Diagrams in manuscripts



Last update December 14, 2014

From this page you can download diagrams of several Greek mathematical works reproduced from the manuscripts: Euclid's Elements, Books 2-4, 6, 11-13, Phaenomena, Optics, as well Arabic translation of Menelaus' Spherics.

Elements III.13: diagram in Heiberg's edition and in codex P

In proposition III.13 of the Elements, Euclid proves (by contradiction) that two circles cannot touch at two points either internally or externally. All the modern editions have a diagram similar to the one in the left, which appears in Heiberg's edition. However, the diagram found in the manuscripts is like the one in the right, which is a faithful reproduction from Vatican manuscript of the ninth century. Here, the second circle ΑΚΓ touching externally at two points to the first circle ΑΒΓΔ is represented by a lunule, which Heiberg (more precisely August, whose diagrams were copied by Heiberg) has changed into a circle.

Reproduced Diagrams from Greek and Arabic manuscripts (February 28, 2013, last update December 14, 2014)


Nathan Sidoli and Chengxin Li The manuscript diagrams of al-Harawī's version of Menelaus' Spherics
Ken Saito The Greek manuscript diagrams of the Elements (Books VI, XI, XII, XIII)



Corrections in Book XII (pp. 141-161) on December 14, 2014.

Correction in page 71 on April 27, 2013.

Download page 71 only

Download Book XII (141-161) only

Diagrams in Greek Mathematical Texts (March 31, 2008. Latest update April 3, 2011)


1. Gregg De Young DRaFT Software --- A User Report (March 2008)
2. Takanori Suzuki The Diagrams of the Phaenomena in Greek and Arabic Manuscripts (March 2008, last update May 2008)
3. Ken Saito The Diagrams of Book II and III of the Elements in Greek Manuscripts (March 2008, last update April 2011)
4. Ken'ichi Takahashi Diagrams in Euclid's Optica --- A Preliminary Examination of Its Various Texts (March 2008, last update May 2008)
5. Ken Saito The Diagrams of Book IV of the Elements in Greek Manuscripts (new in May 2008 version, last update November 2008)
6. Ken Saito The Diagrams of Book VI of the Elements in two Greek Manuscripts newer version exists (new in December 2008 version)
For the diagrams of the Elements Book VI, see the newer report above: "Reproduced Diagrams from Greek and Arabic manuscriptsReproduced Diagrams"


DOWNLOAD PDF File of the latest version 2.03 of Apr. 3, 2011

Changes in ver. 2.03
The diagrams of proposition III.36 in codex F (article 2) has been revised.

Older versions

Diagrams: older versions

The program DRaFT

Most of the diagrams in the report is drawn by the program DRaFT developed in this project, which can be downloaded from this site. Download DRaFT

This research was supported by KAKENHI, or Grants-in-Aid for Scientific Research by Japan Society for the Promotion of Science (JSPS)
Category B, ID 17300287, April 2005 - March 2008
Category B, ID 21300325, April 2009 - March 2013

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