A History of Mathematics From Mesopotamia to Modernity

Understanding the‘ScientificRevolution’ 139

is given by Nicolas Oresme. Oresme has been considered the originator of graphical (coordinate)

representation of quantities before Descartes.^5 A particularly good example of his thinking, and of

what the Scholastics could produce at their best, is given by his discussion of infinite series in his

Quaestiones super Euclidem(Questions on Euclid).

The role of proportion in medieval thought was extremely important, both as a tool of ele-

mentary mathematics and as a philosophical theme; but the treatment of proportion in Euclid,

particularly in book V (the ‘Eudoxan theory’), was a constant problem on account of its difficulty.

A detailed account of this theme (including the various mistranslations and misinterpretations

in the medieval Euclid versions) is given by John E. Murdoch in ‘The Medieval Language of Pro-

portions’—see Murdoch 1963, pp. 237–271. The particular problem of what happened when—

in modern terms—one took successive proportionsq,q^2 ,q^3 ,...and added them had preoccupied

Islamic mathematicians, because of its relation to the ‘method of exhaustion’. The point is as

follows. Euclid’s proposition X.1 states:

If two unequal quantities be given, and if from the greater, greater than half be subtracted, and again from the

remainder, greater than half be taken, and we continue successively in the same way, then it is at last necessary that

there remain a quantity less than the lesser of those given.

In the Islamic tradition, the tendency was to ask: does it have to be ‘greater than half ’.
This was answered by Nas.ir al-D ̄in al-T.us ̄ ̄i in his commentary: you do need (something like)
Euclid’s statement.
There is, then, underlying proposition X.1 the idea that you continue subtracting parts ‘as long
as you need to’, and that at a certain point (if they are greater than halves) you can stop. However,
it would seem that the scholastics were the first to consider the idea of taking an actual infinite
sum; and the result was expressed most clearly by Oresme.
His text is given as Appendix A to this chapter; I have tried to doctor it as little as possible, so as
to clarify exactly what he does say.
First, we should note that Oresme seems to have no doubt that you canphysically addan infinite
sequence of numbers. The numbers will be positive, as the techniques for dealing with negative
numbers had not been developed, so some problems which arise in our general theory are absent.
The sum may be ‘infinite’, whatever that means, or it may be a number; but he has no doubt that it
exists. As far as I know, this is quite original. Since the days when Zeno (the ‘cruel Zeno’ of Valéry’s
poem) devised his paradoxes of the infinite in the fifth centurybce, there had been strong objections
to treating a ‘completed infinity’ as opposed to a ‘potential infinity’ in Greek mathematics, which
were spelt out by Aristotle. Indeed, Oresme deals with the argument from the authority of Aristotle
before proceeding any further.
To consider what the extract says in detail, let us break the taboos on ‘presentism’, and translate
his statements into modern language. The results are as follows:

1. A geometric seriesa+ax+ax^2 +···whose ratioxis≥1 has an infinite sum; one whose ratio
   is<1 has a finite sum. [‘Second, it must be noted that...’.]


2. For example, 1+^12 +(^12 )^2 +··· =2; 1 +^13 +(^13 )^2 +··· =^32 ; and generally, 1+q+q^2 +··· =
   ( 1 −q)−^1 ,ifq<1. [‘The first proposition is...The second proposition is...’.]


3. This is considered in detail by Dijksterhuis (1986, p. 266), who isnotan uncritical supporter of the idea of ‘revolution’; on the
   whole, his verdict is that Oresme’s writings, however novel they were, cannot seriously be considered an anticipation of later ideas.