# Reviews of G H Hardy's *Course of Pure Mathematics*

G H Hardy's

*A Course of Pure Mathematics*has run to over ten editions, with substantial changes being made in the early ones. We present below publishers information, extracts from prefaces and extracts from some reviews of most (but not all) of these editions. We treat the new editions as separate works in our numbering below.

#### 1. A Course of Pure Mathematics (1908), by G H Hardy.

**1.1. From the Preface.**

This book has been designed primarily for the use of first year students at the Universities whose abilities reach or approach something like what is usually described as 'scholarship standard'. I hope that it may be useful to other classes of readers, but it is this class whose wants I have considered first. It is in any case a book for mathematicians: I have nowhere made any attempt to meet the needs of students of engineering or indeed any class of students whose interests are not primarily mathematical.

I regard the book as being really elementary. There are plenty of hard examples (mainly at the ends of the chapters): to these I have added, wherever space permitted, an outline of the solution. But I have done my best to avoid the inclusion of anything that involves really difficult ideas. For instance, I make no use of the 'principle of convergence': uniform convergence, double series, infinite products, are never alluded to: and I prove no general theorems whatever concerning the inversion of limit-operations ... . In the last two chapters I have occasion once or twice to integrate a power-series, but I have confined myself to the very simplest cases and given a special discussion in each instance. Anyone who has read this book will be in a position to read with profit Dr Bromwich's

*Infinite Series*, where a full and adequate discussion of all these points will be found.

**1.2. Review by: Arthur Berry.**

*The Mathematical Gazette*

**5**(87) (1910), 303-305.

The title of Mr Hardy's book is a little perplexing. Anyone who was unfamiliar with the author's other work might expect it to be a compendium of pure mathematics, or of such comparatively elementary parts of the subject as could be pressed into a single volume of reasonable size. Thus it might consist of enunciations and proofs in outline of propositions in elementary plane geometry, geometrical and analytical conics, algebra, trigonometry, and differential and integral calculus. Any such expectation would be at once falsified by an inspection of the book. is, in fact, substantially an elementary treatise on what continental writers analysis, a body of doctrine of which the central idea is the limit. It is remarkable that, as far as I know, there is no English book on the subject, though its contents are dealt with, more or less imperfectly, in the later chapters of books on algebra and trigonometry and in treatises on the differential and integral calculus. Hardy's book is much more elementary in scope than the familiar treatises courses on analysis with which Goursat, Jordan, Picard, Humbert and eminent French mathematicians have enriched our literature. Tannery's

*d'Algebre et d'Analyse*is perhaps the book nearest to Mr Hardy's in scope, though it is wholly dissimilar in many respects.

The mere transference to a single book of matter which is usually found in or four has the advantage of bringing out points of connection which are otherwise liable to be obscured, and from the point of view of the student avoids deal of repetition. A beginner, who has read about the exponential theorem Chrystal's

*Algebra*, may be a little distressed when he is expected to read over again in a different form in Hobson's

*Trigonometry*, and again when later stage he meets a more or less satisfactory treatment in a differential calculus. Moreover, the writer who is mainly interested in obtaining trigonometrical algebraic formulae is very liable to slur over some of the delicate logical points are really involved in quite elementary propositions about series and products. Mr Hardy is primarily interested in the fundamental ideas underlying infinitesimal analysis, so that he is naturally led to treat them fully and carefully, the trigonometrical and other applications as illustrations rather than regarding them as ends in themselves. ...

There is naturally much room for difference of opinion as to the proper content of a book of this sort, where there is no tradition as a guide, and probably any teacher would wish certain things added and others omitted. For my own part, though I am in substantial agreement with Mr Hardy's scheme, I could gladly spare the few pages on elementary analytical conics, which would surely be familiar to any one who is capable of reading the book at all, and I should like a rather fuller treatment of the integration of algebraic functions. Mr Hardy goes so far in his treatment of the integration of rational functions and functions involving a square root of a quadratic, that it seems a pity that he did not add a little more and so complete the subject, as far as is needed for ordinary purposes. I have suggested at the beginning of the review that the chief feature of Mr Hardy's book was a rearrangement into a single book of matter usually treated in. three or four. Far more important is the fact that - to put it shortly - when Mr Hardy sets out to prove something, then, unlike the writers of too many widely read text books, he really does prove it. Probably a really acute critic reading the book would discover here and there gaps in the logic, but I feel sure that these would prove to be few and trivial; and if the book is widely read, I for one shall hope to avoid in the future the many weary hours that have usually to be spent in convincing University students that "proofs" which they have laboriously learned at school are little better than nonsense.

