# Diagrams and traces of oral teaching in Euclid’s *Elements*: labels and references

- 25 Downloads

## Abstract

The analysis of a historical mathematical textbook constitutes a particular challenge when the original had been composed before the invention of the printing press and an even greater one when the very original might even not have been fixed as a manuscript. The analysis of such cases affords a focus on the relation between orality and written text. A particular clue for this relation is given by the diagrams accompanying the text. Diagrams in mediaeval manuscripts have several particularities which have been eliminated in modern editions as inappropriate for mathematical arguments. However, they may have not been so strange in the context of the direct oral teaching in which the teacher drew the diagrams in front of the pupils. The text and the structure of the Elements also contain other particular features preserved also in modern editions of which the following ones are examined: (1) no use of proposition numbers, (2) label assignment in alphabetical order which may result in different assignments in similar propositions, (3) oscillation of the appellation of the objects expressed by two or more letters such as line AB and line BA, (4) long and sometimes incomprehensible protasis (general enunciation at the beginning of each proposition). All these particularities can be interpreted as traces of oral teaching and communication of mathematics in ancient Greece.

## Keywords

Euclid Elements Diagram Oral teaching## Notes

### Acknowledgements

The research on diagrams was supported by Japan Society for the Promotion of Science (Grant nos. JP17300287, JP21300325). The author would like to express his gratitude to Henry Mendell for linguistic corrections and precious suggestions concerning interpretations of documents.

## References

- Asper, M. (2008). The two cultures of mathematics in ancient Greece. In E. Robson & J. Stedall (Eds.),
*Oxford handbook of the history of mathematics*(pp. 107–132). Oxford: Oxford University Press.Google Scholar - August, E. F. (1826–1829).
*Euclidis Elementa ex optimis**libris in usum tinorum*(Vols. 1–2). Berlin.Google Scholar - Cairncross, A., & Henry, W. B. (2015). Euclid, Elements I.4 (diagram), 8–11, 14–25 (without proofs).
*The Oxyrhyunchus Papyri, LXXXII*(*Graeco*-*Roman memoires, No. 103*), 23–38.Google Scholar - Commandino, F. (1572). Euclidis elementorum libri XV, una cum scholiis antiquis. A Federico Commandino Urbinate nuper in latinum conversi, commentariisque quibusdam illustrati. Pisauri
**(Pisauri is locative of Pisaurum (Pesaro))**.Google Scholar - Fowler, D. (1999).
*The mathematics of Plato’s Academy: A new reconstruction*(2nd edn.). Oxford: Oxford University Press.Google Scholar - Frajese, A. (1970).
*Gli Elementi di Euclide. A cura di A. Frajese e L. Maccioni*. Torino: UTET.Google Scholar - Gregory, D. (1703). Euclidis quae supersunt omnia ex recensione Davidis Gregorii. Oxoniae.Google Scholar
- Heath, T. L. (1925).
*The thirteen books of the Elements. 3 Vols*. (2nd edn.) New York: Dover Publications**(Cambridge. Reprint, 1956)**.Google Scholar - Heiberg, J. L. (1883–1916).
*Euclides. Euclidis opera omnia. 8 vols. and a supplement*. Leipzig: Teubner.Google Scholar - Heiberg, J. L., & Stamatis, E. S. (1969–1977).
*Euclides. Euclidis Elementa*(Vols. 1–5). Leipzig: Teubner.Google Scholar - Mogenet, J., & Tihon, A. (1985).
*Le Grand Commentaire de Théon d’Alexandrie aux tables faciles de Ptolémée. Livre I. Histoire du texte, édition critique, traduction. Revues et complétées par Anne Tihon, commentaire par Anne Tihon (Studi e Testi 315)*. Città del Vaticano: Biblioteca Apostolica Vaticana.Google Scholar - Mueller, I. (1981).
*Philosophy of mathematics and deductive structure in Euclid’s Elements*. Cambridge, MA: The MIT Press.Google Scholar - Netz, R. (1999).
*The shaping of deduction in Greek mathematics*. Cambridge: Cambridge University Press.CrossRefGoogle Scholar - Saito, K. (2006). A preliminary study in the critical assessment of diagrams in Greek mathematical works.
*SCIAMVS, 7*, 81–144.Google Scholar - Saito, K. (2018).
*Diagrams in arithmetical**books of Euclid’s Elements*, SCIAMVS, 19**(forthcoming)**.Google Scholar - Saito, K., & Sidoli, N. (2012). Diagrams and arguments in ancient Greek mathematics: Lessons drawn from comparison of the manuscript diagrams with those in modern critical editions. In K. Chemla (Ed.),
*History of mathematical proof in ancient traditions*(pp. 135–162). Cambridge: Cambridge University Press.Google Scholar - Thomas, R. (1989).
*Oral tradition written record in classical Athens*. Cambridge: Cambridge University Press.CrossRefGoogle Scholar