100 bernard vitrac
Questions of authenticity and the logical architecture of the Elements
If the diff erent versions are considered from the point of view of the ‘mate-
rial contents’, the question of authenticity is perhaps the least complex of the
three, at least as far as the fi rst dichotomy is concerned. Th ere exists in the
Greek manuscripts material which I describe as ‘additional’. Th is additional
material includes cases, some portions identifi ed as additions, the double
proofs, and the Lemmas. 77 Th e critical edition of Heiberg, completed in
1888, four years aft er the debate with Klamroth, condemns the lot of this
material as inauthentic. In this regard, the (rather relative) thinness of P
compared with the other Greek manuscripts is one of the criteria which jus-
tifi es its greater antiquity. 78 Now this additional material, to nearly a single
exception,^79 is absent from the medieval Arabic and Arabo-Latin tradition.
However, Heiberg did not alter his position and did not accept this conclu-
sion about the ‘thinness’ of the indirect tradition as a gauge of its purity.
According to Heiberg – and this too is a hypothesis about the nature of the
treatise – the Elements could not be so thin that it suff ered from deductive
lacunae, but such thinness is the case with the medieval versions.
I do not believe that anyone (and certainly not Klamroth or Knorr)
contested the global deductive structure of the Elements. If the Elements is
compared with the geometric treatises of Archimedes or Apollonius, the
local ‘texture’ may not be so diff erent, but the principal variation resides
in the fact that the Elements was edited as if it supposed no previous geo-
metric knowledge. Th e identifi cation of what would be a deductive lacuna
in Euclid is thus a crucial point, but not always a simple one. Indeed, all
the exegetical history of the Euclidean treatise, from antiquity until David
Hilbert, has shown that the logical progression of the Elements , probably
like any geometric text composed in natural language, rests on implicit
presuppositions. 80 Th e identifi cation of the deductive lacunae supposes
that consciously permitted ‘previous knowledge’ is always capable of clearly
being distinguished from ‘implicit presumption’.
Let us take the example of Proposition xii .15. Here it is established that:
Th e bases of equal cones and cylinders are inversely [proportional] to the heights;
and among the cones and cylinders, those in which the bases are inversely [propor-
tional] to the heights are equal,77 For details, see Table 1 of the Appendix.
78 See Table 3 of the Appendix.
79 Th e addition of special cases in Prop. iii .35, 36 and 37.
80 See the beautiful study by Mueller 1981.