These movements were wholly incompatible with the supposition that the journeys of Venus were described by a single motion of the kind regarded as perfect.  It was obvious that the movement was connected in some strange manner with the revolution of the sun, and here was the ingenious method by which Ptolemy sought to render account of it.  Imagine a fixed arm to extend from the earth to the sun, as shown in the accompanying figure (Fig. 1), then this arm will move round uniformly, in consequence of the sun’s movement.  At a point P on this arm let a small circle be described.  Venus is supposed to revolve uniformly in this small circle, while the circle itself is carried round continuously by the movement of the sun.  In this way it was possible to account for the chief peculiarities in the movement of Venus.  It will be seen that, in consequence of the revolution around P, the spectator on the earth will sometimes see Venus on one side of the sun, and sometimes on the other side, so that the planet always remains in the sun’s vicinity.  By properly proportioning the movements, this little contrivance simulated the transitions from the morning star to the evening star.  Thus the changes of Venus could be accounted for by a Combination of the “perfect” movement of P in the circle which it described uniformly round the earth, combined with the “perfect” motion of Venus in the circle which it described uniformly around the moving centre.

In a precisely similar manner Ptolemy rendered an explanation of the fitful apparitions of Mercury.  Now just on one side of the sun, and now just on the other, this rarely-seen planet moved like Venus on a circle whereof the centre was also carried by the line joining the sun and the earth.  The circle, however, in which Mercury actually revolved had to be smaller than that of Venus, in order to account for the fact that Mercury lies always much closer to the sun than the better-known planet.

[Fig. 2.  Ptolemy’s theory of the movement of Mars.]

The explanation of the movement of an outer planet like Mars could also be deduced from the joint effect of two perfect motions.  The changes through which Mars goes are, however, so different from the movements of Venus that quite a different disposition of the circles is necessary.  For consider the facts which characterise the movements of an outer planet such as Mars.  In the first place, Mars accomplishes an entire circuit of the heaven.  In this respect, no doubt, it may be said to resemble the sun or the moon.  A little attention will, however, show that there are extraordinary irregularities in the movement of the planet.  Generally speaking, it speeds its way from west to east among the stars, but sometimes the attentive observer will note that the speed with which the planet advances is slackening, and then it will seem to become stationary.  Some days later the direction of the planet’s movement