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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What are the oldest books regularly used in your field (and which don't feel "outdated")?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
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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
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@Lee Mosher: Would you prefer to call yourself "classical"? :) –  user29720 Dec 29 '12 at 0:08
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Timeless . . . . –  Rodrigo A. Pérez Dec 29 '12 at 3:08
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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
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61 Answers

Meet the Rudins: Baby Rudin (first published in 1953), Papa Rudin (whose oldest copyright I've been able to find dates back to 1966) and Grandaddy Rudin (1973 is the oldest reference I've found).

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Also I would add his book "Functional Analysis". –  Vahid Shirbisheh Dec 28 '12 at 21:59
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Also known as Grandaddy Rudin. –  Nate Eldredge Dec 30 '12 at 18:47
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@Robert: judging by the year it was published, I suppose it should be adolescent Rudin. –  Alberto García-Raboso Jan 7 '13 at 13:36
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A rare example of a family where the granddaddy is the youngest... –  Daniel McLaury Oct 13 '13 at 7:03
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EGA and SGA, both from the 1960s and 1970s, are very widely used in algebraic geometry. Hartshorne's textbook (first published in 1977) is still the main choice for courses on the theory of schemes.

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I think the absolute record (excluding Euclid) belongs to

E. T. Whittaker G. H. Watson, A course of modern analysis.

According to the Jahrbuch database, the first edition was in 1915. Moreover, this 1915 edition was an extended version of a 1902 book, by Whittaker alone.

The last revision was in 1927. The book is still in print, and widely used, not only by mathematicians but by physicists and engineers. Soon we will celebrate the centenary... It has 1056 citations on Mathscinet, by the way, and 8866 on the Google Scholar !

Perhaps this deserves a Guinnes book of records entry as a "textbook longest continuously in print". And I suppose this is a record not only for math but for all sciences... with the exception of Euclid and Ptolemy, of course:-)

If we include not only textbooks but research monographs there are plenty of other examples, even older ones:

H. F. Baker, Abelian functions, was first published in 1897. Rerinted in 1995, and there is a new Russian translation.

Just out of curiosity, look at its current citation rate in Mathscinet:-)

They also reprinted

H. Schubert, Kalkül der abzählenden Geometrie, 1879, in 1979,

and again you can see from Mathscinet that people are using this.

EDIT: A brief inspection of the most cited (and thus most used) books on Mathscinet shows that a very large proportion of the most cited books are 30-40 years old. Which is easy to explain, by the way. Thus on my opinion, such books do not qualify for this list (unless we want to make it infinite).

EDIT2: Today I accidentally found that 3 of the 4 copies of

G. H. Watson, Treatise on the theory of Bessel functions (first edition, 1922)

are checked out from my university library. Mathscinet shows 1157 citations for the last 2 editions.

Another question is old papers which are still highly sited. A typical life span of a paper is much smaller than that of a book. In the list of 100 most cited papers in 2011, I found only two papers published before 1950 (One by Shannon and another by Leray).

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I have an electronic copy of the 1996 reissue of Whittaker and Watson's that details its history: first edition 1902, second edition 1915, third edition 1920, fourth edition 1927. Since then, there were 8 reprints (1935, 1940, 1946, 1950, 1952, 1958, 1962 and 1963) and the 1996 reissue. –  Alberto García-Raboso Dec 28 '12 at 19:25
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I'm amazed no one has mentioned Hardy and Wright's wonderful Introduction to the Theory of Numbers. It was first published in 1938 and is absolutely delightful.

The most recent (6th) edition includes a chapter on elliptic curves.

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@ayanta: Well, the new chapter on elliptic curves was written with an eye towards fitting into the style of the rest of the text. (An assertion that I feel that I'm entitled to state as a fact, rather than as an opinion.) So I guess there might be some who would say that the elliptic curves chapter is also "outdated", despite having been written quite recently! But I have to respectively disagree with your opinion of the book, which I feel is a masterpiece. –  Joe Silverman Dec 29 '12 at 23:37
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Henri Cartan and Samuel Eilenberg published their Homological Algebra in 1956, although it was famously circulated for a long time before that. While that book more or less founded its subject, it is still quite useful.

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"Introduction to commutative algebra" by Atiyah and MacDonald is from 1969. (I learnt commutative algebra from this book at the University of Oslo just a few years ago)

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If computer science counts as math, then The Art of Computer Programming (first volume published 1968) would be a good example of a text that's still in wide use.

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Mac Lane's "Categories for the working mathematician" (1971).

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That depends if you speak of research books or advanced text book. In the second category, I should place

  • Rudin's Real and complex analysis (1966),

  • J.-P. Serre's Cours d'Arithmétique (1970) (hope you will forgive me),

  • Lang's Algebra (1st Edt 1965).

