TUGboat, Volume 24 (2003), No. 2 165

Both are pertinent to the purposes of this paper: it Typography is the aspect of ‘arranging’ or ‘rightly disposing’ ma- terial ‘to aid to the maximum the reader’s compre- hension’ that will be emphasized. Some will argue Maths = Typography? that both these definitions are rather utilitarian and Richard Lawrence omit any feeling for the art and beauty that typog- raphy can bring to the printed document. However Introduction the question of beauty in mathematical typography This paper is written for a conference with the theme is also addressed. ‘Hidden typography’. Broadly the author’s interest So what is hidden typography? In the best is in mathematical printing and typesetting in par- sense it is Beatrice Warde’s crystal goblet typog- ticular. So, as is usual for academic conferences, raphy: invisible or unobtrusive but making the the author’s task is to persuade you, the reader, reader’s task easier and more pleasant. It is design that there is some connection between the confer- that helps the reader to extract meaning from the ence subject and the author’s personal interest. To written word. This is very much the sense relevant see if this can be done it is pertinent to ask a few to the printing of maths. Enough people have trou- questions: ble grappling with the abstraction of maths that it would not be a good idea to add typographical flour- * What is typography and so what is hidden ishes and quirks to its written form. Good ‘crystal typography? goblet’ typography is what the complexity of maths * What is mathematics? typesetting really does need. * Is there any typography in mathematics and is there any hidden typography in it? What is mathematics? The last of these questions is the one that is central The branch of science concerned with number, to the subject of the conference; answers to the other quantity, and space, either as abstract concepts two help to explain the author’s answer to it. So to (pure mathematics) or as applied to physics, engi- start with the conclusion: neering, and other subjects (applied mathematics). (Concise Oxford English Dictionary, 2001) * There is typography in mathematics. Mathematics is its own branch of science (like * The typography in written mathematics is physics or chemistry) and comes in two forms, pure not hidden, it is overlooked. and applied. Both forms are concerned with ‘num- * There is a strong case for saying that written ber, quantity, and space’, one in the abstract, one mathematics is a very highly developed in practical terms. The graphic representation of example of typography: it may even be maths then has to be able to encompass both ab- possible to say ‘Maths = Typography’. stract notions and practical applications if it is to What is typography? be any use to mathematicians and those who use maths (physicists, engineers, etc.). It has to deal The art or process of setting and arranging types with ‘number, quantity, and space’. It also has to and printing from them. (Concise Oxford be able to describe a whole branch of science (part English Dictionary, 10th edition, 2001) will not do). As we know the result is that the print- Typography may be defined as the craft of rightly ing of mathematics is challenging and specialist work disposing printing material in accordance with spe- largely avoided by many. cific purpose; of so controlling the type as to aid Another common view of mathematics, par- to the maximum the reader’s comprehension of the ticularly popular with those who use maths (engi- text. (Stanley Morison, neers, physicists, etc.) is that it is a language that First principles of typography, 1951, CUP) is used to describe physical situations and relation- Here are two definitions of typography, one short ships. Mathematical equations are used to describe and written for a general audience, the other longer the motion of a pendulum, the decay of radioactive and written for an audience wanting to know more. waste, the flow of traffic on congested roads, and the relationship between infinitely large groups of This paper was presented at the 2003 St. Bride Printing objects in multidimensional space. Mathematics is Library Conference on “Hidden Typography”, and appears the language that scientists (physical scientists at here with permission. The texts of all the talks from the conference can be viewed at http://www.stbride.org/ least) use to communicate their ideas and observa- conference2003/. tions. Like mathematics in the dictionary definition 166 TUGboat, Volume 24 (2003), No. 2 above, the physical scientist is interested in ‘number, One of the very earliest mathematical works quantity, and space’. The graphic representation of is the Algebra of Al-Khowarazimi, a ninth-century maths has to reflect this. scholar in Baghdad. Florian Cajori (A history of Having used the standard trick of looking at mathematical notation, 1929, Open Court) quotes a a dictionary definition of the subject it