#### 2. A Course of Pure Mathematics (Second Edition) (1914), by G H Hardy.

**2.1. From the Preface.**

The principal changes made in this edition are as follows. I have inserted in Chapter I a sketch of Dedekind's theory of real numbers, and a proof of Weierstrass's theorem concerning points of condensation; in Chapter IV an account of 'limits of indetermination' and the 'general principle of convergence'; in Chapter V a proof of the 'Heine-Borel Theorem', Heine's theorem concerning uniform continuity, and the fundamental theorem concerning implicit functions; in Chapter VI some additional matter concerning the integration of algebraical functions; and in Chapter VII a section on differentials. I have also rewritten in a more general form the sections which deal with the definition of the definite integral. In order to find space for these insertions I have deleted a good deal of the analytical geometry and formal trigonometry contained in Chapters II and III of the first edition. These changes have naturally involved a large number of minor alterations.

**2.2. Review by: Philip E B Jourdain.**

*Mind*

**25**(100) (1916), 525-533.

It is tolerably well known that, especially during the last fifty years, there has been in mathematics a growing tendency towards a logical treatment of conceptions and a logical deduction of theorems. This is due to the growth of branches of mathematics, such as the theory of aggregates, which are concerned with very much the same things as modern logic as developed by Leibniz, Boole, De Morgan, Frege, Peano, Russell, and Whitehead. One of the results of this mathematico-logical rapprochement is that number of people who are interested in both logic and mathematics is now fairly large; and this is very striking when we reflect that there was hardly a single logician among the mathematicians of Great Britain between the times of John Wallis and Augustus De Morgan. Nowadays, mathematics would hardly be put to shame by another Bishop Berkeley, and probably the only place in Britain where logic and mathematics are still held to be separate sciences is the city of lost causes.

Another result is that some mathematicians have become much more strict in their expositions. In what follows, I will try to give a fairly exhaustive examination of the changes which the last ten years or so have brought about in successive editions of two well-known mathematical text-books in use at Cambridge. This will suggest some reflections on the question as to whether or not teaching should be guided by what may seem to be the spirit of logic into paths which differ from the historical ones that alone seem to lead to great and lasting work. It is a truism that history is irrelevant to logic, that the truth or falsity of a proposition is independent of the way in which so and so discovered it, but it seems as great a mistake to banish from teaching a discussion of the growth of ideas as to try to build a house without scaffolding on the ground that the scaffolding is not a part of the building.

Mr Hardy's book, as the Preface tells us, was 'designed primarily for the use of first-year students at the Universities whose abilities reach or approach something like what is usually described as "scholarship standard,"' and it is, this class whose wants he has considered first: it is in any case a book for mathematicians. This is still more the case in the second edition: whereas in the first edition a great use is made of geometrical intuition instead of strict logical proof, in the second edition the logical aspect of mathematical analysis, is treated at far greater length. It is undoubtedly the case that the first edition offered fewer difficulties to a student, but it cannot be denied that the changes in the second edition have made the book far more useful to mathematicians. In it hints are given to enable a student to omit any parts in which a logical argument is too subtle for him, and a particular merit is that the fact that there are logical difficulties is quite clearly explained. Mr Hardy seems free from any tendency to insert notes mentioning that there is a difficulty which do not say what it is; this process is not unknown among unpractised teachers, and only succeeds in quite pointlessly puzzling, irritating, or even boring a student. ...

For teaching purposes it would seem advisable still to use the first edition for an elementary course, and to use the second edition as a stepping-stone to the theory of analytic functions. Every page breathes Mr Hardy's supreme competence as teacher and mathematician. That he considers it advisable to use modern logic in teaching mathematics is one of the strongest arguments there is for inverting historical order in a course of mathematics.

