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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 *** |