Mathematical proof: a research programme 23
Th is also holds true for most of the material added (propositions, lemmas,
porisms). Th ese remarks seem to support Klamroth’s view. In this respect,
Vitrac considers the indirect tradition as more authentic, a fact which calls
for a re-examination of proofs in the various versions of Euclid’s Elements.
Vitrac suggests, however, that the enlargement and ‘improvement’ of the
Elements could have started in Greek and continued in Arabic and Latin.
Th e extant versions all seem to bear signs of corruption by such activity.
Th e second type of divergence between the sources Vitrac examines
relates to the order in which propositions are arranged. Th is order con-
stitutes a key ingredient in an axiomatic–deductive structure. In fact, the
order does vary according to the version of the text. Th e decisions imple-
mented by any critical edition hence represent an interpretation of Euclid’s
original deductive structure. However, on this count, Vitrac suggests the
provisional conclusion that the indirect tradition more frequently bears
witness to modifi cations of this type.
Th e third kind of divergence which he analyses has perhaps the greatest
impact on our perception of Euclid’s proofs, since it relates to major dif-
ferences between the sources: substitution of proofs, integration of these
substitutions in a set of related proofs, addition or subtraction of cases, and
double proofs, of which Heiberg kept only one according to criteria that
need to be examined. Such cases indicate that proofs and their modifi cation
were the subject of a continuous eff ort, part of which was integrated into the
editions of the Elements available to us today.
In conclusion, before we consult the critical editions of Arabic, Arabo-
Hebrew or Arabo-Latin versions of Euclid’s Elements , it may be diffi cult to
go substantially further in the analysis of the proofs or the deductive system
attributed to Euclid. Most probably, this goal may remain forever out of
reach. However, we can already appreciate the extent to which the textual
decisions made by the philologist aff ect the discourse on the practice of
proof in ancient Greece. Th is remark shows that the discourse on the prac-
tice of proof in ancient Greece may not be as solidly founded as was previ-
ously thought. As Vitrac suggests in his conclusion, rather than holding
to the romantic ideal of some day retrieving the original Elements , it may
be far more reasonable and interesting to consider the various versions of
Euclid’s Elements for which we have evidence. Th is new perspective would
provide us with a better grasp of the various forms that the text took in
history – namely, the forms through which diff erent generations of scholars
read and used the Elements.
Ken Saito and Nathan Sidoli critically examine the work of the philolo-
gist from an entirely diff erent perspective. Th e purpose of their chapter is