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

LEGENDRE

17

All these results had been obtained many years earlier by
Saccheri.
Legendre made various attempts to prove that the sum cannot be less than two right angles, even in a single triangle;
but these efforts all failed, as we now know they were bound to
do. He published several so-called proofs in the successive
editions of his text-book of geometry, the Eltdments de Geomttrie.
All contained some assumption equivalent to the hypothesis
which they were meant to establish.
For example, in one he assumes that there cannot be an
absolute unit of length; * an alternative hypothesis already
noted by Lambert (1728-1777).t
In a second he assumes that from any point whatever, taken
within an angle, we can always draw a straight line which
will cut the two lines bounding the angle.
In a third he shows that the Parallel Postulate would be
true, if a circle can always be drawn through any three points
not in a straight line.
In another [cf. p. 279,14th Ed.] he argues somewhat as follows:
A straight line divides a plane in which it lies into two congruent parts. Thus two rays from a point enclosing an angle
less than two right angles contain an area less than half the
plane. If an infinite straight line lies wholly in the region
bounded by these two rays, it would follow that the area of
half the plane can be enclosed within an area itself less than
half the plane.
Bertrand's well-known " proof " (1778) of the Parallel Postulate j and another similar to it to be found in Crelle's Journal
(1834) fail for the same reason as does Legendre's. They
depend upon a comparison of infinite areas. But a process of
reasoning which is sound for finite magnitudes need not be
valid in the case of infinite magnitudes. If it is to be extended
to such a field, the legitimacy of the extension must be proved.
Lobatschewsky himself dealt with these proofs, and pointed out
the weakness in the argument. First of all, the idea of congruence, as applied to finite areas, is used in dealing with infinite regions, without any exact statement of its meaning in
this connection. Further-and here it seems best to quote his
* See below, p. 90. Also Bonola, loc. cit. ~ 20.
t Cf. Engel u. Stackel, loc. cit. p. 200.
+ f. Frankland, Theories of Parallelism, p. 26 (Cambridge, 1910),
N.-E.G B