SCIENTIFIC LECTURES UCSB LIBRARf 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 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 DY KEGAN PAUL, TRENCH, TRUEBNER & Co., LONDON 1898 COPYRIGHT BY THE OPEN COURT PUBLISHING Co. Pages 1-258 I - g Pages 338-374 f Pages 259-281 in 1896. Pages 282-308 m 1897. Pages 309-337 in 1898. AUTHOR'S PREFACE TO THE FIRST EDITION. T~)OPULAR LECTURES, owing to the knowledge they presup- JL pose, 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 scien- tific 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 pro- cess 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 At- tending 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 Orien- tation," recently delivered and summing up the results of an im- portant 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 foot- notes 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 electro- statics and education will more than realise the hope expressed in the author's Preface. The eighth, ninth, tenth, eleventh, and twelfth lectures are of viii TRANSLA TORS NO TE. a more philosophical character and deal principally with the meth- ods 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. Sig- nificant 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 i 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 be- cause they love the holy ? By such easy questions did the wise Socrates make the market-place of Athens un- safe 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 2 THE FORMS OF LIQUIDS. effaced and new ones introduced. The history of sci- ence is full of examples of this constant change, de- velopment, 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 defi- niteness 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 them- selves 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, THE FORMS OF LIQUIDS. 3 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 ap- plication of the slightest pressure ; a liquid conforms exactly to the shapes of the vessels in which it is con- tained ; 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 weak- est, would possess a character, if it were not too diffi- cult in this world to keep it. So, too, we must sup- pose 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 4 THE FORMS OF LIQUIDS. 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 heav- enly 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.* He employed for his experi- ments oil and a mixture of alcohol and water. By Archimedes's well-known principle, the oil in this mix- ture loses its entire weight. It no longer sinks be- neath 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 * Statique expirimentale et thlorique des liquides, 1873. See also The Sci- ence of Mechanics, p. 384 et seqq.,The Open Court Publishing Co., Chicago, 1893. THE FORMS OF LIQUIDS. 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 as- sumes 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. Still more curious are the phe- nomena 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 im- merse, 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 Fig. i. THE FORMS OF LIQUIDS. can produce all kinds of geometrical figures of oil, for example, a three-sided pyramid, a cylinder (by bring- ing the oil between two wire rings), and so on. In- teresting 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 Fig. 2. 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. 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 THE FORMS OF LIQUIDS. 7 open air. And Plateau, in fact, found that these thin figures, or at least very similar ones, could be pro- duced 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, quad- ratic film in the centre. In the pyramid, a film pro- ceeds from each edge to the centre. These figures are so beautiful that they hardly ad- mit 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 perma- nent form. This is very simply done.* 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 * Compare Mach, Utbtr die Moleculanvirkitng der Fliissiskeiten, Reports of the Vienna Academy, 1862. 8 THE FORMS OF LIQUIDS. 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 ob- tain a small solid cube of water. With a piece of blot- ting paper the superfluous water may be easily removed and the sides of the cube made smooth. Yet another simple method may be devised for ob- serving these figures. A drop of water on a greased glass plate will not run if it is small enough, but will j- be flattened by its weight, which presses it against its support. The smaller the drop the less the flatten- ing. The smaller the drop the nearer it approaches the form of a sphere. On the other hand, a drop sus- pended 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 can- not 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 pro- duced by our various wire models, reveals to us a great multiplicity of form. But great as this multiplicity is, THE FORMS OF LIQUIDS. g the common features of the figures also are easily dis- cernible. 14 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 circum- stances. If, therefore, some extraordinarily intelligent tailor, possessing a knowledge of all the artifices of the higher mathematics, should set himself the task of so cover- ing 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 accom- plish this feat with the greatest saving of material, he io THE FORMS OF LIQUIDS. 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 pri- marily 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 para- graphs 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, be- ing 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 THE FORMS OF LIQUIDS. n 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 repre- sents the quantity of oil. The stretched rubber film, which, clinging to the wire edges, does its utmost to contract, rep- resents the surface of the oil en- deavoring 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 rub- Fi e-3. ber films strike together, assume the form of planes, and meet at four sharp edges in the centre of the pyramid. 