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\RefKeys
 alley apostol cwr flanders gillman knuth munkres SandW TandF
 \endRefKeys

\rightline{\ssi Word-smithing is a much greater percentage of what I am
 supposed to be doing in life}
\rightline{\ssi than I would ever have thought.}
\rightline{\ssq DONALD KNUTH \cite{\knuth, \p.54}}
\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip

\centerline{\titlefont  Writing a Math Phase Two Paper}
\bigskip
\centerline{\smc Steven L. Kleiman}
\centerline{with the collaboration of \smc Glenn P. Tesler}
\centerline{\copyright\today}

\bigskip\medskip

{\narrower \noindent \eightpoint
 {\bf Abstract.} We discuss the kind of writing that's appropriate in a
paper submitted to the math department to complete Phase Two of MIT's
writing requirement.  First, we review the general purpose of the
requirement and the specific way of completing it for the math
department.  Then we consider the writing itself: the organization into
sections, the use of language, and the presentation of mathematics.
Finally, we give a short example of mathematical writing.
\par}

\bigskip\medskip

\sctn Introduction.
 MIT established the writing requirement to ensure that its graduates
can write both a good general essay and a good technical report.
Correspondingly, the requirement has two phases, which engage students
at the beginning and toward the end of their undergraduate careers.
The requirement is governed by an institute committee, the Committee on
the Writing Requirement (CWR).  The requirement is administered by the
Undergraduate Education Office, which works in cooperation with the
individual departments on Phase Two.  The general information given
here about the requirement is taken from the MIT {\it Bulletin\/} and
the CWR's brochure \cite{\cwr}, which are the official sources.

To complete Phase One, students must achieve a suitable score on the
College Board Achievement Test or Advanced Placement Examination, pass
the Freshman Essay Evaluation, pass an appropriate writing subject in
Course 21 and be certified by the instructor, or write a satisfactory
five page paper for any MIT subject, Wellesley exchange subject, or
UROP activity.  In level, format, and style, a Phase One paper should
be like a magazine article for an informed, but general, readership.
Papers are judged on their logical structure, language and tone,
technical accuracy, and mechanics (grammar, spelling, and punctuation)
by the instructor of the subject and by evaluators for the
Undergraduate Education Office.  A paper judged not acceptable may be
revised and resubmitted twice.  Students must complete Phase One by the
middle of their third semester at the Institute.

To complete Phase Two, students must receive a grade of B or better for
the quality of writing in a cooperative subject approved by the
student's major department, receive a grade of B or better in one of
several advanced classes in technical writing, or write a satisfactory
ten page paper for any MIT subject or UROP activity approved by the
major department.  A student with two majors needs only to complete the
requirement in one department.  In level, format, and style, a Phase
Two paper should be like a formal professional report.  Thus a term
paper or laboratory report may have to be reworked substantially before
it is acceptable as a Phase Two paper.  The paper is judged by its
supervisor primarily for the technical content and by departmental
evaluators primarily for the quality of writing.  Students must
complete Phase Two by the end of registration day of their last
semester; otherwise, they must petition their departments and the CWR.
Petitions for permission to enroll in a writing subject are routinely
approved; petitions to submit a late paper are approved only when there
are exceptional circumstances.

In the Department of Mathematics, there is no cooperative subject, and
most students write a paper to satisfy Phase Two.  Between one-half and
three-quarters of the papers begin as term papers for one course,
18.310 Principles of Applied Mathematics.  Every paper must have some
technical mathematics in it.  When the student and the supervisor feel
the paper is ready, the student picks up a cover sheet in the
Undergraduate Mathematics Office, Room 2--108.  The student fills out
the top, and gives it to the supervisor, who must vouch for the paper's
technical accuracy, and may comment on the quality of the writing.  The
student then submits the paper and the cover sheet to the departmental
coordinator.  The paper must be submitted by the start of IAP if the
student intends to graduate the following June.  After a paper is
submitted, it is read for organization and language by the coordinator,
who determines whether the paper is acceptable as is or needs
improvement.  If the paper needs improvement, the department's Writing
TA contacts the student and sets up an appointment to discuss the areas
requiring further work.  The student submits further revisions to the
TA, and when the paper is perfected, it is resubmitted to the
coordinator.  On occasion, the coordinator works directly with the
student.  Thus, not only is the paper improved, but, more importantly,
the student becomes a better writer.  The process is not examinational,
but tutorial.

