MathML: a Key to Math on the Web∗

Patrick D. F. Ion Mathematical Reviews P. O. Box 8604 Ann Arbor, MI 48107, USA [email protected]

Abstract As the Web gains in importance and as the needs of mathematical formalism on the Web are beginning to be met by MathML, it is an opportune time to reflect on the design decisions made by the W3C Math Working Group that resulted in the verbose markup language for transport of math on the Web that MathML turns out to be. The TEX community need not be frightened by the advent of MathML, but may learn to work in the Web environment it provides. The presentation will describe some of the key aspects of MathML, and explain how it has begun to be used (by August 1999 the support for MathML may be expected to be much greater in a practical sense than it is now). Expected future developments and roles will be commented upon.

Introduction The presenter, Ernst Schr¨oder, an algebraist 5 The Math Working Group1 of the World Wide Web and logician from Karlsruhe began his talk by say- Consortium2 certainly believes that MathML, Math ing that if there were any topic that really belonged Markup Language [Ion and Miner, eds., 1997], is a at an International Conference of Mathematicians, key to math on the Web. It is a protocol for transfer- then it was pasigraphy. He was sure that pasigraphy ring mathematical knowledge, and for building tools would take its rightful place on the agenda of all to manipulate it, which shows great promise. There succeeding such conferences. Perhaps it is needless are already several implementations and tools based to say it did not. on MathML. Maybe its greatest strength is that Schr¨oderthen went on to disagree with the 6 MathML is a specification which is open to view, distinguished chair of the session, G. Peano , by and has been adopted as a Recommendation by the saying that he did not think that Leibniz’s problem World-Wide Web Consortium (W3C). of providing an algebra universalis, a symbolic cal- This contribution will discuss some of the con- culus for mathematics, had been solved. Peano had text in which MathML has been developed, and some just begun to publish, in 1894, his treatise [Peano, of the design decisions that went into it will be 1894–1908] intended to provide just that. It is, illustrated in the live presentation. in fact, there that Peano’s axioms for the natural numbers are to be found, along with axiomatizations History and highly symbolic representations for much of arithmetic, algebra, geometry and calculus. In 1897, at the First International Congress of Math- Schr¨oderoffered some of his own considerations ematicians in Z¨urich [Rudio, 1898] there were not on the topic of universal symbolics for math as part many talks3. Striking among their titles is “Uber¨ of logic. He favored a system, near that of C. S. Pasigraphie, ihren gegenw¨artigenZustand und die Peirce, using eighteen special symbols. pasigraphische Bewegung in Italien”4. It is clear that the progress of science in general, ∗ [This article has not yet been edited. – Ed.] and mathematics in particular, depends on there 1 http://www.w3c.org/Math, co-chairs: Angel L. Diaz being a representation of its findings external to (1998–2000), Patrick D. F. Ion (1997–2000), Robert L. Miner (1997-1998) 2 http://www.w3c.org international language using characters (as mathematical 3 I thank R. Keith Dennis for showing me his copy of the symbols) instead of words to express ideas. 5 proceedings in 1997. Perhaps best known today from the so-called Schr¨oder- 4 “Pasigraphy, its present state and the pasigraphic move- Bernstein theorem. 6 ment in Italy” [Schr¨oder,1898] Pasigraphy is an artificial The Italian mathematician who can be considered leader of the pasigraphic movement in Italy.

