Previous to the time of Newton, the doctrine of infinite quantities had been the subject of profound study. The ancients made the first step in this curious inquiry by a rude though ingenious attempt to determine the area of curves. The method of exhaustions which was used for this purpose consisted in finding a given rectilineal area to which the inscribed and circumscribed polygonal figures continually approached by increasing the number of their sides. This area was obviously the area of the curve, and in the case of the parabola it was found by Archimedes to be two-thirds of the area formed by multiplying the ordinate by the abscissa. Although the synthetical demonstration of the results was perfectly conclusive, yet the method itself was limited and imperfect. The celebrated Pappus of Alexandria followed Archimedes in the same inquiries; and in his demonstration of the property of the centre of gravity of a plane figure, by which we may determine the solid formed by its revolution, he has shadowed forth the discoveries of later times. In his curious tract on Stereometry, published in 1615, Kepler made some advances in the doctrine of infinitesimals. Prompted to the task by a dispute with the seller of some casks of wine, he studied the measurement of solids formed by the revolution of a curve round any line whatever. In solving some of the simplest of these problems, he conceived a circle to be formed of an infinite number of triangles having all their vertices in the centre, and their infinitely small bases in the circumference of the circle, and by thus rendering familiar the idea of quantities infinitely great and infinitely small, he gave an impulse to this branch of mathematics. The failure of Kepler, too, in solving some of the more difficult of the problems which he himself proposed roused the attention of geometers, and seems particularly to have attracted the notice of Cavaleri.