328</a>.</li> - -<li class="indx">Evolution, theory of, <a href="#Page_761">761</a>.</li> - -<li class="indx">Exact science, <a href="#Page_456">456</a>.</li> - -<li class="indx">Exceptions, <a href="#Page_132">132</a>, <a href="#Page_644">644</a>, <a href="#Page_728">728</a>;</li> -<li class="isub1">classification of, <a href="#Page_645">645</a>;</li> -<li class="isub1">imaginary, <a href="#Page_647">647</a>;</li> -<li class="isub1">apparent, <a href="#Page_649">649</a>;</li> -<li class="isub1">singular, <a href="#Page_652">652</a>;</li> -<li class="isub1">divergent, <a href="#Page_655">655</a>;</li> -<li class="isub1">accidental, <a href="#Page_658">658</a>;</li> -<li class="isub1">novel, <a href="#Page_661">661</a>;</li> -<li class="isub1">limiting, <a href="#Page_663">663</a>;</li> -<li class="isub1">real, <a href="#Page_666">666</a>;</li> -<li class="isub1">unclassed, <a href="#Page_668">668</a>.</li> - -<li class="indx">Excluded middle, law of, <a href="#Page_6">6</a>.</li> - -<li class="indx">Exclusive alternatives, <a href="#Page_68">68</a>.</li> - -<li class="indx">Exhaustive investigation, <a href="#Page_418">418</a>.</li> - -<li class="indx">Expansion, of bodies, <a href="#Page_478">478</a>;</li> -<li class="isub1">of liquids, <a href="#Page_488">488</a>.</li> - -<li class="indx">Experiment, <a href="#Page_400">400</a>, <a href="#Page_416">416</a>;</li> -<li class="isub1">in probability, <a href="#Page_208">208</a>;</li> -<li class="isub1">test or blind, <a href="#Page_433">433</a>;</li> -<li class="isub1">negative results of, <a href="#Page_434">434</a>;</li> -<li class="isub1">limits of, <a href="#Page_437">437</a>;</li> -<li class="isub1">collective, <a href="#Page_445">445</a>;</li> -<li class="isub1">simplification of, <a href="#Page_422">422</a>;</li> -<li class="isub1">failure in simplification, <a href="#Page_424">424</a>.</li> - -<li class="indx">Experimentalist, character of, <a href="#Page_574">574</a>, <a href="#Page_592">592</a>.</li> - -<li class="indx">Experimentum crucis, <a href="#Page_518">518</a>, <a href="#Page_667">667</a>.</li> - -<li class="indx">Explanation, <a href="#Page_532">532</a>.</li> - -<li class="indx">Extent of meaning, <a href="#Page_26">26</a>;</li> -<li class="isub1">of terms, <a href="#Page_48">48</a>.</li> - -<li class="indx">Extrapolation, <a href="#Page_495">495</a>.</li> - - -<li class="abet"><span class="alpha"><a id="IX_F"></a><a href="#alpha-table">F</a></span></li> -<li class="ifrst">Factorials, <a href="#Page_179">179</a>.</li> - -<li class="indx">Facts, importance of false, <a href="#Page_414">414</a>;</li> -<li class="isub1">conformity with, <a href="#Page_516">516</a>.</li> - -<li class="indx">Fallacies, <a href="#Page_62">62</a>;</li> -<li class="isub1">analysed by indirect method, <a href="#Page_102">102</a>;</li> -<li class="isub1">of observation, <a href="#Page_408">408</a>.</li> - -<li class="indx">Faraday, Michael, measurement of gold-leaf, <a href="#Page_296">296</a>;</li> -<li class="isub1">on gravity, <a href="#Page_342">342</a>, <a href="#Page_589">589</a>;</li> -<li class="isub1">magnetism of gases, <a href="#Page_352">352</a>;</li> -<li class="isub1">vibrating plate, <a href="#Page_419">419</a>;</li> -<li class="isub1">electric poles, <a href="#Page_421">421</a>;</li> -<li class="isub1">circularly polarised light, <a href="#Page_424">424</a>, <a href="#Page_588">588</a>, <a href="#Page_630">630</a>;</li> -<li class="isub1">freezing mixtures, <a href="#Page_427">427</a>;</li> -<li class="isub1">magnetic experiments, <a href="#Page_431">431</a>, <a href="#Page_434">434</a>;</li> -<li class="isub1">lines of magnetic force, <a href="#Page_446">446</a>, <a href="#Page_580">580</a>;</li> -<li class="isub1">errors of experiment, <a href="#Page_465">465</a>;</li> -<li class="isub1">electrolysis, <a href="#Page_502">502</a>;</li> -<li class="isub1">velocity of light, <a href="#Page_520">520</a>;</li> -<li class="isub1">prediction, <a href="#Page_543">543</a>;</li> -<li class="isub1">relations of physical forces, <a href="#Page_547">547</a>;</li> -<li class="isub1">character of, <a href="#Page_578">578</a>, <a href="#Page_587">587</a>;</li> -<li class="isub1">ray vibrations, <a href="#Page_579">579</a>;</li> -<li class="isub1">mathematical power, <a href="#Page_580">580</a>;</li> -<li class="isub1">philosophic reservation of opinion, <a href="#Page_592">592</a>;</li> -<li class="isub1">use of heavy glass, <a href="#Page_609">609</a>;</li> -<li class="isub1">electricity, <a href="#Page_612">612</a>;</li> -<li class="isub1">radiant matter, <a href="#Page_642">642</a>;</li> -<li class="isub1">hydrogen, <a href="#Page_691">691</a>.</li> - -<li class="indx">Fatality, belief in, <a href="#Page_264">264</a>.</li> - -<li class="indx">Ferio, <a href="#Page_56">56</a>.</li> - -<li class="indx">Figurate numbers, <a href="#Page_183">183</a>, <a href="#Page_186">186</a>.</li> - -<li class="indx">Figure of earth, <a href="#Page_459">459</a>, <a href="#Page_565">565</a>.</li> - -<li class="indx">Fizeau, use of Newton’s rings, <a href="#Page_297">297</a>, <a href="#Page_582">582</a>;</li> -<li class="isub1">fixity of properties, <a href="#Page_313">313</a>;</li> -<li class="isub1">velocity of light, <a href="#Page_441">441</a>, <a href="#Page_561">561</a>.</li> - -<li class="indx">Flamsteed, use of wells, <a href="#Page_294">294</a>;</li> -<li class="isub1">standard stars, <a href="#Page_301">301</a>;</li> -<li class="isub1">parallax of pole-star, <a href="#Page_338">338</a>;</li> -<li class="isub1">selection of observations, <a href="#Page_358">358</a>;</li> -<li class="isub1">astronomical instruments, <a href="#Page_391">391</a>;</li> -<li class="isub1">solar eclipses, <a href="#Page_486">486</a>.</li> - -<li class="indx">Fluorescence, <a href="#Page_664">664</a>.</li> - -<li class="indx">Fontenelle on the senses, <a href="#Page_405">405</a>.</li> - -<li class="indx">Forbes, J. D., <a href="#Page_248">248</a>.</li> - -<li class="indx">Force, unit of, <a href="#Page_322">322</a>, <a href="#Page_326">326</a>;</li> -<li class="isub1">emanating, <a href="#Page_464">464</a>;</li> -<li class="isub1">representation of, <a href="#Page_633">633</a>.</li> - -<li class="indx">Formulæ, empirical, <a href="#Page_487">487</a>;</li> -<li class="isub1">rational, <a href="#Page_489">489</a>.</li> - -<li class="indx">Fortia, <i>Traité des Progressions</i>, <a href="#Page_183">183</a>.</li> - -<li class="indx">Fortuitous coincidences, <a href="#Page_261">261</a>.</li> - -<li class="indx">Fossils, <a href="#Page_661">661</a>.</li> - -<li class="indx">Foster, G. C., on classification, <a href="#Page_691">691</a>.</li> - -<li class="indx">Foucault, rotating mirror, <a href="#Page_299">299</a>;</li> -<li class="isub1">pendulum, <a href="#Page_342">342</a>, <a href="#Page_431">431</a>, <a href="#Page_522">522</a>;<span class="pagenum" id="Page_777">777</span></li> -<li class="isub1">on velocity of light, <a href="#Page_441">441</a>, <a href="#Page_521">521</a>, <a href="#Page_561">561</a>.</li> - -<li class="indx">Fourier, Joseph, theory of dimensions, <a href="#Page_325">325</a>;</li> -<li class="isub1">theory of heat, <a href="#Page_469">469</a>, <a href="#Page_744">744</a>.</li> - -<li class="indx">Fowler, Thomas, on method of difference, <a href="#Page_439">439</a>;</li> -<li class="isub1">reasoning from case to case, <a href="#Page_227">227</a>.</li> - -<li class="indx">Frankland, Professor Edward, on spectrum of gases, <a href="#Page_606">606</a>.</li> - -<li class="indx">Franklin’s experiments on heat, <a href="#Page_424">424</a>.</li> - -<li class="indx">Fraunhofer, dark lines of spectrum, <a href="#Page_429">429</a>.</li> - -<li class="indx">Freezing-point, <a href="#Page_546">546</a>.</li> - -<li class="indx">Freezing mixtures, <a href="#Page_546">546</a>.</li> - -<li class="indx">Fresnel, inflexion of light, <a href="#Page_420">420</a>;</li> -<li class="isub1">corpuscular theory, <a href="#Page_521">521</a>;</li> -<li class="isub1">on use of hypothesis, <a href="#Page_538">538</a>;</li> -<li class="isub1">double refraction, <a href="#Page_539">539</a>.</li> - -<li class="indx">Friction, <a href="#Page_417">417</a>;</li> -<li class="isub1">determination of, <a href="#Page_347">347</a>.</li> - -<li class="indx">Function, definitions of, <a href="#Page_489">489</a>.</li> - -<li class="indx">Functions, discovery of, <a href="#Page_496">496</a>.</li> - - -<li class="abet"><span class="alpha"><a id="IX_G"></a><a href="#alpha-table">G</a></span></li> -<li class="ifrst">Galileo, <a href="#Page_626">626</a>;</li> -<li class="isub1">on cycloid, <a href="#Page_232">232</a>, <a href="#Page_235">235</a>;</li> -<li class="isub1">differential method of observation, <a href="#Page_344">344</a>;</li> -<li class="isub1">projectiles, <a href="#Page_447">447</a>, <a href="#Page_466">466</a>;</li> -<li class="isub1">use of telescope, <a href="#Page_522">522</a>;</li> -<li class="isub1">gravity, <a href="#Page_604">604</a>;</li> -<li class="isub1">principle of continuity, <a href="#Page_617">617</a>.</li> - -<li class="indx">Gallon, definition of, <a href="#Page_318">318</a>.</li> - -<li class="indx">Galton, Francis, divergence from mean, <a href="#Page_188">188</a>;</li> -<li class="isub1">works by, <a href="#Page_188">188</a>, <a href="#Page_655">655</a>;</li> -<li class="isub1">on hereditary genius, <a href="#Page_385">385</a>, <a href="#Page_655">655</a>.</li> - -<li class="indx">Galvanometer, <a href="#Page_351">351</a>.</li> - -<li class="indx">Ganières, de, <a href="#Page_182">182</a>.</li> - -<li class="indx">Gases, <a href="#Page_613">613</a>;</li> -<li class="isub1">properties of, <a href="#Page_601">601</a>, <a href="#Page_602">602</a>;</li> -<li class="isub1">perfect, <a href="#Page_470">470</a>;</li> -<li class="isub1">liquefiable, <a href="#Page_665">665</a>.</li> - -<li class="indx">Gauss, pendulum experiments, <a href="#Page_316">316</a>;</li> -<li class="isub1">law of error, <a href="#Page_375">375–6</a>;</li> -<li class="isub1">detection of error, <a href="#Page_396">396</a>;</li> -<li class="isub1">on gravity, <a href="#Page_463">463</a>.</li> - -<li class="indx">Gay Lussac, on boiling point, <a href="#Page_659">659</a>;</li> -<li class="isub1">law of, <a href="#Page_669">669</a>.</li> - -<li class="indx">Genealogical classification, <a href="#Page_680">680</a>, <a href="#Page_719">719</a>.</li> - -<li class="indx">General, terms, <a href="#Page_29">29</a>;</li> -<li class="isub1">truths, <a href="#Page_647">647</a>;</li> -<li class="isub1">notions, <a href="#Page_673">673</a>.</li> - -<li class="indx">Generalisation, <a href="#Page_2">2</a>, <a href="#Page_594">594</a>, <a href="#Page_704">704</a>;</li> -<li class="isub1">mathematical, <a href="#Page_168">168</a>;</li> -<li class="isub1">two meanings of, <a href="#Page_597">597</a>;</li> -<li class="isub1">value of, <a href="#Page_599">599</a>;</li> -<li class="isub1">hasty, <a href="#Page_623">623</a>.</li> - -<li class="indx">Genius, nature of, <a href="#Page_575">575</a>.</li> - -<li class="indx">Genus, <a href="#Page_433">433</a>, <a href="#Page_698">698</a>;</li> -<li class="isub1">generalissimum, <a href="#Page_701">701</a>;</li> -<li class="isub1">natural, <a href="#Page_724">724</a>.</li> - -<li class="indx">Geology, <a href="#Page_667">667</a>;</li> -<li class="isub1">records in, <a href="#Page_408">408</a>;</li> -<li class="isub1">slowness of changes, <a href="#Page_438">438</a>;</li> -<li class="isub1">exceptions in, <a href="#Page_660">660</a>.</li> - -<li class="indx">Geometric mean, <a href="#Page_361">361</a>.</li> - -<li class="indx">Geometric reasoning, <a href="#Page_458">458</a>;</li> -<li class="isub1">certainty of, <a href="#Page_267">267</a>.</li> - -<li class="indx">Giffard’s injector, <a href="#Page_536">536</a>.</li> - -<li class="indx">Gilbert, on rotation of earth, <a href="#Page_249">249</a>;</li> -<li class="isub1">magnetism of silver, <a href="#Page_431">431</a>;</li> -<li class="isub1">experimentation, <a href="#Page_443">443</a>.</li> - -<li class="indx">Gladstone, J. H., <a href="#Page_445">445</a>.</li> - -<li class="indx">Glaisher, J. W. L., on mathematical tables, <a href="#Page_331">331</a>;</li> -<li class="isub1">law of error, <a href="#Page_375">375</a>, <a href="#Page_395">395</a>.</li> - -<li class="indx">Gold, discovery of, <a href="#Page_413">413</a>.</li> - -<li class="indx">Gold-assay process, <a href="#Page_434">434</a>.</li> - -<li class="indx">Gold-leaf, thickness of, <a href="#Page_296">296</a>.</li> - -<li class="indx">Graham, Professor Thomas, on chemical affinity, <a href="#Page_614">614</a>;</li> -<li class="isub1">continuity, <a href="#Page_616">616</a>;</li> -<li class="isub1">nature of hydrogen, <a href="#Page_691">691</a>.</li> - -<li class="indx">Grammar, <a href="#Page_39">39</a>;</li> -<li class="isub1">rules of, <a href="#Page_31">31</a>.</li> - -<li class="indx">Grammatical, change, <a href="#Page_119">119</a>;</li> -<li class="isub1">equivalence, <a href="#Page_120">120</a>.</li> - -<li class="indx">Gramme, <a href="#Page_317">317</a>.</li> - -<li class="indx">Graphical method, <a href="#Page_492">492</a>.</li> - -<li class="indx">Gravesande, on inflection of light, <a href="#Page_420">420</a>.</li> - -<li class="indx">Gravity, <a href="#Page_422">422</a>, <a href="#Page_512">512</a>, <a href="#Page_514">514</a>, <a href="#Page_604">604</a>, <a href="#Page_740">740</a>;</li> -<li class="isub1">determination of, <a href="#Page_302">302</a>;</li> -<li class="isub1">elimination of, <a href="#Page_427">427</a>;</li> -<li class="isub1">law of, <a href="#Page_458">458</a>, <a href="#Page_462">462</a>, <a href="#Page_474">474</a>;</li> -<li class="isub1">inconceivability of, <a href="#Page_510">510</a>;</li> -<li class="isub1">Newton’s theory, <a href="#Page_555">555</a>;</li> -<li class="isub1">variation of, <a href="#Page_565">565</a>;</li> -<li class="isub1">discovery of law, <a href="#Page_581">581</a>;</li> -<li class="isub1">Faraday on, <a href="#Page_589">589</a>;</li> -<li class="isub1">discontinuity in, <a href="#Page_620">620</a>;</li> -<li class="isub1">Aristotle on, <a href="#Page_649">649</a>;</li> -<li class="isub1">Hooke’s experiment, <a href="#Page_436">436</a>.</li> - -<li class="indx">Grimaldi on the spectrum, <a href="#Page_584">584</a>.</li> - -<li class="indx">Grove, Mr. Justice, on ether, <a href="#Page_514">514</a>;</li> -<li class="isub1">electricity, <a href="#Page_615">615</a>.</li> - -<li class="indx">Guericke, Otto von, <a href="#Page_432">432</a>.</li> - - -<li class="abet"><span class="alpha"><a id="IX_H"></a><a href="#alpha-table">H</a></span></li> -<li class="ifrst">Habit, formation of, <a href="#Page_618">618</a>.</li> - -<li class="indx">Halley, trade-winds, <a href="#Page_534">534</a>.</li> - -<li class="indx">Halley’s comet, <a href="#Page_537">537</a>, <a href="#Page_570">570</a>.</li> - -<li class="indx">Hamilton, Sir William, disjunctive propositions, <a href="#Page_69">69</a>;</li> -<li class="isub1">inference, <a href="#Page_118">118</a>;</li> -<li class="isub1">free-will, <a href="#Page_223">223</a>.</li> - -<li class="indx">Hamilton, Sir W. Rowan, on conical refraction, <a href="#Page_540">540</a>;</li> -<li class="isub1">quaternions, <a href="#Page_634">634</a>.</li> - -<li class="indx">Harley, Rev. Robert, on Boole’s logic, <a href="#Page_23">23</a>, <a href="#Page_155">155</a>.</li> - -<li class="indx">Harris, standards of length, <a href="#Page_312">312</a>.</li> - -<li class="indx">Hartley, on logic, <a href="#Page_7">7</a>.</li> - -<li class="indx">Hatchett, on alloys, <a href="#Page_191">191</a>.</li> - -<li class="indx">Haughton, Professor, on tides, <a href="#Page_450">450</a>;</li> -<li class="isub1">muscular exertion, <a href="#Page_490">490</a>.</li> - -<li class="indx">Haüy, on crystallography, <a href="#Page_529">529</a>.</li> - -<li class="indx">Hayward, R. B., <a href="#Page_142">142</a>.</li> - -<li class="indx">Heat, unit of, <a href="#Page_324">324</a>;</li> -<li class="isub1">measurement of, <a href="#Page_349">349</a>;<span class="pagenum" id="Page_778">778</span></li> -<li class="isub1">experiments on, <a href="#Page_444">444</a>;</li> -<li class="isub1">mechanical equivalent of, <a href="#Page_568">568</a>.</li> - -<li class="indx">Heavy glass, <a href="#Page_588">588</a>, <a href="#Page_609">609</a>.</li> - -<li class="indx">Helmholtz, on microscopy, <a href="#Page_406">406</a>;</li> -<li class="isub1">undulations, <a href="#Page_414">414</a>;</li> -<li class="isub1">sound, <a href="#Page_476">476</a>.</li> - -<li class="indx">Hemihedral crystals, <a href="#Page_649">649</a>.</li> - -<li class="indx">Herschel, Sir John, on rotation of plane of polarisation of light, <a href="#Page_129">129</a>, <a href="#Page_630">630</a>;</li> -<li class="isub1">quartz crystals, <a href="#Page_246">246</a>;</li> -<li class="isub1">numerical precision, <a href="#Page_273">273</a>;</li> -<li class="isub1">photometry, <a href="#Page_273">273</a>;</li> -<li class="isub1">light of stars, <a href="#Page_302">302</a>;</li> -<li class="isub1">actinometer, <a href="#Page_337">337</a>;</li> -<li class="isub1">mean and average, <a href="#Page_363">363</a>;</li> -<li class="isub1">eclipses of Jupiter’s satellites, <a href="#Page_372">372</a>;</li> -<li class="isub1">law of error, <a href="#Page_377">377</a>;</li> -<li class="isub1">error in observations, <a href="#Page_392">392</a>;</li> -<li class="isub1">on observation, <a href="#Page_400">400</a>;</li> -<li class="isub1">moon’s influence on clouds, <a href="#Page_410">410</a>;</li> -<li class="isub1">comets, <a href="#Page_411">411</a>;</li> -<li class="isub1">spectrum analysis, <a href="#Page_429">429</a>;</li> -<li class="isub1">collective instances, <a href="#Page_447">447</a>;</li> -<li class="isub1">principle of forced vibrations, <a href="#Page_451">451</a>, <a href="#Page_663">663</a>;</li> -<li class="isub1">meteorological variations, <a href="#Page_489">489</a>;</li> -<li class="isub1">double stars, <a href="#Page_499">499</a>, <a href="#Page_685">685</a>;</li> -<li class="isub1">direct action, <a href="#Page_502">502</a>;</li> -<li class="isub1">use of theory, <a href="#Page_508">508</a>;</li> -<li class="isub1">ether, <a href="#Page_515">515</a>;</li> -<li class="isub1"><i>experimentum crucis</i>, <a href="#Page_519">519</a>;</li> -<li class="isub1">interference of light, <a href="#Page_539">539</a>;</li> -<li class="isub1">interference of sound, <a href="#Page_540">540</a>;</li> -<li class="isub1">density of earth, <a href="#Page_567">567</a>;</li> -<li class="isub1">residual phenomena, <a href="#Page_569">569</a>;</li> -<li class="isub1">helicoidal dissymmetry, <a href="#Page_630">630</a>;</li> -<li class="isub1">fluorescence, <a href="#Page_664">664</a>.</li> - -<li class="indx">Hindenburg, on combinatorial analysis, <a href="#Page_176">176</a>.</li> - -<li class="indx">Hipparchus, used method of repetition, <a href="#Page_289">289</a>;</li> -<li class="isub1">longitudes of stars, <a href="#Page_294">294</a>.</li> - -<li class="indx">Hippocrates, area of lunule, <a href="#Page_480">480</a>.</li> - -<li class="indx">History, science of, <a href="#Page_760">760</a>.</li> - -<li class="indx">Hobbes, Thomas, definition of cause, <a href="#Page_224">224</a>;</li> -<li class="isub1">definition of time, <a href="#Page_307">307</a>;</li> -<li class="isub1">on hypothesis, <a href="#Page_510">510</a>.</li> - -<li class="indx">Hofmann, unit called crith, <a href="#Page_321">321</a>;</li> -<li class="isub1">on prediction, <a href="#Page_544">544</a>;</li> -<li class="isub1">on anomalies, <a href="#Page_670">670</a>.</li> - -<li class="indx">Homogeneity, law of, <a href="#Page_159">159</a>, <a href="#Page_327">327</a>.</li> - -<li class="indx">Hooke, on gravitation, <a href="#Page_436">436</a>, <a href="#Page_581">581</a>;</li> -<li class="isub1">philosophical method, <a href="#Page_507">507</a>;</li> -<li class="isub1">on strange things, <a href="#Page_671">671</a>.</li> - -<li class="indx">Hopkinson, John, <a href="#Page_194">194</a>;</li> -<li class="isub1">method of interpolation, <a href="#Page_497">497</a>.</li> - -<li class="indx">Horrocks, use of mean, <a href="#Page_358">358</a>;</li> -<li class="isub1">use of hypothesis, <a href="#Page_507">507</a>.</li> - -<li class="indx">Hume on perception, <a href="#Page_34">34</a>.</li> - -<li class="indx">Hutton, density of earth, <a href="#Page_566">566</a>.</li> - -<li class="indx">Huxley, Professor Thomas, <a href="#Page_764">764</a>;</li> -<li class="isub1">on hypothesis, <a href="#Page_509">509</a>;</li> -<li class="isub1">classification, <a href="#Page_676">676</a>;</li> -<li class="isub1">mammalia, <a href="#Page_682">682</a>;</li> -<li class="isub1">palæontology, <a href="#Page_682">682</a>.</li> - -<li class="indx">Huyghens, theory of pendulum, <a href="#Page_302">302</a>;</li> -<li class="isub1">pendulum standard, <a href="#Page_315">315</a>;</li> -<li class="isub1">cycloidal pendulum, <a href="#Page_341">341</a>;</li> -<li class="isub1">differential method, <a href="#Page_344">344</a>;</li> -<li class="isub1">distant stars, <a href="#Page_405">405</a>;</li> -<li class="isub1">use of hypothesis, <a href="#Page_508">508</a>;</li> -<li class="isub1">philosophical method of, <a href="#Page_585">585</a>;</li> -<li class="isub1">on analogy, <a href="#Page_639">639</a>.</li> - -<li class="indx">Hybrids, <a href="#Page_727">727</a>.</li> - -<li class="indx">Hydrogen, expansion of, <a href="#Page_471">471</a>;</li> -<li class="isub1">refractive power, <a href="#Page_527">527</a>;</li> -<li class="isub1">metallic nature of, <a href="#Page_691">691</a>.</li> - -<li class="indx">Hygrometry, <a href="#Page_563">563</a>.</li> - -<li class="indx">Hypotheses, use of, <a href="#Page_265">265</a>, <a href="#Page_504">504</a>;</li> -<li class="isub1">substitution of simple hypotheses, <a href="#Page_458">458</a>;</li> -<li class="isub1">working hypotheses, <a href="#Page_509">509</a>;</li> -<li class="isub1">requisites of, <a href="#Page_510">510</a>;</li> -<li class="isub1">descriptive, <a href="#Page_522">522</a>, <a href="#Page_686">686</a>;</li> -<li class="isub1">representative, <a href="#Page_524">524</a>;</li> -<li class="isub1">probability of, <a href="#Page_559">559</a>.</li> - - -<li class="abet"><span class="alpha"><a id="IX_I"></a><a href="#alpha-table">I</a></span></li> -<li class="ifrst">Identical propositions, <a href="#Page_119">119</a>.</li> - -<li class="indx">Identities, simple, <a href="#Page_37">37</a>;</li> -<li class="isub1">partial, <a href="#Page_40">40</a>;</li> -<li class="isub1">limited, <a href="#Page_42">42</a>;</li> -<li class="isub1">simple and partial, <a href="#Page_111">111</a>;</li> -<li class="isub1">inference from, <a href="#Page_51">51</a>, <a href="#Page_55">55</a>.</li> - -<li class="indx">Identity, law of, <a href="#Page_5">5</a>, <a href="#Page_6">6</a>, <a href="#Page_74">74</a>;</li> -<li class="isub1">expression of, <a href="#Page_14">14</a>;</li> -<li class="isub1">propagating power, <a href="#Page_20">20</a>;</li> -<li class="isub1">reciprocal, <a href="#Page_46">46</a>.</li> - -<li class="indx">Illicit process, of major term, <a href="#Page_65">65</a>, <a href="#Page_103">103</a>;</li> -<li class="isub1">of minor term, <a href="#Page_65">65</a>.</li> - -<li class="indx">Immediate inference, <a href="#Page_50">50</a>, <a href="#Page_61">61</a>.</li> - -<li class="indx">Imperfect induction, <a href="#Page_146">146</a>, <a href="#Page_149">149</a>.</li> - -<li class="indx">Inclusion, relation of, <a href="#Page_40">40</a>.</li> - -<li class="indx">Incommensurable quantities, <a href="#Page_275">275</a>.</li> - -<li class="indx">Incompossible events, <a href="#Page_205">205</a>.</li> - -<li class="indx">Independence of small effects, <a href="#Page_475">475</a>.</li> - -<li class="indx">Independent events, <a href="#Page_204">204</a>.</li> - -<li class="indx">Indestructibility of matter, <a href="#Page_465">465</a>.</li> - -<li class="indx">Indexes, classification by, <a href="#Page_714">714</a>;</li> -<li class="isub1">formation of, <a href="#Page_717">717</a>.</li> - -<li class="indx">India-rubber, properties of, <a href="#Page_545">545</a>.</li> - -<li class="indx">Indirect method of deduction, <a href="#Page_49">49</a>, <a href="#Page_81">81</a>;</li> -<li class="isub1">illustrations of, <a href="#Page_98">98</a>;</li> -<li class="isub1">fallacies analysed by, <a href="#Page_102">102</a>;</li> -<li class="isub1">the test of equivalence, <a href="#Page_115">115</a>.</li> - -<li class="indx">Induction, <a href="#Page_11">11</a>, <a href="#Page_121">121</a>;</li> -<li class="isub1">symbolic statement of, <a href="#Page_131">131</a>;</li> -<li class="isub1">perfect, <a href="#Page_146">146</a>;</li> -<li class="isub1">imperfect, <a href="#Page_149">149</a>;</li> -<li class="isub1">philosophy of, <a href="#Page_218">218</a>;</li> -<li class="isub1">grounds of, <a href="#Page_228">228</a>;</li> -<li class="isub1">illustrations of, <a href="#Page_229">229</a>;</li> -<li class="isub1">quantitative, <a href="#Page_483">483</a>;</li> -<li class="isub1">problem of two classes, <a href="#Page_134">134</a>;</li> -<li class="isub1">problem of three classes, <a href="#Page_137">137</a>.</li> - -<li class="indx">Inductive truths, classes of, <a href="#Page_219">219</a>.</li> - -<li class="indx">Inequalities, reasoning by, <a href="#Page_47">47</a>, <a href="#Page_163">163</a>, <a href="#Page_165">165–166</a>.</li> - -<li class="indx">Inference, <a href="#Page_9">9</a>;</li> -<li class="isub1">general formula of, <a href="#Page_17">17</a>;</li> -<li class="isub1">immediate, <a href="#Page_50">50</a>;</li> -<li class="isub1">with two simple identities, <a href="#Page_51">51</a>;</li> -<li class="isub1">from simple and partial identity, <a href="#Page_53">53</a>;<span class="pagenum" id="Page_779">779</span></li> -<li class="isub1">with partial identities, <a href="#Page_55">55</a>;</li> -<li class="isub1">by sum of predicates, <a href="#Page_61">61</a>;</li> -<li class="isub1">by disjunctive propositions, <a href="#Page_76">76</a>;</li> -<li class="isub1">indirect method of, <a href="#Page_81">81</a>;</li> -<li class="isub1">nature of, <a href="#Page_118">118</a>;</li> -<li class="isub1">principle of mathematical, <a href="#Page_162">162</a>;</li> -<li class="isub1">certainty of, <a href="#Page_236">236</a>.</li> - -<li class="indx">Infima species, <a href="#Page_701">701</a>, <a href="#Page_702">702</a>.</li> - -<li class="indx">Infiniteness of universe, <a href="#Page_738">738</a>.</li> - -<li class="indx">Inflection of light, <a href="#Page_420">420</a>.</li> - -<li class="indx">Instantiæ, citantes, evocantes, radii, curriculi, <a href="#Page_270">270</a>;</li> -<li class="isub1">monodicæ, irregulares, heteroclitæ, <a href="#Page_608">608</a>;</li> -<li class="isub1">clandestinæ, <a href="#Page_610">610</a>.