Complete and Unabridged.
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.
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 dififerentials. 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.