will be reversed, and it will be found moving from the east towards the west.  At first it proceeds slowly and then quickens its pace, until a certain speed is attained, which afterwards declines until a second stationary position is reached.  After a due pause the original motion from west to east is resumed, and is continued until a similar cycle of changes again commences.  Such movements as these were obviously quite at variance with any perfect movement in a single circle round the earth.  Here, again, the geometrical sagacity of Ptolemy provided him with the means of representing the apparent movements of Mars, and, at the same time, restricting the explanation to those perfect movements which he deemed so essential.  In Fig. 2 we exhibit Ptolemy’s theory as to the movement of Mars.  We have, as before, the earth at the centre, and the sun describing its circular orbit around that centre.  The path of Mars is to be taken as exterior to that of the sun.  We are to suppose that at a point marked M there is a fictitious planet, which revolves around the earth uniformly, in a circle called the deferent.  This point M, which is thus animated by a perfect movement, is the centre of a circle which is carried onwards with M, and around the circumference of which Mars revolves uniformly.  It is easy to show that the combined effect of these two perfect movements is to produce exactly that displacement of Mars in the heavens which observation discloses.  In the position represented in the figure, Mars is obviously pursuing a course which will appear to the observer as a movement from west to east.  When, however, the planet gets round to such a position as R, it is then moving from east to west in consequence of its revolution in the moving circle, as indicated by the arrowhead.  On the other hand, the whole circle is carried forward in the opposite direction.  If the latter movement be less rapid than the former, then we shall have the backward movement of Mars on the heavens which it was desired to explain.  By a proper adjustment of the relative lengths of these arms the movements of the planet as actually observed could be completely accounted for.

The other outer planets with which Ptolemy was acquainted, namely, Jupiter and Saturn, had movements of the same general character as those of Mars.  Ptolemy was equally successful in explaining the movements they performed by the supposition that each planet had perfect rotation in a circle of its own, which circle itself had perfect movement around the earth in the centre.

It is somewhat strange that Ptolemy did not advance one step further, as by so doing he would have given great simplicity to his system.  He might, for instance, have represented the movements of Venus equally well by putting the centre of the moving circle at the sun itself, and correspondingly enlarging the circle in which Venus revolved.  He might, too, have arranged that the several circles which the outer