ematics was almost entirely geometrical
for accomplishing what a later generation could accomplish by an
algebraic theory of functions. As has been pointed out, the undertaking
of the ancient mathematical astronomer to resolve the motions of
planetary bodies into circular, uniform, continuous, symmetrical
movements is comparable to the theorem of Fourier which allows the
mathematician to replace any one periodic function by a sum of circular
functions. In other words, the astronomy of the Alexandrian period is a
somewhat cumbrous development of the mathematical technique of the time
to enable the astronomer to bring the anomalies of the planetary bodies,
as they increased under observation, within the axioms of a metaphysical
physics. The genius exhibited in the development of the mathematical
technique places the names of Apollonius of Perga, Hipparchus of Nicaea,
and Ptolemy among the great mathematicians of the world, but they never
felt themselves free to attack by their hypotheses the fundamental
assumptions of the ancient metaphysical doctrine of the universe. Thus
it was said of Hipparchus by Adrastus, a philosopher of the first
century A. D., in explaining his preference for the epicycle to the
eccentric as a means of analyzing the motions of the planetary bodies:
"He preferred and adopted the principle of the epicycle as more probable
to his mind, because it ordered the system of the heavens with more
symmetry and with a more intimate dependence with reference to the
center of the universe. Although he guarded himself from assuming the
role of the physicist in devoting himself to the investigations of the
real movements of the stars, and in undertaking to distinguish between
the motions which nature has adopted from those which the appearances
present to our eyes, he assumed that every planet revolved along an
epicycle, the center of which describes a circumference concentric with
the earth." Even mathematical astronomy does not offer an exception to
the scientific method of the ancient world, that of bringing to
consciousness the concepts involved in their world of experience,
organizing these concepts with reference to each, analyzing and
restating them within the limits of their essential accidents, and
assimilating the concrete objects of experience to these typical forms
as more or less complete realizations.

At the beginning of the process of Greek self-conscious reflection and
analysis, the mind ran rio