Th e Elements and uncertainties in Heiberg’s edition 121
that this passage contains some rhetorical exaggerations or stock
phrases about the improvement of a text. If the quest for conciseness
seems hardly debatable, the preface indicates neither the motivations
for the suppressions nor the criteria used to identify the ‘additions’.
It is conceivable that al-Hajjâj knew of other Greek versions, more
concise than the text or texts initially translated, to which the phe-
nomenon of the epitomization had itself already been applied.
(11) We know that at least one abridged version of the Elements had been
produced in antiquity by Aigeias of Hierapolis. Mentioned by Proclus,
he wrote therefore no later than the fi ft h century of the modern era.
Th e diff erence with the second version of al-Hajjâj is that there is no
evidence that it played a role in the transmission of the text. However,
besides the obvious textual enrichment, it is not possible to completely
exclude the intervention of one or several abridged Greek versions.
Th e relative ‘thinness’ of the al-Hajjâj version, as far as can be
known, can indeed be explained in diff erent ways depending on
the portion of text considered. Proposition ii .14, which treats the
quadrature of the triangle (with the associated absence of i .45), and
Propositions xii .5, 7 and 8, which treat pyramids on a triangular base,
proceed from the same attitude, and, in these cases, there are good
reasons to think that the origin of this minimalist treatment has a
Greek origin. 143 For the absence of Proposition iii .37 I have noted that
it was probably an accident of transmission. Th e absence of the bulk
of the additional material, of several Defi nitions in Books v , vi , vii
and xi and of the Porisms in the stereometric books may perhaps be
explained because al-Hajjâj had identifi ed them as additions. Similarly,
several other Propositions missing from his version ( vi .12, viii .11a–
12a, x .16, x .27–28), but present in the Ishâq–Th âbit translation, might
be the result of additions lacking from the Greek or Syriac manuscripts
consulted by al-Hajjâj, or they might have possessed these assertions,
but he judged them to be useless, as they very nearly are.
(12) Th e existence of abridged versions in Greek also made up part of the
hypothesis of Heiberg, and he described the model of manuscript
b in this way for its divergent part ( xi .36– xii .17). 144 M a n u s c r i p t b
is, however, very fl awed. It contains problems in the lettering of the
143 Let us recall that Proposition xii .6 and the Porisms to xii .7 and 8 are missing in manuscript
b. For ii .14, Simplicius seemingly knew two versions of the theorem: the ‘rectangular’ version
in his commentary to the Physics of Aristotle ( CAG , 62. 8 Diels) and the ‘triangular’ version in
his commentary to De cælo : ( CAG , ed. 414.1 Heiberg).
144 See above, pp. 81–2.