Pythagoras invented the two-way counting. Let us represent the single-way counting by the posits _aa_, _ab_, _ac_, _ad_, using these pairs of letters instead of the numbers 1, 2, 3, 4. I put an _a_ in each case first for a reason which will immediately appear. We have a sequence and order. There is no conception of distance necessarily involved. The difference between the posits is one of order not of distance—only 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_ _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. [Illustration: 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. [Illustration: Fig. 16. 1 and 2] It is evident that in fig. 16 four of the points determine a square, which square we may take as the unit of measurement for areas. 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. Thus we may say that the point value of the square shown is one point, for if we take the square in fig. 16 (1) it has four points, but each of these belong equally to four other squares. Hence one fourth of each of them belongs to the square (1) in fig. 16. Thus the point value of the square is one point. The result of counting the points is the same as that arrived at by reckoning the square units enclosed. Hence, if we wish to measure the area of any square we can take the number of points it encloses, count these as one each, and take one-fourth of the number of points at its corners. [Illustration: Fig. 17.] Now draw a diagonal square as shown in fig. 17. It contains one point and the four corners count for one point more; hence its point value is 2. The value is the measure of its area—the size of this square is two of the unit squares.