Th e Elements and uncertainties in Heiberg’s edition 99
in the discussion of problematic places, variant readings of the indirect
medieval tradition ought to be accounted for. Th is was exactly what Knorr
recommended. He even thought that it was possible to reconstruct a Greek
archetype for the whole of the medieval tradition.
In other words, by comparing the diff erent states of the text for each
attested divergence, we ought to be able to identify the least inauthentic
version (or versions). Taking into account the three principal types of struc-
tural variants that we have recognized, this amounts to:
- solving the question of authenticity for each contested textual unit (the
determination of the ‘materiel’ contained there) - selecting a method of presentation (in particular, an order) when several
are known; and - knowing, for the cases of substitution or double proofs, which of the two
is older.
To pronounce such judgements supposes criteria. Th ere are essentially
two of them:
(i) the fi rst concerns the ‘quantity’ of material transmitted by various ver-
sions, and
(ii) the second bears on the form of this material (order of presentation,
modifi cation of proofs).
Th ese criteria rest on the presuppositions that the historians accept regard-
ing the nature of the text of the Elements and on the hypotheses that they
imagine regarding its transmission. According to Klamroth (and Knorr),
the textual history has essentially been an amplifi cation. Th us, for example,
except by accident, a Proposition missing from a ‘thin’ version (contain-
ing less material than another or even several others) will be judged
inauthentic.
As for the transformations of form, if it is not an accident of transmission
but a deliberate alteration of the structure of the text (supposing that it is
possible to discriminate between the two), the criterion, as stated explicitly
by W. Knorr, will be improvement – that is, whether it met with success or
failure, whether it was really justifi ed or invalid, the deliberate modifi ca-
tion of the form (order, proof ) of the text sought to better the composition.
Obviously, this is an optimistic vision of the history of mathematics.
To see how to apply these principles and to understand the nature of
the structural modifi cations that we have called up, it is easiest to produce
some examples. Th e limitations of the aforementioned criteria will appear
more clearly when we examine their application to the proofs (see below,
pp. 111–13).