In the first category, I see

  • Kato's Perturbation theory of linear operators (1966),

  • Courant & Hilbert's Methods of Mathematical Physics (1924),

  • Courant & Friedrich's Supersonic Flow and Shock Waves (1948),

  • V. I. Arnold's Mathematical methods of classical mechanics (1974).

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+1 for Courant & Hilbert! –  Igor Khavkine Dec 28 '12 at 18:36
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@Qfwfq: Well, we used it when I was a junior, so it had already appeared in 1975. But I don't know the original publication date offhand. –  Joe Silverman Dec 29 '12 at 23:29
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+1 for the last four, in particular for Kato and Arnold –  RSG Jan 7 '13 at 11:19
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van der Waerden's Moderne Algebra was first published in 1930, I think. I use the book occasionally for my course, but am not sure which edition.

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The old edition with a chapter on elimination theory is still a good place to learn the subject. Unfortunately it was eliminated in the later editions. –  Abdelmalek Abdesselam Dec 28 '12 at 18:48
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I've also used the Elimination Theory chapter of the first editions a lot! –  Bruno Jan 10 '13 at 7:59
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How about:

G. H. Hardy, J. E. Littlewood, G. Pólya, Inequalities (1934, second edition 1952).

G. Pólya, G. Szegő, Problems and Theorems in Analysis (first German edition in 1925)

G. Szegő, Orthogonal Polynomials (1939)

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+1 for Polya & Szego! –  Alexander Shamov Dec 31 '12 at 10:45
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Abramowitz and Stegun's Handbook of Mathematical Functions (1964) is still used. As the August 2011 Notices article by Boisvert et al. says,

The Handbook remains highly relevant today in spite of its age. In 2009, for example, the Web of Science records more than 2,000 citations to the Handbook. That is more than one published paper every five hours—quite remarkable!

In time it might be superseded by the NIST Handbook of Mathematical Functions (or its online version, the Digital Library of Mathematical Functions), but not yet.

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Ahlfors' Complex Analysis. The 3rd edition is from 1978, but the book itself was written in the 50s. No other book comes close.

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Has anyone else ever noticed something funny about the title of Chapter 1 in that edition? –  Adam Epstein Dec 29 '12 at 19:44
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My own copy has COMPUTER NUMBERS. (Rodrigo, perhaps I showed you when you were in my lecture course?) But I would actually imagine that far more people (not necessarily mathematicians) would find COMPLEX NUMBERS funnier. –  Adam Epstein Jan 4 '13 at 14:07
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The notes of the 1951-2 Artin-Tate seminar on class field theory (published in 1968, and re-issued in LaTeX form a few years ago with a new Introduction by Tate addressing subsequent developments) remains a fundamental reference in algebraic number theory, despite the abundant supply of more recent references on the subject.

One reason is that it is the only reference outside the original research literature where one can find a complete treatment (with proofs) of certain key aspects of the theory such as the Grunwald-Wang phenomenon and Weil groups for class formations (especially the case of number fields, which lacks a bare-hands construction as for local fields and global function fields). Come to think of it, the general notion of Weil groups for class formations emerged from that seminar...The style of the proofs remains generally quite fresh.

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The standard, go to reference in geometric measure theory is still Federer's 1969 classic, Geometric Measure Theory. It is very rarely the first reference one uses since it is rather dense and there are other introductions and expositions, some of them very good.

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The calculus and analysis texts of Michael Spivak and Tom Apostol come to my mind...at least they are still widely used in my land (Colombia) for undergraduate (serious) math courses.

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I'm surprised that nobody has mentioned Serre's Corps locaux (Local Fields), his Cohomologie galoisienne (Galois cohomology) and his Représentations linéaires des groupes finis (Linear representations of finite groups).

Other eternal texts in Number Theory include Artin's Algebraic numbers and algebraic functions and the Artin-Tate notes on Class field theory, Hasse's Zahlentheorie and his Klassenkörperbericht, Hecke's Vorlesungen über die Theorie der Algebraischen Zahlen, Weyl's Algebraic Theory of Numbers, and Hilbert's Zahlbericht.

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Artin's Galois theory (1942) is still a classic. People in automata theory and finite semigroups still use Samuel Eilenberg's two volumes on the subject (1974).

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If one needs to use tools from classical invariant theory or elimination theory then some books that come to mind are:

and there are quite a few more.


For Salmon's book, the 4th edition of 1885 might be best. Indeed, as I learned from a paper by Macauley, it has a discussion (on p. 87) of Cayley's very general formula for the multivariate resultant as the determinant of a complex (see the book by Gelfand, Kapranov and Zelevinsky for a modern account and a reprint of Cayley's paper).

