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

HYPERSPACES 89
To distinguish a ine of ordinary space from all its other
lines, it is necessary and sufficient to tell four independent
facts about it; ordinary space contains oo 4 lines, and you
see that Plicker's famous line geometry (of ordinary
space), which studies configurations composed of lines
(and not of points), is a four-dimensional geometry. Let
us return for a moment to the plane; think of it as a
plenum of circles. Each of its points is the center of an
oo of circles, and it has oo 2 points; and so, ydu see, a plane
has 003 circles; in a plane the circle has three degrees of
freedom,-three coordinates or parameters; a plane of
circles is a three-dimensional space-as rich in circles as in
point-triads-as rich in circles as ordinary space in points.
You can readily show that ordinary space is four-dimensional in spheres, as we have seen it to be in lines, fivedimensional in flat line-pencils (explained before), sixdimensional in circles, and so on and on to your heart's
content.
I venture to believe that the foregoing illustrations
have sufficiently disclosed one of the meanings of the
term "hyperspace": that meaning, namely, according
to which the term signifies an ensemble of geometric, or
spatial, entities, or elements, of such a kind that an
undetermined (or arbitrary) one of them has, in the
ensemble, four or more degrees of freedom. This statement is not designed to be a definition of the meaning,
but only a good-enough description of it. In this meaning
of hyperspace there certainly is nothing to mystify; for,
in order to find examples of such hyperspaces, one is not
obliged to perform the familiar mathematical feat,-which
many good people seem to find difficult or even impossible,
-of going beyond the great domain of Imagination into
the infinitely vaster domain of pure Conception. I have