From the Beginning to Plato

278 FROM THE BEGINNING TO PLATO
Von Fritz [8.46] ascribed the discovery of incommensurability to the
Pythagorean Hippasus of Metapontum on the basis of two texts of Iamblichus
printed under DK 18.4. The first says:
About Hippasus they say that he died at sea for impiety because he
published and described the sphere composed of twelve pentagons [i.e the
dodecahedron] and allowed himself to be credited with the discovery, but
all these things were the discoveries of ‘that man’ (for this is the way they
refer to Pythagoras and not by his name). Mathematics advanced because
of these things, and two people were most of all considered the first
mathematicians of the time, Theodoras of Cyrene and Hippocrates of
Chios.
(Iamblichus [8.53] 78.27–36)
In the other text Iamblichus ([8.51] 132.11–23) does not mention Hippasus, but
says that the divine destroyed at sea the person who revealed the construction of
the dodecahedron. Iamblichus adds that ‘some people say it was the person who
spoke out about irrationality and incommensurability who suffered this’. Von
Fritz claimed that Hippasus discovered irrationality in connection with the
regular pentagon (the face of the dodecahedron) and the star or pentagram, a
Pythagorean symbol formed by connecting alternating vertices of the pentagon. I
sketch, with reference to Figure 8.11, the reasoning von Fritz ascribed to
Hippasus.
Suppose one tries to find the greatest common measure of the side AE and the
diagonal AD. It is clear that AE≈AE′, so that AD −AE≈E′D. But E′D