**2.3. Review by: C.**

*Science Progress in the Twentieth Century (1906-1916)*

**10**(37) (1915), 158-159.

A comparison between the treatises on the

*Differential and Integral Calculus*of Dr Todhunter and the book before us would convince even a superficial observer of the change which has revolutionised mathematical instruction in this country in the last quarter of a century. The writers of text-books at the beginning of that period believed that the subject, as they expounded it, was as firmly established and as logically deduced from its premises as geometry. Their practice had the authority of great names, and the form of their books reflected their confidence. Proofs which had been devised by Euler and Leibnitz seemed above challenge. If cracks were visible in the armour in which these authors did their work, they were not pointed out; the brilliant results achieved sufficed to give sanction to the theory. Under such circumstances it was unnecessary to examine foundations too closely; the theory expounded fitted certain classes of functions, and by established custom the student's attention was confined to such functions. A good deal of comfort was then, as it is to-day in another branch of philosophy, administered to doubting disciples by the use of the mysterious word 'continuous'. Even in universities it was no uncommon thing to hear a lecturer declare that a function was continuous, and therefore could be differentiated. English mathematicians did not concern themselves with functions which declined to obey their rules, indeed, such functions were regarded as freaks. The work of Weierstrass, communicated in the lecture room to his students, made its way slowly across the Channel, while Cantor and Dedekind found few disciples in our midst. But the work of these great thinkers has as certainly changed the light in which every mathematician regards the Infinitesimal Calculus, as Newton's discoveries affected the outlook of the natural philosopher upon the material universe.

Now Mr Hardy's book is almost the first attempt to bring before junior English mathematical students at the outset of their career the rigorous methods by which alone this subject can be safely established. Mr Hardy's task is one of no common difficulty; he can at the best select. It is impossible for him to present the theory in its entirety even to clever undergraduates. No reader can, however, finish the course of Pure Mathematics without knowing a good deal about limits, infinity, functions, continuity, and without wishing to know more of these subjects. Such a reader will have no excuse for thinking that functions which are continuous are necessarily differentiable, and he may even have learnt that continuous functions can be integrated. It is an excellent thing for mathematics and its progress in this country that the book was written, and it is a sign, full of promise, that a new edition has been demanded so soon. In the second edition the author has made certain excisions and some additions, the result of which, as he warns us, is to make the book a little more difficult. ...

Mr Hardy writes well, and never shirks telling us how little he is doing: he has also a pleasant habit of interspersing matter which is not germane to his subject. Such rubbing-posts are very welcome to the restless student in a journey which is necessarily at times tedious. But it is doubtful whether he was well advised in trotting out Mr B Russell's trite paradox on the value of mathematics: we are sure that he was unwise in attempting to dissect it. The wit of a philosopher is often worthy of admiration, but it should always be admired from a distance. To base upon a paradox "highly important truths" is certainly out of place in a treatise in which above all a plea is made for secure foundations.

#### 3. A Course of Pure Mathematics (Third Edition) (1921), by G H Hardy.

**3.1. From the Preface.**

It is curious to note how the character of the criticisms I have had to meet has changed. I was too meticulous and pedantic for my pupils of fifteen years ago: I am altogether too popular for the Trinity scholar of to-day. I need hardly say that I find such criticisms very gratifying, as the best evidence that the book has to some extent fulfilled the purpose with which it was written.

#### 4. A Course of Pure Mathematics (Fourth Edition) (1925), by G H Hardy.

**4.1. Review by: Eric Harold Neville.**

*The Mathematical Gazette*

**13**(183) (1926), 172-174.

Prof Hardy's book needs no words of mine to recommend it, and I shall not be thought blind to its many well-known merits if I take the occasion of welcoming a new edition, at a price, be it noted, only sixpence above the price in 1908, to discuss a detail. The one important change that the author has made, which is to transfer to the text an account of an analytical basis for the theory of the circular functions, prompts the consideration of the relation of elementary geometry to elementary analysis. Every learner for whom such a course as this is designed is acquainted, in some sense or other, with theorems about triangles and circles, and the writer on analysis may accept this knowledge uncritically, ignore it, allow geometrical results to suggest possibilities for analytical investigation, or indicate how the subject can be established on a strict basis in some form relevant to his needs.