The present paper is a primer on mathematical writing, especially the
writing of short papers.  Indeed, this paper itself is intended to be a
model of format and style.  Mathematical writing is primarily a craft,
and anyone can learn it.  The aim is to inform efficiently.  The basic
principles are discussed and illustrated here.  Some of these
principles are simple matters of common sense; others are conventions
that have evolved from experience.  None need be followed slavishly,
but none should be broken thoughtlessly.  When one is broken, the
break may stand out like a sore thumb~--- just as unconventional
spelling does.  However, the writing itself should fade into the
background, leaving the information to be conveyed out front.
Following these principles will not cramp anyone's style; there's
plenty of room for individual variation.  The various principles are
discussed more fully in a number of works, including the following
works on which this primer is based: Alley's down-to-earth book
\cite{\alley}, Flanders' article \cite{\flanders} and Gillman's manual
\cite{\gillman} for authors of articles for MAA journals, the notes
\cite{\knuth} to Knuth's Stanford course on mathematical writing, and
Munkres' brief manual of style \cite{\munkres}.

In Section 2, we discuss the normal way a short mathematical paper is
broken into sections.  We consider the purpose and content of the
individual sections: the abstract, the introduction, the several
sections of the main discussion, the conclusion (which is rare in a
mathematical work), the appendix, and the list of references.  In
Section 3, we deal with ``language,'' that is, the choice of words and
symbols, and the structuring of sentences and paragraphs.  We consider
seven goals of language: precision, clarity, familiarity,
forthrightness, conciseness, fluidity, and imagery.  We discuss the
meaning of these goals and how best to meet them.  Sections 2 and 3 are
based mainly on Alley's book \cite{\alley}.  In Section 4, we deal with
a number of special problems that arise in writing mathematics, such as
the treatment of formulas, the presentation of theorems and proofs, and
the use of symbols. The material is drawn from all five sources cited
above.  In Section 5, we give an illustrative sample of mathematical
writing.  We treat the two fundamental theorems of calculus, for the
most part paraphrasing the treatment in Apostol's book \cite{\apostol,
\pp.202--4}; we state and prove the theorems, and explain their
significance.  Finally, in the appendix, we deal with the use of such
terms as lemma, proposition, and definition, which are common in
treatments of advanced mathematics.

 \sctn Organization.  Most short technical papers are divided up into
about a half dozen sections, which are numbered and titled.  (The pages
too should be numbered for easy reference.)  Most papers have an
abstract, an introduction, a number of sections of discussion, and a
list of references.  On occasion, papers have appendices, which give
special detailed information, or provide necessary general background to
secondary audiences.  In mathematics, few papers have a section of
conclusions and recommendations.  Such a section would discuss the
results from an overall perspective, bring together the loose ends, and
possibly make recommendations for future research.  In mathematical
papers, these issues are almost always incorporated into the
introduction.  Normally, short papers have no formal table of contents.

Sectioning involves more than merely dividing up the material; you have
to decide about what to put where, about what to leave out, and about
what to emphasize.  If you make the wrong decisions, you will lose your
readers.  There is no simple formula for deciding, because the
decisions depend heavily on the subject and the audience.  However, you
must structure your paper in a way that is easy for your readers to
follow, and you must emphasize the key results.

The title is very important.  If it is inexact or unclear, it will not
attract all the intended readers.  A strong title identifies the
general area of the subject and its most distinctive features.  A strong
title contains no distracting secondary details and no formulas.  A
strong title is concise.

The abstract is the most important section.  First it identifies the
subject; it repeats words and phrases from the title to corroborate a
reader's first impression, and it gives details that didn't fit into
the title.  Then it lays out the central issues, and summarizes the
discussion to come. It is drawn completely from the paper.  However, it
includes no general background material.  The abstract is a table of
contents in a paragraph of prose.  It allows readers to decide quickly
about reading on.  While many will decide to stop there, the
potentially interested will continue.  The goal is not to entice all,
but to inform the interested efficiently.  Remember, readers are
busy.  They have to decide quickly whether your paper is worth their
time.  They have to decide whether the subject matter is of interest to
them, and whether the presentation will bog them down.  A well-written
abstract will increase the readership.

