POPULAR LECTURES SCIENTIFIC SUBJECTS. TEXT-BOOKS OF PHYSICS. GANOT'S ELEMENTARY TREATISE on PHYSICS, Kxnerimental and Applied. Translated and Edited from GANOT'S EUmrmts de f'hysique, by E. ATKINSON. Ph.D. F.C.S. Fourteenth Edition, Revised and Enlarged. With 9 Coloured Plates and 1,028 Woodcuts. Crown 8vo. 15*. GANOT'S NATURAL PHILOSOPHY for GENERAL READERS and YOUNG PERSONS : a Course of Physics divested of Mathematical Formulae, expressed in the langnase of daily life. Translated and Edited from GANOT'S Court EUmtniaire de Physiquf, by E. ATKINSON, Ph.D. F.C.S. Seventh Edition. With 37 pagps of new matter, 7 Plates, 569 Woodcuts, and an Appendix of Questions. Crown 8vo. Is. 6d. The ELEMENTS of PHYSICS or NATURAL PHILOSO- PHY. By NEIL ARNOTT, M.D. Edited by A. BAIN. LL.D. and A. S. TAYLOR, M.D. F.R.S. Woodcuts. Crown 8vo. I'ts. 6d. The ELEMENTS of LABORATORY WORK: a Course of Natural Science By A. G. EA.RL, M.A. F.C.S. Science Master at Tonbridge School. With 57 Diagrams and numerous Exercises and Questions. Crown 8vo. 4s. 6d. A CLASS-BOOK of PRACTICAL PHYSICS. Molecular Physics and Sound. By F. GUTHRIE, Ph.D. With 91 Diagrams. Fcp. 8vo. Is. 6d. POPULAR LECTURES on SCIENTIFIC SUBJECTS. By Professor HELMHOLTZ. With 68 Woodcuts. 2 vols. Crown 8vo. 3*. Gd. each. The METHODS of GLASS-BLOWING. For the use of Physical and Chemical Students. By W. A. SHBNSTONR, Lecturer on Chemistry in Clifton College, Bristol. With 42 Illustrations. Crown 8vo. 1*. 6d. A FIRST COURSE of PHYSICAL LABORATORY PRAC- TICE. Containing 264 Experiments. By A. M. WORTHIXGTON, M.A. Head Master of the Royal Naval Engineering School, Devonport. With Illustrations. Crown 8vo. 4*. Gd. ELEMENTARY PHYSICS. By MARK R. WRIGHT, Prin- cipal of the Dav Trai- ing College, Newcastle-on-Tyne. With 242 Illustrations. Crown 8vo. 2*. 6d. London : LONGMANS, GREEN, & CO. New York : 15 East 16th Street. POPULAR LECTURES ON SCIENTIFIC SUBJECTS BY HERMANN VON HELMHOLTZ TRANSLATED BY E. ATKINSON, PH.D., F.C.S. FORMERLY PROFESSOR OF EXPERIMENTAL SCIENCE, STAFF COLLKOB FIRST SERIES WITH AS INTRODUCTION by PROFESSOR TYNDALL LONDON LONGMANS, GREEN, AND CO. AND NEW YORK : 15 EAST 16th STREET 1895 All rights reserved a /?otisms of our times have been forced to think of removing restric- tions on industry, and of conceding to the industrious middle classes a due voice in their councils. But political organisation, the administration of justice, and the moral discipline of individual citizens are no less important conditions of the preponderance of civilised nations ; and so surely as a nation remains in- accessible to the influences of civilisation in these respects, so surely is it on the high road to destruction. The several con- ditions of national prosperity act and react on each other; where the administration of justice is uncertain, where the interests of the majority cannot be asserted by legitimate means, the development of the national resources, and of the power depending upon them, is impossible ; nor, again, is it possible to make good soldiers except out of men who have learnt under just laws to educate the sense of honour that characterises an 24 OX THE RELATION OF independent man, certainly not out of those who nave lived the submissive slaves of a capricious tyrant. Accordingly every nation is interested in the progress of know ledge on the simple ground of self-preservation, even were there no higher .wants of an ideal character to be satisfied; and not merely in the development of the physical sciences, and their technical application, but also in the progress of legal, political, and moral sciences, and of the accessory historical and philological studies. No nation which would be independent and influential can afford to be left behind in the race. Nor has this escaped the notice of the cultivated peoples of Europe. Never before was so large a part of the public resources devoted to universities, schools, and scientific institutions. We in Heidelberg have this year occasion to congratulate ourselves on another rich endowment granted by our government and our parliament. I "svas speaking, at the beginning of my address, of the in- creasing division of labour and the improved organisation among scientific workers. In fact, men of science form, as it were, an organised army labouring on behalf of the whole nation, and generally under its direction and at its expense, to augment the stock of such knowledge as may serve to promote industrial enterprise, to increase wealth, to adorn life, to improve political and social relations, and to further the moral development of indivi- dual citizens. After the immediate practical results of their work we forbear to inquire ; that we leave to the uninstructed. "We are convinced that whatever contributes to the knowledge of the forces of nature or the powers of the human mind is worth cherishing, and may, in its own due time, bear practical fruit, very often where we should least have expected it. Who, when Galvani touched the muscles of a frog with different metals, and noticed their contraction, could have dreamt that eighty years afterwards, in virtue of the self-same process, whose earliest manifestations attracted his attention in his anatomical researches, all Europe would be traversed with wires, flashing intelligence from Madrid to St. Petersburg with the speed of lightning? In the hands of Galvani, and at first even in Volta's, electrical currents were phenomena capable of exerting NATURAL SCIENCE TO GENERAL SCIENCE. 25 only the feeblest forces, and could not be detected except by the most delicate apparatus. Had they been neglected, on the ground that the investigation of them promised DO immediate practical result, we should now be ignorant of the most import- ant and most interesting of the links between the various forces of nature. When young Galileo, then a student at Pisa, noticed one day during divine service a chandelier swinging backwards and forwards, and convinced himself, by counting his pulse, that the duration of the oscillations was independent of the arc through which it moved, who could know that this discovery would eventually put it in our power, by means of the pendulum, to attain an accuracy in the measurement of time till then deemed impossible, and would enable the storm-tossed seaman in the most distant oceans to determine in what degree of longi- tude he was sailing ? Whoever, in the pursuit of science, seeks after immediate practical utility, may generally rest assured that he will seek in vain. All that science can achieve is a perfect knowledge and a perfect understanding of the action of natural and moral forces. Each individual student must be content to find his reward in rejoicing over new discoveries, as over new victories of mind over reluctant matter, or in enjoying the aesthetic beauty of a well-ordered field of knowledge, where the connection and the filiation of every detail is clear to the mind, and where all denotes the presence of a ruling intellect ; he must rest satisfied with the consciousness that he too has contributed something to the increasing fund of knowledge on which the dominion of man over all the forces hostile to intelligence reposes. He will, indeed, not always be permitted to expect from his fellow-men appreciation and reward adequate to the value of his work. It is only too true that many a man to whom a monument has been erected after his death would have been delighted to receive during his lifetime a tenth part of the money spent in doing honour to his memory. At the same time, we must acknowledge that the value of scientific discoveries is now far more fully recog- nised than formerly by public opinion, and that in stances of the authors of great advance in science starving in obscurity have 20 ON THE RELATION OF become rarer and rarer. On the contrary, the governments and peoples of Europe have, as a rule, admitted it to be their duty to recompense distinguished achievements in science by appro' priate appointments or special rewards. The sciences have then, in this respect, all one common aim, to establish the supremacy of intelligence over the world: while the moral sciences aim directly at making the resources of intellectual life more abundant and more interesting, and seek to separate the pure gold of truth from alloy, the physical sciences are striving indirectly towards the same goal, inasmuch as they labour to make mankind more and more independent of the material restraints that fetter their activity. Each student works in his own department, he chooses for himself those tasks for which he is best fitted by his abilities and his training. But each one must be convinced that it is only in connection with others that he can further the great work, and that therefore he is bound, not only to investigate, but to do his utmost to make the resu'ts of his investigation completely and easily accessible. If ho does this, he will derive assistance from others, and will in his turn be able to render them his aid. The annals of science abound in evidence how such mutual services have been exchanged, even between departments of science apparently most remote. Historical chronology is essentially based on astronomical calculations of eclipses, accounts of which are pre- Bsrved in ancient histories. Conversely, many of the important data of astronomy — for instance, the invariabi'ity of the length of the day, and the periods of several comets — rest upon ancient historical notices. Of late years, physiologists, especially Briicke, have actually undertaken to draw up a complete system of all the vocables that can be produced by the organs of speech, and to base upon it propositions for an universal alphabet, adapted to all human languages. Thus physiology has entered the service of comparative philology, and has already succeeded in account- ing for many apparently anomalous substitutions, on the ground that they are governed, not as hitherto supposed, by the laws of euphony, but by similarity between the movements of the mouth that produce them. Again, comparative philology gives us NATURAL SCIENCE TO GENERAL SCIENCE. 