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.

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The productiveness of modern writers has been enormous. "It is difficult," says Professor Cayley, "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,57 "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

undiscovered planetary orb of speculation." In mathematics, as in all true sciences, no subject is considered in itself alone, but always as related to, or an outgrowth of, other things. The development of the notion of continuity plays a leading part in modern research. In geometry the principle of continuity, the idea of correspondence, and the theory of projection constitute the fundamental modern notions. Continuity asserts itself in a most striking way in relation to the circular points at infinity in a plane. In algebra the modern idea finds expression in the theory of linear transformations and invariants, and in the recognition of the value of homogeneity and symmetry.

SYNTHETIC GEOMETRY.

The conflict between geometry and analysis which arose near the close of the last century and the beginning of the present has now come to an end. Neither side has come out victorious. The greatest strength is found to lie, not in the suppression of either, but in the friendly rivalry between the two, and in the stimulating influence of the one upon the other. Lagrange prided himself that in his Mecanique Analytique he had succeeded in avoiding all figures; but since his time mechanics has received much help from geometry.

Modern synthetic geometry was created by several investigators about the same time. It seemed to be the outgrowth of a desire for general methods which should serve as threads of Ariadne to guide the student through the labyrinth of theorems, corollaries, porisms, and problems. Synthetic geometry was first cultivated by Monge, Carnot, and Poncelet in France; it then bore rich fruits at the hands of Möbius and Steiner in Germany and Switzerland, and was finally developed to still

higher perfection by Chasles in France, von Staudt in Germany, and Cremona in Italy.

Augustus Ferdinand Möbius (1790-1868) was a native of Schulpforta in Prussia. He studied at Göttingen under Gauss, also at Leipzig and Halle. In Leipzig he became, in 1815, privat-docent, the next year extraordinary professor of astronomy, and in 1844 ordinary professor. This position he held till his death. The most important of his researches are on geometry. They appeared in Crelle's Journal, and in his celebrated work entitled Der Barycentrische Calcul, Leipzig, 1827. As the name indicates, this calculus is based upon properties of the centre of gravity.58 Thus, that the point S is the centre of gravity of weights a, b, c, d placed at the points A, B, C, D respectively, is expressed by the equation

(a+b+c+d)S= aA+bB+cC+dD.

His calculus is the beginning of a quadruple algebra, and contains the germs of Grassmann's marvellous system. In designating segments of lines we find throughout this work for the first time consistency in the distinction of positive and negative by the order of letters AB, BA. Similarly for triangles and tetrahedra. The remark that it is always possible to give three points A, B, C such weights a, B, y that any fourth point M in their plane will become a centre of mass, led Möbius to a new system of co-ordinates in which the position of a point was indicated by an equation, and that of a line by co-ordinates. By this algorithm he found by algebra many geometric theorems expressing mainly invariantal properties,

for example, the theorems on the anharmonic relation. Möbius wrote also on statics and astronomy. He generalised spherical trigonometry by letting the sides or angles of triangles exceed 180°.

Jacob Steiner (1796-1863), "the greatest geometrician since the time of Euclid," was born in Utzendorf in the Canton of Bern. He did not learn to write till he was fourteen. At eighteen he became a pupil of Pestalozzi. Later he studied at Heidelberg and Berlin. When Crelle started, in 1826, the celebrated mathematical journal bearing his name, Steiner and Abel became leading contributors. In 1832 Steiner published his Systematische Entwickelung der Abhängigkeit geometrischer Gestalten von einander, "in which is uncovered the organism by which the most diverse phenomena (Erscheinungen) in the world of space are united to each other." Through the influence of Jacobi and others, the chair of geometry was founded for him at Berlin in 1834. This position he occupied until his death, which occurred after years of bad health. In his Systematische Entwickelungen, for the first time, is the principle of duality introduced at the outset. This book and von Staudt's lay the foundation on which synthetic geometry in its present form rests. Not only did he fairly complete the theory of curves and surfaces of the second degree, but he made great advances in the theory of those of higher degrees. In his hands synthetic geometry made prodigious progress. New discoveries followed each other so rapidly that he often did not take time to record their demonstrations. In an article in Crelle's Journal on Allgemeine Eigenschaften Algebraischer Curven he gives without proof theorems which were declared by Hesse to be "like Fermat's theorems, riddles to the present and future generations." Analytical proofs of some of them have been given since by others, but Cremona finally proved them all by a synthetic method. Steiner discovered synthetically the two prominent properties of a surface of the third order; viz. that it contains twenty-seven straight lines and a pentahedron which has the double points. for its vertices and the lines of the Hessian of the given sur