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

30

NON-EUCLIDEAN GEOMETRY

[CH. II.

Postulate. He refers to the point more than once; but he
postpones fuller treatment till a later occasion; an occasion
which, so far as the public are concerned, never came. The
last sentences of the Appendix (Halsted's translation) are as
follows:
" It remains finally, (that the thing may be completed in
every respect), to demonstrate the impossibility (apart from
any supposition), of deciding a priori, whether A, or some S
(and which one) exists.* This, however, is reserved for a more
suitable occasion."
~ 15. Bolyai retired from the army in 1833 and lived till
1860. So far as we know he published nothing further, either
in extension of the Appendix or on any other mathematical
subject. From several sources, chiefly notes found among his
papers, we learn that he occupied himself with some of the
problems of the Non-Euclidean Geometry. He carried his
work further in the direction of Solid Geometry. He investigated more fully the relation between the Non-Euclidean
Geometry and Spherical Trigonometry; and he pondered the
question of the possibility or impossibility of proving Euclid's
Hypothesis.
An unpublished version of part of the Appendix exists in
German,t in which he gives clearer expression to his views upon
the last of these topics than is to be found in the corresponding
section of the original. In this version, which dates from 1832,
the first part of ~ 33 reads as follows:
"Now I should briefly state the essential result of this
theory, and what it is in a position to effect:
"I. Whether E or S actually exists, remains here (and, as
the author can prove, for ever) undecided.
"II. Now there is a Plane Trigonometry absolutely true
(i.e. free from every hypothesis), in which, however, (according
to I.), the constant i and its very existence remain wholly
undetermined. With the exception of this unknown everything is determined. But Spherical Trigonometry was
* Bolyai calls Z the system of Geometry resting upon Euclid's
Hypothesis; and S the system founded upon his own definition of
parallels.
t Cf. Stackel, "Untersuchungen aus der absoluten Geometrie aus
Johann Bolyai's Nachlass," Math. u. Naturw. Berichte aus Ungarn,
vol. xviii. p. 280, 1902. Also loc. cit. vol. ii. p. 181.