474 16. THE CALCULUSMost students of calculus know the Maclaurin series as a special case of the
Taylor series. Its discoverer was a Scottish contemporary of Taylor, Colin Maclaurin
(1698-1746), whose treatise on fluxions (1742) contained a thorough and rigorous
exposition of calculus. It was written partly as a response to the philosophical
attacks on the foundations of calculus by the philosopher George Berkeley.
The Italian textbook Istituzioni analitiche ad uso della gioventu italiana (An-
alytic Principles for the Use of Italian Youth) became a standard treatise on ana-
lytic geometry and calculus and was translated into English in 1801. Its author was
Maria Gaetana Agnesi, who was mentioned in Chapter 4 as one of the first women
to achieve prominence in mathematics.
The definitive textbooks of calculus were written by the greatest mathematician
of the eighteenth century, the Swiss scholar Leonhard Euler. In his 1748 Introductio
in analysin infinitorum, a two-volume work, Euler gave a thorough discussion of
analytic geometry in two and three dimensions, infinite series (including the use
of complex variables in such series), and the foundations of a systematic theory of
algebraic functions. The modern presentation of trigonometry was established in
this work. The Introductio was followed in 1755 by Institutiones calculi differentialis
and a three-volume Institutiones calculi integralis (1768-1774), which included the
entire theory of calculus and the elements of differential equations, richly illustrated
with challenging examples. It was from Euler's textbooks that many prominent
nineteenth-century mathematicians such as the Norwegian genius Niels Henrik Abel
first encountered higher mathematics, and the influence of Euler's books can be
traced in their work.
The state of the calculus around 1700. Most of what we now know as calculus—
rules for differentiating and integrating elementary functions, solving simple differ-
ential equations, and expanding functions in power series—was known by the early
eighteenth century and was included in the standard textbooks just mentioned.
Nevertheless, there was much unfinished work. We list here a few of the open
questions:
Nonelementary integrals. Differentiation of elementary functions is an algorithmic
procedure, and the derivative of any elementary function whatsoever, no matter how
complicated, can be found if the investigator has sufficient patience. Such is not the
case for the inverse operation of integration. Many important elementary functions,
such as (sinx)/a: and e~x , are not the derivatives of elementary functions. Since
such integrals turned up in the analysis of some fairly simple motions, such as that
of a pendulum, the problem of these integrals became pressing.
Differential equations. Although integration had originally been associated with
problems of area and volume, because of the importance of differential equations in
mechanical problems the solution of differential equations soon became the major
application of integration. The general procedure was to convert an equation to
a form in which the derivatives could be eliminated by integrating both sides (re-
duction to quadratures). As these applications became more extensive, more and
more cases began to arise in which the natural physical model led to equations that
could not be reduced to quadratures. The subject of differential equations began
to take on a life of its own, independent of the calculus.
Foundational difficulties. The philosophical difficulties connected with the use of in-
finitesimal methods were paralleled by mathematical difficulties connected with the
application of the algebra of finite polynomials to infinite series. These difficulties