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What are the oldest books regularly used in your field (and which don't feel "outdated")?

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I think this should be Community Wiki. –  Alberto García-Raboso Dec 28 '12 at 16:28
Please don't call "Characteristic Classes" old or I will have to call myself old, being born in the same year as the lectures :-/ –  Lee Mosher Dec 28 '12 at 18:28
@Lee Mosher: Would you prefer to call yourself "classical"? :) –  user29720 Dec 29 '12 at 0:08
Timeless . . . . –  Rodrigo A. Pérez Dec 29 '12 at 3:08
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) –  Buschi Sergio Dec 29 '12 at 10:45

61 Answers 61

Spivak's five volume "Comprehensive Introduction to Differential Geometry" still gets a lot of use---particularly the first two volumes.

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Montgomery and Zippin "Topological Transformation Groups" (originally published in 1955) is still the only book to cover the relevant results on topological characterization of Lie groups in full generality (including Lie group actions). I am not sure if this belongs to algebra or topology area-wise, but it is used in my area, geometric group theory.

For pedagogical purposes, I still use "What Is Mathematics?" by Courant and Robbins (originally published in 1941) and "Geometry and Imagination" (1932) by Hilbert and Kohn-Vossen, when a high school student or an undergraduate asks me for suggestions.

My personal definition of an "old book" is the same as Lee Mosher's, so I do not include here Chapters 4-6 of Bourbaki's "Lie groups and Lie algebras" (1968) which I use as a working tool.

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Some volumes of Bourbaki, as Topological Vector spaces or Lie groups are still widely quoted.

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In numerical linear algebra, Gantmacher's The theory of matrices is still a widely read and cited text (see MathSciNet citations). The Russian original dates back to 1953 (thanks @Giuseppe), and the first English translation is from 1959.

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The first Russian edition is dated 1953. –  Giuseppe Tortorella Jan 7 '13 at 11:38

No one suggests Weyl's Classical Groups? It was first published in 1939. I don't know if researchers in representation theory and invariant theory value it nowadays, but it is still frequently cited in random matrix literature.

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My first thought was Atiyah & Macdonald's 'An Introduction to Commutative Algebra' - which has already been mentioned - and 'anything by J.P. Serre' (that's old enough, of course!). It appears that not quite everything in this latter category has been mentioned; notably, 'Algebres de Lie Semi-simple Complexes', first published in 1966. There is also a later English translation, 'Complex Semisimple Lie Algebras' published in 1987.

While not quite an introduction, I find myself referring back to this text often for its streamlined, beautiful exposition (a hallmark of Serre). It also has the best exposition of root systems I've encountered.

Furthermore, another classic text on semisimple Lie algebras (J. Humphreys - 'Introduction to Lie Algebras & Representation Theory') is a 'fleshing out' of Serre's notes. Actually, Humphreys's textbook was first published in 1972 so might squeeze onto this list too?

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In metric geometry Busemann's "The Geometry of Geodesics" (1955) is still wonderful reading. This book is now published by Dover.

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Tate's thesis, Fourier analysis in number fields, and Hecke's zeta-functions, is from 1950 and is certainly still considered a primary on the subject (in addition to being the original resource).

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Many systematic introductions to the foundations of the edifice of Differential Geometry appeared in the sixties, and they are useful references even today. Some of them are:

  • Lang, Introduction to Differentiable Manifolds, 1962;
  • Helgason, Differential Geometry and Symmetric Spaces, 1962;
  • Kobayashi, Nomizu, Foundations of Differential Geometry, 1st Vol 1963, 2nd Vol 1969;
  • Sternberg, Lectures on Differential Geometry, 1964;
  • Bishop, Crittenden, Geometry of Manifolds, 1964;
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I still think the exposition on elliptic functions in Jacobi's Fundamenta Nova (1829) is one of the best I've encountered if you are interested in the functional relationships. A close second for me is Cayley's An elementary treatise on elliptic functions (1895), especially for the number of alternative proofs presented and the numerous relationships detailed. Modern books tend to take the algebraic approach, which is obviously extremely important for understanding the true nature of the relationships here, but for those of us who study the field because of it's incidental use in combinatorics and generating functions, these older books are a wealth.

Also, I have a personal love of Gauss' Disquisitiones Arithmeticae (1798) because it introduced me to number theory at a young age in a way that was very natural and elegant. Again, I appreciated it's approach to forms and related because it was all easily understandable with middle school algebra.

And finally more modern, for me Goldblatt's Topoi: The categorial analysis of logic (1979) is the best introduction to categories one could have, far better in my opinion than even Mac Lane's. That it is also subversive propaganda for constructivism is also a huge bonus.

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A book edited in 79 is not old! –  Mariano Suárez-Alvarez Jan 10 '13 at 1:52

Emil Artin's Geometric Algebra (Interscience, 1957) is definitely immortal.

