Page:A History of Mathematics (1893).djvu/192

 areas, volumes, and centres of gravity. He effected the quadrature of a parabola of any degree $$\scriptstyle{y^m=a^{m-1}x}$$, and also of a parabola $$\scriptstyle{y^m=a^{m-n}x^n}$$. We have already mentioned his quadrature of the cycloid. Roberval is best known for his method of drawing tangents. He was the first to apply motion to the resolution of this important problem. His method is allied to Newton's principle of fluxions. Archimedes conceived his spiral to be generated by a double motion. This idea Roberval extended to all curves. Plane curves, as for instance the conic sections, may be generated by a point acted upon by two forces, and are the resultant of two motions. If at any point of the curve the resultant be resolved into its components, then the diagonal of the parallelogram determined by them is the tangent to the curve at that point. The greatest difficulty connected with this ingenious method consisted in resolving the resultant into components having the proper lengths and directions. Roberval did not always succeed in doing this, yet his new idea was a great step in advance. He broke off from the ancient definition of a tangent as a straight line having only one point in common with a curve,—a definition not valid for curves of higher degrees, nor apt even in curves of the second degree to bring out the properties of tangents and the parts they may be made to play in the generation of the curves. The subject of tangents received special attention also from Fermat, Descartes, and Barrow, and reached its highest development after the invention of the differential calculus. Fermat and Descartes defined tangents as secants whose two points of intersection with the curve coincide; Barrow considered a curve a polygon, and called one of its sides produced a tangent.

A profound scholar in all branches of learning and a mathematician of exceptional powers was Pierre de Fermat (1601–1665). He studied law at Toulouse, and in 1631 was made