1 The Birth of Science - MSRI

3.8 Ptolemaic Astronomy 81
early Hellenistic ideas and Ptolemy’s on such subjects as space and motion,
limiting ourselves here to some observations on the mathematical
model used in the Almagest. Everyone knows that Ptolemy’s planetary
theory is based on the composition of circular motions. This technique had
been used in astronomy by Apollonius of Perga, and it was in fact much
older: the first algorithm of this type that we know about was worked out page 118
by Eudoxus, who described planetary motions as obtained from a succession
of concentric spheres rotating uniformly, each with an axis of rotation
marked by two points fixed on the preceding sphere.
Because many strange notions about this method of decomposing motions
are still widespread, we spell out the two main reasons why it was
supremely well-adapted to the purposes to which it was put.
First, accounting for the observed motion of planets as the composite of
several uniform motions on circular orbits (the first centered on the earth
and called the deferent in medieval terminology, and each of the others,
called epicycles, centered on the point obtained on the preceding circumference)
is equivalent to a modern expansion in Fourier series, and allows
an efficient description of observed data with increasing precision as the
number of epicycles grows. The analogy between Fourier series expansions
and developments in epicycles was observed by Schiaparelli, 131 but
one can imagine that the thought occurred earlier. 132 A formal demonstration
of the equivalence has been given by Giovanni Gallavotti. 133
Second, since the main computational tool of Hellenistic mathematics
was geometric algebra performed with ruler and compass, decomposition
into circular motions was the most efficient possible system for computing
the observable position of planets. 134 If, for example, the motion of a planet
is described as a combination of three uniform circular motions, in order
to compute the position at a given instant it is only necessary to draw three
arcs of circle and measure out three angles obtained by multiplication. 134a
Thus the calculation is reduced to a very few arithmetic operations and six page 119
131
[Schiaparelli], part I, vol. II, p. 11.
132
One can conjecture, in fact, that in this important observation Schiaparelli was preceded by
the mathematicians who developed the idea of Fourier series expansions (the first being Daniel
Bernoulli in the eighteenth century, who also studied planetary motion and surely knew about
developments in epicycles).
133
In [Gallavotti: QPM], which contains an interesting translation into modern mathematical
terms of the main ideas from the systems of Hipparchus, Ptolemy and Copernicus.
134
Just a few epicycles suffice to account for the position of the planets to within the precision
afforded even by modern experimental data.
134a
It’s worth noting that, since multiplications were carried out in sexagesimal notation and angles
were measured, accordingly, in degrees, the result of multiplying an angular velocity by time
is immediately reducible to a “plottable” angle of less than 360 degrees. If the same task is performed
(as today) in decimal arithmetic using the same degree units, every such multiplication
must be followed by an extraction of the remainder under division by 360.
Revision: 1.13 Date: 2002/10/16 19:04:00