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diff --git a/39508-0.txt b/39508-0.txt new file mode 100644 index 0000000..0259186 --- /dev/null +++ b/39508-0.txt @@ -0,0 +1,13979 @@ +The Project Gutenberg EBook of Popular scientific lectures, by Ernst Mach + +This eBook is for the use of anyone anywhere at no cost and with +almost no restrictions whatsoever. You may copy it, give it away or +re-use it under the terms of the Project Gutenberg License included +with this eBook or online at www.gutenberg.org + + +Title: Popular scientific lectures + +Author: Ernst Mach + +Translator: Thomas Joseph McCormack + +Release Date: April 22, 2012 [EBook #39508] + +Language: English + +Character set encoding: UTF-8 + +*** START OF THIS PROJECT GUTENBERG EBOOK POPULAR SCIENTIFIC LECTURES *** + + + + +Produced by Anna Hall, Albert László and the Online +Distributed Proofreading Team at http://www.pgdp.net (This +file was produced from images generously made available +by The Internet Archive) + + + + + + + + + +POPULAR SCIENTIFIC LECTURES. + + + + +BY THE SAME AUTHOR. + + +THE SCIENCE OF MECHANICS. Translated from the Second German Edition + by T. J. McCormack. 250 Cuts and Illustrations. 534 Pages. Half + Morocco, Gilt Top. Price, $2.50. + +CONTRIBUTIONS TO THE ANALYSIS OF THE SENSATIONS. Translated by C. M. + Williams. With Notes and New Additions by the Author. 200 Pages. + 36 Cuts. Price, $1.00. + +POPULAR SCIENTIFIC LECTURES. Translated by T. J. McCormack. Third + Revised and Enlarged Edition. 411 Pages. 59 Cuts. Cloth, $1.50; + Paper, 50 cents. + +THE OPEN COURT PUBLISHING CO., +324 DEARBORN ST., CHICAGO. + + + + + POPULAR SCIENTIFIC LECTURES + + BY ERNST MACH + + FORMERLY PROFESSOR OF PHYSICS IN THE UNIVERSITY OF PRAGUE, NOW + PROFESSOR OF THE HISTORY AND THEORY OF INDUCTIVE SCIENCE IN THE + UNIVERSITY OF VIENNA + + TRANSLATED BY THOMAS J. McCORMACK + + THIRD EDITION, REVISED AND ENLARGED + + WITH FIFTY-NINE CUTS AND DIAGRAMS + + CHICAGO THE OPEN COURT PUBLISHING COMPANY + + FOR SALE BY + + KEGAN PAUL, TRENCH, TRUEBNER & CO., LONDON + + 1898 + + + + + COPYRIGHT + + BY THE OPEN COURT PUBLISHING CO. + + Pages 1-258 } + } in 1894. + Pages 338-374 } + Pages 259-281 in 1896. + Pages 282-308 in 1897. + Pages 309-337 in 1898. + + + + +AUTHOR'S PREFACE TO THE FIRST EDITION. + + +Popular lectures, owing to the knowledge they presuppose, and the time +they occupy, can afford only a _modicum_ of instruction. They must +select for this purpose easy subjects, and restrict themselves to the +exposition of the simplest and the most essential points. Nevertheless, +by an appropriate choice of the matter, the _charm_ and the _poetry_ of +research can be conveyed by them. It is only necessary to set forth the +attractive and the alluring features of a problem, and to show what +broad domains of fact can be illuminated by the light radiating from the +solution of a single and ofttimes unobtrusive point. + +Furthermore, such lectures can exercise a favorable influence by showing +the substantial sameness of scientific and every-day thought. The +public, in this way, loses its shyness towards scientific questions, and +acquires an interest in scientific work which is a great help to the +inquirer. The latter, in his turn, is brought to understand that his +work is a small part only of the universal process of life, and that the +results of his labors must redound to the benefit not only of himself +and a few of his associates, but to that of the collective whole. + +I sincerely hope that these lectures, in the present excellent +translation, will be productive of good in the direction indicated. + + E. MACH. + +PRAGUE, December, 1894. + + + + +TRANSLATOR'S NOTE TO THE THIRD EDITION. + + +The present third edition of this work has been enlarged by the addition +of a new lecture, "On Some Phenomena Attending the Flight of +Projectiles." The additions to the second consisted of the following +four lectures and articles: Professor Mach's Vienna Inaugural Lecture, +"The Part Played by Accident in Invention and Discovery," the lecture on +"Sensations of Orientation," recently delivered and summing up the +results of an important psychological investigation, and two historical +articles (see Appendix) on Acoustics and Sight. + +The lectures extend over a long period, from 1864 to 1898, and differ +greatly in style, contents, and purpose. They were first published in +collected form in English; afterwards two German editions were called +for. + +As the dates of the first five lectures are not given in the footnotes +they are here appended. The first lecture, "On the Forms of Liquids," +was delivered in 1868 and published with that "On Symmetry" in 1872 +(Prague). The second and third lectures, on acoustics, were first +published in 1865 (Graz); the fourth and fifth, on optics, in 1867 +(Graz). They belong to the earliest period of Professor Mach's +scientific activity, and with the lectures on electrostatics and +education will more than realise the hope expressed in the author's +Preface. + +The eighth, ninth, tenth, eleventh, and twelfth lectures are of a more +philosophical character and deal principally with the methods and nature +of scientific inquiry. In the ideas summarised in them will be found one +of the most important contributions to the theory of knowledge made in +the last quarter of a century. Significant hints in psychological +method, and exemplary specimen-researches in psychology and physics, are +also presented; while in physics many ideas find their first discussion +that afterwards, under other names and other authorship, became +rallying-cries in this department of inquiry. + +All the proofs of this translation have been read by Professor Mach +himself. + + T. J. MCCORMACK. + +LA SALLE, ILL., May, 1898. + + + + +TABLE OF CONTENTS. + + + The Forms of Liquids 1 + The Fibres of Corti 17 + On the Causes of Harmony 32 + The Velocity of Light 48 + Why Has Man Two Eyes? 66 + On Symmetry 89 + On the Fundamental Concepts of Electrostatics 107 + On the Principle of the Conservation of Energy 137 + On the Economical Nature of Physical Inquiry 186 + On Transformation and Adaptation in Scientific Thought 214 + On the Principle of Comparison in Physics 236 + On the Part Played by Accident in Invention and Discovery 259 + On Sensations of Orientation 282 + On Some Phenomena Attending the Flight of Projectiles 309 + On Instruction in the Classics and the Mathematico-Physical + Sciences 338 + Appendixes. + I. A Contribution to the History of Acoustics 375 + II. Remarks on the Theory of Spatial Vision 386 + Index 393 + + + + +THE FORMS OF LIQUIDS. + + +What thinkest thou, dear Euthyphron, that the holy is, and the just, and +the good? Is the holy holy because the gods love it, or are the gods +holy because they love the holy? By such easy questions did the wise +Socrates make the market-place of Athens unsafe and relieve presumptuous +young statesmen of the burden of imaginary knowledge, by showing them +how confused, unclear, and self-contradictory their ideas were. + +You know the fate of the importunate questioner. So called good society +avoided him on the promenade. Only the ignorant accompanied him. And +finally he drank the cup of hemlock--a lot which we ofttimes wish would +fall to modern critics of his stamp. + +What we have learned from Socrates, however,--our inheritance from +him,--is scientific criticism. Every one who busies himself with science +recognises how unsettled and indefinite the notions are which he has +brought with him from common life, and how, on a minute examination of +things, old differences are effaced and new ones introduced. The +history of science is full of examples of this constant change, +development, and clarification of ideas. + +But we will not linger by this general consideration of the fluctuating +character of ideas, which becomes a source of real uncomfortableness, +when we reflect that it applies to almost every notion of life. Rather +shall we observe by the study of a physical example how much a thing +changes when it is closely examined, and how it assumes, when thus +considered, increasing definiteness of form. + +The majority of you think, perhaps, you know quite well the distinction +between a liquid and a solid. And precisely persons who have never +busied themselves with physics will consider this question one of the +easiest that can be put. But the physicist knows that it is one of the +most difficult. I shall mention here only the experiments of Tresca, +which show that solids subjected to high pressures behave exactly as +liquids do; for example, may be made to flow out in the form of jets +from orifices in the bottoms of vessels. The supposed difference of kind +between liquids and solids is thus shown to be a mere difference of +degree. + +The common inference that because the earth is oblate in form, it was +originally fluid, is an error, in the light of these facts. True, a +rotating sphere, a few inches in diameter will assume an oblate form +only if it is very soft, for example, is composed of freshly kneaded +clay or some viscous stuff. But the earth, even if it consisted of the +rigidest stone, could not help being crushed by its tremendous weight, +and must perforce behave as a fluid. Even our mountains could not extend +beyond a certain height without crumbling. The earth _may_ once have +been fluid, but this by no means follows from its oblateness. + +The particles of a liquid are displaced on the application of the +slightest pressure; a liquid conforms exactly to the shapes of the +vessels in which it is contained; it possesses no form of its own, as +you have all learned in the schools. Accommodating itself in the most +trifling respects to the conditions of the vessel in which it is placed, +and showing, even on its surface, where one would suppose it had the +freest play, nothing but a polished, smiling, expressionless +countenance, it is the courtier _par excellence_ of the natural bodies. + +Liquids have no form of their own! No, not for the superficial observer. +But persons who have observed that a raindrop is round and never +angular, will not be disposed to accept this dogma so unconditionally. + +It is fair to suppose that every man, even the weakest, would possess a +character, if it were not too difficult in this world to keep it. So, +too, we must suppose that liquids would possess forms of their own, if +the pressure of the circumstances permitted it,--if they were not +crushed by their own weights. + +An astronomer once calculated that human beings could not exist on the +sun, apart from its great heat, because they would be crushed to pieces +there by their own weight. The greater mass of this body would also +make the weight of the human body there much greater. But on the moon, +because here we should be much lighter, we could jump as high as the +church-steeples without any difficulty, with the same muscular power +which we now possess. Statues and "plaster" casts of syrup are +undoubtedly things of fancy, even on the moon, but maple-syrup would +flow so slowly there that we could easily build a maple-syrup man on the +moon, for the fun of the thing, just as our children here build +snow-men. + +Accordingly, if liquids have no form of their own with us on earth, they +have, perhaps, a form of their own on the moon, or on some smaller and +lighter heavenly body. The problem, then, simply is to get rid of the +effects of gravity; and, this done, we shall be able to find out what +the peculiar forms of liquids are. + +The problem was solved by Plateau of Ghent, whose method was to immerse +the liquid in another of the same specific gravity.[1] He employed for +his experiments oil and a mixture of alcohol and water. By Archimedes's +well-known principle, the oil in this mixture loses its entire weight. +It no longer sinks beneath its weight; its formative forces, be they +ever so weak, are now in full play. + +As a fact, we now see, to our surprise, that the oil, instead of +spreading out into a layer, or lying in a formless mass, assumes the +shape of a beautiful and perfect sphere, freely suspended in the +mixture, as the moon is in space. We can construct in this way a sphere +of oil several inches in diameter. + +If, now, we affix a thin plate to a wire and insert the plate in the oil +sphere, we can, by twisting the wire between our fingers, set the whole +ball in rotation. Doing this, the ball assumes an oblate shape, and we +can, if we are skilful enough, separate by such rotation a ring from the +ball, like that which surrounds Saturn. This ring is finally rent +asunder, and, breaking up into a number of smaller balls, exhibits to us +a kind of model of the origin of the planetary system according to the +hypothesis of Kant and Laplace. + +[Illustration: Fig. 1.] + +Still more curious are the phenomena exhibited when the formative forces +of the liquid are partly disturbed by putting in contact with the +liquid's surface some rigid body. If we immerse, for example, the wire +framework of a cube in our mass of oil, the oil will everywhere stick to +the wire framework. If the quantity of oil is exactly sufficient we +shall obtain an oil cube with perfectly smooth walls. If there is too +much or too little oil, the walls of the cube will bulge out or cave in. +In this manner we can produce all kinds of geometrical figures of oil, +for example, a three-sided pyramid, a cylinder (by bringing the oil +between two wire rings), and so on. Interesting is the change of form +that occurs when we gradually suck out the oil by means of a glass tube +from the cube or pyramid. The wire holds the oil fast. The figure grows +smaller and smaller, until it is at last quite thin. Ultimately it +consists simply of a number of thin, smooth plates of oil, which extend +from the edges of the cube to the centre, where they meet in a small +drop. The same is true of the pyramid. + +[Illustration: Fig. 2.] + +The idea now suggests itself that liquid figures as thin as this, and +possessing, therefore, so slight a weight, cannot be crushed or deformed +by their weight; just as a small, soft ball of clay is not affected in +this respect by its weight. This being the case, we no longer need our +mixture of alcohol and water for the production of figures, but can +construct them in the open air. And Plateau, in fact, found that these +thin figures, or at least very similar ones, could be produced in the +air, by dipping the wire nets described in a solution of soap and water +and quickly drawing them out again. The experiment is not difficult. The +figure is formed of itself. The preceding drawing represents to the eye +the forms obtained with cubical and pyramidal nets. In the cube, thin, +smooth films of soap-suds proceed from the edges to a small, quadratic +film in the centre. In the pyramid, a film proceeds from each edge to +the centre. + +These figures are so beautiful that they hardly admit of appropriate +description. Their great regularity and geometrical exactness evokes +surprise from all who see them for the first time. Unfortunately, they +are of only short duration. They burst, on the drying of the solution in +the air, but only after exhibiting to us the most brilliant play of +colors, such as is often seen in soap-bubbles. Partly their beauty of +form and partly our desire to examine them more minutely induces us to +conceive of methods of endowing them with permanent form. This is very +simply done.[2] Instead of dipping the wire nets in solutions of soap, +we dip them in pure melted colophonium (resin). When drawn out the +figure at once forms and solidifies by contact with the air. + +It is to be remarked that also solid fluid-figures can be constructed +in the open air, if their weight be light enough, or the wire nets of +very small dimensions. If we make, for example, of very fine wire a +cubical net whose sides measure about one-eighth of an inch in length, +we need simply to dip this net in water to obtain a small solid cube of +water. With a piece of blotting paper the superfluous water may be +easily removed and the sides of the cube made smooth. + +Yet another simple method may be devised for observing these figures. A +drop of water on a greased glass plate will not run if it is small +enough, but will be flattened by its weight, which presses it against +its support. The smaller the drop the less the flattening. The smaller +the drop the nearer it approaches the form of a sphere. On the other +hand, a drop suspended from a stick is elongated by its weight. The +undermost parts of a drop of water on a support are pressed against the +support, and the upper parts are pressed against the lower parts because +the latter cannot yield. But when a drop falls freely downward all its +parts move equally fast; no part is impeded by another; no part presses +against another. A freely falling drop, accordingly, is not affected by +its weight; it acts as if it were weightless; it assumes a spherical +form. + +A moment's glance at the soap-film figures produced by our various wire +models, reveals to us a great multiplicity of form. But great as this +multiplicity is, the common features of the figures also are easily +discernible. + + "All forms of Nature are allied, though none is the same as the other; + Thus, their common chorus points to a hidden law." + +This hidden law Plateau discovered. It may be expressed, somewhat +prosily, as follows: + +1) If several plane liquid films meet in a figure they are always three +in number, and, taken in pairs, form, each with another, nearly equal +angles. + +2) If several liquid edges meet in a figure they are always four in +number, and, taken in pairs, form, each with another, nearly equal +angles. + +This is a strange law, and its reason is not evident. But we might apply +this criticism to almost all laws. It is not always that the motives of +a law-maker are discernible in the form of the law he constructs. But +our law admits of analysis into very simple elements or reasons. If we +closely examine the paragraphs which state it, we shall find that their +meaning is simply this, that the surface of the liquid assumes the shape +of smallest area that is possible under the circumstances. + +If, therefore, some extraordinarily intelligent tailor, possessing a +knowledge of all the artifices of the higher mathematics, should set +himself the task of so covering the wire frame of a cube with cloth that +every piece of cloth should be connected with the wire and joined with +the remaining cloth, and should seek to accomplish this feat with the +greatest saving of material, he would construct no other figure than +that which is here formed on the wire frame in our solution of soap and +water. Nature acts in the construction of liquid figures on the +principle of a covetous tailor, and gives no thought in her work to the +fashions. But, strange to say, in this work, the most beautiful fashions +are of themselves produced. + +The two paragraphs which state our law apply primarily only to soap-film +figures, and are not applicable, of course, to solid oil-figures. But +the principle that the superficial area of the liquid shall be the least +possible under the circumstances, is applicable to all fluid figures. He +who understands not only the letter but also the reason of the law will +not be at a loss when confronted with cases to which the letter does not +accurately apply. And this is the case with the principle of least +superficial area. It is a sure guide for us even in cases in which the +above-stated paragraphs are not applicable. + +Our first task will now be, to show by a palpable illustration the mode +of formation of liquid figures by the principle of least superficial +area. The oil on the wire pyramid in our mixture of alcohol and water, +being unable to leave the wire edges, clings to them, and the given mass +of oil strives so to shape itself that its surface shall have the least +possible area. Suppose we attempt to imitate this phenomenon. We take a +wire pyramid, draw over it a stout film of rubber, and in place of the +wire handle insert a small tube leading into the interior of the space +enclosed by the rubber (Fig. 3). Through this tube we can blow in or +suck out air. The quantity of air in the enclosure represents the +quantity of oil. The stretched rubber film, which, clinging to the wire +edges, does its utmost to contract, represents the surface of the oil +endeavoring to decrease its area. By blowing in, and drawing out the +air, now, we actually obtain all the oil pyramidal figures, from those +bulged out to those hollowed in. Finally, when all the air is pumped or +sucked out, the soap-film figure is exhibited. The rubber films strike +together, assume the form of planes, and meet at four sharp edges in the +centre of the pyramid. + +[Illustration: Fig. 3.] + +[Illustration: Fig. 4.] + +The tendency of soap-films to assume smaller forms may be directly +demonstrated by a method of Van der Mensbrugghe. If we dip a square wire +frame to which a handle is attached into a solution of soap and water, +we shall obtain on the frame a beautiful, plane film of soap-suds. (Fig. +4.) On this we lay a thread having its two ends tied together. If, now, +we puncture the part enclosed by the thread, we shall obtain a soap-film +having a circular hole in it, whose circumference is the thread. The +remainder of the film decreasing in area as much as it can, the hole +assumes the largest area that it can. But the figure of largest area, +with a given periphery, is the circle. + +[Illustration: Fig. 5.] + +Similarly, by the principle of least superficial area, a freely +suspended mass of oil assumes the shape of a sphere. The sphere is the +form of least surface for a given content. This is evident. The more we +put into a travelling-bag, the nearer its shape approaches the spherical +form. + +The connexion of the two above-mentioned paragraphs with the principle +of least superficial area may be shown by a yet simpler example. Picture +to yourselves four fixed pulleys, _a_, _b_, _c_, _d_, and two movable +rings _f_, _g_ (Fig. 5); about the pulleys and through the rings imagine +a smooth cord passed, fastened at one extremity to a nail _e_, and +loaded at the other with a weight _h_. Now this weight always tends to +sink, or, what is the same thing, always tends to make the portion of +the string _e h_ as long as possible, and consequently the remainder of +the string, wound round the pulleys, as short as possible. The strings +must remain connected with the pulleys, and on account of the rings also +with each other. The conditions of the case, accordingly, are similar to +those of the liquid figures discussed. The result also is a similar one. +When, as in the right hand figure of the cut, four pairs of strings +meet, a different configuration must be established. The consequence of +the endeavor of the string to shorten itself is that the rings separate +from each other, and that now at all points only three pairs of strings +meet, every two at equal angles of one hundred and twenty degrees. As a +fact, by this arrangement the greatest possible shortening of the string +is attained; as can be easily proved by geometry. + +This will help us to some extent to understand the creation of beautiful +and complicated figures by the simple tendency of liquids to assume +surfaces of least superficial area. But the question arises, _Why_ do +liquids seek surfaces of least superficial area? + +The particles of a liquid cling together. Drops brought into contact +coalesce. We can say, liquid particles attract each other. If so, they +seek to come as close as they can to each other. The particles at the +surface will endeavor to penetrate as far as they can into the interior. +This process will not stop, cannot stop, until the surface has become as +small as under the circumstances it possibly can become, until as few +particles as possible remain at the surface, until as many particles as +possible have penetrated into the interior, until the forces of +attraction have no more work to perform.[3] + +The root of the principle of least surface is to be sought, accordingly, +in another and much simpler principle, which may be illustrated by some +such analogy as this. We can _conceive_ of the natural forces of +attraction and repulsion as purposes or intentions of nature. As a +matter of fact, that interior pressure which we feel before an act and +which we call an intention or purpose, is not, in a final analysis, so +essentially different from the pressure of a stone on its support, or +the pressure of a magnet on another, that it is necessarily unallowable +to use for both the same term--at least for well-defined purposes.[4] It +is the purpose of nature, accordingly, to bring the iron nearer the +magnet, the stone nearer the centre of the earth, and so forth. If such +a purpose can be realised, it is carried out. But where she cannot +realise her purposes, nature does nothing. In this respect she acts +exactly as a good man of business does. + +It is a constant purpose of nature to bring weights lower. We can raise +a weight by causing another, larger weight to sink; that is, by +satisfying another, more powerful, purpose of nature. If we fancy we are +making nature serve our purposes in this, it will be found, upon closer +examination, that the contrary is true, and that nature has employed us +to attain her purposes. + +Equilibrium, rest, exists only, but then always, when nature is brought +to a halt in her purposes, when the forces of nature are as fully +satisfied as, under the circumstances, they can be. Thus, for example, +heavy bodies are in equilibrium, when their so-called centre of gravity +lies as low as it possibly can, or when as much weight as the +circumstances admit of has sunk as low as it can. + +The idea forcibly suggests itself that perhaps this principle also holds +good in other realms. Equilibrium exists also in the state when the +purposes of the parties are as fully satisfied as for the time being +they can be, or, as we may say, jestingly, in the language of physics, +when the social potential is a maximum.[5] + +You see, our miserly mercantile principle is replete with +consequences.[6] The result of sober research, it has become as fruitful +for physics as the dry questions of Socrates for science generally. If +the principle seems to lack in ideality, the more ideal are the fruits +which it bears. + +But why, tell me, should science be ashamed of such a principle? Is +science[7] itself anything more than--a business? Is not its task to +acquire with the least possible work, in the least possible time, with +the least possible thought, the greatest possible part of eternal truth? + + + FOOTNOTES: + + [Footnote 1: _Statique expérimentale et théorique des liquids_, + 1873. See also _The Science of Mechanics_, p. 384 et seqq., The Open + Court Publishing Co., Chicago, 1893.] + + [Footnote 2: Compare Mach, _Ueber die Molecularwirkung der + Flüssigkeiten_, Reports of the Vienna Academy, 1862.] + + [Footnote 3: In almost all branches of physics that are well worked + out such maximal and minimal problems play an important part.] + + [Footnote 4: Compare Mach, _Vorträge über Psychophysik_, Vienna, + 1863, page 41; _Compendium der Physik für Mediciner_, Vienna, 1863, + page 234; and also _The Science of Mechanics_, Chicago, 1893, pp. 84 + and 464.] + + [Footnote 5: Like reflexions are found in Quételet, _Du système + sociale_.] + + [Footnote 6: For the full development of this idea see the essay "On + the Economical Nature of Physical Inquiry," p. 186, and the chapter + on "The Economy of Science," in my _Mechanics_ (Chicago: The Open + Court Publishing Company, 1893), p. 481.] + + [Footnote 7: Science may be regarded as a maximum or minimum + problem, exactly as the business of the merchant. In fact, the + intellectual activity of natural inquiry is not so greatly different + from that exercised in ordinary life as is usually supposed.] + + + + +THE FIBRES OF CORTI. + + +Whoever has roamed through a beautiful country knows that the tourist's +delights increase with his progress. How pretty that wooded dell must +look from yonder hill! Whither does that clear brook flow, that hides +itself in yonder sedge? If I only knew how the landscape looked behind +that mountain! Thus even the child thinks in his first rambles. It is +also true of the natural philosopher. + +The first questions are forced upon the attention of the inquirer by +practical considerations; the subsequent ones are not. An irresistible +attraction draws him to these; a nobler interest which far transcends +the mere needs of life. Let us look at a special case. + +For a long time the structure of the organ of hearing has actively +engaged the attention of anatomists. A considerable number of brilliant +discoveries has been brought to light by their labors, and a splendid +array of facts and truths established. But with these facts a host of +new enigmas has been presented. + +Whilst in the theory of the organisation and functions of the eye +comparative clearness has been attained; whilst, hand in hand with this, +ophthalmology has reached a degree of perfection which the preceding +century could hardly have dreamed of, and by the help of the +ophthalmoscope the observing physician penetrates into the profoundest +recesses of the eye, the theory of the ear is still much shrouded in +mysterious darkness, full of attraction for the investigator. + +Look at this model of the ear. Even at that familiar part by whose +extent we measure the quantity of people's intelligence, even at the +external ear, the problems begin. You see here a succession of helixes +or spiral windings, at times very pretty, whose significance we cannot +accurately state, yet for which there must certainly be some reason. + +[Illustration: Fig. 6.] + +The shell or concha of the ear, _a_ in the annexed diagram, conducts the +sound into the curved auditory passage _b_, which is terminated by a +thin membrane, the so-called tympanic membrane, _e_. This membrane is +set in motion by the sound, and in its turn sets in motion a series of +little bones of very peculiar formation, _c_. At the end of all is the +labyrinth _d_. The labyrinth consists of a group of cavities filled with +a liquid, in which the innumerable fibres of the nerve of hearing are +imbedded. By the vibration of the chain of bones _c_, the liquid of the +labyrinth is shaken, and the auditory nerve excited. Here the process of +hearing begins. So much is certain. But the details of the process are +one and all unanswered questions. + +To these old puzzles, the Marchese Corti, as late as 1851, added a new +enigma. And, strange to say, it is this last enigma, which, perhaps, has +first received its correct solution. This will be the subject of our +remarks to-day. + +Corti found in the cochlea, or snail-shell of the labyrinth, a large +number of microscopic fibres placed side by side in geometrically +graduated order. According to Kölliker their number is three thousand. +They were also the subject of investigation at the hands of Max Schultze +and Deiters. + +A description of the details of this organ would only weary you, besides +not rendering the matter much clearer. I prefer, therefore, to state +briefly what in the opinion of prominent investigators like Helmholtz +and Fechner is the peculiar function of Corti's fibres. The cochlea, it +seems, contains a large number of elastic fibres of graduated lengths +(Fig. 7), to which the branches of the auditory nerve are attached. +These fibres, called the fibres, pillars, or rods of Corti, being of +unequal length, must also be of unequal elasticity, and, consequently, +pitched to different notes. The cochlea, therefore, is a species of +pianoforte. + +[Illustration: Fig. 7.] + +What, now, may be the office of this structure, which is found in no +other organ of sense? May it not be connected with some special +property of the ear? It is quite probable; for the ear possesses a very +similar power. You know that it is possible to follow the individual +voices of a symphony. Indeed, the feat is possible even in a fugue of +Bach, where it is certainly no inconsiderable achievement. The ear can +pick out the single constituent tonal parts, not only of a harmony, but +of the wildest clash of music imaginable. The musical ear analyses every +agglomeration of tones. + +The eye does not possess this ability. Who, for example, could tell from +the mere sight of white, without a previous experimental knowledge of +the fact, that white is composed of a mixture of other colors? Could it +be, now, that these two facts, the property of the ear just mentioned, +and the structure discovered by Corti, are really connected? It is very +probable. The enigma is solved if we assume that every note of definite +pitch has its special string in this pianoforte of Corti, and, +therefore, its special branch of the auditory nerve attached to that +string. But before I can make this point perfectly plain to you, I must +ask you to follow me a few steps into the dry domain of physics. + +Look at this pendulum. Forced from its position of equilibrium by an +impulse, it begins to swing with a definite time of oscillation, +dependent upon its length. Longer pendulums swing more slowly, shorter +ones more quickly. We will suppose our pendulum to execute one +to-and-fro movement in a second. + +This pendulum, now, can be thrown into violent vibration in two ways; +either by a _single_ heavy impulse, or by a _number_ of properly +communicated slight impulses. For example, we impart to the pendulum, +while at rest in its position of equilibrium, a very slight impulse. It +will execute a very small vibration. As it passes a third time its +position of equilibrium, a second having elapsed, we impart to it again +a slight shock, in the same direction with the first. Again after the +lapse of a second, on its fifth passage through the position of +equilibrium, we strike it again in the same manner; and so continue. You +see, by this process the shocks imparted augment continually the motion +of the pendulum. After each slight impulse, the pendulum reaches out a +little further in its swing, and finally acquires a considerable +motion.[8] + +But this is not the case under all circumstances. It is possible only +when the impulses imparted synchronise with the swings of the pendulum. +If we should communicate the second impulse at the end of half a second +and in the same direction with the first impulse, its effects would +counteract the motion of the pendulum. It is easily seen that our little +impulses help the motion of the pendulum more and more, according as +their time accords with the time of the pendulum. If we strike the +pendulum in any other time than in that of its vibration, in some +instances, it is true, we shall augment its vibration, but in others +again, we shall obstruct it. Our impulses will be less effective the +more the motion of our own hand departs from the motion of the pendulum. + +What is true of the pendulum holds true of every vibrating body. A +tuning-fork when it sounds, also vibrates. It vibrates more rapidly when +its sound is higher; more slowly when it is deeper. The standard _A_ of +our musical scale is produced by about four hundred and fifty vibrations +in a second. + +I place by the side of each other on this table two tuning-forks, +exactly alike, resting on resonant cases. I strike the first one a sharp +blow, so that it emits a loud note, and immediately grasp it again with +my hand to quench its note. Nevertheless, you still hear the note +distinctly sounded, and by feeling it you may convince yourselves that +the other fork which was not struck now vibrates. + +I now attach a small bit of wax to one of the forks. It is thrown thus +out of tune; its note is made a little deeper. I now repeat the same +experiment with the two forks, now of unequal pitch, by striking one of +them and again grasping it with my hand; but in the present case the +note ceases the very instant I touch the fork. + +What has happened here in these two experiments? Simply this. The +vibrating fork imparts to the air and to the table four hundred and +fifty shocks a second, which are carried over to the other fork. If the +other fork is pitched to the same note, that is to say, if it vibrates +when struck in the same time with the first, then the shocks first +emitted, no matter how slight they may be, are sufficient to throw the +second fork into rapid sympathetic vibration. But when the time of +vibration of the two forks is slightly different, this does not take +place. We may strike as many forks as we will, the fork tuned to _A_ is +perfectly indifferent to their notes; is deaf, in fact, to all except +its own; and if you strike three, or four, or five, or any number +whatsoever, of forks all at the same time, so as to make the shocks +which come from them ever so great, the _A_ fork will not join in with +their vibrations unless another fork _A_ is found in the collection +struck. It picks out, in other words, from all the notes sounded, that +which accords with it. + +The same is true of all bodies which can yield notes. Tumblers resound +when a piano is played, on the striking of certain notes, and so do +window panes. Nor is the phenomenon without analogy in other provinces. +Take a dog that answers to the name "Nero." He lies under your table. +You speak of Domitian, Vespasian, and Marcus Aurelius Antoninus, you +call upon all the names of the Roman Emperors that occur to you, but the +dog does not stir, although a slight tremor of his ear tells you of a +faint response of his consciousness. But the moment you call "Nero" he +jumps joyfully towards you. The tuning-fork is like your dog. It answers +to the name _A_. + +You smile, ladies. You shake your heads. The simile does not catch your +fancy. But I have another, which is very near to you: and for punishment +you shall hear it. You, too, are like tuning-forks. Many are the hearts +that throb with ardor for you, of which you take no notice, but are +cold. Yet what does it profit you! Soon the heart will come that beats +in just the proper rhythm, and then your knell, too, has struck. Then +your heart, too, will beat in unison, whether you will or no. + +The law of sympathetic vibration, here propounded for sounding bodies, +suffers some modification for bodies incompetent to yield notes. Bodies +of this kind vibrate to almost every note. A high silk hat, we know, +will not sound; but if you will hold your hat in your hand when +attending your next concert you will not only hear the pieces played, +but also feel them with your fingers. It is exactly so with men. People +who are themselves able to give tone to their surroundings, bother +little about the prattle of others. But the person without character +tarries everywhere: in the temperance hall, and at the bar of the +public-house--everywhere where a committee is formed. The high silk hat +is among bells what the weakling is among men of conviction. + +A sonorous body, therefore, always sounds when its special note, either +alone or in company with others, is struck. We may now go a step +further. What will be the behaviour of a group of sonorous bodies which +in the pitch of their notes form a scale? Let us picture to ourselves, +for example (Fig. 8), a series of rods or strings pitched to the notes +_c d e f g_.... On a musical instrument the accord _c e g_ is struck. +Every one of the rods of Fig. 8 will see if its special note is +contained in the accord, and if it finds it, it will respond. The rod +_c_ will give at once the note _c_, the rod _e_ the note _e_, the rod +_g_ the note _g_. All the other rods will remain at rest, will not +sound. + +[Illustration: Fig. 8.] + +We need not look about us long for such an instrument. Every piano is an +instrument of this kind, with which the experiment mentioned may be +executed with splendid success. Two pianos stand here by the side of +each other, both tuned alike. We will employ the first for exciting the +notes, while we will allow the second to respond; after having first +pressed upon the loud pedal, so as to render all the strings capable of +motion. + +Every harmony struck with vigor on the first piano is distinctly +repeated on the second. To prove that it is the same strings that are +sounded in both pianos, we repeat the experiment in a slightly changed +form. We let go the loud pedal of the second piano and pressing on the +keys _c e g_ of that instrument vigorously strike the harmony _c e g_ on +the first piano. The harmony _c e g_ is now also sounded on the second +piano. But if we press only on one key _g_ of one piano, while we strike +_c e g_ on the other, only _g_ will be sounded on the second. It is +thus always the like strings of the two pianos that excite each other. + +The piano can reproduce any sound that is composed of its musical notes. +It will reproduce, for example, very distinctly, a vowel sound that is +sung into it. And in truth physics has proved that the vowels may be +regarded as composed of simple musical notes. + +You see that by the exciting of definite tones in the air quite definite +motions are set up with mechanical necessity in the piano. The idea +might be made use of for the performance of some pretty pieces of +wizardry. Imagine a box in which is a stretched string of definite +pitch. This is thrown into motion as often as its note is sung or +whistled. Now it would not be a very difficult task for a skilful +mechanic to so construct the box that the vibrating cord would close a +galvanic circuit and open the lock. And it would not be a much more +difficult task to construct a box which would open at the whistling of a +certain melody. Sesame! and the bolts fall. Truly, we should have here a +veritable puzzle-lock. Still another fragment rescued from that old +kingdom of fables, of which our day has realised so much, that world of +fairy-stories to which the latest contributions are Casselli's +telegraph, by which one can write at a distance in one's own hand, and +Prof. Elisha Gray's telautograph. What would the good old Herodotus have +said to these things who even in Egypt shook his head at much that he +saw? [Greek: emoi men ou pista], just as simple-heartedly as then, when +he heard of the circumnavigation of Africa. + +A new puzzle-lock! But why invent one? Are not we human beings ourselves +puzzle-locks? Think of the stupendous groups of thoughts, feelings, and +emotions that can be aroused in us by a word! Are there not moments in +all our lives when a mere name drives the blood to our hearts? Who that +has attended a large mass-meeting has not experienced what tremendous +quantities of energy and motion can be evolved by the innocent words, +"Liberty, Equality, Fraternity." + +But let us return to the subject proper of our discourse. Let us look +again at our piano, or what will do just as well, at some other +contrivance of the same character. What does this instrument do? +Plainly, it decomposes, it analyses every agglomeration of sounds set up +in the air into its individual component parts, each tone being taken up +by a different string; it performs a real spectral analysis of sound. A +person completely deaf, with the help of a piano, simply by touching the +strings or examining their vibrations with a microscope, might +investigate the sonorous motion of the air, and pick out the separate +tones excited in it. + +The ear has the same capacity as this piano. The ear performs for the +mind what the piano performs for a person who is deaf. The mind without +the ear is deaf. But a deaf person, with the piano, does hear after a +fashion, though much less vividly, and more clumsily, than with the +ear. The ear, thus, also decomposes sound into its component tonal +parts. I shall now not be deceived, I think, if I assume that you +already have a presentiment of what the function of Corti's fibres is. +We can make the matter very plain to ourselves. We will use the one +piano for exciting the sounds, and we shall imagine the second one in +the ear of the observer in the place of Corti's fibres, which is a model +of such an instrument. To every string of the piano in the ear we will +suppose a special fibre of the auditory nerve attached, so that this +fibre and this alone, is irritated when the string is thrown into +vibration. If we strike now an accord on the external piano, for every +tone of that accord a definite string of the internal piano will sound +and as many different nervous fibres will be irritated as there are +notes in the accord. The simultaneous sense-impressions due to different +notes can thus be preserved unmingled and be separated by the attention. +It is the same as with the five fingers of the hand. With each finger I +can touch something different. Now the ear has three thousand such +fingers, and each one is designed for the touching of a different +tone.[9] Our ear is a puzzle-lock of the kind mentioned. It opens at the +magic melody of a sound. But it is a stupendously ingenious lock. Not +only one tone, but every tone makes it open; but each one differently. +To each tone it replies with a different sensation. + +More than once it has happened in the history of science that a +phenomenon predicted by theory, has not been brought within the range of +actual observation until long afterwards. Leverrier predicted the +existence and the place of the planet Neptune, but it was not until +sometime later that Galle actually found the planet at the predicted +spot. Hamilton unfolded theoretically the phenomenon of the so-called +conical refraction of light, but it was reserved for Lloyd some time +subsequently to observe the fact. The fortunes of Helmholtz's theory of +Corti's fibres have been somewhat similar. This theory, too, received +its substantial confirmation from the subsequent observations of V. +Hensen. On the free surface of the bodies of Crustacea, connected with +the auditory nerves, rows of little hairy filaments of varying lengths +and thicknesses are found, which to some extent are the analogues of +Corti's fibres. Hensen saw these hairs vibrate when sounds were excited, +and when different notes were struck different hairs were set in +vibration. + +I have compared the work of the physical inquirer to the journey of the +tourist. When the tourist ascends a new hill he obtains of the whole +district a different view. When the inquirer has found the solution of +one enigma, the solution of a host of others falls into his hands. + +Surely you have often felt the strange impression experienced when in +singing through the scale the octave is reached, and nearly the same +sensation is produced as by the fundamental tone. The phenomenon finds +its explanation in the view here laid down of the ear. And not only this +phenomenon but all the laws of the theory of harmony may be grasped and +verified from this point of view with a clearness before undreamt of. +Unfortunately, I must content myself to-day with the simple indication +of these beautiful prospects. Their consideration would lead us too far +aside into the fields of other sciences. + +The searcher of nature, too, must restrain himself in his path. He also +is drawn along from one beauty to another as the tourist from dale to +dale, and as circumstances generally draw men from one condition of life +into others. It is not he so much that makes the quests, as that the +quests are made of him. Yet let him profit by his time, and let not his +glance rove aimlessly hither and thither. For soon the evening sun will +shine, and ere he has caught a full glimpse of the wonders close by, a +mighty hand will seize him and lead him away into a different world of +puzzles. + +Respected hearers, science once stood in an entirely different relation +to poetry. The old Hindu mathematicians wrote their theorems in verses, +and lotus-flowers, roses, and lilies, beautiful sceneries, lakes, and +mountains figured in their problems. + +"Thou goest forth on this lake in a boat. A lily juts forth, one palm +above the water. A breeze bends it downwards, and it vanishes two palms +from its previous spot beneath the surface. Quick, mathematician, tell +me how deep is the lake!" + +Thus spoke an ancient Hindu scholar. This poetry, and rightly, has +disappeared from science, but from its dry leaves another poetry is +wafted aloft which cannot be described to him who has never felt it. +Whoever will fully enjoy this poetry must put his hand to the plough, +must himself investigate. Therefore, enough of this! I shall reckon +myself fortunate if you do not repent of this brief excursion into the +flowered dale of physiology, and if you take with yourselves the belief +that we can say of science what we say of poetry, + + "Who the song would understand, + Needs must seek the song's own land; + Who the minstrel understand + Needs must seek the minstrel's land." + + FOOTNOTES: + + [Footnote 8: This experiment, with its associated reflexions, is due + to Galileo.] + + [Footnote 9: A development of the theory of musical audition + differing in many points from the theory of Helmholtz here + expounded, will be found in my _Contributions to the Analysis of the + Sensations_ (English translation by C. M. Williams), Chicago, The + Open Court Publishing Company, 1897.] + + + + +ON THE CAUSES OF HARMONY. + + +We are to speak to-day of a theme which is perhaps of somewhat more +general interest--_the causes of the harmony of musical sounds_. The +first and simplest experiences relative to harmony are very ancient. Not +so the explanation of its laws. These were first supplied by the +investigators of a recent epoch. Allow me an historical retrospect. + +Pythagoras (586 B. C.) knew that the note yielded by a string of steady +tension was converted into its octave when the length of the string was +reduced one-half, and into its fifth when reduced two-thirds; and that +then the first fundamental tone was consonant with the two others. He +knew generally that the same string under fixed tension gives consonant +tones when successively divided into lengths that are in the proportions +of the simplest natural numbers; that is, in the proportions of 1:2, +2:3, 3:4, 4:5. + +Pythagoras failed to reveal the causes of these laws. What have +consonant tones to do with the simple natural numbers? That is the +question we should ask to-day. But this circumstance must have appeared +less strange than inexplicable to Pythagoras. This philosopher sought +for the causes of harmony in the occult, miraculous powers of numbers. +His procedure was largely the cause of the upgrowth of a numerical +mysticism, of which the traces may still be detected in our +oneirocritical books and among some scientists, to whom marvels are more +attractive than lucidity. + +Euclid (300 B. C.) gives a definition of consonance and dissonance that +could hardly be improved upon, in point of verbal accuracy. The +consonance ([Greek: symphônia]) of two tones, he says, is the mixture, +the blending ([Greek: krasis]) of those two tones; dissonance ([Greek: +diaphônia]), on the other hand, is the incapacity of the tones to blend +([Greek: amixia]), whereby they are made harsh for the ear. The person +who knows the correct explanation of the phenomenon hears it, so to +speak, reverberated in these words of Euclid. Still, Euclid did not know +the true cause of harmony. He had unwittingly come very near to the +truth, but without really grasping it. + +Leibnitz (1646-1716 A. D.) resumed the question which his predecessors +had left unsolved. He, of course, knew that musical notes were produced +by vibrations, that twice as many vibrations corresponded to the octave +as to the fundamental tone, etc. A passionate lover of mathematics, he +sought for the cause of harmony in the secret computation and comparison +of the simple numbers of vibrations and in the secret satisfaction of +the soul at this occupation. But how, we ask, if one does not know that +musical notes are vibrations? The computation and the satisfaction at +the computation must indeed be pretty secret if it is unknown. What +queer ideas philosophers have! Could anything more wearisome be imagined +than computation as a principle of æsthetics? Yes, you are not utterly +wrong in your conjecture, yet you may be sure that Leibnitz's theory is +not wholly nonsense, although it is difficult to make out precisely what +he meant by his secret computation. + +The great Euler (1707-1783) sought the cause of harmony, almost as +Leibnitz did, in the pleasure which the soul derives from the +contemplation of order in the numbers of the vibrations.[10] + +Rameau and D'Alembert (1717-1783) approached nearer to the truth. They +knew that in every sound available in music besides the fundamental note +also the twelfth and the next higher third could be heard; and further +that the resemblance between a fundamental tone and its octave was +always strongly marked. Accordingly, the combination of the octave, +fifth, third, etc., with the fundamental tone appeared to them +"natural." They possessed, we must admit, the correct point of view; but +with the simple naturalness of a phenomenon no inquirer can rest +content; for it is precisely this naturalness for which he seeks his +explanations. + +Rameau's remark dragged along through the whole modern period, but +without leading to the full discovery of the truth. Marx places it at +the head of his theory of composition, but makes no further application +of it. Also Goethe and Zelter in their correspondence were, so to speak, +on the brink of the truth. Zelter knew of Rameau's view. Finally, you +will be appalled at the difficulty of the problem, when I tell you that +till very recent times even professors of physics were dumb when asked +what were the causes of harmony. + +Not till quite recently did Helmholtz find the solution of the question. +But to make this solution clear to you I must first speak of some +experimental principles of physics and psychology. + +1) In every process of perception, in every observation, the attention +plays a highly important part. We need not look about us long for proofs +of this. You receive, for example, a letter written in a very poor hand. +Do your best, you cannot make it out. You put together now these, now +those lines, yet you cannot construct from them a single intelligible +character. Not until you direct your attention to groups of lines which +really belong together, is the reading of the letter possible. +Manuscripts, the letters of which are formed of minute figures and +scrolls, can only be read at a considerable distance, where the +attention is no longer diverted from the significant outlines to the +details. A beautiful example of this class is furnished by the famous +iconographs of Giuseppe Arcimboldo in the basement of the Belvedere +gallery at Vienna. These are symbolic representations of water, fire, +etc.: human heads composed of aquatic animals and of combustibles. At a +short distance one sees only the details, at a greater distance only the +whole figure. Yet a point can be easily found at which, by a simple +voluntary movement of the attention, there is no difficulty in seeing +now the whole figure and now the smaller forms of which it is composed. +A picture is often seen representing the tomb of Napoleon. The tomb is +surrounded by dark trees between which the bright heavens are visible as +background. One can look a long time at this picture without noticing +anything except the trees, but suddenly, on the attention being +accidentally directed to the bright background, one sees the figure of +Napoleon between the trees. This case shows us very distinctly the +important part which attention plays. The same sensuous object can, +solely by the interposition of attention, give rise to wholly different +perceptions. + +If I strike a harmony, or chord, on this piano, by a mere effort of +attention you can fix every tone of that harmony. You then hear most +distinctly the fixed tone, and all the rest appear as a mere addition, +altering only the quality, or acoustic color, of the primary tone. The +effect of the same harmony is essentially modified if we direct our +attention to different tones. + +Strike in succession two harmonies, for example, the two represented in +the annexed diagram, and first fix by the attention the upper note _e_, +afterwards the base _e_-_a_; in the two cases you will hear the same +sequence of harmonies differently. In the first case, you have the +impression as if the fixed tone remained unchanged and simply altered +its _timbre_; in the second case, the whole acoustic agglomeration seems +to fall sensibly in depth. There is an art of composition to guide the +attention of the hearer. But there is also an art of hearing, which is +not the gift of every person. + +[Music: Fig. 9.] + +The piano-player knows the remarkable effects obtained when one of the +keys of a chord that is struck is let loose. Bar 1 played on the piano +sounds almost like bar 2. The note which lies next to the key let loose +resounds after its release as if it were freshly struck. The attention +no longer occupied with the upper note is by that very fact insensibly +led to the upper note. + +[Music: Fig. 10.] + +Any tolerably cultivated musical ear can perform the resolution of a +harmony into its component parts. By much practice we can go even +further. Then, every musical sound heretofore regarded as simple can be +resolved into a subordinate succession of musical tones. For example, if +I strike on the piano the note 1, (annexed diagram,) we shall hear, if +we make the requisite effort of attention, besides the loud fundamental +note the feebler, higher overtones, or harmonics, 2 ... 7, that is, the +octave, the twelfth, the double octave, and the third, the fifth, and +the seventh of the double octave. + +[Music: Fig. 11.] + +The same is true of every musically available sound. Each yields, with +varying degrees of intensity, besides its fundamental note, also the +octave, the twelfth, the double octave, etc. The phenomenon is +observable with special facility on the open and closed flue-pipes of +organs. According, now, as certain overtones are more or less distinctly +emphasised in a sound, the _timbre_ of the sound changes--that peculiar +quality of the sound by which we distinguish the music of the piano from +that of the violin, the clarinet, etc. + +On the piano these overtones can be very easily rendered audible. If I +strike, for example, sharply note 1 of the foregoing series, whilst I +simply press down upon, one after another, the keys 2, 3, ... 7, the +notes 2, 3, ... 7 will continue to sound after the striking of 1, +because the strings corresponding to these notes, now freed from their +dampers, are thrown into sympathetic vibration. + +As you know, this sympathetic vibration of the like-pitched strings with +the overtones is really not to be conceived as sympathy, but rather as +lifeless mechanical necessity. We must not think of this sympathetic +vibration as an ingenious journalist pictured it, who tells a gruesome +story of Beethoven's F minor sonata, Op. 2, that I cannot withhold from +you. "At the last London Industrial Exhibition nineteen virtuosos played +the F minor sonata on the same piano. When the twentieth stepped up to +the instrument to play by way of variation the same production, to the +terror of all present the piano began to render the sonata of its own +accord. The Archbishop of Canterbury, who happened to be present, was +set to work and forthwith expelled the F minor devil." + +Although, now, the overtones or harmonics which we have discussed are +heard only upon a special effort of the attention, nevertheless they +play a highly important part in the formation of musical _timbre_, as +also in the production of the consonance and dissonance of sounds. This +may strike you as singular. How can a thing which is heard only under +exceptional circumstances be of importance generally for audition? + +But consider some familiar incidents of your every-day life. Think of +how many things you see which you do not notice, which never strike your +attention until they are missing. A friend calls upon you; you cannot +understand why he looks so changed. Not until you make a close +examination do you discover that his hair has been cut. It is not +difficult to tell the publisher of a work from its letter-press, and yet +no one can state precisely the points by which this style of type is so +strikingly different from that style. I have often recognised a book +which I was in search of from a simple piece of unprinted white paper +that peeped out from underneath the heap of books covering it, and yet I +had never carefully examined the paper, nor could I have stated its +difference from other papers. + +What we must remember, therefore, is that every sound that is musically +available yields, besides its fundamental note, its octave, its twelfth, +its double octave, etc., as overtones or harmonics, and that these are +important for the agreeable combination of several musical sounds. + +2) One other fact still remains to be dealt with. Look at this +tuning-fork. It yields, when struck, a perfectly smooth tone. But if you +strike in company with it a second fork which is of slightly different +pitch, and which alone also gives a perfectly smooth tone, you will +hear, if you set both forks on the table, or hold both before your ear, +a uniform tone no longer, but a number of shocks of tones. The rapidity +of the shocks increases with the difference of the pitch of the forks. +These shocks, which become very disagreeable for the ear when they +amount to thirty-three in a second, are called "beats." + +Always, when one of two like musical sounds is thrown out of unison with +the other, beats arise. Their number increases with the divergence from +unison, and simultaneously they grow more unpleasant. Their roughness +reaches its maximum at about thirty-three beats in a second. On a still +further departure from unison, and a consequent increase of the number +of beats, the unpleasant effect is diminished, so that tones which are +widely apart in pitch no longer produce offensive beats. + +To give yourselves a clear idea of the production of beats, take two +metronomes and set them almost alike. You can, for that matter, set the +two exactly alike. You need not fear that they will strike alike. The +metronomes usually for sale in the shops are poor enough to yield, when +set alike, appreciably unequal strokes. Set, now, these two metronomes, +which strike at unequal intervals, in motion; you will readily see that +their strokes alternately coincide and conflict with each other. The +alternation is quicker the greater the difference of time of the two +metronomes. + +If metronomes are not to be had, the experiment may be performed with +two watches. + +Beats arise in the same way. The rhythmical shocks of two sounding +bodies, of unequal pitch, sometimes coincide, sometimes interfere, +whereby they alternately augment and enfeeble each other's effects. +Hence the shock-like, unpleasant swelling of the tone. + +Now that we have made ourselves acquainted with overtones and beats, we +may proceed to the answer of our main question, Why do certain relations +of pitch produce pleasant sounds, consonances, others unpleasant sounds, +dissonances? It will be readily seen that all the unpleasant effects of +simultaneous sound-combinations are the result of beats produced by +those combinations. Beats are the only sin, the sole evil of music. +Consonance is the coalescence of sounds without appreciable beats. + +[Illustration: Fig. 12.] + +To make this perfectly clear to you I have constructed the model which +you see in Fig. 12. It represents a claviatur. At its top a movable +strip of wood _aa_ with the marks 1, 2 ... 6 is placed. By setting this +strip in any position, for example, in that where the mark 1 is over the +note _c_ of the claviatur, the marks 2, 3 ... 6, as you see, stand over +the overtones of _c_. The same happens when the strip is placed in any +other position. A second, exactly similar strip, _bb_, possesses the +same properties. Thus, together, the two strips, in any two positions, +point out by their marks all the tones brought into play upon the +simultaneous sounding of the notes indicated by the marks 1. + +The two strips, placed over the same fundamental note, show that also +all the overtones of those notes coincide. The first note is simply +intensified by the other. The single overtones of a sound lie too far +apart to permit appreciable beats. The second sound supplies nothing +new, consequently, also, no new beats. Unison is the most perfect +consonance. + +Moving one of the two strips along the other is equivalent to a +departure from unison. All the overtones of the one sound now fall +alongside those of the other; beats are at once produced; the +combination of the tones becomes unpleasant: we obtain a dissonance. If +we move the strip further and further along, we shall find that as a +general rule the overtones always fall alongside each other, that is, +always produce beats and dissonances. Only in a few quite definite +positions do the overtones partially coincide. Such positions, +therefore, signify higher degrees of euphony--they point out _the +consonant intervals_. + +These consonant intervals can be readily found experimentally by cutting +Fig. 12 out of paper and moving _bb_ lengthwise along _aa_. The most +perfect consonances are the octave and the twelfth, since in these two +cases the overtones of the one sound coincide absolutely with those of +the other. In the octave, for example, 1_b_ falls on 2_a_, 2_b_ on 4_a_, +3_b_ on 6_a_. Consonances, therefore, are simultaneous +sound-combinations not accompanied by disagreeable beats. This, by the +way, is, expressed in English, what Euclid said in Greek. + +Only such sounds are consonant as possess in common some portion of +their partial tones. Plainly we must recognise between such sounds, also +when struck one after another, a certain affinity. For the second sound, +by virtue of the common overtones, will produce partly the same +sensation as the first. The octave is the most striking exemplification +of this. When we reach the octave in the ascent of the scale we actually +fancy we hear the fundamental tone repeated. The foundations of harmony, +therefore, are the foundations of melody. + +Consonance is the coalescence of sounds without appreciable beats! This +principle is competent to introduce wonderful order and logic into the +doctrines of the fundamental bass. The compendiums of the theory of +harmony which (Heaven be witness!) have stood hitherto little behind the +cook-books in subtlety of logic, are rendered extraordinarily clear and +simple. And what is more, all that the great masters, such as +Palestrina, Mozart, Beethoven, unconsciously got right, and of which +heretofore no text-book could render just account, receives from the +preceding principle its perfect verification. + +But the beauty of the theory is, that it bears upon its face the stamp +of truth. It is no phantom of the brain. Every musician can hear for +himself the beats which the overtones of his musical sounds produce. +Every musician can satisfy himself that for any given case the number +and the harshness of the beats can be calculated beforehand, and that +they occur in exactly the measure that theory determines. + +This is the answer which Helmholtz gave to the question of Pythagoras, +so far as it can be explained with the means now at my command. A long +period of time lies between the raising and the solving of this +question. More than once were eminent inquirers nearer to the answer +than they dreamed of. + +The inquirer seeks the truth. I do not know if the truth seeks the +inquirer. But were that so, then the history of science would vividly +remind us of that classical rendezvous, so often immortalised by +painters and poets. A high garden wall. At the right a youth, at the +left a maiden. The youth sighs, the maiden sighs! Both wait. Neither +dreams how near the other is. + +I like this simile. Truth suffers herself to be courted, but she has +evidently no desire to be won. She flirts at times disgracefully. Above +all, she is determined to be merited, and has naught but contempt for +the man who will win her too quickly. And if, forsooth, one breaks his +head in his efforts of conquest, what matter is it, another will come, +and truth is always young. At times, indeed, it really seems as if she +were well disposed towards her admirer, but that admitted--never! Only +when Truth is in exceptionally good spirits does she bestow upon her +wooer a glance of encouragement. For, thinks Truth, if I do not do +something, in the end the fellow will not seek me at all. + +This one fragment of truth, then, we have, and it shall never escape us. +But when I reflect what it has cost in labor and in the lives of +thinking men, how it painfully groped its way through centuries, a +half-matured thought, before it became complete; when I reflect that it +is the toil of more than two thousand years that speaks out of this +unobtrusive model of mine, then, without dissimulation, I almost repent +me of the jest I have made. + +And think of how much we still lack! When, several thousand years hence, +boots, top-hats, hoops, pianos, and bass-viols are dug out of the earth, +out of the newest alluvium as fossils of the nineteenth century; when +the scientists of that time shall pursue their studies both upon these +wonderful structures and upon our modern Broadways, as we to-day make +studies of the implements of the stone age and of the prehistoric +lake-dwellings--then, too, perhaps, people will be unable to comprehend +how we could come so near to many great truths without grasping them. +And thus it is for all time the unsolved dissonance, for all time the +troublesome seventh, that everywhere resounds in our ears; we feel, +perhaps, that it will find its solution, but we shall never live to see +the day of the pure triple accord, nor shall our remotest descendants. + +Ladies, if it is the sweet purpose of your life to sow confusion, it is +the purpose of mine to be clear; and so I must confess to you a slight +transgression that I have been guilty of. On one point I have told you +an untruth. But you will pardon me this falsehood, if in full repentance +I make it good. The model represented in Fig. 12 does not tell the whole +truth, for it is based upon the so-called "even temperament" system of +tuning. The overtones, however, of musical sounds are not tempered, but +purely tuned. By means of this slight inexactness the model is made +considerably simpler. In this form it is fully adequate for ordinary +purposes, and no one who makes use of it in his studies need be in fear +of appreciable error. + +If you should demand of me, however, the full truth, I could give you +that only by the help of a mathematical formula. I should have to take +the chalk into my hands and--think of it!--reckon in your presence. This +you might take amiss. Nor shall it happen. I have resolved to do no more +reckoning for to-day. I shall reckon now only upon your forbearance, and +this you will surely not gainsay me when you reflect that I have made +only a limited use of my privilege to weary you. I could have taken up +much more of your time, and may, therefore, justly close with Lessing's +epigram: + + "If thou hast found in all these pages naught that's worth the thanks, + At least have gratitude for what I've spared thee." + + FOOTNOTES: + + [Footnote 10: Sauveur also set out from Leibnitz's idea, but arrived + by independent researches at a different theory, which was very near + to that of Helmholtz. Compare on this point Sauveur, _Mémoires de + l'Académie des Sciences_, Paris, 1700-1705, and R. Smith, + _Harmonics_, Cambridge, 1749. (See _Appendix_, p. 346.)] + + + + +THE VELOCITY OF LIGHT. + + +When a criminal judge has a right crafty knave before him, one well +versed in the arts of prevarication, his main object is to wring a +confession from the culprit by a few skilful questions. In almost a +similar position the natural philosopher seems to be placed with respect +to nature. True, his functions here are more those of the spy than the +judge; but his object remains pretty much the same. Her hidden motives +and laws of action is what nature must be made to confess. Whether a +confession will be extracted depends upon the shrewdness of the +inquirer. Not without reason, therefore, did Lord Bacon call the +experimental method a questioning of nature. The art consists in so +putting our questions that they may not remain unanswered without a +breach of etiquette. + +Look, too, at the countless tools, engines, and instruments of torture +with which man conducts his inquisitions of nature, and which mock the +poet's words: + + "Mysterious even in open day, + Nature retains her veil, despite our clamors; + That which she doth not willingly display + Cannot be wrenched from her with levers, screws, and hammers." + +Look at these instruments and you will see that the comparison with +torture also is admissible.[11] + +This view of nature, as of something designedly concealed from man, that +can be unveiled only by force or dishonesty, chimed in better with the +conceptions of the ancients than with modern notions. A Grecian +philosopher once said, in offering his opinion of the natural science of +his time, that it could only be displeasing to the gods to see men +endeavoring to spy out what the gods were not minded to reveal to +them.[12] Of course all the contemporaries of the speaker were not of +his opinion. + +Traces of this view may still be found to-day, but upon the whole we are +now not so narrow-minded. We believe no longer that nature designedly +hides herself. We know now from the history of science that our +questions are sometimes meaningless, and that, therefore, no answer can +be forthcoming. Soon we shall see how man, with all his thoughts and +quests, is only a fragment of nature's life. + +Picture, then, as your fancy dictates, the tools of the physicist as +instruments of torture or as engines of endearment, at all events a +chapter from the history of those implements will be of interest to you, +and it will not be unpleasant to learn what were the peculiar +difficulties that led to the invention of such strange apparatus. + +Galileo (born at Pisa in 1564, died at Arcetri in 1642) was the first +who asked what was the velocity of light, that is, what time it would +take for a light struck at one place to become visible at another, a +certain distance away.[13] + +The method which Galileo devised was as simple as it was natural. Two +practised observers, with muffled lanterns, were to take up positions in +a dark night at a considerable distance from each other, one at _A_ and +one at _B_. At a moment previously fixed upon, _A_ was instructed to +unmask his lantern; while as soon as _B_ saw the light of _A_'s lantern +he was to unmask his. Now it is clear that the time which _A_ counted +from the uncovering of his lantern until he caught sight of the light of +_B_'s would be the time which it would take light to travel from _A_ to +_B_ and from _B_ back to _A_. + +[Illustration: Fig. 13.] + +The experiment was not executed, nor could it, in the nature of the +case, have been a success. As we now know, light travels too rapidly to +be thus noted. The time elapsing between the arrival of the light at _B_ +and its perception by the observer, with that between the decision to +uncover and the uncovering of the lantern, is, as we now know, +incomparably greater than the time which it takes light to travel the +greatest earthly distances. The great velocity of light will be made +apparent, if we reflect that a flash of lightning in the night +illuminates instantaneously a very extensive region, whilst the single +reflected claps of thunder arrive at the observer's ear very gradually +and in appreciable succession. + +During his life, then, the efforts of Galileo to determine the velocity +of light remained uncrowned with success. But the subsequent history of +the measurement of the velocity of light is intimately associated with +his name, for with the telescope which he constructed he discovered the +four satellites of Jupiter, and these furnished the next occasion for +the determination of the velocity of light. + +The terrestrial spaces were too small for Galileo's experiment. The +measurement was first executed when the spaces of the planetary system +were employed. Olaf Römer, (born at Aarhuus in 1644, died at Copenhagen +in 1710) accomplished the feat (1675-1676), while watching with Cassini +at the observatory of Paris the revolutions of Jupiter's moons. + +[Illustration: Fig. 14.] + +Let _AB_ (Fig. 14) be Jupiter's orbit. Let _S_ stand for the sun, _E_ +for the earth, _J_ for Jupiter, and _T_ for Jupiter's first satellite. +When the earth is at _E₁_ we see the satellite enter regularly into +Jupiter's shadow, and by watching the time between two successive +eclipses, can calculate its time of revolution. The time which Römer +noted was forty-two hours, twenty-eight minutes, and thirty-five +seconds. Now, as the earth passes along in its orbit towards E₂, the +revolutions of the satellite grow apparently longer and longer: the +eclipses take place later and later. The greatest retardation of the +eclipse, which occurs when the earth is at _E₂_, amounts to sixteen +minutes and twenty-six seconds. As the earth passes back again to _E₁_, +the revolutions grow apparently shorter, and they occur in exactly the +time that they first did when the earth arrives at _E₁_. It is to be +remarked that Jupiter changes only very slightly its position during one +revolution of the earth. Römer guessed at once that these periodical +changes of the time of revolution of Jupiter's satellite were not +actual, but apparent changes, which were in some way connected with the +velocity of light. + +Let us make this matter clear to ourselves by a simile. We receive +regularly by the post, news of the political status at our capital. +However far away we may be from the capital, we hear the news of every +event, later it is true, but of all equally late. The events reach us in +the same succession of time as that in which they took place. But if we +are travelling away from the capital, every successive post will have a +greater distance to pass over, and the events will reach us more slowly +than they took place. The reverse will be the case if we are approaching +the capital. + +At rest, we hear a piece of music played in the same _tempo_ at all +distances. But the _tempo_ will be seemingly accelerated if we are +carried rapidly towards the band, or retarded if we are carried rapidly +away from it.