the arc p'"p" being compensated by the smallness of the angle at which "E"" is inclined to the line of sight EP"", 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

visible, would appear to move more rapidly when in conjunction than at any other time. A planet is, however, always invisible when at or near conjunction, since it then occupies the same region of the celestial sphere as the sun, and is therefore lost in his superior light. When the planet next becomes visible, however, the effect of its swifter motion in conjunction is seen in the change of its position among the zodiacal constellations.

Passing from conjunction to opposition, the planet goes through the same changes in a reverse order. Its progressive motion gradually diminishes till it becomes stationary; thence it retrogrades through opposition to its next station; and so on continually, the total result of its motion in each synodical revolution being a progression from east to west, or in the order of the signs of the zodiac.

Let us now consider what inferences may be drawn from the nature of Saturn's apparent motions on the celestial sphere. The first point to be noticed is the slowness of his retrogression when in opposition. Now, referring to fig. 1, Plate IV., it becomes clear that this must arise from one of two causes. If PP were very nearly equal to EE, E'P' would be very nearly parallel to E'r', or, in other words, the angle E O E' would be very small. Hence the slowness of Saturn's retrogression might arise from his velocity being very nearly equal to that of the earth. But again, from the sun; p, P their respective periods; v, v their respective mean angular velocities about the sun. We have from Kepler's third law

now, on the supposition of circular orbits, the retrograde velocity of a superior planet in opposition would be proportional to, and the progressive velocity in conjunc

D-d'

; but from the proportion deduced above, it follows

1

1

=

that is, (since (D+d)2 - D+ d + 2D d1, or is greater than D +d,) the apparent velocity in conjunction is greater than the apparent velocity in opposition.

if (the rest of the figure remaining unchanged) we make the orbit PP'p" very large, and suppose the planet's velocity in this large orbit to be no greater than in the smaller one, the angle EOE would obviously become very small, for the farther PP' is removed from E E' the more nearly will E'P' and E P approach to parallelism. Thus, then, Saturn's slow motion in opposition might arise from the fact that his orbit is very large compared with that of the earth. Hence, within a week of Saturn's discovery, enough might be known to show that either his velocity in his orbit is very nearly equal to that of the earth in hers, or that he must move at an immense distance from the earth.

Another fact revealed by observation points to the true cause of Saturn's slow retrograde motion. After opposition he retrogrades for nearly two months and a half. In this time the earth has completed nearly a quarter of her orbit, and therefore her path has become inclined at a very small angle to the line of sight to Saturn. Since, then, it is only when the earth's path is thus inclined that her superior velocity is so far compensated by inclination that Saturn appears to be stationary, it is clear that the earth's motion must be much swifter than Saturn's. Hence we have only one possible explanation left of the slowness of Saturn's retrograde motion; namely, that it is due to his vast distance from the earth.

Having arrived at this conclusion, let us see how the ancient astronomers might apply the results of observations of the new planet (even those taken during only the first few months after discovery) to obtain more definite notions of his distance, and thence to determine his period.

Let a small circle E E'E" (fig. 2, Plate IV.) be described to represent the orbit of the earth about s the sun; and with the same centre let P P', part of a large circle, be described to represent Saturn's orbit, of which as yet nothing is supposed to be known but that it is large compared with the earth's orbit. Let E, P be the positions of the earth, and Saturn when the latter is in opposition, so that SEP is a straight line. Let E' be the position of the earth when Saturn is stationary, that is, two months and a half after opposition; thus ES E' is an angle of

about 75°.

Through E let the straight line E'P' be drawn, inclined to SP at an angle containing the same number of degrees, minutes, and seconds, as the arc on the celestial sphere through which Saturn has been observed to move during the two months and a half following opposition: thus E'K' is the line of sight from the earth to the planet when the earth is at E'; and r', the point in which E'K' meets the planet's orbit, must be the position of the planet at that time. Produce P'E' to a convenient distance from E', to B; through B draw BA in a direction perpendicular to SP, and draw E'A touching the circle E E'E" in E'. We have, then, the following facts to guide us:-the earth and Saturn both lie in the line B K'; the earth is leaving this line in the direction E'A; Saturn is leaving it in the direction of the tangent to PP'P' at P', a direction approximately parallel to B A.* Now the planet appears stationary when at p'-in other words, the rates of departure of Saturn and the earth from the line E'P' are exactly equal. If, then, we suppose a point to move from B in direction BA (which is parallel to Saturn's line of motion at p') with Saturn's velocity, and another point to start from E' at the same moment in direction E'A with the earth's velocity, then, since the rates of departure of these two points from the line B E are exactly equal, they would arrive at the point A at the same instant. Hence the velocities of these moving points-which velocities are, by our supposition, the velocities of Saturn and the earth respectively-are respectively proportional to BA, E'A, the spaces they pass over in equal times. We arrive, then, at the important result, that Saturn's velocity in his orbit: the earth's velocity in hers :: the line BA the line E'A, very approximately. If the figure is constructed with proper care, we have only to measure the lines B A and E'A to determine the value of this proportion; or we can employ a very simple trigonometrical calculation for this purpose.† Either method leads to the result

* The tangent at P is parallel to BA; since, then, the arc PP' is very small, the tangent at P', which is perpendicular to s P', is inclined at a very small angle to the tangent at P, and is therefore very nearly parallel to B A, and for our purpose may be considered as actually parallel to BA.

In the triangle ABE' the angle BAE' is equal to the known angle ESE', and the angle A E'B is the complement of the angle B E'S, which is the sum of two known angles; viz.

that A E' is about 3 times as large as B A, or Saturn's velocity is to that of the earth in the proportion of 11 to 34, very nearly. We can now determine Saturn's distance in either of two ways. The orbit PP'P", assigned to Saturn in the above investigation, was simply a circle, large compared with E E'E", and it is to he observed that the dimensions of this circle had nothing to do with the formation of the triangle AB E', on which the determination of Saturn's velocity was made to depend; except that, knowing the planet's orbit to be large, we were able to assert that the direction of its motion at P' was very nearly parallel to B A. But we can apply the result just obtained to see whether P P'p" correctly represents Saturn's orbit. For the arc PP' passed over by Saturn should bear to the arc E E' passed over, in the same time, by the earth, the proportion, above determined, of 11 to 34. In our figure PP does not bear this proportion to E E', being too large. The radius SP is therefore too small, and we must select such a radius in place of SP that the arc intercepted between SP and E'P' may be of the requisite length, viz. 11ths of EE'. It will be found that for this purpose SP should be about 9 times as great

as s E.

We may confirm the correctness of this result by applying a second method to determine Saturn's distance. Observation shows that, when near opposition, Saturn retrogrades daily over an arc of about 4′ 43′′ on the celestial sphere. Now Saturn is advancing from P with 11ths of the velocity with which the earth advances from E. Therefore Saturn's motion, as observed from the earth, is the same as if he were retrograding with 24ths of the earth's velocity. If, then, he were at the same distance from the earth as the sun is, he would appear to pass daily over an arc equal to 23ths of the arc passed over daily by the sun, since the sun's apparent motion is due to the whole of the earth's motion. Now the sun passes daily over an arc of about 59′ 8'.* Thus Saturn, if he were at the sun's distance, would pass over nearly 40′ 0′′ daily. But Saturn actually passes over 4′ 43′′, or about ths

the angle ES E' and the angle of inclination of E'P' to S P. Thus the proportion that B A bears to A E' can be determined.

* The arc passed over daily by the sun is 524 of 360° approximately.