In France, where geometry began to be cultivated with greatest success, Roberval, Fermat, Pascal, employed the Method of Indivisibles and made new improvements in it. Giles Persone de Roberval (1602–1675), for forty years professor of mathematics at the College of France in Paris, claimed for himself the invention of the Method of Indivisibles. Since his complete works were not published until after his death, it is difficult to settle questions of priority. Montucla and Chasles are of the opinion that he invented the
method independent of and earlier than the Italian geometer, though the work of the latter was published much earlier than Roberval's. Marie finds it difficult to believe that the
Frenchman borrowed nothing whatever from the Italian, for both could not have hit independently upon the word Indivisibles, which is applicable to infinitely small quantities, as conceived by Cavalieri, but not as conceived by Roberval. Roberval and Pascal improved the rational basis of the Method of Indivisibles, by considering an area as made up of an indefinite number of rectangles instead of lines, and a solid as composed of indefinitely small solids instead of surfaces. Roberval applied the method to the finding of
areas, volumes, and centres of gravity. He effected the quadrature of a parabola of any degree , and also of a parabola . 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
councillor for the parliament of Toulouse. His leisure time was mostly devoted to mathematics, which he studied with irresistible passion. Unlike Descartes and Pascal, he led a