Nghiên cứu ba font chữ cho toán học: Times 4-line, AMS Euler và Cambria Math

Three typefaces for mathematics The development of Times 4-line Mathematics Series 569, AMS Euler, and Cambria Math by Daniel Rhatigan Dissertation submitted in partial fulfilment of the requirements for the MA in Typeface Design, University of Reading, 2007 Typeset with Arno Pro (Adobe) and Whitney (Hoefler & Frere-Jones). All figures shown at 100% scale unless indicated otherwise. Daniel Rhatigan, Three typefaces for mathematics September 2007 The development of Times 4-line Mathematics Series 569, AMS Euler, and Cambria Math Abstract This paper examines the issues involved in the design of typefaces for mathematics. After a brief discussion of some of the typographic and technical requirements of maths composition, three case studies in the development of maths types are presented: Times 4-line Mathematics Series 569, a complement to the Times New Roman text types as set with Monotype equipment; AMS Euler, an experimental design intended to contrast against non-mathematical typefaces set with TEX; and Cambria Math, designed in concert with a new text face to take advantage of new Microsoft solutions for screen display and maths composition.

In all three cases, the typefaces were created to show the capabili- ties of new technological solutions for setting maths. The technical advances inherent in each font are shown to be as central to its function as its visual characteristics. By looking at each typeface and technology in turn, and then comparing and contrasting the issues that are addressed in each case, it becomes apparent that even though certain challenges are overcome with technical advances, the need to consider the specific behaviours of type in a maths setting remains constant. Contents Introduction 1 1 The difficulties of typesetting mathematics 5 1.2 Technical concerns 7 2 The typefaces 9 2.1 Times 4-line Mathematics Series 569 9 2.1 Monotype 4-line Mathematics 9 2.2 Development of Times New Roman and Series 569 13 2.3 Characteristics of the design 15 2.4 Updates for newer technologies 17 2.1 METAFONT and TEX 19 2.2 Development of AMS Euler 21 2.3 Characteristics of the design 25 2.4 Updates for newer technologies 27 2.1 ClearType and the Microsoft maths engine 29 2.2 Development of the Cambria family 31 2.3 Characteristics of the design 35 3 Comparing the typefaces 39 3.1 Design of italics 39 3.2 Relation of maths to text 41 3.3 Range of character styles 43 3.2 The influence of Times New Roman 45 3.3 Composing the page 49 3.4 Typesetting roles 51 Conclusion 55 Appendix: Correspondence 57 Works Cited 67 Works Consulted 69 Acknowledgements 71 Introduction The difficulty of printing mathematical notation has always been a challenge for printers, who have urged mathematicians to use nota- tion that is easier to print, as well as mathematicians, who have sought new means of expression with the type and techniques available. It is not enough to talk about simply choosing a typeface for setting maths when the subject matter has such complex, specific requirements.

The most effective typefaces for maths are those that have been ­created or adapted especially for it, anticipating both the typographic and the technical issues involved. This dissertation will examine the design and the function of alphabetic characters of three typefaces created specifically for mathematics — Times 4-line Mathematics Series 569, AMS Euler, and Cambria Math. It will be helpful to look closely at the development of typefaces that shed light on different aspects of the issues involved in setting maths, and see how their designs evolved in response to the available means of composing mathematical work. Each typeface represents a distinct period of technical development, each in some sense reacting to the typefaces and technologies preceding it.

Times 4-line Mathematics Series 569 was created by the Monotype Corporation in the United Kingdom for use with its hot-metal com- position equipment. This typeface, a variation of the popular Times New Roman, was specifically developed for use with Monotype’s 4-line system, used to mechanise many aspects of maths composition without sacrificing the quality of the finished product. AMS Euler was designed by typographer and calligrapher Hermann Zapf in collaboration with Donald E Knuth, a mathematician and com- puter scientist who developed pioneering software for creating digital typefaces and composing text and mathematics with computers. Euler was a test of the capabilities of these new tools, both technically and aesthetically.

Cambria Math was designed by Jelle Bosma to take advantage of two major technologies from Microsoft: one a new way to improve the appearance of text displayed on screens, the other a sophisticated new method of creating and typesetting equations.  ‘Mathematical work: It’s not just a case of ‘follow copy’’. The British & Colonial Printer. ‘Maths = typography?’ TUGboat.

165–167   The first section presents an overview of the typographic and technical issues that make maths composition so challenging. The second consists of an analysis of each of typeface in turn, including the its history, the details of its design, any features specific to its use for mathematics, and how it addresses the issues presented by the typeset- ting technologies for which it was developed. The third section will compare and contrast the three typefaces in terms of their design and their context.  Figure 1 A sample of text set with 4-line maths shows the the variety of characters, styles, and sizes which are often used together in maths.

Characters in equations must stand alone in strings of symbols, while the characters in the text surrounding the equations are grouped into words. Information sheet no. London: The Monotype Corporation, Ltd. 4 Figure 2 The equation above (set with Cambria Math) is shown at full size and magnified 3µ.