[14] + +[Illustration: Fig. 15.] + +Picture to yourself a cross, say the sails of a wind-mill (Fig. 15), in +uniform rotation about its centre. Clearly, the rotation of the cross +will appear to you more slowly executed if you are carried very rapidly +away from it. For the post which in this case conveys to you the light +and brings to you the news of the successive positions of the cross will +have to travel in each successive instant over a longer path. + +Now this must also be the case with the rotation (the revolution) of the +satellite of Jupiter. The greatest retardation of the eclipse (16½ +minutes), due to the passage of the earth from _E₁_ to _E₂_, or to its +removal from Jupiter by a distance equal to the diameter of the orbit of +the earth, plainly corresponds to the time which it takes light to +traverse a distance equal to the diameter of the earth's orbit. The +velocity of light, that is, the distance described by light in a second, +as determined by this calculation, is 311,000 kilometres,[15] or 193,000 +miles. A subsequent correction of the diameter of the earth's orbit, +gives, by the same method, the velocity of light as approximately +186,000 miles a second. + +The method is exactly that of Galileo; only better conditions are +selected. Instead of a short terrestrial distance we have the diameter +of the earth's orbit, three hundred and seven million kilometres; in +place of the uncovered and covered lanterns we have the satellite of +Jupiter, which alternately appears and disappears. Galileo, therefore, +although he could not carry out himself the proposed measurement, found +the lantern by which it was ultimately executed. + +Physicists did not long remain satisfied with this beautiful discovery. +They sought after easier methods of measuring the velocity of light, +such as might be performed on the earth. This was possible after the +difficulties of the problem were clearly exposed. A measurement of the +kind referred to was executed in 1849 by Fizeau (born at Paris in 1819). + +I shall endeavor to make the principle of Fizeau's apparatus clear to +you. Let _s_ (Fig. 16) be a disk free to rotate about its centre, and +perforated at its rim with a series of holes. Let _l_ be a luminous +point casting its light on an unsilvered glass, _a_, inclined at an +angle of forty-five degrees to the axis of the disk. The ray of light, +reflected at this point, passes through one of the holes of the disk and +falls at right angles upon a mirror _b_, erected at a point about five +miles distant. From the mirror _b_ the light is again reflected, passes +once more through the hole in _s_, and, penetrating the glass plate, +finally strikes the eye, _o_, of the observer. The eye, _o_, thus, sees +the image of the luminous point _l_ through the glass plate and the hole +of the disk in the mirror _b_. + +[Illustration: Fig. 16.] + +If, now, the disk be set in rotation, the unpierced spaces between the +apertures will alternately take the place of the apertures, and the eye +o will now see the image of the luminous point in _b_ only at +interrupted intervals. On increasing the rapidity of the rotation, +however, the interruptions for the eye become again unnoticeable, and +the eye sees the mirror _b_ uniformly illuminated. + +But all this holds true only for relatively small velocities of the +disk, when the light sent through an aperture in _s_ to _b_ on its +return strikes the aperture at almost the same place and passes through +it a second time. Conceive, now, the speed of the disk to be so +increased that the light on its return finds before it an unpierced +space instead of an aperture, it will then no longer be able to reach +the eye. We then see the mirror _b_ only when no light is emitted from +it, but only when light is sent to it; it is covered when light comes +from it. In this case, accordingly, the mirror will always appear dark. + +If the velocity of rotation at this point were still further increased, +the light sent through one aperture could not, of course, on its return +pass through the same aperture but might strike the next and reach the +eye by that. Hence, by constantly increasing the velocity of the +rotation, the mirror _b_ may be made to appear alternately bright and +dark. Plainly, now, if we know the number of apertures of the disk, the +number of rotations per second, and the distance _sb_, we can calculate +the velocity of light. The result agrees with that obtained by Römer. + +The experiment is not quite as simple as my exposition might lead you to +believe. Care must be taken that the light shall travel back and forth +over the miles of distance _sb_ and _bs_ undispersed. This difficulty +is obviated by means of telescopes. + +If we examine Fizeau's apparatus closely, we shall recognise in it an +old acquaintance: the arrangement of Galileo's experiment. The luminous +point _l_ is the lantern _A_, while the rotation of the perforated disk +performs mechanically the uncovering and covering of the lantern. +Instead of the unskilful observer _B_ we have the mirror _b_, which is +unfailingly illuminated the instant the light arrives from _s_. The disk +_s_, by alternately transmitting and intercepting the reflected light, +assists the observer _o_. Galileo's experiment is here executed, so to +speak, countless times in a second, yet the total result admits of +actual observation. If I might be pardoned the use of a phrase of +Darwin's in this field, I should say that Fizeau's apparatus was the +descendant of Galileo's lantern. + +A still more refined and delicate method for the measurement of the +velocity of light was employed by Foucault, but a description of it here +would lead us too far from our subject. + +The measurement of the velocity of sound is easily executed by the +method of Galileo. It was unnecessary, therefore, for physicists to rack +their brains further about the matter; but the idea which with light +grew out of necessity was applied also in this field. Koenig of Paris +constructs an apparatus for the measurement of the velocity of sound +which is closely allied to the method of Fizeau. + +The apparatus is very simple. It consists of two electrical clock-works +which strike simultaneously, with perfect precision, tenths of seconds. +If we place the two clock-works directly side by side, we hear their +strokes simultaneously, wherever we stand. But if we take our stand by +the side of one of the works and place the other at some distance from +us, in general a coincidence of the strokes will now not be heard. The +companion strokes of the remote clock-work arrive, as sound, later. The +first stroke of the remote work is heard, for example, immediately after +the first of the adjacent work, and so on. But by increasing the +distance we may produce again a coincidence of the strokes. For example, +the first stroke of the remote work coincides with the second of the +near work, the second of the remote work with the third of the near +work, and so on. If, now, the works strike tenths of seconds and the +distance between them is increased until the first coincidence is noted, +plainly that distance is travelled over by the sound in a tenth of a +second. + +We meet frequently the phenomenon here presented, that a thought which +centuries of slow and painful endeavor are necessary to produce, when +once developed, fairly thrives. It spreads and runs everywhere, even +entering minds in which it could never have arisen. It simply cannot be +eradicated. + +The determination of the velocity of light is not the only case in which +the direct perception of the senses is too slow and clumsy for use. The +usual method of studying events too fleet for direct observation +consists in putting into reciprocal action with them other events +already known, the velocities of all of which are capable of comparison. +The result is usually unmistakable, and susceptible of direct inference +respecting the character of the event which is unknown. The velocity of +electricity cannot be determined by direct observation. But it was +ascertained by Wheatstone, simply by the expedient of watching an +electric spark in a mirror rotating with tremendous known velocity. + +[Illustration: Fig. 17.] + +[Illustration: Fig. 18.] + +If we wave a staff irregularly hither and thither, simple observation +cannot determine how quickly it moves at each point of its course. But +let us look at the staff through holes in the rim of a rapidly rotating +disk (Fig. 17). We shall then see the moving staff only in certain +positions, namely, when a hole passes in front of the eye. The single +pictures of the staff remain for a time impressed upon the eye; we think +we see several staffs, having some such disposition as that represented +in Fig. 18. If, now, the holes of the disk are equally far apart, and +the disk is rotated with uniform velocity, we see clearly that the staff +has moved slowly from _a_ to _b_, more quickly from _b_ to _c_, still +more quickly from _c_ to _d_, and with its greatest velocity from _d_ to +_e_. + +A jet of water flowing from an orifice in the bottom of a vessel has the +appearance of perfect quiet and uniformity, but if we illuminate it for +a second, in a dark room, by means of an electric flash we shall see +that the jet is composed of separate drops. By their quick descent the +images of the drops are obliterated and the jet appears uniform. Let us +look at the jet through the rotating disk. The disk is supposed to be +rotated so rapidly that while the second aperture passes into the place +of the first, drop 1 falls into the place of 2, 2 into the place of 3, +and so on. We see drops then always in the same places. The jet appears +to be at rest. If we turn the disk a trifle more slowly, then while the +second aperture passes into the place of the first, drop 1 will have +fallen somewhat lower than 2, 2 somewhat lower than 3, etc. Through +every successive aperture we shall see drops in successively lower +positions. The jet will appear to be flowing slowly downwards. + +[Illustration: Fig. 19.] + +Now let us turn the disk more rapidly. Then while the second aperture is +passing into the place of the first, drop 1 will not quite have reached +the place of 2, but will be found slightly above 2, 2 slightly above 3, +etc. Through the successive apertures we shall see the drops at +successively higher places. It will now look as if the jet were flowing +upwards, as if the drops were rising from the lower vessel into the +higher. + +You see, physics grows gradually more and more terrible. The physicist +will soon have it in his power to play the part of the famous lobster +chained to the bottom of the Lake of Mohrin, whose direful mission, if +ever liberated, the poet Kopisch humorously describes as that of a +reversal of all the events of the world; the rafters of houses become +trees again, cows calves, honey flowers, chickens eggs, and the poet's +own poem flows back into his inkstand. + + * * * * * + +You will now allow me the privilege of a few general remarks. You have +seen that the same principle often lies at the basis of large classes of +apparatus designed for different purposes. Frequently it is some very +unobtrusive idea which is productive of so much fruit and of such +extensive transformations in physical technics. It is not otherwise here +than in practical life. + +The wheel of a waggon appears to us a very simple and insignificant +creation. But its inventor was certainly a man of genius. The round +trunk of a tree perhaps first accidentally led to the observation of the +ease with which a load can be moved on a roller. Now, the step from a +simple supporting roller to a fixed roller, or wheel, appears a very +easy one. At least it appears very easy to us who are accustomed from +childhood up to the action of the wheel. But if we put ourselves vividly +into the position of a man who never saw a wheel, but had to invent one, +we shall begin to have some idea of its difficulties. Indeed, it is +even doubtful whether a single man could have accomplished this feat, +whether perhaps centuries were not necessary to form the first wheel +from the primitive roller.[16] + +History does not name the progressive minds who constructed the first +wheel; their time lies far back of the historic period. No scientific +academy crowned their efforts, no society of engineers elected them +honorary members. They still live only in the stupendous results which +they called forth. Take from us the wheel, and little will remain of the +arts and industries of modern life. All disappears. From the +spinning-wheel to the spinning-mill, from the turning-lathe to the +rolling-mill, from the wheelbarrow to the railway train, all vanishes. + +In science the wheel is equally important. Whirling machines, as the +simplest means of obtaining quick motions with inconsiderable changes of +place, play a part in all branches of physics. You know Wheatstone's +rotating mirror, Fizeau's wheel, Plateau's perforated rotating disks, +etc. Almost the same principle lies at the basis of all these apparatus. +They differ from one another no more than the pen-knife differs, in the +purposes it serves, from the knife of the anatomist or the knife of the +vine-dresser. Almost the same might be said of the screw. + +It will now perhaps be clear to you that new thoughts do not spring up +suddenly. Thoughts need their time to ripen, grow, and develop in, like +every natural product; for man, with his thoughts, is also a part of +nature. + +Slowly, gradually, and laboriously one thought is transformed into a +different thought, as in all likelihood one animal species is gradually +transformed into new species. Many ideas arise simultaneously. They +fight the battle for existence not otherwise than do the Ichthyosaurus, +the Brahman, and the horse. + +A few remain to spread rapidly over all fields of knowledge, to be +redeveloped, to be again split up, to begin again the struggle from the +start. As many animal species long since conquered, the relicts of ages +past, still live in remote regions where their enemies cannot reach +them, so also we find conquered ideas still living on in the minds of +many men. Whoever will look carefully into his own soul will acknowledge +that thoughts battle as obstinately for existence as animals. Who will +gainsay that many vanquished modes of thought still haunt obscure +crannies of his brain, too faint-hearted to step out into the clear +light of reason? What inquirer does not know that the hardest battle, in +the transformation of his ideas, is fought with himself. + +Similar phenomena meet the natural inquirer in all paths and in the most +trifling matters. The true inquirer seeks the truth everywhere, in his +country-walks and on the streets of the great city. If he is not too +learned, he will observe that certain things, like ladies' hats, are +constantly subject to change. I have not pursued special studies on this +subject, but as long as I can remember, one form has always gradually +changed into another. First, they wore hats with long projecting rims, +within which, scarcely accessible with a telescope, lay concealed the +face of the beautiful wearer. The rim grew smaller and smaller; the +bonnet shrank to the irony of a hat. Now a tremendous superstructure is +beginning to grow up in its place, and the gods only know what its +limits will be. It is not otherwise with ladies' hats than with +butterflies, whose multiplicity of form often simply comes from a slight +excrescence on the wing of one species developing in a cognate species +to a tremendous fold. Nature, too, has its fashions, but they last +thousands of years. I could elucidate this idea by many additional +examples; for instance, by the history of the evolution of the coat, if +I were not fearful that my gossip might prove irksome to you. + + * * * * * + +We have now wandered through an odd corner of the history of science. +What have we learned? The solution of a small, I might almost say +insignificant, problem--the measurement of the velocity of light. And +more than two centuries have worked at its solution! Three of the most +eminent natural philosophers, Galileo, an Italian, Römer, a Dane, and +Fizeau, a Frenchman, have fairly shared its labors. And so it is with +countless other questions. When we contemplate thus the many blossoms of +thought that must wither and fall before one shall bloom, then shall we +first truly appreciate Christ's weighty but little consolatory words: +"Many be called but few are chosen." + +Such is the testimony of every page of history. But is history right? +Are really only those chosen whom she names? Have those lived and +battled in vain, who have won no prize? + +I doubt it. And so will every one who has felt the pangs of sleepless +nights spent in thought, at first fruitless, but in the end successful. +No thought in such struggles was thought in vain; each one, even the +most insignificant, nay, even the erroneous thought, that which +apparently was the least productive, served to prepare the way for those +that afterwards bore fruit. And as in the thought of the individual +naught is in vain, so, also, it is in that of humanity. + +Galileo wished to measure the velocity of light. He had to close his +eyes before his wish was realised. But he at least found the lantern by +which his successor could accomplish the task. + +And so I may maintain that we all, so far as inclination goes, are +working at the civilisation of the future. If only we all strive for the +right, then are we _all_ called and _all_ chosen! + + FOOTNOTES: + + [Footnote 11: According to Mr. Jules Andrieu, the idea that nature + must be tortured to reveal her secrets is preserved in the name + _crucible_--from the Latin _crux_, a cross. But, more probably, + _crucible_ is derived from some Old French or Teutonic form, as + _cruche_, _kroes_, _krus_, etc., a pot or jug (cf. Modern English + _crock_, _cruse_, and German _Krug_).--_Trans._] + + [Footnote 12: Xenophon, Memorabilia iv, 7, puts into the mouth of + Socrates these words: [Greek: oute gar heureta anthrôpois auta + enomizen einai, oute chaoizesthai theois an hêgeito ton zêtounta ha + ekeinoi saphênisai ouk eboulêthêsan].] + + [Footnote 13: Galilei, _Discorsi e dimostrazione matematiche_. + Leyden, 1638. _Dialogo Primo._] + + [Footnote 14: In the same way, the pitch of a locomotive-whistle is + higher as the locomotive rapidly approaches an observer, and lower + when rapidly leaving him than if the locomotive were at + rest.--_Trans._] + + [Footnote 15: A kilometre is 0.621 or nearly five-eighths of a + statute mile.] + + [Footnote 16: Observe, also, the respect in which the wheel is held + in India, Japan and other Buddhistic countries, as the emblem of + power, order, and law, and of the superiority of mind over matter. + The consciousness of the importance of this invention seems to have + lingered long in the minds of these nations.--_Tr._] + + + + +WHY HAS MAN TWO EYES? + + +Why has man two eyes? That the pretty symmetry of his face may not be +disturbed, the artist answers. That his second eye may furnish a +substitute for his first if that be lost, says the far-sighted +economist. That we may weep with two eyes at the sins of the world, +replies the religious enthusiast. + +Odd opinions! Yet if you should approach a modern scientist with this +question you might consider yourself fortunate if you escaped with less +than a rebuff. "Pardon me, madam, or my dear sir," he would say, with +stern expression, "man fulfils no purpose in the possession of his eyes; +nature is not a person, and consequently not so vulgar as to pursue +purposes of any kind." + +Still an unsatisfactory answer! I once knew a professor who would shut +with horror the mouths of his pupils if they put to him such an +unscientific question. + +But ask a more tolerant person, ask me. I, I candidly confess, do not +know exactly why man has two eyes, but the reason partly is, I think, +that I may see you here before me to-night and talk with you upon this +delightful subject. + +Again you smile incredulously. Now this is one of those questions that a +hundred wise men together could not answer. You have heard, so far, only +five of these wise men. You will certainly want to be spared the +opinions of the other ninety-five. To the first you will reply that we +should look just as pretty if we were born with only one eye, like the +Cyclops; to the second we should be much better off, according to his +principle, if we had four or eight eyes, and that in this respect we are +vastly inferior to spiders; to the third, that you are not just in the +mood to weep; to the fourth, that the unqualified interdiction of the +question excites rather than satisfies your curiosity; while of me you +will dispose by saying that my pleasure is not as intense as I think, +and certainly not great enough to justify the existence of a double eye +in man since the fall of Adam. + +But since you are not satisfied with my brief and obvious answer, you +have only yourselves to blame for the consequences. You must now listen +to a longer and more learned explanation, such as it is in my power to +give. + +As the church of science, however, debars the question "Why?" let us put +the matter in a purely orthodox way: Man has two eyes, what _more_ can +he see with two than with one? + +I will invite you to take a walk with me? We see before us a wood. What +is it that makes this real wood contrast so favorably with a painted +wood, no matter how perfect the painting may be? What makes the one so +much more lovely than the other? Is it the vividness of the coloring, +the distribution of the lights and the shadows? I think not. On the +contrary, it seems to me that in this respect painting can accomplish +very much. + +The cunning hand of the painter can conjure up with a few strokes of his +brush forms of wonderful plasticity. By the help of other means even +more can be attained. Photographs of reliefs are so plastic that we +often imagine we can actually lay hold of the elevations and +depressions. + +[Illustration: Fig. 20.] + +But one thing the painter never can give with the vividness that nature +does--the difference of near and far. In the real woods you see plainly +that you can lay hold of some trees, but that others are inaccessibly +far. The picture of the painter is rigid. The picture of the real woods +changes on the slightest movement. Now this branch is hidden behind +that; now that behind this. The trees are alternately visible and +invisible. + +Let us look at this matter a little more closely. For convenience sake +we shall remain upon the highway, I, II. (Fig. 20.) To the right and the +left lies the forest. Standing at I, we see, let us say, three trees (1, +2, 3) in a line, so that the two remote ones are covered by the nearest. +Moving further along, this changes. At II we shall not have to look +round so far to see the remotest tree 3 as to see the nearer tree 2, nor +so far to see this as to see 1. _Hence, as we move onward, objects that +are near to us seem to lag behind as compared with objects that are +remote from us, the lagging increasing with the proximity of the +objects._ Very remote objects, towards which we must always look in the +same direction as we proceed, appear to travel along with us. + +If we should see, therefore, jutting above the brow of yonder hill the +tops of two trees whose distance from us we were in doubt about, we +should have in our hands a very easy means of deciding the question. We +should take a few steps forward, say to the right, and the tree-top +which receded most to the left would be the one nearer to us. In truth, +from the amount of the recession a geometer could actually determine the +distance of the trees from us without ever going near them. It is simply +the scientific development of this perception that enables us to +measure the distances of the stars. + +_Hence, from change of view in forward motion the distances of objects +in our field of vision can be measured._ + +Rigorously, however, even forward motion is not necessary. For every +observer is composed really of _two_ observers. Man has _two_ eyes. The +right eye is a short step ahead of the left eye in the right-hand +direction. Hence, the two eyes receive _different_ pictures of the same +woods. The right eye will see the near trees displaced to the left, and +the left eye will see them displaced to the right, the displacement +being greater, the greater the proximity. This difference is sufficient +for forming ideas of distance. + +We may now readily convince ourselves of the following facts: + +1. With one eye, the other being shut, you have a very uncertain +judgment of distances. You will find it, for example, no easy task, with +one eye shut, to thrust a stick through a ring hung up before you; you +will miss the ring in almost every instance. + +2. You see the same object differently with the right eye from what you +do with the left. + +Place a lamp-shade on the table in front of you with its broad opening +turned downwards, and look at it from above. (Fig. 21.) You will see +with your right eye the image 2, with your left eye the image 1. Again, +place the shade with its wide opening turned upwards; you will receive +with your right eye the image 4, with your left eye the image 3. Euclid +mentions phenomena of this character. + +3. Finally, you know that it is easy to judge of distances with both +eyes. Accordingly your judgment must spring in some way from a +co-operation of the two eyes. In the preceding example the openings in +the different images received by the two eyes seem displaced with +respect to one another, and this displacement is sufficient for the +inference that the one opening is nearer than the other. + +[Illustration: Fig. 21.] + +I have no doubt that you, ladies, have frequently received delicate +compliments upon your eyes, but I feel sure that no one has ever told +you, and I know not whether it will flatter you, that you have in your +eyes, be they blue or black, little geometricians. You say you know +nothing of them? Well, for that matter, neither do I. But the facts are +as I tell you. + +You understand little of geometry? I shall accept that confession. Yet +with the help of your two eyes you judge of distances? Surely that is a +geometrical problem. And what is more, you know the solution of this +problem: for you estimate distances correctly. If, then, _you_ do not +solve the problem, the little geometricians in your eyes must do it +clandestinely and whisper the solution to you. I doubt not they are +fleet little fellows. + +What amazes me most here is, that you know nothing about these little +geometricians. But perhaps they also know nothing about you. Perhaps +they are models of punctuality, routine clerks who bother about nothing +but their fixed work. In that case we may be able to deceive the +gentlemen. + +If we present to our right eye an image which looks exactly like the +lamp-shade for the right eye, and to our left eye an image which looks +exactly like a lamp-shade for the left eye, we shall imagine that we see +the whole lamp-shade bodily before us. + +You know the experiment. If you are practised in squinting, you can +perform it directly with the figure, looking with your right eye at the +right image, and with your left eye at the left image. In this way the +experiment was first performed by Elliott. Improved and perfected, its +form is Wheatstone's stereoscope, made so popular and useful by +Brewster. + +By taking two photographs of the same object from two different points, +corresponding to the two eyes, a very clear three-dimensional picture of +distant places or buildings can be produced by the stereoscope. + +But the stereoscope accomplishes still more than this. It can visualise +things for us which we never see with equal clearness in real objects. +You know that if you move much while your photograph is being taken, +your picture will come out like that of a Hindu deity, with several +heads or several arms, which, at the spaces where they overlap, show +forth with equal distinctness, so that we seem to see the one picture +_through_ the other. If a person moves quickly away from the camera +before the impression is completed, the objects behind him will also be +imprinted upon the photograph; the person will look transparent. +Photographic ghosts are made in this way. + +Some very useful applications may be made of this discovery. For +example, if we photograph a machine stereoscopically, successively +removing during the operation the single parts (where of course the +impression suffers interruptions), we obtain a transparent view, endowed +with all the marks of spatial solidity, in which is distinctly +visualised the interaction of parts normally concealed. I have employed +this method for obtaining transparent stereoscopic views of anatomical +structures. + +You see, photography is making stupendous advances, and there is great +danger that in time some malicious artist will photograph his innocent +patrons with solid views of their most secret thoughts and emotions. How +tranquil politics will then be! What rich harvests our detective force +will reap! + + * * * * * + +By the joint action of the two eyes, therefore, we arrive at our +judgments of distances, as also of the forms of bodies. + +Permit me to mention here a few additional facts connected with this +subject, which will assist us in the comprehension of certain phenomena +in the history of civilisation. + +You have often heard, and know from personal experience, that remote +objects appear perspectively dwarfed. In fact, it is easy to satisfy +yourself that you can cover the image of a man a few feet away from you +simply by holding up your finger a short distance in front of your eye. +Still, as a general rule, you do not notice this shrinkage of objects. +On the contrary, you imagine you see a man at the end of a large hall, +as large as you see him near by you. For your eye, in its measurement of +the distances, makes remote objects correspondingly larger. The eye, so +to speak, is aware of this perspective contraction and is not deceived +by it, although its possessor is unconscious of the fact. All persons +who have attempted to draw from nature have vividly felt the difficulty +which this superior dexterity of the eye causes the perspective +conception. Not until one's judgment of distances is made uncertain, by +their size, or from lack of points of reference, or from being too +quickly changed, is the perspective rendered very prominent. + +On sweeping round a curve on a rapidly moving railway train, where a +wide prospect is suddenly opened up, the men upon distant hills appear +like dolls.[17] You have at the moment, here, no known references for +the measurement of distances. The stones at the entrance of a tunnel +grow visibly larger as we ride towards it; they shrink visibly in size +as we ride from it. + +Usually both eyes work together. As certain views are frequently +repeated, and lead always to substantially the same judgments of +distances, the eyes in time must acquire a special skill in geometrical +constructions. In the end, undoubtedly, this skill is so increased that +a single eye alone is often tempted to exercise that office. + +Permit me to elucidate this point by an example. Is any sight more +familiar to you than that of a vista down a long street? Who has not +looked with hopeful eyes time and again into a street and measured its +depth. I will take you now into an art-gallery where I will suppose you +to see a picture representing a vista into a street. The artist has not +spared his rulers to get his perspective perfect. The geometrician in +your left eye thinks, "Ah ha! I have computed that case a hundred times +or more. I know it by heart. It is a vista into a street," he continues; +"where the houses are lower is the remote end." The geometrician in the +right eye, too much at his ease to question his possibly peevish comrade +in the matter, answers the same. But the sense of duty of these punctual +little fellows is at once rearoused. They set to work at their +calculations and immediately find that all the points of the picture are +equally distant from them, that is, lie all upon a plane surface. + +What opinion will you now accept, the first or the second? If you accept +the first you will see distinctly the vista. If you accept the second +you will see nothing but a painted sheet of distorted images. + +It seems to you a trifling matter to look at a picture and understand +its perspective. Yet centuries elapsed before humanity came fully to +appreciate this trifle, and even the majority of you first learned it +from education. + +I can remember very distinctly that at three years of age all +perspective drawings appeared to me as gross caricatures of objects. I +could not understand why artists made tables so broad at one end and so +narrow at the other. Real tables seemed to me just as broad at one end +as at the other, because my eye made and interpreted its calculations +without my intervention. But that the picture of the table on the plane +surface was not to be conceived as a plane painted surface but stood for +a table and so was to be imaged with all the attributes of extension was +a joke that I did not understand. But I have the consolation that whole +nations have not understood it. + +Ingenuous people there are who take the mock murders of the stage for +real murders, the dissembled actions of the players for real actions, +and who can scarcely restrain themselves, when the characters of the +play are sorely pressed, from running in deep indignation to their +assistance. Others, again, can never forget that the beautiful +landscapes of the stage are painted, that Richard III. is only the +actor, Mr. Booth, whom they have met time and again at the clubs. + +Both points of view are equally mistaken. To look at a drama or a +picture properly one must understand that both are _shows_, simply +_denoting_ something real. A certain preponderance of the intellectual +life over the sensuous life is requisite for such an achievement, where +the intellectual elements are safe from destruction by the direct +sensuous impressions. A certain liberty in choosing one's point of view +is necessary, a sort of humor, I might say, which is strongly wanting in +children and in childlike peoples. + +Let us look at a few historical facts. I shall not take you as far back +as the stone age, although we possess sketches from this epoch which +show very original ideas of perspective. But let us begin our +sight-seeing in the tombs and ruined temples of ancient Egypt, where the +numberless reliefs and gorgeous colorings have defied the ravages of +thousands of years. + +A rich and motley life is here opened to us. We find the Egyptians +represented in all conditions of life. What at once strikes our +attention in these pictures is the delicacy of their technical +execution. The contours are extremely exact and distinct. But on the +other hand only a few bright colors are found, unblended and without +trace of transition. Shadows are totally wanting. The paint is laid on +the surfaces in equal thicknesses. + +Shocking for the modern eye is the perspective. All the figures are +equally large, with the exception of the king, whose form is unduly +exaggerated. Near and far appear equally large. Perspective contraction +is nowhere employed. A pond with water-fowl is represented flat, as if +its surface were vertical. + +Human figures are portrayed as they are never seen, the legs from the +side, the face in profile. The breast lies in its full breadth across +the plane of representation. The heads of cattle appear in profile, +while the horns lie in the plane of the drawing. The principle which the +Egyptians followed might be best expressed by saying that their figures +are pressed in the plane of the drawing as plants are pressed in a +herbarium. + +The matter is simply explained. If the Egyptians were accustomed to +looking at things ingenuously with both eyes at once, the construction +of perspective pictures in space could not be familiar to them. They saw +all arms, all legs on real men in their natural lengths. The figures +pressed into the planes resembled more closely, of course, in their eyes +the originals than perspective pictures could. + +This will be better understood if we reflect that painting was developed +from relief. The minor dissimilarities between the pressed figures and +the originals must gradually have compelled men to the adoption of +perspective drawing. But physiologically the painting of the Egyptians +is just as much justified as the drawings of our children are. + +A slight advance beyond the Egyptians is shown by the Assyrians. The +reliefs rescued from the ruined mounds of Nimrod at Mossul are, upon the +whole, similar to the Egyptian reliefs. They were made known to us +principally by Layard. + +Painting enters on a new phase among the Chinese. This people have a +marked feeling for perspective and correct shading, yet without being +very logical in the application of their principles. Here, too, it +seems, they took the first step but did not go far. In harmony with this +immobility is their constitution, in which the muzzle and the bamboo-rod +play significant functions. In accord with it, too, is their language, +which like the language of children has not yet developed into a +grammar, or, rather, according to the modern conception, has not yet +degenerated into a grammar. It is the same also with their music which +is satisfied with the five-toned scale. + +The mural paintings at Herculaneum and Pompeii are distinguished by +grace of representation, as also by a pronounced sense for perspective +and correct illumination, yet they are not at all scrupulous in +construction. Here still we find abbreviations avoided. But to offset +this defect, the members of the body are brought into unnatural +positions, in which they appear in their full lengths. Abridgements are +more frequently observed in clothed than in unclothed figures. + +A satisfactory explanation of these phenomena first occurred to me on +the making of a few simple experiments which show how differently one +may see the same object, after some mastery of one's senses has been +attained, simply by the arbitrary movement of the attention. + +[Illustration: Fig. 22.] + +Look at the annexed drawing (Fig. 22). It represents a folded sheet of +paper with either its depressed or its elevated side turned towards you, +as you wish. You can conceive the drawing in either sense, and in either +case it will appear to you differently. + +If, now, you have a real folded sheet of paper on the table before you, +with its sharp edges turned towards you, you can, on looking at it with +one eye, see the sheet alternately elevated, as it really is, or +depressed. Here, however, a remarkable phenomenon is presented. When you +see the sheet properly, neither illumination nor form presents anything +conspicuous. When you see it bent back you see it perspectively +distorted. Light and shadow appear much brighter or darker, or as if +overlaid thickly with bright colors. Light and shadow now appear devoid +of all cause. They no longer harmonise with the body's form, and are +thus rendered much more prominent. + +In common life we employ the perspective and illumination of objects to +determine their forms and position. Hence we do not notice the lights, +the shadows, and the distortions. They first powerfully enter +consciousness when we employ a different construction from the usual +spatial one. In looking at the planar image of a camera obscura we are +amazed at the plenitude of the light and the profundity of the shadows, +both of which we do not notice in real objects. + +In my earliest youth the shadows and lights on pictures appeared to me +as spots void of meaning. When I began to draw I regarded shading as a +mere custom of artists. I once drew the portrait of our pastor, a friend +of the family, and shaded, from no necessity, but simply from having +seen something similar in other pictures, the whole half of his face +black. I was subjected for this to a severe criticism on the part of my +mother, and my deeply offended artist's pride is probably the reason +that these facts remained so strongly impressed upon my memory. + +You see, then, that many strange things, not only in the life of +individuals, but also in that of humanity, and in the history of general +civilisation, may be explained from the simple fact that man has two +eyes. + +Change man's eye and you change his conception of the world. We have +observed the truth of this fact among our nearest kin, the Egyptians, +the Chinese, and the lake-dwellers; how must it be among some of our +remoter relatives,--with monkeys and other animals? Nature must appear +totally different to animals equipped with substantially different eyes +from those of men, as, for example, to insects. But for the present +science must forego the pleasure of portraying this appearance, as we +know very little as yet of the mode of operation of these organs. + +It is an enigma even how nature appears to animals closely related to +man; as to birds, who see scarcely anything with two eyes at once, but +since their eyes are placed on opposite sides of their heads, have a +separate field of vision for each.[18] + +The soul of man is pent up in the prison-house of his head; it looks at +nature through its two windows, the eyes. It would also fain know how +nature looks through other windows. A desire apparently never to be +fulfilled. But our love for nature is inventive, and here, too, much has +been accomplished. + +Placing before me an angular mirror, consisting of two plane mirrors +slightly inclined to each other, I see my face twice reflected. In the +right-hand mirror I obtain a view of the right side, and in the +left-hand mirror a view of the left side, of my face. Also I shall see +the face of a person standing in front of me, more to the right with my +right eye, more to the left with my left. But in order to obtain such +widely different views of a face as those shown in the angular mirror, +my two eyes would have to be set much further apart from each other than +they actually are. + +[Illustration: Fig. 23.] + +Squinting with my right eye at the image in the right-hand mirror, with +my left eye at the image in the left-hand mirror, my vision will be the +vision of a giant having an enormous head with his two eyes set far +apart. This, also, is the impression which my own face makes upon me. I +see it now, single and solid. Fixing my gaze, the relief from second to +second is magnified, the eyebrows start forth prominently from above the +eyes, the nose seems to grow a foot in length, my mustache shoots forth +like a fountain from my lip, the teeth seem to retreat immeasurably. But +by far the most horrible aspect of the phenomenon is the nose. + +Interesting in this connexion is the telestereoscope of Helmholtz. In +the telestereoscope we view a landscape by looking with our right eye +(Fig. 24) through the mirror _a_ into the mirror _A_, and with our left +eye through the mirror _b_ into the mirror _B_. The mirrors _A_ and _B_ +stand far apart. Again we see with the widely separated eyes of a giant. +Everything appears dwarfed and near us. The distant mountains look like +moss-covered stones at our feet. Between, you see the reduced model of a +city, a veritable Liliput. You are tempted almost to stroke with your +hand the soft forest and city, did you not fear that you might prick +your fingers on the sharp, needle-shaped steeples, or that they might +crackle and break off. + +[Illustration: Fig. 24.] + +Liliput is no fable. We need only Swift's eyes, the telestereoscope, to +see it. + +Picture to yourself the reverse case. Let us suppose ourselves so small +that we could take long walks in a forest of moss, and that our eyes +were correspondingly near each other. The moss-fibres would appear like +trees. On them we should see strange, unshapely monsters creeping about. +Branches of the oak-tree, at whose base our moss-forest lay, would seem +to us dark, immovable, myriad-branched clouds, painted high on the vault +of heaven; just as the inhabitants of Saturn, forsooth, might see their +enormous ring. On the tree-trunks of our mossy woodland we should find +colossal globes several feet in diameter, brilliantly transparent, +swayed by the winds with slow, peculiar motions. We should approach +inquisitively and should find that these globes, in which here and there +animals were gaily sporting, were liquid globes, in fact that they were +water. A short, incautious step, the slightest contact, and woe betide +us, our arm is irresistibly drawn by an invisible power into the +interior of the sphere and held there unrelentingly fast! A drop of dew +has engulfed in its capillary maw a manikin, in revenge for the +thousands of drops that its big human counterparts have quaffed at +breakfast. Thou shouldst have known, thou pygmy natural scientist, that +with thy present puny bulk thou shouldst not joke with capillarity! + +My terror at the accident brings me back to my senses. I see I have +turned idyllic. You must pardon me. A patch of greensward, a moss or +heather forest with its tiny inhabitants have incomparably more charms +for me than many a bit of literature with its apotheosis of human +character. If I had the gift of writing novels I should certainly not +make John and Mary my characters. Nor should I transfer my loving pair +to the Nile, nor to the age of the old Egyptian Pharaohs, although +perhaps I should choose that time in preference to the present. For I +must candidly confess that I hate the rubbish of history, interesting +though it may be as a mere phenomenon, because we cannot simply observe +it but must also _feel_ it, because it comes to us mostly with +supercilious arrogance, mostly unvanquished. The hero of my novel would +be a cockchafer, venturing forth in his fifth year for the first time +with his newly grown wings into the light, free air. Truly it could do +no harm if man would thus throw off his inherited and acquired +narrowness of mind by making himself acquainted with the world-view of +allied creatures. He could not help gaining incomparably more in this +way than the inhabitant of a small town would in circumnavigating the +globe and getting acquainted with the views of strange peoples. + + * * * * * + +I have now conducted you, by many paths and by-ways, rapidly over hedge +and ditch, to show you what wide vistas we may reach in every field by +the rigorous pursuit of a single scientific fact. A close examination of +the two eyes of man has conducted us not only into the dim recesses of +humanity's childhood, but has also carried us far beyond the bourne of +human life. + +It has surely often struck you as strange that the sciences are divided +into two great groups; that the so-called humanistic sciences, belonging +to the so-called "higher education," are placed in almost a hostile +attitude to the natural sciences. + +I must confess I do not overmuch believe in this partition of the +sciences. I believe that this view will appear as childlike and +ingenuous to a matured age as the want of perspective in the old +paintings of Egypt does to us. Can it really be that "higher culture" is +to be gotten only from a few old pots and palimpsests, which are at best +mere scraps of nature, or that more is to be learned from them alone +than from all the rest of nature? I believe that both these sciences are +simply parts of the same science, which have begun at different ends. If +these two ends still act towards each other as the Montagues and +Capulets, if their retainers still indulge in lively tilts, I believe +that after all they are not in earnest. On the one side there is surely +a Romeo, and on the other a Juliet, who, some day, it is hoped, will +unite the two houses with a less tragic sequel than that of the play. + +Philology began with the unqualified reverence and apotheosis of the +Greeks. Now it has begun to draw other languages, other peoples and +their histories, into its sphere; it has, through the mediation of +comparative linguistics, already struck up, though as yet somewhat +cautiously, a friendship with physiology. + +Physical science began in the witch's kitchen. It now embraces the +organic and inorganic worlds, and with the physiology of articulation +and the theory of the senses, has even pushed its researches, at times +impertinently, into the province of mental phenomena. + +In short, we come to the understanding of much within us solely by +directing our glance without, and _vice versa_. Every object belongs to +both sciences. You, ladies, are very interesting and difficult problems +for the psychologist, but you are also extremely pretty phenomena of +nature. Church and State are objects of the historian's research, but +not less phenomena of nature, and in part, indeed, very curious +phenomena. If the historical sciences have inaugurated wide extensions +of view by presenting to us the thoughts of new and strange peoples, the +physical sciences in a certain sense do this in a still greater degree. +In making man disappear in the All, in annihilating him, so to speak, +they force him to take an unprejudiced position without himself, and to +form his judgments by a different standard from that of the petty human. + +But if you should ask me now why man has two eyes, I should answer: + +That he may look at nature justly and accurately; that he may come to +understand that he himself, with all his views, correct and incorrect, +with all his _haute politique_, is simply an evanescent shred of nature; +that, to speak with Mephistopheles, he is a part of the part, and that +it is absolutely unjustified, + + "For man, the microcosmic fool, to see + Himself a whole so frequently." + + FOOTNOTES: + + [Footnote 17: This effect is particularly noticeable in the size of + workmen on high chimneys and church-steeples--"steeple Jacks." When + the cables were slung from the towers of the Brooklyn bridge (277 + feet high), the men sent out in baskets to paint them, appeared, + against the broad background of heaven and water, like + flies.--_Trans._] + + [Footnote 18: See Joh. Müller, _Vergleichende Physiologie des + Gesichtssinnes_, Leipsic, 1826.] + + + + +ON SYMMETRY.[19] + + +An ancient philosopher once remarked that people who cudgelled their +brains about the nature of the moon reminded him of men who discussed +the laws and institutions of a distant city of which they had heard no +more than the name. The true philosopher, he said, should turn his +glance within, should study himself and his notions of right and wrong; +only thence could he derive real profit. + +This ancient formula for happiness might be restated in the familiar +words of the Psalm: + + "Dwell in the land, and verily thou shalt be fed." + +To-day, if he could rise from the dead and walk about among us, this +philosopher would marvel much at the different turn which matters have +taken. + +The motions of the moon and the other heavenly bodies are accurately +known. Our knowledge of the motions of our own body is by far not so +complete. The mountains and natural divisions of the moon have been +accurately outlined on maps, but physiologists are just beginning to +find their way in the geography of the brain. The chemical constitution +of many fixed stars has already been investigated. The chemical +processes of the animal body are questions of much greater difficulty +and complexity. We have our _Mécanique céleste_. But a _Mécanique +sociale_ or a _Mécanique morale_ of equal trustworthiness remains to be +written. + +Our philosopher would indeed admit that we have made great progress. But +we have not followed his advice. The patient has recovered, but he took +for his recovery exactly the opposite of what the doctor prescribed. + +Humanity is now returned, much wiser, from its journey in celestial +space, against which it was so solemnly warned. Men, after having become +acquainted with the great and simple facts of the world without, are now +beginning to examine critically the world within. It sounds absurd, but +it is true, that only after we have thought about the moon are we able +to take up ourselves. It was necessary that we should acquire simple and +clear ideas in a less complicated domain, before we entered the more +intricate one of psychology, and with these ideas astronomy principally +furnished us. + +To attempt any description of that stupendous movement, which, +originally springing out of the physical sciences, went beyond the +domain of physics and is now occupied with the problems of psychology, +would be presumptuous in this place. I shall only attempt here, to +illustrate to you by a few simple examples the methods by which the +province of psychology can be reached from the facts of the physical +world--especially the adjacent province of sense-perception. And I wish +it to be remembered that my brief attempt is not to be taken as a +measure of the present state of such scientific questions. + + * * * * * + +It is a well-known fact that some objects please us, while others do +not. Generally speaking, anything that is constructed according to fixed +and logically followed rules, is a product of tolerable beauty. We see +thus nature herself, who always acts according to fixed rules, +constantly producing such pretty things. Every day the physicist is +confronted in his workshop with the most beautiful vibration-figures, +tone-figures, phenomena of polarisation, and forms of diffraction. + +A rule always presupposes a repetition. Repetitions, therefore, will +probably be found to play some important part in the production of +agreeable effects. Of course, the nature of agreeable effects is not +exhausted by this. Furthermore, the repetition of a physical event +becomes the source of agreeable effects only when it is connected with +a repetition of sensations. + +An excellent example that repetition of sensations is a source of +agreeable effects is furnished by the copy-book of every schoolboy, +which is usually a treasure-house of such things, and only in need of an +Abbé Domenech to become celebrated. Any figure, no matter how crude or +poor, if several times repeated, with the repetitions placed in line, +will produce a tolerable frieze. + +[Illustration: Fig. 25.] + +Also the pleasant effect of symmetry is due to the repetition of +sensations. Let us abandon ourselves a moment to this thought, yet not +imagine when we have developed it, that we have fully exhausted the +nature of the agreeable, much less of the beautiful. + +First, let us get a clear conception of what symmetry is. And in +preference to a definition let us take a living picture. You know that +the reflexion of an object in a mirror has a great likeness to the +object itself. All its proportions and outlines are the same. Yet there +is a difference between the object and its reflexion in the mirror, +which you will readily observe. + +Hold your right hand before a mirror, and you will see in the mirror a +left hand. Your right glove will produce its mate in the glass. For you +could never use the reflexion of your right glove, if it were present to +you as a real thing, for covering your right hand, but only for covering +your left. Similarly, your right ear will give as its reflexion a left +ear; and you will at once perceive that the left half of your body could +very easily be substituted for the reflexion of your right half. Now +just as in the place of a missing right ear a left ear cannot be put, +unless the lobule of the ear be turned upwards, or the opening into the +concha backwards, so, despite all similarity of form, the reflexion of +an object can never take the place of the object itself.[20] + +The reason of this difference between the object and its reflexion is +simple. The reflexion appears as far behind the mirror as the object is +in front of it. The parts of the object, accordingly, which are nearest +the mirror will also be nearest the mirror in the reflexion. +Consequently, the succession of the parts in the reflexion will be +reversed, as may best be seen in the reflexion of the face of a watch or +of a manuscript. + +It will also be readily seen, that if a point of the object be joined +with its reflexion in the image, the line of junction will cut the +mirror at right angles and be bisected by it. This holds true of all +corresponding points of object and image. + +If, now, we can divide an object by a plane into two halves so that each +half, as seen in the reflecting plane of division, is a reproduction of +the other half, such an object is termed symmetrical, and the plane of +division is called the plane of symmetry. + +If the plane of symmetry is vertical, we can say that the body is +vertically symmetrical. An example of vertical symmetry is a Gothic +cathedral. + +If the plane of symmetry is horizontal, we can say that the object is +horizontally symmetrical. A landscape on the shores of a lake with its +reflexion in the water, is a system of horizontal symmetry. + +Exactly here is a noticeable difference. The vertical symmetry of a +Gothic cathedral strikes us at once, whereas we can travel up and down +the whole length of the Rhine or the Hudson without becoming aware of +the symmetry between objects and their reflexions in the water. Vertical +symmetry pleases us, whilst horizontal symmetry is indifferent, and is +noticed only by the experienced eye. + +Whence arises this difference? I say from the fact that vertical +symmetry produces a repetition of the same sensation, while horizontal +symmetry does not. I shall now show that this is so. + +Let us look at the following letters: + + d b + q p + +It is a fact known to all mothers and teachers, that children in their +first attempts to read and write, constantly confound d and b, and q and +p, but never d and q, or b and p. Now d and b and q and p are the two +halves of a _vertically_ symmetrical figure, while d and q, and b and p +are two halves of a _horizontally_ symmetrical figure. The first two are +confounded; but confusion is only possible of things that excite in us +the same or similar sensations. + +Figures of two flower-girls are frequently seen on the decorations of +gardens and of drawing-rooms, one of whom carries a flower-basket in her +right hand and the other a flower-basket in her left. All know how apt +we are, unless we are very careful, to confound these figures with one +another. + +While turning a thing round from right to left is scarcely noticed, the +eye is not at all indifferent to the turning of a thing upside down. A +human face which has been turned upside down is scarcely recognisable as +a face, and makes an impression which is altogether strange. The reason +of this is not to be sought in the unwontedness of the sight, for it is +just as difficult to recognise an arabesque that has been inverted, +where there can be no question of a habit. This curious fact is the +foundation of the familiar jokes played with the portraits of unpopular +personages, which are so drawn that in the upright position of the page +an exact picture of the person is presented, but on being inverted some +popular animal is shown. + +It is a fact, then, that the two halves of a vertically symmetrical +figure are easily confounded and that they therefore probably produce +very nearly the same sensations. The question, accordingly, arises, +_why_ do the two halves of a vertically symmetrical figure produce the +same or similar sensations? The answer is: Because our apparatus of +vision, which consists of our eyes and of the accompanying muscular +apparatus is itself vertically symmetrical.[21] + +Whatever external resemblances one eye may have with another they are +still not alike. The right eye of a man cannot take the place of a left +eye any more than a left ear or left hand can take the place of a right +one. By artificial means, we can change the part which each of our eyes +plays. (Wheatstone's pseudoscope.) But we then find ourselves in an +entirely new and strange world. What is convex appears concave; what is +concave, convex. What is distant appears near, and what is near appears +far. + +The left eye is the reflexion of the right. And the light-feeling retina +of the left eye is a reflexion of the light-feeling retina of the right, +in all its functions. + +The lense of the eye, like a magic lantern, casts images of objects on +the retina. And you may picture to yourself the light-feeling retina of +the eye, with its countless nerves, as a hand with innumerable fingers, +adapted to feeling light. The ends of the visual nerves, like our +fingers, are endowed with varying degrees of sensitiveness. The two +retinæ act like a right and a left hand; the sensation of touch and the +sensation of light in the two instances are similar. + +Examine the right-hand portion of this letter T: namely, T. Instead of +the two retinæ on which this image falls, imagine feeling the object, my +two hands. The T, grasped with the right hand, gives a different +sensation from that which it gives when grasped with the left. But if we +turn our character about from right to left, thus: T, it will give the +same sensation in the left hand that it gave before in the right. The +sensation is repeated. + +If we take a whole T, the right half will produce in the right hand the +same sensation that the left half produces in the left, and _vice +versa_. + +The symmetrical figure gives the same sensation twice. + +If we turn the T over thus: T, or invert the half T thus: L, so long as +we do not change the position of our hands we can make no use of the +foregoing reasoning. + +The retinæ, in fact, are exactly like our two hands. They, too, have +their thumbs and index fingers, though they are thousands in number; and +we may say the thumbs are on the side of the eye near the nose, and the +remaining fingers on the side away from the nose. + +With this I hope to have made perfectly clear that the pleasing effect +of symmetry is chiefly due to the repetition of sensations, and that +the effect in question takes place in symmetrical figures, only where +there is a repetition of sensation. The pleasing effect of regular +figures, the preference which straight lines, especially vertical and +horizontal straight lines, enjoy, is founded on a similar reason. A +straight line, both in a horizontal and in a vertical position, can cast +on the two retinæ the same image, which falls moreover on symmetrically +corresponding spots. This also, it would appear, is the reason of our +psychological preference of straight to curved lines, and not their +property of being the shortest distance between two points. The straight +line is felt, to put the matter briefly, as symmetrical to itself, which +is the case also with the plane. Curved lines are felt as deviations +from straight lines, that is, as deviations from symmetry.[22] The +presence of a sense for symmetry in people possessing only one eye from +birth, is indeed a riddle. Of course, the sense of symmetry, although +primarily acquired by means of the eyes, cannot be wholly limited to the +visual organs. It must also be deeply rooted in other parts of the +organism by ages of practice and can thus not be eliminated forthwith by +the loss of one eye. Also, when an eye is lost, the symmetrical muscular +apparatus is left, as is also the symmetrical apparatus of innervation. + + +It appears, however, unquestionable that the phenomena mentioned have, +in the main, their origin in the peculiar structure of our eyes. It will +therefore be seen at once that our notions of what is beautiful and ugly +would undergo a change if our eyes were different. Also, if this view is +correct, the theory of the so-called eternally beautiful is somewhat +mistaken. It can scarcely be doubted that our culture, or form of +civilisation, which stamps upon the human body its unmistakable traces, +should not also modify our conceptions of the beautiful. Was not +formerly the development of all musical beauty restricted to the narrow +limits of a five-toned scale? + +The fact that a repetition of sensations is productive of pleasant +effects is not restricted to the realm of the visible. To-day, both the +musician and the physicist know that the harmonic or the melodic +addition of one tone to another affects us agreeably only when the added +tone reproduces a part of the sensation which the first one excited. +When I add an octave to a fundamental tone, I hear in the octave a part +of what was heard in the fundamental tone. (Helmholtz.) But it is not my +purpose to develop this idea fully here.[23] We shall only ask to-day, +whether there is anything similar to the symmetry of figures in the +province of sounds. + +Look at the reflexion of your piano in the mirror. + +You will at once remark that you have never seen such a piano in the +actual world, for it has its high keys to the left and its low ones to +the right. Such pianos are not manufactured. + +If you could sit down at such a piano and play in your usual manner, +plainly every step which you imagined you were performing in the upward +scale would be executed as a corresponding step in the downward scale. +The effect would be not a little surprising. + +For the practised musician who is always accustomed to hearing certain +sounds produced when certain keys are struck, it is quite an anomalous +spectacle to watch a player in the glass and to observe that he always +does the opposite of what we hear. + +But still more remarkable would be the effect of attempting to strike a +harmony on such a piano. For a melody it is not indifferent whether we +execute a step in an upward or a downward scale. But for a harmony, so +great a difference is not produced by reversal. I always retain the same +consonance whether I add to a fundamental note an upper or a lower +third. Only the order of the intervals of the harmony is reversed. In +point of fact, when we execute a movement in a major key on our +reflected piano, we hear a sound in a minor key, and _vice versa_. + +It now remains to execute the experiments indicated. Instead of playing +upon the piano in the mirror, which is impossible, or of having a piano +of this kind built, which would be somewhat expensive, we may perform +our experiments in a simpler manner, as follows: + +1) We play on our own piano in our usual manner, look into the mirror, +and then repeat on our real piano what we see in the mirror. In this way +we transform all steps upwards into corresponding steps downwards. We +play a movement, and then another movement, which, with respect to the +key-board, is symmetrical to the first. + +2) We place a mirror beneath the music in which the notes are reflected +as in a body of water, and play according to the notes in the mirror. In +this way also, all steps upwards are changed into corresponding, equal +steps downwards. + +3) We turn the music upside down and read the notes from right to left +and from below upwards. In doing this, we must regard all sharps as +flats and all flats as sharps, because they correspond to half lines and +spaces. Besides, in this use of the music we can only employ the bass +clef, as only in this clef are the notes not changed by symmetrical +reversal. + +You can judge of the effect of these experiments from the examples which +appear in the annexed musical cut. (Page 102.) The movement which +appears in the upper lines is symmetrically reversed in the lower. + +The effect of the experiments may be briefly formulated. The melody is +rendered unrecognisable. The harmony suffers a transposition from a +major into a minor key and _vice versa_. The study of these pretty +effects, which have long been familiar to physicists and musicians, was +revived some years ago by Von Oettingen.[24] + +[Music: Fig. 26. + +(See pages 101 and 103.)] + +Now, although in all the preceding examples I have transposed steps +upward into equal and similar steps downward, that is, as we may justly +say, have played for every movement the movement which is symmetrical to +it, yet the ear notices either little or nothing of symmetry. The +transposition from a major to a minor key is the sole indication of +symmetry remaining. The symmetry is there for the mind, but is wanting +for sensation. No symmetry exists for the ear, because a reversal of +musical sounds conditions no repetition of sensations. If we had an ear +for height and an ear for depth, just as we have an eye for the right +and an eye for the left, we should also find that symmetrical +sound-structures existed for our auditory organs. The contrast of major +and minor for the ear corresponds to inversion for the eye, which is +also only symmetry for the mind, but not for sensation. + +By way of supplement to what I have said, I will add a brief remark for +my mathematical readers. + +Our musical notation is essentially a graphical representation of a +piece of music in the form of curves, where the time is the abscissæ, +and the logarithms of the number of vibrations the ordinates. The +deviations of musical notation from this principle are only such as +facilitate interpretation, or are due to historical accidents. + +If, now, it be further observed that the sensation of pitch also is +proportional to the logarithm of the number of vibrations, and that the +intervals between the notes correspond to the differences of the +logarithms of the numbers of vibrations, the justification will be found +in these facts of calling the harmonies and melodies which appear in the +mirror, symmetrical to the original ones. + + * * * * * + +I simply wish to bring home to your minds by these fragmentary remarks +that the progress of the physical sciences has been of great help to +those branches of psychology that have not scorned to consider the +results of physical research. On the other hand, psychology is beginning +to return, as it were, in a spirit of thankfulness, the powerful +stimulus which it received from physics. + +The theories of physics which reduce all phenomena to the motion and +equilibrium of smallest particles, the so-called molecular theories, +have been gravely threatened by the progress of the theory of the senses +and of space, and we may say that their days are numbered. + +I have shown elsewhere[25] that the musical scale is simply a species of +space--a space, however, of only one dimension, and that, a one-sided +one. If, now, a person who could only hear, should attempt to develop a +conception of the world in this, his linear space, he would become +involved in many difficulties, as his space would be incompetent to +comprehend the many sides of the relations of reality. But is it any +more justifiable for us, to attempt to force the whole world into the +space of our eye, in aspects in which it is not accessible to the eye? +Yet this is the dilemma of all molecular theories. + +We possess, however, a sense, which, with respect to the scope of the +relations which it can comprehend, is richer than any other. It is our +reason. This stands above the senses. It alone is competent to found a +permanent and sufficient view of the world. The mechanical conception of +the world has performed wonders since Galileo's time. But it must now +yield to a broader view of things. A further development of this idea is +beyond the limits of my present purpose. + +One more point and I have done. The advice of our philosopher to +restrict ourselves to what is near at hand and useful in our researches, +which finds a kind of exemplification in the present cry of inquirers +for limitation and division of labor, must not be too slavishly +followed. In the seclusion of our closets, we often rack our brains in +vain to fulfil a work, the means of accomplishing which lies before our +very doors. If the inquirer must be perforce a shoemaker, tapping +constantly at his last, it may perhaps be permitted him to be a +shoemaker of the type of Hans Sachs, who did not deem it beneath him to +take a look now and then at his neighbor's work and to comment on the +latter's doings. + +Let this be my apology, therefore, if I have forsaken for a moment +to-day the last of my specialty. + + FOOTNOTES: + + [Footnote 19: Delivered before the German Casino of Prague, in the + winter of 1871. + + A fuller treatment of the problems of this lecture will be found in + my _Contributions to the Analysis of the Sensations_ (Jena, 1886), + English Translation, Chicago, 1895. J. P. Soret, _Sur la perception + du beau_ (Geneva, 1892), also regards repetition as a principle of + æsthetics. His discussions of the _æsthetical_ side of the subject + are much more detailed than mine. But with respect to the + psychological and physiological foundation of the principle, I am + convinced that the _Contributions to the Analysis of the Sensations_ + go deeper.--MACH (1894).] + + [Footnote 20: Kant, in his _Prolegomena zu jeder künftigen + Metaphysik_, also refers to this fact, but for a different purpose.] + + [Footnote 21: Compare Mach, _Fichte's Zeitschrift für Philosophie_, + 1864, p. 1.] + + [Footnote 22: The fact that the first and second differential + coefficients of a curve are directly seen, but the higher + coefficients not, is very simply explained. The first gives the + position of the tangent, the declination of the straight line from + the position of symmetry, the second the declination of the curve + from the straight line. It is, perhaps, not unprofitable to remark + here that the ordinary method of testing rulers and plane surfaces + (by reversed applications) ascertains the deviation of the object + from symmetry to itself.] + + [Footnote 23: See the lecture _On the Causes of Harmony_.] + + [Footnote 24: A. von Oettingen, _Harmoniesystem in dualer + Entwicklung_. Leipsic and Dorpat, 1866.] + + [Footnote 25: Compare Mach's _Zur Theorie des Gehörorgans_, Vienna + Academy, 1863.] + + + + +ON THE FUNDAMENTAL CONCEPTS OF ELECTROSTATICS.[26] + + +The task has been assigned me to develop before you in a +popular manner the fundamental quantitative concepts of +electrostatics--"quantity of electricity," "potential," "capacity," +and so forth. It would not be difficult, even within the brief +limits of an hour, to delight the eye with hosts of beautiful +experiments and to fill the imagination with numerous and varied +conceptions. But we should, in such a case, be still far from a +lucid and easy grasp of the phenomena. The means would still fail us +for reproducing the facts accurately in thought--a procedure which +for the theoretical and practical man is of equal importance. These +means are the _metrical concepts_ of electricity. + +As long as the pursuit of the facts of a given province of phenomena +is in the hands of a few isolated investigators, as long as every +experiment can be easily repeated, the fixing of the collected facts +by provisional description is ordinarily sufficient. But the case +is different when the whole world must make use of the results +reached by many, as happens when the science acquires broader +foundations and scope, and particularly so when it begins to supply +intellectual nourishment to an important branch of the practical +arts, and to draw from that province in return stupendous empirical +results. Then the facts must be so described that individuals in all +places and at all times can, from a few easily obtained elements, +put the facts accurately together in thought, and reproduce them +from the description. This is done with the help of the metrical +concepts and the international measures. + +The work which was begun in this direction in the period of the +purely scientific development of the science, especially by Coulomb +(1784), Gauss (1833), and Weber (1846), was powerfully stimulated by +the requirements of the great technical undertakings manifested +since the laying of the first transatlantic cable, and brought to a +brilliant conclusion by the labors of the British Association, 1861, +and of the Paris Congress, 1881, chiefly through the exertions of +Sir William Thomson. + +It is plain, that in the time allotted to me I cannot conduct you +over all the long and tortuous paths which the science has actually +pursued, that it will not be possible at every step to remind you of +all the little precautions for the avoidance of error which the +early steps have taught us. On the contrary, I must make shift with +the simplest and rudest tools. I shall conduct you by the shortest +paths from the facts to the ideas, in doing which, of course, it +will not be possible to anticipate all the stray and chance ideas +which may and must arise from prospects into the by-paths which we +leave untrodden. + + * * * * * + +Here are two small, light bodies (Fig. 27) of equal size, freely +suspended, which we "electrify" either by friction with a third body +or by contact with a body already electrified. At once a repulsive +force is set up which drives the two bodies away from each other in +opposition to the action of gravity. This force could accomplish +anew the same mechanical work which was expended to produce it.[27] + +[Illustration: Fig. 27.] + +[Illustration: Fig. 28.] + +Coulomb, now, by means of delicate experiments with the +torsion-balance, satisfied himself that if the bodies in question, +say at a distance of two centimetres, repelled each other with the +same force with which a milligramme-weight strives to fall to the +ground, at half that distance, or at one centimetre, they would +repel each other with the force of four milligrammes, and at double +that distance, or at four centimetres, they would repel each other +with the force of only one-fourth of a milligramme. He found that +the electrical force acts inversely as the square of the distance. + +Let us imagine, now, that we possessed some means of measuring +electrical repulsion by weights, a means which would be supplied, +for example, by our electrical pendulums; then we could make the +following observation. + +The body _A_ (Fig. 28) is repelled by the body _K_ at a distance of +two centimetres with a force of one milligramme. If we touch _A_, +now, with an equal body _B_, the half of this force of repulsion +will pass to the body _B_; both _A_ and _B_, now, at a distance of +two centimetres from _K_, are repelled only with the force of +one-half a milligramme. But both together are repelled still with +the force of one milligramme. Hence, _the divisibility of electrical +force_ among bodies in contact _is a fact_. It is a useful, but by +no means a necessary supplement to this fact, to imagine an +electrical fluid present in the body _A_, with the quantity of which +the electrical force varies, and half of which flows over to _B_. +For, in the place of the new physical picture, thus, an old, +familiar one is substituted, which moves spontaneously in its wonted +courses. + +Adhering to this idea, we define the _unit_ of electrical +quantity, according to the now almost universally adopted +centimetre-gramme-second (C. G. S.) system, as that quantity which +at a distance of one centimetre repels an equal quantity with unit +of force, that is, with a force which in one second would impart to +a mass of one gramme a velocity-increment of a centimetre. +As a gramme mass acquires through the action of gravity a +velocity-increment of about 981 centimetres in a second, +accordingly, a gramme is attracted to the earth with 981, or, in +round numbers, 1000 units of force of the centimetre-gramme-second +system, while a milligramme-weight would strive to fall to the earth +with approximately the unit force of this system. + +We may easily obtain by this means a clear idea of what the unit +quantity of electricity is. Two small bodies, _K_, weighing each a +gramme, are hung up by vertical threads, five metres in length and +almost weightless, so as to touch each other. If the two bodies be +equally electrified and move apart upon electrification to a +distance of one centimetre, their charge is approximately equivalent +to the electrostatic unit of electric quantity, for the repulsion +then holds in equilibrium a gravitational force-component of +approximately one milligramme, which strives to bring the bodies +together. + +Vertically beneath a small sphere suspended from the equilibrated +beam of a balance a second sphere is placed at a distance of a +centimetre. If both be equally electrified the sphere suspended +from the balance will be rendered apparently lighter by the +repulsion. If by adding a weight of one milligramme equilibrium is +restored, each of the spheres contains in round numbers the +electrostatic unit of electrical quantity. + +In view of the fact that the same electrical bodies exert at +different distances different forces upon one another, exception +might be taken to the measure of quantity here developed. What kind +of a quantity is that which now weighs more, and now weighs less, so +to speak? But this apparent deviation from the method of +determination commonly used in practical life, that by weight, is, +closely considered, an agreement. On a high mountain a heavy mass +also is less powerfully attracted to the earth than at the level of +the sea, and if it is permitted us in our determinations to neglect +the consideration of level, it is only because the comparison of a +body with fixed conventional weights is invariably effected at the +same level. In fact, if we were to make one of the two weights +equilibrated on our balance approach sensibly to the centre of the +earth, by suspending it from a very long thread, as Prof. von Jolly +of Munich suggested, we should make the gravity of that weight, its +heaviness, proportionately greater. + +Let us picture to ourselves, now, two different electrical fluids, a +positive and a negative fluid, of such nature that the particles of +the one attract the particles of the other according to the law of +the inverse squares, but the particles of the same fluid repel each +other by the same law; in non-electrical bodies let us imagine the +two fluids uniformly distributed in equal quantities, in electric +bodies one of the two in excess; in conductors, further, let us +imagine the fluids mobile, in non-conductors immobile; having formed +such pictures, we possess the conception which Coulomb developed and +to which he gave mathematical precision. We have only to give this +conception free play in our minds and we shall see as in a clear +picture the fluid particles, say of a positively charged conductor, +receding from one another as far as they can, all making for the +surface of the conductor and there seeking out the prominent parts +and points until the greatest possible amount of work has been +performed. On increasing the size of the surface, we see a +dispersion, on decreasing its size we see a condensation of the +particles. In a second, non-electrified conductor brought into the +vicinity of the first, we see the two fluids immediately separate, +the positive collecting itself on the remote and the negative on the +adjacent side of its surface. In the fact that this conception +reproduces, lucidly and spontaneously, all the data which arduous +research only slowly and gradually discovered, is contained its +advantage and scientific value. With this, too, its value is +exhausted. We must not seek in nature for the two hypothetical +fluids which we have added as simple mental adjuncts, if we would +not go astray. Coulomb's view may be replaced by a totally +different one, for example, by that of Faraday, and the most proper +course is always, after the general survey is obtained, to go back +to the actual facts, to the electrical forces. + +[Illustration: Fig. 29.] + +[Illustration: Fig. 30.] + +We will now make ourselves familiar with the concept of electrical +quantity, and with the method of measuring or estimating it. Imagine +a common Leyden jar (Fig. 29), the inner and outer coatings of which +are connected together by means of two common metallic knobs placed +about a centimetre apart. If the inside coating be charged with the +quantity of electricity +_q_, on the outer coating a distribution of +the electricities will take place. A positive quantity almost +equal[28] to the quantity +_q_ flows off to the earth, while a +corresponding quantity-_q_ is still left on the outer coating. The +knobs of the jar receive their portion of these quantities and when +the quantity _q_ is sufficiently great a rupture of the insulating +air between the knobs, accompanied by the self-discharge of the +jar, takes place. For any given distance and size of the knobs, a +charge of a definite electric quantity _q_ is always necessary for +the spontaneous discharge of the jar. + +Let us insulate, now, the outer coating of a Lane's unit jar _L_, +the jar just described, and put in connexion with it the inner +coating of a jar _F_ exteriorly connected with the earth (Fig. 30). +Every time that _L_ is charged with +_q_, a like quantity +_q_ is +collected on the inner coating of _F_, and the spontaneous discharge +of the jar _L_, which is now again empty, takes place. The number of +the discharges of the jar _L_ furnishes us, thus, with a measure of +the quantity collected in the jar _F_, and if after 1, 2, 3, ... +spontaneous discharges of _L_ the jar _F_ is discharged, it is +evident that the charge of _F_ has been proportionately augmented. + +[Illustration: Fig. 31.] + +Let us supply now, to effect the spontaneous discharge, the jar _F_ +with knobs of the same size and at the same distance apart as those +of the jar _L_ (Fig. 31). If we find, then, that five discharges of +the unit jar take place before one spontaneous discharge of the jar +_F_ occurs, plainly the jar _F_, for equal distances between the +knobs of the two jars, equal striking distances, is able to hold +five times the quantity of electricity that _L_ can, that is, has +five times the _capacity_ of _L_.