P A S P A S edition has been long promised for the excellent collection of Les Grands ticrimins de la France; it has been understood to be under the charge of M. Faugere. Meanwhile., with the exception of the Provinciales (of which there are numerous editions, no one much to be preferred to any other, for the text is undisputed and the book itself contains almost all the exegesis of its own contents necessary), Pascal can be read only at a dis advantage. There are four chief editions of the true Pcnslcs: that of M. Faugere (1844 1, the cditio princeps ; that of M. Havet (1852, 1867, and 1881), on the whole the best; that of M. Victor Rochet (1873), good, but arranged and edited with the deliberate intention of making Pascal first of all an orthodox apologist; and that of M. Molinier (1877-79), a carefully edited and interesting text, the important corrections of which have been introduced into M. Havet s last edition. Unfortunately, none of these can be said to be exclusively satisfactory. The minor works must chiefly be sought in Bossut or reprints of him. Works on Pascal nre innumerable: Sainte-Beuve s Port Royal, Cousin s writings on Pascal and his Jacqueliiie Pascal, and the essays of the editors of the Pensees just mentioned are the most noteworthy. Principal Tulloch has contributed a useful little monograph to the series of Foreign Classics fur English Readers (Edinburgh and London, 1878;. (G. SA.) Pascal as Natiiral 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 capa cities all will admire the forni 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 Des cartes, Fermat, Huygens, Wallis, and Roberval. There are wonderful stories on record of his precocity in mathe matical 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 resume of its results published in 1640 by Pascal himself, under the title Essai pour les Coniques. The method which he fol lowed was that introduced by his contemporary 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 four hundred corollaries, embracing, according to his own account, the conies 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 mathema ticians. 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 M. Cercavi. There has been some dis cussion 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 pro gress of the differential calculus. In yet another branch of pure mathematics Pascal ranks as a founder. The mathematical theory of proba bility 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 Mer6. A complete account of this interesting correspond ence would surpass our present limits ; but the reader may be referred to Todhunter s History of the Thtory of Proba bility (Cambridge and London, 1865) pp. 7-21. It appears that Pascal contemplated publishing a treatise De Alex Geometria; but all that actually appeared was a fragment on the arithmetical triangle ("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 1653, published 1662) 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 in the first place by himself in a tower at Paris, and was afterwards carried out on a grand scale under his instructions by his brother-in-law Perier on the Puy de Dome 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 un familiar in his time. After the lapse of more than two hundred years, we can stil point to much in exact science that is absolutely his ; and we can indicate infinitely more which is due to his inspiration. (u. CH.) PASCHAL L, pope from 817 to 824, a native of Rome, was raised to the pontificate by popular acclamation, 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 relations with the imperial house, however, never became cordial ; and he was also unsuccessful in retaining in Rome itself the popularity to which he had owed his election. He died at Rome while the imperial commissioners were investigating the circum stances under which two important officers of Lothair, the eldest son of Louis, had been seized at the Lateran, blinded, and afterwards beheaded ; Paschal had shielded the murderers but denied all personal complicity in their crime. The Roman people refused him the honour of burial within the church of St Peter, but he now holds a place in the Roman calendar (May 16). Like one or two of his more immediate predecessors he was liberal in his donations to several churches of the city, St Cecilia in Trastevere having been restored and St Maria in Domnica
