Diagrams and arguments in Greek mathematics 149
obtaining among the objects, but rather to convey specifi c mathematical
relationships that are essential to the argument.
Some conspicuous exceptions to this general tendency should be men-
tioned. For example, the diagrams for the rectilinear solids treated in Elem.
xi and xii and the early derivations of the conic sections in the cone, in
Con. i , appear to use techniques of linear perspective to convey a sense
of the three-dimensionality of the objects. In Figure 2.11 , we reproduce
the diagram for Elem. xi .33 from Vatican 190 and that for Con. i .13 from
Vatican 206.
In all of these cases, however, it is possible to represent the three-
dimensionality of the objects simply and without introducing any object
not explicitly named in the proof merely for the sake of the diagram. For
example, in Figure 2.1 above, the plane upon which the perpendicular is
to be constructed does not appear in the manuscript fi gure. Hence, even
in these three-dimensional diagrams, techniques of linear perspective are
used only to the extent that they do not confl ict with the schematic nature
of the diagram. Auxiliary, purely graphical elements are not used, nor is
there any attempt to convey the visual impression of the mathematical
objects through graphical techniques. An example of this is the case of
circles seen at an angle. Although it is not clear that there was a consist-
ent theory of linear perspective in antiquity, ancient artists regularly drew
circles as ovals and Ptolemy, in his Geography , describes the depiction of
circles seen from an angle as represented by ovals, 28 nevertheless, in the
medieval manuscripts such oblique circles are always drawn with two
(^28) Knorr 1992: 280–91; Berggren and Jones 2000: 116.
Figure 2.11 Diagrams for Euclid’s Elements , Book xi, Proposition 33 and Apollonius’
Conica , Book i, Proposition 13.
Vatican 190 Vatican 206