by Barbara Beeton and Richard Palais
1. Introduction
Until about the early 1960s, most published mathematics was typeset professionally by skilled compositors working on Monotype machines. As this form of “hot-metal” composition became less readily available, on account of both cost and the fact that skilled compositors were retiring and not being replaced, “enhanced” typewriters began to be used to prepare less prestigious publications. Phototypesetting (“cold type”) began to appear gradually, although it was more expensive than typewriter-based composition, and generally not as attractive in appearance as professionally prepared Monotype copy.
By the mid-1970s, Monotype composition was essentially dead.
Donald Knuth,
a professor of computer science at Stanford University, was
writing a projected seven-volume survey entitled The Art of Computer
Programming (TAOCP); volume 3 was published in 1973, composed with
Monotype. By then, computer science had advanced to the point where a
revised edition of volume 2 was in order but Monotype composition was
no longer possible; the galleys returned to Knuth by his publisher
were photocomposed. Knuth was distressed: the results looked so awful
that it discouraged him from wanting to write any more. But an
opportunity presented itself in the form of the emerging digital
output devices — images of letters could be constructed of zeros and
ones.1
This was something that he, as a computer scientist, understood. Thus
began the development of
2. The problem
Mathematics as a discipline depends on its own arcane language for communication. Prior to the ubiquitous availability of personal computers, the options for communicating mathematical knowledge were limited to face-to-face contact, preferably with a writing surface handy, although conventions developed to enable intelligible telephone discussion, personal letters (at least bits of which required handwritten notation), or formal publication. The last mode required a highly skilled compositor, working either with traditional hand-set type or with a hot-metal typecaster, or a combination of the two.
The gold standard for typeset mathematics in the midtwentieth century was the Monotype typecaster [e1], [e3]. The audience was relatively small, and the work exacting. Since mathematical notation is essentially multi-level (see Figure 1), the Linotype, the linear-type workhorse for newspapers and most book publishing, was not up to the task. Only a few suppliers would take on such work, and mathematical composition was always considered “penalty copy”.2
Quadratic formula
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Maxwell's equations
\begin{align*}
\vec{\nabla} \cdot \vec{B} &= 0 \\
\vec{\nabla} \times \vec{E} + \frac{\partial B}{\partial t} &= 0 \\
\vec{\nabla} \cdot \vec{E} &= \frac{\rho}{\epsilon_0} \\
\vec{\nabla} \times \vec{B}
- \frac{1}{c^2} \, \frac{\partial E}{\partial t} &= \mu_0 \vec{J}
\end{align*}
Another system of equations
\newcommand{\gammaurad}[1]{%
\frac{\gamma u_{\text{rad}}^{} \bar{\lambda} a_{\text{eff}}^2}{2I_1 {#1}}\,}
\begin{align*}
\frac{d\phi}{dt} &= \gammaurad{\omega \sin \xi} G(\xi, \phi)
- \Omega_{\mathrm{B}} \, , \\
\frac{d\xi}{dt} &= \gammaurad{\omega} F(\xi, \phi)
- \frac{\sin \xi \cos \xi}{\tau_{\text{DG}}^{}} , \\
\frac{d\omega}{dt} &= \gammaurad{}
\bigl[ \gamma H(\xi, \phi)
+ (1 - \gamma) \langle Q_\Gamma^{\text{iso}} \rangle \bigr] \\
&\phantom{{}={}} - \frac{\omega \sin^2 \xi}{\tau_{\text{DG}}^{}}
+ \frac{\omega \sin^2 \xi}{\tau_{\text{drag}}^{}}
- \frac{\omega}{\tau_{\text{drag}}^{}}
\end{align*}
Figure 1. Samples of display math using
3. Analysis of the problem
What Knuth did next is described nicely in his lecture on the occasion of his receiving the Kyoto Prize in 1996 [e6]. Publication of the photoset volume 2 was halted, and Knuth sought out the best examples he could find of the mathematical typesetter’s art. He chose three: Addison-Wesley books, in particular the original TAOCP; the Swedish journal Acta Mathematica, from about 1910; and the Dutch journal Indagationes Mathematicae, from about 1950.
