His illness began by violent toothache; he was kept awake night after night: during these painful vigils, his thoughts recurred to the studies of his youth. He revolved in his head problems proposed by the scientific men of the day.
His attention was now chiefly engaged with the solution of various questions in the higher departments of geometry, especially those connected with the properties of cycloids. He succeeded in solving many problems of great difficulty relating to the quadrature and rectification of segments and arcs of cycloids, and the volumes of solids formed by their revolutions round their axes and ordinates. Except so far as they form part of the history of mathematical science, and illustrate the powers of great minds, such as that of the subject of this memoir, these problems have now lost all their interest. The powerful instruments of investigation supplied by the differential and integral calculus, have reduced their solution to the mere elements of transcendental mathematics. At the epoch when they engaged the attention of Pascal, before the invention of the modern methods, they were questions presenting the most formidable difficulties. To Pascal, however, they were mere matters of mental relaxation, resorted to with a view to divert his attention from his acute bodily sufferings. He entertained, himself, no intention of making them public. It was, however, the wish of several of his companions in religious retirement that they should be made public, were it only to afford a proof that the highest mathematical genius is not incompatible with the deepest and most sincere Christian faith. [Sidenote: 1658. Ætat. 35.] Pascal yielded, and, according to a custom which was then usual, however puerile it may now appear, he, in the first instance, proposed the several questions which he solved as subjects for a prize to the scientific world. Many competitors presented themselves; and others, who, though not competing for the prize, offered partial solutions. Among these were several who have since attained great celebrity, such as Wallis, Huygens, Fermat, and sir Christopher Wren.
The prize, however, was not gained, nor the problems solved. In the beginning of the year 1659, Pascal published his complete solutions of the problems of the cycloid, with some other mathematical tracts. These admirable investigations cannot fail to excite in every mind a deep regret, that a morbid state of moral and religious feeling should ever have diverted Pascal from mathematical and physical research.