In the cases of Mercury and Venus, Ptolemy’s explanation of the motion could with little difficulty be rearranged so as to fit the ideas of Coppernicus. We have seen (chapter II., § 51) that, minor irregularities being ignored, the motion of either of these planets could be represented by means of an epicycle moving on a deferent, the centre of the epicycle being always in the direction of the sun, the ratio of the sizes of the epicycle and deferent being fixed, but the actual dimensions being practically arbitrary. Ptolemy preferred on the whole to regard the epicycles of both these planets as lying between the earth and the sun. The idea of making the sun a centre of motion having once been accepted, it was an obvious simplification to make the centre of the epicycle not merely lie in the direction of the sun, but actually be the sun. In fact, if the planet in question revolved round the sun at the proper distance and at the proper rate, the same appearances would be produced as by Ptolemy’s epicycle and deferent, the path of the planet round the sun replacing the epicycle, and the apparent path of the sun round the earth (or the path of the earth round the sun) replacing the deferent. [Illustration: FIG. 43.—The orbits of Venus and of the earth.] [Illustration: FIG. 44.—The synodic and sidereal periods of Venus.] In discussing the time of revolution of a planet a distinction has to be made, as in the case of the moon (chapter II., § 40), between the synodic and sidereal periods of revolution. Venus, for example, is seen as an evening star at its greatest angular distance from the sun (as at V in fig. 43) at intervals of about 584 days. This is therefore the time which Venus takes to return to the same position relatively to the sun, as seen from the earth, or relatively to the earth, as seen from the sun; this time is called the =synodic period=. But as during this time the line E S has changed its direction, Venus is no longer in the same position relatively to the stars, as seen either from the sun or from the earth. If at first Venus and the earth are at V_{1}, E_{1}; respectively, after 584 days (or about a year and seven months) the earth will have performed rather more than a revolution and a half round the sun and will be at E_{2}; Venus being again at the greatest distance from the sun will therefore be at V_{2}, but will evidently be seen in quite a different part of the sky, and will not have performed an exact revolution round the sun. It is important to know how long the line S V_{1} takes to return to the same position, _i.e._ how long Venus takes to return to the same position with respect to the stars, as seen from the sun, an interval of time known as the =sidereal period=. This can evidently be calculated by a simple rule-of-three sum from the data given. For Venus has in 584 days gained a complete revolution on the earth, or has gone as far as the earth would have gone in 584 + 365 or 949 days (fractions of days being