116 bernard vitrac
(4) To explain this state of aff airs, I see at least two explanations, that
perhaps work in tandem:- Either our diff erent witnesses of the text refl ect a general contamina-
tion^122 and a global criterion – at the scale of the complete treatise- cannot be reached.
- Or the principles that underpin the local criteria are inadequate.
If certain branches of the tradition have epitomized the Elements ,
then the principle used by Klamroth and Knorr that the text of the
Elements grew increasingly amplifi ed proves inadequate. Th ese prin-
ciples may also miss their goals if it is not possible to identify the
motivations of the ancient re-editors when they sought to improve
the form of a mathematical text. We have seen that the criterion of
mathematical refi nement is sometimes diffi cult to use.
(5) Certain characteristics of the preserved versions and diff erent external
confi rmations have convinced us that there has been both contamina-
tion and epitomization. Th us, not only is the text of the version by
Ishâq, as revised by Th âbit, without any additional deductive lacuna
in Books i – x , but the medieval evidence teaches us that the revision
of Th âbit implied the consultation of other manuscripts and, con-
sequently, the collation of alternative proofs. 123 In so doing, various
versions of the Greek or Arabic texts, if not contaminated by, were at
from what, in the manuscripts, is nothing more than an addition to the Porism to vi .20 and
an insertion of a heading [Porism] before the large recapitulation following x .111, although
he did not do this for the summary following x .72! For fi ft een Porisms, there is one or more
Greek manuscripts in which the heading <Porism> is missing. Fift een Porisms are placed
before the standard clause (‘what ought to be proved’), particularly true for P. Eleven are
inserted aft er the clause. Normally, a Porism begins with the expression, ‘From this, it is clear
that’ (‘ἐκ δὴ τούτου φανερὸν ὅτι ...’), but in seven cases ( iv .5, vi .20, ix .11, x .9, x .111, x .114
and xii .17), the formulation is not canonical. Th e possibility of confusion appears in the fact
that ten Porisms retained by Heiberg were amplifi ed by inauthentic additions. If the indirect
tradition is consulted, it ought not to be forgotten that two Porisms from the Greek are related
to substitutions of proof ( iii .31, iv .5) and to an addition ( x .114) which do not exist in this
tradition. Th us, it is not at all surprising that these Porisms did not exist in it. By holding to
comparable cases, the indirect tradition counts eleven Porisms from the Greek, but two exist
in a diff erent form. Th e Porism to x .111 exists as a Proposition and the one to xii .17 appears
as part of a proof (as is also the case in certain Greek manuscripts). Th is ‘πόρισμα’ exists only
in the margin of P and not in the other manuscripts! It may be remarked that neither has
the standard formulation and that the indirect tradition has none of the other fi ve Porisms
‘heterodox’ to the Greek text. For the others, their number decreases (to seven from nine in
i – ix to which could be added three supplementary Porisms from the Ishâq–Th âbit version (to
viii .14, 15; ix .5), to one from four in x , to nil from six in xi – xiii ).
122 Th is is the opinion of Brentjes, at least as concerns the Arabic and Arabo-Latin traditions. See
Brentjes 1996 : 205.
123 S e e E n g r o ff 1980 : 20–39.