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

36

NON-EUCLIDEAN GEOMETRY

[CH. II.

remark that these equations are transformed into the equations
(16) of Spherical Trigonometry by substituting ia, ib, and ic for
the sides a, b, and c. And in ordinary geometry and Spherical
Trigonometry there enter only the relations between lines. It
follows that the ordinary geometry, (Spherical) Trigonometry
and this new geometry must always be in agreement with
one another." *
~ 18. The writings of Lobatschewsky were brought under
the notice of Gauss as early as 1841, and we gather from his
letters how much impressed he was with them. Indeed it
almost appears as if he had thrown himself into the study of
Russian that he might be able to read the numerous papers
which he hears this " clear-sighted mathematician" had
published in that tongue. Through Gauss the elder Bolyai
learnt in 1848 of the Russian's work, and in particular of the
Geometrische Untersuchungen zur Theorie der Parallellinien of
1840. The astonishing news and the volume, which Lobatschewsky had written as a summary of his work, were passed
on from the father to his son. How he received the intelligence
we learn from the following passage in some unpublished Notes
upon Nicolaus Lobatschewsky's Geometrische Untersuchungen:t
"Even if in this remarkable work many other methods are
adopted, yet the spirit and the result so closely resemble the
Appendix to the Tentamen matheseos, which appeared in MarosVasarhely in 1832, that one cannot regard it without astonishment. If Gauss was, as he says, immensely surprised, first by
the Appendix and soon after by the remarkable agreement of
the Hungarian and Russian mathematician, not less so am I.
"The nature of absolute truth can indeed only be the same
in Maros-VAsarhely as in Kamschatka and on the Moon, or, in
a word, anywhere in the world; and what one reasonable
being discovers, that can also quite possibly be discovered by
another."
*The same point is referred to in Lobatschewsky's other works:
cf. (i) Imaginary Geometry (Liebmann's translation, p. 8); (ii) Geometrische Untersuchungen zur Theorie der Parallellinien (Halsted's
translation, p. 163); (iii) Pangeometrie, ~ 8 (quoted by Bonola, loc. cit.
p. 93).
f Cf. Kiirschak u. Stackel, "Johann Bolyai's Bemerkungen fiber
Nicolaus Lobatschewsky's Geometrische Untersuchungen zur Theorie
der Parallellinien," lMacth. u. Naturvu. Berichte aus Ungarn, vol. xviii.
p. 256 (1902). Also, Stackel, loc. cit. vol. i. p. 140.