1006 Monday, 16 August, 9.00 am Preprint: 1999 TEX Users Group Annual Meeting MathML: a Key to Math on the Web

individuals. In this way the evolving knowledge The W3C forms working groups (WG) to study can be shared. The standard reference work on and produce recommendations on subjects concern- the history of mathematical notation is by Florian ing the Web. These range from HTML, Hyper- Cajori [1928/29]. Text Markup Language [Raggett and Jacobs, eds., The ontology of mathematics has changed over 1998], the basic markup language for the Web, to the centuries, so what the objects of mathematics PICS, the Protocol for Internet Content Specifi- are is by no means a given. Very roughly speaking, cation [Consortium, 1997], which allows people to and seen from the present day, for the early Greeks control access to pages based on content ratings. A math was geometry. Then came along a revolution list of all the concerns of the W3C is available at started by Descartes who put forward a successful their main Web site mentioned above. algebraic form of geometry through the introduction Particularly relevant to the present subject are of coordinates. Throughout this time what numbers the working groups centered around XMLTM, an were was really not up for discussion, although acronym from eXtensible Markup Language, there were, for instance, differences drawn between [T. Bray and Sperberg-McQueen, eds., 1998] and rational and irrational numbers, and later between XSL/CSS (eXtensible Style Language/Cascading algebraic and transcendental numbers. At the end Style Sheets, [Stephen Deach, 1999; Bos and H. Wium Lie, of the last century and the beginning of this, a new 1999; B. Bos and Jacobs, eds., 1998]. view held that all math had to be founded upon set Members of the W3C may request representa- theory and logic. And now that is seen as not all tion on any WG of interest to them (one represen- that satisfactory, so that categories, toposes, or the tative and possibly one alternate at most on a WG, new developments of mathematical logic are viewed together entitled to one vote in deliberations), and as fundamental7. there may be invited experts from outside the W3C These remarks have been made because there is in a WG as deemed appropriate. The W3C tries an obvious sense in which the W3C Math Working to work by consensus as much as possible. The Group is just trying to create a modern form of limitations on voting are there so that problems universal symbolic language. It is not intrinsically will be sorted out on technical merit, rather than an easy task. being subject to gross commercial pressures. There are guidelines and procedures, including voting if The W3C – World Wide Web Consortium necessary, developed by the WG that deals with that The advantage of MathML mentioned above, that aspect of the W3C. it is a public specification and a recommendation SGML, HTML and XML adopted by the W3C, stems from the sort of organ- isation the W3C is. The specification for which the W3C is best known The W3C is a consortium of some 340 or so is HTML, presently at version 4.0. This was de- organizational members8, mostly commercial en- veloped by Berners-Lee along with the linking and tities. They include big computer and software transfer protocols of HTTP (Hypertext Transfer pro- companies — such as IBM, Hewlett-Packard, Sun, tocol [Fielding, 1998]). In the beginning this was Microsoft and Netscape, as well as smaller ones — a language developed for a specific purpose, and and others from, for instance, the big aircraft or not necessarily consonant with any other standards. electricity industries — Boeing and Electricit´ede However, it was realised that a few changes to France, to name two. In general, the W3C is an HTML would make it a markup language obey- organization joined by those who wish to understand ing the principles of SGML, Standard Generalized and influence the development of the protocols and Markup Language [International Standards Organi- mechanisms that control the functioning of the Web. zation, 1986; van Herwijnen, 1994]. SGML is an The W3C Director is Tim Berners-Lee, generally ISO international standard that describes a way of recognized as the inventor of the Web. The W3C writing down document markup. It provides a very is fully international in scope, with main offices general framework, both verbose and complicated to in Cambridge, Massachusets, USA (MITLCS), in use. There has been a lobby for some years trying Grenoble, France (INRIA) and in Tokyo, Japan to push SGML standardization as a highly desirable (Keio U); it receives additional support from both means for publishing production, which facilitates DARPA and the European Commission. document re-use amongst other things. 7 See the potted history in, say, a book by Godement However, the difficulties in using SGML stan- [1998]. dards in practice meant that it was unthinkable that 8 http://www.w3.org/Consortium/Member/List those standards be taken straight over to the Web.

