\documentclass[12pt,a4paper]{article}
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\begin{document}
\title{How to Write a Part~III Essay}
\author{T. W. K\"{o}rner\\Trinity Hall}
\date{}
\maketitle
\begin{abstract} These \emph{unofficial} notes replace an earlier set
by Marj Batchelor  which were becoming illegible
through repeated photocopying.
Many of the key pieces of advice
are taken almost word for word from her notes though the
elegant picture of a small crustacean has, I am afraid,
vanished. I should be glad to
have suggestions
for additions, corrections and improvements (by
e-mail to \texttt{twk@dpmms.cam.ac.uk} or otherwise)
both from essay writers and essay markers. These
notes were last revised in 2009. 
\end{abstract}
\tableofcontents
\section{Introduction} It is as foolish to 
write an essay
on essay writing as it is to lecture on lecturing or 
give a course on teaching. We do not learn to write
mathematics by following a set of rules. We learn by imitating
other mathematicians
or avoiding their mistakes\footnote{Clearly we should follow the
example of good expositors and avoid the mistakes of bad ones.
Who are your models of good exposition? Or do you think
all mathematical exposition is of the same standard?}.
Eventually, with practice, we acquire our own voice.

On the other hand, there are a few tricks of the trade
which will convert a ghastly expositor into a bad one,
a bad expositor in to a moderate one and a moderate one
into a good one. I shall try to give some of them. You
may disagree with some or all of what I say. That does
not matter. What matters is that you should think about
the problems of mathematical writing.


\subsection{What is the essay?}
Every Part~III student has the option
of replacing a three hour examination paper by an essay.
In the `standard essay' you are asked to read two or
three mathematics papers and then write a connected account
of their contents. The essay is set and marked by an `assessor'
who also gives you advice on how to tackle it.
Because Part~III covers a wide range
of subjects and is taught by a wide range of people this
standard pattern may be modified in all sorts of ways

The essay option is not compulsory, partly to allow you to abandon
your essay if it goes badly wrong and partly because few
of the staff fancy supervising the essay of an unwilling student.
However, there are many reasons why you should do the essay.

(1) It provides practice in reading and writing mathematics.

(2) With luck, it will help you choose a PhD topic. In any case
it will help get you into a `PhD state of mind'.

(3) It will bring you into contact with a member of the staff.
This is a good thing in itself and may prove very useful when 
you need a reference. If you want to stay on to do a PhD
here a good essay will be an excellent recommendation to
possible research supervisors.

(4) It provides an alternative to the `listen to the lecturer,
study your notes, write out the exam' system of the rest of Part~III.

Inspection of the Part~III mark lists shows that \emph{your essay
will normally be among your best papers}\footnote{This may not be true
for the top 10 candidates who may well find it easier
to get full marks on an exam than on an essay. However, if you
are that good you ought to do the essay on other, educational grounds
and, in fact, the top candidates almost invariably do an essay.}.
The reasons for this are obvious. With an essay you can judge your
progress much more easily than with preparation for an exam.
If you work hard at an essay you can expect to do well at it --- even
if you work hard preparing an exam you may still be unlucky on the day.

Sometimes students complain that they work harder
on their essay than on their other papers. There are
various answers to this.

(1) Perhaps they need to work harder on their other papers.

(2) Both students and assessors are explicitly advised by the Faculty Board
that a good essay should not require more work than a 24 hour examinable
course.

(3) (This is I think the real answer.) Work done studying for an examination
is not directly comparable with work done on an essay. It is easier to
work for five exams and one essay than for six exams because the essay
requires different skills and provides a refreshing change.

\subsection{Essay length}
There is no set length for an essay but the standard advice
is to aim for 5000 to 7000 words\footnote{Unless your assessor
suggests otherwise. This is one of the things you should
discuss at your first meeting. It is unlikely that your
assessor agrees with everything in these notes and
since your task is to please him or her, your assessor's
views are much more important than mine.}. This does not mean that anybody
will be worried if you aim for 7000 words and end up with
8000 but if you aim for  7000 words and end up with
10000  your assessor will certainly feel that your essay is 
too long. Would you be happy if you went to a lecture scheduled
to last 50 minutes and slowly realised as the lecturer droned on
that it was going to last for two hours? After a certain point 
writing more will only involve you in extra
work without any reward. (Indeed, you may actually lose marks.) 

In theory (see Littlewood's essay \emph{Mathematics with a
minimum of raw material} in~\cite{Misc})
two sentences could deserve a College Research Fellowship
(and so, certainly a PhD). In the same way 2000 words, if they
were the right words, could well constitute an $\alpha+$ essay
but, in practice, much less than 5000 words would probably
be considered a bit light-weight. 

When I explain this point students invariably ask
`Do you count formulae?' This shows that they do not
understand what I have just said. A figure like 6000 words
is only a guideline. Nobody is going to actually count
words so nobody cares what you do about formulae and diagrams.
(Most students use too many
formulae and not enough diagrams, but that is another
matter.) I give a method for estimating the number
of words in your essay in Section~\ref{word}.

\subsection{Shakespeare it is not} The essay is not an
exercise in fine English but a preparation for writing
mathematics papers. (The only way to learn to ride a bicycle
is to ride a bicycle.) If English is not your first language
you should rest assured that this will not 
handicap you\footnote{If you do not believe me and you think
that we are more interested in your English than 
your mathematics then you should still do the essay
since you have even less chance of writing good English
in a three hour exam.}. The assessors are interested
in clear thought and clear mathematics however expressed.
For good or ill, English is likely to remain the lingua
franca of mathematics for your lifetime. You may have to write
your mathematics in English in the future. You should seize
the opportunity to practice now.

More generally
\begin{center}\fbox{the less you want to do the essay, 
the more important
it is that you do it.}
\end{center}
The next time you write a substantial piece of mathematics
it will probably be a paper for publication or a PhD thesis
for examination. Take this one available opportunity
for a dress rehearsal.

\subsection{Nor is it research} Sometimes students ask
why the essay could not be replaced by a research
project. This question makes three basic assumptions:

(1) Reading mathematics is easy.

(2) Writing mathematics is easy.

(3) Mathematical research is easy.

\noindent Assumption (3) is clearly false. A PhD is a 
\emph{three year} apprenticeship and the test of a
good PhD is that it should contain one good and original idea.
In my opinion (and in the opinion of most mathematicians
I know) assumptions (1) and (2) are also false.
The essay gives you the opportunity to practice reading
and writing mathematics.

Of course, the assessor will be delighted if you find a
new proof of some result or make a clever application
of the method you describe but such things are not required
for a good essay\footnote{You should also remember that what
seems new to you may not be new to your more knowledgeable
assessor.}.

