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

9, 10]

GAUSS AND SCHWEIKART

21

block which he found therein. That this was not unlike the
obstacle which so far had baffled his own efforts. " However,
I am always hopeful," he added, "that some day, and that
in my own lifetime, a way over this obstacle will be revealed." *
Though John Bolyai afterwards saw how groundless his
suspicions were, he always held that Gauss had treated him
badly in this matter; and it does seem unfortunate that Gauss
did not more effectively use his great influence to rescue from
ill-merited neglect the notable work of the two comparatively
unknown young mathematicians, Bolyai and Lobatschewsky.
Not till years after they had passed away did the scientific
world realise the immense value of their discoveries.
~ 10. Bolyai's discovery was made in 1823, and first published in 1832. Far away in Kasan, Lobatschewsky-one of
the Professors of Mathematics in the local University-not
later than 1829, and probably as early as 1826, had also discovered this new Geometry, of which the Euclidean was a
special case. Thus it is interesting to trace, so far as we can,
Gauss's attitude to the Theory of Parallels at that time. The
chief available authorities are some letters of his which still
survive, and some notes found among his papers.t
In the early years of the nineteenth century he shared the
common belief that a proof of the Euclidean Hypothesis might
possibly be found. But in 1817 we find him writing to Olbers
as follows:
" Wachter has published a little paper on the 'First Principles of Geometry,' of which you will probably get a copy
through Lindenau. Although he has got nearer the root of
the matter than his predecessors, his proof is no more rigorous
than any of the others. I am becoming more and more
convinced that the necessity of our geometry cannot be
proved.."+
In 1819 he learnt from Gerling in Marburg that one of his
colleagues, Schweikart-a Professor of Law, but formerly a
keen student of Mathematics-had informed him that he was
practically certain that Euclid's Postulate could not be proved
without some hypothesis or other; and that it seemed to him
* Gauss, Werke, vol. viii. p. 160.
t See Gauss, Werke, vol. viii.
+ Gauss, Werke, vol. viii. p. 177.