Th e Elements and uncertainties in Heiberg’s edition 91
the category of global diff erences – that is, substitution of proof – be well
defi ned, it is necessary also to propose a typology 57 of changes for which I
will reserve the qualifi er local (see the fi gure 1.1 above).
Let us also give a few explanations or examples for the variations for
which the designation is perhaps not immediately apparent:
- Th ere is a doubling when a Proposition concerning two Cases is replaced
by two distinct, consecutive Propositions. Th is expansion is observed in
the indirect medieval tradition for x .31 and 32, xi .31 and 34. Th e inverse
operation is fusion. Of course, these alterations are not the same as the
substitution of a proof. Th us, the doubling might correspond to a logical
or (in the case of very long proofs) pedagogical concern. Even stylistic
concerns might be represented, but they would not alter the mathemati-
cal content of the proofs. - Th e change of status may, for example, aff ect a Porism (corollary). Th is
is the case of the Porism to Heib. x .72, transformed into an independ-
ent Proposition in the indirect medieval tradition. According to another
example, the (apocryphal) principle that ‘two lines do not contain an area’
is presented as Postulate No. 6 in some of the Greek manuscripts ( PF ), in
the translation by al-Hajjâj 58 and in the work of Adelard, but as Common
Notion No. 9 in another part ( BVb ) of the direct tradition, in the transla-
tion of Ishâq–Th âbit, and in the work by Gerard of Cremona. - Th ere is, for example, a diff erent formulation in Proposition ii .14. Th e
translations of al-Hajjâj and the Adelardian tradition propose to present
the quadrature of a triangle, while the Greek manuscripts, the Ishâq–
Th âbit and Gerard of Cremona translations undertake the quadrature
of an unspecifi ed rectilinear fi gure. Th is is related to another category
of variations represented by the absence of Proposition i .45 in the fi rst
group of witnesses just mentioned. In the same way, the Porism to vi .19
is formulated diff erently in the manuscript P (for a fi gure) and in the
manuscript Th (for a triangle). Here, too, the variant is connected with
the existence of the Porism to vi .20, No. 2 (for a fi gure), found in only the
so-called Th eonine manuscripts. Th e divergences may thus be correlated
at long range. - As for the local variants with some possible logical and pedagogical
purpose, we will see some examples in what follows. Let us specify only
those which approach the category ‘abridged demonstrations’. Th is cat-
egory concerns the use of proofs described as analogical proofs (AP) and
(^57) See this point introduced in Euclid/Vitrac 2001: iv 41–69, in particular the chart on p. 55.
(^58) See De Young 2002–3: 134.