This precise and detailed account of the Undulatory Motion of air by which sound is transmitted was first given by Newton. He further attempted to determine the motions of the separate particles, and to point out the force by which each particle affects the next, so as to continue the progress of the undulation once begun. The motions of each particle must be oscillatory; he assumed the oscillations to be governed by the simplest law of oscillation which had come under the notice of mathematicians, (that of small vibrations of a pendulum;) and he proved that in this manner the forces which are called into play by the contraction and expansion of the parts of the elastic fluid are such as the continuance of the motion requires. Newton's proof of the exact law of Oscillatory Motion of the aërial particles was not considered satisfactory by succeeding mathematicians; for it was found that the same result, the development of forces adequate to continue the motion, would follow if any other law of the motion were assumed. Cramer proved this by a sort of _parody_ on Newton's proof, in which, by the {326} alteration of a few phrases in this formula of demonstration, it was made to establish an entirely different conclusion. But the general conception of an Undulation as presented by Newton was, as from its manifest mechanical truth it could not fail to be, accepted by all mathematicians; and in proportion as the methods of calculating the motions of fluids were further improved, the necessary consequences of this conception, in the communication of sound through air, were traced by unexceptionable reasoning. This was especially done by Euler and Lagrange, whose memoirs on such motions of fluids are some of the most admirable examples which exist, of refined mathematical methods applied to the solution of difficult mechanical problems. But the great step in the formation of the theory of sound was undoubtedly that which we have noticed, the introduction of the Conception of an Undulation such as we have attempted to describe it:--a state, condition, or arrangement of the particles of a fluid, which is transferred from one part of space to another by means of small motions of the particles, altogether distinct from the movement of the Undulation itself. This is a conception which is not obvious to common apprehension. It appears paradoxical at first sight to speak of a large _wave_ (as the tide-wave) running up a river at the rate of twenty miles an hour, while the _stream_ of the river is all the while flowing downwards. Yet this is a very common fact. And the conception of such a motion must be fully mastered by all who would reason rightly concerning the mechanical transmission of impressions through a medium.