[29] + +[Illustration: Fig. 32.] + +We will now replace the unit jar _L_, with which we measure +electricity, so to speak, _into_ the jar _F_, by a Franklin's pane, +consisting of two parallel flat metal plates (Fig. 32), separated +only by air. If here, for example, thirty spontaneous discharges of +the pane are sufficient to fill the jar, ten discharges will be +found sufficient if the air-space between the two plates be filled +with a cake of sulphur. Hence, the capacity of a Franklin's pane of +sulphur is about three times greater than that of one of the same +shape and size made of air, or, as it is the custom to say, the +specific inductive capacity of sulphur (that of air being taken as +the unit) is about 3.[30] We are here arrived at a very simple fact, +which clearly shows us the significance of the number called +_dielectric constant_, or _specific inductive capacity_, the +knowledge of which is so important for the theory of submarine +cables. + +Let us consider a jar _A_, which is charged with a certain quantity +of electricity. We can discharge the jar directly. But we can also +discharge the jar _A_ (Fig. 33) partly into a jar _B_, by connecting +the two outer coatings with each other. In this operation a portion +of the quantity of electricity passes, accompanied by sparks, into +the jar _B_, and we now find both jars charged. + +[Illustration: Fig. 33.] + +[Illustration: Fig. 34.] + +It may be shown as follows that the conception of a constant +quantity of electricity can be regarded as the expression of a pure +fact. Picture to yourself any sort of electrical conductor (Fig. +34); cut it up into a large number of small pieces, and place these +pieces by means of an insulated rod at a distance of one centimetre +from an electrical body which acts with unit of force on an equal +and like-constituted body at the same distance. Take the sum of the +forces which this last body exerts on the single pieces of the +conductor. The sum of these forces will be the quantity of +electricity on the whole conductor. It remains the same, whether we +change the form and the size of the conductor, or whether we bring +it near or move it away from a second electrical conductor, so long +as we keep it insulated, that is, do not discharge it. + +A basis of reality for the notion of electric quantity seems also to +present itself from another quarter. If a current, that is, in the +usual view, a definite quantity of electricity per second, is sent +through a column of acidulated water; in the direction of the +positive stream, hydrogen, but in the opposite direction, oxygen is +liberated at the extremities of the column. For a given quantity of +electricity a given quantity of oxygen appears. You may picture the +column of water as a column of hydrogen and a column of oxygen, +fitted into each other, and may say the electric current is a +chemical current and _vice versa_. Although this notion is more +difficult to adhere to in the field of statical electricity and with +non-decomposable conductors, its further development is by no means +hopeless. + +The concept quantity of electricity, thus, is not so aerial as might +appear, but is able to conduct us with certainty through a multitude +of varied phenomena, and is suggested to us by the facts in almost +palpable form. We can collect electrical force in a body, measure it +out with one body into another, carry it over from one body into +another, just as we can collect a liquid in a vessel, measure it out +with one vessel into another, or pour it from one into another. + +For the analysis of mechanical phenomena, a metrical notion, derived +from experience, and bearing the designation _work_, has proved +itself useful. A machine can be set in motion only when the forces +acting on it can perform work. + +[Illustration: Fig. 35.] + +Let us consider, for example, a wheel and axle (Fig. 35) having the +radii 1 and 2 metres, loaded respectively with the weights 2 and 1 +kilogrammes. On turning the wheel and axle, the 1 kilogramme-weight, +let us say, sinks two metres, while the 2 kilogramme-weight rises +one metre. On both sides the product + +KGR. M. KGR. M. + +1 × 2 = 2 × 1. + +is equal. So long as this is so, the wheel and axle will not move of +itself. But if we take such loads, or so change the radii of the +wheels, that this product (kgr. × metre) on displacement is in +excess on one side, that side will sink. As we see, this product is +characteristic for mechanical events, and for this reason has been +invested with a special name, _work_. + +In all mechanical processes, and as all physical processes present a +mechanical side, in all physical processes, work plays a +determinative part. Electrical forces, also, produce only changes in +which work is performed. To the extent that forces come into play in +electrical phenomena, electrical phenomena, be they what they may, +extend into the domain of mechanics and are subject to the laws +which hold in this domain. The universally adopted measure of work, +now, is the product of the force into the distance through which it +acts, and in the C. G. S. system, the unit of work is the action +through one centimetre of a force which would impart in one second +to a gramme-mass a velocity-increment of one centimetre, that is, in +round numbers, the action through a centimetre of a pressure equal +to the weight of a milligramme. From a positively charged body, +electricity, yielding to the force of repulsion and performing work, +flows off to the earth, providing conducting connexions exist. To a +negatively charged body, on the other hand, the earth under the +same circumstances gives off positive electricity. The electrical +work possible in the interaction of a body with the earth, +characterises the electrical condition of that body. We will call +the work which must be expended on the unit quantity of positive +electricity to raise it from the earth to the body _K_ the +_potential_ of the body _K_.[31] + +We ascribe to the body _K_ in the C. G. S. system the potential +1, +if we must expend the unit of work to raise the positive +electrostatic unit of electric quantity from the earth to that body; +the potential -1, if we gain in this procedure the unit of work; the +potential 0, if no work at all is performed in the operation. + +The different parts of one and the same electrical conductor in +electrical equilibrium have the same potential, for otherwise the +electricity would perform work and move about upon the conductor, +and equilibrium would not have existed. Different conductors of +equal potential, put in connexion with one another, do not exchange +electricity any more than bodies of equal temperature in contact +exchange heat, or in connected vessels, in which the same pressures +exist, liquids flow from one vessel to the other. Exchange of +electricity takes place only between conductors of different +potentials, but in conductors of given form and position a definite +difference of potential is necessary for a spark, which pierces the +insulating air, to pass between them. + +On being connected, every two conductors assume at once the same +potential. With this the means is given of determining the potential +of a conductor through the agency of a second conductor expressly +adapted to the purpose called an electrometer, just as we determine +the temperature of a body with a thermometer. The values of the +potentials of bodies obtained in this way simplify vastly our +analysis of their electrical behavior, as will be evident from what +has been said. + +Think of a positively charged conductor. Double all the electrical +forces exerted by this conductor on a point charged with unit +quantity, that is, double the quantity at each point, or what is the +same thing, double the total charge. Plainly, equilibrium still +subsists. But carry, now, the positive electrostatic unit towards +the conductor. Everywhere we shall have to overcome double the force +of repulsion we did before, everywhere we shall have to expend +double the work. By doubling the charge of the conductor a double +potential has been produced. Charge and potential go hand in hand, +are proportional. Consequently, calling the total quantity of +electricity of a conductor _Q_ and its potential _V_, we can write: +_Q = CV_, where _C_ stands for a constant, the import of which will +be understood simply from noting that _C = Q/V_.[32] But the +division of a number representing the units of quantity of a +conductor by the number representing its units of potential tells us +the quantity which falls to the share of the unit of potential. Now +the number _C_ here we call the capacity of a conductor, and have +substituted, thus, in the place of the old relative determination of +capacity, an absolute determination.[33] + +In simple cases the connexion between charge, potential, and +capacity is easily ascertained. Our conductor, let us say, is a +sphere of radius _r_, suspended free in a large body of air. There +being no other conductors in the vicinity, the charge _q_ will then +distribute itself uniformly upon the surface of the sphere, and +simple geometrical considerations yield for its potential the +expression _V = q/r_. Hence, _q/V = r_; that is, the capacity of a +sphere is measured by its radius, and in the C. G. S. system in +centimetres.[34] It is clear also, since a potential is a quantity +divided by a length, that a quantity divided by a potential must be +a length. + +Imagine (Fig. 36) a jar composed of two concentric conductive +spherical shells of the radii _r_ and _r₁_, having only air between +them. Connecting the outside sphere with the earth, and charging the +inside sphere by means of a thin, insulated wire passing through the +first, with the quantity _Q_, we shall have _V = (r₁-r)/(r₁r)Q_, and +for the capacity in this case _(r₁r)/(r₁-r)_, or, to take a specific +example, if _r = 16_ and _r₁ = 19_, a capacity of about 100 +centimetres. + +[Illustration: Fig. 36.] + +We shall now use these simple cases for illustrating the principle +by which capacity and potential are determined. First, it is clear +that we can use the jar composed of concentric spheres with its +known capacity as our unit jar and by means of this ascertain, in +the manner above laid down, the capacity of any given jar _F_. We +find, for example, that 37 discharges of this unit jar of the +capacity 100, just charges the jar investigated at the same +striking distance, that is, at the same potential. Hence, the +capacity of the jar investigated is 3700 centimetres. The large +battery of the Prague physical laboratory, which consists of sixteen +such jars, all of nearly equal size, has a capacity, therefore, of +something like 50,000 centimetres, or the capacity of a sphere, a +kilometre in diameter, freely suspended in atmospheric space. This +remark distinctly shows us the great superiority which Leyden jars +possess for the storage of electricity as compared with common +conductors. In fact, as Faraday pointed out, jars differ from simple +conductors mainly by their great capacity. + +[Illustration: Fig. 37.] + +For determining potential, imagine the inner coating of a jar _F_, +the outer coating of which communicates with the ground, connected +by a long, thin wire with a conductive sphere _K_ placed free in a +large atmospheric space, compared with whose dimensions the radius +of the sphere vanishes. (Fig. 37.) The jar and the sphere assume at +once the same potential. But on the surface of the sphere, if that +be sufficiently far removed from all other conductors, a uniform +layer of electricity will be found. If the sphere, having the radius +_r_, contains the charge _q_, its potential is _V = q/r_. If the +upper half of the sphere be severed from the lower half and +equilibrated on a balance with one of whose beams it is connected by +silk threads, the upper half will be repelled from the lower half +with the force _P = q²/8r² = 1/8V²_. This repulsion _P_ may be +counter-balanced by additional weights placed on the beam-end, and +so ascertained. The potential is then _V = [sqrt](8P)_.[35] + +That the potential is proportional to the square root of the force +is not difficult to see. A doubling or trebling of the potential +means that the charge of all the parts is doubled or trebled; hence +their combined power of repulsion quadrupled or nonupled. + +Let us consider a special case. I wish to produce the potential 40 +on the sphere. What additional weight must I give to the half sphere +in grammes that the force of repulsion shall maintain the balance in +exact equilibrium? As a gramme weight is approximately equivalent +to 1000 units of force, we have only the following simple example to +work out: _40×40 = 8× 1000.x_, where _x_ stands for the number of +grammes. In round numbers we get _x_ = 0.2 gramme. I charge the jar. +The balance is deflected; I have reached, or rather passed, the +potential 40, and you see when I discharge the jar the associated +spark.[36] + +The striking distance between the knobs of a machine increases with +the difference of the potential, although not proportionately to +that difference. The striking distance increases faster than the +potential difference. For a distance between the knobs of one +centimetre on this machine the difference of potential is 110. It +can easily be increased tenfold. Of the tremendous differences of +potential which occur in nature some idea may be obtained from the +fact that the striking distances of lightning in thunder-storms is +counted by miles. The differences of potential in galvanic batteries +are considerably smaller than those of our machine, for it takes +fully one hundred elements to give a spark of microscopic striking +distance. + + * * * * * + +We shall now employ the ideas reached to shed some light upon +another important relation between electrical and mechanical +phenomena. We shall investigate what is the potential _energy_, or +the _store of work_, contained in a charged conductor, for example, +in a jar. + +If we bring a quantity of electricity up to a conductor, or, to +speak less pictorially, if we generate by work electrical force in a +conductor, this force is able to produce anew the work by which it +was generated. How great, now, is the energy or capacity for work of +a conductor of known charge _Q_ and known potential _V_? + +Imagine the given charge _Q_ divided into very small parts _q_, +_q₁_, _q₂_ ..., and these little parts successively carried up to +the conductor. The first very small quantity _q_ is brought up +without any appreciable work and produces by its presence a small +potential _V__{'}. To bring up the second quantity, accordingly, we +must do the work _q__{'}_V__{'}, and similarly for the quantities +which follow the work _q__{''}_V__{''}, _q__{'''}_V__{'''}, and so +forth. Now, as the potential rises proportionately to the quantities +added until the value _V_ is reached, we have, agreeably to the +graphical representation of Fig. 38, for the total work performed, + +_W = 1/2QV_, + +which corresponds to the total energy of the charged conductor. +Using the equation _Q_ = _CV_, where _C_ stands for capacity, we +also have, + +_W = 1/2CV²_, or _W = Q²/2C_. + +It will be helpful, perhaps, to elucidate this idea by an analogy +from the province of mechanics. If we pump a quantity of liquid, +_Q_, gradually into a cylindrical vessel (Fig. 39), the level of the +liquid in the vessel will gradually rise. The more we have pumped +in, the greater the pressure we must overcome, or the higher the +level to which we must lift the liquid. The stored-up work is +rendered again available when the heavy liquid _Q_, which reaches up +to the level _h_, flows out. This work _W_ corresponds to the fall +of the whole liquid weight _Q_, through the distance _h_/2 or +through the altitude of its centre of gravity. We have + +_W = 1/2Qh_. + +Further, since _Q_ = _Kh_, or since the weight of the liquid and the +height _h_ are proportional, we get also + +_W = 1/2Kh²_ and _W = Q²/2K_. + +[Illustration: Fig. 38.] + +[Illustration: Fig. 39.] + +As a special case let us consider our jar. Its capacity is _C_ = +3700, its potential _V_ = 110; accordingly, its quantity _Q = CV_ = +407,000 electrostatic units and its energy _W = 1/2QV_ = 22,385,000 +C. G. S. units of work. + +The unit of work of the C. G. S. system is not readily appreciable +by the senses, nor does it well admit of representation, as we are +accustomed to work with weights. Let us adopt, therefore, as our +unit of work the gramme-centimetre, or the gravitational pressure of +a gramme-weight through the distance of a centimetre, which in round +numbers is 1000 times greater than the unit assumed above; in this +case, our numerical result will be approximately 1000 times smaller. +Again, if we pass, as more familiar in practice, to the +kilogramme-metre as our unit of work, our unit, the distance being +increased a hundred fold, and the weight a thousand fold, will be +100,000 times larger. The numerical result expressing the work done +is in this case 100,000 times less, being in round numbers 0.22 +kilogramme-metre. We can obtain a clear idea of the work done here +by letting a kilogramme-weight fall 22 centimetres. + +This amount of work, accordingly, is performed on the charging of +the jar, and on its discharge appears again, according to the +circumstances, partly as sound, partly as a mechanical disruption of +insulators, partly as light and heat, and so forth. + +The large battery of the Prague physical laboratory, with its +sixteen jars charged to equal potentials, furnishes, although the +effect of the discharge is imposing, a total amount of work of only +three kilogramme-metres. + +In the development of the ideas above laid down we are not +restricted to the method there pursued; in fact, that method was +selected only as one especially fitted to familiarise us with the +phenomena. On the contrary, the connexion of the physical processes +is so multifarious that we can come at the same event from very +different directions. Particularly are electrical phenomena +connected with all other physical events; and so intimate is this +connexion that we might justly call the study of electricity the +theory of the general connexion of physical processes. + +With respect to the principle of the conservation of energy which +unites electrical with mechanical phenomena, I should like to point +out briefly two ways of following up the study of this connexion. + +A few years ago Professor Rosetti, taking an influence-machine, +which he set in motion by means of weights alternately in the +electrical and non-electrical condition with the same velocities, +determined the mechanical work expended in the two cases and was +thus enabled, after deducting the work of friction, to ascertain the +mechanical work consumed in the development of the electricity. + +I myself have made this experiment in a modified, and, as I think, +more advantageous form. Instead of determining the work of friction +by special trial, I arranged my apparatus so that it was eliminated +of itself in the measurement and could consequently be neglected. +The so-called fixed disk of the machine, the axis of which is +placed vertically, is suspended somewhat like a chandelier by three +vertical threads of equal lengths _l_ at a distance _r_ from the +axis. Only when the machine is excited does this fixed disk, which +represents a Prony's brake, receive, through its reciprocal action +with the rotating disk, a deflexion _[alpha]_ and a moment of +torsion which is expressed by _D = (Pr²/l)[alpha]_, where _P_ is the +weight of the disk.[37] The angle _[alpha]_ is determined by a +mirror set in the disk. The work expended in _n_ rotations is given +by _2n[pi]D_. + +If we close the machine, as Rosetti did, we obtain a continuous +current which has all the properties of a very weak galvanic +current; for example, it produces a deflexion in a multiplier which +we interpose, and so forth. We can directly ascertain, now, the +mechanical work expended in the maintenance of this current. + +If we charge a jar by means of a machine, the energy of the jar +employed in the production of sparks, in the disruption of the +insulators, etc., corresponds to a part only of the mechanical work +expended, a second part of it being consumed in the arc which forms +the circuit.[38] This machine, with the interposed jar, affords in +miniature a picture of the transference of force, or more properly +of work. And in fact nearly the same laws hold here for the +economical coefficient as obtain for large dynamo-machines. + +Another means of investigating electrical energy is by its +transformation into heat. A long time ago (1838), before the +mechanical theory of heat had attained its present popularity, Riess +performed experiments in this field with the help of his electrical +air-thermometer or thermo-electrometer. + +[Illustration: Fig. 40.] + +If the discharge be conducted through a fine wire passing through +the globe of the air-thermometer, a development of heat is observed +proportional to the expression above-discussed _W = 1/2QV_. Although +the total energy has not yet been transformed into measurable heat +by this means, in as much as a portion is left behind in the spark +in the air outside the thermometer, still everything tends to show +that the total heat developed in all parts of the conductor and +along all the paths of discharge is the equivalent of the work +1/2_QV_. + +It is not important here whether the electrical energy is +transformed all at once or partly, by degrees. For example, if of +two equal jars one is charged with the quantity _Q_ at the potential +_V_ the energy present is 1/2_QV_. If the first jar be discharged +into the second, _V_, since the capacity is now doubled, falls to +_V_/2. Accordingly, the energy 1/4_QV_ remains, while 1/4_QV_ is +transformed in the spark of discharge into heat. The remainder, +however, is equally distributed between the two jars so that each on +discharge is still able to transform 1/8_QV_ into heat. + + * * * * * + +We have here discussed electricity in the limited phenomenal form in +which it was known to the inquirers before Volta, and which has been +called, perhaps not very felicitously, "statical electricity." It is +evident, however, that the nature of electricity is everywhere one +and the same; that a substantial difference between statical and +galvanic electricity does not exist. Only the quantitative +circumstances in the two provinces are so widely different that +totally new aspects of phenomena may appear in the second, for +example, magnetic effects, which in the first remained unnoticed, +whilst, _vice versa_, in the second field statical attractions and +repulsions are scarcely appreciable. As a fact, we can easily +show the magnetic effect of the current of discharge of an +influence-machine on the galvanoscope although we could hardly have +made the original discovery of the magnetic effects with this +current. The statical distant action of the wire poles of a galvanic +element also would hardly have been noticed had not the phenomenon +been known from a different quarter in a striking form. + +If we wished to characterise the two fields in their chief and most +general features, we should say that in the first, high potentials +and small quantities come into play, in the second small potentials +and large quantities. A jar which is discharging and a galvanic +element deport themselves somewhat like an air-gun and the bellows +of an organ. The first gives forth suddenly under a very high +pressure a small quantity of air; the latter liberates gradually +under a very slight pressure a large quantity of air. + +In point of principle, too, nothing prevents our retaining the +electrostatical units in the domain of galvanic electricity and in +measuring, for example, the strength of a current by the number of +electrostatic units which flow per second through its cross-section. +But this would be in a double aspect impractical. In the first +place, we should totally neglect the magnetic facilities for +measurement so conveniently offered by the current, and substitute +for this easy means a method which can be applied only with +difficulty and is not capable of great exactness. In the second +place our units would be much too small, and we should find +ourselves in the predicament of the astronomer who attempted to +measure celestial distances in metres instead of in radii of the +earth and the earth's orbit; for the current which by the magnetic +C. G. S. standard represents the unit, would require a flow of some +30,000,000,000 electrostatic units per second through its +cross-section. Accordingly, different units must be adopted here. +The development of this point, however, lies beyond my present +task. + + FOOTNOTES: + + [Footnote 26: A lecture delivered at the International Electrical + Exhibition, in Vienna, on September 4, 1883.] + + [Footnote 27: If the two bodies were oppositely electrified they + would exert attractions upon each other.] + + [Footnote 28: The quantity which flows off is in point of fact less + than _q_. It would be equal to the quantity _q_ only if the inner + coating of the jar were wholly encompassed by the outer coating.] + + [Footnote 29: Rigorously, of course, this is not correct. First, it + is to be noted that the jar _L_ is discharged simultaneously with + the electrode of the machine. The jar _F_, on the other hand, is + always discharged simultaneously with the outer coating of the jar + _L_. Hence, if we call the capacity of the electrode of the machine + _E_, that of the unit jar _L_, that of the outer coating of _L_, + _A_, and that of the principal jar _F_, then this equation would + exist for the example in the text: _(F + A)/(L + E) = 5_. A cause of + further departure from absolute exactness is the residual charge.] + + [Footnote 30: Making allowance for the corrections indicated in the + preceding footnote, I have obtained for the dielectric constant of + sulphur the number 3.2, which agrees practically with the results + obtained by more delicate methods. For the highest attainable + precision one should by rights immerse the two plates of the + condenser first wholly in air and then wholly in sulphur, if the + ratio of the capacities is to correspond to the dielectric constant. + In point of fact, however, the error which arises from inserting + simply a plate of sulphur that exactly fills the space between the + two plates, is of no consequence.] + + [Footnote 31: As this definition in its simple form is apt to give + rise to misunderstandings, elucidations are usually added to it. It + is clear that we cannot lift a quantity of electricity to _K_, + without changing the distribution on _K_ and the potential on _K_. + Hence, the charges on _K_ must be conceived as fixed, and so small a + quantity raised that no appreciable change is produced by it. Taking + the work thus expended as many times as the small quantity in + question is contained in the unit of quantity, we shall obtain the + potential. The potential of a body _K_ may be briefly and precisely + defined as follows: If we expend the element of work _dW_ to raise + the element of positive quantity _dQ_ from the earth to the + conductor, the potential of a conductor _K_ will be given by _V = + dW/dQ_.] + + [Footnote 32: In this article the solidus or slant stroke is used + for the usual fractional sign of division. Where plus or minus signs + occur in the numerator or denominator, brackets or a vinculum is + used.--_Tr._] + + [Footnote 33: A sort of agreement exists between the notions of + thermal and electrical capacity, but the difference between the two + ideas also should be carefully borne in mind. The thermal capacity + of a body depends solely upon that body itself. The electrical + capacity of a body _K_ is influenced by all bodies in its vicinity, + inasmuch as the charge of these bodies is able to alter the + potential of _K_. To give, therefore, an unequivocal significance to + the notion of the capacity (_C_) of a body _K_, _C_ is defined as + the relation _Q_/_V_ for the body _K_ in a certain given position of + all neighboring bodies, and during connexion of all neighboring + conductors with the earth. In practice the situation is much + simpler. The capacity, for example, of a jar, the inner coating of + which is almost enveloped by its outer coating, communicating with + the ground, is not sensibly affected by charged or uncharged + adjacent conductors.] + + [Footnote 34: These formulæ easily follow from Newton's theorem that + a homogeneous spherical shell, whose elements obey the law of the + inverse squares, exerts no force whatever on points within it but + acts on points without as if the whole mass were concentrated at its + centre. The formulæ next adduced also flow from this proposition.] + + [Footnote 35: The energy of a sphere of radius _r_ charged with the + quantity _q_ is 1/2(_q_²/_r_). If the radius increase by the space + _dr_ a loss of energy occurs, and the work done is + 1/2(_q_²/_r_²)_dr_. Letting _p_ denote the uniform electrical + pressure on unit of surface of the sphere, the work done is also + 4_r_²[pi]_pdr_. Hence _p = (1/8r²[pi])(q²/r²)_. Subjected to the + same superficial pressure on all sides, say in a fluid, our half + sphere would be an equilibrium. Hence we must make the pressure _p_ + act on the surface of the great circle to obtain the effect on the + balance, which is _r²[pi]p = 1/8(q²/r²) = 1/8V²_.] + + [Footnote 36: The arrangement described is for several reasons not + fitted for the actual measurement of potential. Thomson's absolute + electrometer is based upon an ingenious modification of the + electrical balance of Harris and Volta. Of two large plane parallel + plates, one communicates with the earth, while the other is brought + to the potential to be measured. A small movable superficial portion + _f_ of this last hangs from the balance for the determination of the + attraction _P_. The distance of the plates from each other being _D_ + we get _V = D[sqrt](8[pi]P/f)_.] + + [Footnote 37: This moment of torsion needs a supplementary + correction, on account of the vertical electric attraction of the + excited disks. This is done by changing the weight of the disk by + means of additional weights and by making a second reading of the + angles of deflexion.] + + [Footnote 38: The jar in our experiment acts like an accumulator, + being charged by a dynamo machine. The relation which obtains + between the expended and the available work may be gathered from the + following simple exposition. A Holtz machine _H_ (Fig. 40) is + charging a unit jar _L_, which after _n_ discharges of quantity _q_ + and potential _v_, charges the jar _F_ with the quantity _Q_ at the + potential _V_. The energy of the unit-jar discharges is lost and + that of the jar _F_ alone is left. Hence the ratio of the available + work to the total work expended is + +_½QV/[½QV + (n/2)qv]_ and as _Q = nq_, also _V/(V + v)_. + + If, now, we interpose no unit jar, still the parts of the machine + and the wires of conduction are themselves virtually such unit jars + and the formula still subsists _V/(V + [sum]v)_, in which [sum]_v_ + represents the sum of all the successively introduced differences of + potential in the circuit of connexion.] + + + + +ON THE PRINCIPLE OF THE CONSERVATION OF ENERGY.[39] + + +In a popular lecture, distinguished for its charming simplicity and +clearness, which Joule delivered in the year 1847,[40] that famous +physicist declares that the living force which a heavy body has acquired +by its descent through a certain height and which it carries with it in +the form of the velocity with which it is impressed, is the _equivalent_ +of the attraction of gravity through the space fallen through, and that +it would be "absurd" to assume that this living force could be destroyed +without some restitution of that equivalent. He then adds: "You will +therefore be surprised to hear that until very _recently_ the universal +opinion has been that living force could be absolutely and irrevocably +destroyed at any one's option." Let us add that to-day, after +forty-seven years, the _law of the conservation of energy_, wherever +civilisation exists, is accepted as a fully established truth and +receives the widest applications in all domains of natural science. + +The fate of all momentous discoveries is similar. On their first +appearance they are regarded by the majority of men as errors. J. R. +Mayer's work on the principle of energy (1842) was rejected by the first +physical journal of Germany; Helmholtz's treatise (1847) met with no +better success; and even Joule, to judge from an intimation of Playfair, +seems to have encountered difficulties with his first publication +(1843). Gradually, however, people are led to see that the new view was +long prepared for and ready for enunciation, only that a few favored +minds had perceived it much earlier than the rest, and in this way the +opposition of the majority is overcome. With proofs of the fruitfulness +of the new view, with its success, confidence in it increases. The +majority of the men who employ it cannot enter into a deep-going +analysis of it; for them, its success is its proof. It can thus happen +that a view which has led to the greatest discoveries, like Black's +theory of caloric, in a subsequent period in a province where it does +not apply may actually become an obstacle to progress by its blinding +our eyes to facts which do not fit in with our favorite conceptions. If +a theory is to be protected from this dubious rôle, the grounds and +motives of its evolution and existence must be examined from time to +time with the utmost care. + +The most multifarious physical changes, thermal, electrical, chemical, +and so forth, can be brought about by mechanical work. When such +alterations are reversed they yield anew the mechanical work in exactly +the quantity which was required for the production of the part reversed. +This is the _principle of the conservation of energy_; "energy" being +the term which has gradually come into use for that "indestructible +something" of which the measure is mechanical _work_. + +How did we acquire this idea? What are the sources from which we have +drawn it? This question is not only of interest in itself, but also for +the important reason above touched upon. The opinions which are held +concerning the foundations of the law of energy still diverge very +widely from one another. Many trace the principle to the impossibility +of a perpetual motion, which they regard either as sufficiently proved +by experience, or as self-evident. In the province of pure mechanics the +impossibility of a perpetual motion, or the continuous production of +_work_ without some _permanent_ alteration, is easily demonstrated. +Accordingly, if we start from the theory that all physical processes are +purely _mechanical_ processes, motions of molecules and atoms, we +embrace also, by this _mechanical_ conception of physics, the +impossibility of a perpetual motion in the _whole_ physical domain. At +present this view probably counts the most adherents. Other inquirers, +however, are for accepting only a purely _experimental_ establishment of +the law of energy. + +It will appear, from the discussion to follow, that _all_ the factors +mentioned have co-operated in the development of the view in question; +but that in addition to them a logical and purely formal factor, +hitherto little considered, has also played a very important part. + + +I. THE PRINCIPLE OF THE EXCLUDED PERPETUAL MOTION. + +The law of energy in its modern form is not identical with the principle +of the excluded perpetual motion, but it is very closely related to it. +The latter principle, however, is by no means new, for in the province +of mechanics it has controlled for centuries the thoughts and +investigations of the greatest thinkers. Let us convince ourselves of +this by the study of a few historical examples. + +[Illustration: Fig. 41.] + +S. Stevinus, in his famous work _Hypomnemata mathematica_, Tom. IV, _De +statica_, (Leyden, 1605, p. 34), treats of the equilibrium of bodies on +inclined planes. + +Over a triangular prism _ABC_, one side of which, _AC_, is horizontal, +an endless cord or chain is slung, to which at equal distances apart +fourteen balls of equal weight are attached, as represented in +cross-section in Figure 41. Since we can imagine the lower symmetrical +part of the cord _ABC_ taken away, Stevinus concludes that the four +balls on _AB_ hold in equilibrium the two balls on _BC_. For if the +equilibrium were for a moment disturbed, it could never subsist: the +cord would keep moving round forever in the same direction,--we should +have a perpetual motion. He says: + + "But if this took place, our row or ring of balls would come once + more into their original position, and from the same cause the + eight globes to the left would again be heavier than the six to the + right, and therefore those eight would sink a second time and these + six rise, and all the globes would keep up, of themselves, _a + continuous and unending motion, which is false_."[41] + +Stevinus, now, easily derives from this principle the laws of +equilibrium on the inclined plane and numerous other fruitful +consequences. + +In the chapter "Hydrostatics" of the same work, page 114, Stevinus sets +up the following principle: "Aquam datam, datum sibi intra aquam locum +servare,"--a given mass of water preserves within water its given place. + +[Illustration: Fig. 42.] + +This principle is demonstrated as follows (see Fig. 42): + + "For, assuming it to be possible by natural means, let us suppose + that A does not preserve the place assigned to it, but sinks down + to D. This being posited, the water which succeeds A will, for the + same reason, also flow down to _D_; _A_ will be forced out of its + place in _D_; and thus this body of water, for the conditions in it + are everywhere the same, _will set up a perpetual motion, which is + absurd_."[42] + +From this all the principles of hydrostatics are deduced. On this +occasion Stevinus also first develops the thought so fruitful for modern +analytical mechanics that the equilibrium of a system is not destroyed +by the addition of rigid connexions. As we know, the principle of the +conservation of the centre of gravity is now sometimes deduced from +D'Alembert's principle with the help of that remark. If we were to +reproduce Stevinus's demonstration to-day, we should have to change it +slightly. We find no difficulty in imagining the cord on the prism +possessed of unending uniform motion if all hindrances are thought away, +but we should protest against the assumption of an accelerated motion or +even against that of a uniform motion, if the resistances were not +removed. Moreover, for greater precision of proof, the string of balls +might be replaced by a heavy homogeneous cord of infinite flexibility. +But all this does not affect in the least the historical value of +Stevinus's thoughts. It is a fact, Stevinus deduces apparently much +simpler truths from the principle of an impossible perpetual motion. + +In the process of thought which conducted Galileo to his discoveries at +the end of the sixteenth century, the following principle plays an +important part, that a body in virtue of the velocity acquired in its +descent can rise exactly as high as it fell. This principle, which +appears frequently and with much clearness in Galileo's thought, is +simply another form of the principle of excluded perpetual motion, as we +shall see it is also in Huygens. + +Galileo, as we know, arrived at the law of uniformly accelerated motion +by _a priori_ considerations, as that law which was the "simplest and +most natural," after having first assumed a different law which he was +compelled to reject. To verify his law he executed experiments with +falling bodies on inclined planes, measuring the times of descent by the +weights of the water which flowed out of a small orifice in a large +vessel. In this experiment he assumes as a fundamental principle, that +the velocity acquired in descent down an inclined plane always +corresponds to the vertical height descended through, a conclusion which +for him is the immediate outcome of the fact that a body which has +fallen down one inclined plane can, with the velocity it has acquired, +rise on another plane of any inclination only to the same vertical +height. This principle of the height of ascent also led him, as it +seems, to the law of inertia. Let us hear his own masterful words in the +_Dialogo terzo_ (_Opere_, Padova, 1744, Tom. III). On page 96 we read: + + "I take it for granted that the velocities acquired by a body in + descent down planes of different inclinations are equal if the + heights of those planes are equal."[43] + +Then he makes Salviati say in the dialogue:[44] + + "What you say seems very probable, but I wish to go further and by + an experiment so to increase the probability of it that it shall + amount almost to absolute demonstration. Suppose this sheet of + paper to be a vertical wall, and from a nail driven in it a ball of + lead weighing two or three ounces to hang by a very fine thread + _AB_ four or five feet long. (Fig. 43.) On the wall mark a + horizontal line _DC_ perpendicular to the vertical _AB_, which + latter ought to hang about two inches from the wall. If now the + thread _AB_ with the ball attached take the position _AC_ and the + ball be let go, you will see the ball first descend through the arc + _CB_ and passing beyond _B_ rise through the arc _BD_ almost to the + level of the line _CD_, being prevented from reaching it exactly by + the resistance of the air and of the thread. From this we may truly + conclude that its impetus at the point _B_, acquired by its descent + through the arc _CB_, is sufficient to urge it through a similar + arc _BD_ to the same height. Having performed this experiment and + repeated it several times, let us drive in the wall, in the + projection of the vertical _AB_, as at _E_ or at _F_, a nail five + or six inches long, so that the thread _AC_, carrying as before the + ball through the arc _CB_, at the moment it reaches the position + _AB_, shall strike the nail _E_, and the ball be thus compelled to + move up the arc _BG_ described about _E_ as centre. Then we shall + see what the same impetus will here accomplish, acquired now as + before at the same point _B_, which then drove the same moving body + through the arc _BD_ to the height of the horizontal _CD_. Now + gentlemen, you will be pleased to see the ball rise to the + horizontal line at the point _G_, and the same thing also happen if + the nail be placed lower as at _F_, in which case the ball would + describe the arc _BJ_, always terminating its ascent precisely at + the line _CD_. If the nail be placed so low that the length of + thread below it does not reach to the height of _CD_ (which would + happen if _F_ were nearer _B_ than to the intersection of _AB_ with + the horizontal _CD_), then the thread will wind itself about the + nail. This experiment leaves no room for doubt as to the truth of + the supposition. For as the two arcs _CB_, _DB_ are equal and + similarly situated, the momentum acquired in the descent of the arc + _CB_ is the same as that acquired in the descent of the arc _DB_; + but the momentum acquired at _B_ by the descent through the arc + _CB_ is capable of driving up the same moving body through the arc + _BD_; hence also the momentum acquired in the descent _DB_ is equal + to that which drives the same moving body through the same arc from + _B_ to _D_, so that in general every momentum acquired in the + descent of an arc is equal to that which causes the same moving + body to ascend through the same arc; but all the momenta which + cause the ascent of all the arcs _BD_, _BG_, _BJ_, are equal since + they are made by the same momentum acquired in the descent _CB_, as + the experiment shows: therefore all the momenta acquired in the + descent of the arcs _DB_, _GB_, _JB_ are equal." + +[Illustration: Fig. 43.] + +The remark relative to the pendulum may be applied to the inclined plane +and leads to the law of inertia. We read on page 124:[45] + + "It is plain now that a movable body, starting from rest at _A_ and + descending down the inclined plane _AB_, acquires a velocity + proportional to the increment of its time: the velocity possessed + at _B_ is the greatest of the velocities acquired, and by its + nature immutably impressed, provided all causes of new acceleration + or retardation are taken away: I say acceleration, having in view + its possible further progress along the plane extended; + retardation, in view of the possibility of its being reversed and + made to mount the ascending plane _BC_. But in the horizontal plane + _GH_ its equable motion, according to its velocity as acquired in + the descent from _A_ to _B_, will be continued _ad infinitum_." + (Fig. 44.) + +[Illustration: Fig. 44.] + +Huygens, upon whose shoulders the mantel of Galileo fell, forms a +sharper conception of the law of inertia and generalises the principle +respecting the heights of ascent which was so fruitful in Galileo's +hands. He employs the latter principle in the solution of the problem of +the centre of oscillation and is perfectly clear in the statement that +the principle respecting the heights of ascent is identical with the +principle of the excluded perpetual motion. + +The following important passages then occur (Hugenii, _Horologium +oscillatorium, pars secunda_). _Hypotheses_: + + "If gravity did not exist, nor the atmosphere obstruct the motions + of bodies, a body would keep up forever the motion once impressed + upon it, with equable velocity, in a straight line."[46] + +In part four of the _Horologium de centro oscillationis_ we read: + + "If any number of weights be set in motion by the force of gravity, + the common centre of gravity of the weights as a whole cannot + possibly rise higher than the place which it occupied when the + motion began. + + "That this hypothesis of ours may arouse no scruples, we will state + that it simply imports, what no one has ever denied, that heavy + bodies do not move _upwards_.--And truly if the devisers of the new + machines who make such futile attempts to construct a perpetual + motion would acquaint themselves with this principle, they could + easily be brought to see their errors and to understand that the + thing is utterly impossible by mechanical means."[47] + +There is possibly a Jesuitical mental reservation contained in the words +"mechanical means." One might be led to believe from them that Huygens +held a non-mechanical perpetual motion for possible. + +The generalisation of Galileo's principle is still more clearly put in +Prop. IV of the same chapter: + + "If a pendulum, composed of several weights, set in motion from + rest, complete any part of its full oscillation, and from that + point onwards, the individual weights, with their common connexions + dissolved, change their acquired velocities upwards and ascend as + far as they can, the common centre of gravity of all will be + carried up to the same altitude with that which it occupied before + the beginning of the oscillation."[48] + +On this last principle now, which is a generalisation, applied to a +system of masses, of one of Galileo's ideas respecting a single mass and +which from Huygens's explanation we recognise as the principle of +excluded perpetual motion, Huygens grounds his theory of the centre of +oscillation. Lagrange characterises this principle as precarious and is +rejoiced at James Bernoulli's successful attempt, in 1681, to reduce the +theory of the centre of oscillation to the laws of the lever, which +appeared to him clearer. All the great inquirers of the seventeenth and +eighteenth centuries broke a lance on this problem, and it led +ultimately, in conjunction with the principle of virtual velocities, to +the principle enunciated by D'Alembert in 1743 in his _Traité de +dynamique_, though previously employed in a somewhat different form by +Euler and Hermann. + +Furthermore, the Huygenian principle respecting the heights of ascent +became the foundation of the "law of the conservation of living force," +as that was enunciated by John and Daniel Bernoulli and employed with +such signal success by the latter in his _Hydrodynamics_. The theorems +of the Bernoullis differ in form only from Lagrange's expression in the +_Analytical Mechanics_. + +The manner in which Torricelli reached his famous law of efflux for +liquids leads again to our principle. Torricelli assumed that the liquid +which flows out of the basal orifice of a vessel cannot by its velocity +of efflux ascend to a greater height than its level in the vessel. + +Let us next consider a point which belongs to pure mechanics, the +history of the principle of _virtual motions_ or _virtual velocities_. +This principle was not first enunciated, as is usually stated, and as +Lagrange also asserts, by Galileo, but earlier, by Stevinus. In his +_Trochleostatica_ of the above-cited work, page 72, he says: + + "Observe that this axiom of statics holds good here: + + "As the space of the body acting is to the space of the body acted + upon, so is the power of the body acted upon to the power of the + body acting."[49] + +Galileo, as we know, recognised the truth of the principle in the +consideration of the simple machines, and also deduced the laws of the +equilibrium of liquids from it. + +Torricelli carries the principle back to the properties of the centre of +gravity. The condition controlling equilibrium in a simple machine, in +which power and load are represented by weights, is that the common +centre of gravity of the weights shall not sink. Conversely, if the +centre of gravity cannot sink equilibrium obtains, because heavy bodies +of themselves do not move upwards. In this form the principle of virtual +velocities is identical with Huygens's principle of the impossibility of +a perpetual motion. + +John Bernoulli, in 1717, first perceived the universal import of the +principle of virtual movements for all systems; a discovery stated in a +letter to Varignon. Finally, Lagrange gives a general demonstration of +the principle and founds upon it his whole _Analytical Mechanics_. But +this general demonstration is based after all upon Huygens and +Torricelli's remarks. Lagrange, as is known, conceives simple pulleys +arranged in the directions of the forces of the system, passes a cord +through these pulleys, and appends to its free extremity a weight which +is a common measure of all the forces of the system. With no difficulty, +now, the number of elements of each pulley may be so chosen that the +forces in question shall be replaced by them. It is then clear that if +the weight at the extremity cannot sink, equilibrium subsists, because +heavy bodies cannot of themselves move upwards. If we do not go so far, +but wish to abide by Torricelli's idea, we may conceive every individual +force of the system replaced by a special weight suspended from a cord +passing over a pulley in the direction of the force and attached at its +point of application. Equilibrium subsists then when the common centre +of gravity of all the weights together cannot sink. The fundamental +supposition of this demonstration is plainly the impossibility of a +perpetual motion. + +Lagrange tried in every way to supply a proof free from extraneous +elements and fully satisfactory, but without complete success. Nor were +his successors more fortunate. + +The whole of mechanics, thus, is based upon an idea, which, though +unequivocal, is yet unwonted and not coequal with the other principles +and axioms of mechanics. Every student of mechanics, at some stage of +his progress, feels the uncomfortableness of this state of affairs; +every one wishes it removed; but seldom is the difficulty stated in +words. Accordingly, the zealous pupil of the science is highly rejoiced +when he reads in a master like Poinsot (_Théorie générale de l'équilibre +et du mouvement des systèmes_) the following passage, in which that +author is giving his opinion of the _Analytical Mechanics_: + + "In the meantime, because our attention in that work was first + wholly engrossed with the consideration of its beautiful + development of mechanics, which seemed to spring complete from a + single formula, we naturally believed that the science was + completed or that it only remained to seek the demonstration of the + principle of virtual velocities. But that quest brought back all + the difficulties that we had overcome by the principle itself. That + law so general, wherein are mingled the vague and unfamiliar ideas + of infinitely small movements and of perturbations of equilibrium, + only grew obscure upon examination; and the work of Lagrange + supplying nothing clearer than the march of analysis, we saw + plainly that the clouds had only appeared lifted from the course of + mechanics because they had, so to speak, been gathered at the very + origin of that science. + + "At bottom, a general demonstration of the principle of virtual + velocities would be equivalent to the establishment of the whole of + mechanics upon a different basis: for the demonstration of a law + which embraces a whole science is neither more nor less than the + reduction of that science to another law just as general, but + evident, or at least more simple than the first, and which, + consequently, would render that useless."[50] + +According to Poinsot, therefore, a proof of the principle of virtual +movements is tantamount to a total rehabilitation of mechanics. + +Another circumstance of discomfort to the mathematician is, that in the +historical form in which mechanics at present exists, dynamics is +founded on statics, whereas it is desirable that in a science which +pretends to deductive completeness the more special statical theorems +should be deducible from the more general dynamical principles. + +In fact, a great master, Gauss, gave expression to this desire in his +presentment of the principle of least constraint (Crelle's _Journal für +reine und angewandte Mathematik_, Vol. IV, p. 233) in the following +words: "Proper as it is that in the gradual development of a science, +and in the instruction of individuals, the easy should precede the +difficult, the simple the complex, the special the general, yet the +mind, when once it has reached a higher point of view, demands the +contrary course, in which all statics shall appear simply as a special +case of mechanics." Gauss's own principle, now, possesses all the +requisites of universality, but its difficulty is that it is not +immediately intelligible and that Gauss deduced it with the help of +D'Alembert's principle, a procedure which left matters where they were +before. + +Whence, now, is derived this strange part which the principle of virtual +motion plays in mechanics? For the present I shall only make this reply. +It would be difficult for me to tell the difference of impression which +Lagrange's proof of the principle made on me when I first took it up as +a student and when I subsequently resumed it after having made +historical researches. It first appeared to me insipid, chiefly on +account of the pulleys and the cords which did not fit in with the +mathematical view, and whose action I would much rather have discovered +from the principle itself than have taken for granted. But now that I +have studied the history of the science I cannot imagine a more +beautiful demonstration. + +In fact, through all mechanics it is this self-same principle of +excluded perpetual motion which accomplishes almost all, which +displeased Lagrange, but which he still had to employ, at least tacitly, +in his own demonstration. If we give this principle its proper place and +setting, the paradox is explained. + +The principle of excluded perpetual motion is thus no new discovery; it +has been the guiding idea, for three hundred years, of all the great +inquirers. But the principle cannot properly be _based_ upon mechanical +perceptions. For long before the development of mechanics the conviction +of its truth existed and even contributed to that development. Its power +of conviction, therefore, must have more universal and deeper roots. We +shall revert to this point. + + +II. MECHANICAL PHYSICS. + +It cannot be denied that an unmistakable tendency has prevailed, from +Democritus to the present day, to explain _all_ physical events +_mechanically_. Not to mention earlier obscure expressions of that +tendency we read in Huygens the following:[51] + + "There can be no doubt that light consists of the _motion_ of a + certain substance. For if we examine its production, we find that + here on earth it is principally fire and flame which engender it, + both of which contain beyond doubt bodies which are in rapid + movement, since they dissolve and destroy many other bodies more + solid than they: while if we regard its effects, we see that when + light is accumulated, say by concave mirrors, it has the property + of combustion just as fire has, that is to say, it disunites the + parts of bodies, which is assuredly a proof of _motion_, at least + in the _true philosophy_, in which the causes of all natural + effects are conceived as _mechanical_ causes. Which in my judgment + must be accomplished or all hope of ever understanding physics + renounced."[52] + +S. Carnot,[53] in introducing the principle of excluded perpetual motion +into the theory of heat, makes the following apology: + + "It will be objected here, perhaps, that a perpetual motion proved + impossible for _purely mechanical actions_, is perhaps not so when + the influence of _heat_ or of electricity is employed. But can + phenomena of heat or electricity be thought of as due to anything + else than to _certain motions of bodies_, and as such must they not + be subject to the general laws of mechanics?"[54] + +These examples, which might be multiplied by quotations from recent +literature indefinitely, show that a tendency to explain all things +mechanically actually exists. This tendency is also intelligible. +Mechanical events as simple motions in space and time best admit of +observation and pursuit by the help of our highly organised senses. We +reproduce mechanical processes almost without effort in our imagination. +Pressure as a circumstance that produces motion is very familiar to us +from daily experience. All changes which the individual personally +produces in his environment, or humanity brings about by means of the +arts in the world, are effected through the instrumentality of +_motions_. Almost of necessity, therefore, motion appears to us as the +most important physical factor. Moreover, mechanical properties may be +discovered in all physical events. The sounding bell trembles, the +heated body expands, the electrified body attracts other bodies. Why, +therefore, should we not attempt to grasp all events under their +mechanical aspect, since that is so easily apprehended and most +accessible to observation and measurement? In fact, no objection _is_ to +be made to the attempt to elucidate the properties of physical events by +mechanical _analogies_. + +But modern physics has proceeded _very far_ in this direction. The point +of view which Wundt represents in his excellent treatise _On the +Physical Axioms_ is probably shared by the majority of physicists. The +axioms of physics which Wundt sets up are as follows: + +1. All natural causes are motional causes. + +2. Every motional cause lies outside the object moved. + +3. All motional causes act in the direction of the straight line of +junction, and so forth. + +4. The effect of every cause persists. + +5. Every effect involves an equal countereffect. + +6. Every effect is equivalent to its cause. + +These principles might be studied properly enough as fundamental +principles of mechanics. But when they are set up as axioms of physics, +their enunciation is simply tantamount to a negation of all events +except motion. + +According to Wundt, all changes of nature are mere changes of place. All +causes are motional causes (page 26). Any discussion of the +philosophical grounds on which Wundt supports his theory would lead us +deep into the speculations of the Eleatics and the Herbartians. Change +of place, Wundt holds, is the _only_ change of a thing in which a thing +remains identical with itself. If a thing changed _qualitatively_, we +should be obliged to imagine that something was annihilated and +something else created in its place, which is not to be reconciled with +our idea of the identity of the object observed and of the +indestructibility of matter. But we have only to remember that the +Eleatics encountered difficulties of exactly the same sort in motion. +Can we not also imagine that a thing is destroyed in _one_ place and in +_another_ an exactly similar thing created? After all, do we really know +_more_ why a body leaves one place and appears in another, than why a +_cold_ body grows _warm_? Granted that we had a perfect knowledge of the +mechanical processes of nature, could we and should we, for that reason, +_put out of the world_ all other processes that we do not understand? On +this principle it would really be the simplest course to deny the +existence of the whole world. This is the point at which the Eleatics +ultimately arrived, and the school of Herbart stopped little short of +the same goal. + +Physics treated in this sense supplies us simply with a diagram of the +world, in which we do not know reality again. It happens, in fact, to +men who give themselves up to this view for many years, that the world +of sense from which they start as a province of the greatest +familiarity, suddenly becomes, in their eyes, the supreme +"world-riddle." + +Intelligible as it is, therefore, that the efforts of thinkers have +always been bent upon the "reduction of all physical processes to the +motions of atoms," it must yet be affirmed that this is a chimerical +ideal. This ideal has often played an effective part in popular +lectures, but in the workshop of the serious inquirer it has discharged +scarcely the least function. What has really been achieved in mechanical +physics is either the _elucidation_ of physical processes by more +familiar _mechanical analogies_, (for example, the theories of light and +of electricity,) or the exact _quantitative_ ascertainment of the +connexion of mechanical processes with other physical processes, for +example, the results of thermodynamics. + + +III. THE PRINCIPLE OF ENERGY IN PHYSICS. + +We can know only from _experience_ that mechanical processes produce +other physical transformations, or _vice versa_. The attention was first +directed to the connexion of mechanical processes, especially the +performance of work, with changes of thermal conditions by the invention +of the steam-engine, and by its great technical importance. Technical +interests and the need of scientific lucidity meeting in the mind of S. +Carnot led to the remarkable development from which thermodynamics +flowed. It is simply _an accident of history_ that the development in +question was not connected with the practical applications of +_electricity_. + +In the determination of the maximum quantity of _work_ that, generally, +a heat-machine, or, to take a special case, a steam-engine, can perform +with the expenditure of a _given_ amount of heat of combustion, Carnot +is guided by mechanical analogies. A body can do work on being heated, +by expanding under pressure. But to do this the body must receive heat +from a _hotter_ body. Heat, therefore, to do work, must pass from a +hotter body to a colder body, just as water must fall from a higher +level to a lower level to put a mill-wheel in motion. Differences of +temperature, accordingly, represent forces able to do work exactly as do +differences of height in heavy bodies. Carnot pictures to himself an +ideal process in which no heat flows away unused, that is, without doing +work. With a given expenditure of heat, accordingly, this process +furnishes the maximum of work. An analogue of the process would be a +mill-wheel which scooping its water out of a higher level would slowly +carry it to a lower level without the loss of a drop. A peculiar +property of the process is, that with the expenditure of the same work +the water can be raised again exactly to its original level. This +property of _reversibility_ is also shared by the process of Carnot. His +process also can be reversed by the expenditure of the same amount of +work, and the heat again brought back to its original temperature level. + +Suppose, now, we had _two_ different reversible processes _A_, _B_, such +that in _A_ a quantity of heat, _Q_, flowing off from the temperature +_t₁_ to the lower temperature _t₂_ should perform the work _W_, but in +_B_ under the same circumstances it should perform a greater quantity of +work _W_ + _W'_; then, we could join _B_ in the sense assigned and _A_ +in the reverse sense into a _single_ process. Here _A_ would reverse the +transformation of heat produced by _B_ and would leave a surplus of work +_W'_, produced, so to speak, from nothing. The combination would present +a perpetual motion. + +With the feeling, now, that it makes little difference whether the +mechanical laws are broken directly or indirectly (by processes of +heat), and convinced of the existence of a _universal_ law-ruled +connexion of nature, Carnot here excludes for the first time from the +province of _general_ physics the possibility of a perpetual motion. +_But it follows, then, that the quantity of work W, produced by the +passage of a quantity of heat Q from a temperature t₁ to a temperature +t₂, is independent of the nature of the substances as also of the +character of the process, so far as that is unaccompanied by loss, but +is wholly dependent upon the temperature t₁, t₂. + +This important principle has been fully confirmed by the special +researches of Carnot himself (1824), of Clapeyron (1834), and of Sir +William Thomson (1849), now Lord Kelvin. The principle was reached +_without any assumption whatever_ concerning the nature of heat, simply +by the exclusion of a perpetual motion. Carnot, it is true, was an +adherent of the theory of Black, according to which the sum-total of the +quantity of heat in the world is constant, but so far as his +investigations have been hitherto considered the decision on this point +is of no consequence. Carnot's principle led to the most remarkable +results. W. Thomson (1848) founded upon it the ingenious idea of an +"absolute" scale of temperature. James Thomson (1849) conceived a Carnot +process to take place with water freezing under pressure and, therefore, +performing work. He discovered, thus, that the freezing point is lowered +0·0075° Celsius by every additional atmosphere of pressure. This is +mentioned merely as an example. + +About twenty years after the publication of Carnot's book a further +advance was made by J. R. Mayer and J. P. Joule. Mayer, while engaged as +a physician in the service of the Dutch, observed, during a process of +bleeding in Java, an unusual redness of the venous blood. In agreement +with Liebig's theory of animal heat he connected this fact with the +diminished loss of heat in warmer climates, and with the diminished +expenditure of organic combustibles. The total expenditure of heat of a +man at rest must be equal to the total heat of combustion. But since +_all_ organic actions, even the mechanical actions, must be set down to +the credit of the heat of combustion, some connexion must exist between +mechanical work and expenditure of heat. + +Joule started from quite similar convictions concerning the galvanic +battery. A heat of association equivalent to the consumption of the zinc +can be made to appear in the galvanic cell. If a current is set up, a +part of this heat appears in the conductor of the current. The +interposition of an apparatus for the decomposition of water causes a +part of this heat to disappear, which on the burning of the explosive +gas formed, is reproduced. If the current runs an electromotor, a +portion of the heat again disappears, which, on the consumption of the +work by friction, again makes its appearance. Accordingly, both the +heat produced and the work produced, appeared to Joule also as +connected with the consumption of material. The thought was therefore +present, both to Mayer and to Joule, of regarding heat and work as +equivalent quantities, so connected with each other that what is lost in +one form universally appears in another. The result of this was a +_substantial_ conception of heat and of work, and _ultimately a +substantial conception of energy_. Here every physical change of +condition is regarded as energy, the destruction of which generates work +or equivalent heat. An electric charge, for example, is energy. + +In 1842 Mayer had calculated from the physical constants then +universally accepted that by the disappearance of one kilogramme-calorie +365 kilogramme-metres of work could be performed, and _vice versa_. +Joule, on the other hand, by a long series of delicate and varied +experiments beginning in 1843 ultimately determined the mechanical +equivalent of the kilogramme-calorie, more exactly, as 425 +kilogramme-metres. + +If we estimate every change of physical condition by the _mechanical +work_ which can be performed upon the _disappearance_ of that condition, +and call this measure _energy_, then we can measure all physical changes +of condition, no matter how different they may be, with the same common +measure, and say: _the sum-total of all energy remains constant_. This +is the form that the principle of excluded perpetual motion received at +the hands of Mayer, Joule, Helmholtz, and W. Thomson in its extension to +the whole domain of physics. + +After it had been proved that heat must _disappear_ if mechanical work +was to be done at its expense, Carnot's principle could no longer be +regarded as a complete expression of the facts. Its improved form was +first given, in 1850, by Clausius, whom Thomson followed in 1851. It +runs thus: "If a quantity of heat _Q'_ is transformed into work in a +reversible process, _another_ quantity of heat _Q_ of the absolute[55] +temperature _T₁_ is lowered to the absolute temperature _T₂_." Here +_Q'_ is dependent only on _Q_, _T₁_, _T₂_, but is independent of the +substances used and of the character of the process, so far as that is +unaccompanied by loss. Owing to this last fact, it is sufficient to find +the relation which obtains for some one well-known physical substance, +say a gas, and some definite simple process. The relation found will be +the one that holds generally. We get, thus, + +_Q'/(Q' + Q) = (T₁-T₂)/T₁_ (1) + +that is, the quotient of the available heat _Q'_ transformed into work +divided by the sum of the transformed and transferred heats (the total +sum used), the so-called _economical coefficient_ of the process, is, + +_(T₁-T₂)/T₁_. + + +IV. THE CONCEPTIONS OF HEAT. + +When a cold body is put in contact with a warm body it is observed that +the first body is warmed and that the second body is cooled. We may say +that the first body is warmed _at the expense of_ the second body. This +suggests the notion of a thing, or heat-substance, which passes from the +one body to the other. If two masses of water _m_, _m'_, of unequal +temperatures, be put together, it will be found, upon the rapid +equalisation of the temperatures, that the respective changes of +temperatures _u_ and _u'_ are inversely proportional to the masses and +of opposite signs, so that the algebraical sum of the products is, + +_mu + m'u' = 0_. + +Black called the products _mu_, _m'u'_, which are decisive for our +knowledge of the process, _quantities of heat_. We may form a very clear +_picture_ of these products by conceiving them with Black as measures of +the quantities of some substance. But the essential thing is not this +picture but the _constancy_ of the sum of these products in simple +processes of conduction. If a quantity of heat disappears at one point, +an equally large quantity will make its appearance at some other point. +The retention of this idea leads to the discovery of specific heat. +Black, finally, perceives that also something else may appear for a +vanished quantity of heat, namely: the fusion or vaporisation of a +definite quantity of matter. He adheres here still to this favorite +view, though with some freedom, and considers the vanished quantity of +heat as still present, but as _latent_. + +The generally accepted notion of a caloric, or heat-stuff, was strongly +shaken by the work of Mayer and Joule. If the quantity of heat can be +increased and diminished, people said, heat cannot be a substance, but +must be a _motion_. The subordinate part of this statement has become +much more popular than all the rest of the doctrine of energy. But we +may convince ourselves that the motional conception of heat is now as +unessential as was formerly its conception as a substance. Both ideas +were favored or impeded solely by accidental historical circumstances. +It does not follow that heat is not a substance from the fact that a +mechanical equivalent exists for quantity of heat. We will make this +clear by the following question which bright students have sometimes put +to me. Is there a mechanical equivalent of electricity as there is a +mechanical equivalent of heat? Yes, and no. There is no mechanical +equivalent of _quantity_ of electricity as there is an equivalent of +_quantity_ of heat, because the same quantity of electricity has a very +different capacity for work, according to the circumstances in which it +is placed; but there _is_ a mechanical equivalent of electrical energy. + +Let us ask another question. Is there a mechanical equivalent of water? +No, there is no mechanical equivalent of quantity of water, but there is +a mechanical equivalent of weight of water multiplied by its distance +of descent. + +When a Leyden jar is discharged and work thereby performed, we do not +picture to ourselves that the quantity of electricity disappears as work +is done, but we simply assume that the electricities come into different +positions, equal quantities of positive and negative electricity being +united with one another. + +What, now, is the reason of this difference of view in our treatment of +heat and of electricity? The reason is purely historical, wholly +conventional, and, what is still more important, is wholly indifferent. +I may be allowed to establish this assertion. + +In 1785 Coulomb constructed his torsion balance, by which he was enabled +to measure the repulsion of electrified bodies. Suppose we have two +small balls, _A_, _B_, which over their whole extent are similarly +electrified. These two balls will exert on one another, at a certain +distance _r_ of their centres, a certain repulsion _p_. We bring into +contact with _B_ now a ball _C_, suffer both to be equally electrified, +and then measure the repulsion of _B_ from _A_ and of _C_ from _A_ at +the same distance _r_. The sum of these repulsions is again _p_. +Accordingly something has remained constant. If we ascribe this effect +to a substance, then we infer naturally its constancy. But the essential +point of the exposition is the divisibility of the electric force _p_ +and not the simile of substance. + +In 1838 Riess constructed his electrical air-thermometer (the +thermoelectrometer). This gives a measure of the quantity of heat +produced by the discharge of jars. This quantity of heat is not +proportional to the quantity of electricity contained in the jar by +Coulomb's measure, but if _Q_ be this quantity and _C_ be the capacity, +is proportional to _Q_²/2_C_, or, more simply still, to the energy of +the charged jar. If, now, we discharge the jar completely through the +thermometer, we obtain a certain quantity of heat, _W_. But if we make +the discharge through the thermometer into a second jar, we obtain a +quantity less than _W_. But we may obtain the remainder by completely +discharging both jars through the air-thermometer, when it will again be +proportional to the energy of the two jars. On the first, incomplete +discharge, accordingly, a part of the electricity's capacity for work +was lost. + +When the charge of a jar produces heat its energy is changed and its +value by Riess's thermometer is decreased. But by Coulomb's measure the +quantity remains unaltered. + +Now let us imagine that Riess's thermometer had been invented before +Coulomb's torsion balance, which is not a difficult feat, since both +inventions are independent of each other; what would be more natural +than that the "quantity" of electricity contained in a jar should be +measured by the heat produced in the thermometer? But then, this +so-called quantity of electricity would decrease on the production of +heat or on the performance of work, whereas it now remains unchanged; +in that case, therefore, electricity would not be a _substance_ but a +_motion_, whereas now it is still a substance. The reason, therefore, +why we have other notions of electricity than we have of heat, is purely +historical, accidental, and conventional. + +This is also the case with other physical things. Water does not +disappear when work is done. Why? Because we measure quantity of water +with scales, just as we do electricity. But suppose the capacity of +water for work were called quantity, and had to be measured, therefore, +by a mill instead of by scales; then this quantity also would disappear +as it performed the work. It may, now, be easily conceived that many +substances are not so easily got at as water. In that case we should be +unable to carry out the one kind of measurement with the scales whilst +many other modes of measurement would still be left us. + +In the case of heat, now, the historically established measure of +"quantity" is accidentally the work-value of the heat. Accordingly, its +quantity disappears when work is done. But that heat is not a substance +follows from this as little as does the opposite conclusion that it is a +substance. In Black's case the quantity of heat remains constant because +the heat passes into no _other_ form of energy. + +If any one to-day should still wish to think of heat as a substance, we +might allow that person this liberty with little ado. He would only have +to assume that that which we call quantity of heat was the energy of a +substance whose quantity remained unaltered, but whose energy changed. +In point of fact we might much better say, in analogy with the other +terms of physics, energy of heat, instead of quantity of heat. + +When we wonder, therefore, at the discovery that heat is motion, we +wonder at something that was never discovered. It is perfectly +indifferent and possesses not the slightest scientific value, whether we +think of heat as a substance or not. The fact is, heat behaves in some +connexions like a substance, in others not. Heat is latent in steam as +oxygen is latent in water. + + +V. THE CONFORMITY IN THE DEPORTMENT OF THE ENERGIES. + +The foregoing reflexions will gain in lucidity from a consideration of +the conformity which obtains in the behavior of all energies, a point to +which I called attention long ago.[56] + +A weight _P_ at a height _H₁_ represents an energy _W₁ = PH₁_. If we +suffer the weight to sink to a lower height _H₂_, during which work is +done, and the work done is employed in the production of living force, +heat, or an electric charge, in short, is transformed, then the energy +_W₂ = PH₂_ is still _left_. The equation subsists + +_W₁/H₁ = W₂/H₂_, (2) + +or, denoting the _transformed_ energy by _W' = W₁-W₂_ and the +_transferred_ energy, that transported to the lower level, by _W = W₂_, + +_W'/(W' + W) = (H₁-H₂)/H₁_, (3) + +an equation in all respects analogous to equation (1) at page 165. The +property in question, therefore, is by no means peculiar to heat. +Equation (2) gives the relation between the energy taken from the higher +level and that deposited on the lower level (the energy left behind); it +says that these _energies_ are proportional to the _heights of the +levels_. An equation analogous to equation (2) may be set up for _every_ +form of energy; hence the equation which corresponds to equation (3), +and so to equation (1), may be regarded as valid for every form. For +electricity, for example, _H₁_, _H₂_ signify the potentials. + +When we observe for the first time the agreement here indicated in the +transformative law of the energies, it appears surprising and +unexpected, for we do not perceive at once its reason. But to him who +pursues the comparative historical method that reason will not long +remain a secret. + +Since Galileo, mechanical work, though long under a different name, has +been a _fundamental concept_ of mechanics, as also a very important +notion in the applied sciences. The transformation of work into living +force, and of living force into work, suggests directly the notion of +energy--the idea having been first fruitfully employed by Huygens, +although Thomas Young first called it by the _name_ of "energy." Let us +add to this the constancy of weight (really the constancy of mass) and +we shall see that with respect to mechanical energy it is involved in +the very definition of the term that the capacity for work or the +potential energy of a weight is proportional to the height of the level +at which it is, in the geometrical sense, and that it decreases on the +lowering of the weight, on transformation, proportionally to the height +of the level. The zero level here is wholly arbitrary. With this, +equation (2) is given, from which all the other forms follow. + +When we reflect on the tremendous start which mechanics had over the +other branches of physics, it is not to be wondered at that the attempt +was always made to apply the notions of that science wherever this was +possible. Thus the notion of mass, for example, was imitated by Coulomb +in the notion of quantity of electricity. In the further development of +the theory of electricity, the notion of work was likewise immediately +introduced in the theory of potential, and heights of electrical level +were measured by the work of unit of quantity raised to that level. But +with this the preceding equation with all its consequences is given for +electrical energy. The case with the other energies was similar. + +_Thermal_ energy, however, appears as a special case. Only by the +peculiar experiments mentioned could it be discovered that heat is an +energy. But the measure of this energy by Black's quantity of heat is +the outcome of fortuitous circumstances. In the first place, the +accidental slight variability of the capacity for heat _c_ with the +temperature, and the accidental slight deviation of the usual +thermometrical scales from the scale derived from _the tensions of +gases_, brings it about that the notion "quantity of heat" can be set up +and that the quantity of heat _ct_ corresponding to a difference of +temperature _t_ is nearly proportional to the energy of the heat. It is +a quite accidental historical circumstance that Amontons hit upon the +idea of measuring temperature by the tension of a gas. It is certain in +this that he did not think of the work of the heat.