Comparing diagrams in Euclid’s Elements

Summary
This poster aims to introduce a new research project on the potential of automated collation for non-textual data such as mathematical diagrams, focusing on the case of Euclid’s
Elements.

Netz (1999) argues that diagrams are crucial to Greek mathematics and necessary to reading the text, but he notes that this fact was little discussed in modern literature. In recent years, however, there has been a growing interest in including diagrams and their manuscript evidence in the preparation of scholarly editions.

The use of diagrams in scholarly editing
Saito (2006), Sidoli (2007), De Young (2009) and Roughan (2016) have all advocated for critically editing the diagrams which accompany ancient mathematical texts. Until now, the editorial practice was often confined to publishing diagrams which follow modern standards rather than manuscript evidence (De Young 2009), as well as trying to benefit the reader (Sidoli 2007). Modern editors print arbitrary parallelograms and triangles, whereas manuscripts have rectangles and isosceles triangles even when the text does not require it. The critical apparatus does not usually offer information regarding the state of diagrams in the manuscripts (Netz 1999, Saito 2012). Netz (2004) has attempted to provide a kind of critical apparatus for diagrams in Archimedes’ work, but Roughan (2016) remarks that this approach is limited for printed edition due to the amount of space it consumes.
The study of diagrams can broaden our knowledge of textual transmission by highlighting elements absent from the text: Raynaud (2014), for instance, has shown how the tradition of diagrams can help create a stemma codicum of Ibn al-Haytham’s
Epistle on the Shape of the Eclipse, an Arabic mathematical treatise from the eleventh century. The diagrams were analysed with regard to stated or unstated characters: unstated characters are aspects of a diagram not described in the text, and therefore much more difficult to correct. Raynaud (2014) concluded that diagrams had a higher concentration of errors and were therefore particularly promising to study manuscript traditions, especially rich and complex traditions.

As a result, automated collation of diagrams may be particularly useful for Euclid’s
Elements. A broad survey of the
Elements witnesses by Roughan (2016) has recorded 477 manuscripts in Latin, Greek and Arabic.

Preliminary to automated collation
Automated textual collation is divided into a series of steps: the first is to split the text into smaller units, called tokens, which are then compared (Dekker et al. 2015). How can diagrams be tokenized for the purpose of automated collation? Scholars have provided lists of a diagram’s features: ‘the features which usually define a technical drawing are its basic components: points, segments, curves, colored surfaces, and various labels’ (Roegel 2015: 1). Sidoli and Li (2013) have divided features between mathematical (labeling, ordering of points, positioning of lines, and differentiation into cases) and visual features (general layout or orientation, direction of curvature, and the presence of extra, unnecessary arcs). A potential implementation of automated collation could make use of CollateX’s flexible input, which permits to encode a reading - here a diagram - as a set of features. Simplified examples of features could include “labels:ABCD”, “labels:none”, “orientation:left” or “orientation:right”.
However, as soon as diagrams become more complex, the number of potential features increases: not only point, lines, and angles but also multiple surfaces would make automated collation more cumbersome. To attempt identifying relevant patterns of differences in diagrams and thus reduce the number of features to be compared, this project proposes to create a database of annotated diagrams with the tool Archetype

https://archetype.ink/
(Accessed November 26, 2018). Archetype is the framework underlying the Digipal project (
http://digipal.eu, accessed 20 April 2019). Although Archetype was first developed to study palaeography, it has a rich annotation environment where a figure can be decomposed into a series of component with particular features.

. Fifty-nine manuscripts freely available online and compiled by Roughan (2016) form the basis of the data sample. The scope of the text would be at first limited to the first two books, for which there is a higher number of manuscript coverage (Roughan 2016). The database currently holds over 600 isolated diagrams from book 1 of the
Elements, including some manuscripts very important to the tradition, such as Vat.gr.190 (the earliest witness closest to Euclid’s original text) or
Bodleian Library MS. D'Orville 301 (the earliest witness to the text as edited by Theon of Alexandria in the 4th century AD).

Future directions for this work could include identification of accidental versus substantial features in diagrams, or the relationship between text and diagrams.

Bibliography

De Young, G. (2009). Diagrams in ancient Egyptian geometry. Survey and assessment.
Historia Mathematica,
36(4): 321–73.

Dekker, R. H., Hulle, D. van, Middell, G., Neyt, V. and Zundert, J. van (2015). Computer-supported collation of modern manuscripts: CollateX and the Beckett Digital Manuscript Project.
Literary and Linguistic Computing,
30(3): 452–70.

Netz, Reviel (1999).
Netz, Reviel. 1999. The Shaping of Deduction in Greek Mathematics: A Study in Cognitive History. Cambridge: Cambridge University Press.

Netz, Reviel (2014).
The Works of Archimedes: Translation and Commentary. Vol. I. Cambridge: Cambridge University Press.

Raynaud, D. (2014). Building the stemma codicum from geometric diagrams.
Archive for History of Exact Sciences,
68(2): 207–39.

Roegel, D. (2015). Editing ancient technical and mathematical figures: Tools and traps.
Research Report from LORIA - Université de Lorraine,
https://hal.inria.fr/hal-01198446 (accessed 26 November, 2018).

Roughan, C. (2016). Digital Texts and Diagrams: Representing the Transmission of Euclid’s Elements.
Digital Classics Online,
2: 32–48.

Saito, K. (2006). A preliminary study in the critical assessment of diagrams in Greek mathematical works.
Sciamvs,
7: 81–144.

Saito, K. (2012). Traditions of the diagram, tradition of the text: A case study.
Synthese,
186(1): 7–20.

Sidoli, N. (2007). What We Can Learn from a Diagram: The Case of Aristarchus’s
On The Sizes and Distances of the Sun and Moon.
Annals of Science,
64(4): 525–47.

Sidoli, N. and Li, C. (2013). The manuscript diagrams of al-Harawī ’ s version of Menelaus ’ Spherics..

Sidoli, N. and Li, C. (2013). The manuscript diagrams of al-Harawī’s version of Menelaus’ Spherics. In Saito, K. (ed),
Reproduced Diagrams from Greek and Arabic Manuscripts. Research Report “Databasing the Manuscript Diagrams of Sources in Ancient and Medieval Mathematics”. Japan Society for the Promotion of Science, pp. 1–68.