112 bernard vitrac
those who deliberately changed the text of the Elements – diff erent criteria
could be used. From the logical, or metamathematical, point of view, the
criteria are:- Render the deductive structure more dense, as the superfi cial proofs have
done, or conversely minimize the structure in order not to introduce
what would eventually become accidental ‘causalities’, that is, links of
dependence, as found among the linear proofs. - Prefer either a type of object language over a second-order language –
that is, a relational terminology, like the theory of proportions – or, on
the contrary, privilege a concise but more general second-order language.
A choice of this kind explains the aliter family of proofs conceived for
Propositions vi .20, 22, 31, x .9, xi .37.^109 Th e same choice exists also in our
families of proofs, but in these instances it acts in the opposite direction
with respect to reinforcing the deductive structure.
It would then be welcome to be able to organize these criteria hierarchi-
cally. Th e deductively minimalist attitude seems well represented in the
Elements. For example, deductive minimalism may safely be assumed to
underpin the decision to postpone as long as possible the intervention of
the parallel postulate in Book i. It appears again in the decision to establish
a number of results from plane geometry before the theory of proportions
is introduced at the beginning of Book v , even though this theory would
have allowed considerable abbreviation. Th e idea that geometry ought to
restrict itself to a minimal number of principles had already been explained
by Aristotle. 110 Deduction is not neglected, but emphasis is placed on the
‘fertility’ of the initial principles, rather than on the possible interaction of
the resultants which are deduced from them.
Th ere are thus diff erent ways to put emphasis on the deductive structure.
Th e case of our proofs from Book x is not unique. Th e ten Propositions
from Book ii and the fi rst fi ve from Book xiii are successively set out in a
quasi-independent manner based on the least number of principles, even if
this means reproducing several times certain portions of the arguments. 111
Remarkably, we know that for the sequences ii .2–10 and xiii .1–5 alternative
proofs had been elaborated, annulling this deductive mutual independence
in order to construct a chain in the case of ii .2–10 or to deduce xiii .1–5 from
certain results from Books ii and v. Even better, thanks to the testimony of
109 See Vitrac 2004 : 18–20.
110 De cælo , iii , 4, 302 b26–30.
111 Similarly in the group El. iii .1, 3, 9, 10 (considering the fi rst proofs of iii .9–10).