Abraham Robinson discovered a rigorous approach to calculus with infinitesimals in 1960 and published it in

Non-standard Analysis, Proceedings of the Royal Academy of Sciences, Amsterdam, ser A, 64, 1961, p.432-440

This solved a 300 year old problem dating to Leibniz and Newton.  Extending the ordered field of (Dedekind) "real" numbers to include infinitesimals is not difficult algebraically, but calculus depends on approximations with transcendental functions.  Robinson used mathematical logic to show how to extend the numbers and all real functions in a way that preserves their properties in a precise sense.  These properties can be used to develop calculus with infinitesimals.  This section of the website is a brief introduction to this rigorous approach to calculus using infinitesimals.

Infinitesimal numbers have always fit basic intuitive approximation when certain quantities are "small enough," but Leibniz, Euler, and many others could not make the approach free of contradiction.  Robinson's discovery offers the possibility of making rigorous foudations of calculus more accessible.    

H. Jerome Keisler developed simpler approaches to Robinson's logic and began using infinitesimals in beginning U. S. calculus courses in 1969.  The experimental and first edition of his book were used widely in the 1970's.

Elementary Calculus: An Infinitesimal Approach, PWS Publishers, 1986, now available free at http://www.math.wisc.edu/~keisler/calc.html

Section 1: Intuitive Proofs with "Small" Quantities (click for details)

Section 1 of this article uses some intuitive approximations to derive a few fundamental results of analysis.  We use approximate equality, only in an intuitive sense " is sufficiently close to ".  Intutive approximation gives compelling arguments for the results, but not technically complete proofs.

Continuity & Extreme Values

Microscopic tangency in one variable

The Fundamental Theorem of Integral Calculus

Continuity of the Derivative

Trig, Polar Coordinates & Holditch's Formula

Leibniz's Formula for Radius of Curvature

Concerning the History of the Calculus

Section 2: Keisler's Axioms (click for details)

Section 2 of this article completes the proofs of Section 1 using Keisler's approach to the logic of infinitesimals.

Small, Medium, and Large Hyperreal Numbers

Keisler's Function Extension Axiom

Continuity & Extreme Values

Microscopic tangency in one variable

The Fundamental Theorem of Integral Calculus

The Local Inverse Function Theorem

Second Differences and Higher Order Smoothness

Advanced Books on the Foundations of Infinitesimal Calculus

Mathematical Background: Foundations of Infinitesimal Calculus

by Keith Stroyan

This is a complete text on the foundations of calculus using infinitesimals.  It is written at an upper undergraduate or beginning graduate level.

Foundations of Infinitesimal Calculus

by H.Jerome Keisler

This monograph is a companion to the online edition of Keisler’s textbook ``Elementary Calculus: An Approach Using Infinitesimals''. It can be used as a quick introduction to the infinitesimal approach to calculus for mathematicians, as background material for instructors, or as a text for an undergraduate seminar.