sity of the atmospheric medium for the transmission of sonorous vibrations; and we may conceive of the possibility of determining by experiment the duration of the propagation, in the air, and then through other media; but the general laws of the vibrations of sonorous bodies escape immediate observation. We should know almost nothing of the whole case if the mathematical theory did not come in to connect the different phenomena of sound, enabling us to substitute for direct observation an equivalent examination of more favourable cases subjected to the same law. For instance, when the analysis of the problem of vibrating chords has shown us that, other things being equal, the number of oscillations is in inverse proportion to the length of the chord, we see that the most rapid vibrations of a very short chord may be counted, since the law enables us to direct our attention to very slow vibrations. The same substitution is at our command in many cases in which it is less direct. Still, it is to be regretted that the process of experimentation has not been further improved.

Divisions.

Acoustics consists of three parts.

We

might perhaps say four, including the timbre (ring or tone) arising from the particular mode of vibration of each resonant body. This quality is so real that we constantly speak of it, both in daily life and in natural history: but it would be wandering out of the department of general physics to inquire what constitutes the ring or tone peculiar to different bodies, such as stones, wood, metals, organized tissues, etc., whose properties lie within the scope of concrete physics. But, if we regard this quality as capable of modification, by changes of circumstances, then we bring it into the domain of acoustics, and recognize its proper position, though we know nothing else about it. That part of the science presents a mere void.

The three parts referred to are, first, the mode of propagation of sounds: next, their degree of intensity; and thirdly, their musical tone. Of these departments, the second is that of which our knowledge is most imperfect.

DIVISIONS: THE PROPAGATION OF SOUND. 279

SECTION I.

PROPAGATION OF SOUND.

Propagation of sound.

As to the first, the propagation of sound, the simplest, most interesting, and best known question is the measurement of the duration, especially when the atmosphere is the medium. Newton enunciated it very simply, apart from all 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.

Effect of atmospheric agitation.

All mathematical calculations about the velocity of sound suppose the atmosphere to be motionless, except in regard to the vibrations 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 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.

Intensity of sounds.

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 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 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 scien

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 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 far the most interesting and satisfactory to us in its existing state.

Theory of
Tones.

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 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 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.