1 The Birth of Science - MSRI

66 3. Other Hellenistic Scientific Theories
object is buoyed up with a force equal to the weight of liquid it displaces.
But, contrary to the impression we generally get in school, Archimedes’
hydrostatics is not limited to this statement. The typical problems that
Archimedes solves in his treatise are finding the waterline for solids in
equilibrium in a homogeneous liquid, and above all whether the equilibrium
position is stable. The most interesting result along these lines
is about an arbitrary floating solid in the shape of a right paraboloid of
revolution (that is, a paraboloid truncated perpendicularly to the rotation
axis). The stability of equilibrium in the upright position is studied
as a function of two parameters: the form factor, which says how “fat” the
paraboloid is, and the density of the solid. The result says essentially that if
the paraboloid is fat (shallow) enough, upright equilibrium is always stable,
whereas if it’s skinny beyond a certain threshold, 71 the density also acquires
a threshold value (which he calculates) below which upright equilibrium
is unstable, the stable positions being those where the paraboloid page 101
is tilted by a certain angle that depends on the density. This study would
be regarded today as an application of bifurcation theory, and according to
Dijksterhuis it “deserves the highest admiration of the present-day mathematician,
both for the high standard of the results obtained, which would
seem to be quite beyond the pale of classical mathematics, and for the ingenuity
of the argument”. 72 Evidently, in present-day opinion the pale of
“classical mathematics” does not even reach as far as the few surviving
works of its most famous luminary.
That Hellenistic scientists were conscious of the “theoretical model” nature
of scientific theories is made clear by the use of not one but two such
models in what we know of Archimedean hydrostatics. Indeed, the first
book of On floating bodies derives from the postulate already quoted the
fact that the surface of the oceans is spherical, whereas in the second book
the surface of the liquid is assumed to be flat. 73 Clearly we are dealing
with two different models, appropriate for phenomena at different scales.
If the communicating tube is not horizontal, the deduction is a bit more involved, but it can be
derived as an exercise by anyone who has read carefully the first few propositions of On floating
bodies, book II.
The principle of communicating vessels has generally been attributed to Heron, who uses it in
the Pneumatica and in the Dioptra. But it was certainly known empirically before Archimedes; Plato
mentions that water will flow through a wool thread from the fuller to the emptier of two cups
(Symposium, 175d, 6–7), implicitly supposing the two cups to be identical and placed on the same
table.
71
Namely, if the height of the segment of paraboloid exceeds three quarters of the latus rectum
of the generating parabola.
72
[Dijksterhuis: Archimedes], p. 380.
73
The surface is implicitly assumed flat from the beginning and Archimedes does not spend a
single word in justifying this assumption as an approximation of the “true” spherical shape.
Revision: 1.13 Date: 2002/10/16 19:04:00