\section{Technical points}\label{Technical points}
\subsection{How to count words}\label{word} 
The standard way to count typed or handwritten words is 
statistical. Take a page of your essay, count the number of
words on it and multiply by the number of pages. Of course this
may be inaccurate by 10\% or 20\% but nobody cares about the
exact length of your essay, your object is to provide
a rough guide for your own use. In later life you may
need to gauge the length of a paper or book. The publisher is
then not interested in the number of words but in the number
of published pages you will use. In accordance with the
modern principle of giving customers what they want
rather than what they need most word processors have
a word counting facility.

\subsection{Meetings with your assessor} You should normally
have a talk with the assessor for your essay once before
starting work on it, once when you have sketched
out a full plan for it and once when you have completed
your first draft. (Of course, individual assessors
may wish to see you on fewer or more occasions or at different
times.) In addition you should contact your assessor
if a really serious problem crops up (for example if the
main proof in the paper you are studying appears to be
fatally flawed). Many assessors will confine their advice
to cases when your project seems to them to be veering
badly off course. If things are going well they will
give you encouragement but nothing else. Other assessors
discuss essays in detail with everybody. (If you go on
to do a PhD you will find that PhD supervisors exhibit
an even wider range of attitudes.)

\subsection{How to contact your assessor}
You know when your essay has reached one of the points
when it needs to be discussed. Your assessor does not.
It is therefore up to you to make contact with your
assessor. Most assessors can be contacted by e-mail.
Close study of the following two possible messages
may help you draft your own.

\begin{sf}
\underline{Message A}\\
Hi I'm Jean and I'm doing your essay. It would be real cool
if we could meet sometime and discuss it.\\
\noindent Thought for the day. No dog is so short that
its legs do not reach the ground.


\underline{Message B}\\
Dear Dr Moreau\\ 
My name is Jean Brun and I am a student at St Judes.
I have completed the first draft
of my essay on `Central Principles' for which you are the assessor.\\
\ \ Could we meet to discuss it? I am free every afternoon except Thursday
and from 11  to 1 on Tuesday, Thursday and Saturday. Would you like
me to place a Xerox of my first draft in your pigeon hole
in the department?\\

\hfill Thank you
\end{sf}

If the assessor does not use e-mail send them a letter or contact
them personally.


\subsection{Unfair means} The nature of the Part~III essay
is such that cheating is unlikely to occur. However,
here are some guidelines for you to observe.

(1) You {\bf are} employing unfair means if you make substantial
use of a source without making it clear that you are doing so,
(I go into the mechanics of acknowledgement in the last section.)

(2) You {\bf are} 
employing unfair means if you make substantial
use of an unpublished source. Thus you may not use an essay
that you wrote for some other purpose or adapt part of
someone else's essay. (It is possible to imagine exceptions to
this rule but you should not go against it without the
{\bf explicit} permission of your assessor.)

(3) You {\bf are} 
employing unfair means if you ask someone other than the assessor
to help you improve or correct the mathematics of your essay.

(4) You {\bf are not} employing unfair means if you ask someone
to read your essay with a view to removing grammatical errors
and misprints of all kinds. (Use someone who is not an expert in the topic.)

(5) You {\bf are not} employing unfair means if you talk about your essay
with other students or give a seminar based on your essay with
questions at the end.

\noindent If you are in any doubt whatsoever as to the propriety
of anything that you wish to do you should {\bf at once} consult
your assessor or a senior member of staff.

\subsection{Making a timetable} Every substantial
piece of work involves crises. The printer may break down
for five days, the book you want may be out of the library
or you may suddenly realise that the proof you worked out
with so much labour has a gaping hole in the middle.
Your timetable must leave room for such crises.

Your timetable should also leave time for reflection.
Every author's manual contains the same advice, often in
the same words. `When you have completed a draft let it
sleep for a time before embarking on correction or
rewriting.' If you attack the same problem over and over
again without rest you will always attack it in the same way.
If you step away from it for a while and then return
you may discover a new method of approach.

Unfortunately you must hand in your essay by a certain date.
(If you fail to do so, the examiners may simply refuse
to consider it.) The obvious and correct advice is
`start as early as possible and aim to finish as soon
as possible'. One plausible timetable would call for
you to read up during the Christmas holidays, to write your
first draft in the second term, revise in the Easter holidays
and hand in at the beginning of the third term leaving you
free to concentrate on examination revision in the five or six week
run up to your exams.
 

\section{How to read a paper} Why are bookshops filled with
`self-help' books? Evidently because  people buy
them. Why do people buy `self-help' books? 
One reason must be to reassure themselves that they
are not alone and that whatever their problem,
from excessive shyness through to poor punctuation,
it is one shared with many other people.  I can give
very little help with your problems in reading
mathematics, but I can assure you that you share them
with most other mathematicians.
\subsection{Why are mathematics papers hard to read?}
You are hardly likely to be doing Part~III if you have
not been able to understand most of what your lecturers
told you in the past. It therefore comes as a shock
when you try to read mathematics papers and find them
hard to follow.  Of course,
the main reason why mathematics papers are hard to read
is that mathematics is hard but there are good reasons
why they are harder to understand than lectures.

(1) A mathematics paper stakes out a claim. Thus the
writer will prove the strongest version of the theorem
that he or she can. Frequently a slightly weaker
theorem is much easier to prove and contains the basic
idea.

(2) A mathematics paper emphasises novelty. A theorem
or its proof is best understood in context but a paper
will concentrate on what is new and not waste time
discussing the known context.

(3) A mathematics paper is a private enterprise
which contributes to a communal good. In a lecture course,
the lecturer is like a conductor
blending the contribution of many
individuals into a harmonious whole. 
A single paper represents the contribution of the double bass
or the triangle.

(4) A mathematics paper contains new mathematics. The
writer may not fully understand what is important
and what is unimportant in what he or she has done.
Similarly he or she may not understand what is truly
difficult and what merely seems so.  
\subsection{How do we read a proof?} It seems to me
that most mathematicians approach the job of reading
a long proof as follows.

(1) What does the theorem mean? One way of trying to
find out is to try it out on a few simple examples.

(2) Let us try and show it is false. By trying to
construct counter-examples we get some idea of how the
theorem works.

(3) Once we have convinced ourselves that there are no
counter-examples let us try and prove it for ourselves.

(4) If we cannot prove it then there must be a counter-example.
Return to (2).

(5) After repeated cycles through (2) and (3) we admit
we cannot do it ourselves and look at what the author says.

(6) The initial steps of the author's proof should, if we have
tried (3) sufficiently often, be familiar but at some point
something new will turn up.

(7) Perhaps this new point is the key? Return to (3).

(8) After repeated cycles through (2) to (7) we reach the 
end of the proof.

