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
Fig. 35.—The equant.
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.
By means of a system of this kind, worked out with great care, and evidently at the cost of enormous labour, Ptolemy was able to represent with very fair exactitude the motions of the planets, as given by the observations in his possession.
It has been pointed out by modern critics, as well as by some mediaeval writers, that the use of the equant (which played also a small part in Ptolemy's lunar theory) was a violation of the principle of employing only uniform circular motions, on which the systems of Hipparchus and Ptolemy were supposed to be based, and that Ptolemy himself appeared unconscious of his inconsistency. It may, however, fairly be doubted whether Hipparchus or Ptolemy ever had an abstract belief in the exclusive virtue of such motions, except as a convenient and easily intelligible way of representing certain more complicated motions, and it is difficult to conceive that Hipparchus would have scrupled any more than his great follower, in using an
