Old books still used - MathOverflow

Old books still used

2012-12-28T16:24:04Z

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What are the oldest books regularly used in your field (and which don't feel "outdated")?

Answer by Abdelmalek Abdesselam for Old books still used

2012-12-28T16:33:39Z

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

Grace and Young "The Algebra of Invariants", 1903.
Elliott "An Introduction to The Algebra of Quantics", 1913.
Salmon "Lessons Introductory to The Modern Higher Algebra", 1876.
Faa di Bruno "Théorie Des Formes Binaires", 1876.
Faa di Bruno "Théorie Générale de l'Elimination", 1859.

and there are quite a few more.

Answer by Alberto García-Raboso for Old books still used

2012-12-28T16:37:19Z

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.

Answer by Alberto García-Raboso for Old books still used

2012-12-28T16:50:27Z

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).

Answer by András Bátkai for Old books still used

2012-12-28T17:09:30Z

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.

Answer by Denis Serre for Old books still used

2012-12-28T17:22:01Z

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).

Answer by Alexandre Eremenko for Old books still used

2012-12-28T17:29:26Z

I think the absolute record (excluding Euclid) belongs to

E. T. Whittaker G. H. Watson, A course of modern analysis. According to Jahrbuch database, the first edition is 1915. Moreover, this 1915 edition was an extended version of a 1902 book, by Whittaker alone.

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

Perhaps this deserves a Guinnes book of record 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).

Answer by Aaron Meyerowitz for Old books still used

2012-12-28T17:30:00Z

I've used Euclid's Elements

Halmos (several)

Answer by ayanta for Old books still used

2012-12-28T17:40:09Z

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.

Answer by Mariano Suárez-Alvarez for Old books still used

2012-12-28T17:49:24Z

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.

Answer by Goldstern for Old books still used

2012-12-28T17:54:47Z

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.

Answer by Robert Bryant for Old books still used

2012-12-28T17:58:04Z

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.

Answer by alvarezpaiva for Old books still used

2012-12-28T19:31:02Z

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

Answer by Benjamin Steinberg for Old books still used

2012-12-28T21:11:45Z

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).

Answer by brunoh for Old books still used

2012-12-28T21:43:01Z

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.

Answer by Timothy Chow for Old books still used

2012-12-28T22:03:35Z

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.

Answer by Joseph Van Name for Old books still used

2012-12-28T22:20:23Z

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)

Answer by Adeel for Old books still used

2012-12-28T22:22:44Z

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

Answer by Amritanshu Prasad for Old books still used

2012-12-29T01:30:55Z

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.

Answer by Daniel McLaury for Old books still used

2012-12-29T02:43:42Z

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.

Answer by Rodrigo A. Pérez for Old books still used

2012-12-29T03:14:54Z

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

Answer by Anthony Quas for Old books still used

2012-12-29T05:49:11Z

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.

Answer by Chandan Singh Dalawat for Old books still used

2012-12-29T07:12:53Z

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.

Answer by Kevin R. Vixie for Old books still used

2012-12-29T07:24:29Z

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.

Answer by Martin for Old books still used

2012-12-29T09:53:38Z

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)

Answer by Ian Morris for Old books still used

2012-12-29T10:54:26Z

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.

Answer by Chandrasekhar for Old books still used

2012-12-29T11:10:55Z

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).

Answer by Misha for Old books still used

2012-12-29T14:22:03Z

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.

Answer by jmbr for Old books still used

2012-12-29T15:13:00Z

Mathematical Foundations of Statistical Mechanics by A. I. Khinchin. The original edition in Russian was published in 1943 according to MathSciNet (MR Number=(17677)).

Answer by Matemáticos Chibchas for Old books still used

2012-12-30T02:09:46Z

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.

Answer by David Corwin for Old books still used

2012-12-30T05:35:24Z

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).

Answer by lostinfinity for Old books still used

2012-12-30T09:39:05Z

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

Answer by lostinfinity for Old books still used

2012-12-30T09:42:41Z

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

Answer by András Salamon for Old books still used

2012-12-30T10:31:50Z

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.

Answer by PuyuWang for Old books still used

2012-12-30T12:37:23Z

Mathematical Analysis By Zorich

Answer by none for Old books still used

2012-12-30T15:01:08Z

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

Answer by njguliyev for Old books still used

2012-12-30T17:37:56Z

"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.

Answer by Ostap Chervak for Old books still used

2012-12-31T10:37:01Z

R. Engelking (1977). General Topology.

Answer by Alexander Shamov for Old books still used

2012-12-31T10:59:28Z

Hardy "Divergent series" (1949)

Naimark "Normed rings" (1968)

Maurin "Methods of Hilbert spaces" (1959)

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

Answer by Federico Poloni for Old books still used

2012-12-31T15:39:39Z

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.

Answer by Jonny Evans for Old books still used

2012-12-31T15:52:17Z

Dickson's "History of the Theory of Numbers" is not only old (1919), but it reviews material which is even older. I found it extremely useful when calculating some family Gromov-Witten invariants in a recent paper with Jarek Kedra - while performing the arithmetic manipulations in Section 8, we would have been lost without the wealth of formulae in Dickson. I've no doubt the material appears elsewhere, but Dickson has a comprehensive and carefully historical approach.

Answer by Fredrik Meyer for Old books still used

2013-01-01T23:33:45Z

"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)

Answer by Mustafa Said for Old books still used

2013-01-02T09:00:52Z

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.

Answer by Zsbán Ambrus for Old books still used

2013-01-02T11:19:54Z

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.

Answer by Gene Ward Smith for Old books still used

2013-01-02T19:53:36Z

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.

Answer by J.J. Green for Old books still used

2013-01-05T12:24:33Z

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

Answer by Dong Wang for Old books still used

2013-01-07T08:57:00Z

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.

Answer by Giuseppe for Old books still used

2013-01-07T10:47:14Z

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;

Answer by ex0du5 for Old books still used

2013-01-10T01:40:43Z

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.

Answer by Ben Crowell for Old books still used

2013-01-10T01:59:09Z

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.

Answer by George Melvin for Old books still used

2013-01-10T05:01:03Z

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?

Answer by Yazdegerd III for Old books still used

2013-01-10T21:54:05Z

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

Answer by Scott Guthery for Old books still used

2013-02-06T16:24:02Z

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.

Answer by Reeve for Old books still used

2013-04-14T21:45:14Z

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.

Answer by Dick Palais for Old books still used

2013-04-15T02:06:26Z

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

Answer by Charlie Frohman for Old books still used

2013-04-15T04:00:08Z

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.

Answer by Henr.L for Old books still used

2013-04-20T10:30:17Z

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...

Answer by S. Carnahan for Old books still used

2013-04-20T11:38:34Z

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.

Answer by unknown (google) for Old books still used

2013-04-21T02:11:21Z

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