\subsection{Understanding} After a recent TV programme
on Andrew Wiles, one of the Arts Fellows at my college
told me that he now felt that he almost understood the
proof of Fermat's last theorem. If you feel that you have understood a proof
except for a few calculations and some technicalities
you are closer in spirit to him than is altogether proper.
You have not understood a theorem until you can
prove it yourself.

\subsection{Some papers are just long proofs} If
a paper is just a long proof then we should treat
it as such. First we must identify the central theorem.
Having done so we try to prove it as before. Now the
point at which we say `Ah, I did not think of that'
may be a reference to a previous lemma. We now know
(or think we know) the point of the lemma and
we try to prove it. By repeated use of this technique
we can identify the structure of lemmas and definitions
which support the theorem and, ultimately, obtain the
full proof of the main theorem.

Much of mathematics is automatic writing, only by trying to
do as much of the proof yourself as you can will you identify
the key steps which are not automatic.
\subsection{But all papers have context} Mathematicians
ask two questions about theorems --- how and why.
How do you prove it and why should you prove it.
Thus, given a theorem, we may ask:

(1) What simpler results does it generalise?

(2) How can you use it prove other things? Can you give
examples?

(3) Does it generalise? If not, what is the obstacle?
What are the counter-examples which demonstrate the obstacle?

(4) What is the next step in the development of the subject?
What are the open questions?

\noindent Even if the paper does not consider these questions your 
essay should do so.

 
\subsection{A possible moral} If you are a great mathematician
like Kolmogorov you may have so many ideas that you
have no time to spend in presenting them.
In any case, if you are a great mathematician, people
will not grudge the work required to read your papers.
If you are a mathematician with nothing to say
then no matter how clearly you write and how inviting
your presentation, people will not read your papers.

If you are a middle ranking mathematician (particularly
if you are starting out in your profession) the number
of people who actually read your papers will depend 
on how clearly written they are as well as on what
they say. Most middle ranking mathematicians
(and quite a lot of high ranking mathematicians)
have papers which they feel have been unfairly 
neglected. Sometimes their opinion is at fault
(we are not the best judges of our own children),
often it is a question of fashion or bad luck,
but sometimes the paper just fails to communicate
its point.

Erd\H{o}s says `Everyone writes. Nobody reads.' 
Creative mathematicians
are more interested in their own ideas than in other people's.
Reading mathematics is hard. You have to write
so as to catch and hold the attention of an
unwilling audience. You have read other people's papers.
Can you do better?

\section{How to write your essay} Much of what follows
is a more or less lightly modified
version of Marj Batchelor's advice.

\subsection{Your purpose} Your only purpose 
should be to teach your readers
a little bit about your subject. You may think that your sole purpose
is to obtain an alpha for your Part~III exam and you may argue that
the only person who is likely to read this essay will be your assessor.
However, this one of the only exercises (perhaps the
only exercise)
you will have in writing mathematics before you have to write
for a real audience in a PhD thesis or journal article
and you need to make the most of it.
Moreover, if you pretend to write your essay for your colleagues
with the sole intent of making it easy for them to understand
the material then you will certainly please your assessor too.

\subsection{Your topic} Unless you have a clear idea of what
you want to explain you will certainly fail to explain it.
Is the centre of your essay a theorem to be proved or a method to
be illustrated? If it is a method, which examples will show it in
its best light, which examples does it fail to cope with, which
other methods should it be compared with? Every talk, article, lecture
course or book must have a focus. In a short talk the focus is usually
achieved by concentrating on a single result. In an essay you may
wish to present a small collection of \emph{related} results. 
However there is \emph{absolutely no point} in writing down everything
you know or even the most difficult thing you know about your topic.
A statue is a block of marble from which material has been removed.
It is better to write up a simple result well, illustrating it with examples 
which you have invented and calculations you have done, than to do a shoddy
job of copying somebody else's highly technical paper.
 
\subsection{The crustacean style} 
Remember that your purpose
is to explain. Consider also that your reader is short of time, tired,
and possibly stupid as well. Your aim is to make it easy for your
reader to find out what she needs in a minimum of time, The
secret of achieving this is to adopt a crustacean rather than a
vertebrate philosophy of presentation. Vertebrates have their
skeleton hidden within. This philosophy is appropriate not only 
to detective novels (`So far we have assumed that Sir Horace could not
have reached Hangdog Hall in time. But the vicar mentioned (page 33)
that the train to Maplethorpe was running late and 
we know from our own experience
(page 77) that the train slows down as it reaches Blackberry
Cutting so it would be perfectly possible for an experienced mountaineer
(pages 56 and 103) like Sir Horace \dots') 
but for any novel in which you do not
want the reader to be aware of how the plot is planned. In this style
it is acceptable, even laudable to bury key pieces of information
in the middle of interior paragraphs. It is a fault if the reader
is consciously aware of the construction details. 

In the crustacean style, however, the structure is on the outside,
and the organisation is evident at a glance. First paragraphs of sections
describe what follows in the section, first sentences of paragraphs
indicate what will follow within the paragraph and important information
is made to stand out visibly on the page. This style is used in front
page reporting and is to be used by you.

Here are some phrases typical of the crustacean style:

`The key point of the proof is to show that $f$ is a 
well defined isomorphism.'

`We first show that $f$ is well defined.'

`Next we show that $f$ is a morphism.'

`Since $f$ is clearly surjective we need only check that $f$ is injective
which we do by looking at the kernel.'

`This completes the proof that $f$ is a well defined isomorphism.'

`The next three lemmas are entirely routine and show that our results on
continuous functions can be extended to distributions.'

`Lemma 3 can be improved to show that the growth is no faster than
polynomial but we only need some bound depending on $n$ alone.'

`This is the only point in the argument where we use Axiom~A.'
 
Often it is better to say `By Theorem 7.3 which says that all snarks 
are boojums' than `By Theorem 7.3' or `Since all snarks are boojums'.
In the same way `We show that $G$ is Abelian' may be less
helpful to the reader than `We show that the group $G$ of translations
is Abelian' or `We show that the group $G$ defined at the start of Section~2
is Abelian'. Not all readers have perfect memories. 
\subsection{Your outline} Once you have decided what your subject
is you must decide how to present it. Which points should you
make in the introduction? Which definitions will you need
and where should they be put? What notation are you going
to use? (If you use $i$ for the identity map and
go on to talk about complex numbers you, or at least your readers,
may have problems.) Which lemmas will you need to prove the central theorem
and in which order should they come? Should the counter-examples
be presented early to show how strong the main theorem is
or late to show which avenues for generalisation are blocked?

