toms of the elements contained in the compound. These atoms are
supposed to be at certain distances from one another. It sometimes
happens that two compound substances differ in their chemical or
physical properties, or both, even though they have like chemical
elements in the same proportion. This phenomenon is called isomerism,
and the generally accepted explanation is that the atoms in isomeric
molecules are differently arranged, or grouped, in space. It is
difficult to imagine how atoms, alike in number, nature, and
relative proportion, can be so grouped as somehow to produce
compounds with different properties, particularly as in
three-dimensional space four is the greatest number of points whose
mutual distances, six in number, are all independent of each other.
In four-dimensional space, however, the _ten_ equal distances
between any two of _five_ points are geometrically independent, thus
greatly augmenting the number and variety of possible arrangements
of atoms.

This just escapes being the kind of proof demanded by science. If
the independence of all the possible distances between the atoms of
a molecule is absolutely required by theoretical chemical research,
then science is really compelled, in dealing with molecules of more
than four atoms, to make use of the idea of a space of more than
three dimensions.

THE ORBITAL MOTION OF SPHERES: CELL SUB-DIVISION

There is in nature another representation of hyper-dimensionality
which, though difficult to demonstrate, is too interesting and
significant to be omitted here.

Imagine a helix, intersected, in its vertical dimension, by a moving
plane. If necessary to assist the mind, suspend a spiral spring
above a pail of water, then raise the pail until the coils, one
after another, become immersed. The spring would represent the helix,
and the surface of the water the moving plane. Concentrating
attention upon this surface, you would see a point--the elliptical
cross-section of the wire where it intersected the plane--moving
round and round in a circle. Next conceive of the wire itself as a
lesser helix of many convolutions, and repeat the experiment. The
point of intersection would then continually return upon its own
track in a series of minute loops forming those lesser loops, which,
moving circle-wise, registered the involvement of the helix in the
plane.

It is easy to go on imagining complicated structures of the nature
of the spiral, and to suppose