Archimedes’ writings: through Heiberg’s veil 199
Palimpsest, in a form which was mostly illegible to Heiberg. Th ere
follows a gap in the text extant in the Palimpsest, followed by four
considerable fragments extant on fo. 165 of the Palimpsest. Two of
those fragments were nearly fully read by Heiberg, and they formed
a basis for an interpretation of the proof as a whole, one which the
much fuller reading we possess today corroborates on the whole.
Its main feature is the following. In this, purely geometrical proof,
Heiberg makes Archimedes follow a route comparable to that used in
the measurement of conoids of revolution in CS. A sequence of prisms
is inscribed inside the curvilinear object; the diff erence between the
sequence of prisms and the curvilinear object is made smaller than
any stated magnitude; and the assumption that the curvilinear fi gure is
not of the volume stated then leads to contradiction. All of this is well
known from elsewhere in Archimedes and Heiberg had many patterns
to follow – especially from CS itself – in his reconstruction of the text
of fos. 158–9 beginning the proof.
In contrast to proposition 14, where the lacuna unread by Heiberg –
no more than about half a column of writing – proved to be much
richer in mathematical meaning than Heiberg imagined, here, fos.
158–9 contain three and a half columns of writing, mostly unread by
Heiberg, and they contain practically no mathematical signifi cance.
Here the surprise is the opposite to that of proposition 14. Heiberg in his
reconstruction rather quickly establishes the geometrical construction
required for inscribing prisms inside the curvilinear object. Archimedes
himself, however, went through what may have been the most detailed
construction in his entire corpus. Th e construction is much slower
than that of the analogous proofs in CS. At the end of these three and
a half columns of writing, Archimedes had not yet reached the explicit
conclusion that the diff erence between the curvilinear object and the
inscribed prisms is smaller than any given magnitude. It appears that
in making the transition from the unorthodox procedures of proposi-
tions 1–14, to the ‘classical’ procedure of proposition 15, Archimedes
made a deliberate eff ort to make proposition 15 as ‘classical’ as possi-
ble – as explicit and precise as possible. (One of course is reminded of
how Heiberg tends, elsewhere, to doubt passages where Archimedes is
especially explicit and transparent. Would he have excised a good deal
of proposition 15, had he been able to read more of it?)
Archimedes’ motives are diffi cult to judge but the eff ect most cer-
tainly was to emphasize the gap between the two parts of the treatise,
the unorthodox and the orthodox. Th is gap was somewhat smoothed