“Having thus completed his researches respecting elastic and incompressible fluids, Pascal seems to have resumed with a fatal enthusiasm his mathematical studies: but, unfortunately for science, several of the works which he composed have been lost. Others, however, have been preserved, which entitle him to a high rank amongst the greatest mathematicians of the age. Of these, his ‘Traité du Triangle Arithmétique,’ his ‘Tractatus de Numericis Ordinibus,’ and his ‘Problemata de Cycloide,’ are the chief. By means of the _Arithmetical Triangle_, an invention equally ingenious and original, he succeeded in solving a number of theorems which it would have been difficult to demonstrate in any other way, and in finding the coefficients of different terms of a binomial raised to an even and positive power. The same principles enabled him to lay the foundation of the doctrine of probabilities, an important branch of mathematical science, which Huyghens, a few years afterwards, improved, and which the Marquis la Place and M. Poisson have so greatly extended. These treatises, with the exception of that on the Cycloid, were composed and printed in the year 1654, but were not published till 1668, after the death of the author.” Pascal’s discoveries as to the cycloid belong to a later period of his life, after he had long forsaken the scientific studies which engrossed him at this time, and had become an inmate of Port Royal. But, as we have already said, it is well to complete our view of his scientific labours in a single chapter. 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.”