Archimedes’ writings: through Heiberg’s veil 169
himself knew Greek mathematics only via editions such as Heiberg’s. Little
could he guess that the ancient manuscripts for Archimedes had just the
kind of diagrams he considered logically viable!
Heiberg goes three-dimensional
A group of propositions early in Sphere and Cylinder i involves the com-
parison of cones or cylinders with the pyramids or prisms they enclose:
propositions 7–12. Proposition 12 selects a diagram focused on the base
alone, but the diagrams of propositions 7–11 require that we look at the
entire solid construction. Th e manuscripts’ diagrams (with a single excep-
tion, on which more below) produce a representation with a markedly ‘fl at’
eff ect, whereas Heiberg produces several times a partly perspectival image
with a three-dimensional eff ect.
Th e fi gure for i .9 (see Figure 3.3 ) may be taken as an example. What is
the view selected by the manuscripts’ diagram? Perhaps we may think of it
as a view from above, slightly slanted so as to make the vertex Δ appear to
fall not on the centre of the circle but somewhat below. Th e view selected
by Heiberg’s diagram is much ‘lower’, so that the point Δ appears higher
above the plane of the base circle, allowing the pyramid to emerge out and
produce an illusionistic three-dimensional eff ect. Th e net result is that
Heiberg’s fi gure impresses the eye with the picture of an external object; the
manuscripts’ diagram is reduced to a mere schema of interconnected lines.
Th is defi nitely should not be understood as a mark of poor draughts-
manship on the part of the manuscripts. Indeed, the one exception is
telling: i .11 has a clear three-dimensional representation of a cylinder,
and here the motivation is clear: since the proposition refers in detail to
both the top and bottom bases of the cylinder, a view from ‘above’, where
the bases coincide or nearly coincide, would have been useless. It turns
out, therefore, that once the view from above was excluded, the manu-
scripts were capable of producing a lower view, with its consequent three-
dimensional illusionistic eff ect. Strikingly and decisively, we note that
the manuscripts’ diagrams for i .11 represent the bases by almond-shapes
(standardly used elsewhere for the representation of conic sections). 4 Th is
is a deliberate foreshortening eff ect – which Heiberg himself eschews
in his own diagram. Clearly, Heiberg has established a certain compro-
mise between three-dimensional representation and geometric fi delity, to
(^4) Th is practice is commented upon, for the Arabic tradition, in Toomer 1990 : lxxxv, and it is
indeed widespread in the various manuscript traditions of Greek mathematics.