27 information about the relationships, the separations, and the migrations of tribes in pi-ehistoric times, and of the d-::gree of civilisation which they had reached at the time when they parted. For the names of objects to which they had already learnt to give distinctive appellations reappear as words common to their later languages. So thatthe study of languages actually gives us historical data for periods respecting which no other historical evidence exists.1 Yet again I may notice the help which not only the sculptor, but the archaeologist, concerned with the investigation of ancient statues, derives from anatomy. And if I may be permitted to refer to my own most recent studies, I would mention that it is possible, by reference to physical r.coustics and to the physiological theory of the sensation of hearing, to account for the elementary principles on which our musical system is constructed, a problem essentially within the sphere of aesthetics. In fact, it is a general principle that the physiology of the organs of sense is most intimately connected with psychology, inasmuch as physiology traces in our sensations the results of mental processes which do not fall within the sphere of consciousness, and must therefore have remained inac- cessible to us. I have been able to quote only some of the most striking instances of this interdependence of different sciences, and such as could be explained in a few words. Naturally, too, I have tried to choose them from the most widely separated sciences. But far wider is of course the influence which allied sciences exert upon each other. Of that I need not speak, for each of you knows it from his own experience. In conclusion, I would say, let each of us think of himself, not as a man seeking to gratify his own thirst for knowledge, or to promote his own private advantage, or to shine by his own abilities, but rather as a fellow-labourer in one great com- mon work bearing upon the highest interests of humanity. Then assuredly we shall not fail of our reward in the approval of our own conscience and the esteem of our fallow-citizens. 1 See, for example, Mommseu's Rome, Book I. ch. ii. — TR. 28 ON THE RELATION OF NATURAL SCIENCE. To keep up these relations between all seai-chers after truth and all branches of knowledge, to animate them all to vigorous co- operation towards their common end, is the great office of the Universities. Therefore is it necessary that the four faculties should 'ever go hand in hand, and in this conviction will we strive, so far as in us lies, to press onward to the fulfilment of our in-eat mission, ON GOETHE'S SCIENTIFIC EESEAECHES. A Lecture delivered before the German Society of Kionigsberg, in the Sprinff of 1853. IT could not but be that Goethe, whose comprehensive genius was most strikingly apparent in that sober clearness with which he grasped and reproduced with lifelike freshness the realities of nature and human life in their minutest details, should, by those very qualities of his mind, be drawn towards the study of physical science. And in that department, he was not content with acquiring what others could teach him, but he soon at- tempted, as so original a mind was sure to do, to strike out an in- dependent and a very characteristic line of thought. He directed his energies not only to the descriptive but also to the experi- mental sciences; the chief results being his botanical and osteological treatises on the one hand, and his theory of colour on the other. The first germs of these researches belong for the most part to the last decade of the eighteenth century, though some of them were not completed nor published till later. Since that time science has not only made great progress but has widely extended its range. It has assumed in some respects an entirely new aspect, it has opened out new fields of research and undergone many changes in its theoretical views. I. shall attempt in the following Lecture to sketch the rela- tion of Goethe's researches to the present standpoint of science, and to bring out the guiding idea that is common to them all. 30 oy GOETHE'S sciE?rnFic RESEARCHES. The peculiar character of the descriptive sciences — botany, zoology, anatomy, and the like — is a necessary result of tbe work imposed upon them. They undertake to collect and sift an enormous mass of facts, and, above all, to bring them into a logical order or system. Up to this point their work is only the dry task of a lexicographer ; their system is nothing more than a muniment-room in which the accumulation of papers is so arranged that any one can find what he wants at any moment. The more intellectual part of their work and their real interest only begins when they attempt to feel after the scattered traces of law and order in the disjointed, heterogeneous mass, and out of it to construct for themselves an orderly system, accessible at a glance, in which every detail has its due place, and gains additional interest from its connection with the whole. In such studies, both the organising capacity and theinsighu of our poet found a congenial sphere — the epoch was moreover propitious to him. He found ready to his hand a sufficieiit store of logically arranged materials in botany and comparative anatomy, copious and systematic enough to admit cf a compre- hensive view, and to indicate the way to some happy glimpse of an all-pervading law ; while his contemporaries, if they mado any efforts in this direction, wandered without a compass, or else they were so absorbed in the dry registration of facts, that they scarcely ventured to think of anything beyond. It was reserved for Goethe to introduce two ideas of infinite fruit- fulness. The first was the conception that the differences in the anatomy of different animals are to be looked upon as variations from a common phase or type, induced by differences of habit, locality, cr food. The observation which led him to this fertile conception was by no means a striking one ; it is to be found in a monograph on the intermaxillary bone, written as early as 1786. It was known that in most vertebrate animals (that is, mammalia, birds, amphibia, and fishes) the upper jaw consists of two bones, the upper jaw-bone and the intermaxillary bone. The former always contains in the mammalia the molar and the canine teeth, the latter the incisors. Man, who is dis- ON GOETHE'S SCIENTIFIC RESEARCHES. 31 tinguished from all other animals by the absence of the projecting snout, has, on the contrary, on each side only one bone, the upper jaw-bone, containing all the teeth. This being so, Goethe discovered in the human skull faint traces of the sutures which in animals unite the upper and middle jaw-bones, and concluded from it that man had originally possessed an intermaxillary bone, which had subsequently coalesced with the upper jaw-bone. This obscure fact opened up to him a source of the most intense interest in the field of osteology, generally so much decried as the driest of studies. That details of structure should be the same in man and in animals when the parts continue to perform similar functions had involved nothing extraordinary. In fact, Camper had already attempted, on this principle, to trace similarities of structure even between man and fishes. But the persistence of this similarity, at least in a rudimentary form, even in a case when it evidently does not correspond to any of the icquirements of the complete human structure, and consequently needs to be adapted to them by the coalescence of two parts originally separate, was what struck Goethe's far-seeing eye, and suggested to him a far more comprehensive view than had hitherto been taken. Further studies soon convinced him of the universality of his newly discovered principle, so that in 1795 and 1796 he was able to define more clearly the idea that had struck him in 1786, and to commit it to wi iting in his ' Sketch of a General Intrp- duction to Comparative Anatomy.' He there lays down with the utmost confidence and precision that all differences in the structure of animals must be looked upon as variations of a single primitive type, induced by the coalescence, the alteration, the increase, the diminution, or even the complete removal of single parts of the stru.ture; the very principle, in fact, which has become the leading idea of comparative anatomy in its present stage. Nowhere ha? it been better or more clearly ex- pressed than in Goathe's writings. Subsequent authorities have made but few essential alterations in his theory. The