notion of a constant force that (omitting the resistance of the air,) this equality must take place, for the force which will gradually destroy the whole velocity in a certain time in ascending, will, in the same time, generate again the same velocity by the same gradations inverted; and therefore the same space will be passed over in the same time in the descent and in the ascent.

Another difficulty arose from a necessary consequence of the laws of falling bodies thus established, namely, that in acquiring its motion, a body passes through every intermediate degree of velocity, from the smallest conceivable, up to that which it at last acquires. When a body falls from rest, it begins to fall with no velocity; the velocity increases with the time; and in one thousandth part of a second, the body has only acquired one thousandth part of the velocity which it has at the end of one second.

This is certain, and manifest on consideration; yet there was at first much difficulty raised on the subject of this assertion; and disputes took place concerning the velocity with which a body begins to fall. On this subject also Descartes did not form clear notions. He writes to a correspondent, "I have been revising my notes on Galileo, in which I have not said expressly that falling bodies do not pass through every degree of slowness, but I said that this cannot be known without knowing what weight is, which comes to the same thing; as to your example, I grant that it proves that every degree of velocity

VOL. II.

D

is infinitely divisible, but not that a falling body actually passes through all these divisions."

The principles of the motion of falling bodies being thus established by Galileo, the deduction of the principal mathematical consequences was, as is usual, effected with great rapidity, and is to be found in his works and in those of his scholars and successors. The motion of bodies falling freely was, however, in such treatises, generally combined with the motion of bodies falling upon inclined planes; a part of the theory of which we have still to speak.

The notion of accelerating force and its operation, once formed, was naturally applied in other cases than that of bodies falling freely. The different velocities with which heavy and light bodies fall were explained by the different resistance of the air, which diminishes the accelerating force'; and it was boldly asserted, that in a vacuum a lock of wool and a piece of lead would fall equally quick. It was also maintained" that any falling body, however large and heavy, would always have its velocity in some degree diminished by the air in which it falls, and would at last be reduced to a state of uniform motion, as soon as the resistance upwards became equal to the accelerating force downwards. Though the law of progress of a body to this limiting velocity was not made out till the " Principia" of Newton appeared, the views on which Galileo made this

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9

Galileo, iii. 43.

10 iii. 54.

assertion are perfectly sound, and show that he had clearly conceived the nature and operation of accelerating and retarding force.

When uniform accelerating forces had once been mastered, there remained only mathematical difficulties in the treatment of variable forces. A variable force was measured by the limit of the increment of the velocity, compared with that of the time; just as a variable velocity was measured by the limit of the increment of the space compared with that of the time.

With this introduction of the notion of limits, we are, of course, led to the higher geometry, either in its geometrical or its analytical form. The general laws of bodies falling by the action of any variable forces, were given by Newton in the seventh Section of the "Principia." The subject is there, according to Newton's preference of geometrical methods, treated by means of the quadrature of curves, the doctrine of limits being exhibited in a peculiar manner in the first Section of the work, in order to -prepare the way for such applications of it. Leibnitz, the Bernouillis, Euler, and since their time, many other mathematicians, have treated such questions by means of the analytical method of limits, the Differential Calculus. The rectilinear motion of bodies acted upon by variable forces is, of course, a simpler problem than their curvilinear motion, to which we have now to proceed. But it may be remarked that Newton, having established the laws of curvilinear

motion independently, has, in a great part of his seventh Section, deduced the simpler case of the rectilinear motion from the more complex problem, by reasonings of great ingenuity and beauty.

Sect. 3.-Establishment of the Second Law of Motion. Curvilinear Motions.

A SLIGHT degree of distinctness in men's mechanical notions enabled them to perceive, as we have already explained, that a body which traces a curved line must be urged by some force, by which it is constantly made to deviate from the rectilinear path, which it would pursue if acted upon by no force. Thus, when a body is made to describe a circle, as when a stone is whirled round in a sling, we find that the string does exert such a force on the stone, for it is stretched by the effort, and if it be too slender, it may thus be broken. This centrifugal force of bodies moving in circles was noticed even by the ancients. This effect of force to produce curvilinear motion also appears in the paths described by projectiles. We have already seen that though Tartalea did not perceive this correctly, Rivius, about the same time, did.

To see that the transverse force would produce a curve was one step; to determine what the curve is was another step, which involved the discovery of the second law of motion. This step was made by Galileo. In his Dialogues on Motion, he asserts

that a body projected horizontally will retain a uniform motion in the horizontal direction, and will have, compounded with this, a uniformly accelerated motion downwards; that is, the motion of a body falling vertically from rest, and will thus describe the curve called a parabola.

The second law of motion consists of this assertion in a general form ;-that in all cases the motion which the force would produce is compounded with the motion which the body previously has. This was not obvious; for Cardan had maintained", that "if a body is moved by two motions at once, it will come to the place resulting from their composition slower than by either of them." The proof of the truth of the law to Galileo's mind was, so far as we collect from the dialogue itself, the simplicity of the supposition, and his clear perception of the causes which, in some cases, produced an obvious deviation in practice from this theoretical result. For it may be observed, that the curvilinear paths ascribed to military projectiles by Rivius and Tartalea, and by other writers who followed them, as Digges and Norton in our own country, though utterly different from the theoretical form, the parabola, do, in fact, approach nearer the true paths of a cannon or musketball than a parabola would do: and this approximation more especially exists in that which at first sight appears most absurd in the old theory; namely,