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

22

NON-EUCLIDEAN GEOMETRY

[CH. 1.

not improbable that our geometry was only a special case of a
more general one. At the same time Gerling sent him, at
Schweikart's request, a Memorandum, which the latter had
given him, desiring to know Gauss's opinion upon it.
This Memorandum is as follows:
" Marburg, December, 1818.
"There are two kinds of geometry-a geometry in the
strict sense-the Euclidean; and an astral geometry.
" Triangles in the latter have the property that the sum of
their three angles is not equal to two right angles.
" This being assumed, we can prove rigorously:
(a) That the sum of the three angles of a triangle is less
than two right angles;
(b) That the sum becomes always less, the greater the area
of the triangle;
(c) That the altitude of an isosceles right-angled triangle
continually grows, as the sides increase, but it can
never become greater than a certain length, which I
call the Constant.
" Squares have, therefore, the following form (Fig. 11):
FIG. 11.
" If this Constant were for us the radius of the earth (so
that every line drawn in the universe from one fixed star to
another, distant 90~ from the first, would be a tangent to the
surface of the earth), it would be infinitely great in comparison
with the spaces which occur in daily life.
*Gauss, Werke, vol. viii. p. 180.