Th e Elements and uncertainties in Heiberg’s edition 103
- One in Proposition iv .10 of the Adelardo-Hajjajian tradition, connected
to the absence of iii .37, probably due to an accident in transmission,
namely, the mutilation of the end of a Greek (or possibly Syriac?) scroll
containing Book iii.
Books i – x , perhaps the only ones to have been both translated by Ishâq
and reworked by Th âbit, 85 contain no supplementary deductive lacunae.
In other words, the deductive lacunae which appear there already existed
in the Greek text which served as their model. Th e most striking case is
that of the Lemmas designed to fi ll what can be regarded as a ‘deductive
leap’, especially in Book x. 86 In fact, there are in (some manuscripts of ) the
Ishâq–Th âbit and Gerard of Cremona translations a number of additions
that fulfi l the same role of completion. 87 When compared with the direct
tradition, they are presented as additions, mathematically useful, but well
distinguished from the Euclidean text. Th ose who composed our Greek
manuscripts had no such scruples.
Th e addition of the so-called missing propositions and part of the addi-
tional material (Lemmas of deductive completion, some of the Porisms)
serve with a certain fl uidity the obvious intention of improving the proofs
and reinforcing the deductive structure. Th e second part of the Porism
to x .6 allows the resolution of the same problem as the lemma { x .29/30}.
Th e Proposition xi .38 vulgo is clearly a lemma to xii .17. Th e Proposition
was probably inspired by a marginal scholium and then moved to the
end of Book xi. 88 Th e textual variants of xii .6 suggest that perhaps it was
initially introduced as a Porism to xii .5 and eventually transformed into a
Proposition. For the other additional Porisms, it would certainly be exces-
sive to speak about a deductive lacuna to be fi lled. However, v .7 Por. and
v .19 Por explicitly justify the use of inversion and conversion of ratios. Th e
Porisms to vi .20, ix .11, xi .35 serve to make explicit a deductive dependence
on the Propositions x .6 Por., ix .12 and xi .36, respectively. Our examples,
found in Books x – xii , show that this work of enrichment began in the
Greek tradition, but the Arabic and Arabo-Latin versions tell us that the
85 See below, pp. 116–19.
86 I have called them the ‘lemmas of deductive completion’ in order to distinguish them from
lemmas with only a pedagogical use. See the list given in Euclid/Vitrac 1998: iii 391. To these
might be added Lemma xii .4/5.
87 See Euclid/Vitrac 1998: iii 392–4.
88 See Euclid/Vitrac 2001: iv 229–30.
lacuna in the proofs of Propositions xii .10–11. Here the properties established previously for
pyramids and prisms are shown for cones and cylinders, by using the method of exhaustion.
To do this, the pyramids are considered as having polygonal bases with an arbitrary number of
sides, inscribed in the circular bases of the cones and cylinders.