- THE EARLIEST GREEK GEOMETRY 289
"arranged many of the theorems of Eudoxus"; (2) an anonymous scholium (com-
mentary) on Euclid's Book 5, which asserts that the book is the creation "of a
certain Eudoxus, [the student] of the teacher Plato" (Allman, 1889, p. 132).
His central observation is a very simple one: Suppose that D and S are respec-
tively the diagonal and side of a square (or pentagon). Even though there are no
integers m and ç such that mD = nS, so that the ratio D : S cannot be defined as
ç : m for any integers, it remains true that for every pair of integers m and ç there
is a trichotomy: Either mD < nS or mD — nS or mD > nS. That fact makes it
possible at least to define what is meant by saying that the ratio of D to S is the
same for all squares. We simply define the proportion D\ : Si :: D2 • S2 for two
different squares to mean that whatever relation holds between mD\ and nS\ for
a given pair of integers m and n, that same relation holds between m£>2 and nSi.
Accordingly, as defined by Euclid at the beginning of Book 5, "A relation that two
magnitudes of the same kind have due to their sizes is a ratio." As a definition, this
statement is somewhat lacking, but we may paraphrase it as follows: "the relative
size of one magnitude in terms of a second magnitude of the same kind is the ratio
of the first to the second." We think of size as resulting from measurement and
relative size as the result of dividing one measurement by another, but Euclid keeps
silent on both of these points. Then, "Two magnitudes are said to have a ratio to
each other if they are capable of exceeding each other when multiplied." That is,
some multiple of each is larger than the other. Thus, the periphery of a circle and
its diameter can have a ratio, but the periphery of a circle and its center cannot.
Although the definition of ratio would be hard to use, fortunately there is no need
to use it. What is needed is equality of ratios, that is, proportion. That definition
follows from the trichotomy just mentioned. Here is the definition given in Book 5
of Euclid, with the material in brackets added from the discussion just given to
clarify the meaning:
Magnitudes are said to be in the same ratio, the first to the second
[Di : S\] and the third to the fourth [D% : S2], when, if any equimul-
tiples whatever be taken of the first and third [mD\ and mDg] and
any equimultiples whatever of the second and fourth [nSi and nS2],
the former equimultiples alike exceed, are alike equal to, or are alike
less than the latter equimultiples taken in corresponding order [that
is, mDi > nS\ and m£>2 > nS2, or mD\ = nS\ and m£>2 = nS%, or
mD\ < nS\ and m.D 2 < nSy.Let us now look again at our conjectured early Pythagorean proof of Euclid's
Proposition 1 of Book 6 of the Elements. How much change is required to make
this proof rigorous? Very little. Where we have assumed that 3BC = 2CD, it is
only necessary to consider the cases 3BC > 2CD and SBC < 2CD and show with
the same figure that 3ABC > 2ACD and 3ABC < 2ACD respectively, and that is
done by using the trivial corollary of Proposition 38 of Book 1: // two triangles have
equal altitudes and unequal bases, the one with the larger base is larger. Eudoxus has
not only shown how proportion can be defined so as to apply to incommensurables,
he has done so in a way that fits together seamlessly with earlier proofs that apply
only in the commensurable case. If only the fixes for bugs in modern computer
programs were so simple and effective!