Introduction

In his autobiography [12], Winston Churchill remembered his struggles with

Latin at school. ' ... even as a schoolboy I questioned the aptness of the

Classics for the prime structure of our education. So they told me how Mr

Gladstone read Homer for fun, which I thought served him right.' 'Naturally'

he says 'I am in favour of boys learning English. I would make them all learn

English; and then I would let the clever ones learn Latin as an honour, and

Greek as a treat.'

This book is intended for those students who might find rigorous anal-

ysis a treat. The content of this book is summarised in Appendix J and

corresponds more or less (more rather than less) to a recap at a higher

level of the first course in analysis followed by the second course in analysis

at Cambridge in 2004 together with some material from various methods

courses (and thus corresponds to about 60 to 70 hours of lectures). Like

those courses, it aims to provide a foundation for later courses in functional

analysis, differential geometry and measure theory. Like those courses also,

it assumes complementary courses such as those in mathematical methods

and in elementary probability to show the practical uses of calculus and

strengthen computational and manipulative skills. In theory, it starts more

or less from scratch, but the reader who finds the discussion of Section 1.1

baffling or the e, 5 arguments of Section 1.2 novel will probably find this

book unrewarding. I assume a fair degree of algebraic fluency and, from

Chapter 4 onwards, some exposure to linear algebra.

This book is about mathematics for its own sake. It is a guided tour of a

great but empty Opera House. The guide is enthusiastic but interested only

in sight-lines, acoustics, lighting and stage machinery. If you wish to see the

XI