T'

earth describes a complete revolution, T: T'::1:

= the part T

of the circumference described by the earth in the time T'. But during the same time the planet describes a whole circumference.

T'

Therefore, 1- is what the planet gains on the earth in one

T

revolution. In order to a new conjunction the planet must gain an entire circumference; therefore, denoting the synodical period by S, the gain in one revolution will be to the time in which it is acquired, as a whole circumference is to the time in which that is gained, which is the synodical period. That is,

From this formula we may find the synodical period of Mercury or Venus by substituting the numbers denoted by the letters. 365.256 × 87.969 277.287

Thus,

of Mercury.

115.877, which is the synodical period

By a similar computation, the synodical revolution of Venus will be found to be about 584 days.

305. The motion of an inferior planet is direct in passing through its superior conjunction, and retrograde in passing through its inferior conjunction. Thus Venus, while going from B through D to A, (Fig. 60,) moves in the order of the signs, or from west to east, and would appear to traverse the celestial vault B'S'A' from right to left; but in passing from A through C to B, her course would be retrograde, returning on the same arc from left to right. If the earth were at rest, therefore, (and the sun, of course, at rest,) the inferior planets would appear to oscillate backwards and forwards across the sun. But, it must be recollected, that the earth is moving in the same direction with the planet, as respects the signs, but with a slower motion. This modifies the motions of the planet, accelerating it in the superior and retarding it in the inferior conjunctions. Thus in figure 60, Venus while moving through BDA would seem to move in the heavens from B' to A' were the earth at rest; but meanwhile the earth changes its position from E to E', by which means the planet is not seen

at A' but at A", being accelerated by the arc A'A" in consequence of the earth's motion. On the other hand, when the planet is passing through its inferior conjunction ACB, it appears to move backwards in the heavens from A' to B' if the earth is at rest, but from A' to B" if the earth has in the mean time moved from E to E', being retarded by the arc B'B". Although the motions of the earth have the effect to accelerate the planet in the superior conjunction, and to retard it in the inferior, yet, on account of the greater distance, the apparent motion of the planet is much slower in the superior than in the inferior conjunction.

306. When passing from the superior to the inferior conjunc tion, or from the inferior to the superior conjunction, through the greatest elongations, the inferior planets are stationary.

If the earth were at rest, the stationary points would be at the greatest elongations as at A and B, for then the planet would be moving directly towards or from the earth, and would be seen for some time in the same place in the heavens; but the earth itself is moving nearly at right angles to the line of the planet's motion, that is, the line which is drawn from the earth to the planet through the point of greatest elongation; hence a direct motion is given to the planet by this cause. When the planet, however, has passed this line, by its superior velocity it soon overcomes this tendency of the earth to give it a relative motion eastward, and becomes retrograde as it approaches the inferior conjunction. Its stationary point obviously lies between its place of greatest elongation, and the place where its motion becomes retrograde. Mercury is stationary at an elongation of from 15° to 20° from the sun; and Venus at about 29°.*

307. Mercury and Venus exhibit to the telescope phases similar to those of the moon.

When on the side of their inferior conjunctions, these planets appear horned, like the moon in her first and last quarters; and when on the side of their superior conjunctions they appear gib'bous. At the moment of superior conjunction, the whole enlightened orb of the planet is turned towards the earth, and the *Herschel, p. 242.-Woodhouse, 557.

appearance would be that of the full moon, but the planet is too near the sun to be commonly visible.

These different phases show that these bodies are opake, and shine only as they reflect to us the light of the sun; and the same remark applies to all the planets.

308. The distance of an inferior planet from the sun, may be found by observations at the time of its greatest elongation.

Thus if E be the place of the earth, and B that of Venus at the time of her greatest elongation, the angle SBE will be known, being a right angle. Also the angle SEB is known from observation. Hence the ratio of SB to SE becomes known; or, since SE is given, being the distance of the earth from the sun, SB the radius of the orbit of the planet is determined. If the orbits were both circles, this method would be very exact; but being elliptical, we obtain the mean value of the radius SB by observing its greatest elongation in different parts of its orbit.*

309. The orbit of Mercury is the most eccentric, and the most inclined of all the planets; while that of Venus varies but little from a circle, and lies much nearer to the ecliptic.

