Th e Elements and uncertainties in Heiberg’s edition 97
It is possible to imagine (at least) two scenarios: either these post- factum
explanations are inauthentic, or the translator (or the editor Th âbit), noting
the divergence among the Greek manuscripts and the defi ciency of the
proposed explanations, refrained from retaining one or the other. In other
words, he has ‘cleaned up’ the text.
Th e mathematical defi ciency of the explanation in P is obvious. It allows
the points ABCD to be co-planar. In order to prove the co-planarity of
lines ABC and ABD starting from the fact that they are secant (they even
have a segment in common), one would have to use xi .2 – which in turn
invokes xi .1! Th us, and this is Heiberg’s reading, an argument akin to lectio
diffi cilior may be implemented and the text of the Th eonine manuscripts
may be declared an improvement. Hence, his editorial decision. Th is
scenario is hardly likely.
In fact, in certain manuscripts of the Th family, particularly V , there exists
a scholium proposing a proof of the impossibility of two straight lines having
a common segment, that is the concluding point of our indirect proof: 72
For two straight lines, there is no common segment. Th us, for the two straight
lines ABC and ABD, let AB be a common segment, and on the straight line ABC,
let B be taken as the centre and let BA be the radius and let circle AEZ be drawn.
Th en, since B is the centre of the circle AEZ and since a straight line ABC has been
drawn through the point B, line ABC is thus a diameter of the circle AEZ. Now, the
diameter cuts the circle in two. Th us AEC is a semi-circle. Th en, since point B is the
centre of circle AEZ and since straight line ABD passes through point B, line ABD
is thus a diameter of circle AEZ. However, ABC has also been demonstrated to be
a diameter of the same AEZ. Now semi-circles of the same circle are equal to each
other. Th erefore, the semi-circle AEC is equal to semi-circle AED, the smallest to
the largest. Th is is impossible. Th us, for the two straight lines, there is no common
segment. Th erefore, [they are completely] distinct. From that starting point, it is
no longer possible to continuously prolong the lines by any given line, but [only]
a [given] line and, that because, as has been shown, [namely] that for two straight
lines, there is no common segment.
Th is scholium does not exist in P , but its absence may be explained if it
is the origin of the post-factum explanation, albeit in severely abbreviated
form, inserted in the text of the manuscript. Th us, there was no longer
need to recopy the aforementioned scholium. It is likely that the explana-
tions appearing in the Th eonine manuscripts come from the insertion of an
abridgment of some (another) scholium into the text. Th ere is even a chance
that we know the source of these marginal annotations. In his commentary
to Proposition i .1, Proclus reports an objection by the Epicurean Zenon of
(^72) Cf. EHS: v, 2, 243.27–244.22.