may be in- translation of an example from this work: structive to consider the popular views of its users What must be the amount of a square, which, and practitioners. A physicist uses a variety of when twenty-one dirhems are added to it, becomes strange machines to investigate the rules that gov- equal to the equivalent of ten roots of that square? ern the physical world and records observations as Solution: Halve the number of the roots; the moiety mathematical relationships. A biologist grows then is five. Multiply this by itself; the product is experiments on living things in order to understand twenty-five. Subtract from this the twenty-one more about them perhaps using statistics to sup- which are connected with the square; the remainder is four. Extract its root; it is two. Subtract this port arguments and observations. An engineer de- from the moiety of the rots, which is five; the signs and builds machinery to exploit the discover- remainder is three. This is the root of the square ies of other scientists and uses approximate equa- which you required and the square is nine. Or you tions to predict how the machinery will behave. may add the root to the moiety of the roots; the The mathematician sits and thinks and scribbles sum is seven; this is the root of the square which and rearranges equations on paper or blackboard. you sought for, and the square itself is forty-nine. The mathematician has no machinery or plants or In modern notation the statement of the problem animals to work with. The mathematician’s only and its solution is: prop in this populist view is the piece of paper or x2 + 21 = 10x blackboard, or more specifically equations written √ on these. Mathematics is in some sense the written Solution: x = 10/2√± [(10/2)2 − 21] equations on these surfaces. If this is indeed so, then = 5 ± √(25 − 21) it can be appreciated that the optimal arrangement = 5 ± 4 of the symbols in the equations is of some conse- = 5 ± 2 quence. = 7, 3 So we have three views of mathematics: it is Even if the reader can not follow the mathe- a branch of science dealing with the description of matical notation, it should be apparent that the ver- the abstract and real and quantity, number, and sion written using symbols is potentially much eas- space; it is a language; and it is written equations. ier to comprehend. It is enormously more compact. Properly, and not just as a result of undue deference This compactness and its consequences for intelligi- to the majesty of the Oxford English Dictionary, it bility were commented on a long time ago. William is only the first of these. But mathematics’ very Oughtred (quoted by Cajori), an English mathe- singular distinction is that it is dependent on its own matician promoting his own work in 1647 noted: language to communicate it. That language is only . . . Which treatise being not written in the usuall easily communicated in its written form (equations synthetical manner, nor with verbous expressions, on paper or blackboard). While the individual but in the inventive way of Analitice, and with sym- components of an equation can be read out loud and boles or notes of things instead of words, seemed their relative positions can be described, it is not too unto many very hard; though indeed it was but their owne diffidence, being scared by the newness far-fetched to say that maths is an unpronouncable, of the delivery; and not any difficulty in it selfe. even a silent, language. So that its written form, For this specious and symbolicall manner, neither equations, has to be able to communicate matters racketh the memory with multiplicity of words, nor of ‘quantity, number, and space’. chargeth the phantasie with comparing and laying things together; but plainly presenteth to the eye Printing’s influence on mathematics the whole course and processe of every operation Before pursuing the intellectual argument that writ- and argumentation. ten maths involves a lot of typography, it is instruc- The essence here is Oughtred’s observation that by tive and interesting to look at the influence of print- writing ‘with symboles or notes of things instead of ing on the development of maths. It is also instruc- words’ the argument is ‘plainly presenteth to the tive to see the consequences of trying to write math- eye’. The ‘symboles’ used by early mathematicians ematics without symbols to understand why the lan- are dictated by what the printer had available. So guage of mathematics is necessary. in one of the earliest printed maths books, Cardan’s Ars magna (1545, quoted in Cajori) the author TUGboat, Volume 24 (2003), No. 2 167 contents himself with using abbreviations set in measure) arisen from the character of the printed the text roman type to express unknowns. Vieta and written letters employed for so many genera- in 1591 (quoted by Cajori) uses single text roman tions, by inheritance and accumulation. It became capitals for unknowns. It was Ren´eDescartes in