[57] But the numbers +standing for temperature, thus, are made proportional to the tensions of +gases, that is, to the work done by gases, with otherwise equal changes +of volume. It thus happens that _temperature heights_ and _level heights +of work_ are proportional to one another. + +If properties of the thermal condition varying greatly from the tensions +of gases had been chosen, this relation would have assumed very +complicated forms, and the agreement between heat and the other energies +above considered would not subsist. It is very instructive to reflect +upon this point. A _natural law_, therefore, is not implied in the +conformity of the behavior of the energies, but this conformity is +rather conditioned by the uniformity of our modes of conception and is +also partly a matter of good fortune. + + +VI. THE DIFFERENCES OF THE ENERGIES AND THE LIMITS OF THE PRINCIPLE OF +ENERGY. + +Of every quantity of heat _Q_ which does work in a reversible process +(one unaccompanied by loss) between the absolute temperatures _T₁_, +_T₂_, only the portion + +_(T₁-T₂)/T₁_ + +is transformed into work, while the remainder is transferred to the +lower temperature-level _T₂_. This transferred portion can, upon the +reversal of the process, with the same expenditure of work, again be +brought back to the level _T₁_. But if the process is not reversible, +then more heat than in the foregoing case flows to the lower level, and +the surplus can no longer be brought back to the higher level _T₂_ +without some _special_ expenditure. W. Thomson (1852), accordingly, drew +attention to the fact, that in all non-reversible, that is, in all real +thermal processes, quantities of heat are lost for mechanical work, and +that accordingly a dissipation or waste of mechanical energy is taking +place. In all cases, heat is only partially transformed into work, but +frequently work is wholly transformed into heat. Hence, a tendency +exists towards a diminution of the _mechanical_ energy and towards an +increase of the _thermal_ energy of the world. + +For a simple, closed cyclical process, accompanied by no loss, in which +the quantity of heat _Q₁_ is taken from the level _T₁_, and the quantity +_Q₂_ is deposited upon the level _T₂_, the following relation, agreeably +to equation (2), exists, + +_-(Q₁/T₁) + (Q₂/T₂) = 0_. + +Similarly, for any number of compound reversible cycles Clausius finds +the algebraical sum + +_[sum]Q/T = 0_, + +and supposing the temperature to change continuously, + +_[integral]dQ/T = 0_ (4) + +Here the elements of the quantities of heat deducted from a given level +are reckoned negative, and the elements imparted to it, positive. If the +process is not reversible, then expression (4), which Clausius calls +_entropy_, increases. In actual practice this is always the case, and +Clausius finds himself led to the statement: + +1. That the energy of the world remains constant. + +2. That the entropy of the world tends toward a maximum. + +Once we have noted the above-indicated conformity in the behavior of +different energies, the _peculiarity_ of thermal energy here mentioned +must strike us. Whence is this peculiarity derived, for, generally every +energy passes only partly into another form, which is also true of +thermal energy? The explanation will be found in the following. + +Every transformation of a special kind of energy _A_ is accompanied with +a fall of potential of that particular kind of energy, including heat. +But whilst for the other kinds of energy a transformation and therefore +a loss of energy on the part of the kind sinking in potential is +connected with the fall of the potential, with heat the case is +different. Heat can suffer a fall of potential without sustaining a loss +of energy, at least according to the customary mode of estimation. If a +weight sinks, it must create perforce kinetic energy, or heat, or some +other form of energy. Also, an electrical charge cannot suffer a fall of +potential without loss of energy, i. e., without transformation. But +heat can pass with a fall of temperature to a body of greater capacity +and the same thermal energy still be preserved, so long as we regard +_every quantity_ of heat as energy. This it is that gives to heat, +besides its property of energy, in many cases the character of a +material _substance_, or quantity. + +If we look at the matter in an unprejudiced light, we must ask if there +is any scientific sense or purpose in still considering as energy a +quantity of heat that can no longer be transformed into mechanical work, +(for example, the heat of a closed equably warmed material system). The +principle of energy certainly plays in this case a wholly superfluous +rôle, which is assigned to it only from habit.[58] To maintain the +principle of energy in the face of a knowledge of the dissipation or +waste of mechanical energy, in the face of the increase of entropy is +equivalent almost to the liberty which Black took when he regarded the +heat of liquefaction as still present but latent.[59] It is to be +remarked further, that the expressions "energy of the world" and +"entropy of the world" are slightly permeated with scholasticism. Energy +and entropy are _metrical_ notions. What meaning can there be in +applying these notions to a case in which they are not applicable, in +which their values are not determinable? + +If we could really determine the entropy of the world it would represent +a true, absolute measure of time. In this way is best seen the utter +tautology of a statement that the entropy of the world increases with +the time. Time, and the fact that certain changes take place only in a +definite sense, are one and the same thing. + + + +VII. THE SOURCES OF THE PRINCIPLE OF ENERGY. + +We are now prepared to answer the question, What are the sources of the +principle of energy? All knowledge of nature is derived in the last +instance from experience. In this sense they are right who look upon the +principle of energy as a result of experience. + +Experience teaches that the sense-elements [alpha beta gamma delta ...] +into which the world may be decomposed, are subject to change. It tells +us further, that certain of these elements are _connected_ with other +elements, so that they appear and disappear together; or, that the +appearance of the elements of one class is connected with the +disappearance of the elements of the other class. We will avoid here the +notions of cause and effect because of their obscurity and +equivocalness. The result of experience may be expressed as follows: +_The sensuous elements of the world ([alpha beta gamma delta ...]) show +themselves to be interdependent._ This interdependence is best +represented by some such conception as is in geometry that of the mutual +dependence of the sides and angles of a triangle, only much more varied +and complex. + +As an example, we may take a mass of gas enclosed in a cylinder and +possessed of a definite volume ([alpha]), which we change by a pressure +([beta]) on the piston, at the same time feeling the cylinder with our +hand and receiving a sensation of heat ([gamma]). Increase of pressure +diminishes the volume and increases the sensation of heat. + +The various facts of experience are not in all respects alike. Their +common sensuous elements are placed in relief by a process of +abstraction and thus impressed upon the memory. In this way the +expression is obtained of the features of _agreement_ of extensive +groups of facts. The simplest sentence which we can utter is, by the +very nature of language, an abstraction of this kind. But account must +also be taken of the _differences_ of related facts. Facts may be so +nearly related as to contain the same kind of a [alpha beta gamma ...], +but the relation be such that the [alpha beta gamma ...] of the one +differ from the [alpha beta gamma ...] of the other only by the number +of equal parts into which they can be divided. Such being the case, if +rules can be given for deducing _from one another_ the numbers which are +the measures of these [alpha beta gamma ...], then we possess in such +rules the _most general_ expression of a group of facts, as also that +expression which corresponds to all its differences. This is the goal of +quantitative investigation. + +If this goal be reached what we have found is that between the [alpha +beta gamma ...] of a group of facts, or better, between the numbers +which are their measures, a number of equations exists. The simple fact +of change brings it about that the number of these equations must be +smaller than the number of the [alpha beta gamma ...]. If the former be +smaller by one than the latter, then one portion of the [alpha beta +gamma ...] is _uniquely_ determined by the other portion. + +The quest of relations of this last kind is the most important function +of special experimental research, because we are enabled by it to +complete in thought facts that are only partly given. It is self-evident +that only experience can ascertain that between the [alpha beta gamma +...] relations exist and of what kind they are. Further, only experience +can tell that the relations that exist between the [alpha beta gamma +...] are such that changes of them can be reversed. If this were not the +fact all occasion for the enunciation of the principle of energy, as is +easily seen, would be wanting. In experience, therefore, is buried the +ultimate well-spring of all knowledge of nature, and consequently, in +this sense, also the ultimate source of the principle of energy. + +But this does not exclude the fact that the principle of energy has also +a logical root, as will now be shown. Let us assume on the basis of +experience that one group of sensuous elements [alpha beta gamma ...] +determines _uniquely_ another group [lambda mu nu ...]. Experience +further teaches that changes of [alpha beta gamma ...] can be +_reversed_. It is then a logical consequence of this observation, that +every time that [alpha beta gamma ...] assume the same values this is +also the case with [lambda mu nu ...]. Or, that purely _periodical_ +changes of [alpha beta gamma ...] can produce no _permanent_ changes of +[lambda mu nu ...]. If the group [lambda mu nu ...] is a mechanical +group, then a perpetual motion is excluded. + +It will be said that this is a vicious circle, which we will grant. But +psychologically, the situation is essentially different, whether I think +simply of the unique determination and reversibility of events, or +whether I exclude a perpetual motion. The attention takes in the two +cases different directions and diffuses light over different sides of +the question, which logically of course are necessarily connected. + +Surely that firm, logical setting of the thoughts noticeable in the +great inquirers, Stevinus, Galileo, and the rest, which, consciously or +instinctively, was supported by a fine feeling for the slightest +contradictions, has no other purpose than to limit the bounds of thought +and so exempt it from the possibility of error. In this, therefore, the +logical root of the principle of excluded perpetual motion is given, +namely, in that universal conviction which existed even before the +development of mechanics and co-operated in that development. + +It is perfectly natural that the principle of excluded perpetual motion +should have been first developed in the simple domain of pure mechanics. +Towards the transference of that principle into the domain of general +physics the idea contributed much that all physical phenomena are +mechanical phenomena. But the foregoing discussion shows how little +essential this notion is. The issue really involved is the recognition +of a general interconnexion of nature. This once established, we see +with Carnot that it is indifferent whether the mechanical laws are +broken directly or circuitously. + +The principle of the excluded perpetual motion is very closely related +to the modern principle of energy, but it is not identical with it, for +the latter is to be deduced from the former only by means of a definite +_formal conception_. As may be seen from the preceding exposition, the +perpetual motion can be excluded without our employing or possessing the +notion of _work_. The modern principle of energy results primarily from +a _substantial_ conception of work and of every change of physical +condition which by being reversed produces work. The strong need of such +a conception, which is by no means necessary, but in a formal sense is +very convenient and lucid, is exhibited in the case of J. R. Mayer and +Joule. It was before remarked that this conception was suggested to both +inquirers by the observation that both the production of heat and the +production of mechanical work were connected with an expenditure of +substance. Mayer says: "Ex nihilo nil fit," and in another place, "The +creation or destruction of a force (work) lies without the province of +human activity." In Joule we find this passage: "It is manifestly +_absurd_ to suppose that the powers with which God has endowed matter +can be destroyed." + +Some writers have observed in such statements the attempt at a +_metaphysical_ establishment of the doctrine of energy. But we see in +them simply the formal need of a simple, clear, and living grasp of the +facts, which receives its development in practical and technical life, +and which we carry over, as best we can, into the province of science. +As a fact, Mayer writes to Griesinger: "If, finally, you ask me how I +became involved in the whole affair, my answer is simply this: Engaged +during a sea voyage almost exclusively with the study of physiology, I +discovered the new theory for the sufficient reason that I _vividly felt +the need of it_." + +The substantial conception of work (energy) is by no means a necessary +one. And it is far from true that the problem is solved with the +recognition of the need of such a conception. Rather let us see how +Mayer gradually endeavored to satisfy that need. He first regards +quantity of motion, or momentum, _mv_, as the equivalent of work, and +did not light, until later, on the notion of living force (_mv²/2_). In +the province of electricity he was unable to assign the expression which +is the equivalent of work. This was done later by Helmholtz. The formal +need, therefore, is _first_ present, and our conception of nature is +subsequently gradually _adapted_ to it. + +The laying bare of the experimental, logical, and formal root of the +present principle of energy will perhaps contribute much to the removal +of the mysticism which still clings to this principle. With respect to +our formal need of a very simple, palpable, substantial conception of +the processes in our environment, it remains an open question how far +nature corresponds to that need, or how far we can satisfy it. In one +phase of the preceding discussions it would seem as if the substantial +notion of the principle of energy, like Black's material conception of +heat, has its natural limits in facts, beyond which it can only be +artificially adhered to. + + FOOTNOTES: + + [Footnote 39: Published in Vol. 5, No. I, of _The Monist_, October, + 1894, being in part a re-elaboration of the treatise _Ueber die + Erhaltung der Arbeit_, Prague, 1872.] + + [Footnote 40: _On Matter, Living Force, and Heat_, Joule: + _Scientific Papers_, London, 1884, I, p. 265.] + + [Footnote 41: "Atqui hoc si sit, globorum series sive corona eundem + situm cum priore habebit, eademque de causa octo globi sinistri + ponderosiores erunt sex dextris, ideoque rursus octo illi + descendent, sex illi ascendent, istique globi ex sese _continuum et + aeternum motum efficient, quod est falsum_."] + + [Footnote 42: "A igitur, (si ullo modo per naturam fieri possit) + locum sibi tributum non servato, ac delabatur in _D_; quibus positis + aqua quae ipsi _A_ succedit eandem ob causam deffluet in _D_, eademque + ab alia istinc expelletur, atque adeo aqua haec (cum ubique eadem + ratio sit) _motum instituet perpetuum, quod absurdum fuerit_."] + + [Footnote 43: "Accipio, gradus velocitatis ejusdem mobilis super + diversas planorum inclinationes acquisitos tunc esse aequales, cum + eorundum planorum elevationes aequales sint."] + + [Footnote 44: "Voi molto probabilmente discorrete, ma oltre al veri + simile voglio con una esperienza crescer tanto la probabilità, che + poco gli manchi all'agguagliarsi ad una ben necessaria + dimostrazione. Figuratevi questo foglio essere una parete eretta + all'orizzonte, e da un chiodo fitto in essa pendere una palla di + piombo d'un'oncia, o due, sospesa dal sottil filo _AB_ lungo due, o + tre braccia perpendicolare all'orizzonte, e nella parete segnate una + linea orizontale _DC_ segante a squadra il perpendicolo _AB_, il + quale sia lontano dalla parete due dita in circa, trasferendo poi il + filo _AB_ colla palla in _AC_, lasciata essa palla in libertà, la + quale primieramente vedrete scendere descrivendo l'arco _CBD_, e di + tanto trapassare il termine _B_, che scorrendo per l'arco _BD_ + sormonterà fino quasi alla segnata parallela _CD_, restando di per + vernirvi per piccolissimo intervallo, toltogli il precisamente + arrivarvi dall'impedimento dell'aria, e del filo. Dal che possiamo + veracemente concludere, che l'impeto acquistato nel punto _B_ dalla + palla nello scendere per l'arco _CB_, fu tanto, che bastò a + risospingersi per un simile arco _BD_ alla medesima altezza; fatta, + e più volte reiterata cotale esperienza, voglio, che fiechiamo nella + parete rasente al perpendicolo _AB_ un chiodo come in _E_, ovvero in + _F_, che sporga in fuori cinque, o sei dita, e questo acciocchè il + filo _AC_ tornando come prima a riportar la palla _C_ per l'arco + _CB_, giunta che ella sia in _B_, inoppando il filo nel chiodo _E_, + sia costretta a camminare per la circonferenza _BG_ descritta in + torno al centro _E_, dal che vedremo quello, che potrà far quel + medesimo impeto, che dianzi concepizo nel medesimo termine _B_, + sospinse l'istesso mobile per l'arco _ED_ all'altezza + dell'orizzonale _CD_. Ora, Signori, voi vedrete con gusto condursi + la palla all'orizzontale nel punto _G_, e l'istesso accadere, + l'intoppo si metesse più basso, come in _F_, dove la palla + descriverebbe l'arco _BJ_, terminando sempre la sua salita + precisamente nella linea _CD_, e quando l'intoppe del chiodo fusse + tanto basso, che l'avanzo del filo sotto di lui non arivasse + all'altezza di _CD_ (il che accaderebbe, quando fusse più vicino al + punto _B_, che al segamento dell' _AB_ coll'orizzontale _CD_), + allora il filo cavalcherebbe il chiodo, e segli avolgerebbe intorno. + Questa esperienza non lascia luogo di dubitare della verità del + supposto: imperocchè essendo li due archi _CB_, _DB_ equali e + similmento posti, l'acquisto di momento fatto per la scesa nell'arco + _CB_, è il medesimo, che il fatto per la scesa dell'arco _DB_; ma il + momento acquistato in _B_ per l'arco _CB_ è potente a risospingere + in su il medesimo mobile per l'arco _BD_; adunque anco il momento + acquistato nella scesa _DB_ è eguale a quello, che sospigne + l'istesso mobile pel medesimo arco da _B_ in _D_, sicche + universal-mente ogni memento acquistato per la scesa d'un arco è + eguale a quello, che può far risalire l'istesso mobile pel medesimo + arco: ma i momenti tutti che fanno resalire per tutti gli archi + _BD_, _BG_, _BJ_ sono eguali, poichè son fatti dal istesso medesimo + momento acquistato per la scesa _CB_, come mostra l'esperienza: + adunque tutti i momenti, che si acquistano per le scese negli archi + _DB_, _GB_, _JB_ sono eguali."] + + [Footnote 45: "Constat jam, quod mobile ex quiete in _A_ descendens + per _AB_, gradus acquirit velocitatis juxta temporis ipsius + incrementum: gradum vero in _B_ esse maximum acquisitorum, et suapte + natura immutabiliter impressum, sublatis scilicet causis + accelerationis novae, aut retardationis: accelerationis inquam, si + adhuc super extenso plano ulterius progrederetur; retardationis + vero, dum super planum acclive _BC_ fit reflexio: in horizontali + autem _GH_ aequabilis motus juxta gradum velocitatis ex _A_ in _B_ + acquisitae in infinitum extenderetur."] + + [Footnote 46: "Si gravitas non esset, neque aër motui corporum + officeret, unumquodque eorum, acceptum semel motum continuaturum + velocitate aequabili, secundum lineam rectam."] + + [Footnote 47: "Si pondera quotlibet, vi gravitatis suae, moveri + incipiant; non posse centrum gravitatis ex ipsis compositae altius, + quam ubi incipiente motu reperiebatur, ascendere. + + "Ipsa vero hypothesis nostra quominus scrupulum moveat, nihil aliud + sibi velle ostendemus, quam, quod nemo unquam negavit, gravia nempe + sursum non ferri.--Et sane, si hac eadem uti scirent novorum operum + machinatores, qui motum perpetuum irrito conatu moliuntur, facile + suos ipsi errores deprehenderent, intelligerentque rem eam mechanica + ratione haud quaquam possibilem esse."] + + [Footnote 48: "Si pendulum e pluribus ponderibus compositum, atque e + quiete dimissum, partem quamcunque oscillationis integrae + confecerit, atque inde porro intelligantur pondera ejus singula, + relicto communi vinculo, celeritates acquisitas sursum convertere, + ac quousque possunt ascendere; hoc facto centrum gravitatis ex + omnibus compositae, ad eandem altitudinem reversum erit, quam ante + inceptam oscillationem obtinebat."] + + [Footnote 49: "Notato autem hic illud staticum axioma etiam locum + habere: + + "Ut spatium agentis ad spatium patientis + Sic potentia patientis ad potentiam agentis."] + + [Footnote 50: "Cependant, comme dans cet ouvrage on ne fut d'abord + attentif qu'à considérer ce beau développement de la mécanique qui + semblait sortir tout entière d'une seule et même formule, on crut + naturellement que la science etait faite, et qu'il ne restait plus + qu'à chercher la démonstration du principe des vitesses virtuelles. + Mais cette recherche ramena toutes les difficultés qu'on avait + franchies par le principe même. Cette loi si générale, où se mêlent + des idées vagues et étrangères de mouvements infinement petits et de + perturbation d'équilibre, ne fit en quelque sorte que s'obsurcir à + l'examen; et le livre de Lagrange n'offrant plus alors rien de clair + que la marche des calculs, on vit bien que les nuages n'avaient paru + levé sur le cours de la mécanique que parcequ'ils étaient, pour + ainsi dire, rassemblés à l'origine même do cette science. + + "Une démonstration générale du principe des vitesses virtuelles + devait au fond revenir a établir le mécanique entière sur une autre + base: car la demonstration d'une loi qui embrasse toute une science + ne peut être autre chose qua la reduction de cette science à une + autre loi aussi générale, mais évidente, ou du moins plus simple que + la première, et qui partant la rende inutile."] + + [Footnote 51: _Traité de la lumière_, Leyden, 1690, p. 2.] + + [Footnote 52: "L'on ne sçaurait douter que la lumière ne consiste + dans le _mouvement_ de certaine matière. Car soit qu'on regarde sa + production, on trouve qu'içy sur la terre c'est principalement le + feu et la flamme qui l'engendrent, lesquels contient sans doute des + corps qui sont dans un mouvement rapide, puis qu'ils dissolvent et + fondent plusieurs autres corps des plus solides: soit qu'on regarde + ses effets, on voit que quand la lumière est ramasseé, comme par des + miroires concaves, elle a la vertu de brûler comme le feu. + c-est-à-dire qu'elle desunit les parties des corps; ce qui marque + assurément du _mouvement_, au moins dans la _vraye Philosophie_, + dans laquelle on conçoit la cause de tous les effets naturels par + des raisons de _mechanique_. Ce qu'il faut faire à mon avis, ou bien + renoncer à tout espérance de jamais rien comprendre dans la + Physique."] + + [Footnote 53: _Sur la puissance motrice du feu_. (Paris, 1824.)] + + [Footnote 54: "On objectra peut-être ici que le mouvement perpétuel, + démontré impossible par les _seules actions mécaniques_, ne l'est + peut-être pas lorsqu'on emploie l'influence soit de la _chaleur_, + soit de l'électricité; mais pent-on concevoir les phénomènes de la + chaleur et de l'électricité comme dus à autre chose qu'à des + _mouvements quelconques des corps_ et comme tels ne doivent-ils pas + être soumis aux lois générales de la mécanique?"] + + [Footnote 55: By this is meant the temperature of a Celsius scale, + the zero of which is 273° below the melting-point of ice.] + + [Footnote 56: I first drew attention to this fact in my treatise + _Ueber die Erhaltung der Arbeit_, Prague, 1872. Before this, Zeuner + had pointed out the analogy between mechanical and thermal energy. I + have given a more extensive development of this idea in a + communication to the _Sitzungsberichte der Wiener_ _Akademie_, + December, 1892, entitled _Geschichte und Kritik des Carnot'schen + Wärmegesetzes_. Compare also the works of Popper (1884), Helm + (1887), Wronsky (1888), and Ostwald (1892).] + + [Footnote 57: Sir William Thomson first consciously and + intentionally introduced (1848, 1851) a _mechanical_ measure of + temperature similar to the electric measure of potential.] + + [Footnote 58: Compare my _Analysis of the Sensations_, Jena, 1886: + English translation, Chicago, 1897.] + + [Footnote 59: A better terminology appears highly desirable in the + place of the usual misleading one. Sir William Thomson (1852) + appears to have felt this need, and it has been clearly expressed by + F. Wald (1889). We should call the work which corresponds to a + vanished quantity of heat its mechanical substitution-value; while + that work which can be _actually_ performed in the passage of a + thermal condition _A_ to a condition _B_, alone deserves the name of + the _energy-value_ of this change of condition. In this way the + _arbitrary_ substantial conception of the processes would be + preserved and misapprehensions forestalled.] + + + + +THE ECONOMICAL NATURE OF PHYSICAL INQUIRY.[60] + + +When the human mind, with its limited powers, attempts to mirror in +itself the rich life of the world, of which it is itself only a small +part, and which it can never hope to exhaust, it has every reason for +proceeding economically. Hence that tendency, expressed in the +philosophy of all times, to compass by a few organic thoughts the +fundamental features of reality. "Life understands not death, nor death +life." So spake an old Chinese philosopher. Yet in his unceasing desire +to diminish the boundaries of the incomprehensible, man has always been +engaged in attempts to understand death by life and life by death. + +Among the ancient civilised peoples, nature was filled with demons and +spirits having the feelings and desires of men. In all essential +features, this animistic view of nature, as Tylor[61] has aptly termed +it, is shared in common by the fetish-worshipper of modern Africa and +the most advanced nations of antiquity. As a theory of the world it has +never completely disappeared. The monotheism of the Christians never +fully overcame it, no more than did that of the Jews. In the belief in +witchcraft and in the superstitions of the sixteenth and seventeenth +centuries, the centuries of the rise of natural science, it assumed +frightful pathological dimensions. Whilst Stevinus, Kepler, and Galileo +were slowly rearing the fabric of modern physical science, a cruel and +relentless war was waged with firebrand and rack against the devils that +glowered from every corner. To-day even, apart from all survivals of +that period, apart from the traces of fetishism which still inhere in +our physical concepts,[62] those very ideas still covertly lurk in the +practices of modern spiritualism. + + +By the side of this animistic conception of the world, we meet from time +to time, in different forms, from Democritus to the present day, another +view, which likewise claims exclusive competency to comprehend the +universe. This view may be characterised as the _physico-mechanical_ +view of the world. To-day, that view holds, indisputably, the first +place in the thoughts of men, and determines the ideals and the +character of our times. The coming of the mind of man into the full +consciousness of its powers, in the eighteenth century, was a period of +genuine disillusionment. It produced the splendid precedent of a life +really worthy of man, competent to overcome the old barbarism in the +practical fields of life; it created the _Critique of Pure Reason_, +which banished into the realm of shadows the sham-ideas of the old +metaphysics; it pressed into the hands of the mechanical philosophy the +reins which it now holds. + +The oft-quoted words of the great Laplace,[63] which I will now give, +have the ring of a jubilant toast to the scientific achievements of the +eighteenth century: "A mind to which were given for a single instant all +the forces of nature and the mutual positions of all its masses, if it +were otherwise powerful enough to subject these problems to analysis, +could grasp, with a single formula, the motions of the largest masses as +well as of the smallest atoms; nothing would be uncertain for it; the +future and the past would lie revealed before its eyes." In writing +these words, Laplace, as we know, had also in mind the atoms of the +brain. That idea has been expressed more forcibly still by some of his +followers, and it is not too much to say that Laplace's ideal is +substantially that of the great majority of modern scientists. + +Gladly do we accord to the creator of the _Mécanique céleste_ the sense +of lofty pleasure awakened in him by the great success of the +Enlightenment, to which we too owe our intellectual freedom. But to-day, +with minds undisturbed and before _new_ tasks, it becomes physical +science to secure itself against self-deception by a careful study of +its character, so that it can pursue with greater sureness its true +objects. If I step, therefore, beyond the narrow precincts of my +specialty in this discussion, to trespass on friendly neighboring +domains, I may plead in my excuse that the subject-matter of knowledge +is common to all domains of research, and that fixed, sharp lines of +demarcation cannot be drawn. + +The belief in occult magic powers of nature has gradually died away, but +in its place a new belief has arisen, the belief in the magical power of +science. Science throws her treasures, not like a capricious fairy into +the laps of a favored few, but into the laps of all humanity, with a +lavish extravagance that no legend ever dreamt of! Not without apparent +justice, therefore, do her distant admirers impute to her the power of +opening up unfathomable abysses of nature, to which the senses cannot +penetrate. Yet she who came to bring light into the world, can well +dispense with the darkness of mystery, and with pompous show, which she +needs neither for the justification of her aims nor for the adornment of +her plain achievements. + +The homely beginnings of science will best reveal to us its simple, +unchangeable character. Man acquires his first knowledge of nature +half-consciously and automatically, from an instinctive habit of +mimicking and forecasting facts in thought, of supplementing sluggish +experience with the swift wings of thought, at first only for his +material welfare. When he hears a noise in the underbrush he constructs +there, just as the animal does, the enemy which he fears; when he sees a +certain rind he forms mentally the image of the fruit which he is in +search of; just as we mentally associate a certain kind of matter with a +certain line in the spectrum or an electric spark with the friction of a +piece of glass. A knowledge of causality in this form certainly reaches +far below the level of Schopenhauer's pet dog, to whom it was ascribed. +It probably exists in the whole animal world, and confirms that great +thinker's statement regarding the will which created the intellect for +its purposes. These primitive psychical functions are rooted in the +economy of our organism not less firmly than are motion and digestion. +Who would deny that we feel in them, too, the elemental power of a long +practised logical and physiological activity, bequeathed to us as an +heirloom from our forefathers? + +Such primitive acts of knowledge constitute to-day the solidest +foundation of scientific thought. Our instinctive knowledge, as we shall +briefly call it, by virtue of the conviction that we have consciously +and intentionally contributed nothing to its formation, confronts us +with an authority and logical power which consciously acquired knowledge +even from familiar sources and of easily tested fallibility can never +possess. All so-called axioms are such instinctive knowledge. Not +consciously gained knowledge alone, but powerful intellectual instinct, +joined with vast conceptive powers, constitute the great inquirer. The +greatest advances of science have always consisted in some successful +formulation, in clear, abstract, and communicable terms, of what was +instinctively known long before, and of thus making it the permanent +property of humanity. By Newton's principle of the equality of pressure +and counterpressure, whose truth all before him had felt, but which no +predecessor had abstractly formulated, mechanics was placed by a single +stroke on a higher level. Our statement might also be historically +justified by examples from the scientific labors of Stevinus, S. Carnot, +Faraday, J. R. Mayer, and others. + +All this, however, is merely the soil from which science starts. The +first real beginnings of science appear in society, particularly in the +manual arts, where the necessity for the communication of experience +arises. Here, where some new discovery is to be described and related, +the compulsion is first felt of clearly defining in consciousness the +important and essential features of that discovery, as many writers can +testify. The aim of instruction is simply the saving of experience; the +labor of one man is made to take the place of that of another. + +The most wonderful economy of communication is found in language. Words +are comparable to type, which spare the repetition of written signs and +thus serve a multitude of purposes; or to the few sounds of which our +numberless different words are composed. Language, with its helpmate, +conceptual thought, by fixing the essential and rejecting the +unessential, constructs its rigid pictures of the fluid world on the +plan of a mosaic, at a sacrifice of exactness and fidelity but with a +saving of tools and labor. Like a piano-player with previously prepared +sounds, a speaker excites in his listener thoughts previously prepared, +but fitting many cases, which respond to the speaker's summons with +alacrity and little effort. + +The principles which a prominent political economist, E. Hermann,[64] +has formulated for the economy of the industrial arts, are also +applicable to the ideas of common life and of science. The economy of +language is augmented, of course, in the terminology of science. With +respect to the economy of written intercourse there is scarcely a doubt +that science itself will realise that grand old dream of the +philosophers of a Universal Real Character. That time is not far +distant. Our numeral characters, the symbols of mathematical analysis, +chemical symbols, and musical notes, which might easily be supplemented +by a system of color-signs, together with some phonetic alphabets now in +use, are all beginnings in this direction. The logical extension of what +we have, joined with a use of the ideas which the Chinese ideography +furnishes us, will render the special invention and promulgation of a +Universal Character wholly superfluous. + +The communication of scientific knowledge always involves description, +that is, a mimetic reproduction of facts in thought, the object of which +is to replace and save the trouble of new experience. Again, to save the +labor of instruction and of acquisition, concise, abridged description +is sought. This is really all that natural laws are. Knowing the value +of the acceleration of gravity, and Galileo's laws of descent, we +possess simple and compendious directions for reproducing in thought all +possible motions of falling bodies. A formula of this kind is a complete +substitute for a full table of motions of descent, because by means of +the formula the data of such a table can be easily constructed at a +moment's notice without the least burdening of the memory. + +No human mind could comprehend all the individual cases of refraction. +But knowing the index of refraction for the two media presented, and the +familiar law of the sines, we can easily reproduce or fill out in +thought every conceivable case of refraction. The advantage here +consists in the disburdening of the memory; an end immensely furthered +by the written preservation of the natural constants. More than this +comprehensive and condensed report about facts is not contained in a +natural law of this sort. In reality, the law always contains less than +the fact itself, because it does not reproduce the fact as a whole but +only in that aspect of it which is important for us, the rest being +either intentionally or from necessity omitted. Natural laws may be +likened to intellectual type of a higher order, partly movable, partly +stereotyped, which last on new editions of experience may become +downright impediments. + +When we look over a province of facts for the first time, it appears to +us diversified, irregular, confused, full of contradictions. We first +succeed in grasping only single facts, unrelated with the others. The +province, as we are wont to say, is not _clear_. By and by we discover +the simple, permanent elements of the mosaic, out of which we can +mentally construct the whole province. When we have reached a point +where we can discover everywhere the same facts, we no longer feel lost +in this province; we comprehend it without effort; it is _explained_ for +us. + +Let me illustrate this by an example. As soon as we have grasped the +fact of the rectilinear propagation of light, the regular course of our +thoughts stumbles at the phenomena of refraction and diffraction. As +soon as we have cleared matters up by our index of refraction we +discover that a special index is necessary for each color. Soon after we +have accustomed ourselves to the fact that light added to light +increases its intensity, we suddenly come across a case of total +darkness produced by this cause. Ultimately, however, we see everywhere +in the overwhelming multifariousness of optical phenomena the fact of +the spatial and temporal periodicity of light, with its velocity of +propagation dependent on the medium and the period. This tendency of +obtaining a survey of a given province with the least expenditure of +thought, and of representing all its facts by some one single mental +process, may be justly termed an economical one. + +The greatest perfection of mental economy is attained in that science +which has reached the highest formal development, and which is widely +employed in physical inquiry, namely, in mathematics. Strange as it +may sound, the power of mathematics rests upon its evasion of +all unnecessary thought and on its wonderful saving of mental +operations. Even those arrangement-signs which we call numbers are a +system of marvellous simplicity and economy. When we employ the +multiplication-table in multiplying numbers of several places, and so +use the results of old operations of counting instead of performing the +whole of each operation anew; when we consult our table of logarithms, +replacing and saving thus new calculations by old ones already +performed; when we employ determinants instead of always beginning +afresh the solution of a system of equations; when we resolve new +integral expressions into familiar old integrals; we see in this simply +a feeble reflexion of the intellectual activity of a Lagrange or a +Cauchy, who, with the keen discernment of a great military commander, +substituted for new operations whole hosts of old ones. No one will +dispute me when I say that the most elementary as well as the highest +mathematics are economically-ordered experiences of counting, put in +forms ready for use. + +In algebra we perform, as far as possible, all numerical operations +which are identical in form once for all, so that only a remnant of work +is left for the individual case. The use of the signs of algebra and +analysis, which are merely symbols of operations to be performed, is due +to the observation that we can materially disburden the mind in this way +and spare its powers for more important and more difficult duties, by +imposing all mechanical operations upon the hand. One result of this +method, which attests its economical character, is the construction of +calculating machines. The mathematician Babbage, the inventor of the +difference-engine, was probably the first who clearly perceived this +fact, and he touched upon it, although only cursorily, in his work, _The +Economy of Manufactures and Machinery_. + +The student of mathematics often finds it hard to throw off the +uncomfortable feeling that his science, in the person of his pencil, +surpasses him in intelligence,--an impression which the great Euler +confessed he often could not get rid of. This feeling finds a sort of +justification when we reflect that the majority of the ideas we deal +with were conceived by others, often centuries ago. In great measure it +is really the intelligence of other people that confronts us in science. +The moment we look at matters in this light, the uncanniness and magical +character of our impressions cease, especially when we remember that we +can think over again at will any one of those alien thoughts. + +Physics is experience, arranged in economical order. By this order not +only is a broad and comprehensive view of what we have rendered +possible, but also the defects and the needful alterations are made +manifest, exactly as in a well-kept household. Physics shares with +mathematics the advantages of succinct description and of brief, +compendious definition, which precludes confusion, even in ideas where, +with no apparent burdening of the brain, hosts of others are contained. +Of these ideas the rich contents can be produced at any moment and +displayed in their full perceptual light. Think of the swarm of +well-ordered notions pent up in the idea of the potential. Is it +wonderful that ideas containing so much finished labor should be easy to +work with? + +Our first knowledge, thus, is a product of the economy of +self-preservation. By communication, the experience of _many_ persons, +individually acquired at first, is collected in _one_. The communication +of knowledge and the necessity which every one feels of managing his +stock of experience with the least expenditure of thought, compel us to +put our knowledge in economical forms. But here we have a clue which +strips science of all its mystery, and shows us what its power really +is. With respect to specific results it yields us nothing that we could +not reach in a sufficiently long time without methods. There is no +problem in all mathematics that cannot be solved by direct counting. But +with the present implements of mathematics many operations of counting +can be performed in a few minutes which without mathematical methods +would take a lifetime. Just as a single human being, restricted wholly +to the fruits of his own labor, could never amass a fortune, but on the +contrary the accumulation of the labor of many men in the hands of one +is the foundation of wealth and power, so, also, no knowledge worthy of +the name can be gathered up in a single human mind limited to the span +of a human life and gifted only with finite powers, except by the most +exquisite economy of thought and by the careful amassment of the +economically ordered experience of thousands of co-workers. What strikes +us here as the fruits of sorcery are simply the rewards of excellent +housekeeping, as are the like results in civil life. But the business of +science has this advantage over every other enterprise, that from _its_ +amassment of wealth no one suffers the least loss. This, too, is its +blessing, its freeing and saving power. + +The recognition of the economical character of science will now help us, +perhaps, to understand better certain physical notions. + +Those elements of an event which we call "cause and effect" are certain +salient features of it, which are important for its mental reproduction. +Their importance wanes and the attention is transferred to fresh +characters the moment the event or experience in question becomes +familiar. If the connexion of such features strikes us as a necessary +one, it is simply because the interpolation of certain intermediate +links with which we are very familiar, and which possess, therefore, +higher authority for us, is often attended with success in our +explanations. That _ready_ experience fixed in the mosaic of the mind +with which we meet new events, Kant calls an innate concept of the +understanding (_Verstandesbegriff_). + +The grandest principles of physics, resolved into their elements, differ +in no wise from the descriptive principles of the natural historian. The +question, "Why?" which is always appropriate where the explanation of a +contradiction is concerned, like all proper habitudes of thought, can +overreach itself and be asked where nothing remains to be understood. +Suppose we were to attribute to nature the property of producing like +effects in like circumstances; just these like circumstances we should +not know how to find. Nature exists once only. Our schematic mental +imitation alone produces like events. Only in the mind, therefore, does +the mutual dependence of certain features exist. + +All our efforts to mirror the world in thought would be futile if we +found nothing permanent in the varied changes of things. It is this that +impels us to form the notion of substance, the source of which is not +different from that of the modern ideas relative to the conservation of +energy. The history of physics furnishes numerous examples of this +impulse in almost all fields, and pretty examples of it may be traced +back to the nursery. "Where does the light go to when it is put out?" +asks the child. The sudden shrivelling up of a hydrogen balloon is +inexplicable to a child; it looks everywhere for the large body which +was just there but is now gone. + +Where does heat come from? Where does heat go to? Such childish +questions in the mouths of mature men shape the character of a century. + +In mentally separating a body from the changeable environment in which +it moves, what we really do is to extricate a group of sensations on +which our thoughts are fastened and which is of relatively greater +stability than the others, from the stream of all our sensations. +Absolutely unalterable this group is not. Now this, now that member of +it appears and disappears, or is altered. In its full identity it never +recurs. Yet the sum of its constant elements as compared with the sum of +its changeable ones, especially if we consider the continuous character +of the transition, is always so great that for the purpose in hand the +former usually appear sufficient to determine the body's identity. But +because we can separate from the group every single member without the +body's ceasing to be for us the same, we are easily led to believe that +after abstracting all the members something additional would remain. It +thus comes to pass that we form the notion of a substance distinct from +its attributes, of a thing-in-itself, whilst our sensations are regarded +merely as symbols or indications of the properties of this +thing-in-itself. But it would be much better to say that bodies or +things are compendious mental symbols for groups of sensations--symbols +that do not exist outside of thought. Thus, the merchant regards the +labels of his boxes merely as indexes of their contents, and not the +contrary. He invests their contents, not their labels, with real value. +The same economy which induces us to analyse a group and to establish +special signs for its component parts, parts which also go to make up +other groups, may likewise induce us to mark out by some single symbol a +whole group. + +On the old Egyptian monuments we see objects represented which do not +reproduce a single visual impression, but are composed of various +impressions. The heads and the legs of the figures appear in profile, +the head-dress and the breast are seen from the front, and so on. We +have here, so to speak, a mean view of the objects, in forming which the +sculptor has retained what he deemed essential, and neglected what he +thought indifferent. We have living exemplifications of the processes +put into stone on the walls of these old temples, in the drawings of our +children, and we also observe a faithful analogue of them in the +formation of ideas in our own minds. Only in virtue of some such +facility of view as that indicated, are we allowed to speak of a body. +When we speak of a cube with trimmed corners--a figure which is not a +cube--we do so from a natural instinct of economy, which prefers to add +to an old familiar conception a correction instead of forming an +entirely new one. This is the process of all judgment. + +The crude notion of "body" can no more stand the test of analysis than +can the art of the Egyptians or that of our little children. The +physicist who sees a body flexed, stretched, melted, and vaporised, cuts +up this body into smaller permanent parts; the chemist splits it up into +elements. Yet even an element is not unalterable. Take sodium. When +warmed, the white, silvery mass becomes a liquid, which, when the heat +is increased and the air shut out, is transformed into a violet vapor, +and on the heat being still more increased glows with a yellow light. If +the name sodium is still retained, it is because of the continuous +character of the transitions and from a necessary instinct of economy. +By condensing the vapor, the white metal may be made to reappear. +Indeed, even after the metal is thrown into water and has passed into +sodium hydroxide, the vanished properties may by skilful treatment still +be made to appear; just as a moving body which has passed behind a +column and is lost to view for a moment may make its appearance after a +time. It is unquestionably very convenient always to have ready the name +and thought for a group of properties wherever that group by any +possibility can appear. But more than a compendious economical symbol +for these phenomena, that name and thought is not. It would be a mere +empty word for one in whom it did not awaken a large group of +well-ordered sense-impressions. And the same is true of the molecules +and atoms into which the chemical element is still further analysed. + +True, it is customary to regard the conservation of weight, or, more +precisely, the conservation of mass, as a direct proof of the constancy +of matter. But this proof is dissolved, when we go to the bottom of it, +into such a multitude of instrumental and intellectual operations, that +in a sense it will be found to constitute simply an equation which our +ideas in imitating facts have to satisfy. That obscure, mysterious lump +which we involuntarily add in thought, we seek for in vain outside the +mind. + +It is always, thus, the crude notion of substance that is slipping +unnoticed into science, proving itself constantly insufficient, and ever +under the necessity of being reduced to smaller and smaller +world-particles. Here, as elsewhere, the lower stage is not rendered +indispensable by the higher which is built upon it, no more than the +simplest mode of locomotion, walking, is rendered superfluous by the +most elaborate means of transportation. Body, as a compound of light and +touch sensations, knit together by sensations of space, must be as +familiar to the physicist who seeks it, as to the animal who hunts its +prey. But the student of the theory of knowledge, like the geologist and +the astronomer, must be permitted to reason back from the forms which +are created before his eyes to others which he finds ready made for +him. + +All physical ideas and principles are succinct directions, frequently +involving subordinate directions, for the employment of economically +classified experiences, ready for use. Their conciseness, as also the +fact that their contents are rarely exhibited in full, often invests +them with the semblance of independent existence. Poetical myths +regarding such ideas,--for example, that of Time, the producer and +devourer of all things,--do not concern us here. We need only remind the +reader that even Newton speaks of an _absolute_ time independent of all +phenomena, and of an absolute space--views which even Kant did not shake +off, and which are often seriously entertained to-day. For the natural +inquirer, determinations of time are merely abbreviated statements of +the dependence of one event upon another, and nothing more. When we say +the acceleration of a freely falling body is 9·810 metres per second, we +mean the velocity of the body with respect to the centre of the earth is +9·810 metres greater when the earth has performed an additional 86400th +part of its rotation--a fact which itself can be determined only by the +earth's relation to other heavenly bodies. Again, in velocity is +contained simply a relation of the position of a body to the position of +the earth.[65] Instead of referring events to the earth we may refer +them to a clock, or even to our internal sensation of time. Now, because +all are connected, and each may be made the measure of the rest, the +illusion easily arises that time has significance independently of +all.[66] + +The aim of research is the discovery of the equations which subsist +between the elements of phenomena. The equation of an ellipse expresses +the universal _conceivable_ relation between its co-ordinates, of which +only the real values have _geometrical_ significance. Similarly, the +equations between the elements of _phenomena_ express a universal, +mathematically conceivable relation. Here, however, for many values only +certain directions of change are _physically_ admissible. As in the +ellipse only certain _values_ satisfying the equation are realised, so +in the physical world only certain _changes_ of value occur. Bodies are +always accelerated towards the earth. Differences of temperature, left +to themselves, always grow less; and so on. Similarly, with respect to +space, mathematical and physiological researches have shown that the +space of experience is simply an _actual_ case of many conceivable +cases, about whose peculiar properties experience alone can instruct us. +The elucidation which this idea diffuses cannot be questioned, despite +the absurd uses to which it has been put. + +Let us endeavor now to summarise the results of our survey. In the +economical schematism of science lie both its strength and its weakness. +Facts are always represented at a sacrifice of completeness and never +with greater precision than fits the needs of the moment. The +incongruence between thought and experience, therefore, will continue to +subsist as long as the two pursue their course side by side; but it will +be continually diminished. + +In reality, the point involved is always the completion of some partial +experience; the derivation of one portion of a phenomenon from some +other. In this act our ideas must be based directly upon sensations. We +call this measuring.[67] The condition of science, both in its origin +and in its application, is a _great relative stability_ of our +environment. What it teaches us is interdependence. Absolute forecasts, +consequently, have no significance in science. With great changes in +celestial space we should lose our co-ordinate systems of space and +time. + +When a geometer wishes to understand the form of a curve, he first +resolves it into small rectilinear elements. In doing this, however, he +is fully aware that these elements are only provisional and arbitrary +devices for comprehending in parts what he cannot comprehend as a whole. +When the law of the curve is found he no longer thinks of the elements. +Similarly, it would not become physical science to see in its +self-created, changeable, economical tools, molecules and atoms, +realities behind phenomena, forgetful of the lately acquired sapience of +her older sister, philosophy, in substituting a mechanical mythology for +the old animistic or metaphysical scheme, and thus creating no end of +suppositious problems. The atom must remain a tool for representing +phenomena, like the functions of mathematics. Gradually, however, as the +intellect, by contact with its subject-matter, grows in discipline, +physical science will give up its mosaic play with stones and will seek +out the boundaries and forms of the bed in which the living stream of +phenomena flows. The goal which it has set itself is the _simplest_ and +_most economical_ abstract expression of facts. + + * * * * * + +The question now remains, whether the same method of research which till +now we have tacitly restricted to physics, is also applicable in the +psychical domain. This question will appear superfluous to the physical +inquirer. Our physical and psychical views spring in exactly the same +manner from instinctive knowledge. We read the thoughts of men in their +acts and facial expressions without knowing how. Just as we predict the +behavior of a magnetic needle placed near a current by imagining +Ampère's swimmer in the current, similarly we predict in thought the +acts and behavior of men by assuming sensations, feelings, and wills +similar to our own connected with their bodies. What we here +instinctively perform would appear to us as one of the subtlest +achievements of science, far outstripping in significance and ingenuity +Ampère's rule of the swimmer, were it not that every child unconsciously +accomplished it. The question simply is, therefore, to grasp +scientifically, that is, by conceptional thought, what we are already +familiar with from other sources. And here much is to be accomplished. A +long sequence of facts is to be disclosed between the physics of +expression and movement and feeling and thought. + +We hear the question, "But how is it possible to explain feeling by the +motions of the atoms of the brain?" Certainly this will never be done, +no more than light or heat will ever be deduced from the law of +refraction. We need not deplore, therefore, the lack of ingenious +solutions of this question. The problem is not a problem. A child +looking over the walls of a city or of a fort into the moat below sees +with astonishment living people in it, and not knowing of the portal +which connects the wall with the moat, cannot understand how they could +have got down from the high ramparts. So it is with the notions of +physics. We cannot climb up into the province of psychology by the +ladder of our abstractions, but we can climb down into it. + +Let us look at the matter without bias. The world consists of colors, +sounds, temperatures, pressures, spaces, times, and so forth, which now +we shall not call sensations, nor phenomena, because in either term an +arbitrary, one-sided theory is embodied, but simply _elements_. The +fixing of the flux of these elements, whether mediately or immediately, +is the real object of physical research. As long as, neglecting our own +body, we employ ourselves with the interdependence of those groups of +elements which, including men and animals, make up _foreign_ bodies, we +are physicists. For example, we investigate the change of the red color +of a body as produced by a change of illumination. But the moment we +consider the special influence on the red of the elements constituting +our body, outlined by the well-known perspective with head invisible, we +are at work in the domain of physiological psychology. We close our +eyes, and the red together with the whole visible world disappears. +There exists, thus, in the perspective field of every sense a portion +which exercises on all the rest a different and more powerful influence +than the rest upon one another. With this, however, all is said. In the +light of this remark, we call _all_ elements, in so far as we regard +them as dependent on this special part (our body), _sensations_. That +the world is our sensation, in this sense, cannot be questioned. But to +make a system of conduct out of this provisional conception, and to +abide its slaves, is as unnecessary for us as would be a similar course +for a mathematician who, in varying a series of variables of a function +which were previously assumed to be constant, or in interchanging the +independent variables, finds his method to be the source of some very +surprising ideas for him.[68] + +If we look at the matter in this unbiassed light it will appear +indubitable that the method of physiological psychology is none other +than that of physics; what is more, that this science is a part of +physics. Its subject-matter is not different from that of physics. It +will unquestionably determine the relations the sensations bear to the +physics of our body. We have already learned from a member of this +academy (Hering) that in all probability a sixfold manifoldness of the +chemical processes of the visual substance corresponds to the sixfold +manifoldness of color-sensation, and a threefold manifoldness of the +physiological processes to the threefold manifoldness of +space-sensations. The paths of reflex actions and of the will are +followed up and disclosed; it is ascertained what region of the brain +subserves the function of speech, what region the function of +locomotion, etc. That which still clings to our body, namely, our +thoughts, will, when those investigations are finished, present no +difficulties new in principle. When experience has once clearly +exhibited these facts and science has marshalled them in economic and +perspicuous order, there is no doubt that we shall _understand_ them. +For other "understanding" than a mental mastery of facts never existed. +Science does not create facts from facts, but simply _orders_ known +facts. + +Let us look, now, a little more closely into the modes of research of +physiological psychology. We have a very clear idea of how a body moves +in the space encompassing it. With our optical field of sight we are +very familiar. But we are unable to state, as a rule, how we have come +by an idea, from what corner of our intellectual field of sight it has +entered, or by what region the impulse to a motion is sent forth. +Moreover, we shall never get acquainted with this mental field of view +from self-observation alone. Self-observation, in conjunction with +physiological research, which seeks out physical connexions, can put +this field of vision in a clear light before us, and will thus first +really reveal to us our inner man. + +Primarily, natural science, or physics, in its widest sense, makes us +acquainted with only the firmest connexions of groups of elements. +Provisorily, we may not bestow too much attention on the single +constituents of those groups, if we are desirous of retaining a +comprehensible whole. Instead of equations between the primitive +variables, physics gives us, as much the easiest course, equations +between _functions_ of those variables. Physiological psychology teaches +us how to separate the visible, the tangible, and the audible from +bodies--a labor which is subsequently richly requited, as the division +of the subjects of physics well shows. Physiology further analyses the +visible into light and space sensations; the first into colors, the last +also into their component parts; it resolves noises into sounds, these +into tones, and so on. Unquestionably this analysis can be carried much +further than it has been. It will be possible in the end to exhibit the +common elements at the basis of very abstract but definite logical acts +of like form,--elements which the acute jurist and mathematician, as it +were, _feels_ out, with absolute certainty, where the uninitiated hears +only empty words. Physiology, in a word, will reveal to us the true real +elements of the world. Physiological psychology bears to physics in its +widest sense a relation similar to that which chemistry bears to physics +in its narrowest sense. But far greater than the mutual support of +physics and chemistry will be that which natural science and psychology +will render each other. And the results that shall spring from this +union will, in all likelihood, far outstrip those of the modern +mechanical physics. + +What those ideas are with which we shall comprehend the world when the +closed circuit of physical and psychological facts shall lie complete +before us, (that circuit of which we now see only two disjoined parts,) +cannot be foreseen at the outset of the work. The men will be found who +will see what is right and will have the courage, instead of wandering +in the intricate paths of logical and historical accident, to enter on +the straight ways to the heights from which the mighty stream of facts +can be surveyed. Whether the notion which we now call matter will +continue to have a scientific significance beyond the crude purposes of +common life, we do not know. But we certainly shall wonder how colors +and tones which were such innermost parts of us could suddenly get lost +in our physical world of atoms; how we could be suddenly surprised that +something which outside us simply clicked and beat, in our heads should +make light and music; and how we could ask whether matter can feel, that +is to say, whether a mental symbol for a group of sensations can feel? + +We cannot mark out in hard and fast lines the science of the future, but +we can foresee that the rigid walls which now divide man from the world +will gradually disappear; that human beings will not only confront each +other, but also the entire organic and so-called lifeless world, with +less selfishness and with livelier sympathy. Just such a presentiment as +this perhaps possessed the great Chinese philosopher Licius some two +thousand years ago when, pointing to a heap of mouldering human bones, +he said to his scholars in the rigid, lapidary style of his tongue: +"These and I alone have the knowledge that we neither live nor are +dead." + + FOOTNOTES: + + [Footnote 60: An address delivered before the anniversary meeting of + the Imperial Academy of Sciences, at Vienna, May 25, 1882.] + + [Footnote 61: _Primitive Culture._] + + [Footnote 62: Tylor, _loc. cit._] + + [Footnote 63: _Essai philosophique sur les probabilités_. 6th Ed. + Paris, 1840, p. 4. The necessary consideration of the initial + velocities is lacking in this formulation.] + + [Footnote 64: _Principien der Wirthschaftslehre_, Vienna, 1873.] + + [Footnote 65: It is clear from this that all so-called elementary + (differential) laws involve a relation to the whole.] + + [Footnote 66: If it be objected, that in the case of perturbations + of the velocity of rotation of the earth, we could be sensible of + such perturbations, and being obliged to have some measure of time, + we should resort to the period of vibration of the waves of sodium + light,--all that this would show is that for practical reasons we + should select that event which best served us as the _simplest_ + common measure of the others.] + + [Footnote 67: Measurement, in fact, is the definition of one + phenomenon by another (standard) phenomenon.] + + [Footnote 68: I have represented the point of view here taken for + more than thirty years and developed it in various writings + (_Erhaltung der Arbeit_, 1872, parts of which are published in the + article on _The Conservation of Energy_ in this collection; _The + Forms of Liquids_, 1872, also published in this collection; and the + _Bewegungsempfindungen_, 1875). The idea, though known to + philosophers, is unfamiliar to the majority of physicists. It is a + matter of deep regret to me, therefore, that the title and author of + a small tract which accorded with my views in numerous details and + which I remember having caught a glance of in a very busy period + (1879-1880), have so completely disappeared from my memory that all + efforts to obtain a clue to them have hitherto been fruitless.] + + + + +ON TRANSFORMATION AND ADAPTATION IN SCIENTIFIC THOUGHT.[69] + + +It was towards the close of the sixteenth century that Galileo with a +superb indifference to the dialectic arts and sophistic subtleties of +the Schoolmen of his time, turned the attention of his brilliant mind to +nature. By nature his ideas were transformed and released from the +fetters of inherited prejudice. At once the mighty revolution was felt, +that was therewith effected in the realm of human thought--felt indeed +in circles far remote and wholly unrelated to the sphere of science, +felt in strata of society that hitherto had only indirectly recognised +the influence of scientific thought. + +And how great and how far-reaching that revolution was! From the +beginning of the seventeenth century till its close we see arising, at +least in embryo, almost all that plays a part in the natural and +technical science of to-day, almost all that in the two centuries +following so wonderfully transformed the facial appearance of the earth, +and all that is moving onward in process of such mighty evolution +to-day. And all this, the direct result of Galilean ideas, the direct +outcome of that freshly awakened sense for the investigation of natural +phenomena which taught the Tuscan philosopher to form the concept and +the law of falling bodies from the _observation_ of a falling stone! +Galileo began his investigations without an implement worthy of the +name; he measured time in the most primitive way, by the efflux of +water. Yet soon afterwards the telescope, the microscope, the barometer, +the thermometer, the air-pump, the steam engine, the pendulum, and the +electrical machine were invented in rapid succession. The fundamental +theorems of dynamical science, of optics, of heat, and of electricity +were all disclosed in the century that followed Galileo. + +Of scarcely less importance, it seems, was that movement which was +prepared for by the illustrious biologists of the hundred years just +past, and formally begun by the late Mr. Darwin. Galileo quickened the +sense for the simpler phenomena of _inorganic_ nature. And with the same +simplicity and frankness that marked the efforts of Galileo, and without +the aid of technical or scientific instruments, without physical or +chemical experiment, but solely by the power of thought and observation, +Darwin grasps a new property of _organic_ nature--which we may briefly +call its _plasticity_.[70] With the same directness of purpose, Darwin, +too, pursues his way. With the same candor and love of truth, he points +out the strength and the weakness of his demonstrations. With masterly +equanimity he holds aloof from the discussion of irrelevant subjects and +wins alike the admiration of his adherents and of his adversaries. + +Scarcely thirty years have elapsed[71] since Darwin first propounded the +principles of his theory of evolution. Yet, already we see his ideas +firmly rooted in every branch of human thought, however remote. +Everywhere, in history, in philosophy, even in the physical sciences, we +hear the watchwords: heredity, adaptation, selection. We speak of the +struggle for existence among the heavenly bodies and of the struggle for +existence in the world of molecules.[72] + +The impetus given by Galileo to scientific thought was marked in every +direction; thus, his pupil, Borelli, founded the school of exact +medicine, from whence proceeded even distinguished mathematicians. And +now Darwinian ideas, in the same way, are animating all provinces of +research. It is true, nature is not made up of two distinct parts, the +inorganic and the organic; nor must these two divisions be treated +perforce by totally distinct methods. Many _sides_, however, nature has. +Nature is like a thread in an intricate tangle, which must be followed +and traced, now from this point, now from that. But we must never +imagine,--and this physicists have learned from Faraday and J. R. +Mayer,--that progress along paths once entered upon is the _only_ means +of reaching the truth. + +It will devolve upon the specialists of the future to determine the +relative tenability and fruitfulness of the Darwinian ideas in the +different provinces. Here I wish simply to consider the growth of +natural _knowledge_ in the light of the theory of evolution. For +knowledge, too, is a product of organic nature. And although ideas, as +such, do not comport themselves in all respects like independent organic +individuals, and although violent comparisons should be avoided, still, +if Darwin reasoned rightly, the general imprint of evolution and +transformation must be noticeable in ideas also. + +I shall waive here the consideration of the fruitful topic of the +transmission of ideas or rather of the transmission of the aptitude for +certain ideas.[73] Nor would it come within my province to discuss +psychical evolution in any form, as Spencer[74] and many other modern +psychologists have done, with varying success. Neither shall I enter +upon a discussion of the struggle for existence and of natural selection +among scientific theories.[75] We shall consider here only such +processes of transformation as every student can easily observe in his +own mind. + + * * * * * + +The child of the forest picks out and pursues with marvellous acuteness +the trails of animals. He outwits and overreaches his foes with +surpassing cunning. He is perfectly at home in the sphere of his +peculiar experience. But confront him with an unwonted phenomenon; place +him face to face with a technical product of modern civilisation, and he +will lapse into impotency and helplessness. Here are facts which he +does not comprehend. If he endeavors to grasp their meaning, he +misinterprets them. He fancies the moon, when eclipsed, to be tormented +by an evil spirit. To his mind a puffing locomotive is a living monster. +The letter accompanying a commission with which he is entrusted, having +once revealed his thievishness, is in his imagination a conscious being, +which he must hide beneath a stone, before venturing to commit a fresh +trespass. Arithmetic to him is like the art of the geomancers in the +Arabian Nights,--an art which is able to accomplish every imaginable +impossibility. And, like Voltaire's _ingénu_, when placed in our social +world, he plays, as we think, the maddest pranks. + +With the man who has made the achievements of modern science and +civilisation his own, the case is quite different. He sees the moon pass +temporarily into the shadow of the earth. He feels in his thoughts the +water growing hot in the boiler of the locomotive; he feels also the +increase of the tension which pushes the piston forward. Where he is not +able to trace the direct relation of things he has recourse to his +yard-stick and table of logarithms, which aid and facilitate his thought +without predominating over it. Such opinions as he cannot concur in, are +at least known to him, and he knows how to meet them in argument. + +Now, wherein does the difference between these two men consist? The +train of thought habitually employed by the first one does not +correspond to the facts that he sees. He is surprised and nonplussed at +every step. But the thoughts of the second man follow and anticipate +events, his thoughts have become adapted or accommodated to the larger +field of observation and activity in which he is located; he conceives +things as they are. The Indian's sphere of experience, however, is quite +different; his bodily organs of sense are in constant activity; he is +ever intensely alert and on the watch for his foes; or, his entire +attention and energy are engaged in procuring sustenance. Now, how can +such a creature project his mind into futurity, foresee or prophesy? +This is not possible until our fellow-beings have, in a measure, +relieved us of our concern for existence. It is then that we acquire +freedom for observation, and not infrequently too that narrowness of +thought which society helps and teaches us to disregard. + +If we move for a time within a fixed circle of phenomena which recur +with unvarying uniformity, our thoughts gradually adapt themselves to +our environment; our ideas reflect unconsciously our surroundings. The +stone we hold in our hand, when dropped, not only falls to the ground in +reality; it also falls in our thoughts. Iron-filings dart towards a +magnet in imagination as well as in fact, and, when thrown into a fire, +they grew hot in conception as well. + +The impulse to complete mentally a phenomenon that has been only +partially observed, has not its origin in the phenomenon itself; of this +fact, we are fully sensible. And we well know that it does not lie +within the sphere of our volition. It seems to confront us rather as a +power and a law imposed from without and controlling both thought and +facts. + +The fact that we are able by the help of this law to prophesy and +forecast, merely proves a sameness or uniformity of environment +sufficient to effect a mental adaptation of this kind. A necessity of +fulfilment, however, is not contained in this compulsory principle which +controls our thoughts; nor is it in any way determined by the +possibility of prediction. We are always obliged, in fact, to await the +completion of what has been predicted. Errors and departures are +constantly discernible, and are slight only in provinces of great rigid +constancy, as in astronomy. + +In cases where our thoughts follow the connexion of events with ease, +and in instances where we positively forefeel the course of a +phenomenon, it is natural to fancy that the latter is determined by and +must conform to our thoughts. But the belief in that mysterious agency +called _causality_, which holds thought and event in unison, is +violently shaken when a person first enters a province of inquiry in +which he has previously had no experience. Take for instance the strange +interaction of electric currents and magnets, or the reciprocal action +of currents, which seem to defy all the resources of mechanical science. +Let him be confronted with such phenomena and he will immediately feel +himself forsaken by his power of prediction; he will bring nothing with +him into this strange field of events but the hope of soon being able +to adapt his ideas to the new conditions there presented. + +A person constructs from a bone the remaining anatomy of an animal; or +from the visible part of a half-concealed wing of a butterfly he infers +and reconstructs the part concealed. He does so with a feeling of +highest confidence in the accuracy of his results; and in these +processes we find nothing preternatural or transcendent. But when +physicists adapt their thoughts to conform to the dynamical course of +events in time, we invariably surround their investigations with a +metaphysical halo; yet these latter adaptations bear quite the same +character as the former, and our only reason for investing them with a +metaphysical garb, perhaps, is their high practical value.