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 I2 THE FORMS OF LIQUIDS. 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. 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 para- graphs with the principle of least superficial area may be shown by a yet simpler example. Picture to your- selves four fixed pulleys, a, b, c, d t and two movable THE FORMS OF LIQUIDS. 13 rings/, g (Fig 5); about the pulleys and through the rings imagine a smooth cord passed, fastened at one extremity to a nail ?, 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 por- tion of the string e h as long as possible, and conse- quently 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, ac- cordingly, are similar to those of the liquid figures dis- cussed. 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 j 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 I 4 THE FORMS OF LIQUIDS. 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, can- not stop, until the surface has become as small as un- der 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.* 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 an- alogy 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 in- tention or purpose, is not, in a final analysis, so essen- tially different from the pressure of a stone on its sup- port, 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, f It is the pur- pose 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 pur- * In almost all branches of physics that are well worked out such maximal and minimal problems play an important part. t Compare Mach, VortrSge Uber Psychophysik, Vienna, 1863, page 41 ; Cam- pendiMmderPhysikfiirMediciner, Vienna, 1863, page 234 ; and also The Science of Mechanics, Chicago, 1893, pp. 84 and 464. THE FORMS OF LIQUIDS. 15 poses, 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 par- ties 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.* You see, our miserly mercantile principle is replete with consequences, f The result of sober research, it * Like reflexions are found in Quetelet, Du systtme sociale. tFor 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. 16 THE FORMS OF LIQUIDS, 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* 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 ? * 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 coun- try 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 subse- quent 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 hear- ing 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 func- 18 THE FIBRES OF CORTI. tions of the eye comparative clearness has been at- tained ; whilst, hand in hand with this, ophthalmology has reached a degree of perfection which the preced- ing 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 mys- terious darkness, full of attraction for the investi- gator. Look at this model of the ear. Even at that fami- liar 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 signi- ficance we cannot accurately state, yet for which there must certainly be some reason. 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 forma- tion, 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 Fig. e. innumerable fibres of the nerve of hear- ing 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 THE FIBRES OF CORTI. 19 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. Accord- ing to Kolliker 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 Fi f?- 7- notes. The cochlea, therefore, is a species of piano- forte. What, now, may be the office of this structure, which is found in no other organ of sense? May it 20 THE FIBRES OF CORTI. 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 fol- low the individual voices of a symphony. Indeed, the feat is possible even in a fugue of Bach, where it is cer- tainly 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, with- out 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 audi- tory 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 exe- cute one to-and-fro movement in a second. THE FIBRES OF CORTI. 21 This pendulum, now, can be thrown into violent vibration in two ways ; either by a single heavy im- pulse, 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 pen- dulum reaches out a little further in its swing, and finally acquires a considerable motion.* But this is not the case under all circumstances. It is possible only when the impulses imparted syn- chronise 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, ac- cording 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 This experiment, with its associated reflexions, is due to Galileo. 22 THE FIBRES OF CORTL 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 hun- dred 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 othet fork is pitched to the same note, that is to say, if it THE FIBRES OF CORTL 23 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 vibra- tion 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 j 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 pro- vinces. 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 oc- cur 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 24 THE FIBRES OF CORTI. 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 - j who are themselves able to give tone to their surround- ings, 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 pic- THE FIBRES OF CORTI. c d ef ga b c de f Fig. 8. ture to ourselves, for example (Fig. 8), a series of rods or strings pitched to the notes c defg. .... 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. We need not look about us long for such an instrument. Every piano is an instrument of this kind, with which the experi- ment mentioned may be executed with splendid suc- cess. Two pianos stand here by the side of each other, both tuned alike. We will employ the first for excit- ing the notes, while we will allow the second to re- spond ; 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 eg of that instrument vigorously strike the harmony ce g on the first piano. The har- mony ceg is now also sounded on the second piano. But if we press only on one key g of one piano, while we strike ceg on the other, only g will be sounded on 26 THE FIBRES OF CORTI. 