The introduction is the place where readers settle into the ``story,''
and often make the final decision about reading the whole paper.  Start
strong; don't waste words or time.  Your readers have just read your
title and abstract, and they've gained a general idea of your subject
and treatment.  However, they are probably still wondering what exactly
your subject is and how you'll present it.  A strong introduction
answers these questions with clarity and precision.  It identifies the
subject precisely and instills interest in it by giving details that
did not fit into the title or abstract, such as how the subject arose
and where it is headed, how it relates to other subjects and why it is
important.  A strong introduction touches on all the significant points,
and no more.  A strong introduction gives enough background material
for understanding the paper as a whole, and no more.  Put background
material pertinent to a particular section in that section, weaving it
unobtrusively into the text.  A strong introduction discusses the
relevant literature, citing a good survey or two.  Finally, a strong
introduction describes the organization of the paper, making explicit
references to the section numbers.  It summarizes the
contents  in more detail than the abstract, and it says
what can be found in each section.  It gives a ``road map,'' which
indicates the route to be followed and the prominent features along the
way.  This road map is placed at the end of the introduction to ease
the transition into the next section. 

The body discusses the various aspects of the subject individually.  In
writing the body, your hardest job is developing a strategy for
parceling out the information.  Every paper requires its own strategy,
which must be worked out by trial and error.  There are, however, a few
guidelines.  First, present the material in small digestible portions.
Second, beware of jumping haphazardly from one detail to another, and of
illogically making some details specific and others generic.  Third, if
possible, follow a sequential path through the subject.  If such a path
simply doesn't exist, then break the subject down into logical units,
and present them in the order most conducive to understanding.  If the
units are independent, then order them according to their importance to
the primary audience.

There are three main reasons for dividing the body into sections:
(1)~the division indicates the strategy of your presentation; (2)~it
allows readers to quickly and easily find the information that interests
them; and (3)~it gives readers restful white space, allowing them to
stop and reflect on what was said.  Make the introduction and the
several sections of the body roughly equal in length.  When you title a
section, strive for precision and clarity; then readers will have an
easier time finding particular information.  Don't simply insert a
title, as is often done in newspaper articles, to recapture interest;
rather, wind down the discussion in the first section in preparation for
a break, and then restart the discussion in the next section, after the
title.  When you refer to Section 3, remember to capitalize the word
``Section''; it is considered a proper name.  Don't use subsections in a
short paper; they make the flow too choppy.

Each main point should be accented via stylistic repetition,
illustration, or language.  Stylistic repetition is the selective
repetition of something important; for example, you should talk about
the important points once in the abstract, a second time in the
introduction, and a third time in the body.  When appropriate, repeat
an important point in a figure or diagram.  Finally, accent an
important point with a linguistic device: italics, boldface, or
quotations marks; a one sentence paragraph; or a short sentence at the
end of a long paragraph.  In particular, set a technical term in
italics or boldface~--- or enclose it in quotation marks if it is only
moderately technical~--- once, at the time it is being defined.  Do not
underline when italics or boldface is available.  Use headings such as
{\bf Table~1-1}, {\bf Figure~1-2}, and {\bf Theorem~5-2}, and refer to
them as Table~1-1, Figure~1-2, and Theorem~5-2; note that the
references are capitalized and set in roman.  When you employ
linguistic devices, be consistent: always use the same device for the
same job.

The list of references contains bibliographical information about each
source cited.  The style of the list is different in technical and
nontechnical writing; so is the style of citation.  In fact, there are
several different styles used in technical writing, but they are
relatively minor variations of each other.  The style used in this
paper is commonly used in contemporary mathematical writing.  The
citation is treated somewhat like a parenthetical remark within a
sentence.  Footnotes are not used; neither are the abbreviations ``loc.
cit.,'' ``op. cit.,'' and ``ibid.''  The reference key, usually a
numeral, is enclosed in square brackets.  Within the brackets and after
the reference key, place~--- as a service~--- specific page numbers,
section numbers, or equation numbers, preceded by a comma
\cite{\gillman, Section 3.7, \p.9}.  If the citation comes at the end
of a sentence, put the period after the citation, not before the
brackets or inside them.

 \sctn Language.  In the subject of writing, the word ``language''
means the choice of words and symbols, and their arrangement in
phrases.  It means the structuring of sentences and paragraphs, and the
use of examples and analogies.  When you write, watch your language.
When it falters, your readers stumble; if they stumble too often,
they'll lose their patience and stop reading.  Write, rewrite, then
rewrite again, improving your language as you go; there is no short
cut!

Alley \cite{\alley, \pp.25--130} identifies seven goals of language:
two primary goals~--- precision and clarity~--- and five secondary
goals~--- familiarity, forthrightness, conciseness, fluidity, and
imagery.  These goals often reinforce one another.  For example,
clarity and forthrightness promote conciseness; precision and
familiarity promote clarity.  We will now consider these goals
individually.

