Th e Elements and uncertainties in Heiberg’s edition 113
the Persian commentator an-Nayrîzî, we know that the author of the fi rst
suggestion was Heron of Alexandria. It is thus tempting, as Heiberg did
in his Paralipomena of 1903, to attribute to him the other alteration (in
xiii ) that shares the same spirit. 112 If, in order to strengthen the deductive
structure, it is appropriate to argue about segments rather than the surfaces
described thereon as in the case of Books ii and xiii , 113 it will be noted that
the opposite is the case in the example from Book x which has just been dis-
cussed. Reinforcement of the aforementioned structure is realized through
the introduction of surfaces. For us to attribute it to Heron, it is necessary
to be sure that the parameter most important to him was indeed the den-
sifi cation of the deductive structure. Without any external confi rmation or
other historical information, as in the case of Books ii and iii , this scenario
remains a stimulating hypothesis, but only a hypothesis! 114
Conclusions: contributions and limitations of the
indirect tradition
From the study of a better-known indirect tradition, several lessons may
be drawn. Newly available information confi rms certain results of the
Klamroth–Heiberg debate. Consideration of a greater number of versions
of the Elements than Heiberg or Klamroth could have used reinforces the
existence of a dichotomy between the direct and indirect traditions.
(1) Although they agree (albeit with opposite interpretations of the fact),
the ‘thinness’ of the indirect tradition is not so marked as Klamroth
and Heiberg would have us believe, especially in Books i – x. Th e most
complete inventory of variants, probably Greek in origin, which we
have now (by induction or thanks to information transmitted by Arab
scholars or copyists), has several consequences:
- It puts into perspective the diff erent textual dichotomies. For example,
No. 3 ( P / Th ), within the Greek direct tradition, is quite modest with
112 See Heiberg 1903 : 59. I have espoused the same hypothesis in Euclid/Vitrac 2001: iv 399–400.
113 Th e insertion of iii .10 aliter , explicitly attributed to Heron by an-Nayrîzî, has the same eff ect of
strengthening the deductive structure.
114 A single thing seems likely. Th e version of Euclid which Pappus had – if he is indeed the
author of the second table of contents of Book x contained in the fi rst Book of Commentary to
the aforementioned book transmitted under his name – contained the linear proofs. In eff ect,
Propositions x .60–65 and x .66–70 were inverted (similarly for x .97–102/103–107) and this
fact precludes the existence of superfi cial proofs for x .68–70.