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

10

NON-EUCLIDEAN GEOMETRY

[CH. I.

The sides of the triangle BB'B" are bisected at M and N.
Therefore the line bisecting B'B" at right angles is also
perpendicular to MN.
But this line bisects L B'A'B", since A'B'=A'B".
Now produce A'B" to C", so that B"C"= BC = B'C'.
Join C' C", MC" and MC.
The triangles MAC and MA'C" are congruent, and it follows
that MC and MC" are in one straight line.
Since A'C'=A'C", the line bisecting C'C" at right angles
coincides with the line bisecting B'B" at right angles.
Therefore MN and MP are perpendicular to the same
straight line.
Therefore MNP are collinear.
Proceeding to the points A, B, D, A', B', D' we have a
corresponding result, and in this way our, theorem is proved.
~ 5. From the Commentary of Proclus * it is known that not
long after Euclid's own time his Parallel Postulate was the
subject of controversy. The questions in dispute remained
unsolved till the nineteenth century, though many mathematicians of eminence devoted much time and thought to
their investigation. Three separate problems found a place
in this discussion:
(i) Can the Parallel Postulate be deduced from the other
assumptions on which Euclid's Geometry is based?
(ii) If not, is it an assumption demanded by the facts of
experience, so that the system of propositions deduced from
the fundamental assumptions will describe the space in which
we live?
(iii) And finally, are both it and assumptions incompatible
with it consistent with the other assumptions, so that the
adoption of the Euclidean Hypothesis can be regarded as an
arbitrary specialisation of a more general system, accepted not
because it is more true than the others, but because the
Geometry founded upon it is simpler and more convenient?
There can be little doubt that Euclid himself was convinced
that the first of these questions must be answered in the
negative. The place he assigned to the Parallel Postulate and
* Cf. Friedlein, Procli Diadochi in primumr Euclidis elementorum
librum commentarii (Leipzig, 1873). Also Heath's Euclid, vol. i.
Introduction, chapter iv.