the surface of the globe would be inexplicable, even con- CONCLUDING REMARKS. 273 sidering the interior heat, if the siirrounding space had not a determinate temperature differing but little from that which we should find at the poles, if we could pre- cisely estimate it. It is remarkable that, of the two new thermological causes discovered by Fourier, one may be directly observed at the equator and the other at the poles; whilst, for all the intermediate points, our observa- tion must be guided and interpreted by mathematical analysis.
New as this difficult inquiry is, our progress in it depends only on the perfecting of the observations which Fourier's theory has marked out for us. Wlieu the data of the problem thus become better known, this theory will enable us to lay hold of some certain evidences of the ancient thermological state of our globe, as well as of its future modifications. We have already learned one fact of high importance; that the periodical state of the earth's surface has become essentially fixed, and cannot undergo any but imperceptible changes by the continuous cooling of the interior mass through future ages. This rapid sketch will suffice to show what a sudden scientific con- sistency has been given, by the labours of one man of genius, to this fundamental portion of natural history, which, before Fourier's time, was made up of vague and arbitrary opinions, mingled with incomplete and inco- herent observations, out of which no exact general view could possibly arise.
274 CHAPTEE IV.
ACOUSTICS.
' I ""HIS science liad to pass, like all the rest, J- through the theological and metaphy- sical stages; but it assumed its positive character about the same time with Barology, and as completely, though our knowledge of it is, as yet, very scaiity, in comparison with what we have learned of gravity. The exact in- formation which was obtained in the middle of the seven- teenth century about the elementary mechanical properties of the atmosphere, opened up a clear conception of the production and transmission of sonorous vibrations. The analysis of the phenomena of sound shows us that the doctrine of vibrations offers the exact expression of an in- contestable reality. Besides its philosophical interest, and the direct importance of the j^henomena of Acoustics, this department of Physics aj^peals to special attention in two principal relations, arising from its use in perfecting our fundamental ideas regarding inorganic bodies, and Man himself.
Relation to -^J studying sonorous vibrations, we obtain tlie study of some insight into the interior mechanical con- inorganic stitution of natural bodies, manifested by the bodies. modifications undergone by the vibratory motions of their molecules. Acoustics affords the best, if not the only means for this inquiry; and the small present amount of our acquisitions seems to me no reason why we should not obtain abundant results when the study of acoustics is more advanced. It has already revealed to us some delicate properties of natural bodies which could not have been perceived in any other way. For instance, the capacity to contract habits, — a faculty which seemed to be- long exclusively to living beings (I mean the power of con- tracting fixed dispositions, according to a prolonged series SCIENTIFIC RELATIONS OF ACOUSTICS. 275 of uniform impressions), — is clearly shown to exist, in a greater or smaller degree, iu inorganic apparatus. By vibratory motions, also, two mechanical structures, placed apart, act remarkably upon each other, as in the case of two clocks placed upon the same pedestal.
On the other hand, acoustics forms a basis to physiology for the analysis of the two Ph\'siolo"v elementary functions which are most impor- tant to the establishment of social relations, — hearing and the utterance of sound. Putting aside, in this place, all the nervous phenomena of the case, it is clear that the in- quiry rests on a knowledge of the general laws of acoustics, which regulate the mode of vibration of all auditory apjm- ratus. It is remarkably so with regard to the production of the voice, — a phenomenon of the same character with that of the action of any other sonorous instrument, except for its extreme complication, through the organic variations which affect the vocal system. Yet, it is not to physicists that the study of these two great phenomena belongs. The anatomists and physiologists ought not to surrender it to them, but to derive from physics all the ideas necessary for conducting the research themselves: for physicists are not prepai'ed with the anatomical data of the problem, nor yet to supply a sound physiological interpretation of the results obtained. Science has indeed suffered from the prejudices which have grown out of the introduction into physics of superficial theories of hearing and phonation, from physical inquirers having intruded upon the province of the phy- siologists.