[76] + +Let us consider for a moment what takes place when the field of +observation to which our ideas have been adapted and now conform, +becomes enlarged. We had, let us say, always seen heavy bodies sink when +their support was taken away; we had also seen, perhaps, that the +sinking of heavier bodies forced lighter bodies upwards. But now we see +a lever in action, and we are suddenly struck with the fact that a +lighter body is lifting another of much greater weight. Our customary +train of thought demands its rights; the new and unwonted event likewise +demands its rights. From this conflict between thought and fact the +_problem_ arises; out of this partial contrariety springs the question, +"Why?" With the new adaptation to the enlarged field of observation, the +problem disappears, or, in other words, is solved. In the instance +cited, we must adopt the habit of always considering the mechanical work +performed. + +The child just awakening into consciousness of the world, knows no +problem. The bright flower, the ringing bell, are all new to it; yet it +is surprised at nothing. The out and out Philistine, whose only thoughts +lie in the beaten path of his every-day pursuits, likewise has no +problems. Everything goes its wonted course, and if perchance a thing go +wrong at times, it is at most a mere object of curiosity and not worth +serious consideration. In fact, the question "Why?" loses all warrant in +relations where we are familiar with every aspect of events. But the +capable and talented young man has his head full of problems; he has +acquired, to a greater or less degree, certain habitudes of thought, and +at the same time he is constantly observing what is new and unwonted, +and in his case there is no end to the questions, "Why?" + +Thus, the factor which most promotes scientific thought is the gradual +widening of the field of experience. We scarcely notice events we are +accustomed to; the latter do not really develop their intellectual +significance until placed in contrast with something to which we are +unaccustomed. Things that at home are passed by unnoticed, delight us +when abroad, though they may appear in only slightly different forms. +The sun shines with heightened radiance, the flowers bloom in brighter +colors, our fellow-men accost us with lighter and happier looks. And, +returning home, we find even the old familiar scenes more inspiring and +suggestive than before. + +Every motive that prompts and stimulates us to modify and transform our +thoughts, proceeds from what is new, uncommon, and not understood. +Novelty excites wonder in persons whose fixed habits of thought are +shaken and disarranged by what they see. But the element of wonder never +lies in the phenomenon or event observed; its place is in the person +observing. People of more vigorous mental type aim at once at an +_adaptation of thought_ that will conform to what they have observed. +Thus does science eventually become the natural foe of the wonderful. +The sources of the marvellous are unveiled, and surprise gives way to +calm interpretation. + +Let us consider such a mental transformative process in detail. The +circumstance that heavy bodies fall to the earth appears perfectly +natural and regular. But when a person observes that wood floats upon +water, and that flames and smoke rise in the air, then the contrary of +the first phenomenon is presented. An olden theory endeavors to explain +these facts by imputing to substances the power of volition, as that +attribute which is most familiar to man. It asserted that every +substance seeks its proper place, heavy bodies tending downwards and +light ones upwards. It soon turned out, however, that even smoke had +weight, that it, too, sought its place below, and that it was forced +upwards only because of the downward tendency of the air, as wood is +forced to the surface of water because the water exerts the greater +downward pressure. + +Again, we see a body thrown into the air. It ascends. How is it that it +does not seek its proper place? Why does the velocity of its "violent" +motion decrease as it rises, while that of its "natural" fall increases +as it descends. If we mark closely the relation between these two facts, +the problem will solve itself. We shall see, as Galileo did, that the +decrease of velocity in rising and the increase of velocity in falling +are one and the same phenomenon, viz., an increase of velocity towards +the earth. Accordingly, it is not a place that is assigned to the body, +but an increase of velocity towards the earth. + +By this idea the movements of heavy bodies are rendered perfectly +familiar. Newton, now, firmly grasping this new way of thinking, sees +the moon and the planets moving in their paths upon principles similar +to those which determine the motion of a projectile thrown into the air. +Yet the movements of the planets were marked by peculiarities which +compelled him once more to modify slightly his customary mode of +thought. The heavenly bodies, or rather the parts composing them, do not +move with constant accelerations towards each other, but "attract each +other," directly as the mass and inversely as the square of the +distance. + +This latter notion, which includes the one applying to terrestrial +bodies as a special case, is, as we see, quite different from the +conception from which we started. How limited in scope was the original +idea and to what a multitude of phenomena is not the present one +applicable! Yet there is a trace, after all, of the "search for place" +in the expression "attraction." And it would be folly, indeed, for us to +avoid, with punctilious dread, this conception of "attraction" as +bearing marks of its pedigree. It is the historical base of the +Newtonian conception and it still continues to direct our thoughts in +the paths so long familiar to us. Thus, the happiest ideas do not fall +from heaven, but spring from notions already existing. + +Similarly, a ray of light was first regarded as a continuous and +homogeneous straight line. It then became the path of projection for +minute missiles; then an aggregate of the paths of countless different +kinds of missiles. It became periodic; it acquired various sides; and +ultimately it even lost its motion in a straight line. + +The electric current was conceived originally as the flow of a +hypothetical fluid. To this conception was soon added the notion of a +chemical current, the notion of an electric, magnetic, and anisotropic +optical field, intimately connected with the path of the current. And +the richer a conception becomes in following and keeping pace with +facts, the better adapted it is to anticipate them. + +Adaptive processes of this kind have no assignable beginning, inasmuch +as every problem that incites to new adaptation, presupposes a fixed +habitude of thought. Moreover, they have no visible end; in so far as +experience never ceases. Science, accordingly, stands midway in the +evolutionary process; and science may advantageously direct and promote +this process, but it can never take its place. That science is +inconceivable the principles of which would enable a person with no +experience to construct the world of experience, without a knowledge of +it. One might just as well expect to become a great musician, solely by +the aid of theory, and without musical experience; or to become a +painter by following the directions of a text-book. + +In glancing over the history of an idea with which we have become +perfectly familiar, we are no longer able to appreciate the full +significance of its growth. The deep and vital changes that have been +effected in the course of its evolution, are recognisable only from the +astounding narrowness of view with which great contemporary scientists +have occasionally opposed each other. Huygens's wave-theory of light was +incomprehensible to Newton, and Newton's idea of universal gravity was +unintelligible to Huygens. But a century afterwards both notions were +reconcilable, even in ordinary minds. + +On the other hand, the original creations of pioneer intellects, +unconsciously formed, do not assume a foreign garb; their form is their +own. In them, childlike simplicity is joined to the maturity of manhood, +and they are not to be compared with processes of thought in the average +mind. The latter are carried on as are the acts of persons in the state +of mesmerism, where actions involuntarily follow the images which the +words of other persons suggest to their minds. + +The ideas that have become most familiar through long experience, are +the very ones that intrude themselves into the conception of every new +fact observed. In every instance, thus, they become involved in a +struggle for self-preservation, and it is just they that are seized by +the inevitable process of transformation. + +Upon this process rests substantially the method of explaining by +hypothesis new and uncomprehended phenomena. Thus, instead of forming +entirely new notions to explain the movements of the heavenly bodies and +the phenomena of the tides, we imagine the material particles composing +the bodies of the universe to possess weight or gravity with respect to +one another. Similarly, we imagine electrified bodies to be freighted +with fluids that attract and repel, or we conceive the space between +them to be in a state of elastic tension. In so doing, we substitute +for new ideas distinct and more familiar notions of old +experience--notions which to a great extent run unimpeded in their +courses, although they too must suffer partial transformation. + +The animal cannot construct new members to perform every new function +that circumstances and fate demand of it. On the contrary it is obliged +to make use of those it already possesses. When a vertebrate animal +chances into an environment where it must learn to fly or swim, an +additional pair of extremities is not grown for the purpose. On the +contrary, the animal must adapt and transform a pair that it already +has. + +The construction of hypotheses, therefore, is not the product of +artificial scientific methods. This process is unconsciously carried on +in the very infancy of science. Even later, hypotheses do not become +detrimental and dangerous to progress except when more reliance is +placed on them than on the facts themselves; when the contents of the +former are more highly valued than the latter, and when, rigidly +adhering to hypothetical notions, we overestimate the ideas we possess +as compared with those we have to acquire. + +The extension of our sphere of experience always involves a +transformation of our ideas. It matters not whether the face of nature +becomes actually altered, presenting new and strange phenomena, or +whether these phenomena are brought to light by an intentional or +accidental turn of observation. In fact, all the varied methods of +scientific inquiry and of purposive mental adaptation enumerated by John +Stuart Mill, those of observation as well as those of experiment, are +ultimately recognisable as forms of one fundamental method, the method +of change, or variation. It is through change of circumstances that the +natural philosopher learns. This process, however, is by no means +confined to the investigator of nature. The historian, the philosopher, +the jurist, the mathematician, the artist, the æsthetician,[77] all +illuminate and unfold their ideas by producing from the rich treasures +of memory similar, but different, cases; thus, they observe and +experiment in their thoughts. Even if all sense-experience should +suddenly cease, the events of the days past would meet in different +attitudes in the mind and the process of adaptation would still +continue--a process which, in contradistinction to the adaptation of +thoughts to facts in practical spheres, would be strictly theoretical, +being an adaptation of thoughts to thoughts. + +The method of change or variation brings before us like cases of +phenomena, having partly the same and partly different elements. It is +only by comparing different cases of refracted light at changing angles +of incidence that the common factor, the constancy of the refractive +index, is disclosed. And only by comparing the refractions of light of +different colors, does the difference, the inequality of the indices of +refraction, arrest the attention. Comparison based upon change leads the +mind simultaneously to the highest abstractions and to the finest +distinctions. + +Undoubtedly, the animal also is able to distinguish between the similar +and dissimilar of two cases. Its consciousness is aroused by a noise or +a rustling, and its motor centre is put in readiness. The sight of the +creature causing the disturbance, will, according to its size, provoke +flight or prompt pursuit; and in the latter case, the more exact +distinctions will determine the mode of attack. But man alone attains to +the faculty of voluntary and conscious comparison. Man alone can, by his +power of abstraction, rise, in one moment, to the comprehension of +principles like the conservation of mass or the conservation of energy, +and in the next observe and mark the arrangement of the iron lines in +the spectrum. In thus dealing with the objects of his conceptual life, +his ideas unfold and expand, like his nervous system, into a widely +ramified and organically articulated tree, on which he may follow every +limb to its farthermost branches, and, when occasion demands, return to +the trunk from which he started. + +The English philosopher Whewell has remarked that two things are +requisite to the formation of science: facts and ideas. Ideas alone lead +to empty speculation; mere facts can yield no organic knowledge. We see +that all depends upon the capacity of adapting existing notions to fresh +facts. + +Over-readiness to yield to every new fact prevents fixed habits of +thought from arising. Excessively rigid habits of thought impede freedom +of observation. In the struggle, in the compromise between judgment and +prejudgment (prejudice), if we may use the term, our understanding of +things broadens. + +Habitual judgment, applied to a new case without antecedent tests, we +call prejudgment or prejudice. Who does not know its terrible power! But +we think less often of the importance and utility of prejudice. +Physically, no one could exist, if he had to guide and regulate the +circulation, respiration, and digestion of his body by conscious and +purposive acts. So, too, no one could exist intellectually if he had to +form judgments on every passing experience, instead of allowing himself +to be controlled by the judgments he has already formed. Prejudice is a +sort of reflex motion in the province of intelligence. + +On prejudices, that is, on habitual judgments not tested in every case +to which they are applied, reposes a goodly portion of the thought and +work of the natural scientist. On prejudices reposes most of the conduct +of society. With the sudden disappearance of prejudice society would +hopelessly dissolve. That prince displayed a deep insight into the power +of intellectual habit, who quelled the loud menaces and demands of his +body-guard for arrears of pay and compelled them to turn about and +march, by simply pronouncing the regular word of command; he well knew +that they would be unable to resist that. + +Not until the discrepancy between habitual judgments and facts becomes +great is the investigator implicated in appreciable illusion. Then +tragic complications and catastrophes occur in the practical life of +individuals and nations--crises where man, placing custom above life, +instead of pressing it into the service of life, becomes the victim of +his error. The very power which in intellectual life advances, fosters, +and sustains us, may in other circumstances delude and destroy us. + + * * * * * + +Ideas are not all of life. They are only momentary efflorescences of +light, designed to illuminate the paths of the will. But as delicate +reagents on our organic evolution our ideas are of paramount importance. +No theory can gainsay the vital transformation which we feel taking +place within us through their agency. Nor is it necessary that we should +have a proof of this process. We are immediately assured of it. + +The transformation of ideas thus appears as a part of the general +evolution of life, as a part of its adaptation to a constantly widening +sphere of action. A granite boulder on a mountain-side tends towards the +earth below. It must abide in its resting-place for thousands of years +before its support gives way. The shrub that grows at its base is +farther advanced; it accommodates itself to summer and winter. The fox +which, overcoming the force of gravity, creeps to the summit where he +has scented his prey, is freer in his movements than either. The arm of +man reaches further still; and scarcely anything of note happens in +Africa or Asia that does not leave an imprint upon his life. What an +immense portion of the life of other men is reflected in ourselves; +their joys, their affections, their happiness and misery! And this too, +when we survey only our immediate surroundings, and confine our +attention to modern literature. How much more do we experience when we +travel through ancient Egypt with Herodotus, when we stroll through the +streets of Pompeii, when we carry ourselves back to the gloomy period of +the crusades or to the golden age of Italian art, now making the +acquaintance of a physician of Molière, and now that of a Diderot or of +a D'Alembert. What a great part of the life of others, of their +character and their purpose, do we not absorb through poetry and music! +And although they only gently touch the chords of our emotions, like the +memory of youth softly breathing upon the spirit of an aged man, we have +nevertheless lived them over again in part. How great and comprehensive +does self become in this conception; and how insignificant the person! +Egoistical systems both of optimism and pessimism perish with their +narrow standard of the import of intellectual life. We feel that the +real pearls of life lie in the ever changing contents of consciousness, +and that the person is merely an indifferent symbolical thread on which +they are strung.[78] + +We are prepared, thus, to regard ourselves and every one of our ideas as +a product and a subject of universal evolution; and in this way we shall +advance sturdily and unimpeded along the paths which the future will +throw open to us.[79] + + FOOTNOTES: + + [Footnote 69: Inaugural Address, delivered on assuming the Rectorate + of the University of Prague, October 18, 1883. + + The idea presented in this essay is neither new nor remote. I have + touched upon it myself on several occasions (first in 1867), but + have never made it the subject of a formal disquisition. Doubtless, + others, too, have treated it; it lies, so to speak, in the air. + However, as many of my illustrations were well received, although + known only in an imperfect form from the lecture itself and the + newspapers, I have, contrary to my original intention, decided to + publish it. It is not my intention to trespass here upon the domain + of biology. My statements are to be taken merely as the expression + of the fact that no one can escape the influence of a great and + far-reaching idea.] + + [Footnote 70: At first sight an apparent contradiction arises from + the admission of both heredity and adaptation; and it is undoubtedly + true that a strong disposition to heredity precludes great + capability of adaptation. But imagine the organism to be a plastic + mass which retains the form transmitted to it by former influences + until new influences modify it; the _one_ property of _plasticity_ + will then represent capability of adaptation as well as power of + heredity. Analogous to this is the case of a bar of magnetised steel + of high coercive force: the steel retains its magnetic properties + until a new force displaces them. Take also a body in motion: the + body retains the velocity acquired in (_inherited_ from) the + interval of time just preceding, except it be changed in the next + moment by an accelerating force. In the case of the body in motion + the _change_ of velocity (_Abänderung_) was looked upon as a matter + of course, while the discovery of the principle of _inertia_ (or + persistence) created surprise; in Darwin's case, on the contrary, + _heredity_ (or persistence) was taken for granted, while the + principle of _variation_ (_Abänderung_) appeared novel. + + Fully adequate views are, of course, to be reached only by a study + of the original facts emphasised by Darwin, and not by these + analogies. The example referring to motion, if I am not mistaken, I + first heard, in conversation, from my friend J. Popper, Esq., of + Vienna. + + Many inquirers look upon the stability of the species as something + settled, and oppose to it the Darwinian theory. But the stability of + the species is itself a "theory." The essential modifications which + Darwin's views also are undergoing will be seen from the works of + Wallace [and Weismann], but more especially from a book of W. H. + Rolph, _Biologische Probleme_, Leipsic, 1882. Unfortunately, this + last talented investigator is no longer numbered among the living.] + + [Footnote 71: Written in 1883.] + + [Footnote 72: See Pfaundler, _Pogg. Ann., Jubelband_, p. 182.] + + [Footnote 73: See the beautiful discussions of this point in + Hering's _Memory as a General Function of Organised Matter_ (1870), + Chicago, The Open Court Publishing Co., 1887. Compare also Dubois, + _Ueber die Uebung_, Berlin, 1881.] + + [Footnote 74: Spencer, _The Principles of Psychology_. London, + 1872.] + + [Footnote 75: See the article _The Velocity of Light_, page 63.] + + [Footnote 76: I am well aware that the endeavor to confine oneself + in natural research to _facts_ is often censured as an exaggerated + fear of metaphysical spooks. But I would observe, that, judged by + the mischief which they have wrought, the metaphysical, of all + spooks, are the least fabulous. It is not to be denied that many + forms of thought were not originally acquired by the individual, but + were antecedently formed, or rather prepared for, in the development + of the species, in some such way as Spencer, Haeckel, Hering, and + others have supposed, and as I myself have hinted on various + occasions.] + + [Footnote 77: Compare, for example, _Schiller, Zerstreute + Betrachtungen über verschiedene ästhetische Gegenstände_.] + + [Footnote 78: We must not be deceived in imagining that the + happiness of other people is not a very considerable and essential + portion of our own. It is common capital, which cannot be created by + the individual, and which does not perish with him. The formal and + material limitation of the _ego_ is necessary and sufficient only + for the crudest practical objects, and cannot subsist in a broad + conception. Humanity in its entirety may be likened to a + polyp-plant. The material and organic bonds of individual union + have, indeed, been severed; they would only have impeded freedom of + movement and evolution. But the ultimate aim, the psychical + connexion of the whole, has been attained in a much higher degree + through the richer development thus made possible.] + + [Footnote 79: C. E. von Baer, the subsequent opponent of Darwin and + Haeckel, has discussed in two beautiful addresses (_Das allgemeinste + Gesetz der Natur in aller Entwickelung_, and _Welche Auffassung der + lebenden Natur ist die richtige, und wie ist diese Auffassung auf + die Entomologie anzuwenden_?) the narrowness of the view which + regards an animal in its existing state as finished and complete, + instead of conceiving it as a phase in the series of evolutionary + forms and regarding the species itself as a phase of the development + of the animal world in general.] + + + + +ON THE PRINCIPLE OF COMPARISON IN PHYSICS.[80] + + +Twenty years ago when Kirchhoff defined the object of mechanics as the +"description, in complete and very simple terms, of the motions +occurring in nature," he produced by the statement a peculiar +impression. Fourteen years subsequently, Boltzmann, in the life-like +picture which he drew of the great inquirer, could still speak of the +universal astonishment at this novel method of treating mechanics, and +we meet with epistemological treatises to-day, which plainly show how +difficult is the acceptance of this point of view. A modest and small +band of inquirers there were, however, to whom Kirchhoff's few words +were tidings of a welcome and powerful ally in the epistemological +field. + +Now, how does it happen that we yield our assent so reluctantly to the +philosophical opinion of an inquirer for whose scientific achievements +we have only words of praise? One reason probably is that few inquirers +can find time and leisure, amid the exacting employments demanded for +the acquisition of new knowledge, to inquire closely into that +tremendous psychical process by which science is formed. Further, it is +inevitable that much should be put into Kirchhoff's rigid words that +they were not originally intended to convey, and that much should be +found wanting in them that had always been regarded as an essential +element of scientific knowledge. What can mere description accomplish? +What has become of explanation, of our insight into the causal connexion +of things? + + * * * * * + +Permit me, for a moment, to contemplate not the results of science, but +the mode of its _growth_, in a frank and unbiassed manner. We know of +only _one_ source of _immediate revelation_ of scientific facts--_our +senses_. Restricted to this source alone, thrown wholly upon his own +resources, obliged to start always anew, what could the isolated +individual accomplish? Of a stock of knowledge so acquired the science +of a distant negro hamlet in darkest Africa could hardly give us a +sufficiently humiliating conception. For there that veritable miracle of +thought-transference has already begun its work, compared with which the +miracles of the spiritualists are rank monstrosities--_communication by +language_. Reflect, too, that by means of the magical characters which +our libraries contain we can raise the spirits of the "the sovereign +dead of old" from Faraday to Galileo and Archimedes, through ages of +time--spirits who do not dismiss us with ambiguous and derisive +oracles, but tell us the best they know; then shall we feel what a +stupendous and indispensable factor in the formation of science +_communication_ is. Not the dim, half-conscious _surmises_ of the acute +observer of nature or critic of humanity belong to science, but only +that which they possess clearly enough to _communicate_ to others. + +But how, now, do we go about this communication of a newly acquired +experience, of a newly observed fact? As the different calls and +battle-cries of gregarious animals are unconsciously formed signs for a +common observation or action, irrespective of the causes which produce +such action--a fact that already involves the germ of the concept; so +also the words of human language, which is only more highly specialised, +are names or signs for universally known facts, which all can observe or +have observed. If the mental representation, accordingly, follows the +new fact at once and _passively_, then that new fact must, of itself, +immediately be constituted and represented in thought by facts already +universally known and commonly observed. Memory is always ready to put +forward for _comparison_ known facts which resemble the new event, or +agree with it in certain features, and so renders possible that +elementary internal judgment which the mature and definitively +formulated judgment soon follows. + +Comparison, as the fundamental condition of communication, is the most +powerful inner vital element of science. The zoölogist sees in the +bones of the wing-membranes of bats, fingers; he compares the bones of +the cranium with the vertebræ, the embryos of different organisms with +one another, and the different stages of development of the same +organism with one another. The geographer sees in Lake Garda a fjord, in +the Sea of Aral a lake in process of drying up. The philologist compares +different languages with one another, and the formations of the same +language as well. If it is not customary to speak of comparative physics +in the same sense that we speak of comparative anatomy, the reason is +that in a science of such great experimental activity the attention is +turned away too much from the _contemplative_ element. But like all +other sciences, physics lives and grows by comparison. + + * * * * * + +The manner in which the result of the comparison finds expression in the +communication, varies of course very much. When we say that the colors +of the spectrum are red, yellow, green, blue, and violet, the +designations employed may possibly have been derived from the technology +of tattooing, or they may subsequently have acquired the significance of +standing for the colors of the rose, the lemon, the leaf, the +corn-flower, and the violet. From the frequent repetition of such +comparisons, however, made under the most manifold circumstances, the +inconstant features, as compared with the permanent congruent features, +get so obliterated that the latter acquire a fixed significance +independent of every object and connexion, or take on as we say an +_abstract_ or _conceptual_ import. No one thinks at the word "red" of +any other agreement with the rose than that of color, or at the word +"straight" of any other property of a stretched cord than the sameness +of direction. Just so, too, numbers, originally the names of the fingers +of the hands and feet, from being used as arrangement-signs for all +kinds of objects, were lifted to the plane of abstract concepts. A +verbal report (communication) of a fact that uses only these purely +abstract implements, we call a _direct description_. + +The direct description of a fact of any great extent is an irksome task, +even where the requisite notions are already completely developed. What +a simplification it involves if we can say, the fact _A_ now considered +comports itself, not in _one_, but in _many_ or in _all_ its features, +like an old and well-known fact _B_. The moon comports itself as a heavy +body does with respect to the earth; light like a wave-motion or an +electric vibration; a magnet, as if it were laden with gravitating +fluids, and so on. We call such a description, in which we appeal, as it +were, to a description already and elsewhere formulated, or perhaps +still to be precisely formulated, an _indirect description_. We are at +liberty to supplement this description, gradually, by direct +description, to correct it, or to replace it altogether. We see, thus, +without difficulty, that what is called a _theory_ or a _theoretical +idea_, falls under the category of what is here termed indirect +description. + + * * * * * + +What, now, is a theoretical idea? Whence do we get it? What does it +accomplish for us? Why does it occupy a higher place in our judgment +than the mere holding fast to a fact or an observation? Here, too, +memory and comparison alone are in play. But instead of _a single_ +feature of resemblance culled from memory, in this case _a great system_ +of resemblances confronts us, a well-known physiognomy, by means of +which the new fact is immediately transformed into an old acquaintance. +Besides, it is in the power of the idea to offer us more than we +actually see in the new fact, at the first moment; it can extend the +fact, and enrich it with features which we are first induced to _seek_ +from such suggestions, and which are often actually found. It is this +_rapidity_ in extending knowledge that gives to theory a preference over +simple observation. But that preference is wholly _quantitative_. +Qualitatively, and in real essential points, theory differs from +observation neither in the mode of its origin nor in its last results. + +The adoption of a theory, however, always involves a danger. For a +theory puts in the place of a fact _A_ in thought, always a _different_, +but simpler and more familiar fact _B_, which in _some_ relations can +mentally represent _A_, but for the very reason that it is different, in +other relations cannot represent it. If now, as may readily happen, +sufficient care is not exercised, the most fruitful theory may, in +special circumstances, become a downright obstacle to inquiry. Thus, the +emission-theory of light, in accustoming the physicist to think of the +projectile path of the "light-particles" as an undifferentiated +straight-line, demonstrably impeded the discovery of the periodicity of +light. By putting in the place of light the more familiar phenomena of +sound, Huygens renders light in many of its features a familiar event, +but with respect to polarisation, which lacks the longitudinal waves +with which alone he was acquainted, it had for him a doubly strange +aspect. He is unable thus to grasp in abstract thought the fact of +polarisation, which is before his eyes, whilst Newton, merely by +adapting to the observation his thoughts, and putting this question, +"_Annon radiorum luminis diversa sunt latera?_" abstractly grasped +polarisation, that is, directly described it, a century before Malus. On +the other hand, if the agreement of the fact with the idea theoretically +representing it, extends further than its inventor originally +anticipated, then we may be led by it to unexpected discoveries, of +which conical refraction, circular polarisation by total reflexion, +Hertz's waves offer ready examples, in contrast to the illustrations +given above. + +Our insight into the conditions indicated will be improved, perhaps, by +contemplating the development of some theory or other more in detail. +Let us consider a magnetised bar of steel by the side of a second +unmagnetised bar, in all other respects the same. The second bar gives +no indication of the presence of iron-filings; the first attracts them. +Also, when the iron-filings are absent, we must think of the magnetised +bar as in a different condition from that of the unmagnetised. For, that +the mere presence of the iron-filings does not induce the phenomenon of +attraction is proved by the second unmagnetised bar. The ingenuous man, +who finds in his will, as his most familiar source of power, the best +facilities for comparison, conceives a species of _spirit_ in the +magnet. The behavior of a warm body or of an _electrified_ body suggests +similar ideas. This is the point of view of the oldest theory, +_fetishism_, which the inquirers of the early Middle Ages had not yet +overcome, and which in its last vestiges, in the conception of forces, +still flourishes in modern physics. We see, thus, the _dramatic_ element +need no more be absent in a scientific description, than in a thrilling +novel. + +If, on subsequent examination, it be observed that a cold body, in +contact with a hot body, warms itself, so to speak, _at the expense_ of +the hot body; further, that when the substances are the same, the cold +body, which, let us say, has twice the mass of the other, gains only +half the number of degrees of temperature that the other loses, a wholly +new impression arises. The demoniac character of the event vanishes, for +the supposed spirit acts not by caprice, but according to fixed laws. In +its place, however, _instinctively_ the notion of a _substance_ is +substituted, part of which flows over from the one body to the other, +but the total amount of which, representable by the sum of the products +of the masses into the respective changes of temperature, remains +constant. Black was the first to be _powerfully_ struck with this +resemblance of thermal processes to the motion of a substance, and under +its guidance discovered the specific heat, the heat of fusion, and the +heat of vaporisation of bodies. Gaining strength and fixity, however, +from these successes, this notion of substance subsequently stood in the +way of scientific advancement. It blinded the eyes of the successors of +Black, and prevented them from seeing the manifest fact, which every +savage knows, that heat is _produced_ by friction. Fruitful as that +notion was for Black, helpful as it still is to the learner to-day in +Black's special field, permanent and universal validity as a _theory_ it +could never maintain. But what is essential, conceptually, in it, viz., +the constancy of the product-sum above mentioned, retains its value and +may be regarded as a _direct description_ of Black's facts. + +It stands to reason that those theories which push themselves forward +unsought, instinctively, and wholly of their own accord, should have the +greatest power, should carry our thoughts most with them, and exhibit +the staunchest powers of self-preservation. On the other hand, it may +also be observed that when critically scrutinised such theories are +extremely apt to lose their cogency. We are constantly busied with +"substance," its modes of action have stamped themselves indelibly upon +our thoughts, our vividest and clearest reminiscences are associated +with it. It should cause us no surprise, therefore, that Robert Mayer +and Joule, who gave the final blow to Black's substantial conception of +heat, should have re-introduced the same notion of substance in a more +abstract and modified form, only applying to a much more extensive +field. + +Here, too, the psychological circumstances which impart to the new +conception its power, lie clearly before us. By the unusual redness of +the venous blood in tropical climates Mayer's attention is directed to +the lessened expenditure of internal heat and to the proportionately +lessened _consumption of material_ by the human body in those climates. +But as every effort of the human organism, including its mechanical +work, is connected with the consumption of material, and as work by +friction can engender heat, therefore heat and work appear in kind +equivalent, and between them a proportional relation must subsist. Not +_every_ quantity, but the appropriately calculated _sum_ of the two, as +connected with a proportionate consumption of material, appears +_substantial_. + +By exactly similar considerations, relative to the economy of the +galvanic element, Joule arrived at his view; he found experimentally +that the sum of the heat evolved in the circuit, of the heat consumed in +the combustion of the gas developed, of the electro-magnetic work of +the current, properly calculated,--in short, the sum of all the effects +of the battery,--is connected with a proportionate consumption of zinc. +Accordingly, this sum itself has a substantial character. + +Mayer was so absorbed with the view attained, that the indestructibility +of _force_, in our phraseology _work_, appeared to him _a priori_ +evident. "The creation or annihilation of a force," he says, "lies +without the province of human thought and power." Joule expressed +himself to a similar effect: "It is manifestly absurd to suppose that +the powers with which God has endowed matter can be destroyed." Strange +to say, on the basis of such utterances, not Joule, but Mayer, was +stamped as a metaphysician. We may be sure, however, that both men were +merely giving expression, and that half-unconsciously, to a powerful +_formal_ need of the new simple view, and that both would have been +extremely surprised if it had been proposed to them that their principle +should be submitted to a philosophical congress or ecclesiastical synod +for a decision upon its validity. But with all agreements, the attitude +of these two men, in other respects, was totally different. Whilst Mayer +represented this _formal_ need with all the stupendous instinctive force +of genius, we might say almost with the ardor of fanaticism, yet was +withal not wanting in the conceptive ability to compute, prior to all +other inquirers, the mechanical equivalent of heat from old physical +constants long known and at the disposal of all, and so to set up for +the new doctrine a programme embracing all physics and physiology; +Joule, on the other hand, applied himself to the exact verification of +the doctrine by beautifully conceived and masterfully executed +experiments, extending over all departments of physics. Soon Helmholtz +too attacked the problem, in a totally independent and characteristic +manner. After the professional virtuosity with which this physicist +grasped and disposed of all the points unsettled by Mayer's programme +and more besides, what especially strikes us is the consummate critical +lucidity of this young man of twenty-six years. In his exposition is +wanting that vehemence and impetuosity which marked Mayer's. The +principle of the conservation of energy is no self-evident or _a priori_ +proposition for him. What follows, on the assumption that that +proposition obtains? In this hypothetical form, he subjugates his +matter. + +I must confess, I have always marvelled at the æsthetic and ethical +taste of many of our contemporaries who have managed to fabricate out of +this relation of things, odious national and personal questions, instead +of praising the good fortune that made _several_ such men work together +and of rejoicing at the instructive diversity and idiosyncrasies of +great minds fraught with such rich consequences for us. + +We know that still another theoretical conception played a part in the +development of the principle of energy, which Mayer held aloof from, +namely, the conception that heat, as also the other physical processes, +are due to motion. But once the principle of energy has been reached, +these auxiliary and transitional theories discharge no essential +function, and we may regard the principle, like that which Black gave, +as a contribution to the _direct description_ of a widely extended +domain of facts. + +It would appear from such considerations not only advisable, but even +necessary, with all due recognition of the helpfulness of theoretic +ideas in research, yet gradually, as the new facts grow familiar, to +substitute for indirect description _direct_ description, which contains +nothing that is unessential and restricts itself absolutely to the +abstract apprehension of facts. We might almost say, that the +descriptive sciences, so called with a tincture of condescension, have, +in respect of scientific character, outstripped the physical expositions +lately in vogue. Of course, a virtue has been made of necessity here. + +We must admit, that it is not in our power to describe directly every +fact, on the moment. Indeed, we should succumb in utter despair if the +whole wealth of facts which we come step by step to know, were presented +to us all at once. Happily, only detached and unusual features first +strike us, and such we bring nearer to ourselves by _comparison_ with +every-day events. Here the notions of the common speech are first +developed. The comparisons then grow more manifold and numerous, the +fields of facts compared more extensive, the concepts that make direct +description possible, proportionately more general and more abstract. + +First we become familiar with the motion of freely falling bodies. The +concepts of force, mass, and work are then carried over, with +appropriate modifications, to the phenomena of electricity and +magnetism. A stream of water is said to have suggested to Fourier the +first distinct picture of currents of heat. A special case of vibrations +of strings investigated by Taylor, cleared up for him a special case of +the conduction of heat. Much in the same way that Daniel Bernoulli and +Euler constructed the most diverse forms of vibrations of strings from +Taylor's cases, so Fourier constructs out of simple cases of conduction +the most multifarious motions of heat; and that method has extended +itself over the whole of physics. Ohm forms his conception of the +electric current in imitation of Fourier's. The latter, also, adopts +Fick's theory of diffusion. In an analogous manner a conception of the +magnetic current is developed. All sorts of stationary currents are thus +made to exhibit common features, and even the condition of complete +equilibrium in an extended medium shares these features with the +dynamical condition of equilibrium of a stationary current. Things as +remote as the magnetic lines of force of an electric current and the +stream-lines of a frictionless liquid vortex enter in this way into a +peculiar relationship of similarity. The concept of potential, +originally enunciated for a restricted province, acquires a +wide-reaching applicability. Things as dissimilar as pressure, +temperature, and electromotive force, now show points of agreement in +relation to ideas derived by definite methods from that concept: viz., +fall of pressure, fall of temperature, fall of potential, as also with +the further notions of liquid, thermal, and electric strength of +current. That relationship between systems of ideas in which the +dissimilarity of every two homologous concepts as well as the agreement +in logical relations of every two homologous pairs of concepts, is +clearly brought to light, is called an _analogy_. It is an effective +means of mastering heterogeneous fields of facts in unitary +comprehension. The path is plainly shown in which _a universal physical +phenomenology_ embracing all domains, will be developed. + +In the process described we attain for the first time to what is +indispensable in the direct description of broad fields of fact--the +wide-reaching _abstract concept_. And now I must put a question smacking +of the school-master, but unavoidable: What is a concept? Is it a hazy +representation, admitting withal of mental visualisation? No. Mental +visualisation accompanies it only in the simplest cases, and then merely +as an adjunct. Think, for example, of the "coefficient of +self-induction," and seek for its visualised mental image. Or is, +perhaps, the concept a mere word? The adoption of this forlorn idea, +which has been actually proposed of late by a reputed mathematician +would only throw us back a thousand years into the deepest +scholasticism. We must, therefore, reject it. + +The solution is not far to seek. We must not think that sensation, or +representation, is a purely passive process. The lowest organisms +respond to it with a simple reflex motion, by engulfing the prey which +approaches them. In higher organisms the centripetal stimulus encounters +in the nervous system obstacles and aids which modify the centrifugal +process. In still higher organisms, where prey is pursued and examined, +the process in question may go through extensive paths of circular +motions before it comes to relative rest. Our own life, too, is enacted +in such processes; all that we call science may be regarded as parts, or +middle terms, of such activities. + +It will not surprise us now if I say: the definition of a concept, and, +when it is very familiar, even its name, is an _impulse_ to some +accurately determined, often complicated, critical, comparative, or +constructive _activity_, the usually sense-perceptive result of which is +a term or member of the concept's scope. It matters not whether the +concept draws the attention only to one certain sense (as sight) or to a +phase of a sense (as color, form), or is the starting-point of a +complicated action; nor whether the activity in question (chemical, +anatomical, and mathematical operations) is muscular or technical, or +performed wholly in the imagination, or only intimated. The concept is +to the physicist what a musical note is to a piano-player. A trained +physicist or mathematician reads a memoir like a musician reads a score. +But just as the piano-player must first learn to move his fingers singly +and collectively, before he can follow his notes without effort, so the +physicist or mathematician must go through a long apprenticeship before +he gains control, so to speak, of the manifold delicate innervations of +his muscles and imagination. Think of how frequently the beginner in +physics or mathematics performs more, or less, than is required, or of +how frequently he conceives things differently from what they are! But +if, after having had sufficient discipline, he lights upon the phrase +"coefficient of self-induction," he knows immediately what that term +requires of him. Long and thoroughly practised _actions_, which have +their origin in the necessity of comparing and representing facts by +other facts, are thus the very kernel of concepts. In fact, positive and +philosophical philology both claim to have established that all roots +represent concepts and stood originally for muscular activities alone. +The slow assent of physicists to Kirchhoff's dictum now becomes +intelligible. They best could feel the vast amount of individual labor, +theory, and skill required before the ideal of direct description could +be realised. + + * * * * * + +Suppose, now, the ideal of a given province of facts is reached. Does +description accomplish all that the inquirer can ask? In my opinion, it +does. Description is a building up of facts in thought, and this +building up is, in the experimental sciences, often the condition of +actual execution. For the physicist, to take a special case, the +metrical units are the building-stones, the concepts the directions for +building, and the facts the result of the building. Our mental imagery +is almost a complete substitute for the fact, and by means of it we can +ascertain all the fact's properties. We do not know that worst which we +ourselves have made. + +People require of science that it should _prophesy_, and Hertz uses that +expression in his posthumous _Mechanics_. But, natural as it is, the +expression is too narrow. The geologist and the palæontologist, at times +the astronomer, and always the historian and the philologist, prophesy, +so to speak, _backwards_. The descriptive sciences, like geometry and +mathematics, prophesy neither forward or backwards, but seek from given +conditions the conditioned. Let us say rather: _Science completes in +thought facts that are only partly given_. This is rendered possible by +description, for description presupposes the interdependence of the +descriptive elements: otherwise nothing would be described. + +It is said, description leaves the sense of causality unsatisfied. In +fact, many imagine they understand motions better when they picture to +themselves the pulling forces; and yet the _accelerations_, the facts, +accomplish more, without superfluous additions. I hope that the science +of the future will discard the idea of cause and effect, as being +formally obscure; and in my feeling that these ideas contain a strong +tincture of fetishism, I am certainly not alone. The more proper course +is, _to regard the abstract determinative elements of a fact as +interdependent_, in a purely logical way, as the mathematician or +geometer does. True, by comparison with the will, forces are brought +nearer to our feeling; but it may be that ultimately the will itself +will be made clearer by comparison with the accelerations of masses. + +If we are asked, candidly, when is a fact _clear_ to us, we must say +"when we can reproduce it by very _simple_ and very familiar +intellectual operations, such as the construction of accelerations, or +the geometrical summation of accelerations, and so forth." The +requirement of _simplicity_ is of course to the expert a different +matter from what it is to the novice. For the first, description by a +system of differential equations is sufficient; for the second, a +gradual construction out of elementary laws is required. The first +discerns at once the connexion of the two expositions. Of course, it is +not disputed that the _artistic_ value of materially equivalent +descriptions may not be different. + +Most difficult is it to persuade strangers that the grand universal laws +of physics, such as apply indiscriminately to material, electrical, +magnetic, and other systems, are not essentially different from +descriptions. As compared with many sciences, physics occupies in this +respect a position of vantage that is easily explained. Take, for +example, anatomy. As the anatomist in his quest for agreements and +differences in animals ascends to ever higher and higher +_classifications_, the individual facts that represent the ultimate +terms of the system, are still so different that they must be _singly_ +noted. Think, for example, of the common marks of the Vertebrates, of +the class-characters of Mammals and Birds on the one hand and of Fishes +on the other, of the double circulation of the blood on the one hand and +of the single on the other. In the end, always _isolated_ facts remain, +which show only a _slight_ likeness to one another. + +A science still more closely allied to physics, chemistry, is often in +the same strait. The abrupt change of the qualitative properties, in all +likelihood conditioned by the slight stability of the intermediate +states, the remote resemblance of the co-ordinated facts of chemistry +render the treatment of its data difficult. Pairs of bodies of different +qualitative properties unite in different mass-ratios; but no connexion +between the first and the last is to be noted, at first. + +Physics, on the other hand, reveals to us wide domains of _qualitatively +homogeneous_ facts, differing from one another only in the number of +equal parts into which their characteristic marks are divisible, that +is, differing only _quantitatively_. Even where we have to deal with +qualities (colors and sounds), quantitative characters of those +qualities are at our disposal. Here the classification is so simple a +task that it rarely impresses us as such, whilst in infinitely fine +gradations, in a _continuum of facts_, our number-system is ready +beforehand to follow as far as we wish. The co-ordinated facts are here +extremely similar and very closely affined, as are also their +descriptions which consist in the determination of the numerical +measures of one given set of characters from those of a different set by +means of familiar mathematical operations--methods of derivation. Thus, +the common characteristics of all descriptions can be found here; and +with them a succinct, comprehensive description, or a rule for the +construction of all single descriptions, is assigned,--and this we call +_law_. Well-known examples are the formulæ for freely falling bodies, +for projectiles, for central motion, and so forth. If physics apparently +accomplishes more by its methods than other sciences, we must remember +that in a sense it has presented to it much simpler problems. + +The remaining sciences, whose facts also present a physical side, need +not be envious of physics for this superiority; for all its acquisitions +ultimately redound to their benefit as well. But also in other ways this +mutual help shall and must change. Chemistry has advanced very far in +making the methods of physics her own. Apart from older attempts, the +periodical series of Lothar Meyer and Mendelejeff are a brilliant and +adequate means of producing an easily surveyed system of facts, which by +gradually becoming complete, will take the place almost of a continuum +of facts. Further, by the study of solutions, of dissociation, in fact +generally of phenomena which present a continuum of cases, the methods +of thermodynamics have found entrance into chemistry. Similarly we may +hope that, at some future day, a mathematician, letting the +fact-continuum of embryology play before his mind, which the +palæontologists of the future will supposedly have enriched with more +intermediate and derivative forms between Saurian and Bird than the +isolated Pterodactyl, Archæopteryx, Ichthyornis, and so forth, which we +now have--that such a mathematician shall transform, by the variation of +a few parameters, as in a dissolving view, one form into another, just +as we transform one conic section into another. + +Reverting now to Kirchhoff's words, we can come to some agreement +regarding their import. Nothing can be built without building-stones, +mortar, scaffolding, and a builder's skill. Yet assuredly the wish is +well founded, that will show to posterity the complete structure in its +finished form, bereft of unsightly scaffolding. It is the pure logical +and æsthetic sense of the mathematician that speaks out of Kirchhoff's +words. Modern expositions of physics aspire after his ideal; that, too, +is intelligible. But it would be a poor didactic shift, for one whose +business it was to train architects, to say: "Here is a splendid +edifice; if thou wouldst really build, go thou and do likewise". + +The barriers between the special sciences, which make division of work +and concentration possible, but which appear to us after all as cold and +conventional restrictions, will gradually disappear. Bridge upon bridge +is thrown over the gaps. Contents and methods, even of the remotest +branches, are compared. When the Congress of Natural Scientists shall +meet a hundred years hence, we may expect that they will represent a +unity in a higher sense than is possible to-day, not in sentiment and +aim alone, but in method also. In the meantime, this great change will +be helped by our keeping constantly before our minds the fact of the +intrinsic relationship of all research, which Kirchhoff characterised +with such classical simplicity. + + FOOTNOTES: + + [Footnote 80: An address delivered before the General Session of the + German Association of Naturalists and Physicians, at Vienna, Sept. + 24, 1894.] + + + + +THE PART PLAYED BY ACCIDENT IN INVENTION AND DISCOVERY.[81] + + +It is characteristic of the naïve and sanguine beginnings of thought in +youthful men and nations, that all problems are held to be soluble and +fundamentally intelligible on the first appearance of success. The sage +of Miletus, on seeing plants take their rise from moisture, believed he +had comprehended the whole of nature, and he of Samos, on discovering +that definite numbers corresponded to the lengths of harmonic strings, +imagined he could exhaust the nature of the world by means of numbers. +Philosophy and science in such periods are blended. Wider experience, +however, speedily discloses the error of such a course, gives rise to +criticism, and leads to the division and ramification of the sciences. + +At the same time, the necessity of a broad and general view of the world +remains; and to meet this need philosophy parts company with special +inquiry. It is true, the two are often found united in gigantic +personalities. But as a rule their ways diverge more and more widely +from each other. And if the estrangement of philosophy from science can +reach a point where data unworthy of the nursery are not deemed too +scanty as foundations of the world, on the other hand the thorough-paced +specialist may go to the extreme of rejecting point-blank the +possibility of a broader view, or at least of deeming it superfluous, +forgetful of Voltaire's apophthegm, nowhere more applicable than here, +_Le superflu--chose très nécessaire_. + +It is true, the history of philosophy, owing to the insufficiency of its +constructive data, is and must be largely a history of error. But it +would be the height of ingratitude on our part to forget that the seeds +of thoughts which still fructify the soil of special research, such as +the theory of irrationals, the conceptions of conservation, the doctrine +of evolution, the idea of specific energies, and so forth, may be traced +back in distant ages to philosophical sources. Furthermore, to have +deferred or abandoned the attempt at a broad philosophical view of the +world from a full knowledge of the insufficiency of our materials, is +quite a different thing from never having undertaken it at all. The +revenge of its neglect, moreover, is constantly visited upon the +specialist by his committal of the very errors which philosophy long ago +exposed. As a fact, in physics and physiology, particularly during the +first half of this century, are to be met intellectual productions +which for naïve simplicity are not a jot inferior to those of the Ionian +school, or to the Platonic ideas, or to that much reviled ontological +proof. + +Latterly, there has been evidence of a gradual change in the situation. +Recent philosophy has set itself more modest and more attainable ends; +it is no longer inimical to special inquiry; in fact, it is zealously +taking part in that inquiry. On the other hand, the special sciences, +mathematics and physics, no less than philology, have become eminently +philosophical. The material presented is no longer accepted +uncritically. The glance of the inquirer is bent upon neighboring +fields, whence that material has been derived. The different special +departments are striving for closer union, and gradually the conviction +is gaining ground that philosophy can consist only of mutual, +complemental criticism, interpenetration, and union of the special +sciences into a consolidated whole. As the blood in nourishing the body +separates into countless capillaries, only to be collected again and to +meet in the heart, so in the science of the future all the rills of +knowledge will gather more and more into a common and undivided stream. + +It is this view--not an unfamiliar one to the present generation--that I +purpose to advocate. Entertain no hope, or rather fear, that I shall +construct systems for you. I shall remain a natural inquirer. Nor expect +that it is my intention to skirt all the fields of natural inquiry. I +can attempt to be your guide only in that branch which is familiar to +me, and even there I can assist in the furtherment of only a small +portion of the allotted task. If I shall succeed in rendering plain to +you the relations of physics, psychology, and the theory of knowledge, +so that you may draw from each profit and light, redounding to the +advantage of each, I shall regard my work as not having been in vain. +Therefore, to illustrate by an example how, consonantly with my powers +and views, I conceive such inquiries should be conducted, I shall treat +to-day, in the form of a brief sketch, of the following special and +limited subject--of _the part which accidental circumstances play in the +development of inventions and discoveries_. + + * * * * * + +When we Germans say of a man that he was not the inventor of +gunpowder,[82] we impliedly cast a grave suspicion on his abilities. But +the expression is not a felicitous one, as there is probably no +invention in which deliberate thought had a smaller, and pure luck a +larger, share than in this. It is well to ask, Are we justified in +placing a low estimate on the achievement of an inventor because +accident has assisted him in his work? Huygens, whose discoveries and +inventions are justly sufficient to entitle him to an opinion in such +matters, lays great emphasis on this factor. He asserts that a man +capable of inventing the telescope without the concurrence of accident +must have been gifted with superhuman genius.[83] + +A man living in the midst of civilisation finds himself surrounded by a +host of marvellous inventions, considering none other than the means of +satisfying the needs of daily life. Picture such a man transported to +the epoch preceding the invention of these ingenious appliances, and +imagine him undertaking in a serious manner to comprehend their origin. +At first the intellectual power of the men capable of producing such +marvels will strike him as incredible, or, if we adopt the ancient view, +as divine. But his astonishment is considerably allayed by the +disenchanting yet elucidative revelations of the history of primitive +culture, which to a large extent prove that these inventions took their +rise very slowly and by imperceptible degrees. + +A small hole in the ground with fire kindled in it constituted the +primitive stove. The flesh of the quarry, wrapped with water in its +skin, was boiled by contact with heated stones. Cooking by stones was +also done in wooden vessels. Hollow gourds were protected from the fire +by coats of clay. Thus, from the burned clay accidentally originated the +enveloping pot, which rendered the gourd superfluous, although for a +long time thereafter the clay was still spread over the gourd, or +pressed into woven wicker-work before the potter's art assumed its final +independence. Even then the wicker-work ornament was retained, as a sort +of attest of its origin. + +We see, thus, it is by accidental circumstances, or by such as lie +without our purpose, foresight, and power, that man is gradually led to +the acquaintance of improved means of satisfying his wants. Let the +reader picture to himself the genius of a man who could have foreseen +without the help of accident that clay handled in the ordinary manner +would produce a useful cooking utensil! The majority of the inventions +made in the early stages of civilisation, including language, writing, +money, and the rest, could not have been the product of deliberate +methodical reflexion for the simple reason that no idea of their value +and significance could have been had except from practical use. The +invention of the bridge may have been suggested by the trunk of a tree +which had fallen athwart a mountain-torrent; that of the tool by the use +of a stone accidentally taken into the hand to crack nuts. The use of +fire probably started in and was disseminated from regions where +volcanic eruptions, hot springs, and burning jets of natural gas +afforded opportunity for quietly observing and turning to practical +account the properties of fire. Only after that had been done could the +significance of the fire-drill be appreciated, an instrument which was +probably discovered from boring a hole through a piece of wood. The +suggestion of a distinguished inquirer that the invention of the +fire-drill originated on the occasion of a religious ceremony is both +fantastic and incredible. And as to the use of fire, we should no more +attempt to derive that from the invention of the fire-drill than we +should from the invention of sulphur matches. Unquestionably the +opposite course was the real one.[84] + +Similar phenomena, though still largely veiled in obscurity, mark the +initial transition of nations from a hunting to a nomadic life and to +agriculture.[85] We shall not multiply examples, but content ourselves +with the remark that the same phenomena recur in historical times, in +the ages of great technical inventions, and, further, that regarding +them the most whimsical notions have been circulated--notions which +ascribe to accident an unduly exaggerated part, and one which in a +psychological respect is absolutely impossible. The observation of steam +escaping from a tea-kettle and of the clattering of the lid is supposed +to have led to the invention of the steam-engine. Just think of the gap +between this spectacle and the conception of the performance of great +mechanical work by steam, for a man totally ignorant of the +steam-engine! Let us suppose, however, that an engineer, versed in the +practical construction of pumps, should accidentally dip into water an +inverted bottle that had been filled with steam for drying and still +retained its steam. He would see the water rush violently into the +bottle, and the idea would very naturally suggest itself of founding on +this experience a convenient and useful atmospheric steam-pump, which by +imperceptible degrees, both psychologically possible and immediate, +would then undergo a natural and gradual transformation into Watt's +steam-engine. + +But granting that the most important inventions are brought to man's +notice accidentally and in ways that are beyond his foresight, yet it +does not follow that accident alone is sufficient to produce an +invention. The part which man plays is by no means a passive one. Even +the first potter in the primeval forest must have felt some stirrings of +genius within him. In all such cases, the inventor is obliged _to take +note_ of the new fact, he must discover and grasp its advantageous +feature, and must have the power to turn that feature to account in the +realisation of his purpose. He must _isolate_ the new feature, impress +it upon his memory, unite and interweave it with the rest of his +thought; in short, he must possess the capacity _to profit by +experience_. + +The capacity to profit by experience might well be set up as a test of +intelligence. This power varies considerably in men of the same race, +and increases enormously as we advance from the lower animals to man. +The former are limited in this regard almost entirely to the reflex +actions which they have inherited with their organism, they are almost +totally incapable of individual experience, and considering their simple +wants are scarcely in need of it. The ivory-snail (_Eburna spirata_) +never learns to avoid the carnivorous Actinia, no matter how often it +may wince under the latter's shower of needles, apparently having no +memory for pain whatever.[86] A spider can be lured forth repeatedly +from its hole by touching its web with a tuning-fork. The moth plunges +again and again into the flame which has burnt it. The humming-bird +hawk-moth[87] dashes repeatedly against the painted roses of the +wall-paper, like the unhappy and desperate thinker who never wearies of +attacking the same insoluble chimerical problem. As aimlessly almost as +Maxwell's gaseous molecules and in the same unreasoning manner common +flies in their search for light and air stream against the glass pane of +a half-opened window and remain there from sheer inability to find their +way around the narrow frame. But a pike separated from the minnows of +his aquarium by a glass partition, learns after the lapse of a few +months, though only after having butted himself half to death, that he +cannot attack these fishes with impunity. What is more, he leaves them +in peace even after the removal of the partition, though he will bolt a +strange fish at once. Considerable memory must be attributed to birds of +passage, a memory which, probably owing to the absence of disturbing +thoughts, acts with the precision of that of some idiots. Finally, the +susceptibility to training evinced by the higher vertebrates is +indisputable proof of the ability of these animals to profit by +experience. + +A powerfully developed _mechanical_ memory, which recalls vividly and +faithfully old situations, is sufficient for avoiding definite +particular dangers, or for taking advantage of definite particular +opportunities. But more is required for the development of _inventions_. +More extensive chains of images are necessary here, the excitation by +mutual contact of widely different trains of ideas, a more powerful, +more manifold, and richer connexion of the contents of memory, a more +powerful and impressionable psychical life, heightened by use. A man +stands on the bank of a mountain-torrent, which is a serious obstacle to +him. He remembers that he has crossed just such a torrent before on the +trunk of a fallen tree. Hard by trees are growing. He has often moved +the trunks of fallen trees. He has also felled trees before, and then +moved them. To fell trees he has used sharp stones. He goes in search of +such a stone, and as the old situations that crowd into his memory and +are held there in living reality by the definite powerful interest which +he has in crossing just this torrent,--as these impressions are made to +pass before his mind in the _inverse order_ in which they were here +evoked, he invents the bridge. + +There can be no doubt but the higher vertebrates adapt their actions in +some moderate degree to circumstances. The fact that they give no +appreciable evidence of advance by the accumulation of inventions, is +satisfactorily explained by a difference of degree or intensity of +intelligence as compared with man; the assumption of a difference of +kind is not necessary. A person who saves a little every day, be it ever +so little, has an incalculable advantage over him who daily squanders +that amount, or is unable to keep what he has accumulated. A slight +quantitative difference in such things explains enormous differences of +advancement. + +The rules which hold good in prehistoric times also hold good in +historical times, and the remarks made on invention may be applied +almost without modification to discovery; for the two are distinguished +solely by the use to which the new knowledge is put. In both cases the +investigator is concerned with some _newly observed_ relation of new or +old properties, abstract or concrete. It is observed, for example, that +a substance which gives a chemical reaction _A_ is also the cause of a +chemical reaction _B_. If this observation fulfils no purpose but that +of furthering the scientist's insight, or of removing a source of +intellectual discomfort, we have a discovery; but an invention, if in +using the substance giving the reaction _A_ to produce the desired +reaction _B_, we have a practical end in view, and seek to remove a +source of material discomfort. The phrase, _disclosure of the connexion +of reactions_, is broad enough to cover discoveries and inventions in +all departments. It embraces the Pythagorean proposition, which is a +combination of a geometrical and an arithmetical reaction, Newton's +discovery of the connexion of Kepler's motions with the law of the +inverse squares, as perfectly as it does the detection of some minute +but appropriate alteration in the construction of a tool, or of some +appropriate change in the methods of a dyeing establishment. + +The disclosure of new provinces of facts before unknown can only be +brought about by accidental circumstances, under which are _remarked_ +facts that commonly go unnoticed. The achievement of the discoverer here +consists in his _sharpened attention_, which detects the uncommon +features of an occurrence and their determining conditions from their +most evanescent marks,[88] and discovers means of submitting them to +exact and full observation. Under this head belong the first disclosures +of electrical and magnetic phenomena, Grimaldi's observation of +interference, Arago's discovery of the increased check suffered by a +magnetic needle vibrating in a copper envelope as compared with that +observed in a bandbox, Foucault's observation of the stability of the +plane of vibration of a rod accidentally struck while rotating in a +turning-lathe, Mayer's observation of the increased redness of venous +blood in the tropics, Kirchhoff's observation of the augmentation of the +_D_-line in the solar spectrum by the interposition of a sodium lamp, +Schönbein's discovery of ozone from the phosphoric smell emitted on the +disruption of air by electric sparks, and a host of others. All these +facts, of which unquestionably many were _seen_ numbers of times before +they were _noticed_, are examples of the inauguration of momentous +discoveries by accidental circumstances, and place the importance of +strained attention in a brilliant light. + +But not only is a significant part played in the beginning of an inquiry +by co-operative circumstances beyond the foresight of the investigator; +their influence is also active in its prosecution. Dufay, thus, whilst +following up the behavior of _one_ electrical state which he had +assumed, discovers the existence of _two_. Fresnel learns by accident +that the interference-bands received on ground glass are seen to better +advantage in the open air. The diffraction-phenomenon of two slits +proved to be considerably different from what Fraunhofer had +anticipated, and in following up this circumstance he was led to the +important discovery of grating-spectra. Faraday's induction-phenomenon +departed widely from the initial conception which occasioned his +experiments, and it is precisely this deviation that constitutes his +real discovery. + +Every man has pondered on some subject. Every one of us can multiply the +examples cited, by less illustrious ones from his own experience. I +shall cite but one. On rounding a railway curve once, I accidentally +remarked a striking apparent inclination of the houses and trees. I +inferred that the direction of the total resultant _physical_ +acceleration of the body reacts _physiologically_ as the vertical. +Afterwards, in attempting to inquire more carefully into this +phenomenon, and this only, in a large whirling machine, the collateral +phenomena conducted me to the sensation of angular acceleration, +vertigo, Flouren's experiments on the section of the semi-circular +canals etc., from which gradually resulted views relating to sensations +of direction which are also held by Breuer and Brown, which were at +first contested on all hands, but are now regarded on many sides as +correct, and which have been recently enriched by the interesting +inquiries of Breuer concerning the _macula acustica_, and Kreidel's +experiments with magnetically orientable crustacea.[89] Not disregard of +accident but a direct and purposeful employment of it advances research. + +The more powerful the psychical connexion of the memory pictures +is,--and it varies with the individual and the mood,--the more apt is +the same accidental observation to be productive of results. Galileo +knows that the air has weight; he also knows of the "resistance to a +vacuum," expressed both in weight and in the height of a column of +water. But the two ideas dwelt asunder in his mind. It remained for +Torricelli to vary the specific gravity of the liquid measuring the +pressure, and not till then was the air included in the list of +pressure-exerting fluids. The reversal of the lines of the spectrum was +seen repeatedly before Kirchhoff, and had been mechanically explained. +But it was left for his penetrating vision to discern the evidence of +the connexion of this phenomenon with questions of heat, and to him +alone through persistent labor was revealed the sweeping significance of +the fact for the mobile equilibrium of heat. Supposing, then, that such +a rich organic connexion of the elements of memory exists, and is the +prime distinguishing mark of the inquirer, next in importance certainly +is that _intense interest_ in a definite object, in a definite idea, +which fashions advantageous combinations of thought from elements before +disconnected, and obtrudes that idea into every observation made, and +into every thought formed, making it enter into relationship with all +things. Thus Bradley, deeply engrossed with the subject of aberration, +is led to its solution by an exceedingly unobtrusive experience in +crossing the Thames. It is permissible, therefore, to ask whether +accident leads the discoverer, or the discoverer accident, to a +successful outcome in scientific quests. + +No man should dream of solving a great problem unless he is so +thoroughly saturated with his subject that everything else sinks into +comparative insignificance. During a hurried meeting with Mayer in +Heidelberg once, Jolly remarked, with a rather dubious implication, that +if Mayer's theory were correct water could be warmed by shaking. Mayer +went away without a word of reply. Several weeks later, and now +unrecognised by Jolly, he rushed into the latter's presence exclaiming: +"Es ischt aso!" (It is so, it is so!) It was only after considerable +explanation that Jolly found out what Mayer wanted to say. The incident +needs no comment.