Archimedes’ writings: through Heiberg’s veil 171
some diagrams in the manuscripts take a vertical arrangement (even though
this arrangement is not determined by the geometrical situation). I do think
the manuscripts avoid the three-dimensional representation, among other
things, because of their preference for the horizontal over the vertical; what
I wish to stress is that this shows how little weight they allow the pictorial
quality of the diagram – so that the minor consideration of a preferred ori-
entation trumps over that of the three-dimensional representation.
Note now that our discussion touches on a small stretch of text, but
this is in fact in itself meaningful. Th e Archimedean corpus is sometimes
dedicated to purely plane fi gures ( Spiral Lines , Planes in Equilibrium ,
Measurement of Circle , Stomachion , Quadrature of Parabola ) but, even
in the several cases where Archimedes studies solid objects, these are
studied essentially via some plane section passing through them ( Floating
Bodies ii , Method , Conoids and Spheroids , Sphere and Cylinder ii ). Sphere
and Cylinder i forms an exception because of its mathematical theme of the
comparison of curved, concave surfaces – one which calls for a direct three-
dimensional treatment. 7 Now consider i .12, where Archimedes’ treatment
of the three-dimensional cone is mediated via the plane base (where
two lines form tangents to the circle of the base). Such is the standard
Archimedean diagram. In the manuscripts, the diagrams of i .12 and of
i .9 are closely aligned together, displaying a similar confi guration of criss-
crossing lines; whereas Heiberg’s diagrams open up a chasm between the
two situations, the solid picture of i .9 marked against the planar view of i .12
(see Figure 3.4 ). I would venture to say as much: that by making i .9 appear
more solid , Heiberg simultaneously makes i .12 appear more planar. If i .9
is designed to bring to mind a picture of what a pyramid looks like, then
i .12 should be seen to be designed so as to bring to mind a picture of what
a circle looks like. But if i .9 is a mere schematic representation of lines in
confi guration, then the same must be said of i .12 as well: it is not a picture of
a two-dimensional fi gure. It is, instead, a geometrically valid way of provid-
ing information, visually, about such a fi gure.
Th is, of course, is an interpretation that goes beyond the evidence. Th e
facts on three-dimensional representation are simple: such representation
is avoided as far as possible by the manuscripts, but is produced, wherever
(^7) Among the lost works by Archimedes, the Centres of Weights of Solids may well have been
based on planar sectional treatment – which Archimedes invariably pursues in the closely
related Method (where various spheres, conoids and prisms are represented by planar cuts).
One wonders how Archimedes’ treatment of semi-regular solids was handled: the account
in Pappus (Hultsch 1876: 350–8) carries no diagrams and is based on a purely numerical
characterization of the fi gures.