After Barology, there is no science which admits of the application of mathematical Matlieruatics doctrines and methods so well as Acoustics. In the most general view, the phenomena of sound evidently belong to the theory of very minute oscillations of any system of molecules round a situation of stable equilibrium; for, in order to the sound being produced, there must be an abrupt perturbation in the molecular equilibrium; and this transient derangement must be fol- lowed T)y a quick return to the jDrimitive state. Once pro- duced, in the body directly shaken, the vibrations may be transmitted at considerable intervals, by means of an elastic 276 POSITIVE PHILOSOPHY.
medium, by exciting a gradual succession of expansions and contractions wliicli are in evident analogy with the waves formed ou the surface of a liquid, and have giveu occasion to the term sonorous vndulations. In the air, in particular, so elastic as it is, the vibration must propagate itself, not only in the direction of the primitive concussion, but in all directions, in the same degree. The transmitted vibra- tions, we must observe, are always necessarily isochronal with the primitive vibrations, though their amplitude may be widely different.
It is clear from the outset that the science of acoustics becomes, almost from its origin, subject to the laws of rational Mechanics. Since the time of Newton, who was the first to attemj^t to determine the rate of propagation of sound in the air, acoustics has always been more or less mixed up with the labovir of geometers to develope abstract Mechanics. It was from simple considerations of acoustics that Daniel Bernouilli derived the general principle relat- ing to the necessary and sej^arate co-existence, or indepen- dency, of small and various oscillations occasioned at the same time in any system, by distinct concussions. The phenomena of sound afford the best realization of that law, without which it would be impossible to explain the com- monest phenomenon of acoustics, — the simultaneous exis- tence of numerous and distinct sounds, such as we are every moment hearing.
Though the connection of acoiistics with rational Me- chanics is almost as direct and complete as that of Barology, this mathematical character is far less manageable in the one case than the other. The most important questions in barology are immediately connected with the clearest and most primitive mechanical theories; whereas the mathe- matical study of sonorous vibrations depends on that diffi- cult and delicate dynamical theory, — the theory of the per- turbations of equilibrium, and the differential equations which it furnishes relative to the highest and most imper- fect part of the integral calculus. Vibratory motion of one dimension is the only one, even in regard to solids, whose mathematical theory is complete. Of such motion of three dimensions we are, as yet, wholly ignorant.
To form any idea of the difficulties of the case, we must SCIENTIFIC RELATIONS OF ACOUSTICS. 277 rein ember that vibratory motions must occasion certain physical modifications of another nature in the molecular constitution of bodies; and that these changes, though affecting the vibratory result, are too minute and transient to be appreciable. The only attempt that has been made to analyse such a complication is in the case of the thermo- logical effects which result from the vibratory motion. La- place used this case to explain the difference between the velocity of sound in the air as determined by experiment and that prescribed by the dynamic formula, which indi- cated a variation of about one-sixth. This difference is accounted for by the heat disengaged by the comj^ression of the atmospheric strata, which must make their elas- ticity vary in a greater j^roportion than their density, thereby accelerating the propagation of the vibratory motion. It is true, a great gap is left here; since, as it is impossible to measure this disengagement of heat, we must assign to it conjecturally the value which compensates for the difference or the two velocities. But we learn from this procedure of Laplace the necessity of combining thermological considerations with the dynamical theory of vibratory motions. The modification is less marked in the case of liquids; and still less in that of solids; but we are too far behind with our comparative expei'iments to be able to judge whether the modification is or is not too incon- siderable for notice.
Notwithstanding the eminent difiiculties of the mathe- matical theory of sonorous vibrations, we owe to it such progress as has yet been made in acoustics. The forma- tion of the differential equations proper to the jjlienomena is, independent of their integration, a very important accp;i- sition, on account of the approximations which mathe- matical analysis allows between questions, otherwise hetero- geneous, which lead to similar equations. This fundamental property, whose value we have so often to recognize, applies remarkably in the present case; and especially since the creation of mathematical thermology, whose principal equa- tions are strongly analogous to those of vibratory motion. — This means of investigation is all the more valuable on account of the difficulties in the way of direct inquiry into the phenomena of sound. We may decide upon the neces- 278 posrrrvK piiilosophv.
