II (CF I, II) (also a single work divided into two rolls); Stomakhion (Stom.); Cattle Problem
(Bov.).
DC contains obvious mistakes, alongside some inspired mathematics, and it is often
assumed that the extant version is a corruption of an original work by Archime ̄de ̄s (Knorr
1989, part 3). Similar doubts were raised concerning PE I (Berggren). Still, the presence of
Doric dialect in PE, CF, as well as some ancient testimonies connecting the contents of DC,
Stom., and Bov. to Archime ̄de ̄s, make us believe that indeed all the works extant in Greek
are by Archime ̄de ̄s himself – even if in corrupt form. To this should be added the treatise
On Polyhedra (Poly.) which P (Coll. 5.19, pp. 2.352–358 H.) describes in detail.
Ancient testimony mentions several more works (in a few cases, inside the works of
Archime ̄de ̄s himself), but evidence is meager and allows no firm conclusions. The mention
of a work on Optics, though, is especially intriguing, as this field – otherwise not very well
attested in Archime ̄de ̄s’ time – would provide ample opportunities for Archime ̄de ̄s’ genius in
mathematical physics (Knorr 1985 connects Archime ̄de ̄s and the Catoptrics ascribed to
Euclid).
Many Arabic treatises are ascribed to Archime ̄de ̄s, but most go beyond the Greek corpus.
Four of these are usually taken to be authentic (even if in a more or less mediated form:
Sesiano): Construction of the Regular Heptagon (Hept.); Tangent Circles (Tan.); Lemmas (Lem.);
Assumptions (Assum.).
With the exception of Bov. (surviving through collections of epigrams), all Greek works
are transmitted through one or more of three early Byzantine MSS. One of these, a collec-
tion of various mechanical works, only some by Archime ̄de ̄s, was lost and is known only
through Moerbeke’s Latin translations of PE and CF, partly based on this MS. The two
remaining MSS seem to form a kind of “collected works of Archime ̄de ̄s” (Medieval scien-
tific MSS are arranged typically by subject matter rather than author, so that those
“collected works” are an exception testifying to Archime ̄de ̄s’ stature). One of these two,
containing SC I, SC II, DC, CS, SL, PE I-II, Aren., QP, became lost in the Renaissance,
after serving as another source for Moerbeke’s translation as well as for numerous copies.
The second, “The Archime ̄de ̄s Palimpsest,” contains PE II (end only), CF I-II, Meth., SL,
SC I, SC II, DC, Stom. (beginning only). This 10th c. Byzantine MS, turned in the 13th c.
into a prayer book, was rediscovered by Heiberg in 1906. Again lost, it resurfaced in an
auction in 1998 and has been recently the subject of intensive study, giving rise to notable
changes in the received text.
- Major Areas of Discovery. The most significant body of Archime ̄de ̄s’ work con-
cerns measuring curvilinear objects, based on the technique of “exhaustion,” which
prefigures the calculus (SC I, SC II: sphere; CS: conoids of revolution; SL: spirals, QP:
parabolic segment, Meth.: various figures). In the most general terms, the method of
exhaustion works as follows (Fig.). The curvilinear object is bound (from above, below, or
both) by a complex rectilinear object whose difference, or ratio, of volume or area, from the
given curvilinear object, can be made indefinitely small. Typically, this involves dividing the
curvilinear object into an indefinitely large number of sections, each of which is circum-
scribed or inscribed by a respective section of the complex rectilinear object. Certain meas-
urements are then made for the rectilinear objects and, based on these measurements, one
shows through contradiction that the curvilinear object must possess the specified measure
(or else for instance it can be made smaller than a rectilinear object it circumscribes, etc.).
E had already applied the same technique for the measurement of the cone (Elements
12.10) – a result ascribed to E K, partly on the authority of Archime ̄de ̄s
ARCHIME ̄DE ̄S OF SURAKOUSAI