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 com- posed of its musical notes. It will reproduce, for ex- ample, 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 wiz- ardry. 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 con- struct 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. Se- same ! and the bolts fall. Truly, we should have here a veritable puzzle-lock. Still another fragment res- cued 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 ? THE FIBRES OF CORTI. 27 v ov niGra, 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 at- tended 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 dis- course. 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 28 THE FIBRES OF CORTI. clumsily, than with the ear. The ear, thus, also de- composes 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 vibra- tion. 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 thou- sand such fingers, and each one is designed for the touching of a different tone.* 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 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. THE FIBRES OF CORTI. 29 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 observa- tion 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 some- what similar. This theory, too, received its substan- tial confirmation from the subsequent observations of V. Hensen. On the free surface of the bodies of Crusta- cea, connected with the auditory nerves, rows of lit- tle 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 as- cends a new hill he obtains of the whole district a different view. When the inquirer has found the so- lution of one enigma, the solution of a host of others falls into his hands. Surely you have often felt the strange impression ex- 36 THE FIBRES OF CORTI. perienced 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 the- ory 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 cir- cumstances 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 aim- lessly 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 en- tirely 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 THE FIBRES OF CORTI. 31 it downwards, and it vanishes two palms from its pre- vious spot beneath the surface. Quick, mathemati- cian, 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 can- not be described to him who has never felt it. Who- ever 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." 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 sup- plied 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 pro- portions 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 nat- ural numbers ? That is the question we should ask ON THE CAUSES OF HARMONY. 33 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 (GV^JL- cpoovia) of two tones, he says, is the mixture, the blending (xpaffiS') of those two tones ; dissonance (Siatpwvia'}, on the other hand, is the incapacity of the tones to blend (a//z/or), 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 un- wittingly 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 vi- brations, that twice as many vibrations corresponded to the octave as to the fundamental tone, etc. A pas- sionate 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 34 ON THE CAUSES OF HARMONY. 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 computa- tion as a principle of aesthetics ? 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.* 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 fundamen- tal tone and its octave was always strongly marked. Accordingly, the combination of the octave, fifth, third, etc., with the fundamental tone appeared to them "nat- ural." 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 pre- * 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, Mimoires de P Academic ties Sciences, Paris, 1700-1705, and R. Smith, Harmonics, Cambridge, 1749. (See Appendix, p. 346.) ON THE CAUSES OF HARMONY. 35 cisely this naturalness for which he seeks his explana- tions. Rameau's remark dragged along through the whole modern period, but without leading to the full discov- ery of the truth. Marx places it at the head of his theory of composition, but makes no further applica- tion of it. Also Goethe and Zelter in their correspon- dence 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 phys- ics were dumb when asked what were the causes of harmony. Not till quite recently did Helmholtz find the so- lution of the question. But to make this solution clear to you I must first speak of some experimental prin- ciples of physics and psychology. i) In every process of perception, in every obser- vation, 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 char- acter. 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 3 6 ON THE CA USES OF HARMONY. 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 atVienna. These are symbolic representations of water, fire, etc. : hu- man heads composed of aquatic animals and of com- bustibles. At a short distance one sees only the de- tails, at a greater distance only the whole figure. Yet a point can be easily found at which, by a simple vol- untary 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 sur- rounded by dark trees between which the bright heav- ens 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 acciden- tally directed to the bright background, one sees the figure of Napoleon between the trees. This case shows us very distinctly the important part which at- tention plays. The same sensuousi 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 pri- mary tone. The effect of the same harmony is essen- ON THE CAUSES OF HARMONY. 