In connection with the Bolyais is natural to mention the work of Nicolai Ivanovitch Lobachevsky 1 . Although the son of a Russian peasant, he early showed a remarkable genius. He studied at the University of Kazan and when only twenty-one became professor of mathematics in that institution. In 1826 he made known through his lectures his concep tion of a geometry which should not depend upon the Euclidean postulate of parallel lines. These ideas were published in 1829, and in various later works 2.

Of the independent discovery of the non-Euclidean geometry by Lobachevsky and Bolyai there can be no doubt. The subject was in the general intellectual atmosphere of the time. Gauss, who was considering the question as early as 1792, had doubtless st imulated the elder Bolyai to study the problem, and no doubt had been stimulated in return. Both Lobachevsky and the younger Bolyai had been influenced by the Gottingen school. Each in his own way had attacked the question, and each had worked out his t heory at about the same time (1825-1826); Lobachevsky published his theory first (1829), but Janos Bolyai published his independently (1832) 3.

(Smith, pp. 529-530)