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 (6 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

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best 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 an ordinate. The programme was put forth in the name of Amos Dettonville, the anagram of Pascal's assumed name as the writer of the 'Provincial Letters.' Huyghens, Sluzsius, a canon of the Cathedral of Liège, and Wren, the architect of St Paul's, sent in partial solutions of the problems-those of Wren especially attracting the interest of both Fermat and Roberval. But Wallis, of Oxford, and Lallouère, a Jesuit of Toulouse, were the only two competitors who treated all the problems proposed. It was held that they had not completely succeeded in solving them; and Dettonville published his own solution in an elaborate letter addressed to M. Carcavi, and in a treatise on the subject. Carcavi was an old friend of Pascal's father as well as of himself; and being a lawyer as well as a mathematician, the arrangement of the affair seems to have been intrusted to him. This did not save him, however, from attacks by the disappointed candidates, who accused him of unfairness; and Leibnitz has given his decision that both Wallis and Lallouère, in the treatises which they published, — which did not, however, appear till after Pascal's, had succeeded in solving the problems. Upon such a point we cannot pretend to judge; but it may be safely said that the design of the Duc de Roannez was hardly realised in the issue.

F.C.-III.

D

It was sufficiently proved, indeed, that Pascal, in the midst of all his austerities and devotional exercises, was the same Pascal who had held his own both with Descartes and with the Jesuits. But the life of thought which survived in him no sooner touched the outer world of intellectual ambition, than it flamed forth into something of the passion of controversy which his pen had already kindled in another direction. Religion is best vindicated, not in the strifes of science, but by the beauty of its own activities.

Pascal's labours on the cycloid may be said to bring to a close his scientific career. There is still one invention, however, of a very practical kind, associated with the very last months of his life. Amongst the letters of Madame Périer, there is one of date March 24, 1662, addressed to M. Arnauld de Pompone1 —a nephew of the great Arnauld -in which she gives a lively description of the success of an experiment "dans l'affaire des carrosses." The affair was nothing less than the trial on certain routes in Paris of what is now known as an 66 omnibus ;" and the idea of such conveyances for the public-" carrosses à cinq sols," as they were called-is attributed to Pascal. It is certain that the privilege of running "carrosses à cinq sols" was granted to Pascal's friend, the Duc de Roannez, and to other noblemen, by royal patent, in January 1662, -and that the experiment, as described by Madame Périer, was made with great success in the following March, and that Pascal had an active interest in the undertaking. His sister tells that he had mortgaged his share of its first year's profits in order to provide

1 Faugère, Lettres, &c., p. 80.

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for the poor at Blois;1 and a note from his own hand, appended to his sister's letter, shows with what eagerness he entered into the affair and hailed its success. It is singular to connect the name of Pascal, and that, too, during the last sad months of his life, with so world-wide a commonplace as the omnibus.

1 Vie de Pascal.

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CHAPTER III.

PASCAL IN THE WORLD.

PASCAL'S health, we have seen, was very delicate. His labours to perfect his arithmetical machine had seriously impaired it. The attack of partial paralysis, described by his niece, seems to have taken place in the early summer of 1647. As soon as he was able, he removed to Paris, where we find him settled with his younger sister in September of the same year. It was on the twentyfifth of this month that Jacqueline writes from Paris of Descartes's memorable visits. One of the motives of his change of residence was no doubt to consult the best physicians of the day; and Descartes, who, amongst his other numerous gifts, had some skill in medicine, made his second visit to him partly as a physician. "He came in part," says Jacqueline, "to consult as to my brother's illness." He appears to have given him very sound advice, which, unfortunately, Pascal did not follow-"to lie in bed as much as he could, and take strong soup." On the contrary, he was "bled, bathed, and purged," after the usual medical routine of the time, apparently without any good effects, or any alleviation of his sufferings.