ON THE FIRST PART OF PLATO 'S PARMENIDES. 13 For suppose we have realised the incommensurability of the " side and diagonal," and wish to reconcile our discovery with the Pythagorean view of extension. We can only do so by assuming that side and diagonal are respectively multiples of mutually incommensurable unit lines. Thus if the side of the square consists say of n unit lines each equal to x, the diagonal must consist of n units each equal to x >/ 2 ; then comparing the side-unit with the diagonal- unit, the latter exceeds the former by a quantity viz. x (V 2-1), which we shall have to regard as composed of still minuter units incommensurable with x, or, as Par- menides is made to say, " by a part of magnitude less than magnitude itself". The same set of ideas underlie his next paradox ; " Can one quantity be equal to another by something less than equality itself ?" On the Pythagorean theory the equality of two lines would be due to the fact that each contained the same number of units. Each side of the square is equal to each of the rest because each contains n times the unit line. But, we may understand the opponent to rejoin, it is tactily assumed that not only the number of units in each side but also the individual units themselves are equal, other- wise the resulting lines will be unequal. And on Pythagorean principles the equality of the units can mean nothing but that they in turn are composed of an equal number of more ultimate units, and this is inconsistent with their supposed indivisible character. The last of Parmenides' supplementary arguments against the Pythagorean position is harder to understand, and this is perhaps why certain persons of whom Proclus speaks wished to reject the passage (131 d-e) as spurious, though it is not easy to see why, if not genuine, it should have been inserted. I venture with some diffidence to suggest the following as approximately representing Plato's meaning in this obscure sentence. On the Pythagorean view the point or unit is of course a minimum of extension and may thus fairly be taken to be signified by the expression " the small itself ". But the point, as we have already seen, is itself, for the Pythagoreans, a quantum, and it therefore contains parts, each of which is, in Plato's words, " smaller than the small itself". Now suppose you add one of these parts to one of two equal magnitudes, what will happen ? The magnitude so augmented will not become larger than the other, for it can ex hypothesi only be larger if it contains a greater number of units ; it will not remain equal, for equality means com- position, out of the same number of equal units ; thus nothing