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Since you answered before this was turned into CW by the questioner, your answer stayed in normal mode. Typically moderators would take care of this, but since your answer is the only one affected in this case, I thought it could be more efficient if you turned your answer into CW manually (edit and tick the box). –  quid Dec 28 '12 at 18:18
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done........... –  Abdelmalek Abdesselam Dec 28 '12 at 18:37
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@Abdelmalek Abdesselam: Can it really be that modern books on computational commutative algebra have not adequately replaced the need to look at a book on "modern higher algebra" from 1876 (or some of the others that you list)? This sounds very surprising. What are examples of things found in such old books that are not available in more recent references? –  user30180 Dec 29 '12 at 6:00
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@Ayanta: Despite the eloquence of your rethorical question, what you said is simply wrong. For instance anything involving the classical symbolic method in relation with specific invariants coming from elimination theory is not really accounted for nor "adequately replaced" in the recent commutative algebra literature. To form an accurate and informed opinion you need to have a look at the books I mentioned especially Grace and Young if you only have time to look at one. –  Abdelmalek Abdesselam Dec 31 '12 at 12:31
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Sz. Nagy-Foias: Harmonic Analysis of Operators in Hilbert Space (1970) is a still widely used and lively book (though there is a new updated edition in 2012).

T. Kato's Perturbation Theory book (1967) is also definitely in this category, though there is a 1980 second edition and a 1995 reprint.

Nelson Dunford, Jacob T. Schwartz: Linear Operators (1958,1963, 1971). I still take this book regularly into my hands.

An other reference on differential equations is

J. L. Lions, E. Magenes: Non-Homogeneous Boundary Value Problems, 1972. It is still "the" reference.

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Maybe Dunford-Schwartz (1957) could be added here? –  Theo Buehler Dec 28 '12 at 17:20
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+1 for Kato. I've had my copy for six years so far and I've learned something new from it in every year. –  Ian Morris Dec 29 '12 at 10:58
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Gaston Darboux' magnum opus Leçons sur la Théorie générale des Surfaces et les Applications géométriques du Calcul infinitésimal (first edition 1890, I think; there is a second edition dating from around 1915) is still read by many differential geometers, and, as far as I know, it is still in print via the AMS Chelsea series.

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I used G. H. Hardy's A Course of Pure Mathematics (First edition 1908) when I taught undergraduate real analysis not so long ago. The care with which concepts are explained and the number of interesting problems and examples is, in my opinion, unmatched by newer books.

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N. G de Bruijn's Asymptotic methods in analysis is still the best reference for the topic. The current 1981 Dover reprint edition is largely unchanged since the 1958 first edition.

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I've used Euclid's Elements

Halmos (several)

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Most good books in general topology are old. Here are some good topology books that I often refer to.

rings of continuous functions by Gillman and Jerison (1960)

Uniform Spaces by John Isbell (1964)

General Topology by Stephen Willard (1970)

Topology by James Dugundji (1966)

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It would probably be a good idea to remove the generic derogatory comment... –  Mariano Suárez-Alvarez Dec 28 '12 at 23:19
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I would add Hurewicz and Wallman "Dimension theory" here as well. –  Misha Dec 29 '12 at 13:49
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I am not a general topologist, but I am truly amazed that among "old good books on general topology", Kuratowski (first edition 1933) was not mentioned. –  Alexandre Eremenko Dec 30 '12 at 4:26
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Kelley's book on topology is one I consult reasonably often. –  Todd Trimble Jan 1 '13 at 23:55
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My choice of books would be:

  • Theory of Riemann-Zeta Function by E.C. Titchmarsh, (Oxford University Press)

  • Theory of Functions by E.C. Titchmarsh (Oxford University Press, 1952).

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I was just looking at HSM Coxeter's Regular Polytopes (1948) pretty recently, and it is still wonderful.

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My own field, ergodic theory, is relatively young in that some concepts now regarded as fundamental -- Kolmogorov-Sinai entropy, for example -- were not fully formulated until around 1960. Nonetheless there are a couple of old books still in use and receiving citations:

E. Hopf, Ergodentheorie, 1937;

R. Halmos, Ergodic theory, 1957.

If the 1960s are sufficiently long ago to constitute "old" then there are many old references in probability which remain in heavy use, for example:

P. Billingsley, Convergence of probability measures, 1968;

L. Breiman, Probability, 1968;

and one of the classics of the field:

W. Feller, Introduction to probability theory and its applications, 1950.

Outside my own field, a much-cited number theory text which no-one has yet mentioned:

A. Khinchin, Continued fractions, 1936.

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I used to be puzzled why so many people cite Feller, since there are so many newer books on the topic. And then I actually read Feller, and was enlightened. –  arsmath Jan 2 '13 at 21:31
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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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Mathematical Foundations of Statistical Mechanics by A. I. Khinchin. The original edition in Russian was published in 1943 according to MathSciNet (MR Number=(17677)).

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