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

840 MATHEMATICAL PILOSOPHY
now have the analogical clue. Following it you see immediately that in a 4-dimensional space (S4) an equation
ax+by+cz+dw+e =O of first degree in four variables
(x, y, z, w) represents an ordinary space (Sa)-named a
lineoid by my colleague, Professor F. N. Cole; that two
such equations together represent a plane (S2)-the plane
common to the two lineoids; that three such equations
(if independent) represent a line (Si)-the line common to
the three lineoids; and, finally, that four such equations
(if independent) represent a point (So)-the common
point of the four lineoids. You are already, you see, in
the midst of astonishing things: you see that an S4-a
hyperspace of the lowest dimensionality-contains a fourfold infinity (oo4) of lineoids (spaces like our own); you
see that any two of these have a plane for their intersection, that any three independent lineoids (in S4) have
a line in common, and that four of them have one point
in common and only one. I spoke of showing you a
" minor wonder." It is that in S4 two planes (unless they
happen to be in a same lineoid) have one and only one
point in common. To see that this statement is true,
consider four independent equations like the last of the
foregoing; two of them, as we have seen, represent a
plane; the other two represent another plane. What
points have the planes in common? The answer is:
those points whose coordinates (x, y, z, w) satisfy the
four equations. But, as you know, such a system of
equations is satisfied by only one set of values. Hence
the proposition. There are many other near-lying marvels in S4. One of them is that you can pass from the
inside to the outside of an ordinary sphere without going
through its surface. Another one is this: if in ordinary
space you wish to make a prison bounded by planes, you
have to use at least four planes; while in S4 the analogous