All mathematical calculations about the VAYvct of atvelocity of sound suppose the atmosphere to iii()s])lieric be motionless, except iu regard to the vibra- aj,ntatu)n. tions under notice, and it is one of the interesting points of the case to ascertain what effect is produced by agitations of the air. The result of experiments for this ])urpose is that, within the limits of the common winds, there is no perceptible effect on the velocity of sound when the direction of the atmos})heric current is perpendicular to that iu which the sound is ])ropagated; and that when the two directions coincide, the velocity is slightly accelerated if the directions agree, and retarded if they are opposed: but the amount and, of course, the law of this slight perturbation are unknown. — It is only in regard to the air that the velocity of sound has been effectually studied.
INTENSITY OF SOUNDS.
We cannot pretend to be any wiser about soumfs^ ^ ^ *^^® intensity of sounds, — which is the second j^art of acoustics. Not only have the phenomena never been analysed or estimated, but the labours of the student have added nothing essential to the results of popular experience about the influences which regulate the intensity of sound; such as the extent of vibrating surfaces, the distance of the resonant body, and so on. These subjects have therefore no right to figure in our programmes of physical science; and to expatiate upon them is to misconceive the character of science, which can never be anything else than a special carrying out of universal reason and experience, and which therefore has for its starting-point the aggregate of the ideas spontaneously acquired by the generality of men in regard to the subjects in question. If we did but attend to this truth, we should simplify our scientific expositions not a little, by stripping them of a m.ultitude of superfluous details which only obscure the additions that science is able to make to the fimdamental mass of human knowledge.
With regard to the intensity of sound, the only scientific inquiry, — a very easy one, — which has been accomplished, relates to the effect of the density of the atmospheric medium on the force of sounds. Here acoustics confirms and explains the common observation on the attenuation of sound in proportion to the rarity of the air, without informing us whether the weakening of the sound is in exact proportion to the rarefaction of the medium, as it is natural to suppose. In my opinion, we know nothing yet of a matter usually understood to be settled, — the mode of decrease of sound, in proportion to the distance of the sounding body; as to which science has added nothing to ordinary experience. It is commonly su2:)posed that the decrease is in an inverse ratio to the square of the distance. This would be a very important law if we could establish it: but it is at present only a conjecture; and I prefer admitting our ignorance to attemjjting to conceal a scien- THEORY OF TONES, 281 tific void, by arbitrarily extending to this case the mathematical formula which belongs to gravitation. A natural prejudice may dispose us to find it again here; but we have no proof of its presence.
It would be strange if we had any notion of the law of the case, when we have not yet any fixed ideas as to the way in which intensity of sound may be estimated; nor even as to the exact meaning of the term. We have no instrument which, can fulfil, with regard to the theory of sound, the same office as the pendulum and the barometer with regard to gravity, or the thermometer and electrometer with regard to heat and electricity. We do not even discerii any clear principle by which to conceive of a sonometer. While the science is in this state, it is mucli too soon to hazard any numerical law of the variations in intensity of sound.
THEORY OF TONES.
The third department of acoustics, — the theory of tones, — is by far the most interest- Tones'^ ^ ing and satisfactory to us in its existing state.
The laws which determine the musical nature of different sounds, that is, their precise degree of acuteness or gravity, marked by the number of vibrations executed in a given time, are accurately known only in the elementary case of a series of linear, even rectilinear, vibrations j^roduced either in a metallic rod, fixed at one end and free at the other, or in a column of air filling a very narrow cylindrical pipe. It is by a combination of experiment and of mathematical theory that this case is understood. It is the most important for the analysis of the commonest inorganic instruments, but not for the study of the mechanism of hearing and utterance. With regard to stretched chords, the established mathematical theory is that the number of vibrations in a given time is in the direct ratio of the square root of the tension of the chord, and in the inverse ratio of the product of its length by its thickness. In straight and homogeneous metallic rods this number is in proportion to the relation of their thickness to the square of their length.
