The fourth dimension

THE FIRST CHAPTER IN THE HISTORY OF FOUR SPACE 29

between the posits is one of order not of distanceonly when identified with a number of equal material things in juxtaposition does the notion of distance arise.

Now, besides the simple series I can have, starting from aa, ba, ca, da, from ab, bb, cb, db, and so on, and forming

a scheme: da db de dd

ca cb co cd ba bb be bd aa ab ae ad

This complex or manifold gives a two-way order. 1 can

represent it by a set of points, if I am on my guard s * ¢ » against assuming any relation of distance. « « + e Pythagoras studied this two-fold way of e e + » counting in reference to material bodies, and » » e e discovered that most remarkable property of the combination of number and matter that bears his name.

The Pythagorean property of an extended material system can be exhibited in a manner which will be of use to us afterwards, and which therefore I will employ now instead of using the kind of figure which he himself employed.

Consider a two-fold field of points arranged in regular rows. Such a field will be presupposed in the following

argument,

Fig. 15.

- © + «© ¢ + It is evident that in fig. 16 four

° ia) . » of the points determine a square,

. e KH + which square we may take as the

. eo. * unit of measurement for areas. 2 But we can also measure areas

in another way. Fig. 16 (1) shows four points determining a square. But four squares also meet in a point, fig. 16 (2).

Hence a point at the corner of a square belongs equally to four squares.

s 1 Fig, 16,