The standard advice with which I have no reason to
disagree says that you should start by writing down
a paragraph in the style of an undergraduate syllabus.

\begin{quotation}
Inversion theorems of classical Fourier Analysis for ${\mathbb R}$
and ${\mathbb T}$. Definition of a Locally Compact Abelian Group.
Statement (without proof) of existence of Haar measure.
Definition of character. Inversion theorem corresponds
to existence of `sufficiently many' characters.
Proof (follow Rudin) of inversion theorem for LCA group
(giving parallels with classical case). Statement structure
theorem and brief sketch proof.
\end{quotation}

Next write out the statements of your main definitions, examples,
lemmas and theorems in the order that you intend to give them.
You have now decided your strategy leaving your tactics (the
proofs and the connecting discussions) for the first draft.
At this point you should consult the assessor to check that
your plans are reasonable.

Do remember that the logical order is not necessarily the
pedagogic order. Your object is to keep the reader interested
and to ensure that he or she can understand the general
sweep of the argument without bothering with all the details.
For example if your topic is Theorem~A and Corollary~B
but the proof of Theorem~A depends on Lemmas~1, 2 and~3
which have long and complicated proofs, it may be best to
order your outline in the following way.

Section 1. Introduction

Section 2. Statements of Lemmas 1, 2 and 3.

Section 3. Statement and proof of Theorem A.

Section 4. Statement and proof of Corollary B.

Section 5. Discussion of Corollary B.

Sections 6, 7, 8, \dots Proofs of Lemmas 1, 2 and 3.

In this way the reader learns as much as is needed to get to
the point in as brief a way as possible. Details of proofs are
sentenced to the end of the paper to be consulted if needed.
This technique is particularly important when giving
talks. \emph{Always put your important points at the beginning
of your talk.} Observation shows that halfway through the
average seminar most of the audience are asleep, catching
up on correspondence, thinking about their own mathematics
or trying to prove your result by a slicker method. In addition
all talks take 50\% longer to give than expected (even
after allowing for this rule) so if you leave important results
until last you will have to engage in an undignified scramble to reach
them.


\subsection{Navigation} Remember that few people read
mathematics papers straight through.
You must make it easy for your
reader to skip bits (with the intention, of course, of coming 
back to them later) or to refer back to some previous
point without rereading the whole essay. Good layout
will help but the necessary signposts should be incorporated
into your prose. If you make sure that new information is never
buried in the middle of a paragraph your readers can hop
about your essay with the confidence that all they need
will be evident.

\subsection{The introduction} For many mathematicians the
introduction is the place where they dump a survey of the literature
and all the definitions and trivial remarks they can find.
If you are sure that everybody will have to read your
paper or you expect that nobody will, this is the easiest
way to construct a paper. If you think that you have
something worthwhile to say but are modest enough to
doubt whether your potential readers know this, you
will use your introduction as an advertisement and a map.

You would like to address your reader as follows:-
\begin{quotation}Since
you have read my title and my abstract I can assume
that you are interested to hear what I have to offer.
My main theorem is the following. You may need the following
two definitions to understand it. It is important
because it does so and so. You may also be interested
in the following corollary and in the lemma I use to
establish the theorem. If you are not interested in these
then I am afraid you will not be interested in the rest
and we part company with no hard feelings. 

If you are
interested, a proper respect for my predecessors means
that I must explain briefly how my work depends on theirs.
Now let me sketch the plan of my paper. Section~2 contains
definitions and preliminary computations. Section~3 is devoted
to the proof of the key lemma in the case of the real line.
Section~4 extends the lemma to general locally compact
Abelian groups. The details are very technical and not
needed if we only wish to prove the main theorem for
standard groups like the line and the circle. Section~5
contains the proof of the main theorem. Section~6 contains
examples showing that our main theorem is, in some sense,
best possible.
\end{quotation}

Academic convention prevents you from being quite as direct
as this but the crustacean style requires you to get
as close as possible. As a rule of thumb the introduction
should occupy about 1/7 of a mathematical paper but
in a short paper it may well occupy rather
more and in a long paper rather less.

If you only wish to write your introduction once
you will have to make it the last thing you write.
It is very rare for mathematical paper to
turn out as planned and the introduction describes
what you have written and not what you wished to write.
My personal preference is to write the introduction first
and then rewrite it repeatedly as the exposition progresses.
This requires more work but makes sure that
you keep the general plan constantly in mind.
\subsection{The conclusion} The introduction had to be
an advertisement to tempt the customer to buy and
a map to help the explorer navigate. At the end of
your essay the customer has already bought 
the goods and the
explorer has completed the journey. There is no
logical reason why we should trouble with
an ending just as there is no logical reason why
we should say thank you at the end of a transaction
or goodbye at the end of a meeting but there are human
reasons why we should.

Your reader has worked hard to understand your
work. Can you not reward him or her with some final
insight or a tantalising open question so that
your essay ends on a high note? Or, if this is not possible,
should you not review the path you have taken together?
Remember the preacher's advice `First I tell them what I'm
going to say. Then I tell them. Then I tell them what
I've said.'

What applies to the essay as a whole applies
to its component sections.
I am tempted to reverse my advice
on the crustacean style.  Last paragraphs of sections
summarise the section, last sentences of paragraphs
summarise the paragraph. This advice should not be taken literally
but I am sure that most mathematics lecture courses
would be improved if 
mathematicians took more trouble with 
the beginning and ending
of each lecture.
 
\subsection{A note on omissions} In a talk it may well be desirable
to omit long calculations and long lists of conditions which
are not required to understand the central idea of the proof.
In a paper you have to present the complete argument however
repulsive the details. Similarly, in your essay you must give 
proofs in full although you may well precede such a proof by
an example or a proof of a simple case to help the reader.
In talking about the background to your paper (for example
if the purpose of your essay is to use a technique to do
some calculations) or in describing developments of your
results you may well choose to summarise and give a reference
to the literature. Think of your topic as as a tree, you
can \emph{assume} the roots and \emph{prune} the branches
but you cannot remove a section of the trunk.

Of course, there are exceptions to this rule. There are branches
of mathematics where full proofs of major results do not exist 
or are just too complicated and some where, to the outsider
at least, it appears that practitioners leap from assertion to assertion
like chamois on a mountainside. However, if you do decide to omit
a proof on the grounds that it is too hard for the reader examine
your conscience and ask whether you do not 
mean that it is too hard for you.
Such omissions are precisely the kind of thing you should discuss
with your assessor.

\subsection{Layout is important} The layout of 
a paper is how it looks on
the page --- margins, indentation, how many lines are skipped
between paragraphs, which headings are underlined and so on.
Here the general principle is 
\begin{center}
\fbox{do what is necessary to make the important bits stand out.}
\end{center}

If you write your paper by hand most problems of layout will
settle themselves. If you use \LaTeX\ then the system will
settle most problems for you. If you are a \TeX\ fan then you do
not need (or, what comes to almost the same thing, you think
you do not need) any advice. If you use a typewriter or a non-\TeX\ based
word processing system then you need all the help you can get.

