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,[]57 "can in no case be discounted,