Hinton - The Fourth Dimension.pdf

SECOND CHAPTER IN THE HISTORY OF FOUR SPACE 55
But a connection arises in this way. It is evident that,
instead of taking a simple shear as I have done, and
defining it as that change of the arrangement of the
particles of a solid which they will undergo without
offering any resistance due to their mutual action, I
might take a complex motion, composed of a shear and
a rotation together, or some other kind of deformation.
Let us suppose such an alteration picked out and
defined as the one which means simple rotation, then
the type, according to which all bodies will alter by this
rotation, is fixed.
Looking at the movements of this kind, we should say
that the objects were altering their shape as well as
rotating. But to the inhabitants of that world they
would seem to be unaltered, and our figures in their
motions would seem to them to alter.
In such a world the features of geometry are different.
We have seen one such difference in the case of our illustration
of the world of shear, where the square on the
hypothenuse was equal to the difference, not the sum, of
the squares on the sides.
In our illustration we have the same laws of parallel
lines as in our ordinary rotation world, but in general the
laws of parallel lines are different.
In one of these worlds of a different constitution of
matter through one point there can be two parallels to
a given line. In another of them there can be none, that
is, although a line be drawn parallel to another it will
meet it after a time.
Now it was precisely in this respect of parallels that
Lobatchewsky and Bolyai discovered these different
worlds. They did not think of them as worlds of matter,
but they discovered that space did not necessarily mean
that our law of parallels is true. They made the
distinction between laws of space and laws of matter,