Hinton - The Fourth Dimension.pdf

48
THE FOURTH DIMENSION
the end of a line would come when it turned about a
point keeping one extremity fixed at the point.
We can solve this problem in a particular case. If we
can find a square lying slantwise amongst the dots which is
equal to one which goes regularly, we shall know that the
two sides are equal, and that the slanting side is equal to the
straight-way side. Thus the volume and shape of a figure
remaining unchanged will be the test of its having rotated
about the point, so that we can say that its side in its first
position would turn into its side in the second position.
Now, such a square can be found in the one whose side
is five units in length.
E
A
D
B
C
F
H G
Fig. 23.
In fig. 23, in the square on AB, there are—
9 points interior . . . . . 9
4 at the corners . . . . . 1
4 sides with 3 on each side, considered as
11⁄2 on each side, because belonging
equally to two squares. . . 6
The total is 16. There are 9 points in the square
on BC.