Th e Elements and uncertainties in Heiberg’s edition 107
When the various versions are considered, 97 the inversions are not the
result of happenstance in binding or in later inexpert replacement of lost
pages. As in our example, they leave practically all the deductive structure
intact and they even improve it. Of course, not all the examples are equally
simple, and the same principle clearly cannot be applied to the inversions
in the Defi nitions, for which it seems that a criterion, which I call ‘aesthetic’
for lack of anything better, has prevailed. Th e evidence is divided but at this
stage in my work, it seems to me that the preliminary conclusions about the
orders confl ict with what can be determined about the content. 98 N a m e l y ,
for problems regarding order, notably in Books v – x , the indirect tradition
received the greatest number of improvements!
Although changes in order may be limited, they are interesting because
they have an advantage with respect to the authenticity or alteration of
proofs. Such changes are hardly conducive to contamination. Admittedly,
we have several remarks by Th âbit ibn Qurra affi rming that he had found a
diff erent order of presentation in another manuscript, 99 but no one saw fi t
to reproduce the Propositions twice in each of the orders. In contrast, for
the problems of authenticity, the contamination between textual families
concerns the whole text, beginning particularly with the margins of the
manuscripts. As for the substitutions of proofs, we will see that they are the
cause, at least in part, of the phenomenon of double proofs.
From the substitution of proof to the phenomenon of double proofs:
the example of x .105
Th e Propositions (Heib.) x .66–70 and 103–107 establish that the twelve types
of irrational lines obtained through addition and subtraction distinguished
by Euclid are stable with respect to commensurability. In the Greek version,
97 Th ings are a little diff erent at the level of individual manuscripts which have not been
preserved though the accidents of transmission.
98 For example, in the Greek, the order of the Propositions (Heib.) vii .21–22 (each the converse
of the other) runs opposite to the order in medieval indirect tradition. Th e inversion has no
infl uence on the deductive structure, but the proof of (Heib.) vii .21 uses vii. 20. It is probable
this time that the inversion was made in the direct tradition, in order to make the two
connected deductive theorems consecutive.
99 For example, in Book vi which was just discussed. In (at least) three mss of the Ishâq–Th âbit
version, the following gloss appears aft er (Ishâq–Th âbit) vi .9 = (Heib.) vi .13. ‘Th âbit says: we
have found, in certain Greek manuscripts, in the place of this Proposition, that which we have
made the thirteenth.’ Undoubtedly, the existence of two distinct orders ought to be understood
as having been observed by the Editor among the Greek manuscripts which he consulted.
(Th us, the change is Greek in origin.) Th e editor retained the better order (which was that
already in al-Hajjâj). See Engroff 1980 : 29, who mentions two mss. Th e gloss also exists in ms
Tehran Malik 3586 (the oldest preserved copy of the Ishâq–Th âbit version), fo.75a. I thank
A. Djebbar for this information.