Blaise 1623-1662 Pascal

Natural Philosopher and Mathematician.

Pascal was a great mathematician in an age which produced Descartes, Fermat, Huygens, Wallis and Roberval. His precocity in mathematics is established by the fact that he had completed before he was six teen years of age a work on the conic sections, in which he had laid down a series of propositions, discovered by himself, of such importance that they may be said to form the foundations of the modern treatment of that subject. This work (like many others by the same master hand) was never published. We know some thing of what it contained from a report by Leibnitz, who had seen it in Paris, and from a resume of its results published in 1640 by Pascal himself, under the title Essai pour les coniques. The method was that of Girard Desargues, viz., the transformation of geometrical figures by conical or optical projection. In this way he established the famous theorem that the intersections of the three pairs of opposite sides of a hexagon inscribed in a conic are collinear. This proposition, which he called the mystic hexagram, he made the keystone of his theory; from it alone he deduced more than 400 corollaries, embracing, according to his own account, the conics of Apollonius, and other results innumerable.

Pascal also worked on the infinitesimal calculus, then in the embryonic form of Cavalieri's method of indivisibles. The cycloid was a famous curve in those days; it had been discussed by Galileo, Descartes, Fermat, Roberval and Torricelli, who had in turn exhausted their skill upon it. Pascal solved the hitherto refractory problem of the general quadrature of the cycloid, and proposed and solved a variety of others relating to the centre of gravity of the curve and its segments, and to the volume and centre of gravity of solids of revolution generated in various ways by means of it. He published a number of these theorems with out demonstration as a challenge to contemporary mathematicians. Solutions were furnished by Wallis, Huygens, Wren and others; and Pascal published his own in the form of letters from Amos Dettonville (his assumed name as challenger) to Pierre de Car cavy. His initiative led to a great extension of our knowledge of the properties of the cycloid, and indirectly hastened the progress of the differential calculus.

The

mathematical theory of probability and the allied theory of the combinatorial analysis were in effect created by the cor respondence between Pascal and Fermat, concerning certain ques tions as to the division of stakes in games of chance, which had been propounded to the former by the gaming philosopher De Mere. Of the treatise De aleae geometria all that actually appeared was a fragment on the arithmetical triangle (Traite du triangle arithmetique, "Properties of the Figurate Numbers"), printed in 1654, but not published till 1665, after his death.

Pascal's work as a natural philosopher was not less remarkable than his discoveries in pure mathematics. His experiments and his treatise (written before 1651, published 1663) on the equilib rium of fluids entitle him to rank with Galileo and Stevinus as one of the founders of the science of hydrodynamics. The idea of the pressure of the air and the invention of the instrument for measuring it were both new when he made his famous experiment, showing that the height of the mercury column in a barometer decreases when it is carried upwards through the atmosphere. This experiment was made by himself in a tower at Paris, and was carried out on a grand scale under his instructions by his brother in-law Florin Perier on the Puy de Dome in Auvergne. Its suc cess greatly helped to break down the old prejudices, and to bring home to the minds of ordinary men the truth of the new ideas propounded by Galileo and Torricelli.

Whether we look at his pure mathematical or at his physical researches we see the strongest marks of a great original genius creating new ideas, and seizing upon, mastering, and pursuing farther everything that was fresh and unfamiliar in his time. We can still point to much in exact science that is absolutely his; and we can indicate infinitely more which is due to his inspiration.

BIBLIOGRAPHY.-Editions of Pascal's works are those by Bossut (5 vols. 1779), Renouard (1803), Lefevre (1819), and Lahure (1858). The standard edition is by Brunschwig and Boutroux in the Grands icrivains de la France (14 vols., Paris, 1908-14). There is also a separate edition of the Pensies by Brunschwig (1904). Among other modern editions of the Pensees may be mentioned that of Molinier (1877-79), a carefully edited and interesting text, the important corrections of which have been introduced into Havet's last edition, and that of G. Michelant (Freiburg, 1896). The main original source for the biogra phy of Pascal is P. Faugere, Lettres, opuscules et memoires de Mme. Perier et de Jacqueline, soeurs de Pascal, et de Marguerite Perier, sa niece (Paris, 1645). Works on Pascal are innumerable: Sainte-Beuve's Port Royal, Cousin's writings on Pascal and his Jacqueline Pascal, and the essays of the editors of the Pensees first mentioned are the most noteworthy. Recent handlings are, in French, E. Boutroux's Pascal (Paris, 1903), and, in English, R. H. Soltau, Pascal: The Man and the Message (1927), which contains a selected bibliography.