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des infiniment petits (Infinitesimal Analysis) and has ever since been known as
L'Hospital's rule. Like the followers of Newton, who had to answer the objections
of Bishop Berkeley (see Section 3 below) Leibniz' followers encountered objections
from Michel Rolle (1652-1719), which were answered by Johann Bernoulli with the
claim that Rolle didn't understand the subject.The priority dispute. One of the better-known and less edifying incidents in the
history of mathematics is the dispute between the disciples of Newton and those
of Leibniz over the credit for the invention of the calculus. Although Newton
had discovered the calculus by the early 1670s and had described it in a paper
sent to James Collins, the librarian of the Royal Society, he did not publish his
discoveries until 1687. Leibniz made his discoveries a few years later than Newton
but published some of them earlier, in 1684. Newton's vanity was wounded in 1695
when he learned that Leibniz was regarded on the Continent as the discoverer of the
calculus, even though Leibniz himself made no claim to this honor. In 1699 a Swiss
immigrant to England, Nicolas Fatio de Duillier (1664-1753), suggested that Leibniz
had seen Newton's paper when he had visited London and talked with Collins
in 1673. (Collins died in 1683, before his testimony in the matter was needed.)
This unfortunate rumor poisoned relations between Newton and Leibniz and their
followers. In 1711-1712 a committee of the Royal Society (of which Newton was
President) investigated the matter and reported that it believed Leibniz had seen
certain documents that in fact he had not seen. Relations between British and
Continental mathematicians reached such a low ebb that Newton deleted certain
laudatory references to Leibniz from the third edition of his Principia. This dispute
confirmed the British in the use of the clumsy Newtonian notation for more than a
century, a notation far inferior to Leibniz's elegant and intuitive symbolism. But in
the early nineteenth century the impressive advances made by Continental scholars
such as Euler, Lagrange, and Laplace won over the British mathematicians, and
scholars such as William Wallace (1768-1843) rewrote the theory of fluxions in
terms of the theory of limits. Wallace asserted that there was never any need to
introduce motion and velocity into this theory, except as illustrations, and that
indeed Newton himself used motion only for illustration, recasting his arguments
in terms of limits when rigor was needed (see Panteki, 1987, and Craik, 1999).
Eventually, even the British began using the term integral instead of fluent and
derivative instead of fluxion, and these Newtonian terms became mathematically
part of a dead language.
Certain relevant facts were concealed by the terms in which the priority dispute
was cast. One of these is the extent to which Fermat, Descartes, Cavalieri, Pascal,
Roberval, and others had developed the techniques in isolated cases that were to
be unified by the calculus as we know it now. In any case, Newton's teacher Isaac
Barrow had the insight into the connection between subtangents and area before
either Newton or Leibniz thought of it. Barrow's contributions were shunted aside
in the heat of the dispute; their significance has been pointed out by Feingold
(1993).
Early textbooks on calculus. The secure place of calculus in the mathematical cur-
riculum was established by the publication of a number of excellent textbooks. One
of the earliest was the Analyse des infiniment petits, mentioned above, which was
published by the Marquis de 1'Hospital in 1696.