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this is really a question about math and not books. I am mainly wondering if reading really old calculus books is still beneficial for undrgraduate students today. I was told that the material covered won't be of much benefit, things like curves, various mechanical integration methods, etc., and that I would be better off studying a 'calculus with an intro to analysis' type book like Apostol or Spivak's calculus. Is this true? Some specific examples of old books I have in mind are:

Edwards: https://archive.org/details/anelementarytre01edwagoog

Todhunter: https://archive.org/details/atreatiseondiff06todhgoog

Williamson: https://archive.org/details/anelementarytre20willgoog

One thing's for sure: The problems in these books are much harder than in modern books, which is very appealing to me coming from an olympiad background.

So they aren't as far back as say Cauchy, but still are fairly old. I would still be interested however in knowing if something like Cauchy's Calcul Differentiel et Integral (I can read french!) is worth studying today; I know that Clerk Maxwell studied it at Edinburgh University for instance (before "going up" to Cambridge): https://archive.org/details/leonsdecalculdi02goog

Thanks

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    $\begingroup$ G.H Hardy's A course in Pure Math is wonderful $\endgroup$ – Taylor Ted Aug 3 '15 at 10:01
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    $\begingroup$ Georges Valiron's Cours d'analyse is another very good book, especially on complex variables (dates back to the 50s). $\endgroup$ – Bernard Aug 3 '15 at 10:39
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    $\begingroup$ On a tangent, the Mathematics Bookshelf at Project Gutenberg has a number of gratis pre-1923 books typeset in LaTeX (typos fixed, some modernized notation), including Calculus Made easy (MacMillan 2nd Ed. 1914) and A Course of Pure Mathematics (Cambridge 3rd Ed. 1921). A few of these have been converted to HTML5 by the Sayahna Foundation from the Gutenberg sources. $\endgroup$ – Andrew D. Hwang Aug 3 '15 at 11:55
  • $\begingroup$ Thanks @TaylorTed for mentioning Hardy's book. Its simply the best and no modern text can beat it. It is designed for self study. See a review paramanand.blogspot.com/2005/11/… $\endgroup$ – Paramanand Singh Aug 4 '15 at 8:45
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To answer your question: yes, it still can be beneficial to read old textbooks or publications, especially from a history of math point of view it is even necessary to read the original works. You get an idea on how the original ideas have been developed. Very interesting - no question about it.

However, you have to take into account that these old books most likely:

  • lack of course on modern references
  • are not aligned with the modern curriculum
  • use often a different notation
  • are often hard to come by

Maybe a good compromise would be to read a history of math book, for example History of Topology and parallel a modern book to compare the ideas and get appropriate modern references.

EDIT (due to a comment of Andrew D. Hwang)

Andrew mentioned the Gutenberg project with a big collection of old and mildly adapted math publications, I find this reference needs definitely to be in this answer, great reference!

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Ramunajan and Dirac were both fans of Edwards Calculus. (I assume the 1896 differential volume you link).

What's funny is to read the review of Edwards Integral Calculus (1920s) by Hardy. He is totally pissed at Edwards for using 19th century thinking and for emphasizing Tripos tricks (you would like it, with your background). But Hardy couldn't help but respect it given the number of tricks contained. Check it out...is badass.

Thre interesting diffyQ books are Ince Ordinary Differential Equations and Forsyth Treatise on diffyQ and Forsyth Theory of DiffyQ (6 volumes!). The Forsyth volumes still gets referenced even today and arguably has some developments (within the exercises) that were published as findings as late as the 1940s and 1980s (silly physicists!)

Forsyth was a badass who could read and translate both the language and content of German and French analysts from the late 1800s (even Lie). But he also knew every trick in the book for dealing with a diffyQ. Even solved some elliptical coordinate thingamajig that comes from Einstein General Relativity (short paper commenting on Einstein's paper!)

By the way, the Treatise book was his more accessible textbook versus encyclopedic diffyq book. It was intended for a first diffyQ course for good students (but now would be considered more of a second course). He eschewed Lie group methods for diffyQs (knowing them) and was criticized by a reviewer for it. Imaging having that in current first course!

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  • $\begingroup$ indeed, as you said, Hardy was totally against tripos! he wanted to abolish them, it is really a very interesting history. www-groups.dcs.st-and.ac.uk/history/Extras/Hardy_Tripos.html $\endgroup$ – iadvd Aug 16 '17 at 2:53
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    $\begingroup$ @iadvd: IMHO Hardy did a great service to the mathematical community by making a case against Tripos. Most of the problems in tripos are laborious and do not require insight on part of the problem solver. The idea that mathematics is just a bag of tricks has to be ditched seriously. $\endgroup$ – Paramanand Singh Aug 16 '17 at 7:13
  • $\begingroup$ @ParamanandSingh I totally agree! indeed I would like to recommend the following great book for a very detailed description of the reasons defended by Hardy: maa.org/press/books/the-g-h-hardy-reader $\endgroup$ – iadvd Aug 16 '17 at 7:24

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