102 bernard vitrac
from their absence, as Heiberg has done, that there is a deductive ‘lacuna’
in the proof of xii .15. However, from the point of view of the history of the
text, the question immediately arises about whether or not the insertion
of Propositions xii .13–14 represents an addition aimed at fi lling a lacuna
perceived in the original proof of xii .15. Let us add that the assertions of
Heiberg on this subject are oft en a little hasty because the status of authen-
ticity cannot be judged independently of the status of the proofs.
For example, the indirect tradition does not contain Proposition x .13 (‘If
two magnitudes be commensurable and one of them be incommensurable
with any magnitude, the remaining one will also be incommensurable with
the same’). Heiberg suggests that the absence of this Proposition introduces
deductive lacunae in several Propositions which exist in the Arabic transla-
tions. In these Propositions, the Greek text explicitly uses x .13. However,
in fact, when the proofs in the aforementioned translations are examined,
they are formulated a little diff erently than in Greek and x .12 (‘Magnitudes
commensurable with the same magnitude are commensurable with one
another too’) is employed in place of x .13. Consequently, there is not a
deductive lacuna! 82 By consulting the indirect tradition of Greek citations
in Pappus, the idea may be supported that x .13 did not exist in his version
of the Elements. 83 Th us, the most natural conclusion is that x .13 is eff ec-
tively an inauthentic addition and its addition has allowed reconsideration
of the proofs of the other Propositions.
Th rough a simple comparison of the diff erent versions, I have examined
each of the Propositions whose authenticity has been called into question.
My conclusion regarding this point – the details would exceed the scope
of this essay – is that the real deductive lacunae, proper to the indirect
tradition, are, so far as can be judged, far from numerous:- Two in Book xii , 84 with the provision that in any event the stereometric
Books constitute a particular case in the transmission of the Elements
(see below).
82 See Vitrac 2004 : 25–6.
83 See Euclid/Vitrac 1998: iii 384–5.
84 Th e second is due to the absence, this time in b as well as in the indirect medieval tradition,
of Proposition xii .6 and the Porisms to xii .7–8 which generalize the results established for
pyramids on a triangular base to pyramids on an unspecifi ed polygonal base, respectively
in Propositions xii .5, 7 and 8. Th ere also, Euclid may have considered this generalization
as intuitively obvious given the decomposition of all polygons into triangles and the rule
concerning proportions established in (Heib.) v .12: ‘If any number of magnitudes be
proportional, as one of the antecedents is to one of the consequents, so will all the antecedents
be to all the consequents.’ Th e non-thematization of pyramids on an unspecifi ed polygonal
base is comparable to what we have seen above regarding ii .14 (triangle unspecifi ed rectilinear
fi gure) in only the Adelardo-Hajjajian tradition. Th e diff erence is that it introduces a deductive