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', E E′ 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 The amount of the planet's retrograde motion during the interval is measured by the angle contained between the lines EP, E'P', that is, by the angle EO E' 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.

P'.

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 EEE"; thus if E", P", and E"", "", be respectively contemporaneous positions of the earth and planet, the angle between the line "P"" and the arc E""E" is less than the angle between the line E′′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

the arc-p'"p" being compensated by the smallness of the angle at which E'E' is inclined to the line of sight E''P'", compared with the inclination of p"p" to the same line. Thus the planet is seen in the same direction at the end of this interval as at the beginning, or is stationary.*

After this, it is plain that the planet will appear to advance, for at first the earth's path becomes inclined at a smaller angle to the line of sight to the planet, till it coincides with that line; and afterwards, as the earth passes on to conjunction, its motion (considered with reference to the line of sight at any instant) is in a contrary direction to that of the planet, and therefore adds to the planet's apparent motion on the celestial sphere.

If the planet were visible when in conjunction, its motion would appear swifter then than at any other time. Thus let esp be the line along which conjunction takes place, the earth being at e and the planet at p ; and e', p' the positions of the earth and planet after a short interval of time: then ep is the direction in which the planet would be seen when in conjunction, e'p' that in which it would be seen at the end of the interval of time. Now ep and e'p' meet within the orbit of the planet at o; the angle eoe' or pop' measures the arc on the celestial sphere passed over by the planet, and the whole motions both of the planet and the earth conspire to increase this angle; whereas in any other positions, either the difference of these motions, or only parts of them, affect the angle between the lines of sight at the beginning and end of any corresponding interval of time. And although the angle is diminished when a superior planet is in conjunction through the effect of increased distance (ep plainly exceeding EP by twice the radius of the earth's orbit, or by twice SE), yet, under the actual relations of the velocities and distances of the planets,† increase prevails over decrease, and a superior planet, if

It must be remembered that the arcs E"E"" and p''p"" are supposed to be very small. A planet is not actually stationary during any finite period of time; retrograde motion merges into progressive, progressive into retrograde, at a definite instant, before which (by however small an interval of time) the motion is of one kind, afterwards of the contrary; it is only at that instant that there is no motion, progressive or retrograde.

If d, D be respectively the mean distances of the earth and of a superior planet