though they might remain in doubt for a few days as to his planetary nature, these doubts would soon be set at rest by obvious alterations in his bearing with respect to the neighbouring fixed stars. Let us suppose, by way of illustration, that Saturn is describing that part of his orbit indicated in Plate II., and that he is first observed when in opposition near the bright star Regulus. The time then is mid-winter. The heavens-in the eastern clime of those first astronomers--are sparkling as if set with myriads of varied gems. The greater Lion, with the brilliant stars Deneb and Regulus, is in the south-east, slowly rising to the meridian. Near Regulus, about a degree and a half to the north, a strange star is seen, whose dull yellow light contrasts strangely with the dazzling white of the fixed star. As they rise together to culmination, and then sink towards the south-western horizon, the closest observation can detect no change in their relative positions. On the following night the stranger is again seen close to Regulus. But now its position seems slightly changed:-it is no longer due north of Regulus, but has moved slightly westward. The change of position is, however, so small as to be scarcely perceptible. It is not until several days have elapsed that the wandering nature of the stranger is certainly established. It passes to the north-west of Regulus, and continues to move slowly westward. As Mars and Jupiter both appear to move from east to west when in opposition, though their real motion is from west to east, it remains doubtful whether the new planet moves from east to west, as he appears to do, or, like the other planets, from west to Careful observation soon shows that Saturn's westward motion is gradually diminishing; yet, when six weeks (the time east. * The bright star below the zodiac, near the centre of the map. Regulus now souths at midnight in the middle of March. Four thousand years ago he passed the meridian at midnight two months earlier. The position of the ecliptic was also different. Regulus, now nearly half a degree to the north, was then south of the ecliptic. Saturn's path on the celestial sphere has, in like manner, undergone several changes; for instance, the positions of the nodes and of the perihelion, and the inclination of the orbit to the ecliptic, have varied, and other changes have taken place which need not at present be dwelt upon. in which Mars retrogrades after opposition) have passed, Saturn is still moving westward, nor has his retrograde motion ceased when two months (the corresponding period of Jupiter's retrogression) have elapsed. For yet another fortnight he retrogrades, and then begins to move slowly along his advancing arc. Although he has thus been retrograding for nearly two months and a half from opposition, he has passed over an arc of only three degrees on the celestial sphere. Saturn's progressive motion, slow at first, gradually increases as he approaches conjunction, when, becoming an evening star, his light is dimmed, and finally lost, in the light of the sun. About a month after conjunction, he again becomes visible as a morning star. His apparent motion is still progressive, but gradually decreases until he becomes stationary. He then slowly retrogrades for nearly five months, passing through opposition; becomes stationary again, then advances, and so on continually, advancing during seven months and a half and retrograding during five, but on the whole slowly traversing the zodiac from west to east, or in the order of the signs. Having ascertained that the strange orb is a planet, let us see how the ancient astronomers could approximately determine the distance and period of the new planet from its apparent motions. We may proceed on the supposition that they were acquainted with the true system of the world. It would obviously be a waste of time to consider, at any length, methods belonging to a false system; there are also good reasons for supposing that the true system was actually known to ancient astronomers.* For the sake of simplicity, the paths of Saturn and the earth are supposed to lie in the same plane, and to be circles about the sun as centre. In the first place, what inferences may be deduced from Saturn's slow retrograde motion when in opposition, his long period of retrogression, and the small arc passed over by him in that period? To answer these questions it will be necessary to recall to the reader's mind the cause of the retrograde motion of a planet in opposition. *See Note A, Appendix I., Chaldæan Astronomy. Let s (fig. 1, Plate IV.) represent the sun, E E'E' the earth's orbit, PP'p" part of the orbit of a superior planet. When the earth is at E let the planet be at P, so that (SEP being a straight line) the planet is in opposition when at P. Starting from these positions, suppose that the earth and the planet, in the same interval of time, pass respectively over the arcs E E' and PP', EE' being greater than PP'. Then it is obvious that the line E'P' is inclined to the line EP, and that if these two lines are produced they will meet beyond P. Let them be produced, beyond their point of intersection o, to K and K' respectively. Now the observer on the earth sees the planet in the direction E K when the earth is at E, and in the direction E'K' when the earth is at E'; thus the planet appears to have moved in the direction K K', while it has actually moved in the contrary direction, namely, from P to P'. The amount of the planet's retrograde motion during the interval is measured by the angle contained between the lines E P, E'P', that is, by the angle EOE or KOK'; and, vice versâ, if the retrograde arc passed over by the planet on the celestial sphere be measured, the angle E O E' becomes known with an exactness proportioned to the accuracy of the instruments used in effecting the measurement and the skill of the observer employing them. Let us now carry the earth and planet forward in their orbits. It is obvious that the path of the earth becomes more and more inclined to the line of sight to the planet the farther the earth is carried on in the arc EE'E"; thus if E", P'', and E"", "", be respectively contemporaneous positions of the earth and planet, the angle between the line E'''p'"' and the arc E'"E" is less than the angle between the line "P" and the arc E′′E', and this angle again is less than the angle between the line E'P' and the arc E'E. Hence the effect of the earth's superior velocity, so far as it operates in changing the direction of the line of sight to the planet, gradually diminishes, until at length the earth reaches a position, as at E'', such that the effect of the direction of its motion exactly counterbalances its superior velocity, and the planet appears to be stationary. If E'E" and p'p"" are small arcs passed over by the earth and planet in the same time at this period, the line ""P" is parallel to the line "P", the superiority in length of the arc E"E" over |