Th e Elements and uncertainties in Heiberg’s edition 95
(b) Th en the two textual families distinguished by Heiberg diverge:
P (^) ἐπειδήπερ ἐὰν κέντρῳ τῷ Β καὶ
διαστήματι τῷ ΑΒ κύκλον γράψωμεν,
αἱ διάμετροι ἀνίσους ἀπολήψονται
τοῦ κύκλου περιφερείας.
Because, if we describe a circle with the
centre B and distance AB, the
diameters will cut unequal arcs of the
circle
BF Vb εὐθεῖα γαρ εὐθεῖᾳ οὐ συμβάλλει κατὰ
πλείομα σημεῖα ἢ καθ' ἕν· εἰ δὲ μή,
ἐφαρμόσουσιν αλλήλαις αἱ εὐθεῖαι.
for a straight line does not meet a
straight line in more points than one;
otherwise the lines will coincide.
(c) Th e general conclusion follows, then the closing of the theorem:
Εὐθείας ἄρα γραμμῆς μέρος μέν τι οὐκ
ἒστιν ἐν τῷ ὑποκειμένῳ ἐπιπέδῳ, μέρος
δέ ἐν μετεωροτέρῳ·
Th erefore, it is not the case that some part
of a straight line is in a subjacent plane and
another part is in a higher plane.
ὅπερ ἔδει δεῖξαι.^66 Which is what was to be proved.
Conforming to the general rule which he follows, Heiberg has retained
the reading of P in his text, and he consigns the reading of the Th eonine
manuscripts in his apparatus criticus. 67 From the stylistic point of view, one
can see that:
- Th e two variants are what I call post-factum explanations because they
have the form ‘ q , because p ’, rather than ‘if p , then q ’. Th e ‘cause’ ( p ) is
stated aft er the fact ( q ) of which it is supposed to be the cause. 68 - Th e variant P is introduced by the conjunction ‘ἐπειδήπερ’, which is suf-
fi cient to arouse suspicions about its authenticity. 69 Moreover, I call what
appears here an ‘active, personal, conjugated form’ (‘γράψωμεν’) since
the normal Euclidean form of conjugation in the portion of the deduc-
tive argument is the middle voice, 70 which reinforces the suspicion of
inauthenticity.
(^66) See EHS: iv: 4.8–5.3.
(^67) Th is same variant appears in the margin of P, but by a later hand, followed by the addition:
‘οὕτως ἐν ἄλλοις εὕρηται, ἔπειτα τὸ ̇ εὐθείας ἄρα γραμμῆς’ (alternatively, this is found in
other [copies]: ‘Of a straight line ...’).
(^68) See Euclid/Vitrac 2001: iv 50, 56, 67–9.
(^69) Th ere exist, in the text of Book xii as edited by Heiberg, about fi ft een passages introduced
by the conjunction ‘ἐπειδήπερ’, all of which contain elementary explanations found neither
in manuscript b, nor in the Arabo-Latin translations by Adelard of Bath and by Gerard of
Cremona. In the whole of the Elements, 38 instances occur. As already indicated by Knorr 1996:
241–2, we know that there are relatively late interpolations in manuscripts used by Heiberg. A
posteriori, we can see that Heiberg considered seven of these passages interpolations on the basis
of criteria other than their absence in manuscript b and the indirect tradition.
(^70) See Euclid/Vitrac, 2001: iv 47.