most important of these is, that we no longer undertake to construct a common type for the whole animal kingdom, but are content 32 ON GOETHE'S SCIENTIFIC RESEARCHES. with one for each of Cuvier's great divisions. The industry of Goethe's successors has accumulated a well-sifted stock of facts, infinitely more copious than what he could command, and has followed up successfully into the minutest details what he could only indicate in a general way. The second leading conception which science owes to Goethe enunciated the existence of an analogy between the different parts of one and the same organic being, similar to that which we have just pointed out as subsisting between corresponding parts of different species. In most organisms we see a great repetition of single parts. This is most striking in the veget- able kingdom ; each plant has a great number of similar stem leaves, similar petals, similar stamens, and so on. According to Goethe's own account, the idea first occurred to him while look- ing at a fan-palm at Padua. He was struck by the immense variety of changes of form which the successively developed stem-leaves exhibit, by the way in which the first simple root leaflets are replaced by a series of more and more divided leaves, till we come to the most complicated. He afterwards succeeded in discovering the transformation of stem-leaves into sepals and petals, and of sepals and petals into stamens, nectaries, and ovaries, and thus he was led to the doctrine of the metamorphosis of plants, which he published in 1790. Just as the anterior extremity of vertebrate animals takes different forms, becoming in man and in apes an arm, in other animals a paw with claws, or a forefoot with a hoof, or a fin, or a wing, but always retains the same divisions, the same position, and the same connection with the trunk, so the leaf appears as a cotyledon, stem-leaf, sepal, petal, stamen, nectary, ovary, 34 ON GOETHE'S SCIENTIFIC RESEARCHES. subject of controversy, but the principle has maintained ita ground. Goethe's views, however, on the existence of a common type in the animal kingdom do not seem to have exercised any direct influence on the progress of science. The doctrine of the meta- morphosis of plants was introduced into botany as his distinct and recognised property; but his views on osteology were at first disputed by anatomists, and only subsequently attracted attention when the science had, apparently on independent grounds, found its way to the same discovery. He himself com- plains that his first ideas of a common type had encountered nothing but contradiction and scepticism at the time when he was working them out in his own mind, and that even men of the freshest and most original intellect, like the two Von Humboldts, had listened to them with something like impatience. But it is almost a matter of course that in any natural or physical science, theoretical ideas attract the attention of its cultivators only when they are advanced in connection with the whole of the evidence on which they rest, and thus justify their title to recognition. Be that as it may, Goethe is entitled to the credit of having caught the first glimpse of the guiding ideas to which the sciences of botany and anatomy were tending, and by which their present form is determined. But great as is the respect which Goethe has secured by his achievements in the descriptive natural sciences, the denuncia- tion heaped by all physicists on his researches in their depart- ment, and especially on his ' theory of colour,' is at least as uncom- pi'omising. This is not the place to plunge into the controversy that raged on the subject, and so I shall only attempt to state clearly the points at issue, and to explain what principle was involved, and what is the latent significance of the dispute. To this end it is of some importance to go back to the history of the origin of the theory, and to its simplest form, because at that stage of the controversy the points at issue are obvious, and admit of easy and distinct statement, unincumbered by disputes about the correctness of detached facts and complicated theories. Goethe himself describes very gracefully, in the confession at ON GOETHE'S SCIENTIFIC RESEARCHES. 35 the end of his ' Theory of Colour.' how he came to take up the subject. Finding himself unable to grasp the aesthetic principles involved in effects of colour, he resolved to resume the study of the physical theory, which he had been taught at the university, and to repeat for himself the experiments connected with it. With that view he borrowed a prism of Hofrath Biitter, of Jena, but was prevented by other occupations from carrying out his plan, and kept it by him for a long time unused. The owner of the prism, a very orderly man, after several times asking in vain, sent a messenger with instructions to bring it back directly. Goethe took it out of the case, and thought he would take one more peep through it. To make certain of seeing something, he turned it towards a long white wall, under the impression that as there was plenty of light there he could not fail to see a brilliant example of the resolution of light into different colours; a supposition, by the way, which shows how little Newton's theory of the phenomena was then present to his mind. Of course he was disappointed. On the white wall he saw no colours ; they only appeared where it was bounded by darker objects. Accordingly he made the observation — which, it should be added, is fully accounted for by Newton's theory — that colour can only be seen through a prism where a dark object and a bright one have the same boundary. Struck by this observation, which was quite new to him, and convinced that it was irreconcilable with Newton's theory, he induced the owner of the prism to relent, and devoted himself to the question with the utmost zeal and interest. He prepared sheets of paper with black and white spaces, and studied the phenomenon under every variety of condition, until he thought he had sufficiently proved his rules. He next attempted to explain his supposed discovery to a neighbour, who was a physicist, and was dis- agreeably surprised to be assured by him that the experiments were well known, and fully accounted for in Newton's theory. Every other natural philosopher whom he consulted told him exactly the same, including even the brilliant Lichtenberg, whom he tried for a long time to convert, but in vain. I£e "tudied Newton's writings, and fancied he had found some D2 30 ON GOETHE'S SCIENTIFIC RESEARCHES. fallacies in them which accounted for the error. Unable to con- vince any of his acquaintances, he at last resolved to appear before the har of public opinion, and in 1791 and 1792 published the first and second parts of his 'Contributions to Physical Optics.' In that work he describes the appearances presented by white discs on a black ground, black discs on a white ground, and coloured discs on a black or white ground, when examined through a prism. As to the results of the experiments, there is no dispute whatever between him and the. physicists. He de- scribes the phenomena he saw with great truth to nature; the style is lively, and the arrangement such as to make a conspectus of them easy and inviting ; in short, in thus as in all other cases where facts are to be described, he proves himself a master. At the same time he expresses his conviction that the facts he has adduced are calculated to refute Newton's theory. There are two points especially which he considers fatal to it : first, that the centre of a broad white surface remains white when seen through a prism; and secondly, that even a black streak on a white ground can be entirely decomposed into colours. Newton's theory is based on the hypothesis that there exists light of different kinds, distinguished from one another by the sensation of colour which they produce in the eye. Thus there is red, orange, yellow, green, blue, and violet light, and light of all intermediate colours. Different kinds of light, or differently coloured lights, produce, when mixed, derived colours, which to a certain extent resemble the original colours from which they are derived ; to a certain extent form new tints. White is a mixture of all the before-named coloiirs in certain definite pro- portions. But the primitive colours can always be reproduced by analysis from derived colours, or from white, while themselves incapable of analysis or change. The cause of the colours of transparent and opaque bodies is, that when white light falls upon them they destroy some of its constituents and send to the eye other constituents, but no longer mixed in the right proportions to produce white light. Thus a piece of red glass looks red because it transmits only red rays. Consequently aD ON GOETHE'S SCIENTIFIC RESEARCHES. 