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What are the oldest books regularly used in your field (and which don't feel "outdated")?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

66 Answers 66

I'm surprised that no one has mentioned Emil Artin's beautiful monograph on the Gamma Function. The economy and elegance are unsurpassed.

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For ordinary differential equations there is:

  • Theory of ordinary differential equations by Coddington and Levinson, McGraw-Hill Book Company, 1955

I'm not sure it is used in courses, but it is certainly still cited frequently, for example as a reference for Carathéodory type differential equations where the vector field is only integrable in time.

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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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H.S. Hall and S.R. Knight, Higher Algebra

First edition 1891 (or so), recent edition 2001 (for example). Subtitled a Sequel to Elementary Algebra for Schools, and so betrays the fact it's not really like Lurie's book of the same title.

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My father used this book when he studied at Brooklyn College in the 1930's. Luckily, he still owned it when I was growing up. I spent many happy hours poring over it when I was 8 or 9. – Victor Miller Sep 19 '15 at 15:43

"Projective Geometry" by Coxeter (1963), "Finite Geometries" by Dembowski (1968) and "Projective Planes" by Hughes and Piper (1973), still serve as great textbooks for these topics.

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I wonder why there is a downvote at the time of this comment. I particularly think that "Finite Geometries" by Dembowski is still referred to today... (+1). – knsam May 18 '15 at 3:18
@the person who downvoted this: it will be much more constructive to write a comment to explain your issue with this answer. – Anurag May 18 '15 at 3:43

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. Reprinted 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

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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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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Daniel Quillen's "Homotopical algebra", 1967.

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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). – user9072 Dec 28 '12 at 18:18
done........... – Abdelmalek Abdesselam Dec 28 '12 at 18:37
@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
@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

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

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

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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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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 Jan 7 '13 at 11:38

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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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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According to wikipedia (see the book has been continually revised since 1804 and at this point contains essentially none of the 19th century content. – Andy Putman Feb 6 '13 at 16:58

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

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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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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: . 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: . 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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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

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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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
@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
+1 for the last four, in particular for Kato and Arnold – RSG Jan 7 '13 at 11:19
I took math 1c-2c I (math for prospective PhD's :-)) from Lang in 1964-65. At the end of the year he had the surviving members of the class pack up envelopes of authors copies of the newly published Algebra to be sent out to his friends/contacts. After an afternoon's work of this he presented each of us with a copy, which I still have on my bookshelf. – Victor Miller Sep 19 '15 at 15:47

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

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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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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. – user9072 Jan 2 '13 at 20:04

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