The first course leads to the royal road of pictures and plausibility, which it is the very object of Prof Hardy's teaching to discredit. When we consider what the second course involves, we may doubt both whether it could be adopted successfully at the undergraduate stage and whether it is really the analyst's ideal. The first point need not be laboured: to use familiar trigonometrical language and yet to know that none of the implications of that language is being utilised demands a certain maturity. As to the second point, the domain of the complex variable, however approached, is a two-dimensional continuum, and the mathematician is not going to say that when he calls himself a geometer in studying this continuum he necessarily accepts a standard of reasoning which as an analyst he regards as unworthy of confidence.

If geometry is not repudiated, retrospective analytical proofs still form a delightful exercise, and it is particularly important that a generation whose teachers are so alive to the difficulties of inverting a complex integral that the Jacobian elliptic functions have practically disappeared from the elementary programme, should at least see how for a real variable the circular functions would be demanded for integration even if they were not otherwise known. The conclusions I wish to emphasise are that sooner or later the student must learn that geometry is a rigorous branch of mathematics, and that in developing analysis seriously, just as we say enough of the theory of real numbers to make clear the nature of the problem and of its solution without giving an elaborate formal proof of the distributive law, so at the appropriate time we can explain that deductions from elementary geometry are valid without interrupting our work to make a premature study of foundations. All that I have said is in agreement with Prof Hardy's procedure.

#### 5. A Course of Pure Mathematics (Fifth Edition) (1928), by G H Hardy.

**5.1. Review by: Arthur Berry.**

*The Mathematical Gazette*

**14**(200) (1929), 428-429.

I am very glad to welcome a fifth edition of Prof Hardy's book, not so much for the sake of the changes and corrections made, as for the evidence afforded by its publication that so admirable a book is still being sold (and presumably read) on a considerable scale.

The most important change is the addition of an appendix on Landau's symbols O, o, ~, which are now in common use in higher analysis, and may profitably be introduced at a fairly early stage to simplify and abbreviate the discussion of problems of limits and approximations when what we want to prove is (loosely stated) that certain "remainders" are negligible as compared with what we retain.

It is an interesting illustration of the difficulty of ensuring accuracy in detail, however accurate the mind of the writer may be, that a sentence in the preface, calling attention to a correction in the text, itself requires correction, as the reference should be to § 9, not to p. 9. I notice also that of three similar examples (App. I. Exx. 1, 6, 7) which caught my eye in the first edition as (trivially) incorrect, owing to an ambiguity of sign, only Ex. 6 has been corrected.

An interval of just twenty years separates the present from the first edition. Though the general plan of the book has not been substantially changed, it has become perceptibly harder. Dedekind's theory of real numbers, the Heine-Borel theorem, uniform continuity, upper and lower "bounds", the elements of the theory of implicit functions, and other topics have replaced some comparatively commonplace matter belonging to analytical geometry and trigonometry, most of the changes having taken place between the first and second editions. The changes correspond not unfairly, though perhaps with some exaggeration, to the progress in the analytical knowledge and skill of the competent young mathematician which has taken place in the last twenty years, much of which is undoubtedly due to Prof Hardy's book and to the teaching of younger analysts who have been taught or influenced by him. It is refreshing to compare the grasp of fundamental analytical ideas of the better mathematical students whom I now meet with the slipshod notions and methods of my undergraduate contemporaries and of my earlier pupils. On the other hand, as the capacity of the human mind is probably fairly constant and the time available for mathematical study does not appear to have increased appreciably, there has been almost inevitably some loss to correspond with this important advance; and I seem to notice an appreciable diminution in the power of manipulation and in knowledge of straightforward parts of analysis which the average mathematician has to use in his ordinary work. This is, however, in part due to the lack of a satisfactory book, at any rate in English, dealing with parts of the differential and integral calculus which lie outside of the scope of Prof Hardy's book. When a student inquires where he can read about, say, Jacobians, or maxima and minima of two variables, or the evaluation of "definite integrals" by real methods, or double integrals, to mention only a few interesting and important topics, I have to refer him either to an old-fashioned text-book, sadly lacking in rigour but full of information, or to a French book which he may be unable to read, or (perhaps more frequently) may imagine that he cannot read.

