The elements of non-Euclidean plane geometry and trigonometry, by H. S. Carslaw.

CHAPTER II.

THE WORK OF BOLYAI, LOBATSCHEWSKY, AND RIEMANN,
THE FOUNDERS OF THE NON-EUCLIDEAN GEOMETRIES.
~ 13. John Bolyai (1802-1860).
As we have already seen, John Bolyai, a Hungarian officer
in the Austrian army, had in 1823 built up a consistent system
of geometry in which the Parallel Postulate of Euclid was
replaced by another, contradictory to the former. His discovery was published in 1832 as an Appendix to his father's
work: Tentamen juventutem studiosam in elementa mnatheseos
purae, elementaris ac sublimioris, methodo intuitiva, evidentiaque huic propria, introducendi. This work is usually referred
to as the Tentamen. The title of the Appendix contributed by
the son, and placed at the end of vol. i. of the Tentamen, is:
Appendix. Scientiam spatii absolute veram exhibens: a veritate
aut falsitate Axiomatis XI Euclidei (a priori haud unquam
decidenda) independentem: adjecta ad casum falsitatis, quadratura circuli geometrica. Auctore Johanne Bolyai de eadem,
Geometrarum in Exercitu Caesareo Regio Austriaco Castrensium
Capitaneo.
If we omit the title page, a page explaining the notation,
and two pages of errata, the Appendix contains only twentyfour pages.
Bolyai's discovery was made as early as 1823, when he was
but 21 years old. At the time, we find him writing to his
father as follows: *
" I have resolved to publish a work on the theory of parallels,
as soon as I shall have put the material in order, and my cir* Stackel u. Engel, " Gauss die beiden Bolyai und die nichteuklidische
Geometrie, Math. Ann. vol. xlix. p. 155 (1897). Also Stdckel, loc. cit.
vol. i. p. 85.