Th e Elements and uncertainties in Heiberg’s edition 115
tradition has the benefi t of many more improvements, and the Greek
tradition seems to have been very conservative in this area.
(3) Furthermore, the conclusions drawn from the results of the compari-
son of versions change according to the book or group of books being
studied. For example, interaction between Euclid and the Nicomachean
tradition has had an impact on the text of the arithmetical Books.
If x .68–70 and 105–107 and xiii .1–5 are judged by the criteria of
improvement, the medieval versions (particular Adelard’s) are more
sophisticated than the Greek text, at least as far as the contents are
concerned. At the end of Book iii (and perhaps also in response to an
initiative by Heron), the medieval versions are also more sophisticated
with regard to the material contents, 118 although the opposite is much
more frequent.
Along the same lines, the mathematically insuffi cient proofs (accord-
ing to the criteria of the ancients) in the Elements are four in number if
the direct and indirect traditions are combined: viii .22–23, ix .19 and
xii .17. If, as Knorr argues, we assume the errors are from Euclid and
not textual corruptions, we arrive then, by applying his criteria, at the
following conclusions:
- For viii .22–23, the original proofs are those common to both the
Greek and to the Hajjajian tradition; the proofs presented by the
Ishâq–Th âbit version are improvements. 119 - For ix .19, the original proof is that of manuscript P ; those of Th and
of the indirect tradition are improvements. - For xii .17, the original proof is that of the indirect tradition; those of
b as well as of P and Th are improvements. 120
Th e type of statements must also be taken into account. Th e
Defi nitions occupy a privileged place in philosophical exegesis. Th e
Porisms are particularly prone to the vagaries of transmission because
they may easily be confused with additions. 121
118 Th ere is the addition of the case of fi gures in the Propositions (Heib.) iii .25, 33, 35, 36;
iv .5. Th e copyists ascribe them to the version of al-Hajjâj, and even to his second version if
al-Karâbîsî is to be believed. See Brentjes 2000 : 48, 50. Other cases are also added in iii .37
without al-Hajjâj being mentioned.
119 See De Young 1991 : 657–9.
120 For my part, contrary to Knorr, I believe that the criterion of improvement does not apply for
ix .19 or xii .17. I also believe that the proofs of P in one case and the proofs of the indirect
tradition in the other are corrupt. For ix .19, see Vitrac 2004 : 10–12. For xii .17, see Euclid/
Vitrac 2001: iv 369–71.
121 Heiberg 1884 : 20 observed that with the Defi nitions and Corollaries (Porisms) ‘die Araber ...
sehr frei verfahren haben’. In fact, it is not even simple to say exactly how many Porisms
there are in the Greek text. Heiberg identifi es 30 of them as such but makes a second Porism