174 reviel netz
circle. A square perhaps could still do, but this off ers a very special case
of the 4 n -sided regular polygon: considered as a division of the sphere, it
reduces to a system of two cones, without any truncated cones. Th e octagon
already brings in a truncated cone, but this is the limiting truncated cone
lying directly on the diameter of the sphere. Only with the dodecagon do
we begin to see the general case of a division of the sphere based on 4 n -
sided regular polygons, with a limiting truncated cone lying directly on the
diameter, another truncated cone next to it, and fi nally a non-truncated
cone away from the diameter ( Figure 3.6 ). Of course, a regular dodecagon
is nearly impossible to distinguish, visually, from a circle, but the entire
point of avoiding a limiting case for the diagram is the desire to limit the
extent to which the visual impression of the diagram creates false expecta-
tions. Th e same desire, then, accounts for the radical, non-iconic represen-
tation itself: no one is going to base an argument concerning polygons on
the visual impression made by the curved arcs. Indeed, the visual impres-
sion as such does not play into the argument. What matters, for the argu-
ment, is the similarity of the polygons and the purely topological structure
they determine – a circle nested precisely between two polygons, triggering
Archimedes’ results on concave surfaces.
Th is diagrammatic practice is not isolated: it defi nes the character of
Archimedes’ SC i. As soon as the structure of a polygon inscribed inside
the circle is introduced, in proposition 21, and right through the ensuing
argument, the manuscripts systematically deploy such representations
based on curved lines – in fi ft een propositions altogether ( i .21, 23–6, 28, 30,Figure 3.6 Th e general case of a division of the sphere.