Text and Figure in Theodosius of Bithynia: An XML-Based Approach

The multilingual and multicultural transmission of ancient Greek mathematics to modern Europe is well illustrated by the case of Theodosius of Bithynia: in the Byzantine east, his work on Spherics was copied in Greek; in the Islamic world, it was translated into and commented on in Arabic; the first Latin translations in Europe were made from Arabic sources in the twelfth century, and were the basis for the first printed edition of 1529; before the end of the sixteenth century, new Latin translations had been published based directly on Arabic and Greek sources, and the first Greek edition was printed in 1558. In this paper I will present work on a corpus of texts representing key moments in the transmission and interpretation of Theodosius's Spherics (including a new English translation that is the first directly from Greek).

Textual content of manuscripts and printed editions is represented with TEI-conformant XML. The markup uses the canonical reference system familiar from today's printed editions of Theodosius's work, and uniquely identifies named entities across versions, so that versions can be readily aligned and compared. The TEI texts are made accessible to the internet through the TextServer protocol recently developed by a team of classicists for working with corpora of TEI-conformant texts (http://shot.holycross.edu/projects/TextServer). Applications using this protocol allow users to see the automatically generated equivalent of an apparatus criticus showing differences among versions in a single language; they also allow users to compare passages by canonical reference across languages, or search for named entities across versions regardless of language. Other applications depending on a multilingual corpus of aligned texts could easily be developed as well.

The textual content is only one part of a Greek mathematical text, however: our manuscript traditions and printed editions uniformly include figures, too. Traditions of visual presentation may vary, but all our sources agree that figures with labelled elements referred to in the text are an essential part of a formal proof. Modern editors may on occasion make this point explicitly -- in Renaissance Latin translations, the figures can even be called “demonstrationes,” that is, “proofs” -- but in all cases the claim is implicit in the organization and form of the text.

I have therefore developed an XML DTD for describing the semantic content of geometric figures in terms of their logical structure, and the properties of the geometric elements in them. I argue that this is analogous to the principle of semantic markup of texts: the particular visual form of a figure can be considered a presentational variant comparable to the page layout, selection of type face, and other presentational choices made in the copying or editing of a specific version of a text.

The DTD organizes figures in the same scheme used in the text's canonical reference system: “proposition 1 of book 1 of the Spherics” represents the same notion in the TEI XML of the text, and in the XML for the figure. In addition, the DTD identifies specific elements with the same identifiers used in the text. Just as these identifiers make unambiguous in the text that, for example, “line AB” and “line BA” refer to the same element, they also make explicit that those elements correspond to a specific element in the XML description of the figure. Thus, computational manipulation or analysis of the text can be associated with the figure, and vice versa.

Combining a logical representation of the figures with a TEI XML representation of the text can open new questions about Greek mathematical texts to computational analysis, which I will illustrate with several examples, implemented as XSLT transformations within a Cocoon pipeline. One simple transformation of the figure represents it with a series of English statements (rather than graphically). This output can be paralleled with the corresponding text: the logic of textual presentation can be directly confronted with the logic of the figure. Other transformations create graphics from the logical representation of the figure. (At present, I create SVG graphics that mimic the two-dimensional conventions of Greek manuscript figures; I would like to develop other visualizations of the figures' logic, including OpenGL models in three dimensions for Theodosius's spherical elements.) Corresponding elements in the text and diagram can be highlighted similarly (e.g., with the same color) to make clear their relation. This view recalls the famous edition of Euclid by Oliver Byrne (see page images online at http://www.math.ubc.ca/people/faculty/cass/Euclid/euclid.html), but unlike Byrne's printed text, can be walked through one step at a time.

I believe that these transformations will have obvious pedagogical benefits in helping readers understand the relation of text and figure, as Byrne aimed at doing for Euclid.

At the same time, they suggest new approaches to scholarly questions about the transmission of Theodosius's works. Renaissance translations of Theodosius boast about their “new and improved” diagrams, when they introduce new features like three-dimensional perspective views, or multi-step views of a construction: the Latin translators literally see themselves as translators of the figures as well as the text. With a formal representation of the figure's logic, it should be possible to test automatically: 1) whether the editor's figure accurately reflects the semantics of the text 2) whether the semantics of two different versions' figures are equivalent or not, independently of whether their visual appearance is identical

A systematic, computational approach to these issues should further clarify the relation of text and figure in the textual tradition of Theodosius, and illustrate a method that could be applied to ancient mathematical and scientific texts more broadly.