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 V1 (fig. 44) such that V1 E1 touches the circle in which Venus moves, and the angle E1 V1 S is then (by a known property of a circle) a right angle. The angle S E1 V1 being observed, the shape of the triangle S E1 V1 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.
Fig. 45.—The epicycle of Jupiter.
87. In the case of the superior planets. Mars, Jupiter, and Saturn, it was much more difficult to recognise that their motions could be explained by supposing them to revolve round the sun, since the centre of the epicycle did not always lie in the direction of the sun, but might be anywhere in the ecliptic. One peculiarity, however, in the motion of any of the superior planets might easily have suggested their motion round the sun, and was either completely overlooked by Ptolemy or not recognised by him as important. It is possible that it was one of the clues which led Coppernicus to his system. This peculiarity is that the radius of the epicycle of the planet, j J, is always parallel to the line E S joining the earth and sun, and consequently performs a complete revolution in a year. This connection between the motion of the planet and that of the sun received no explanation from Ptolemy’s theory. Now if we draw E J′ parallel to j J and equal to it in length, it is easily seen[55] that the line J′ J is equal and parallel to E j, that consequently J describes a circle round J′ just as j round E. Hence the motion of the planet can equally well be represented by supposing it to move in an epicycle (represented by the large dotted circle in the figure) of which J′ is the centre and J′ J the radius, while the centre of the epicycle, remaining always in the direction of the sun, describes a deferent (represented by the small circle round E) of which the earth is the centre. By this method of representation the motion of the superior planet is exactly like that of an inferior planet, except that its epicycle is larger than its deferent; the same reasoning as before shows that the motion can be represented simply by supposing the centre J′ of the epicycle to be actually the sun. Ptolemy’s epicycle and deferent are therefore capable of being replaced, without affecting the position of the planet in the sky, by a motion of the planet in a circle round the sun, while the sun moves round the earth, or, more simply, the earth round the sun.
Fig. 46.—The relative sizes of the orbits of the earth and of a superior planet.
The synodic period of a superior planet could best be determined by observing when the planet was in opposition, i.e. when it was (nearly) opposite the sun, or, more accurately (since a planet does not move exactly in the ecliptic), when the longitudes of the planet and sun differed by 180° (or two right angles, chapter II., [§ 43]). The sidereal period could then be deduced nearly as in the case of an inferior planet, with this difference, that the superior planet moves more slowly than the earth, and therefore loses one complete revolution in each synodic period; or the sidereal period might be found as before by observing when oppositions occurred nearly in the same part of the sky.[56] Coppernicus thus obtained very fairly accurate values for the synodic and sidereal periods, viz. 780 days and 687 days respectively for Mars, 399 days and about 12 years for Jupiter, 378 days and 30 years for Saturn (cf. fig. 40).
The calculation of the distance of a superior planet from the sun is a good deal more complicated than that of Venus or Mercury. If we ignore various details, the process followed by Coppernicus is to compute the position of the planet as seen from the sun, and then to notice when this position differs most from its position as seen from the earth, i.e. when the earth and sun are farthest apart as seen from the planet. This is clearly when (fig. 46) the line joining the planet (P) to the earth (E) touches the circle described by the earth, so that the angle S P E is then as great as possible. The angle P E S is a right angle, and the angle S P E is the difference between the observed place of the planet and its computed place as seen from the sun; these two angles being thus known, the shape of the triangle S P E is known, and therefore also the ratio of its sides. In this way Coppernicus found the average distances of Mars, Jupiter, and Saturn from the sun to be respectively about 1-1∕2, 5, and 9 times that of the earth; the corresponding modern figures are 1·5, 5·2, 9·5.