Being precise means using the right word.  However, finding the right
word can be difficult.  Consult a dictionary, not a thesaurus, because
the dictionary explains the differences among words.  For example, the
{\it American Heritage Dictionary\/} is a good choice, because it has
many notes on usage.  Consult a book on usage, such as {\it Webster's
Dictionary of English Usage}.  Always consider a word's connotations
(associated meanings) along with its denotations (explicit meanings);
the wrong connotations can trip up your readers by suggesting
unintended ideas.  For example, the word ``adequate'' means enough for
what is required, but it gives you the feeling that there's not quite
enough; its connotation is the exact opposite of its denotation.
Strong writing does not require using synonyms, contrary to popular
belief.  Indeed, by repeating a word, you often strengthen the bond
between two thoughts.  Moreover, few words are exact synonyms, and
often, using an exact synonym adds nothing to the discussion.

Being precise means giving specific and concrete details.  Without the
details, readers stop and wonder needlessly.  On the other hand,
readers remember by means of the details. Being precise does not mean
giving all the details, but giving the informative details.  Giving the
wrong details or giving the right ones at the wrong time makes the
writing boring and hard to follow.  Being specific does not mean
eradicating general statements.  General statements are important,
particularly in summaries.  However, specific examples, illustrations,
and analogies add meaning to the general statements.  

Being clear means using no wrong words.  An ambiguous phrase or sentence
will disrupt the continuity and diminish the authority of an entire
section.  A common mistake is to use overly complex prose.  Don't string
adjectives before nouns, lest they lose their strength and precision;
instead, use prepositional phrases and dependent clauses, or use two
sentences.  Keep your sentences simple and to the point.  It may help to
keep most of them short, but you need some longer ones to keep your
writing from sounding choppy and to provide variety and emphasis.

A pronoun normally refers to the previous noun.  Unfortunately, it is
common to abuse pronouns, particularly ``it,'' ``this,'' and ``which.''
Make sure the reference is immediately clear, especially when you refer
broadly to a preceding phrase, topic, or idea.  It is also common to
use a plural pronoun such as ``their'' to refer back to a singular, but
indefinite, antecedent such as a ``reader.''  This usage is still
considered unacceptable in formal writing; reformulate your sentence if
necessary.  The pronouns ``that'' and ``which'' are not always
interchangeable.  Either may be used to introduce a restrictive clause,
but use ``that'' ordinarily.  Only ``which'' may be used to introduce a
descriptive clause, and the clause must be set off with commas.  Strunk
and White in their classic guide to style
\cite{\SandW, \p.47} recommend ``which-hunting.''

Punctuation is used to eliminate ambiguities in language, and to smooth
the flow of the text.  A simple misuse of punctuation can weaken your
authority.  Learn how to punctuate properly, and use a handbook like
{\it The Chicago Manual of Style.}  In optional cases, use the punctuation
if it promotes clarity at all, but strive for consistency through out
the paper.  Here are a few rules.

Use periods only to end sentences.  (A complete sentence within
parentheses should begin with a capital letter and end with a
punctuation mark, unless the sentence is part of another and would end
with a period.)  Avoid abbreviations that require periods; for example,
use ``that is'' instead of ``i.e.''  Always use commas to separate three
or more items in a list and to set off contrasted elements (they often
begin with ``but'' or ``not'').  Most of the time, use a comma after an
introductory phrase.  Use colons to introduce lists, explanations, and
displays. Do not use colons in continuing statements: if a statement is
stopped at the colon, then the words should form a complete sentence.
Use a semicolon to join two sentences to indicate that they are closely
linked in content; however, if you insert a conjunction (not an adverb),
then use a comma.  Use a dash as a comma of extra strength~--- but use
it sparingly.  Place closing quotation marks ('') after commas and
periods; it is a matter of appearance, not logic.  Enclose parenthetical
material in parentheses; footnotes and endnotes are generally
discouraged in technical reports.

To inform, you must use language familiar to your readers.  Define
unfamiliar words, and familiar words used in unfamiliar ways.  If the
definition is short, then include it in the same sentence, preceding it
by ``or'' or setting it off by commas or parentheses.  If the
definition is complex or technical, then expand it in a sentence or
two.  Do not use words like ``capability,'' ``utilize,'' and
``implement''; they offer no precision, clarity, or continuity and
smack of pseudo-intellectualism.  Beware of words like ``interface'';
they are precise in some contexts, yet imprecise and pretentious in
others.