</li> - -<li class="indx">Instruments of measurement, <a href="#Page_284">284</a>.</li> - -<li class="indx">Insufficient enumeration, <a href="#Page_176">176</a>.</li> - -<li class="indx">Integration, <a href="#Page_123">123</a>.</li> - -<li class="indx">Intellect, etymology of, <a href="#Page_5">5</a>.</li> - -<li class="indx">Intension of logical terms, <a href="#Page_26">26</a>, <a href="#Page_48">48</a>;</li> -<li class="isub1">of propositions, <a href="#Page_47">47</a>.</li> - -<li class="indx">Interchangeable system, <a href="#Page_20">20</a>.</li> - -<li class="indx">Interpolation, <a href="#Page_495">495</a>;</li> -<li class="isub1">in meteorology, <a href="#Page_497">497</a>.</li> - -<li class="indx">Inverse, process, <a href="#Page_12">12</a>;</li> -<li class="isub1">operation, <a href="#Page_122">122</a>, <a href="#Page_689">689</a>;</li> -<li class="isub1">problem of two classes, <a href="#Page_134">134</a>;</li> -<li class="isub1">problem of three classes, <a href="#Page_137">137</a>;</li> -<li class="isub1">problem of probability, <a href="#Page_240">240</a>, <a href="#Page_251">251</a>;</li> -<li class="isub1">rules of inverse method, <a href="#Page_257">257</a>;</li> -<li class="isub1">simple illustrations, <a href="#Page_253">253</a>;</li> -<li class="isub1">general solution, <a href="#Page_255">255</a>.</li> - -<li class="indx">Iodine, the substance X, <a href="#Page_523">523</a>.</li> - -<li class="indx">Iron, properties of, <a href="#Page_528">528</a>, <a href="#Page_670">670</a>.</li> - -<li class="indx"><i>Is</i>, ambiguity of verb, <a href="#Page_16">16</a>, <a href="#Page_41">41</a>.</li> - -<li class="indx">Isomorphism, <a href="#Page_662">662</a>.</li> - -<li class="indx">Ivory, <a href="#Page_375">375</a>.</li> - - -<li class="abet"><span class="alpha"><a id="IX_J"></a><a href="#alpha-table">J</a></span></li> -<li class="ifrst">James, Sir H., on density of earth, <a href="#Page_567">567</a>.</li> - -<li class="indx">Jenkin, Professor Fleming, <a href="#Page_328">328</a>.</li> - -<li class="indx">Jevons, W. S., on use of mean, <a href="#Page_361">361</a>;</li> -<li class="isub1">on pedesis or molecular movement of microscopic particles, <a href="#Page_406">406</a>, <a href="#Page_549">549</a>;</li> -<li class="isub1">cirrous clouds, <a href="#Page_411">411</a>;</li> -<li class="isub1">spectrum analysis, <a href="#Page_429">429</a>;</li> -<li class="isub1">elevated rain-gauges, <a href="#Page_430">430</a>;</li> -<li class="isub1">experiments on clouds, <a href="#Page_447">447</a>;</li> -<li class="isub1">on muscular exertion, <a href="#Page_490">490</a>;</li> -<li class="isub1">resisting medium, <a href="#Page_570">570</a>;</li> -<li class="isub1">anticipations of the electric telegraph, <a href="#Page_671">671</a>.</li> - -<li class="indx">Jones, Dr. Bence, Life of Faraday, <a href="#Page_578">578</a>.</li> - -<li class="indx">Jordanus, on the mean, <a href="#Page_360">360</a>.</li> - -<li class="indx">Joule, <a href="#Page_545">545</a>;</li> -<li class="isub1">on thermopile, <a href="#Page_299">299</a>, <a href="#Page_300">300</a>;</li> -<li class="isub1">mechanical equivalent of heat, <a href="#Page_325">325</a>, <a href="#Page_347">347</a>, <a href="#Page_568">568</a>;</li> -<li class="isub1">temperature of air, <a href="#Page_343">343</a>;</li> -<li class="isub1">rarefaction, <a href="#Page_444">444</a>;</li> -<li class="isub1">on Thomson’s prediction, <a href="#Page_543">543</a>;</li> -<li class="isub1">molecular theory of gases, <a href="#Page_548">548</a>;</li> -<li class="isub1">friction, <a href="#Page_549">549</a>;</li> -<li class="isub1">thermal phenomena of fluids, <a href="#Page_557">557</a>.</li> - -<li class="indx">Jupiter, satellites of, <a href="#Page_372">372</a>, <a href="#Page_458">458</a>, <a href="#Page_638">638</a>, <a href="#Page_656">656</a>;</li> -<li class="isub1">long inequality of, <a href="#Page_455">455</a>;</li> -<li class="isub1">figure of, <a href="#Page_556">556</a>.</li> - - -<li class="abet"><span class="alpha"><a id="IX_K"></a><a href="#alpha-table">K</a></span></li> -<li class="ifrst">Kames, Lord, on bifurcate classification, <a href="#Page_697">697</a>.</li> - -<li class="indx">Kant, disjunctive propositions, <a href="#Page_69">69</a>;</li> -<li class="isub1">analogy, <a href="#Page_597">597</a>;</li> -<li class="isub1">doctrine of space, <a href="#Page_769">769</a>.</li> - -<li class="indx">Kater’s pendulum, <a href="#Page_316">316</a>.</li> - -<li class="indx">Keill, law of emanating forces, <a href="#Page_464">464</a>;</li> -<li class="isub1">axiom of simplicity, <a href="#Page_625">625</a>.</li> - -<li class="indx">Kepler, on star-discs, <a href="#Page_390">390</a>;</li> -<li class="isub1">comets, <a href="#Page_408">408</a>;</li> -<li class="isub1">laws of, <a href="#Page_456">456</a>;</li> -<li class="isub1">refraction, <a href="#Page_501">501</a>;</li> -<li class="isub1">character of, <a href="#Page_578">578</a>.</li> - -<li class="indx">Kinds of things, <a href="#Page_718">718</a>.</li> - -<li class="indx">King Charles and the Royal Society, <a href="#Page_647">647</a>.</li> - -<li class="indx">Kirchhoff, on lines of spectrum, <a href="#Page_245">245</a>.</li> - -<li class="indx">Kohlrausch, rules of approximate calculation, <a href="#Page_479">479</a>.</li> - - -<li class="abet"><span class="alpha"><a id="IX_L"></a><a href="#alpha-table">L</a></span></li> -<li class="ifrst">Lagrange, formula for interpolation, <a href="#Page_497">497</a>;</li> -<li class="isub1">accidental discovery, <a href="#Page_531">531</a>;</li> -<li class="isub1">union of algebra and geometry, <a href="#Page_633">633</a>.</li> - -<li class="indx">Lambert, <a href="#Page_15">15</a>.</li> - -<li class="indx">Lamont, <a href="#Page_452">452</a>.</li> - -<li class="indx">Language, <a href="#Page_8">8</a>, <a href="#Page_628">628</a>, <a href="#Page_643">643</a>.</li> - -<li class="indx">Laplace, on probability, <a href="#Page_200">200</a>, <a href="#Page_216">216</a>;</li> -<li class="isub1">principles of inverse method, <a href="#Page_242">242</a>;</li> -<li class="isub1">solution of inverse problem, <a href="#Page_256">256</a>;</li> -<li class="isub1">planetary motions, <a href="#Page_249">249</a>, <a href="#Page_250">250</a>;</li> -<li class="isub1">conjunctions of planets, <a href="#Page_293">293</a>;</li> -<li class="isub1">observation of tides, <a href="#Page_372">372</a>;</li> -<li class="isub1">atmospheric tides, <a href="#Page_367">367</a>;</li> -<li class="isub1">law of errors, <a href="#Page_378">378</a>;</li> -<li class="isub1">dark stars, <a href="#Page_404">404</a>;</li> -<li class="isub1">hyperbolic comets, <a href="#Page_407">407</a>;</li> -<li class="isub1">his works on probability, <a href="#Page_395">395</a>;</li> -<li class="isub1">velocity of gravity, <a href="#Page_435">435</a>;</li> -<li class="isub1">stability of planetary system, <a href="#Page_448">448</a>, <a href="#Page_746">746</a>;</li> -<li class="isub1">form of Jupiter, <a href="#Page_556">556</a>;</li> -<li class="isub1">corpuscular theory, <a href="#Page_521">521</a>;</li> -<li class="isub1">ellipticity of earth, <a href="#Page_565">565</a>;</li> -<li class="isub1">velocity of sound, <a href="#Page_571">571</a>;</li> -<li class="isub1">analogy, <a href="#Page_597">597</a>;</li> -<li class="isub1">law of gravity, <a href="#Page_615">615</a>;</li> -<li class="isub1">inhabitants of planets, <a href="#Page_640">640</a>;</li> -<li class="isub1">laws of motion, <a href="#Page_706">706</a>;</li> -<li class="isub1">power of science, <a href="#Page_739">739</a>.</li> - -<li class="indx">Lavoisier, mistaken inference of, <a href="#Page_238">238</a>;</li> -<li class="isub1">pyrometer, <a href="#Page_287">287</a>;</li> -<li class="isub1">on experiments, <a href="#Page_423">423</a>;</li> -<li class="isub1">prediction of, <a href="#Page_544">544</a>;</li> -<li class="isub1">theory, <a href="#Page_611">611</a>;</li> -<li class="isub1">on acids, <a href="#Page_667">667</a></li> - -<li class="indx">Law, <a href="#Page_3">3</a>;</li> -<li class="isub1">of simplicity, <a href="#Page_33">33</a>, <a href="#Page_72">72</a>, <a href="#Page_161">161</a>;<span class="pagenum" id="Page_780">780</span></li> -<li class="isub1">commutativeness, <a href="#Page_35">35</a>, <a href="#Page_160">160</a>;</li> -<li class="isub1">disjunctive relation, <a href="#Page_71">71</a>;</li> -<li class="isub1">unity, <a href="#Page_72">72</a>, <a href="#Page_157">157</a>, <a href="#Page_162">162</a>;</li> -<li class="isub1">identity, <a href="#Page_74">74</a>;</li> -<li class="isub1">contradiction, <a href="#Page_74">74</a>, <a href="#Page_82">82</a>;</li> -<li class="isub1">duality, <a href="#Page_73">73</a>, <a href="#Page_74">74</a>, <a href="#Page_81">81</a>, <a href="#Page_97">97</a>, <a href="#Page_169">169</a>;</li> -<li class="isub1">homogeneity, <a href="#Page_159">159</a>;</li> -<li class="isub1">error, <a href="#Page_374">374</a>;</li> -<li class="isub1">continuity, <a href="#Page_615">615</a>;</li> -<li class="isub1">of Boyle, <a href="#Page_619">619</a>;</li> -<li class="isub1">natural, <a href="#Page_737">737</a>.</li> - -<li class="indx">Laws, of thought, <a href="#Page_6">6</a>;</li> -<li class="isub1">empirical mathematical, <a href="#Page_487">487</a>;</li> -<li class="isub1">of motion, <a href="#Page_617">617</a>;</li> -<li class="isub1">of botanical nomenclature, <a href="#Page_727">727</a>;</li> -<li class="isub1">natural hierarchy of, <a href="#Page_742">742</a>.</li> - -<li class="indx">Least squares, method of, <a href="#Page_386">386</a>, <a href="#Page_393">393</a>.</li> - -<li class="indx">Legendre, on geometry, <a href="#Page_275">275</a>;</li> -<li class="isub1">rejection of observations, <a href="#Page_391">391</a>;</li> -<li class="isub1">method of least squares, <a href="#Page_377">377</a>.</li> - -<li class="indx">Leibnitz, <a href="#Page_154">154</a>, <a href="#Page_163">163</a>;</li> -<li class="isub1">on substitution, <a href="#Page_21">21</a>;</li> -<li class="isub1">propositions, <a href="#Page_42">42</a>;</li> -<li class="isub1">blunder in probability, <a href="#Page_213">213</a>;</li> -<li class="isub1">on Newton, <a href="#Page_515">515</a>;</li> -<li class="isub1">continuity, <a href="#Page_618">618</a>.</li> - -<li class="indx">Leslie, differential thermometer, <a href="#Page_345">345</a>;</li> -<li class="isub1">radiating power, <a href="#Page_425">425</a>;</li> -<li class="isub1">on affectation of accuracy, <a href="#Page_482">482</a>.</li> - -<li class="indx">Letters, combinations of, <a href="#Page_193">193</a>.</li> - -<li class="indx">Leverrier, on solar parallax, <a href="#Page_562">562</a>.</li> - -<li class="indx">Lewis, Sir G. C., on time, <a href="#Page_307">307</a>.</li> - -<li class="indx">Life is change, <a href="#Page_173">173</a>.</li> - -<li class="indx">Light, intensity of, <a href="#Page_296">296</a>;</li> -<li class="isub1">unit, <a href="#Page_324">324</a>;</li> -<li class="isub1">velocity, <a href="#Page_535">535</a>, <a href="#Page_560">560</a>, <a href="#Page_561">561</a>;</li> -<li class="isub1">science of, <a href="#Page_538">538</a>;</li> -<li class="isub1">total reflection, <a href="#Page_650">650</a>;</li> -<li class="isub1">waves of, <a href="#Page_637">637</a>;</li> -<li class="isub1">classification of, <a href="#Page_731">731</a>.</li> - -<li class="indx">Lighthouses, Babbage on, <a href="#Page_194">194</a>.</li> - -<li class="indx">Limited identities, <a href="#Page_42">42</a>;</li> -<li class="isub1">inference of 59.</li> - -<li class="indx">Lindsay, Prof. T. M., <a href="#Page_6">6</a>, <a href="#Page_21">21</a>.</li> - -<li class="indx">Linear variation, <a href="#Page_474">474</a>.</li> - -<li class="indx">Linnæus on synopsis, <a href="#Page_712">712</a>;</li> -<li class="isub1">genera and species, <a href="#Page_725">725</a>.</li> - -<li class="indx">Liquid state, <a href="#Page_601">601</a>, <a href="#Page_614">614</a>.</li> - -<li class="indx">Locke, John, on induction, <a href="#Page_121">121</a>;</li> -<li class="isub1">origin of number, <a href="#Page_157">157</a>;</li> -<li class="isub1">on probability, <a href="#Page_215">215</a>;</li> -<li class="isub1">the word power, <a href="#Page_221">221</a>.</li> - -<li class="indx">Lockyer, J. Norman, classification of elements, <a href="#Page_676">676</a>.</li> - -<li class="indx">Logarithms, <a href="#Page_148">148</a>;</li> -<li class="isub1">errors in tables, <a href="#Page_242">242</a>.</li> - -<li class="indx">Logic, etymology of name, <a href="#Page_5">5</a>.</li> - -<li class="indx">Logical abacus, <a href="#Page_104">104</a>.</li> - -<li class="indx">Logical alphabet, <a href="#Page_93">93</a>, <a href="#Page_116">116</a>, <a href="#Page_173">173</a>, <a href="#Page_417">417</a>, <a href="#Page_701">701</a>;</li> -<li class="isub1">table of, <a href="#Page_94">94</a>;</li> -<li class="isub1">connection with arithmetical triangle, <a href="#Page_189">189</a>;</li> -<li class="isub1">in probability, <a href="#Page_205">205</a>.</li> - -<li class="indx">Logical conditions, numerical meaning of, <a href="#Page_171">171</a>.</li> - -<li class="indx">Logical machine, <a href="#Page_107">107</a>.</li> - -<li class="indx">Logical relations, number of, <a href="#Page_142">142</a>.</li> - -<li class="indx">Logical slate, <a href="#Page_95">95</a>.</li> - -<li class="indx">Logical truths, certainty of, <a href="#Page_153">153</a>.</li> - -<li class="indx">Lottery, the infinite, <a href="#Page_2">2</a>.</li> - -<li class="indx">Lovering, Prof., on ether, <a href="#Page_606">606</a>.</li> - -<li class="indx">Lubbock and Drinkwater-Bethune, <a href="#Page_386">386</a>, <a href="#Page_395">395</a>.</li> - -<li class="indx">Lucretius, rain of atoms, <a href="#Page_223">223</a>, <a href="#Page_741">741</a>;</li> -<li class="isub1">indestructibility of matter, <a href="#Page_622">622</a>.</li> - - -<li class="abet"><span class="alpha"><a id="IX_M"></a><a href="#alpha-table">M</a></span></li> -<li class="ifrst">Machine, logical, <a href="#Page_107">107</a>.</li> - -<li class="indx">Macleay, system of classification, <a href="#Page_719">719</a>.</li> - -<li class="indx">Magnetism of gases, <a href="#Page_352">352</a>.</li> - -<li class="indx">Mallet, on earthquakes, <a href="#Page_314">314</a>.</li> - -<li class="indx">Malus, polarised light, <a href="#Page_530">530</a>.</li> - -<li class="indx">Mammalia, characters of, <a href="#Page_681">681</a>.</li> - -<li class="indx">Manchester Literary and Philosophical Society, papers quoted, <a href="#Page_137">137</a>, <a href="#Page_143">143</a>, <a href="#Page_168">168</a>.</li> - -<li class="indx">Mansel, on disjunctive propositions, <a href="#Page_69">69</a>.</li> - -<li class="indx">Mars, white spots of, <a href="#Page_596">596</a>.</li> - -<li class="indx">Maskelyne, on personal error, <a href="#Page_347">347</a>;</li> -<li class="isub1">deviation of plumbline, <a href="#Page_369">369</a>;</li> -<li class="isub1">density of earth, <a href="#Page_566">566</a>.</li> - -<li class="indx">Mass, unit of, <a href="#Page_317">317</a>, <a href="#Page_325">325</a>.</li> - -<li class="indx">Mathematical science, <a href="#Page_767">767</a>;</li> -<li class="isub1">incompleteness of, <a href="#Page_754">754</a>.</li> - -<li class="indx">Matter, uniform properties of, <a href="#Page_603">603</a>;</li> -<li class="isub1">variable properties, <a href="#Page_606">606</a>.</li> - -<li class="indx">Matthiessen, <a href="#Page_528">528</a>.</li> - -<li class="indx">Maximum points, <a href="#Page_371">371</a>.</li> - -<li class="indx">Maxwell, Professor Clerk, on the balance, <a href="#Page_304">304</a>;</li> -<li class="isub1">natural system of standards, <a href="#Page_311">311</a>, <a href="#Page_319">319</a>;</li> -<li class="isub1">velocity of electricity, <a href="#Page_442">442</a>;</li> -<li class="isub1">on Faraday, <a href="#Page_580">580</a>;</li> -<li class="isub1">his book on <i>Matter and Motion</i>, <a href="#Page_634">634</a>.</li> - -<li class="indx">Mayer, proposed repeating circle, <a href="#Page_290">290</a>;</li> -<li class="isub1">on mechanical equivalent of heat, <a href="#Page_568">568</a>, <a href="#Page_572">572</a>.</li> - -<li class="indx">Mean, etymology of, <a href="#Page_359">359–360</a>;</li> -<li class="isub1">geometric, <a href="#Page_362">362</a>;</li> -<li class="isub1">fictitious, <a href="#Page_363">363</a>;</li> -<li class="isub1">precise, <a href="#Page_365">365</a>;</li> -<li class="isub1">probable, <a href="#Page_385">385</a>;</li> -<li class="isub1">rejection of, <a href="#Page_389">389</a>;</li> -<li class="isub1">method of, <a href="#Page_357">357</a>, <a href="#Page_554">554</a>.</li> - -<li class="indx">Mean error, <a href="#Page_387">387</a>.</li> - -<li class="indx">Meaning, of names, <a href="#Page_25">25</a>;</li> -<li class="isub1">of propositions, <a href="#Page_47">47</a>.</li> - -<li class="indx">Measurement, of phenomena, <a href="#Page_270">270</a>;</li> -<li class="isub1">methods of, <a href="#Page_282">282</a>;</li> -<li class="isub1">instruments, <a href="#Page_284">284</a>;</li> -<li class="isub1">indirect, <a href="#Page_296">296</a>;</li> -<li class="isub1">accuracy of, <a href="#Page_303">303</a>;</li> -<li class="isub1">units and standards of, <a href="#Page_305">305</a>;</li> -<li class="isub1">explained results of, <a href="#Page_554">554</a>;</li> -<li class="isub1">agreement of modes of, <a href="#Page_564">564</a>.</li> - -<li class="indx">Mediate statements, <a href="#Page_144">144</a>.</li> - -<li class="indx">Melodies, possible number of, <a href="#Page_191">191</a>.<span class="pagenum" id="Page_781">781</span></li> - -<li class="indx">Melvill, Thomas, on the spectrum, <a href="#Page_429">429</a>.</li> - -<li class="indx"><i>Membra dividentia</i>, <a href="#Page_68">68</a>.</li> - -<li class="indx">Metals, probable character of new, <a href="#Page_258">258</a>;</li> -<li class="isub1">transparency, <a href="#Page_548">548</a>;</li> -<li class="isub1">classification, <a href="#Page_675">675</a>;</li> -<li class="isub1">density, <a href="#Page_706">706</a>.</li> - -<li class="indx">Method, indirect, <a href="#Page_98">98</a>;</li> -<li class="isub1">of avoidance of error, <a href="#Page_340">340</a>;</li> -<li class="isub1">differential, <a href="#Page_344">344</a>;</li> -<li class="isub1">correction, <a href="#Page_346">346</a>;</li> -<li class="isub1">compensation, <a href="#Page_350">350</a>;</li> -<li class="isub1">reversal, <a href="#Page_354">354</a>;</li> -<li class="isub1">means, <a href="#Page_357">357</a>;</li> -<li class="isub1">least squares, <a href="#Page_377">377</a>, <a href="#Page_386">386</a>, <a href="#Page_393">393</a>;</li> -<li class="isub1">variations, <a href="#Page_439">439</a>;</li> -<li class="isub1">graphical, <a href="#Page_492">492</a>;</li> -<li class="isub1">Baconian, <a href="#Page_507">507</a>.</li> - -<li class="indx">Meteoric streams, <a href="#Page_372">372</a>.</li> - -<li class="indx">Meteoric cycle, <a href="#Page_537">537</a>.</li> - -<li class="indx">Metre, <a href="#Page_349">349</a>;</li> -<li class="isub1">error of, <a href="#Page_314">314</a>.</li> - -<li class="indx">Metric system, <a href="#Page_318">318</a>, <a href="#Page_323">323</a>.</li> - -<li class="indx">Michell, speculations, <a href="#Page_212">212</a>;</li> -<li class="isub1">on double stars, <a href="#Page_247">247</a>;</li> -<li class="isub1">Pleiades, <a href="#Page_248">248</a>;</li> -<li class="isub1">torsion balance, <a href="#Page_566">566</a>.</li> - -<li class="indx">Middle term undistributed, <a href="#Page_64">64</a>.</li> - -<li class="indx">Mill, John Stuart, disjunctive propositions, <a href="#Page_69">69</a>;</li> -<li class="isub1">induction, <a href="#Page_121">121</a>, <a href="#Page_594">594</a>;</li> -<li class="isub1">music, <a href="#Page_191">191</a>;</li> -<li class="isub1">probability, <a href="#Page_200">200–201</a>, <a href="#Page_222">222</a>;</li> -<li class="isub1">supposed reform of logic, <a href="#Page_227">227</a>;</li> -<li class="isub1">deductive method, <a href="#Page_265">265</a>, <a href="#Page_508">508</a>;</li> -<li class="isub1">elimination of chance, <a href="#Page_385">385</a>;</li> -<li class="isub1">joint method of agreement and difference, <a href="#Page_425">425</a>;</li> -<li class="isub1">method of variations, <a href="#Page_484">484</a>;</li> -<li class="isub1">on collocations, <a href="#Page_740">740</a>;</li> -<li class="isub1">erroneous tendency of his philosophy, <a href="#Page_752">752</a>.</li> - -<li class="indx">Miller, Prof. W. H., kilogram, <a href="#Page_318">318</a>.</li> - -<li class="indx">Mind, powers of, <a href="#Page_4">4</a>;</li> -<li class="isub1">phenomena of, <a href="#Page_672">672</a>.</li> - -<li class="indx">Minerals, classification of, <a href="#Page_678">678</a>.</li> - -<li class="indx">Minor term, illicit process of, <a href="#Page_65">65</a>.</li> - -<li class="indx">Mistakes, <a href="#Page_7">7</a>.</li> - -<li class="indx"><i>Modus, tolendo ponens</i>, <a href="#Page_77">77</a>;</li> -<li class="isub1"><i>ponendo tollens</i>, <a href="#Page_78">78</a>.</li> - -<li class="indx">Molecular movement, or pedesis, <a href="#Page_406">406</a>.</li> - -<li class="indx">Molecules, number of, <a href="#Page_195">195</a>.</li> - -<li class="indx">Momentum, <a href="#Page_322">322</a>, <a href="#Page_326">326</a>.</li> - -<li class="indx">Monro, C. J., correction by, <a href="#Page_172">172</a>;</li> -<li class="isub1">on Comte, <a href="#Page_753">753</a>.</li> - -<li class="indx">Monstrous productions, <a href="#Page_657">657</a>.</li> - -<li class="indx">Moon, supposed influence on clouds, <a href="#Page_410">410</a>;</li> -<li class="isub1">atmosphere of, <a href="#Page_434">434</a>;</li> -<li class="isub1">motions, <a href="#Page_485">485</a>;</li> -<li class="isub1">fall towards earth, <a href="#Page_555">555</a>.</li> - -<li class="indx">Morse alphabet, <a href="#Page_193">193</a>.</li> - -<li class="indx">Mother of pearl, <a href="#Page_419">419</a>.</li> - -<li class="indx">Müller, Max, on etymology of intellect, <a href="#Page_5">5</a>.</li> - -<li class="indx">Multiplication in logic, <a href="#Page_161">161</a>.</li> - -<li class="indx">Murphy, J. J., on disjunctive relation, <a href="#Page_71">71</a>.</li> - -<li class="indx">Murray, introduced use of ice, <a href="#Page_343">343</a>.</li> - -<li class="indx">Muscular susurrus, <a href="#Page_298">298</a>.</li> - -<li class="indx">Music, possible combinations of, <a href="#Page_191">191</a>.</li> - - -<li class="abet"><span class="alpha"><a id="IX_N"></a><a href="#alpha-table">N</a></span></li> -<li class="ifrst">Names, <a href="#Page_25">25</a>;</li> -<li class="isub1">of persons, ships, &c., <a href="#Page_680">680</a>.</li> - -<li class="indx">Nature, <a href="#Page_1">1</a>;</li> -<li class="isub1">laws of, <a href="#Page_737">737</a>;</li> -<li class="isub1">uniformity of, <a href="#Page_745">745</a>.</li> - -<li class="indx">Nebular theory, <a href="#Page_427">427</a>.</li> - -<li class="indx">Negation, <a href="#Page_44">44</a>.</li> - -<li class="indx">Negative arguments, <a href="#Page_621">621</a>.</li> - -<li class="indx">Negative density, <a href="#Page_642">642</a>.</li> - -<li class="indx">Negative premises, <a href="#Page_63">63</a>, <a href="#Page_103">103</a>.</li> - -<li class="indx">Negative propositions, <a href="#Page_43">43</a>.</li> - -<li class="indx">Negative results of experiment, <a href="#Page_434">434</a>.</li> - -<li class="indx">Negative terms, <a href="#Page_14">14</a>, <a href="#Page_45">45</a>, <a href="#Page_54">54</a>, <a href="#Page_74">74</a>.</li> - -<li class="indx">Neil on use of hypothesis, <a href="#Page_509">509</a>.</li> - -<li class="indx">Neptune, discovery of, <a href="#Page_537">537</a>, <a href="#Page_660">660</a>.</li> - -<li class="indx">Newton, Sir Isaac, binomial theorem, <a href="#Page_231">231</a>;</li> -<li class="isub1">spectrum, <a href="#Page_262">262</a>, <a href="#Page_418">418</a>, <a href="#Page_420">420</a>, <a href="#Page_424">424</a>, <a href="#Page_583">583</a>;</li> -<li class="isub1">rings of, <a href="#Page_288">288</a>, <a href="#Page_470">470</a>;</li> -<li class="isub1">velocity of sound, <a href="#Page_295">295</a>;</li> -<li class="isub1">wave-lengths, <a href="#Page_297">297</a>;</li> -<li class="isub1">use of pendulum, <a href="#Page_303">303</a>;</li> -<li class="isub1">on time, <a href="#Page_308">308</a>;</li> -<li class="isub1">definition of matter, <a href="#Page_316">316</a>;</li> -<li class="isub1">pendulum experiment, <a href="#Page_348">348</a>, <a href="#Page_443">443</a>, <a href="#Page_604">604</a>;</li> -<li class="isub1">centrobaric bodies, <a href="#Page_365">365</a>;</li> -<li class="isub1">on weight, <a href="#Page_422">422</a>;</li> -<li class="isub1">achromatic lenses, <a href="#Page_432">432</a>;</li> -<li class="isub1">resistance of space, <a href="#Page_435">435</a>;</li> -<li class="isub1">absorption of light, <a href="#Page_445">445</a>;</li> -<li class="isub1">planetary motions, <a href="#Page_249">249</a>, <a href="#Page_457">457</a>, <a href="#Page_463">463</a>, <a href="#Page_466">466</a>, <a href="#Page_467">467</a>;</li> -<li class="isub1">infinitesimal calculus, <a href="#Page_477">477</a>;</li> -<li class="isub1">as an alchemist, <a href="#Page_505">505</a>;</li> -<li class="isub1">his knowledge of Bacon’s works, <a href="#Page_507">507</a>;</li> -<li class="isub1"><i>hypotheses non fingo</i>, <a href="#Page_515">515</a>;</li> -<li class="isub1">on vortices, <a href="#Page_517">517</a>;</li> -<li class="isub1">theory of colours, <a href="#Page_518">518</a>;</li> -<li class="isub1">corpuscular theory of light, <a href="#Page_520">520</a>;</li> -<li class="isub1">fits of easy reflection, &c., <a href="#Page_523">523</a>;</li> -<li class="isub1">combustible substances, <a href="#Page_527">527</a>;</li> -<li class="isub1">gravity, <a href="#Page_555">555</a>, <a href="#Page_650">650</a>;</li> -<li class="isub1">density of earth, <a href="#Page_566">566</a>;</li> -<li class="isub1">velocity of sound, <a href="#Page_571">571</a>;</li> -<li class="isub1">third law of motion, <a href="#Page_622">622</a>;</li> -<li class="isub1">his rules of philosophising, <a href="#Page_625">625</a>;</li> -<li class="isub1">fluxions, <a href="#Page_633">633</a>;</li> -<li class="isub1">theory of sound, <a href="#Page_636">636</a>;</li> -<li class="isub1">negative density, <a href="#Page_642">642</a>;</li> -<li class="isub1">rays of light having sides, <a href="#Page_662">662</a>.</li> - -<li class="indx">Newtonian Method, <a href="#Page_581">581</a>.</li> - -<li class="indx">Nicholson, discovery of electrolysis, <a href="#Page_530">530</a>.</li> - -<li class="indx"><i>Ninth Bridgewater Treatise</i> quoted, <a href="#Page_743">743</a>, <a href="#Page_757">757</a>.</li> - -<li class="indx">Nipher, Professor, on muscular exertion, <a href="#Page_490">490</a>.