[90] + +A person deadened to sensory impressions and given up solely to the +pursuit of his own thoughts, may also light on an idea that will divert +his mental activity into totally new channels. In such cases it is a +psychical accident, an intellectual experience, as distinguished from a +physical accident, to which the person owes his discovery--a discovery +which is here made "deductively" by means of mental copies of the world, +instead of experimentally. _Purely_ experimental inquiry, moreover, does +not exist, for, as Gauss says, virtually we always experiment with our +thoughts. And it is precisely that constant, corrective interchange or +intimate union of experiment and deduction, as it was cultivated by +Galileo in his _Dialogues_ and by Newton in his _Optics_, that is the +foundation of the benign fruitfulness of modern scientific inquiry as +contrasted with that of antiquity, where observation and reflexion +ofttimes pursued their respective courses like two strangers. + +We have to wait for the appearance of a favorable physical accident. The +movement of our thoughts obeys the law of association. In the case of +meagre experience the result of this law is simply the mechanical +reproduction of definite sensory experiences. On the other hand, if the +psychical life is subjected to the incessant influences of a powerful +and rich experience, then every representative element in the mind is +connected with so many others that the actual and natural course of the +thoughts is easily influenced and determined by insignificant +circumstances, which accidentally are decisive. Hereupon, the process +termed imagination produces its protean and infinitely diversified +forms. Now what can we do to guide this process, seeing that the +combinatory law of the images is without our reach? Rather let us ask, +what influence can a powerful and constantly recurring idea exert on the +movement of our thoughts? According to what has preceded, the answer is +involved in the question itself. The _idea_ dominates the thought of the +inquirer, not the latter the former. + +Let us see, now, if we can acquire a profounder insight into the process +of discovery. The condition of the discoverer is, as James has aptly +remarked, not unlike the situation of a person who is trying to remember +something that he has forgotten. Both are sensible of a gap, and have +only a remote presentiment of what is missing. Suppose I meet in a +company a well-known and affable gentleman whose name I have forgotten, +and who to my horror asks to be introduced to some one. I set to work +according to Lichtenberg's rule, and run down the alphabet in search of +the initial letter of his name. A vague sympathy holds me at the letter +_G_. Tentatively I add the second letter and am arrested at _e_, and +long before I have tried the third letter _r_, the name "Gerson" sounds +sonorously upon my ear, and my anguish is gone. While taking a walk I +meet a gentleman from whom I receive a communication. On returning home, +and in attending to weightier affairs, the matter slips my mind. +Moodily, but in vain, I ransack my memory. Finally I observe that I am +going over my walk again in thought. On the street corner in question +the self-same gentleman stands before me and repeats his communication. +In this process are successively recalled to consciousness all the +percepts which were connected with the percept that was lost, and with +them, finally, that, too, is brought to light. In the first case--where +the experience had already been made and is permanently impressed on our +thought--a _systematic_ procedure is both possible and easy, for we know +that a name must be composed of a limited number of sounds. But at the +same time it should be observed that the labor involved in such a +combinatorial task would be enormous if the name were long and the +responsiveness of the mind weaker. + +It is often said, and not wholly without justification, that the +scientist has solved a _riddle_. Every problem in geometry may be +clothed in the garb of a _riddle_. Thus: "What thing is that _M_ which +has the properties _A_, _B_, _C_?" "What circle is that which touches +the straight lines _A_, _B_, but touches _B_ in the point _C_?" The +first two conditions marshal before the imagination the group of circles +whose centres lie in the line of symmetry of _A_, _B_. The third +condition reminds us of all the circles having centres in the straight +line that stands at right angles to _B_ in _C_. The _common_ term, or +common terms, of the two groups of images solves the riddle--satisfies +the problem. Puzzles dealing with things or words induce similar +processes, but the memory in such cases is exerted in many directions +and more varied and less clearly ordered provinces of ideas are +surveyed. The difference between the situation of a geometer who has a +construction to make, and that of an engineer, or a scientist, +confronted with a problem, is simply this, that the first moves in a +field with which he is thoroughly acquainted, whereas the two latter are +obliged to familiarise themselves with this field subsequently, and in a +measure far transcending what is commonly required. In this process the +mechanical engineer has at least always a definite goal before him and +definite means to accomplish his aim, whilst in the case of the +scientist that aim is in many instances presented only in vague and +general outlines. Often the very formulation of the riddle devolves on +him. Frequently it is not until the aim has been reached that the +broader outlook requisite for systematic procedure is obtained. By far +the larger portion of his success, therefore, is contingent on luck and +instinct. It is immaterial, so far as its character is concerned, +whether the process in question is brought rapidly to a conclusion in +the brain of one man, or whether it is spun out for centuries in the +minds of a long succession of thinkers. The same relation that a word +solving a riddle bears to that riddle is borne by the modern conception +of light to the facts discovered by Grimaldi, Römer, Huygens, Newton, +Young, Malus, and Fresnel, and only by the help of this slowly developed +conception is our mental vision enabled to embrace the broad domain of +facts in question. + +A welcome complement to the discoveries which the history of +civilisation and comparative psychology have furnished, is to be found +in the confessions of great scientists and artists. Scientists _and_ +artists, we might say, for Liebig boldly declared there was no essential +difference between the two. Are we to regard Leonardo da Vinci as a +scientist or as an artist? If the artist builds up his work from a few +motives, the scientist discovers the motives which permeate reality. If +scientists like Lagrange or Fourier are in a certain measure artists in +the presentation of their results, on the other hand, artists like +Shakespeare or Ruysdael are scientists in the insight which must have +preceded their creations. + +Newton, when questioned about his methods of work, could give no other +answer but that he was wont to ponder again and again on a subject; and +similar utterances are accredited to D'Alembert and Helmholtz. +Scientists and artists both recommend persistent labor. After the +repeated survey of a field has afforded opportunity for the +interposition of advantageous accidents, has rendered all the traits +that suit with the mood or the dominant thought more vivid, and has +gradually relegated to the background all things that are inappropriate, +making their future appearance impossible; then from the teeming, +swelling host of fancies which a free and high-flown imagination calls +forth, suddenly that particular form arises to the light which +harmonises perfectly with the ruling idea, mood, or design. Then it is +that that which has resulted slowly as the result of a gradual +selection, appears as if it were the outcome of a deliberate act of +creation. Thus are to be explained the statements of Newton, Mozart, +Richard Wagner, and others, when they say that thoughts, melodies, and +harmonies had poured in upon them, and that they had simply retained the +right ones. Undoubtedly, the man of genius, too, consciously or +instinctively, pursues systematic methods wherever it is possible; but +in his delicate presentiment he will omit many a task or abandon it +after a hasty trial on which a less endowed man would squander his +energies in vain. Thus, the genius accomplishes[91] in a brief space of +time undertakings for which the life of an ordinary man would far from +suffice. We shall hardly go astray if we regard genius as only a slight +deviation from the average mental endowment--as possessing simply a +greater sensitiveness of cerebral reaction and a greater swiftness of +reaction. The men who, obeying their inner impulses, make sacrifices for +an idea instead of advancing their material welfare, may appear to the +full-blooded Philistine as fools; yet we shall scarcely adopt Lombroso's +view, that genius is to be regarded as a disease, although it is +unfortunately true that the sensitive brains and fragile constitutions +succumb most readily to sickness. + +The remark of C. G. J. Jacobi that mathematics is slow of growth and +only reaches the truth by long and devious paths, that the way to its +discovery must be prepared for long beforehand, and that then the truth +will make its long-deferred appearance as if impelled by some divine +necessity[92]--all this holds true of every science. We are astounded +often to note that it required the combined labors of many eminent +thinkers for a full century to reach a truth which it takes us only a +few hours to master and which once acquired seems extremely easy to +reach under the right sort of circumstances. To our humiliation we learn +that even the greatest men are born more for life than for science. The +extent to which even they are indebted to accident--to that singular +conflux of the physical and the psychical life in which the continuous +but yet imperfect and never-ending adaptation of the latter to the +former finds its distinct expression--that has been the subject of our +remarks to-day. Jacobi's poetical thought of a divine necessity acting +in science will lose none of its loftiness for us if we discover in this +necessity the same power that destroys the unfit and fosters the fit. +For loftier, nobler, and more romantic than poetry is the truth and the +reality. + + FOOTNOTES: + + [Footnote 81: Inaugural lecture delivered on assuming the + Professorship of the History and Theory of Inductive Science in the + University of Vienna, October 21, 1895.] + + [Footnote 82: The phrase is, _Er hat das Pulver nicht erfunden_.] + + [Footnote 83: "Quod si quis tanta industria exstitisset, ut ex + naturae principiis at geometria hanc rem eruere potuisset, eum ego + supra mortalium sortem ingenio valuisse dicendum crederem. Sed hoc + tantum abest, ut fortuito reperti artificii rationem non adhuc satis + explicari potuerint viri doctissimi."--Hugenii Dioptrica (de + telescopiis).] + + [Footnote 84: I must not be understood as saying that the fire-drill + has played no part in the worship of fire or of the sun.] + + [Footnote 85: Compare on this point the extremely interesting + remarks of Dr. Paul Carus in his _Philosophy of the Tool_, Chicago, + 1893.] + + [Footnote 86: Möbius, _Naturwissenschaftlicher Verein für + Schleswig-Holstein_, Kiel, 1893, p. 113 et seq.] + + [Footnote 87: I am indebted for this observation to Professor + Hatscheck.] + + [Footnote 88: Cf. Hoppe, _Entdecken und Finden_, 1870.] + + [Footnote 89: See the lecture "Sensations of Orientation," p. 282 et + seq.] + + [Footnote 90: This story was related to me by Jolly, and + subsequently repeated in a letter from him.] + + [Footnote 91: I do not know whether Swift's academy of schemers in + Lagado, in which great discoveries and inventions were made by a + sort of verbal game of dice, was intended as a satire on Francis + Bacon's method of making discoveries by means of huge synoptic + tables constructed by scribes. It certainly would not have been + ill-placed.] + + [Footnote 92: "Crescunt disciplinae lente tardeque; per varios + errores sero pervenitur ad veritatem. Omnia praeparata esse debent + diuturno et assiduo labore ad introitum veritatis novae. Jam illa + certo temporis momento divina quadam necessitate coacta emerget." + + Quoted by Simony, _In ein ringförmiges Band einen Knoten zu machen_, + Vienna, 1881, p. 41.] + + + + +ON SENSATIONS OF ORIENTATION.[93] + + +Through the co-operation of a succession of inquirers, among whom are +particularly to be mentioned Goltz of Strassburg and Breuer of Vienna, +considerable advances have been made during the last twenty-five years +in our knowledge of the means by which we ascertain our position in +space and the direction of our motion, or orient ourselves, as the +phrase goes. I presume that you are already acquainted with the +physiological part of the processes with which our sensations of +movement, or, more generally speaking, our sensations of orientation, +are connected. Here I shall consider more particularly the physical side +of the matter. In fact, I was originally led to the consideration of +these questions by the observation of extremely simple and perfectly +well-known physical facts, before I had any great acquaintance with +physiology and while pursuing unbiasedly my natural thoughts; and I am +of the conviction that the way which I have pursued, and which is +entirely free from hypotheses, will, if you will follow my exposition, +be that of easiest acquisition for the most of you. + +No man of sound common sense could ever have doubted that a pressure or +force is requisite to set a body in motion in a given direction and that +a contrary pressure is required to stop suddenly a body in motion. +Though the law of inertia was first formulated with anything like +exactness by Galileo, the facts at the basis of it were known long +previously to men of the stamp of Leonardo da Vinci, Rabelais, and +others, and were illustrated by them with appropriate experiments. +Leonardo knew that by a swift stroke with a ruler one can knock out from +a vertical column of checkers a single checker without over-throwing the +column. The experiment with a coin resting on a piece of pasteboard +covering a goblet, which falls into the goblet when the pasteboard is +jerked away, like all experiments of the kind, is certainly very old. + +With Galileo the experience in question assumes greater clearness and +force. In the famous dialogue on the Copernican system which cost him +his freedom, he explains the tides in an unfelicitous, though in +principle correct manner, by the analogue of a platter of water swung to +and fro. In opposition to the Aristotelians of his time, who believed +the descent of a heavy body could be accelerated by the superposition +of another heavy body, he asserted that a body could never be +accelerated by one lying upon it unless the first in some way impeded +the superposed body in its descent. To seek to press a falling body by +means of another placed upon it, is as senseless as trying to prod a man +with a lance when the man is speeding away from one with the same +velocity as the lance. Even this little excursion into physics can +explain much to us. You know the peculiar sensation which one has in +falling, as when one jumps from a high springboard into the water, and +which is also experienced in some measure at the beginning of the +descent of elevators and swings. The reciprocal gravitational pressure +of the different parts of our body, which is certainly felt in some +manner, vanishes in free descent, or, in the case of the elevator, is +diminished on the beginning of the descent. A similar sensation would be +experienced if we were suddenly transported to the moon where the +acceleration of gravity is much less than upon the earth. I was led to +these considerations in 1866 by a suggestion in physics, and having also +taken into account the alterations of the blood-pressure in the cases in +question, I found I coincided without knowing it with Wollaston and +Purkinje. The first as early as 1810 in his Croonian lecture had touched +on the subject of sea-sickness and explained it by alterations of the +blood-pressure, and later had laid similar considerations at the basis +of his explanation of vertigo (1820-1826).[94] + +Newton was the first to enunciate with perfect generality that a body +can change the velocity and direction of its motion only by the action +of a force, or the action of a second body. A corollary of this law +which was first expressly deduced by Euler is that a body can never be +set _rotating_ or made to cease rotating of itself but only by forces +and other bodies. For example, turn an open watch which has run down +freely backwards and forwards in your hand. The balance-wheel will not +fully catch the rapid rotations, it does not even respond fully to the +elastic force of the spring which proves too weak to carry the wheel +entirely with it. + +Let us consider now that whether we move ourselves by means of our legs, +or whether we are moved by a vehicle or a boat, at first only a part of +our body is directly moved and the rest of it is afterwards set in +motion by the first part. We see that pressures, pulls, and tensions are +always produced between the parts of the body in this action, which +pressures, pulls, and tensions give rise to sensations by which the +forward or rotary movements in which we are engaged are made +perceptible.[95] But it is quite natural that sensations so familiar +should be little noticed and that attention should be drawn to them only +under special circumstances when they occur unexpectedly or with unusual +strength. + +[Illustration: Fig. 45.] + +Thus my attention was drawn to this point by the sensation of falling +and subsequently by another singular occurrence. I was rounding a sharp +railway curve once when I suddenly saw all the trees, houses, and +factory chimneys along the track swerve from the vertical and assume a +strikingly inclined position. What had hitherto appeared to me perfectly +natural, namely, the fact that we distinguish the vertical so perfectly +and sharply from every other direction, now struck me as enigmatical. +Why is it that the same direction can now appear vertical to me and now +cannot? By what is the vertical distinguished for us? (Compare Figure +45.) + +The rails are raised on the convex or outward side of the track in order +to insure the stability of the carriage as against the action of the +centrifugal force, the whole being so arranged that the combination of +the force of gravity with the centrifugal force of the train shall give +rise to a force perpendicular to the plane of the rails. + +Let us assume, now, that under all circumstances we somehow sense the +direction of the total resultant mass-acceleration whencesoever it may +arise as the vertical. Then both the ordinary and the extraordinary +phenomena will be alike rendered intelligible.[96] + +I was now desirous of putting the view I had reached to a more +convenient and exact test than was possible on a railway journey where +one has no control over the determining circumstances and cannot alter +them at will. I accordingly had the simple apparatus constructed which +is represented in Figure 46. + +In a large frame _BB_, which is fastened to the walls, rotates about a +vertical axis _AA_ a second frame _RR_, and within the latter a third +one _rr_, which can be set at any distance and position from the axis, +made stationary or movable, and is provided with a chair for the +observer. + +[Illustration: Fig. 46. + +From Mach's _Bewegungsempfindungen_, Leipsic, Engelmann, 1875.] + +The observer takes his seat in the chair and to prevent disturbances of +judgment is enclosed in a paper box. If the observer together with the +frame _rr_ be then set in uniform rotation, he will feel and see the +beginning of the rotation both as to direction and amount very +distinctly although every outward visible or tangible point of reference +is wanting. If the motion be uniformly continued the sensation of +rotation will gradually cease entirely and the observer will imagine +himself at rest. But if _rr_ be placed outside the axis of rotation, at +once on the rotation beginning, a strikingly apparent, palpable, +actually visible inclination of the entire paper box is produced, slight +when the rotation is slow, strong when the rotation is rapid, and +continuing as long as the rotation lasts. It is absolutely impossible +for the observer to escape perceiving the inclination, although here +also all outward points of reference are wanting. If the observer, for +example, is seated so as to look towards the axis, he will feel the box +strongly tipped backwards, as it necessarily must be if the direction of +the total resultant force is perceived as the vertical. For other +positions of the observer the situation is similar.[97] + +Once, while performing one of these experiments, and after rotating so +long that I was no longer conscious of the movement, I suddenly caused +the apparatus to be stopped, whereupon I immediately felt and saw myself +with the whole box rapidly flung round in rotation in the opposite +direction, although I knew that the whole apparatus was at rest and +every outward point of reference for the perception of motion was +wanting. Every one who disbelieves in sensations of movement should be +made acquainted with these phenomena. Had Newton known them and had he +ever observed how we may actually imagine ourselves turned and displaced +in space without the assistance of stationary bodies as points of +reference, he would certainly have been confirmed more than ever in his +unfortunate speculations regarding absolute space. + +The sensation of rotation in the opposite direction after the apparatus +has been stopped, slowly and gradually ceases. But on accidentally +inclining my head once during this occurrence, the axis of apparent +rotation was also observed to incline in exactly the same manner both as +to direction and as to amount. It is accordingly clear that the +acceleration or retardation of rotation is felt. The acceleration +operates as a stimulus. The sensation, however, like almost all +sensations, though it gradually decreases, lasts perceptibly longer than +the stimulus. Hence the long continued apparent rotation after the +stopping of the apparatus. The organ, however, which causes the +persistence of this sensation must have its seat in the _head_, since +otherwise the axis of apparent rotation could not assume the same motion +as the head. + +If I were to say, now, that a light had flashed upon me in making these +last observations, the expression would be a feeble one. I ought to say +I experienced a perfect illumination. My juvenile experiences of vertigo +occurred to me. I remembered Flourens's experiments relative to the +section of the semi-circular canals of the labyrinths of doves and +rabbits, where this inquirer had observed phenomena similar to vertigo, +but which he preferred to interpret, from his bias to the acoustic +theory of the labyrinth, as the expression of painful auditive +disturbances. I saw that Goltz had nearly but not quite hit the bull's +eye with his theory of the semi-circular canals. This inquirer, who, +from his happy habit of following his own natural thoughts without +regard for tradition, has cleared up so much in science, spoke, as early +as 1870, on the ground of experiments, as follows: "It is uncertain +whether the semi-circular canals are auditive organs or not. In any +event they form an apparatus which serves for the preservation of +equilibrium. They are, so to speak, the sense-organs of equilibrium of +the head and indirectly of the whole body." I remembered the galvanic +dizziness which had been observed by Ritter and Purkinje on the passage +of a current through the head, when the persons experimented upon +imagined they were falling towards the cathode. The experiment was +immediately repeated, and sometime later (1874) I was enabled to +demonstrate the same objectively with fishes, all of which placed +themselves sidewise and in the same direction in the field of the +current as if at command.[98] Müller's doctrine of specific energies now +appeared to me to bring all these new and old observations into a +simple, connected unity. + +[Illustration: Fig. 47. + +The labyrinth of a dove (stereoscopically reproduced), from R. Ewald, +_Nervus Octavus_, Wiesbaden, Bergmann, 1892.] + +Let us picture to ourselves the labyrinth of the ear with its three +semi-circular canals lying in three mutually perpendicular planes (Comp. +Fig. 47), the mysterious position of which inquirers have endeavored to +explain in every possible and impossible way. Let us conceive the nerves +of the ampullæ, or the dilated extensions of the semi-circular canals, +equipped with a capacity for responding to every imaginable stimulus +with a sensation of rotation just as the nerves of the retina of the eye +when excited by pressures, by electrical or chemical stimuli always +respond with the sensation of light; let us picture to ourselves, +further, that the usual excitation of the ampullæ nerves is produced by +the inertia of the contents of the semi-circular canals, which contents +on suitable rotations in the plane of the semi-circular canal are left +behind in the motion, or at least have a tendency to remain behind and +consequently exert a pressure. It will be seen that on this supposition +all the single facts which without the theory appear as so many +different individual phenomena, become from this single point of view +clear and intelligible. + +I had the satisfaction, immediately after the communication in which I +set forth this idea,[99] of seeing a paper by Breuer appear[100] in +which this author had arrived by entirely different methods at results +that agreed in all essential points with my own. A few weeks later +appeared the researches of Crum Brown of Edinburgh, whose methods were +even still nearer mine. Breuer's paper was far richer in physiological +respects than mine, and he had particularly gone into greater detail in +his investigation of the collateral effects of the reflex motions and +orientation of the eyes in the phenomena under consideration.[101] In +addition certain experiments which I had suggested in my paper as a test +of the correctness of the view in question had already been performed by +Breuer. Breuer has also rendered services of the highest order in the +further elaboration of this field. But in a physical regard, my paper +was, of course, more complete. + +In order to portray to the eye the behavior of the semi-circular canals, +I have constructed here a little apparatus. (See Fig. 48.) The large +rotatable disc represents the osseous semi-circular canal, which is +continuous with the bones of the head; the small disc, which is free to +rotate on the axis of the first, represents the mobile and partly liquid +contents of the semi-circular canal. On rotating the large disc, the +small disc as you see remains behind. I have to turn some time before +the small disc is carried along with the large one by friction. But if I +now stop the large disc the small disc as you see continues to rotate. + +[Illustration: Fig. 48. + +Model representing the action of the semi-circular canals.] + +Simply assume now that the rotation of the small disc, say in the +direction of the hands of a watch, would give rise to a sensation of +rotation in the opposite direction, and conversely, and you already +understand a good portion of the facts above set forth. The explanation +still holds, even if the small disc does not perform appreciable +rotations but is checked by a contrivance similar to an elastic spring, +the tension of which disengages a sensation. Conceive, now, three such +contrivances with their mutually perpendicular planes of rotation joined +together so as to form a single apparatus; then to this apparatus as a +whole, no rotation can be imparted without its being indicated by the +small mobile discs or by the springs which are attached to them. +Conceive both the right and the left ear equipped with such an +apparatus, and you will find that it answers all the purposes of the +semi-circular canals, which you see represented stereoscopically in Fig. +47 for the ear of a dove. + +Of the many experiments which I have made on my own person, and the +results of which could be predicted by the new view according to the +behavior of the model and consequently according to the rules of +mechanics, I shall cite but one. I fasten a horizontal board in the +frame _RR_ of my rotatory apparatus, lie down upon the same with my +right ear upon the board, and cause the apparatus to be uniformly +rotated. As soon as I no longer perceive the rotation, I turn around +upon my left ear and immediately the sensation of rotation again starts +up with marked vividness. The experiment can be repeated as often as one +wishes. A slight turn of the head even is sufficient for reviving the +sensation of rotation which in the perfectly quiescent state at once +disappears altogether. + +We will imitate the experiment on the model. I turn the large disc until +finally the small disc is carried along with it. If, now, while the +rotation continues uniform, I burn off a little thread which you see +here, the small disc will be flipped round by a spring into its own +plane 180°, so as now to present its opposite side to you, when the +rotation at once begins in the opposite direction. + +We have consequently a very simple means for determining whether one is +actually the subject or not of uniform and imperceptible rotations. If +the earth rotated much more rapidly than it really does, or if our +semi-circular canals were much more sensitive, a Nansen sleeping at the +North Pole would be waked by a sensation of rotation every time he +turned over. Foucault's pendulum experiment as a demonstration of the +earth's rotation would be superfluous under such circumstances. The only +reason we cannot prove the rotation of the earth with the help of our +model, lies in the small angular velocity of the earth and in the +consequent liability to great experimental errors.[102] + +Aristotle has said that "The sweetest of all things is knowledge." And +he is right. But if you were to suppose that the _publication_ of a new +view were productive of unbounded sweetness, you would be mightily +mistaken. No one disturbs his fellow-men with a new view unpunished. Nor +should the fact be made a subject of reproach to these fellow-men. To +presume to revolutionise the current way of thinking with regard to any +question, is no pleasant task, and above all not an easy one. They who +have advanced new views know best what serious difficulties stand in +their way. With honest and praiseworthy zeal, men set to work in search +of everything that does not suit with them. They seek to discover +whether they cannot explain the facts better or as well, or +approximately as well, by the traditional views. And that, too, is +justified. But at times some extremely artless animadversions are heard +that almost nonplus us. "If a sixth sense existed it could not fail to +have been discovered thousands of years ago." Indeed; there was a time, +then, when only seven planets could have existed! But I do not believe +that any one will lay any weight on the philological question whether +the set of phenomena which we have been considering should be called a +sense. The phenomena will not disappear when the name disappears. It was +further said to me that animals exist which have no labyrinth, but which +can yet orientate themselves, and that consequently the labyrinth has +nothing to do with orientation. We do not walk forsooth with our legs, +because snakes propel themselves without them! + +But if the promulgator of a new idea cannot hope for any great pleasure +from its publication, yet the critical process which his views undergo +is extremely helpful to the subject-matter of them. All the defects +which necessarily adhere to the new view are gradually discovered and +eliminated. Over-rating and exaggeration give way to more sober +estimates. And so it came about that it was found unpermissible to +attribute all functions of orientation exclusively to the labyrinth. In +these critical labors Delage, Aubert, Breuer, Ewald, and others have +rendered distinguished services. It can also not fail to happen that +fresh facts become known in this process which could have been predicted +by the new view, which actually were predicted in part, and which +consequently furnish a support for the new view. Breuer and Ewald +succeeded in electrically and mechanically exciting the labyrinth, and +even single parts of the labyrinth, and thus in producing the movements +that belong to such stimuli. It was shown that when the semi-circular +canals were absent vertigo could not be produced, when the entire +labyrinth was removed the orientation of the head was no longer +possible, that without the labyrinth galvanic vertigo could not be +induced. I myself constructed as early as 1875 an apparatus for +observing animals in rotation, which was subsequently reinvented in +various forms and has since received the name of "cyclostat."[103] In +experiments with the most varied kinds of animals it was shown that, for +example, the larvæ of frogs are not subject to vertigo until their +semi-circular canals which at the start are wanting are developed (K. +Schäfer). A large percentage of the deaf and dumb are afflicted with +grave affections of the labyrinth. The American psychologist, William +James, has made whirling experiments with many deaf and dumb subjects, +and in a large number of them found that susceptibility to giddiness is +wanting. He also found that many deaf and dumb people on being ducked +under water, whereby they lose their weight and consequently have no +longer the full assistance of their muscular sense, utterly lose their +sense of position in space, do not know which is up and which is down, +and are thrown into the greatest consternation,--results which do not +occur in normal men. Such facts are convincing proof that we do not +orientate ourselves entirely by means of the labyrinth, important as it +is for us. Dr. Kreidl has made experiments similar to those of James and +found that not only is vertigo absent in deaf and dumb people when +whirled about, but that also the reflex movements of the eyes which are +normally induced by the labyrinth are wanting. Finally, Dr. Pollak has +found that galvanic vertigo does not exist in a large percentage of the +deaf and dumb. Neither the jerking movements nor the uniform movements +of the eyes were observed which normal human beings exhibit in the +Ritter and Purkinje experiment. + +After the physicist has arrived at the idea that the semi-circular +canals are the organ of sensation of rotation or of angular +acceleration, he is next constrained to ask for the organs that mediate +the sensation of acceleration noticed in forward movements. In +searching for an organ for this function, he of course is not apt to +select one that stands in no anatomical and spatial relation with the +semi-circular canals. And in addition there are physiological +considerations to be weighed. The preconceived opinion once having been +abandoned that the _entire_ labyrinth is auditory in its function, there +remains after the cochlea is reserved for sensations of tone and the +semi-circular canals for the sensation of angular acceleration, the +vestibule for the discharge of additional functions. The vestibule, +particularly the part of it known as the sacculus, appeared to me, by +reason of the so-called otoliths which it contains, eminently adapted +for being the organ of sensation of forward acceleration or of the +position of the head. In this conjecture I again closely coincided with +Breuer. + +That a sensation of position, of direction and amount of +mass-acceleration exists, our experience in elevators as well as of +movement in curved paths is sufficient proof. I have also attempted to +produce and destroy suddenly great velocities of forward movement by +means of various contrivances of which I shall mention only one here. +If, while enclosed in the paper box of my large whirling apparatus at +some distance from the axis, my body is in uniform rotation which I no +longer feel, and I then loosen the connexions of the frame _rr_ with _R_ +thus making the former moveable and I then suddenly stop the larger +frame, my forward motion is abruptly impeded while the frame _rr_ +continues to rotate. I imagine now that I am speeding on in a straight +line in a direction opposite to that of the checked motion. +Unfortunately, for many reasons it cannot be proved convincingly that +the organ in question has its seat in the head. According to the opinion +of Delage, the labyrinth has nothing to do with this particular +sensation of movement. Breuer, on the other hand, is of the opinion that +the organ of forward movement in man is stunted and the persistence of +the sensation in question is too brief to permit our instituting +experiments as obvious as in the case of rotation. In fact, Crum Brown +once observed while in an irritated condition peculiar vertical +phenomena in his own person, which were all satisfactorily explained by +an abnormally long persistence of the sensation of rotation, and I +myself in an analogous case on the stopping of a railway train felt the +apparent backward motion in striking intensity and for an unusual length +of time. + +There is no doubt whatever that we feel changes of vertical +acceleration, and it will appear from the following extremely probable +that the otoliths of the vestibule are the sense-organ for the +_direction_ of the mass-acceleration. It will then be incompatible with +a really logical view to regard the latter as incapable of sensing +horizontal accelerations. + +In the lower animals the analogue of the labyrinth is shrunk to a little +vesicle filled with a liquid and containing tiny crystals, auditive +stones, or otoliths, of greater specific gravity, suspended on minute +hairs. These crystals appear physically well adapted for indicating both +the direction of gravity and the direction of incipient movements. That +they discharge the former function, Delage was the first to convince +himself by experiments with lower animals which on the removal of the +otoliths utterly lost their bearings and could no longer regain their +normal position. Loeb also found that fishes without labyrinths swim now +on their bellies and now on their backs. But the most remarkable, most +beautiful, and most convincing experiment is that which Dr. Kreidl +instituted with crustaceans. According to Hensen, certain Crustacea on +sloughing spontaneously introduce fine grains of sand as auditive stones +into their otolith vesicle. At the ingenious suggestion of S. Exner, Dr. +Kreidl constrained some of these animals to put up with iron filings +(_ferrum limatum_). If the pole of an electro-magnet be brought near the +animal, it will at once turn its back away from the pole accompanying +the movement with appropriate reflex motions of the eye the moment the +current is closed, exactly as if gravity had been brought to bear upon +the animal in the same direction as the magnetic force.[104] This, in +fact, is what should be expected from the function ascribed to the +otoliths. If the eyes be covered with asphalt varnish, and the auditive +sacs removed, the crustaceans lose their sense of direction utterly, +tumble head over heels, lie on their side or their back indifferently. +This does not happen when the eyes only are covered. For vertebrates, +Breuer has demonstrated by searching investigations that the otoliths, +or better, statoliths, slide in three planes parallel to the planes of +the semi-circular canals, and are consequently perfectly adapted for +indicating changes both in the amount and the direction of the +mass-acceleration.[105] + +I have already remarked that not every function of orientation can be +ascribed exclusively to the labyrinth. The deaf and dumb who have to be +immersed in water, and the crustaceans who must have their eyes closed +if they are to be perfectly disorientated, are proof of this fact. I saw +a blind cat at Hering's laboratory which to one who was not a very +attentive observer behaved exactly like a seeing cat. It played nimbly +with objects rolling on the floor, stuck its head inquisitively into +open drawers, sprang dexterously upon chairs, ran with perfect accuracy +through open doors, and never bumped against closed ones. The visual +sense had here been rapidly replaced by the tactual and auditive senses. +And it appears from Ewald's investigations that even after the +labyrinths have been removed, animals gradually learn to move about +again quite in the normal fashion, presumably because the eliminated +function of the labyrinth is now performed by some part of the brain. A +certain peculiar weakness of the muscles alone is perceptible which +Ewald ascribes to the absence of the stimulus which is otherwise +constantly emitted by the labyrinth (the labyrinth-tonus). But if the +part of the brain which discharges the deputed function be removed, the +animals are again completely disorientated and absolutely helpless. + +It may be said that the views enunciated by Breuer, Crum Brown and +myself in 1873 and 1874, and which are substantially a fuller and richer +development of Goltz's idea, have upon the whole been substantiated. At +least they have exercised a helpful and stimulative influence. New +problems have of course arisen in the course of the investigation which +still await solution, and much work remains to be done. At the same time +we see how fruitful the renewed co-operation of the various special +departments of science may become after a period of isolation and +invigorating labor apart. + +I may be permitted, therefore, to consider the relation between hearing +and orientation from another and more general point of view. What we +call the auditive organ is in the lower animals simply a sac containing +auditive stones. As we ascend the scale, 1, 2, 3 semi-circular canals +gradually develop from them, whilst the structure of the otolith organ +itself becomes more complicated. Finally, in the higher vertebrates, and +particularly in the mammals, a part of the latter organ (the lagena) +becomes the cochlea, which Helmholtz explained as the organ for +sensations of tone. In the belief that the entire labyrinth was an +auditive organ, Helmholtz, contrary to the results of his own masterly +analysis, originally sought to interpret another part of the labyrinth +as the organ of noises. I showed a long time ago (1873) that every tonal +stimulus by shortening the duration of the excitation to a few +vibrations, gradually loses its character of pitch and takes on that of +a sharp, dry report or noise.[106] All the intervening stages between +tones and noises can be exhibited. Such being the case, it will hardly +be assumed that one organ is suddenly and at some given point replaced +in function by another. On the basis of different experiments and +reasonings S. Exner also regards the assumption of a special organ for +the sensing of noises as unnecessary. + +If we will but reflect how small a portion of the labyrinth of higher +animals is apparently in the service of the sense of hearing, and how +large, on the other hand, the portion is which very likely serves the +purposes of orientation, how much the first anatomical beginnings of the +auditive sac of lower animals resemble that part of the fully developed +labyrinth which does not hear, the view is irresistibly suggested which +Breuer and I (1874, 1875) expressed, that the auditive organ took its +development from an organ for sensing movements by adaptation to weak +periodic motional stimuli, and that many apparatuses in the lower +animals which are held to be organs of hearing are not auditive organs +at all.[107] + +This view appears to be perceptibly gaining ground. Dr. Kreidl by +skilfully-planned experiments has arrived at the conclusion that even +fishes do not hear, whereas E. H. Weber, in his day, regarded the +ossicles which unite the air-bladder of fishes with the labyrinth as +organs expressly designed for conducting sound from the former to the +latter.[108] Störensen has investigated the excitation of sounds by the +air-bladder of fishes, as also the conduction of shocks through Weber's +ossicles. He regards the air-bladder as particularly adapted for +receiving the noises made by other fishes and conducting them to the +labyrinth. He has heard the loud grunting tones of the fishes in South +American rivers, and is of the opinion that they allure and find each +other in this manner. According to these views certain fishes are +neither deaf nor dumb.[109] The question here involved might be solved +perhaps by sharply distinguishing between the sensation of hearing +proper, and the perception of shocks. The first-mentioned sensation may, +even in the case of many vertebrates, be extremely restricted, or +perhaps even absolutely wanting. But besides the auditive function, +Weber's ossicles may perfectly well discharge some other function. +Although, as Moreau has shown, the air-bladder itself is not an organ of +equilibrium in the simple physical sense of Borelli, yet doubtless some +function of this character is still reserved for it. The union with the +labyrinth favors this conception, and so a host of new problems rises +here before us. + +I should like to close with a reminiscence from the year 1863. +Helmholtz's _Sensations of Tone_ had just been published and the +function of the cochlea now appeared clear to the whole world. In a +private conversation which I had with a physician, the latter declared +it to be an almost hopeless undertaking to seek to fathom the function +of the other parts of the labyrinth, whereas I in youthful boldness +maintained that the question could hardly fail to be solved, and that +very soon, although of course I had then no glimmering of how it was to +be done. Ten years later the question was substantially solved. + +To-day, after having tried my powers frequently and in vain on many +questions, I no longer believe that we can make short work of the +problems of science. Nevertheless, I should not consider an +"ignorabimus" as an expression of modesty, but rather as the opposite. +That expression is a suitable one only with regard to problems which are +wrongly formulated and which are therefore not problems at all. Every +real problem can and will be solved in due course of time without +supernatural divination, entirely by accurate observation and close, +searching thought. + + FOOTNOTES: + + [Footnote 93: A lecture delivered on February 24, 1897, before the + _Verein zur Verbreitung naturwissenschaftlicher Kenntnisse in + Wien_.] + + [Footnote 94: Wollaston, _Philosophical Transactions, Royal + Society_, 1810. In the same place Wollaston also describes and + explains the creaking of the muscles. My attention was recently + called to this work by Dr. W. Pascheles.--Cf. also Purkinje, _Prager + medicin_. _Jahrbücher_, Bd. 6, Wien, 1820.] + + [Footnote 95: Similarly many external forces do not act at once on + all parts of the earth, and the internal forces which produce + deformations act at first immediately only upon limited parts. If + the earth were a feeling being, the tides and other terrestrial + events would provoke in it similar sensations to those of our + movements. Perhaps the slight alterations of the altitude of the + pole which are at present being studied are connected with the + continual slight deformations of the central ellipsoid occasioned by + seismical happenings.] + + [Footnote 96: For the popular explanation by unconscious inference + the matter is extremely simple. We regard the railway carriage as + vertical and unconsciously infer the inclination of the trees. Of + course the opposite conclusion that we regard the trees as vertical + and infer the inclination of the carriage, unfortunately, is equally + clear on this theory.] + + [Footnote 97: It will be observed that my way of thinking and + experimenting here is related to that which led Knight to the + discovery and investigation of the geotropism of plants. + _Philosophical Transactions_, January 9, 1806. The relations between + vegetable and animal geotropism have been more recently investigated + by J. Loeb.] + + [Footnote 98: This experiment is doubtless related to the + galvanotropic experiment with the larvæ of frogs described ten years + later by L. Hermann. Compare on this point my remarks in the + _Anzeiger der Wiener Akademie_, 1886, No. 21. Recent experiments in + galvanotropism are due to J. Loeb.] + + [Footnote 99: _Wiener Akad._, 6 November, 1873.] + + [Footnote 100: _Wiener Gesellschaft der Aerzte_, 14 November, 1874.] + + [Footnote 101: I have made a contribution to this last question in + my _Analysis of the Sensations_, (1886), English translation, 1897.] + + [Footnote 102: In my _Grundlinien der Lehre von den + Bewegungsempfindungen_, 1875, the matter occupying lines 4 to 13 of + page 20 from below, which rests on an error, is, as I have also + elsewhere remarked, to be stricken out. For another experiment + related to that of Foucault, compare my _Mechanics_, p. 303.] + + [Footnote 103: _Anzeiger der Wiener Akad._, 30 December, 1875.] + + [Footnote 104: The experiment was specially interesting for me as I + had already attempted in 1874, although with very little confidence + and without success, to excite electromagnetically my own labyrinth + through which I had caused a current to pass.] + + [Footnote 105: Perhaps the discussion concerning the peculiarity of + cats always falling on their feet, which occupied the Parisian + Academy, and, incidentally, Parisian society a few years ago, will + be remembered here. I believe that the questions which arose are + disposed of by the considerations advanced in my + _Bewegungsempfindungen_ (1875). I also partly gave, as early as + 1866, the apparatus conceived by the Parisian scientists to + illustrate the phenomena in question. One difficulty was left + untouched in the Parisian debate. The otolith apparatus of the cat + can render it no service in _free_ descent. The cat, however, while + at rest, doubtless knows its position in space and is instinctively + conscious of the amount of movement which will put it on its feet.] + + [Footnote 106: See the Appendix to the English edition of my + _Analysis of the Sensations_, Chicago, 1897.] + + [Footnote 107: Compare my _Analysis of Sensations_, p. 123 ff.] + + [Footnote 108: E. H. Weber, _De aure et auditu hominis et + animalium_, Lipsiae, 1820.] + + [Footnote 109: Störensen, _Journ. Anat. Phys._, London, Vol. 29 + (1895).] + + + + +ON SOME PHENOMENA ATTENDING THE FLIGHT OF PROJECTILES.[110] + + + "I have led my ragamuffins where they were peppered."--_Falstaff._ + + "He goes but to see a noise that he heard."--_Midsummer Night's + Dream._ + +To shoot, in the shortest time possible, as many holes as possible in +one another's bodies, and not always for exactly pardonable objects and +ideals, seems to have risen to the dignity of a duty with modern men, +who, by a singular inconsistency, and in subservience to a diametrically +contrary ideal, are bound by the equally holy obligation of making these +holes as small as possible, and, when made, of stopping them up and of +healing them as speedily as possible. Since, then, shooting and all that +appertains thereto, is a very important, if not the most important, +affair of modern life, you will doubtless not be averse to giving your +attention for an hour to some experiments which have been undertaken, +not for advancing the ends of war, but for promoting the ends of +science, and which throw some light on the phenomena attending the +flight of projectiles. + +Modern science strives to construct its picture of the world not from +speculations but so far as possible from facts. It verifies its +constructs by recourse to observation. Every newly observed fact +completes its world-picture, and every divergence of a construct from +observation points to some imperfection, to some lacuna in it. What is +seen is put to the test of, and supplemented by, what is thought, which +is again naught but the result of things previously seen. It is always +peculiarly fascinating, therefore, to subject to direct verification by +observation, that is, to render palpable to the senses, something which +we have only theoretically excogitated or theoretically surmised. + +In 1881, on hearing in Paris the lecture of the Belgian artillerist +Melsens, who hazarded the conjecture that projectiles travelling at a +high rate of speed carry masses of compressed air before them which are +instrumental in producing in bodies struck by the projectiles certain +well-known facts of the nature of explosions, the desire arose in me of +experimentally testing his conjecture and of rendering the phenomenon, +if it really existed, perceptible. The desire was the stronger as I +could say that all the means for realising it existed, and that I had in +part already used and tested them for other purposes. + +And first let us get clear regarding the difficulties which have to be +surmounted. Our task is that of observing a bullet or other projectile +which is rushing through space at a velocity of many hundred yards a +second, together with the disturbances which the bullet causes in the +surrounding atmosphere. Even the opaque solid body itself, the +projectile, is only exceptionally visible under such circumstances--only +when it is of considerable size and when we see its line of flight in +strong perspective abridgement so that the velocity is apparently +diminished. We see a large projectile quite clearly when we stand behind +the cannon and look steadily along its line of flight or in the less +pleasant case when the projectile is speeding towards us. There is, +however, a very simple and effective method of observing swiftly moving +bodies with as little trouble as if they were held at rest at some point +in their path. The method is that of illumination by a brilliant +electric spark of extremely short duration in a dark room. But since, +for the full intellectual comprehension of a picture presented to the +eye, a certain, not inconsiderable interval of time is necessary, the +method of instantaneous photography will naturally also be employed. The +pictures, which are of extremely minute duration, are thus permanently +recorded and can be examined and analysed at one's convenience and +leisure. + +With the difficulty just mentioned is associated still another and +greater difficulty which is due to the air. The atmosphere in its usual +condition is generally not visible even when at rest. But the task +presented to us is to render visible masses of air which in addition +are moving with a high velocity. + +To be visible, a body must either emit light itself, must shine, or must +affect in some way the light which falls upon it, must take up that +light entirely or partly, absorb it, or must have a deflective effect +upon it, that is, reflect or refract it. We cannot see the air as we can +a flame, for it shines only exceptionally, as in a Geissler's tube. The +atmosphere is extremely transparent and colorless; it cannot be seen, +therefore, as a dark or colored body can, or as chlorine gas can, or +vapor of bromine or iodine. Air, finally, has so small an index of +refraction and so small a deflective influence upon light, that the +refractive effect is commonly imperceptible altogether. + +A glass rod is visible in air or in water, but it is almost invisible in +a mixture of benzol and bisulphuret of carbon, which has the same mean +index of refraction as the glass. Powdered glass in the same mixture has +a vivid coloring, because owing to the decomposition of the colors the +indices are the same for only one color which traverses the mixture +unimpeded, whilst the other colors undergo repeated reflexions.[111] + +Water is invisible in water, alcohol in alcohol. But if alcohol be mixed +with water the flocculent streaks of the alcohol in the water will be +seen at once and _vice versa_. And in like manner the air, too, under +favorable circumstances, may be seen. Over a roof heated by the burning +sun, a tremulous wavering of objects is noticeable, as there is also +over red-hot stoves, radiators, and registers. In all these cases tiny +flocculent masses of hot and cold air, of slightly differing +refrangibility, are mingled together. + +In like manner the more highly refracting parts of non-homogeneous +masses of glass, the so-called striæ or imperfections of the glass, are +readily detectible among the less refracting parts which constitute the +bulk of the same. Such glasses are unserviceable for optical purposes, +and special attention has been devoted to the investigation of the +methods for eliminating or avoiding these defects. The result has been +the development of an extremely delicate method for detecting optical +faults--the so-called method of Foucault and Toepler--which is suitable +also for our present purpose. + +[Illustration: Fig. 49.] + +Even Huygens when trying to detect the presence of striæ in polished +glasses viewed them under oblique illumination, usually at a +considerable distance, so as to give full scope to the aberrations, and +had recourse for greater exactitude to a telescope. But the method was +carried to its highest pitch of perfection in 1867 by Toepler who +employed the following procedure: A small luminous source _a_ (Fig. 49) +illuminates a lens _L_ which throws an image _b_ of the luminous source. +If the eye be so placed that the image falls on the pupil, the entire +lens, if perfect, will appear equally illuminated, for the reason that +all points of it send out rays to the eye. Coarse imperfections of form +or of homogeneity are rendered visible only in case the aberrations are +so large that the light from many spots passes by the pupil of the eye. +But if the image _b_ be partly intercepted by the edge of a small slide, +then those spots in the lens as thus partly darkened will appear +brighter whose light by its greater aberrations still reaches the eye in +spite of the intercepting slide, while those spots will appear darker +which in consequence of aberration in the other direction throw their +light entirely upon the slide. This artifice of the intercepting slide +which had previously been employed by Foucault for the investigation of +the optical imperfections of mirrors enhances enormously the delicacy of +the method, which is still further augmented by Toepler's employment of +a telescope behind the slide. Toepler's method, accordingly, enjoys all +the advantages of the Huygens and the Foucault procedure combined. It is +so delicate that the minutest irregularities in the air surrounding the +lens can be rendered distinctly visible, as I shall show by an example. +I place a candle before the lens _L_ (Fig. 50) and so arrange a second +lens _M_ that the flame of the candle is imaged upon the screen _S_. As +soon as the intercepting slide is pushed into the focus, _b_, of the +light issuing from _a_, you see the images of the changes of density and +the images of the movements induced in the air by the flame quite +distinctly upon the screen. The distinctness of the phenomenon as a +whole depends upon the position of the intercepting slide _b_. The +removal of _b_ increases the illumination but decreases the +distinctness. If the luminous source _a_ be removed, we see the image of +the candle flame only upon the screen _S_. If we remove the flame and +allow _a_ to continue shining, the screen _S_ will appear uniformly +illuminated. + +[Illustration: Fig. 50.] + +After Toepler had sought long and in vain to render the irregularities +produced in air by sound-waves visible by this principle, he was at last +conducted to his goal by the favorable circumstances attending the +production of electric sparks. The waves generated in the air by +electric sparks and accompanying the explosive snapping of the same, are +of sufficiently short period and sufficiently powerful to be rendered +visible by these methods. Thus we see how by a careful regard for the +merest and most shadowy indications of a phenomenon and by slight +progressive and appropriate alterations of the circumstances and the +methods, ultimately the most astounding results can be attained. +Consider, for example, two such phenomena as the rubbing of amber and +the electric lighting of modern streets. A person ignorant of the myriad +minute links that join these two things together, will be absolutely +nonplussed at their connexion, and will comprehend it no more than the +ordinary observer who is unacquainted with embryology, anatomy, and +paleontology will understand the connexion between a saurian and a bird. +The high value and significance of the co-operation of inquirers through +centuries, where each has but to take up the thread of work of his +predecessors and spin it onwards, is rendered forcibly evident by such +examples. And such knowledge destroys, too, in the clearest manner +imaginable that impression of the marvellous which the spectator may +receive from science, and at the same time is a most salutary +admonishment to the worker in science against superciliousness. I have +also to add the sobering remark that all our art would be in vain did +not nature herself afford at least some slight guiding threads leading +from a hidden phenomenon into the domain of the observable. And so it +need not surprise us that once under particularly favorable +circumstances an extremely powerful sound-wave which had been caused by +the explosion of several hundred pounds of dynamite threw a directly +visible shadow in the sunlight, as Boys has recently told us. If the +sound-waves were absolutely without influence upon the light, this could +not have occurred, and all our artifices would then, too, be in vain. +And so, similarly, the phenomenon accompanying projectiles which I am +about to show you was once in a very imperfect manner incidentally seen +by a French artillerist, Journée, while that observer was simply +following the line of flight of a projectile with a telescope, just as +also the undulations produced by candle flames are in a weak degree +directly visible and in the bright sunlight are imaged in shadowy waves +upon a uniform white background. + +_Instantaneous illumination_ by the electric spark, the method of +rendering visible small optical differences or striæ, which may hence be +called the _striate_, or _differential_, method,[112] invented by +Foucault and Toepler, and finally the _recording_ of the image by a +_photographic_ plate,--these therefore are the chief means which are to +lead us to our goal. + +I instituted my first experiments in the summer of 1884 with a +target-pistol, shooting the bullet through a striate field as described +above, and taking care that the projectile whilst in the field should +disengage an illuminating electric spark from a Leyden jar or Franklin's +pane, which spark produced a photographic impression of the projectile +upon a plate, especially arranged for the purpose. I obtained the image +of the projectile at once and without difficulty. I also readily +obtained, with the still rather defective dry plate which I was using, +exceedingly delicate images of the sound-waves (spark-waves). But no +atmospheric condensation produced by the projectile was visible. I now +determined the velocity of my projectile and found it to be only 240 +metres per second, or considerably less than the velocity of sound +(which is 340 metres per second). I saw immediately that under such +circumstances no noticeable compression of the air could be produced, +for any atmospheric compression must of necessity travel forward at the +same speed with sound (340 metres per second) and consequently would be +always ahead of and speeding away from the projectile. + +I was so thoroughly convinced, however, of the existence of the supposed +phenomenon at a velocity exceeding 340 metres per second, that I +requested Professor Salcher, of Fiume, an Austrian port on the Gulf of +Quarnero, to undertake the experiment with projectiles travelling at a +high rate of speed. In the summer of 1886 Salcher in conjunction with +Professor Riegler conducted in a spacious and suitable apartment placed +at their disposal by the Directors of the Royal Imperial Naval Academy, +experiments of the kind indicated and conforming in method exactly to +those which I had instituted, with the precise results expected. The +phenomenon, in fact, accorded perfectly with the _a priori_ sketch of it +which I had drafted previously to the experiment. As the experimenting +was continued, new and unforeseen features made their appearance. + +It would be unfair, of course, to expect from the very first experiments +faultless and highly distinct photographs. It was sufficient that +success was secured and that I had convinced myself that further labor +and expenditure would not be vain. And on this score I am greatly +indebted to the two gentlemen above mentioned. + +The Austrian Naval Department subsequently placed a cannon at Salcher's +disposal in Pola, an Adriatic seaport, and I myself, together with my +son, then a student of medicine, having received and accepted a +courteous invitation from Krupp, repaired to Meppen, a town in Hanover, +where we conducted with only the necessary apparatus several experiments +on the open artillery range. All these experiments furnished tolerably +good and complete pictures. Some little progress, too, was made. The +outcome of our experience on both artillery ranges, however, was the +settled conviction that really good results could be obtained only by +the most careful conduct of the experiments in a laboratory especially +adapted to the purpose. The expensiveness of the experiments on a large +scale was not the determining consideration here, for the size of the +projectile is indifferent. Given the same velocity and the results are +quite similar, whether the projectiles are large or small. On the other +hand, in a laboratory the experimenter has perfect control over the +initial velocity, which, provided the proper equipment is at hand, can +be altered at will simply by altering the charge and the weight of the +projectile. The requisite experiments were accordingly conducted by me +in my laboratory at Prague, partly in conjunction with my son and partly +afterwards by him alone. The latter are the most perfect and I shall +accordingly speak in detail here of these only. + +[Illustration: Fig. 51.] + +Picture to yourself an apparatus for detecting optical striæ set up in a +dark room. In order not to make the description too complicated, I shall +give the essential features only of the apparatus, leaving out of +account altogether the minuter details which are rather of consequence +for the technical performance of the experiment than for its +understanding. We suppose the projectile speeding on its path, +accordingly, through the field of our differential optical apparatus. +On reaching the centre of the field (Fig. 51) the projectile disengages +an illuminating electric spark _a_, and the image of the projectile, so +produced, is photographically impressed upon the plate of the camera +behind the intercepting slide _b_. In the last and best experiments the +lens _L_ was replaced by a spherical silvered-glass mirror made by K. +Fritsch (formerly Prokesch) of Vienna, whereby the apparatus was +naturally more complicated than it appears in our diagram. The +projectile having been carefully aimed passes in crossing the +differential field between two vertical isolated wires which are +connected with the two coatings of a Leyden jar, and completely filling +the space between the wires discharges the jar. In the axis of the +differential apparatus the circuit has a second gap _a_ which furnishes +the illuminating spark, the image of which falls on the intercepting +slide _b_. The wires in the differential field having occasioned +manifold disturbances were subsequently done away with. In the new +arrangement the projectile passes through a ring (see dotted line, Fig. +51), to the air in which it imparts a sharp impulse which travels +forward in the tube _r_ as a sound-wave having the approximate velocity +of 340 metres per second, topples over through the aperture of an +electric screen the flame of a candle situated at the other opening of +the tube, and so discharges the jar. The length of the tube _r_ is so +adjusted that the discharge occurs the moment the projectile enters the +centre of the now fully clear and free field of vision. We will also +leave out of account the fact that to secure fully the success of the +experiment, a large jar is first discharged by the flame, and that by +the agency of this first discharge the discharge of a second small jar +having a spark of very short period which furnishes the spark really +illuminating the projectile is effected. Sparks from large jars have an +appreciable duration, and owing to the great velocity of the projectiles +furnish blurred photographs only. By carefully husbanding the light of +the differential apparatus, and owing to the fact that much more light +reaches the photographic plate in this way than would otherwise reach +it, we can obtain beautiful, strong, and sharp photographs with +incredibly small sparks. The contours of the pictures appear as very +delicate and very sharp, closely adjacent double lines. From their +distance from one another, and from the velocity of the projectile, the +duration of the illumination, or of the spark, is found to be 1/800000 +of a second. It is evident, therefore, that experiments with mechanical +snap slides can furnish no results worthy of the name. + +[Illustration: Fig. 52.] + +Let us consider now first the picture of a projectile in the rough, as +represented in Figure 52, and then let us examine it in its photographic +form as seen in Figure 53. The latter picture is of a shot from an +Austrian Mannlicher rifle. If I were not to tell you what the picture +represented you would very likely imagine it to be a bird's eye view of +a boat _b_ moving swiftly through the water. In front you see the +bow-wave and behind the body a phenomenon _k_ which closely resembles +the eddies formed in the wake of a ship. And as a matter of fact the +dark hyperboloid arc which streams from the tip of the projectile really +is a compressed wave of air exactly analogous to the bow-wave produced +by a ship moving through the water, with the exception that the wave of +air is not a surface-wave. The air-wave is produced in atmospheric space +and encompasses the projectile in the form of a shell on all sides. The +wave is visible for the same reason that the heated shell of air +surrounding the candle flame of our former experiments is visible. And +the cylinder of friction-heated air which the projectile throws off in +the form of vortex rings really does answer to the water in the wake of +a vessel. + +[Illustration: Fig. 53. Photograph of a blunted projectile.] + +Now just as a slowly moving boat produces no bow-wave, but the bow-wave +is seen only when the boat moves with a speed which is greater than the +velocity of propagation of surface-waves in water, so, in like manner, +no wave of compression is visible in front of a projectile so long as +the speed of the projectile is less than the velocity of sound. But if +the speed of the projectile reaches and exceeds the velocity of sound, +then the head-wave, as we shall call it, augments noticeably in power, +and is more and more extended, that is, the angle made by the contours +of the wave with the direction of flight is more and more diminished, +just as when the speed of a boat is increased a similar phenomenon is +noticed in connexion with the bow-wave. In fact, we can from an +instantaneous photograph so taken approximately estimate the speed with +which the projectile is travelling. + +The explanation of the bow-wave of a ship and that of the head-wave of a +body travelling in atmospheric space both repose upon the same +principle, long ago employed by Huygens. Conceive a number of pebbles to +be cast into a pond of water at regular intervals in such wise that all +the spots struck are situate in the same straight line, and that every +spot subsequently struck lies a short space farther to the right. The +spots first struck will furnish then the wave-circles which are widest, +and all of them together will, at the points where they are thickest, +form a sort of cornucopia closely resembling the bow-wave. (Fig. 54.) +The resemblance is greater the smaller the pebbles are, and the more +quickly they succeed each other. If a rod be dipped into the water and +quickly carried along its surface, the falling of the pebbles will then +take place, so to speak, uninterruptedly, and we shall have a real +bow-wave. If we put the compressed air-wave in the place of the +surface-waves of the water, we shall have the head-wave of the +projectile. + +[Illustration: Fig. 54.] + +You may be disposed to say now, it is all very pretty and interesting to +observe a projectile in its flight, but of what practical use is it? + +It is true, I reply, one cannot _wage war_ with photographed +projectiles. And I have likewise often had to say to medical students +attending my lectures on physics, when they inquired for the practical +value of some physical observation, "You cannot, gentlemen, cure +diseases with it." I had also once to give my opinion regarding how much +physics should be taught at a school for millers, supposing the +instruction there to be confined _exactly_ to what was necessary for a +miller. I was obliged to reply: "A miller always _needs_ exactly as much +physics as he _knows_." Knowledge which one does not possess one cannot +use. + +Let us forego entirely the consideration that as a general thing every +scientific advance, every new problem elucidated, every extension or +enrichment of our knowledge of facts, affords a better foundation for +practical pursuits. Let us rather put the special question, Is it not +possible to derive some really practical knowledge from our theoretical +acquaintance with the phenomena which take place in the space +surrounding a projectile? + +No physicist who has ever studied waves of sound or photographed them +will have the least doubt regarding the sound-wave character of the +atmospheric condensation encompassing the head of a flying projectile. +We have therefore, without ado, called this condensation the head-wave. + +Knowing this, it follows that the view of Melsens according to which the +projectile carries along with it masses of air which it forces into the +bodies struck, is untenable. A forward-moving sound-wave is not a +forward-moving mass of matter but a forward-moving form of motion, just +as a water-wave or the waves of a field of wheat are only forward-moving +forms of motion and not movements of masses of water or masses of wheat. + +By interference-experiments, on which I cannot touch here but which will +be found roughly represented in Figure 55, it was found that the +bell-shaped head-wave in question is an extremely thin shell and that +the condensations of the same are quite moderate, scarcely exceeding +two-tenths of an atmosphere. There can be no question, therefore, of +explosive effects in the body struck by the projectile through so slight +a degree of atmospheric compression. The phenomena attending wounds from +rifle balls, for example, are not to be explained as Melsens and Busch +explain them, but are due, as Kocher and Reger maintain, to the effects +of the impact of the projectile itself. + +[Illustration: Fig. 55.] + +A simple experiment will show how insignificant is the part played by +the friction of the air, or the supposed conveyance of the air along +with the moving projectile. If the photograph of the projectile be +taken while passing through a flame, i. e., a visible gas, the flame +will be seen to be, not torn and deformed, but smoothly and cleanly +perforated, like any solid body. Within and around the flame the +contours of the head-wave will be seen. The flickering, the extinction +of the flame, etc., take place only after the projectile has travelled +on a considerable distance in its path, and is then affected by the +powder gases which hurry after the bullet or by the air preceding the +powder-gases. + +The physicist who examines the head-wave and recognises its sound-wave +character also sees that the wave in question is of the same kind with +the short sharp waves produced by electric sparks, that it is a +_noise_-wave. Hence, whenever any portion of the head-wave strikes the +ear it will be heard as a report. Appearances point to the conclusion +that the projectile carries this report along with it. In addition to +this report, which advances with the velocity of the projectile and so +usually travels at a speed greater than the velocity of sound, there is +also to be heard the report of the exploding powder which travels +forward with the ordinary velocity of sound. Hence two explosions will +be heard, each distinct in time. The circumstance that this fact was +long misconstrued by practical observers but when actually noticed +frequently received grotesque explanations and that ultimately my view +was accepted as the correct one, appears to me in itself a sufficient +justification that researches such as we are here speaking of are not +utterly superfluous even in practical directions. That the flashes and +sounds of discharging artillery are used for estimating the distances of +batteries is well known, and it stands to reason that any unclear +theoretical conception of the facts here involved will seriously affect +the correctness of practical calculations. + +It may appear astonishing to a person hearing it for the first time, +that a single shot has a double report due to two different velocities +of propagation. But the reflexion that projectiles whose velocity is +less than the velocity of sound produce no head-waves (because every +impulse imparted to the air travels forward, that is, ahead, with +exactly the velocity of sound), throws full light when logically +developed upon the peculiar circumstance above mentioned. If the +projectile moves faster than sound, the air ahead of it cannot recede +from it quickly enough. The air is condensed and warmed, and thereupon, +as all know, the velocity of sound is augmented until the head-wave +travels forward as rapidly as the projectile itself, so that there is no +need whatever of any additional augmentation of the velocity of +propagation. If such a wave were left entirely to itself, it would +increase in length and soon pass into an ordinary sound-wave, travelling +with less velocity. But the projectile is always behind it and so +maintains it at its proper density and velocity. Even if the projectile +penetrates a piece of cardboard or a board of wood, which catches and +obstructs the head-wave, there will, as Figure 56 shows, immediately +appear at the emerging apex a newly formed, not to say newly born, +head-wave. We may observe on the cardboard the reflexion and diffraction +of the head-wave, and by means of a flame its refraction, so that no +doubt as to its nature can remain. + +[Illustration: Fig. 56.] + +Permit me, now, to illustrate the most essential of the points that I +have just adduced, by means of a few rough drawings taken from older and +less perfect photographs. + +In the sketch of Figure 57 you see the projectile, which has just left +the barrel of the rifle, touch a wire and disengage the illuminating +spark. At the apex of the projectile you already see the beginnings of +a powerful head-wave, and in front of the wave a transparent fungiform +cluster. This latter is the air which has been forced out of the barrel +by the projectile. Circular sound-waves, noise-waves, which are soon +overtaken by the projectile, also issue from the barrel. But behind the +projectile opaque puffs of powder-gas rush forth. It is scarcely +necessary to add that many other questions in ballistics may be studied +by this method, as, for example, the movement of the gun-carriage. + +[Illustration: Fig. 57.] + +A distinguished French artillerist, M. Gossot, has applied the views of +the head-wave here given in quite a different manner. The practice in +measuring the velocity of projectiles is to cause the projectile to pass +through wire screens placed at different points in its path, and by the +tearing of these screens to give rise to electro-magnetic time-signals +on falling slabs or rotating drums. Gossot caused these signals to be +made directly by the impact of the head-wave, did away thus with the +wire screens, and carried the method so far as to be able to measure the +velocities of projectiles travelling in high altitudes, where the use of +wire screens was quite out of the question. + +The laws of the resistance of fluids and of air to bodies travelling in +them form an extremely complicated problem, which can be reasoned out +very simply and prettily as a matter of pure philosophy but practice +offers not a few difficulties. The same body having the velocity 2, 3, 4 +... displaces in the same interval 2, 3, 4 ... times the same mass of +air, or the same mass of fluid, and imparts to it _in addition_ 2, 3, 4 +... times the same velocity. But for this, plainly, 4, 9, 16 ... times +the original force is required. Hence, the resistance, it is said, +increases with the square of the velocity. This is all very pretty and +simple and obvious. But practice and theory are at daggers' points here. +Practice tells us that when we increase the velocity, the law of the +resistance changes. For every portion of the velocity the law is +different. + +The studies of the talented English naval architect, Froude, have thrown +light upon this question. Froude has shown that the resistance is +conditioned by a combination of the most multifarious phenomena. A ship +in motion is subjected to the friction of the water. It causes eddies +and it generates in addition waves which radiate outward from it. Every +one of these phenomena are dependent upon the velocity in some different +manner, and it is consequently not astonishing that the law of the +resistance should be a complicated one. + +The preceding observations suggest quite analogous reflexions for +projectiles. Here also we have friction, the formation of eddies, and +the generation of waves. Here, also, therefore, we should not be +surprised at finding the law of the resistance of the air a complicated +one, nor puzzled at learning that in actuality the law of resistance +changes as soon as the speed of the projectile exceeds the velocity of +sound, for this is the precise point at which one important element of +the resistance, namely, the formation of waves, first comes into play. + +No one doubts that a pointed bullet pierces the air with less resistance +than a blunt bullet. The photographs themselves show that the head-wave +is weaker for a pointed projectile. It is not impossible, similarly, +that forms of bullets will be invented which generate fewer eddies, +etc., and that we shall study these phenomena also by photography. I am +of opinion from the few experiments which I have made in this direction +that not much more can be done by changing the form of the projectile +when the velocity is very great, but I have not gone into the question +thoroughly. Researches of the kind we are considering can certainly not +be detrimental to practical artillery, and it is no less certain that +experiments by artillerists on a large scale will be of undoubted +benefit to physics. + +No one who has had the opportunity of studying modern guns and +projectiles in their marvellous perfection, their power and precision, +can help confessing that a high technical and scientific achievement has +found its incarnation in these objects. We may surrender ourselves so +completely to this impression as to forget for a moment the terrible +purposes they serve. + +Permit me, therefore, before we separate, to say a few words on this +glaring contrast. The greatest man of war and of silence which the +present age has produced once asserted that perpetual peace is a dream, +and not a beautiful dream at that. We may accord to this profound +student of mankind a judgment in these matters and can also appreciate +the soldier's horror of stagnation from all too lengthy peace. But it +requires a strong belief in the insuperableness of mediæval barbarism to +hope for and to expect no great improvement in international relations. +Think of our forefathers and of the times when club law ruled supreme, +when within the same country and the same state brutal assaults and +equally brutal self-defence were universal and self-evident. This state +of affairs grew so oppressive that finally a thousand and one +circumstances compelled people to put an end to it, and the cannon had +most to say in accomplishing the work. Yet the rule of club law was not +abolished so quickly after all. It had simply passed to other clubs. We +must not abandon ourselves to dreams of the Rousseau type. Questions of +law will in a sense forever remain questions of might. Even in the +United States where every one is as a matter of principle entitled to +the same privileges, the ballot according to Stallo's pertinent remark +is but a milder substitute for the club. Nor need I tell you that many +of our own fellow-citizens are still enamored of the old original +methods. Very, very gradually, however, as civilisation progresses, the +intercourse of men takes on gentler forms, and no one who really knows +the good old times will ever honestly wish them back again, however +beautifully they may be painted and rhymed about. + +In the intercourse of the nations, however, the old club law still +reigns supreme. But since its rule is taxing the intellectual, the +moral, and the material resources of the nations to the utmost and +constitutes scarcely less a burden in peace than in war, scarcely less a +yoke for the victor than for the vanquished, it must necessarily grow +more and more unendurable. Reason, fortunately, is no longer the +exclusive possession of those who modestly call themselves the upper ten +thousand. Here, as everywhere, the evil itself will awaken the +intellectual and ethical forces which are destined to mitigate it. Let +the hate of races and of nationalities run riot as it may, the +intercourse of nations will still increase and grow more intimate. By +the side of the problems which separate nations, the great and common +ideals which claim the exclusive powers of the men of the future appear +one after another in greater distinctness and in greater might. + + FOOTNOTES: + + [Footnote 110: A lecture delivered on Nov. 10, 1897.] + + [Footnote 111: Christiansen, _Wiedemann's Annalen_, XXIII. S. 298, + XXIV., p. 439 (1884-1885).] + + [Footnote 112: The German phrase is _Schlierenmethode_, by which + term the method is known even by American physicists. It is also + called in English the "shadow-method." But a term is necessary which + will cover all the derivatives, and so we have employed + alternatively the words _striate_ and _differential_. The etymology + of _schlieren_, it would seem, is uncertain. Its present use is + derived from its technological signification in glass-manufacturing, + where by _die Schlieren_ are meant the wavy streaks and + imperfections in glass. Hence its application to the method for + detecting small optical _differences_ and faults generally. + Professor Crew of Evanston suggests to the translator that + _schlieren_ may be related to our _slur_ (L. G., _slüren_, to trail, + to draggle), a conjecture which is doubtless correct and agrees both + with the meaning of _schlieren_ as given in the large German + dictionaries and with the intransitive use of our own verb _slur_, + the faults in question being conceived as "trailings," "streakings," + etc.