A History of Mathematics From Mesopotamia to Modernity

138 A History ofMathematics

say that, however learned Albert was, he did not know very much about Greek geometry; it also

shows that his ignorance, and his determination to proceed by what could be called common sense,

led him to a new way of thinking about the problem. It is not strictly ‘modern’, but it is a break with

ancient tradition.

Finally, he defines squaring the circle ‘in the fifth way with respect to sense and to intellect’ as

the usual problem—to find a square whose area is equal to that of the circle. With respect to sense,

because you cannot perceive the difference; with respect to intellect, because you can prove they

are the same. Again we see the very specific nature of scholastic reasoning, and how odd it seems

when applied to geometry. Albert ‘proves’ that this is possible, by using

Fact 1. Archimedes’s result that the area equals half the radius times the circumference (well

known, and often used);

Fact 2. Archimedes’s ‘formula’ that the circumference is three and one-seventh times the dia-

meter (used by Archimedes as an approximation, but as we have seen quoted at least from

Heron’s time onwards as if it were exact).

Although thisisa mathematical argument, if a wrong one, the whole idea of settling the question

by such a sequence of pros and cons seems to us exotic and ‘unmathematical’; and it is easy to see

why later generations were to consider the mathematicians of the Middle Ages, by and large, as

simply confused. However, in Albert’s favour, it should be said that the Greeks had failed to produce

a conclusive ‘answer’ to the circle-squaring problem, and that the idea of posing the alternative—

not to square it, but to prove that it couldnotbe squared—was a new one, and (however poor his

arguments were) pointed in the right direction.

This poses again the question of what might be regarded as revolutionary in science. We have

a scientific practice which is unlike any that has preceded it, so it seems reasonable to describe the

change (from Greek and Islamic mathematics, say, to that of Albert) as revolutionary; and if the

revolution in some sense goes backwards, with a great deal of loss of content and sophistication,

this is partly because the questions being studied are different. Science does notonlyprogress—this

is a modern myth, and later we shall see some alternative myths which were peddled in the sixteenth

century. And without being completely relativist, it is clear that different societies have different

ideas of what their object of study is. Our view that they are confused and/or wrong-headed should

be tempered by an honest attempt to see what they were trying to do.

Exercise 1.Given the two ‘facts’ above, how do you use them to square a circle?

4. Oresme and series

Zénon! Cruel Zénon! Zénon d’Élée!
M’as-tu percé de cette flèche ailée
Qui vibre, vole, et qui ne vole pas!
Zeno, Zeno, cruel philosopher Zeno,
Have you then pierced me with your feathered arrow
That hums and flies, yet does not fly! (Valéry 1920)

The scholastic tradition in mathematics was, as we shall see, not the only one in the Middle
Ages, but it was important. One of the best examples of new work produced by this approach