- NUMBER SYSTEMS 201
Side é—é—é—é—é
Diagonal é—é—é—é—é—é—i—
Side é é é é é é
Diagonal é—é—é—é—\—é—é—é—éFIGURE 1. Diagonal and side of a regular pentagon. If a unit is
chosen that divides the side into equal parts, it cannot divide the
diagonal into equal parts, and vice versa.pentagon. Therefore, in this case the Euclidean algorithm will never produce an
equal pair of lines. We know, however, that it must produce an equal pair if a
common measure exists. We conclude that no common measure can exist for the
side and diagonal of a pentagon.
The argument just presented was originally given by von Fritz (1945). Knorr
(1975, pp. 22-36) argues against this approach, however, pointing out that the
simple arithmetic relation d^2 = 2s^2 satisfied by the diagonal and side of a square
can be used in several ways to show that d and s could not both be integers, no
matter what length is chosen as unit. Knorr prefers a reconstruction closer to
the argument given in Plato's Meno, in which the problem of doubling a square is
discussed.^12 Knorr points out that when discussing irrationals, Plato and Aristotle
always invoke the side and diagonal of a square, never the pentagon or the related
problem of dividing a line in mean and extreme ratio, which they certainly knew
about.
Whatever the argument used, the Greeks discovered the existence of incom-
mensurable pairs of line segments before the time of Plato. For Pythagorean meta-
physics this discovery was disturbing: Number, it seems, is not adequate to explain
all of nature. A legend arose that the Pythagoreans attempted to keep secret the
discovery of this paradox.^13 However, scholars believe that the discovery of in-
commensurables came near the end of the fifth century BCE, when the original
Pythagorean group was already defunct.
(^12) In Chapter 9 we invoke the same passage to speculate on the origin of the Pythagorean theorem.
(^13) The legend probably arose from a passage in Chapter 18, Section 88 of the Life of Pythagoras
by Iamblichus. Iamblichus says that a certain Hippasus perished at sea, a punishment for his
impiety because he published "the sphere of the 12 pentagons" (probably the radius of the sphere
circumscribed about a dodecahedron), talcing credit as if he had discovered it, when actually
everything was a discovery of That Man (Pythagoras, who was too august a personage to be
called by name). Apparently, new knowledge was to be kept in-house as a secret of the initiated
and attributed in a mystical sense to Pythagoras.
A