180 A History ofMathematics
showed that it worked because two errors in calculation happened to cancel. (It just happened that
one of the advantages of the early calculus was that this cancellation of errors had the status of
a general rule.) Attacking the new scientists of what was becoming called the ‘Enlightenment’
for transferring their belief from theology to fictions in mathematics (he was particularly incensed
against Halley, who was believed to have died an atheist), he asked for example:Whether mathematicians, who are so delicate in religious points, are strictly scrupulous in their own science? Whether
they do not submit to authority, take things upon trust, and believe points inconceivable? Whether they have not
their mysteries, and what is more, their repugnancies and contradictions? (Berkeley, in Fauvel and Gray 18.A.1,
pp. 557–8)The point was a good one, but it was largely ignored. Eighteenth-century mathematicians divided
into those who genuinely believed that infinitesimals existed, and those who justified the methods
of the calculus by their results in the belief that a sound foundation would arrive sooner or later.
D’Alembert, the key Enlightenment figure and editor of theEncyclopédie, advised the student in
theological terms: ‘Allez en avant, et la foi vous viendra’ (Go forward, faith will come to you). Faith
and science, indeed, had neatly exchanged places.
9. The calculus in practice
So far, the story seems to be a purely ‘internal’ one, with problems within mathematics being
argued over and solved in competing ways by professors (or diplomats, or men of independent
means) in universities or coffee-houses. To go a little further, we need to consider the kinds of
problems which were of interest, and what constituted a ‘solution’; as well as the ways in which
mathematicians—who are always trying to justify their bizarre activity to themselves and to the
outside world—thought of the possible practical outcomes of their work. Some light is thrown
on this by the often provocative essays of Henk Bos (1991); from whom I draw one particularly
interesting example.
One of the classical challenge problems which the mathematicians of the 1690s set one
another—in the interests both of propaganda for the calculus and of internal competition—was
the description of the curve which a heavy chain follows hanging under gravity (Fig. 6). History
books will in general tell you that the problem was solved by Leibniz, l’Hôpital and the Bernoullis
using the calculus; but reference to their papers (which are neither easy to access nor to read, being
in Latin in Gerhardt (1962, vol. V) shows that the story is somewhat more complicated. The naive
reader would assume that the ‘solution’ was an equation for the catenary in the formy=f(x),
wherexis horizontal andyvertical; and reference to a ‘modern’ textbook, if they still deal with the
topic, will show that this is how it is now expressed:y=acosh(x
a)
=a(
1
2
(ex/a+e−x/a))
However, as Bos shows, despite the allegiance of all parties to Descartes’s geometry, the idea that a
curve was specified by its equation was not the norm. You could describe a curve ‘by quadrature’
(give they-coordinate, say, as the area under another curve—obviously difficult, since you have
to measure a curved area); or ‘by rectification’ (give it as the arc-length of a curve—much better,
since you only have to stretch a string along the curve and measure it).