stage filled with spectacle and the air filled with music you must come at
another time and with a different guide.
Although I hope this book may be useful to others, I wrote it for students
to read either before or after attending the appropriate lectures. For this
reason, I have tried to move as rapidly as possible to the points of difficulty,
show why they are points of difficulty and explain clearly how they are
overcome. If you understand the hardest part of a course then, almost
automatically, you will understand the easiest. The converse is not true.
In order to concentrate on the main matter in hand, some of the simpler
arguments have been relegated to exercises. The student reading this book
before taking the appropriate course may take these results on trust and
concentrate on the central arguments which are given in detail. The stu-
dent reading this book after taking the appropriate course should have no
difficulty with these minor matters and can also concentrate on the central
arguments. I think that doing at least some of the exercises will help stu-
dents to 'internalise' the material, but I hope that even students who skip
most of the exercises can profit from the rest of the book.
I have included further exercises in Appendix K. Some are standard,
some form commentaries on the main text and others have been taken or
adapted from the Cambridge mathematics exams. None are 'makeweights',
they are all intended to have some point of interest. Sketches of some solu-
tions are available from the home pages given on page xiii. I have tried to
keep to standard notations, but a couple of notational points are mentioned
in the index under the heading 'notation'.
I have not tried to strip the subject down to its bare bones. A skeleton
is meaningless unless one has some idea of the being it supports, and that
being in turn gains much of its significance from its interaction with other
beings, both of its own species and of other species. For this reason, I have
included several sections marked by a ^?. These contain material which is
not necessary to the main argument but which sheds light on it. Ideally,
the student should read them but not study them with anything like the
same attention which she devotes to the unmarked sections. There are
two sections marked W which contain some, very simple, philosophical
discussion. It is entirely intentional that removing the appendices and the
sections marked with a ? more than halves the length of the book.
It is an honour to publish with the AMS; I am grateful to the editorial
staff for making it a pleasure. I thank Dr Gunther Leobacher for com-
puter generating the figures for this book. I owe particular thanks to Jorge
Aarao, Brian Blank, Johan Grundberg, Jonathan Partington, Ralph Sizer,
Thomas Ward and an anonymous referee, but I am deeply grateful to the
many other people who pointed out errors in and suggested improvements