Archimedes’ writings: through Heiberg’s veil 167
Th e manuscripts’ diagram avoids this diffi culty altogether. Th e three
equal circles – in fl agrant violation of the textual requirement that Λ > Θ,
Λ > K – imply that the diagram carries no metrical consequences (at least
so far as these three circles are concerned) and therefore the diagram itself
leaves the metrical relationship between K and Θ indeterminate.
Th is is a systematic feature of the manuscripts’ diagrams. Th ere are
twenty-four cases where a system of homogeneous, unequal magnitudes
(typically all circles, or all lines) is represented by equal magnitudes set side
by side, as well as fi ve cases where a system of homogeneous unequal mag-
nitudes is represented by magnitudes some of which (in contradiction to
the text) are represented equally. Th ere are only four cases where a system
of unequal magnitudes is allowed to be represented by a diagram where all
traces are appropriately unequal.
Th e consequence of this convention is clear: the ancient diagrams are not
read off as metrical. As a corollary, they are read more for their confi gura-
tional information. Th is is obvious from the comparison with Heiberg: in
the latter’s diagram of i .16, the readers’ expectation clearly is not that the
three circles should indeed all be concentric. Indeed, the reader must under-
stand that such fi gures are pure magnitudes and do not stand to each other
in any spatial, confi gurational sense. While the conical surface ABΓ does
indeed envelope the smaller surfaces ΔBE, AΔEΓ, no such envelopment is
understood between the three circles K, Λ and Θ that merely represent three
magnitudes manipulated in the course of the proposition. Now, this does not
make Heiberg’s diagram false. It simply highlights what Heiberg’s reader – in
direct opposition to the reader of the ancient diagrams – is supposed to edit
away in his reading of the diagram. Heiberg’s reader is supposed to edit away
a certain piece of confi gurational information (the circles merely appear to
envelop each other), whereas the ancient reader was supposed to edit away
a certain piece of metrical information (the circles merely appear equal).
One can say that both representational systems foreground one dimension
of information, overruling the other dimension: Th e metrical dimension of
information is foregrounded in Heiberg and overrules the confi gurational
dimension; the confi gurational dimension of information is foregrounded
in the ancient diagram and overrules the metrical dimension.
Th is may serve to elucidate the following. Interestingly, the fi ve cases
where the ancient diagrams represent unequals by unequals – proposi-
tions SC i .15, 33, 34, 44 – all involve lines. Consider the typical case of i .15
(see Figure 3.2 ). B is the radius of the circle A, Γ – the side of a cone set
up on that circle, E – a mean proportional between the two. Th e metrical
relationship B < E < Γ is indeed determined. Further, the circle Δ is drawn