A History of Mathematics/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 object-less 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 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.