Throughout the mechanical sciences the Principle of the Superposition of Small Motions is of fundamental importance,17 and it may be thus explained. Suppose that two forces, acting from the points B and C, are simultaneously moving a body A. Let the force acting from B be such that in one second it would move A to p, and similarly let the second force, acting alone, move A to r. The question arises, then, whether their joint action will urge A to q along the diagonal of the parallelogram. May we say that A will move the distance Ap in the direction AB, and Ar in the direction AC, or, what is the same thing, along the parallel line pq? In strictness we cannot say so; for when A has moved towards p, the force from C will no longer act along the line AC, and similarly the motion of A towards r will modify the action of the force from B. This interference of one force with the line of action of the other will evidently be greater the larger is the extent of motion considered; on the other hand, as we reduce the parallelogram Apqr, compared with the distances AB and AC, the less will be the interference of the forces. Accordingly mathematicians avoid all error by considering the motions as infinitely small, so that the interference becomes of a still higher order of infinite smallness, and may be entirely neglected. By the resources of the differential calculus it is possible to calculate the motion of the particle A, as if it went through an infinite number of infinitely small diagonals of parallelograms. The great discoveries of Newton really arose from applying this method of calculation to the movements of the moon round the earth, which, while constantly tending to move onward in a straight line, is also deflected towards the earth by gravity, and moves through an elliptic curve, composed as it were of the infinitely small diagonals of infinitely numerous parallelograms. The mathematician, in his investigation of a curve, always treats it as made up of