110 bernard vitrac
If the principle of improvement advanced by Knorr is applied, we are led
to think that the linear proofs of the Greek are authentic, with the superfi -
cial proofs clearly being ameliorations from a mathematical point of view.
Th is attempt at strengthening the deductive structure and generalizing
was begun in Greek, as demonstrated by the proofs aliter to x .105–106. It
is likely that there was also a proof aliter to x .107 which has disappeared.
Th e opposite hardly makes any sense. Its disappearance is probably due to
codicological reasons.
However, the question of knowing who produced the alternative proofs
for the Propositions of the fi rst triad remains unanswered. A likely hypoth-
esis is that the same editor is responsible for the parallel modifi cation of the
two triads and he happened to be a Greek. But it could also be imagined that
it was a contribution from the indirect tradition, occurring as the result of an
initiative by al-Hajjâj. Th is latter explanation is the interpretation of Gregg
De Young. 105 Th e examples of at-Tûsî and pseudo-Tûsî show that improve-
ments continued into the medieval tradition, but it should not be forgotten
that these were authors of recensions, not translators. As for the structure
for the Ishâq–Th âbit version, it may be explained in diff erent ways – either
by the existence of a Greek model combining the two approaches or by an
attempt at compromise on the part of the editor Th âbit. In the fi rst case, there
would have been at least three diff erent states of the text. In the second case,
Th âbit would have combined the fi rst (linear) triad from the translation of
Ishâq (considered closer to the Greek) and the second (superfi cial) triad pre-
sented in the earlier translation! In neither of these scenarios does recourse
to the indirect tradition simplify the identifi cation of the oldest proofs.
Whatever scenario is chosen, it must be admitted that there was a sub-
stitution of proofs in one branch of the tradition. Th e substitution occurred
in the model(s) of al-Hajjâj, if the superfi cial proofs are considered later
improvements, but in the Greek, if the opposite explanation is adopted.
Th is fact is not surprising. 106 In the situations in which the Greek tradition
contains double proofs, the medieval versions contain only one of them.
(Th is is confi rmed by the remarks of Th âbit and Gerard when they make
such comments as ‘in another copy, we have found ...’ and thus, probably,
in Greek models of which we have no evidence.)
It is possible to take a lesson from this example. Th e existence of double
proofs in the Byzantine manuscripts could be explained, for the majority of105 See De Young 1991 : 660–1.
106 It is noted for i .44p; ii .14; iii .7p, 8p, 25, 31, 33p, 35, 36; iv .5; v .5, 18; vi .9p, 20p, 31; viii .11p-
12p; 22–23; x .1, 6, 14, 26p, 27–28, 29–30, 68–70, 105–107, 115; xi .30, xiii .5. Th e note ‘p’
signifi es that the variant pertains only to a portion of the proof.