Greeks and‘Origins’ 39
with a historicist sense of difference. The classical works were using methods which are alien to us
to achieve ends, which in the main, we no longer have.^3
With this in mind, let us consider what we know of Greek methods, and how far conjecture,
hearsay, and the like can help us.
Exercise 4.What steps would be needed to deduce a formula for the area of a parallelogram from
proposition I.35 above?
4. The problem of material
Acting as many inventors are known to have done in the case of their discoveries, they have perhaps feared that
their method being so very easy and simple, would if made public, diminish, not increase public esteem. Instead they
have chosen to propound, as being the fruits of their skill, a number of sterile truths, deductively demonstrated with
great show of logical subtlety, with a view to winning an amazing admiration, thus dwelling indeed on the results
obtained by way of their method, but without disclosing the method itself —a disclosure which would have completely
undermined that amazement. (DescartesRules for the Direction of the Mind(1968a), Rule 4)
Our Greek sources, with much more material at their disposal than we have, had little difficulty
telling stories of how mathematics developed, however inventive we might find them. By the
seventeenth century when Descartes wrote, it was a different matter. New translations of the
ancient Greek writers clarified both their importance and the excessive difficulty of their work, in
comparison with the algebraic methods which could be seen as an alternative. Never one to defer
to any older authority, Descartes imagined the Greeks using easy methods of discovery, similar to
his own prescriptions for solving problems, and then making the results look hard so as to mystify
posterity. Drawing on hints in Greek authors, particularly Pappus, he opposed ‘analysis’ (a method
of discovery, in which you consider the properties of the thing you want to find and deduce what it
must be like) to the classical Euclidean ‘synthesis’ (a method of disclosure, in which you state what
the result must be, without explanation, and prove you are right). His claim was that the Greeks
had used analysis, as he would do in his own work, but destroyed the works in which it was used.
By such arguments, he not only propagandized for his own work, but started the fertile genealogy
of speculation about why the Greeks had chosen to do mathematics in such a strange way. The
tradition of ‘scholarly’ history, which began in the late nineteenth and early twentieth century
with Paul Tannéry (1887) and Sir Thomas Heath (1921), had more respect for the Greeks and less
of an axe to grind, but little more material to go on.^4 In his pioneering work, Tannéry described the
problem of sources:
These writings [Euclid, Archimedes, etc.] cannot teach us the history of science; they leave us ignorant about its
origin, of its first developments, just as, since important works have been lost, they give us no way of gauging, without
having recourse to conjectures, the direction of research in higher geometry and the level of understanding which
was reached.
The history of Greek geometry must therefore appeal to other sources; it must subject those sources to a methodical
critique such as one applies in other similar cases. This is the aim which I have set myself. (Tannéry 1887)
- It is enough to recall the classical Greek problem of ‘squaring the circle’, that is, constructing a square whose area is equal to that
of a given circle. A great deal of work was expended on this in ancient times (see Knorr 1986, for example). Grossly misunderstood
by some medieval writers (see chapter 5), it is now—for reasons which are as much to do with our different perspective as with our
greater knowledge—not a concern for mathematics. - The only exception was Heath’s discovery of an unknown manuscript of Archimedes’ ‘The Method’, published for the first time
in 1906. This work, which will be discussed in the next chapter, in some ways confirms Descartes’s suspicions.