#### 6. A Course of Pure Mathematics (Seventh edition) (1938), by G H Hardy.

**6.1. From the Preface.**

The changes in this edition are more important than in any since the second. The book has been reset, and this has given me the opportunity of altering it freely.

I have cancelled what was Appendix II (on the '0, o, ~' notation), and incorporated its contents in the appropriate places in the text. I have rewritten the parts of Chs VI and VII which deal with the elementary properties of differential coefficients. Here I have found de la Vallée-Poussin's

*Cours d'analyse*the best guide, and I am sure that this part of the book is much improved. These important changes have naturally involved many minor emendations.

I have inserted a large number of new examples from the papers for the Mathematical Tripos during the last twenty years, which should be useful to Cambridge students. These were collected for me by Mr E R Love, who has also read all the proofs and corrected many errors.

The general plan of the book is unchanged. I have often felt tempted, re-reading it in detail for the first time for twenty years, to make much more drastic changes both in substance and in style. It was written when analysis was neglected in Cambridge, and with an emphasis and enthusiasm which seem rather ridiculous now. If I were to rewrite it now I should not write (to use Prof Littlewood's simile) like 'a missionary talking to cannibals', but with decent terseness and restraint; and, writing more shortly, I should be able to include a great deal more. The book would then be much more like a

*Traité d'analyse*of the standard pattern.

It is perhaps fortunate that I have no time for such an undertaking, since I should probably end by writing a much better but much less individual book, and one less useful as an introduction to the books on analysis of which, even in England, there is now no lack.

**6.2. Review by: Ralph Palmer Agnew.**

*The American Mathematical Monthly*

**45**(9) (1938), 613.

This is the seventh edition of a book first published in 1908. The subject matter, fundamental in analysis, is largely that of the elementary calculus. The thorough competence with which the author gives discussions and definitions, and the sensible rigor with which he gives proofs, have made this book one of the "best sellers" among mathematics books. It is lamentable that this book is "too hard" for a text in an ordinary first course in calculus. For a superior student in mathematics, it should be more intelligible and inspiring than is the conventional mediocre elementary calculus text based on a definition of limit so vague as to make a researcher in analysis shudder ... The new edition contains a few new sections, some sections which have been rewritten, and numerous additional problems of which many are recent Tripos examination questions. These Tripos problems are sufficiently difficult to sharpen the wits of a superior undergraduate student.

**6.3. Review by: Thomas Arthur Alan Broadbent.**

*The Mathematical Gazette*

**22**(249) (1938), 194-195.

"When Mr Hardy sets out to prove something, then, unlike the writers of too many widely read text books, he really does prove it. ... I shall hope to avoid in the future the many weary hours that have usually to be spent in convincing University students that 'proofs' which have been laboriously learned at school are little better than nonsense" (Arthur Berry,

*Gazette*, V, p. 304, on the first (1908) edition).

"One result, he adds, of all these alterations is to make the book more difficult, and we have the comforting assurance that 'it is no longer necessary to apologise for treating mathematical analysis as a serious subject worthy of study for its own sake' " (W J Greenstreet,

*Gazette*, VIII, p. 60, on the second (1914) edition).

"... the course is an unsurpassed education in exact thinking, which has played and will long continue to play a notable part in setting a high logical standard for the early training of mathematicians in this country" (E H Neville,

*Gazette*, XIII, p. 174, on the fourth (1925) edition).

"It is refreshing to compare the grasp of fundamental analytical ideas of the better mathematical students whom I now meet with the slipshod notions and methods of my undergraduate contemporaries and of my earlier pupils" (Arthur Berry,

*Gazette*, XIV, p. 428, on the fifth (1928) edition).

Comment on these quotations would be superfluous; praise of Professor Hardy's book an impertinence. We all know what a change has taken place in the teaching of analysis in in England since 1908 and the main cause of that change.

