# diagrams

Keywords: diagrams, Euclid, geometry, mathematical papyri, ancient mathematics

Illustrations were extremely rare in ancient literary texts. They were only occasionally used in medical texts, principally Apollonius of Citium, Dioscorides, and perhaps Soranus; references survive to illustrations in lost works on biology by Aristotle.^{1} Illustrations, or diagrams, were mandatory in the exact sciences—the unique genre of illustrated text in antiquity. Such diagrams were formed by a network of straight and curved lines (certainly drawn with ruler and perhaps by compass as well; the few extant arcs on papyri are drawn freehand). In the extant literature, diagrams are always labelled by letters of the alphabet, standing typically at the intersection points of the lines. The diagrams are crucial to the logical development of the text and encode some of the information the text takes for granted, in a nonverbal way. At the same time, diagrams are drawn schematically so that the apparent metrical relations of the diagram are not meant to represent the metrical relations of the object studied. Thus, diagrams encode topological rather than metrical properties. The foundational study of these attributes of diagrams is Reviel Netz’s study of 1999.^{2}

The diagrams of ancient mathematical texts survive primarily as medieval copies, though a few survive on papyrus. Many of the surviving mathematical papyri pertain to the “scribal” traditions of practical calculation rather than the “academic” geometrical tradition; hence numerical and calculating tables are better represented among the visual elements of mathematical papyri than geometrical diagrams (Figure 1).

Surviving mathematical papyri with tables and other images include a set of division tables (P. Cair. [inv.] 10758), a collection of stereometric problems with diagrams (P. Vindob. [inv.] 19996), and four papyri of Euclid’s *Elements* (P. Herc. 1061, P. Oxy. I 29, P. Fay. 9, P. Berol [inv.] 17469). Worp et al. give additional details on some papyri in the Vienna collection with geometrical diagrams.^{3}

Medieval copies of mathematical diagrams survive in much greater numbers, and many works exhibit considerable variations in the transmission of their diagrams. Notable variations include the relative lengths and positions of elements in the diagram, changes in perspective techniques, different orientations of the entire diagram, and different relative positions of diagram and text (Figure 2). The major editions of the late 19th and early 20th centuries largely ignored the variations in manuscript evidence for diagrams, but more recent work insists on the need to develop critical editions of diagrams capable of tracking their variations. Saito and Sidoli provide a brief history of such editions alongside a detailed account of the most important variant characteristics of manuscript diagrams.^{4}

These characteristics include “overspecification” (cases where the diagram is drawn with more regularity than the proof’s generality suggests), metrical inaccuracy, the wide variety of strategies for representing three-dimensional objects, and the use of a single diagram for multiple cases of a proposition. Saito’s work on Greek geometrical diagrams includes an extensive suite of examples and discussion, as well as offering the “DRaFT” program developed to improve the fidelity between diagrams in editions of those texts and their manuscript originals.

## Links to Digital Materials

fol. 325v Elementa Book XII, Proposition 12, on relationship of cones and cylinders with diagrams of their bases.

fol. 46r Euclid, Elements; written by Stephanus Clericus in 888, and owned by Arethas of Patras. Two diagrams of circles.

Cod. Par. gr. 2347 includes Euclid’s Elements alongside other works attributed to him; the left-hand margin contains the diagram for the construction of Elements Prop. I.1 and the right-hand margin Prop. I.2. Bibliothèque nationale de France, Département des manuscrits.

Cod. Par. gr. 2466: The lower margin contains diagrams for the construction of Euclid’s *Elements* Prop. I.47, the “Pythagorean Theorem,” in different hands. Bibliothèque nationale de France, Département des Manuscrits.

Ken Saito’s work on Greek geometrical diagrams, including the software he developed for reproducing manuscript illustrations, is available at Greek Math.org.

## Bibliography

Catton, Philip, and Clemency Montelle. “To Diagram, to Demonstrate: To Do, To See, and To Judge in Greek Geometry.” *Philosophia Mathematica* 20.1 (February 1, 2012): 25–57.Find this resource:

Chemla, Karine, ed. *The History of Mathematical Proof in Ancient Traditions*. Cambridge, U.K.: Cambridge University Press, 2012.Find this resource:

Fowler, David. “Further Arithmetical Tables.” *Zeitschrift Für Papyrologie Und Epigraphik* 105 (1995): 225–228.Find this resource:

Fowler, David. *The Mathematics of Plato’s Academy: A New Reconstruction*. 2d ed. Oxford and New York: Clarendon, 1999.Find this resource:

Jones, Alexander. “Mathematics, Science, and Medicine in the Papyri.” In *The Oxford Handbook of Papyrology*. Edited by Roger S. Bagnall, 338–357. Oxford and New York: Oxford University Press, 2009.Find this resource:

