Preface
There is an element of truth in the old saying that the Euler textbook
Introductio in 4nalysin Infinitorum (Lausannae, 1748) was the first great
calculus textbook, and that all elementary calculus textbooks published
since that time have been copied from Euler or have been copied from
books that were copied from Euler. Euler, the greatest mathematician
of his day and in many respects the greatest mathematician of all time,
held sway when, except where the geometry of Euclid was involved, it
was not the fashion to try to base mathematical work upon accurately
formulated basic concepts. Problems were the important things, and
meaningful formulations of axioms, postulates, definitions, hypotheses,
conclusions, and theorems either were not written or played minor roles.
Through most of the first half of the twentieth century, elementary
textbooks in our subject taught unexplained but "well motivated" intui-
tive ideas along with their problems. Enthusiasm for this approach to
calculus waned when it was realized that students were not nourished by
stews in which problems, motivations, fuzzy definitions, and fuzzy theo-
rems all boiled together while something approached something else with-
out ever quite getting there. About the middle of the twentieth century,
precise formulations of basic concepts began to occupy minor but increas-
ingly important roles.
So far as calculus is concerned, this book attaches primary importance
to basic concepts. These concepts comprise the solid foundation upon
which advanced as well as elementary applications of calculus are based.
Applications, including those that have great historical interest, occupy
secondary roles. With this shift in our emphasis, we can remove the
mystery from old mathematics and learn modern mathematics when we
sometimes spend a day or two studying basic concepts and attaining
mastery of ideas, language, and notation that are used. The mathe-
matical counterparts of hydrogen and electrons are important, and we
study them before trying to construct the mathematical counterparts of
carbohydrates and television receivers.
This book contains just 76 sections, of which only a half dozen can be
omitted without destroying the continuity of the course. In a three-
term course meeting thrice weekly for fifteen weeks each term, times for
reviews, tests, and occasional excursions remain when two sections are
covered each week.
With few or no exceptions, each section presents each student an oppor-
tunity to make a thoroughly sound investment of time that will pay divi-
dends in personal satisfaction, intellectual enlightenment, and scientific
power. The material of the section is guaranteed to be worthy of study,
it being stoutly maintained that nobody should study inept material.
Each student is expected to read the text and problems of each section
V