Here are a couple of general principles.

(1) All definitions and notational conventions should be clearly
set out. Leaving spaces before and after definitions
greatly increases their visibility. The same goes for
lemmas, examples and theorems.

(2) When in doubt, display. Most mathematicians include some very
short formulae in their text. `If $x=2$ then $\int_{3}^{x}f(t)\,dt$
is negative.' However, even moderately short formulae like
$A=\bigcup_{i=1}^{\infty}(A_{i}\cap B)$ may become ugly and difficult
to read. It is much better to use a displayed equation
\[A=\bigcup_{i=1}^{\infty}(A_{i}\cap B).\]
If you are writing for a journal the printer may be anxious to
save space but, so far as your essay is concerned,
paper is cheap. (Nowadays, even publishers may approve of displayed
equations since they make the job of electronic typesetting
much easier.) Next time you have to attend a seminar with
overheads covered with minute detail  (or even better when
you are preparing an overhead) remember
\begin{center}
SPACE aids  LEGIBILITY.\ \  LEGIBILITY aids  COMPREHENSION.
\end{center}

\subsection{But layout is no substitute for clear explanation}
Few blessings are unalloyed and \TeX\ is no exception.
Mathematicians now spend hours
discussing the difference between 
\[\int_{0}^{1}g(x)dx\ \mbox{and}\ \int_{0}^{1}g(x)\,dx\]
or trying to move a subscript 2.2 ems to the right. This
is a harmless way of wasting time like train spotting
or building a model of Canterbury Cathedral out of matchsticks
but has no positive benefits. One reason is that most
mathematicians have the visual sense of a dead codfish.
(However,  many mathematicians smell rather less and a few
have better conversation.) The more fundamental reason is that,
provided a certain standard of legibility is attained, content
is more important than presentation.

Let me reiterate my advice on a crustacean style. Clarity
depends on strategic choices. For the essay as a whole you
must find an order of lemmas, examples and explanations
which carries the reader along the path of understanding.
Within each proof you must find an order of steps
which, if possible, will show your readers why the theorem
is true, or if this is not possible, at least convince
them that it is true. These are the strategic decisions
a mathematician
must make --- not fiddling tactical decisions about 
the spacing in formulae.
I do not say that if you make the right strategic decisions
then the essay will write itself (though this is, I think, almost
the case) but I do say that if you make the wrong strategic
decisions no amount of tactical brilliance whether
in fine writing or fine printing will make your essay
readable.

\subsection{Stylistic points} Although your chief concern
should be strategy, here are some `tactical' points of style.
They are adapted from the \emph{Notes for Contributors} 
of the London
Mathematical Society. Not only are they good sense but they
are the kind of points referees tend to insist on.

(i) A paper should be written in clear, unambiguous and 
grammatical language.
Thus a `sentence' like `Let $x=3$, then $x^{3}=27$.' is unacceptable
because it has two main verbs. A necessary, though not a sufficient
condition of grammatical correctness
is that your paper should sound OK when read aloud.

(ii) Words such as `assume', `suppose', `show', `imply', \dots
should usually be followed by `that'.

(iii) Where `if' is used to introduce a conditional clause,
it should usually be followed by `then' at the appropriate
point, as in `If $x=3$, then $x^{3}=27$'.

(iv) Sentences should begin with words, not mathematical symbols.
Formulae should never be separated merely by punctuation.
Either place at least one word between the formulae or
display as a vertical list. Thus you should replace\\
`$f$ is defined by $f(x)=0$ $[x<0]$, $f(0)=\frac{1}{2}$, $f(x)=1$
$[x>0]$'\\by\\
`We set $f(x)=0$ for $x<0$ and $f(x)=1$ for $x>0$. 
We take $f(0)=\frac{1}{2}$.'\\
or better,\\`We set
\begin{alignat*}{2}
f(x)&=0&&\qquad\text{if $x<0$,}\\
f(0)&=\tfrac{1}{2},&&\\
f(x)&=1&&\qquad\text{if $x>0$.'}
\end{alignat*}
The last suggestion takes up more space but can be read at a glance.

(v) You already know from your lectures and texts that most
abbreviations are for the benefit of the writer and not the reader.
The use of LC saves the writer's wrist but leaves the reader
wondering `locally compact' or perhaps `locally convex'
or perhaps `L\"{o}wner-Carleson'. Of course, if the reader has been
following with attention he or she will remember that 20
pages back (or was it 25?) you gave a list of abbreviations
but you should do the work not the reader. Mathematicians
have caused themselves much misery by ill chosen notation
and most abbreviations are, almost by definition, ill chosen.

(vi) Do not write things which look like nonsense even if
close textual study shows that they are not. Your reader
has enough trouble without adding artificial difficulties.
The London Mathematical Society \emph{Notes} give the following
examples of `unnecessarily disturbing usages':

The number of prime divisors of 30=3.

$\exists 0\leq i\leq n$ with $f(i)>0$.

Let $f(g)$ be the left (right) quotient.

Let $A\ni a$.

Consider the open interval $]a,b[$.

\noindent Not all of these usages may disturb you but
all of them disturb some people.
In an undergraduate lecture course the lecturer has
a captive audience who have to fall in with his
or her notational conventions. Writers of 
mathematical papers have no such power over their audience.
The reader
who does not like your style or is bored with your 
content can just stop reading. Since readers will not labour
to understand your meaning, you must labour to make
your meaning clear to them.
\section{Starting, keeping going and stopping}
\subsection{Word processing} You do not need to use a
word processor. Your assessor will be just as happy with a reasonable
handwritten manuscript.

However, many of you will choose to use a mathematical word
processing system. The standard advice, with which
I firmly concur, is \emph{do not use word processing for your
first draft}. There are several reasons for this advice.

(1) The mechanics of mathematical word processing will distract
you from the much more important task of mathematical essay
writing.

(2) Word processing encourages you to forget the global
shape of your work and concentrate on the local. This may
lead to logical mistakes like the omission of lemmas
and, in long proofs, to circular or incomplete arguments.
It may also encourage large disparities in the amount
of space allotted to the various parts of the essay.
 
(3) More generally, word processing encourages prolixity,
repetition and slack construction.

\noindent With experience you may learn how to avoid these
problems and compose directly at the keyboard, but you do not
yet have experience.

\subsection{The first draft} You may find it easiest
to get started with an ordinary block of of lined
paper (wide spaced for preference), several soft
pencils (easy to erase), a good pencil sharpener
and a very large, very good eraser\footnote{Instead of a wastepaper
basket you should have a very large file in which you place
every sheet of paper that you discard. Sometimes
first thoughts turn out to be better than second.}.
Skip every other line. This is partly a psychological gimmick 
to get you past page~1 in a hurry but its main purpose
is to make changes easier at a later stage.