Kadunz, Gert. “Schrift Und Diagramm: Mittel Beim Lernen von Mathematik.” *Journal Für Mathematik-Didaktik* 27.3–4 (2006): 220–239.Find this resource:

Mau, Jürgen, and Wolfgang Müller. “Mathematische Ostraka Aus Der Berliner Sammlung.” *Archiv Für Papyrusforschung Und Verwandte Gebiete* 1960.17 (2009): 1–10.Find this resource:

Netz, Reviel. *The Shaping of Deduction in Greek Mathematics: A Study in Cognitive History*. New York: Cambridge University Press, 1999.Find this resource:

Saito, Ken. “Reading Greek Mathematics.” In *The Oxford Handbook of the History of Mathematics*. Edited by Eleanor Robson and Jacqueline A. Stedall, 801–826. Oxford and New York: Oxford University Press, 2009.Find this resource:

Saito, Ken. “Traditions of the Diagram, Tradition of the Text: A Case Study.” *Synthese* 186.1 (May 2012): 7–20.Find this resource:

Sidoli, Nathan, and Ken Saito. “Diagrams and Arguments in Ancient Greek Mathematics: Lessons Drawn from Comparisons of the Manuscript Diagrams with Those in Modern Critical Editions.” In *The History of Mathematical Proof in Ancient Traditions*. Edited by Karine Chemla, 135–162. Cambridge, U.K.: Cambridge University Press, 2012.Find this resource:

Sidoli, Nathan. “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 (October 2007): 525–547.Find this resource:

Stückelberger, Alfred. *Bild und Wort: Das Illustrierte Fachbuch in der Antiken Naturwissenschaft, Medizin und Technik*. Mainz, Germany: P. von Zabern, 1994.Find this resource:

van Leeuwen, Joyce. “Thinking and Learning from Diagrams in the Aristotelian Mechanics.” *Nuncius* 29.1 (January 1, 2014): 53–87.Find this resource:

*Visualization, Explanation and Reasoning Styles in Mathematics*. Dordrecht: Springer, 2005.Find this resource:

Weitzmann, Kurt. *Illustrations in Roll and Codex: A Study of the Origin and Method of Text Illustration*. Princeton, NJ: Princeton University Press, 1947.Find this resource:

Worp, K. A., E. M. Bruins, and P. J. Sijpesteijn. “A Greek Mathematical Papyrus.” *Janus* 61 (1974): 297–312.Find this resource:

Worp, K. A., E. M. Bruins, and P. J. Sijpesteijn. “Fragments of Mathematics on Papyrus.” *Chronique d’Égypte* 52 (1977): 105–111.Find this resource:

## Notes:

(1.)
See Alfred Stückelberger, *Bild und Wort: Das Illustrierte Fachbuch in der Antiken Naturwissenschaft, Medizin und Technik* (Mainz, Germany: P. von Zabern, 1994), for an illustrated review of the different traditions of technical images from antiquity.

(2.)
Reviel Netz, *The Shaping of Deduction in Greek Mathematics: A Study in Cognitive History* (New York: Cambridge University Press, 1999).

(3.)
For the context of mathematical papyri in practice and details of the surviving materials, see David Fowler, *The Mathematics of Plato’s Academy: A New Reconstruction*, 2d ed. (Oxford and New York: Clarendon, 1999), 204–217, 229–278; David Fowler, “Further Arithmetical Tables,” *Zeitschrift Für Papyrologie Und Epigraphik* 105 (1995): 225–228; and Alexander Jones, “Mathematics, Science, and Medicine in the Papyri,” in *The Oxford Handbook of Papyrology*, ed. Roger S. Bagnall (Oxford and New York: Oxford University Press, 2009), 342–343. For details on mathematical ostraka, see Mau Jürgen and Wolfgang Müller, “Mathematische Ostraka Aus Der Berliner Sammlung,” *Archiv Für Papyrusforschung Und Verwandte Gebiete* 1960.17 (2009): 1–10. See also K. A. Worp et al., “A Greek Mathematical Papyrus,” *Janus* 61 (1974): 297–312, and K. A. Worp et al., “Fragments of Mathematics on Papyrus,” *Chronique d’Égypte* 52 (1977): 105–111.

(4.)
Nathan Sidoli and Ken Saito, “Diagrams and Arguments in Ancient Greek Mathematics: Lessons Drawn from Comparisons of the Manuscript Diagrams with Those in Modern Critical Editions,” in *The History of Mathematical Proof in Ancient Traditions*, ed. Karine Chemla (Cambridge, U.K.: Cambridge University Press, 2012).