A Short History of Astronomy by Arthur Berry (best novels for students .TXT) π
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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 72Β·3 and 38Β·7.
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 Ptolemyβs theory. Now if we draw E Jβ² parallel to j J and equal to it in length, it is easily seen55 that the line Jβ² J is equal and parallel to E j, that consequently J describes a circle round Jβ² 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 Jβ² is the centre and Jβ² 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 Jβ² of the epicycle to be actually the sun. Ptolemyβs 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.
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 180Β° (or two right angles, chapter II., Β§ 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 1-1β2, 5, and 9 times that of the earth; the corresponding modern figures are 1Β·5, 5Β·2, 9Β·5.
88. The explanation of the stationary points of the planets (chapter I., Β§ 14) is much simplified by the ideas of Coppernicus. If we take first an inferior planet, say Mercury (fig. 47), then when it lies between the earth and sun, as at M (or as on Sept. 5 in fig. 7), both the earth and Mercury are moving in the same direction, but a comparison of the sizes of the paths of Mercury and the earth, and of their respective times of performing complete circuits, shews that Mercury is moving faster than the earth. Consequently to the observer at E, Mercury appears to be moving from left to right (in the figure), or from east to west; but this is contrary to the general direction of motion of the planets, i.e. Mercury appears to be retrograding. On the other hand, when Mercury appears at the greatest distance from the sun, as at M1 and M2, its own motion is directly towards or away from the earth, and is therefore imperceptible; but the earth is moving towards the observerβs right, and therefore Mercury appears to be moving towards the left, or from west to east. Hence between M1 and M its motion has changed from direct to retrograde, and therefore at some intermediate point, say m1, (about Aug, 23 in fig. 7), Mercury appears for the moment to be stationary, and similarly it appears to be stationary again when at some point m2 between M and M2 (about Sept. 13 in fig. 7).
In the case of a superior planet, say Jupiter, the argument is nearly the same. When in opposition at J (as on Mar. 26 in fig. 6), Jupiter moves more slowly than the earth, and in the same direction, and therefore appears to be moving in the opposite direction to the earth, i.e. as seen from E (fig. 48), from left to right, or from east to west, that is in the retrograde direction. But when Jupiter is in either of the positions J1 or J (in which the earth appears to the observer on Jupiter to be at its greatest distance from the sun), the motion of the earth itself being directly to or from Jupiter produces no effect on the apparent motion of Jupiter (since any displacement directly to or from the observer makes no difference in the objectβs place on the celestial sphere); but Jupiter itself is actually moving towards the left, and therefore the motion of Jupiter appears to be also from right to left, or from west to east. Hence, as before, between J1 and J and between J and J2 there must be points j1, j2 (Jan. 24 and May 27, in fig. 6) at which Jupiter appears for the moment to be stationary.
The actual discussion of the stationary points given by Coppernicus is a good deal more elaborate and more technical than the outline given here, as he not only shews that the stationary points must exist, but shews how to calculate their exact positions.
89. So far the theory of the planets has only been sketched very roughly, in order to bring into prominence the essential differences between the Coppernican and the Ptolemaic explanations of their motions, and no account has been taken of the minor irregularities for which Ptolemy devised his system of equants, eccentrics, etc., nor of the motion in latitude, i.e. to and from the ecliptic. Coppernicus, as already mentioned, rejected the equant, as being productive of an irregularity βunworthyβ of the celestial bodies, and constructed for each planet a fairly complicated system of epicycles. For the motion in latitude discussed in Book VI. he supposed the orbit of each planet round the sun to be inclined to the ecliptic at a small angle, different for each planet, but found it necessary, in order that his theory should agree with observation, to introduce the wholly imaginary complication of a regular increase and decrease in the inclinations of the orbits of the planets to the ecliptic.
The actual details of the epicycles employed are of no great interest now, but it may be worth while to notice that for the motions of the moon, earth, and five other planets Coppernicus required altogether 34 circles, viz. four for the moon, three for the earth, seven for Mercury (the motion of which is peculiarly irregular), and five for each of the other planets; this number being a good deal less than that required in most versions of Ptolemyβs system: Fracastor (chapter III., Β§ 69), for example, writing in 1538, required 79 spheres, of which six were required for the fixed stars.
90. The planetary theory of Coppernicus necessarily suffered from one of the essential defects of the system of epicycles.
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