Greeks and‘Origins’ 49
challenge is to reconstruct what the two did. The main fact known about Democritus is that
he was an ‘atomist’, that is, regarded the universe as made up of atoms; and this has led to a
great deal of speculation on his possible use of the infinitely small. For Eudoxus there is a more
accepted reconstruction, based on the supposition that his work was much like Euclid’s—a
supposition which in turn owes something to Archimedes’s passing remark.- In Plato’sTheaetetus, Theaetetus claims that his teacher Theodorus was drawing diagrams to
show that a square of 3 square feet was not commensurable in respect of side with a square of
1 square foot. This relates to the problem in theMenofor the square of 8 square feet; the idea
is that the square root of three is not a fraction. Theaetetus claims that Theodorus continued
with 5, 7, up to 17, and then for some reason could go no further. This has given rise to a
great number of reconstructions of the (geometric) method which Theodorus could have used
which would have worked for 17 but not for 19. Euclid’s proof of the same result^10 is of no
help; it is in his extremely difficult book X, and would work for any number you like.
Again, it can be seen that mathematics is peculiarly susceptible to this kind of ‘history’. A brilliant
example is Fowler’s (1999), which gives a plausible version of how Greek mathematicians thought
of ratios in the fourth centurybce, with a great deal of mathematics and scholarly apparatus to
back it up; aside from giving a great deal of information on what we knowfor certain(manuscripts,
papyrus fragments, etc.) about mathematics in Plato’s time, Fowler constructs a detailed model of
how it could have worked. Reviewers have been respectful (see, for example, Berggren (2003)), but
have usually expressed natural reservations.
9. On ratios
It seems necessary to insert something here to clear up what, in classical Greek terms, a ‘ratio’
was—even if the experts are not altogether agreed on it. Euclid’s book V starts by saying that a ratio
is something which two quantities may have (simplifying, if they are of the same kind, say both
lengths, or both times,...). He then sets out a complicated criterion for two ratios to be equal. This
was much disliked, when it was not simply misunderstood, by his Islamic and medieval successors;
it is popularly believed that Isaac Barrow in the seventeenth century was the first mathematician
(after the Greeks?) to understand the theory.^11
I have already given some simple examples. The ratio of the diagonal of a square to its side is
one; the golden ratio (see Appendix B) is another. A third is the ratio of circumference to diameter.
And so on. Here, for the record, is the definition—it is a good example of what is really difficult in
Euclid, and we shall leave it to you to spend some time thinking about its interpretation. (Or look at
the commentary in editions of Euclid, websites, etc.)
Definition V.5.Magnitudes are said tobe in the same ratio, the first to the second and the third to the fourth , when, if
any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the third and fourth, the
former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in
corresponding order. (Fauvel and Gray 3.C3, pp. 123–4)
- That is, ifxis a whole number but is not a square, then its square root is not a fraction.
- For the medieval theory, see in particular Murdoch (1963). In particular, Murdoch claims that the eleventh-century Islamic
mathematician al-Jayy ̄an ̄ididhave a correct understanding of the Euclidean theory (the only writer between the Greeks and Isaac
Barrow?).