Metageometry
The theories which are generally connected with the names of Lobatchewsky and Bolyai bear a singular and curious relation to the subject of higher space.
In order to show what this relation is, I must ask the reader to be at the pains to count carefully the sets of points by which I shall estimate the volumes of certain figures.
No mathematical processes beyond this simple one of counting will be necessary.
Fig. 19.
Let us suppose we have before us in [fig. 19] a plane covered with points at regular intervals, so placed that every four determine a square.
Now it is evident that as four points determine a square, so four squares meet in a point.
Fig. 20.