Mathematical philosophy, a study of fate and freedom; lectures for educated laymen, by Cassius J. Keyser.

856 MATHEMATICAL PHILOSOPHY
on Modern Mathematics, edited by J. W. A. Young).
We have seen in what respects the bases, the postulate
systems, of the three geometries are alike and in what
respects they are unlike. We naturally pass to a comparison of the superstructures-to a comparison, that is,
of the theorematic contents of the geometries. Do the
geometries intersect? Have they, that is, any theorems
in common? The answer is obvious: they have in common such and only such theorems as are deducible from
the assumptions that are common to the three systems
thereof. One such theorem is this: The summit angles
of a birectangular quadrilateral are equal; in other words,
if ABCD be a quadrilateral having right angles at A and
B and having the side AC equal to the side BD, then the
angle at C is equal to that at D. You may wish to make
a list of such common theorems as an exercise. More
striking are the theorems in which the geometries differ;
such differences are of course due to the differences in the
postulate systems. Let us notice some of them. The
one most commonly mentioned relates to the sum of the
angles of a triangle. In the parabolic geometry that sum
is constant (the same for all triangles) and is exactly two
right angles, as you know; in the hyperbolic and elliptic
geometries the sum is variable (depending upon the triangle's size); in the former geometry it is always less
than two right angles and decreases as the triangle's area
increases; in the latter, the sum is always greater than
two right angles and increases with the area.
Again: If two lines of a plane are perpendicular to a
third, then, in parabolic geometry, the two are parallel;
in the hyperbolic, they are not parallel nor do they intersect; in the elliptic, they meet at a point whose distance