37 colour is derived solely from a change in the proportions in which light is mixed, and is, therefore, a property of light, not of the coloured bodies, which only furnish an occasion for its manifestation. A prism refracts transmitted light; that is to say, deflects it BO that it makes a certain angle with its original direction; the rays of simple light of different colours have, according to Newton, different refrangibilities, and therefore, after refraction in the prism, pursue different courses and separate from each other. Accordingly a luminous point of infinitely small dimen- sions appears, when seen through the pi-ism, to be first displaced, and, secondly, extended into a coloured line, the so-called pris^ matic spectrum, Avhich shows what are called the primary colours in the order above-named. If, however, you look at a broader luminous surface, the spectra of the points near the middle are superposed, as may be seen from a simple geometrical investigation, in such proportions as to give white light, except at the edges, where certain of the colours are free. This white surface appears displaced, as the luminous point did; but in- stead of being coloured throughout, it has on one side a margin of blue and violet, on the other a margin of red and yellow. A black patch between two bright surfaces may be entirely covered by their coloured edges; and when these spectra meet in the middle, the red of the one and the violet of the other combine to form purple. Thus the colours into which, at first sight, it seems as if the black were analysed are in reality due, not to the black strip, but to the white on each side of it. It is evident that at the first moment Goethe did not recol- lect Newton's theory well enough to be able to find out the physical explanation of the facts I have just glanced at. It was afterwards laid before him again and again, and that in a thoroughly intelligible form, for he speaks about it several times in terms that show he understood it quite correctly. But he is still so dissatisfied with it that he persists in his assertion that the facts just cited are of a nature to convince any one who observes them of the absolute incorrectness of Newton's theory. Neither here nor in his later controversial writings does he ever 425182 38 oy GOETEE'S SCIENTIFIC RESEARCHES. clearly state in what he conceives the insufficiency of the ex- planation to consist. He merely repeats again and again that it is quite absurd. And yet I cannot see how any one, whatever his views about colour, can deny that the theory is perfectly consistent with itself; and that if the hypothesis from which it starts be granted, it explains the observed facts completely and even simply. Newton himself mentions these spurious spectra in several passages of his optical works, without going into any special elucidation of the point, considering, of course, that the explanation follows at once from his hypothesis. And he seems to have had good reason to think so ; for Goethe no sooner began to call the attention of his scientific friends to the pheno- mena than all with one accord, as he himself tells us, met his difficulties with this explanation from Newton's principles, which, though not actually in his writings, instantly suggested itself to every one who knew them. A reader who tries to realise attentively and thoroughly every step in this part of the controversy is ant to experience at this point an uncomfortable, almost a painful, feeling to see a man of extraordinary abilities persistently declaring that there is an obvious absurdity lurking in a few inferences appai'ently quite clear and simple. He searches and searches, and at last unable, with all his efforts, to find any such absurdity, or even the ap- pearance of it, he gets into a state of mind in which his own ideas are, so to speak, crystallisad. But it is just this obvious, flat contradiction that makes Goethe's point of view in 1792 so interesting and so important. At this point he has not as yet developed any theory of his own; there is nothing under dis- cussion but a few easily grasped facts, as to the correctness of which both parties are agreed, and yet both hold distinctly opposite views; neither of them even understands what his opponent is driving at. On the one side are a number of phy- sicists, who, by a long series of the ablest investigations, the most elaborate calculations, and the most ingenious inventions, have Drought optics to such perfection that it, and it alone, nmong the physical sciences, was beginning almost to rival astronomy in accuracy. Some of them have made the pheno- ON GOETHE'S SCIENTIFIC RESEARCHES. 39 ir.ena the subject of direct investigation ; all of them, thanks to the accuracy with which it is possible to calculate beforehand the result of every variety in the construction and combination of instruments, have had the opportunity of putting the infer- ences deduced from Newton's views to the test of experiment, and all, without exception, agree in accepting them. On the other aide is a man whose remarkable mental endowments, and AT hose singular talent for seeing through whatever obscures reality, we have had occasion to recognise, not only in poetry, but also in the descriptive parts of the natural sciences ; and this man assures us with the utmost zeal that the physicists are wrong : he is so convinced of the correctness of his own view, that he cannot explain the contradiction except by assuming narrowness or malice on their part, and finally declares that he cannot help looking upon his own achievement in the theory of colour as far more valuable than anything he has accomplished in poetry.1 So flat a contradiction leads us to suspect that there must be behind some deeper antagonism of principle, some difference of organisation between his mind and theirs, to prevent them from understanding each other. I will try to indicate in the following pages what I conceive to be the grounds of this anta- gonism. Goethe, though he exercised his powers in many spheres of intellectual activity, is nevertheless, par excellence, a poet. Now in poetry, as in every other art, the essential thing is to make the material of the art, be it words, or music, or colour, the direct vehicle of an idea. In a perfect work of art, the idea must be present and dominate the whole, almost unknown to the poet himself, not as the result of a long intellectual process, but as inspired by a direct intuition of the inner eye, or by an outburst of excited feeling. An idea thus embodied in a work of art, and dre?sei in the garb of reality, does indeed make a vivid impression by appeal- ing directly to the senses, but loses, of course, that universality and that intelligibility which it would have had if presented in 1 See Eckermann's Conversation* 40 ON GOETHE'S SCIENTIFIC RESEARCHES. the form of an abstract notion. The poet, feeling how the charm of his works is involved in an intellectual process of this type, seeks to apply it to other materials. Instead of trying to arrange the phenomena of nature under definite conceptions, independent of intuition, he sits down to contemplate them as he would a work of art, complete in itself, and certain to yield up its central idea, sooner or later, to a sufficiently susceptible student. Accordingly, when he sees the skull on the Lido, which suggests to him the vertebral theory of the cranium, he remarks that it serves to revive his old belief, already confirmed by experience, that Nature has no secrets from the attentive observer. So again in his first conversation with Schiller on the 'Metamorphosis of Plants.' To Schiller, as a follower of Kant, the idea is the goal, ever to be sought, but ever unattain- able, and therefore never to be exhibited as realised in a phe- nomenon. Goethe, on the other hand, as a genuine poet, conceives that he finds in the phenomenon the direct expression of the idea. He himself tells us that nothing brought out more sharply the separation between himself and Schiller. This, too, is the secret of his affinity with the natural philosophy of Schelling and Hegel, which likewise proceeds from the assumption that Nature shows us by direct intuition the several steps by which a conception is developed. Hence too the ardour with which Hegel and his school defended Goethe's scientific views. Moreover, this view of Nature accounts for the war which Goethe continued to wage against complicated experi- mental researches. Just as a genuine work of art cannot bear retouching by a strange hand, so he would have us believe Nature resists the interference of the experimenter who torturea her and disturbs her ; and, in revenge, misleads the impertinent kill-joy by a distorted image of herself. Accordingly, in his attack upon Newton he often sneers at spectra, tortured through a number of narrow slits and glasses, and commends the experiments that can be made in the open air under a bright sun, not merely as particularly easy and parti- cularly enchanting, but also as particularly convincing ! The poetic turn of mind is very marked even in his morphological ON GOETHE'S SCIENTIFIC RESEARCHES. 