During the present century very remarkable generalisations have been made, which reach to the very root of two of the oldest branches of mathematics,–-elementary algebra and geometry. In algebra the laws of operation have been extended; in geometry the axioms have been searched to the
bottom, and the conclusion has been reached that the space defined by Euclid's axioms is not the only possible non-contradictory space. Euclid proved (I. 27) that "if a straight line falling on two other straight lines make the alternate angles equal to one another, the two straight lines shall be parallel to one another." Being unable to prove that in every other case the two lines are not parallel, he assumed this to be true in what is generally called the 12th "axiom," by some
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
Göttingen, where he became intimate with Gauss, then nineteen
years old. Gauss used to say that Bolyai was the only
man who fully understood his views on the metaphysics of mathematics. Bolyai became professor at the Reformed College of Maros-Vásárhely, where for forty-seven years he had for his pupils most of the present professors of Transylvania. The first publications of this remarkable genius were dramas and poetry. Clad in