It could be difficult to discern a (shown in orange) from alpha (shown in blue) without a clear difference in their shapes, since they are not set within words that provide additional context, such as mathematical or μαθηματικὴ. John Hudson and Ross Mills, eds. Mathematical typesetting: Mathematical and scientific typesetting solutions from Microsoft. Redmond, WA: The Microsoft Corporation (2007) p.

54  1  The difficulty of typesetting mathematics 1.1  Typeface requirements Mathematics is a field of study that employs its own vocabulary and conventions, and in many ways it has a language and writing system of its own. Although its notation uses many familiar characters, it uses them as symbols rather than words. However, maths is set alongside the words that convey its meaning, and publishing maths requires the ability to mix the symbolic and the verbal languages with one another. Setting complex mathematics requires the use of a wide array of characters that must work in harmony. Numbers are mixed with alpha- betic characters of Latin and Greek origin in a variety of styles — italic, bold, fraktur, serif and serif forms — each with a specific semantic function.

All these characters are then mixed with mathematical opera- tors and other symbols that often conflict with the scale or texture of the other characters. Setting maths, then, requires access to a vast set of unique characters, preferably ones that have been designed to work with one another.) Setting the material so that it can be read and understood eas- ily requires a delicate balance between characters that blend with one another yet can also be easily distinguished. A reader’s ability to recognise words by their overall shape makes it easier to compensate for individual characters whose shape may be unusual or ambiguous. In mathematics, however, each letter, number, or ­symbol — as well as each of its stylistic variants — must be perfectly legible in isolation, since these are read not as parts of words but as discrete signifiers of an equation’s meaning.) If the Greek letter α (alpha) is incorrectly set within the word mathematics (like so: mαthemαtics) the word’s meaning is not obscured by the error.

In maths, where letters are used to represent numerical values, confusing the similar letter shapes within x = a + 4 and χ = α + 4 changes the result of the calculation. A reliable typeface for mathematics must offer enough visual cues to help the reader avoid such confusion in the event that the text surrounding the equations does not. ‘Setting mathematics’, The Monotype Recorder. 3  John Hudson and Ross Mills, eds.

Mathematical typesetting: Mathematical and scientific typesetting solutions from Microsoft. Redmond, WA: The Microsoft Corporation (2007) p. 13  Figure 3 This diagram shows the complex arrangement of characters and spaces required to compose mathematics with metal type. Not only are numerous type styles and non- standard symbols mixed together, but characters are set on a wide variety of body sizes.

The black rectangles indicate the many sizes of spaces that must be arranged to hold the characters in place. Before the introduction of the 4-line system, expressions like these were composed entirely by hand out of characters cast in multiple batches, requiring the time and skill of a highly trained compositor. T W Chaundy, P R Barrett, and Charles Batey. The printing of ­mathematics.

London: Oxford University Press (1954) p.2  Technical concerns A suitable typeface for maths is only a partial solution without a way to compose the material properly. Most typesetting solutions have diffi- culty with the spatial arrangement required for maths. Equations often work in two dimensions, with frequent use of stacked components, oversize symbols, and multiple levels of superior and inferior position- ing. With so much variation along a vertical axis and with individual characters rather than words as the basic unit of meaning, horizontal spacing that has been designed for text use is rarely helpful.

Compositors working with metal type often tried to ease the proc- ess by urging authors to use alternate forms of notation that could be set more easily, but the clarity of the subject matter often depended on notation that was difficult to set. Even if there were room in the matrix case for all the symbols needed at one time, the frequent use of complex positioning and various sizes of symbols require type set on alternate body sizes and fitted together like a puzzle. This wide variety of type styles and sizes made it costly to set text with even moderately complex mathematics, since so much time and effort went into com- posing the material by hand at the make-up stage.) Filmsetting and photocomposition systems made it easier to posi- tion characters in equations since they were not constrained by the very physical problem of fitting unyielding pieces of lead together, but they still presented the problems of making the full set of necessary characters available at once. In practice, these methods of typesetting still required their own version of hand-setting, except pieces of film or photographic paper were assembled rather than pieces of metal type. Digital typesetting tools remove the physical limitations of fitting pieces of type together, but a piece of software that sets text quickly and efficiently may still have difficulties with maths.

Word processing and desktop publishing applications set type in a manner that mimics physical typesetting methods: characters are set in a line, and those lines are set one after another. If the software is not written to arrange equations as easily it can arrange text, a user can only set maths by resorting to a ­digital equivalent of hand-setting: placing characters one by one to build an equation slowly and consciously. As with hand- ­setting ­methods for metal or film, this is slow and ultimately expensive. London: The Monotype Corporation, Ltd.

6–8   Figure 4 2  The Typefaces A specimen sheet of the matrices available for Times 4-Line Mathematics Series 569 shortly after its release. New pattern drawings 2.1  Times 4-line Mathematics Series 569 were made for all the glyphs in the series to ensure consistency within After World War II, an explosion of scientific and technical publishing the series, for both aesthetic and placed a strain on traditional methods for composing mathematics.