41 researches. If we only examine what has really been accom- plished by the help of the ideas which he contributed to science, we shall be struck by the very singular relation which they bear to it. !No one will refuse to be convinced if you lay before him the series of transformations by which a leaf passes into a etainen, an arm into a fin or a wing, a vertebra into the occipital bone. The idea that all the parts of a flower are modified leaves reveals a connecting law which surprises us into acquiescence. But now try and define the leaf-like organ, determine its essential characteristics, so as to include all the forms that we have named. You will find yourself in a difficult}-, for all distinctive marks vanieh, and you have nothing left, except that a leaf in the wider sense of the term is a lateral appendage of the axis of a plant. Try then to express the proposition 'the parts of the flower are modified leaves ' in the language of scientific defi- nition, and it reads, ' the parts of the flower are lateral appen- dnges of the axis.' To see this does not require a Goethe. So igain it has been objected, and not unjustly, to the vertebral theory, that it must extend the notion of a vertebra so much that nothing is left but the bare fact — a vertebra is a bone. We are equally perplexed if we try to express in clear scientific language what we mean by saying that such and such a part of one animal corresponds to such and such a part of another. We ilo not mean that their physiological use is the same, for the name piece which in bird serves as the lower jaw, becomes in mammals a tiny tympana! bone. Nor would the shape, the position, or the connection of the part in question with other parts serve to identify it in all cases. But yet it has been found possible in most cases, by following the intermediate steps, to determine with tolerable certainty which parts correspond to each other. Goethe himself said this very clearly : he says, in speaking of the vertebral thory of the sk\ill, ' Such an aper^i, such an intuition, conception, representation, notion, idea, or whatever you choose to call it, always retains something esoteric and indefinable, struggle as you will against it ; as a general principle, it may be enunciated, but cannot be proved ; in detail it may be exhibited, but can never be put in a cut and 42 ON GOETHE'S SCIENTIFIC RESEARCHES. dry form.' And so, or nearly so, the problem stands to this day. The difference may be brought out still more clearly if wo consider how physiology, which investigates the relations of vital processes as cause and effect, would have to treat this idea of a common type of animal structure. The science might ask, Is it, on the one hand, a correct view, that during the geological periods that have passed over the earth, one species has been developed from another, so that, for example, the breast-fin of the fish has gradually changed into an arm or a wing ? Or again, shall we say that the different species of animals were created equally perfect — that the points of resemblance between them are to be ascribed to the fact that in all vertebrate animals the first steps in development from the egg can only be effected by Nature in one way, almost identical in all cases, and that the later analogies of structure are determined by these features, common to all embryos 1 Probably the majority of observers incline to the latter view,1 for the agreement between the embryos of different vertebrate animals, in the earlier stages, is very striking. Thus even young mammals have occasionally rudimentary gills on the side of the neck, like fishes. It seems, in fact, that what are in the mature animals corresponding parts originate in the same way during the process of development, so that scientific men have lately begun to make use of embryology as a sort of check on the theoretical views of comparative ana- tomy. It is evident that by the application of the physiological views just suggested, the idea of a common type would acquire definiteness and meaning as a distinct scientific conception. Goethe did much : he saw by a happy intuition that there was a law, and he followed up the indications of it writh great shrewdness. But what law it was he did not see ; nor did he even try to find it out. That was not in his line. Moreover, even in the present condition of science, a definite view on the question is impossible ; the very form in which it should be proposed is scarcely yet settled. And therefore we readily admit that in this department Goethe did all that was possible at the time when he lived. I said just now that he treated nature like a work of 1 This was writteu before the appearance of Darwin's Origin of Species. ON GOETHE'S SCIENTIFIC RESEARCHES. 43 art. In his studies on morphology, he reminds one of a spectator at a play, with strong artistic sympathies. His delicate instinct makes him feel how all the details fall into their places, and work harmoniously together, and how some common purpose governs the whole ; and yet while this exquisite order and sym- metry give him intense pleasure he cannot formulate the dominant idea. That is reserved for the scientific critic of the drama, while the artistic spectator feels perhaps, as Goethe did in the presence of natural phenomena, an antipathy to such dissection, fearing, though without reason, that his pleasure may be spoilt by it. Goethe's point of view in the Theory of Colour is much the same. We havo seen that he rebels against the physical theory just at the point where it gives complete and consistent expla- nations from principles once accepted. Evidently it is not the insufficiency of the theory to explain individual cases that is a stumbling-block to him. He takes offence at the assumption made for the sake of explaining the phenomena, which seem to him so absurd, that he looks upon the interpretation as no inter- pretation at all Above all, the idea that white light could be composed of coloured light seems to have been quite inconceiv- able to him ; at the very beginning of the controversy, he rails at the disgusting Newtonian white of the natural philosophers, an expression which seems to show that this was the assumption that most annoyed him. Again, in his later attacks on Newton, which were not published till after his Theory of Colour was completed, he rather strives to show that Newton's facts might be explained on his own hypothesis, and that therefore Newton's hypothesis was not fully proved, than attempts to prove that hypothesis inconsistent with itself or with the facts. Nay, he seems to consider the obviousness of his own hypothesis so overwhelming, that it need only be brought forward to upset Newton's entirely. There are only a few passages where he disputes the experiments described by Newton. Some of them, apparently, he could not succeed in refuting, because the result is not equally easy to observe in all positions of the lenses used, and because he waa 44 ON GOETHE'S SCIENTIFIC RESEARCHES. unacquainted with the geometrical relations by which the most favourable positions of them are determined. In other experi- ments on the separation of simple coloured light by means of prisms alone, Goethe's objections are not quite groundless, inas- much as the isolation of single colours cannot by this means be so effectually carried out, that after refraction through another prism there are no traces of other tints at the edges. A com- plete isolation of light of one colour can only be effected by very carefully arranged apparatus, consisting of combined prisms and lenses, a set of experiments which Goethe postponed to a supplement, and finally left unnoticed. When he complains of the complication of these contrivances, we need only think of the laborious and roundabout methods which chemists must often adopt to obtain certain elementary bodies in a pui-e form ; and we need not be surprised to find that it is impossible to solve a similar problem in the case of light in the open air in a garden, and with a single prism in one's hand.1 Goethe must, consistently with his theory, deny in toto the possibility of isolating pure light of one colour. Whether he ever experi- mented with the proper apparatus to solve the problem remains doubtful, as the supplement in which he promised to detail these experiments was never published. To give some idea of the passionate way in which Goethe, usually so temperate and even courtier-like, attacks Newton, I quote from a few pages of the controversial part of his work the following expressions, which he applies to the propositions of this consummate thinker in physical and astronomical science — ' incredibly impudent' ; ' mere twaddle' ; ' ludicrous ex- planation' ; ' admirable for school-children in a go-cart' j ' but I see nothing will do but lying, and plenty of it.'2 1 I venture to add that 1 am acquainted -with the impossibility of decom- posing or changing simple coloured light, the two principles which form the basis of Newton's theory, not merely by hearsay, but from actual observation, having been under the necessity in one of my own researches of obtaining light of one colour in a state of the greatest possible purity. (See Poggendorff's Annalen, vol. Ixxxvi. p. 501, on Sir D. Brewster's New Analysis of Sunlight.) 2 Something parallel to this extraordinary proceeding of Goethe's may be found in Hobbes's attack on Wallis. — TB. ON GOETHE'S SCIENTIFIC RESEARCHES. 