\subsection{The daily task} Decide how much you
expect to write in a day. Now halve it. Now halve it again.
This is your daily task, say 250~words or 1~page
or 1000~keystrokes. These quantities may seem ridiculously
small but if you write 250~words \emph{each day} for three weeks
you will have a 5000~word essay, if you write one
handwritten page \emph{each day} for 4~months you will have 
something the length of a PhD thesis and if you type
1000~keystrokes \emph{each day} for 2~years you will
have a substantial book.

Now comes the difficult bit. You must do your daily task
\emph{each day}. If it takes you an hour, congratulations,
you have the rest of the day free for other things.
If it takes all afternoon,
you have the evening free. If it takes longer, you must
cancel your candle lit dinner, make excuses to your drinking
club and leave your opera seat unoccupied; completing your
daily task takes priority even if it takes until midnight.

Never do more than your daily task. Experience shows
that if you do three times as much one day you will
take four days off as a reward. By working on your
essay \emph{each day} you keep it constantly in front
of your subconscious so that it can work
while you do not. 

You may be worried by the thought of leaving your 
essay in the middle of some argument. Let me
quote from Littlewood's \emph{The Mathematician's
Art of Work}~\cite{Littlewood}. 
\begin{quotation} Most people need half an hour or
so before being able to concentrate fully. I once came
across some wise advice on this, and have taken it.
The natural impulse towards the end of a day's work
is to finish the immediate job: this is of course right
if stopping would mean doing work all over again.
But try to end in the middle of something; in a job 
of writing out, stop in the middle of a sentence. The
usual recipe for warming-up is to run over the latter
part of the previous day's work; this dodge is a further
improvement.
\end{quotation}

\subsection{What to do when you are stuck} Here are some suggestions.

(1) Make a note of what ought to follow and skip to the next section.

(2) Go back to some section which you have previously skipped
and work on that bit.

(3) Go to a blackboard and explain the point to an invisible but patient
audience.

(4) Is your organisation wrong? 
Perhaps you should have proved some lemma
earlier? Go back and insert it. 
(Rather than rewriting use scissors and paste
to rearrange your manuscript.) Perhaps you can break down one horrible,
complicated section into three simpler ones.

(5) Perhaps you are pursuing a side issue. Is the result or argument
essential for your essay? If it is you have no choice but to press on.
If not, leave it out. What the writer does not enjoy writing the reader
will not enjoy reading.

(6) If nothing seems to work you may have to face an
unpleasant truth. It turns out to be very difficult
indeed to give a clear account of woolly or incorrect
mathematics. It may be that you have not reached the
required stage of understanding which would permit you
to write up. The only remedy is to stop writing and start
thinking. 

As you might expect this happens quite frequently when
mathematicians write up their research. In these cases
the correct remedy may be to let the paper rest for a week
and then reconsider matters. You are writing to a deadline
and though this remedy remains the best it will only
be possible if you have left plenty of slack in your 
timetable.  

\subsection{Transferring the first draft to your word processor}
If you are going to use a word processor you will
need to transfer your handwritten
document to the computer. How long this will take you
depends on your temperament, your typing skills and
your familiarity with mathematical word processing.
However, I recommend that you plan on the assumption
that you can transfer your work only twice as fast as you can
write it. (Thus if it takes you 20 days to write your
first draft you should allow 10 days for the transfer.)
If you are already familiar with mathematical word processing
this may be a very pessimistic estimate (but you will already
have the experience to make a better one). If you are not,
or if you revise as you type, I do not think it
will be far wrong.

\subsection{Standard advice on word processing} Make a backup
copy at the end of each session so that when (I do not say if)
you delete your file by mistake you only lose 24 hours'
work. Never eat at a place named Mom's, never play cards
with a man named Doc and never, never use an automatic
replacement facility\footnote{Since the effect
of replacing sin by cos is to produce `cosce the effect
of replacing cos by cos'.} 
or spelling `corrector'\footnote{On the other hand a
spell checker is invaluable when used with a good dictionary.}.
Modern programs allow you to view your document on screen
without printing it out --- help save the forests.


\subsection{Revision} Once you have your first draft
on the word processor print it out onto \emph{real} paper.
The go back to your room or the library with  \emph{real}
scissors and paste, plenty of extra paper and four or five
fine pens of various vivid colours. Decide what revisions
are necessary (be ruthless) and go back to the computer
and make them. Let your essay sleep for a week and then
repeat the process. Continue until you are happy with
your work or you run out of time. `A work of art is never finished,
it is merely abandoned.' (But see the note~\ref{stop} on stopping.)

If you do not use a word processor your tools will be
the xerox machine, a pen and black ink, snowpak and
sellotape but the principle remains the same.

\subsection{Revise, rewrite or reject} Much revision consists
of adding or removing punctuation, changing a word
here and there, inserting an extra step in an argument 
and so on. If you find yourself doing more than this
you should rewrite the offending passage from 
scratch\footnote{But keep a copy of the original in case
you change your mind.}. It will not take as long as you
think it will and the new version will flow better
than a hacked about original. Piecemeal revision 
is also more likely to produce inconsistencies.    

Remember that there is an alternative to revision and rewriting.
It is called omission. If something does not fit in with
the flow of the essay or reads badly however often you rewrite
it then, perhaps, it should not form part of the essay. 
The hardest thing for an author to do is to leave out 
a passage over which he or she has sweated blood --- but
often it is the right thing to do. How many lectures have you
heard which were too short and how many too long? How many
books on your shelves are too short and how many too long?

\subsection{Know yourself} Halmos is one of the great
mathematical expositors. Here is some advice from his essay
\emph{How To Write Mathematics}~\cite{Selected}.
\begin{quotation} In the first draft \dots I recommend
that you spill your heart, write quickly, violate
all rules, write with hate or with pride, be snide, be confused,
be `funny' if you must, be unclear, be ungrammatical
--- just keep on writing. When you come to rewrite, however,
and however often that may be necessary, do not edit
but rewrite. It is tempting to use a red pencil to 
indicate insertions, deletions, and permutations,
but in my experience it leads to catastrophic blunders.
Against human impatience, against the all too human
partiality everyone feels towards his own words,
a red pencil is much too feeble a weapon. You are
faced with a first draft that any reader except yourself
would find all but unbearable; you must be merciless about 
changes of all kinds and, especially, about wholesale
omissions. Rewrite means write again --- every word.
\end{quotation}

This is attractive advice until you realise that
Halmos rewrote (that is wrote again --- every word)
each of his books three times and followed this
by a massive revision. (This is an underestimate, parts
of his books were rewritten six or seven times.)

