1 The Birth of Science - MSRI

64 3. Other Hellenistic Scientific Theories
chanics was based on the slow accretion of craftsmen’s experience. The
qualitative leap made possible by science lay in that now one could compute
the mechanical advantage theoretically, and so for the first time design
a machine from first principles. This leap surely took place as early as
the third century B.C. Pappus 65 and Plutarch 66 tell us that Archimedes had page 98
solved the problem of lifting a given weight with a given force; in other
words, he know how to design a machine with a specified mechanical advantage.
There is no reason to doubt these sources, since the theoretical
bases of such a solution are given by Archimedes in his extant work and
several applications of his designs were reported by various authors. We
also know that the same period witnessed the introduction, perhaps due
to Archimedes himself, of a piece of technology still used today for many
problems of this type: the gear. 67
Hellenistic mechanics is closely connected to geometry. Diogenes Laertius
states that Archytas (first half of the fourth century B.C.) was the first
not only to introduce concepts from mechanics in the study of geometry
(using lines generated by moving figures to construct the two proportional
means between magnitudes), but also to treat mechanical questions using
mathematical principles. 68
The close link between geometry and mechanics, understood as two
scientific theories, is clear in Archimedes. First of all, his On the equilibrium
of plane figures, which founds the study of simple machines, borrows
from geometry not only the general form of the deductive scheme, but
also many particular technical results. Much more surprising to us today
is that Archimedes uses the laws of mechanics to discover theorems of geometry.
In his Quadrature of the parabola, the rigorous proof we reproduced
in Section 2.7 is preceded by a heuristic discussion based on the principle
of the lever. Likewise, the volume of the sphere is found by imagining
the balancing of a spherical and a cylindrical object, each placed on one
plate of a balance. This procedure is explained systematically in the treatise
The method, where Archimedes expounds the two distinct methods he
uses, respectively, for discovering mathematical results and for proving page 99
them rigorously. The geometric method is used only as a second step, to
prove propositions already identified as plausible. For the discovery of
propositions he uses instead the “mechanical” method, which he consid-
65
Pappus, Collectio, VIII, 1068, 20 (ed. Hultsch).
66
Plutarch, Vita Marcelli, xiv, 7.
67
See Section 4.1.
68
Diogenes Laertius, Vitae philosophorum, VIII, 83. The construction given by Archytas for the
two proportional means is reported by Eutocius in his commentary to Archimedes’ On the sphere
and cylinder (pp. 62–64 in [Archimedes/Mugler], vol. IV). Plato reproached Archytas for having
contaminated geometry with mechanics (Plutarch, Quaestionum convivalium libri iii, 718E–F).
Revision: 1.13 Date: 2002/10/16 19:04:00