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Rudin's Principles of Mathematical Analysis, and Herstein's Topics in Algebra if not heavily used, are the ideal that many people strive to in teaching introductory analysis and abstract algebra to undergraduates.

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Probability Theory, by Feller. Volumes I and II. Oldies but goldies

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I would like to add the nine volumes of the "Treatise on Analysis" of Jean Dieudonné (in French, "Éléments d'Analyse") which is quite thorough with beautiful exercises (unfortunately some of them contain errors or wrong hints) and give a broad view of contemporary aspects of Analysis, still useful nowadays especially the ninth & last volume (they were published in the 70s and 80s I think). Written with a flavor of Bourbaki, it gives the right level of generality (not too much, usually using only locally compact metrizable groups) and the numerous exercises really help to master maim results and methods of proof.

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I've always wondered, though, how much these amazing books were actually used (specially the later volumes) –  Mariano Suárez-Alvarez Dec 28 '12 at 22:09

Barry Simon and Michael Reed's classic volume on Functional Analysis (1981) is my one of my favorites.

Ayoub, "An Introduction to the Analytic Theory of Numbers," (1963) is out of print but one of the best books on the subject.

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"Differential and integral calculus" (Russian) by G. M. Fichtenholz was first published in 1948. Recently (in 2009) its $9^{th}$ edition was published and this book is still used as the main calculus textbook at some universities.

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R. Engelking (1977). General Topology.

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Hardy "Divergent series" (1949)

Naimark "Normed rings" (1968)

Maurin "Methods of Hilbert spaces" (1959)

Hille & Phillips "Functional analysis and semigroups" (1957)

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G.N. Watson's "A Treatise on the Theory of Bessel Functions" (1922),

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Keisler's "Calculus: An Approach Using Infinitesimals" is a very cool freshman calc book using NSA. It dates back to 1976, and is available for free online: http://www.math.wisc.edu/~keisler/calc.html . Although I'm not aware of anyone who's using Keisler in the classroom today, it's under a Creative Commons license, and there is a newer book by Guichard and Koblitz that incorporates a bunch of material from Keisler: http://www.whitman.edu/mathematics/multivariable/ . In the world of the digital commons, it's a little hard to define how old a book is. It's like asking how old a bacterium is. Bacteria are in some sense immortal. They just evolve.

Another wonderful old calc book that is still in print is Calculus Made Easy, by Silvanus Thompson, 1910.

I noticed that another answer to this question got heavily downvoted for referring to a book published in the 1980's. The question was: 'What are the oldest books regularly used in your field (and which don't feel "outdated")?' It didn't specify what "used" meant -- used in research, teaching, personal study, ...? The lower you get on the educational totem pole, the shorter the half-life of a book. Someone posted that they liked Disquisitiones Arithmeticae, but that doesn't mean it's being used for teaching number theory to undergrad math majors. For freshman calc, it is extremely unusual for anybody to use anything more than 5 years old. The community college where I teach has an explicit rule forbidding the use of books of more than about that age.

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Cassels and Frohlich (editors) on class field theory is regularly reprinted.

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It is reprinted, but as a participant in a long struggle by various people to get it reprinted, I have to say that "regularly" is the wrong adverb here. –  Timothy Chow Dec 30 '12 at 23:53

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.

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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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Two old books by David Mumford are not mentioned above (unless I am wrong):

1) Introduction to Algebraic Geometry (preliminary version of first 3 Chapters)

 (published and distributed by Harvard math. dpt., bound in red !, and containing 444 pages.)

  At that time (around end of 1960's ), this book was the unique good way to be introduced to theory of schemes . The EGA's were not helpful.
  In 1988, it became "The Red Book of Varieties and Schemes "(Springer). He is still excellent for learning ,and teaching , schemes.

2) The classical and fundamental: Geometric Invariant Theory (Springer, 1965),

  It has two enlarged editions : 1982, 1994.
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Mathematical Analysis By Zorich

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I'm really surprised at the suggestion that a book first published in the 1980ies should be a serious contender for "oldest books regularly used" in classical analysis –  Martin Dec 30 '12 at 15:17

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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It seems these were already mentioned in Abdelmalek Abdesselam's answer. –  quid Jan 2 '13 at 20:04

My field is dominated by older books, it seems. Gilmer's Multiplicative Ideal Theory came out in 1972 and it's nearly unmatched in the content it covers. We're currently using Kaplansky's Commutative Rings book for the Commutative Algebra course I'm taking at UCR; Atiyah and Macdonald's book is also considered a standard reference for those kinds of courses, and it came out in 1969. And, of course, you can't forget Bourbaki. I'm also partial to Zariski and Samuel's Commutative Algebra texts over other texts in the field, which came out in 1958 and 1961.

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When I was an undergrad, at the turn of the millenium, I took a complex analysis class that used (an English translation of) Knopp's 1936 Funktionentheorie.

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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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