<span class="pagenum" id="Page_782">782</span></li> - -<li class="indx">Noble, Captain, chronoscope, <a href="#Page_308">308</a>, <a href="#Page_616">616</a>.</li> - -<li class="indx">Nomenclature, laws of botanical, <a href="#Page_727">727</a>.</li> - -<li class="indx">Non-observation, arguments from, <a href="#Page_411">411</a>.</li> - -<li class="indx">Norwood’s measurement of a degree, <a href="#Page_272">272</a>.</li> - -<li class="indx">Nothing, <a href="#Page_32">32</a>.</li> - -<li class="indx">Number, nature of, <a href="#Page_153">153</a>, <a href="#Page_156">156</a>;</li> -<li class="isub1">concrete and abstract, <a href="#Page_159">159</a>, <a href="#Page_305">305</a>.</li> - -<li class="indx">Numbers, prime, <a href="#Page_123">123</a>;</li> -<li class="isub1">of Bernoulli, <a href="#Page_124">124</a>;</li> -<li class="isub1">figurate, <a href="#Page_183">183</a>;</li> -<li class="isub1">triangular, &c., <a href="#Page_185">185</a>.</li> - -<li class="indx">Numerical abstraction, <a href="#Page_158">158</a>.</li> - - -<li class="abet"><span class="alpha"><a id="IX_O"></a><a href="#alpha-table">O</a></span></li> -<li class="ifrst">Observation, <a href="#Page_399">399</a>;</li> -<li class="isub1">mental conditions, <a href="#Page_402">402</a>;</li> -<li class="isub1">instrumental and sensual conditions, <a href="#Page_404">404</a>;</li> -<li class="isub1">external conditions, <a href="#Page_407">407</a>.</li> - -<li class="indx">Obverse statements, <a href="#Page_144">144</a>.</li> - -<li class="indx">Ocean, depth of, <a href="#Page_297">297</a>.</li> - -<li class="indx">Odours, <a href="#Page_732">732</a>.</li> - -<li class="indx">Oersted, on electro-magnetism, <a href="#Page_530">530</a>, <a href="#Page_535">535</a>.</li> - -<li class="indx"><i>Or</i>, meaning of, <a href="#Page_70">70</a>.</li> - -<li class="indx">Order, of premises, <a href="#Page_114">114</a>;</li> -<li class="isub1">of terms, <a href="#Page_33">33</a>.</li> - -<li class="indx">Orders of combinations, <a href="#Page_194">194</a>.</li> - -<li class="indx">Original research, <a href="#Page_574">574</a>.</li> - -<li class="indx">Oscillation, centre of, <a href="#Page_364">364</a>.</li> - -<li class="indx">Ostensive instances, <a href="#Page_608">608</a>.</li> - -<li class="indx">Ozone, <a href="#Page_663">663</a>.</li> - - -<li class="abet"><span class="alpha"><a id="IX_P"></a><a href="#alpha-table">P</a></span></li> -<li class="ifrst">π, value of, <a href="#Page_234">234</a>, <a href="#Page_529">529</a>.</li> - -<li class="indx">Pack of cards, arrangement of, <a href="#Page_241">241</a>.</li> - -<li class="indx">Paley on design, <a href="#Page_762">762</a>, <a href="#Page_763">763</a>.</li> - -<li class="indx">Parallax, of stars, <a href="#Page_344">344</a>;</li> -<li class="isub1">of sun, <a href="#Page_560">560</a>.</li> - -<li class="indx">Parallel forces, <a href="#Page_652">652</a>.</li> - -<li class="indx">Paralogism, <a href="#Page_62">62</a>.</li> - -<li class="indx">Parity of reasoning, <a href="#Page_268">268</a>.</li> - -<li class="indx">Partial identities, <a href="#Page_40">40</a>, <a href="#Page_55">55</a>, <a href="#Page_57">57</a>, <a href="#Page_111">111</a>;</li> -<li class="isub1">induction of, <a href="#Page_130">130</a>.</li> - -<li class="indx">Particular quantity, <a href="#Page_56">56</a>.</li> - -<li class="indx">Particulars, reasoning from, <a href="#Page_227">227</a>.</li> - -<li class="indx">Partition, <a href="#Page_29">29</a>.</li> - -<li class="indx">Pascal, <a href="#Page_176">176</a>;</li> -<li class="isub1">arithmetical machine, <a href="#Page_107">107</a>;</li> -<li class="isub1">arithmetical triangle, <a href="#Page_182">182</a>;</li> -<li class="isub1">binomial formula, <a href="#Page_182">182</a>;</li> -<li class="isub1">error in probabilities, <a href="#Page_213">213</a>;</li> -<li class="isub1">barometer, <a href="#Page_519">519</a>.</li> - -<li class="indx">Passive state of steel, <a href="#Page_659">659</a>.</li> - -<li class="indx">Pedesis, or molecular movement of microscopic particles, <a href="#Page_406">406</a>, <a href="#Page_612">612</a>.</li> - -<li class="indx">Peirce, Professor, <a href="#Page_23">23</a>;</li> -<li class="isub1">on rejection of observations, <a href="#Page_391">391</a>.</li> - -<li class="indx">Pendulum, <a href="#Page_290">290</a>, <a href="#Page_302">302</a>, <a href="#Page_315">315</a>;</li> -<li class="isub1">faults of, <a href="#Page_311">311</a>;</li> -<li class="isub1">vibrations, <a href="#Page_453">453</a>, <a href="#Page_454">454</a>;</li> -<li class="isub1">cycloidal, <a href="#Page_461">461</a>.</li> - -<li class="indx">Perfect induction, <a href="#Page_146">146</a>, <a href="#Page_149">149</a>.</li> - -<li class="indx">Perigon, <a href="#Page_306">306</a>.</li> - -<li class="indx">Permutations, <a href="#Page_173">173</a>, <a href="#Page_178">178</a>;</li> -<li class="isub1">distinction from combinations, <a href="#Page_177">177</a>.</li> - -<li class="indx">Personal error, <a href="#Page_347">347</a>.</li> - -<li class="indx">Photometry, <a href="#Page_288">288</a>.</li> - -<li class="indx">Physiology, exceptions in, <a href="#Page_666">666</a>.</li> - -<li class="indx">Planets, conjunctions of, <a href="#Page_181">181</a>, <a href="#Page_187">187</a>, <a href="#Page_657">657</a>;</li> -<li class="isub1">discovery of, <a href="#Page_412">412</a>;</li> -<li class="isub1">motions, <a href="#Page_457">457</a>;</li> -<li class="isub1">perturbations of, <a href="#Page_657">657</a>;</li> -<li class="isub1">classification, <a href="#Page_683">683</a>;</li> -<li class="isub1">system of, <a href="#Page_748">748</a>.</li> - -<li class="indx">Plants, classification of, <a href="#Page_678">678</a>.</li> - -<li class="indx">Plateau’s experiments, <a href="#Page_427">427</a>.</li> - -<li class="indx">Plato on science, <a href="#Page_595">595</a>.</li> - -<li class="indx">Plattes, Gabriel, <a href="#Page_434">434</a>, <a href="#Page_438">438</a>.</li> - -<li class="indx">Pliny on tides, <a href="#Page_451">451</a>.</li> - -<li class="indx">Plumb-line, divergence of, <a href="#Page_461">461</a>.</li> - -<li class="indx">Plurality, <a href="#Page_29">29</a>, <a href="#Page_156">156</a>.</li> - -<li class="indx">Poinsot, on probability, <a href="#Page_214">214</a>.</li> - -<li class="indx">Poisson, on principle of the inverse method, <a href="#Page_244">244</a>;</li> -<li class="isub1">work on Probability, <a href="#Page_395">395</a>;</li> -<li class="isub1">Newton’s rings, <a href="#Page_470">470</a>;</li> -<li class="isub1">simile of ballot box, <a href="#Page_524">524</a>.</li> - -<li class="indx">Polarisation, <a href="#Page_653">653</a>;</li> -<li class="isub1">discovery of, <a href="#Page_530">530</a>.</li> - -<li class="indx">Pole-star, <a href="#Page_652">652</a>;</li> -<li class="isub1">observations of, <a href="#Page_366">366</a>.</li> - -<li class="indx">Poles, of magnets, <a href="#Page_365">365</a>;</li> -<li class="isub1">of battery, <a href="#Page_421">421</a>.</li> - -<li class="indx">Political economy, <a href="#Page_760">760</a>.</li> - -<li class="indx">Porphyry, on the Predicables, <a href="#Page_698">698</a>;</li> -<li class="isub1">tree of, <a href="#Page_702">702</a>.</li> - -<li class="indx">Port Royal logic, <a href="#Page_22">22</a>.</li> - -<li class="indx">Positive philosophy, <a href="#Page_760">760</a>, <a href="#Page_768">768</a>.</li> - -<li class="indx">Pouillet’s pyrheliometer, <a href="#Page_337">337</a>.</li> - -<li class="indx">Powell, Baden, <a href="#Page_623">623</a>;</li> -<li class="isub1">on planetary motions, <a href="#Page_660">660</a>.</li> - -<li class="indx">Power, definition of, <a href="#Page_224">224</a>.</li> - -<li class="indx">Predicables, <a href="#Page_698">698</a>.</li> - -<li class="indx">Prediction, <a href="#Page_536">536</a>, <a href="#Page_739">739</a>;</li> -<li class="isub1">in science of light, <a href="#Page_538">538</a>;</li> -<li class="isub1">theory of undulations, <a href="#Page_540">540</a>;</li> -<li class="isub1">other sciences, <a href="#Page_542">542</a>;</li> -<li class="isub1">by inversion of cause and effect, <a href="#Page_545">545</a>.</li> - -<li class="indx">Premises, order of, <a href="#Page_114">114</a>.</li> - -<li class="indx">Prime numbers, <a href="#Page_123">123</a>, <a href="#Page_139">139</a>;</li> -<li class="isub1">formula for, <a href="#Page_230">230</a>.</li> - -<li class="indx"><i>Principia</i>, Newton’s, <a href="#Page_581">581</a>, <a href="#Page_583">583</a>.</li> - -<li class="indx">Principle, of probability, <a href="#Page_200">200</a>;</li> -<li class="isub1">inverse method, <a href="#Page_242">242</a>;</li> -<li class="isub1">forced vibrations, <a href="#Page_451">451</a>;</li> -<li class="isub1">approximation, <a href="#Page_471">471</a>;</li> -<li class="isub1">co-existence of small vibrations, <a href="#Page_476">476</a>;<span class="pagenum" id="Page_783">783</span></li> -<li class="isub1">superposition of small effects, <a href="#Page_476">476</a>.</li> - -<li class="indx">Probable error, <a href="#Page_555">555</a>.</li> - -<li class="indx">Probability, etymology of, <a href="#Page_197">197</a>;</li> -<li class="isub1">theory of, <a href="#Page_197">197</a>;</li> -<li class="isub1">principles, <a href="#Page_200">200</a>;</li> -<li class="isub1">calculations, <a href="#Page_203">203</a>;</li> -<li class="isub1">difficulties of theory, <a href="#Page_213">213</a>;</li> -<li class="isub1">application of theory, <a href="#Page_215">215</a>;</li> -<li class="isub1">in induction, <a href="#Page_219">219</a>;</li> -<li class="isub1">in judicial proceedings, <a href="#Page_216">216</a>;</li> -<li class="isub1">works on, <a href="#Page_394">394</a>;</li> -<li class="isub1">results of law, <a href="#Page_656">656</a>.</li> - -<li class="indx">Problems, to be worked by reader, <a href="#Page_126">126</a>;</li> -<li class="isub1">inverse problem of two classes, <a href="#Page_135">135</a>;</li> -<li class="isub1">of three classes, <a href="#Page_137">137</a>.</li> - -<li class="indx">Proclus, commentaries of, <a href="#Page_232">232</a>.</li> - -<li class="indx">Proctor, R. A., star-drifts, <a href="#Page_248">248</a>.</li> - -<li class="indx">Projectiles, theory of, <a href="#Page_466">466</a>.</li> - -<li class="indx">Proper names, <a href="#Page_27">27</a>.</li> - -<li class="indx">Properties, generality of, <a href="#Page_600">600</a>;</li> -<li class="isub1">uniform, <a href="#Page_603">603</a>;</li> -<li class="isub1">extreme instances, <a href="#Page_607">607</a>;</li> -<li class="isub1">correlation, <a href="#Page_681">681</a>.</li> - -<li class="indx">Property, logical, <a href="#Page_699">699</a>;</li> -<li class="isub1">peculiar, <a href="#Page_699">699</a>.</li> - -<li class="indx">Proportion, simple, <a href="#Page_501">501</a>.</li> - -<li class="indx">Propositions, <a href="#Page_36">36</a>;</li> -<li class="isub1">negative, <a href="#Page_43">43</a>;</li> -<li class="isub1">conversion of, <a href="#Page_46">46</a>;</li> -<li class="isub1">twofold meaning, <a href="#Page_47">47</a>;</li> -<li class="isub1">disjunctive, <a href="#Page_66">66</a>;</li> -<li class="isub1">equivalence of, <a href="#Page_115">115</a>;</li> -<li class="isub1">identical, <a href="#Page_119">119</a>;</li> -<li class="isub1">tautologous, <a href="#Page_119">119</a>.</li> - -<li class="indx">Protean verses, <a href="#Page_175">175</a>.</li> - -<li class="indx">Protoplasm, <a href="#Page_524">524</a>, <a href="#Page_764">764</a>.</li> - -<li class="indx">Prout’s law, <a href="#Page_263">263</a>, <a href="#Page_464">464</a>.</li> - -<li class="indx">Provisional units, <a href="#Page_323">323</a>.</li> - -<li class="indx">Proximate statements, <a href="#Page_144">144</a>.</li> - -<li class="indx">Pyramidal numbers, <a href="#Page_185">185</a>.</li> - -<li class="indx">Pythagoras, on duality, <a href="#Page_95">95</a>;</li> -<li class="isub1">on the number seven, <a href="#Page_262">262</a>, <a href="#Page_624">624</a>.</li> - - -<li class="abet"><span class="alpha"><a id="IX_Q"></a><a href="#alpha-table">Q</a></span></li> -<li class="ifrst">Quadric variation, <a href="#Page_474">474</a>.</li> - -<li class="indx">Qualitative, reasoning, <a href="#Page_48">48</a>;</li> -<li class="isub1">propositions, <a href="#Page_119">119</a>.</li> - -<li class="indx">Quantification of predicate, <a href="#Page_41">41</a>.</li> - -<li class="indx">Quantitative, reasoning, <a href="#Page_48">48</a>;</li> -<li class="isub1">propositions, <a href="#Page_119">119</a>;</li> -<li class="isub1">questions, <a href="#Page_278">278</a>;</li> -<li class="isub1">induction, <a href="#Page_483">483</a>.</li> - -<li class="indx">Quantities, continuous, <a href="#Page_274">274</a>;</li> -<li class="isub1">incommensurable, <a href="#Page_275">275</a>.</li> - -<li class="indx">Quaternions, <a href="#Page_160">160</a>, <a href="#Page_634">634</a>.</li> - -<li class="indx">Quetelet, <a href="#Page_188">188</a>;</li> -<li class="isub1">experiment on probability, <a href="#Page_208">208</a>;</li> -<li class="isub1">on mean and average, <a href="#Page_363">363</a>;</li> -<li class="isub1">law of error, <a href="#Page_378">378</a>, <a href="#Page_380">380</a>;</li> -<li class="isub1">verification of law of error, <a href="#Page_385">385</a>.</li> - - -<li class="abet"><span class="alpha"><a id="IX_R"></a><a href="#alpha-table">R</a></span></li> -<li class="ifrst">Radian, <a href="#Page_306">306</a>.</li> - -<li class="indx">Radiant matter, <a href="#Page_642">642</a>.</li> - -<li class="indx">Radiation of heat, <a href="#Page_430">430</a>.</li> - -<li class="indx">Radiometer, <a href="#Page_435">435</a>.</li> - -<li class="indx">Rainbow, theory of, <a href="#Page_526">526</a>, <a href="#Page_533">533</a>.</li> - -<li class="indx">Rainfall, variation of, <a href="#Page_430">430</a>.</li> - -<li class="indx">Ramean tree, <a href="#Page_703">703</a>, <a href="#Page_704">704</a>.</li> - -<li class="indx">Ramsden’s balance, <a href="#Page_304">304</a>.</li> - -<li class="indx">Rankine, on specific heat of air, <a href="#Page_557">557</a>;</li> -<li class="isub1">reconcentration of energy, <a href="#Page_751">751</a>.</li> - -<li class="indx">Rational formulæ, <a href="#Page_489">489</a>.</li> - -<li class="indx">Rayleigh, Lord, on graphical method, <a href="#Page_495">495</a>.</li> - -<li class="indx">Reasoning, arithmetical, <a href="#Page_167">167</a>;</li> -<li class="isub1">numerically definite, <a href="#Page_168">168</a>;</li> -<li class="isub1">geometrical, <a href="#Page_458">458</a>.</li> - -<li class="indx">Recorde, Robert, <a href="#Page_15">15</a>.</li> - -<li class="indx">Reduction, of syllogisms, <a href="#Page_85">85</a>;</li> -<li class="isub1"><i>ad absurdum</i>, <a href="#Page_415">415</a>;</li> -<li class="isub1">of observations, <a href="#Page_552">552</a>, <a href="#Page_572">572</a>.</li> - -<li class="indx">Reflection, total, <a href="#Page_650">650</a>.</li> - -<li class="indx">Refraction, atmospheric, <a href="#Page_340">340</a>, <a href="#Page_356">356</a>, <a href="#Page_500">500</a>;</li> -<li class="isub1">law of, <a href="#Page_501">501</a>;</li> -<li class="isub1">conical, <a href="#Page_540">540</a>;</li> -<li class="isub1">double, <a href="#Page_585">585</a>.</li> - -<li class="indx">Regnault, dilatation of mercury, <a href="#Page_342">342</a>;</li> -<li class="isub1">measurement of heat, <a href="#Page_350">350</a>;</li> -<li class="isub1">exact experiment, <a href="#Page_397">397</a>;</li> -<li class="isub1">on Boyle’s law, <a href="#Page_468">468</a>, <a href="#Page_471">471</a>;</li> -<li class="isub1">latent heat of steam, <a href="#Page_487">487</a>;</li> -<li class="isub1">graphical method, <a href="#Page_494">494</a>;</li> -<li class="isub1">specific heat of air, <a href="#Page_557">557</a>.</li> - -<li class="indx">Reid, on bifurcate classification, <a href="#Page_697">697</a>.</li> - -<li class="indx">Reign of law, <a href="#Page_741">741</a>, <a href="#Page_759">759</a>.</li> - -<li class="indx">Rejection of observations, <a href="#Page_390">390</a>.</li> - -<li class="indx">Relation, sign of, <a href="#Page_17">17</a>;</li> -<li class="isub1">logic of, <a href="#Page_22">22</a>;</li> -<li class="isub1">logical, <a href="#Page_35">35</a>;</li> -<li class="isub1">axiom of, <a href="#Page_164">164</a>.</li> - -<li class="indx">Repetition, method of, <a href="#Page_287">287</a>, <a href="#Page_288">288</a>.</li> - -<li class="indx">Representative hypotheses, <a href="#Page_524">524</a>.</li> - -<li class="indx">Reproduction, modes of, <a href="#Page_730">730</a>.</li> - -<li class="indx">Reservation of judgment, <a href="#Page_592">592</a>.</li> - -<li class="indx">Residual effects, <a href="#Page_558">558</a>;</li> -<li class="isub1">phenomena, <a href="#Page_560">560</a>, <a href="#Page_569">569</a>.</li> - -<li class="indx">Resisting medium, <a href="#Page_310">310</a>, <a href="#Page_523">523</a>, <a href="#Page_570">570</a>.</li> - -<li class="indx">Resonance, <a href="#Page_453">453</a>.</li> - -<li class="indx">Reusch, on substitution, <a href="#Page_21">21</a>.</li> - -<li class="indx">Reversal, method of, <a href="#Page_354">354</a>.</li> - -<li class="indx">Revolution, quantity of, <a href="#Page_306">306</a>.</li> - -<li class="indx">Robertson, Prof. Croom, <a href="#Page_27">27</a>, <a href="#Page_101">101</a>.</li> - -<li class="indx">Robison, electric curves, <a href="#Page_446">446</a>.</li> - -<li class="indx">Rock-salt, <a href="#Page_609">609</a>.</li> - -<li class="indx">Rœmer, divided circle, <a href="#Page_355">355</a>;</li> -<li class="isub1">velocity of light, <a href="#Page_535">535</a>.</li> - -<li class="indx">Roscoe, Prof., photometrical researches, <a href="#Page_273">273</a>;</li> -<li class="isub1">solubility of salts, <a href="#Page_280">280</a>;</li> -<li class="isub1">constant flame, <a href="#Page_441">441</a>;</li> -<li class="isub1">absorption of gases, <a href="#Page_499">499</a>;</li> -<li class="isub1">vanadium, <a href="#Page_528">528</a>;</li> -<li class="isub1">atomic weight of vanadium, <a href="#Page_392">392</a>, <a href="#Page_649">649</a>.</li> - -<li class="indx">Rousseau on geometry, <a href="#Page_233">233</a>.</li> - -<li class="indx">Rules, of inference, <a href="#Page_9">9</a>, <a href="#Page_17">17</a>;</li> -<li class="isub1">indirect method of inference, <a href="#Page_89">89</a>;<span class="pagenum" id="Page_784">784</span></li> -<li class="isub1">for calculation of combinations, <a href="#Page_180">180</a>;</li> -<li class="isub1">of probabilities, <a href="#Page_203">203</a>;</li> -<li class="isub1">of inverse method, <a href="#Page_257">257</a>;</li> -<li class="isub1">for elimination of error, <a href="#Page_353">353</a>.</li> - -<li class="indx">Rumford, Count, experiments on heat, <a href="#Page_343">343</a>, <a href="#Page_350">350</a>, <a href="#Page_467">467</a>.</li> - -<li class="indx">Ruminants, Cuvier on, <a href="#Page_683">683</a>.</li> - -<li class="indx">Russell, Scott, on sound, <a href="#Page_541">541</a>.</li> - - -<li class="abet"><span class="alpha"><a id="IX_S"></a><a href="#alpha-table">S</a></span></li> -<li class="ifrst">Sample, use of, <a href="#Page_9">9</a>.</li> - -<li class="indx">Sandeman, on perigon, <a href="#Page_306">306</a>;</li> -<li class="isub1">approximate arithmetic, <a href="#Page_481">481</a>.</li> - -<li class="indx">Saturn, motions of satellites, <a href="#Page_293">293</a>;</li> -<li class="isub1">rings, <a href="#Page_293">293</a>.</li> - -<li class="indx">Schehallien, attraction of, <a href="#Page_369">369</a>, <a href="#Page_566">566</a>.</li> - -<li class="indx">Schottus, on combinations, <a href="#Page_179">179</a>.</li> - -<li class="indx">Schwabe, on sun-spots, <a href="#Page_452">452</a>.</li> - -<li class="indx">Science, nature of, <a href="#Page_1">1</a>, <a href="#Page_673">673</a>.</li> - -<li class="indx">Selenium, <a href="#Page_663">663</a>, <a href="#Page_670">670</a>.</li> - -<li class="indx">Self-contradiction, <a href="#Page_32">32</a>.</li> - -<li class="indx">Senior’s definition of wealth, <a href="#Page_75">75</a>.</li> - -<li class="indx">Senses, fallacious indications of, <a href="#Page_276">276</a>.</li> - -<li class="indx">Seven, coincidences of number, <a href="#Page_262">262</a>;</li> -<li class="isub1">fallacies of, <a href="#Page_624">624</a>.</li> - -<li class="indx">Sextus, fatality of name, <a href="#Page_264">264</a>.</li> - -<li class="indx">Sieve of Eratosthenes, <a href="#Page_82">82</a>, <a href="#Page_123">123</a>, <a href="#Page_139">139</a>.</li> - -<li class="indx">Similars, substitution of, <a href="#Page_17">17</a>.</li> - -<li class="indx">Simple identity, <a href="#Page_37">37</a>, <a href="#Page_111">111</a>;</li> -<li class="isub1">inference of, <a href="#Page_58">58</a>;</li> -<li class="isub1">contrapositive, <a href="#Page_86">86</a>;</li> -<li class="isub1">induction of, <a href="#Page_127">127</a>.</li> - -<li class="indx">Simple statement, <a href="#Page_143">143</a>.</li> - -<li class="indx">Simplicity, law of, <a href="#Page_33">33</a>, <a href="#Page_58">58</a>, <a href="#Page_72">72</a>.</li> - -<li class="indx">Simpson, discovery of property of chloroform, <a href="#Page_531">531</a>.</li> - -<li class="indx">Simultaneity of knowledge, <a href="#Page_34">34</a>.</li> - -<li class="indx">Singular names, <a href="#Page_27">27</a>;</li> -<li class="isub1">terms, <a href="#Page_129">129</a>.</li> - -<li class="indx">Siren, <a href="#Page_10">10</a>, <a href="#Page_298">298</a>, <a href="#Page_421">421</a>.</li> - -<li class="indx">Slate, the logical, <a href="#Page_95">95</a>.</li> - -<li class="indx">Smeaton’s experiments, on water-wheels, <a href="#Page_347">347</a>;</li> -<li class="isub1">windmills, <a href="#Page_401">401</a>, <a href="#Page_441">441</a>.</li> - -<li class="indx">Smee, Alfred, logical machines, <a href="#Page_107">107</a>.</li> - -<li class="indx">Smell, delicacy of, <a href="#Page_437">437</a>.</li> - -<li class="indx">Smithsonian Institution, <a href="#Page_329">329</a>.</li> - -<li class="indx">Smyth, Prof. Piazzi, <a href="#Page_452">452</a>.</li> - -<li class="indx">Socrates, on the sun, <a href="#Page_611">611</a>.</li> - -<li class="indx">Solids, <a href="#Page_602">602</a>.</li> - -<li class="indx">Solubility of salts, <a href="#Page_279">279</a>.</li> - -<li class="indx"><i>Some</i>, the adjective, <a href="#Page_41">41</a>, <a href="#Page_56">56</a>.</li> - -<li class="indx">Sorites, <a href="#Page_60">60</a>.</li> - -<li class="indx">Sound, observations on, <a href="#Page_356">356</a>;</li> -<li class="isub1">undulations, <a href="#Page_405">405</a>, <a href="#Page_421">421</a>;</li> -<li class="isub1">velocity of, <a href="#Page_571">571</a>;</li> -<li class="isub1">classification of sounds, <a href="#Page_732">732</a>.</li> - -<li class="indx">Space, relations of, <a href="#Page_220">220</a>.</li> - -<li class="indx">Species, <a href="#Page_698">698</a>;</li> -<li class="isub1">infima, <a href="#Page_701">701</a>;</li> -<li class="isub1">natural, <a href="#Page_724">724</a>.</li> - -<li class="indx">Specific gravities, <a href="#Page_301">301</a>;</li> -<li class="isub1">heat of air, <a href="#Page_557">557</a>.</li> - -<li class="indx">Spence, on boiling point, <a href="#Page_546">546</a>.</li> - -<li class="indx">Spencer, Herbert, nature of logic, <a href="#Page_4">4</a>, <a href="#Page_7">7</a>;</li> -<li class="isub1">sign of equality, <a href="#Page_15">15</a>;</li> -<li class="isub1">rhythmical motion, <a href="#Page_448">448</a>;</li> -<li class="isub1">abstraction, <a href="#Page_705">705</a>;</li> -<li class="isub1">philosophy of, <a href="#Page_718">718</a>, <a href="#Page_761">761</a>, <a href="#Page_762">762</a>.</li> - -<li class="indx">Spectroscope, <a href="#Page_437">437</a>.</li> - -<li class="indx">Spectrum, <a href="#Page_583">583</a>.</li> - -<li class="indx">Spiritualism, <a href="#Page_671">671</a>.</li> - -<li class="indx">Spontaneous generation, <a href="#Page_432">432</a>.</li> - -<li class="indx">Standards of measurement, <a href="#Page_305">305</a>;</li> -<li class="isub1">the bar, <a href="#Page_312">312</a>;</li> -<li class="isub1">terrestrial, <a href="#Page_314">314</a>;</li> -<li class="isub1">pendulum, <a href="#Page_315">315</a>;</li> -<li class="isub1">provisional, <a href="#Page_318">318</a>;</li> -<li class="isub1">natural system, <a href="#Page_319">319</a>.</li> - -<li class="indx">Stars, discs of, <a href="#Page_277">277</a>;</li> -<li class="isub1">motions of, <a href="#Page_280">280</a>, <a href="#Page_474">474</a>;</li> -<li class="isub1">variations of, <a href="#Page_281">281</a>;</li> -<li class="isub1">approach or recess, <a href="#Page_298">298</a>;</li> -<li class="isub1">standard stars, <a href="#Page_301">301</a>;</li> -<li class="isub1">apparent diameter, <a href="#Page_390">390</a>;</li> -<li class="isub1">variable, <a href="#Page_450">450</a>;</li> -<li class="isub1">proper motions, <a href="#Page_572">572</a>;</li> -<li class="isub1">Bruno on, <a href="#Page_639">639</a>;</li> -<li class="isub1">new, <a href="#Page_644">644</a>;</li> -<li class="isub1">pole-star, <a href="#Page_652">652</a>;</li> -<li class="isub1">conflict with wandering stars, <a href="#Page_748">748</a>.</li> - -<li class="indx">Stas, M., his balance, <a href="#Page_304">304</a>;</li> -<li class="isub1">on atomic weights, <a href="#Page_464">464</a>.</li> - -<li class="indx">Statements, kinds of, <a href="#Page_144">144</a>.</li> - -<li class="indx">Statistical conditions, <a href="#Page_168">168</a>.</li> - -<li class="indx">Stevinus, on inclined plane, <a href="#Page_622">622</a>.</li> - -<li class="indx">Stewart, Professor Balfour, on resisting medium, <a href="#Page_570">570</a>;</li> -<li class="isub1">theory of exchanges, <a href="#Page_571">571</a>.</li> - -<li class="indx">Stifels, arithmetical triangle, <a href="#Page_182">182</a>.</li> - -<li class="indx">Stokes, Professor, on resistance, <a href="#Page_475">475</a>;</li> -<li class="isub1">fluorescence, <a href="#Page_664">664</a>.</li> - -<li class="indx">Stone, E. J., heat of the stars, <a href="#Page_370">370</a>;</li> -<li class="isub1">temperature of earth’s surface, <a href="#Page_452">452</a>;</li> -<li class="isub1">transit of Venus, <a href="#Page_562">562</a>.</li> - -<li class="indx">Struve on double stars, <a href="#Page_247">247</a>.</li> - -<li class="indx">Substantial terms, <a href="#Page_28">28</a>.</li> - -<li class="indx">Substantives, <a href="#Page_14">14</a>.</li> - -<li class="indx">Substitution of similars, <a href="#Page_17">17</a>, <a href="#Page_45">45</a>, <a href="#Page_49">49</a>, <a href="#Page_104">104</a>, <a href="#Page_106">106</a>;</li> -<li class="isub1">anticipations of, <a href="#Page_21">21</a>.</li> - -<li class="indx">Substitutive weighing, <a href="#Page_345">345</a>.