Archimedes’ writings: through Heiberg’s veil 175
32–3, 37–42). I fi nd it hard to see how a scribe, asked to copy a manuscript
where polygons are represented by polygons, would produce a manuscript
where polygons are represented by a system of curved lines. Th is lectio dif-
fi cilior argument is the best I have for showing that, if not introduced by a
scribe, such diagrammatic practice is likely authorial. Perhaps our simplest
hypothesis is that the diagrams as a whole derive, largely speaking, from
Archimedes himself.
Th e texture of Archimedes’ diagrams: summary
Whether by Archimedes or not, the non-iconic character of the repre-
sentation of polygons in SC i is a striking example of how schematic the
manuscripts’ diagrams are – and how Heiberg has turned such schematic
representations into pictures. Th is is of course consistent with the manu-
scripts’ preference for a ‘fl at’ representation as against Heiberg’s pictorial
pyramids, as well as with the much wider manuscript practice of metrical
simplifi cation, typically that of representing unequal magnitudes by equal
fi gures.
Heiberg has clearly transformed the manuscripts’ schematic diagrams into
pictorially ‘correct’ ones. By so doing, however, he has also constructed dia-
grams of a diff erent logical character. If diagrams are expected to be pictori-
ally correct, then one is expected to read them for some metrical information;
and if so, the information one gathers from the diagrams is potentially false
(since no metrical drawing can answer the infi nite precision demanded
by mathematics) as well as potentially overdetermined (since a particular
metrical confi guration may introduce constraints that are less general than
the case required by the proposition). Th e schematic and more ‘topologic-
al’ character of the manuscripts’ diagrams, on the other hand, makes them
logically useful. One can rely on the manuscripts’ diagrams as part of the
argument, without thereby compromising the logical validity of the proof.
A major claim of my book (N1999) was that diagrams play a role in Greek
mathematical reasoning. 9 I have suggested there – following Poincaré – that
the diagrams may have been used as if they were merely topological. My
consequent study of the palaeography of Greek diagrams has revealed a
striking and more powerful result: the diagrams, at least as preserved by
early Byzantine manuscripts, simply were topological. Heiberg’s choice to
obscure this character of the diagrams was not only philologically but also
philosophically motivated. Clearly, he did not perceive diagrams to form
(^9) N1999, especially chapters 1, 2, 5.