92 bernard vitrac
potential proofs (PP) introduced by the formulae: ‘So also for the same
reasons ...’ (= ‘διὰ τὰ αὐτὰ δὴ καὶ ...’) (AP), ‘Similarly we will prove
(alternatively, it will be shown) that ...’ (= ‘ὁμοίως δὴ δείξομεν (alter-
natively, δειχθήσεται) ὅτι...’) (PP). Th ese phrases refer to the desire to
shorten the text. Th e fi rst is the equivalent of our mutatis mutandis ; it
allows the omission of a completely similar argument with a particular
fi gure or elements from a diff erent fi gure. Th e second is a false ‘prophecy’.
It is invoked precisely not to have to prove in detail what it introduces.
Th e ‘abbreviated’ proofs are not uncommon in the Elements (they
number about 250), but in certain cases, it is easy to imagine that a later
editor has used this Euclidean stylistic convention to abridge his text. It is
rather striking that the Arabo-Latin versions are on the whole much more
concise than the Greek text and sometimes have complete proofs, where the
latter uses one of the formulae just cited. In Proposition xii .6, the version
carried by manuscript P uses a potential proof (‘δειχθήσεται’), whereas that
of the so-called Th eonine manuscripts advances an analogical proof (‘διὰ
τὰ αὐτὰ δὴ’). Th e appearance of these formulae is therefore not independ-
ent of the transmission of the text. 59Quantitative aspect
Th e 220 structural modifi cations in my database include: more than 60
Defi nitions out of about 130, 8 of 11 Common Notions, 29 of 35 Porisms,
41 of 42 Lemmas and additions, 173 Propositions of 474 (actually, 465 in
the Greek tradition) which is a little more than a third of the total. 60 Th ese
modifi cations are very unequally distributed through the Books, depend-
ing on the type of textual units. Taking a cue from medieval scholars, I have
grouped together the principal global variations according to three (not
completely, but almost) independent criteria:
(a) Th e presence or absence of certain portions of the text (35 Defi nitions,
8 Common Notions, 27 Porisms, 41 Lemmas and additions, 25
Propositions).
(b) A change in the order of presentation. Th ere are roughly 30 which
relate to about 30 Defi nitions and more than 60 Propositions.
(c) Th e (structural) alteration of proofs. For now, I have listed about 80
which concern a little fewer than 100 Propositions. 61(^59) For other examples, see the references given in Euclid/Vitrac 2001: iv 46–7, n. 51, 53.
(^60) Some relate to a group of Propositions, for a total greater than 220.
(^61) See Vitrac 2004: 40–2.