</li> - -<li class="indx"><i>Sui generis</i>, <a href="#Page_629">629</a>, <a href="#Page_728">728</a>.</li> - -<li class="indx">Sulphur, <a href="#Page_670">670</a>.</li> - -<li class="indx">Summum genus, <a href="#Page_93">93</a>, <a href="#Page_701">701</a>.</li> - -<li class="indx">Sun, distance, <a href="#Page_560">560</a>;</li> -<li class="isub1">variations of spots, <a href="#Page_452">452</a>.</li> - -<li class="indx">Superposition, of small effects, <a href="#Page_450">450</a>;</li> -<li class="isub1">small motions, <a href="#Page_476">476</a>.</li> - -<li class="indx">Swan, W., on sodium light, <a href="#Page_430">430</a>.<span class="pagenum" id="Page_785">785</span></li> - -<li class="indx">Syllogism, <a href="#Page_140">140</a>;</li> -<li class="isub1">moods of, <a href="#Page_55">55</a>, <a href="#Page_84">84</a>, <a href="#Page_85">85</a>, <a href="#Page_88">88</a>, <a href="#Page_105">105</a>, <a href="#Page_141">141</a>;</li> -<li class="isub1">numerically definite, <a href="#Page_168">168</a>.</li> - -<li class="indx">Symbols, use of, <a href="#Page_13">13</a>, <a href="#Page_31">31</a>, <a href="#Page_32">32</a>;</li> -<li class="isub1">of quantity, <a href="#Page_33">33</a>.</li> - -<li class="indx">Synthesis, <a href="#Page_122">122</a>;</li> -<li class="isub1">of terms, <a href="#Page_30">30</a>.</li> - - -<li class="abet"><span class="alpha"><a id="IX_T"></a><a href="#alpha-table">T</a></span></li> -<li class="ifrst">Table-turning, <a href="#Page_671">671</a>.</li> - -<li class="indx">Tacit knowledge, <a href="#Page_43">43</a>.</li> - -<li class="indx">Tacquet on combinations, <a href="#Page_179">179</a>.</li> - -<li class="indx">Tait, P. G., <a href="#Page_375">375</a>;</li> -<li class="isub1">theory of comets, <a href="#Page_571">571</a>.</li> - -<li class="indx">Talbot on the spectrum, <a href="#Page_429">429</a>.</li> - -<li class="indx">Tartaglia on projectiles, <a href="#Page_466">466</a>.</li> - -<li class="indx">Tastes, classification of, <a href="#Page_732">732</a>.</li> - -<li class="indx">Tautologous propositions, <a href="#Page_119">119</a>.</li> - -<li class="indx">Teeth, use in classification, <a href="#Page_710">710</a>.</li> - -<li class="indx">Temperature, variations of, <a href="#Page_453">453</a>.</li> - -<li class="indx">Tension of aqueous vapour, <a href="#Page_500">500</a>.</li> - -<li class="indx">Terms, <a href="#Page_24">24</a>;</li> -<li class="isub1">abstract, <a href="#Page_27">27</a>;</li> -<li class="isub1">substantial, <a href="#Page_28">28</a>;</li> -<li class="isub1">collective, <a href="#Page_29">29</a>;</li> -<li class="isub1">synthesis of, <a href="#Page_30">30</a>;</li> -<li class="isub1">negative, <a href="#Page_45">45</a>.</li> - -<li class="indx">Terrot, Bishop, on probability, <a href="#Page_212">212</a>.</li> - -<li class="indx">Test experiments, <a href="#Page_347">347</a>, <a href="#Page_433">433</a>.</li> - -<li class="indx">Tetractys, <a href="#Page_95">95</a>.</li> - -<li class="indx">Thales, predicted eclipse, <a href="#Page_537">537</a>.</li> - -<li class="indx">Theory, results of, <a href="#Page_534">534</a>;</li> -<li class="isub1">facts known by, <a href="#Page_547">547</a>;</li> -<li class="isub1">quantitative, <a href="#Page_551">551</a>;</li> -<li class="isub1">of exchanges, <a href="#Page_571">571</a>;</li> -<li class="isub1">freedom of forming, <a href="#Page_577">577</a>;</li> -<li class="isub1">of evolution, <a href="#Page_761">761</a>.</li> - -<li class="indx">Thermometer, differential, <a href="#Page_345">345</a>;</li> -<li class="isub1">reading of, <a href="#Page_390">390</a>;</li> -<li class="isub1">change of zero, <a href="#Page_390">390</a>.</li> - -<li class="indx">Thermopile, <a href="#Page_300">300</a>.</li> - -<li class="indx">Thomas, arithmetical machine, <a href="#Page_107">107</a>.</li> - -<li class="indx">Thomson, Archbishop, <a href="#Page_50">50</a>, <a href="#Page_61">61</a>.</li> - -<li class="indx">Thomson, James, prediction by, <a href="#Page_542">542</a>;</li> -<li class="isub1">on gaseous state, <a href="#Page_654">654</a>.</li> - -<li class="indx">Thomson, Sir W., lighthouse signals, <a href="#Page_194">194</a>;</li> -<li class="isub1">size of atoms, <a href="#Page_195">195</a>;</li> -<li class="isub1">tides, <a href="#Page_450">450</a>;</li> -<li class="isub1">capillary attraction, <a href="#Page_614">614</a>;</li> -<li class="isub1">magnetism, <a href="#Page_665">665</a>;</li> -<li class="isub1">dissipation of energy, <a href="#Page_744">744</a>.</li> - -<li class="indx">Thomson and Tait, chronometry, <a href="#Page_311">311</a>;</li> -<li class="isub1">standards of length, <a href="#Page_315">315</a>;</li> -<li class="isub1">the crowbar, <a href="#Page_460">460</a>;</li> -<li class="isub1">polarised light, <a href="#Page_653">653</a>.</li> - -<li class="indx">Thomson, Sir Wyville, <a href="#Page_412">412</a>.</li> - -<li class="indx">Thunder-cloud, <a href="#Page_612">612</a>.</li> - -<li class="indx">Tides, <a href="#Page_366">366</a>, <a href="#Page_450">450</a>, <a href="#Page_476">476</a>, <a href="#Page_541">541</a>;</li> -<li class="isub1">velocity of, <a href="#Page_298">298</a>;</li> -<li class="isub1">gauge, <a href="#Page_368">368</a>;</li> -<li class="isub1">atmospheric, <a href="#Page_367">367</a>, <a href="#Page_553">553</a>.</li> - -<li class="indx">Time, <a href="#Page_220">220</a>;</li> -<li class="isub1">definition of, <a href="#Page_307">307</a>.</li> - -<li class="indx">Todhunter, Isaac, <i>History of the Theory of Probability</i>, <a href="#Page_256">256</a>, <a href="#Page_375">375</a>, <a href="#Page_395">395</a>;</li> -<li class="isub1">on insoluble problems, <a href="#Page_757">757</a>.</li> - -<li class="indx">Tooke, Horne, on cause, <a href="#Page_226">226</a>.</li> - -<li class="indx">Torricelli, cycloid, <a href="#Page_235">235</a>;</li> -<li class="isub1">his theorem, <a href="#Page_605">605</a>;</li> -<li class="isub1">on barometer, <a href="#Page_666">666</a>.</li> - -<li class="indx">Torsion balance, <a href="#Page_272">272</a>, <a href="#Page_287">287</a>.</li> - -<li class="indx">Transit of Venus, <a href="#Page_294">294</a>, <a href="#Page_348">348</a>, <a href="#Page_562">562</a>.</li> - -<li class="indx">Transit-circle, <a href="#Page_355">355</a>.</li> - -<li class="indx">Tree of Porphyry, <a href="#Page_702">702</a>;</li> -<li class="isub1">of Ramus, <a href="#Page_703">703</a>.</li> - -<li class="indx">Triangle, arithmetical, <a href="#Page_93">93</a>, <a href="#Page_182">182</a>.</li> - -<li class="indx">Triangular numbers, <a href="#Page_185">185</a>.</li> - -<li class="indx">Trigonometrical survey, <a href="#Page_301">301</a>;</li> -<li class="isub1">calculations of, <a href="#Page_756">756</a>.</li> - -<li class="indx">Trisection of angles, <a href="#Page_414">414</a>.</li> - -<li class="indx">Tuning-fork, <a href="#Page_541">541</a>.</li> - -<li class="indx">Tycho Brahe, <a href="#Page_271">271</a>;</li> -<li class="isub1">on star discs, <a href="#Page_277">277</a>;</li> -<li class="isub1">obliquity of earth’s axis, <a href="#Page_289">289</a>;</li> -<li class="isub1">circumpolar stars, <a href="#Page_366">366</a>;</li> -<li class="isub1">Sirius, <a href="#Page_390">390</a>.</li> - -<li class="indx">Tyndall, Professor, on natural constants, <a href="#Page_328">328</a>;</li> -<li class="isub1">magnetism of gases, <a href="#Page_352">352</a>;</li> -<li class="isub1">precaution in experiments, <a href="#Page_431">431</a>;</li> -<li class="isub1">use of imagination, <a href="#Page_509">509</a>;</li> -<li class="isub1">on Faraday, <a href="#Page_547">547</a>;</li> -<li class="isub1">magnetism, <a href="#Page_549">549</a>, <a href="#Page_607">607</a>;</li> -<li class="isub1">scope for discovery, <a href="#Page_753">753</a>.</li> - -<li class="indx">Types, of logical conditions, <a href="#Page_140">140</a>, <a href="#Page_144">144</a>;</li> -<li class="isub1">of statements, <a href="#Page_145">145</a>;</li> -<li class="isub1">classification by, <a href="#Page_722">722</a>.</li> - - -<li class="abet"><span class="alpha"><a id="IX_U"></a><a href="#alpha-table">U</a></span></li> -<li class="ifrst">Ueberweg’s logic, <a href="#Page_6">6</a>.</li> - -<li class="indx">Ultimate statements, <a href="#Page_144">144</a>.</li> - -<li class="indx">Undistributed, attribute, <a href="#Page_40">40</a>;</li> -<li class="isub1">middle term, <a href="#Page_64">64</a>, <a href="#Page_103">103</a>.</li> - -<li class="indx">Undulations, of light, <a href="#Page_558">558</a>;</li> -<li class="isub1">analogy in theory of, <a href="#Page_635">635</a>.</li> - -<li class="indx">Undulatory theory, <a href="#Page_468">468</a>, <a href="#Page_520">520</a>, <a href="#Page_538">538</a>, <a href="#Page_540">540</a>;</li> -<li class="isub1">inconceivability of, <a href="#Page_510">510</a>.</li> - -<li class="indx">Unique objects, <a href="#Page_728">728</a>.</li> - -<li class="indx">Unit, definition of, <a href="#Page_157">157</a>;</li> -<li class="isub1">groups, <a href="#Page_167">167</a>;</li> -<li class="isub1">of measurement, <a href="#Page_305">305</a>;</li> -<li class="isub1">arcual, <a href="#Page_306">306</a>;</li> -<li class="isub1">of time, <a href="#Page_307">307</a>;</li> -<li class="isub1">space, <a href="#Page_312">312</a>;</li> -<li class="isub1">density, <a href="#Page_316">316</a>;</li> -<li class="isub1">mass, <a href="#Page_317">317</a>;</li> -<li class="isub1">subsidiary, <a href="#Page_320">320</a>;</li> -<li class="isub1">derived, <a href="#Page_321">321</a>;</li> -<li class="isub1">provisional, <a href="#Page_323">323</a>;</li> -<li class="isub1">of heat, <a href="#Page_325">325</a>;</li> -<li class="isub1">magnetical and electrical units, <a href="#Page_326">326</a>, <a href="#Page_327">327</a>.</li> - -<li class="indx">Unity, law of, <a href="#Page_72">72</a>.</li> - -<li class="indx">Universe, logical, <a href="#Page_43">43</a>;</li> -<li class="isub1">infiniteness of, <a href="#Page_738">738</a>;</li> -<li class="isub1">heat-history of, <a href="#Page_744">744</a>, <a href="#Page_749">749</a>;</li> -<li class="isub1">possible states of, <a href="#Page_749">749</a>.</li> - -<li class="indx">Uranus, anomalies of, <a href="#Page_660">660</a>.</li> - - -<li class="abet"><span class="alpha"><a id="IX_V"></a><a href="#alpha-table">V</a></span></li> -<li class="ifrst">Vacuum, Nature’s abhorrence of, <a href="#Page_513">513</a>.</li> - -<li class="indx">Vapour densities, <a href="#Page_548">548</a>.</li> - -<li class="indx">Variable, variant, <a href="#Page_440">440</a>, <a href="#Page_441">441</a>, <a href="#Page_483">483</a>.<span class="pagenum" id="Page_786">786</span></li> - -<li class="indx">Variation, linear, elliptic, &c., <a href="#Page_474">474</a>;</li> -<li class="isub1">method of, <a href="#Page_439">439</a>.</li> - -<li class="indx">Variations, logical, <a href="#Page_140">140</a>;</li> -<li class="isub1">periodic, <a href="#Page_447">447</a>;</li> -<li class="isub1">combined, <a href="#Page_450">450</a>;</li> -<li class="isub1">integrated, <a href="#Page_452">452</a>;</li> -<li class="isub1">simple proportional, <a href="#Page_501">501</a>.</li> - -<li class="indx">Variety, of nature, <a href="#Page_173">173</a>;</li> -<li class="isub1">of nature and art, <a href="#Page_190">190</a>;</li> -<li class="isub1">higher orders of, <a href="#Page_192">192</a>.</li> - -<li class="indx">Velocity, unit of, <a href="#Page_321">321</a>.</li> - -<li class="indx">Venn, Rev. John, logical problem by, <a href="#Page_90">90</a>;</li> -<li class="isub1">on Boole, <a href="#Page_155">155</a>;</li> -<li class="isub1">his work on <i>Logic of Chance</i>, <a href="#Page_394">394</a>.</li> - -<li class="indx">Venus, <a href="#Page_449">449</a>;</li> -<li class="isub1">transits of, <a href="#Page_294">294</a>.</li> - -<li class="indx">Verses, Protean, <a href="#Page_175">175</a>.</li> - -<li class="indx">Vibrations, law of, <a href="#Page_295">295</a>;</li> -<li class="isub1">principle of forced, <a href="#Page_451">451</a>;</li> -<li class="isub1">co-existence of small, <a href="#Page_476">476</a>.</li> - -<li class="indx">Vital force, <a href="#Page_523">523</a>.</li> - -<li class="indx">Voltaire on fossils, <a href="#Page_661">661</a>.</li> - -<li class="indx">Vortices, theory of, <a href="#Page_513">513</a>, <a href="#Page_517">517</a>.</li> - -<li class="indx">Vulcan, supposed planet, <a href="#Page_414">414</a>.</li> - - -<li class="abet"><span class="alpha"><a id="IX_W"></a><a href="#alpha-table">W</a></span></li> -<li class="ifrst">Wallis, <a href="#Page_124">124</a>, <a href="#Page_175">175</a>.</li> - -<li class="indx">Water, compressibility of, <a href="#Page_338">338</a>;</li> -<li class="isub1">properties of, <a href="#Page_610">610</a>.</li> - -<li class="indx">Watt’s parallel motion, <a href="#Page_462">462</a>.</li> - -<li class="indx">Waves, <a href="#Page_599">599</a>, <a href="#Page_635">635</a>;</li> -<li class="isub1">nature of, <a href="#Page_468">468</a>;</li> -<li class="isub1">in canals, <a href="#Page_535">535</a>;</li> -<li class="isub1">earthquake, <a href="#Page_297">297</a>.</li> - -<li class="indx">Weak arguments, effect of, <a href="#Page_211">211</a>.</li> - -<li class="indx">Wells, on dew, <a href="#Page_425">425</a>.</li> - -<li class="indx">Wenzel, on neutral salts, <a href="#Page_295">295</a>.</li> - -<li class="indx">Whately, disjunctive propositions, <a href="#Page_69">69</a>;</li> -<li class="isub1">probable arguments, <a href="#Page_210">210</a>.</li> - -<li class="indx">Wheatstone, cipher, <a href="#Page_124">124</a>;</li> -<li class="isub1">galvanometer, <a href="#Page_286">286</a>;</li> -<li class="isub1">revolving mirror, <a href="#Page_299">299</a>, <a href="#Page_308">308</a>;</li> -<li class="isub1">kaleidophone, <a href="#Page_445">445</a>;</li> -<li class="isub1">velocity of electricity, <a href="#Page_543">543</a>.</li> - -<li class="indx">Whewell, on tides, <a href="#Page_371">371</a>, <a href="#Page_542">542</a>;</li> -<li class="isub1">method of least squares, <a href="#Page_386">386</a>.</li> - -<li class="indx">Whitworth, Sir Joseph, <a href="#Page_304">304</a>, <a href="#Page_436">436</a>.</li> - -<li class="indx">Whitworth, Rev. W. A., on <i>Choice and Chance</i>, <a href="#Page_395">395</a>.</li> - -<li class="indx">Wilbraham, on Boole, <a href="#Page_206">206</a>.</li> - -<li class="indx">Williamson, Professor A. W., chemical unit, <a href="#Page_321">321</a>;</li> -<li class="isub1">prediction by, <a href="#Page_544">544</a>.</li> - -<li class="indx">Wollaston, the goniometer, <a href="#Page_287">287</a>;</li> -<li class="isub1">light of moon, <a href="#Page_302">302</a>;</li> -<li class="isub1">spectrum, <a href="#Page_429">429</a>.</li> - -<li class="indx">Wren, Sir C., on gravity, <a href="#Page_581">581</a>.</li> - - -<li class="abet"><span class="alpha"><a id="IX_X"></a><a href="#alpha-table">X</a></span></li> -<li class="ifrst">X, the substance, <a href="#Page_523">523</a>.</li> - - -<li class="abet"><span class="alpha"><a id="IX_Y"></a><a href="#alpha-table">Y</a></span></li> -<li class="ifrst">Yard, standard, <a href="#Page_397">397</a>.</li> - -<li class="indx">Young, Dr. Thomas, tension of aqueous vapour, <a href="#Page_500">500</a>;</li> -<li class="isub1">use of hypotheses, <a href="#Page_508">508</a>;</li> -<li class="isub1">ethereal medium, <a href="#Page_515">515</a>.</li> - - -<li class="abet"><span class="alpha"><a id="IX_Z"></a><a href="#alpha-table">Z</a></span></li> -<li class="ifrst">Zero point, <a href="#Page_368">368</a>.</li> - -<li class="indx">Zodiacal light, <a href="#Page_276">276</a>.</li> - -<li class="indx">Zoology, <a href="#Page_666">666</a>.</li> -</ul> - - - -<p class="tac fs60 mt6em">LONDON: R. 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KING AND CO.</p> - - -<div class="footnotes"><h3>FOOTNOTES:</h3> - -<div class="footnote"> - -<p><a id="Footnote_1" href="#FNanchor_1" class="label">1</a> -Since the above was written Mr. Harley has read an account of Stanhope’s -logical remains at the Dublin Meeting (1878) of the British -Association. The paper will be printed in <i>Mind</i>. (Note added November, -1878.)</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_2" href="#FNanchor_2" class="label">2</a> -Leibnitii <i>Opera Philosophica quæ extant</i>. Erdmann, Pars I. Berolini, -1840, p. 94.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_3" href="#FNanchor_3" class="label">3</a> Erdmann, p. 102.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_4" href="#FNanchor_4" class="label">4</a> -Ibid. p. 98.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_5" href="#FNanchor_5" class="label">5</a> -Erdmann, p. 100.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_6" href="#FNanchor_6" class="label">6</a> -Fifth Edition, 1860, p. 158.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_7" href="#FNanchor_7" class="label">7</a> -Section 120.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_8" href="#FNanchor_8" class="label">8</a> -See his “Remarks on Boole’s Mathematical Analysis of Logic.” -<i>Report of the 36th Meeting of the British Association, Transactions of the -Sections</i>, pp. 3–6.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_9" href="#FNanchor_9" class="label">9</a> -Hamilton’s Lectures, vol. iv. p. 319.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_10" href="#FNanchor_10" class="label">10</a> -Ibid. p. 326.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_11" href="#FNanchor_11" class="label">11</a> -<i>Pure Logic, or the Logic of Quality apart from Quantity; with -Remarks on Boole’s System, and on the Relation of Logic and Mathematics.</i> -London, 1864, p. 3.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_12" href="#FNanchor_12" class="label">12</a> -<i>La Philosophie Positive</i>, Mai-Juin, 1877, tom. xviii. p. 456.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_13" href="#FNanchor_13" class="label">13</a> -<i>Inventum Novum Quadrati Logici</i>, &c., Gissæ Hassorum, 1714, -8vo.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_14" href="#FNanchor_14" class="label">14</a> -See <i>Ueberweg’s System of Logic</i>, &c., translated by Lindsay, p. 302.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_15" href="#FNanchor_15" class="label">15</a> -Since the above was written M. Liard has republished this exposition -as one chapter of an interesting and admirably lucid account of the -progress of logical science in England. After a brief but clear introduction, -treating of the views of Herschel, Mill, and others concerning -Inductive Logic, M. Liard describes in succession the logical systems of -George Bentham, Hamilton, De Morgan, Boole, and that contained in -the present work. The title of the book is as follows:—<i>Les Logiciens -Anglais Contemporains</i>. Par Louis Liard, Professeur de Philosophie à -la Faculté des Lettres de Bordeaux. Paris: Librairie Germer Baillière. -1878. (Note added November, 1878.)</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_16" href="#FNanchor_16" class="label">16</a> -<i>Spectator</i>, September 19, 1874, p. 1178. A second portion of the -review appeared in the same journal for September 26, 1874, p. 1204.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_17" href="#FNanchor_17" class="label">17</a> -<i>Mind</i>: a Quarterly Review of Psychology and Philosophy. No. II. -April 1876. Vol. I. p. 206.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_18" href="#FNanchor_18" class="label">18</a> -Portions of this work have already been published in my articles, -entitled “John Stuart Mill’s Philosophy Tested,” printed in the <i>Contemporary -Review</i> for December, 1877, vol. xxxi. p. 167, and for January and -April, 1878, vol. xxxi. p. 256, and vol. xxxii. p. 88. (Note added in -November, 1878.)</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_19" href="#FNanchor_19" class="label">19</a> -<i>Mind</i>, vol. i. p. 222.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_20" href="#FNanchor_20" class="label">20</a> -<i>Fortnightly Review</i>, New Series, April 1875, p. 480. Lecture reprinted -by the Sunday Lecture Society, p. 24.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_21" href="#FNanchor_21" class="label">21</a> -Sir W. Thomson’s words are as follows (<i>Cambridge Mathematical -Journal</i>, Nov. 1842, vol. iii. p. 174). “When <i>x</i> is negative, the state -represented cannot be the result of any <i>possible</i> distribution of temperature -which has previously existed.” There is no limitation in the -sentence to the laws of conduction, but, as the whole paper treats of the -results of conduction in a solid, it may no doubt be understood that there -is a <i>tacit</i> limitation. See also a second paper on the subject in the same -journal for February, 1844, vol. iv. p. 67, where again there is no expressed -limitation.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_22" href="#FNanchor_22" class="label">22</a> -Pp. 25–26. The parentheses are in the original, and show Professor -Tait’s corrections in the verbatim reports of his lectures. The subject is -treated again on pp. 168–9.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_23" href="#FNanchor_23" class="label">23</a> -<i>Theory of Heat</i> 1871, p. 245.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_24" href="#FNanchor_24" class="label">24</a> -<i>The Senses and the Intellect</i>, Second Ed., pp. 5, 325, &c.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_25" href="#FNanchor_25" class="label">25</a> -Max Müller, <i>Lectures on the Science of Language</i>, Second Series, -vol. ii. p. 63; or Sixth Edition, vol. ii. p. 67. The view of the etymological -meaning of “intellect” is given above on the authority of Professor -Max Müller. It seems to be opposed to the ordinary opinion, according -to which the Latin <i>intelligere</i> means to choose between, to see a difference -between, to discriminate, instead of to unite.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_26" href="#FNanchor_26" class="label">26</a> -Hartley on Man, vol. i. p. 359.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_27" href="#FNanchor_27" class="label">27</a> -<i>Principles of Psychology</i>, Second Ed., vol. ii. p. 86.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_28" href="#FNanchor_28" class="label">28</a> -<i>Pure Logic, or the Logic of Quality apart from Quantity</i>, 1864, -pp. 10, 16, 22, 29, 36, &c.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_29" href="#FNanchor_29" class="label">29</a> -Brewster, <i>Treatise on New Philosophical Instruments</i>, p. 273. -Concerning this method see also Whewell, <i>Philosophy of the Inductive -Sciences</i>, vol. ii. p. 355; Tomlinson, <i>Philosophical Magazine</i>, Fourth -Series, vol. xl. p. 328; Tyndall, in Youmans’ <i>Modern Culture</i>, p. 16.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_30" href="#FNanchor_30" class="label">30</a> -<i>Formal Logic</i>, p. 38.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_31" href="#FNanchor_31" class="label">31</a> -Hallam’s <i>Literature of Europe</i>, First Ed., vol. ii. p. 444.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_32" href="#FNanchor_32" class="label">32</a> -<i>Outline of a New System of Logic</i>, London, 1827, pp. 133, &c.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_33" href="#FNanchor_33" class="label">33</a> -<i>An Investigation of the Laws of Thought</i>, pp. 27, &c.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_34" href="#FNanchor_34" class="label">34</a> -<i>Formal Logic</i>, pp. 82, 106. In his later work, <i>The Syllabus of a -New System of Logic</i>, he discontinued the use of the sign.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_35" href="#FNanchor_35" class="label">35</a> -<i>Principles of Psychology</i>, Second Ed., vol. ii. pp. 54, 55.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_36" href="#FNanchor_36" class="label">36</a> -<i>Pure Logic, or the Logic of Quality</i>, p. 14.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_37" href="#FNanchor_37" class="label">37</a> -<i>Pure Logic</i>, pp. 18, 19.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_38" href="#FNanchor_38" class="label">38</a> -Ueberweg’s <i>System of Logic</i>, transl. by Lindsay, pp. 442–446, -571, 572. The anticipations of the principle of substitution to be -found in the works of Leibnitz, Reusch, and perhaps other German -logicians, will be noticed in the preface to this second edition.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_39" href="#FNanchor_39" class="label">39</a> -<i>Substitution of Similars</i> (1869), p. 9.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_40" href="#FNanchor_40" class="label">40</a> -<i>Port-Royal Logic</i>, transl. by Spencer Baynes, pp. 212–219. -Part III. chap. x. and xi.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_41" href="#FNanchor_41" class="label">41</a> -<i>Description of a Notation for the Logic of Relatives, resulting -from an Amplification of the Conceptions of Boole’s Calculus of Logic.</i> -By C. S. Peirce. <i>Memoirs of the American Academy</i>, vol. ix. Cambridge, -U.S., 1870.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_42" href="#FNanchor_42" class="label">42</a> -<i>On the Syllogism No IV., and on the Logic of Relations.</i> By -Augustus De Morgan. <i>Transactions of the Cambridge Philosophical -Society</i>, vol. x. part ii., 1860.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_43" href="#FNanchor_43" class="label">43</a> -<i>Observations on Boole’s Laws of Thought.</i> By the late R. Leslie -Ellis; communicated by the Rev. Robert Harley, F.R.S. <i>Report of -the British Association</i>, 1870. <i>Report of Sections</i>, p. 12. Also, <i>On -Boole’s Laws of Thought</i>. By the Rev. Robert Harley, F.R.S., <i>ibid.</i> -p. 14.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_44" href="#FNanchor_44" class="label">44</a> -Jevons’ <i>Elementary Lessons in Logic</i>, pp. 41–43; <i>Pure Logic</i>, p. 6. -See also J. S. Mill, <i>System of Logic</i>, Book I. chap. ii. section 5, and -Shedden’s <i>Elements of Logic</i>, London, 1864, pp. 14, &c. Professor -Robertson objects (<i>Mind</i>, vol. i. p. 210) that I confuse <i>singular</i> and -<i>proper</i> names; if so, it is because I hold that the same remarks apply -to proper names, which do not seem to me to differ logically from -singular names.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_45" href="#FNanchor_45" class="label">45</a> -Professor Robertson has criticised my introduction of “Substantial -Terms” (<i>Mind</i>, vol. i. p. 210), and objects, perhaps correctly, that the -distinction if valid is extra-logical. I am inclined to think, however, -that the doctrine of terms is, strictly speaking, for the most part -extra-logical.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_46" href="#FNanchor_46" class="label">46</a> -<i>Mathematical Analysis of Logic</i>, Cambridge, 1847, p. 17. <i>An -Investigation of the Laws of Thought</i>, London, 1854, p. 31.