Jargon is vocabulary particular to a certain group, and it consists of
abbreviations and slang terms.  Jargon is not inherently bad.  Indeed,
it is useful in internal memos and reports.  However, jargon alienates
external readers and may even mislead them.  So beware.  Clich\'es are
figurative expressions that have been overused and have taken on
undesirable connotations.  Most are imprecise and unclear.  Avoid them,
or be laughed at.  In addition, avoid numerals because they slow down
the reading.  Write numbers out if they can be expressed in one or two
words and are used as adjectives, unless they are accompanied by units,
a percentage sign, or a monetary sign.  For instance, write, ``The
equation has two roots,'' and ``One root is 2.'' Don't
begin a sentence with a numeral or a symbol; reformulate the sentence
if necessary.

Be forthright: write in an unhesitating, straightforward, and friendly
style, ridding your language of needless and bewildering formality.
Be wary of awkward and inefficient passive constructions.  Often the
passive voice is used simply to avoid the first person.  However, the
pronoun ``we'' is now generally considered acceptable in contexts where
it means the author and reader together, or the author with the reader
looking on.  (Still, ``we'' should not be used as a formal equivalent of
``I,'' and ``I'' should be used rarely, if at all.)  For instance,
don't write, ``By solving the equation, it is found that the roots are
real.''  Instead write, ``Solving the equation, we find the roots are
real,'' or ``Solving the equation yields {\it real\/} roots.''  It is
acceptable, but less desirable, to write, ``Solving the equation, one
finds the roots are real''; the personal pronoun ``one'' is a sign of
formality, and is uncommon in mathematical writing today.  Beware of
dangling participles.  It is wrong to write, ``Solving the equation,
the roots are real,'' because ``the roots'' cannot solve the equation.

Concise writing is vigorous.  Conciseness comes from reducing sentences
to their simplest forms.  For instance, don't write, ``In order to find
the solution of the equation, we can use one of two alternative
methods.''  Instead, write, ``To solve the equation, we can use one of
two methods,'' thus eliminating empty words (``in order to''), reducing
fat phrases (``to find the solution of''), and eliminating needless
repetition (``alternative'').  Concise writing is simple and efficient,
thus beautiful.

%Conciseness comes from eliminating pretentious diction, thus being clear
%and forthright.

The flow of a paper is disturbed by weak transitions between sentences
and paragraphs.  To smooth the flow, start a sentence where the
preceding one left off.  Use connective words and phrases.  Avoid gaps
in the logic, and give ample details.  Don't take needless jumps when
deriving equations.  Use parallel wording when discussing parallel
concepts.  Don't raise questions implicitly, and leave them unanswered.
Pay attention to the tense, voice, and mode of verbs.

Many papers stagnate because they lack variety.  The sentences begin
the same way, run the same length, and are of the same type.  The
paragraphs have the same length and structure.  Don't worry about
varying your sentences and paragraphs at first; wait until you polish
your writing.  Remember though, if you have to choose between fluidity
and clarity, then you must choose clarity.

The very structure of a sentence conveys meaning.  Readers expect the
stress to lie at the beginning and end.  They take a breath at the
beginning, but will run out of breath before the end if the structure
is too complex, for instance, if the subject is too far from the verb.

Most people think and remember images, not abstractions, and images are
clarified by illustrations.  Illustrations also give readers rest
stops, so complex ideas can soak in.  Moreover, illustrations can make
a paper more palatable and less intimidating.  However, the use of
illustrations can be overdone; it must fit the audience and the subject.

Illustrations cannot stand alone; they must be introduced in the text.
Assign them titles, like Figure 5-1 or Table 5-1, for reference.  Assign
them captions that tell, independently of the text, what they are and
how they differ from one another, without being overly specific.  In
addition, clearly label the parts of your illustrations: label the axes
of graphs with words, not symbols; identify any unusual symbols of your
diagrams in the text.  Don't put too much information into one
illustration, because papers without white space tire readers.  For the
same reason, use adequate borders.  Smooth the transitions between your
words and pictures.  First of all, match the information in your text
and illustrations.  Secondly, place the illustrations closely after~---
never before~--- their references in the text.

\sctn Mathematics.  There are some special problems in mathematical writing
because it tends to involve many abstract symbols and formal arguments.
Here are some guidelines to keep in mind.

Formulas are difficult to read because readers have to stop and work
through the meaning of each term.  Don't merely list a sequence of
formulas with no discernible goal, but give a running commentary.
Define all the terms as they are introduced, state any assumptions
about their validity, and give examples to provide a feeling for them.
Similarly, motivate and explain formal statements.  Don't simply call a
statement ``important,'' ``interesting,'' or ``remarkable,'' but explain
why it is so.

