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A Course of Pure Mathematics: Centenary Edition

G. H. Hardy
Cambridge University Press
Publication Date: 
Number of Pages: 
Cambridge Mathematical Library
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The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Luiz Henrique de Figueiredo
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This is a special reissue of the 10th edition of Hardy's classic, first published in 1908. The main addition is an interesting foreword by T. W. Körner, which describes the huge influence of the book on the teaching and development of mathematics, especially in Britain. The book was the first textbook in English on analysis; the Encyclopaedia Britannica says that it "transformed university teaching."

The book contains a presentation of analysis as the foundation for calculus in the precise but lively style that is Hardy's hallmark. However, the book does show its age. The notation and terminology are slightly different from the ones we use today: sets are called aggregates or classes, sequences are called functions of a positive integral variable, closed intervals are denoted by (a,b), epsilon-delta arguments are delta-epsilon arguments (given delta, find epsilon!), sup arguments are replaced by equivalent Dedekind cut arguments (the real numbers are constructed from the rationals using "sections"). Although these glitches are not serious, they will probably be confusing for beginners.

So, who is likely to profit from reading this book? Certainly students and teachers interested in classics written by one of the best writers of his era (those who have read Hardy and Wright's book on number theory, a new edition of which is forthcoming, will recognize the force of the prose, even if it may seem somewhat heavy nowadays). And certainly anyone looking for challenging, interesting exercises not usually found in modern calculus books. (In the Cambridge tradition, exercises are called examples!) Nevertheless, students looking for a careful presentation of rigorous calculus will probably profit much more by reading and working through Spivak's Calculus, a modern classic.

Hardy said that "young men should prove theorems, old men should write books." We have been fortunate that he wrote such good books (and he was not at all old when he wrote them). It is thus fitting to celebrate Hardy's writings with this centenary edition.

Further Reading

For some more comments about Hardy's book, see Allen Stenger's review of An Introduction to Mathematical Analysis by Robert A. Rankin.

It may also be instructive to read older reviews of earlier editions:

Arthur Berry, Review of "A Course of Pure Mathematics " by G. H. Hardy. The Mathematical Gazette, Vol. 5, No. 87 (Jul., 1910 ), pp. 303-305.

E. H. Neville, Review of "A Course of Pure Mathematics " by G. H. Hardy. The Mathematical Gazette, Vol. 13, No. 183 (Jul., 1926), pp. 172-174.

Both of these are available through JSTOR.

Luiz Henrique de Figueiredo is a researcher at IMPA in Rio de Janeiro, Brazil. His main interests are numerical methods in computer graphics, but he remains an algebraist at heart. He is also one of the designers of the Lua language.

Foreword by T. W. Körner; 1. Real variables; 2. Functions of real variables; 3. Complex numbers; 4. Limits of functions of a positive integral variable; 5. Limits of functions of a continuous variable: continuous and discontinuous functions; 6. Derivatives and integrals; 7. Additional theorems in the differential and integral calculus; 8. The convergence of infinite series and infinite integrals; 9. The logarithmic, exponential, and circular functions; 10. The general theory of the logarithmic, exponential, and circular functions; Appendices; Index.