Every book that Halmos wrote was a success and some 
were outstanding. The problem with his advice is that
most mathematicians dislike writing, hate revising 
and consider rewriting a confession of failure.
The word processor encourages endless minor changes
but discourages root and branch revision. Under these circumstances
my advice is as follows.

\begin{quotation} Revision, however thorough you believe
it to be, leaves most things unaltered. You may
haul large chunks of prose from one place to another
but the patterns embedded in those chunks remain unchanged
and the overall structure is altered only in the crudest
way. However hard you wield the red pencil most of your sentences
will retain the the form you originally gave them.

Thus the last draft that you write out in full (and for most
mathematicians this will also be the first) must be as perfect
as you can make it. If you are dissatisfied with anything,
rewrite it before starting revision. Once you start revision
you will find that what you thought was perfect requires
a multitude of large and small corrections but at
least you can concentrate on the job in hand.
\end{quotation}

Be a Halmos if you can, but, if you cannot, 
recognise your limitations and act accordingly.

\subsection{Abandoning} From time to time you will be
filled with disgust and despair at what you have
written. You may be right but it is much more
probable that you are wrong since almost everybody
goes through such a phase from time to time.
(It gets worse when you are doing a PhD.)

Even if you are right, it probably makes sense
to continue. Provided that your essay shows evidence
of hard work and thought it should receive a decent mark.
The alternative is to try and work up a course that
you did not much like (otherwise you would already
have decided to take it for examination) to that
fairly high level at which you can be confident
of a decent mark in your exam. In any case you should
go to your assessor for a second opinion before
deciding to abandon.

\subsection{Stopping}\label{stop} You are writing an essay, not
a rocket guidance system. Your essay does not have to
be perfect. The maximum mark you can get for an
essay is 100 and a decent essay will get 65.
Once you have got the mathematics and the exposition
settled, repeated minor changes will have little effect on
your mark\footnote{And \TeX tual changes will have no
effect whatever.}. On the other hand working on
your weaker exam subjects can boost your total marks
very substantially. Once your final version is complete
leave it and get on with your exam revision.

\section{Sources} By comparison with practitioners in
other disciplines, mathematicians are very lax in acknowledging 
sources. Undergraduate texts customarily contain no 
references and even graduate texts announce
that `No attempt has been made to trace the sources
of the various theorems proved here'. Mathematics
papers normally open with a cluster of references
intended to establish the importance of the problem
treated and the originality of the approach adopted
but otherwise only contain references to results required
but not proved in the paper itself (and these references
will be further limited to results which it cannot be assumed
that the reader will know).

I do not suggest that you should improve on the prevailing
standards. I do, however, point out that these standards
are so low that you should on no account fall below them
now or in the future.

\subsection{Form of acknowledgement}
In addition to the kind of references you find in most
mathematical papers
the nature of a Part~III essay requires that you
state all the major sources that you have used and
indicate how you have used them. One way of doing this is to
include a statement at the end of your introduction
along the following lines.

\begin{quotation}
I have used the book of Kahane and Salem~[21] and
Kahane's \emph{S{\'e}ries de Fourier Absolument 
Convergentes}~[18]
as general sources of information. Sections~1 to~3 are
based on~[14], Section~4 is based on the
treatment of Malliavin's theorem in~[15]
whilst section~5 is a composite from both sources. The
proof of Theorem~6 is taken from~[16] but I have modified
it substantially. The proofs of Lemmas~3 and~4 are my own,
though they may well exist elsewhere in the literature.
\end{quotation}  


\subsection{Secondary references} Suppose that the paper of Y~[17]
you consult
says `X has shown~[18] that all left principles are right
principles'.
The obvious thing to do is to repeat the reference. However
the obvious thing to do may be the wrong thing.
\emph{Never cite a reference that you have not read.}
It may not exist, it may not say what you think it
says and, in any case,
by citing it you claim to have done something you
have not done. The best thing to do is to get hold of the
reference, verify that it does what Y says it does and
then cite it. If you cannot get hold of the reference
then what you do depends on whether you know by other means
that all left principles are right principles. If you know that
the result is true you are simply acknowledging priority
and you can write `X (see citation in~[17])
has shown that all left principles are right
principles' where~[17] is your reference to Y's paper. 
If you do not know whether the result is true you should
consider very carefully whether to include any mention
of it at all. If you do decide to refer to it you must proceed with
the utmost caution. The words `X claims to have proved ....'
may annoy X. Perhaps honesty `In~[17] Y reports a proof
of X that left principles are right principles but I have not 
been able to obtain the original paper~[18]' is the best
policy. 

\subsection{Data} A rose by any other name would smell as sweet,
and, presumably, the \emph{Air on a G String} would sound
as sweet if we did not know that it was by Bach. 
The fact that Stokes's Theorem was first discovered
by Kelvin (and first published as a Cambridge examination
problem) does not affect its truth. We acknowledge the
authorship of poems, music and proofs out of courtesy
(and in the hope that others will extend the same courtesy
to us). 

Data, on the other hand,
is like evidence in court, valueless without a provenance.
Compare the effect of `Someone said that the knave 
of hearts stole the tarts' with
`X said that the knave of hearts stole the tarts',
`X told me that he saw the knave of hearts steal the tarts',
and `I saw the knave of hearts steal the tarts'.
For this reason \emph{data must always be accompanied by references}
such as `The graph of wing span against speed is taken from~[19].'

From the point of view of your essay points where you quote data
may well be points where
you have the opportunity to add 
something. At its simplest you may be able
to gather data from other papers but if your essay is about 
a statistical technique
you could try it out on data from another source or obtained by
Monte Carlo simulation. If you are comparing two numerical methods
then instead of using run times provided by others you could do
your own testing on your own machine.
 
\subsection{When found, make a note of}
Whenever you use a reference for the first
time, record it in some fixed place (a file card in a file box, a notebook
or a file in your computer). Otherwise, you may find that, when you
write up, you cannot remember where you found the result you need.

Since the time scale for the essay is so short and the number
of references that you are likely to use is small the advice
just given
may not be necessary now but if you go on to write a PhD
it is vital. In the 1930's many mathematicians carried notebooks
in which they jotted down problems, ideas, references and so on
as they occurred to them.
I suspect that this custom died out because it is easy to carry
a large notebook in a suit pocket and hard to carry one in
the back pocket of a pair of jeans, (`Is that a Nachschrift in your pocket
or are you just pleased to see me?') Perhaps the custom should be revived
with filofaxes.
\section{Further advice} 
\subsection{Books on writing} There are several good books
about writing. You should start with \emph{The Elements of Style}
by Strunk and White~\cite{Strunk}. 
It is American and aimed at general journalism
but has the virtue of being very short indeed. If you
want something longer and British then \emph{The Complete
Plain Words} by Sir Ernest Gowers\footnote{Great-grandfather
of our own Professor W.~T.~Gowers.}~\cite{Gowers} is worth reading.
\subsection{And what they say} 
Here is some of the advice given by such authorities.

