Mathematical proof: a research programme 25
proofs according to his vision. In conclusion, we understand better how we
were mistaken, when we took Heiberg’s words for Archimedes’ writings as
the manuscripts bear witness to them.
To start with, Netz examines the diagrams of the critical edition. Clearly,
like cases analysed by Saito and Sidoli, the diagrams used by Heiberg diff er
markedly from the evidence contained in the manuscripts, and Heiberg
drew the diagrams according to his own understanding of what the original
diagrams might have looked like. Yet Netz argues that the manuscripts rep-
resent a coherent and perfectly valid practice with diagrams. Further, three
criteria allow him to discern how the ancient diagrams, drawn within the
context of this practice, systematically diff er from those which Heiberg sub-
stituted. Note that one of Netz’s criteria relates to a feature already discussed
by Saito and Sidoli: Heiberg tended to picture elements of the diagram as
unequal that the manuscripts, in contrast to the discourse, drew as equal.
Interestingly, the two chapters suggest slightly diff erent interpretations of
this ancient element of practice. Th e broader analysis developed by Netz
further leads him to restore an ancient and consistent regime of conceiving
and using diagrams which Heiberg’s critical edition concealed and replaced
with another more modern usage, for which there exists no ancient evidence.
In addition, Netz argues that, in relation to this transformation, the role of
the diagrams in the text underwent a dramatic shift : although the ancient
evidence preserves diagrams that were an integral component of the argu-
mentative text, Heiberg turned the diagrams into mere ‘aids’, dispensable ele-
ments for reading a discursive text that was ‘logically self enclosed’. Th is fi rst
conclusion thus identifi es one way in which the critical edition distorted the
texts of Archimedes’ proofs with respect to the extant manuscripts.
Th e second systematic intervention by Heiberg which Netz analyses is
the bracketing of words, sentences and passages in Archimedes’ writings,
despite the fact that the manuscripts all agree on the wording of these pas-
sages. In other words, by rejecting portions as belonging to the original text,
Heiberg modifi ed the received text of Archimedes’ writings in conformity
with the representation that he had formed for Archimedes as a sharp con-
trast to Euclid. While, for Heiberg, Euclid was characterized by the careful
expression of the full-fl edged argument, Archimedes’ style was, in his
view, to focus on the main line of the proof, leaving aside ‘obvious’ details.
Accordingly, Heiberg designated many passages of the received text as pos-
sible interpolations. Heiberg thus made Archimedes’ style more coherent
than what the manuscript evidence shows. Netz brings to light Heiberg’s
uneven pattern of bracketing and suggests factors which account for it.
What is important for us here are the conclusions that Netz’s analysis allows