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_47" href="#FNanchor_47" class="label">47</a> -<i>Pure Logic</i>, p. 15.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_48" href="#FNanchor_48" class="label">48</a> -“Velut si dicam, Sol, Sol, Sol, non tres soles effecerim, sed uno -toties prædicaverim.”</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_49" href="#FNanchor_49" class="label">49</a> -Book i., Part iv., Section 5.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_50" href="#FNanchor_50" class="label">50</a> -<i>Laws of Thought</i>, p. 29. It is pointed out in the preface to this -Second Edition that Leibnitz was acquainted with the Laws of -Simplicity and of Commutativeness.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_51" href="#FNanchor_51" class="label">51</a> -<i>Prior Analytics</i>, i. cap. xxvii. 3.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_52" href="#FNanchor_52" class="label">52</a> -<i>Encyclopædia Britannica</i>, Eighth Ed. art. Logic, sect. 37, note. -8vo. reprint, p. 79.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_53" href="#FNanchor_53" class="label">53</a> -De Morgan, <i>On the Root of any Function</i>. Cambridge Philosophical -Transactions, 1867, vol. xi. p. 25.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_54" href="#FNanchor_54" class="label">54</a> -<i>Syllabus of a proposed System of Logic</i>, §§ 122, 123.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_55" href="#FNanchor_55" class="label">55</a> -<i>Elementary Lessons in Logic</i>, p. 86.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_56" href="#FNanchor_56" class="label">56</a> -<i>Outline of the Laws of Thought</i>, § 87.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_57" href="#FNanchor_57" class="label">57</a> -<i>Treatise on Natural Philosophy</i>, vol. i. p. 161.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_58" href="#FNanchor_58" class="label">58</a> -<i>Treatise on Natural Philosophy</i>, vol. i. p. 6.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_59" href="#FNanchor_59" class="label">59</a> -Todhunter’s <i>Plane Co-ordinate Geometry</i>, chap. ii. pp. 11–14.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_60" href="#FNanchor_60" class="label">60</a> -An explanation of this and other technical terms of the old logic -will be found in my <i>Elementary Lessons in Logic</i>, Sixth Edition, -1876; Macmillan.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_61" href="#FNanchor_61" class="label">61</a> -<i>Elementary Lessons in Logic</i>, pp. 67, 79.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_62" href="#FNanchor_62" class="label">62</a> -<i>Pure Logic</i>, p. 19.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_63" href="#FNanchor_63" class="label">63</a> -<i>An Outline of the Necessary Laws of Thought</i>, Fifth Ed. p. 161.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_64" href="#FNanchor_64" class="label">64</a> -Mansel’s <i>Aldrich</i>, p. 103, and <i>Prolegomena Logica</i>, p. 221.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_65" href="#FNanchor_65" class="label">65</a> -<i>Elements of Logic</i>, Book II. chap. iv. sect. 4.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_66" href="#FNanchor_66" class="label">66</a> -Aldrich, <i>Artis Logicæ Rudimenta</i>, p. 104.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_67" href="#FNanchor_67" class="label">67</a> -<i>Examination of Sir W. Hamilton’s Philosophy</i>, pp. 452–454.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_68" href="#FNanchor_68" class="label">68</a> -<i>Pure Logic</i>, pp 76, 77.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_69" href="#FNanchor_69" class="label">69</a> -<i>Pure Logic</i>, p. 65. See also the criticism of this point by De -Morgan in the <i>Athenæum</i>, No. 1892, 30th January, 1864; p. 155.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_70" href="#FNanchor_70" class="label">70</a> -Boole’s <i>Laws of Thought</i>, p. 106. Jevons’ <i>Pure Logic</i>, p. 69.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_71" href="#FNanchor_71" class="label">71</a> -<i>On the Syllogism</i>, No. iii. p. 12. Camb. Phil. Trans. vol. x, -part i.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_72" href="#FNanchor_72" class="label">72</a> -See Horsley, <i>Philosophical Transactions</i>, 1772; vol. lxii. p. 327. -Montucla, <i>Histoire des Mathematiques</i>, vol. i. p. 239. <i>Penny -Cyclopædia</i>, article “Eratosthenes.”</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_73" href="#FNanchor_73" class="label">73</a> -Euclid, Book x. Prop. 117.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_74" href="#FNanchor_74" class="label">74</a> -<i>Philosophical Magazine</i>, December 1852; Fourth Series, vol. iv. -p. 435, “On Indirect Demonstration.”</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_75" href="#FNanchor_75" class="label">75</a> -<i>Philosophical Magazine</i>, Dec. 1852; p. 437.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_76" href="#FNanchor_76" class="label">76</a> -<i>Mind</i>; a Quarterly Review of Psychology and Philosophy; -October, 1876, vol. i. p. 487.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_77" href="#FNanchor_77" class="label">77</a> -Whewell, <i>History of the Inductive Sciences</i>, vol. i. p. 222.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_78" href="#FNanchor_78" class="label">78</a> -<i>Formal Logic</i>, p. 124. As Professor Croom Robertson has -pointed out to me, the second and third premises may be thrown -into a single proposition, D = D<i>e</i>BC ꖌ DE<i>bc</i>.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_79" href="#FNanchor_79" class="label">79</a> -Pp. 55–59, 81–86.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_80" href="#FNanchor_80" class="label">80</a> -See his work called <i>The Process of Thought adapted to Words and -Language, together with a Description of the Relational and Differential -Machines</i>. Also <i>Philosophical Transactions</i>, [1870] vol. 160, -p. 518.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_81" href="#FNanchor_81" class="label">81</a> -<i>Philosophical Transactions</i> [1870], vol. 160, p. 497. <i>Proceedings -of the Royal Society</i>, vol. xviii. p. 166, Jan. 20, 1870. <i>Nature</i>, vol, i. -p. 343.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_82" href="#FNanchor_82" class="label">82</a> -<i>Syllabus of a proposed system of Logic</i>, §§ 57, 121, &c. <i>Formal -Logic</i>, p. 66.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_83" href="#FNanchor_83" class="label">83</a> -Lectures on Metaphysics, vol. iv. p. 369.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_84" href="#FNanchor_84" class="label">84</a> -Bowen, <i>Treatise on Logic</i>, Cambridge, U.S., 1866; p. 362.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_85" href="#FNanchor_85" class="label">85</a> -The contents of this and the following section nearly correspond -with those of a paper read before the Manchester Literary and -Philosophical Society on December 26th, 1871. See Proceedings of -the Society, vol. xi. pp. 65–68, and Memoirs, Third Series, vol. v. -pp. 119–130.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_86" href="#FNanchor_86" class="label">86</a> -<i>Proceedings of the Manchester Literary and Philosophical Society</i>, -6th February, 1877, vol. xvi., p. 113.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_87" href="#FNanchor_87" class="label">87</a> -Montucla. <i>Histoire des Mathématiques</i>, vol. iii. p. 373.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_88" href="#FNanchor_88" class="label">88</a> -<i>British Quarterly Review</i>, No. lxxxvii, July 1866.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_89" href="#FNanchor_89" class="label">89</a> -<i>Mind</i>, October 1876, vol. i. p. 484.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_90" href="#FNanchor_90" class="label">90</a> -<i>Pure Logic</i>, Appendix, p. 82, § 192.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_91" href="#FNanchor_91" class="label">91</a> -<i>Elementary Lessons in Logic</i> (Macmillan), p. 123. It is pointed -out in the preface to this Second Edition, that the views here given -were partially stated by Leibnitz.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_92" href="#FNanchor_92" class="label">92</a> -<i>Syllabus of a Proposed System of Logic</i>, p. 29.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_93" href="#FNanchor_93" class="label">93</a> -It has been pointed out to me by Mr. C. J. Monroe, that section 14 -(p. 339) of this paper is erroneous, and ought to be cancelled. The -problem concerning the number of paupers illustrates the answer -which should have been obtained. Mr. A. J. Ellis, F.R.S., had -previously observed that my solution in the paper of De Morgan’s -problem about “men in the house” did not answer the conditions -intended by De Morgan, and I therefore give in the text a more -satisfactory solution.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_94" href="#FNanchor_94" class="label">94</a> -Montucla, <i>Histoire</i>, &c., vol. iii. p. 388.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_95" href="#FNanchor_95" class="label">95</a> -Wallis, <i>Of Combinations</i>, &c., p. 119.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_96" href="#FNanchor_96" class="label">96</a> -James Bernoulli, <i>De Arte Conjectandi</i>, translated by Baron -Maseres. London, 1795, pp. 35, 36.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_97" href="#FNanchor_97" class="label">97</a> -<i>Arithmeticæ Theoria.</i> Ed. Amsterd. 1704. p. 517.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_98" href="#FNanchor_98" class="label">98</a> -Rees’s <i>Cyclopædia</i>, art. <i>Cipher</i>.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_99" href="#FNanchor_99" class="label">99</a> -<i>Œuvres Complètes de Pascal</i> (1865), vol. iii. p. 302. Montucla -states the name as De Gruières, <i>Histoire des Mathématiques</i>, vol. iii. -p. 389.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_100" href="#FNanchor_100" class="label">100</a> -<i>Histoire des Mathématiques</i>, vol. iii. p. 378.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_101" href="#FNanchor_101" class="label">101</a> -Bernoulli, <i>De Arte Conjectandi</i>, translated by Francis Maseres. -London, 1795, p. 75.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_102" href="#FNanchor_102" class="label">102</a> -Wallis’s <i>Algebra</i>, Discourse of Combinations, &c., p. 109.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_103" href="#FNanchor_103" class="label">103</a> -<i>Œuvres Complètes</i>, vol. iii. p. 251.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_104" href="#FNanchor_104" class="label">104</a> -See also Galton’s Lecture at the Royal Institution, 27th February, -1874; Catalogue of the Special Loan Collection of Scientific Instruments, -South Kensington, Nos. 48, 49; and Galton, <i>Philosophical -Magazine</i>, January 1875.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_105" href="#FNanchor_105" class="label">105</a> -Wallis, <i>Of Combinations</i>, p. 116, quoting Vossius.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_106" href="#FNanchor_106" class="label">106</a> -<i>Philosophical Transactions</i> (1803), vol. xciii. p. 193.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_107" href="#FNanchor_107" class="label">107</a> -Hofmann’s <i>Introduction to Chemistry</i>, p. 36.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_108" href="#FNanchor_108" class="label">108</a> -<i>Works</i>, edited by Shaw, vol. i. pp. 141–145, quoted in Rees’s -<i>Encyclopædia</i>, art. <i>Cipher</i>.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_109" href="#FNanchor_109" class="label">109</a> -<i>Nature</i>, vol. i. p. 553.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_110" href="#FNanchor_110" class="label">110</a> -<i>Formal Logic</i>, p. 172.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_111" href="#FNanchor_111" class="label">111</a> -<i>Philosophical Magazine</i>, 4th Series, vol. i. p. 355.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_112" href="#FNanchor_112" class="label">112</a> -<i>Transactions of the Royal Society of Edinburgh</i>, vol. xxi. part 4.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_113" href="#FNanchor_113" class="label">113</a> -<i>Philosophical Magazine</i>, 4th Series, vol. vii. p. 465; vol. viii. -p. 91.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_114" href="#FNanchor_114" class="label">114</a> -<i>Memoirs of the Manchester Literary and Philosophical Society</i>, -3rd Series, vol. iv. p. 347.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_115" href="#FNanchor_115" class="label">115</a> -<i>Letters on the Theory of Probabilities</i>, translated by Downes, 1849, -pp. 36, 37.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_116" href="#FNanchor_116" class="label">116</a> -<i>Encyclopædia Metropolitana</i>, art. <i>Probabilities</i>, p. 396.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_117" href="#FNanchor_117" class="label">117</a> -<i>Elements of Logic</i>, Book III. sections 11 and 18.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_118" href="#FNanchor_118" class="label">118</a> -<i>Encyclopædia Metropolitana</i>, art. <i>Probabilities</i>, p. 400.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_119" href="#FNanchor_119" class="label">119</a> -<i>Philosophical Transactions</i> (1767). Abridg. vol. xii. p. 435.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_120" href="#FNanchor_120" class="label">120</a> -<i>Transactions of the Edinburgh Philosophical Society</i>, vol. xxi. -p. 375.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_121" href="#FNanchor_121" class="label">121</a> -Montucla, <i>Histoire des Mathématiques</i>, vol. iii. p. 386.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_122" href="#FNanchor_122" class="label">122</a> -Leibnitz <i>Opera</i>, Dutens’ Edition, vol. vi. part i. p. 217. Todhunter’s -<i>History of the Theory of Probability</i>, p. 48. To the latter -work I am indebted for many of the statements in the text.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_123" href="#FNanchor_123" class="label">123</a> -<i>Positive Philosophy</i>, translated by Martineau, vol. ii. p. 120.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_124" href="#FNanchor_124" class="label">124</a> -<i>System of Logic</i>, bk. iii. chap. 18, 5th Ed. vol. ii. p. 61.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_125" href="#FNanchor_125" class="label">125</a> -Montucla, <i>Histoire</i>, vol. iii. p. 405; Todhunter, p. 263.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_126" href="#FNanchor_126" class="label">126</a> -<i>Essay concerning Human Understanding</i>, bk. iv. ch. 14. § 1.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_127" href="#FNanchor_127" class="label">127</a> -<i>Philosophical Magazine</i>, 4th Series, vol. i. p. 354.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_128" href="#FNanchor_128" class="label">128</a> -<i>Essay concerning Human Understanding</i>, bk. ii. chap. xxi.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_129" href="#FNanchor_129" class="label">129</a> -<i>De Rerum Natura</i>, bk. ii. ll. 216–293.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_130" href="#FNanchor_130" class="label">130</a> -<i>Cambridge Philosophical Transactions</i> (1830), vol. iii. pp. -369–372.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_131" href="#FNanchor_131" class="label">131</a> -<i>Observations on the Nature and Tendency of the Doctrine of -Mr. Hume, concerning the Relation of Cause and Effect.</i> Second ed. -p. 44.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_132" href="#FNanchor_132" class="label">132</a> -Ibid. p. 97.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_133" href="#FNanchor_133" class="label">133</a> -<i>System of Logic</i>, bk. II. chap, iii.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_134" href="#FNanchor_134" class="label">134</a> -<i>Inductive Logic</i>, pp. 13, 14.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_135" href="#FNanchor_135" class="label">135</a> -Bain, <i>Deductive Logic</i>, pp. 208, 209.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_136" href="#FNanchor_136" class="label">136</a> -<i>System of Logic.</i> Introduction, § 4. Fifth ed. pp. 8, 9.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_137" href="#FNanchor_137" class="label">137</a> -Ibid. bk. II. chap. iii. § 5, pp. 225, &c.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_138" href="#FNanchor_138" class="label">138</a> -These are the figurate numbers considered in pages 183, 187, &c.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_139" href="#FNanchor_139" class="label">139</a> -<i>Commercium Epistolicum.</i> <i>Epistola ad Oldenburgum</i>, Oct. 24, -1676. Horsley’s <i>Works of Newton</i>, vol. iv. p. 541. See De Morgan -in <i>Penny Cyclopædia</i>, art. “Binomial Theorem,” p. 412.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_140" href="#FNanchor_140" class="label">140</a> -Bk. ii. chap. iv.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_141" href="#FNanchor_141" class="label">141</a> -<i>Philosophical Transactions</i> (1866), vol. 146, p. 334.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_142" href="#FNanchor_142" class="label">142</a> -<i>Budget of Paradoxes</i>, p. 257.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_143" href="#FNanchor_143" class="label">143</a> -<i>Proceedings of the Royal Society</i> (1872–3), vol. xxi. p. 319.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_144" href="#FNanchor_144" class="label">144</a> -<i>Life of Galileo</i>, Society for the Diffusion of Useful Knowledge, -p. 102.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_145" href="#FNanchor_145" class="label">145</a> -Professor Bowen has excellently stated this view. <i>Treatise on -Logic.</i> Cambridge, U.S.A., 1866, p. 354.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_146" href="#FNanchor_146" class="label">146</a> -Roscoe’s <i>Spectrum Analysis</i>, 1st edit., p. 98.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_147" href="#FNanchor_147" class="label">147</a> -Euler’s <i>Letters to a German Princess</i>, translated by Hunter. -2nd ed., vol. ii. pp. 17, 18.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_148" href="#FNanchor_148" class="label">148</a> -Lavoisier’s <i>Chemistry</i>, translated by Kerr. 3rd ed., pp. 114, -121, 123.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_149" href="#FNanchor_149" class="label">149</a> -Euler’s <i>Letters</i>, vol. ii. p. 21.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_150" href="#FNanchor_150" class="label">150</a> -Lardner, <i>Edinburgh Review</i>, July 1834, p. 277.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_151" href="#FNanchor_151" class="label">151</a> -<i>Mémoires par divers Savans</i>, tom. vi.; quoted by Todhunter in -his <i>History of the Theory of Probability</i>, p. 458.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_152" href="#FNanchor_152" class="label">152</a> -Poisson, <i>Recherches sur la Probabilité des Jugements</i>, Paris, 1837, -pp. 82, 83.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_153" href="#FNanchor_153" class="label">153</a> -Kirchhoff’s <i>Researches on the Solar Spectrum</i>. First part, translated -by Roscoe, pp. 18, 19.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_154" href="#FNanchor_154" class="label">154</a> -<i>Edinburgh Review</i>, No. 185, vol. xcii. July 1850, p. 32; Herschel’s -<i>Essays</i>, p. 421; <i>Transactions of the Cambridge Philosophical Society</i>, -vol. i. p. 43.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_155" href="#FNanchor_155" class="label">155</a> -Evans’ <i>Ancient Stone Implements of Great Britain</i>. London, -1872 (Longmans).</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_156" href="#FNanchor_156" class="label">156</a> -Herschel, <i>Outlines of Astronomy</i>, 1849, p. 565; but Todhunter, -in his <i>History of the Theory of Probability</i>, p. 335, states that the -calculations do not agree with those published by Struve.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_157" href="#FNanchor_157" class="label">157</a> -<i>Philosophical Transactions</i>, 1767, vol. lvii. p. 431.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_158" href="#FNanchor_158" class="label">158</a> -<i>Philosophical Magazine</i>, 3rd Series, vol. xxxvii. p. 401, December -1850; also August 1849.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_159" href="#FNanchor_159" class="label">159</a> -<i>History</i>, &c., p. 334.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_160" href="#FNanchor_160" class="label">160</a> -<i>Essai Philosophique</i>, p. 57.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_161" href="#FNanchor_161" class="label">161</a> -<i>Proceedings of the Royal Society</i>; 20 January, 1870; <i>Philosophical -Magazine</i>, 4th Series, vol. xxxix. p. 381.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_162" href="#FNanchor_162" class="label">162</a> -<i>Principia</i>, bk. ii. General scholium.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_163" href="#FNanchor_163" class="label">163</a> -<i>Essai Philosophique</i>, p. 55. Laplace appears to count the rings of -Saturn as giving two independent movements.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_164" href="#FNanchor_164" class="label">164</a> -Lubbock, <i>Essay on Probability</i>, p. 14. De Morgan, <i>Encyc. -Metrop.</i> art. <i>Probability</i>, p. 412. Todhunter’s <i>History of the Theory -of Probability</i>, p. 543. Concerning the objections raised to these -conclusions by Boole, see the <i>Philosophical Magazine</i>, 4th Series, -vol. ii. p. 98. Boole’s <i>Laws of Thought</i>, pp. 364–375.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_165" href="#FNanchor_165" class="label">165</a> -Laplace, <i>Essai Philosophique</i>, pp. 55, 56.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_166" href="#FNanchor_166" class="label">166</a> -Chambers’ <i>Astronomy</i>, 2nd ed. pp. 346–49.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_167" href="#FNanchor_167" class="label">167</a> -<i>Traité élémentaire du Calcul des Probabilités</i>, 3rd ed. (1833), -p. 148.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_168" href="#FNanchor_168" class="label">168</a> -<i>Laws of Thought</i>, pp. 368–375.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_169" href="#FNanchor_169" class="label">169</a> -De Morgan’s <i>Essay on Probabilities</i>, Cabinet Cyclopædia, p. 67.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_170" href="#FNanchor_170" class="label">170</a> -<i>Essay on Probabilities</i>, p. 128.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_171" href="#FNanchor_171" class="label">171</a> -J. S. Mill, <i>System of Logic</i>, 5th edition, bk. iii. chap. xviii. § 3.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_172" href="#FNanchor_172" class="label">172</a> -Todhunter’s <i>History</i>, pp. 472, 598.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_173" href="#FNanchor_173" class="label">173</a> -Todhunter’s <i>History</i>, pp. 378, 379.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_174" href="#FNanchor_174" class="label">174</a> -<i>Philosophical Transactions</i>, [1763], vol. liii. p. 370, and [1764], -vol. liv. p. 296. Todhunter, pp. 294–300.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_175" href="#FNanchor_175" class="label">175</a> -Newton’s <i>Opticks</i>, Bk. I., Part ii. Prop. 3; <i>Nature</i>, vol. i. p. 286.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_176" href="#FNanchor_176" class="label">176</a> -Aristotle’s <i>Metaphysics</i>, xiii. 6. 3.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_177" href="#FNanchor_177" class="label">177</a> -Possunt autem omnes testes et uno annulo signare testamentum -Quid enim si septem annuli una sculptura fuerint, secundum quod -Pomponio visum est?—<i>Justinian</i>, ii. tit. x. 5.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_178" href="#FNanchor_178" class="label">178</a> -See Wills on <i>Circumstantial Evidence</i>, p. 148.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_179" href="#FNanchor_179" class="label">179</a> -<i>Memoirs of the Royal Astronomical Society</i>, vol. iv. p. 290, quoted -by Lardner, <i>Edinburgh Review</i>, July 1834, p. 278.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_180" href="#FNanchor_180" class="label">180</a> -Baily, <i>British Association Catalogue of Stars</i>, pp. 7, 23.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_181" href="#FNanchor_181" class="label">181</a> -<i>Outlines of Astronomy</i>, 4th ed. sect. 781, p. 522. <i>Results of -Observations at the Cape of Good Hope</i>, &c., p. 37.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_182" href="#FNanchor_182" class="label">182</a> -See De Morgan, <i>Study of Mathematics</i>, in U.K.S. Library, p. 81.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_183" href="#FNanchor_183" class="label">183</a> -Loomis, <i>On the Aurora Borealis</i>. Smithsonian Transactions, -quoting Parry’s Third Voyage, p. 61.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_184" href="#FNanchor_184" class="label">184</a> -Watts’ <i>Dictionary of Chemistry</i>, vol. ii. p. 790.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_185" href="#FNanchor_185" class="label">185</a> -<i>Philosophical Transactions</i>, (1856) vol. 146, Part i. p. 297.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_186" href="#FNanchor_186" class="label">186</a> -Airy, <i>On Tides and Waves</i>, Encyclopædia Metropolitana, p. 345. -Scott Russell, <i>British Association Report</i>, 1837, p. 432.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_187" href="#FNanchor_187" class="label">187</a> -<i>Hugenii Cosmotheoros</i>, pp. 117, 118. Laplace’s <i>Système</i>, translated, -vol. i. p. 67.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_188" href="#FNanchor_188" class="label">188</a> -Grant’s <i>History of Physical Astronomy</i>, p. 129.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_189" href="#FNanchor_189" class="label">189</a> -Baily’s <i>Account of Flamsteed</i>, p. lix.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_190" href="#FNanchor_190" class="label">190</a> -Jamin, <i>Cours de Physique</i>, vol. i. p. 152.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_191" href="#FNanchor_191" class="label">191</a> -Faraday, <i>Chemical Researches</i>, p. 393.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_192" href="#FNanchor_192" class="label">192</a> -<i>Proceedings of the Royal Society</i>, 30th November, 1866.