nts out, found all he
wanted in the one.

As a mathematician Anaximander must have been familiar in various
aspects with the functions of the Infinite or Indefinable in the
organisation of thought. To the student of Euclid, for example, the
impossibility of adequately defining any of the fundamental elements of
the science of geometry--the point, the line, the surface--is a
familiar fact. In so far as a science of geometry is possible at all,
the exactness, which is its essential characteristic, is only {11}
attainable by starting from data which are in themselves impossible, as
of a point which has no magnitude, of a line which has no breadth, of a
surface which has no thickness. So in the science of abstract number
the fundamental assumptions, as that 1=1, _x=x_, etc., are contradicted
by every fact of experience, for in the world as we know it, absolute
equality is simply impossible to discover; and yet these fundamental
conceptions are in their development most powerful instruments for the
extension of man's command over his own experiences. Their
completeness of abstraction from the accidents of experience, from the
differences, qualifications, variations which contribute so largely to
the personal interests of life, this it is which makes the abstract
sciences demonstrative, exact, and universally applicable. In so far,
therefore, as we are permitted to grasp the conception of a perfectly
abstract existence prior to, and underlying, and enclosing, all
separate existences, so far also do we get to a conception which is
demonstrative, exact, and universally applicable throughout the whole
world of knowable objects.

Such a conception, however, by its absolute emptiness of content, does
not afford any means in itself of progression; somehow and somewhere a
principle of movement, of development, of concrete reality, must be
found or assumed, to link this ultimate abstraction of existence to the
multifarious phenomena {12} of existence as known. And it was,
perhaps, because Anaximander failed to work out this aspect of the
question, that in the subsequent leaders of the school _movement_,
rather than mere existence, was the principle chiefly insisted upon.

Before passing, however, to these successors of Anaximander, some
opinions of his which we have not perhaps the means of satisfactorily
correlating with his general conception, but which are not without
their individual interest, may here be noted. [14] Th