himself (in the introduction to Meth.). Archime ̄de ̄s has put his signature on this technique,
through his wide ranging and elegant application of it. It was following on his lead that
modern mathematicians strove to transform this technique into the project of measuring,
systematically, all curvilinear objects – a project giving rise to the calculus.
In both PE and CF, Archime ̄de ̄s applies mathematics rigorously to the physical world.
Starting from simple assumptions – e.g. that equal weights balance at equal distances (PE) or
that the columns of a liquid at rest all press down with equal force (CF) – he derives by pure
logic the main principles of Statics and Hydrostatics, respectively – Archime ̄de ̄s’ Laws of
the Lever and Buoyancy. From these, Archime ̄de ̄s derives special results such as the finding
of centers of weight, ultimately deriving complex results of a more geometrical character,
once again having to do with curvilinear objects: e.g. the center of weight of a parabolic
segment (PE II) and the hydrostatic properties of certain conoids of revolution (CF II). The
rigorous application of mathematics to physics appears to have been original to Archime ̄de ̄s
and served as major inspiration to the scientific revolution.
Meth. combines Archime ̄de ̄s’ interest in measuring curvilinear objects with mathematical
physics, in subtle and surprising ways. In most propositions of this treatise, plane figures (or
solid figures) are sliced by parallel lines (or parallel planes), and results are obtained for the
center of gravity of each of the resulting linear segments (or planar segments). Those results
are then summed up as a result for the center of gravity of the plane or solid figure as a
whole, giving rise to its measurement. This technique prefigures Cavalieri’s indivisibles
(1635). Unfortunately this treatise was discovered only with the appearance of the Palimp-
sest in 1906 and so did not contribute to the scientific revolution. In 2001, a new reading
revealed Archime ̄de ̄s’ use of actual infinity, in the course of applying proportion theory to a
geometrical arrangement involving indivisibles (Netz, Saito and Tchernetska). This appears
to be unique in the extant Greek corpus.
While Archime ̄de ̄s’ most remarkable achievements are qualitative in character (in either
pure geometry or in mathematical physics) many of his works involve detailed calculation.
His bounds for the value of π (to use the modern notation), 3^1 / 7 ≥ π ≥ 310 / 71 , are obtained in
DC based on a whole set of numerical results, including an approximation of √3. Aren.
states the number of grains of sand it takes to fill up the universe; Bov. is a staggeringly
difficult numerical puzzle; Poly., at least in the form reported by Pappos, appears to have
been primarily a numerical study in the faces, edges and vertices of solids; finally, it has been
suggested recently that Stom. formed a study in geometrical combinatorics, counting the
number of ways in which a certain jigsaw puzzle can be put together (Netz, Acerbi and
Wilson).
- Scientific Personality. The subject matters chosen by Archime ̄de ̄s all revolve
around the surprising: curved objects are equal to straight ones; physical objects obey geo-
metrical laws; apparently impossible calculations are executed. Such results are always
shown through elegant and surprising routes. In a typical work, Archime ̄de ̄s builds up an
arsenal of apparently unrelated results which then unexpectedly combine to yield the
main result of the treatise. No allusion is ever made, within the works themselves, to any
extra-mathematical interests, and it appears that Archime ̄de ̄s saw himself (at least in his
persona of a scientific author) as a pure mathematician, dedicated to the pure pursuit of
proofs.
Archime ̄de ̄s’ manner of proof is more difficult to study. While a substantial part of
Archime ̄de ̄s’ work does survive, the works as transmitted contain what appears to be obvi-
ous later glosses, in places quite substantial. It is thus to some extent a matter of conjecture
ARCHIME ̄DE ̄S OF SURAKOUSAI