tion de l'analyse à la géométrie. The last two contain most of his miscellaneous papers.

Monge was an inspiring teacher, and he gathered around him a large circle of pupils, among which were Dupin, Servois, Brianchion, Hachette, Biot, and Poncelet.

Charles Dupin (1784-1873), for many years professor of mechanics in the Conservatoire des Arts et Métiers in Paris, published in 1813 an important work on Développements de géométrie, in which is introduced the conception of conjugate tangents of a point of a surface, and of the indicatrix.55 It contains also the theorem known as "Dupin's theorem." Surfaces of the second degree and descriptive geometry were successfully studied by Jean Nicolas Pierre Hachette (1769– 1834), who became professor of descriptive geometry at the Polytechnic School after the departure of Monge for Rome and Egypt. In 1822 he published his Traité de géométrie descriptive. Descriptive geometry, which arose, as we have seen, in technical schools in France, was transferred to Germany at the foundation of technical schools there. G. Schreiber, professor in Karlsruhe, was the first to spread Monge's geometry in Germany by the publication of a work thereon in 1828-1829.54 In the United States descriptive geometry was introduced in 1816 at the Military Academy in West Point by Claude Crozet, once a pupil at the Polytechnic School in Paris. Crozet wrote the first English work on the subject.2

Lazare Nicholas Marguerite Carnot (1753-1823) was born at Nolay in Burgundy, and educated in his native province. He entered the army, but continued his mathematical studies, and wrote in 1784 a work on machines, containing the earliest proof that kinetic energy is lost in collisions of bodies. With the advent of the Revolution he threw himself into politics, and when coalesced Europe, in 1793, launched against France a million soldiers, the gigantic task of organising fourteen

armies to meet the enemy was achieved by him. He was banished in 1796 for opposing Napoleon's coup d'état. The refugee went to Geneva, where he issued, in 1797, a work still frequently quoted, entitled, Réflexions sur la Métaphysique du Calcul Infinitésimal. He declared himself as an "irreconcilable enemy of kings." After the Russian campaign he offered to fight for France, though not for the empire. On the restoration he was exiled. He died in Magdeburg. His Géométrie de position, 1803, and his Essay on Transversals, 1806, are important contributions to modern geometry. While Monge revelled mainly in three-dimensional geometry, Carnot confined himself to that of two. By his effort to explain the meaning of the negative sign in geometry he established a "geometry of position," which, however, is different from the "Geometrie der Lage" of to-day. He invented a class of general theorems on projective properties of figures, which have since been pushed to great extent by Poncelet, Chasles, and others.

Jean Victor Poncelet (1788-1867), a native of Metz, took part in the Russian campaign, was abandoned as dead on the bloody field of Krasnoi, and taken prisoner to Saratoff. Deprived there of all books, and reduced to the remembrance of what he had learned at the Lyceum at Metz and the Polytechnic School, where he had studied with predilection the works of Monge, Carnot, and Brianchion, he began to study mathematics from its elements. He entered upon original researches which afterwards made him illustrious. While in prison he did for mathematics what Bunyan did for literature, produced a much-read work, which has remained of great value down to the present time. He returned to France in 1814, and in 1822 published the work in question, entitled, Traité des Propriétés projectives des figures. In it he investigated the properties of figures which remain un

altered by projection of the figures. The projection is not effected here by parallel rays of prescribed direction, as with Monge, but by central projection. Thus perspective projection, used before him by Desargues, Pascal, Newton, and Lambert, was elevated by him into a fruitful geometric method. In the same way he elaborated some ideas of De Lahire, Servois, and Gergonne into a regular method - the method of "reciprocal polars." To him we owe the Law of Duality as a consequence of reciprocal polars. As an independent principle it is due to Gergonne. Poncelet wrote much on applied mechanics. In 1838 the Faculty of Sciences was enlarged by his election to the chair of mechanics.

While in France the school of Monge was creating modern geometry, efforts were made in England to revive Greek geometry by Robert Simson (1687-1768) and Matthew Stewart (1717-1785). Stewart was a pupil of Simson and Maclaurin, and succeeded the latter in the chair at Edinburgh. During the eighteenth century he and Maclaurin were the only prominent mathematicians in Great Britain. His genius was illdirected by the fashion then prevalent in England to ignore higher analysis. In his Four Tracts, Physical and Mathematical, 1761, he applied geometry to the solution of difficult astronomical problems, which on the Continent were approached analytically with greater success. He published, in 1746, General Theorems, and in 1763, his Propositiones geometricæ more veterum demonstratæ. The former work contains sixty-nine theorems, of which only five are accompanied by demonstrations. It gives many interesting new results on the circle and the straight line. Stewart extended some theorems on transversals due to Giovanni Ceva (1648-1737), an Italian, who published in 1678 at Mediolani a work containing the theorem now known by his name.

RECENT TIMES.

NEVER more zealously and successfully has mathematics been cultivated than in this century. Nor has progress, as in previous periods, been confined to one or two countries. While the French and Swiss, who alone during the preceding epoch carried the torch of progress, have continued to develop mathematics with great success, from other countries whole armies of enthusiastic workers have wheeled into the front rank. Germany awoke from her lethargy by bringing forward Gauss, Jacobi, Dirichlet, and hosts of more recent men; Great Britain produced her De Morgan, Boole, Hamilton, besides champions who are still living; Russia entered the arena with her Lobatchewsky; Norway with Abel; Italy with Cremona; Hungary with her two Bolyais; the United States with Benjamin Peirce.

The productiveness of modern writers has been enormous. "It is difficult," says Professor Cayley,56 "to give an idea of the vast extent of modern mathematics. This word 'extent' is not the right one: I mean extent crowded with beautiful detail, not an extent of mere uniformity such as an objectless plain, but of a tract of beautiful country seen at first in the distance, but which will bear to be rambled through and studied in every detail of hillside and valley, stream, rock, wood, and flower." It is pleasant to the mathematician to think that in his, as in no other science, the achievements of

every age remain possessions forever; new discoveries seldom disprove older tenets; seldom is anything lost or wasted.

If it be asked wherein the utility of some modern extensions of mathematics lies, it must be acknowledged that it is at present difficult to see how they are ever to become applicable to questions of common life or physical science. But our inability to do this should not be urged as an argument against the pursuit of such studies. In the first place, we know neither the day nor the hour when these abstract developments will find application in the mechanic arts, in physical science, or in other branches of mathematics. For example, the whole subject of graphical statics, so useful to the practical engineer, was made to rest upon von Staudt's Geometrie der Lage; Hamilton's "principle of varying action ” has its use in astronomy; complex quantities, general integrals, and general theorems in integration offer advantages in the study of electricity and magnetism. "The utility of such researches," says Spottiswoode," "can in no case be discounted, or even imagined beforehand. Who, for instance, would have supposed that the calculus of forms or the theory of substitutions would have thrown much light upon ordinary equations ; or that Abelian functions and hyperelliptic transcendents would have told us anything about the properties of curves; or that the calculus of operations would have helped us in any way towards the figure of the earth?" A second reason in favour of the pursuit of advanced mathematics, even when there is no promise of practical application, is this, that mathematics, like poetry and music, deserves cultivation for its own sake.

The great characteristic of modern mathematics is its generalising tendency. Nowadays little weight is given to isolated theorems, "except as affording hints of an unsuspected new sphere of thought, like meteorites detached from some