37 tially 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 im- pression as if the fixed tone re- mained unchanged and simply al- tered 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. The piano-player knows the remarkable effects ob- tained when one of the keys of a chord that is struck ivy g i \o i' ^g= 7^~H g^ ^ f^T^T is let loose. Bar i 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. 3 8 ON THE CAUSES OF HARMONY. 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 suc- cession of musical tones. For example, if I strike on the piano the note i, (an- nexed 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. The same is true of every musically available sound. Each yields, with varying degrees of inten- sity, 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 over- tones 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 i 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 ON THE CAUSES OF HARMONY. 39 striking of i, 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 mechani- cal 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 im- portant 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 circum- stances 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 40 ON THE CAUSES OF HARMONY. 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 cover- ing 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 per- fectly 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 ON THE CAUSES OF HARMONY. 41 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, some- times coincide, sometimes interfere, whereby they al- ON THE CAUSES OF HARMONY, ternately 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 unpleas- ant sounds, dissonances ? It will be readily seen that all the unpleasant effects of simultaneous sound-com- binations 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 with- out appreciable beats. Fig. 12. To make this perfectly clear to you I have con- structed the model which you see in Fig. 12. It rep- resents a claviatur. At its top a movable strip of wood aa with the marks i, 2 .... 6 is placed. By setting this strip in any position, for example, in that where the mark i 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 similiar strip, bb, possesses the same properties. Thus, together, the two strips, in any two positions, point out by their ON THE CA USES OF HARMONY. 43 marks all the tones brought into play upon the simulta- neous sounding of the notes indicated by the marks i. 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 sup- plies 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 over- tones 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 disso- nance. If we move the strip further and further along, we shall find that as a general rule the overtones al- ways 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 ex- perimentally 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, i /'falls on 2 a, o.b on 4 a, $b on 6 a. Consonances, therefore, are simultaneous sound-combinations not 44 ON THE CAUSES OF HARMONY. 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 com- mon 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 in- troduce 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 ren- der 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. ON THE CAUSES OF HARMONY. 45 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 ex- actly 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 paint- ers 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 de- termined 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 al- ways 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 46 ON THE CA USES OF HARMONY. 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 1 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, sev- eral thousand years hence, boots, top-hats, hoops, pia- nos, 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 un- able 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 solu- tion, 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 ; ON THE CAUSES OF HARMONY. 47 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 false- hood, 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 consider- ably 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 math- ematical 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 Les- sing'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." THE VELOCITY OF LIGHT. WHEN a criminal judge has a right crafty knave before him, one well versed in the arts of pre- varication, his main object is to wring a confession from the culprit by a few skilful questions. In almost a simi- lar 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 de- pends upon the shrewdness of the inquirer. Not with- out reason, therefore, did Lord Bacon call the ex- perimental method a questioning of nature. The art consists in so putting our questions that they may not remain unanswered without a breach of eti- quette. Look, too, at the countless tools, engines, and in- struments of torture with which man conducts his inquisitions of nature, and which mock the poet's words : THE VELOCITY OF LIGHT. 49 " 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.* 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 concep- tions 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.f 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. * 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 crucke, kroes, krus, etc., a pot or jug (cf. Modorn English crock, cruse, and German Krug). Trans. t Xenophon, Memorabilia iv, 7, puts into the mouth of Socrates these words : OVTE yap Evpera avftpum>i avra tv6pun rival, OVTS xaoi&aticu deoif av f/yEiTo rov (.r/rovvra a EKEIVOI oat/vtaai ova 5 o THE VELOCITY OF LIGHT. Picture, then, as your fancy dictates, the tools oi 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 diffi- culties that led to the invention of such strange appa- ratus. 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.* 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 dis- A B tance from each other, one at Fig. 13. 