In this new edition the 0, o notations have been removed from an appendix and incorporated in the text. Part of the treatment of differentiation has been rewritten; the author says, "Here I have found de la Vallée Poussin's '

*Cours d'analyse*' the best guide." Examples taken from recent Tripos papers have been added.

The book is probably as fascinating to the pupils of today as it was to those of twenty or thirty years ago, but for a different reason; thanks to the influence of the book itself, there will be less sense of adventure, and more of the firm and masterly filling in of outlines already familiar. But whatever the difference in emphasis, the debt of English students of analysis to the author is beyond computation.

#### 7. A Course of Pure Mathematics (Tenth Edition) (1952), by G H Hardy.

**7.1. From the Preface by J E Littlewood.**

The changes in the present edition are as follows:

1. An index has been added. Hardy had begun a revision of an index compiled by Professor S Mitchell; this has been completed, as far as possible on Hardy's lines, by Dr T M Flett.

2. The original proof of the Heine-Borel Theorem (pp. 197-199) has been replaced by two alternative proofs due to Professor A S Besicovitch.

3. The 'Implicit Function Theorem' (p. 203) has now a revised statement and proof due to Professor A S Besicovitch.

4. Example 24, p. 394 has been added to.

**7.2. Review by: V.**

*Current Science*

**28**(11) (1959), 465.

'

*A Course of Pure Mathematics*' by G H Hardy has established itself as a standard text-book for students of the honours level for a long time and is such a popular book that it hardly needs any introductory review. The present volume is a paper bound tenth edition of the book, the appearance of the first edition being in the year 1908. Much progress has been made in the subject since the book was first published and Analysis is now being steadily dominated by ideas of Measure Theory and Topology. The book, however, has a charm and vitality of its own and its usefulness to University students still remains undiminished. The present edition of the book will be warmly welcomed by all.

#### 8. A Course of Pure Mathematics (Tenth Edition, Student edition) (1959), by G H Hardy.

**8.1. Review by: Cyrus Colton MacDuffee.**

*Science, New Series*

**130**(3368) (1959), 157.

This book occupies a special niche in my heart, since it was used in the first course in mathematics that I attended as a graduate student, in 1917. That text was the first edition (1908), and the book under review is the tenth edition (1959). The book has been revised several times, but only in detail, to include newer concepts and proofs. The chapter titles and illustrations are intact, and so is the original flavour. Hardy always felt it necessary to defend the study of pure mathematics against those for whom mathematics is merely a tool; as this attitude was particularly prevalent in England in 1908, this book was written more or less in a spirit of evangelism. This tenth edition, now in its third printing, was brought out after Hardy's death in 1947 by several of his former colleagues at Cambridge University, among them J E Littlewood, and it is greatly to their credit that the enthusiastic style of the original has been preserved.

The book corresponds most closely with texts of advanced calculus in our American hierarchy of course titles, but one can learn much algebra and real and complex variable theory from it. It is 50 years old, and its hair is beginning to grey in places, but it is a fascinating book. With its wealth of problems, it is well suited to the needs of a student who must work by himself, without lectures. This, you must agree, is high praise.

#### 9. A Course of Pure Mathematics: Centenary Edition (2008), by G H Hardy.

**9.1. From the Publisher.**

There are few textbooks of mathematics as well-known as Hardy's Pure Mathematics. Since its publication in 1908, this classic book has inspired successive generations of budding mathematicians at the beginning of their undergraduate courses. In its pages, Hardy combines the enthusiasm of the missionary with the rigour of the purist in his exposition of the fundamental ideas of the differential and integral calculus, of the properties of infinite series and of other topics involving the notion of limit. Celebrating 100 years in print with Cambridge, this edition includes a Foreword by T W Körner, describing the huge influence the book has had on the teaching and development of mathematics worldwide. Hardy's presentation of mathematical analysis is as valid today as when first written: students will find that his economical and energetic style of presentation is one that modern authors rarely come close to.

