36 A History ofMathematics
quite easy to spectacularly difficult. This will give some idea of the achievements of Greek mathe-
matics, of its range, and of the limits within which it operated. In Fauvel and Gray you will also
find some useful extracts from the very late (c.450ce) commentator Proclus, for whom see later.
With regard to secondary literature, the situation is better, since the question of the earliest
Greeks, their aims, and achievements, has seemed so important—we will discuss why later. There
are sections on the Greeks in both van der Waerden (1961) and Neugebauer (1952). They are
sometimes dated, and in van der Waerden’s case, given to conjecture using unreliable ancient
sources. The major modern works which cover the question can be dauntingly detailed, but if you
find yourself developing, as one does, an interest in Greek mathematics, they will draw you in.
The central works are probably Knorr’s (1975) attempt to account for the form of theElements
and Fowler’s (1999) more informal but very scholarly reconstruction of what geometry might
have been likebeforeEuclid. Netz (referred to by Berggren above) (1999) gives an intriguingly
different approach, by a consideration of actual Greek proofs and how they work, followed by a
‘sociology’ of Greek mathematicians based on the very fragmentary evidence we have about them.
We shall refer to other texts when they are useful; but these will do for the present. You have a
reasonable chance, given the prestige of Greek mathematics, of finding some or all of them in your
library.
3. An example
Being in a Gentleman’s Library, Euclid’sElementslay open, and ’twas the47 El. libri I[Pythagoras’ theorem]. He
[Thomas Hobbes] read the Proposition.By G—, sayd he (he would now and then sweare an emphaticall Oath by way
of emphasis)this is impossible! So he reads the Demonstration of it, which referred him back to such a Proposition;
which proposition he read. That referred him back to another, which he also read.Et sic deinceps[and so on] that at
last he was demonstratively convinced of that trueth. This made him in love with Geometry. ( J. Aubrey,Brief Lives,
quoted in Fauvel and Gray 3.F.2)While theMenois a very illuminating discussion on a mathematical subject, it is too informal
to be a good illustration of the mainstream Greek mathematics which is our primary concern.
Socrates’ arguments make no attempt to go back to first principles, and the points he makes about
areas of triangles are treated as obvious (which they are) rather than justified in painful detail. The
mathematical argument of theMeno, if not its philosophical one, would have been easily accessible
to an Egyptian.
To see how ‘classical’ Greek mathematics claims to work, it is best to start, at least, with Euclid’s
Elements. (For texts see the bibliography.) This is a strange and complex work—some would say a
composite, or scissors-and-paste compilation of previous works; but it has been the most read and
commented of all mathematical works in history, so it deserves a central position in any account.
For that reason, we shall privilege it over the harder works of Archimedes and Apollonius, the other
main classics. It is also a sensible idea in the first place to consider it in itself as a text rather than
speculating on its origins; such speculation is natural in a history which focuses on ‘discovery’, but
other histories are available. A whole book could be written about theElements, and many have;
the most recent and scholarly are Knorr (1975) (already mentioned) and Artmann (1999). The
work, as its title suggests, is supposed to give the student the essentials of mathematics, carefully
deduced from ‘first principles’, statements which are either in some sense obvious, or which the
reader/student can reasonably allow to be true. (‘All right angles are equal’, for example.) Here, as