The History of Mathematical Proof in Ancient Traditions

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Th e Elements and uncertainties in Heiberg’s edition 119


hand. 135 It is interesting to note that in Proposition (Heib.) xiii .11
(= IsTh 14), the manuscript Petersb. 2145, as well as Tehran Malik 3586
and Rabat 1101, record ‘asamm!’ 136 Th is does not necessarily mean that
Ishâq did not translate Books xi – xiii , 137 but it at least suggests that at
some moment of transmission, the stereometric books existed only in
a single version. 138 Th is homogeneity, recorded by Tûsî, might even be
the cause of the glosses inserted in the three manuscripts of the Ishâq–
Th âbit version that I just mentioned. 139
(8) Two consequences may be drawn from the preceding considerations.
First, Knorr’s hypothesis that the indirect tradition derived from a
single Greek archetype, based only on the stereometric books – in fact
only on the portion xi .36– xii .17 – is challenged. Second, I have said
that there are, in the versions of al-Hajjâj and Ishâq–Th âbit, three and
two deductive lacunae respectively. Th ose of Ishâq–Th âbit occur in
Book xii. But, if the hypothesis of Klamroth or one of his variations is
adopted, we know the translation of Ishâq–Th âbit only for Books i – x.
Th e translation here is without deductive lacunae, which, considering
the work of the Reviser, is to be expected. As for the translation of al-
Hajjâj, the evidence of the preface in the Leiden manuscript suggests
that it could scarcely be other than an epitome!
(9) Th ese consequences being noted, it ought not to be forgotten that
it is thanks to the indirect tradition itself that we have been able to
determine some of its limitations. Th e medieval versions, notably
those of Ishâq–Th âbit and Gerard of Cremona, are more attentive to
problems of textual origin than the Greek manuscripts and thereby
more informative about the divergences between versions observed
by their authors. Th e contamination is clearly not the doing of medi-
eval scholars only. Th e subject of double proofs demonstrates this.
Th e abundance of additional material and local alterations of the


135 See Rommevaux, Djebbar and Vitrac 2001 : 259, 288–9.
136 I thank A. Djebbar for this information.
137 It is possible to doubt such an abstention by Ishâq given that there are two series of defi nitions
for Book xi in Tehran Malik 3586, the latter being attributed to Hunayn ibn Ishâq and,
probably, there was some confusion here between the father and the son (see Brentjes 2000 :
54). However, Ishâq may well have brought his translation to an end with the Defi nitions for
Book xi , which have been (piously) conserved, though he did not translate what followed.
Th us, one again arrives at the thesis of Klamroth.
138 Although she disagrees with the thesis of Klamroth, Brentjes pointed out that in regard to
Defi nition xi , the fi rst version of Tehran Malik 3586 (the Ishâq–Th âbit version) and the version
given by al-Karâbîsî, who, (according to Brentjes), follows Hajjajian version, have minuscule
diff erences. See Brentjes 2000 : 53. Th is seems to me to concur with the preceding remark.
139 See above, nn. 131 – 132.

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