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

10, 11] GAUSS, SCHWEIKART, AND TAURINUS 23

"The Euclidean geometry holds only on the assumption that
the Constant is infinite. Only in this case is it true that the
three angles of every triangle are equal to two right angles;
and this can easily be proved, as soon as we admit that the
Constant is infinite."
This document is of peculiar importance, as it is in all
probability the earliest statement of the Non-Euclidean Geometry. From a passage in a letter of Gerling's,* we learn that
Schweikart made his discovery when in Charkow. As he left
that place for Marburg in 1816, he seems by that date to have
advanced further than the stage which Gauss had reached
in 1817, according to the letter quoted above.
To Gerling, Gauss replied as follows: t
"... Schweikart's Memorandum has given me the greatest
pleasure, and I beg you to convey to him my hearty congratulations upon it. It could almost have been written by
myself. (Es ist mir fast alles aus der Seele geschrieben)....
I would only further add that I have extended the Astral
Geometry so far, that I can fully solve all its problems as soon
as the Constant = C is given, e.g. not only is the Defectt of the
angles of a plane triangle greater, the greater the area, but it is
exactly proportional to it; so that the area has a limit which
it can never reach; and this limit is the area of the triangle
formed by three lines asymptotic in pairs...."
From Bolyai's papers it appears that at this date he was
attempting to prove the truth of the Parallel Postulate. Also
in 1815-17 Lobatschewsky was working on the same traditional
lines.
~ 11. The above Memorandum is the only work of Schweikart's on the Astral Geometry that is known. Like Gauss, he
seems not to have published any of his researches. However,
at his instigation, and encouraged by Gauss, his nephew
Taurinus devoted himself to the subject. In 1825 he published a Theorie der Parallllinien, containing a treatment of
Parallels on Non-Euclidean Lines, the rejection of the Hypothesis of the Obtuse Angle, and some investigations resembling
those of Saccheri and Lambert on the Hypothesis of the Acute
Angle. For various reasons he decided that the Hypothesis of
* Cf. Gauss, Werke, vol. viii. p. 238.

t G~auss, Werk-e, vol. viii. p. 181.

+ See p. 54.