--_Trans._] + + + + +ON INSTRUCTION IN THE CLASSICS AND THE SCIENCES.[113] + + +Perhaps the most fantastic proposition that Maupertuis,[114] the +renowned president of the Berlin Academy, ever put forward for the +approval of his contemporaries was that of founding a city in which, to +instruct and discipline young students, only Latin should be spoken. +Maupertuis's Latin city remained an idle wish. But for centuries Latin +and Greek _institutions_ exist in which our children spend a goodly +portion of their days, and whose atmosphere constantly surrounds them, +even when without their walls. + +For centuries instruction in the ancient languages has been zealously +cultivated. For centuries its necessity has been alternately championed +and contested. More strongly than ever are authoritative voices now +raised against the preponderance of instruction in the classics and in +favor of an education more suited to the needs of the time, especially +for a more generous treatment of mathematics and the natural sciences. + +In accepting your invitation to speak here on the relative educational +value of the classical and the mathematico-physical sciences in colleges +and high schools, I find my justification in the duty and the necessity +laid upon every teacher of forming from his own experiences an opinion +upon this important question, as partly also in the special circumstance +that in my youth I was personally under the influence of school-life for +only a short time, just previous to my entering the university, and had, +therefore, ample opportunity to observe the effects of widely different +methods upon my own person. + +Passing now, to a review of the arguments which the advocates of +instruction in the classics advance, and of what the adherents of +instruction in the physical sciences in their turn adduce, we find +ourselves in rather a perplexing position with respect to the arguments +of the first named. For these have been different at different times, +and they are even now of a very multifarious character, as must be where +men advance, in favor of an institution that exists and which they are +determined to retain at any cost, everything they can possibly think of. +We shall find here much that has evidently been brought forward only to +impress the minds of the ignorant; much, too, that was advanced in good +faith and which is not wholly without foundation. We shall get a fair +idea of the reasoning employed by considering, first, the arguments that +have grown out of the historical circumstances connected with the +original introduction of the classics, and, lastly, those which were +subsequently adduced as accidental afterthoughts. + + * * * * * + +Instruction in Latin, as Paulsen[115] has minutely shown, was introduced +by the Roman Church along with Christianity. With the Latin language +were also transmitted the scant and meagre remnants of ancient science. +Whoever wished to acquire this ancient education, then the only one +worthy of the name, for him the Latin language was the only and +indispensable means; such a person had to learn Latin to rank among +educated people. + +The wide-spread influence of the Roman Church wrought many and various +results. Among those for which all are glad, we may safely count the +establishment of a sort of _uniformity_ among the nations and of a +regular international intercourse by means of the Latin language, which +did much to unite the nations in the common work of civilisation, +carried on from the fifteenth to the eighteenth century. The Latin +language was thus long the language of scholars, and instruction in +Latin the road to a liberal education--a shibboleth still employed, +though long inappropriate. + +For scholars as a class, it is to be regretted, perhaps, that Latin has +ceased to be the medium of international communication. But the +attributing of the loss of this function by the Latin language to its +incapacity to accommodate itself to the numerous new ideas and +conceptions which have arisen in the course of the development of +science is, in my opinion, wholly erroneous. It would be difficult to +find a modern scientist who had enriched science with as many new ideas +as Newton has, yet Newton knew how to express those ideas very correctly +and precisely in the Latin language. If this view were correct, it would +also hold true of every living language. Originally every language has +to adapt itself to new ideas. + +It is far more likely that Latin was displaced as the literary vehicle +of science by the influence of the nobility. By their desire to enjoy +the fruits of literature and science, through a less irksome medium +than Latin, the nobility performed for the people at large an +undeniable service. For the days were now past when acquaintance with +the language and literature of science was restricted to a caste, and in +this step, perhaps, was made the most important advance of modern times. +To-day, when international intercourse is firmly established in spite of +the many languages employed, no one would think of reintroducing +Latin.[116] + +The facility with which the ancient languages lend themselves to the +expression of new ideas is evidenced by the fact that the great majority +of our scientific ideas, as survivals of this period of Latin +intercourse, bear Latin and Greek designations, while in great measure +scientific ideas are even now invested with names from these sources. +But to deduce from the existence and use of such terms the necessity of +still learning Latin and Greek on the part of all who employ them is +carrying the conclusion too far. All terms, appropriate and +inappropriate,--and there are a large number of inappropriate and +monstrous combinations in science,--rest on convention. The essential +thing is, that people should associate with the sign the precise idea +that is designated by it. It matters little whether a person can +correctly derive the words _telegraph_, _tangent_, _ellipse_, _evolute_, +etc., if the correct idea is present in his mind when he uses them. On +the other hand, no matter how well he may know their etymology, his +knowledge will be of little use to him if the correct idea is absent. +Ask the average and fairly educated classical scholar to translate a few +lines for you from Newton's _Principia_, or from Huygens's _Horologium_, +and you will discover at once what an extremely subordinate rôle the +mere knowledge of language plays in such things. Without its associated +thought a word remains a mere sound. The fashion of employing Greek and +Latin designations--for it can be termed nothing else--has a natural +root in history; it is impossible for the practice to disappear +suddenly, but it has fallen of late considerably into disuse. The terms +_gas_, _ohm_, _Ampère_, _volt_, etc., are in international use, but they +are not Latin nor Greek. Only the person who rates the unessential and +accidental husk higher than its contents, can speak of the necessity of +learning Latin or Greek for such reasons, to say nothing of spending +eight or ten years on the task. Will not a dictionary supply in a few +seconds all the information we wish on such subjects?[117] + +It is indisputable that our modern civilisation took up the threads of +the ancient civilisation, that at many points it begins where the latter +left off, and that centuries ago the remains of the ancient culture were +the only culture existing in Europe. Then, of course, a classical +education really was the liberal education, the higher education, the +ideal education, for it was the _sole_ education. But when the same +claim is now raised in behalf of a classical education, it must be +uncompromisingly contested as bereft of all foundation. For our +civilisation has gradually attained its independence; it has lifted +itself far above the ancient civilisation, and has entered generally new +directions of progress. Its note, its characteristic feature, is the +enlightenment that has come from the great mathematical and physical +researches of the last centuries, and which has permeated not only the +practical arts and industries but is also gradually finding its way into +all fields of thought, including philosophy and history, sociology and +linguistics. Those traces of ancient views that are still discoverable +in philosophy, law, art, and science, operate more as hindrances than +helps, and will not long stand before the development of independent and +more natural views. + +It ill becomes classical scholars, therefore, to regard themselves, at +this day, as the educated class _par excellence_, to condemn as +uneducated all persons who do not understand Latin and Greek, to +complain that with such people profitable conversations are not to be +carried on, etc. The most delectable stories have got into circulation, +illustrative of the defective education of scientists and engineers. A +renowned inquirer, for example, is said to have once announced his +intention of holding a free course of university lectures, with the word +"frustra"; an engineer who spent his leisure hours in collecting insects +is said to have declared that he was studying "etymology." It is true, +incidents of this character make us shudder or smile, according to our +mood or temperament. But we must admit, the next moment, that in giving +way to such feelings we have merely succumbed to a childish prejudice. A +lack of tact but certainly no lack of education is displayed in the use +of such half-understood expressions. Every candid person will confess +that there are many branches of knowledge about which he had better be +silent. We shall not be so uncharitable as to turn the tables and +discuss the impression that classical scholars might make on a scientist +or engineer, in speaking of science. Possibly many ludicrous stories +might be told of them, and of far more serious import, which should +fully compensate for the blunders of the other party. + +The mutual severity of judgment which we have here come upon, may also +forcibly bring home to us how really scarce a true liberal culture is. +We may detect in this mutual attitude, too, something of that narrow, +mediæval arrogance of caste, where a man began, according to the special +point of view of the speaker, with the scholar, the soldier, or the +nobleman. Little sense or appreciation is to be found in it for the +_common_ task of humanity, little feeling for the need of mutual +assistance in the great work of civilisation, little breadth of mind, +little truly liberal culture. + +A knowledge of Latin, and partly, also, a knowledge of Greek, is still a +necessity for the members of a few professions by nature more or less +directly concerned with the civilisations of antiquity, as for lawyers, +theologians, philologists, historians, and generally for a small number +of persons, among whom from time to time I count myself, who are +compelled to seek for information in the Latin literature of the +centuries just past.[118] But that all young persons in search of a +higher education should pursue for this reason Latin and Greek to such +excess; that persons intending to become physicians and scientists +should come to the universities defectively educated, or even +miseducated; and that they should be compelled to come only from schools +that do _not_ supply them with the proper preparatory knowledge is going +a little bit too far. + + * * * * * + +After the conditions which had given to the study of Latin and Greek +their high import had ceased to exist, the traditional curriculum, +naturally, was retained. Then, the different effects of this method of +education, good and bad, which no one had thought of at its +introduction, were realised and noted. As natural, too, was it that +those who had strong interests in the preservation of these studies, +from knowing no others or from living by them, or for still other +reasons, should emphasise the _good_ results of such instruction. They +pointed to the good effects as if they had been consciously aimed at by +the method and could be attained only through its agency. + +One real benefit that students might derive from a rightly conducted +course in the classics would be the opening up of the rich literary +treasures of antiquity, and intimacy with the conceptions and views of +the world held by two advanced nations. A person who has read and +understood the Greek and Roman authors has felt and experienced more +than one who is restricted to the impressions of the present. He sees +how men placed in different circumstances judge quite differently of the +same things from what we do to-day. His own judgments will be rendered +thus more independent. Again, the Greek and Latin authors are +indisputably a rich fountain of recreation, of enlightenment, and of +intellectual pleasure after the day's toil, and the individual, not less +than civilised humanity generally, will remain grateful to them for all +time. Who does not recall with pleasure the wanderings of Ulysses, who +does not listen joyfully to the simple narratives of Herodotus, who +would ever repent of having made the acquaintance of Plato's Dialogues, +or of having tasted Lucian's divine humor? Who would give up the glances +he has obtained into the private life of antiquity from Cicero's +letters, from Plautus or Terence? To whom are not the portraits of +Suetonius undying reminiscences? Who, in fact, would throw away _any_ +knowledge he had once gained? + +Yet people who draw from these sources only, who know only this culture, +have surely no right to dogmatise about the value of some other culture. +As objects of research for individuals, this literature is extremely +valuable, but it is a different question whether it is equally valuable +as the almost exclusive means of education of our youth. + +Do not other nations and other literatures exist from which we ought to +learn? Is not nature herself our first school-mistress? Are our highest +models always to be the Greeks, with their narrow provinciality of mind, +that divided the world into "Greeks and barbarians," with their +superstitions, with their eternal questioning of oracles? Aristotle with +his incapacity to learn from facts, with his word-science; Plato with +his heavy, interminable dialogues, with his barren, at times childish, +dialectics--are they unsurpassable?[119] The Romans with their apathy, +their pompous externality, set off by fulsome and bombastic phrases, +with their narrow-minded, philistine philosophy, with their frenzied +sensuality, with their cruel and bestial indulgence in animal and man +baiting, with their outrageous maltreatment and plundering of their +subjects--are they patterns worthy of imitation? Or shall, perhaps, our +science edify itself with the works of Pliny who cites midwives as +authorities and himself stands on their point of view? + +Besides, if an acquaintance with the ancient world really were attained, +we might come to some settlement with the advocates of classical +education. But it is words and forms, and forms and words only, that are +supplied to our youth; and even collateral subjects are forced into the +strait-jacket of the same rigid method and made a science of words, +sheer feats of mechanical memory. Really, we feel ourselves set back a +thousand years into the dull cloister-cells of the Middle Ages. + +This must be changed. It is possible to get acquainted with the views of +the Greeks and Romans by a shorter road than the intellect deadening +process of eight or ten years of declining, conjugating, analysing, and +extemporisation. There are to-day plenty of educated persons who have +acquired through good translations vivider, clearer, and more just views +of classical antiquity than the graduates of our gymnasiums and +colleges.[120] + +For us moderns, the Greeks and the Romans are simply two objects of +archæological and historical research like all others. If we put them +before our youth in fresh and living pictures, and not merely in words +and syllables, the effect will be assured. We derive a totally different +enjoyment from the Greeks when we approach them after a study of the +results of modern research in the history of civilisation. We read many +a chapter of Herodotus differently when we attack his works equipped +with a knowledge of natural science, and with information about the +stone age and the lake-dwellers. What our classical institutions +_pretend_ to give can and actually will be given to our youth with much +more fruitful results by competent _historical_ instruction, which must +supply, not names and numbers alone, nor the mere history of dynasties +and wars, but be in every sense of the word a true history of +civilisation. + +The view still widely prevails that although all "higher, ideal +culture," all extension of our view of the world, is acquired by +philological and in a lesser degree by historical studies, still the +mathematics and natural sciences should not be neglected on account of +their usefulness. This is an opinion to which I must refuse my assent. +It were strange if man could learn more, could draw more intellectual +nourishment, from the shards of a few old broken jugs, from inscribed +stones, or yellow parchments, than from all the rest of nature. True, +man is man's first concern, but he is not his sole concern. + +In ceasing to regard man as the centre of the world; in discovering that +the earth is a top whirled about the sun, which speeds off with it into +infinite space; in finding that in the fixed stars the same elements +exist as on earth; in meeting everywhere the same processes of which the +life of man is merely a vanishingly small part--in such things, too, is +a widening of our view of the world, and edification, and poetry. There +are here perhaps grander and more significant facts than the bellowing +of the wounded Ares, or the charming island of Calypso, or the +ocean-stream engirdling the earth. He only should speak of the relative +value of these two domains of thought, of their poetry, who knows both. + +The "utility" of physical science is, in a measure, only a _collateral_ +product of that flight of the intellect which produced science. No one, +however, should underrate the utility of science who has shared in the +realisation by modern industrial art of the Oriental world of fables, +much less one upon whom those treasures have been poured, as it were, +from the fourth dimension, without his aid or understanding. + +Nor may we believe that science is useful only to the practical man. Its +influence permeates all our affairs, our whole life; everywhere its +ideas are decisive. How differently does the jurist, the legislator, or +the political economist think, who knows, for example, that a square +mile of the most fertile soil can support with the solar heat annually +consumed only a definite number of human beings, which no art or science +can increase. Many economical theories, which open new air-paths of +progress, air-paths in the literal sense of the word, would be made +impossible by such knowledge. + + * * * * * + +The eulogists of classical education love to emphasise the cultivation +of taste which comes from employment with the ancient models. I candidly +confess that there is something absolutely revolting in this to me. To +form the taste, then, our youths must sacrifice ten years of their life! +Luxury takes precedence over necessity. Have the future generations, in +the face of the difficult problems, the great social questions, which +they must meet, and that with strengthened mind and heart, no more +important duties to fulfil than these? + +But let us assume that this end were desirable. Can taste be formed by +rules and precepts? Do not ideals of beauty change? Is it not a +stupendous absurdity to force one's self artificially to admire things +which, with all their historical interest, with all their beauty in +individual points, are for the most part foreign to the rest of our +thoughts and feelings, provided we have such of _our own_. A nation that +is truly such, has its own taste and will not go to others for it. And +every individual perfect man has his own taste.[121] + +And what, after all, does this cultivation of taste consist in? In the +acquisition of the personal literary style of a few select authors! What +should we think of a people that would force its youth a thousand years +from now, by years of practice, to master the tortuous or bombastic +style of some successful lawyer or politician of to-day? Should we not +justly accuse them of a woful lack of taste? + +The evil effects of this imagined cultivation of the taste find +expression often enough. The young _savant_ who regards the composition +of a scientific essay as a rhetorical exercise instead of a simple and +unadorned presentation of the facts and the truth, still sits +unconsciously on the school-bench, and still unwittingly represents the +point of view of the Romans, by whom the elaboration of speeches was +regarded as a serious scientific (!) employment. + + * * * * * + +Far be it from me to underrate the value of the development of the +instinct of speech and of the increased comprehension of our own +language which comes from philological studies. By the study of a +foreign language, especially of one which differs widely from ours, the +signs and forms of words are first clearly distinguished from the +thoughts which they express. Words of the closest possible +correspondence in different languages never coincide absolutely with the +ideas they stand for, but place in relief slightly different aspects of +the same thing, and by the study of language the attention is directed +to these shades of difference. But it would be far from admissible to +contend that the study of Latin and Greek is the most fruitful and +natural, let alone the _only_, means of attaining this end. Any one who +will give himself the pleasure of a few hours' companionship with a +Chinese grammar; who will seek to make clear to himself the mode of +speech and thought of a people who never advanced as far as the analysis +of articulate sounds, but stopped at the analysis of syllables, to whom +our alphabetical characters, therefore, are an inexplicable puzzle, and +who express all their rich and profound thoughts by means of a few +syllables with variable emphasis and position,--such a person, perhaps, +will acquire new, and extremely elucidative ideas upon the relation of +language and thought. But should our children, therefore, study Chinese? +Certainly not. No more, then, should they be burdened with Latin, at +least in the measure they are. + +It is a beautiful achievement to reproduce a Latin thought in a modern +language with the maximum fidelity of meaning and expression--for the +_translator_. Moreover, we shall be very grateful to the translator for +his performance. But to demand this feat of every educated man, without +consideration of the sacrifice of time and labor which it entails, is +unreasonable. And for this very reason, as classical teachers admit, +that ideal is never perfectly attained, except in rare cases with +scholars possessed of special talents and great industry. Without +slurring, therefore, the high importance of the study of the ancient +languages as a profession, we may yet feel sure that the instinct for +speech which is part of every liberal education can, and must, be +acquired in a different way. Should we, indeed, be forever lost if the +Greeks had not lived before us? + +The fact is, we must carry our demands further than the representatives +of classical philology. We must ask of every educated man a fair +scientific conception of the nature and value of language, of the +formation of language, of the alteration of the meaning of roots, of the +degeneration of fixed forms of speech to grammatical forms, in brief, of +all the main results of modern comparative philology. We should judge +that this were attainable by a careful study of our mother tongue and of +the languages next allied to it, and subsequently of the more ancient +tongues from which the former are derived. If any one object that this +is too difficult and entails too much labor, I should advise such a +person to place side by side an English, a Dutch, a Danish, a Swedish, +and a German Bible, and to compare a few lines of them; he will be +amazed at the multitude of suggestions that offer themselves.[122] In +fact, I believe that a really progressive, fruitful, rational, and +instructive study of languages can be conducted only on this plan. Many +of my audience will remember, perhaps, the bright and encouraging +effect, like that of a ray of sunlight on a gloomy day, which the meagre +and furtive remarks on comparative philology in Curtius's Greek grammar +wrought in that barren and lifeless desert of verbal quibbles. + + * * * * * + +The principal result obtained by the present method of studying the +ancient languages is that which comes from the student's employment with +their complicated grammars. It consists in the sharpening of the +attention and in the exercise of the judgment by the practice of +subsuming special cases under general rules, and of distinguishing +between different cases. Obviously, the same result can be reached by +many other methods; for example, by difficult games of cards. Every +science, the mathematics and the physical sciences included, accomplish +as much, if not more, in this disciplining of the judgment. In addition, +the matter treated by those sciences has a much higher intrinsic +interest for young people, and so engages spontaneously their attention; +while on the other hand they are elucidative and useful in other +directions in which grammar can accomplish nothing. + +Who cares, so far as the matter of it is concerned, whether we say +_hominum_ or _hominorum_ in the genitive plural, interesting as the fact +may be for the philologist? And who would dispute that the intellectual +need of causal insight is awakened not by grammar but by the natural +sciences? + +It is not our intention, therefore, to gainsay in the least the good +influence which the study of Latin and Greek grammar _also_ exercises on +the sharpening of the judgment. In so far as the study of words as such +must greatly promote lucidity and accuracy of expression, in so far as +Latin and Greek are not yet wholly indispensable to many branches of +knowledge, we willingly concede to them a place in our schools, but +would demand that the disproportionate amount of time allotted to them, +wrongly withdrawn from other useful studies, should be considerably +curtailed. That in the end Latin and Greek will not be employed as the +universal means of education, we are fully convinced. They will be +relegated to the closet of the scholar or professional philologist, and +gradually make way for the modern languages and the modern science of +language. + +Long ago Locke reduced to their proper limits the exaggerated notions +which obtained of the close connexion of thought and speech, of logic +and grammar, and recent investigators have established on still surer +foundations his views. How little a complicated grammar is necessary for +expressing delicate shades of thought is demonstrated by the Italians +and French, who, although they have almost totally discarded the +grammatical redundancies of the Romans, are yet not surpassed by the +latter in accuracy of thought, and whose poetical, but especially whose +scientific literature, as no one will dispute, can bear favorable +comparison with the Roman. + +Reviewing again the arguments advanced in favor of the study of the +ancient languages, we are obliged to say that in the main and as +applied to the present, they are wholly devoid of force. In so far as +the aims which this study theoretically pursues are still worthy of +attainment, they appear to us as altogether too narrow, and are +surpassed in this only by the means employed. As almost the sole, +indisputable result of this study we must count the increase of the +student's skill and precision in expression. One inclined to be +uncharitable might say that our gymnasiums and classical academies turn +out men who can speak and write, but, unfortunately, have little to +write or speak about. Of that broad, liberal view, of that famed +universal culture, which the classical curriculum is supposed to yield, +serious words need not be lost. This culture might, perhaps, more +properly be termed the contracted or lopsided culture. + + * * * * * + +While considering the study of languages we threw a few side glances at +mathematics and the natural sciences. Let us now inquire whether these, +as branches of study, cannot accomplish much that is to be attained in +no other way. I shall meet with no contradiction when I say that without +at least an elementary mathematical and scientific education a man +remains a total stranger in the world in which he lives, a stranger in +the civilisation of the time that bears him. Whatever he meets in +nature, or in the industrial world, either does not appeal to him at +all, from his having neither eye nor ear for it, or it speaks to him in +a totally unintelligible language. + +A real understanding of the world and its civilisation, however, is not +the only result of the study of mathematics and the physical sciences. +Much more essential for the preparatory school is the _formal_ +cultivation which comes from these studies, the strengthening of the +reason and the judgment, the exercise of the imagination. Mathematics, +physics, chemistry, and the so-called descriptive sciences are so much +alike in this respect, that, apart from a few points, we need not +separate them in our discussion. + +Logical sequence and continuity of ideas, so necessary for fruitful +thought, are _par excellence_ the results of mathematics; the ability to +follow facts with thoughts, that is, to observe or collect experiences, +is chiefly developed by the natural sciences. Whether we notice that the +sides and the angles of a triangle are connected in a definite way, that +an equilateral triangle possesses certain definite properties of +symmetry, or whether we notice the deflexion of a magnetic needle by an +electric current, the dissolution of zinc in diluted sulphuric acid, +whether we remark that the wings of a butterfly are slightly colored on +the under, and the fore-wings of the moth on the upper, surface: +indiscriminately here we proceed from _observations_, from individual +acts of immediate intuitive knowledge. The field of observation is more +restricted and lies closer at hand in mathematics; it is more varied and +broader but more difficult to compass in the natural sciences. The +essential thing, however, is for the student to learn to make +observations in all these fields. The philosophical question whether our +acts of knowledge in mathematics are of a special kind is here of no +importance for us. It is true, of course, that the observation can be +practised by languages also. But no one, surely, will deny, that the +concrete, living pictures presented in the fields just mentioned possess +different and more powerful attractions for the mind of the youth than +the abstract and hazy figures which language offers, and on which the +attention is certainly not so spontaneously bestowed, nor with such good +results.[123] + +Observation having revealed the different properties of a given +geometrical or physical object, it is discovered that in many cases +these properties _depend_ in some way upon one another. This +interdependence of properties (say that of equal sides and equal angles +at the base of a triangle, the relation of pressure to motion,) is +nowhere so distinctly marked, nowhere is the necessity and permanency of +the interdependence so plainly noticeable, as in the fields mentioned. +Hence the continuity and logical consequence of the ideas which we +acquire in those fields. The relative simplicity and perspicuity of +geometrical and physical relations supply here the conditions of natural +and easy progress. Relations of equal simplicity are not met with in +the fields which the study of language opens up. Many of you, doubtless, +have often wondered at the little respect for the notions of cause and +effect and their connexion that is sometimes found among professed +representatives of the classical studies. The explanation is probably to +be sought in the fact that the analogous relation of motive and action +familiar to them from their studies, presents nothing like the clear +simplicity and determinateness that the relation of cause and effect +does. + +That perfect mental grasp of all possible cases, that economical order +and organic union of the thoughts which comes from it, which has grown +for every one who has ever tasted it a permanent need which he seeks to +satisfy in every new province, can be developed only by employment with +the relative simplicity of mathematical and scientific investigations. + +When a set of facts comes into apparent conflict with another set of +facts, and a problem is presented, its solution consists ordinarily in a +more refined distinction or in a more extended view of the facts, as may +be aptly illustrated by Newton's solution of the problem of dispersion. +When a new mathematical or scientific fact is _demonstrated_, or +_explained_, such demonstration also rests simply upon showing the +connexion of the new fact with the facts already known; for example, +that the radius of a circle can be laid off as chord exactly six times +in the circle is explained or proved by dividing the regular hexagon +inscribed in the circle into equilateral triangles. That the quantity of +heat developed in a second in a wire conveying an electric current is +quadrupled on the doubling of the strength of the current, we explain +from the doubling of the fall of the potential due to the doubling of +the current's intensity, as also from the doubling of the quantity +flowing through, in a word, from the quadrupling of the work done. In +point of principle, explanation and direct proof do not differ much. + +He who solves scientifically a geometrical, physical, or technical +problem, easily remarks that his procedure is a _methodical_ mental +quest, rendered possible by the economical order of the province--a +simplified purposeful quest as contrasted with unmethodical, +unscientific guess-work. The geometer, for example, who has to construct +a circle touching two given straight lines, casts his eye over the +relations of symmetry of the desired construction, and seeks the centre +of his circle solely in the line of symmetry of the two straight lines. +The person who wants a triangle of which two angles and the sum of the +sides are given, grasps in his mind the determinateness of the form of +this triangle and restricts his search for it to a certain group of +triangles of the _same form_. Under very different circumstances, +therefore, the simplicity, the intellectual perviousness, of the +subject-matter of mathematics and natural science is felt, and promotes +both the discipline and the self-confidence of the reason. + +Unquestionably, much more will be attained by instruction in the +mathematics and the natural sciences than now is, when more natural +methods are adopted. One point of importance here is that young students +should not be spoiled by premature abstraction, but should be made +acquainted with their material from living pictures of it before they +are made to work with it by purely ratiocinative methods. A good stock +of geometrical experience could be obtained, for example, from +geometrical drawing and from the practical construction of models. In +the place of the unfruitful method of Euclid, which is only fit for +special, restricted uses, a broader and more conscious method must be +adopted, as Hankel has pointed out.[124] Then, if, on reviewing +geometry, and after it presents no substantial difficulties, the more +general points of view, the principles of scientific method are placed +in relief and brought to consciousness, as Von Nagel,[125] J. K. +Becker,[126] Mann,[127] and others have well done, fruitful results will +be surely attained. In the same way, the subject-matter of the natural +sciences should be made familiar by pictures and experiment before a +profounder and reasoned grasp of these subjects is attempted. Here the +emphasis of the more general points of view is to be postponed. + +Before my present audience it would be superfluous for me to contend +further that mathematics and natural science are justified constituents +of a sound education,--a claim that even philologists, after some +resistance, have conceded. Here I may count upon assent when I say that +mathematics and the natural sciences pursued alone as means of +instruction yield a richer education in matter and form, a more general +education, an education better adapted to the needs and spirit of the +time,--than the philological branches pursued alone would yield. + +But how shall this idea be realised in the curricula of our intermediate +educational institutions? It is unquestionable in my mind that the +German _Realschulen_ and _Realgymnasien_, where the exclusive classical +course is for the most part replaced by mathematics, science, and modern +languages, give the _average_ man a more timely education than the +gymnasium proper, although they are not yet regarded as fit preparatory +schools for future theologians and professional philologists. The German +gymnasiums are too one-sided. With these the first changes are to be +made; of these alone we shall speak here. Possibly a _single_ +preparatory school, suitably planned, might serve all purposes. + +Shall we, then, in our gymnasiums fill out the hours of study which +stand at our disposal, or are still to be wrested from the classicists, +with as great and as varied a quantity of mathematical and scientific +matter as possible? Expect no such proposition from me. No one will +suggest such a course who has himself been actively engaged in +scientific thought. Thoughts can be awakened and fructified as a field +is fructified by sunshine and rain. But thoughts cannot be juggled out +and worried out by heaping up materials and the hours of instruction, +nor by any sort of precepts: they must grow naturally of their own free +accord. Furthermore, thoughts cannot be accumulated beyond a certain +limit in a single head, any more than the produce of a field can be +increased beyond certain limits. + +I believe that the amount of matter necessary for a useful education, +such as should be offered to _all_ the pupils of a preparatory school, +is very small. If I had the requisite influence, I should, in all +composure, and fully convinced that I was doing what was best, first +greatly curtail in the lower classes the amount of matter in both the +classical and the scientific courses; I should cut down considerably the +number of the school hours and the work done outside the school. I am +not with many teachers of opinion that ten hours work a day for a child +is not too much. I am convinced that the mature men who offer this +advice so lightly are themselves unable to give their attention +successfully for as long a time to any subject that is new to them, (for +example, to elementary mathematics or physics,) and I would ask every +one who thinks the contrary to make the experiment upon himself. +Learning and teaching are not routine office-work that can be kept up +mechanically for long periods. But even such work tires in the end. If +our young men are not to enter the universities with blunted and +impoverished minds, if they are not to leave in the preparatory schools +their vital energy, which they should there gather, great changes must +be made. Waiving the injurious effects of overwork upon the body, the +consequences of it for the mind seem to me positively dreadful. + +I know of nothing more terrible than the poor creatures who have learned +too much. Instead of that sound powerful judgment which would probably +have grown up if they had learned nothing, their thoughts creep timidly +and hypnotically after words, principles, and formulæ, constantly by the +same paths. What they have acquired is a spider's web of thoughts too +weak to furnish sure supports, but complicated enough to produce +confusion. + +But how shall better methods of mathematical and scientific education be +combined with the decrease of the subject-matter of instruction? I +think, by abandoning systematic instruction altogether, at least in so +far as that is required of _all_ young pupils. I see no necessity +whatever that the graduates of our high schools and preparatory schools +should be little philologists, and at the same time little +mathematicians, physicists, and botanists; in fact, I do not see the +possibility of such a result. I see in the endeavor to attain this +result, in which every instructor seeks for his own branch a place apart +from the others, the main mistake of our whole system. I should be +satisfied if every young student could come into living contact with +and pursue to their ultimate logical consequences merely a _few_ +mathematical or scientific discoveries. Such instruction would be mainly +and naturally associated with selections from the great scientific +classics. A few powerful and lucid ideas could thus be made to take root +in the mind and receive thorough elaboration. This accomplished, our +youth would make a different showing from what they do to-day.[128] + +What need is there, for example, of burdening the head of a young +student with all the details of botany? The student who has botanised +under the guidance of a teacher finds on all hands, not indifferent +things, but known or unknown things, by which he is stimulated, and his +gain made permanent. I express here, not my own, but the opinion of a +friend, a practical teacher. Again, it is not at all necessary that all +the matter that is offered in the schools should be learned. The best +that we have learned, that which has remained with us for life, outlived +the test of examination. How can the mind thrive when matter is heaped +on matter, and new materials piled constantly on old, undigested +materials? The question here is not so much that of the accumulation of +positive knowledge as of intellectual discipline. It seems also +unnecessary that _all_ branches should be treated at school, and that +exactly the same studies should be pursued in all schools. A single +philological, a single historical, a single mathematical, a single +scientific branch, pursued as common subjects of instruction for all +pupils, are sufficient to accomplish all that is necessary for the +intellectual development. On the other hand, a wholesome mutual stimulus +would be produced by this greater variety in the positive culture of +men. Uniforms are excellent for soldiers, but they will not fit heads. +Charles V. learned this, and it should never be forgotten. On the +contrary, teachers and pupils both need considerable latitude, if they +are to yield good results. + +With John Karl Becker I am of the opinion that the utility and amount +for individuals of every study should be precisely determined. All that +exceeds this amount should be unconditionally banished from the lower +classes. With respect to mathematics, Becker,[129] in my judgment, has +admirably solved this question. + +With respect to the upper classes the demand assumes a different form. +Here also the amount of matter obligatory on all pupils ought not to +exceed a certain limit. But in the great mass of knowledge that a young +man must acquire to-day for his profession it is no longer just that ten +years of his youth should be wasted with mere preludes. The upper +classes should supply a truly useful preparation for the professions, +and should not be modelled upon the wants merely of future lawyers, +ministers, and philologists. Again, it would be both foolish and +impossible to attempt to prepare the same person properly for all the +different professions. In such case the function of the schools would +be, as Lichtenberg feared, simply to select the persons best fitted for +being drilled, whilst precisely the finest special talents, which do not +submit to indiscriminate discipline, would be excluded from the contest. +Hence, a certain amount of liberty in the choice of studies must be +introduced in the upper classes, by means of which it will be free for +every one who is clear about the choice of his profession to devote his +chief attention either to the study of the philologico-historical or to +that of the mathematico-scientific branches. Then the matter now treated +could be retained, and in some branches, perhaps, judiciously +extended,[130] without burdening the scholar with many branches or +increasing the number of the hours of study. With more homogeneous work +the student's capacity for work increases, one part of his labor +supporting the other instead of obstructing it. If, however, a young man +should subsequently choose a different profession, then it is _his_ +business to make up what he has lost. No harm certainly will come to +society from this change, nor could it be regarded as a misfortune if +philologists and lawyers with mathematical educations or physical +scientists with classical educations should now and then appear. + + * * * * * + +The view is now wide-spread that a Latin and Greek education no longer +meets the general wants of the times, that a more opportune, a more +"liberal" education exists. The phrase, "a liberal education," has been +greatly misused. A truly liberal education is unquestionably very rare. +The _schools_ can hardly offer such; at best they can only bring home to +the student the necessity of it. It is, then, his business to acquire, +as best he can, a more or less liberal education. It would be very +difficult, too, at any one time to give a definition of a "liberal" +education which would satisfy every one, still more difficult to give +one which would hold good for a hundred years. The educational ideal, in +fact, varies much. To one, a knowledge of classical antiquity appears +not too dearly bought "with early death." We have no objection to this +person, or to those who think like him, pursuing their ideal after their +own fashion. But we may certainly protest strongly against the +realisation of such ideals on our own children. Another,--Plato, for +example,--puts men ignorant of geometry on a level with animals.[131] +If such narrow views had the magical powers of the sorceress Circe, many +a man who perhaps justly thought himself well educated would become +conscious of a not very flattering transformation of himself. Let us +seek, therefore, in our educational system to meet the wants of the +present, and not establish prejudices for the future. + +But how does it come, we must ask, that institutions so antiquated as +the German gymnasiums could subsist so long in opposition to public +opinion? The answer is simple. The schools were first organised by the +Church; since the Reformation they have been in the hands of the State. +On so large a scale, the plan presents many advantages. Means can be +placed at the disposal of education such as no private source, at least +in Europe, could furnish. Work can be conducted upon the same plan in +many schools, and so experiments made of extensive scope which would be +otherwise impossible. A single man with influence and ideas can under +such circumstances do great things for the promotion of education. + +But the matter has also its reverse aspect. The party in power works for +its own interests, uses the schools for its special purposes. +Educational competition is excluded, for all successful attempts at +improvement are impossible unless undertaken or permitted by the State. +By the uniformity of the people's education, a prejudice once in vogue +is permanently established. The highest intelligences, the strongest +wills cannot overthrow it suddenly. In fact, as everything is adapted to +the view in question, a sudden change would be physically impossible. +The two classes which virtually hold the reins of power in the State, +the jurists and theologians, know only the one-sided, predominantly +classical culture which they have acquired in the State schools, and +would have this culture alone valued. Others accept this opinion from +credulity; others, underestimating their true worth for society, bow +before the power of the prevalent opinion; others, again, affect the +opinion of the ruling classes even against their better judgment, so as +to abide on the same plane of respect with the latter. I will make no +charges, but I must confess that the deportment of medical men with +respect to the question of the qualification of graduates of your +_Realschulen_ has frequently made that impression upon me. Let us +remember, finally, that an influential statesman, even within the +boundaries which the law and public opinion set him, can do serious harm +to the cause of education by considering his own one-sided views +infallible, and in enforcing them recklessly and inconsiderately--which +not only _can_ happen, but has, repeatedly, happened.[132] The monopoly +of education by the State[133] thus assumes in our eyes a somewhat +different aspect. And to revert to the question above asked, there is +not the slightest doubt that the German gymnasiums in their present +form would have ceased to exist long ago if the State had not supported +them. + +All this must be changed. But the change will not be made of itself, nor +without our energetic interference, and it will be made slowly. But the +path is marked out for us, the will of the people must acquire and exert +upon our school legislation a greater and more powerful influence. +Furthermore, the questions at issue must be publicly and candidly +discussed that the views of the people may be clarified. All who feel +the insufficiency of the existing _régime_ must combine into a powerful +organisation that their views may acquire impressiveness and the +opinions of the individual not die away unheard. + +I recently read, gentlemen, in an excellent book of travels, that the +Chinese speak with unwillingness of politics. Conversations of this sort +are usually cut short with the remark that they may bother about such +things whose business it is and who are paid for it. Now it seems to me +that it is not only the business of the State, but a very serious +concern of all of us, how our children shall be educated in the public +schools at _our_ cost. + + FOOTNOTES: + + [Footnote 113: An address delivered before the Congress of Delegates + of the German Realschulmännerverein, at Dortmund, April 16, 1886. + The full title of the address reads: "On the Relative Educational + Value of the Classics and the Mathematico-Physical Sciences in + Colleges and High Schools." + + Although substantially contained in an address which I was to have + made at the meeting of Natural Scientists at Salzburg in 1881 + (deferred on account of the Paris Exposition), and in the + Introduction to a course of lectures on "Physical Instruction in + Preparatory Schools," which I delivered in 1883, the invitation of + the German Realschulmännerverein afforded me the first opportunity + of putting my views upon this subject before a large circle of + readers. Owing to the place and circumstances of delivery, my + remarks apply of course, primarily, only to German schools, but, + with slight modifications, made in this translation, are not without + force for the institutions of other countries. In giving here + expression to a strong personal conviction formed long ago, it is a + matter of deep satisfaction to me to find that they agree in many + points with the views recently advanced in independent form by + Paulsen (_Geschichte des gelehrten Unterrichts_, Leipsic, 1885) and + Frary (_La question du latin_, Paris, Cerf, 1885). It is not my + desire nor effort here to say much that is new, but merely to + contribute my mite towards bringing about the inevitable revolution + now preparing in the world of elementary instruction. In the opinion + of experienced educationists the first result of that revolution + will be to make Greek and mathematics alternately optional subjects + in the higher classes of the German Gymnasium and in the + corresponding institutions of other countries, as has been done in + the splendid system of instruction in Denmark. The gap between the + German classical Gymnasium and the German Realgymnasium, or between + classical and scientific schools generally, can thus be bridged + over, and the remaining inevitable transformations will then be + accomplished in relative peace and quiet. (Prague, May, 1886.)] + + [Footnote 114: Maupertuis, _Oeuvres_, Dresden, 1752, p. 339.] + + [Footnote 115: F. Paulsen, _Geschichte des gelehrten Unterrichts_, + Leipsic, 1885.] + + [Footnote 116: There is a peculiar irony of fate in the fact that + while Leibnitz was casting about for a new vehicle of universal + linguistic intercourse, the Latin language which still subserved + this purpose the best of all, was dropping more and more out of use, + and that Leibnitz himself contributed not the least to this result.] + + [Footnote 117: As a rule, the human brain is too much, and wrongly, + burdened with things which might be more conveniently and accurately + preserved in books where they could be found at a moment's notice. + In a recent letter to me from Düsseldorf, Judge Hartwich writes: + + "A host of words exist which are out and out Latin or Greek, yet are + employed with perfect correctness by people of good education who + never had the good luck to be taught the ancient languages. For + example, words like 'dynasty.' ... The child learns such words as + parts of the common stock of speech, or even as parts of his + mother-tongue, just as he does the words 'father,' 'mother,' + 'bread,' 'milk.' Does the ordinary mortal know the etymology of + these Saxon words? Did it not require the almost incredible industry + of the Grimms and other Teutonic philologists to throw the merest + glimmerings of light upon the origin and growth of our own + mother-tongue? Besides, do not thousands of people of so-called + classical education use every moment hosts of words of foreign + origin whose derivation they do not know? Very few of them think it + worth while to look up such words in the dictionaries, although they + love to maintain that people should study the ancient languages for + the sake of etymology alone."] + + [Footnote 118: Standing remote from the legal profession I should + not have ventured to declare that the study of Greek was not + necessary for the jurists; yet this view was taken in the debate + that followed this lecture by professional jurists of high standing. + According to this opinion, the preparatory education obtained in the + German Realgymnasium would also be sufficient for the future jurists + and insufficient only for theologians and philologists. [In England + and America not only is Greek not necessary, but the law-Latin is so + peculiar that even persons of _good_ classical education cannot + understand it.--_Tr._]] + + [Footnote 119: In emphasising here the weak sides of the writings of + Plato and Aristotle, forced on my attention while reading them in + German translations, I, of course, have no intention of underrating + the great merits and the high historical importance of these two + men. Their importance must not be measured by the fact that our + speculative philosophy still moves to a great extent in their paths + of thought. The more probable conclusion is that this branch has + made very little progress in the last two thousand years. Natural + science also was implicated for centuries in the meshes of the + Aristotelian thought, and owes its rise mainly to having thrown off + those fetters.] + + [Footnote 120: I would not for a moment contend that we derive + exactly the same profit from reading a Greek author in a translation + as from reading him in the original; but the difference, the excess + of gain in the second case, appears to me, and probably will to most + men who are not professional philologists, to be too dearly bought + with the expenditure of eight years of valuable time.] + + [Footnote 121: "The temptation," Judge Hartwich writes, "to regard + the 'taste' of the ancients as so lofty and unsurpassable appears to + me to have its chief origin in the fact that the ancients were + unexcelled in the representation of the nude. First, by their + unremitting care of the human body they produced splendid models; + and secondly, in their gymnasiums and in their athletic games they + had these models constantly before their eyes. No wonder, then, that + their statues still excite our admiration! For the form, the ideal + of the human body has not changed in the course of the centuries. + But with intellectual matters it is totally different; they change + from century to century, nay, from decennium to decennium. It is + very natural now, that people should unconsciously apply what is + thus so easily seen, namely, the works of sculpture, as a universal + criterion of the highly developed taste of the ancients--a fallacy + against which people cannot, in my judgment, be too strongly + warned."] + + [Footnote 122: English: "In the beginning God created the heaven and + the earth. And the earth was without form and void; and darkness was + upon the face of the deep. And the spirit of God moved upon the face + of the waters."--Dutch: "In het begin schiep God den hemel en de + aarde. De aarde nu was woest en ledig, en duisternis was op den + afgrond; en de Geest Gods zwefde op de wateren."--Danish: "I + Begyndelsen skabte Gud Himmelen og Jorden. Og Jorden var ode og tom, + og der var morkt ovenover Afgrunden, og Guds Aand svoevede ovenover + Vandene."--Swedish: "I begynnelsen skapade Gud Himmel och Jord. Och + Jorden war öde och tom, och mörker war pä djupet, och Gods Ande + swäfde öfwer wattnet."--German: "Am Anfang schuf Gott Himmel und + Erde. Und die Erde war wüst und leer, und es war finster auf der + Tiefe; und der Geist Gottes schwebte auf dem Wasser."] + + [Footnote 123: Compare Herzen's excellent remarks, _De + l'enseignement secondaire dans la Suisse romande_, Lausanne, 1886.] + + [Footnote 124: _Geschichte der Mathematik_, Leipsic, 1874.] + + [Footnote 125: _Geometrische Analyse_, Ulm, 1886.] + + [Footnote 126: In his text-books of elementary mathematics] + + [Footnote 127: _Abhandlungen aus dem Gebiete der Mathematik_, + Würzburg, 1883.] + + [Footnote 128: My idea here is an appropriate selection of readings + from Galileo, Huygens, Newton, etc. The choice is so easily made + that there can be no question of difficulties. The contents would be + discussed with the students, and the original experiments performed + with them. Those scholars alone should receive this instruction in + the upper classes who did not look forward to systematical + instruction in the physical sciences. I do not make this proposition + of reform here for the first time. I have no doubt, moreover, that + such radical changes will only be slowly introduced.] + + [Footnote 129: _Die Mathematik als Lehrgegenstand des Gymnasiums_, + Berlin, 1883.] + + [Footnote 130: Wrong as it is to burden future physicians and + scientists with Greek for the sake of the theologians and + philologists, it would be just as wrong to compel theologians and + philologists, on account of the physicians, to study such subjects + as analytical geometry. Moreover, I cannot believe that ignorance of + analytical geometry would be a serious hindrance to a physician that + was otherwise well versed in quantitative thought. No special + advantage generally is observable in the graduates of the Austrian + gymnasiums, all of whom have studied analytical geometry. [Refers to + an assertion of Dubois-Reymond.]] + + [Footnote 131: Compare M. Cantor, _Geschichte der Mathematik_, + Leipsic, 1880, Vol. I. p. 193.] + + [Footnote 132: Compare Paulsen, _l. c._, pp. 607, 688.] + + [Footnote 133: It is to be hoped that the Americans will jealously + guard their schools and universities against the influence of the + State.] + + + + +APPENDIX. + +I. + +A CONTRIBUTION TO THE HISTORY OF ACOUSTICS.[134] + + +While searching for papers by Amontons, several volumes of the Memoirs +of the Paris Academy for the first years of the eighteenth century, fell +into my hands. It is difficult to portray the delight which one +experiences in running over the leaves of these volumes. One sees as an +actual spectator almost the rise of the most important discoveries and +witnesses the progress of many fields of knowledge from almost total +ignorance to relatively perfect clearness. + +I propose to discuss here the fundamental researches of Sauveur in +Acoustics. It is astonishing how extraordinarily near Sauveur was to the +view which Helmholtz was the first to adopt in its full extent a hundred +and fifty years later. + +The _Histoire de l'Académie_ for 1700, p. 131, tells us that Sauveur had +succeeded in making music an object of scientific research, and that he +had invested the new science with the name of "acoustics." On five +successive pages a number of discoveries are recorded which are more +fully discussed in the volume for the year following. + +Sauveur regards the _simplicity_ of the ratios obtaining between the +rates of vibration of consonances as something universally known.[135] +He is in hope, by further research, of determining the chief rules of +musical composition and of fathoming the "metaphysics of the agreeable," +the main law of which he asserts to be the union of "simplicity with +multiplicity." Precisely as Euler[136] did a number of years later, he +regards a consonance as more perfect according as the ratio of its +vibrational rates is expressed in smaller whole numbers, because the +smaller these whole numbers are the oftener the vibrations of the two +tones coincide, and hence the more readily they are apprehended. As the +limit of consonance, he takes the ratio 5:6, although he does not +conceal the fact that practice, sharpened attention, habit, taste, and +even prejudice play collateral rôles in the matter, and that +consequently the question is not a purely scientific one. + +Sauveur's ideas took their development from his having instituted at +all points more exact quantitative investigations than his predecessors. +He is first desirous of determining as the foundation of musical tuning +a fixed note of one hundred vibrations which can be reproduced at any +time; the fixing of the notes of musical instruments by the common +tuning pipes then in use with rates of vibration unknown, appearing to +him inadequate. According to Mersenne (_Harmonie Universelle_, 1636), a +given cord seventeen feet long and weighted with eight pounds executes +eight visible vibrations in a second. By diminishing its length then in +a given proportion we obtain a proportionately augmented rate of +vibration. But this procedure appears too uncertain to Sauveur, and he +employs for his purpose the beats (_battemens_), which were known to the +organ-makers of his day, and which he correctly explains as due to the +alternate coincidence and non-coincidence of the same vibrational phases +of differently pitched notes.[137] At every coincidence there is a +swelling of the sound, and hence the number of beats per second will be +equal to the difference of the rates of vibration. If we tune two of +three organ-pipes to the remaining one in the ratio of the minor and +major third, the mutual ratio of the rates of vibration of the first two +will be as 24: 25, that is to say, for every 24 vibrations to the lower +note there will be 25 to the higher, and one beat. If the two pipes give +together four beats in a second, then the higher has the fixed tone of +100 vibrations. The open pipe in question will consequently be five feet +in length. We also determine by this procedure the absolute rates of +vibration of all the other notes. + +It follows at once that a pipe eight times as long or 40 feet in length +will yield a vibrational rate of 12½, which Sauveur ascribes to the +lowest audible tone, and further also that a pipe 64 times as small will +execute 6,400 vibrations, which Sauveur took for the highest audible +limit. The author's delight at his successful enumeration of the +"imperceptible vibrations" is unmistakably asserted here, and it is +justified when we reflect that to-day even Sauveur's principle, slightly +modified, constitutes the simplest and most delicate means we have for +exactly determining rates of vibration. Far more important still, +however, is a second observation which Sauveur made while studying +beats, and to which we shall revert later. + +Strings whose lengths can be altered by movable bridges are much easier +to handle than pipes in such investigations, and it was natural that +Sauveur should soon resort to their use. + +One of his bridges accidentally not having been brought into full and +hard contact with the string, and consequently only imperfectly impeding +the vibrations, Sauveur discovered the harmonic overtones of the string, +at first by the unaided ear, and concluded from this fact that the +string was divided into aliquot parts. The string when plucked, and +when the bridge stood at the third division for example, yielded the +twelfth of its fundamental note. At the suggestion of some +academician[138] probably, variously colored paper riders were placed at +the nodes (_noeuds_) and ventral segments (_ventres_), and the division +of the string due to the excitation of the overtones (_sons +harmoniques_) belonging to its fundamental note (_son fondamental_) thus +rendered visible. For the clumsy bridge the more convenient feather or +brush was soon substituted. . While engaged in these investigations +Sauveur also observed the sympathetic vibration of a string induced by +the excitation of a second one in unison with it. He also discovered +that the overtone of a string can respond to another string tuned to its +note. He even went further and discovered that on exciting one string +the overtone which it has in common with another, differently pitched +string can be produced on that other; for example, on strings having for +their vibrational ratio 3:4, the fourth of the lower and the third of +the higher may be made to respond. It follows indisputably from this +that the excited string yields overtones simultaneously with its +fundamental tone. Previously to this Sauveur's attention had been drawn +by other observers to the fact that the overtones of musical instruments +can be picked out by attentive listening, particularly in the +night.[139] He himself mentions the simultaneous sounding of the +overtones and the fundamental tone.[140] That he did not give the proper +consideration to this circumstance was, as will afterwards be seen, +fatal to his theory. + +While studying beats Sauveur makes the remark that they are +_displeasing_ to the ear. He held the beats were distinctly audible only +when less than six occurred in a second. Larger numbers were not +distinctly perceptible and gave rise accordingly to no disturbance. He +then attempts to reduce the difference between consonance and dissonance +to a question of beats. Let us hear his own words.[141] + + "Beats are unpleasing to the ear because of the unevenness of the + sound, and it may be held with much plausibility that the reason + why octaves are so pleasing is that we never hear their beats.[142] + + "In following out this idea, we find that the chords whose beats we + cannot hear are precisely those which the musicians call + consonances and that those whose beats are heard are the + dissonances, and that when a chord is a dissonance in one octave + and a consonance in another, it beats in the one and does not beat + in the other. Consequently it is called an imperfect consonance. It + is very easy by the principles of M. Sauveur, here established, to + ascertain what chords beat and in what octaves, above or below the + fixed note. If this hypothesis be correct, it will disclose the + true source of the rules of composition, hitherto unknown to + science, and given over almost entirely to judgment by the ear. + These sorts of natural judgment, marvellous though they may + sometimes appear, are not so but have very real causes, the + knowledge of which belongs to science, provided it can gain + possession thereof."[143] + +Sauveur thus correctly discerns in beats the cause of the disturbance +of consonance, to which all disharmony is "probably" to be referred. It +will be seen, however, that according to his view all distant intervals +must necessarily be consonances and all near intervals dissonances. He +also overlooks the absolute difference in point of principle between his +old view, mentioned at the outset, and his new view, rather attempting +to obliterate it. + +R. Smith[144] takes note of the theory of Sauveur and calls attention to +the first of the above-mentioned defects. Being himself essentially +involved in the old view of Sauveur, which is usually attributed to +Euler, he yet approaches in his criticism a brief step nearer to the +modern theory, as appears from the following passage.[145] + + "The truth is, this gentleman confounds the distinction between + perfect and imperfect consonances, by comparing imperfect + consonances which beat because the succession of their short + cycles[146] is periodically confused and interrupted, with perfect + ones which cannot beat, because the succession of their short + cycles is never confused nor interrupted. + + "The _fluttering roughness_ above mentioned is perceivable in all + other perfect consonances, in a smaller degree in proportion as + their cycles are shorter and simpler, and their pitch is higher; + and is of a _different kind_ from the _smoother beats_ and + undulations of _tempered consonances_; because we can alter the + rate of the latter by altering the temperament, but not of the + former, the consonance being perfect at a given pitch: And because + a judicious ear can often hear, at the same time, both the + flutterings and the beats of a tempered consonance; sufficiently + distinct from each other. + + "For nothing gives greater offence to the hearer, though ignorant + of the cause of it, than those rapid, piercing beats of high and + loud sounds, which make imperfect consonances with one another. And + yet a few slow beats, like the slow undulations of a close shake + now and then introduced, are far from being disagreeable." + +Smith is accordingly clear that other "roughnesses" exist besides the +beats which Sauveur considered, and if the investigations had been +continued on the basis of Sauveur's idea, these additional roughnesses +would have turned out to be the beats of the overtones, and the theory +thus have attained the point of view of Helmholtz. + +Reviewing the differences between Sauveur's and Helmholtz's theories, we +find the following: + +1. The theory according to which consonance depends on the frequent and +regular coincidence of vibrations and their ease of enumeration, appears +from the new point of view inadmissible. The simplicity of the ratios +obtaining between the rates of vibration is indeed a _mathematical_ +characteristic of consonance as well as a _physical_ condition thereof, +for the reason that the coincidence of the overtones as also their +further physical and physiological consequences is connected with this +fact. But no _physiological_ or _psychological_ explanation of +consonance is given by this fact, for the simple reason that in the +acoustic nerve-process nothing corresponding to the periodicity of the +sonant stimulus is discoverable. + +2. In the recognition of beats as a disturbance of consonance, both +theories agree. Sauveur's theory, however, does not take into account +the fact that clangs, or musical sounds generally, are composite and +that the disturbance in the consonances of distant intervals principally +arise from the beats of the overtones. Furthermore, Sauveur was wrong in +asserting that the number of beats must be less than six in a second in +order to produce disturbances. Even Smith knows that very slow beats are +not a cause of disturbance, and Helmholtz found a much higher number +(33) for the maximum of disturbance. Finally, Sauveur did not consider +that although the number of beats increases with the recession from +unison, yet their _strength_ is diminished. On the basis of the +principle of specific energies and of the laws of sympathetic vibration +the new theory finds that two atmospheric motions of like amplitude but +different periods, _a_ sin(_rt_) and _a_ sin[(_r_ + [rho])(_t_ + +[tau])], cannot be communicated with the same amplitude to the same +nervous end-organ. On the contrary, an end-organ that reacts best to the +period _r_ responds more weakly to the period _r_ + [rho], the two +amplitudes bearing to each other the proportion _a_: [phi]_a_. Here +[phi] decreases when [rho] increases, and when [rho] = 0 it becomes +equal to 1, so that only the portion of the stimulus [phi]_a_ is subject +to beats, and the portion (1-[phi])_a_ continues smoothly onward without +disturbance. + +If there is any moral to be drawn from the history of this theory, it is +that considering how near Sauveur's errors were to the truth, it +behooves us to exercise some caution also with regard to the new theory. +And in reality there seems to be reason for doing so. + +The fact that a musician will never confound a more perfectly consonant +chord on a poorly tuned piano with a less perfectly consonant chord on a +well tuned piano, although the roughness in the two cases may be the +same, is sufficient indication that the degree of roughness is not the +only characteristic of a harmony. As the musician knows, even the +harmonic beauties of a Beethoven sonata are not easily effaced on a +poorly tuned piano; they scarcely suffer more than a Raphael +drawing executed in rough unfinished strokes. The _positive +physiologico-psychological_ characteristic which distinguishes one +harmony from another is not given by the beats. Nor is this +characteristic to be found in the fact that, for example, in sounding a +major third the fifth partial tone of the lower note coincides with the +fourth of the higher note. This characteristic comes into consideration +only for the investigating and abstracting reason. If we should regard +it also as characteristic of the sensation, we should lapse into a +fundamental error which would be quite analogous to that cited in (1). + +The _positive physiological_ characteristics of the intervals would +doubtless be speedily revealed if it were possible to conduct aperiodic, +for example galvanic, stimuli to the single sound-sensing organs, in +which case the beats would be totally eliminated. Unfortunately such an +experiment can hardly be regarded as practicable. The employment of +acoustic stimuli of short duration and consequently also free from +beats, involves the additional difficulty of a pitch not precisely +determinable. + + FOOTNOTES: + + [Footnote 134: This article, which appeared in the Proceedings of + the German Mathematical Society of Prague for the year 1892, is + printed as a supplement to the article on "The Causes of Harmony," + at page 32.] + + [Footnote 135: The present exposition is taken from the volumes for + 1700 (published in 1703) and for 1701 (published in 1704), and + partly also from the _Histoire de l'Académie_ and partly from the + _Mémoires_. Sauveur's later works enter less into consideration + here.] + + [Footnote 136: Euler, _Tentamen novae theoriae musicae_, Petropoli, + 1739.] + + [Footnote 137: In attempting to perform his experiment of beats + before the Academy, Sauveur was not quite successful. _Histoire de + l'Académie_, Année 1700, p. 136.] + + [Footnote 138: _Histoire de l'Académie_, Année 1701, p. 134.] + + [Footnote 139: _Ibid._, p. 298.] + + [Footnote 140: _Histoire de l'Académie_, Année 1702, p. 91.] + + [Footnote 141: From the _Histoire de l'Académie_, Année 1700, p. + 139.] + + [Footnote 142: Because all octaves in use in music offer too great + differences of rates of vibration.] + + [Footnote 143: "Les battemens ne plaisent pas à l'Oreille, à cause + de l'inégalité du son, et l'on peut croire avec beaucoup d'apparence + que ce qui rend les Octaves si agréables, c'est qu'on n'y entend + jamais de battemens. + + "En suivant cette idée, on trouve que les accords dont on ne peut + entendre les battemens, sont justement ceux que les Musiciens + traitent de Consonances, et que ceux dont les battemens se font + sentir, sont les Dissonances, et que quand un accord est Dissonance + dans une certaine octave et Consonance dans une autre, c'est qu'il + bat dans l'une, et qu'il ne bat pas dans l'autre. Aussi est il + traité de Consonance imparfaite. Il est fort aisé par les principes + de Mr. Sauveur qu'on a établis ici, de voir quels accords battent, + et dans quelles Octaves au-dessus on au-dessous du son fixe. Si + cette hypothèse est vraye, elle découvrira la véritable source des + Règles de la composition, inconnue jusqu'à présent à la Philosophie, + qui s'en remettait presque entièrement au jugement de l'Oreille. Ces + sortes de jugemens naturels, quelque bisarres qu'ils paroissent + quelquefois, ne le sont point, ils ont des causes très réelles, dont + la connaissance appartient à la Philosophie, pourvue qu'elle s'en + puisse mettre en possession."] + + [Footnote 144: _Harmonics or the Philosophy of Musical Sounds_, + Cambridge, 1749. I saw this book only hastily in 1864 and drew + attention to it in a work published in 1866. I did not come into its + actual possession until three years ago and then only did I learn + its exact contents.] + + [Footnote 145: _Harmonics_, pp. 118 and 243.] + + [Footnote 146: "Short cycle" is the period in which the same phases + of the two co-operant tones are repeated.] + + + + +II. + +REMARKS ON THE THEORY OF SPATIAL VISION.[147] + + +According to Herbart, spatial vision rests on reproduction-series. In +such an event, of course, and if the supposition is correct, the +magnitudes of the residua with which the percepts or representations are +coalesced (the helps to coalescence) are of cardinal influence. +Furthermore, since the coalescences must first be fully perfected before +they make their appearance, and since upon their appearance the +inhibitory ratios are brought into play, ultimately, then, if we leave +out of account the accidental order of time in which the percepts are +given, everything in spatial vision depends on the oppositions and +affinities, or, in brief, on the qualities of the percepts, which enter +into series. + +Let us see how the theory stands with respect to the special facts +involved. + +1. If intersecting series only, running anteriorly and posteriorly, are +requisite for the production of spatial sensation, why are not analogues +of them found in all the senses? + +2. Why do we measure differently colored objects and variegated objects +with one and the same spatial measure? How do we recognise differently +colored objects as the same in size? Where do we get our measure of +space from and what is it? + +3. Why is it that differently colored figures of the same form reproduce +one another and are recognised as the same? + +Here are difficulties enough. Herbart is unable to solve them by his +theory. The unprejudiced student sees at once that his "inhibition by +reason of form" and "preference by reason of form" are absolutely +impossible. Think of Herbart's example of the red and black letters. + +The "help to coalescence" is a passport, so to speak, made out to the +name and person of the percept. A percept which is coalesced with +another cannot reproduce all others qualitatively different from it for +the simple reason that the latter are in like manner coalesced with one +another. Two qualitatively different series certainly do not reproduce +themselves because they present the same order of degree of coalescence. + +If it is certain that only things simultaneous and things which are +alike are reproduced, a basic principle of Herbart's psychology which +even the most absolute empiricists will not deny, nothing remains but to +modify the theory of spatial perception or to invent in its place a new +principle in the manner indicated, a step which hardly any one would +seriously undertake. The new principle could not fail to throw all +psychology into the most dreadful confusion. + +As to the modification which is needed there can be hardly any doubt as +to how in the face of the facts and conformably to Herbart's own +principles it is to be carried out. If two differently colored figures +of equal size reproduce each other and are recognised as equal, the +result can be due to nothing but to the existence in both series of +presentations of a presentation or percept which is qualitatively _the +same_. The colors are different. Consequently, like or equal percepts +must be connected with the colors which are yet independent of the +colors. We have not to look long for them, for they are the like effects +of the muscular feelings of the eye when confronted by the two figures. +We might say we reach the vision of space by the registering of +light-sensations in a schedule of graduated muscle-sensations.