140 A History ofMathematics
This is, at any rate, my interpretation of the way Oresme describes the summation of
geometric series; it is hard to be sure not only because the language used is obscure but
because, since there is no proof, we cannot see how the result was arrived at.- On the other hand, it is possible for a decreasing series to have an infinite sum, in particular
the ‘harmonic series’ 1+^12 +^13 +···has an infinite sum. The proof is the usual one. [‘The
third proposition is...’]. On this basis, Oresme is generally credited with discovering that what
is now called the harmonic series (whose terms decrease to zero) has an infinite sum; and,
unless an earlier candidate turns up, this seems right. He did understand that if you continued
to add( 1 /n)’s you would get sums which were bigger than any assigned number. In other
words, this is a particular instance where presentism gives a reasonably accurate version. We
could say that here we have a fourteenth-century mathematician finding out facts about the
convergence of series in a modern way.
Having established this to our satisfaction, we would still be left with some puzzling questions.
To begin with, what exactly was Oresme trying to do, and why does the context of his work look so
different? And, second, why did no one else deal with similar questions? Why were his results not
reproduced for so long? Certainly there seems to be no record of a general acceptance that it is all
right to use infinite sums, or of any similar use of them until much later.
The answer to all of these questions seems to lie in the framework of the discussion; the old-
style scholasticquaestio. Unlike his early modern successors, Oresme was not concerned with
series as the answers to problems in calculation. (Newton’sMethod of fluxions and infinite series
is an obvious example of the later approach.) Rather, he wanted to know the answers to some
questions about ‘quantity’, and the paradox—which is already present in a concealed form in
the method of exhaustion, or in Euclid X.1—that an infinite number of finite quantities can
have a finite sum. The Greeks would not have put it like this; the scholastics, for whom the
infinite was attractive precisely because it was so fertile in contradictions and paradoxes, would.
What, asked Oresme, are the conditions for an addition ‘by proportional parts’ to be possible? The
question goes back to Zeno’s paradox of Achilles and the tortoise. What was new about Oresme’s
treatment is that he gave precise conditions in terms of the ratios, and even summed the series.
And, of course, that in going on to examine the possibility of a series which is ‘by ratios of
lesser proportionality’—decreasing—becoming infinite, he came up with the simple example of
the harmonic series.
Exercise 2.(a) What does Euclid’s statement in proposition X.1, quoted above, mean? (b) Why can you
not use proportions less than a half in general? (c)What has this to do, if anything, with sums of series?5. The calculating tradition
Forsooth, a great arithmetician
One Michael Cassio, a Florentine...(Shakespeare,Othello,Act I, Scene 1)The claim of Duhem and his successors that the discoveries of the scientific revolution were in the
main developments of earlier work by the scholastics has had the positive effect of drawing attention
to what it was that the scholastics actually did. However, like most priority claims, it makes for bad