The fourth dimension

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 > * ° . . s . . . Fig. 23. In fig. 23, in the square on AB, there are9 points interior ° . . . - 9 4 at the corners A i

4 sides with 3 on each stb. aetacil as 1} on each side, because pelweeice equally to two squares The total is 16. There are 9 points in the square on BC.