**9.2. From the Foreword by Thomas William Körner.**

My copy of Hardy's

*Pure Mathematics*is the eighth edition, printed in 1941. It must have been one of the first books that my father bought as an almost penniless refugee student in England, and the pencilled notations show that he read most of it. It was the first real mathematics book that I attempted to read and, though much must have passed over my head, I can still feel the thrill of reading the construction of the real numbers by Dedekind cuts. One hundred years after it was first published, CUP is issuing this Centenary edition, not as an act of piety, but because

*A Course In Pure Mathematics*remains an excellent seller, bought and read by every new generation of mathematicians.

During most of the nineteenth century, mathematics stood supreme among the subjects studied at Cambridge. Exposure to the absolute truths of mathematics was an essential part of an intellectual education. The most able students could measure themselves against their opponents in mathematical examinations (the Tripos) which tested speed, accuracy and problem-solving abilities to the utmost. However, it was a system directed entirely towards the teaching of undergraduates. In Germany and France there were research schools in centres like Berlin, Göttingen and Paris. In England, major mathematicians like Henry Smith and Cayley remained admired but isolated.

An education that produced Maxwell, Kelvin, Rayleigh and Stokes cannot be dismissed out of hand, but any mathematical school which concentrates on teaching and examining runs the risk of becoming old-fashioned. (Think of the concours for the Grandes Écoles in our day.) It is possible that, even in applied mathematics, the Cambridge approach was falling behind Europe. It is certain that, with a few notable but isolated exceptions, pure mathematical research hardly existed in Britain. Hardy took pleasure in repeating the judgement of an unnamed European colleague that the characteristics of English mathematics had been 'occasional flashes of insight, isolated achievements sufficient to show that the ability is really there, but for the most part, amateurism, ignorance, incompetence and triviality.'

When Hardy arrived as a student at Cambridge, reform was very much in the air.

*"I had of course found at school, as every future mathematician does, that I could often do things much better than my teachers; and even at Cambridge I found, though naturally much less frequently, that I could sometimes do things better than the college lecturers. But I was really quite ignorant, even when I took the Tripos, of the subjects on which I have spent my life; and I still thought of mathematics as essentially a 'competitive' subject. My eyes were first opened by Professor Love, who taught me for a few terms and gave me my first serious conception of analysis. But the great debt which I owe to him - he was after all primarily an applied mathematician - was his advice to read Jordan's famous 'Cours d'analyse'; and I shall never forget the astonishment with which I read this remarkable work, the first inspiration for so many mathematicians of my generation, and learnt for the first time as I read it what mathematics really meant."*

[

*A Mathematician's Apology*]

Ever since Newton, mathematicians had struggled with the problem of putting the calculus on as sound a footing as Euclid's geometry. But what were the fundamental axioms on which the calculus was to be founded? How should concepts like a differentiable function be defined? Which theorems were 'obvious' and which 'subtle'? Until these questions were answered, all calculus textbooks would have to mix accurate argument with hand waving. Sometimes the author would be aware of gaps and resort to rhetoric 'Persist and faith will come to you.' More frequently, author and reader would sleepwalk hand in hand through the difficulty - most lecturers will be aware how fatally easy it is to convince an audience of an erroneous proof provided you are convinced of it yourself.

The first edition of Jordan's work (1882-87) belonged to this old tradition but the second edition (1893-96) wove together the work of rigorisers like Weierstrass to produce a complete and satisfactory account of the calculus. The impact on Cambridge of Jordan and of the new 'continental' analysis was immense. Young and Hobson, men who had expected to spend their lifetimes in the comfortable routine of undergraduate teaching, suddenly threw themselves into research and, still more remarkably, became great mathematicians.

This impact can be read in three books which are still in print today. We read them now in various revised editions, but all were first written by young men determined to challenge a century of tradition. The first was Whittaker's

*A Course of Modern Analysis*(1902) (later editions by Whittaker and Watson), which showed that the special functions which formed the crown jewels of the old analysis were best treated by modern methods. The second was Hobson's

*The Theory of Functions of a real variable*(1907), which set out the new analysis for professional mathematicians. The third is the present text, first published in 1908 and intended for 'first year students ... whose abilities reach or approach ... scholarship standard'.