This essential difference between the laws of these two kinds of vibrations is owing to the flexibility of the one sounding body and the rigidity of the other. Observation pointed it out first, and especially with regard to the effect of thickness. These laws relate to ordinary vibrations, which take place transversely; but there are vibrations in a longitudinal direction much more acute, which are not affected by thickness, and in which the difference between strings and rods disappears, the vibrations varying reciprocally to the length; a result which might be anticipated from the inextensibility of the string being equivalent to the rigidity of the rod. A third order of vibrations arises from the twisting of metallic rods, when the direction becomes more or less oblique. It ought to be observed however that recent experiments have shown that these three kinds are not radically distinct, as they caii be mutually transformed by varying the direction in which the sounds are pi'opagated. As for the sounds yielded by a slender column of air, the number of vibrations is in inverse proportion to the length of each column, if the mechanical state of the air is undisturbed: otherwise, it varies as the square root of the relation between the elasticity of the air and its density. Hence it is that changes of temperature which alter this relation in the same direction have here an action absolutely inverse to that which they produce on strings or rods: and thus it is explained by acoustics why it is impossible, as musicians have always found it, to maintain through a changing temperature the harmony at first established between stringed and wind instruments.
Thus far the resonant line has been supposed to vibrate through its whole length. But if, as usually happens, the slightest obstacle to the vibrations occurs at any point, the sound undergoes a radical modification, the law of which could not have been mathematically discovered, but has been clearly apprehended by the great acoustic experimentalist, Sauveur. He has established that the sound produced coincides with that which would be yielded by a similar but shorter chord, equal in length to that of the greatest common measure between the two parts of the whole string.
COMPOSITION OF SOUNDS. 283 The same discovery explains another fundamental law, which we owe to the same philosopher, — that of the series of harmonic sounds which always accompanies the principal sound of every resonant string, their acuteness increasing with the natural series of whole numbers; the truth of which is easily tested by a delicate ear or by experiment. The phenomenon is, if not explained, exactly represented by referring it to the preceding case; though we cannot conceive how the spontaneous division of the string takes place, nor how so many vibratory motions, so nearly simultaneous, agree as they do.
These are the laws of simple sounds. Of the important theory of the composition of of^soimds^^^ sounds we have yet very imperfect notions. It is supjDosed to be indicated by the experiment of the musician Tartini, with regard to resulting sounds. He showed that the precisely simultaneous production of any two sounds, sufficiently marked and intense, occasions a single sound, graver than the other two, according to an invariable and simj^le rule. Interesting as this fact is, it relates to jDhysiology, and not to acoustics. It is a j^heuomenon of the nerves; a sort of normal hallucination of the sense of hearing, analogous to optical illusions.
The vibrations of resonant surfaces have exhibited some curious phenomena to observation, though the mathematical theory of the case is still in its infancy: and M. Savart's observations on the vibratory motions of stretched membranes must cast much light on the auditory mechanism, in I'egard to the effects of degrees of tension, the hygrometrical state, etc.
The study of the most general and most complicated case, that of a mass which vibrates in three dimensions, is scarcely begun, except with some hollow and regular solids. Yet this analysis is above all important, as without it it is clearly impossible to complete the explanation of any real instrument; even of those in which the principal sound is produced by simple lines, the vibrations of which must always be more or less modified by the masses which are connected with them. We may say that the state of acoustics is such that we cannot explain the fundamental properties of any musical instrument whatever. Daniel Bernoiiilli worked at the theory of wind instruments; a subject which may apjDear very simple, but which really requires the highest perfection of the science, even putting aside those extraordinary effects, far ti'anscending scientific analysis, which the art of a musician may obtain from any instrument whatever, and restricting ourselves to influences which may be clearly defined and durably characterized.
Imperfect as is our review of Acoustics, I hope we now understand something of its general chai-acter, the importance of its laws, as far as we know them, the connection of its parts, the development that they have obtained, and the intervals which are left void, to be filled uj) by future knowledge.
OPTICS.
