3 Greeks, practical and theoretical
1. Introduction, and an example
But, unlike Euclid, who attempts to prove musical propositions through mathematical theorems, Nicomachus seeks
to show their validity by measurement of the lengths of strings. (Entry ‘Nicomachus of Gerasa’ inDictionary of
Scientific Biography)
The leading Greek geometricians were all master carpenters. Euclid, the author of theBook of Principles, was a
carpenter and known as such. The same was the case with Apollonius, the author of the book onConic Sections, and
Menelaus and others. (Ibn Khald ̄un 1958, II, p. 365)
The complaints of the previous chapter about the poverty of documentary evidence for Greek
mathematics before Euclid need to be modified for the later period, say from 300bceto 600ce.
There is indeed a variety of material, but it is quite heterogeneous, and scattered in time and space.
We are often vague about the dates of writers, and we have to guess about their communication;
and still it seems that the survival of material is determined mainly by chance. Our first quotation
describes the work of Nicomachus of Gerasa ( Jerash, in Palestine), whose arithmetical, philosoph-
ical, and musical works were treated as important in the Islamic and European Middle Ages, and
so survived although modern authorities consider him a desperately poor arithmetician. At least,
as the quotation shows, he sometimes had a practical approach which was quite different from
what we consider ‘typically’ Greek. The description of Euclid and Apollonius as carpenters runs
contrary to all the information we have from other sources, but is it simply folklore? We have no
way of knowing. How many ways were there, indeed, of being a Greek mathematician, and did
they change over the 900-year period which we are considering? Did they interact? How did the
Romans, generally portrayed as an uncultured master-race with no interest in science, contribute
to the way mathematics was done, in the period when they dominated the Greek world (say from
100 bceto 400ce, when the ‘Roman’ part of the empire collapsed and the ‘Greek’ survived)?
For if often (e.g. with the Babylonians) we can say: ‘At this time, mathematics was used in a differ-
ent way from our own; the following methods were used, with the following ends in view’, with the
Greeks the situation is more complicated, and less well understood. The heritage which was passed
on as important, particularly from the sixteenth century, was that of Euclid, Archimedes, and those
who followed their models: axioms, theorems, and proofs. The ideology which went with it, which
I have referred to in the last chapter as due (in part) to Plato, is that mathematics should not deal
with the real world, or with applied problems. The reality is certainly more complicated; and even
accepting that we have lost many records of carpenters, tax-gatherers, architects, and engineers—
which we must suppose existed—there is quite a complexity and variety in what remains, even if
some of it is what appears to us rather low-level mathematics.