The Printing of Mathematics by Chaundry, Barrett, & Batey

How they used to print math

In the dark ages before $\LaTeX$, mathematical books and papers used to be printed by letterpress. This book explains how mathematical type was set as of the mid 1950s and what constraints the process imposed on mathematical communication. It's written not primarily for other printers, but rather for working mathematicians, so that they can use the proper house style and be considerate of the compositors when preparing their manuscripts for Oxford University Press.

In Chaundry's age, most of the typesetting at OUP was done with a monotype machine. Here's a great video explaining how it works, courtesy of the National Print Museum in Dublin. The first step was for a highly trained operator to read the manuscript and transcribe it to a perforated roll of paper, basically like a piano roll. This was done with a glorified typewriter that used puffs of compressed air to punch holes in the roll corresponding to the keys that the operator depressed. This keyboard machine also kept track of how many ems of width the operator had typed on the current line so that he would know (1) when he had run out of space and needed to start a new line, and (2) how many ems wide each of the spaces had to be in order to full justify the line. At the end of each line, the operator would retroactively tell the machine how wide to make each of the spaces he had typed so far.

The resulting roll of paper was then fed into a second machine, the caster, which casts the required type character-by-character out of molten lead alloyed with some tin and antimony. The caster would read the roll in reverse order, beginning with the last thing the operator had typed. This makes a lot of sense when you consider that the last keystroke of each line—the one that sets the space width—is actually the first piece of information that the caster needs to queue up all the spaces correctly.

^[1]

The monotype machine can set type far faster than a compositor working by hand, but its key weakness is that it can only produce one uniform body size—read text height. It can print inline math like $a^2$ or $5/2$ or $\sum_{i,j}a_ib_j$ just fine, but not something like

$$\int_0^t \frac{e^{\lambda x}}{f(x)}\,dx \;\;\text{or} \;\;\sqrt{\left\{\frac{a}{b}\right\}}.$$

Display math has to be assembled by hand by a compositor, which is far slower and more expensive than running the monotype machine. So the main theme of this handbook is that mathematical authors should try their utmost to minimize display math, and if they must include it, they should at least have the consideration to use as few unusual or fiddly symbols as possible. The gravest sin of all was neologism. Perhaps the author thinks some idea they want to express simply must be notated with an entirely new symbol, like $\overgroup{AB}$ or $\circledast$, but "editor and publisher will look askance at the expense (20s. or more for every sort); and the whole triumvirate will disapprove on grounds of delay (anything up to twelve months)."^[2] Why so much time and expense? The inventive mathematician's new coinage wouldn't be in the default matrix of the caster machine, so it would have to be custom engraved by a skilled craftsman and specially fitted into the machine just for that one book or paper.

This entire system of mathematical typesetting was quite unsatisfactory. For one, the large quantities of skilled labor required made the process expensive, and especially so after many of the craftsmen with the skills to set math had been killed or disabled in World War 2.^[3] Chaundry claims that by his day, the cost of printing math had gotten so high that it was starting to limit the spread of scientific ideas.

A greater and more abiding anxiety is that of the steeply rising costs of mathematical printing. Learned bodies charged with the maintenance of mathematical periodicals are faced with difficult economic problems: are they to raise prices at each increase of costs or must they steadily cut down their acceptance of material? Similar questions confront the publishers of mathematical books.^[4]

Also, the process was slow. Today I can open up Overleaf, spew some $\LaTeX$ into an editor, click compile, and in a few seconds, a fully typeset pdf pops out on the other side. The physical latency between completion of a manuscript and delivery of a typeset copy is negligible, but this was not so in the fifties. Chaundry doesn't say precisely how long it would take to turn a mathematical manuscript into a proof, but judging by the complexity of the process, it probably took on the order of months.^[5]

[1]	A monotype machine, seen by the author at the OUP museum. The keyboard is on the right and the caster on the left.

[2]

pg 16

[3]

pg v

[4]

pg 21

[5]	Source: my best guess after reading the book, vibe-checked by Gemini 2.0 and Sonnet 3.7.

Typographical suggestions

Chaundry makes a bunch of typographical suggestions, most of them oriented toward making the printer's life easier. Some of them seem quite elegant to me, even thought they've outlived their original rationale. For instance, Chaundry suggests that we use $\exp_a(x)$ for $a^x$ by analogy with $\log_a(x)$, basically because superscripts are annoying to print. He also suggests replacing a sum like

$$\sum_{\substack{n\leq x \\ n + \kappa_1 = a \\ a> x^\alpha}} \qquad \text{with}\qquad \sum_1 \;\cdots\; (\text {where}\; n\leq x, n + \kappa_1 = a, a> x^\alpha\; \text{in}\, \textstyle\sum_1).$$

Instead of writing something minima and maxima like $\min \mu$ or $\max \delta$, we could instead write $\mu_\flat$ or $\delta^\sharp$ to save on characters.^[6] I'm confused about why this is an improvement from the printer's point of view, since it requires the caster matrix to contain musical symbols even when it's being used to print math.

