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 analysis 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 2003 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, 8 arguments of section 1.2 novel will probably find this book unrewarding.
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 stage filled with spectacle and the air filled with music you must come at another time and with a different guide.