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

24

NON-EUCLIDEAN GEOMETRY

[CH. I.

the Acute Angle must also be rejected, though he recognised the
logical possibility of the propositions which followed from it.
Again, it is from a letter which Gauss wrote to Taurinus in
1824, before the publication of his book, that we obtain the
fullest information of his views: *
" Your kind letter of the 30th October with the accompanying little theorem I have read not without pleasure, all the
more as up till now I have been accustomed to find not even a
trace of real geometrical insight in the majority of the people
who make new investigations upon the so-called Theory of
Parallels. In criticism of your work I have nothing (or not
much) more to say than that it is incomplete. It is true that
your treatment of the proof that the sum of the angles of a
plane triangle cannot be greater than 180~ is still slightly
lacking in geometrical precision. But there is no difficulty in
completing this; and there is no doubt that that impossibility can be established in the strictest possible fashion. The
position is quite different with regard to the second part, that
the sum of the angles cannot be smaller than 180~. This is
the real hitch, the obstacle, where all goes to pieces. I imagine
that you have not occupied yourself with this question for
long. It has been before me for over thirty years, and I don't
believe that anyone can have occupied himself more with this
second part than I, even though I have never published anything upon it. The assumption that the sum of the three
angles is smaller than 180~ leads to a peculiar Geometry, quite
distinct from our Euclidean, which is quite consistent. For
myself I have developed it quite satisfactorily, so that I can
solve every problem in it, with the exception of the determination of a Constant, which there is no means of settling a priori.
The greater we take this Constant, the nearer does the geometry
approach the Euclidean, and when it is given an infinite value
the two coincide. The theorems of that Geometry appear
almost paradoxical, and to the ignorant, absurd. When considered more carefully and calmly, one finds that they contain
nothing in itself impossible. For example, the three angles of
a triangle can become as small as we please, if only we may take
the sides large enough; however, the area of a triangle cannot
exceed a definite limit, no matter how great the sides are
taken, nor can it reach that limit. All my attempts to find a
* Cf. Gauss, TJerke, vol. viii. p. 186. This letter is reproduced in
facsimile in Engel u. Stackel's Theorie der Parallellinien (Leipzig, 1895).