The motion being thus represented geometrically, it became merely a matter of not very difficult calculation to construct a table from which the position of the sun for any day in the year could be easily deduced. This was done by computing the so-called =equation of the centre=, the angle C S E of fig. 17, which is the excess of the actual longitude of the sun over the longitude which it would have had if moving uniformly.
Owing to the imperfection of the observations used (Hipparchus estimated that the times of the equinoxes and solstices could only be relied upon to within about half a day), the actual results obtained were not, according to modern ideas, very accurate, but the theory represented the sun’s motion with an accuracy about as great as that of the observations. It is worth noticing that with the same theory, but with an improved value of the eccentricity, the motion of the sun can be represented so accurately that the error never exceeds about 1′, a quantity insensible to the naked eye.
The theory of Hipparchus represents the variations in the distance of the sun with much less accuracy, and whereas in fact the angular diameter of the sun varies by about 1∕30th part of itself, or by about 1′ in the course of the year, this variation according to Hipparchus should be about twice as great. But this error would also have been quite imperceptible with his instruments.
[Illustration: FIG. 19.—The epicycle and the deferent.]
Hipparchus saw that the motion of the sun could equally well be represented by the other device suggested by Apollonius, the =epicycle=. The body the motion of which is to be represented is supposed to move uniformly round the circumference of one circle, called the epicycle, the centre of which in turn moves on another circle called the =deferent=. It is in fact evident that if a circle equal to the eccentric, but with its centre at E (fig. 19), be taken as the deferent, and if S′ be taken on this so that E S′ is parallel to C S, then S′ S is parallel and equal to E C; and that therefore the sun S, moving uniformly on the eccentric, may equally well be regarded as lying on a circle of radius S′ S, the centre S′ of which moves on the deferent. The two constructions lead in fact in this particular problem to exactly the same result, and Hipparchus chose the eccentric as being the simpler.