Preprint: 1999 TEX Users Group Annual Meeting Monday, 16 August, 9.00 am 1007 Patrick D. F. Ion

True, SGML adoption would have provided a great There are many other technical differences from wealth of possibilities for Web publishing. But a SGML which allow XML parsers to be much easier Web browser would essentially incorporate a full to. Many simplifications have come out of years of SGML parser, and worse still it would have to deal experience with SGML. with the DSSSL (Document Style Semantics and Again of great importance, perhaps more than Specification Language) [10179:1996(E), 1996] for its technical quality, is that XML is a Recommenda- actual display. tion of the W3C. The corporate members of W3C To use SGML strictly every document should will support it in its place on the Web. Browsers are be parsed against an explicit DTD (Document Type being made with with XML support (e. g., by both Definition) to be sure that it is a correct instance of Microsoft and Netscape). Tools, such as editors, are some general class of documents and markup. This being produced to enable XML document creation. would require a level of rigor in SGML Web page Associated specifications such as XSL and CSS for programming in clear conflict with the use, and use- formatting [see, for an overview, W3C Style, 1999], fulness, of the Web. An SGML Web, it was feared, a DOM (Document Object Model) [V. Apparao, would rapidly age. Pages might not be produced 1998] for the Web, the RDF (Resource Description because it would be too difficult to provide the level Framework) [Lassila and Swick, eds., 1999] model of correctness required for complex SGML browsers for providing metadata9, as well as specifications for to function; also standards would be difficult to name spaces, linking, database queries, and many observe in writing pages and in writing software, and other ancillary developments are underway at the the increase of entropy toward a chaotic breakdown W3C, using XML. The fundamental HTML is being of the Web would be rapid. Thus a cut-down, or redone as XHTML [W3C HTML Working Group, simplified, version of SGML began to be developed 1999]. for use with the World Wide Web, under the aegis of the W3C. The W3C Math Working Group XMLis a restricted form of SGML as a speci- It has been rather paradoxical that the mathemat- fication for markup languages. The most obvious ical formulae of science have been so difficult to thing unchanged is the tagging structure. As in represent on the Web. Tim Berners-Lee, after all, SGML, elements of the document are marked up was a scientist at CERN, a massive international with tags, to form phrases like element center of physics in Geneva, when he came up with content. An element need not have any what became HTML and the Web. explicit content, and then is of the form . In May 1997, the W3C chartered a Math Work- There is a simple syntactic difference from SGML; ing Group to consider how to facilitate math on XML elements which are not containers have tags the Web. Originally called the HTML-Math WG, ending with the tokens />. Then again, in XML, it contained representatives of diverse backgrounds. markup has to be complete; it is not allowed, as it There were people from computer corporations and can be in an SGML markup specification, to shorten from publishing, computer algebra people10 and in- the markup by leaving out those end tags which vited experts from organizations not members11 of can be inferred as necessary because some overall the W3C. containing element has finished. For instance, para- Dave Raggett, the Math WG’s W3C staff con- graphs have to ended explicitly with a

, and tact and author of the HTML 3.2 reference specifi- cannot be assumed to have done so just because cation, was himself an early proponent of adding another one has started with a

. some math capabilities to HTML. In fact, there The next important matter for the Web is was some confusion over math support following that the language design is intended to allow the HTML 3.2 [Raggett, 1997]. Books appeared with processing, at least to some reasonable level, of sections explaining simple extensions to HTML for pages without a declared DTD. An XML document math that were little more than suggestions how can be well-formed without being an instance of something might be done. The extensions were not some DTD, although being well-formed essentially means it would be possible to construct a DTD which has the document as a valid instance. 9 Also of importance to math on the Web, especially in So a browser which can parse XML can be education, and for which there is an interest group. allowed to process a free-form page, if XML syntax 10 IBM, Hewlett-Packard, Adobe, Elsevier Science, Wol- is respected. fram Research, Maplesoft, SoftQuad, .... 11 American Mathematical Society, Geometry Center, Stilo Technologies, (and later Design Science) . . .

