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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