Again, according to the argument of Zeno, there must be some magnitude without parts, since it is impossible to touch an infinite number of things in a finite time, when touching each of them, and that which moves must first reach half-way, and half clearly belongs to that which is not without parts. But if anything travelling along a line touches an infinite series in a finite time, secondly if the faster it travels the greater the space it covers in the same time, and lastly if the movement of thought is the quickest movement, then even thought must touch an infinite series one by one in a finite time. If, then, thought touching the series one by one is counting, then it must be possible to count an infinite series in finite time. If this is impossible, then there must exist an indivisible line.

The next argument, we are told, is used by the mathematicians to prove that the indivisible line must exist, if we admit that “commensurate” lines are those which are measured by the same unit, and all the lines measured are “commensurate.” For there must be some length by which they are all measured. And this must be incapable of division. For if it is divisible, then its parts can also be expressed in the terms of some unit. For they are commensurate with the whole. So that the measurement of each part would be double its half; since this is impossible the unit of measurement must itself be indivisible.

Again, just as the lines built up from the unit of measurement are all composed of units without parts, so also must those be which are once measured by it. The same thing will also happen in plane figures; for all the squares on rational lines are commensurable with each other, so that their unit of measurement