168 reviel netz
around the radius E. It thus follows also that A < Δ. Th e diagram displays
the inequality between the lines B < E < Γ but not the equally determined
inequality between the circles A < Δ. Th ere are six other cases, however,
where unequal lines are represented by equal diagram traces. Th e rule then
appears to be that the manuscripts’ diagrams have a very strong prefer-
ence to mark unequal plane fi gures as equal, but only a tendency to mark
unequal line segments as unequal. Why should that be the case? Clearly,
lines are less confi gurationally charged than plane fi gures are. Th e represen-
tation of a system of line traces does not suggest so powerfully a confi gura-
tion made of those lines in spatial arrangement, and it is easier to read as a
purely quantitative representation (indeed, such lines form the principle of
representation used by Greek mathematicians when dealing with numbers
or with general magnitudes, whose signifi cance is purely quantitative, as in
Euclid’s Elements v , vii – ix ). Th e principle is clear, then: the more the dia-
grams are taken to convey confi gurational meaning, the less metrical they
are made. Lines – whose non-confi gurational character is easy to establish –
may sometimes take metrical characteristics; but with plane fi gures, metri-
cal characteristics are altogether avoided.
Th e upshot of this is obvious: diagrams which mostly carry confi gura-
tional information, to the exclusion of the metrical, can also be rigorous. As
Poincaré pointed out long ago, diagrams may be geometrically correct, to
the extent that they are taken to be purely topological. 3 Of course, Poincaré(^3) Poincaré 1913 : 60. Needless to say, topology or ‘analysis situs’ (as Poincaré would say) meant
something diff erent a century ago: in particular, this to Poincaré had absolutely nothing to do
with Set Th eory and instead had everything to do with a study of spatial relations abstracted
away from any metrical conditions – which of course makes ‘topology’ even more obviously
relevant to the study of schematic diagrams.
Figure 3.2 A reconstruction of Archimedes’ diagram for Sphere and Cylinder i.15.
A Δ
BEΓ