Display an important formula by centering it on a line by itself, and
give a reference number in the margin if it's especially important or
if you need to refer to it.  Also display any formula that's more than
a quarter of a line long or that would be broken badly between lines.
Punctuate the display with commas, a period, or whatever as you would
if you had not displayed it.

Be clear about the status of every assertion; indicate whether it is a
conjecture, the previous theorem, or the next corollary.  If it is not a
standard result and you omit its proof, then give a reference,
preferably in the text just before the statement (If you give the
reference in the statement, then do so after the heading like this: {\bf
Theorem 5-1 {\rm\cite{\apostol, \p.202}}}.) Tell whether the omitted
proof is hard or easy to help readers decide whether to try to work it
out for themselves.  If the theorem has a name, use it: say ``by the
First Fundamental Theorem,'' not ``by Theorem~5-1.''  State a theorem
before proving it.  Keep the statement concise; put definitions and
discussion elsewhere.

Prefer a conceptual proof to a computational one; ideas are easier to
communicate, understand, and remember.  Omit the details of purely
routine computations and arguments~--- ones with no unexpected tricks
and no new ideas.  Beware of any proof by contradiction; often there's a
simpler direct argument.  Finally, when the proof has ended, say so
outright; for instance, say, ``The proof is now complete.'' In addition,
surround the proof~--- and the statement as well~--- with some extra
white space.

Here are some more guidelines:

\nxtpt Separate symbols in different formulas with words.
 \bad Consider $S_q$, $q=1,\dots,n$.
 \good   Consider $S_q$ for $q=1,\dots,n$.

 \nxtpt Don't use such symbols as $\exists$, $\forall$, $\wedge$,
$\Rightarrow$, $\approx$, $=$, $>$ in text; replace them by
words.  They may, of course, be used in formulas placed in text.
 \bad Let $S$ be the set of all numbers of absolute value $<1$.
 \good Let $S$ be the set of all numbers of absolute value less than 1.
 \good Let $S$ be the set of all numbers $x$ such that $|x|<1$.

\nxtpt Don't start a sentence with a symbol.
 \bad  $ax^2+bx+c=0$ has real roots if $b^2-4ac\ge0$.
 \good The quadratic equation $ax^2+bx+c=0$ has real roots if $b^2-4ac\ge0$.

\nxtpt Beware of using symbols to convey too much information all at once.
 \verybad If $\Delta=b^2-4ac \ge0$, then the roots are real.
 \bad If $\Delta=b^2-4ac$ is nonnegative, then the roots are real.
 \good Set $\Delta=b^2-4ac$.  If  $\Delta\ge0$, then the roots are real. 

\nxtpt If you  introduce a condition with  ``if,'' then introduce the
conclusion with  ``then.''
 \verybad If $\Delta\ge0$, $ax^2+bx+c=0$ has real roots.
 \bad If $\Delta\ge0$, the roots are real.
  \good  If  $\Delta\ge0$, then the roots are real.

\nxtpt Don't set off by commas any symbol or formula used in text in
apposition to a noun.
 \bad If the discriminant, $\Delta$, is nonnegative, then the roots are real.
 \good If the discriminant $\Delta$ is nonnegative, then the roots are real.

\nxtpt Use consistent notation.  Don't say ``$A_j$ where $1\le j\le
n$'' one place and ``$A_k$ where $1\le k\le n$'' another.

 \nxtpt Keep the notation simple.  For example, don't write ``$x_i$ is
an element of X'' if ``$x$ is an element of X'' will do.

\nxtpt Precede a theorem, algorithm, and the like with a complete sentence.
 \bad We now have the following \hfil\break
 	 {\bf Theorem 4-1.} $H(x)$ is continuous.
 \good We can now prove the following result.  \hfil\break 
	 {\bf Theorem 4-1.} The function $H(x)$ defined by Formula (4-1)
is continuous.

