continually approach each other they must at length meet. Here
is a demonstration contradicting an axiom; and no man has ever yet shown
the possibilities of reconciling them, nor yet of denying either side of
the contradiction.

Again: it is a fundamental axiom, contained in the definition of a
circle, that it must have a center; but the non-existence of this center
is mathematically demonstrable, as follows: Let the diameter of the
circle be bisected into two equal parts; the center must be in one, or
the other, of these parts, or between them. It can not be in one of
these parts, for they are equal; and, therefore, if it is in the one, it
must also be in the other, and thus the circle would have two centers,
which is absurd. Neither can it be between them, for they are in
contact. Therefore the center must be a point, destitute of extension,
something which does not occupy or exist in space. But as all existences
exist in space, and this supposed center does not, it can not be an
existence; therefore it is a non-existence.

In like manner it has been mathematically demonstrated,[325] that
motion, or any change in the rate of progress in a moving body, is
impossible; because in passing from any one degree of rapidity to
another, all the intermediate degrees must be passed through. As when a
train of cars moving four miles an hour strikes a train at rest, the
resulting instantaneous motion is two miles an hour; and the first train
must therefore be moving at the rate of four, and at the rate of two
miles an hour at the same time, which is impossible. And so the ancients
demonstrated the impossibility of motion.

Thus the non-existence of the most undeniable truths, and the
impossibilities of the most common facts are mathematically
demonstrable; and the proper refutation of such reasoning is, not the
scientific, but the common sensible; as when Plato refuted the
demonstration of the impossibility of motion, by getting up and walking
across the floor. In the hyperbola we have the mathematical
demonstration of the error of an axiom. In the infinite inch we behold
an absurdity mathematically demonstrated. So that it appears we can give
mathematical demonstration in support of untruth, impossibilities and
absurdities; and our reason can not discover the error of the reasoning!
Alas, for poor humanity, if an endless destiny depended upon such
scientific certainty! Yet mathematical reasoning about abstract truth is
univers