e by a point of a circle or sphere, which
itself is carried along at the same time, is some kind of cycloid; if
the centre of the tracing circle travels along a straight line, we get
the ordinary cycloid, the curve traced in air by a nail on a
coach-wheel; but if the centre of the tracing circle be carried round
another circle the curve described is called an epicycloid. By such
curves the planetary stations and retrogressions could be explained. A
large sphere would have to revolve once for a "year" of the particular
planet, carrying with it a subsidiary sphere in which the planet was
fixed; this latter sphere revolving once for a "year" of the earth. The
actual looped curve thus described is depicted for Jupiter and Saturn in
the annexed diagram (fig. 10.)

It was long ago perceived that real material spheres were
unnecessary; such spheres indeed, though possibly transparent to
light, would be impermeable to comets: any other epicyclic gearing
would serve, and as a mere description of the motion it is simpler
to think of a system of jointed bars, one long arm carrying a
shorter arm, the two revolving at different rates, and the end of
the short one carrying the planet. This does all that is needful
for the first approximation to a planet's motion. In so far as the
motion cannot be thus truly stated, the short arm may be supposed
to carry another, and that another, and so on, so that the
resultant motion of the planet is compounded of a large number of
circular motions of different periods; by this device any required
amount of complexity could be attained. We shall return to this at
greater length in Lecture III.

The main features of the motion, as shown in the diagram, required
only two arms for their expression; one arm revolving with the
average motion of the planet, and the other revolving with the
apparent motion of the sun, and always pointing in the same
direction as the single arm supposed to carry the sun. This last
fact is of course because the motion to be represented does not
really belong to the planet at all, but to the earth, and so all
the main epicyclic motions for the superior planets were the same.
As for the inferior planets (Mercury and Venus) they only appear
to oscillate like the bob of a pendulum about the sun, and so it is
very obvious that they must be really revolvin