Chapter 13

Fig. 43.—The orbits of Venus and of the earth.

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 (chapterII.,§ 40), between the synodic and sidereal periods of revolution. Venus, for example, is seen as an evening starat its greatest angular distance from the sun (as atVin 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 thesynodic period. But as during this time the lineE Shas 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 atV1,E1; 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 atE2; Venus being again at the greatest distance from the sun will therefore be atV2, 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 lineS V1takes 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 thesidereal 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 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 pointV1(fig. 44) such thatV1E1touches the circle in which Venus moves, and the angleE1V1Sis then (by a known property of a circle) a right angle. The angleS E1V1being observed, the shape of the triangleS E1V1is 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 torevolve 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,jJ, is always parallel to the lineE Sjoining 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 drawE J′parallel tojJand equal to it in length, it is easily seen55that the lineJ′ Jis equal and parallel toEj, that consequentlyJdescribes a circle roundJ′just asjroundE. 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 whichJ′is the centre andJ′ Jthe radius, while the centre of the epicycle, remaining always in the direction of the sun, describes a deferent (represented by the small circle roundE) 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 centreJ′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 amotion 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, chapterII.,§ 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 thereforelosesone 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.56Coppernicus thus obtained very fairly accuratevalues 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 angleS P Eis then as great as possible. The angleP E Sis a right angle, and the angleS P Eis 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 triangleS P Eis 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.

88. The explanation of the stationary points of the planets (chapterI.,§ 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 atM(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 atE, 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 atM1andM2, its own motion is directly towards or away from the earth, and is therefore imperceptible; but the earth is moving towards the observer’s right, and therefore Mercury appears to be moving towards the left,or from west to east. Hence betweenM1andMits motion has changed from direct to retrograde, and therefore at some intermediate point, saym1, (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 pointm2betweenMandM2(about Sept. 13 in fig. 7).

Fig. 47.—The stationary points of Mercury.

In the case of a superior planet, say Jupiter, the argument is nearly the same. When in opposition atJ(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 fromE(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 positionsJ1orJ(in which the earth appears to the observer on Jupiter to be at its greatest distancefrom 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 object’s 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, betweenJ1andJand betweenJandJ2there must be pointsj1,j2(Jan. 24 and May 27, in fig. 6) at which Jupiter appears for the moment to be stationary.

Fig. 48.—The stationary points of Jupiter.

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 shewsthat 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 “unworthy” 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 Ptolemy’s system: Fracastor (chapterIII.,§ 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. It is, in fact, always possible to choose a system of epicycles in such a way as to makeeitherthe direction of any bodyorits distance vary in any required manner, but not to satisfy both requirements at once. In the case of the motion of the moon round the earth, or of the earth round the sun, cases in which variations in distance could not readily be observed, epicycles might therefore be expected to give a satisfactory result, at any rate until methods ofobservation were sufficiently improved to measure with some accuracy the apparent sizes of the sun and moon, and so check the variations in their distances. But any variation in the distance of the earth from the sun would affect not merely the distance, but also the direction in which a planet would be seen; in the figure, for example, when the planet is atPand the sun atS, the apparent position of the planet, as seen from the earth, will be different according as the earth is atEorE′. Hence the epicycles and eccentrics of Coppernicus, which had to be adjusted in such a way that they necessarily involved incorrect values of the distances between the sun and earth, gave rise to corresponding errors in the observed places of the planets. The observations which Coppernicus used were hardly extensive or accurate enough to show this discrepancy clearly; but a crucial test was thus virtually suggested by means of which, when further observations of the planets had been made, a decision could be taken between an epicyclic representation of the motion of the planets and some other geometrical scheme.

Fig. 49.—The alteration in a planet’s apparent position due to an alteration in the earth’s distance from the sun.

91. The merits of Coppernicus are so great, and the partwhich he played in the overthrow of the Ptolemaic system is so conspicuous, that we are sometimes liable to forget that, so far from rejecting the epicycles and eccentrics of the Greeks, he used no other geometrical devices, and was even a more orthodox “epicyclist” than Ptolemy himself, as he rejected the equants of the latter.57Milton’s famous description (Par. Lost, VIII. 82-5) of

“The SphereWith Centric and Eccentric scribbled o’er,Cycle and Epicycle, Orb in Orb,”

“The Sphere

With Centric and Eccentric scribbled o’er,

Cycle and Epicycle, Orb in Orb,”

applies therefore just as well to the astronomy of Coppernicus as to that of his predecessors; and it was Kepler (chapterVII.), writing more than half a century later, not Coppernicus, to whom the rejection of the epicycle and eccentric is due.

92. One point which was of importance in later controversies deserves special mention here. The basis of the Coppernican system was that a motion of the earth carrying the observer with it produced an apparent motion of other bodies. The apparent motions of the sun and planets were thus shewn to be in great part explicable as the result of the motion of the earth round the sun. Similar reasoning ought apparently to lead to the conclusion that the fixed stars would also appear to have an annual motion. There would, in fact, be a displacement of the apparent position of a star due to the alteration of the earth’s position in its orbit, closely resembling the alteration in the apparent position of the moon due to the alteration of the observer’s position on the earth which had long been studied under the name of parallax (chapterII.,§ 43). As such a displacement had never been observed, Coppernicus explained the apparent contradiction by supposing the fixed stars sofar off that any motion due to this cause was too small to be noticed. If, for example, the earth moves in six months fromEtoE′, the change in direction of a star atS′is the angleE′ S′ E, which is less than that of a nearer star atS; and by supposing the starS′sufficiently remote, the angleE′ S′ Ecan be made as small as may be required. For instance, if the distance of the star were 300 times the distanceE E′,i.e.600 times as far from the earth as the sun is, the angleE S′ E′would be less than 12′, a quantity which the instruments of the time were barely capable of detecting.58But more accurate observations of the fixed stars might be expected to throw further light on this problem.

Fig. 50.—Stellar parallax.

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