45 Thus, in the theory of colour, Goethe remains faithful to his principle, that Nature must reveal her secrets of her own free will ; that she is but the transparent representation of the ideal world. Accordingly, he demands, as a preliminary to the investigation of physical phenomena, that the observed facts shall be so arranged that one explains the other, and that thus we may attain an insight into their connection without ever having to trust to anything but our senses. This demand of his looks most attractive, but is essentially wrong in principle. For a natural phenomenon is not considered in physical science to be fully explained until you have traced it back to the ultimate forces which are concerned in its production and its maintenance. Now, as we can never become cognisant of forces qua forces, but only of their effects, we are compelled in every explanation of natural phenomena to leave the sphere of sense, and to pass to things which are not objects of sense, and are denned only by abstract conceptions. When we find a stove warm, and then observe that a fire is burning in it, we say, though somewhat inaccurately, that the former sensation is explained by the latter. But in reality this is equivalent to say- ing, we are always accustomed to find heat where fire is bum- ing ; now, a fire is burning in the stove, therefore we shall find heat there. Accordingly we bring our single fact under a more general, better-known fact, rest satisfied with it, and call it falsely an explanation. Evidently, however, the generality of the observation does not necessarily imply an insight into causes; such an insight is only obtained when we can make out what forces are at work in the fire, and how the effects depend upon them. But this step into the region of abstract conceptions, which must necessarily be taken if we wish to penetrate to the causes of phenomena, scares the poet away. In writing a poem he has been accustomed to look, as it were, right into the subject, and to reproduce his intuition without formulating any of the steps that led him to it. And his success is proportionate to the vividness of the intuition. Such is the fashion in which he would have Nature attacked. But the natural philosopher in- sists on transporting him into a world of invisible atoms and 46 ON GOETHE'S SCIENTIFIC RESEARCHES. movements, of attractive and repulsive forces, whose intricate actions and reactions, though governed by strict laws, can scarcely be taken in ai a glance. To him the impiessions of sense are not an irrefragable authority ; he examines what claim they have to be trusted ; he asks whether things which they pronounce alike are really alike, and whether things which they pronounce different are really different ; and often finds that he must answer, no ! The result of such examination, as at present understood, is that the organs of sense do indeed give us informa- tion about external effects produced on them, but convey thorte effects to our consciousness in a totally different form, so that the character of a sensuous perception depends not so much on the properties of the object perceived as on those of the organ by which we I'eceive the information. All that the optic nerve conveys to us, it conveys under the form of a sensation of light, whether it be the rays of the sun, or a blow in the eye, or an electric current passing through it. Again, the auditory nerve translates everything into phenomena of sound, the nerves of the skin into sensations of temperature or touch. The same electric current whose existence is indicated by the optic nerve as a flash of light, or by the organ of taste as an acid flavour, excites in the nerves of the skin the sensation of burning. The same ray of sunshine, which is called light when it falls on the eye, \re call heat when it falls on the skin. But on the other hand, in spite of their different effects upon our organisation, the daylight which enters through our windows, and the heat radiated by an iron stove, do not in reality differ more or less from each other than the red and blue constituents of light. In fact, just as in the Undulatory Theory the red rays are distinguished from the blue rays only by their longer period of vibration, and their smaller refrangibility, so the dark heat rays of the stove have a still longer period and still smaller refrangibility than the red lays of light, but are in every other respect exactly similar to them. All these rays, whether luminous or non-luminous, have heating properties, but only a certain number of them, to which for that reason we give the name of light, can penetrate through the transparent part of the eye to the optic nerve, and excite a ox GOETHE'S SCIENTIFIC RESEARCHES. 47 sensation of light. Perhaps the relation between our senses and the external world may be best enunciated as follows : our sen- sations are for us only symbols of the objects of the external world, and correspond to them only in some such way as written characters or articulate words to the things they denote. They give us, it is true, information respecting the properties of things without us, but no better information than we give a blind man about colour by verbal descriptions. We see that science has arrived at an estimate of the senses very different from that which was present to the poet's mind. A.nd Newton's assertion that white was composed of all the colours of the spectrum was the first gerni of the scientific view which has subsequently been developed. For at that time there were none of those galvanic observations which paved the way to a knowledge of the functions of the nerves in the production of sensations. Natural philosophers asserted that white, to the ^ye the simplest and purest of all our sensations of colour, was compounded of less pure and complex materials. It seems to have flashed upon the poet's mind that all his principles were unsettled by the results of this assertion, and that is why the hypothesis seems to him so unthinkable, so ineffably absurd. We must look upon his theoiy of colour as a forlorn hope, as a desperate attempt to rescue from the attacks of science the belief in the direct truth of our sensations. And this will ac- count for the enthusiasm with which he strives to elaborate and to defend his theory, for the passionate irritability with which he attacks his opponent, for the overweening importance which he attaches to these researches in comparison with his other achieve- ments, and for his inaccessibility to conviction or compromise. If we now turn to Goethe's own theories on the subject, we must, en the grounds above stated, expect to find that he cannot, without being untrue to his own principle, give us anything deserving to be called a scientific explanation of the phenomena, and that is exactly what happens. He starts with the proposition that all colours are darker than white, that they have something of shade in them (on the physical theoiy, white compounded of all colours must necessarily be brighter than 48 ON GOETHE'S SCIENTIFIC RESEARCHES. any of its constituents). The direct mixture of dark and light, of black and white, gives grey ; the colours must therefore owe their existence to some form of the co-operation of light and shade. Goethe imagines he has discovered it in the phenomena presented by slightly opaque or hazy media. Such media usually jj look blue when the light falls on them and they are seen in front of a dark object, but yellow when a bright object is looked at through them. Thus in the daytime the air looks blue against the dark background of the sky, and the sun, when viewed, as is the case at sunset, through a thick and hazy stratum of air, appears yellow. The physical explanation of this phenomenon, which, however, is not exhibited by all such media, as, for instance, by plates of unpolished glass, would lead us too far from the subject. According to Goethe, the semi-opaque medium imparts to the light something corporeal, something of the nature of shade, such as is requisite, he would say, for the formation of colour. This conception alone is enough to perplex any one who looks upon it as a physical explanation. Does he mean to say that material particles mingle with the light and fly away with it ? But this is Goethe's fundamental experiment, this is the typical phenomenon under which he tries to reduce all the phenomena of colour, especially those connected with the prismatic spectritrn. He looks upon all transparent bodies as slightly hazy, and assumes that the prism imparts to the image which it shows to an observer something of its own opacity. Here, again, it is hard to get a definite conception of what is meant. Goethe seems to have thought that a prism never gives perfectly defined images, but only indistinct, half- obliterated ones, for he puts them all in the same class with the double images which are exhibited by parallel plates of glass and by Iceland spar. The images formed by a prism are, it is true, indistinct in compound light, but they are perfectly defined when simple light is used. If you examine, he says, a bright surface on a dark ground through a prism, the image is displaced and blurred by the prism. The anterior edge is pushed forward over the dark background, and consequently 9 hazy light on a dark ground appears blue, while the other edge ON GOETHE'S SCIENTIFIC RESEARCHES. 