Top Five Tips on Book Writing

Snoopy writing

I’ve written four books, and am currently writing and editing a fifth (The Princeton Companion to Applied Mathematics). I am also an editor of two SIAM book series and chair the SIAM Book Committee. Based on this experience here are my top five tips about writing an (academic) book. These cover high level issues. In a subsequent post I will give some more specific tips relating to writing and typesetting a book or thesis.

1. Identify Your Audience

Book publishers ask prospective authors to complete a proposal form, one part of which asks who is the audience for the book. This is a crucial question that should be answered before a book is written, as the answer will influence the book in many ways.

As an example, you might be contemplating writing a book about the numerical solution of a certain class of equations and intend to include computer code. Your audience might be

  • readers in mathematics or a related subject who wish to learn about numerical methods for solving the equations and are most concerned with the theory or algorithms,
  • readers whose primary interest is in solving the equations and who wish to have lots of sample code that they can run,
  • readers in the previous class who also need to learn the language in which the examples are written.

The choice of content, and how the book is presented, will depend very much on which audience you are writing for.

2. Revise, Revise, Revise

Just like a paper, a book draft needs to go through multiple revisions, and you must not be afraid to make major changes at any stage. You may receive constructive criticisms from reviewers of your book proposal, but reviewers may not have time to read the complete manuscript carefully and you should not assume that they have found all errors, typos, and areas for improvement.

3. Take Time to Choose Your Publisher

Given the huge effort that goes into writing a book you should take the time to find the right publisher. Discuss your book with several publishers and compare what they can offer in the way of

  • format (hardback, paperback, electronic) and, if more than one format, the timescale in which each is made available,
  • if the publisher has branches in more than one country, how price and publication schedule will differ between the countries,
  • whether you are allowed to make a PDF version of the book freely available on your website, if this interests you,
  • willingness to allow you to choose the book design (page size, font, cover, etc.),
  • use of colour (which increases the cost),
  • royalties (including a possible advance),
  • pricing,
  • the publisher’s policy on translations,
  • copy editing (see the next section),
  • time from delivering a completed manuscript to publication,
  • marketing (will the book be advertised at all, and if so how?), and
  • how long your book is guaranteed to stay in print.

It is perfectly acceptable to submit a proposal to several publishers and see what they are willing to offer. However, it is only fair and proper to make clear to a publisher that you are talking to other publishers and, once you have set the wheels of a publisher’s review process in motion, to wait for an offer before making a decision to go with another publisher.

I am always surprised when I hear of authors who approach only one publisher, or who go with the first publisher to express an interest in the book. As in many contexts, it is best to make an informed choice from among the available options.

4. Ensure Your Book is Copy Edited

If you are an inexperienced writer, or your first language is not English, the benefits of copy editing are obvious. But even an experienced author finds it virtually impossible to think about all the little details that a copy editor will check for, such as correctness and consistency of spelling, notation, punctuation (notably the serial comma), citations, and references. For example, I sometimes mix US and UK spellings and don’t want to have to worry about finding and correcting my occasional lapses. A good copy editor will also suggest minor improvements of the text that might escape even the best writers.

Unfortunately, not all publishers copy edit all books nowadays. Notable exceptions that always do copy edit (and, as I know from experience, work to the highest standards in every respect) are Princeton University Press and SIAM.

If your publisher has a Style Manual it obviously makes sense to follow its guidelines in order to minimize changes at the copy editing stage. Here is a link to the SIAM Style Manual.

5. Think Twice Before Co-Authoring a Book

It might seem an attractive proposition to share authorship of a book: surely having n co-authors reduces the work by a factor 1/n? Unfortunately it often does not work out like that, despite best intentions. In fact, n co-authors can easily take n times as long to write a book as any one of them would. One of the biggest difficulties is timescale: one author may be willing and able to finish a book in a year but another may need twice that period to make their contribution. Indeed it is rare for the co-authors to be matched in the amount of effort they can put into the book; this is clearly problematic if initial expectations are not realized. Other potential problems are potentially differing opinions on content, notation, level, length, and almost anything else associated with a book.

Successful authorship teams often have a track record of co-authoring papers together. Although it is no guarantee that a much larger book project will run smoothly, experience with writing papers together will at least have given a good indication of where disagreements are likely to lie.

The Spotlight Factor

In my Handbook of Writing for the Mathematical Sciences I described the spotlight factor, originally introduced by Tompa in 1989. The spotlight factor is defined for the first author of a paper in which there are n authors listed alphabetically, and it is assumed that the paper is from a community where it is the custom to order authors alphabetically.

The spotlight factor is the probability that if n-1 coauthors are chosen independently at random they will all have surnames later in the alphabet than the first author. This definition is not precise, since it is not clear what is the sample space of all possible names, so it is better to regard the spotlight factor as being defined by the formula given by Tompa, which is implemented in the MATLAB function below.