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_193" href="#FNanchor_193" class="label">193</a> -Herschel, <i>Physical Geography</i>, § 40.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_194" href="#FNanchor_194" class="label">194</a> -<i>Principia</i>, bk. iii. Prop. 37, <i>Corollaries</i>, 2 and 3. Motte’s -translation, vol. ii. p. 310.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_195" href="#FNanchor_195" class="label">195</a> -Roscoe’s <i>Spectrum Analysis</i>, 1st ed. p. 296.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_196" href="#FNanchor_196" class="label">196</a> -<i>Philosophical Transactions</i> (1859), vol. cxlix. p. 94.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_197" href="#FNanchor_197" class="label">197</a> -Watts’ <i>Dictionary of Chemistry</i>, vol. ii. p. 393.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_198" href="#FNanchor_198" class="label">198</a> -<i>Philosophical Transactions</i> (1859), vol. cxlix. p. 119, &c.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_199" href="#FNanchor_199" class="label">199</a> -Baily’s <i>Account of Flamsteed</i>, pp. 378–380.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_200" href="#FNanchor_200" class="label">200</a> -Herschel’s <i>Astronomy</i>, § 817, 4th. ed. p. 553.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_201" href="#FNanchor_201" class="label">201</a> -<i>Principia</i>, bk. ii. Sect. 6. Prop. 31. Motte’s Translation, vol. ii. -p. 107.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_202" href="#FNanchor_202" class="label">202</a> -Ibid. bk. i. Law iii. Corollary 6. Motte’s Translation, vol. i. p. 33.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_203" href="#FNanchor_203" class="label">203</a> -Thomson and Tait’s <i>Natural Philosophy</i>, vol. i. p. 333.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_204" href="#FNanchor_204" class="label">204</a> -<i>Philosophical Transactions</i>, (1856), vol. cxlvi. pp. 330, 331.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_205" href="#FNanchor_205" class="label">205</a> -<i>First Annual Report of the Mint</i>, p. 106.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_206" href="#FNanchor_206" class="label">206</a> -Jevons, in Watts’ <i>Dictionary of Chemistry</i>, vol. i. p. 483.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_207" href="#FNanchor_207" class="label">207</a> -British Association, Glasgow, 1856. <i>Address of the President of -the Mechanical Section</i>.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_208" href="#FNanchor_208" class="label">208</a> -<i>Pelicotetics, or the Science of Quantity; an Elementary Treatise on -Algebra, and its groundwork Arithmetic.</i> By Archibald Sandeman, -M. A. Cambridge (Deighton, Bell, and Co.), 1868, p. 304.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_209" href="#FNanchor_209" class="label">209</a> -De Morgan’s <i>Trigonometry and Double Algebra</i>, p. 5.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_210" href="#FNanchor_210" class="label">210</a> -<i>English Works of Thos. Hobbes</i>, Edit. by Molesworth, vol. i. p. 95.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_211" href="#FNanchor_211" class="label">211</a> -<i>Confessions</i>, bk. xi. chapters 20–28.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_212" href="#FNanchor_212" class="label">212</a> -Sir G. C. Lewis gives many curious particulars concerning the -measurement of time in his <i>Astronomy of the Ancients</i>, pp. 241, &c.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_213" href="#FNanchor_213" class="label">213</a> -<i>Principia</i>, bk. i. <i>Scholium to Definitions</i>. Translated by Motte, -vol. i. p. 9. See also p. 11.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_214" href="#FNanchor_214" class="label">214</a> -Rankine, <i>Philosophical Magazine</i>, Feb. 1867, vol. xxxiii. p. 91.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_215" href="#FNanchor_215" class="label">215</a> -<i>Treatise on Natural Philosophy</i>, vol. i. p. 179.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_216" href="#FNanchor_216" class="label">216</a> -<i>Proceedings of the Manchester Philosophical Society</i>, 28th Nov. -1871, vol. xi. p. 33.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_217" href="#FNanchor_217" class="label">217</a> -<i>The Elements of Natural Philosophy</i>, part i. p. 119.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_218" href="#FNanchor_218" class="label">218</a> -See Harris’ <i>Essay upon Money and Coins</i>, part. ii. [1758] p. 127.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_219" href="#FNanchor_219" class="label">219</a> -<i>Philosophical Magazine</i>, (1868), 4th Series, vol. xxxvi. p. 32.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_220" href="#FNanchor_220" class="label">220</a> -<i>Proceedings of the Royal Society</i>, 20th June, 1872, vol. xx. p. 438.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_221" href="#FNanchor_221" class="label">221</a> -Kater’s <i>Treatise on Mechanics</i>, Cabinet Cyclopædia, p. 154.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_222" href="#FNanchor_222" class="label">222</a> -Grant’s <i>History of Physical Astronomy</i>, p. 156.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_223" href="#FNanchor_223" class="label">223</a> -Clerk Maxwell’s <i>Theory of Heat</i>, p. 79.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_224" href="#FNanchor_224" class="label">224</a> -<i>Treatise on Electricity and Magnetism</i>, vol. i. p. 3.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_225" href="#FNanchor_225" class="label">225</a> -<i>Chemistry for Students</i>, by A. W. Williamson. Clarendon Press -Series, 2nd ed. Preface p. vi.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_226" href="#FNanchor_226" class="label">226</a> -<i>Introduction to Chemistry</i>, p. 131.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_227" href="#FNanchor_227" class="label">227</a> -<i>Philosophical Transactions</i> (1859), vol. cxlix. p. 884, &c.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_228" href="#FNanchor_228" class="label">228</a> -<i>Théorie Analytique de la Chaleur</i>, Paris; 1822, §§ 157–162.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_229" href="#FNanchor_229" class="label">229</a> -Tyndall’s <i>Sound</i>, 1st ed. p. 26.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_230" href="#FNanchor_230" class="label">230</a> -British Association, Cambridge, 1833. Report, pp. 484–490.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_231" href="#FNanchor_231" class="label">231</a> -<i>Smithsonian Miscellaneous Collections</i>, vol. xii., the Constants of -Nature, part. i. Specific gravities compiled by F. W. Clarke, 8vo. -Washington, 1873.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_232" href="#FNanchor_232" class="label">232</a> -J. W. L. Glaisher, <i>Philosophical Magazine</i>, 4th Series, vol. xlii. -p. 421.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_233" href="#FNanchor_233" class="label">233</a> -Stokes, <i>Philosophical Transactions</i> (1852), vol. cxlii. p. 529.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_234" href="#FNanchor_234" class="label">234</a> -<i>Admiralty Manual of Scientific Enquiry</i>, 2nd ed. p. 299.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_235" href="#FNanchor_235" class="label">235</a> -Pouillet, <i>Taylor’s Scientific Memoirs</i>, vol. iv. p. 45.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_236" href="#FNanchor_236" class="label">236</a> -Baily’s <i>Account of the Rev. John Flamsteed</i>, p. 58.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_237" href="#FNanchor_237" class="label">237</a> -Jamin, <i>Cours de Physique</i>, vol. ii. pp. 15–28.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_238" href="#FNanchor_238" class="label">238</a> -<i>Philosophical Magazine</i>, 1851, 4th Series, vol. ii. <i>passim</i>.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_239" href="#FNanchor_239" class="label">239</a> -Hearn, <i>Philosophical Transactions</i>, 1847, vol. cxxxvii. pp. 217–221.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_240" href="#FNanchor_240" class="label">240</a> -<i>The Correlation of Physical Forces</i>, 3rd ed. p. 159.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_241" href="#FNanchor_241" class="label">241</a> -<i>Collected Works of Sir H. Davy</i>, vol. ii. pp. 12–14. <i>Elements of -Chemical Philosophy</i>, p. 94.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_242" href="#FNanchor_242" class="label">242</a> -<i>Nicholson’s Journal</i>, vol. i. p. 241; quoted in <i>Treatise on Heat</i>, -Useful Knowledge Society, p. 24.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_243" href="#FNanchor_243" class="label">243</a> -Clerk Maxwell, <i>Theory of Heat</i>, p. 228. <i>Proceedings of the -Manchester Philosophical Society</i>, Nov. 26, 1867, vol. vii. p. 35.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_244" href="#FNanchor_244" class="label">244</a> -Leslie, <i>Inquiry into the Nature of Heat</i>, p. 10.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_245" href="#FNanchor_245" class="label">245</a> -Jevons, Watts’ <i>Dictionary of Chemistry</i>, vol. i. pp. 513–515.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_246" href="#FNanchor_246" class="label">246</a> -<i>Philosophical Transactions</i>, vol. li. p. 100.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_247" href="#FNanchor_247" class="label">247</a> -<i>Philosophical Magazine</i>, 3rd Series, vol. xxvi. p. 372.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_248" href="#FNanchor_248" class="label">248</a> -<i>Greenwich Observations for</i> 1866, p. xlix.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_249" href="#FNanchor_249" class="label">249</a> -<i>Philosophical Transactions</i>, 1856, p. 309.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_250" href="#FNanchor_250" class="label">250</a> -Penny <i>Cyclopædia</i>, art. <i>Transit</i>, vol. xxv. pp. 129, 130.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_251" href="#FNanchor_251" class="label">251</a> -Ibid. art. <i>Observation</i>, p. 390.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_252" href="#FNanchor_252" class="label">252</a> -<i>Nature</i>, vol. i. p. 85.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_253" href="#FNanchor_253" class="label">253</a> -<i>Nature</i>, vol. i. p 337. See references to the Memoirs describing -the method.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_254" href="#FNanchor_254" class="label">254</a> -<i>Principia</i>, Book I. Law III. Corollary VI. Scholium. Motte’s -translation, vol. i. p. 33.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_255" href="#FNanchor_255" class="label">255</a> -Graham’s <i>Chemical Reports and Memoirs</i>, Cavendish Society, -pp. 247, 268, &c.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_256" href="#FNanchor_256" class="label">256</a> -Regnault’s <i>Cours Elémentaire de Chimie</i>, 1851, vol i. p. 141.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_257" href="#FNanchor_257" class="label">257</a> -Tyndall’s <i>Faraday</i>, pp. 114, 115.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_258" href="#FNanchor_258" class="label">258</a> -See, for instance, the Compensated Sympiesometer, <i>Philosophical -Magazine</i>, 4th Series, vol. xxxix. p. 371.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_259" href="#FNanchor_259" class="label">259</a> -Grant, <i>History of Physical Astronomy</i>, pp. 146, 147.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_260" href="#FNanchor_260" class="label">260</a> -Quetelet, <i>Sur la Physique du Globe</i>, p. 174. Jamin, <i>Cours de -Physique</i>, vol. i. p. 504.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_261" href="#FNanchor_261" class="label">261</a> -Baily’s <i>Account of Flamsteed</i>, p. 376.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_262" href="#FNanchor_262" class="label">262</a> -<i>The Transit of Venus across the Sun</i>, by Horrocks, London, 1859, -p. 146.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_263" href="#FNanchor_263" class="label">263</a> -De Morgan, Supplement to the <i>Penny Cyclopædia</i>, art. <i>Old -Appellations of Numbers</i>.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_264" href="#FNanchor_264" class="label">264</a> -<i>Penny Cyclopædia</i>, art. <i>Mean</i>.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_265" href="#FNanchor_265" class="label">265</a> -Jevons, <i>Journal of the Statistical Society</i>, June 1865, vol. xxviii, -p. 296.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_266" href="#FNanchor_266" class="label">266</a> -<i>Letters on the Theory of Probabilities</i>, transl. by Downes, Part ii.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_267" href="#FNanchor_267" class="label">267</a> -Herschel’s <i>Essays</i>, &c. pp. 404, 405.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_268" href="#FNanchor_268" class="label">268</a> -<i>On the Theory of Errors of Observations, Cambridge Philosophical -Transactions</i>, vol. x. Part ii. 416.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_269" href="#FNanchor_269" class="label">269</a> -Thomson and Tait, <i>Treatise on Natural Philosophy</i>, vol. i. p. 394.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_270" href="#FNanchor_270" class="label">270</a> -<i>Essai Philosophique sur les Probabilités</i>, pp. 49, 50.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_271" href="#FNanchor_271" class="label">271</a> -Grant, <i>History of Physical Astronomy</i>, p. 163.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_272" href="#FNanchor_272" class="label">272</a> -Gauss, Taylor’s <i>Scientific Memoirs</i>, vol. ii. p. 43, &c.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_273" href="#FNanchor_273" class="label">273</a> -<i>Proceedings of the Royal Society</i>, vol. xviii. p. 159 (Jan. 13, 1870). -<i>Philosophical Magazine</i> (4th Series), vol. xxxix. p. 376.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_274" href="#FNanchor_274" class="label">274</a> -Airy <i>On Tides and Waves</i>, Encycl. Metrop. pp. 364*-366*.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_275" href="#FNanchor_275" class="label">275</a> -<i>Outlines of Astronomy</i>, 4th edition, § 538.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_276" href="#FNanchor_276" class="label">276</a> -<i>Philosophical Magazine</i>, 3rd Series, vol. xxxvii. p. 324.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_277" href="#FNanchor_277" class="label">277</a> -<i>Letters on the Theory of Probabilities</i>, by Quetelet, translated by -O. G. Downes, Notes to Letter XXVI. pp. 286–295.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_278" href="#FNanchor_278" class="label">278</a> -<i>On the Law of Facility of Errors of Observations, and on the -Method of Least Squares</i>, Memoirs of the Royal Astronomical Society, -vol. xxxix. p. 75.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_279" href="#FNanchor_279" class="label">279</a> -<i>Méthode des Moindres Carrés. Mémoires sur la Combinaison des -Observations, par Ch. Fr. Gauss. Traduit en Français par J. -Bertrand</i>, Paris, 1855, pp. 6, 133, &c.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_280" href="#FNanchor_280" class="label">280</a> -De Morgan, <i>Penny Cyclopædia</i>, art. <i>Least Squares</i>.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_281" href="#FNanchor_281" class="label">281</a> -<i>Edinburgh Review</i>, July 1850, vol. xcii. p. 17. Reprinted <i>Essays</i>, -p. 399. This method of demonstration is discussed by Boole, <i>Transactions -of Royal Society of Edinburgh</i>, vol. xxi. pp. 627–630.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_282" href="#FNanchor_282" class="label">282</a> -<i>Letters on the Theory of Probabilities</i>, Letter XV. and Appendix, -note pp. 256–266.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_283" href="#FNanchor_283" class="label">283</a> -Encke, <i>On the Method of Least Squares</i>, Taylor’s <i>Scientific -Memoirs</i>, vol. ii. pp. 338, 339.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_284" href="#FNanchor_284" class="label">284</a> -Quetelet, <i>Letters on the Theory of Probabilities</i>, translated by -Downes, Letter XIX. p. 88. See also Galton’s <i>Hereditary Genius</i>, -p. 379.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_285" href="#FNanchor_285" class="label">285</a> -<i>System of Logic</i>, bk. iii. chap. 17, § 3. 5th ed. vol. ii. p. 56.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_286" href="#FNanchor_286" class="label">286</a> -<i>Philosophy of the Inductive Sciences</i>, 2nd ed. vol. ii. pp. 408, 409.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_287" href="#FNanchor_287" class="label">287</a> -<i>Essay on Probability</i>, Useful Knowledge Society, 1833, p. 41.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_288" href="#FNanchor_288" class="label">288</a> -Taylor’s <i>Scientific Memoirs</i>, vol. ii. p. 333.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_289" href="#FNanchor_289" class="label">289</a> -<i>Philosophical Transactions</i>, 1873, p. 83.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_290" href="#FNanchor_290" class="label">290</a> -Taylor’s <i>Scientific Memoirs</i>, vol. ii. pp. 330, 347, &c.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_291" href="#FNanchor_291" class="label">291</a> -Quetelet, <i>Letters</i>, &c. p. 116.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_292" href="#FNanchor_292" class="label">292</a> -Baily, <i>Account of Flamsteed</i>, p. 56.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_293" href="#FNanchor_293" class="label">293</a> -Gould’s <i>Astronomical Journal</i>, Cambridge, Mass., vol. ii. p. 161.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_294" href="#FNanchor_294" class="label">294</a> -Philadelphia (London, Trübner) 1863. Appendix, vol. ii. p. 558.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_295" href="#FNanchor_295" class="label">295</a> -Bakerian Lecture, <i>Philosophical Transactions</i> (1868), vol. clviii. -p. 6.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_296" href="#FNanchor_296" class="label">296</a> -<i>Results of Observations at the Cape of Good Hope</i>, p. 283.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_297" href="#FNanchor_297" class="label">297</a> -<i>The Logic of Chance</i>, an Essay on the Foundations and Province -of the Theory of Probability, with especial reference to its Logical -Bearings and its Application to Moral and Social Science. (Macmillan), -1876.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_298" href="#FNanchor_298" class="label">298</a> -Gauss, translated by Bertrand, p. 25.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_299" href="#FNanchor_299" class="label">299</a> -Jamin, <i>Cours de Physique</i>, vol. ii. p. 60.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_300" href="#FNanchor_300" class="label">300</a> -<i>Preliminary Discourse on the Study of Natural Philosophy</i>, p. 77.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_301" href="#FNanchor_301" class="label">301</a> -Lavoisier’s <i>Elements of Chemistry</i>, translated by Kerr, 3rd ed. -p. 148.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_302" href="#FNanchor_302" class="label">302</a> -Babbage, <i>Economy of Manufactures</i>, p. 194.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_303" href="#FNanchor_303" class="label">303</a> -<i>System of the World</i>, translated by Harte, vol. ii. p. 335.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_304" href="#FNanchor_304" class="label">304</a> -This curious phenomenon, which I propose to call <i>pedesis</i>, or the <i>pedetic -movement</i>, from πηδόω, to jump, is carefully described in my paper published -in the <i>Quarterly Journal of Science</i> for April, 1878, vol. viii. (N.S.) -p. 167. See also <i>Proceedings of the Literary and Philosophical Society -of Manchester</i>, 25th January, 1870, vol. ix. p. 78, <i>Nature</i>, 22nd August, -1878, vol. xviii. p. 440, or the <i>Quarterly Journal of Science</i>, vol. viii. -(N.S.) p. 514.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_305" href="#FNanchor_305" class="label">305</a> -Maxwell, <i>Theory of Heat</i>, p. 301.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_306" href="#FNanchor_306" class="label">306</a> -Laplace, <i>Essai Philosophique</i>, p. 59. Todhunter’s <i>History</i>, -pp. 491–494.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_307" href="#FNanchor_307" class="label">307</a> -Chambers’ <i>Astronomy</i>, 1st ed. p. 203.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_308" href="#FNanchor_308" class="label">308</a> -<i>Essay on Probabilities</i>, Cabinet Cyclopædia, p. 121.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_309" href="#FNanchor_309" class="label">309</a> -<i>Philosophical Magazine</i>, 4th Series (1867), vol. xxxiv. p. 64.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_310" href="#FNanchor_310" class="label">310</a> -See <i>Notes to Measures of Double Stars</i>, 1204, 1336, 1477, 1686, -1786, 1816, 1835, 1929, 2081, 2186, pp. 265, &c. See also Herschel’s -<i>Familiar Lectures on Scientific Subjects</i>, p. 147, and <i>Outlines of -Astronomy</i>, 7th ed. p. 285.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_311" href="#FNanchor_311" class="label">311</a> -Jevons, <i>On the Cirrous Form of Cloud</i>, Philosophical Magazine, -July, 1857, 4th Series, vol. xiv. p. 22.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_312" href="#FNanchor_312" class="label">312</a> -<i>Astronomy</i>, 4th ed. p. 358.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_313" href="#FNanchor_313" class="label">313</a> -Babbage, <i>Ninth Bridgewater Treatise</i>, p. 67.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_314" href="#FNanchor_314" class="label">314</a> -Cuvier, <i>Essay on the Theory of the Earth</i>, translation, p. 61, &c.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_315" href="#FNanchor_315" class="label">315</a> -Murchison’s <i>Siluria</i>, 1st ed. p. 432.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_316" href="#FNanchor_316" class="label">316</a> -Darwin’s <i>Fertilisation of Orchids</i>, p. 48.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_317" href="#FNanchor_317" class="label">317</a> -Peacock, <i>Algebre</i>, vol. ii. p. 344.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_318" href="#FNanchor_318" class="label">318</a> -Ibid, p. 359. Serret, <i>Algèbre Supérieure</i>, 2nd ed. p. 304.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_319" href="#FNanchor_319" class="label">319</a> -<i>Treatise on Optics</i>, by Brewster, Cab. Cyclo. p. 117.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_320" href="#FNanchor_320" class="label">320</a> -<i>Opticks</i>, 3rd. ed. p. 25.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_321" href="#FNanchor_321" class="label">321</a> -<i>Experimental Researches in Electricity</i>, vol. i. pp. 133, 134.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_322" href="#FNanchor_322" class="label">322</a> -Ibid. vol i. pp. 127, 162, &c.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_323" href="#FNanchor_323" class="label">323</a> -<i>Principia</i>, bk. iii. Prop. vi. Corollary i.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_324" href="#FNanchor_324" class="label">324</a> -Lavoisier’s <i>Chemistry</i>, translated by Kerr, p. 103.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_325" href="#FNanchor_325" class="label">325</a> -Cuvier’s <i>Animal Kingdom</i>, introduction, pp. 1, 2.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_326" href="#FNanchor_326" class="label">326</a> -<i>Experimental Researches in Electricity</i>, vol. iii. p. 4.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_327" href="#FNanchor_327" class="label">327</a> -<i>Philosophical Magazine</i>, 4th Series, vol. ix. p. 327.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_328" href="#FNanchor_328" class="label">328</a> -<i>Inquiry into the Nature of Heat</i>, p. 95.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_329" href="#FNanchor_329" class="label">329</a> -Herschel, <i>Preliminary Discourse</i>, p. 161.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_330" href="#FNanchor_330" class="label">330</a> -<i>System of Logic</i>, bk. iii. chap. viii. § 4, 5th ed. vol. i. p. 433.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_331" href="#FNanchor_331" class="label">331</a> -<i>Essayes of Natural Experiments made in the Accademia del -Cimento.</i> Englished by Richard Waller, 1684, p. 40, &c.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_332" href="#FNanchor_332" class="label">332</a> -Plateau, <i>Taylor’s Scientific Memoirs</i>, vol. iv. pp. 16–43.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_333" href="#FNanchor_333" class="label">333</a> -<i>Philosophical Transactions</i> [1826], vol. cxvi. pp. 388, 389. Works -of Sir Humphry Davy, vol. v. pp. 1–12.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_334" href="#FNanchor_334" class="label">334</a> -<i>National Review</i>, July, 1861, p. 13.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_335" href="#FNanchor_335" class="label">335</a> -His published works are contained in <i>The Edinburgh Physical -and Literary Essays</i>, vol. ii. p. 34; <i>Philosophical Transactions</i> [1753], -vol. xlviii. p. 261; see also Morgan’s Papers in <i>Philosophical Transactions</i> -[1785], vol. lxxv. p. 190.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_336" href="#FNanchor_336" class="label">336</a> -<i>Edinburgh Journal of Science</i>, vol. v. p. 79.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_337" href="#FNanchor_337" class="label">337</a> -<i>Encyclopædia Metropolitana</i>, art. <i>Light</i>, § 524; Herschel’s -<i>Familiar Lectures</i>, p. 266.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_338" href="#FNanchor_338" class="label">338</a> -Talbot, <i>Philosophical Magazine</i>, 3rd Series, vol. ix. p. 1 (1836); -Brewster, <i>Transactions of the Royal Society of Edinburgh</i> [1823], -vol. ix. pp. 433, 455; Swan, ibid. [1856] vol. xxi. p. 411; <i>Philosophical -Magazine</i>, 4th Series, vol. xx. p. 173 [Sept. 1860]; Roscoe, <i>Spectrum -Analysis</i>, Lecture III.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_339" href="#FNanchor_339" class="label">339</a> -Balfour Stewart, <i>Elementary Treatise on Heat</i>, p. 192.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_340" href="#FNanchor_340" class="label">340</a> -British Association, Liverpool, 1870. <i>Report on Rainfall</i>, p. 176.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_341" href="#FNanchor_341" class="label">341</a> -<i>Philosophical Magazine.