appropriately the size and rapidity of motion of the epicycle. It is moreover evident that with this arrangement the apparent motion of Jupiter will vary considerably, as the two motions—that on the epicycle and that of the centre of the epicycle on the deferent—are sometimes in the same direction, so as to increase one another’s effect, and at other times in opposite directions. Thus, when Jupiter is most distant from the earth, that is at J_{3}, the motion is most rapid, at J_{1} and J_{2} the motion as seen from the earth is nearly the same as that of _j_; while at J_{4} the two motions are in opposite directions, and the size and motion of the epicycle having been chosen in the way indicated above, it is found in fact that the motion of the planet in the epicycle is the greater of the two motions, and that therefore the planet when in this position appears to be moving from east to west (from left to right in the figure), as is actually the case. As then at J_{1} and J_{2} the planet appears to be moving from west to east, and at J_{4} in the opposite direction, and sudden changes of motion do not occur in astronomy, there must be a position between J_{1} and J_{4}, and another between J_{4} and J_{2}, at which the planet is just reversing its direction of motion, and therefore appears for the instant at rest. We thus arrive at an explanation of the stationary points (chapter I., § 14). An exactly similar scheme explains roughly the motion of Mercury and Venus, except that the centre of the epicycle must always be in the direction of the sun. [Illustration: FIG. 34.—Jupiter’s epicycle and deferent.] Hipparchus, as we have seen (§ 41), found the current representations of the planetary motions inaccurate, and collected a number of fresh observations. These, with fresh observations of his own, Ptolemy now employed in order to construct an improved planetary system. As in the case of the moon, he used as deferent an eccentric circle (centre C), but instead of making the centre _j_ of the epicycle move uniformly in the deferent, he introduced a new point called an =equant= (E′), situated at the same distance from the centre of the deferent as the earth but on the opposite side, and regulated the motion of _j_ by the condition that the apparent motion _as seen from the equant_ should be uniform; in other words, the angle A E′ _j_ was made to increase uniformly. In the case of Mercury (the motions of which have been found troublesome by astronomers of all periods), the relation of the equant to the centre of the epicycle was different, and the latter was made to move in a small circle. The deviations of the planets from the ecliptic (chapter I., §§ 13, 14) were accounted for by tilting up the planes of the several deferents and epicycles so that they were inclined to the ecliptic at various small angles. [Illustration: FIG. 35.—The equant.]