Th e Elements and uncertainties in Heiberg’s edition 87
vii – ix from the so-called Ishâq–Th âbit version. 44 A second manuscript of
the commentary by an-Nayrîzî made it possible to complete the evidence
from the (mutilated) Codex Leidensis regarding the principles in Book i. 45
Several other commentaries (al-Mahânî, 46 a l - F a r â b î ,^47 Ibn al-Haytham,^48
al-Jayyâni,^49 ‘Umar al-Khayyâm^50 ) have also been edited, translated and
analysed. Th e wealth of materials since made available is exceptional. It
is obvious that the history of the text of the Elements during the Middle
Ages and perhaps even from the beginning of the Renaissance ought to be
entirely rewritten.
Th is is clearly not what I propose to do in the remainder of this chapter,
as this task surpasses my competence. I will adopt a more limited perspec-
tive and focus on more modest aims. What does this renewed knowledge
about the indirect tradition teach us about the history of the text in an-
tiquity, more particularly about the redaction of mathematical proofs?
What are the limits?
In so doing, I attempt to explore the consequences of the hypotheses
put forth by Knorr. In his striking 1996 study, knowing that I was in the
process of carrying out an annotated French translation (which was then
partially published), he suggested that I compare the Greek text established
by Heiberg with that of two Arabo-Latin translations, the fi rst attributed to
Adelard of Bath and the second ascribed to Gerard of Cremona, the former
composed around 1140, and the latter about 1180.
Knorr was convinced that these versions transmitted to us a text less
altered than the one contained in the Greek manuscripts. He believed that it
was possible to reconstitute a Greek archetype from the group of medieval
(^44) Engroff 1980; De Young 1981.
(^45) See Arnzen 2002. See also the new partial edition of the Latin translation by Gerard of Cremona,
initially published as vol. ix of EHM: Tummers 1994. Th e preserved Arab and Latin versions of
the text of an-Nayrîzî may be described as passably divergent. See Brentjes 2001b: 17–55.
(^46) Risâla li-al-Mâhânî fî al-mushkil min amr al-nisba (Épitre d’al-Mâhânî sur la diffi culté relative
à la question du rapport). Edition and French translation in Vahabzadeh 1997. Reprinted, with
English translation, in Vahabzadeh 2002: 31–52; Tafsîr al-maqâla al-‘âshira min kitâb Uqlîdis
(Explication du Dixième Livre de l’ouvrage d’Euclide). Edition and French translation in Ben
Miled 2005: 286–92.
(^47) Sharh al-mustaglaq min musâdarât al-maqâla al-ûlâ wa-l-hâmisa min Uqlîdis. Th e text was
translated into Hebrew by Moses ibn Tibbon. See Freudenthal 1988: 104–219.
(^48) Sharh musâdarât Uqlîdis. Partial edition, English translation and commentaries in Sude 1974.
(^49) Maqâla fi sharh al-nisba (Commentaire sur le rapport). Facsimile of manuscript Algier
1466/3, fos. 74r–82r and English translation in Plooij 1950. Edition and French translation in
Vahabzadeh 1997.
(^50) Risâla fî sharh mâ ashkala min musâdarât Kitâb Uqlîdis (Épitre sur les problèmes posés par
certaines prémisses problématiques du Livre d’Euclide). French translation in Djebbar 1997 and
2002: 79–136. Edition of Arabic text with French translation in Rashed and Vahabzadeh 1999:
271–390.