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It's a commonplace to state that while other sciences (like biology) may always need the newest books, we mathematicians also use to use older books. While this is a qualitative remark, I would like to get a quantitative result. So what are some "old" books that are still used?

Coming from (algebraic) topology, the first things which come to my mind are the works by Milnor. Frequently used (also as a topic for seminars) are his Characteristic Classes (1974, but based on lectures from 1957), his Morse Theory (1963) and other books and articles by him from the mid sixties.

An older book, which is sometimes used, is Steenrod's The Topology of Fibre Bundles from 1951, but this feels a bit dated already. Books older than that in topology are usually only read for historical reasons.

As I have only very limited experience in other fields (except, perhaps, in algebraic geometry), my question is:

What are the oldest books regularly used in your field (and which don't feel "outdated")?

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    $\begingroup$ Please don't call "Characteristic Classes" old or I will have to call myself old, being born in the same year as the lectures :-/ $\endgroup$ – Lee Mosher Dec 28 '12 at 18:28
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    $\begingroup$ @Lee Mosher: Would you prefer to call yourself "classical"? :) $\endgroup$ – user29720 Dec 29 '12 at 0:08
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    $\begingroup$ Timeless . . . . $\endgroup$ – Rodrigo A. Pérez Dec 29 '12 at 3:08
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    $\begingroup$ Although the question doesn't ask this exactly, it would be interesting to know what is the oldest textbook that someone still prescribes as the main textbook for a course. This would be more significant than just using an old book for occasional reference. $\endgroup$ – Brendan McKay Dec 29 '12 at 4:07
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    $\begingroup$ E. Spanier "ALgebric TOpology", "Eilenberg Steenrod "ALgebric TOpology", GOdement "Topologie Algébrique et Théorie des Faisceaux ", COurant-Hilbert "Methods of Mathematical Physics"... "the problem of contemporary authors, is to being con-temporary" (Ennio Flaiano) $\endgroup$ – Buschi Sergio Dec 29 '12 at 10:45

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In classical invariant theory, both "The Algebra of Invariants" by Grace and Young and "An introduction to the algebra of quantics" by Elliott are still much in use. The latest edition of Grace and Young is 1903 and of Elliott 1913.

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    $\begingroup$ It seems these were already mentioned in Abdelmalek Abdesselam's answer. $\endgroup$ – user9072 Jan 2 '13 at 20:04
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I would like to mention about M. Postnikov's geometry series, Lectures on geometry which I always refer to when I need some coherent view inbetween geometry and analysis.

Sometimes I may refer to Hopf and Alexanderoff's Topologie in order to gain some authority...

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O. Kellogg, Foundation of Potential Theory

The first edition of Kellogg's Foundation of Potential Theory was published in 1929. Btw he was a student of David Hilbert.

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Most of the textbooks I use are quite new. The old books are the exception.

The oldest book about mathematics I use is Hajós György: Bevezetés a geometriába, a textbook on elementary geometry (in the sense of Euclid). The first edition is from 1950, I have a copy published in 1960. (Edit: it seems there's a German translation.)

I'm also using Knuth's The Art of Computer Programming, does that count as old now? The translation of the first volume is based on the second edition, of which the original was published in 1973. (Edit: the above was accurate when I wrote this post. Since then, I actually bought the third edition versions in original English, of which the first volume was published in 1997, so it no longer counts as an old book.)

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P.A. MacMahon, Combinatory analysis, vols 1 and 2, Cambridge University Press, 1915–16.

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Daniel Quillen's "Homotopical algebra", 1967.

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Bonnesen and Fenchel, "Theorie Der Konvexen Korper" Springer, Berlin 1934 not available in English translation until 1987 although Eggleston's "Convexity" 1958 draws heavily on it.

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An Elementary Treatise on Coordinate geometry of Three Dimensions. Macmillan 1910, reprinted upto 1950 or later. Apart from classical setting of analytical geometries contains early differential geometry with theory of Invariants. Found the book in pavement shop (Cost: 0.1 $ !)

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I would add 1/ Regarding logic and set theory: (i) The consistency of the continuum hypothesis, Kurt Gödel, Princeton University Press, 1940 (ii) Set theory and the indeeandance of the continuum hypothesis, Paul Cohen, Walter Benjamin, 1966 2/ Regarding analysis: Leçons d'analyse fonctionnelle, Frederic Riesz et Bela Nagy, Gauthier-Villars, 1955

Gérard Lang

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    $\begingroup$ I don't think either of the examples in set theory are still "being used", other than for purely historical purposes. There are much improved presentations in more modern texts. In particular, no one uses Cohen's approach to forcing. $\endgroup$ – Andrés E. Caicedo Dec 17 '18 at 14:25
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Nathaniel Bowditch is generally regarded as a nineteenth century American mathematician . His American Practical Navigator has been in continous print since 1804. It is still in use today judging from the comments on Amazon. But perhaps this isn't what was meant by a mathematics book and perhaps navigation isn't to be considered applied mathematics.

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    $\begingroup$ According to wikipedia (see en.wikipedia.org/wiki/American_Practical_Navigator) the book has been continually revised since 1804 and at this point contains essentially none of the 19th century content. $\endgroup$ – Andy Putman Feb 6 '13 at 16:58
  • $\begingroup$ Apart from the details of its revision, I think historically "navigation" was very much a topic in mathematics: spherical trigonometry and all that! Bowditch was certainly a mathematician by U.S. standards, such as they were! :) $\endgroup$ – paul garrett May 17 '15 at 21:55
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