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

4,5] EUCLID'S PARALLEL POSTULATE

11

his refusal to use it earlier than I. 29 are evidence that with
him it had only the value of an hypothesis. It seems at least
very probable that he realised the advantage of proving
without that postulate such theorems as could be established
independently; just as he refrained from using the method
of superposition, when other methods were available and
sufficient for the demonstration.
But the followers of Euclid were not so clear sighted. Fruitless attempts to prove the Parallel Postulate lasted well into
the nineteenth century. Indeed it will be surprising if the
use of the vicious direction-theory of parallels, advocated at
present in some influential quarters in England, does not raise
another crop of so-called demonstrations-the work of those
who are ignorant of the real foundations on which the Theory
of Parallels rests.
The assumption involved in the second question had also
an effect on the duration of the controversy. Had it not been
for the mistake which identified Geometry-the logical doctrine-with Geometry-the experimental science-the Parallel
Postulate would not so long have been regarded as a blemish
upon the body of Geometry. However, it is now admitted
that Geometry is a subject in which the assertions are that
such and such consequences follow from such and such premises. Whether entities such as the premises describe actually
exist is another matter. Whenever we think of Geometry as
a representation of the properties of the external world in
which we live, we are thinking of a branch of Applied Mathematics. That the Euclidean Geometry does describe those
properties we know perfectly well. But we also know that it
is not the only system of Geometry which will describe them.
To this point we shall return in the last pages of this book.
In the answer to the third question the solution of the
problem was found. This discovery will always be associated
with the names of Lobatschewsky and Bolyai. They were the
first to state publicly, and to establish rigorously, that a consistent system of Geometry can be built upon the assumptions,
explicit and implicit, of Euclid, when his Parallel Postulate is
omitted, and another, incompatible with it, put in its place.
The geometrical system constructed upon these foundations is
as consistent as that of Euclid. Not only so, by a proper
choice of a parameter entering into it, this system can be made
to describe and agree with the external relations of things.