The smallest spotlight factor I have found is the value 0.0244 for Zielinski, for the paper

Pawel Zielinski and Krystyna Zietak, The Polar Decomposition—Properties, Applications and Algorithms, Applied Mathematics, Ann. Pol. Math. Soc. 38, 23-49, 1995

This beats the best factor of 0.0251 reported by Tompa in a 1990 follow-up paper.

Can you do better?

Here is a MATLAB M-file to compute the spotlight factor, preceded by an example of its usage:

>> spotlight('zielinski',1)
ans =
function s = spotlight(x, k)
%SPOTLIGHT   Tompa's spotlight factor of authorship.
%   SPOTLIGHT(X, K) is the spotlight factor for the author whose 
%   last name is specified in the string X, with K coauthors.
%   Mixed upper and lower case can be used.
%   Smaller spotlight factors correspond to rarer events.

%   Reference:
%   Martin Tompa, Figures of Merit, SIGACT News 20 (1), 62-71, 1989

if ~ischar(x), error('First argument must be a string.'), end
if nargin < 2, error('Must give two arguments.'), end

x = double(upper(x)) - double('A') + 1;
x( find(x < 0 | x > 26) ) = 0;  % Handle punctuation and spaces.

s = 0;

% Ideally use Horner's rule, but the following is clearer.

for i=1:length(x)
    t = x(i);
    s = s + t/27^i;

s = (1 - s)^k;

The Life of James Joseph Sylvester

Following my previous post about the James Joseph Sylvester Bicentenary and my article Sylvester’s Influence on Applied Mathematics I now give a brief, very selective, overview of Sylvester’s life. Some of this material was used in an after-dinner speech that I gave at the Householder Symposium XIX on Numerical Linear Algebra at Spa, Belgium on June 11, 2014.

The Sylvester Medal of the Royal Society.

I’ve drawn on many sources for this post, but the most important is the 2006 biography by Karen Parshall, James Joseph Sylvester. Jewish Mathematician in a Victorian World. That title brings out two key points: that Sylvester was Jewish, which hindered his career, as we will see, and that he lived much of his life in Victorian England, when almost everything that today we take for granted when doing our research did not exist.

Thumbnail Sketch of The Man

Sylvester was born in London in 1814. He was short, mercurial, absent-minded, temperamental, fluent in French, German, Italian, Latin and Greek, and loved poetry but was not very good at it. He was a man of remarkable tenacity, as his career on both sides of the Atlantic shows.

Career Outline

I’ll give a brief outline of Sylvester’s unusual career, with its many ups and downs, then go on to discuss some specific events in his life.

First Spell in UK

  • Sylvester was a student at University College London (UCL) under De Morgan, age 14. He was withdrawn by his family after attempting to stab a fellow pupil.
  • He was a student at Cambridge, but was not able to take the degree because he was Jewish.
  • He held the chair of natural philosophy at University College London (UCL) for three years.

First Sojourn in USA

Sylvester became Professor of Mathematics at the University of Virginia in 1841. He left after four months after an altercation with an unruly student, because he was felt that the faculty did not back him up in a subsequent inquiry.

While in New York he applied for a position at Columbia University. According to R. L. Cooke (quoted in James Joseph Sylvester. Life and Work in Letters)

After leaving Virginia he sought a position at Columbia University, with a recommendation from one of America’s leading scientists, Joseph Henry. In a wonderful irony … the selection committee informed him that his rejection was in no way connected with the fact that he was British, only the fact that he was Jewish.

Rest of Career (age 29–).

  • Sylvester Worked for the next decade as an actuary for the Equity and Law Life Assurance Society in London and trained for the Bar. He founded the Institute of Actuaries. This is when he met Cayley, who became his best friend. For this ten-year period he was doing mathematics in his spare time.
  • He was appointed Chair at the Royal Military Academy, Woolwich and spent 15 years there.
  • He was appointed Chair at the newly founded Johns Hopkins University, Baltimore, at the age of 61. He negotiated a salary of $5000 payable in gold, plus an annual housing allowance of $1000 also payable in gold.
  • His final position was as the Savilian Professor of Geometry at New College, Oxford in 1883, which he took up at the age of 69.

The Neologist

Sylvester introduced many terms that are still in use today, including matrix (1850), canonical form (1851), Hessian (1851), and Jacobian (1852). Another notable example is the term latent root, which Sylvester introduced in 1883, with two charming similes:

“It will be convenient to introduce here a notion (which plays a conspicuous part in my new theory of multiple algebra), namely that of the latent roots of a matrix—latent in a somewhat similar sense as vapour may be said to be latent in water or smoke in a tobacco-leaf.”

The term has fallen out of use in linear algebra and matrix theory, but it can still be found in use through “the latent root criterion” in, for example (to pick two articles found with a Google search) Differentiating with brand personality in economy hotel segment in Journal of Vacation Marketing (2014) and GHOSTS: A travel barrier to tourism recovery in Annals of tourism research (2011).


