these an additional _annual_ motion to enable us to represent truly
the apparent motions. This way of looking at the apparent motions
shows why each planet, when nearest to the earth, seems to move for a
time in a retrograde direction. The attempts of Ptolemy and others of
his time to explain the retrograde motion in this way were only
approximate. Let us suppose each planet to have a bar with one end
centred at the earth. If at the other end of the bar one end of a
shorter bar is pivotted, having the planet at its other end, then the
planet is given an annual motion in the secondary circle (the
epicycle), whose centre revolves round the earth on the primary circle
(the _deferent_), at a uniform rate round the excentric. Ptolemy
supposed the centres of the epicycles of Mercury and Venus to be on a
bar passing through the sun, and to be between the earth and the
sun. The centres of the epicycles of Mars, Jupiter, and Saturn were
supposed to be further away than the sun. Mercury and Venus were
supposed to revolve in their epicycles in their own periodic times and
in the deferent round the earth in a year. The major planets were
supposed to revolve in the deferent round the earth in their own
periodic times, and in their epicycles once in a year.

It did not occur to Ptolemy to place the centres of the epicycles of
Mercury and Venus at the sun, and to extend the same system to the
major planets. Something of this sort had been proposed by the
Egyptians (we are told by Cicero and others), and was accepted by
Tycho Brahe; and was as true a representation of the relative motions
in the solar system as when we suppose the sun to be fixed and the
earth to revolve.

The cumbrous system advocated by Ptolemy answered its purpose,
enabling him to predict astronomical events approximately. He improved