Robcrval, in France, opened to himself the same career of discovery as Cavalerius had done in Italy. He began by studying the writings of Archimedes ; and his method of resolving problems, relating to curvilineal areas, differs from that of Cavalerius, only in its terms. His ambition to obtain a triumph over his rivals, induced him to conceal his discoveries, rintil he was anticipated by Cavalerius's book, and thus justly punished for his selfishness. He found a method of determining the tangents of curves, which, however, was inferior to another discovered by Descartes: Roberval's methed, in many cases, only sub stitutes one difficulty for another, but Descartes applies to all algebraic curves, and in every case accomplishes the desired purpose.
The obligations which philosophy and mathematics lie under to Descartes, have been generally acknowledged ; but there is a feature in his character which gives him a higher claim than any other geometer of his time to the gratitude of posterity, and that is, his eagerness to dis seminate the knowledge of science, as well as to extend its boundaries. Instead of hoarding his discoveries, or concealing their source, as others had done by tedious syn thetic demonstrations in the manner of the ancients, he gave them with that clearness and simplicity which ought always to characterize the style of works on science.
Fermat possessed earlier than Descartes a method of tangents. But he only published it after Descartes had made his known, and he joined to it a method de maxillas et minimis. These are more simple than Descartes' me thods, but their author, far from imitating the frankness of this philosopher, only in a manner indicated them, conceal ing, at least in the case of the method de maximis et minimis, his analysis, and the mode of demonstration. By a multi tude of discoveries, several of which, relative to numbers, have exercised the most celebrated analysts of this and the preceding age, Fermat gave proofs of a great genius. He has been considered as equal to Descartes, but the latter philosopher probably contributed more to the propagation of science, by his communicative character, and the simple manner in which he has presented his researches.
Huygens first demonstrated Fermat's two rules. Slus sius afterwards found a simple method of drawing tangents, which at bottom was but the enunciation of t/ie calculus re quired by Fermat's method ; but disengaged from whatever was useless : and lastly, Barrow contrived his characteris tic triangle, which in fact is the same as the triangle that measures the fluxions of the abscissa, the ordinate, and curve ; and thus the method of finding the tangents of al gebraic curves attained its last degree of simplicity.
While these improvements in the theory of tangents were going on, Gregory de S. Vincent, Roberval, and Pas cal, made some progress towards a general solution of the problem of quadratures. This, however, was done by the method of the ancients, and that of indivisibles, and so does not bear directly on the history of the fluxional calculus, if we except the consideration of polygons of scales of Gregory de S. Vincent, or of a series of rectangles inscribed in, and circumscribed about a curve, which may have suggested the application of the fluxional calculus to quadratures.
It is in the arithmetic of infinites of Wallis, that we sec the first application of algebraic calculation to quadratures, and this was founded on the method of indivisibles. Wallis considered series, and sought to express their sum by their first and last terms. Ile thus succeeded in finding the sum when the number of terms was infinite, and the last term may be reckoned as nothing. Considering, then, surfaces as formed of a series of lines, the terms of which follow a certain law, he found the expression for the surface by summing the series. The area of a triangle, for example, was determined by summing an arithmetical progression.
Wallis demonstrated, by his method, the fundamental rule for the quadrature of curves, the ordinate of which is proportional to any power whatever of the abscissa. This enabled him to square any curve, having its ordinate ex presseu by a series of nominals. His method of interpo lation, by which the area of a curve was found, when its equation was in a manner comprehended between the equa tions of two other curves, to which his first method was applicable, deserves particular attention, because it was the germ of Newton's most beautiful discoveries, and is at present the most important part of the theory of series. This method led him to a remarkable expression for the area of a circle. Wallis must be allowed to have con tributed greatly to the progress of analysis, both by his own discoveries, and his having introduced the doctrine of series, which led to all the great discoveries of that period.