1) Omit needless words.

2) Keep your sentences short.

3) Keep your paragraphs short.

4) Use active rather than passive forms of verbs. (Replace
`It has been shown by Brown that the method works when $n<6$.'
by `Brown showed that the method works when $n<6$.')
The imperative is also useful, particularly in proofs.
(`Take $n=6$.')

5) Prefer the familiar word to the far-fetched.

6) Prefer the concrete word to the abstract.

7) Prefer the single word to the circumlocution.

\noindent The final piece of advice is mine. 

8) Use `we' when you are doing something
with the reader. `We see that $F$ is closed.' `We now check
that $G$ is Abelian.' Use `I' when expressing an opinion
or (in later life) talking about something you have done.
`I think method $A$ is more transparent.' `In an earlier
paper~[13] I showed that all primes (with at most one exception)
are odd.'

\subsection{A digression} It is clear that mathematics papers
meant for an international audience should use simple
English with as little decoration as possible. What
is more interesting is that, for the last 200 years,
this has been the preferred style when the English write
for the English. Good written English is modelled on
good spoken English. Good English style should be
invisible as a pane of glass leaving the reader free
to concentrate on content.

The fact that there is no `high style' in English 
has many causes but it is worth noting that many of
those who successfully advocated the simple style
linked it with an inclusive rather than an exclusive
view of society. 400 years ago Ascham wrote (quoted in~\cite{Gowers})
\begin{quotation} He that will write well in any
tongue, must follow the counsel of Aristotle, to
speak as the common people do, to think as the wise do;
and so should every man understand him, and the judgement
of wise men allow him.
\end{quotation}

\subsection{Mathematical writing} Few mathematicians have had
the courage to give advice on writing mathematics. 
I have already quoted from Halmos's marvellous
essay \emph{How To Write Mathematics}. This appeared
in an excellent collection of essays on
mathematical writing~\cite{Cobly} and is reprinted
along with his essay \emph{How To Talk Mathematics}
in Volume~2 of his selected works~\cite{Selected}.
Yet more good advice will be found in his
`automathography' \emph{I Want to be a Mathematician}~\cite{want}.
All young mathematicians should read Littlewood's
\emph{The Mathematician's Art of Work} reprinted in the
second edition of his \emph{A Mathematician's Miscellany}~\cite{Misc}
and in his collected works~\cite{Littlewood}.

The book \emph{Writing Mathematics Well} by Gillman~\cite{Gillman}
is good on the nuts and bolts but lacks the zest of
my previous recommendations. The book 
\emph{Mathematical Writing}~\cite{MAA Notes}
in the MAA Notes ought to have been a masterpiece since
it was based on lectures by some of the greatest mathematical
expositors of the present time. Unfortunately the lectures were reported
second hand, translated into Californian mellow speak, 
diluted by student comments, homogenised without being edited
and then issued without index, bibliography or helpful table of
contents. (This is only my opinion, you may disagree and, in any case,
there are some useful pieces of advice buried here and there.) 

The recent book of S.~G.~Krantz entitled \emph{A Primer of
Mathematical Writing}~\cite{Krantz primer} is like his previous
\emph{How to Teach Mathematics}~\cite{Krantz how}
full of excellent advice not only on the subject in hand
but also on many related topics. Taken together they
form something like a \emph{Rough Guide to a Life in Mathematics}
dealing with some of the matters which more up-market
guides prefer to ignore. N.~J.~Higham's 
\emph{Handbook of Writing for the
Mathematical Sciences}~\cite{Higham} is another good book
and (unlike both Krantz's book and the present essay)
is written from the standpoint of the applied mathematician.
\subsection{Valediction} The problem with advice, however
well meant, is that it can make a task seem harder than it is.
Most of those who do Part~III take the essay option and,
as far as can be judged,
most of those who take the option do it well,
find it useful and enjoy it.

A good essay will show that you have read some papers,
understood some hard mathematics and can communicate
what you have learnt clearly and enthusiastically.
I wish you good fortune in your enterprise.
\begin{thebibliography}{10}
\bibitem{Gillman} L. Gillman \emph{Writing Mathematics Well.}
AMS (1987).
\bibitem{Gowers} E. Gowers \emph{The Complete Plain Words.}
(3rd Edition, revised by S.~Greenbaum and J.~ Whitcut.) 
Penguin (1987).
\bibitem{want} P.R. Halmos \emph{I Want to be a Mathematician.}
Springer (1985).
\bibitem{Selected} P. R. Halmos
\emph{Selecta.} (Two volumes.) Springer (1983).
\bibitem{Higham} N.~J.~Higham
\emph{Handbook of Writing for the Mathematical Sciences.}
SIAM (Philadelphia), 1993.
\bibitem{MAA Notes} D. E. Knuth, T. Larrabee, P. M. Roberts
\emph{Mathematical Writing.} MAA Notes, {\bf 14}, Washington (1989).
\bibitem{Krantz how} S. G. Krantz \emph{How to Teach Mathematics.}
AMS (Providence, R. I.), 1993.
\bibitem{Krantz primer} S. G. Krantz \emph{A Primer of Mathematical
Writing.} AMS (Providence R. I.),1996.
\bibitem{Misc}
J. E. Littlewood
\emph{A Mathematician's Miscellany.} 
(2nd Edition, Editor B.~Bollob{\'{a}}s.)
CUP (1986)
\bibitem{Littlewood}
J. E. Littlewood \emph{Collected Works.} (Two volumes.) OUP (1982).
\bibitem{Cobly} I. N. Steenrod,  P. R. Halmos and others
\emph{How to Write Mathematics.} AMS (1973).
\bibitem{Strunk} W. Strunk \emph{The Elements of Style.}
(3rd Edition, revised by E.~B.~White.) MacMillan (1979).
\end{thebibliography}

\vspace{2\baselineskip}

\begin{footnotesize}
\noindent
[Printed out \today. These notes are
written in \LaTeX2e and can
 be accessed via my web home page
\begin{center}
{\tt http://www.dpmms.cam.ac.uk/\~{}twk/}.
\end{center}
Also available:

`Dr K\"{o}rner's Helpful Guide For Mathematicians Seeking A
Cambridge Research Fellowship',

`In Praise of Lectures' (how to listen to a mathematics lecture),

`An Unofficial Guide To Part~III',

`A Supervisor's Primer'.]
\end{footnotesize}

\end{document}