sity of the ii1,in()sj)liei'ic- iiKMlimn for the traiisinissioii of soiioroua vibrations; and wc may conceive of the j)ossil)ility of deterniinin*!- by experiment tlie duration of the propa<fa- tion, in tlie air, and then t]irou<>h other media; but the genera,! laws of the vibrations of sonorous bodies escape inuntHliate obs(n-vation. Wo should know almost nothing of ilic wholes case it" ilic malhcmatical theory did not come in to connect the diJTcrcnt ]>h('nomcna of sound, enabling us to sul)stitute for direct observation an equivalent ex- amination of more favourable cases subjected to the same law. For instance, when the analysis of the problem of vibraiiug chords has shown us that, other things being e(pial, the number of oscillations is in inverse })roportion to the length of the chord, we see that the most rapid vibra- tions of a very short chord may be counted, since the law t'nal)les us to direct our attention to very slow vibrations. The sa.mo substitution is at our command in many cases in which it is less direi-t. Still, it is to be regretted that the ])rocess ol' ('xp<'riHientaii()n has not been further improved.
Acoustics consists of Ihree parts. We Divisions. might perhaps say four, including the timbre (ring or tone) arising from the j)articular mode of vibration of each restnumt body. This (piality is so real thati w(> constantly spi>ak- of it, both in daily life and in natiu'al history: but it would 1k' wandering out of the de])a.rtment of general physics to inquire what constitutes tlu^ ring i)r tone peculiar to different bodies, such as stones, wood, metals, organized tissues, etc., whose properties lie within the scope of concrete -[^hysics. But, if Ave regard this (piality as capable of moditicalion. by changes of eir- ciiius(anc(>s, then we bring it into the domain of acoustics, and r(>cogni/e its ]U"oper position, tluuigh we know nothing elsi^ a,bout it. 'I'lial part of the science juvsents a mere void.
'J^he thre«' parts referrinl to are, first, the mode of propa- galion of sounds: next, their degree of intensity; and thirdly, their nuisical tone. Of these departments, the sci'ond is that of which our lau)wledge is most imperfect.
DIVlSIO^■s: Till': i'kopa(;ation of sound. 279 SECTION I.
PROPAGATION OP SOUND.
As to the first, the pi'opagation of souucl, the simplest, most interesting-, and best known ^J ^^Imnd ^"" question is the measurement of the duration, especially Avhen the atmosphere is the medium. Newtou enunciated it very simply, apart from all modifying causes: — that the velocity of sound is that acquired by a gravitating body falling fronx a height eqiuil to half the weight of the atmosplu're, — su]>posing the atmosphere homogeneous. lu an analogous way, we may calcvdate the velocity of sound in the different gases, according to their respective densities and elasticities. By this law the speed of sound in the air must be regarded as independent of atmosi>heric vicissitudes, since, by Mariotte's rules, the density and elasticity of the air alw^ays vary in proportion; and their mutual relation alone influences the velocity iu question. Of Laplace's rectification of Newton's formula, we took notice just now. — One impoi'tant result of this law is the necessary identity of the velocity of different sounds, notwithstanding their varying degi'ees of intensity or of acuteuess. If any inequality existed, we should be able to establish it, from the irregularity which must take place in musical intervals at a certain distance.
All mathematical calculations about the VAYvct of at- velocity 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 direc- tion 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 perturba- tion are unknown. — It is only in regard to the air that the velocity of sound has been effectually studied.
280 POSITIVE PHILOSOPHY.
SECTION II. 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 pheno- mena 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 sub- jects have therefore no right to figure in our programmes of physical science; and to expatiate upon them is to miscon- ceive 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 inform- ing 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 de- crease 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 mathe- matical 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 electro- meter with regard to heat and electricity. We do not even discerii any clear principle by which to conceive of a sono- meter. While the science is in this state, it is mucli too soon to hazard any numerical law of the variations in intensity of sound.
SECTION III.
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 mathe- matical 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 282 POSITIVE PHILOSOPHY.
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 recipro- cally 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 be- comes 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 mutu- ally 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 experimen- talist, Sauveur. He has established that the sound pro- duced 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 simul- taneous, 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^heuo- menon 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 mem- branes must cast much light on the auditory mechanism, in I'egard to the effects of degrees of tension, the hygrome- trical 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 284 POSITIVE PHILOSOPHY.
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 impor- tance 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 know- ledge.
285 CHAPTER V.
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 in- tention, 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 diffe- rent 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 contem- poraries; 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 de- 286 POSITIVE PHILOSOPHY.
velopment 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 mani- fested the true positive spirit.