Archimedes’ writings: through Heiberg’s veil 201
related passage that was easier to read – just as he reconstructed
proposition 15 on the basis of CS, and proposition 14 on the basis
of propositions 1–11. How else would you reconstruct, if not on the
basis of what you have available? But this immediately suggests that
the act of reconstruction has, automatically, a signifi cant consequence:
if reconstruction is necessarily based on what one has available,
reconstruction necessarily tends to homogenize the text. Hence 14
appears like 1–11; 15 appears like CS; 6 appears like 9. Th e Method as a
whole loses something of its internal variety and of its diff erence from
other parts of the corpus.
In truth, of course, the Method is all about diff erence. It is diff erent
from the rest of the corpus; it highlights internal variety, where the
original procedure contrasts with ‘classical’, geometrical approaches.
Aft er all, what is the point of supplying three separate proofs of the
same result (propositions 12–13, proposition 14, proposition 15) if not
to highlight the diff erence between all of them? Th is can be seen at all
levels. I have concentrated on the global forms of marking diff erence,
but one can fi nd such forms at a more local level. We may return to
proposition 14 to take a closer look at its unfolding. Th e proposition
falls into three parts: (a) a geometrical passage showing that a certain
proportion holds, (b) a proportion theory passage showing that this
proportion may be summed up for sets of infi nite multitude and (c)
an arithmetical passage calculating the numerical value of the segment
of the cylinder measured. Heiberg did not read (b) at all, and had to
reconstruct large parts of (a). Th e only part he could read in full was (c),
which is indeed surprisingly careful and detailed. Heiberg’s reconstruc-
tion ignored (b), and produced a careful and detailed development of
(a). In Heiberg’s reading, therefore, the proposition unfolded in an
uninterrupted progression of careful geometrical argument, followed
by a transition based directly on the method of indivisibles (and thus
merely reduplicating propositions 1–11) leading to another careful,
arithmetical argument.
Following Netz et al. ( 2001 –2), we now know that the structure
of the proof is much more unwieldy. Remarkably, passage (a) hardly
possesses any argument. Th e diffi cult and remarkable geometrical con-
clusion required by Archimedes is thrust upon the reader as a given.
Th is is then followed by the subtle and diffi cult argument of (b), leading
fi nally to the much simpler passage (c) which now, in context, is truly
startling in its slow development of such an obvious claim. Archimedes
fi rst states a diffi cult result as obvious, then outlines the most diffi cult