To develop rules for proper spacing in mathematics, he writes ([e7], pp. 364–365)
I looked at all of the mathematics formulas closely. I measured them, using the TV cameras at Stanford, to find out how far they dropped the subscripts and raised the superscripts, what styles of type they used, how they balanced fractions, and everything. I made detailed measurements, and I asked myself, “What is the smallest number of rules that I need to do what they were doing?” I learned that I could boil it down into a recursive construction that uses only seven types of objects in the formulas.
4. Growing pains
The initial implementation of \gamma
,
as would the structural
components of a document, e.g.,
\chapter
or
\section
,
as opposed to the prevailing compositor’s approach of marking changes by
font and type size. (The latter approach is still evident in the
design of many word processing programs, although it’s usually hidden
from the person entering the text.)
In January 1978, Knuth delivered the Josiah Willard Gibbs lecture to the annual meeting of the American Mathematical Society (AMS). The lecture, entitled “Mathematical Typography” [e2], began “Mathematical books and journals do not look as beautiful as they used to.” Armed with copious examples, both good and bad, and a firm sense of how best to present mathematical notation so that it is intelligible (at least to those who are familiar with its use), Knuth presented a view of how computers can serve to replace the vanishing expertise of traditional compositors and restore the appearance of technical publications to their former glory. In addition to the discussion of proper presentation of mathematical notation, the lecture introduced a companion tool, Metafont, for production of the needed fonts.
The chair of the AMS Board of Trustees, Richard Palais,
was in the audience. Since the AMS was one of the publishers suffering from the
technological transition,
As one of the AMS representatives, Beeton gathered a number of “good
bad examples” that she knew would be encountered in production
because they already had. This turned out to be good preparation:
several of these examples turned up later in The
The
Contributing to
By the end of the 1980s, a growing user population in Europe was
becoming increasingly frustrated with the difficulties in handling
non-English texts.
5. Communicating mathematics
The basic
Document structuring
While AMS-
Fonts
Font development has been driven by the availability of personal
computers and laser printers and the growth of the World Wide Web, as
well as by the desire for variation in type styles available for
One font family that originated in the need for robust output from
low-resolution laser printers is Lucida by Kris Holmes and Charles
Bigelow. Bigelow was on the Stanford faculty during part of the
Desire to give mathematicians the ability to communicate on the Web was the driving force behind the STIX project.10 In the first phase of this project, a comprehensive list of math symbols was compiled from lists submitted by the STIpub member organizations and submitted for addition to Unicode. The bulk of additions became available with Unicode 4.0 in 2003, comprising several thousand symbols, including several variant alphabets (e.g., Fraktur and script) needed to discriminate between different variables as defined in mathematical contexts.
Version 1 of the STIX fonts (based on Times) was released in 2012, and final polishing of version 2 is underway.
Possibly influenced by the STIX work with Unicode,11
Microsoft added mathematics support to Word 200712
along with the newly designed Cambria font
[e10].
Cambria is the
first OpenType font (OTF) to make use of the OTF Math table. Indeed,
the OTF Math table was created specifically for Cambria, and many of
its parameters are recognizable as parallel to the
The Web
XML was developed as a Web-aware application of SGML. Even for SGML,
there had been an effort to standardize the names of math symbols as a
“public entity set”, and this drew heavily on the names assigned for
Since MathML is not as easily comprehended by humans as
Non-technical applications
Since
Remaining limitations
One area that has not yet seen a satisfactory method of presentation
is accessibility — the ability to translate
6. Conclusion
The most lasting effect of
Since the input is plain text, it is not affected by (most) upgrades to the processing system, and it is hardware independent; the same input will yield the same output, modulo the availability of identical fonts. Knuth’s original goal of creating a system that would enable him to typeset his life’s work, TAOCP, with the same high quality shown by the first edition of volume 1 and remain consistent regardless of how many years have elapsed has been achieved admirably.
Unless something totally unforeseen materializes that is simpler to use and produces results of equally high quality without the need to unlearn the basics of mathematical discourse itself, the situation is likely to remain very much the same in the coming decades.
Authors
Barbara Beeton is a long-time employee of the American Mathematical
Society, where she has been involved in technical support of
typesetting ever since installation of the first computer. She is a
founding member of the
Richard Palais was the Founding Chair of the