racial; cultivated in youth, it was intensified in the 1637 who finally established the use of lower case adult, and again transmitted to posterity. German italic letters for unknowns. He also started the letters must go. useful distinction of using letters near the beginning Rejecting Germans and Greeks, I formerly used ordinary Roman letters to mean the same as Max- of the alphabet for unknown constants and letters well’s corresponding Germans. They are plain at the end of the alphabet for unknown variables. enough, of course; but, as before mentioned, are Significantly by this date it was reasonable to expect open to objection. Finally, I found salvation in a printer to have matching roman and italic types Clarendons, and introduced the use of this kind of that could be set together. Mathematical setting is type so called, I believe for vectors (Phil. Mag,. Au- notorious for the diversity of sorts it exploits. In gust, 1886), and have found it thoroughly suitable. early mathematical works the choice of these sorts It is always in stock; it is very neat; it is perfectly is limited to what the printer has. For example in legible (sometimes alarmingly so), and is suitable early printed books on ‘algebra’ much use is made for use in formulae along with other types, Roman of a capital R with a scratched tail to denote root, a or italic, as the case may be, contrasting and also sort ordinarily deployed in liturgical work to denote harmonising well with them. ‘Response’ (e.g. Cardan, 1545). Sometimes block letters have been used; but it is sufficient merely to look at a mixed formula One of the more amusingly documented discov- containing them to see that they are not quite eries made by an author seeking out unexploited cor- suitable. ners of the printer’s stocks is the eventual use of bold to denote vector quantities. Electromagnetic theory This rather long quote illustrates several points was undergoing rapid development in the late nine- about mathematicians and mathematics. teenth century and this called for the development * Mathematicians are quite passionate about of notation in mathematics capable of distinguish- how their maths is presented ing quantities which possessed both direction and * Mathematicians do have a keen appreciation size (vectors) from other non-directional quantities of the utility of notation (scalars). The first attempt used greek type, but this failed because of confusion with other greek sym- * Mathematicians have a real (if idiosyncratic) bols. Maxwell in 1873 promoted German (Fraktur) idea of the aesthetics of maths printing type for the job. Oliver Heaviside (a populariser of It is also clear from this quote what a real influence Maxwell’s work) was the one who started using bold the printer can have on advancing mathematical (Electromagnetic theory, 1893, Ernest Benn Ltd): notation and maths and science themselves.

Maxwell employed German or Gothic type. This The beauty of mathematics was an unfortunate choice, being itself sufficient to The somewhat eccentric writings of Oliver Heavi- prejudice readers against vectorial analysis. Per- side do show that mathematicians have a very keen haps a few readers who were educated at a commer- cial academy where the writing of German letters sense of what works in the notation they use to con- was taught might be able to manage the German vey ideas. Beyond this utilitarian view of notation, vector without much difficulty; but for others it is mathematicians also appreciate wider principles in- a work of great pains to form German letters leg- volved in the proper written display of their work. ibly. Nor is the reading of the printed letters an This appreciation is linked to an appreciation of the easy matter. Some of them are so much alike that maths itself. A mathematician will be very pleased a close scrutiny of them with a glass is needed to if s/he is able to simplify an argument or equation distinguish them unless one is lynx-eyed. This is a and render the mathematical content more compre- fatal objection. But, irrespective of this, the flour- hensible. Mathematicians speak of the ‘beauty’ of ishing ornamental character of the letters is against a well-presented and succinct proof or newly found legibility. In fact, the German type is so thoroughly link between branches of mathematics. The quest unpractical that the Germans themselves are giving it up in favour of the plain Roman characters, which for such ‘beauty’ keeps mathematicians busy try- he who runs may read. It is a relic of mediaeval ing to refine existing proofs. The proof of the Four monkery, and is quite unsuited to the present day. Colour Map problem (any map can be coloured us- Besides there can be little doubt that the prevalent ing only four colours without the same colour ad- shortsightedness of the German nation has (in great joining itself) reported a few years ago is acclaimed, 168 TUGboat, Volume 24 (2003), No. 2 but relying as it does on thousands of