1008 Monday, 16 August, 9.00 am Preprint: 1999 TEX Users Group Annual Meeting MathML: a Key to Math on the Web in any Recommendation accepted by the W3C. Rec- with a TEX-like syntax? That turns out not to be ommendations are issued only after a long careful re- simple at all. Finally, after much discussion, the WG view process, including a period of six weeks during decided that it would develop a markup language which W3C members may vote for or against accep- which accorded with XML. MathML 1.0 is written tance. They normally follow several working drafts as an XML application, one of the first at more than which have been available for public comment. It a toy level. is a very open process. If there is enough support The reason is it was realized that general ac- from the W3C members then a draft is put forward ceptance of a new math specification would happen as a Recommendation. The math suggestions were only if it embedded easily into the technology of dropped because the matter needed more careful the internet, which was coming to be dominated by consideration. But this shows the level of interest XML and its relatives. The Math WG wanted to latent in the community. come up with something that really met the goal of The Math WG’s original objectives were far- facilitating the use of math on the Web. reaching: To develop an open specification for math Many on the WG had considerable experience to be used with HTML that: with TEX, and could consider it as a natural para- Is suitable for teaching, and scientific commu- digm for a math language for the Web. IBM, • nication; drawing on its extensive experience with Scratchpad TM Is easy to learn and to edit by hand for basic and then Axiom , could have itself proposed a • math notation, such as arithmetic, polynomials language for math. Wolfram Research could offer in TM and rational functions, trigonometric expres- Mathematica a very rich language for expressing sions, univariate calculus, sequences and series, math in ASCII characters suitable for easy Web and simple matrices; transmission. But there are disadvantages to such foundations Is well suited to template and other math edit- for math on the Web as those just mentioned. A • ing techniques; specification to be generally used for math commu- Insofar as possible, allows conversion to and • nication should be publicly developed and not pro- from other math formats, both presentational prietary. Math is almost entirely treated as public and semantic, such as TEX and computer al- property: one cannot, in principle, patent mathe- gebra systems. Output formats may include matical facts. Agreement is needed from a broad graphical displays, speech synthesizers, comp- spectrum of interested parties that the language uter algebra systems input, other math layout provided is expressive enough for their purposes, languages such as T X, plain text displays (e.g. E for instance, from several symbolic algebra systems. VT100 emulators), and print media, including Finally, a specification is more likely to be deployed braille. It is recognized that conversion to and if those who will use it have been involved in its from other notational systems or media may development. It is also more likely to be realistic if lose information in the process; people who will implement it have contributed to its Allows the passing of information intended for • development. specific renderers; So the Math WG decided to fall in line with Supports efficient browsing for lengthy expres- the evolving standards of the Web. This was a • sions; decision with many implications for our later work, Provides for extensibility, for example through and not that easy to take. The primary goal was • contexts, macros, new rendering schemas or to create something that would be powerful, usable new symbols. Some extensions may necessitate and adopted. And this aim has continued to drive the use of new renderers. the WG throughout. The above goals were endorsed by an earlier Input Considerations group meeting in October 1996 at the Boston W3C. Plainly they have not all been accomplished yet, Next the WG set about making an XML application. but much progress has been made. Point 2, ease This had the unfortunate corollary that the markup of learning and hand editing, at least, cannot be developed would not directly meet the need for an achieved directly with a satisfactorily general and easy input syntax for math on Web pages, not even expressive markup language. for simple math. Many ideas were considered early on by the After heavy discussion, it was eventually real- WG. TEX is clearly important for communication of ized that the question of input styles was not one mathematics nowadays. Why not just extend HTML that could be solved initially. There are too many

Preprint: 1999 TEX Users Group Annual Meeting Monday, 16 August, 9.00 am 1009 Patrick D. F. Ion