</i>, Dec. 1861. 4th Series, vol. xxii. p. 421.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_342" href="#FNanchor_342" class="label">342</a> -<i>Experimental Researches in Electricity</i>, vol. iii. p. 84, &c.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_343" href="#FNanchor_343" class="label">343</a> -<i>Lectures on Heat</i>, p. 21.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_344" href="#FNanchor_344" class="label">344</a> -Baily, <i>Memoirs of the Royal Astronomical Society</i>, vol. xiv. pp. -29, 30.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_345" href="#FNanchor_345" class="label">345</a> -Grant, <i>History of Physical Astronomy</i>, p. 531.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_346" href="#FNanchor_346" class="label">346</a> -<i>Philosophical Transactions</i>, abridged by Lowthorp, 4th edition, -vol. i. p. 202.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_347" href="#FNanchor_347" class="label">347</a> -Jevons in Watts’ <i>Dictionary of Chemistry</i>, vol. ii. pp. 936, 937.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_348" href="#FNanchor_348" class="label">348</a> -<i>Discovery of Subterraneal Treasure.</i> London, 1639, p. 48.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_349" href="#FNanchor_349" class="label">349</a> -Laplace, <i>System of the World</i>, translated by Harte, vol. ii. p. 322.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_350" href="#FNanchor_350" class="label">350</a> -<i>Principia</i>, bk. ii. sect. 6, Prop. xxxi. Motte’s translation, vol. ii. -p. 108.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_351" href="#FNanchor_351" class="label">351</a> -<i>Essayes of Natural Experiments</i>, &c. p. 117.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_352" href="#FNanchor_352" class="label">352</a> -Hooke’s <i>Posthumous Works</i>, p. 182.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_353" href="#FNanchor_353" class="label">353</a> -<i>Principia</i>, bk. iii. Prop. vii. Corollary 1.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_354" href="#FNanchor_354" class="label">354</a> -Keill’s <i>Introduction to Natural Philosophy</i>, 3rd ed., London, -1733, pp. 48–54.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_355" href="#FNanchor_355" class="label">355</a> -<i>Discovery of Subterraneal Treasure</i>, 1639, p. 52.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_356" href="#FNanchor_356" class="label">356</a> -<i>Elements of Inductive Logic</i>, 1st edit. p. 175.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_357" href="#FNanchor_357" class="label">357</a> -<i>Philosophical Transactions</i>, vol. li. p. 138; abridgment, vol. xi. -p. 355.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_358" href="#FNanchor_358" class="label">358</a> -See Bunsen and Roscoe’s researches, in <i>Philosophical Transactions</i> -(1859), vol. cxlix. p. 880, &c., where they describe a constant flame of carbon monoxide gas.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_359" href="#FNanchor_359" class="label">359</a> -Humboldt’s <i>Cosmos</i> (Bohn), vol. i. p. 7.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_360" href="#FNanchor_360" class="label">360</a> -Gilbert, <i>De Magnete</i>, p. 109.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_361" href="#FNanchor_361" class="label">361</a> -<i>Principia</i>, bk. iii. Prop. vi.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_362" href="#FNanchor_362" class="label">362</a> -<i>Philosophical Magazine</i>, 3rd Series, vol. xxvi. p. 375.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_363" href="#FNanchor_363" class="label">363</a> -<i>Opticks</i>, 3rd edit. p. 159.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_364" href="#FNanchor_364" class="label">364</a> -Watts, <i>Dictionary of Chemistry</i>, vol. iii. p. 637.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_365" href="#FNanchor_365" class="label">365</a> -<i>Faraday’s Life</i>, by Bence Jones, vol. ii. p. 5.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_366" href="#FNanchor_366" class="label">366</a> -<i>Preliminary Discourse</i>, &c., p. 185.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_367" href="#FNanchor_367" class="label">367</a> -<i>Philosophical Magazine</i>, July, 1857, 4th Series, vol. xiv. p. 24.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_368" href="#FNanchor_368" class="label">368</a> -<i>First Principles</i>, 3rd edit. chap. x. p. 253.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_369" href="#FNanchor_369" class="label">369</a> -Laplace, <i>System of the World</i>, vol. i. pp. 50, 54, &c.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_370" href="#FNanchor_370" class="label">370</a> -Herschel’s <i>Outlines of Astronomy</i>, 4th edit. pp. 555–557.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_371" href="#FNanchor_371" class="label">371</a> -Humboldt’s <i>Cosmos</i> (Bohn), vol. iii. p. 229.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_372" href="#FNanchor_372" class="label">372</a> -<i>Encyclopædia Metropolitana</i>, art. <i>Sound</i>, § 323; <i>Outlines of -Astronomy</i>, 4th edit., § 650. pp. 410, 487–88; <i>Meteorology, Encyclopædia -Britannica</i>, Reprint, p. 197.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_373" href="#FNanchor_373" class="label">373</a> -<i>Philosophical Transactions</i>, (1739), vol. xli. p. 126.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_374" href="#FNanchor_374" class="label">374</a> -<i>Principia</i>, bk. iii. Prop. 15.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_375" href="#FNanchor_375" class="label">375</a> -Lockyer’s <i>Lessons in Elementary Astronomy</i>, p. 301.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_376" href="#FNanchor_376" class="label">376</a> -<i>Treatise on Natural Philosophy</i>, vol. i. pp. 337, &c.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_377" href="#FNanchor_377" class="label">377</a> -<i>An Introduction to Natural Philosophy</i>, 3rd edit. 1733, p. 5.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_378" href="#FNanchor_378" class="label">378</a> -Watts, <i>Dictionary of Chemistry</i>, vol. i. p. 455.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_379" href="#FNanchor_379" class="label">379</a> -<i>Philosophical Transactions</i>, (1866) vol. clvi. p. 809.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_380" href="#FNanchor_380" class="label">380</a> -<i>Experimental Researches in Electricity</i>, vol. i. p. 246.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_381" href="#FNanchor_381" class="label">381</a> -Hutton’s <i>Mathematical Dictionary</i>, vol. ii. pp. 287–292.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_382" href="#FNanchor_382" class="label">382</a> -<i>Principia</i>, bk. iii. Prop. 13.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_383" href="#FNanchor_383" class="label">383</a> -Jamin, <i>Cours de Physique</i>, vol. i. pp. 282, 283.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_384" href="#FNanchor_384" class="label">384</a> -Lloyd’s <i>Lectures on the Wave Theory</i>, pp. 22, 23.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_385" href="#FNanchor_385" class="label">385</a> -Tait’s <i>Thermodynamics</i>, p. 10.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_386" href="#FNanchor_386" class="label">386</a> -Lloyd’s <i>Lectures on the Wave Theory</i>, pp. 82, 83.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_387" href="#FNanchor_387" class="label">387</a> -Jamin, <i>Cours de Physique</i>, vol. i. pp. 283–288.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_388" href="#FNanchor_388" class="label">388</a> -Joule and Thomson, <i>Philosophical Transactions</i>, 1854, vol. cxliv. -p. 337.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_389" href="#FNanchor_389" class="label">389</a> -The properties of a perfect gas have been described by Rankine, -<i>Transactions of the Royal Society of Edinburgh</i>, vol. xxv. p. 561.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_390" href="#FNanchor_390" class="label">390</a> -Thomson and Tait’s <i>Natural Philosophy</i>, vol. i. p. 60.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_391" href="#FNanchor_391" class="label">391</a> -Challis, <i>Notes on the Principles of Pure and Applied Calculation</i>, -1869, p. 83.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_392" href="#FNanchor_392" class="label">392</a> -<i>An Introduction to Physical Measurements</i>, translated by Waller -and Procter, 1873, p. 10.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_393" href="#FNanchor_393" class="label">393</a> -<i>Cambridge Philosophical Transactions</i> (1865), vol. xi. Part I.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_394" href="#FNanchor_394" class="label">394</a> -Sandeman, <i>Pelicotetics</i>, p. 214.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_395" href="#FNanchor_395" class="label">395</a> -<i>The Science and Art of Arithmetic for the Use of Schools.</i> -(Whitaker and Co.)</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_396" href="#FNanchor_396" class="label">396</a> -<i>Principles of Approximate Calculations</i>, by J. J. Skinner, C.E. -(New York, Henry Holt), 1876.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_397" href="#FNanchor_397" class="label">397</a> -Leslie, <i>Inquiry into the Nature of Heat</i>, p. 505.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_398" href="#FNanchor_398" class="label">398</a> -<i>System of Logic</i>, bk. iii. chap. viii § 6.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_399" href="#FNanchor_399" class="label">399</a> -Laplace, <i>System of the World</i>, translated by Harte, vol. ii. p. 366.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_400" href="#FNanchor_400" class="label">400</a> -<i>Chemical Reports and Memoirs</i>, Cavendish Society, p. 294.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_401" href="#FNanchor_401" class="label">401</a> -Jamin, <i>Cours de Physique</i>, vol. ii. p. 38.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_402" href="#FNanchor_402" class="label">402</a> -<i>On Tides and Waves</i>, Encyclopædia Metropolitana, p. 366*.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_403" href="#FNanchor_403" class="label">403</a> -<i>Encyclopædia Britannica</i>, art. <i>Meteorology</i>. Reprint, §§ 152–156.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_404" href="#FNanchor_404" class="label">404</a> -Lagrange, <i>Leçons sur le Calcul des Fonctions</i>, 1806, p. 4.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_405" href="#FNanchor_405" class="label">405</a> -Haughton, <i>Principles of Animal Mechanics</i>, 1873, pp. 444–450. -Jevons, <i>Nature</i>, 30th of June, 1870, vol. ii. p. 158. See also the -experiments of Professor Nipher, of Washington University, St. -Louis, in <i>American Journal of Science</i>, vol. ix. p. 130, vol. x. p. 1; -<i>Nature</i>, vol. xi. pp. 256, 276.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_406" href="#FNanchor_406" class="label">406</a> -Jamin, <i>Cours de Physique</i>, vol. ii. p. 50.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_407" href="#FNanchor_407" class="label">407</a> -<i>Philosophical Transactions</i>, 1826, p. 544.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_408" href="#FNanchor_408" class="label">408</a> -Jamin, <i>Cours de Physique</i>, vol. ii. p. 24, &c.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_409" href="#FNanchor_409" class="label">409</a> -J. W. Strutt, <i>On a correction sometimes required in curves professing -to represent the connexion between two physical magnitudes</i>. -Philosophical Magazine, 4th Series, vol. xlii. p. 441.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_410" href="#FNanchor_410" class="label">410</a> -Herschel: Lacroix’ <i>Differential Calculus</i>, p. 551.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_411" href="#FNanchor_411" class="label">411</a> -<i>Cours complet de Météorologie</i>, Note A, p. 449.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_412" href="#FNanchor_412" class="label">412</a> -<i>On the Calculation of Empirical Formulæ. The Messenger of -Mathematics</i>, New Series, No. 17, 1872.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_413" href="#FNanchor_413" class="label">413</a> -Watts’ <i>Dictionary of Chemistry</i>, vol. ii. p. 790.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_414" href="#FNanchor_414" class="label">414</a> -<i>Quarterly Journal of the Chemical Society</i>, vol. viii. p. 15.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_415" href="#FNanchor_415" class="label">415</a> -<i>Results of Observations at the Cape of Good Hope</i>, p. 293.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_416" href="#FNanchor_416" class="label">416</a> -Jamin, <i>Cours de Physique</i>, vol. ii. p. 138.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_417" href="#FNanchor_417" class="label">417</a> -<i>Preliminary Discourse</i>, &c., p. 152.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_418" href="#FNanchor_418" class="label">418</a> -Tyndall, <i>On Cometary Theory</i>, Philosophical Magazine, April -1869. 4th Series, vol. xxxvii. p. 243.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_419" href="#FNanchor_419" class="label">419</a> -See <i>Philosophical Transactions</i>, abridged by Lowthorp. 4th edit. -vol. i. p. 130. I find that opinions similar to those in the text have -been briefly expressed by De Morgan in his remarkable preface to -<i>From Matter to Spirit</i>, by C.D., pp. xxi. xxii.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_420" href="#FNanchor_420" class="label">420</a> -Horrocks, <i>Opera Posthuma</i> (1673), p. 276.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_421" href="#FNanchor_421" class="label">421</a> -Young’s <i>Works</i>, vol. i. p. 593.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_422" href="#FNanchor_422" class="label">422</a> -Boyle’s <i>Physical Examen</i>, p. 84.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_423" href="#FNanchor_423" class="label">423</a> -Young’s <i>Works</i>, vol. i. p. 415.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_424" href="#FNanchor_424" class="label">424</a> -<i>Familiar Lectures on Scientific Subjects</i>, p. 282.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_425" href="#FNanchor_425" class="label">425</a> -Young’s <i>Works</i>, vol. i. p. 417.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_426" href="#FNanchor_426" class="label">426</a> -<i>Principia</i>, bk. iii. Prop. 43. General Scholium.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_427" href="#FNanchor_427" class="label">427</a> -Ibid. bk. ii. Sect. ix. Prop. 53.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_428" href="#FNanchor_428" class="label">428</a> -Brewster’s <i>Life of Newton</i>, 1st edit. chap. vii.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_429" href="#FNanchor_429" class="label">429</a> -<i>Discourse on the Study of Natural Philosophy</i>, p. 151.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_430" href="#FNanchor_430" class="label">430</a> -Ibid. p. 229.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_431" href="#FNanchor_431" class="label">431</a> -<i>Novum Organum</i>, bk. ii. Aphorism 36.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_432" href="#FNanchor_432" class="label">432</a> -<i>Principia</i>, bk. i. Sect. xiv. Prop. 96. Scholium. <i>Opticks</i>, Prop. vi. -3rd edit. p. 70.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_433" href="#FNanchor_433" class="label">433</a> -Airy’s <i>Mathematical Tracts</i>, 3rd edit. pp. 286–288.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_434" href="#FNanchor_434" class="label">434</a> -Jamin, <i>Cours de Physique</i>, vol. iii. p. 372.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_435" href="#FNanchor_435" class="label">435</a> -Young’s <i>Lectures on Natural Philosophy</i> (1845), vol. i. p. 361.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_436" href="#FNanchor_436" class="label">436</a> -Paris, <i>Life of Davy</i>, p. 274.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_437" href="#FNanchor_437" class="label">437</a> -<i>Opus Majus.</i> Edit. 1733. Cap. x. p. 460.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_438" href="#FNanchor_438" class="label">438</a> -Newton’s <i>Opticks</i>. Third edit. p. 249.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_439" href="#FNanchor_439" class="label">439</a> -Brewster. <i>Treatise on New Philosophical Instruments</i>, p. 266, &c.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_440" href="#FNanchor_440" class="label">440</a> -Roscoe, Bakerian Lecture, <i>Philosophical Transactions</i> (1868), -vol. clviii. p. 6.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_441" href="#FNanchor_441" class="label">441</a> -<i>Life of Faraday</i>, vol. ii. p. 104.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_442" href="#FNanchor_442" class="label">442</a> -Watts, <i>Dictionary of Chemistry</i>, vol. ii, p. 39, &c.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_443" href="#FNanchor_443" class="label">443</a> -De Morgan’s <i>Budget of Paradoxes</i>, p. 291.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_444" href="#FNanchor_444" class="label">444</a> -<i>Life of Faraday</i>, vol. ii p. 396.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_445" href="#FNanchor_445" class="label">445</a> -<i>Experimental Researches in Electricity</i>, 1st Series, pp. 24–44.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_446" href="#FNanchor_446" class="label">446</a> -Airy, <i>On Tides and Waves</i>, Encyclopædia Metropolitana, p. 348*</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_447" href="#FNanchor_447" class="label">447</a> -Lib. i. cap. 74.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_448" href="#FNanchor_448" class="label">448</a> -Taylor’s <i>Scientific Memoirs</i>, vol. v. p. 241.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_449" href="#FNanchor_449" class="label">449</a> -Airy’s <i>Mathematical Tracts</i>, 3rd edit. p. 312.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_450" href="#FNanchor_450" class="label">450</a> -Young’s <i>Works</i>, vol. i. p. 412.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_451" href="#FNanchor_451" class="label">451</a> -Lloyd’s <i>Wave Theory</i>, Part ii. pp. 52–58. Babbage, <i>Ninth -Bridgewater Treatise</i>, p. 104, quoting Lloyd, <i>Transactions of the -Royal Irish Academy</i>, vol. xvii. Clifton, <i>Quarterly Journal of Pure -and Applied Mathematics</i>, January 1860.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_452" href="#FNanchor_452" class="label">452</a> -<i>Encyclopædia Metropolitana</i>, art. <i>Sound</i>, p. 753.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_453" href="#FNanchor_453" class="label">453</a> -Tyndall’s <i>Sound</i>, pp. 261, 273.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_454" href="#FNanchor_454" class="label">454</a> -Whewell’s <i>History of the Inductive Sciences</i>, vol. ii. p. 471. -Herschel’s <i>Physical Geography</i>, § 77.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_455" href="#FNanchor_455" class="label">455</a> -Maxwell’s <i>Theory of Heat</i>, p. 174. <i>Philosophical Magazine</i>, -August 1850. Third Series, vol. xxxvii. p. 123.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_456" href="#FNanchor_456" class="label">456</a> -<i>Philosophical Transactions</i>, 1858, vol. cxlviii. p. 127.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_457" href="#FNanchor_457" class="label">457</a> -Tyndall’s <i>Faraday</i>, pp. 73, 74; <i>Life of Faraday</i>, vol. ii. pp. 82, 83.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_458" href="#FNanchor_458" class="label">458</a> -Tait’s <i>Thermodynamics</i>, p. 77.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_459" href="#FNanchor_459" class="label">459</a> -<i>On the Analytical Forms called Trees, with Application to the -Theory of Chemical Combinations.</i> Report of the British Association, -1875, p. 257.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_460" href="#FNanchor_460" class="label">460</a> -Hofmann’s <i>Introduction to Chemistry</i>, pp. 224, 225.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_461" href="#FNanchor_461" class="label">461</a> -<i>Philosophical Transactions</i> (1855), vol. cxlv. pp. 100, &c.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_462" href="#FNanchor_462" class="label">462</a> -<i>Proceedings of the Manchester Philosophical Society</i>, Feb. 1870.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_463" href="#FNanchor_463" class="label">463</a> -Balfour Stewart, <i>Elementary Treatise on Heat</i>, 1st edit. p. 198.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_464" href="#FNanchor_464" class="label">464</a> -Jevons, <i>Proceedings of the Manchester Literary and Philosophical -Society</i>, 25th January, 1870, vol. ix. p. 78.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_465" href="#FNanchor_465" class="label">465</a> -<i>Philosophical Transactions</i>, vol. cxlvi. p. 249.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_466" href="#FNanchor_466" class="label">466</a> -Grant’s <i>History of Physical Astronomy</i>, p. 162.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_467" href="#FNanchor_467" class="label">467</a> -<i>Philosophical Transactions</i> (1854), vol. cxliv. p. 364.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_468" href="#FNanchor_468" class="label">468</a> -<i>Monthly Notices of the Royal Astronomical Society</i>, vol. xxviii. -p. 264.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_469" href="#FNanchor_469" class="label">469</a> -It would seem to be absurd to repeat the profuse expenditure of -1874 at the approaching transit in 1882. The aggregate sum spent in -1874 by various governments and individuals can hardly be less than -£200,000, a sum which, wisely expended on scientific investigations, -would give a hundred important results.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_470" href="#FNanchor_470" class="label">470</a> -<i>Philosophical Transactions</i> (1856), vol. cxlvi. p. 342.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_471" href="#FNanchor_471" class="label">471</a> -<i>Monthly Notices of the Royal Astronomical Society</i>, for 8th Nov. -1844, No. X. vol. vi. p. 89.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_472" href="#FNanchor_472" class="label">472</a> -<i>Philosophical Magazine</i>, 2nd Series, vol. xxvi. p. 61.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_473" href="#FNanchor_473" class="label">473</a> -Clausius in <i>Philosophical Magazine</i>, 4th Series, vol. ii. p. 119.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_474" href="#FNanchor_474" class="label">474</a> -Watts’ <i>Dictionary of Chemistry</i>, vol. iii. p. 129.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_475" href="#FNanchor_475" class="label">475</a> -<i>Preliminary Discourse</i>, §§ 158, 174. <i>Outlines of Astronomy</i>, 4th -edit. § 856.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_476" href="#FNanchor_476" class="label">476</a> -<i>Proceedings of the Manchester Literary and Philosophical Society</i>, -28th November, 1871, vol. xi. p. 33. Since the above remarks were -written, Professor Balfour Stewart has pointed out to me his paper -in the <i>Proceedings of the Manchester Literary and Philosophical -Society</i> for 15th November, 1870 (vol. x. p. 32), in which he shows -that a body moving in an enclosure of uniform temperature would -probably experience resistance independently of the presence of a -ponderable medium, such as gas, between the moving body and the -enclosure. The proof is founded on the theory of the dissipation of -energy, and this view is said to be accepted by Professors Thomson and -Tait. The enclosure is used in this case by Professor Stewart simply -as a means of obtaining a proof, just as it was used by him on a -previous occasion to obtain a proof of certain consequences of the -Theory of Exchanges. He is of opinion that in both of these -cases when once the proof has been obtained, the enclosure may be -dispensed with. We know, for instance, that the relation between the -inductive and absorptive powers of bodies—although this relation -may have been proved by means of an enclosure, does not depend -upon its presence, and Professor Stewart thinks that in like manner -two bodies, or at least two bodies possessing heat such as the sun -and the earth in motion relative to each other, will have the differential -motion retarded until perhaps it is ultimately destroyed.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_477" href="#FNanchor_477" class="label">477</a> -<i>British Association Catalogue of Stars</i>, p. 49.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_478" href="#FNanchor_478" class="label">478</a> -<i>Experimental Researches in Chemistry and Physics</i>, p. 372. -<i>Philosophical Magazine</i>, 3rd Series, May 1846, vol. xxviii. p. 350.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_479" href="#FNanchor_479" class="label">479</a> -See also <i>Nature</i>, September 18, 1873; vol. viii. p. 398.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_480" href="#FNanchor_480" class="label">480</a> -<i>Theory of Political Economy</i>, pp. 3–14.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_481" href="#FNanchor_481" class="label">481</a> -<i>Principia</i>, bk. i. Prop. iv.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_482" href="#FNanchor_482" class="label">482</a> -<i>Opticks</i>, bk. i. part ii. Prop. 3. 3rd ed. p. 115.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_483" href="#FNanchor_483" class="label">483</a> -<i>Experimental Inquiry into the Nature of Heat.</i> Preface, p. xv.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_484" href="#FNanchor_484" class="label">484</a> -Bence Jones, <i>Life of Faraday</i>, vol. i. p. 362.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_485" href="#FNanchor_485" class="label">485</a> -Ibid. vol. ii. p. 199.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_486" href="#FNanchor_486" class="label">486</a> -See also his more formal statement in the <i>Experimental Researches -in Electricity</i>, 24th Series, § 2702, vol. iii. p. 161.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_487" href="#FNanchor_487" class="label">487</a> -Printed in <i>Modern Culture</i>, edited by Youmans, p. 219.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_488" href="#FNanchor_488" class="label">488</a> -<i>Life of Faraday</i>, vol. i. p. 225.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_489" href="#FNanchor_489" class="label">489</a> -Aristotle’s <i>Rhetoric</i>, Liber I. 2. 11.