472 16. THE CALCULUSproof; they contain the basic results on differentiation and integration of elemen-
tary functions, including the Taylor series expansions of logarithms, exponentials,
and trigonometric functions. Although the language seems slightly archaic, one can
easily recognize a core of standard calculus here.
Later reflections on the calculus. Like Newton, Leibniz was forced to answer objec-
tions to the new methods of the calculus. In the Acta eruditorum of 1695 Leibniz
published (in Latin) a "Response to certain objections raised by Herr Bernardo
Niewentiit regarding differential or infinitesimal methods." These objections were
three: (1) that certain infinitely small quantities were discarded as if they were zero
(this principle was set forth as fundamental in the following year in the textbook of
calculus by the Marquis de l'Hospital); (2) the method could not be applied when
the exponent is a variable; and (3) the higher-order differentials were inconsistent
with Leibniz' claim that only geometry could provide a foundation. In answer to
the first objection Leibniz attempted to explain different orders of infinitesimals,
pointing out that one could neglect all but the lowest orders in a given equation.
To answer the second, he used the binomial theorem to demonstrate how to handle
the differentials dx, dy, dz when y^1 = z. To answer the third, Leibniz said that one
should not think of d(dx) as a quantity that fails to yield a (finite) quantity even
when multiplied by an infinite number. He pointed out that if ÷ varies geometri-
cally when y varies arithmetically, then dx = (xdy)/a, and ddx = (dxdy)/a, which
makes perfectly good sense.
2.3. The disciples of Newton and Leibniz. Newton and Leibniz had disciples
who carried on their work. Among Newton's followers was Roger Cotes (1682
1716), who oversaw the publication of a later edition of Newton's Principia and
defended Newton's inverse square law of gravitation in a preface to that work.
He also fleshed out the calculus with some particular results on plane loci and
considered the extension of functions defined by power series to complex values,
deriving the important formula éö — log(cos0 + isin ö), where i = Another
of Newton's followers was Brook Taylor (1685-1731), who developed a calculus of
finite differences that mirrors in many ways the "continuous" calculus of Newton
and Leibniz and is of both theoretical and practical use today. Taylor is famous
for the infinite power series representation of functions that now bears his name. It
appeared in his 1715 treatise on finite differences. We have already seen, however,
that many particular "Taylor series" were known to Newton and Leibniz; Taylor's
merit is to have recognized a general way of producing such a series in terms of
the derivatives of the generating function. This step, however, was also taken by
Johann Bernoulli.
Leibniz also had a group of active and intelligent followers who continued to
develop his ideas. The most prominent of these were the Bernoulli brothers Jakob
(1654-1705) and Johann (1667-1748), citizens of Switzerland, between whom rela-
tions were not always cordial. They investigated problems that arose in connection
with calculus and helped to systematize, extend, and popularize the subject. In ad-
dition, they pioneered new mathematical subjects such as the calculus of variations,
differential equations, and the mathematical theory of probability. A French noble-
man, the Marquis de l'Hospital (1661-1704), took lessons from Johann Bernoulli
and paid him a salary in return for the right to Bernoulli's mathematical discov-
eries. As a result, Bernoulli's discovery of a way of assigning values to what are
now called indeterminate forms appeared in L'Hospital's 1696 textbook Analyse