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HISTORICAL VIEW OF THE

tory view of the subject; for, in the beginning of hisseventh book, he expressly says (in anticipation ofobjections which might be brought against the apparentlooseness of his reasoning),—• “ There is no necessity tosuppose the continuous quantities made up of theseindivisible parts, but only that they will observe thesame ratios as those parts do.” Nevertheless, themethod of indivisibles exposed him to many attacks :but the essential principle was ingeniously explainedand defended by Pascal.

Cavalieri pursued his method into many valuable solu-tions and results, especially as regards the areas of sphericaltriangles, and deduced some important theorems; thoughso great was the impulse now beginning to be given tothe mathematical sciences, that many geometers in othercountries were pursuing the same track. These, inseveral instances, laid claim to the same inventions;which, probably, they had really arrived at independentlyand simultaneously.

Galileo had, before this time, invented the cycloid, orcurve traced out by a point in the circumference of awheel as it rolls along a straight track. In 1639,Galileo mentions to his friend Toricelli that he hadthought of it forty years before that time. He madevarious efforts to determine its area, but wanted thatgeometrical skill which was necessary for the discovery.It also baffled the attempts of Cavalieri; but shortlyafterwards, solutions were given both by Toricelli, andby Roberval, a French mathematician of great originalityand invention; each claiming it as his own discovery.The area is three times that of the generating circle, orrolling wheel. A controversy arose on the question ofpriority ; but it is most probable that each arrived at theresult independently.

Roberval improved upon the method of quadratureproposed by Cavalieri; aresearch in which he was followedby Dr. Wallis. Fermat applied analogous methods forfinding the maxima and minima of the ordinates of acurve ; as also their tangents : and investigations of the