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

188 MATIIEMATICAL PHILOSOPHY
X1', X2', X3a, X4' be their respective transforms. Consider
the expression, or function,
(X -X2) (X3 -X4)
(X2 -X3) (X4 X1)
being very important, this function has a special name;
it is called the anharmonic ratio of i, X2, X3, X4 taken in the
order as written, and may be denoted by the symbol
R(xix2x3x4); so we may write
(X 1-X2) (X3 -X4)
R(XîX2X3X4) =, —
(X2 -X3) (X4-X1)
Now, the transform of R(xlx2x3x4) obviously is
R(x1'x2'x3'x4'). How are these two anharmonic ratios
related? To find the answer it is sufficient to replace
X1t, X2', X3, X4' of the transform ratio by their respective
values
ax +b ax2 +b ax + b ax4 +b
cxi +d' cx2+d' cx3 +d' cx4+d'
and to simplify the result; by doing so, which is easy, you
will find that R(xlx2x3x4) =R(xl'x2'x3'x4'). We have here,
as you see, an exceedingly beautiful specimen of mathematical invariance: namely, under each and all of the
threefold infinity of transformations of form (I), each of
them converting the entire class of real numbers into
itself, the anharmonic ratio, (R(xlx2x3x4), of any ordered
set of four numbers, remains absolutely unchanged.
The invariant in question, though it here appears as
a function of pure numbers, lies, as I have intimated, at
the heart of Projective Geometry. We may see the
thing in geometric light readily as follows. Suppose Fig.
22 to be in a projective plane; let x be the distance from