different communities of users of mathematical for- The LATEX is certainly simple. The equation number mulas to satisfy them all. Making a lower-level lan- can be considered to have been provided by some guage, which input mechanisms could write, would extra-mathematical mechanism. The MathML cod- encourage development of tools which could be of ing of this display seems verbose by contrast: real help to the many not served by something as MathML Presentation coding for equation (1) complicated as TEX or LATEX sources. The keyboard input could be accepted by applications tailored to $$the\; needs\; of\; their\; user\; communities\; —\; high\; school,research\; scientists,\; computer\; algebra\; users.\; .\; .\; .\; Infact,\; if\; a\; symbolic\; algebra\; system,\; or,\; for\; instance,x^2-Microsoft\; Word\; aided\; by\; a\; template\; input\; system,can\; write\; out\; MathML\; markup\; for\; equations,\; then79their\; users\; do\; not\; have\; to\; learn\; a\; new\; input\; envi-ronment.xSeveral\; tools\; are\; already\; under\; development.The\; makers\; of\; Mathematica\; and\; Maple\; have\; pledged+their\; intentions\; to\; support\; MathML\; in\; due\; course,1601and\; Design\; Science\; has\; already\; announced\; thatits\; template\; editor\; MathType\; 4.0\; will\; write\; out=MathML\; [Topping,\; 1999].\; IBM’s\; techexplorer0[Sutor\; and\; Dooley,\; 1998]\; already\; accepts\; and\; writes$$ out some MathML in its currently released version. First note that the punctuational period is not en- Conversion of Legacy Documents coded here; this is intentional. The period, and its space away from the equation, should be part of the The WG chose a layered approach to facilitating overall document, not part of the math. MathML is math on the Web. Once an expressive transport for the math and not for the rest of the document; protocol is agreed upon then developers can set T X is a system that can handle both but then about making their own compatible tools. In partic- E tends to mix contexts, as here. T X is also mis- ular, one can make converters of legacy documents E used when math mode is employed for special non- into editions using MathML, especially converters of mathematical layouts, just as HTML table mecha- material encoded in TEX. Conversion will never be nisms are employed to do math. People should be completely automatic, but there are several efforts inventive, but there is more to the semantics of the underway to provide converters. For instance, two situation than just the displayed form13. talks here will discuss the problems [Gurarie and A mathematical expression is a enclosed within Rahtz, 1999; Lovell, 1999], and the American Math- top-level tagging with . Almost all the tag ematical Society, Society for Industrial and Applied

1010 Monday, 16 August, 9.00 am Preprint: 1999 TEX Users Group Annual Meeting MathML: a Key to Math on the Web

specification is easier to create. , and nus sign, and ⁢ again. To render mean math operator, identifier and number respec- and a routine has to do some tively; identifiers are the sort of thing conventionally geometry, adjusting lengths of lines to cover subex- set in math italic, variables and so on; exactly what pressions and providing an appropriate size for the numbers are could be a problem, but essentially we initial part of the square root sign. MathML provides are thinking of digit strings in some ordinary script. many schemas for placement of symbols in their provides a grouping construction, which al- conventional mathematical relationships; it even ex- lows an infix notation for operations; the operator tends TEX with, for instance, built-in constructions here is made explicit with the non-printing character of pre-super- and pre-sub-scripts. entity ⁢, which is useful for speech The entity ± suggests that a ren- rendering and line-breaking with continuation signs. derer will have to have a whole series of fonts avail- denotes a special element whose children will able to it containing representations of the charac- be treated differently: the second is a superscript to ters; and there will be many of them if the whole of the first. Of course, the parse tree (fragment), and mathematical usage is to be supported. thus the expression, is represented by the sequence The Math WG works with the STIX Project set of tokens read in the ordinary manner with conden- up by the STIPUB group of publishers to identify sation of the white space; the pretty-printing is for those characters in use in scientific publishing. It this exposition. is hoped that arrangements will be made to find All this markup provides enough information as places for them in Unicode [Consortium, 1996], and input to a screen renderer, or even to a composition Unicode Technical Committee is considering pro- system, that it should be able to present the equa- posals in this vein. Fonts, preferably publicly and tion correctly. There are many other elements with freely available, are then the next priority; STIPUB specialized functions. Let us look at the quadratic intends to support their creation, and other people, formula particularly Taco Hoekwater, are beginning to pro- b √b2 4ac duce the font already. x = − ± − (2) 2a which might be used to find roots of equation (1). Presentation and Content Markup MathML Presentation coding for equation (2) The last considerations of the previous section bring $$us\; back\; to\; the\; ideas\; mentioned\; at\; the\; start\; about\; thesignificance\; of\; signs\; and\; the\; semantics\; of\; formulas.x=So\; far\; we\; have\; seen\; Presentation\; markup\; concerned\frac{-blas.\; MathML,\; because\; of\; the\; strong\; interests\; of\; some\pm of\; the\; Math\; WG\; members\; in\; symbolic\; computation,\sqrt{attempts\; to\; provide\; a\; markup\; language\; in\; whichmore\; of\; the\; semantics\; of\; math\; can\; be\; expressed;\; this{b}^{2}is\; called\; Content\; markup.\; Thus\; the\; corresponding-Content\; markup\; for\; the\; two\; expressions\; above\; is4different\; (see\; the\; next\; column).aWe\; see\; that\; equation\; (1)\; as\; a\; whole\; has\; beencidentified\; as\; an\; equality\; relationship\; by\; the\; sur-rounding\; element()element.\; It\; is\; an\; equality\; between\; the\; result}}{number,\; identified\; as\; such\; by.\; The\; Content2number\; elementhas\; to\; be\; distinguished\; fromathe\mathrm{we\; have\; already\; met;\; assumptions\; about}()}matics\; processing\; systems.\; The\; type\; of\; functionbeing\; applied\; is\; shown\; by;\; it\; applies\; to\; a$$ sequence of elements, two s and a number. There we see new items: the fraction builder The functions applied are and , , the square root , and a new char- which are respectively binary and unary functions. acter entity ±, which is the plus or mi-

