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Infinitesimal Calculus

James M. Henle and Eugene M. Kleinberg
Dover Publications
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The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Allen Stenger
, on

This is a concise, clear, and mostly complete textbook on the theory of calculus, as developed through the ideas of non-standard analysis rather than the epsilon-delta approach. It covers all the common theorems of calculus, but does not cover any calculus techniques or applications and has only a modest number of examples. The depth of coverage is roughly that of a rigorous introductory analysis book, such as Ross’s Elementary Analysis. This is a Dover 2003 unaltered reprint of the 1979 MIT Press edition.

The hyperreal number system is a extension of the reals that embeds them in a very careful way so that most of their properties carry over, but such that there are also infinitesimal and infinite quantities. Roughly the first third of the book covers the development of the hyperreal number system and the equivalence of logical statements about real numbers and functions with the formally-identical statements about hyperreal numbers and functions. This moves along quickly, but is mostly about mathematical logic rather than numbers or calculus, and it’s easy to lose sight of where we want to go with this. Most the rest of the book develops the standard theorems of single-variable calculus based on hyperreal numbers. There are also two chapters at the end that develop the topology of the real line and show the equivalence of the non-standard definitions with the standard definitions (and hence the equivalence of the two approaches to calculus). The whole book is only 144 pages, and packs a tremendous amount of information into them. (One weakness is the absence of a symbol index; the book does use several non-standard symbols, and forgetting their meaning will quickly derail you.)

The development is not completely rigorous, but it gives an almost-complete development of the subject. It can be studied at two levels: a lower one which develops the hyperreals (and has some handwaving) and an upper level based on axioms (called here a definition; it opens Chapter 3), that is completely rigorous if you take it as your starting point.

The publisher claims the book can be read by anyone with a good knowledge of high-school math, but I think this is an exaggeration: to understand what’s going on, you should have had a course in calculus techniques first, and a course in real analysis would help sharpen your critical examination of the definitions and proofs. The book probably doesn’t fit anywhere in current curricula, but would work well as a special course for upper-division undergraduates.

Very Unusual Feature: There are two side-by-side narratives, with the main one in the body to the right of each page and a supplemental one in the very wide left margin (about 1/3 of the page width). The marginal narrative gives additional comments and explanations, but sometimes takes on a life of its own, even giving its own exercises! It also contains a lot of quotes about mathematics from philosophers and other non-mathematicians. The writing itself is clear and lively, but the two narratives makes the flow hard to follow sometimes.

The best-known book at this level is probably Keisler’s Elementary Calculus: An Infinitesimal Approach. It is intended as an introductory calculus book and so is aimed lower than the present book. It covers all the techniques, but is skimpy on the development of the hyperreals. In particular, it never shows you an actual infinitesimal, while the present book gives many examples. Keisler’s book is very good at the usage and applications of infinitesimals, particularly in setting up models of physical problems.

Bottom line: a very good way to understand the non-standard analysis approach, both for mathematicians and for interested students. This approach really has never taken off, I think because when you include the underpinnings, it’s really not any simpler than the conventional approach, but still it is intriguing.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.


1 Introduction
2 Language and Structure
3 The Hyperreal Numbers
4 The Hyperreal Line
5 Continuous Functions
6 Integral Calculus
7 Differential Calculus
8 The Fundamental Theorem
9 Infinite Sequences and Series
10 Infinite Polynomials
11 The Topology of the Real Line
12 Standard Calculus and Sequences of Functions
  Appendix A Defining Quasi-big Sets
  Appendix B The Proof of Theorem 3.1
  Subject Index
  Name Index