Th e Elements and uncertainties in Heiberg’s edition 109
proportions. Deductively, the linear proofs may be characterized as mini-
malist.- Th e superfi cial proofs introduce areas, which, from the point of view of
the linguistic style used, might seem more geometric than the proofs
using the theory of proportions, which is a second-order language. But,
in fact, they strengthen the deductive structure because they establish
new connections by using Propositions (Heib.) x .57–59 + 63–65 + 66
(resp. 94–96 + 100–102 + 103). In addition, these superfi cial proofs –
like the linear ones – present results expressed for commensurability in
length but the former proofs may be immediately generalized to com-
mensurability in power.
Th e fi rst anomaly occurs in x .107. No alternative proof exists, although
this Proposition, along with two others, constitutes a triad of quite similar
Propositions. Alternative proofs are no longer known for the parallel triad
of x .68–70, which concerns the irrationals produced by addition, whereas
the other triad x .105–107 treats the corresponding irrationals produced by
subtraction. 102 However, in the indirect Arabic and Arabo-Latin tradition,
there is a textual family in which these two triads of Propositions have (only)
superfi cial proofs. Th is is the case in Arabic, with the recension of Avicenna,
and in Latin, with the translation of Adelard I. Evidence from the copyist
of the manuscript Esc. 907 establishes a link between the superfi cial proofs
and the translation of al-Hajjâj. 103 Th e Ishâq–Th âbit version is less coher-
ent. It contains the linear proofs of the Greek tradition in the triad x .68–70
and the superfi cial proofs for the triad x .105–107. In the manuscript from
the Escorial and the translation of Gerard of Cremona, which agree on this
point, the situation is nearly the inverse to the Greek translation. Th ere are
only the superfi cial proofs for x .105–107 (like the indirect tradition), but
they present proofs of this type as aliter for the fi rst triad, whereas the Greek
texts includes them only for (two Propositions of ) the second triad!
Let us add that the same type of substitution (and thus, generaliza-
tion) is possible in Propositions (Heib.) x .67 + 104 which concern the two
corresponding types of bimedials and apotomes of a bimedial. 104 S u c h
substitution is precisely what is found in the recensions of at-Tûsî and
pseudo-Tûsî, but not in the Arabic or Arabo-Latin translations.
102 On the plan of Book x , see Euclid/Vitrac 1998: iii 63–8.
103 See De Young 1991 : 659.
104 However, this is not possible for Prop. Heib. x .66 (binomials) and 103 (apotomes) because,
in this case, it is required to show that the order (from one to six) of the straight lines
commensurable in length is the same. Th is crucial point is required for the superfi cial proofs
concerning the other ten types of irrationals.