The idea of such a text may appear equally absurd to those who deal with the mass university system of today and to those whose view of the old university system is moulded by

*Brideshead Revisited*or

*Sinister Street*. However, although most of the students in Cambridge came from well-off backgrounds, some came from poorer backgrounds and

*needed*to distinguish themselves whilst some of their richer companions

*wished*to distinguish themselves. Most of the mathematically able students came from a limited number of schools where they often received an outstanding mathematical education. (Read, for example, Littlewood's account of his mathematical education in A

*Mathematician's Miscellany*.)

Hardy's intended audience was small and it is not surprising that CUP made him pay £15 out of his own pocket for corrections. This audience was, however, an audience fully accustomed through the study of Euclidean geometry both to follow and, even more importantly, to construct long chains of reasoning. It was also trained in fast and accurate manipulations, both in algebra and calculus, within a problem-solving context. The modern writer of a first course in analysis must address an audience with much less experience of proof, substantially lower algebraic fluency and little experience of applying calculus to interesting problems in mechanics and geometry. Spivak's

*Calculus*is outstanding, but Hardy can illustrate his text with much richer exercises. (The reader should note that questions like Example 1.1, which appear to be simple statements are, in fact, invitations to prove those statements.)

Cambridge and Oxford used Hardy's

*Pure Mathematics,*and the two universities dominated the British mathematical scene. (Before World War II, almost every mathematics professor in the British Isles was Cambridge or Oxford trained.) For the next 70 years, Hardy's book defined the first analysis course in Britain. Analysis texts could have borne titles like: 'Hardy made easier', 'An introduction to Hardy' or 'Hardy slimmed down'. Burkill's

*First Course in Analysis*represents an outstanding example in the latter class.

In the last 40 years, Hardy's model has been put under strain from two different directions. The expansion of the university system has brought more students into mathematics, but the new students are less well prepared and less willing to study mathematics for its own sake. It is clear that 'Hardy diluted' cannot be appropriate for such students. On the other hand, the frontiers of mathematics have continued to advance and an analysis course for future researchers must prepare them to meet such things as manifolds and infinite dimensional spaces. Dieudonné's

*Foundations of Modern Analysis*and Kolmogorov and Fomin's

*Introductory Real Analysis*represent two very different but equally inspiring approaches to the problem.

As new topics enter the syllabus, old ones have to be removed. Today's best students get 'Hardy stripped to the bone' followed by a course on metric and topological spaces. They are taught speedily and efficiently, but some things have been lost. The TGV carries you swiftly across France, but isolates you from the land and its people. We claim to give our students the experience of mathematics, but provide plenty of 'routine exercises' and relegate 'the more difficult proofs' to appendices. Perhaps later generations of mathematicians may judge our teaching as harshly as Hardy and his generation judged the teaching of their Cambridge predecessors.

**9.3. Review by: Luiz Henrique de Figueiredo.**

*Mathematical Association of America*.

https://www.maa.org/press/maa-reviews/a-course-of-pure-mathematics-centenary-edition

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 recognise 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.

#### 10. A Course of Pure Mathematics (Wildside Press reprint of the Third Edition) (2009), by G H Hardy.

**10.1. From the Publisher.**

G H Hardy's text is a good single volume refresher course for work in analysis and more advanced algebra, including number theory. Not quite as modern as Birkhoff and Mac Lane's text, or Manes' work, this volume forms the underpinnings of both works. If you have a good understanding of the preliminary work required in algebra and geometry, Hardy can be read directly and with pleasure. If you have a desire to understand the basis of what is presented in most first-year calculus texts, then Hardy's text is for you.

#### 11. A Course of Pure Mathematics (Dover reprint of the Third Edition) (2018), by G H Hardy.

**11.1. Review by: Fernando Q Gouvea.**

*Mathematical Association of America.*

https://www.maa.org/press/maa-reviews/a-course-of-pure-mathematics

The Dover reprint is based on the third edition of 1921.

In the preface to this edition, Hardy noted the change "in the character of the criticisms" of his book:

I was too meticulous and pedantic for the pupils of fifteen years ago: I am altogether too popular for the Trinity scholar of to-day. I need hardly say that I find such criticisms very gratifying, as the best evidence that the book has to some extent fulfilled the purpose with which it was written.Hardy was a delightful writer.

Last Updated July 2020