Sylvester did a great deal of editorial work. He was an editor of the Quarterly Journal of Mathematics for 23 years. He founded the American Journal of Mathematics in 1878 when he was at Johns Hopkins University. This was the first mathematics research journal in the USA, and indeed Sylvester set up the first mathematics research department in the country. As Editor-in-Chief he experienced some of the problems that subsequent journal editors have suffered from.

  • He had to work very hard to secure high quality contributions, e.g., from his friend Cayley and from students and colleagues at Johns Hopkins, in addition to his own papers.
  • He solicited Alfred Kempe’s proof of the four color theorem. After Sylvester had accepted the paper his managing editor, William Story, realized there was a gap in the reasoning, due to overlooked cases, and wrote a note the accompany the paper in which he unsuccessfully tried to patch the proof. This all happened while Sylvester was in England and he was very unhappy with the incident.


Even though Sylvester was an editor himself, he was also the author from hell! He was notorious for what his biographer Parshall calls “an impatience with bibliographic research”—something that led him into disputes with other mathematicians.

MacFarlane states that

Sylvester never wrote a paper without foot-notes, appendices, supplements; and the alterations and corrections in his proofs were such that the printers found their task well-nigh impossible. … Sylvester read only what had an immediate bearing on his own researches, and did little, if any, work as a referee.

The title of one particular paper illustrates this point:

J. J. Sylvester, Explanation of the Coincidence of a Theorem Given by Mr
Sylvester in the December Number of This Journal, With One Stated by
Professor Donkin in the June Number of the Same, Philosophical Magazine
(Fourth Series) 1, 44-46, 1851

Secular Equation Paper

Out of Sylvester’s hundreds of papers, one in particular stands out as notable to me: “On the Equation to the Secular Inequalities in the Planetary Theory”, Philosophical Magazine 16, 267-269, 1883, for the following reasons.

  • The title has virtually nothing to do with the paper.
  • This is the paper in which Sylvester defines the term latent roots—but as if a totally new concept, even though the concept of matrix eigenvalue was already known.
  • He states a theorem about a sum of products of latent roots of a product AB being expressible in terms of sums of products of minors of A and B.
  • He gives the first general definition of function of a matrix (later refined by Buchheim).
  • He discusses the special case of pth roots.

The paper is short (3 pages), no proper introduction is given to these concepts, and no proofs are given. In short, a brilliant but infuriating paper!

Baltimore Summer

In these days of ubiquitous air conditioning it is interesting to note one of the things that made it difficult for Sylvester to do research. Parshall writes, of Sylvester in Baltimore,

“He could not concentrate on his research on matrices in the debilitating summer heat and humidity”.


Sylvester’s enthusiasm for matrices is illustrated by his attempt to teach the theory of substitutions out of a new book by Netto. Sylvester

“lectured about three times, following the text closely and stopping sharp at the end of the hour. Then he began to think about matrices again. `I must give one lecture a week on those,’ he said. He could not confine himself to the hour, nor to the one lecture a week. Two weeks were passed, and Netto was forgotten entirely and never mentioned again.” (Parshall, p. 271, quoting Ellery W. Davis).

Compare this with the following quote about E. T. Bell (famous for his book Men of Mathematics, 1937), from Constance Reid’s book about Bell:

Bell’s method of teaching was to read a sentence aloud and announce that he didn’t believe it. `By the time we students convinced him that it was true,’ concedes Highberg, `we pretty well understood it ourselves.’

Inaugural Lecture at Oxford, 12 December 1885

There are many ways in which we are more fortunate today than mathematicians of Sylvester’s time. But there were some advantages to those times. From his inaugural lecture, published as On the Method of Reciprocants as Containing an Exhaustive Theory of the Singularities of Curves (Nature, 1886)

It is now two years and seven days since a message by the Atlantic cable containing the single word “elected” reached me in Baltimore informing me that I had been appointed Savilian Professor of Geometry in Oxford, so that for three weeks I was in the unique position of filling the post and drawing the pay of Professor of Mathematics in each of two Universities:


Emile Picard recounted how Sylvester, on a visit to Paris, asked him if in six weeks he could learn the theory of elliptic functions. Picard said yes, so Sylvester asked if a young geometer could be assigned to give him lessons several times per week. This began, but from the second lesson reciprocants and matrices started to compete with elliptic functions and in the ensuing several lessons Sylvester taught the young geometer about his latest research and they remained on that topic.

What Can We Learn from Sylvester’s Life?

If I had to draw two pieces of advice from Sylvester’s life story I would choose the following.

  • You are never too old to take on a major challenge (he took up the chair at Johns Hopkins University at the age of 61).
  • If you want to be remembered, define some new terms and have some theorems named after you!