174 A History ofMathematics
However, it is only after three pages of this exposition of ‘rules’ for the calculus, which tells you
how to get results with no idea of why they should be true, that Leibniz is prepared to assert that
dx,dy, and so on can be taken infinitely small. In this sense, his paper is less open than Newton’s
early writings; among the reasons usually claimed are:- that the fact that the calculus worked was more important for him than the meaning of the
symbols which it used; - that he was not entirely happy with the infinitely small himself, and was prepared to deny that
he was using it if he could find another defence.
A particular viewpoint drawing on Leibniz’s philosophical work holds that as a rationalist he was
committed to a strong belief in the power of written signs to achieve results in the world. One of
his projects in the late 1670s was for a ‘Universal Characteristic’, which would allow all meanings
in all languages to be unambiguously expressed, and so finally put an end to human strife (since
there would be nothing to argue about).It was now clear to Leibniz that in order to discover the alphabet of human thought and realise the universal charac-
teristic, it would be necessary to analyse all concepts and reduce them to simple elements by means of definitions, then
to represent the simple concepts by appropriate symbols and invent symbols for their combinations, and finally...to
demonstrate all known truths by reducing them to simple, evident principles. (Aiton 1985, p. 78)Compared with this ambitious programme, which of course was never undertaken, the calculus
seems a minor achievement. It might be thought that his calculus works in opposition to the
rational aim of the characteristic; its status as ‘marks on paper which perform a function’ is quite
divorced from its doubtful meaning. Still, in some way for Leibniz, that was its beauty. If the
scientific community could accept that it worked, they would have agreed on a common project
for the betterment of mankind. In fact, the heart of the paper—and this is perhaps the answer
to the question about his aim—lies in his use of the word ‘algorithm’. The reader is being told
how to follow a set of mechanical rules (indeed, Leibniz had also invented a calculating machine)
which will make it possible to solve with ease a vast number of previously unsolved problems.
The justification of the procedure, which was present in Leibniz’s unpublished notes from his time
10 years before in Paris, is secondary to its exposition as method.
Here there is perhaps another parallel between Newton and Leibniz which helps to explain their
delays in publishing and the confusion of what they wrote: that both were still unsure about how
much they needed the infinitely small, or whether given time and application, which neither of them
had, they could dispense with it. Leibniz’s attempts to explain infinitesimals in later years are many,
and often contradictory—sometimes they exist, sometimes they are convenient fictions. A new
generation of less sophisticated mathematicians had to adopt and promote the methods, often
without much encouragement from their supposed patrons, before they became general currency.
Once we arrive at the infinitesimal ‘arguments’ in the 1684 paper, they are the ones—already in
use before Leibniz’ time—which were to become standard, however strange they may seem to us.
A curve ‘is’ a polygon with an infinite number of angles, and its tangent ‘is’ simply one of the sides
of this polygon, produced (see Fig. 3 again). The first of these ideas, as we have seen, goes back to
Nicholas of Cusa (see Chapter 6) if not earlier. They could be used, as they were later, to justify the
rules for differentiation, and much else.
Why (apart from the hurried publication already mentioned) did Leibniz produce his invention,
which he had spent some time developing, in such an unsatisfactory form? The statements which