178 A History ofMathematics
Fig. 5Newton’s picture forPrincipiaI, proposition 1.This argument does not use Newton’s version of the calculus; it is a more careful version of
the infinitesimal arguments which were being used by many of his predecessors (Pascal, Huygens,
Wallis, and others). This does not make it any better, however. It is true that I have omitted the intro-
ductory material, in particular Cor. IV, Lem. III, which states that the limit of a polygon is a curve;
and that this contains the theory justifying the passage to a limit. Such infinitesimal geometry gives
the arguments in thePrincipiaa superficial robustness. What Leibniz’s (and Newton’s) calculus has
which thePrincipiadoes not is the security, and the ease of calculation supplied by algebra. In this
respect, Newton’s decision to turnawayfrom Descartes’s algebraic methods made things harder
for him, and for his readers.^9 And this choice of a method of exposition made it quite unclear to
what extent the two obscure new works, Leibniz’s calculus and Newton’s physics, were related.8. The arrival of the calculus
By 1700 the calculus, or the method of fluxions, as it was now being called in England, had become
a success story. This could hardly have been guessed from the beginnings as they have been sketched
above. Both Newton and Leibniz had acquired a circle of interested students who attempted to find
out what they could about the new methods, and to publicize them. In mathematical terms, Leibniz
had some unexpected good fortune. Two Swiss brothers, Jakob and Johann Bernoulli, understood
very early that something was to be gained from understanding his theory. Later, they were to claim
that they toiledaway tomaster it in a period of weeks; in fact, it took more like three years during
which they wrote abjectly to Leibniz requesting some clarification:Of this method of yours, if you could deign to impart some ray of light (which I earnestly beseech) to me—as much as
you can spare given your very weighty affairs—by doing so you would make me not a mere admirer of your inventions
but also a worthy esteemer and a publicist. (Letter of Jakob Bernoulli to Leibniz, quoted in Roero 1989)- Many stories circulate about the perplexity of thePrincipia’s immediate audience. John Locke (who was not a mediocre
mathematician, in terms of the culture of the time) had to be given an outline by Huygens; ‘he confessed that there were “very few
that [had] Mathematicks enough to understand his Demonstrations” ’. (Iliffe, 1995, p. 173.)