been the construction of the forms of logic. The radical
difference between the demonstration of a theorem of geometry and the
reasoning of every-day life which the masses of men must have practised
from the beginning, and which few even to-day ever get beyond, is so
evident at a glance that I need not dwell upon it. The principal
feature of this advance is that, by one of those antinomies of human
intellect of which examples are not wanting even in our own time, the
development of abstract ideas preceded the concrete knowledge of
natural phenomena. When we reflect that in the geometry of Euclid the
science of space was brought to such logical perfection that even
to-day its teachers are not agreed as to the practicability of any
great improvement upon it, we cannot avoid the feeling that a very
slight change in the direction of 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