Page:Early Greek philosophy by John Burnet, 3rd edition, 1920.djvu/304

290 It is enough to say that the Pythagoreans meant by the Unlimited the res extensa.

As the Unlimited is spatial, the Limit must be spatial too, and we should expect to find that the point, the line, and the surface were regarded as forms of the Limit. That was the later doctrine; but the characteristic feature of Pythagoreanism is just that the point was not regarded as a limit, but as the first product of the Limit and the Unlimited, and was identified with the arithmetical unit instead of with zero. According to this view, then, the point has one dimension, the line two, the surface three, and the solid four. In, other words, the Pythagorean points have magnitude, their lines breadth, and their surfaces thickness. The whole theory, in short, turns on the definition of the point as a unit "having position" (μονὰς θέσιν ἔχουσα). It was out of such elements that it seemed possible to construct a world.

146. This way of regarding the point, the line, and the surface is closely bound up with the practice of representing numbers by dots arranged in symmetrical patterns, which we have seen reason for attributing to the Pythagoreans (§ 47). Geometry had already made considerable advances, but the old view of quantity as a sum of units had not been revised, and so, the point was identified with 1 instead of with 0. That is the answer to Zeller's contention that to regard the Pythagorean numbers as spatial is to ignore the fact that the doctrine was originally arithmetical rather than geometrical. Our interpretation takes full account of that