from AS. Hence, as BV is parallel to AV, the angle VBb, which is the planet's geocentric longitude when at b, is less than VAa, its geocentric longitude when at a. The apparent motion of the planet is, therefore, retrograde at the period of inferior conjunction.

Let E and e be the places of the earth and planet, at the time of superior conjunction, and F and ƒ their places an hour afterwards. Then, it is evident that VFf, the geocentric longitude of the planet at f, is greater than VEe, its geocentric longitude at e. The apparent motion is, therefore, direct at the period of superior conjunction.

As the direction of a from A, or b from B is directly opposite to that of A from a, or B from b, it follows that when the motion of the planet appears to be retrograde as seen from the earth, the motion of the earth must appear retrograde as seen from the planet; and the same must apply to the direct motions. Hence, regarding ace as the orbit of the earth, and ACE as that of a superior planet, it is obvious that the apparent motion of the superior planet must be retrograde, at the period of its opposition, and direct at the period of conjunction.

It will appear from the next article, that the period during which the apparent motion of a planet is retrograde, is much shorter than that during which it is direct.

285. A planet sometimes appears to be stationary. Let C and c be two corresponding places of the earth and planet, and D and d their places an hour afterwards. Then, if the places C and c be such that Dd is parallel to Cc, the geocentric longitudes VDd and VCc will be equal, and, therefore, the planet must at that time appear to be stationary.

To find SCc, the angle of elongation when the planet appears stationary, put x = SCC and y = CCG; and regarding the earth's distance

SC 1, let a Sc = the

=

planet's distance. Then, SDd Snc

=

=

=

SCc+ CSD = x

+ M, and Ddk CkK Sck+GSK CCG + GSK =

=

=

=

y+m. Hence, from the triangle SCc and SDd, we have,

sin

y:

sin x 1: a sin (y + m): sin (x + M),

(S)

or, sin y sin y cos m+cos y sin m :: sin x: sin x cos M+cos x sin M. But, as M and m are both very small, we may regard cos M = 1 and cos m=1.

Hence,

sin y sin y+cos y sin m:: sinx: sin x + cos x sin M,

sin y sin x cos y sin m : cos x sin M :: cos y

Hence, cos y

or,

ava. cosx:: sin y sin x :: 1: a (S) cos'y a3cos*x :: sin3y: sin'x :: 1:a2.

Consequently, a cos* x=a' cos3 y,(T), and sin' x-a' sin' y (U) Therefore, (T and U), a3—a3sin3 x=a*—aʼ sin2 y—a2—sin2 x (1-a) sin' x — a2 — a3

or,

=

2

(V)

Again, (U and T), 1—cos2 x=a2—a2 cos3 y=a' —a3 cos3 x

The upper sign appertains to an inferior, and the lower, to a superior planet.

From (S), by dividing the first and third terms by cosy, and the second and fourth by cos x, we have,

The angles x and y being found, the angle CSc, which is equal to their difference, is known. Now, CSc is the difference between the angular motion of the earth and that of the planet, during the interval from inferior conjunction or from opposition, to the time the planet is stationary. Hence, if we put t = this interval, and d = the difference of the daily motions of the earth and planet, we have, CSC d'

d: CSC 1 day : t=

Now, it is evident, that the planet must appear stationary at a like interval, prior to the conjunction or opposition, and that during the period from the prior, to the subsequent stationary positions, its apparent motion must be retrograde. Hence, 2t expresses the period during which the motion is retrograde. The value of 2t computed for each of the planets, is found to be much less than half the synodic revolution.

Scholium. The times of the stationary positions of the planets, and the periods of their retrogradations, computed as above, are found to agrec, nearly, with those obtained from observation; and when more accurately computed by taking into view the inclinations and elliptical forms of their orbits, the agreement is complete. As these computations are founded on the arrangement of the sun, earth, and planets according to the Copernican System, this agreement is a confirmation of the truth of that system.

286. Real Distances of the Planets from the Sun. From the sun's horizontal parallax, the earth's distance from the sun becomes known (96). This distance, multiplied by the numbers respectively which denote the relative distances of the planets, obtained on the assumption that the earth's distance is a unit (274, schol.), gives the real distances of the planets.

287. Apparent and real diameters of the Planets. The apparent diameter of a planet is determined by measurements with a micrometer or heliometer. Then the planet's distance from the earth, at the time of observation, being computed (281), the real diameter becomes known (99 H).

An inferior planet is nearer the earth at inferior conjunction than at superior, by the whole diameter of its orbit; and a superior planet is nearer at opposition, than at conjunction, by the diameter of the earth's orbit. Hence, as the apparent diameter is inversely as the distance, the apparent diameters of several of the planets are very variable.

The greatest apparent diameter of Venus is about six times that of the least, and the greatest of Mars, about five times that of the least.

288. Rotations of the Planets. All the planets on which sufficiently accurate telescopic observations can be made to ascertain the fact, are found to revolve on their axes in the same direction as the earth's rotation; that is, from west to east.

289. Curious relation in the distances of the planets. If we take 10 to represent the earth's distance from the sun, the distances of the planets, beginning with Mercury, will be nearly represented by this series; 4, 4+ 3.2°, 4 + 3,2', 4 +3.2, &c. This appears from the following table, the last column of which gives the true distances in whole numbers, on the supposition that the earth's distance is 10.

By examining the table it will be seen that in the distances of the planets, excluding those recently discovered, there is an abrupt increase from Mars to Jupiter. This break in the series of distances, led some astronomers to suppose there must be a planet occupying an intermediate position. It was with some surprise that in the early part of the present century, four very small planetary bodies were discovered, all of which revolve round the sun at nearly the same distance; this distance being that at which it was supposed one planet might be found.

290. Orbits of the Satellites. From numerous observations of the positions of the satellites with reference to their respective primary planets, it is found that they revolve round their primaries in elliptical orbits, most of which differ but little from circles. It is also found that they conform in their motions, to Kepler's laws relative to the planets.

The satellites have not received particular names, but are termed 1st satellite, 2nd satellite, &c., according to their distances from the planet, beginning with the

nearest.

CHAPTER XV.

INFERIOR PLANETS, MERCURY AND VENUS.-TRANSITS.SUN'S PARALLAX.

292. Greatest Elongations of Mercury and Venus. Mercury and Venus have their orbits so far within that of the earth, that their elongations are never great. They seem to accompany the sun, being seen in the west soon after sunset, or in the east awhile before sunrise.

Let S, Fig. 50, be the place of the sun, ABC the orbit of Mercury, which we will here suppose to coincide with the plane of the ecliptic, FG a part of the earth's orbit, and A and a corresponding positions of the planet and earth, when the former is at its greatest elongation, at which time the angle aAS is a right angle. As the distances of the planet and earth, from the sun, both vary, the greatest elongation must also vary. Its value will evidently be greatest, when SA is greatest, and at the same time Sa least, that is, when at the time of greatest elongation, Mercury is at the aphelion of her orbit and the earth in perihelion; and least, when the positions are reversed. With the least value of SA and greatest of Sa,