182 reviel netz
First, this is an argument headed by the particle gar , usually translated
‘for’: having established a claim, the text moves on to off er further grounds
for it. Heiberg’s tendency, especially in the books on Sphere and Cylinder ,
was to excise a great proportion of gar statements. Th ere are altogether 155
occurrences of the particle in the text of SC i , ii outside of the introductions,
but of these 58 occur not in the context of a backwards-looking argument
but in the context of some meta-mathematical formulaic expression using
a gar , such as the heading of the reductio mode of reasoning: ‘for if pos-
sible’, ei gar dunaton. Remaining are 95 occurrences. Of these 54 are inside
Heiberg’s brackets; only 41 are considered genuine. Th e 54 excised gar s
represent fewer than 50 excisions (a few long passages excised by Heiberg
include more than a single gar ), all of them constituting at least a phrase
(Heiberg never excises a gar alone – which of course would have produced
an asyndeton). Heiberg excised altogether 124 phrases and passages from
the text of SC i , ii , and so we see that about 40 per cent of these excisions are
claimed by gar s. Note however that many of the remaining excisions have a
similar logical character, even while using a connector other than gar : e.g. a
dēlon , ‘clearly’ phrase in SC i .34 130.20–1, or even an ara , ‘therefore’ phrase
in SC i .32 120.8. In most cases, the excision is motivated by the elementary
character of the claim made.
Th is can be seen from the distribution of excisions of gar between the
two treatises. Of the 68 gar s in SC i , Heiberg excises 45 or about two-thirds;
of the 27 gar s in SC ii , Heiberg excises 9, that is a third. Th e major diff erence
between the two treatises is that SC ii is usually much more complex than
most of SC i. 13 Th e rule then begins to emerge: Heiberg excises gar s in the
context of relatively simple mathematics.
Going back fi nally to our example from SC i .4, we can now see one
reason why Heiberg chose to bracket it: in this example, the text looks back
to explain why a certain construction is possible. Th is condition, however,
is relatively simple: in constructing a right-angled triangle, the hypotenuse
must be greater than the side. Heiberg’s view was that Archimedes could
well have just taken such a condition for granted.
For this, Heiberg had something of a corroboration. Here I pass to the
second ground for Heiberg’s excision: his search for consistency. In the
preceding proposition 3, Archimedes requires an analogous construc-
tion, and there the text does not provide an explicit backwards-looking
argument, merely stating ( i 14.8) ‘for this is possible’ (this is bracketed by13 As a comparison: in the advanced treatise Spiral Lines , Heiberg brackets 2 out of 33 gar s –
which forms, however, a large part of his overall editorial intervention in that treatise.