During an access of severe toothache which, in 1658, deprived him of sleep, his thoughts fastened on certain problems connected with the cycloid. Fermat, Roberval, and Torricelli had all been occupied with the subject, and made some definite progress in ascertaining its properties. But much still remained to be done, and especially to resolve the problems connected with it in a “general and uniform manner.” “Pascal,” says Bossut, “devised within eight days, and in the midst of cruel sufferings, a method which embraced all the problems—a method founded upon the summation of certain series, of which he had given the elements in his writings accompanying his ‘Traité du Triangle Arithmétique.’ From this discovery there was only a step to that of the Differential and Integral Calculus; and it may be confidently presumed that, if Pascal had proceeded with his mathematical studies, he would have anticipated Leibnitz and Newton in the glory of their great invention.”
Having communicated the result of his geometrical meditation to the Duc de Roannez and some of his other religious friends, they conceived the design of making it subservient to the triumph of religion. Pascal himself was an illustrious example that the highest mathematical genius and the humblest Christian piety might be united; but in order to give éclat to such an example, his friends proposed to propound publicly the questions solved by the great Port Royalist in his moments of suffering, and to offer prizes for the page 49best solutions given of them. This they did in June 1658. A programme was published making the offer of prizes of forty and twenty pistoles, for the best determination of the area and the centre of gravity of any segment of the cycloid, and the dimensions and centres of gravity of solids and half and quarter solids which the same curve would generate by revolving round an abscissa and