From the Beginning to Plato

GREEK ARITHMETIC, GEOMETRY AND HARMONICS 251
The last associate of Plato mentioned by Proclus is a Philip of Mende, standardly
assumed to be Philip of Opus. 2 Proclus concludes his history when he introduces
Euclid:
Those who have written histories bring the development of the science up
to this point. Euclid is not much younger than these people; he brought the
elements together, and he gave an order to many propositions of Eudoxus
and perfected many of Theaetetus’s; moreover, he gave irrefutable proofs
to propositions which had been demonstrated rather loosely by his
predecessors.
([8.74], 66.8–68.10)
The chronology of Archytas, Theaetetus and Eudoxus is very obscure, but a
certain consensus has emerged, based importantly on assumptions about the
relationships among their mathematical achievements. Theaetetus is thought to
have died in 369 before he was fifty. Eudoxus is said to have lived 53 years; his
death year is now generally put around the time of Plato’s (348–7) or shortly
thereafter. Archytas is thought to be an approximate contemporary of Plato, and
so born in the 420s. The important issue is not the exact dates, but the assumed
intellectual ordering: Archytas, Theaetetus, Eudoxus.
PART ONE:
THE FOURTH CENTURY
(1)
The Contents of Euclid’s Elements
The oldest Greek scientific text relevant to arithmetic, geometry, and solid
geometry is Euclid’s Elements. I give a brief description of its contents.
Although the proofs of Books I and II make use of the possibility of drawing a
circle with a given radius, the propositions are all concerned with straight lines
and rectilineal angles and figures. The focus of Book III is the circle and its
properties, and in Book IV Euclid treats rectilineal figures inscribed in or
circumscribed about circles. In Books I–IV no use is made of the concept of
proportionality (x:y :: z:w) and in consequence none—or virtually none—is
made of similarity. It seems clear that Euclid chose to postpone the introduction
of proportion, even at the cost of making proofs more complicated than they
need to have been. Indeed, he sometimes proves essentially equivalent
propositions, first independently of the concept of proportion and then —after he
has introduced the concept—using it. Moreover, sometimes the proportion-free
proof looks like a reworked version of the proof using proportion. 3
Book V is a logical tour de force in which Euclid gives a highly abstract
definition of proportionality for what he calls magnitudes (megethē) and