Greeks,Practical andTheoretical 61One suspects that Plutarch had not read the works which he describes as ‘smooth and rapid’, since
later generations have found them impressive but difficult. The geometrical core, which includes
theMeasurement of a ParabolaandOn the Sphere and the Cylindercarries on, with great ingenuity,
from the harder parts of Euclid; we shall not deal with them here, but there are good extracts
in Fauvel and Gray (see also Archimedes 2002). There is, however, more to Archimedes than
these works suggest, and some of his other surviving works contradict Plutarch’s image of the
‘pure’ mathematician. TheStaticsandOn Floating Bodiesare the most serious works of theoretical
physics, outside the framework of Aristotle’s thought, in the Greek tradition; and as such, they had
a great influence in the Renaissance, particularly on Galileo—see Chapter 6. Further evidence of
a mechanical tendency in Archimedes is provided by the strange document called the ‘Method’.
Extravagant claims have been made for this manuscript,^2 for example, that it contains a version of
the calculus, and that the course of history would have been changed if it had not been ‘lost’. There
is no need for such exaggeration;The Methodis, so far as we know, a very unusual work which had
no imitators, and for good reason. In his introductory letter to Eratosthenes, Archimedes describes
what he is doing, and why:
Seeing moreover in you, as I say, an earnest student, a man of considerable eminence in philosophy, and an admirer
[of mathematical inquiry], I thought fit to write out for you and explain in detail in the same book the peculiarity of
a certain method, by which it will be possible for you to get a start to enable you to investigate some of the problems
in mathematics by means of mechanics. This procedure is, I am persuaded, no less useful even for the proof of the
theorems themselves; for certain things first became clear to me by a mechanical method, although they had to
be demonstrated by geometry afterwards because their investigation by the same method did not furnish an actual
demonstration. (From Archimedes tr. Heath, in Fauvel and Gray 4.A9 (a))The ‘Method’ referred to consists of measuring the areas of bodies (e.g. a segment of a parabola)
by ‘balancing’ them against simpler bodies (e.g. a triangle), using a division into infinitely thin
slices. (See Fauvel and Gray 4.A9(a) for an example.) Two things are striking here: first, the use of
weighing as a guide to understanding, presumably inspired by the work in theStatics—this is the
‘applied side’ of Archimedes; and second, the insistence, in the letter quoted above, that this is not a
proof, but that a proof has to be constructed once you have found the answer. (And, in some sense,
that it clarifies why the answer is what it is.) I should stress that the fact thatTheMethodis an applied
work does not make it an easy read; if it had been, perhaps it would have been preserved and quoted
more. To describe it as ‘lost’ is only partly accurate; someone in the ninth century, and various
others before that, must have known it and thought it of enough interest to be worth copying.
However, it had no influence on the later traditions, either through Byzantium or the Islamic
world, so far as we know; and this although some Islamic mathematicians had a great respect for
Archimedes and worked hard to reconstruct alleged works of his which they did not have.
In contrast, one work of Archimedes had tremendous influence, and still does. This was his
Measurement of a Circle. It is very short—it is thought that it is only part of a longer work of which
the rest has been lost; but what remained was found immensely useful by much more simple-
minded mathematicians. The three theorems which it contains are worth quoting in full, as a
typically Greek way of approaching what we would call the problem of calculatingπ:
Proposition 1. The area of any circle is equal to a right-angled triangle in which one of the sides about
the right angle is equal to the radius and the other to the circumference of the circle.- Discovered by Heiberg in Istanbul in 1906, then lost again, but recently rediscovered, sold at Christie’s for $2 m., and subjected
to modern scientific reading methods.