hours of com- on plain work, tho’ less profitable to them than the puter calculations, it is regarded as very inelegant. former; because it is disagreeable and injures the There are many mathematicians busy trying to sim- habit of an expeditious Compositor. plify it. The writing of equations that are compact It is not only challenging, it is also expensive. In and convey meaning easily is central to ‘beautiful’ the 1970s and 1980s publishers sought cheaper al- mathematics. ternatives to hot-metal typesetting and used strike- The written language of mathematics is easily on systems (IBM, Varityper) extensively. Mathe- given the attributes of compactness and beauty by maticians grumbled, but mostly accepted arguments mathematicians. It is not surprising to see a long about costs. One however was so appalled by the and close association between mathematicians and standards of typesetting from Varitypers that he re- their printers, the mathematicians exploiting all the belled, spent time studying typography and typeset- available special sorts and skills that the printer can ting, exploited the just-available raster-scan type- provide. There is such density of meaning in the setters, and wrote his own type design and type- choice of letter, its style (roman, bold, italic), its setting programs. That is Donald Knuth, Professor typeface (serif, sans serif, script, outline), its al- of Computer Science at Stanford. His type design phabet (latin, greek, hebrew), its size, its position program is Metafont and the typesetting program relative to other characters (subscript, superscript), is TEX. This is not the place to go into the work- that the printer’s job is very challenging. The chal- ings of TEX, but it is interesting to remark that the lenge is to follow the very detailed requirements of program is a very good example of what computer the mathematical author without any understand- scientists call an ‘expert system’ that is modelled ing of the content. to some degree on the workings of the Monotype The curious thing from a typographer’s point of sytem that it replaces. Typographers hated it be- view is how little a typographer can contribute once cause of its associated typeface, Computer Modern the mathematician has made all the choices neces- Roman which Knuth modelled on the maths books sary to convey the mathematical meaning. It is not of his youth (set in Modern Series 7). Typeset- too controversial to suggest that the mathematician ters couldn’t understand its input coding: this was is in fact his/her own typographer, at least in the not modelled on the mechanical requirements of the matter of writing equations. Monotype system, but is written to make sense to mathematicians. The program was also free. It has Maths = typography? been hugely successful to the point that it is the only If mathematicians define the typography of the word-processing system any self-respecting mathe- equations they write and ‘beautiful’ mathematics matician will use. Typesetters have then found the is well-presented equations whose meaning shines economic necessity of dealing with it. Several are through the density of notation that they bear, then able to do something about its associated typeface, perhaps mathematics is just a highly refined typo- which pleases typographers. graphical game? Going back to Stanley Morison’s One significant aspect of TEX that is illustra- definition of typography as ‘so controlling the type tive of the theme of this paper is a little-remarked as to aid to the maximum the reader’s comprehen- feature of it. Knuth wrote an output program ab- sion of the text’, then that is exactly what a mathe- breviated to ‘dvi’ which allows TEX to be printed matician does. Given that mathematics’ only phys- on anything from a 64-pin dot-matrix printer to a ical reality (its only props) are written equations, laser raster typesetting machine. In all cases the rel- then perhaps maths really is typography. ative positioning of the characters of the equations are perfectly maintained irrespective of the printer Modern mathematical typography used. A mathematician can email a dvi file to a col- Mathematical typesetting has always challenged the league on the other side of the world know exactly printer (Smith: The printer’s grammar 1755): what that colleague will see (‘dvi’ stands for device independent). The exact arrangement of characters Gentlemen [authors] should be very exact in their is vital to the meaning of the equations. The exact Copy, and Compositors as careful in folowing it, typography is vital. For Knuth typography is maths. that no alterations may ensue after it is composed; since changing and altering work of this nature is  Richard Lawrence more troublesome to a Compositor than can be 18 Bloomfield Avenue imagined by one that has no tolerable knowledge of Bath BA2 3AB, UK Printing. Hence it is, that very few Compositors are [email protected] fond of Algebra, and rather chuse to be employed