Preprint: 1999 TEX Users Group Annual Meeting Monday, 16 August, 9.00 am 1011 Patrick D. F. Ion

MathML Content coding for equation (1) MathML Content coding for equation (2)

$$\[ (=x(, \left(\frac{}{x(\pm , 2, (), , b(), (\times 79x()x()b\times 1601\times 2)(), 0, ())}\right)()))() \]4This\; alternative\; way\; of\; marking\; up\; the\; expressionais\; intended\; to\; assist\; computer\; algebra\; systems\; andcsearch\; engines\; looking\; for\; mathematical\; expressions\; given\; the\; semantics.\; Content\; markup\; does\; not\; nec-2essarily\; bring\; with\; it\; an\; immediately\; clear\; visually\; rendered\; form;\; or\; at\; least\; transformation\; rules\; need\; to\; be\; supplied\; to\; produce\; an\; appropriate\; Presenta-2abut\; it\; is\; naturally\; not\; that\; easy\; to\; combine\; the\; two.\; For\; the\; quadratic\; formula\; we\; have\; something$$ similar: a relationship of equality between results of a cascade of function applications (see the next Conclusion column). The entity ± occurs again, but this time made into a function label by being the Looking at these simple examples only scratches content of an element. Otherwise, the only new the surface of a markup language with about 150 items are function labels: , . primitive elements, and ten times that many iden- In an attempt to allow the capture of much of tified primitive character entities. The devil is in the semantics of elementary math, some 75 or so the details, as usual. Examples of formulas can be Content elements are provided in MathML. This discussed much faster speaking in front of overheads number may change during the revision of MathML than on the printed page. to version 2.0 which the present Math WG is under- The WG is issuing a corrected version, MathML taking. There remains discussion as to how best to 1.01, in July 1999, and will provide a major revision capture content semantics and how much to include. and extension, MathML 2.0 by January 2000. Better The watchword thus far has been what is known in integration with all the new standards from W3C the US as K–14 math education14. an exact range specified by any educational norms. The final 14 K–14 means two college years beyond Kindergarten level may correspond in Europe to matters covered in lyc´ee, through grade 12 in the US educational system. This is not Gymnasium or college, say.