\sctn Example.
 As an example of mathematical writing, we discuss the two fundamental
theorems of calculus.  Our discussion is based on that in Apostol's
book \cite{\apostol, \pp.202--7}.  The First Fundamental Theorem says
that the process of differentiation reverses that of integration.
This statement is remarkable because the two processes appear to be so
different: differentiation gives us the slope of a curve; integration,
the area under the curve.  Here is a precise statement of the theorem.
	\medbreak
{\bf Theorem 5-1} {\rm (First Fundamental Theorem of Calculus).}\enspace
{\it Let $f$ be a function defined and continuous on the closed
interval $[a,\,b]$ and let $c$ be in $[a,\,b]$.  Then for each $x$ in
the open interval $(a,\,b)$, we have
	$${d\over dx}\int_c^x f(t)\,dt = f(x).$$
 } 
%	\medbreak
 {\it Proof.} Take a positive number $h$ such that $x+h\le b$.  Then
  $$\int_c^{x+h}f(t)\,dt - \int_c^xf(t)\,dt = \int_x^{x+h}f(t)\,dt.$$
 By hypothesis, $f$ is continuous.  Hence there is some $z$ in
$[x,\,x+h]$ for which this last integral is equal to $h\,f(z)$ by the
Mean Value Theorem for integrals \cite{\apostol, \p.154} (which is not
hard to derive from the Intermediate Value Theorem).  (The setup is
shown in Figure 5-1; the Mean Value Theorem says that the area under the
graph of $f$  is equal to the area of the rectangle.)  Therefore,
 $${1\over h}\biggl(\int_c^{x+h}f(t)\,dt - \int_c^xf(t)\,dt\biggr)
   =f(z) .$$
 Now, $x\le z \le x+h$.  Hence, as $h$ approaches $0$, the difference
quotient on the left approaches $f(x)$.  A similar argument holds for
negative $h$.  Thus the derivative of the integral exists and is equal
to $f(x)$.  The proof is now complete.
%	\medbreak

\input epsf.tex
\goodbreak\midinsert
	$$\epsfysize=2.5truein\epsffile{figure.ps}$$
\centerline{{\bf Figure 5-1.}\enspace Geometric setup of the proof of
the First Fundamental Theorem.}
\endinsert

The First Fundamental Theorem says that, given a continuous function
$f$, there exists a function $F$ (namely, $F(x)=\int_c^xf(t)\,dt$)
whose derivative is equal to $f$:
	$$F'(x) = f(x).$$
 Such a function $F$ is called an {\it integral\/} (or a {\it
primitive\/} or an {\it antiderivative\/}) of $f$.  Integrals are not
unique: if $F$ is an integral of $f$, then obviously so is $F+C$
for any constant $C$.  On the other hand, there is no further
ambiguity: any two integrals $F$ and $G$ of $f$ differ by a
constant, because their difference $F-G$ has vanishing derivative,
    $$(F-G)'(x) = F'(x)-G'(x) = f(x)-f(x) = 0 \hbox{ for every }x,$$
 and hence $F-G$ is constant as a simple consequence of the Mean Value
Theorem for derivatives (see  \cite{\apostol, Theorem 4.7(c), \p.187}).

When we combine the First Fundamental Theorem with the fact that an
integral is unique up to an additive constant, we obtain the
following theorem.
	\medbreak {\bf Theorem 5-2} {\rm (Second Fundamental Theorem of
Calculus).}\enspace {\it Let $f$ be a function defined and continuous
on the open interval $I$, and let $F$ be an integral of $f$ on
$I$.  Then for each $c$ and $x$ in $I$,
	$$\int_c^x f(t)\,dt = F(x) - F(c). \eqno \hbox{\rm(5-1)}$$}
 	\medbreak
 {\it Proof.} Set $G(x)=\int_c^x f(t)\,dt$.  By the First Fundamental
Theorem, $G$ is an integral of $f$.  Now, any two integrals differ by a
constant.  Hence $G(x)-F(x)=C$ for some constant $C$.  Taking $x=c$
yields $-F(c)=C$ because $G(c)=0$.  Thus $G(x)-F(x)=-F(c)$, and
Equation (5-1) follows.  The proof is now complete.
 \medbreak

The Second Fundamental Theorem is a powerful statement.  It says that
we can compute the value of a definite integral merely by subtracting
two values of any integral of the integrand.  In practice, integrals
are often found by reading a differentiation formula in reverse.  For
example, the integrals in Table 5-1 were found in this way.  The
notation in the table is standard \cite{\TandF, \p.178}: the equation
		       $$\int f(x)\,dx =F(x)+C$$
is read,  ``The integral of  $f(x)\,dx$ is equal to $F(x)$ plus $C$.''