OPTICS.
THE emancipation of natural jMiilosophy from theological and metaj^hysical influence has thus far gone on by means of a succession of partial efforts, each isolated in intention, though all converging to a final end, amidst the entire unconsciousness of those flypothesis on who were brmgmg that result to pass. Such Lio]it.
an incoherence is a valuable evidence of the force of that instinct which universally characterizes modern intelligence; but it is an evil, in as far as it has retarded and embarrassed and even introduced hesitation into the course of our libei'ation. No one having hitherto conceived of the positive philosophy as a whole, and the conditions of positivity not having been analysed, much less prescribed, with the modifications appropriate to different orders of researches, it has followed that the founders of natural philosophy have remained under theological and metaphysical influences in all departments but the one in which they were working, even while their own labours were preparing the overthrow of those influences. It is certain that no thinker has approached Descartes in the clearness and completeness with which he apprehended the true character of modern philosophy; no one exercised so intentionally an action so direct, extensive, and effectual on this transformation, though the action might be transitory; and no one was so independent of the spirit of his contemporaries; yet Descartes, who overthrew the whole ancient philosophy about inorganic phenomena, and the physical phenomena of the organic, was led away by the tendency of his age in a contrary direction, when he strove to put new life into the old theological and metaphysical conceptions of the moral nature of man. If it was so with Descartes, who is one of the chief types of the progress of the general development of humanity, we cannot be surprised that men of a more sj^ecial genius, who have been occupied rather with the development of science than of the human mind, should have followed a metaphysical direction in some matters, while in others not very remote they have manifested the true positive spirit.
These observations are particularly applicable to the philosophical history of Optics, — the department of Physics in which an imperfect positivism maintains the strongest consistence,— chiefly through the mathematical labours which are connected with it. The founders of this science are those who have done most towards laying the foundations of the Positive Philoso^ihy, — Descartes, Huyghens, and Newton; yet each one of them was led away by the old spirit of the absolute to create a chimerical hypothesis on the nature of light. That Newton should have done this is the most remarkable, considering how his doctrine of gravitation had raised the conception of modern philosophy above the point at which Descartes had left it, by establishing the radical inanity of all research into the nature and mode of production of phenomena, and by showing that the great end of scientific effort is the reduction of a system of i3articular facts to one singular and general fact. Newton himself, whose favourite saying was, " O! Physics, beware of Metaphysics! " allowed himself to be seduced by old habits of philosophizing to personify light as a substance distinct from and independent of the luminous body: a conception as metaphysical as it would have been to imagine gravity to have an existence separate from that of the gravitating body.
After what has been said about the philosophical theory of hypotheses, there can be no occasion to expose the fictitious character of the respective doctrines of philosophers on the nature of light. Each one has exposed the untenableness of those of others; and each explorer has confined himself to the evidence which favovxred his own conception. Euler brought fatal objections against the doctrine of emission; yet, at the present day, our instructors conceal the fact that the advocates of the emission doctrine have offered equally fatal objections to that of undulations. To take the most simple instance — Has the fact of i)ropagation in all HYPOTHESES ABOUT LIGHT. 287 directions, characteristic of the vibratory motion, ever been reconciled with the common phenomenon of ni<?ht; that is, of darkness produced by the interposition of an opaque body? Does not the fundamental objection of the Newtonians about this matter hold its ground against the system of Descartes and Huyghens, untouched at this hour as it was above a century ago, after all the subterfuges that have been in use ever since? The case is made clearer by the fact that there are phenomena which the two theories will suit equally well. If the laws of reflection and refraction issue with equal ease from the hypotheses of emission and undulation, it is pretty clear that our business is with the laws, and not with the hypotheses. The mathematical labours expended on the opposite theories will not have been thrown away; they will show, in a veiy short time, that the analytical apparatus is no certain instrument of truth, as it has served the purpose of both hypotheses equally well; as it would, quite as easily, of many others, if the progress of positivity was not excluding, more and more, this vicious method of philosophizing. It is true, the most enlightened advocates of both systems are ready to give up the reality of emission and of imdulation, and hold to them only as a matter of logical convenience, — as a rallying-point of ideas. But if we can pass from the one hypothesis to the other without affecting the science at all, it is clear that such an artifice is needless. We must admit, as we before said, that the combination of scientific ideas would be extremely difficult to minds trained under the prevalent habits of thought, if they were suddenly deprived of such a mode of connection as they here contend for; but it is not the less true that the next generation of scientific thinkers would combine their ideas more easily, and much more perfectly, if they Avere trained to regard directly the relations of phenomena, without being troubled by artifices like these, which only obscure scientific realities.
