It is, I think, only farther necessary to observe, that, when the resistance opposed to the moving body is not uniform but variable, according to any law, it is not simply either the time or the space which is proportional to the velocity or to the square of the velocity, but functions of those quantities. These functions are obtained from the in tegration of certain fiuxionary expressions, in which the measures above described are applied, the resistance being regarded as uniform for an infinitely small portion of the time, or of the space.
Many years after the period I am now treating of, the controversy about the via diva seemed to revive in England, on the occasion of an Essay on Mechanical Force, by the late Mr. Smeaton, an able engineer, who, to great practical, skill, and much ex perience, added no inconsiderable knowledge of the mathematics.' The reality of the via viva, then, under certain conditions, is to be considered as a matter completely established. Another inquiry concerning the nature of this force, which also gave rise to considerable debate, was, whether, in the communication of motion, and in the various changes through which moving bodies pass, the quantity of the via viva remains always the same ? It had been observed, in the col lision of elastic bodies, that the via viva, or the sum made up by multiplying each body into the square of its velocity, and adding the products together, was the same after collision that it was before it, and it was concluded with some precipitation, by those who espoused the Leibnitian theory, that a similar result always took place in the real phenomena of nature. Other instances were cited; and it was observed, that a particular view of this principle which presented itself to Huygens, had en abled him to find the centre of oscillation of a compound pendulum, at a time when the state of mechanical science was scarcely prepared' for so difficult an investigation. The proposition, however, is true only when all the changes are gradual, and rigor ously subjected to the law of continuity. Thus, in the collision of bodies imperfectly elastic (a case which continually occurs in nature), the force which, during the recoil, accelerates the separation of the bodies, does not restore to them the whole velocity they had lost, and the via viva, after the collision, is always less than it was before it. The cases in which the whole amount of the via viva is rigorously preserved, may always be brought under the thirty-ninth proposition of the first book of the Princi pia, where the principle of this theory is placed on its true foundation.
So far as General Principles are concerned, the preceding are the chief mechanical improvements which belong to the period so honourably distinguished by the names of Newton and Leibnitz. The application of these principles to the solution of par ticular problems would afford materials for more ample discussion than suits -the na ture of a historical outline. Such problems as that of finding the centre of oscillation, —the nature of the catenarian curve,—the determination of the line of swiftest descent,— , curve, or that into which an elastic spring forms itself when a force is applied to bend it,—all these were problems of the greatest interest, and were now resolved for the first time ; the science of mechanics being sufficient, by means of the composition of forces, to find out the fluxionary or differential equations which expressed the nature of the gradual changes which in all these cases were produced, and the calculus being now sufficiently powerful to infer the properties of the finite from those of the infinitesi mal quantities.
The doctrine of Hydrostatics was cultivated in England by Cotes. The properties of the atmosphere, or of elastic fluids, were also experimentally investigated; and the barometer, after the ingenuity of Pascal had proved that the mercury stood lower the higher up into the atmosphere the instrument was carried, was at length brought to be a measure of the height of mountains. Mariotte appears to have been the first who proposed this use of it, and who discovered that, while the height from the ground increases in arithmetical, the density of the atmosphere, and the column of mercury, in the barometer, decrease in geometrical progression. Halley, who seems also to have come of himself to the same conclusion, proved its truth by strict geometrical reasoning, and showed, that logarithms are easily applicable on this principle to the problem of finding the height of mountains. This was in the year 1685. Newton two years afterwards gave a demonstration of the same, ex tended to the case when gravity is not constant, but varies as any power of the dis tance from a given centre.