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

14, 15]

BOLYAI'S LATER YEARS

31

developed absolutely and completely in ~ 26; so that the
ordinary familiar Spherical Trigonometry is not in the least
dependent upon Axiom XI. and is unconditionally true.
"III. By means of these two trigonometries and several
subsidiary theorems (to be found in the text of ~ 32) one is able
to solve all the problems of Solid Geometry and Mechanics,
which the so-called Analysis in its present development has in
its power (a statement which requires no further qualification),
and this can be done downright without the help of Axiom XI.
(on which until now everything rested as chief-foundationstone), and the whole theory of space can be treated in the
above-mentioned sense, from now on, with the analytical
methods (rightly praised within suitable limits) of the new
(science).
" Taking now into consideration the demonstration of the
impossibility of deciding between E and S (a proof which the
author likewise possesses), the nature of Axiom XI. is at
length fully determined; the intricate problem of parallels
completely solved; and the total eclipse completely dispelled,
which has so unfortunately reigned till the present (for minds
thirsting for the truth), an eclipse which has robbed so many of
their delight in science, and of their strength and time.
" Also, in the author, there lives the perfectly purified conviction (such as he expects too from every thoughtful reader)
that by the elucidation of this subject one of the most important
and brilliant contributions has been made to the real victory
of knowledge, to the education of the intelligence, and consequently to the uplifting of the fortunes of men."
His proof of the impossibility of proving the Euclidean
Hypothesis seems to have rested upon the conviction that
the Non-Euclidean Trigonometry would not lead to any contradiction. The following sentences are to be found among
his papers:
"We obtain by the analysis of a system of points on a
plane obviously quite the same formulae as on the sphere;
and since continued analysis on the sphere cannot lead to any
contradiction (for Spherical Trigonometry is absolute), it is
therefore clear that in the same way no contradiction could
ever enter into any treatment of the system of points in a
plane." *
*Cf, Stickel, loc. cit, vol. i. p. 121,