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 | dc | dd |
| ca | cb | cc | cd |
| ba | bb | bc | bd |
| aa | ab | ac | ad |
This complex or manifold gives a two-way order. I can represent it by a set of points, if I am on my guard against assuming any relation of distance.
Fig. 15.
Pythagoras studied this two-fold way of counting in reference to material bodies, and 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. 16.