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 's would be the time which it would take light to travel from A to B and from B back to A. The experiment was not executed, nor could it, in the nature of the case, have been a success. As we * Galilei, Discorsi e dimostrazione matematiche. Leyden, 1638. Dialogo THE VELOCITY OF LIGHT. 51 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 be- tween 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 exten- sive region, whilst the single reflected claps of thunder arrive at the observer's ear very gradually and in ap- preciable succession. During his life, then, the efforts of Galileo to de- termine the velocity of light remained uncrowned with success. But the subsequent history of the measure- ment of the velocity of light is intimately associated with his name, for with the telescope which he con- structed he discovered the four satellites of Jupiter, and these furnished the next occasion for the deter- mination 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 em- ployed. Olaf Romer, (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. Let AB (Fig. 14) be Jupiter's orbit. Let S stand for the sun, E for the earth, /for Jupiter, and T for 52 THE VELOCITY OF LIGHT. Jupiter's first satellite. When the earth is at E^ 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 Romer noted was forty-two hours, twenty- eight minutes, and thirty-five seconds. Now, as the earth passes along in its orbit towards JE 2 , the revolu- tions of the satellite grow apparently longer and longer : 1 9 / Fig. 14. the eclipses take place later and later. The greatest retardation of the eclipse, which occurs when the earth is at E z , amounts to sixteen minutes and twenty-six seconds. As the earth passes back again to E lt the revolutions grow apparently shorter, and they occur in exactly the time that they first did when the earth arrives at E^ . It is to be remarked that Jupiter changes only very slightly its position during one revolution of the earth. Romer guessed at once that these period- ical changes of the time of revolution of Jupiter's satel- THE VELOCITY OF LIGHT. 53 lite 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 sim- ile. 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 re- verse 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.* Picture to yourself a cross, say the sails of a wind-mill (Fig. 15), in uniform Fg- 5- 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. *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. 54 THE VELOCITY OF LIGHT. Now this must also be the case with the rotation (the revolution) of the satellite of Jupiter. The great- est retardation of the eclipse (16^ minutes), due to the passage of the earth from i to 2 , or to its re- moval 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,* or 193,000 miles. A subsequent correction of the diam- eter 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 dis- appears. 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 meth- ods 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 *A kilometre is 0.621 or nearly five-eighths of a statute mile. THE VELOCITY OF LIGHT. 55 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 / 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 Fig. 16. 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, penetrat- ing the glass plate, finally strikes the eye, o, of the ob- server. The eye, o, thus, sees the image of the lumi- nous point / through the glass plate and the hole of the disk in the mirror b. 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, 56 THE VELOCITY OF LIGHT. 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 ve- locities 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 in- creased 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 num- ber of rotations per second, and the distance sb, we can calculate the velocity of light. The result agrees with that obtained by Romer. The experiment is not quite as simple as my ex- position might lead you to believe. Care must be taken that the light shall travel back and forth over THE VELOCITY OF LIGHT. 57 the miles of distance sb and bs tmdispersed. 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 / is the lantern A, while the rotation of the perforated disk per- forms 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 descen- dant 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 unneces- sary, therefore, for physicists to rack their brains fur- ther 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 meas- urement of the velocity of sound which is closely allied to the method of Fizeau. 58 THE VELOCITY OF LIGHT. 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 gen- eral a coincidence of the strokes will now not be heard. The companion strokes of the remote clock-work ar- rive, 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 dis- tance is travelled over by the sound in a tenth of a second. We meet frequently the phenomenon here pre- sented, that a thought which centuries of slow and painful endeavor are necessary to produce, when once developed, fairly thrives. It spreads and runs every- where, 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 THE VELOCITY OF LIGHT. 59 is too slow and clumsy for use. The usual method of studying events too fleet for direct observation con- sists 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 di- rect observation. But it was ascertained by Wheatstone, simply by the expedient of watching an electric spark in a mirror rotating with tremendous known velocity. 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 Flg> l8 ' time impressed upon the eye ; we think we see several staffs, having some such disposition as that represented in Fig. 1 8. If, now, the holes of the disk are equally far apart, and the disk is rotated with uniform velo- city, we see clearly that the staff has moved slowly from a to b, more quickly from b to