INTENSITY OF SOUNDS.

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modifying causes :-that the velocity of sound is that acquired by a gravitating body falling from a height equal to half the weight of the atmosphere, supposing the atmosphere homogeneous. In an analogous way, we may calculate 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 atmospheric vicissitudes, since, by Mariotte's rules, the density and elasticity of the air always vary in proportion; and their mutual relation alone influences the velocity in question. Of Laplace's rectification of Newton's formula, we took notice just now. One important result of this law is the necessary identity of the velocity of different sounds, notwithstanding their varying degrees of intensity or of acuteness. 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 velocity of sound suppose the atmosphere to be motionless, except in regard to Effect of atthe vibrations under notice, and it is one of the inter- mospheric esting points of the case to ascertain what effect is agitation. produced by agitations of the air. The result of experiments for this purpose is that, within the limits of the common winds, there is no perceptible effect on the velocity of sound when the direction of the atmospheric current is perpendicular to that in which the sound is propagated; 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.

SECTION II.

INTENSITY OF SOUNDS.

We cannot pretend to be any wiser about the intensity of sounds,. -which is the second part of acoustics. Not only Intensity of have the phenomena never been analysed or estimated, sounds. 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 multitude of superfluous details which only obscure the additions that science is, able to make to the fundamental 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 supposed 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 attempting to conceal a scientific 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 discern any clear principle by which to conceive of a sonometer. While the science is in this state, it is much 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 Theory of far the most interesting and satisfactory to us in its Tones. 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 produced 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 impor

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tant 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 can be mutually transformed by varying the direction in which the sounds are propagated. 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. 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 Composition the composition of sounds we have yet very imperfect of sounds. notions. It is supposed 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 simple rule. Interesting as this fact is, it relates to physiology, and not to acoustics. It is a phenomenon 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 regard 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 Bernouilli worked at the theory of wind instruments; a subject which may appear very simple, but which really requires the highest perfection of the science, even putting aside those extraordinary effects, far transcending 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 character, 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 up by future knowledge.

CHAPTER V.

OPTICS.

Hypothesis on the nature

of Light.

THE emancipation of natural philosophy from theological and metaphysical 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 who were bringing that result to pass. Such 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 liberation. 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 special 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