the means of making figures infallibly exact, he employed his thoughts about it and with a piece of charcoal drew figures upon the tiles of the pavement, trying the methods of drawing, for example, an exact circle or equilateral triangle. He gave names of his own to these figures and then formed axioms, and, in short, came to make perfect demonstrations. In this way he arrived unaided at the theorem that the sum of the three angles of a triangle is equal to two right angles. His father caught him in the act of studying this theorem, and was so astonished at the sublimity and force of his genius as to weep for joy. The father now gave
him Euclid's Elements, which he, without assistance, mastered easily. His regular studies being languages, the boy employed only his hours of amusement on the study of geometry, yet he had so ready and lively a penetration that, at the age of sixteen, he wrote a treatise upon conics, which passed for such a surprising effort of genius, that it was said nothing equal to it in strength had been produced since the time of Archimedes. Descartes refused to believe that it was written by one so young as Pascal. This treatise was never published, and is now lost. Leibniz saw it in Paris and reported on a portion
of its contents. The precocious youth made vast progress in all the sciences, but the constant application at so tender an age greatly impaired his health. Yet he continued working, and at nineteen invented his famous machine for performing arithmetical operations mechanically. This continued strain from overwork resulted in a permanent indisposition, and he would sometimes say that from the time he was eighteen, he never passed a day free from pain. At the age of twenty-four he resolved to lay aside the study of the human sciences and to consecrate his talents to religion. His Provincial Letters against the Jesuits are celebrated. But at times he returned to the favourite study of his youth. Being kept awake one night by a toothache, some thoughts undesignedly came into his head concerning the roulette or cycloid; one idea followed another; and he thus discovered
properties of this curve even to demonstration. A correspondence between him and Fermat on certain problems was the beginning of the theory of probability. Pascal's illness increased, and he died at Paris at the early age of thirty-nine years.30 By him the answer to the objection to Cavalieri's Method of Indivisibles was put in the clearest form. Like
Roberval, he explained "the sum of right lines" to mean "the sum of infinitely small rectangles." Pascal greatly advanced
the knowledge of the cycloid. He determined the area of a section produced by any line parallel to the base; the volume generated by it revolving around its base or around the axis; and, finally, the centres of gravity of these volumes, and also of half these volumes cut by planes of symmetry. Before publishing his results, he sent, in 1658, to all mathematicians that famous challenge offering prizes for the first two solutions of these problems. Only Wallis and A. La Louère competed
for them. The latter was quite unequal to the task; the former, being pressed for time, made numerous mistakes: neither got a prize. Pascal then published his own solutions, which produced a great sensation among scientific men. Wallis, too, published his, with the errors corrected. Though not competing for the prizes, Huygens, Wren, and Fermat solved
some of the questions. The chief discoveries of Christopher Wren (1632–1723), the celebrated architect of St. Paul's Cathedral in London, were the rectification of a cycloidal arc