I am a fan of the dot notation for derivatives, but Chaundry is not. "Dotted symbols are in general to be discouraged except perhaps in traditional devotion to Newton."^[7] The issue is that there aren't enough keys on the monotype keyboard for all of the dotted characters, which forces the compositor to insert them manually at the end.

[6]

pg 42

[7]

pg 36

What was different when text was expensive?

Chaundry repeatedly exhorts authors to write economically and to proofread their manuscripts carefully, lest their mistakes waste the printer's time and money. "It is a sobering experience for an author to contemplate the chain of processes that he sets in motion when he sends a manuscript for publication."^[8] The information contained in the manuscript passes through all of the four elements on its way to the reader. The bits are first converted into blasts of air, thence into fiery molten lead, which being cooled by water, can eventually receive inks made from oil mined deep in the earth. Serious stuff. Again: "Authors should remember too that to destroy an elaborate formula that has cost the compositor time and effort to construct can be very discouraging to him."^[9] Even the white space on a letter-pressed page is expensive. "Where the printed page shows the blankness of space (or what the printer calls ‘white’) there is none the less solid metal, invisible only because it does not reach up to the printing face."^[10]

How does the experience of writing change when text is so dear and errors so hard to correct? Does the pressure of having to get it right the first time force authors to think less sloppily? Does the pressure reduce their creativity? How about the experience of reading? Our instincts as readers are calibrated for an age when printed words are dirt cheap, and we know not to trust something just because it's typeset. But printing used to be a costly signal of quality. If someone had successfully recorded their thoughts in print, it meant that they (or whoever paid for the printing) cared strongly about the content and had put serious effort into getting it right. I still apply this heuristic to some extent. I know that internet text is cheap—indeed, essentially free—so my prior is to trust it a bit less than text that I know cost the author or the publisher something to distribute. One is much less likely to shitpost in manually composed letterpressed print than in pixels.

We can also ask a parallel question for symbols. How was scientific writing encumbered by the cost of introducing new notations and symbols? There's a perspective in the history and philosophy of math and science that takes notations (and usually also diagrams) very seriously. Better notations allow scientists to think useful thoughts that they otherwise could not have thought were they limited to worse notations—or more conservatively, the better notations make useful thoughts more salient.^[11]

This should be fairly obvious to anyone who's studied vectors and matrices with and without Einstein notation. Whenever you have to prove an identity involving cross products, your first step is usually to translate $\vec a\times \vec b = \epsilon_{ijk}\,\vec e_ia_jb_k.$ One of my friends recently claimed to me that the British devotion to Newton notation over Leibniz notation substantially delayed the study of analysis in England in the eighteenth and nineteenth centuries. I find it hard to believe that this effect was so large, but it's certainly true that Newton notation makes some important facts less salient. For instance, the Fundamental Theorem of Calculus just looks like it should be true in Leibniz notation:

$$\int_a^b\frac{df}{dt}\,dt = \int_a^b\frac{df}{\cancel {dt}}\,\cancel{dt} = \int_a^bdf = f(b) - f(a).$$

Let's grant that new notations really do facilitate scientific progress. Well then, because the twenty six Roman letters plus the Greek alphabet and standard mathematical symbols make up such a constrained and overloaded notational vocabulary, we should expect that when new symbols are expensive to create, scientific progress should be (somewhat) restrained. And when new symbols suddenly became dirt cheap, we should have seen a Cambrian explosion of illuminating new notations. I can have a bright idea for a new mathematical symbol, draw it on my tablet, and import it into $\LaTeX$ with \newcommand, all for zero dollars and maybe five minutes of my time. So what have scientists done with this incredible notational freedom? What are the best new coinages of the post-Knuth era?

[8]

pg 22

[9]

pg 71

[10]

pg 22

[11]

Chapter 6 in James R Brown's Philosophy of Mathematics makes a philosophical case for the importance of notations. Florian Cajoli's History of Mathematical Notations makes an implicit historical argument to the same effect. It's a wonder that premodern mathematicians made any progress at all considering the cursed notations they were saddled with.

Thanks to Malaika Aiyar for recommending The Printing of Mathematics to me.