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 the 11th "axiom." But this so-called axiom is far from axiomatic. After centuries of desperate but fruitless attempts to prove Euclid's assumption, the bold idea dawned upon the minds of several mathematicians that a geometry might be built up without assuming the parallel-axiom. While Legendre still endeavoured to establish the axiom by rigid proof, Lobatchewsky brought out a publication which assumed the contradictory of that axiom, and which was the first of a series of articles destined to clear up obscurities in the fundamental concepts, and to greatly extend the field of geometry.

Nicholaus Ivanovitch Lobatchewsky (1793–1856) was born at Makarief, in Nischni-Nowgorod, Russia, studied at Kasan, and from 1827 to 1846 was professor and rector of the University of Kasan. His views on the foundation of geometry were first made public in a discourse before the physical and mathematical faculty at Kasan, and first printed in the Kasan Messenger for 1829, and then in the Gelehrte Schriften der Universität Kasan, 1836–1838, under the title, "New Elements of Geometry, with a complete theory of Parallels." Being in the Russian language, the work remained unknown to foreigners, but even at home it attracted no notice. In 1840 he published a brief statement of his researches in Berlin. Lobatchewsky constructed an "imaginary geometry," as he called it, which has been described by Clifford as "quite simple, merely Euclid without the vicious assumption." A remarkable part of this geometry is this, that through a point an indefinite number of lines can be drawn in a plane, none of which cut a given line in the same plane. A similar system of geometry was deduced independently by the Bolyais in Hungary, who called it "absolute geometry."

Wolfgang Bolyai de Bolya (1775–1856) was born in Szekler-Land, Transylvania. After studying at Jena, he went to