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will be part of version 2.0, and extensibility issues will be further addressed. Cajori, Florian. A History of Mathematical Nota- In the meantime, the promising new develop- tion, 2 vol. Open Court Publishing, La Salle, ment is that the largest browser companies are be- Illinois, 1928/29. Reprinted (Cajori, 1993). ginning to implement MathML rendering in an es- sentially native way. The Math WG is also working Cajori, Florian. A History of Mathematical Nota- on providing a test suite to verify compliance with tion, 2 vol. Dover, New York, 1993. 1013 the specification issued, and to help those who build tools using MathML. There is a great deal of activity DSSSL. Document Style Semantics and of MathML development in process. Specification Language (DSSSL). ISO/IEC 10179:1996(E), 1996. Available at ftp: Acknowledgements //ftp.ornl.gov/pub/sgml/WG8/DSSSL/. The activity of a W3C WG is very much a collab- 1008 orative one. Aside from the numerous people who have expressed their views on how math should be Fielding, R., J. Gettys, J. Mogul, H. Frystyk on the Web, I would like very much to thank all Nielsen, T. Berners-Lee and others. “HTTP Ver- the members of the W3C Math WG — Stephen sion 1.1”. Technical report, Internet Engineering Buswell, David Carlisle, St´ephaneDalmas, Stan De- Task Force (IETF), 1998. 1007 vitt, Ben Hinkle, Angel Diaz, Sam Dooley, Stephen Hunt, Roger Hunter, Doug Lovell, Robert Miner, Godement, Roger. Analyse Math´ematique I. Barry MacKichan, Ivor Philips, Nico Poppelier, T. Springer, Berlin, Heidelberg and New York etc., V. Raman, Dave Raggett, Murray Sargent III, Neil 1998. Soiffer, Paul Topping, Stephen Watt, the current members, and the past members, Stephen Glim, Gurarie, Eitan and S. Rahtz. “LATEX to Brenda Hunt, Arnaud Le Hors, Bruce Smith, Robert XML/MathML (Workshop)”. Tugboat 20(3), ??– Sutor, Ron Whitney, Lauren Wood, Ka-Ping Yee, ??, 1999. 1010 Ralph Youngen, for the pleasure and stimulation of working with them. Together they have achieved Ion, Patrick D. F. and R. L. Miner, eds. something very worthwhile. “Mathematical Markup Language (MathML) In addition, we are all very grateful to Barbara 1.0 Specification”. Technical report, World Beeton for her stalwart efforts in trying to get the Wide Web Consortium, http://www.w3c.org/ characters of math into Unicode, and thus onto the TR/REC-MathML-19970430, 1997. The latest ver- Web. And I am grateful to the American Mathe- sion of MathML is available at http://www.w3c. matical Society for supporting my efforts here, and org/TR/REC-MathML. 1006 to the IHES for superlative conditions during a long visit there. ISO 8879:1986. Information Processing – Text and References Office Systems – Standard Generalized Markup Language (SGML). International Standards Organization, ftp://ftp.ornl.gov/pub/sgml/ B. Bos, H. Wium Lie, C. Lilley and I. Jacobs, WG8/DSSSL/, 1986. Consult http://www.iso. eds. “CSS, level 2 Recommendation”. Technical ch/cate/d16387.html for information about report, World Wide Web Consortium, http: the standard. 1007 //www.w3c.org/TR/REC-CSS2-19990512, 1998. The latest version of CSS2 is available at http: Lassila, O. and R. Swick, eds. “Resource De- //www.w3c.org/TR/REC-CSS2. 1007 scription Framework (RDF) Model and Syn- tax Specification”. Technical report, World Bos, B. and e. H. Wium Lie. “CSS, level Wide Web Consortium, http://www.w3c.org/ 1 Recommendation”. Technical report, World TR/REC-rdf-syntax-19990222, 1999. The lat- Wide Web Consortium, http://www.w3c.org/ est version of RDF is available at http://www. TR/REC-CSS1-19990111, 1999. The latest ver- w3c.org/TR/REC-rdf-syntax. 1008 sion of CSS1 is available at http://www.w3c. org/TR/REC-CSS1. 1007 Lovell, Doug. “TEXML brings TEX to Web Future ”. Tugboat 20(3), ??–??, 1999. 1010

Preprint: 1999 TEX Users Group Annual Meeting Monday, 16 August, 9.00 am 1013 Patrick D. F. Ion

