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The Historical Development of the Calculus

C. H. Edwards, Jr.
Springer Verlag
Publication Date: 
Number of Pages: 
Springer Study Edition
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Allen Stenger
, on

This is a very detailed history of the calculus, starting out with Babylonian calculations of area and going through the increase in rigor and the arithmetization of analysis at the end of the 19th Century. It thus covers not only what most people consider calculus (essentially the work of Newton and Leibniz), but a lot of analytic geometry and real analysis as well. There’s a short section at the end about 20th-century developments, including the Lebesgue integral and nonstandard analysis. This is a 1979 book, that had a corrected reprinting in 1982 and has been reprinted again in 1994.

The most striking feature of the book is that it shows you the drawings and calculations (in modern notation) that were used in developing calculus. It’s not just about concepts but about methods too, and gives the whole picture of how the researchers worked.

Probably the part of calculus that has changed the most from antiquity to the present is the handling of infinity and limits, and this book is especially careful in presenting those changes. Archimedes actually had nearly all the ingredients of 17th-century calculus, including the sum of a finite geometric progression. But instead of taking the limit, he used proofs by contradiction based on approximation by finite sums (p. 22). The book lists on p. 75 the three calculus essentials that the Greeks did not have: limits, a systematic approach to calculating areas, and the realization of the inverse relationship between areas and tangents (i.e., the Fundamental Theorem of Calculus).

The treatment of the Newton–Leibniz controversy is also especially careful, and each researcher gets a separate chapter with a complete exposition of his approach, with a brief comparison section at the end (pp. 265–267). There’s an especially interesting chapter on Napier and logarithms, showing that Napier had a completely different view of logarithms than we do today.

One peculiarity of the book is that there is little or no mention of applications to physics (there are many to geometry). I believe this is because Edwards is skeptical that physics was really a driver in the development of calculus. See for example his discussion on pp. 224–225 about whether Newton really worked out his Principia Mathematica by calculus before covering up his work with geometric proofs.

Another popular book on the subject is Boyer’s The History of the Calculus and Its Conceptual Development. This is a very different type of book, not least because it is nearly all prose with few diagrams or formulas. As indicated by its previous title, The Concepts of the Calculus, Boyer’s work is about the ideas of calculus and not about its methods. Boyer’s work also has a much more antique writing style, despite being only 40 years older; it is an example of the heroic age of math history, especially as practiced by Eric Temple Bell.

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. Area Number and Limit Concepts in Antiquity

2. Archimedes

3. Twilight, Darkness and Dawn

4. Early Indivisibles and Infinitesimal Techniques

5. Early Tangent Constructions

6. Napier's Wonderful Logarithms

7. The Arithmetic of the Infinite

8. The Calculus According to Newton

9. The Calculus According to Leibniz

10. The Age of Euler

11. The Calculus According to Cauchy, Riemann and Weierstrass

12. Postscript: The Twentieth Century