\medbreak
\def\nx#1 #2={\hbox{#1.\enspace}\int#2\,dx=}
\centerline{\bf Table 5-1}
\centerline{\bf A brief table of integrals}
 $$\vbox{\vskip-\smallskipamount\hrule\openup3pt
   \halign{$#$\hfil\cr
   	\noalign{\medskip}
	\nx1 x^a = {\textstyle x^{a+1}\over a+1} +C,
				\hbox{\rm\ if }a\not=-1\cr
	\nx2 x^{-1} = \ln x +C \cr
	\nx3 \sin x = -\cos x +C \cr
	\nx4 \cos x = \sin x + C \cr
	\nx5 e^x = e^x + C \cr}
    \medskip\hrule}$$

 \noindent A longer table of integrals is found on the endpapers of the
calculus textbook \cite{\TandF}. 

\bigbreak \medskip
\centerline{\smc   Appendix. Advanced mathematics}
\medskip

% {\bf  Appendix.\enspace Advanced mathematics.}\enspace
 In many treatments of advanced mathematics, the key results are stated
formally as theorems, propositions, corollaries, and lemmas.  However,
these four terms are often used carelessly, robbing them of some useful
information they have to convey: the nature of the result.

A {\it theorem\/} is a major result, one of the main goals of the work.
Use the term ``theorem'' sparingly.  Call a minor result a {\it
proposition\/} if it is of independent interest.  Call a minor result a
{\it corollary\/} if it follows with relatively little proof from a
theorem, a proposition, or another corollary.  Sometimes a result could
properly be called either a proposition or a corollary.  If so, then
call it a proposition if it relatively important, and call it a
corollary if it is relatively unimportant.  Call a subsidiary statement
a {\it lemma\/} if it is used in the proof of a theorem, a proposition,
or another lemma.  Thus a lemma never has a corollary, although a lemma
may be used, on occasion, in deriving a corollary.  Normally, a lemma
is stated and proved before it is used.

 The terms ``definition'' and ``remark'' are also often abused.  A
formal {\it definition\/} should simply introduce some terminology or
notation; there should be no accompanying discussion of the new terms
or symbols.  A formal {\it remark\/} should be a brief comment made in
passing; the main discussion should be logically independent of the
content of the remark.  Often it is better to weave definitions and
remarks into the general discussion rather than setting them apart
formally.

Typographically, the statements of theorems, propositions, corollaries,
and lemmas are traditionally set in italics, and the headings
themselves are set in boldface or in caps and small caps ({\bf Theorem}
or {\smc Theorem}, and so forth).  The texts of definitions and remarks
are set as ordinary text; so are the texts of proofs, examples, and the
like.  These headings are traditionally set in italics, boldface, or
small caps.  (There is also a tradition of treating definitions
typographically like theorems, but this tradition is less common today
and less desirable.)  All these formal statements and texts are usually
set off from the rest of the discussion by putting some extra white
space before and after them.

Assign sequential reference numbers to these headings, irrespective
of their different natures, and use a hierarchical scheme whose first
component is the section number.  Thus ``Corollary 3-6'' refers to the
sixth prominent statement in Section 3, and indicates that the
statement is a corollary.  If the statement is the second corollary of
the third proposition in the paper, then it may be more logical to name
the statement ``Corollary 2,'' but doing so may make the statement
considerably more difficult to locate.

\bigbreak \medskip
\centerline{\smc References}
\bigskip
\references

alley
 M. Alley, ``The Craft of Scientific Writing,'' Prentice-Hall, 1987

apostol
 T. M. Apostol,  ``Calculus,'' Volume I, Second Edition, Blaisdell,
1967 

cwr
 Committee on the Writing Requirement, ``Guide to the MIT Writing
Requirement,'' Undergraduate Education Office, Room 20C--105, MIT, 1989

flanders
 H. Flanders,  {\it Manual for\/ {\rm Monthly} Authors}, Amer. Math.
Monthly {\bf 78} (1971), 1-10 

gillman
 L. Gillman, ``Writing Mathematics Well,'' Math. Assoc. Amer., 1987

knuth
 D. E. Knuth, T. Larrabee, P. M. Roberts,  ``Mathematical writing,''
MAA Notes Series {\bf 14}, Math Association of America, 1989

munkres
 J. R. Munkres, ``Manual of style for mathematical writing,''
Undergraduate Mathematics Office, Room 2--108, MIT, 1986

SandW
 W. Strunk, Jr., and E. B. White,  ``The Elements of Style,'' Macmillan
Paperbacks Edition, 1962

TandF
 G. B. Thomas and R. L. Finney,  ``Calculus and Analytic Geometry,''
Fifth edition, Addison-Wesley, 1982

\endreferences

\bye