The history of Optics, regarded as a whole, seems to show that these hypotheses have not sensibly aided the progress of the theory of light, since all our important acquisitions have been entirely independent of them. This is true not only of the laws of reflection and refraction, which were discovered before these hypotheses were created.
but with regard to all the other leading truths of Optics. The hypothesis of emission no moi'e suggested to Newton the notion of the unequal refrangibilitj of the different colours, than that of undulation disclosed to Huyghens the law of double refraction proper to certain substances. Grreat discoverers like these observe a connection of facts, and then create a hypothesis to account for the connection; and then those who come after them conclude that the chimerical conceptions must be inseparable from the immortal discoveries. There is a use, as I have before asserted, in these imaginaiy conceptions, which, in regard to their one function, are indispensable. They serve, transiently, to develope the scientific spirit by carrying us over from the metaphysical to the positive system. They can do this and nothing more, and they accomplished their task some time ago. Their action can henceforth be only injurious, and especially in the case of Optics, as any one may see who will inquire into the state of this science, — particularly since the almost universal adoption of the imdulatory in the place of the emissive system.
Excessive ten- One more error must be noticed before we dency to sys- leave the subject of the unscientific pursuit teniatize. ^f Optics. Some enlightened students imagine that the science acquires a satisfactoi-y rationality by being attached to the fundamental laws of universal mechanics. The emission doctrine, if it means anything, must suppose luminous phenomena to be in analogy with those of ordinary motion; and if the doctrine of undulation means anything, it means that the phenomena of light and sound are alike in their vibratory agitation; and thus the one party likens optics to barology and the other to acoustics. But not only is nothing gained by the svipposition, but if either was the case, there would be no room for imagination or for argument. The connection would be at once apparent to all eyes on the simple view of the phenomena. Such a reference of phenomena to those general laws has never been a matter of question or of conjecture. The only difficulty has been to know those laws well enough to admit of the application. No one doubted the mechanical nature of the principal effects of gravity and sound long before the progress of rational dynamics admitted of their TENDE^'CY TO SYSTEMATIZE. 289 exact analysis. The application powerfully tended, as we have seen, to the perfecting of barology and acoustics; but this was precisely because there was nothing forced or hypothetical about it. It is otherwise with Optics. Notwithstanding all arbitrary suppositions, the phenomena of light will always constitute a category siii goieris, necessarily irreducible to any other: a light will be for ever heterogeneous to a motion or a sound.
Again, physiological considerations discredit this confusion of ideas, by the characteristics which distinguish the sense of sight from those of hearing, and of touch or pressure. If we could abolish such distinctions as these by gratuitous hypotheses, there is no saying where we should stop in our wanderings. A chemical 2:)hilosoi>her might make a type of the senses of taste and smell, aiad proceed to explain colours and tones by likening them to flavours and scents. It does not require a wilder imagination to do this, than to issue as a supposition, now become classical, that sounds and colours are radically alike. It is much better to leave such a pursuit of scientific unity, and to admit that the categories of hetereogeneous phenomena are more numerous than a vicious systematizing tendency would suppose. Natural philosophy would no doubt be more perfect if it were otherwise; but co-ordination is of no use unless it rests on real and fundamental assimilation.
