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

18

NON-EUCLIDEAN GEOMETRY

[CH. I.

own words: " when we are dealing with areas extending to
infinity, we must in this case, as in all other parts of mathematics, understand by the ratio of two of these infinitely great
numbers, the limit to which this tends when the numerator and
denominator of the fraction continually increase." *
It is not a little surprising that at the present day mathematicians of distinction have been found quoting Bertrand's
argument with approval.t
~ 8. Both Legendre and Saccheri, in their discussion of
these hypotheses, make use of the axiom that the length of the
straight line is infinite, and they also assume the Postulate of
Archimedes. Hilbert + showed that the Euclidean Geometry
could be built up without the Postulate of Archimedes. Dehn ~
investigated what effect the rejection of the Postulate of
Archimedes would have on the results obtained by Saccheri
and Legendre. He found that the sum of the angles of a triangle can be greater than two right angles in this case. In
other words, the Hypothesis of the Obtuse Angle is possible.
Again, he showed that without the Postulate of Archimedes
we can deduce from the angle-sum in a single triangle being
two right angles, that the angle-sum in every triangle is two
right angles. But his most important discovery was that,
when the Postulate of Archimedes is rejected, the Parallel
Postulate does not follow from the sum of the angles of a
triangle being equal to two right angles. He proved that
there is a Non-Archimedean Geometry in which the angle-sum
in every triangle is two right angles, and the Parallel Postulate
does not hold.
His discovery has been referred to in this place because it
shows that the Euclidean Hypothesis is superior to the others,
which have been suggested as equivalent to it. Upon the
Euclidean Hypothesis, without the aid of the Postulate of
Archimedes, the Euclidean Geometry can be based. If we
* Cf. Lobatschewsky, XNew Principles of Geometry with a Complete
Theory of Parallels, Engel's translation, p. 71, in Engel u. Stdickel's
Urkunden zur Geschichte der nichtekliclischen Geometrie, I. (Leipzig,
1898).
t Cf. Frankland, The Mathematical Gazette, vol. vii. p. 136 (1913)
and p. 332 (1914); lVature, Sept. 7, 1911, and Oct. 5, 1911.
+ Cf. loc. cit. chapter iii.
~Cf. Math, Ann. vol. liii. p. 404 (1900),