**The Bolyais.**The rise of non-Euclidean geometry is so closely connected with the name of Bolyai that a few words are appropriate concerning the family. The name is found in the Magyar records as early as the 13th century, the family belonging to
the landed gentry, its estate lying in Bolya, a small town in Hungary. Farkas 1 Bolyai, a professor in a college 2 at Maros-Vasarhely 3, had a son Farkas Bolyai 4, who, after finishing his prepara
tory work, went to Gottingen (1796), where Kastner was closing his somewhat mediocre career and Gauss was beginning his brilliant one. Here he and Gauss became very intimate, exch
anging ideas, taking long walks together, and together indulging in the few social recreations that they allowed themselves. Circumstances compelled Bolyai to return to his home in 1799, much against his personal desires. In 1804 he became professor of
mathematics, physics, and chemistry in the college at Maros-Vasarhely, and here he remained until 1851. Here he wrote to Gauss two letters 5 on geometry, and here he published
(1830) a little work in the Magyar language on arithmetic 6 and also (1832) his work on elementary mathematics 7. The letters outline a book on geometry and show that he was interested in the subject of parallels. The book itself includes both algebra and geometry, and raises the question of the validity of Euclid's postulate of parallels. The gener
al ideas of this book were in his mind when he went to Gottingen, 1796, and for a generation he had pondered upon the foundations of geometry. Among them appears the principle of the permanence of equivalent forms 8, which English writers assign to Peacock (1830) and the Germans to Hermann Hankel (1867).

To the work of Farkas there was an appendix 9 written by his son, Janos Bolyai 10, of whom the f ather had written to Gauss (1816) that this boy of fourteen already had a good knowledge of the differential and integral calculus and could apply it to mechanics, to the tautochronism of the cycloid, and to other lines of work, and that he knew Latin and astronomy. Janos went to the engineering school in Vienna when he was sixteen, and at twenty-one entered the army. In 1825 or 1826 he worked out the theory of parallels which he set forth in the appendix (1832) above mentioned, and in this is a clear d iscussion of the validity of Euclid's postulate of parallels and a presentation of a non-Euclidean geometry 11.

The last years of Farkas Bolyai were unhappy ones, owing to the loss of his wife and the estrangement of his son. He wrote several other works 12, however, including some on poetry.

(Smith, pp. 527-529.)