143

The eccentricity of the orbit of Mercury is nearly its semimajor axis, while that of Venus is only ; the inclination of Mercury's orbit is 70, while that of Venus is less than 310. Mercury, on account of his different distances from the earth, varies much in his apparent diameter, which is only 5′′ in the apogee, but 12" in the perigee. The inclination of his orbit to his equator being very great, the changes of his seasons must be proportionally great.

310. The most favorable time for determining the sidereal revolution of a planet, is when its conjunction takes place at one of its nodes; for then the sun, the earth, and the planet, being in the same straight line, it is referred to its true place in the heavens, whereas, in other positions, its apparent place is more or less affected by perspective.

Herschel, p. 239.

The new planets are of course excepted.

Baily's Tables.

311. An inferior planet is brightest at a certain point between its greatest elongation and inferior conjunction.

Its maximum brilliancy would happen at the inferior conjunction, (being then nearest to us,) if it shined by its own light; but in that position, its dark side is turned towards us. Still, its maximum cannot be when most of the illuminated side is towards us; for then, being at the superior conjunction, it is at its greatest distance from us. The maximum must therefore be somewhere, between the two. Venus gives her greatest light

when about 40° from the sun.

312. Mercury and Venus both revolve on their axes, in nearly the same time with the earth.

The diurnal period of Mercury is 24h. 5m. 28s., and that of Venus 23h. 21m. 7s. The revolutions on their axes have been determined by means of some spot or mark seen by the telescope, as the revolution of the sun on his axis is ascertained by means of his spots.

313. Venus is regarded as the most beautiful of the planets, and is well known as the morning and evening star. The most ancient nations did not indeed recognize the evening and morning star as one and the same body, but supposed they were different planets, and accordingly gave them different names, calling the morning star Lucifer, and the evening star Hesperus. At her period of greatest splendor, Venus casts a shadow, and is sometimes visible in broad daylight. Her light is then estimated as equal to that of twenty stars of the first magnitude.* At her period of greatest elongation, Venus is visible from three to four hours after the setting or before the rising of the sun.

314. Every eight years, Venus forms her conjunctions with the sun in the same part of the heavens.

For, since the synodical period of Venus is 584 days, and her sidereal period 224.7,

224.7: 360°:584: 935.6-the arc of longitude described by Venus between the first and second conjunctions. Deducting

* Francœur, Uranography, p. 125.

720°, or two entire circumferences, the remainder, 2150.6, shows how far the place of the second conjunction is in advance of the first. Hence, in five synodical revolutions, or 2920 days, the same point must have advanced 215°.6×5=1078°, which is nearly three entire circumferences, so that at the end of five synodical revolutions, occupying 2920 days, or 8 years, the conjunction of Venus takes place nearly in the same place in the heavens as at first.

Whatever appearances of this planet, therefore, arise from its positions with respect to the earth and the sun, they are repeated every eight years in nearly the same form.

TRANSITS OF THE INFERIOR PLANETS.

315. The Transit of Mercury or Venus, is its passage across the sun's disk, as the moon passes over it in a solar eclipse.

As a transit takes place only when the planet is in inferior conjunction, at which time her motion is retrograde (Art. 305,) it is always from left to right, and the planet is seen projected on the solar disk in a black round spot. Were the orbits of the inferior planets coincident with the plane of the carth's orbit a transit would occur to some part of the earth at every inferior conjunction. But the orbit of Venus makes an angle of 31° with the ecliptic, and Mercury an angle of 70; and, moreover, the apparent diameter of each of these bodies is very small, both of which circumstances conspire to render a transit a comparatively rare occurrence, since it can happen only when the sun, at the time of an inferior conjunction, chances to be at or extremely near the planet's node. The nodes of Mercury lie in longitude 46° and 226°, points which the sun passes through in May and November. It is only in these months, therefore, that transits of Mercury can occur. For a similar reason, those of Venus occur only in June and December. Since, however, the nodes of both planets have a small retrograde motion, the months in which transits occur will change in the course of ages.

316. The intervals between successive transits, may be found in the following manner. The formula which gives the synodical