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_490" href="#FNanchor_490" class="label">490</a> -<i>Essai Philosophique sur les Probabilités</i>, p. 86.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_491" href="#FNanchor_491" class="label">491</a> -Kant’s <i>Logik</i>, § 84, Königsberg, 1800, p. 207.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_492" href="#FNanchor_492" class="label">492</a> -<i>Syllabus of a Proposed System of Logic</i>, p. 34.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_493" href="#FNanchor_493" class="label">493</a> -<i>Principia</i>, bk. iii. Prop. VI. Motte’s translation, vol. ii. p. 220.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_494" href="#FNanchor_494" class="label">494</a> -Professor Lovering has pointed out how obscure and uncertain -the ideas of scientific men about this ether are, in his interesting -Presidential Address before the American Association at Hartford, -1874. <i>Silliman’s Journal</i>, October 1874, p. 297. <i>Philosophical -Magazine</i>, vol. xlviii. p. 493.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_495" href="#FNanchor_495" class="label">495</a> -<i>Novum Organum</i>, bk. ii. Aphorisms, 24, 25.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_496" href="#FNanchor_496" class="label">496</a> -Ibid. Aph. 28.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_497" href="#FNanchor_497" class="label">497</a> -<i>Philosophical Transactions</i> (1856) vol. cxlvi. p. 246.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_498" href="#FNanchor_498" class="label">498</a> -<i>Philosophical Magazine</i>, 4th Series, January 1870, vol. xxxix. p. 2.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_499" href="#FNanchor_499" class="label">499</a> -<i>Novum Organum</i>, bk. ii. Aphorism 25.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_500" href="#FNanchor_500" class="label">500</a> -Faraday’s <i>Experimental Researches in Chemistry and Physics</i>, -p. 93.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_501" href="#FNanchor_501" class="label">501</a> -<i>Memorabilia</i>, iv. 7.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_502" href="#FNanchor_502" class="label">502</a> -<i>Experimental Researches in Electricity</i>, Series xii. vol. i. p. 420.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_503" href="#FNanchor_503" class="label">503</a> -<i>Life of Faraday</i>, vol. ii. p. 7.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_504" href="#FNanchor_504" class="label">504</a> -<i>Nature</i>, vol. ii. p. 278.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_505" href="#FNanchor_505" class="label">505</a> -<i>Journal of the Chemical Society</i>, vol. viii. p. 51.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_506" href="#FNanchor_506" class="label">506</a> -<i>Correlation of Physical Forces</i>, 3rd edit. p. 184.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_507" href="#FNanchor_507" class="label">507</a> -<i>Philosophical Magazine</i>, 4th Series, vol. xlii. p. 451.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_508" href="#FNanchor_508" class="label">508</a> -Grove, <i>Correlation of Physical Forces</i>, 3rd edit. p. 118.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_509" href="#FNanchor_509" class="label">509</a> -Ibid. pp. 166, 199, &c.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_510" href="#FNanchor_510" class="label">510</a> -<i>Philosophical Transactions</i>, 1861. <i>Chemical and Physical Researches</i>, -p. 598.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_511" href="#FNanchor_511" class="label">511</a> -<i>Life of Sir W. Hamilton</i>, p. 439.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_512" href="#FNanchor_512" class="label">512</a> -Powell’s <i>History of Natural Philosophy</i>, p. 201. <i>Novum -Organum</i>, bk. ii. Aphorisms 5–7.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_513" href="#FNanchor_513" class="label">513</a> -Thomson and Tait, <i>Treatise on Natural Philosophy</i>, vol. i. pp. -346–351.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_514" href="#FNanchor_514" class="label">514</a> -<i>Philosophical Transactions</i> (1740), vol. xli. p. 454.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_515" href="#FNanchor_515" class="label">515</a> -<i>Principia</i>, bk. i. Law iii. Corollary 6.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_516" href="#FNanchor_516" class="label">516</a> -Helmholtz, Taylor’s <i>Scientific Memoirs</i> (1853), vol. vi. p. 118.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_517" href="#FNanchor_517" class="label">517</a> -<i>Lucretius</i>, bk. i. lines 232–264.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_518" href="#FNanchor_518" class="label">518</a> -<i>Novum Organum</i>, bk. 1 Aphorism 104.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_519" href="#FNanchor_519" class="label">519</a> -<i>The Unity of Worlds and of Nature</i>, 2nd edit. p. 116.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_520" href="#FNanchor_520" class="label">520</a> -<i>Principia</i>, bk. iii, <i>ad initium</i>.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_521" href="#FNanchor_521" class="label">521</a> -Keill, <i>Introduction to Natural Philosophy</i>, p. 89.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_522" href="#FNanchor_522" class="label">522</a> -Jeremiæ Horroccii <i>Opera Posthuma</i> (1673), pp. 26, 27.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_523" href="#FNanchor_523" class="label">523</a> -Young’s <i>Works</i>, vol. ii. p. 564.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_524" href="#FNanchor_524" class="label">524</a> -<i>Essay on Logic</i>, <i>Works</i>, vol. viii. p. 276.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_525" href="#FNanchor_525" class="label">525</a> -<i>Life of Faraday</i>, by Bence Jones, vol. ii. p. 206.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_526" href="#FNanchor_526" class="label">526</a> -Lacroix, <i>Traité Élémentaire de Calcul Différentiel et de Calcul -Intégral</i>, 5<sup>me</sup> édit. p. 699.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_527" href="#FNanchor_527" class="label">527</a> -<i>Histoire des Mathématiques</i>, vol. i. p. 298.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_528" href="#FNanchor_528" class="label">528</a> -See Goodwin, <i>Cambridge Philosophical Transactions</i> (1845), vol. -viii. p. 269. O’Brien, “On Symbolical Statics,” <i>Philosophical -Magazine</i>, 4th Series, vol. i. pp. 491, &c. See also Professor Clerk -Maxwell’s delightful <i>Manual of Elementary Science</i>, called <i>Matter -and Motion</i>, published by the Society for Promoting Christian -Knowledge. In this admirable little work some of the most advanced -results of mechanical and physical science are explained according to -the method of quaternions, but with hardly any use of algebraic -symbols.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_529" href="#FNanchor_529" class="label">529</a> -Birch, <i>History of the Royal Society</i>, vol. iii. p. 262, quoted by -Young, <i>Works</i>, vol. i. p. 246.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_530" href="#FNanchor_530" class="label">530</a> -<i>Opticks</i>, Query 28, 3rd edit. p. 337.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_531" href="#FNanchor_531" class="label">531</a> -Rankine, <i>Philosophical Transactions</i> (1856), vol. cxlvi. p. 282.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_532" href="#FNanchor_532" class="label">532</a> -<i>Cosmotheoros</i> (1699), p. 16.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_533" href="#FNanchor_533" class="label">533</a> -Laplace, <i>System of the World</i>, vol. ii. p. 316.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_534" href="#FNanchor_534" class="label">534</a> -<i>Cosmotheoros</i> (1699), p. 17.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_535" href="#FNanchor_535" class="label">535</a> -Ibid. p. 36.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_536" href="#FNanchor_536" class="label">536</a> -<i>System of the World</i>, vol. ii. p. 326. <i>Essai Philosophique</i>, p. 87.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_537" href="#FNanchor_537" class="label">537</a> -<i>Principia</i>, bk. ii. Section ii. Prop. x.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_538" href="#FNanchor_538" class="label">538</a> -De Morgan, <i>Cambridge Philosophical Transactions</i>, vol. xi. -Part ii. p. 246.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_539" href="#FNanchor_539" class="label">539</a> -<i>Life of Faraday</i>, vol. i. p. 216.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_540" href="#FNanchor_540" class="label">540</a> -Babbage, <i>The Exposition of 1851</i>, p. 1.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_541" href="#FNanchor_541" class="label">541</a> -Daubeny’s <i>Atomic Theory</i>, p. 76.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_542" href="#FNanchor_542" class="label">542</a> -<i>Bakerian Lecture, Philosophical Transactions</i> (1868), vol. clviii. -p. 2.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_543" href="#FNanchor_543" class="label">543</a> -<i>Principia</i>, bk. ii. Prop. 20. Corollaries, 5 and 6.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_544" href="#FNanchor_544" class="label">544</a> -<i>Treatise on Natural Philosophy</i>, vol. i. p. 50.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_545" href="#FNanchor_545" class="label">545</a> -Maxwell’s <i>Theory of Heat</i>, (1871), p. 175.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_546" href="#FNanchor_546" class="label">546</a> -Galton, on the Height and Weight of Boys. <i>Journal of the -Anthropological Institute</i>, 1875, p. 174.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_547" href="#FNanchor_547" class="label">547</a> -Grant’s <i>History of Physical Astronomy</i>, p. 116.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_548" href="#FNanchor_548" class="label">548</a> -<i>Discourse to the Royal Society</i>, 28th May, 1684.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_549" href="#FNanchor_549" class="label">549</a> -Robert Hooke’s <i>Posthumous Works</i>, p. 365.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_550" href="#FNanchor_550" class="label">550</a> -<i>Experimental Researches in Electricity</i>, vol. ii. pp. 240–245.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_551" href="#FNanchor_551" class="label">551</a> -Murchison’s <i>Silurian System</i>, vol. ii. p. 733, &c.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_552" href="#FNanchor_552" class="label">552</a> -<i>Philosophical Transactions</i> (1872), vol. clxii. No. 23.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_553" href="#FNanchor_553" class="label">553</a> -<i>Philosophical Transactions</i> (1852), vol. cxlii. pp. 465, 548, &c.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_554" href="#FNanchor_554" class="label">554</a> -<i>Philosophical Magazine</i>, 4th Series, vol. i. p. 182.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_555" href="#FNanchor_555" class="label">555</a> -Maxwell, <i>Theory of Heat</i>, p. 123.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_556" href="#FNanchor_556" class="label">556</a> -<i>Prior Analytics</i>, ii. 2, 8, and elsewhere.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_557" href="#FNanchor_557" class="label">557</a> -Hofmann’s <i>Introduction to Chemistry</i>, p. 198.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_558" href="#FNanchor_558" class="label">558</a> -Stewart’s <i>Elementary Treatise on Heat</i>, p. 80.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_559" href="#FNanchor_559" class="label">559</a> -Jevons, <i>Proceedings of the Manchester Literary and Philosophical -Society</i>, 6th March, 1877, vol. xvi. p. 164. See also Mr. W. E. -A. Axon’s note on the same subject, ibid. p. 166.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_560" href="#FNanchor_560" class="label">560</a> -<i>A Treatise on Logic, or, the Laws of Pure Thought</i>, by Francis -Bowen, Professor of Moral Philosophy in Harvard College, Cambridge, -United States, 1866, p. 315.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_561" href="#FNanchor_561" class="label">561</a> -<i>Proceedings of the Royal Society</i>, November, 1873, vol. xxi. p. 512.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_562" href="#FNanchor_562" class="label">562</a> -<i>Lectures on the Elements of Comparative Anatomy</i>, 1864, p. 1.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_563" href="#FNanchor_563" class="label">563</a> -<i>Essai sur la Philosophie des Sciences</i>, p. 9.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_564" href="#FNanchor_564" class="label">564</a> -<i>Lectures on the Elements of Comparative Anatomy, and on the -Classification of Animals</i>, 1864, p. 3.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_565" href="#FNanchor_565" class="label">565</a> -<i>Ossemens Fossiles</i>, 4th edit. vol. i. p. 164. Quoted by Huxley, -<i>Lectures</i>, &c., p. 5.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_566" href="#FNanchor_566" class="label">566</a> -Chambers, <i>Descriptive Astronomy</i>, 1st edit. p. 23.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_567" href="#FNanchor_567" class="label">567</a> -<i>Philosophical Magazine</i>, 4th Series, vol. xxxix. p. 396; vol. xl. -p. 183; vol. xli. p. 44. See also Proctor, <i>Popular Science Review</i>, -October 1874, p. 350.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_568" href="#FNanchor_568" class="label">568</a> -Humboldt, <i>Cosmos</i> (Bohn), vol. iii. p. 224.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_569" href="#FNanchor_569" class="label">569</a> -Baily, British <i>Association Catalogue</i>, p. 48.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_570" href="#FNanchor_570" class="label">570</a> -<i>Outlines of Astronomy</i>, § 850, 4th edit. p. 578.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_571" href="#FNanchor_571" class="label">571</a> -<i>Life of Faraday</i>, vol. ii. p. 87.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_572" href="#FNanchor_572" class="label">572</a> -<i>Proceedings of the Royal Society</i>, vol. xvii. p. 212. <i>Chemical and -Physical Researches</i>, reprint, by Young and Angus Smith, p. 290.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_573" href="#FNanchor_573" class="label">573</a> -<i>Essai sur la Nomenclature et la Classification</i>, Paris, 1823, pp. -107, 108.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_574" href="#FNanchor_574" class="label">574</a> -George Bentham, <i>Outline of a New System of Logic</i>, p. 115.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_575" href="#FNanchor_575" class="label">575</a> -<i>Outline of a New System of Logic</i>, 1827, p. 117.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_576" href="#FNanchor_576" class="label">576</a> -<i>Porphyrii Isagoge</i>, Caput ii. 24.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_577" href="#FNanchor_577" class="label">577</a> -Jevons, <i>Elementary Lessons in Logic</i>, p. 104.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_578" href="#FNanchor_578" class="label">578</a> -<i>Chrestomathia; being a Collection of Papers, &c.</i> London, 1816, -Appendix V.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_579" href="#FNanchor_579" class="label">579</a> -<i>The Classification of the Sciences</i>, &c., 3rd edit. p. 7. <i>Essays: -Scientific, Political, and Speculative</i>, vol. iii. p. 13.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_580" href="#FNanchor_580" class="label">580</a> -Owen, <i>Essay on the Classification and Geographical Distribution -of the Mammalia</i>, p. 20.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_581" href="#FNanchor_581" class="label">581</a> -Dana’s <i>Mineralogy</i>, vol. i. p. 123; quoted in Watts’ <i>Dictionary -of Chemistry</i>, vol. ii. p. 166.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_582" href="#FNanchor_582" class="label">582</a> -<i>Instructions for the Discrimination of Minerals by Simple Chemical -Experiments</i>, by Franz von Kobell, translated from the German -by R. C. Campbell. Glasgow, 1841.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_583" href="#FNanchor_583" class="label">583</a> -Edition of 1866, p. lxiii.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_584" href="#FNanchor_584" class="label">584</a> -<i>Philosophia Botanica</i> (1770), § 154, p. 98.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_585" href="#FNanchor_585" class="label">585</a> -<i>Philosophical Magazine</i>, 3rd Series (1845), vol. xxvi. p. 522. See -also De Morgan’s evidence before the Royal Commission on the British -Museum in 1849, Report (1850), Questions, 5704*-5815*, 6481–6513. -This evidence should be studied by every person who wishes -to understand the elements of Bibliography.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_586" href="#FNanchor_586" class="label">586</a> -<i>English Cyclopædia, Arts and Sciences</i>, vol. v. p. 233.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_587" href="#FNanchor_587" class="label">587</a> -Swainson, “Treatise on the Geography and Classification of -Animals,” <i>Cabinet Cyclopædia</i>, p. 201.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_588" href="#FNanchor_588" class="label">588</a> -Darwin, <i>Fertilisation of Orchids</i>, p. 159.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_589" href="#FNanchor_589" class="label">589</a> -<i>Descent of Man</i>, vol. i. p. 214.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_590" href="#FNanchor_590" class="label">590</a> -<i>Laws of Botanical Nomenclature</i>, p. 16.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_591" href="#FNanchor_591" class="label">591</a> -Waterhouse, quoted by Woodward in his <i>Rudimentary Treatise -of Recent and Fossil Shells</i>, p. 61.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_592" href="#FNanchor_592" class="label">592</a> -Bentham’s <i>Handbook of the British Flora</i> (1866), p. 25.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_593" href="#FNanchor_593" class="label">593</a> -<i>Philosophia Botanica</i> (1770), § 157, p. 99.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_594" href="#FNanchor_594" class="label">594</a> -<i>Ibid.</i> § 159, p. 100.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_595" href="#FNanchor_595" class="label">595</a> -<i>Amœnitates Academicæ</i> (1744), vol. i. p. 70. Quoted in <i>Edinburgh -Review</i>, October 1868, vol. cxxviii. pp. 416, 417.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_596" href="#FNanchor_596" class="label">596</a> -<i>Descent of Man</i>, vol. i. p. 228.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_597" href="#FNanchor_597" class="label">597</a> -Agassiz, <i>Essay on Classification</i>, p. 219.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_598" href="#FNanchor_598" class="label">598</a> -<i>Ibid.</i> p. 249.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_599" href="#FNanchor_599" class="label">599</a> -<i>Philosophia Botanica</i>, § 155, p. 98.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_600" href="#FNanchor_600" class="label">600</a> -<i>Laws of Botanical Nomenclature</i>, by Alphonse Decandolle, translated -from the French, 1868, p. 19.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_601" href="#FNanchor_601" class="label">601</a> -Darwin, <i>The Variation of Animals and Plants</i>, vol. ii. pp. 293, -359, &c.; quoting Paget, <i>Lectures on Pathology</i>, 1853, pp. 152, 164.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_602" href="#FNanchor_602" class="label">602</a> -<i>Ibid.</i> vol. ii. p. 372.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_603" href="#FNanchor_603" class="label">603</a> -<i>Théorie Analytique des Probabilités</i>, quoted by Babbage, <i>Ninth -Bridgewater Treatise</i>, p. 173.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_604" href="#FNanchor_604" class="label">604</a> -<i>First Bridgewater Treatise</i> (1834), pp. 16–24.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_605" href="#FNanchor_605" class="label">605</a> -<i>System of Logic</i>, 5th edit. bk. III. chap. V. § 7; chap. XVI. § 3.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_606" href="#FNanchor_606" class="label">606</a> -<i>System of Logic</i>, vol. i. p. 384.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_607" href="#FNanchor_607" class="label">607</a> -<i>Ninth Bridgewater Treatise</i>, p. 140.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_608" href="#FNanchor_608" class="label">608</a> -<i>Ibid.</i> pp. 34–43.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_609" href="#FNanchor_609" class="label">609</a> -Professor Clifford, in his most interesting lecture on “The First -and Last Catastrophe” (<i>Fortnightly Review</i>, April 1875, p. 480, reprint -by the Sunday Lecture Society, p. 24), objects that I have -erroneously substituted “known laws of nature” for “known laws -of conduction of heat.” I quite admit the error, without admitting -all the conclusions which Professor Clifford proceeds to draw; but I -maintain the paragraph unchanged, in order that it may be discussed -in the Preface.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_610" href="#FNanchor_610" class="label">610</a> -Tait’s <i>Thermodynamics</i>, p. 38. <i>Cambridge Mathematical Journal</i>, -vol. iii. p. 174.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_611" href="#FNanchor_611" class="label">611</a> -Clerk Maxwell’s <i>Theory of Heat</i>, p. 245.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_612" href="#FNanchor_612" class="label">612</a> -Maxwell’s <i>Theory of Heat</i>, p. 92.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_613" href="#FNanchor_613" class="label">613</a> -<i>Report of the British Association</i> (1852), Report of Sections, p. 12.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_614" href="#FNanchor_614" class="label">614</a> -Mr. C. J. Monroe objects that in this statement I do injustice -to Comte, who, he thinks, did impress upon his readers the inadequacy -of our mental powers compared with the vastness of the subject -matter of science. The error of Comte, he holds, was in maintaining -that science had been carried about as far as it is worth while to -carry it, which is a different matter. In either case, Comte’s position -is so untenable that I am content to leave the question undecided.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_615" href="#FNanchor_615" class="label">615</a> -<i>Fragments of Science</i>, p. 362.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_616" href="#FNanchor_616" class="label">616</a> -<i>Familiar Lectures on Scientific Subjects</i>, p. 458.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_617" href="#FNanchor_617" class="label">617</a> -<i>Philosophical Magazine</i>, 3rd Series, vol. xxvi. p. 406.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_618" href="#FNanchor_618" class="label">618</a> -<i>History of the Theory of Probability</i>, p. 398.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_619" href="#FNanchor_619" class="label">619</a> -<i>Trigonometry and Double Algebra</i>, chap. ix.</p> - -</div> - -<div class="footnote"> - -<p><a id="Footnote_620" href="#FNanchor_620" class="label">620</a> -Agassiz, <i>Essay on Classification</i>, p. 75.</p> - -</div> -</div> - -<div class="transnote mt3em"> -<a id="Spelling_corrections"></a> -<p>Return to <a href="#Transcribers_notes">transcriber’s notes</a></p> - -<p class="tn"><b>Spelling corrections</b>:<br> -acording → according<br> -aklaline → alkaline<br> -an an → an<br> -aws → laws<br> -beween → between<br> -BOOK III → BOOK IV<br> -errror → error<br> -incapadle → incapable<br> -interpretion → interpretation<br> -justifed → justified<br> -longtitude → longitude<br> -Marriotte → Mariotte<br> -melecules → molecules<br> -Meropolitana → Metropolitana<br> -necesssarily → necessarily<br> -nnmber → number<br> -or → of<br> -probabilty → probability<br> -quantites → quantities<br> -secresy → secrecy<br> -sucession → succession<br> -suficiently → sufficiently<br> -telecope → telescope<br> -verifiy → verify -</p> - -<p>Return to <a href="#Transcribers_notes">transcriber’s notes</a></p> -</div> - - -<div style='text-align:center'>*** END OF THE PROJECT GUTENBERG EBOOK 74864 ***</div> -</body> -</html> - diff --git a/old/74864-h/images/121x6br.png b/old/74864-h/images/121x6br.png Binary files differdeleted file mode 100644 index 1edbaa5..0000000 --- a/old/74864-h/images/121x6br.png +++ /dev/null diff --git a/old/74864-h/images/31x8bl.png b/old/74864-h/images/31x8bl.png Binary files differdeleted file mode 100644 index da8f11c..0000000 --- a/old/74864-h/images/31x8bl.png +++ /dev/null diff --git a/old/74864-h/images/31x8br.png b/old/74864-h/images/31x8br.png Binary files differdeleted file mode 100644 index 4787039..0000000 --- a/old/74864-h/images/31x8br.png +++ /dev/null diff --git a/old/74864-h/images/a002.png b/old/74864-h/images/a002.png Binary files differdeleted file mode 100644 index f072f71..0000000 --- a/old/74864-h/images/a002.png +++ /dev/null diff --git a/old/74864-h/images/a004.jpg b/old/74864-h/images/a004.jpg Binary files differdeleted file mode 100644 index 3937453..0000000 --- a/old/74864-h/images/a004.jpg +++ /dev/null diff --git a/old/74864-h/images/cover.jpg b/old/74864-h/images/cover.jpg Binary files differdeleted file mode 100644 index da751fe..0000000 --- a/old/74864-h/images/cover.jpg +++ /dev/null diff --git a/old/74864-h/images/p018.jpg b/old/74864-h/images/p018.jpg Binary files differdeleted file mode 100644 index 866a4cb..0000000 --- a/old/74864-h/images/p018.jpg +++ /dev/null diff --git a/old/74864-h/images/p073.jpg b/old/74864-h/images/p073.jpg Binary files differdeleted file mode 100644 index 3db24d1..0000000 --- a/old/74864-h/images/p073.jpg +++ /dev/null diff --git a/old/74864-h/images/p185.jpg b/old/74864-h/images/p185.jpg Binary files differdeleted file mode 100644 index a55a14c..0000000 --- a/old/74864-h/images/p185.jpg +++ /dev/null diff --git a/old/74864-h/images/p232.jpg b/old/74864-h/images/p232.jpg Binary files differdeleted file mode 100644 index 218aaa9..0000000 --- a/old/74864-h/images/p232.jpg +++ /dev/null diff --git a/old/74864-h/images/p381.jpg b/old/74864-h/images/p381.jpg Binary files differdeleted file mode 100644 index df2a6f3..0000000 --- a/old/74864-h/images/p381.jpg +++ /dev/null diff --git a/old/74864-h/images/p477.jpg b/old/74864-h/images/p477.jpg Binary files differdeleted file mode 100644 index 614aae2..0000000 --- a/old/74864-h/images/p477.jpg +++ /dev/null diff --git a/old/74864-h/images/p493.jpg b/old/74864-h/images/p493.jpg Binary files differdeleted file mode 100644 index f945fda..0000000 --- a/old/74864-h/images/p493.jpg +++ /dev/null |
