Chapter 1
Introduction
1.1 Historical Prologue
The singular concept which characterizes calculus and simultaneously sets
it apart from arithmetic, algebra, geometry, and trigonometry is the notion
of a limit. The idea of a limit originated with the ancient Greek philosophers
and mathematicians. However, they failed to fully develop and exploit this
concept. It was not until the latter half of the seventeenth century, when
the English mathematician Isaac Newton (1642-1727) and the German math-
ematician Gottfried Wilhelm Leibniz (1646-1716) independently and almost
simultaneously invented differential and integral calculus, that the concept of
a limit was revived and developed more fully.
Calculus, as presently taught, begins with differential calculus, continues
with the consideration of integral calculus, and then analyzes the relation-
ship between the two. Historically, however, integral calculus was developed
much earlier than differential calculus. The idea of integration arose first in
conjunction with attempts by the ancient Greeks to compute areas of plane
figures, volumes of solids, and arc lengths of plane curves.
Archimedes was born about 287 B.C. in the Greek city-state of Syracuse
on the island of Sicily. He was the first person to determine the area and the
circumference of a circle. Archimedes determined the volume of a sphere and
the surface areas of a sphere, cylinder, and cone. In addition, he calculated
areas of ellipses, parabolic segments, and sectors of spirals. However, some-
what surprisingly, no Greek mathematician continued the work of Archimedes,
and the ideas which he had advanced regarding integration lay dormant until
about the beginning of the seventeenth century. Using the present day theory
of limits, Archimedes' ingenious method of equilibrium can be shown to be
equivalent to our definition of integration.
Early in the seventeenth century a significant development, which would
effect calculus dramatically, was taking place in mathematics- the inven-
tion of analytic geometry. Credit for this innovation is given to both Rene
Descartes (1596-1650) and Pierre de Fermat (c. 1601-1665). In 1637 , Descartes
published the philosophical treatise on universal science, A Discourse on the
Method of Rightly Conducting the Reason and Seeking for Truth in the Sci-
ences. The third and last appendix to his Discourse is titled La geometrie.
In La geometrie, Descartes discusses finding normals to algebraic curves-
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