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Boyer 1939 is a nice readable survey of the history of the calculus, but it's showing its age. Discussing the notion of instantaneous velocity, he has:

Mathematics knows no minimum interval of continuous magnitudes [such as distance and time.] Attempts to supply a logical definition of such an infinitesimal minimum which shall be consistent with the body of mathematics as a whole have failed. [p. 7]

This was accurate as of 1939, but is out of date given what we've learned from NSA and SIA. Boyer's account reads somewhat as the glorious march to victory of the limit concept.

Bell 2005 is more modern, but a review by Ehrlich (can't find a reference, but the text comes up in google) complains that it comes off as a sales job for SIA. (It's also extremely expensive.)

Is there a modern source on this topic that doesn't have an ax to grind?

Bell, The continuous and the infinitesimal in mathematics and philosophy, 2005.

Boyer, The History of the Calculus and its Conceptual Development, 1939.


Was the early calculus inconsistent?

history of calculus of several variables

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Anyting by Grattan-Guinness? – Yemon Choi Jun 7 '14 at 2:48
Also, I thought all historians nowadays avow that everyone has an ax(e) to grind... – Yemon Choi Jun 7 '14 at 2:49
We've certainly heard a lot about NSA recently, but I assume you mean NonStandard Analysis. What is SIA? – Donu Arapura Jun 7 '14 at 15:45
@DonuArapura: SIA is smooth infinitesimal analysis: – Ben Crowell Jun 7 '14 at 17:56
Might as well provide a link to Ehrlich's review of Bell's book: – Todd Trimble Aug 19 '14 at 13:05
up vote 11 down vote accepted

A History of Analysis, edited by Hans Niels Janhke, American Mathematical Society (2003) is a superb, all together scholarly collection of essays that cover a wide range of topics in the history of analysis.

Three other noteworthy works are:

The Development of Newtonian Calculus in Britain, 1700-1800, Niccolò Guicciardini, Cambridge University Press (1990).

The Historical Development of the Calculus, C. H. Edwards, Jr., Springer (1979).

The Origins of the Infinitesimal Calculus, Margaret Baron, Dover (1987).

The work by Edwards touches on NSA and the first paper in the collection edited by Janhke very briefly touches on NSA and SIA. Some historical background on NSA can also be found in Joseph Dauban's Biography of Abraham Robinson.

To my knowledge, careful histories of the development of NSA and SIA remain to be written. However, some papers, including the following three by me have been preparing the way by recounting the history of the development of non-Archimedean mathematics.

Hahn’s Über die nichtarchimedischen Grössensysteme and the Origins of the Modern Theory of Magnitudes and Numbers to Measure Them, in From Dedekind to Gödel: Essays on the Development of the Foundations of Mathematics, edited by Jaakko Hintikka, Kluwer Academic Publishers, 1995, pp. 165-213.

The Rise non-Archimedean Mathematics and the Roots of a Misconception I: the Emergence of Non-Archimedean Systems of Magnitudes, Archive for History of Exact Sciences 60 (2006), pp. 1-121. (I hope to publish Part II in the not too distant future)

The Absolute Arithmetic Continuum and the Unification of All Numbers Great and Small, The Bulletin of Symbolic Logic 18 (1) 2012, pp. 1-45.

Additional papers on the subject have been written by Mikhail Katz and a host of co-authors.

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Grattan-Guinness edited a fine volume entitled

From the calculus to set theory, 1630–1910. An introductory history. Edited by I. Grattan-Guinness. Reprint of the 1980 original. Princeton Paperbacks. Princeton University Press, Princeton, NJ, 2000. viii+306 pp. ISBN: 0-691-07082-2

For those interested specifically in the history of the limit concept (as well as the shadow), there is a piece in the current issue of Notices AMS: (see also

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