49 is covered by the image of the black surface which comes after it, and, consequently, being a light image behind a hazy dark colour, appears yellowish-red. But why the anterior edge appears in front of the ground, the posterior edge behind it, and not vice versd, he does not explain. Let us analyse this explanation, and try to grasp clearly the conception of an optical image. When I see a bright object reflected in a mirror, the reason is that the light which proceeds from it is thrown back exactly as if it came from an object of the same kind behind the mirror. The eye of the observer receives the impression accordingly, and therefore he imagines he really sees the object. Every one knows there is nothing real behind the mirror to correspond to the image — that no light can penetrate thither, but that what is called the image is simply a geometrical point, in which the reflected rays, if produced backwards, would intersect. And, accordingly, no one expects the image to produce any real effect behind the mirror. In the same way the prism shows us images of objects which occupy a different position from the objects themselves ; that is to say, the light which an object sends to the prism is refracted by it, so that it appears to come from an object lying to one side, called the image. This image, again; is not real ; it is, as in the case of reflection, the geometrical point in which the refracted rays intersect when produced back- wards. And yet, according to Goethe, this image is to produce real effects by its displacement; the displaced patch of light makes, he says, the dark space behind it appear blue, just as an imperfectly transparent body would, and so again the displaced dark patch makes the bright space behind appear reddish-yellow. That Goethe really treats the image as an actual object in the place it appears to occupy is obvious enough, especially as he is compelled to assume, in the course of his explanation, that the blue and red edges of the bright space are respectively before and behind the dark image which, like it, is displaced by the prism. He does, in fact, remain loyal to the appearance pre- sented to the senses, and treats a geometrical locus as if it were a material object. Again, he does not scruple at one time to make red and blue destroy each other, as, for example, in the I. E 50 ON GOETHE'S SCIENTIFIC RESEARCHES. blue edge of a red surface seen through the prism, and at another to construct out of them a beautiful purple, as when the blue and red edges of two neighbouring white surfaces meet in a black ground. And when he comes to Newton's more complicated experiments, he is driven to still more mar- vellous expedients. As long as you treat his explanations as a pictorial way of representing the physical processes, you may acquiesce in them, and even frequently find them vivid and characteristic, but as physical elucidations of the phenomena they are absolutely irrational. In conclusion, it must be obvious to evpry one that the theoretical part of the Theory of Colour is not natural philo- sophy at all ; at the same time we can, to a certain extent, see that the poet wanted to introduce a totally different method into the study of Nature, and more or less understand how he rame to do so. Poetry is concerned solely with the ' beautiful show which makes it possible to contemplate the ideal ; how that show is produced is a matter of indifference. Even nature is, in the poet's eyes, but the sensible expression of the spiritual. The natural philosopher, on the other haud, tries to discover the levers, the cords, and the pulleys which work behind the scenes, and shift them. Of course the sight of the machinery spoils the beautiful show, and therefore the poet would gladly talk it out of existence, and ignoring cords and pulleys as the chimeras of a pedant's brain, he would have us believe that the scenes shift themselves, or are governed by the idea of the drama. And it is just characteristic of Goethe that he, and he alone among poets, must needs break a lance with natural philosophers. Other poets are either so entirely carried away by the fire of their enthusiasm that they do not trouble themselves about the disturbing influences of the outer world, or else they rejoice in the triumphs of mind over matter, even on that unpropitious battlefield. But Goethe, whom no intensity of subjective feeling could blind to the realities around him, cannot rest satisfied until he has stamped reality itself with the image and super- scription of poetry. This constitutes the peculiar beauty of his poetry, and at the same time fully accounts for his resolute ON GOETHE'S SCIENTIFIC RESEARCHES. 51 hostility to the machinery that every moment threatens to disturb his poetic repose, and for his determination to attack the enemy in his own camp. But we cannot triumph over the machinery of matter by ignoring it; we can triumph over it only by subordinating it to the aims of our moral intelligence. We must familiarise ourselves with its levers and pulleys, fatal though it be to poetic contemplation, in order to be able to govern them after our own will, and therein lies the complete justification of physical investigation, and its vast importance for the advance of human civilisation. From what I have said it will be apparent that Goethe did follow the same line of thought in all his contributions to science, but that the problems he encountered were of diametrically opposite characters. And, perhaps, when it is understood how the self-same characteristic of his intellect, which in one branch of science won for him immortal renown, entailed upon him egregious failure in the other, it will tend to dissipate, in the minds of many worshippers of the great poet, a lingering pre- judice against natural philosophers, whom they suspect of being blinded by narrow professional pride to the loftiest inspirations of genius. 63 ON THE PHYSIOLOGICAL CAUSES OF HAEMONT IN MUSIC, A Lecture delivered in Bonn during the Winter nf 1857. LADIES AND GENTLEMEN, — In the native town of Beethoven, the mightiest among the heroes of harmony, no subject seemed to me better adapted for a popular audience than music itself. Following, therefore, the direction of my researches during the last few years, I will endeavour to explain to you what physics and physiology have to say regarding the most cherished art of the Rhenish land — music and musical relations. Music has hitherto withdrawn itself from scientific treatment more than any other art. Poetry, painting, and sculpture borrow at least the material for their delineations from the world of experience. They portray nature and man. Not only can their material be critically investigated in respect to its correctness and truth to nature, but scientific art-criticism, however much enthusiasts may have disputed its right to do so, has actually succeeded in making some progress in investigating the causes of that aesthetic pleasure which it is the intention of these arts to excite. In music, on the other hand, it seems at first sight as if those were still in the right who reject all ' anatomisation of pleasurable sensations.' This art, borrowing no part of its material from the experience of our senses, not attempting to describe, and 54 ON THE PHYSIOLOGICAL CAUSES OF only exceptionally to imitate the outer world, necessarily with- draws from scientific consideration the chief points of attack which other arts present, and hence seems to be as incompre- hensible and wonderful as it is certainly powerful in its effects. "We are, therefore, obliged, and we purpose, to confine ourselves, in the first place, to a consideration of the material of the art, musical sounds or sensations. It always struck me as a wonder- ful and peculiarly interesting mystery, that in the theory of musical sounds, in the physical and technical foundations of music, which above all other arts seems in its action on the mind as the most immaterial, evanescent, and tender creator of incalculable and indescribable states of consciousness, that here in especial the science of purest and strictest thought — mathe- matics— .should prove pre-eminently fertile. Thorough bass is a kind of applied mathematics. In considering musical intervals, divisions of time, and so forth, numerical fractions, and some- times even logarithms, play a prominent part. Mathematics and music ! the most glaring possible opposite.? of human thought! and yet connected, mutually sustained! It is as if they would demonstrate the hidden consensus of all the actions of our mind, which in the revelations of genius makes us forefeet unconscious utterances of a mysteriously active intelligence. When I considered physical acoustics from a physiological point of view, and thus more closely followed up the part which the ear plays in the perception of musical sounds, much became clear of which the connection had not been previously evident. I will attempt to inspire you with some of the interest which these questions have awakened in niy own mind, by endeavour- ing to exhibit a few of the results of physical and physiological acoustics. The short space of time at my disposal obliges me to confine my attention to one particular point; but I shall select the most important of all, which will best show you the significance and results of scientific investigation in this field ; I mean the foundation of concord. It is an acknowledged fact that the numbers of the vibrations of concordant tones bear to each HARMONY IX MUSIC. 