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

HYPERSPACES 827
space there are, you see, oo 3 points, and a point has therein
three degrees of freedom and only three.
We are going very soon to see very clearly one of the
three meanings of the term n-dimensional space, n greater
than three. Do not fail to note that thus far we have
regarded the line, the plane, and ordinary space as fields,
or plena, or spaces, of points; the point, that is, has been
taken for element; but nothing constrains us to elect the
point to that position; we can geometrize just as well,
sometimes better, with some other entity taken as element;
we may choose for element the point pair or point triad,
and so on; in the case of the plane, we may take the
line or the circle or something else for element; in the
case of ordinary space we may take for element any of
the foregoing entities or a plane, for example, or a sphere,
and so on. It is true that, from time immemorial until
a little less than a century ago, the point was exclusively
employed as geometric or spatial element, but there is
nothing in the ten commandments nor even in the Volstead Act to prevent the use of something else. The
point's ages-old monopoly was broken up mainly by
Julius Plucker (I80o-I868)-one of the greatest of
geometricians and a distinguished physicist besides,who geometrized the plane in terms of its lines and
geometrized ordinary space in terms of its planes and its
lines-thus emancipating geometry forever from its old
bondage to points. A geometry in which the point-pair
is taken for element will deal with the properties of
configurations composed, not of points, but of point-pairs.
Now, in a line a point-pair has two coordinates, two
degrees of freedom; in a plane it has four; in ordinary
space it has six; a line has oc2 point-pairs; a plane, oo4
of them; and ordinary space, oo 6; and so you see that,