198 reviel netz
is measurable, the intersection of the triangular prism and the cylinder
is measured as well.
All this makes sense and we can therefore even understand why
Heiberg was content: his reading, though lacunose, was mathemati-
cally sound. He did remark on the lacuna ‘Quid in tanta lacuna fuerit
dictum, non exputo’ 28 – ‘I do not guess what were to be written in such
a long lacuna’. Th is comment may be prudent, but it accompanies a text
that, otherwise, is meant to be read as mathematically meaningful. In
other words, the implication would be that the missing lacuna was no
more than ornament that does not impinge on the mathematical con-
tents of proposition 14, and it was certainly in this way that proposition
14 was read through the twentieth century. 29
Th e upshot of this reading is indeed to make the proposition less
important, because it contains nothing new. It applies the method of
indivisibles – previously applied in the Method – by assuming that a
certain property obtained for n -dimensional objects is inherited by the
n +1-dimensional objects they constitute. It diff ers from the previous
propositions in a merely negative way – it does not apply geometrical
mechanics – and therefore it makes no contribution to our understand-
ing of Archimedes’ mathematical procedures.
Th is understanding of proposition 14 was revolutionized by the
readings of Netz et al. ( 2001 –2), where the lacuna was fi nally read. It is
clear that this lacuna adds much more than ornament. Indeed, it forms
the mathematical heart of the proof. Archimedes applies certain results
concerning the summation of sets of proportions developed elsewhere,
results that call for counting the number of objects in the sets involved,
with the number of objects in this set equal to the number of objects
in that set. And this – even though the sets involved are infi nite! Th us,
Archimedes does no less than count (by the statement of numeri-
cal equality) infi nite sets. Th e proof is therefore not a mere negative
variation on the previous proofs; to the contrary, it opens up a unique
avenue, completely unlike anything else extant from Greek math-
ematics. Heiberg’s minimal interpretation of the text is thus refuted.
Th ough, of course, this is not to blame Heiberg: what else could he do?
(2) Th e next example comes from the fi nal, fragmentary proposition 15.
Th e fi rst page of this proposition survives on fos. 158–9 of the28 Heiberg 1913: 499, n. 1.
29 See in particular Sato 1986 , Knorr 1996 , texts rare for paying any attention to proposition
14, both assuming that the text extant in Heiberg can be taken to represent Archimedes’ own
reasoning.