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Rh comprehensible, and he need not be taken as a deserter from one region to the other. To those who hold that all intellectual exercise outside the sphere of religion is impious or that all intellectual exercise inside that sphere is futile, he must remain an enigma.

(G. S A .)

Pascal as Natural Philosopher and Mathematician.—Great as is Pascal’s reputation as a philosopher and man of letters, it may be fairly questioned whether his claim to be remembered by posterity as a mathematician and physicist is not even greater. In his two former capacities all will admire the form of his work, while some will question the value of his results; but in his two latter capacities no one will dispute either. He was a great mathematician in an age which produced Descartes, Fermat, Huygens, Wallis and Roberval. There are wonderful stories on record of his precocity in mathematical learning, which is sufficiently established by the well-attested fact that he had completed before he was sixteen 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. Owing partly to the youth of the author, partly to the difficulty in publishing scientific works in those days, and partly no doubt to the continual struggle on his part to devote his mind to what appeared to his conscience more important labour, this work (like many others by the same master hand) was never published. We know something of what it contained from a report by Leibnitz, who had seen it in Paris, and from a résumé of its results published in 1640 by Pascal himself, under the title Essai pour les coniques. The method which he followed was that introduced by his contemporary 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 distinguished himself by his skill in 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 without 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 Carcavy. There has been some discussion as to the fairness of the treatment accorded by Pascal to his rivals, but no question of the fact that 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.

In yet another branch of pure mathematics Pascal ranks as a founder. The mathematical theory of probability and the allied theory of the combinatorial analysis were in effect created by the correspondence between Pascal and Fermat, concerning certain questions as to the division of stakes in games of chance, which had been propounded to the former by the gaming philosopher De Méré. A complete account of this interesting correspondence would surpass our present limits; but the reader may be referred to Todhunter’s History of the Theory of Probability (Cambridge and London, 1865), pp. 7–21. It appears that Pascal contemplated publishing a treatise De aleae geometria; but all that actually appeared was a fragment on the arithmetical triangle (Traité du triangle arithmétique, “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 equilibrium 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 Périer on the Puy de Dôme in Auvergne. Its success 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 receive the same impression of Pascal; 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. (G. C H .)

PASCAL, JACQUELINE (1625-1661), sister of Blaise Pascal, was born at Clermont-Ferrand, France, on the 4th of October 1625. She was a genuine infant prodigy, composing verses when only eight years, and a five-act comedy at eleven. In 1646 the influence of her brother converted her to Jansenism. In 1652, she took the veil, despite the strong opposition of her brother, and subsequently was largely instrumental in the latter's own final conversion. She vehemently opposed the attempt to compel the assent of the nuns to the Papal bulls condemning Jansenism, but was at last compelled to yield her own. This blow, however, hastened her death, which occurred at Paris on the 4th of October 1661.

PASCHAL (Paschalis), the name of two popes, and one anti-pope.

, pope from 817 to 824, a native of Rome, was raised to the pontificate by the acclamation of the clergy, shortly after the death of Stephen IV., and before the sanction of the emperor (Louis the Pious) had been obtained — a circumstance for which it was one of his first cares to apologize. His