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 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. 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.