omitted for simplicity); hence Venus goes in 584 × 365∕949 days as far as the earth in 365 days, _i.e._ Venus completes a revolution in 584 × 365∕949 or 225 days. This is therefore the sidereal period of Venus. The process used by Coppernicus was different, as he saw the advantage of using a long period of time, so as to diminish the error due to minor irregularities, and he therefore obtained two observations of Venus at a considerable interval of time, in which Venus occupied very nearly the same position both with respect to the sun and to the stars, so that the interval of time contained very nearly an exact number of sidereal periods as well as of synodic periods. By dividing therefore the observed interval of time by the number of sidereal periods (which being a whole number could readily be estimated), the sidereal period was easily obtained. A similar process shewed that the synodic period of Mercury was about 116 days, and the sidereal period about 88 days. The comparative sizes of the orbits of Venus and Mercury as compared with that of the earth could easily be ascertained from observations of the position of either planet when most distant from the sun. Venus, for example, appears at its greatest distance from the sun when at a point V_{1} (fig. 44) such that V_{1} E_{1} touches the circle in which Venus moves, and the angle E_{1} V_{1} S is then (by a known property of a circle) a right angle. The angle S E_{1} V_{1} being observed, the shape of the triangle S E_{1} V_{1} is known, and the ratio of its sides can be readily calculated. Thus Coppernicus found that the average distance of Venus from the sun was about 72 and that of Mercury about 36, the distance of the earth from the sun being taken to be 100; the corresponding modern figures are 72·3 and 38·7. [Illustration: FIG. 45.—The epicycle of Jupiter.]