A Course of Pure Mathematics

G. H. Hardy, M.A., F.R.S.

FELLOW OF NEW COLLEGE
SAVILIAN PROFESSOR OF GEOMETRY IN THE UNIVERSITY OF OXFORD
LATE FELLOW OF TRINITY COLLEGE, CAMBRIDGE

Third Edition

Cambridge at the University Press

1921

Read the preface

Preface to the third edition

No extensive changes have been made in this edition. The most important are in §§ 80–82, which I have rewritten in accordance with suggestions made by Mr S. Pollard.

The earlier editions contained no satisfactory account of the genesis of the circular functions. I have made some attempt to meet this objection in § 158 and Appendix III. Appendix IV is also an addition.

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.

G. H. H.

August 1921

Extract from the preface to the second edition

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.

G. H. H.

October 1914

Extract from the preface to the first edition

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—I never even define $\dfrac{\partial^{2} f}{\partial x\, \partial y}$ and $\dfrac{\partial^{2} f}{\partial y\, \partial x}$. 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.

G. H. Hardy, M.A., F.R.S.

Read the preface

CONTENTS