— Physicists must then abstain from fancifidly connecting the phenomena of light and those of motion. All that Optics can admit of mathematical treatment is with relation, not to mechanics, but to geometry, which is eminently applicable to it, from the evidently geometrical character of the principal laws of light. The only case in which we can conceive of a direct application of analysis is in certain optical researches in which observation would immediately furnish some numerical relations: and in no case must the positive study of light give place to a dynamical analysis. These are the two directions in which geometers may aid the progress of Optical science, which they have only too effectually impeded l)y prolonging the influence of antiscientific hypotheses through inappropriate and ill-conceived analyses.
The genius of Fourier released us from the necessity of I. u applying the doctrine of hypotheses, as previously laid down, to the case of thermology: and neither barology nor acoustics required it. As to electrology, there are abundance of chimerical conceptions preponderant in that department: but their absurdities are so obvious, that almost all their advocates acknowledge them. It is in Optics that the plausibility and consistence of such chimeras give them the most importance; and I have therefore chosen that department as the ground on which they should be judged.
We will now pass from these useless hypotheses to the real knowledge that we are in possession of about the.. theory of light. The whole of Optics is OiiYicT"^ ^ naturally divided into four departments, as light, whether homogeneous or coloured, is direct, reflected, refracted, or diifracted. These elementary effects usually co-exist in ordinary phenomena; but they are distinct, and must therefore be sejiarately considered. These four parts comprehend all optical phenomena which are rigorously universal; but we must add, as an indispensable complement, two other sections, relating to double refraction and polarization. These orders of phenomena are pi'oper to certain bodies; but, besides that they are a remarkable modification of fundamental phenomena, they appear in more and more bodies, as the study proceeds, and their conditions refer more to general circumstances of structure than to incidents of substance. For these reasons they ought to be exactly analysed. As for the rest, it is not our business to classify the application of these six departments either to natural historv, as in the niatterr*^ beautiful Newtonian theory of the rainbow, or to the arts, as in the analysis of optical instruments. These applications serve as the best measure of the degree of perfection of the science; but they do not enter into the field of optical philosophy, with which alone we are concerned.
For the same reasons which have led us to visfon^ ^ condemn theories of hearing and utterance, in connection with Physics, we must now refuse to include am.oni^ optical phenomena the theory of vision, which certainly belongs to jihysiology, When phy- THEORIES OF VISION AND COLOUR. 291 sicists undertake the study of it, they bring only one of the special qualifications necessary, being otherwise on a level with the multitude; and, however important their one qualification may be, it cannot fulfil all the conditions. It is in consequence of so many conditions being unfulfilled, that the explanations hitherto offered have been so incomplete, and therefore illusory. There is scarcely a single law of vision which can be regarded as established on a sound basis, even where the simplest and commonest phenomena are in question. The elementary faculty of seeing distinctly at unequal distances remains without any satisfactory explanation, though physicists have attempted to refer it to almost every part of the ocular apparatus in succession. This humbling ignorance is no doubt owing to scientific men, both physiologists and physicists, having left the theory of sensations in the hands of the metaphysicians, who have got nothing out of it but some deceptive ideology: but before this time we should have approached to something like positive solutions, but for the liad organization of scientific labour among us. If, from the time of these questions beginning to assume a positive character, anatomists and physiologists had occupied themselves with a theory of vision grounded on the materials furnished by Optical science, instead of looking to physicists for solutions which they could not furnish, our condition in regard to this important subject would be somewhat less deplorable than it is.
Another study which must be excluded -f, r from Optics, and from all natural philosophy, louToHjoflies is the theory of the colour of bodies. I need not explain that I am not referring to the admirable Newtonian experiments on the decomposition of light, which have supplied a fundamental idea, common to all the departments of Optics. I refer to the attempts made to ascertain, now through the theory of emission, and now through that of undulation, the inexplicable primitive phenomenon of the elementary colour proper to every substance. The so-called explanations, about the supposed faculty of reflecting or transmitting such and such a kind of rays, or of exciting such and such an order of ethereal vibrations, -in virtue of certain supposed arrangements of