[148] + +A few considerations will show the likelihood of the rôle of the +muscle-sensations. The muscular apparatus of _one_ eye is unsymmetrical. +The two eyes together form a system which is vertical in symmetry. This +already explains much. + +1. The _position_ of a figure influences its view. According to the +position in which objects are viewed different muscle-sensations come +into play and the impression is altered. To recognise inverted letters +as such long experience is required. The best proof of this are the +letters d, b, p, q, which are represented by the same figure in +different positions and yet are always distinguished as different.[149] + +2. It will not escape the attentive observer that for the same reasons +and even with the same figure and in the same position the fixation +point is also decisive. The figure seems to change _during_ the act of +vision. For example, an eight-pointed star constructed by successively +joining in a regular octagon the first corner with the fourth, the +fourth with the seventh, etc., skipping in every case two corners, +assumes alternately, according to where we suffer the centre of vision +to rest, a predominantly architectonic or a freer and more open +character. Vertical and horizontal lines are always differently +apprehended from what oblique lines are. + +[Illustration: Fig. 58.] + +3. The reason why we prefer vertical symmetry and regard it as something +special in its kind, whereas we do not recognise horizontal symmetry at +all immediately, is due to the vertical symmetry of the muscular +apparatus of the eye. The left-hand side _a_ of the accompanying +vertically-symmetrical figure induces in the left eye the same muscular +feelings as the right-hand side _b_ does in the right eye. The pleasing +effect of symmetry has its cause primarily in the repetition of muscular +feelings. That a repetition actually occurs here, sometimes sufficiently +marked in character as to lead to the confounding of objects, is proved +apart from the theory by the fact which is familiar to every one _quem +dii oderunt_ that children frequently reverse figures from the right to +the left, but never from above downwards; for example, write [epsilon] +instead of 3 until they finally come to notice the slight difference. +Figure 50 shows how pleasing the repetition of muscular feelings may be. +As will be readily understood, vertical and horizontal lines exhibit +relations similar to symmetrical figures which are immediately disturbed +when oblique positions are chosen for the lines. Compare what Helmholtz +says regarding the repetition and coincidence of partial tones. + +[Illustration: Fig. 59.] + +I may be permitted to add a general remark. It is a quite universal +phenomenon in psychology that certain qualitatively quite different +series of percepts mutually awaken and reproduce one another and in a +certain aspect produce the appearance of sameness or similarity. We say +of such series that they are of like or of similar form, naming their +abstracted likeness _form_. + + 1. Of spatial figures we have already spoken. + + 2. We call two melodies like melodies when they present the same + succession of pitch-ratios; the absolute pitch (or key) may be as + different as can be. We can so select the melodies that not even + two partial tones of the notes in each are common. Yet we recognise + the melodies as alike. And, what is more, we notice the form of the + melody more readily and recognise it again more easily than the key + (the absolute pitch) in which it was played. + + 3. We recognise in two different melodies the same rhythm no matter + how different the melodies may be otherwise. We know and recognise + the rhythm more easily even than the absolute duration (the tempo). + +These examples will suffice. In all these and in all similar cases the +recognition and likeness cannot depend upon the qualities of the +percepts, for these are different. On the other hand recognition, +conformably to the principles of psychology, is possible only with +percepts which are the same in quality. Consequently there is no other +escape than to imagine the qualitatively unlike percepts of the two +series as necessarily connected with other percepts which are +qualitatively alike. + +Since in differently colored figures of like form, like muscular +feelings are necessarily induced if the figures are recognised as alike, +so there must necessarily lie at the basis of all forms also, and we +might even say at the basis of all abstractions, percepts of a peculiar +quality. And this holds true for space and form as well as for time, +rhythm, pitch, the form of melodies, intensity, etc. But whence is +psychology to derive all these qualities? Have no fear, they will all be +found, as were the sensations of muscles for the theory of space. The +organism is at present still rich enough to meet all the requirements of +psychology in this direction, and it is even time to give serious ear to +the question of "corporeal resonance" which psychology so loves to dwell +on. + +Different psychical qualities appear to bear a very intimate mutual +relation to one another. Special research on the subject, as well also +as the demonstration that this remark may be generally employed in +physics, will follow later.[150] + + FOOTNOTES: + + [Footnote 147: This article, designed to illustrate historically + that on Symmetry, at page 89, first appeared in Fichte's + _Zeitschrift für Philosophie_, for 1865.] + + [Footnote 148: Comp. Cornelius, _Ueber das Sehen_; Wundt, _Theorie + der Sinneswahrnehmung_.] + + [Footnote 149: Comp. Mach, _Ueber das Sehen von Lagen and Winkeln_. + _Sitzungsb. der Wiener Akademie_, 1861.] + + [Footnote 150: Comp. Mach, _Zur Theorie des Gehörorgans_. + _Sitsungsber, der Wiener Akad._, 1863.--_Ueber einige Erscheinungen + der physiolog. Akustik._ _Ibid._, 1864.] + + + + +INDEX. + + + Absolute, temperature, 162; + time, 204; + forecasts, have no signification in science, 206. + + Abstract, meaning of the term, 240. + + Abstraction, 180, 200, 208, 231. + + Acceleration, organ for forward, 299 et seq. + + Accelerations, 204, 216, footnote, 225-226, 253. + + Accident, logical and historical, in science, 160, 168, 170, 213; + in inventions and discoveries, 262 et seq. + + Accord, the pure triple, 46. + + Accumulators, electrical, 125 et seq.; + 132, footnote. + + Acoustic color, 36. + + Acoustics, Sauveur on, 375 et seq. + + Action and reaction, importance of the principle of, 191. + + Adaptation, in organic and inorganic matter, 216, 229; + in scientific thought, 214-235. + + Æsthetics, computation as a principle of, 34; + researches in, 89, footnote; + repetition, a principle of, 91. + + Africa, 186, 234, 237. + + Agreeable effects, due to repetition of sensations, 92, 97 et seq. + + Agriculture, transition to, 265. + + Air-gun, 135. + + Alcohol and water, mixture of oil and, in Plateau's experiments, 4. + + Algebra, economy of, 196. + + Alien thoughts in science, 196. + + All, the, 88. + + Amontons, 174, 346. + + Ampère, the word, 314. + + Ampère's swimmer, 207. + + Analogies, mechanical, 157, 160; + generally, 236-258. + + Analogy, defined, 250. + + Analysis, 188. + + Analytical geometry, not necessary to physicians, 370, footnote. + + Anatomic structures, transparent stereoscopic views of, 74. + + Anatomy, character of research in, 255. + + Andrieu, Jules, 49, footnote. + + Animals, the psychical activity of, 190, 231; + the language of, 238; + their capacity for experience, 266 et seq. + + Animism, 186, 187, 243, 254. + + Anisotropic optical fields, 227. + + Apparatus for producing movements of rotation, 287 et seq. + + Arabesque, an inverted, 95. + + Arabian Nights, 219. + + Arago, 270. + + Aral, the Sea of, 239. + + Archæopteryx, 257. + + Archimedes, 4, 237. + + Arcimboldo, Giuseppe, 36. + + Area, principle of least superficial, 10 et seq. + + Ares, the bellowing of the wounded, 272. + + Aristotelians, 283. + + Aristotle, 348, 296. + + Art, development of, 28 et seq. + + Artillery, practical, 334-335. + + Artistic value of scientific descriptions, 254. + + Arts, practical, 108. + + Ascent, heights of, 143-151. + + Asia, 234. + + Assyrians, the art of, 79. + + Astronomer, measures celestial by terrestrial distances, 136. + + Astronomy, antecedent to psychology, 90; + rigidity of its truths, 221. + + Atomic theories, 104. + + Atoms, 207. + + Attention, the rôle of, in sensuous perception, 35 et seq. + + Attraction, generally, 226; + of liquid particles, 13-14; + in electricity, 109 et seq. + + Aubert, 298. + + Audition. See _Ear_. + + Austrian gymnasiums, 370, footnote. + + Axioms, instinctive knowledge, 190. + + + Babbage, on the economy of machinery, 196. + + Bach, 20. + + Backwards, prophesying, 253. + + Bacon, Lord, 48, 280. + + Baer, C. E. von, 235. + + Balance, electrical, 127, footnote; + torsion, 109, 168. + + Balloon, a hydrogen, 199. + + Barbarism and civilisation, 335 et seq. + + Bass-clef, 101. + + Bass, fundamental, 44. + + Beats, 40-45, 377 et seq. + + Beautiful, our notions of, variable, 99. + + Beauty, objects of, in nature, 91. + + Becker, J. K., 364, 369. + + Beethoven, 39, 44. + + Beginnings of science, 189, 191. + + Belvedere Gallery at Vienna, 36. + + Bernoulli, Daniel, on the conservation of living force, 149; + on the vibrations of strings, 249. + + Bernoulli, James, on the centre of oscillation, 149. + + Bernoulli, John, on the conservation of living force, 149; + on the principle of virtual velocities, 151. + + Bible, parallel passages from, for language study, 356. + + Binocular vision, 66 et seq. + + Black, his theory of caloric, 138, 162; + on quantity of heat, 166, 174; + on latent heat, 167, 178; + researches in heat generally, 244. + + Blind cat, 303. + + Bodies, heavy, seek their places, 224 et seq.; + rotating, 285. + + Body, a mental symbol for groups of sensations, 200-203; + the human, our knowledge of, 90. + + Boltzmann, 236. + + Booth, Mr., 77. + + Borelli, 217. + + Boulder, a granite, 233. + + Bow-wave of ships and moving projectiles, 323 et seq. + + Boys, 317. + + Bradley, 273. + + Brahman, the, 63. + + Brain, localisation of functions in, 210. + + Breuer, 272, 282 et seq., 293, 298, 300, 301, 303, 306. + + Brewster, his stereoscope, 73. + + Bridge, invention of the, 264, 268. + + British Association, 108. + + Brooklyn Bridge, 75, footnote. + + Brown, Crum, 293, 301. + + Building, our concepts directions for, 253; + facts the result of, 253; + science compared to, 257. + + Building-stones, metrical units are, 253. + + Busch, 328. + + Business of a merchant, science compared to the, 16. + + Butterfly, a, 22. + + + Calculating machines, their economical character, 196. + + Caloric, theory of, stood in the way of scientific advancement, 138, 167. + + Calypso, the island of, 351. + + Canterbury, Archbishop of, 39. + + Cantor, M., 361, footnote. + + Capacity, electrical, 116 et seq., 123; + thermal, 123; + specific inductive, 117. + + Capulets and Montagues, 87. + + Cards, difficult games of, 357. + + Carnot, S., excludes perpetual motion in heat, 156, 162; + his mechanical view of physics, 156; + on thermodynamics, 160 et seq.; + his principle, 162; + also, 191. + + Carus, Dr. Paul, 265, footnote. + + Casselli's telegraph, 26. + + Cassini, 51. + + Cauchy, character of the intellectual activity of a, 195. + + Causal insight, awakened by science, 357. + + Causality, 157-159, 190, 198 et seq., 221 et seq., 237, 253, 254. + + Cause and effect, 198 et seq. See also _Causality_. + + Centimetre-gramme-second system, 111. + + Centre of gravity, must lie as low as possible for equilibrium to + subsist, 15; + Torricelli's principle of, 150 et seq. + + Centre of oscillation, 149. + + Change, method of, in science, 230. + + Changeable character of bodies, 202. + + Changes, physical, how they occur, 205. + + Character, a Universal Real, 192. + + Character, like the forms of liquids, 3; + persons of, 24. + + Charles the Fifth, 369. + + Chemical, elements, 202; + symbols, 192; + current, 118. + + Chemistry, character of research in, 255; + the method of thermodynamics in, 257. + + Child, a, modes of thought of, 223; + looking into a moat, 208. + + Child of the forest, his interpretation of new events, 218-219. + + Childish questions, 199-200. + + Children, the drawings of, 201-202. + + Chinese language, economy of, 192; + study of, 354. + + Chinese philosopher, an old, 186. + + Chinese, speak with unwillingness of politics, 374; + the art of, 79-80. + + Chosen, many are called but few are, 65. + + Christ, saying of, 65. + + Christianity, Latin introduced with, 311. + + Christians and Jews, monotheism of the, 187. + + Church and State, 88. + + Cicero, 318. + + Circe, 372. + + Circle, the figure of least area with given periphery, 12. + + Circular polarisation, 242. + + Civilisation and barbarism, 335 et seq. + + Civilisation, some phenomena of, explained by binocular vision, 74. + + Civilised man, his modes of conception and interpretation, 219. + + Clapeyron, 162. + + Class-characters of animals, 255. + + Classical, culture, the good and bad effects of, 347; + scholars, not the only educated people, 345. + + Classics, on instruction in, 338-374; + the scientific, 368. + + Classification in science, 255. + + Clausius, on thermodynamics, 165; + on reversible cycles, 176. + + Claviatur, Mach's, 42-43. + + Club-law, 335. + + Cochlea, the, a species of piano-forte, 19. + + Cockchafer, 86. + + Coefficient of self-induction, 250, 252. + + Colophonium, solution of, 7. + + Color, acoustic, 36. + + Color-sensation, 210. + + Color-signs, their economy, 192. + + Colors, origin of the names of, 239. + + Column, body moving behind a, 202. + + Communication, its functions, import and fruits, 197, 238 et seq.; + by language, 237; + high importance of, 191 et seq. + + Comparative physics, 239. + + Comparison in science, 231, 238 et seq. + + Computation, a principle of æsthetics, 34. + + Concepts, abstract, defined, 250-252; + metrical, in electricity, 107 et seq. + + Conceptual, meaning of the term, 240. + + Conceptual thought, 192. + + Concha, 18. + + Condensers, electrical, 125 et seq. 132, footnote. + + Conductors and non-conductors. See _Electrical_, etc. + + Conformity in the deportment of the energies, 171-175. + + Confusion of objects, cause of, 95. + + Conic sections, 257. + + Conical refraction, 29, 242. + + Conservation of energy, 137 et seq. See _Energy_. + + Conservation of weight or mass, 203. + + Consonance, connexion of the simple natural numbers with, 33; + Euclid's definition of, 33; + explanation of, 42; + scientific definition of, 44; + and dissonance reduced to beats, 376, 370, 383. + + Consonant intervals, 43. + + Constancy of matter, 203. + + Constant, the dielectric, 117. + + Constants, the natural, 193. + + Continuum of facts, 256 et seq. + + Cornelius, 388, footnote. + + Corti, the Marchese, his discovery of minute rods in the labyrinth of + the ear, 19. + + Coulomb, his electrical researches, 108, 109, 113; + his notion of quantity of electricity, 173; + his torsion-balance, 168. + + Crew, Prof. Henry, 317, footnote. + + Criticism, Socrates the father of scientific, 1, 16. + + _Critique of Pure Reason_, Kant's, 188. + + Crucible, derivation of the word, 49, footnote. + + Crustacea, auditory filaments of, 29, 272, 302. + + Cube of oil, 5. + + Culture, ancient and modern, 344. + + Currents, chemical, 118; + electrical, 118; + galvanic, 132; + measurement of electrical, 135-136; + of heat, 244, 249-250; + strength of, 250. + + Curtius, 356. + + Curved lines, their asymmetry, 98. + + Curves, how their laws are investigated, 206. + + Cycles, reversible, Clausius on, 176. + + Cyclical processes, closed, 175. + + Cyclops, 67. + + Cyclostat, 298. + + Cylinder, of oil, 6; + mass of gas enclosed in a, 179. + + + D'Alembert, on the causes of harmony, 34; + his principle, 142, 149, 154; + also 234, 279. + + Danish schools, 338, footnote. + + Darwin, his study of organic nature, 215 et seq.; + his methods of research, 216. + + Deaf and dumb, not subject to giddiness, 299. + + Deaf person, with a piano, analyses sounds, 27. + + Death and life, 186. + + Definition, compendious, 197. + + Deiters, 19. + + Delage, 298, 301, 302. + + Democritus, his mechanical conception of the world, 155, 187. + + Demonstration, character of, 362. + + Deportment of the energies, conformity in the, 171-175. + + Derivation, laws only methods of, 256. + + Descent, Galileo's laws of, 193; + generally, 143 et seq., 204, 215. + + Description, 108, 191, 236, 237; + a condition of scientific knowledge, 193; + direct and indirect, 240; + in physics, 197, 199. + + Descriptive sciences, their resemblance to the abstract, 248. + + Determinants, 195. + + Diderot, 234. + + Dielectric constant, the, 117. + + Difference-engine, the, 196. + + Differential coefficients, their relation to symmetry, 98. + + Differential laws, 204. + + Differential method, for detecting optical imperfections, 317. + + Diffraction, 91, 194. + + Diffusion, Fick's theory of, 249. + + Discharge of Leyden jars, 114 et seq. + + Discoveries, the gist of, 270, 375. + + Discovery and invention, distinction between, 269. + + Dissonance, explanation of, 42; + definition of, 33, 44. See _Consonance_. + + Distances, estimation of, by the eye, 68 et seq. + + Dogs, like tuning-forks, 23; + their mentality, 190. + + Domenech, Abbé, 92. + + Dramatic element in science, 243. + + Drop of water, on a greased plate, 8; + on the end of a stick, 8; + in free descent, 8. + + Dubois, 218. + + Dubois-Reymond, 370, footnote. + + Dufay, 271. + + Dynamics, foundations of, 153 et seq. + + + Ear, researches in the theory of, 17 et seq.; + diagram of, 18; + its analysis of sounds, 20 et seq.; + a puzzle-lock, 28; + reflected in a mirror, 93; + no symmetry in its sensation, 103. + + Earth, its oblateness not due to its original fluid condition, 2; + rotation of, 204; + internal disturbances of, 285. + + Economical, nature of physical inquiry, 186; + procedure of the human mind, 186; + order of physics, 197; + schematism of science, 206; + tools of science, 207; + coefficient of dynamos, 133. + + Economy, of the actions of nature, 15; + the purpose of science, 16; + of language, 191 et seq.; + of the industrial arts, 192; + of mathematics, 195-196; + of machinery, 196; + of self-preservation, our first knowledge derived from, 197; + generally, 186 et seq., 269. + + Education, higher, 86; + liberal, 341 et seq., 371. + + Efflux, liquid, 150. + + Ego, its nature, 234-235. + + Egypt, 234. + + Egyptians, art of, 78 et seq., 201. + + Eighteenth century, the scientific achievements of, 187, 188. + + Eleatics, on motion, 158. + + Electrical, attraction and repulsion, 109 et seq., 168; + capacity, 116 et seq.; + force, 110, 119, 168; + spark, 117, 127, 132, 133, 190; + energy, measurement of, 128 et seq., 169; + currents, conceptions of, 118, 132, 135-136, 226-227, 249, 250; + fluids, 112 et seq., 228; + pendulums, 110; + levels, 173; + potential, 121 et seq.; + quantity, 111, 118, 119. + + Electricity, as a substance and as a motion, 170; + difference between the conceptions of heat and, 168 et seq., + rôle of work in, 120 et seq.; + galvanic, 134. + See _Electrical_. + + Electrometer, W. Thomson's absolute, 127, footnote. + + Electrometers, 122, 127. + + Electrostatic unit, 111. + + Electrostatics, concepts of, 107 et seq. + + Elements, interdependence of the sensuous, 179; + of bodies, 202; + of phenomena, equations between, 205; + of sensations, 200; + used instead of sensations, 208-209. + + Ellipse, equation of, 205; + the word, 342. + + Embryology, possible future state of, 257. + + Energies, conformity in the deportment of, 171-175; + differences of, 175. + + Energy, a metrical notion, 178; + conservation of, 137 et seq.; + defined, 139; + metaphysical establishment of the doctrine of, 183; + kinetic, 177; + potential, 128 et seq.; + substantial conception of, 164, 185, 244 et seq.; + conservation of, in electrical phenomena, 131 et seq.; + limits of principle of, 175; + principle of, in physics, 160-166; + sources of principle of, 179, 181; + thermal, 177; + Thomas Young on, 173. + + Energy-value of heat, 178, footnote. + + Enlightenment, the, 188. + + Entropy, a metrical notion, 178. + + Environment, stability of our, 206. + + Equations for obtaining facts, 180; + between the elements of phenomena, 205. + + Equilibrium, conditions of, in simple machines, 151; + figures of liquid, 4 et seq.; + general condition of, 15; + in the State, 15. + + Etymology, the word, misused for entomology, 316. + + Euclid, on consonance and dissonance, 33; + his geometry, 364. + + Euler, on the causes of harmony, 34; + impression of the mathematical processes on, 196; + on the vibrations of strings, 249, 285, 376. + + Euler and Hermann's principle, 149. + + Euthyphron, questioned by Socrates, 1. + + Evolute, the word, 342. + + Evolution, theory of, as applied to ideas, 216 et seq. + + Ewald, 298, 304. + + Excluded perpetual motion, logical root of the principle of, 182. + + Exner, S., 302, 305. + + Experience, communication of, 191; + our ready, 199; + the principle of energy derived from, 179; + the wellspring of all knowledge of nature, 181; + incongruence between thought and, 206. + + Experimental research, function of, 181. + + Explanation, nature of, 194, 237, 362. + + Eye, cannot analyse colors, 20; + researches in the theory of the, 18 et seq.; + loss of, as affecting vision, 98. + + Eyes, purpose of, 66 et seq.; + their structure symmetrical not identical, 96. + + + Face, human, inverted, 95. + + Facts and ideas, necessary to science, 231. + + Facts, description of, 108; + agreement of, 180; + relations of, 180; + how represented, 206; + reflected in imagination, 220 et seq.; + the result of constructions, 253; + a continuum of, 256 et seq.; + equations for obtaining, 180. + + Falling bodies, 204, 215; + Galileo on the law of, 143 et seq., 284. + + Falling, cats, 303, footnote. + + Falstaff, 309. + + Familiar intermediate links of thought, 198. + + Faraday, 191, 217, 237; + his conception of electricity, 114, 271. + + Fechner, theory of Corti's fibres, 19 et seq. + + Feeling, cannot be explained by motions of atoms, 208 et seq. + + Fetishism, 186, 243, 254; + in our physical concepts, 187. + + Fibres of Corti, 17 et seq. + + Fick, his theory of diffusion, 249. + + Figures, symmetry of, 92 et seq. + + Figures of liquid equilibrium, 4 et seq. + + Fire, use of, 264. + + Fishes, 306. + + Fixed note, determining of a, 377. + + Fizeau, his determination of the velocity of light, 55 et seq. + + Flats, reversed into sharps, 101. + + Flouren's experiments, 272, 290. + + Flower-girl, the baskets of a, 95. + + Fluids, electrical, 112 et seq. + + Force, electric, 110, 119, 168; + unit of 111; + living, 137, 149, 184; + generally 253. + See the related headings. + + Forces, will compared to, 254. + + Foreseeing events, 220 et seq. + + Formal conceptions, rôle of, 183. + + Formal need of a clear view of facts, 183, 246; + how far it corresponds to nature, 184. + + Formative forces of liquids, 4. + + Forms of liquids, 3 et seq. + + Forward movement, sensation of, 300. + + Forwards, prophesying, 253. + + Foucault, 57, 70, 296. + + Foucault and Toepler, method of, for detecting optical faults, 313 + et seq., 320. + + Foundation of scientific thought, primitive acts of knowledge, the, 190. + + Fourier, on processes of heat, 249, 278. + + Fox, a, 234. + + Franklin's pane, 116. + + Frary, 338, footnote. + + Fraunhofer, 271. + + Freezing-point, lowered by pressure, 162. + + Fresnel, 271. + + Fritsch, 321. + + Frogs, larvæ of, not subject to vertigo, 298. + + Froude, 333. + + Frustra, misuse of the word, 345. + + Future, science of the, 213. + + + Galileo, on the motion of pendulums, 21; + his attempted measurement of the velocity of light, 50 et seq.; + his exclusion of a perpetual motion, 143; + on velocities acquired in free descent, 143-147; + on the law of inertia, 146-147; + on virtual velocities, 150; + on work, 172; + his laws of descent, 193; + on falling bodies, 225; + great results of his study of nature, 214 et seq.; + his rude scientific implements, 215; + selections from his works for use in instruction, 368; + also 105, 182, 187, 237, 272, 274, 283. + + Galle, observes the planet Neptune, 29. + + Galvanic, electricity, 134; + current, 132; + dizziness, 291; + vertigo, 298. + + Galvanoscope, 135. + + Galvanotropism, 291. + + Garda, Lake, 239. + + Gas, the word, 264; + mass of, enclosed in a cylinder, 179. + + Gases, tensions of, for scales of temperature, 174. + + Gauss, on the foundations of dynamics, 154; + his principle, 154; + also, 108, 274. + + Genius, 279, 280. + + Geography, comparison in, 239. + + Geometers, in our eyes, 72. + + Geotropism, 289. + + German schools and gymnasiums, 372, 373, 338, footnote. + + Ghosts, photographic, 73. + + Glass, invisible in a mixture of the same refrangibility, 312; + powdered, visible in a mixture of the same refrangibility, 312. + + Glove, in a mirror, 93. + + Goethe, quotations from, 9, 31, 49, 88; + on the cause of harmony, 35. + + Goltz, 282, 291. + + Gossot, 332. + + Gothic cathedral, 94. + + Gravitation, discovery of, 225 et seq. + + Gravity, how to get rid of the effects of, in liquids, 4; + also 228. + + Gray, Elisha, his telautograph, 26. + + Greased plate, drop of water on a, 8. + + Great minds, idiosyncrasies of, 247. + + Greek language, scientific terms derivedfrom, 342-343; + common words derived from, 343, footnote; + still necessary for some professions, 346; + its literary wealth, 347-348; + narrowness and one-sidedness of its literature, 348-349; + its excessive study useless, 349-350; + its study sharpens the judgment, 357-358; + a knowledge of it not necessary to a liberal education, 371. + + Greeks, their provinciality and narrow-mindedness, 349; + now only objects of historical research, 350. + + Griesinger, 184. + + Grimaldi, 270. + + Grimm, 344, footnote. + + Grunting fishes, 306. + + + Habitudes of thought, 199, 224, 227, 232. + + Haeckel, 222, 235. + + Hamilton, deduction of the conical refraction of light, 29. + + Hankel, 364. + + Harmonics, 38, 40. + + Harmony, on the causes of, 32 et seq.; + laws of the theory of, explained, 30; + the investigation of the ancients concerning, 32; + generally, 103. + See _Consonance_. + + Harris, electrical balance of, 127, footnote. + + Hartwich, Judge, 343, 353, footnote. + + Hat, a high silk, 24. + + Hats, ladies', development of, 64. + + Head-wave of a projectile, 323 et seq. + + Hearing and orientation, relation between, 304 et seq. + + Heat, a material substance, 177; + difference between the conceptions of electricity and, 168 et seq.; + substantial conception of, 243 et seq.; + Carnot on, 156, 160 et seq.; + Fourier on the conduction of, 249; + not necessarily a motion, 167, 170, 171; + mechanical equivalent of, 164, 167; + of liquefaction, 178; + quantity of, 166; + latent, 167, 178, 244; + specific, 166, 244; + the conceptions of, 160-171; + machine, 160; + a measure of electrical energy, 133 et seq.; + mechanical theory of, 133; + where does it come from? 200. + + Heavy bodies, sinking of, 222. + + Heights of ascent, 143-151. + + Helm, 172. + + Helmholtz, applies the principle of energy to electricity, 184; + his telestereoscope, 84; + his theory of Corti's fibres, 19 et seq.; + on harmony, 35, 99; + on the conservation of energy, 165, 247; + his method of thought, 247; + also 138, 305, 307, 375, 383. + + Hensen, V., on the auditory function of the filaments of Crustacea, 29, + 302. + + Herbart, 386 et seq. + + Herbartians, on motion, 158. + + Herculaneum, art in, 80. + + Heredity, in organic and inorganic matter, 216, footnote. + + Hering, on development, 222; + on vision, 210. + + Hermann, E., on the economy of the industrial arts, 192. + + Hermann, L., 291. + + Herodotus, 26, 234, 347, 350. + + Hertz, his waves, 242; + his use of the phrase "prophesy," 253. + + Herzen, 361, footnote. + + Hindu mathematicians, their beautiful problems, 30. + + Holtz's electric machine, 132. + + Horse, 63. + + Household, physics compared to a well-kept, 197. + + Housekeeping in science and civil life, 198. + + Hudson, the, 94. + + Human beings, puzzle-locks, 27. + + Human body, our knowledge of, 90. + + Human mind, must proceed economically, 186. + + Humanity, likened to a polyp-plant, 235. + + Huygens, his mechanical view of physics, 155; + on the nature of light and heat, 155-156; + his principle of the heights of ascent, 149; + on the law of inertia and the motion of a compound pendulum, 147-149; + on the impossible perpetual motion, 147-148; + on work, 173; + selections from his works for use in instruction, 368; + his view of light, 227-228, 262. + + Huygens, optical method for detecting imperfections in optical glasses + 313. + + Hydrogen balloon, 199. + + Hydrostatics, Stevinus's principle of, 141. + + Hypotheses, their rôle in explanation, 228 et seq. + + + Ichthyornis, 257. + + Ichthyosaurus, 63. + + Idea? what is a theoretical, 241. + + Idealism, 209. + + Ideas, a product of organic nature, 217 et seq.; + and facts, necessary to science, 231; + not all of life, 233; + their growth and importance, 233; + a product of universal evolution, 235; + the history of, 227 et seq.; + in great minds, 228; + the rich contents of, 197; + their unsettled character in common life, their clarification in + science, 1-2. + + Ideography, the Chinese, 192. + + Imagery, mental, 253. + + Imagination, facts reflected in, 220 et seq. + + Inclined plane, law of, 140-141. + + Incomprehensible, the, 186. + + Indian, his modes of conception and interpretation, 218 et seq. + + Individual, a thread on which pearls are strung, 234-235. + + Industrial arts, economy of the, E. Hermann on, 192. + + Inertia, law of, 143 et seq., 146 et seq., 216, footnote, 283 et seq. + + Innate concepts of the understanding, Kant on, 199. + + Innervation, visual, 99. + + Inquirer, his division of labor, 105; + compared to a shoemaker, 105-106; + what constitutes the great, 191; + the true, seeks the truth everywhere, 63 et seq.; + the, compared to a wooer, 45. + + Instinctive knowledge, 189, 190. + + Instruction, aim of, the saving of experience, 191; + in the classics, mathematics, and sciences, 338-374; + limitation of matter of, 365 et seq. + + Insulators, 130. + + Integrals, 195. + + Intellectual development, conditions of, 286 et seq. + + Intentions, acts of nature compared to, 14-15. + + Interconnexion of nature, 182. + + Interdependence, of properties, 361; + of the sensuous elements of the world, 179. + + Interference experiments with the head-wave of moving projectiles, + 327-328. + + International intercourse, established by Latin, 341. + + International measures, 108. + + Invention, discovery and, distinction between, 269. + + Inventions, requisites for the development of, 266, 268 et seq. + + Iron-filings, 220, 243. + + Italian art, 234. + + + Jacobi, C. G. J., on mathematics, 280. + + James, W., 275, 299. + + Java, 163. + + Jews and Christians, monotheism of the, 187. + + Jolly, Professor von, 112, 274. + + Joule, J. P., on the conservation of energy, 163-165, 167, 183; + his conception of energy, 245; + his metaphysics, 183, 246; + his method of thought, 247; + also 137, 138. + + Journée, 317. + + Judge, criminal, the natural philosopher compared to a, 48. + + Judgment, essentially economy of thought, 201-202; + sharpened by languages and sciences, 357-358; + also 232-233, 238. + + Juliet, Romeo and, 87. + + Jupiter, its satellites employed in the determination of the velocity of + light, 51 et seq. + + Jurisprudence, Latin and Greek unnecessary for the study of, 346, + footnote. + + + Kant, his hypothesis of the origin of the planetary system, 5; + his _Critique of Pure Reason_, 188; + on innate concepts of the understanding, 199; + on time, 204; + also footnote, 93. + + Kepler, 187, 270. + + Kinetic energy, 177. + + Kirchhoff, his epistemological ideas, 257-258; + his definition of mechanics, 236, 258, 271, 273. + + Knight, 289. + + Knowledge, a product of organic nature, 217 et seq., 235; + instinctive, 190; + made possible by economy of thought, 198; + our first, derived from the economy of self-preservation, 197; + the theory of, 203; + our primitive acts of the foundation of science, 190. + + Kocher, 328. + + Koenig, measurement of the velocity of sound, 57 et seq. + + Kölliker, 19. + + Kopisch, 61. + + Kreidl, 299, 302, 306; + his experiments, 272. + + Krupp, 319. + + + Labels, the value of, 201. + + Labor, the accumulation of, the foundation of wealth and power, 198; + inquirer's division of, 105, 258. + + Labyrinth, of the ear, 18, 291, 305. + + Lactantius, on the study of moral and physical science, 89. + + Ladder of our abstraction, the, 208. + + Ladies, their eyes, 71; + like tuning-forks, 23-24. + + Lagrange, on Huygens's principle, 149; + on the principle of virtual velocities, 150-155; + character of the intellectual activity of a, 195, 278. + + Lake-dwellers, 46, 271. + + Lamp-shade, 70. + + Lane's unit jar, 115. + + Language, knowledge of the nature of, demanded by a liberal education, + 356; + relationship between, and thought, 358; + communication by 237; + economy of, 191 et seq.; + human its character, 238; + of animals, 238; + instruction in, 338 et seq.; + its methods, 192. + + Laplace, on the atoms of the brain, 188; + on the scientific achievements of the eighteenth century, 188; + his hypothesis of the origin of the planetary system, 5. + + Latent heat, 167, 178, 244. + + Latin city of Maupertuis, 339. + + Latin, instruction in, 311 et seq.; + introduced with the Christian Church, 340; + the language of scholars, the medium of international intercourse, + its power, utility, and final abandonment, 341-347; + the wealth of its literature, 348; + the excessive study of, 346, 349, 354, 355; + its power to sharpen the judgment, 357-358. + + Lavish extravagance of science, 189. + + Law, a, defined, 256; + a natural, not contained in the conformity of the energies, 175. + + Law-maker, motives of not always discernible, 9. + + Layard, 79. + + Learning, its nature, 366 et seq. + + Least superficial area, principle of, accounted for by the mutual + attractions of liquid particles, 13-14; + illustrated by a pulley arrangement, 12-13; + also 9 et seq. + + Leibnitz, on harmony, 33; + on international intercourse, 342, footnote. + + Lessing, quotation from, 47. + + Letters of the alphabet, their symmetry, 94, 97. + + Level heights of work, 172-174. + + Lever, a, in action, 222. + + Leverrier, prediction of the planet Neptune, 29. + + Leyden jar, 114. + + Liberal education, a, 341 et seq., 359, 371. + + Libraries, thoughts stored up in, 237. + + Lichtenberg, on instruction, 276, 370. + + Licius, a Chinese philosopher, 213. + + Liebig, 163, 278. + + Life and death, 186. + + Light, history of as elucidating how theories obstruct research, 242; + Huygens's and Newton's views of, 227-228; + its different conceptions, 226; + rectilinear propagation of, 194; + rôle of, in vision, 81; + spatial and temporal periodicity of, explains optical phenomena, 194; + numerical velocity of, 58; + where does it go to? 199; + generally, 48 et seq. + + Like effects in like circumstances, 199. + + Likeness, 388, 391. + + Lilliput, 84. + + Lines, straight, their symmetry, 98; + curved, their asymmetry, 98; + of force, 249. + + Links of thought, intermediate, 198. + + Liquefaction, latent heat of, 178. + + Liquid, efflux, law of, 150; + equilibrium, figures of, 4 et seq.; + the latter produced in open air, 7-8; + their beauty and multiplicity of form, 7, 8; + made permanent by melted colophonium, 7. + + Liquids, forms of, 1-16; + difference between, and solids, 2; + their mobility and adaptiveness of form, 3; + the courtiers _par excellence_ of the natural bodies, 3; + possess under certain circumstances forms of their own, 3. + + Living force, 137, 184; + law of the conservation of, 149. + + Lloyd, observation of the conical refraction of light, 29. + + Lobster, of Lake Mohrin, the, 61. + + Localisation, cerebral, 210. + + Locke, on language and thought, 358. + + Locomotive, steam in the boiler of, 219. + + Loeb, J., 289, 291, 302. + + Logarithms, 195, 219; + in music, 103-104. + + Logical root, of the principle of energy, 181; + of the principle of excluded perpetual motion, 182. + + Lombroso, 280. + + Lucian, 347. + + + _Macula acustica_, 272. + + Magic lantern, 96. + + Magic powers of nature, 189. + + Magical power of science, belief in the, 189. + + Magnet, a, 220; + will compared to the pressure of a, 14; + coercive force of a, 216. + + Magnetic needle, near a current, 207. + + Magnetised bar of steel, 242-243. + + Major and minor keys in music, 100 et seq. + + Malus, 242. + + Man, a fragment of nature's life, 49; + his life embraces others, 234. + + Mann, 364. + + Manuscript in a mirror, 93. + + Maple syrup, statues of, on Moon, 4. + + Marx, 35. + + Material, the relations of work with heat and the consumption of, 245 + et seq. + + Mathematical methods, their character, 197-198. + + Mathematics, economy of, 195; + on instruction in, 338-374; + C. G. J. Jacobi on, 280. + + Matter, constancy of, 203; + its nature, 203; + the notion of, 213. + + Maupertuis, his Latin city, 338. + + Maximal and minimal problems, their rôle in physics, 14, footnote. + + Mayer, J. R., his conception of energy, 245, 246; + his methods of thought, 247; + on the conservation of energy, 163, 164, 165, 167, 183, 184; + his metaphysical utterances, 183, 246; + also 138, 184, 191, 217, 271, 274. + + Measurement, definition of, 206. + + Measures, international, 108. + + Mécanique céleste, 90, 188; + sociale, and morale, the, 90. + + Mechanical, conception of the world, 105, 155 et seq., 188, 207; + energy, W. Thomson on waste of, 175; + analogies between ---- and thermal energy, 17 et seq.; + equivalent of heat, electricity, etc., 164, 167 et seq.; + mythology, 207; + phenomena, physical events as, 182; + philosophy, 188; + physics, 155-160, 212; + substitution-value of heat, 178, footnote. + + Mechanics, Kirchhoff's definition of, 236. + + Medicine, students of, 326. + + Melody, 101. + + Melsens, 310, 327. + + Memory, a treasure-house for comparison, 230; + common elements impressed upon the, 180; + its importance, 238; + science disburdens the, 193. + + Mendelejeff, his periodical series, 256. + + Mental, adaptation, 214-235; + completion of phenomena, 220; + imagery, 253; + imitation, our schematic, 199; + processes, economical, 195; + reproduction, 198; + visualisation, 250. + + Mephistopheles, 88. + + Mercantile principle, a miserly, at the basis of science, 15. + + Mersenne, 377. + + Mesmerism, the mental state of ordinary minds, 228. + + Metaphysical establishment of doctrine of energy, 183. + + Metaphysical spooks, 222. + + Metrical, concepts of electricity, 107 et seq.; + notions, energy and entropy are, 178; + units, the building-stones of the physicist, 253. + + Metronomes, 41. + + Meyer, Lothar, his periodical series, 256. + + Middle Ages, 243, 349. + + Midsummer Night's Dream, 309. + + Mill, John Stuart, 230. + + Millers, school for, 326. + + Mill-wheel, doing work, 161. + + Mimicking facts in thought, 189, 193. + + Minor and major keys in music, 100 et seq. + + Mirror, symmetrical reversion of objects in, 92 et seq. + + Miserly mercantile principle at the basis of science, 15. + + Moat, child looking into, 208. + + Modern scientists, adherents of the mechanical philosophy, 188. + + Molecular theories, 104. + + Molecules, 203, 207. + + Molière, 234. + + Momentum, 184. + + Monocular vision, 98. + + Monotheism of the Christians and Jews, 187. + + Montagues and Capulets, 87. + + Moon, eclipse of, 219; + lightness of bodies on, 4; + the study of the, 90, 284. + + Moreau, 307. + + Mosaic of thought, 192. + + Motion, a perpetual, 181; + quantity of, 184; + the Eleatics on, 158; + Wundt on, 158; + the Herbartians on, 158. + + Motions, natural and violent, 226; + their familiar character, 157. + + Mountains of the earth, would crumble if very large, 3; + weight of bodies on, 112. + + Mozart, 44, 279. + + Müller, Johann, 291. + + Multiplication-table, 195. + + Multiplier, 132. + + Music, band of, its _tempo_ accelerated and retarded, 53; + the principle of repetition in, 99 et seq.; + its notation, mathematically illustrated, 103-104. + + Musical notes, reversion of, 101 et seq.; + their economy, 192. + + Musical scale, a species of one-dimensional space, 105. + + Mystery, in physics, 222; + science can dispense with, 189. + + Mysticism, numerical, 33; + in the principle of energy, 184. + + Mythology, the mechanical, of philosophy, 207. + + + Nagel, von, 364. + + Nansen, 296. + + Napoleon, picture representing the tomb of, 36. + + Nations, intercourse and ideas of, 336-337. + + Natural constants, 193. + + Natural law, a, not contained in the conformity of the energies, 175. + + Natural laws, abridged descriptions, 193; + likened to type, 193. + + Natural motions, 225. + + Natural selection in scientific theories, 63, 218. + + Nature, experience the well-spring of all knowledge of, 181; + fashions of, 64; + first knowledge of, instinctive, 189; + general interconnexion of, 182; + has many sides, 217; + her forces compared to purposes, 14-15; + likened to a good man of business, 15; + the economy of her actions, 15; + how she appears to other animals, 83 et seq.; + inquiry of, viewed as a torture, 48-49; + view of, as something designedly concealed from man, 49; + like a covetous tailor, 9-10; + magic powers of, 189; + our view of, modified by binocular vision, 82; + the experimental method a questioning of, 48. + + Negro hamlet, the science of a, 237. + + Neptune, prediction and discovery of the planet, 29. + + New views, 296 et seq. + + Newton, describes polarisation, 242; + expresses his wealth of thought in Latin, 341; + his discovery of gravitation, 225 et seq.; + his solution of dispersion, 362; + his principle of the equality of pressure and counterpressure, 191; + his view of light, 227-228; + on absolute time, 204; + selections from his works for use in instruction, 368; + also 270, 274, 279, 285, 289. + + Nobility, they displace Latin, 342. + + Notation, musical, mathematically illustrated, 103-104. + + Numbers, economy of, 195; + their connexion with consonance, 32. + + Numerical mysticism, 33. + + Nursery, the questions of the, 199. + + + Observation, 310. + + Observation, in science, 261. + + Ocean-stream, 272. + + Oettingen, Von, 103. + + Ohm, on electric currents, 249. + + Ohm, the word, 343. + + Oil, alcohol, water, and, employed in Plateau's experiments, 4; + free mass of, assumes the shape of a sphere, 12; + geometrical figures of, 5 et seq. + + One-eyed people, vision of, 98. + + Ophthalmoscope, 18. + + Optic nerves, 96. + + Optimism and pessimism, 234. + + Order of physics, 197. + + Organ, bellows of an, 135. + + Organic nature, results of Darwin's studies of, 215 et seq. + See _Adaptation_ and _Heredity_. + + Oriental world of fables, 273. + + Orientation, sensations of, 282 et seq. + + Oscillation, centre of, 147 et seq. + + Ostwald, 172. + + Otoliths, 301 et seq. + + Overtones, 28, 40, 349. + + Ozone, Schöbein's discovery of, 271. + + + Painted things, the difference between real and, 68. + + Palestrina, 44. + + Parameter, 257. + + Partial tones, 390. + + Particles, smallest, 104. + + Pascheles, Dr. W., 285. + + Paulsen, 338, 340, 373. + + Pearls of life, strung on the individual as on a thread, 234-235. + + Pencil surpasses the mathematician in intelligence, 196. + + Pendulum, motion of a, 144 et seq., + increased motion of, due to slight impulses, 21; + electrical, 110. + + Percepts, of like form, 390. + + Periodical, changes, 181; + series, 256. + + Permanent, changes, 181, 199; + elements of the world, 194. + + Perpetual motion, a, 181; + defined, 139; + impossibility of, 139 et seq.; + the principle of the, excluded, 140 et seq.; + excluded from general physics, 162. + + Personality, its nature, 234-235. + + Perspective, 76 et seq.; + contraction of, 74 et seq.; + distortion of, 77. + + Pessimism and optimism, 234. + + Pharaohs, 85. + + Phenomenology, a universal physical, 250. + + Philistine, modes of thought of, 223. + + Philology, comparison in, 239. + + Philosopher, an ancient, on the moral and physical sciences, 89. + + Philosophy, its character at all times, 186; + mechanical, 155 et seq., 188, 207, 259 et seq. + + Phonetic alphabets, their economy, 192. + + Photography, by the electric spark, 318 et seq. + + Photography of projectiles, 309-337. + + Photography, stupendous advances of, 74. + + Physical, concepts, fetishism in our, 187; + ideas and principles, their nature, 204; + inquiry, the economical nature of, 186; + research, object of 207, 209. + + Physical phenomena, as mechanical phenomena, 182; + relations between, 205. + + Physico-mechanical view of the world, 155, 187, 188, 207 et seq. + + Physics, compared to a well-kept household, 197; + economical experience, 197; + the principles of, descriptive, 199; + the methods of, 209; + its method characterised, 211; + comparison in, 239; + the facts of, qualitatively homogeneous, 255; + how it began, 37; + helped by psychology, 104; + study of its own character, 189; + the goal of, 207, 209. + + Physiological psychology, its methods, 211 et seq. + + Physiology, its scope, 212. + + Piano, its mirrored counterpart, 100 et seq.; + used to illustrate the facts of sympathetic vibration, 25 et seq. + + Piano-player, a speaker compared to, 192. + + Picture, physical, a, 110. + + Pike, learns by experience, 267. + + Pillars of Corti, 19. + + Places, heavy bodies seek their, 224 et seq. + + Planetary system, origin of, illustrated, 5. + + Plasticity of organic nature, 216. + + Plateau, his law of free liquid equilibrium, 9; + his method of getting rid of the effects of gravity, 4. + + Plates of oil, thin, 6. + + Plato, 347, 371. + + Plautus, 347. + + Playfair, 138. + + Pleasant effects, cause of, 94 et seq. + + Pliny, 349. + + Poetry and science, 30, 31, 351. + + Poinsot, on the foundations of mechanics, 152 et seq. + + Polarisation, 91; + abstractly described by Newton, 242. + + Politics, Chinese speak with unwillingness of, 374. + + Pollak, 299. + + Polyp plant, humanity likened to a, 235. + + Pompeii, 234; + art in, 80. + + Popper J., 172, 216. + + Potential, social, 15; + electrical, 121 et seq.; + measurement of, 126; + fall of, 177; + swarm of notions in the idea of, 197; + its wide scope, 250. + + Pottery, invention of, 263. + + Prediction, 221 et seq. + + Prejudice, the function, power, and dangers of, 232-233. + + Preparatory schools, the defects of the German, 346-347; + what they should teach, 364 et seq. + + Pressure of a stone or of a magnet, will compared to, 14; + also 157. + + Primitive acts of knowledge the foundation of scientific thought, 190. + + Problem, nature of a, 223. + + Problems which are wrongly formulated, 308. + + Process, Carnot's, 161 et seq. + + Projectiles, the effects of the impact of, 310, 327-328; + seen with the naked eye, 311, 317; + measuring the velocity of, 332; + photography of, 309-337. + + Prony's brake, 132. + + Proof, nature of, 284. + + Prophesying events, 220 et seq. + + Psalms, quotation from the, 89. + + Pseudoscope, Wheatstone's, 96. + + Psychology, preceded by astronomy, 90; + how reached, 91 et seq.; + helps physical science, 104; + its method the same as that of physics, 207 et seq. + + Pully arrangement, illustrating principle of least superficial area, + 12-13. + + Purkinje, 284, 285, 291, 299. + + Purposes, the acts of nature compared to, 14-15; + nature pursues no, 66. + + Puzzle-lock, a, 26. + + Puzzles, 277. + + Pyramid of oil, 6. + + Pythagoras, his discovery of the laws of harmony, 32, 259. + + + Quality of tones, 36. + + Quantitative investigation, the goal of, 180. + + Quantity of electricity, 111, 118, 119, 167-170, 173; + of heat, 166, 167-171, 174, 177, 244; + of motion, 184. + + Quests made of the inquirer, not by him, 30. + + Quételet, 15, footnote. + + + Rabelais, 283. + + Raindrop, form of, 3. + + Rameau, 34. + + Reaction and action, principle of, 191. + + Reactions, disclosure of the connexion of, 270 et seq. + + Realgymnasien, 365. + + Realschulen, 365, 373. + + Reason, stands above the senses, 105. + + Reflex action, 210. + + Reflexion, produces symmetrical reversion of objects, 93 et seq. + + Refraction, 29, 193, 194, 208, 230, 231. + + Reger, 328. + + Reliefs, photographs of, 68. + + Repetition, its rôle in æsthetics, 89, footnote, 91 et seq., 97, 98 + et seq., 390. + + Reproduction of facts in thought, 189, 193, 198, 253. + + Repulsion, electric, 109 et seq., 168. + + Research, function of experimental 181; + the aim of, 205. + + Resemblances between facts, 255. + + Resin, solution of, 7. + + Resistance, laws of, for bodies travelling in air and fluids, 333 et seq. + + Resonance, corporeal, 392. + + Response of sonorous bodies, 25. + + Retina, the corresponding spots of 98; + nerves of compared to fingers of a hand, 96 et seq. + + Reversible processes, 161 et seq., 175, 176, 181, 182. + + Rhine, the, 94. + + Richard the Third, 77. + + Riddles, 277. + + Riders, 379. + + Riegler, 319. + + Riess, experiment with the thermo-electrometer, 133 et seq., 169. + + Rigid connexions, 142. + + Rind of a fruit, 190. + + Rings of oil, illustrating formation of rings of Saturn, 5. + + Ritter, 291, 299. + + Rods of Corti, 19. + + Rolph, W. H., 216. + + Roman Church, Latin introduced with the, 340 et seq. + + Romans, their provinciality and narrow-mindedness, 270. + + Romeo and Juliet, 87. + + Römer, Olaf, 51 et seq. + + Roots, the nature of, in language, 252. + + Rosetti, his experiment on the work required to develop electricity, 131. + + Rotating bodies, 285. + + Rotation, apparatus of, in physics, 59 et seq.; + sensations of, 288 et seq. + + Rousseau, 336. + + Rubber pyramid, illustrating the principle of least superficial area, + 10-11. + + Ruysdael, 279. + + + Sachs, Hans, 106. + + Salcher, Prof. 319. + + Salviati, 144. + + Saturn, rings of, their formation illustrated, 5. + + Saurians, 257. + + Sauveur, on acoustics, 34, 375 et seq. + + Savage, modes of conception and interpretation of a, 218 et seq. + + Schäfer, K., 298. + + _Schlierenmethode_, 317. + + Schönbein's discovery of ozone, 271. + + School-boy, copy-book of, 92. + + Schoolmen, 214. + + Schools, State-control of, 372 et seq. + + Schopenhauer, 190. + + Schultze, Max, 19. + + Science, a miserly mercantile principle at its basis, 15; + compared to a business, 16; + viewed as a maximum or minimum problem, 16, footnote; + its process not greatly different from the intellectual activity of + ordinary life, 16, footnote; + economy of its task, 16; + relation of, to poetry, 30, 31, 351; + the church of, 67; + beginnings of, 189, 191; + belief in the magical power of, 189; + can dispense with mystery, 189; + lavish extravagance of, 189; + economy of the terminology of, 192; + partly made up of the intelligence of others, 196; + stripped of mystery, 197; + its true power, 197; + the economical schematism of, 206; + the object of, 206; + the tools of, 207; + does not create facts, 211; + of the future, 213; + revolution in, dating from Galileo, 214 et seq.; + the natural foe of the marvellous, 224; + characterised, 227; + growth of, 237; + dramatic element in, 243; + described, 251; + its function, 253; + classification in, 255, 259 et seq.; + the way of discovery in, 316. + See also _Physics_. + + Sciences, partition of the, 86; + the barriers and relations between the 257-258; + on instruction in the, 338-374. + + Scientific, criticism, Socrates the father of, 1, 16; + discoveries, their fate, 138; + knowledge, involves description, 193; + thought, transformation and adaptation in, 214-235; + thought, advanced by new experiences, 223 et seq.; + thought, the difficulty of, 366; + terms, 342-343; + founded on primitive acts of knowledge, 190. + + Scientists, stories about their ignorance, 342. + + Screw, the, 62. + + Sea-sickness, 284. + + Secret computation, Leibnitz's, 33. + + Seek their places, bodies, 226. + + Self-induction, coefficient of, 250, 252. + + Self-observation, 211. + + Self-preservation, our first knowledge derived from the economy of, 197; + struggle for, among ideas, 228. + + Semi-circular canals, 290 et seq. + + Sensation of rounding a railway curve, 286. + + Sensations, analysed, 251; + when similar, produce agreeable effects, 96; + their character, 200; + defined, 209; + of orientation, 282 et seq. + + Sense-elements, 179. + + Senses, theory of, 104; + the source of our knowledge of facts, 237. + + Seventh, the troublesome, 46. + + Shadow method, 313 et seq., 317 footnote. + + Shadows, rôle of, in vision, 81. + + Shakespeare, 278. + + Sharps, reversed into flats, 101. + + Shell, spherical, law of attraction for a, 124, footnote. + + Shoemaker, inquirer compared to, 105-106. + + Shooting, 309. + + Shots, double report of, 229 et seq. + + Similarity, 249. + + Simony, 280. + + Simplicity, a varying element in description, 254. + + Sines, law of the, 193. + + Sinking of heavy bodies, 222. + + Sixth sense, 297. + + Smith, R., on acoustics, 34, 381, 383. + + Soap-films, Van der Mensbrugghe's experiment with, 11-12. + + Soapsuds, films and figures of, 7. + + Social potential, 15. + + Socrates, the father of scientific criticism, 1, 16. + + Sodium, 202. + + Sodium-light, vibrations of, as a measure of time, 205. + + Solidity, conception of, by the eye, 71 et seq.; + spatial, photographs of, 73. + + Solids, and liquids, their difference merely one of degree, 2. + + Sonorous bodies, 24 et seq. + + Soret, J. P., 89. + + Sounds, symmetry of, 99 et seq.; + generally, 22-47, 212. + + Sound-waves rendered visible, 315 et seq. + + Sources of the principle of energy, 179 et seq. + + Space, 205; + sensation of, 210. + + Spark, electric, 117, 127, 132, 133, 190. + + Spatial vision, 386. + + Species, stability of, a theory, 216. + + Specific energies, 291. + + Specific heat, 166, 244. + + Specific inductive capacity, 117. + + Spectral analysis of sound, 27. + + Spectrum, mental associations of the, 190. + + Speech, the instinct of, cultivated by languages, 354. + + Spencer, 218, 222. + + Sphere, a soft rotating, 2; + the figure of least surface, 12; + electrical capacity of, 123 et seq. + + Spherical shell, law of attraction for 124, footnote. + + Spiders, the eyes of, 67. + + Spirits, as explanation of the world 186, 243. + + Spiritualism, modern, 187. + + Spooks, metaphysical, 222. + + Squinting, 72. + + Stability of our environment, 206. + + Stallo, 336. + + Stars, the fixed, 90. + + State, benefits and evils of its control of the schools, 372 et seq.; + the Church and, 88. + + Statical electricity, 134. + + Stationary currents, 249. + + Statoliths, 303. + + Steam-engine, 160, 265. + + Steeple-jacks, 75. + + Stereoscope, Wheatstone and Brewster's, 73. + + Stevinus, on the inclined plane, 140; + on hydrostatics, 141; + on the equilibrium of systems, 142; + discovers the principle of virtual velocities, 150; + characterisation of his thought, 142; + also 182, 187, 191. + + Stone Age, 46, 321. + + Störensen, 306. + + Stove, primitive, 263. + + Straight line, a, its symmetry, 98. + + Straight, meaning of the word, 240. + + Street, vista into a, 75. + + Striae, in glass, 313. + + Striate method, for detecting optical imperfections, 317. + + Striking distance, 115, 127. + + Strings, vibrations of, 249. + + Struggle for existence among ideas, 217. + + Substance, heat conceived as a, 177, 243 et seq.; + electricity as a, 170; + the source of our notion of, 199; + rôle of the notion of, 203, 244 et seq.; + energy conceived as a, 164, 185, 244 et seq. + + Substitution-value of heat, 178, footnote. + + Suetonius, 348. + + Sulphur, specific inductive capacity of, 117. + + Sun, human beings could not exist on, 3. + + Swift, 84, 280. + + Swimmer, Ampère's, 207. + + Symmetry, definition of, 92; + figures of, 92 et seq.; + plane of, 94; + vertical and horizontal, 94; + in music, 99 et seq. + + Sympathetic vibration, 22 et seq., 379. + + + Tailor, nature like a covetous, 9-10. + + Tangent, the word, 263. + + Taste, doubtful cultivation of, by the classics, 352-353; + of the ancients, 353. + + Taylor, on the vibration of strings, 249. + + Teaching, its nature, 366 et seq. + + Telegraph, the word, 263. + + Telescope, 262. + + Telestereoscope, the, 84. + + Temperament, even, in tuning, 47. + + Temperature, absolute, 162; + differences of, 205; + differences of, viewed as level surfaces, 161; + heights of, 174; + scale of, derived from tensions of gases, 174. + + Terence, 347. + + Terms, scientific, 342-343. + + Thales, 259. + + Theories, their scope, function, and power, 241-242; + must be replaced by direct description, 248. + + Thermal, energy, 174, 177; + capacity, 123, footnote. + + Thermodynamics, 160 et seq. + + Thermoelectrometer, Riess's, 133, 169. + + Thing-in-itself, the, 200. + + Things, mental symbols for groups of sensations, 200-201. + + Thomson, James, on the lowering of the freezing-point of water by + pressure, 162. + + Thomson, W., his absolute electrometer, 127, footnote; + on thermodynamics, 162; + on the conservation of energy, 165; + on the mechanical measures of temperature, 174, footnote; + on waste of mechanical energy, 175; + also 108, 173, footnote. + + Thought, habitudes of, 199, 224, 227, 232; + relationship between language and, 329; + incongruence between experience and, 206; + luxuriance of a fully developed, 58; + transformation in scientific, 214-235. + + Thoughts, their development and the struggle for existence among them, + 63; + importance of erroneous, 65; + as reproductions of facts, 107. + + Thread, the individual a, on which pearls are strung, 234-235. + + Tides, 283. + + Timbre, 37, 38, 39. + + Time, 178, 204, 205, footnote. + + Toepler and Foucault, method of, for detecting optical faults, 313 + et seq., 320. + + Tone-figures, 91. + + Tones, 22-47, 99 et seq., 212. + + Torsion, moment of, 132. + + Torsion-balance, Coulomb's, 109, 168. + + Torricelli, on virtual velocities, 150; + his law of liquid efflux, 150; + on the atmosphere, 273. + + Tourist, journey of, work of the inquirer compared to, 17, 29, 30. + + Transatlantic cable, 108. + + Transformation and adaptation in scientific thought, 214-235. + + Transformation of ideas, 63. + + Transformative law of the energies, 172. + + Translation, difficulties of, 354. + + Tree, conceptual life compared to a, 231. + + Triangle, mutual dependence of the sides and angles of a, 179. + + Triple accord, 46. + + Truth, wooed by the inquirer, 45; + difficulty of its acquisition, 46. + + Tumblers, resounding, 23. + + Tuning-forks, explanation of their motion, 22 et seq. + + Tylor, 186. + + Tympanum, 18. + + Type, natural laws likened to, 193; + words compared to, 191. + + + Ulysses, 347. + + Understanding, what it means, 211. + + Uniforms, do not fit heads, 369. + + Unique determination, 181-182. + + Unison, 43. + + Unit, electrostatic, 111. + See _Force_ and _Work_. + + United States, 336. + + Universal Real Character, a, 192. + + Utility of physical science, 351. + + + Variation, the method of, in science, 230; + in biology, 216. + + Velocity, of light, 48 et seq.; + of the descent of bodies, 143 et seq.; + meaning of, 204; + virtual, 149-155. + + _Verstandesbegriffe_, 199. + + Vertical, perception of the, 272, 286 et seq.; + symmetry, 389. + + Vertigo, 285, 290. + + Vestibule of the ear, 300. + + Vibration, 22 et seq. + + Vibration-figures, 91. + + Vinci, Leonardo da, 278, 283. + + Violent motions, 225. + + Virtual velocities, 149-155. + + Visibility, general conditions of, 312. + + Vision, symmetry of our apparatus of, 96. + See _Eye_. + + Visual nerves, 96. + + Visualisation, mental, 250. + + Volt, the word, 343. + + Volta, 127, footnote, 134. + + Voltaire, 260. + + Voltaire's ingènu, 219. + + Vowels, composed of simple musical notes, 26. + + + Wagner, Richard, 279. + + Wald, F., 178, footnote. + + Wallace, 216. + + War, and peace, reflexions upon, 309, 335 et seq. + + Waste of mechanical energy, W. Thomson on, 175. + + Watches, experiment with, 41; + in a mirror, 93. + + Water, jet of, resolved into drops, 60; + free, solid figures of, 8; + objects reflected in, 94, 191; + possible modes of measurement of, 170. + + Watt, 266. + + Wealth, the foundation of, 198. + + Weapons, modern, 335. + + Weber, 108, 306. + + Weight of bodies, varies with their distance from the centre of the + earth, 112. + + Weismann, 216. + + Wheatstone, his stereoscope, 73; + his pseudoscope, 96; + also 59. + + Wheel, history and importance of, 61 et seq. + + Whewell, on the formation of science, 231. + + Whole, the, 204, footnote. + + Why, the question, 199, 223. + + Will, Schopenhauer on the, 190; + man's most familiar source of power, 243; + used to explain the world, 186; + forces compared to, 254; + compared to pressure, 14. + + Windmill, a rotating, 53. + + Wire frames and nets, for constructing liquid figures of equilibrium, + 4 et seq. + + Witchcraft, 187. + + Wollaston, 284, 285. + + Wonderful, science the natural foe of the, 224. + + Woods, the relative distance of trees in, 68. + + Wooer, inquirer compared to a, 45. + + Words and sounds, 343. + + Words, compared to type, 191. + + Work, of liquid forces of attraction, 14; + in electricity, 173; + measure of, 119 et seq., 130, 223; + relation of, with heat, 162, 245 et seq.; + amount required to develop electricity, 131 et seq.; + produces various physical changes, 139; + substantial conception of, 183-184. + See _Energy_. + + World, the, what it consists of, 208. + + World-particles, 203. + + Wronsky, 172. + + Wundt, on causality and the axioms of physics, 157-159; 359 footnote. + + + Xenophon, 49, footnote. + + + Young, Thomas, on energy, 173. + + + Zelter, 35. + + Zeuner, 171. + + Zoölogy, comparison in, 239. + + + + +THE WORKS OF ERNST MACH. + +THE SCIENCE OF MECHANICS. + +A CRITICAL AND HISTORICAL EXPOSITION OF ITS PRINCIPLES. + +By DR. ERNST MACH. + +PROFESSOR OF THE HISTORY AND THEORY OF INDUCTIVE SCIENCE IN THE +UNIVERSITY OF VIENNA. + +Translated from the Second German Edition By THOMAS J. McCORMACK. + + +250 Cuts. 534 Pages. Half Morocco, Gilt Top, Marginal Analyses. + +Exhaustive Index. Price $2.50. + + + + +TABLE OF CONTENTS. + + +STATICS. + + The Lever. + + The Inclined Plane. + + The Composition of Forces. + + Virtual Velocities. + + Statics in Their Application to Fluids. + + Statics in Their Application to Gases. + + +DYNAMICS. + + Galileo's Achievements. + + Achievements of Huygens. + + Achievements of Newton. + + Principle of Reaction. + + Criticism of the Principle of Reaction and of the Concept of Mass. + + Newton's Views of Time, Space, and Motion. + + Critique of the Newtonian Enunciations. + + Retrospect of the Development of Dynamics. + + +THE EXTENSION OF THE PRINCIPLES OF MECHANICS. + + Scope of the Newtonian Principles. + + Formulæ and Units of Mechanics. + + Conservation of Momentum, Conservation of the Centre of Gravity, + and Conservation of Areas. + + Laws of Impact. + + D'Alembert's Principle. + + Principle of _Vis Viva_. + + Principle of Least Constraint. + + Principle of Least Action. + + Hamilton's Principle. + + Hydrostatic and Hydrodynamic Questions. + + +FORMAL DEVELOPMENT OF MECHANICS. + + The Isoperimetrical Problems. + + Theological, Animistic, and Mystical Points of View in Mechanics. + + Analytical Mechanics. + + The Economy of Science. + + +THE RELATION OF MECHANICS TO OTHER DEPARTMENTS OF KNOWLEDGE. + + Relations of Mechanics to Physics. + + Relations of Mechanics to Physiology. + + + + +PRESS NOTICES. + + +"The appearance of a translation into English of this remarkable book +should serve to revivify in this country [England] the somewhat +stagnating treatment of its subject, and should call up the thoughts +which puzzle us when we think of them, and that is not sufficiently +often.... Professor Mach is a striking instance of the combination of +great mathematical knowledge with experimental skill, as exemplified not +only by the elegant illustrations of mechanical principles which abound +in this treatise, but also from his brilliant experiments on the +photography of bullets.... A careful study of Professor Mach's work, and +a treatment with more experimental illustration, on the lines laid down +in the interesting diagrams of his _Science of Mechanics_, will do much +to revivify theoretical mechanical science, as developed from the +elements by rigorous logical treatment."--Prof. A. G. Greenhill, in +_Nature_, London. + +"Those who are curious to learn how the principles of mechanics have +been evolved, from what source they take their origin, and how far they +can be deemed of positive and permanent value, will find Dr. Mach's able +treatise entrancingly interesting.... The book is a remarkable one in +many respects, while the mixture of history with the latest scientific +principles and absolute mathematical deductions makes it exceedingly +attractive."--_Mechanical World_, Manchester and London, England. + +"Mach's Mechanics is unique. It is not a text-book, but forms a useful +supplement to the ordinary text-book. The latter is usually a skeleton +outline, full of mathematical symbols and other abstractions. Mach's +book has 'muscle and clothing,' and being written from the historical +standpoint, introduces the leading contributors in succession, tells +what they did and how they did it, and often what manner of men they +were. Thus it is that the pages glow, as it were, with a certain +humanism, quite delightful in a scientific book.... The book is +handsomely printed, and deserves a warm reception from all interested in +the progress of science."--_The Physical Review_, New York and London. + +"Mr. T. J. McCormack, by his effective translation, where translation +was no light task, of this masterly treatise upon the earliest and most +fundamental of the sciences, has rendered no slight service to the +English speaking student. The German and English languages are generally +accounted second to none in their value as instruments for the +expression of scientific thought; but the conversion bodily of an +abstruse work from one into the other, so as to preserve all the meaning +and spirit of the original and to set it easily and naturally into its +new form, is a task of the greatest difficulty, and when performed so +well as in the present instance, merits great commendation. Dr. Mach has +created for his own works the severest possible standard of judgment. To +expect no more from the books of such a master than from the elementary +productions of an ordinary teacher in the science would be undue +moderation. Our author has lifted what, to many of us, was at one time a +course of seemingly unprofitable mental gymnastics, encompassed only at +vast expenditure of intellectual effort, into a study possessing a deep +philosophical value and instinct with life and interest. 'No profit +grows where is no pleasure ta'en,' and the emancipated collegian will +turn with pleasure from the narrow methods of the text-book to where the +science is made to illustrate, by a treatment at once broad and deep, +the fundamental connexion between all the physical sciences, taken +together."--_The Mining Journal_, London, England. + +"As a history of mechanics, the work is admirable."--_The Nation_, New +York. + +"An excellent book, admirably illustrated."--_The Literary World_, +London, England. + +"Sets forth the elements of its subject with a lucidity, clearness, and +force unknown in the mathematical text-books ... is admirably fitted to +serve students as an introduction on historical lines to the principles +of mechanical science."--_Canadian Mining and Mechanical Review_, +Ottawa, Can. + +"A masterly book.... To any one who feels that he does not know as much +as he ought to about physics, we can commend it most heartily as a +scholarly and able treatise ... both interesting and profitable."--A. M. +Wellington, in _Engineering News_, New York. + +"The book as a whole is unique, and is a valuable addition to any +library of science or philosophy.... Reproductions of quaint old +portraits and vignettes give piquancy to the pages. The numerous +marginal titles form a complete epitome of the work; and there is that +invaluable adjunct, a good index. Altogether the publishers are to be +congratulated upon producing a technical work that is thoroughly +attractive in its make-up."--Prof. D. W. Hering, in _Science_. + +"There is one other point upon which this volume should be commended, +and that is the perfection of the translation. It is a common fault that +books of the greatest interest and value in the original are oftenest +butchered or made ridiculous by a clumsy translator. The present is a +noteworthy exception."--_Railway Age_. + +"The book is admirably printed and bound.... The presswork is +unexcelled by any technical books that have come to our hands for some +time, and the engravings and figures are all clearly and well +executed."--_Railroad Gazette_. + + + + +TESTIMONIALS OF PROMINENT EDUCATORS. + + +"I am delighted with Professor Mach's _Science of Mechanics_."--_M. E. +Cooley_, Professor of Mechanical Engineering, Ann Arbor, Mich. + +"You have done a good service to science in publishing Mach's _Science +of Mechanics_ in English. I shall take every opportunity to recommend it +to young students as a source of much interesting information and +inspiration."--_M. I. Pupin_, Professor of Mechanics, Columbia College, +New York. + +"Mach's _Science of Mechanics_ is an admirable ... book."--_Prof. E. A. +Fuertes_, Director of the College of Civil Engineering of Cornell +University, Ithaca, N. Y. + +"I congratulate you upon producing the work in such good style and in so +good a translation. I bought a copy of it a year ago, very shortly after +you issued it. The book itself is deserving of the highest admiration; +and you are entitled to the thanks of all English-speaking physicists +for the publication of this translation."--_D. W. Hering_, Professor of +Physics, University of the City of New York, New York. + +"I have read Mach's _Science of Mechanics_ with great pleasure. The book +is exceedingly interesting."--_W. F. Magie_, Professor of Physics, +Princeton University, Princeton, N. J. + +"The _Science of Mechanics_ by Mach, translated by T. J. McCormack, I +regard as a most valuable work, not only for acquainting the student +with the history of the development of Mechanics, but as serving to +present to him most favorably the fundamental ideas of Mechanics and +their rational connexion with the highest mathematical developments. It +is a most profitable book to read along with the study of a text-book of +Mechanics, and I shall take pleasure in recommending its perusal by my +students."--_S. W. Robinson_, Professor of Mechanical Engineering, Ohio +State University, Columbus, Ohio. + +"I am delighted with Mach's 'Mechanics.' I will call the attention to it +of students and instructors who have the Mechanics or Physics to study +or teach."--_J. E. Davies_, University of Wisconsin, Madison, Wis. + +"There can be but one opinion as to the value of Mach's work in this +translation. No instructor in physics should be without a copy of +it."--_Henry Crew_, Professor of Physics in the Northwestern University, +Evanston, Ill. + + + + +POPULAR SCIENTIFIC LECTURES. + +A PORTRAYAL OF THE SPIRIT AND METHODS OF SCIENCE. + +By DR. ERNST MACH. + +PROFESSOR OF THE HISTORY AND THEORY OF INDUCTIVE SCIENCE IN THE +UNIVERSITY OF VIENNA. + +Translated by THOMAS J. McCORMACK. + +_Third Edition, Revised Throughout and Greatly Enlarged._ + + +Cloth, Gilt Top. Exhaustively Indexed. Pages, 415. Cuts, 59. Price, +$1.50. + + + + +TITLES OF THE LECTURES. + + + The Forms of Liquids. + + The Fibres of Corti. + + On the Causes of Harmony. + + On the Velocity of Light. + + Why Has Man Two Eyes? + + On Symmetry. + + On the Fundamental Concepts of Static Electricity. + + On the Principle of the Conservation of Energy. + + On the Economical Nature of Physical Inquiry. + + On the Principle of Comparison in Physics. + + On the Part Played by Accident in Invention and Discovery. + + On Sensations of Orientation. + + On the Relative Educational Value of the Classics and the + Mathematico-Physical Sciences. + + A Contribution to the History of Acoustics. + + Remarks on the Theory of Spatial Vision. + + On Transformation and Adaptation in Scientific Thought. + + +PRESS NOTICES. + +"A most fascinating volume, treating of phenomena in which all are +interested, in a delightful style and with wonderful clearness. For +lightness of touch and yet solid value of information the chapter 'Why +Has Man Two Eyes?' has scarcely a rival in the whole realm of popular +scientific writing."--_The Boston Traveller_. + +"Truly remarkable in the insight they give into the relationship of the +various fields cultivated under the name of Physics.... A vein of humor +is met here and there reminding the reader of Heaviside, never offending +one's taste. These features, together with the lightness of touch with +which Mr. McCormack has rendered them, make the volume one that may be +fairly called rare. The spirit of the author is preserved in such +attractive, really delightful, English that one is assured nothing has +been lost by translation."--Prof. Henry Crew, in _The Astrophysical +Journal_. + +"A very delightful and useful book.... The author treats some of the +most recondite problems of natural science, in so charmingly untechnical +a way, with such a wealth of bright illustration, as makes his meaning +clear to the person of ordinary intelligence and education.... This is a +work that should find a place in every library, and that people should +be encouraged to read."--_Daily Picayune_, New Orleans. + +"In his translation Mr. McCormack has well preserved the frank, simple, +and pleasing style of this famous lecturer on scientific topics. +Professor Mach deals with the live facts, the salient points of science, +and not with its mysticism or dead traditions. He uses the simplest of +illustrations and expresses himself clearly, tersely, and with a +delightful freshness that makes entertaining reading of what in other +hands would be dull and prosy."--_Engineering News_, N. Y. + +"The general reader is led by plain and easy steps along a delightful +way through what would be to him without such a help a complicated maze +of difficulties. Marvels are invented and science is revealed as the +natural foe to mysteries."--_The Chautauquan_. + +"The beautiful quality of the work is not marred by abstruse discussions +which would require a scientist to fathom, but is so simple and so clear +that it brings us into direct contact with the matter treated."--_The +Boston Post_. + +"A masterly exposition of important scientific truths."--_Scotsman_, +Edinburgh. + +"These lectures by Dr. Mach are delightfully simple and frank; there is +no dryness or darkness of technicalities, and science and common life do +not seem separated by a gulf.... The style is admirable, and the whole +volume seems gloriously alive and human."--_Providence Journal_, R. I. + +"The non-scientific reader who desires to learn something of modern +scientific theories, and the reasons for their existence, cannot do +better than carefully study these lectures. The English is excellent +throughout, and reflects great credit on the translator."--_Manufacturer +and Builder_. + +"We like the quiet, considerate intelligence of these +lectures."--_Independent_, New York. + +"Professor Mach's lectures are so pleasantly written and illumined with +such charm of illustration that they have all the interest of lively +fiction."--_New York Com. Advertiser_. + +"The literary and philosophical suggestiveness of the book is very +rich." _Hartford Seminary Record_. + +"All are presented so skilfully that one can imagine that Professor +Mach's hearers departed from his lecture-room with the conviction that +science was a matter for abecedarians. Will please those who find the +fairy tales of science more absorbing than fiction."--_The Pilot_, +Boston. + +"Professor Mach ... is a master in physics.... His book is a good one +and will serve a good purpose, both for instruction and +suggestion."--Prof. A. E. Dolbear, in _The Dial_. + +"The most beautiful ideas are unfolded in the exposition."--_Catholic +World_, New York. + + + + +THE ANALYSIS OF THE SENSATIONS + +By DR. ERNST MACH. + +PROFESSOR OF THE HISTORY AND THEORY OF INDUCTIVE SCIENCE IN THE +UNIVERSITY OF VIENNA. + + +Pages, 208. Illustrations, 37. Indexed. + +(Price, Cloth, $1.25.) + + + + +CONTENTS. + + + Introductory: Antimetaphysical. + + The Chief Points of View for the Investigation of the Senses. + + The Space-Sensations of the Eye. + + Space-Sensation, Continued. + + The Relations of the Sight-Sensations to One Another and to the + Other Psychical Elements. + + The Sensation of Time. + + The Sensation of Sound. + + Influence of the Preceding Investigations on the Mode of Conceiving + Physics. + + +"A wonderfully original little book. Like everything he writes a work of +genius."--_Prof. W. James_ of Harvard. + +"I consider each work of Professor Mach a distinct acquisition to a +library of science."--_Prof. D. W. Hering_, New York University. + +"There is no work known to the writer which, in its general scientific +bearings, is more likely to repay richly thorough study. We are all +interested in nature in one way or another, and our interests can only +be heightened and clarified by Mach's wonderfully original and wholesome +book. It is not saying too much to maintain that every intelligent +person should have a copy of it,--and should study that copy."--_Prof. +J. E. Trevor_, Cornell. + +"Students may here make the acquaintance of some of the open questions +of sensation and at the same time take a lesson in the charm of +scientific modesty that can hardly be excelled."--_Prof. E. C. Sanford_, +Clark University. + +"It exhibits keen observation and acute thought, with many new and +interesting experiments by way of illustration. Moreover, the style is +light and even lively--a rare merit in a German prose work, and still +rarer in a translation of one."--_The Literary World_, London. + + +CHICAGO: The Open Court Publishing Company 324 DEARBORN STREET. + +LONDON: Kegan Paul, Trench, Trübner, & Company. + + + + +CATALOGUE OF PUBLICATIONS + +OF THE + +OPEN COURT PUBLISHING CO. + + +COPE, E. D. + + THE PRIMARY FACTORS OF ORGANIC EVOLUTION. + + 121 cuts. Pp., xvi, 547. Cloth, $2.00, net. + +MÜLLER, F. MAX. + + THREE INTRODUCTORY LECTURES ON THE SCIENCE OF THOUGHT. + + With a correspondence on "Thought Without Words," between F. Max + Müller and Francis Galton, the Duke of Argyll, George J. Romanes + and others. 128 pages. Cloth, 75 cents. Paper, 25 cents. + + THREE LECTURES ON THE SCIENCE OF LANGUAGE. + + The Oxford University Extension Lectures, with a Supplement, "My + Predecessors." 112 pages. 2nd Edition. Cloth, 75 cents. Paper, 25c. + +ROMANES, GEORGE JOHN. + + DARWIN AND AFTER DARWIN. + + An Exposition of the Darwinian Theory and a Discussion of + Post-Darwinian Questions. Three Vols., $4.00. Singly, as follows: + + 1. THE DARWINIAN THEORY. 460 pages. 125 illustrations. Cloth, + $2.00. + + 2. POST-DARWINIAN QUESTIONS. Heredity and Utility. Pp. 338. $1.50. + + 3. POST-DARWINIAN QUESTIONS. Isolation and Physiological Selection. + Pp. 181. $1.00. + + AN EXAMINATION OF WEISMANNISM. + + 236 pages. Cloth, $1.00. Paper, 35c. + + THOUGHTS ON RELIGION. + + Edited by Charles Gore, M. A., Canon of Westminster. Third Edition, + Pages, 184. Cloth, gilt top, $1.25. + +RIBOT, TH. + + THE PSYCHOLOGY OF ATTENTION. + + THE DISEASES OF PERSONALITY. + + THE DISEASES OF THE WILL. + + Authorised translations. Cloth, 75 cents each. Paper, 25 cents. + _Full set, cloth, $1.75, net._ + +MACH, ERNST. + + THE SCIENCE OF MECHANICS. + + A CRITICAL AND HISTORICAL EXPOSITION OF ITS PRINCIPLES. Translated + by T. J. MCCORMACK. 250 cuts. 534 pages. 1/2 m., gilt top. $2.50. + + POPULAR SCIENTIFIC LECTURES. + + Third Edition. 415 pages. 59 cuts. Cloth, gilt top. Net, $1.50. + + THE ANALYSIS OF THE SENSATIONS. + + Pp. 208. 37 cuts. Cloth, $1.25, net. + +GOODWIN, REV. T. A. + + LOVERS THREE THOUSAND YEARS AGO. + + As Indicated by the Song of Solomon. Pp. 41. Boards, 50c. + +HOLYOAKE, G. J. + + ENGLISH SECULARISM. A CONFESSION OF BELIEF. + + Pp. 146. Cloth, 50c., net. + +CORNILL, CARL HEINRICH. + + THE PROPHETS OF ISRAEL. + + Popular Sketches from Old Testament History. Pp., 200. Cloth, + $1.00. + + THE RISE OF THE PEOPLE OF ISRAEL. + + See _Epitomes of Three Sciences_, below. + +BINET, ALFRED. + + THE PSYCHIC LIFE OF MICRO-ORGANISMS. + + Authorised translation. 135 pages. Cloth, 75 cents; Paper, 25 + cents. + + ON DOUBLE CONSCIOUSNESS. + + Studies in Experimental Psychology. 93 pages. Paper, 15 cents. + +WAGNER, RICHARD + + A PILGRIMAGE TO BEETHOVEN. + + A Novelette. Frontispiece, portrait of Beethoven. Pp. 40. Boards, + 50c. + +WEISMANN, AUGUST. + + GERMINAL SELECTION. AS A SOURCE OF DEFINITE VARIATION. + + Pp. 73. Paper, 25c. + +NOIRÉ, LUDWIG. + + ON THE ORIGIN OF LANGUAGE. Pp. 57. Paper, 15c. + +FREYTAG, GUSTAV. + + THE LOST MANUSCRIPT. A Novel. + + 2 vols. 953 pages. Extra cloth, $4.00. One vol., cl., $1.00; paper, + 75c. + + MARTIN LUTHER. + + Illustrated. Pp. 130. Cloth, $1.00. Paper, 25c. + +HERING, EWALD. + + ON MEMORY, and THE SPECIFIC ENERGIES OF THE NERVOUS SYSTEM. Pp. 50. + Paper, 15c. + +TRUMBULL, M. M. + + THE FREE TRADE STRUGGLE IN ENGLAND. + + Second Edition. 296 pages. Cloth, 75 cents; paper, 25 cents. + + WHEELBARROW: ARTICLES AND DISCUSSIONS ON THE LABOR QUESTION. + + With portrait of the author. 303 pages. Cloth, $1.00; paper, 35 + cents. + + EARL GREY ON RECIPROCITY AND CIVIL SERVICE REFORM. + + With Comments by Gen. M. M. Trumbull. Price, 10 cents. + +GOETHE AND SCHILLER'S XENIONS. + + Selected and translated by Paul Carus. 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ROMANES. 50 cents. + + 26. _The Philosophy of Ancient India._ By PROF. RICHARD GARBE. 25c. + + 27. _Martin Luther._ By GUSTAV FREYTAG. 25c. + + 28. _English Secularism._ By GEORGE JACOB HOLYOAKE. 25c. + + 29. _On Orthogenesis._ By TH. EIMER. 25c. + + 30. _Chinese Philosophy._ By PAUL CARUS. 25c. + + 31. _The Lost Manuscript._ By GUSTAV FREYTAG. 60c. + + * * * * * + +THE OPEN COURT PUBLISHING CO. + +324 DEARBORN STREET, CHICAGO, ILL. + +LONDON: Kegan Paul, Trench, Trübner & Co + + + + +THE OPEN COURT + +A MONTHLY MAGAZINE + +Devoted to the Science of Religion, the Religion of Science, and the +Extension of the Religious Parliament Idea. + + +_THE OPEN COURT_ does not understand by religion any creed or dogmatic +belief, but man's world-conception in so far as it regulates his +conduct. + +The old dogmatic conception of religion is based upon the science of +past ages; to base religion upon the maturest and truest thought of the +present time is the object of _The Open Court_. 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