Elementary Calculus: An Infinitesimal Approach
h-jerome-keisler‘s full-sequence calculus textbook, and the first source in this corpus that argues with another one. First edition 1976 from Prindle, Weber & Schmidt; second edition 1986; free online since September 2002 and revised since, with Dover publishing a third edition in 2012 on the agreement that the free version stays up. Licensed CC BY-NC-SA. The held file is 992 scanned pages, running to page 902 plus tables, answers and index.
The book covers the standard three-or-four-semester sequence. What makes it unusual is that it builds the whole thing on infinitesimals rather than on limits. See infinitesimal-calculus.
The argument
The preface to the first edition states it without hedging:
“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 to make infinitesimals rigorous.”
Keisler’s history: Leibniz and Newton built calculus on infinitesimals in the 1670s and it was used that way for two hundred years, until Weierstrass produced the first rigorous treatment in the 1870s and the ε-δ definition of limit became the standard course. Robinson’s 1960 result closed the gap the other way, giving infinitesimals a precise treatment — “Robinson’s achievement will probably rank as one of the major mathematical advances of the twentieth century.”
He claims three advantages: it is closer to the intuition that produced the calculus, derivative and integral become easier to understand and use, and the student gets both approaches rather than one. And he is explicit that nothing is being removed — “This book contains all the ordinary calculus topics, including the traditional limit definition, plus one extra tool — the infinitesimals.” Limits are not dropped; they are deferred to Chapter 5, motivated by approximation problems, after derivative, continuity and integral have already been built in chapters 1–4.
One more restraint worth noting: Robinson’s own work used mathematical logic, and this book does not. The theory is presented, in Keisler’s words, simply. The logical machinery lives in the companion instructor’s volume, Foundations of Infinitesimal Calculus.
What’s in it
Fourteen chapters, and the shape is ordinary even though the foundation isn’t:
| Ch | Subject |
|---|---|
| 1 | Real and hyperreal numbers — the real line, the hyperreal line, infinitesimal/finite/infinite numbers, standard parts |
| 2–4 | Differentiation, continuous functions, integration — all built with infinitesimals |
| 5 | Limits, approximation, analytic geometry, conic sections, Newton’s method |
| 6–8 | Applications of the integral, trigonometric functions, exponential and logarithmic functions |
| 9 | Infinite series through Taylor’s formula and Taylor series |
| 10 | Vectors, ending with hyperreal vectors |
| 11–13 | Partial differentiation, multiple integrals, vector calculus (gradients, line integrals, Green’s, Stokes and Gauss) |
| 14 | Differential equations |
Chapter 14 was new in the second edition, and §14.4 is the clearest demonstration of the book’s
premise: infinitesimals give a simple proof that every differential equation y′ = f(t,y) with
continuous f has a solution — a result Keisler says is “beyond the scope of a traditional
elementary calculus course, but is within reach with infinitesimals.” The approach isn’t only a
gentler on-ramp; at least once, it reaches further.
It is the only source here with evidence for its pedagogy
Every text in this corpus makes a claim about how its subject should be taught. This is the one that went and checked. From the first-edition preface: an early draft ran as a one-semester course at Wisconsin in 1969, a two-semester experimental version was published in 1971, and the approach was tested at five schools in a controlled experiment by Sister Kathleen Sullivan between 1972 and 1974, with results in her 1974 Wisconsin PhD thesis and summarized in the American Mathematical Monthly. Keisler reports the results “show the viability of the infinitesimal approach.”
Read that carefully: it is the author reporting someone else’s study of his own method, in his own preface, and the hub does not hold the thesis or the article. It is second-hand and interested. It is still more than corral-vector-calculus, mcmullen-probability-theory or thibos-fourier-analysis offer for their own pedagogical choices, which is nothing.
Tier, age and what has dated
- T1. A commercially published textbook, twice, from a research mathematician working near his own specialism, now in a third edition from Dover. The mathematics is checkable and the exercises come with answers to selected problems.
- The mathematics has not dated at all; the artifact has. The held PDF is a scan of the 1986 second edition with no text layer — searching it means reading pages as images. The scanned copyright page still says “Revised May, 2009” while the file itself is named for and was modified in June 2026, so the front matter was never re-scanned after later corrections. Treat page-level details as 1986 unless checked.
- The licence link in the scan points at CC BY-NC-SA 2.0; the author’s current page states 3.0 Unported. Minor, and the direction of travel is clear.
- The URL that arrived carried a Facebook click tracker. Canonical page:
people.math.wisc.edu/~hkeisler/calc.html. Held atraw/keisler-elementary-calculus.pdf(28.8 MB — it needed a resumed download to arrive intact).
Cross-spoke context
../machine-learning-wiki— Chapter 13’s directional derivatives and gradients are the same material that classical-ml-algorithms rests on, and this is now the second source here covering them.../optimization-algorithms-wiki— Newton’s method arrives in Chapter 5 as a limit application, the same classical baseline corral-vector-calculus supplies from the other foundation.../philosophy-wiki— the infinitesimal was a live philosophical scandal for two centuries (Berkeley’s “ghosts of departed quantities”), and Robinson’s resolution came out of model theory. Noted as an adjacency; this book makes no philosophical argument and that spoke holds no page on it.
Related
infinitesimal-calculus · vector-calculus · mathematics · h-jerome-keisler · university-of-wisconsin-madison · corral-vector-calculus · synthesis