Mollin, R. A. “Prime-producing quadratics”. Amer. T. Bray, J. Paoli and C. M. Sperberg-McQueen, Math. Monthly 104(6), 529–544, 1997. MR eds. “Extensible Markup Language (XML) 1.0.”. 98h:11113. 1998. The latest version of XML is available at http://www.w3c.org/TR/REC-xml. 1007 Peano, Giuseppe, 1858–1932. Formulaire des Math´ematiques, t. I–V. Bocca Fr`eres, Ch. Topping, Paul R. “Using MathType to Create TEX Clausen, Turin, 1894–1908. 1006 and MathML Equations”. Tugboat 20(3), ??–??, 1999. 1010 PICSTM. “Platform for Internet Content Selec- tion (PICSTM)”. http://www.w3c.org, 1997. Unicode. The Unicode Standard: Version 2.0. For more information see http://www.w3.org/ The Unicode Consortium, 1996. The specifi- PICS/. 1007 cation also takes into consideration the corri- genda at http://www.unicode.org/unicode/ Raggett, D., A. Le Hors and I. Jacobs, eds. uni2errata/bidi.htm. For more information, “HyperText Markup Language (HTML) 4.0 consult the Unicode Consortium’s home page at Specification”. Technical report, World Wide http://www.unicode.org/. 1011 Web Consortium, http://www.w3c.org/TR/ REC-html40-19980424, 1998. The latest version V. Apparao, S. Byrne et al., eds. “Document Object of HTML 4.0 is available at http://www.w3c. Model (DOM) Level 1 Specification”. Technical org/TR/REC-html40. 1007 report, World Wide Web Consortium, http:// www.w3c.org/TR/REC-DOM-Level-1-19981001, Raggett, D., ed. “HTML 3.2 Reference Specifica- 1998. The latest version of DOM1 is available at tion”. Technical report, World Wide Web Con- http://www.w3c.org/TR/REC-DOM-Level-1. sortium, http://www.w3c.org/TR/REC-html32, 1008 1997. The latest version, HTML 4.0, is available at http://www.w3c.org/TR/REC-html40. 1008 van Herwijnen, Eric. Practical SGML. Kluwer Aca- demic Publishers Group, Norwell and Dordrecht, Rudio, Ferdinand, 1856-1929., editor. Verhand- 1994. 1007 lungen des ersten internationalen Mathematiker- TM Kongresses in Z¨urichvom 9. bis 11. August 1897 W3C HTML Working Group. “XHTML 1.0: (Congr`es International des Math´ematiciens; The Extensible HyperText Markup Language; International Congress of Mathematicians), A Reformulation of HTML 4.0 in XML Leipzig. B. G. Teubner, 1898. 1006 1.0”. Technical report, World Wide Web Consortium, http://www.w3.org/TR/1999/ xhtml1-19990505, 1999. This is a Working Schr¨oder,(Friedrich Wilhelm Karl) Ernst, 1841- Draft; the latest version of XHTML is available 1902. “Uber¨ Pasigraphie, ihren gegenw¨artigen at http://www.w3c.org/TR/xhtml1. 1008 Zustand und die pasigraphische Bewegung in Italien”. In Verhandlungen des ersten interna- tionalen Mathematiker-Kongresses, pages 147– W3C Style. “Home page of the W3C Style Ac- tivity”. http://www.w3.org/Style/Activity, 162. 1898. 1006 1999. This includes XSL and CSS work. 1008 Stephen Deach, ed. “Extensible Stylesheet Lan- guage (XSL) Specification”. Technical report, World Wide Web Consortium, http://www. w3c.org/TR/1999/WD-xsl-19990421/, 1999. The latest version of the XSL Working Draft is available at http://www.w3c.org/TR/WD-xsl. 1007

Sutor, Robert S. and S. S. Dooley. “TeX and LATEX on the Web via IBM techexplorer”. Tug- boat 19(2), 157–161, 1998. 1010

1014 Monday, 16 August, 9.00 am Preprint: 1999 TEX Users Group Annual Meeting