the intellectual activity of the Greeks would have led to the
beginning of natural science. But it would seem that the very
purity and perfection which was aimed at in their system of
geometry stood in the way of any extension or application of its
methods and spirit to the field of nature. One example of this is
worthy of attention. In modern teaching the idea of magnitude as
generated by motion is freely introduced. A line is described by a
moving point; a plane by a moving line; a solid by a moving plane.
It may, at first sight, seem singular that this conception finds
no place in the Euclidian system. But we may regard the omission
as a mark of logical purity and rigor. Had the real or supposed
advantages of introducing motion into geometrical conceptions been
suggested to Euclid, we may suppose him to have replied that the
theorems of space are independent of time; that the idea of motion
necessarily implies time, and that, in consequence, to avail
ourselves of it would be to introduce an extraneous element into
geometry.

It is quite possible that the contempt of the ancient philosophers
for the practical application of their science, which has
continued in some form to our own time, and which is not
altogether unwholesome, was a powerful factor in the same
direction. The result was that, in keeping geometry pure from
ideas which did not belong to it, it failed to form what might
otherwise have been the basis of physical science. Its founders
missed the discovery that methods similar to those of geometric
demonstration could be extended into other and wider fields than
that of space. Thus not only the development of applied geometry
but the reduction of other conceptions to a rigorous mathematical
form was indefinitely postponed.