PREFACE TO THE
FIRST EDITION
The calculus was originally developed using the intuitive concept of an infinitesimal, or an infinitely small number. But for the past
one hundred years infinitesimals have been banished from the calculus course for reasons of mathematical rigor. Students have had to learn the subject without the original intuition. This calculus book is based on the work of Abraham Robinson, who in 1960
found a way tomake infinitesimals rigorous. While the traditional course begins with the difficult limit concept, this course begins with the more easily understood infinitesimals. It is aimed at the average beginning calculus student and covers the usual
three or four semester sequence.
The infinitesimal approach has three important advantages for thestudent. First, it is closer to the intuition which originally led tothe
calculus. Second, the central concepts of derivative and integralbecome easier for the student to understand and use. Third, itteaches both the infinitesimal and traditional approaches, giving thestudent an extra tool which may become increasingly important
in thefuture.
Beforedescribing this book, I would like to put Robinson’s work inhistorical perspective. In the 1670’s, Leibniz and Newton developedthe calculus
based on the intuitive notion of infinitesimals.Infinitesimals were used for another two hundred years, until thefirst rigorous treatment of the calculus was perfected by Weierstrassin the 1870’s. The standard calculus course of today is still basedon the
“ε, δdefinition” of limit given by Weierstrass. In 1960 Robinson solveda three hundred year old problem by giving a precise treatment of thecalculus using infinitesimals. Robinson’s achievement
will probablyrank as one of the major mathematical advances of the twentiethcentury.
Recently,infinitesimals have had exciting applications outside mathematics,notably in the fields of economics and physics. Since it is quitenatural
to use infinitesimals in modelling physical and social processes, such applications seem certain to grow in variety andimportance. This is a unique opportunity to find new uses formathematics, but at present few people are prepared by training totake advantage
of this opportunity.
Becausethe approach to calculus in new,some instructors may need additionalbackground material. An instructor’s volume, “Foundations ofInfinitesimalCalculus,”
gives the necessary background anddevelops the theory in detail. The instructor’s volume is keyed tothis book but is self-contained and is intended for the generalmathematical public.
Thisbook contains all the ordinary calculus topics, including thetraditional limit definition, plus one extra tool- theinfinitesimals. Thus the student
will be prepared for more advancedcourses as they are now taught. In Chapters 1 through 4 the basicconcepts of derivative, continuity, and integral are developedquickly using infinitesimals. The traditional limit concept is putoff until Chapter 5, where it
is motivated by approximation problems.The later chapters develop transcendental functions, series, vectors,partial derivatives, and multiple integrals. The theory differs fromthe traditional course, but the notation and methods for solvingpractical problems
are the same. There is a variety of applicationsto both natural and social sciences.
Ihave included the following innovation for instructors who wish tointroduce the transcendental functions early. At the end of chapter 2on derivatives,
there is a section beginning an alternate track ontranscendental functions, and each of Chapters 3 through 6 havealternate track problem sets on transcendental functions. Thisalternate track can be used to provide greater variety in the earlyproblems, or can
be skipped in order to reach the integral as soon aspossible. In Chapters 7 and 8 the transcendental functions aredeveloped anew at a more leisurely pace.
Thebook is written for average students. The problems preceded by asquare box go somewhat beyond the examples worked out in the text andare intended
for the more adventuresome.
Iwas originally led to write this book when it became clear thatRobinson’s infinitesimal calculus could be made available tocollege freshmen. The
theory is simply presented; for example,Robinson’s work used mathematical logic, but this book does not. Ifirst used an early draft of this book in a one-semester course atthe University of Wisconsin in 1969. In 1971 a two-semesterexperimental version was
published. It has been used at severalcolleges and at Nicolet High School near Milwaukee, and was tested atfive schools in a controlled experiment by Sister Kathleen Sullivanin 1972-1974. The results (in her 1974 PH.D. thesis at the Universityof Wisconsin)
show the viability of infinitesimal approach and willbe summarized in an article in the American Mathematical Monthly.
Iam indebted to many colleagues and students who have given meencouragement and advice, and have carefully read and used variousstages of the manuscript.
Special thanks are due to Jon Barwise,University of Wisconsin; G. R. Blakley, Texas A &M University;Kenneth A. Bowen, Syracuse University; William P. Francis, MichiganTechnological University; A. W. M. Glass, Bowling Green University;Peter Loeb, University
of Illinois at Urbana; Eugene Madison andKeith Stroyan, University of Iowa; Mark Nadel, Notre Dame University;Sister Kathleen Sullivan, Barat College; and Frank Wattenberg,University of Massachusetts.
J. keisler