Roberval was one of those mathematicians who, just before the invention of the infinitesimal calculus, occupied their attention with problems which are only soluble, or can be most easily solved, by some method involving limits or **infinitesimals**, and in the solution of which accordingly the calculus is always now employed.

Each of the four co-ordinates, n', x', y are functions of %, rj, x, y; and if it be assumed that the field of view and the aperture be infinitely small, then, n, x, y are of the same order of **infinitesimals**; consequently by expanding ', ii', x', y in ascending powers of E, rt, x, y, series are obtained in which it is only necessary to consider the lowest powers.

A notion related to that of **infinitesimals** is presented in the Greek " method of exhaustion "; the more perfect conception, however, only dates from the 17th century, when it led to the infinitesimal calculus.

Weierstrass, by strictly banishing all **infinitesimals**, has at last shown that we live in an unchanging world, and that the arrow at every moment of its flight is truly at rest."

He studied the properties of the cycloid, and attempted the problem of its quadrature; and in the "**infinitesimals**," which he was one of the first to introduce into geometrical demonstrations, was contained the fruitful germ of the differential calculus.