1 The Birth of Science - MSRI

4.4 Sailing and Navigation 97
The most interesting surviving pages on the relationship between mathematics
and military technology are probably those of Philo of Byzantium.
Talking about the construction of catapults, he says:
Later, through the analysis of former mistakes and the observation of
subsequent experiments, the fundamental principle of the construction
was reduced to a constant element, the diameter of the circle
holding the spring. This was first done by Alexandrian technicians,
who benefited from large subsidies from fame-seeking kings who
supported craftmanship and technique. That everything cannot be
accomplished through pure thought and the methods of mechanics,
but much is found also by experiment, is proved especially by what
I’m about to say. 56
Thus Hellenistic scientists had already enunciated explicitly the relationship
between mathematics and experiments that is usually considered
typical of the Galilean method.
Soon after this passage Philo gives the formula for the diameter of the page 135
opening that the spring (tension rope) goes through, and hence the diameter
of the spring itself, as a function of the weight of the projectile that one
wishes to throw a given distance; the diameter is proportional to the cube
root of the weight, the proportionality constants being given by Philo. The
famous problem of the doubling of the cube (extraction of cube roots) thus
reveals its practical interest in the task of “calibrating” catapults. An ingenious
instrument, the mesolabe, was designed by Eratosthenes to perform
the extraction. 57
4.4 Sailing and Navigation
Hellenistic civilization, like Hellenic civilization before it, was that of a
group of harbor cities connected by sea. Therefore sailing and navigation
techniques were crucial to its economy. What were these techniques? Did
they have any relation to science?
We will restrict ourselves to a few observations on two forms of sailing
that were long considered to have been unknown in antiquity: windward
sailing and open-seas sailing.
Did the “Ancients” know how to sail into the wind? Given that Greek
and Latin have specific expressions for this action ( , facere
56 Philo of Byzantium, Belopoeica, 50, 21–29 = [Marsden: TT], pp. 107–109.
57 The description Eratosthenes gives of his mesolabe was preserved by Eutocius (together with
other solutions of the problem of doubling the cube), in his Commentary on the Archimedean treatise
on the sphere and the cylinder = [Archimedes/Mugler], vol. IV, pp. 64–69. Eratosthenes mentions the
usefulness of his instrument in designing catapults.
Revision: 1.14 Date: 2002/10/24 04:25:47