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Maths = Typography? 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.
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