55 other ratios expressible by small whole numbers. But why ? What have the ratios of small whole numbers to do with con cord? This is an old lidd e, propounded by Pythagoras, and hitherto unsolved. Let us see whether the means at the com- mand of modern science will furnish the answer. First of all, what is a musical tone? Common experience teaches us that all sounding bodies are in a state of vibration. This vibration can be seen and felt ; and in the case of loud sounds we feel the trembling of the air even without touching the sounding bodies. Physical science has ascertained that any series of impulses which produce a vibration of the air will, if repeated with sufficient rapidity, generate sound. This sound becomes a musical tone, when such rapid im- pulses recur with perfect regularity and in precisely equal times. Irregular agitation of the air generates only noise. The pitch of a musical tone depends on the number of impulses which take place in a given time ; the more there are in the same time the higher or sharper is the tone. And, as before remarked, there is found to be a close relationship between the well-known harmonious musical intervals and the number of the vibrations of the air. If twice as many vibrations are performed in the same time for one tone as for another, the first is the octave above the second. If the numbers of vibrations in the same time are as 2 to 3, the two tones form a fifth ; if they are as 4 to 5, the two tones form a major third. If you observe that the numbers of the vibrations which generate the tones of the major chord C E G c are in the ratio of the numbers 4:5:6:8, you can deduce from these all other relations of musical tones, by imagining a new major chord, having the same relations of the numbers of vibrations, to be formed upon each of the above-named tones. The num- bers of vibrations within the limits of audible tones which would be obtained by executing the calculation thus indicated are extraordinarily different. Since the octave above any tone has twice as many vibrations as the tone itself, the second octave above will have four times, the third has eight times as many. Our modern pianofortes have seven octaves. Their highest fG ON THE PHYSIOLOGICAL CAUSES OF tones, therefore, perform 128 vibrations in the time that theii lowest tone makes one single vibration. The deepest Cj which our pianos usually possess answers to the sixteen-foot open pipe of the organ — musicians call it the ' contra-C ' — and makes thirty-three vibrations in one second of time. This is very nearly the limit of audibility. You will have observed that these tones have a dull, bad quality of sound on the piano, and that it is difficult to determine their pitch and the accuracy of their tuning. On the organ the contra-C is somewhat more powerful than on the piano, but even here some uncertainty is felt in judging of its pitch. On larger organs there is a whole octave of tones below the contra-C, reaching to the next lower C, with 16^ vibrations in a second. But the ear can scarcely separate these tones from an obscure drone ; and the deeper they are the more plainly can it distinguish the sepa- rate impulses of the air to which they are due. Hence they are used solely in conjunction with the next higher octaves, to strengthen their notes, and produce an impression of greater depth. With the exception of the organ, all musical instruments, however diverse the methods in which their sounds are pro- duced, have their limit of depth at about the same point in the scale as the piano ; not because it would be impossible to produce slower impulses of the air of sufficient power, but because the ear refuses its office, and hears slower impulses separately, without gathering them up into single tones. The often-repeated assertion of the French physicist Savart, that he heard tones of eight vibrations in a second, upon a peculiarly constructed instrument, seems due to an error. Ascending the scale from the contra-C, pianofortes usually have a compass of seven octaves, up to the so-called five-accented c, which has 4,224 vibrations in a second. Among orchestral instruments it is only the piccolo flute which can reach as high, and this will give even one tone higher. The violin usually mounts no higher than the e below, which has 2,640 vibrations — of course we excapt the gymnastics of heaven-scaling virtuosi, who are ever striving to excruciate their audience by some new HARMONY IN MUSIC. 57 impossibility. Such performers may aspire to three whole octaves lying above the five-accented c, and very painful to the ear, for their existence has been established by Despretz, who, by exciting small tuning-forks with a violin bow, obtained and heard the eight-accented c, having 32,770 vibrations in a second. Here the sensation of tone seemed to have reached its upper limit, and the intervals were really undistinguishable in the later octaves. The musical pitch of a tone depends entirely on the number of vibrations of the air in a second, and not at all upon the mode in which they are produced. It is quite indifferent whether they are generated by the vibrating strings of a piano or violin, the vocal chords of the human larynx, the metal tongues of the harmonium, the reeds of the clarionet, oboe, and bassoon, the trembling lips of the trumpeter, or the air cut by a sharp edge in organ pipes and flutes. A tone of the same number of vibrations has always the same pitch, by whichever one of these instruments it is pro- duced. That which distinguishes the note A of a piano, for example, from the equally high A of the violin, flute, clarionet, or trumpet, is called the quality of the tone, and to this we shall have to recur presently. As an interesting example of these assertions, I beg to show you a peculiar physical instrument for producing musical tones, called the siren, Fig. 1, which is especially adapted to establish the properties resulting from the ratios of the numbers of vibrations. In order to produce tones upon this instrument, the portvents g0 and gj are connected by means of flexible tubes with a bellows. The air enters into round brass boxes, a0 and a^ and escapes by the per- forated covers of these boxes at c0 and c,. But the holes for the escape of air are not perfectly free. Immediately before the covers of both boxes there are two other perforated discs, fastened to a per- pendicular axis k, which turns with great readiness. In the figure, only the perforated disc can be seen at c0, and immediately below it is the similarly perforated cover of the box. In the upper box, c]? only the edge of the disc is visible. If then the holes of the disc are precisely opposite to those of the cover, the ah- can escape freely. But if the disc is made to revolve, so that some of its unperforated HARMONY IN MUSIC. 59 portions stand before the holes of the box, the air cannot escape at all. On turning the disc rapidly, the vent-holes of the box are alter- nately opened and closed. During the opening, air escapes ; during the closure, no air can pass. Hence the continuous stream of air from the bellows is converted into a series of discontinuous puffs, which, when they follow one another with sufficient rapidity, gather them- selves together into a tone. Each of the revolving discs of this instrument (_which is more complicated in its construction than any one of the kind hitherto 'uade, and hence admits of a much greater number of combinations of tone) has four concentric circles of holes, the lower set having 8, 10, 12, 18, and the upper set 9, 12, 15, and 16 holes respectively. The series of holes in the covers of the boxes are precisely the same as those in the discs, but under each of them lies a perforated ring, which can be so arranged, by means of the stops i i i i, that the corresponding holes of the cover can either communicate freely with the inside of the box, or are entirely cut off from it; We are thus enabled to use any one of the eight series of holes singly, or com- bined two and two, or three and three together, in any arbitrary manner. The round boxes, h0 h0 and hj hn of which halves only are drawn in the figure, serve by their resonance to soften the harshness of the tone. The holes in the boxes and discs are cut obliquely, so that when the air enters the boxes through one or more of the series of holes, the wind itself drives the discs round with a perpetually increasing velocity. On beginning to blow the instrument, we first hear separate im- pulses of the air, escaping as puffs, as often as the holes of the disc pass in front of those of the box. These puffs of air follow one an- other more and more quickly, as the velocity of the revolving discs increases, just like the puffs of steam of a locomotive on beginning to move with the train. They next produce a whirring and whizzing, which constantly becomes more rapid. At last we hear a dull drone, which, as the velocity further increases, gradually gains in pitch and strength. Suppose that the discs have been brought to a velocity of 33 re- volutions in a second, and that the series with 8 holes has been opened. At each revolution of the disc all these 8 holes will pass before each separate hole of the cover. Hence there will be 8 puffs for each revolution of the disc, or S times 33, that is, 20-i pulls in a 60 ON THE PHYSIOLOGICAL CAUSES OF second. This gives us the once-accented c' of our musical scale [that is, ' middle c,' written on the leger line between the bass and treble staves]. But on opening the series of 16 holes instead, we have twice as many, or 16 times 33, that is, 528 vibrations in a second. We hear exactly the octave above the first