From the Beginning to Plato

254 FROM THE BEGINNING TO PLATO
characterizations of the edges of the regular solids apparently provide the main
motivation for the elaborate classification undertaken in Book X.
(2)
Eudoxus and Books V and XII of Euclid’s Elements
Two scholia associate Book V with Eudoxus. The more interesting one says:
This book is said to belong to Eudoxus of Cnidus, the mathematician who
lived at the time of Plato, but it is nevertheless ascribed to Euclid and not
wrongly. For why shouldn’t a thing be assigned to one person as far as its
discovery is concerned, even though it is agreed by everyone that it is
Euclid’s as far as the arrangement of things with respect to elementhood
and with respect to the relations of implication with other things of what
has been arranged?
([8.30] 5:282.13–20)
It is difficult to make any precise determination of the roles assigned to Eudoxus
and Euclid by this scholium, but it would seem that at least some equivalent of
the Book V definition of proportionality and an indication of its viability should
be ascribed to Eudoxus, and some non-trivial reorganization to Euclid.
Book XII is also closely connected with Eudoxus. For in the preface to On
Sphere and Cylinder I ([8.22] 1:4.2–13) Archimedes ascribes to him proofs of
equivalents of two propositions from Book XII, and in the prefatory letter to The
Method ([8.22] 2:430.1–9) he contrasts Eudoxus being the first person to prove
these propositions with Democritus being the first to assert one or both of them
without proof 5 . Moreover, in the preface to the Metrica Heron ([8.45] 3:2.13–
18) credits Eudoxus with the first proofs of one of these propositions and of XII.
2. In the prefatory letter to Quadrature of the Parabola ([8.22] 2:264.5–25)
Archimedes connects the proof of the equivalents of several propositions in Book
XII with what he there calls a lemma and is more or less equivalent to X.1. It
seems clear that Eudoxus was responsible for some considerable part of the
contents of Book XII, although again we have no way of knowing how much
Euclid contributed to its formulation. Since the treatment of proportion in Book
V and the use of exhaustion in Book XII can be said to represent the outstanding
logical and conceptual achievements of the Elements, there is good reason to
compare Eudoxus’ accomplishments with foundational work of the nineteenth
century by people such as Weierstrass, Cantor, Dedekind, and Frege. 6