The History of Mathematics: A Brief Course

The great watershed in the history of mathematics is the invention of the calcu-
lus. It synthesized nearly all the algebra and geometry that had come before and
generated problems that led to much of the mathematics studied today. Although
calculus is an amalgam of algebra and geometry, it soon developed results that were
indispensible in other areas of mathematics. Even theories whose origins seem to
be independent of all forms of geometry—combinatorics, for example—turn out to
involve concepts such as generating functions, for which the calculus is essential.
Elements of the calculus had existed from the earliest times in the form of
infinitesimal methods in geometry, and such techniques were refined in the early
seventeenth century. Although strict boundaries in history tend to be artificial
constructions, there is such a boundary between analytic geometry and calculus.
That boundary is the introduction of infinitesimal methods. As approximate, easy-
to-remember formulas, we can write: algebra + geometry = analytic geometry, and
analytic geometry 4- infinitesimals = calculus.
The introduction of infinitesimals into a geometry that had only recently strug-
gled back to the level of rigor achieved by Archimedes raised alarms in certain quar-
ters, but the new methods led to spectacular advances in theoretical physics and
geometry that have continued to the present time. Like the Pythagoreans, modern
mathematicians faced the twin challenges of extending the range of applicability of
their mathematics while making it more rigorous. The responses to these challenges
led to the modern subject of analysis. Starting from a base of real numbers, repre-
sented as ratios of line segments and written in the symbolic language of modern
algebra, mathematicians extended their formulas to complex numbers, opening up
a host of new applications and creating the beautiful subject of analytic function
theory (complex analysis). At the same time, they were examining the hidden as-
sumptions in their methods and making their limiting processes more rigorous by
introducing the appropriate definitions of integrals, derivatives, and series, leading
to the subject of functions of a real variable (real analysis).
Part 6 consists of two chapters. The creation of calculus and its immediate
outgrowths, differential equations and the calculus of variations, is described in
Chapter 16, while the further development of these subjects into modern analysis
is the theme of Chapter 17.