Deferent and Epicycle - Mathematical Formalism

Mathematical Formalism

According to the historian of science Norwood Russell Hanson:

There is no bilaterally-symmetrical, nor excentrically-periodic curve used in any branch of astrophysics or observational astronomy which could not be smoothly plotted as the resultant motion of a point turning within a constellation of epicycles, finite in number, revolving around a fixed deferent.

Any path—periodic or not, closed or open—can be represented with an infinite number of epicycles.

This is because epicycles can be represented as a complex Fourier series; so, with a large number epicycles, very complicated paths can be represented in the complex plane.

Let the complex number

,

where and are constants, is an imaginary number, and is time, correspond to a deferent centered on the origin of the complex plane and revolving with a radius and angular velocity

,

where is the period.

If is the path of an epicycle, then the deferent plus epicycle is represented as the sum

.

Generalizing to epicycles yields

,

which is a complex Fourier series. Finding the coefficients to represent a time-dependent path in the complex plane, is the goal of reproducing an orbit with deferent and epicycles, and this is a way of "saving the phenomena" (σώζειν τα φαινόμενα).

This parallel was noted by Giovanni Schiaparelli. Pertinent to the Copernican Revolution debate of "saving the phenomena" versus offering explanations, one can understand why Thomas Aquinas, in the 13th century, wrote:

Reason may be employed in two ways to establish a point: firstly, for the purpose of furnishing sufficient proof of some principle . Reason is employed in another way, not as furnishing a sufficient proof of a principle, but as confirming an already established principle, by showing the congruity of its results, as in astronomy the theory of eccentrics and epicycles is considered as established, because thereby the sensible appearances of the heavenly movements can be explained; not, however, as if this proof were sufficient, forasmuch as some other theory might explain them.