over, if r and R be the radii of the respective orbits, we have also

r: R:: sin. SEP: sin. (SEP+ESP)

from which two relations it is easy to deduce the values of the two angles SEP and ESP; the former of which is the apparent elongation of the planet from the sun, the latter the difference of heliocentric longitudes of the earth and planet.

(476.) When we regard the orbits as other than circles (which they really are), the problem becomes somewhat complex-too much so to be here entered upon. It will suffice to state the results which experience verifies, and which assigns the stationary points of Mercury at from 15° to 20° of elongation from the sun, according to circumstances; and of Venus, at an elongation never varying much from 29°. The former continues to retrograde during about 22 days; the latter, about 42.

(477.) We have said that some of the planets exhibit phases like the moon. This is the case with both Mercury and Venus; and is readily explained by a consideration of their orbits, such as we have above supposed them. In fact, it requires little more than mere inspection of the figure annexed, to show, that to a spectator

B

junction A; gibbous (i.e. more than half full, like the moon between the first and second quarter) between that point and the points B C of its greatest elongation; half-mooned at these points; and crescent-shaped, or horned, between these

and the inferior conjunction D. As it approaches this point, the crescent ought to thin off till it vanishes altogether, rendering the planet invisible, unless in those cases where it transits the sun's disc, and appears on it as a black spot. All these phenomena are exactly conformable to observation.

(478.) The variation in brightness of Venus in different parts of its apparent orbit is very remarkable. This arises from two causes: 1st, the varying proportion of its visible illuminated area to its whole disc; and, 2dly, the varying angular diameter, or whole apparent magnitude of the disc itself. As it approaches its inferior conjunction from its greater elongation, the half-moon becomes a crescent, which thins off; but this is more than compensated, for some time, by the increasing apparent magnitude, in consequence of its diminishing distance. Thus the total light received from it goes on increasing, till at length it attains a maximum, which takes place when the planet's elongation is about 40°.

(479.) The transits of Venus are of very rare occurrence, taking place alternately at the very unequal but regularly recurring intervals of 8, 122, 8, 105, 8, 122, &c., years in succession, and always in June or December. As astronomical phænomena, they are extremely important; since they afford the best and most exact means we possess of ascertaining the sun's distance, or its parallax. Without going into the niceties of calculation of this problem, which, owing to the great multitude of circumstances to be attended to, are extremely intricate, we shall here explain its principle, which, in the abstract, is very simple and obvious. Let E be the earth, V Venus, and S the sun, and CD the portion of Venus's relative orbit which she describes while in the act of transiting the sun's disc. Suppose A B two spectators at opposite extremities of that diameter of the earth which is perpendicular to the ecliptic, and, to avoid complicating the case, let us lay out of consideration the earth's rotation, and suppose A, B, to retain that situation during the whole time of the transit. Then, at any moment when the spectator at A sees the center of Venus projected at a on the sun's disc, he at B will see it projected at b. If then one or other spectator could suddenly

transport himself from A to B, he would see Venus suddenly displaced on the disc from a to b; and if he had any means of noting accurately the place of the points on the disc, either by micrometrical measures from its edge, or by other means, he might ascertain the angular measure of a b as seen from the earth. Now, since A V a, B V b, are straight lines, and

E
B

D

therefore make equal angles on each side V, a b will be to A B as the distance of Venus from the sun is to its distance from the earth, or as 68 to 27, or nearly as 2 to 1; ab therefore occupies on the sun's disc a space 2 times as great as the earth's diameter; and its angular measure is therefore equal to about 2 times the earth's apparent diameter at the distance of the sun, or (which is the same thing) to five times the sun's horizontal parallax (art. 298.). Any error, therefore, which may be committed in measuring a b, will entail only one fifth of that error on the horizontal parallax concluded from it.

(480.) The thing to be ascertained, therefore, is, in fact, neither more nor less than the breadth of the zone P Q R S, pqrs, included between the extreme apparent paths of the center of Venus across the sun's disc, from its entry on one side to its quitting it on the other. The whole business of the observers at A, B, therefore, resolves itself into this; —to ascertain, with all possible care and precision, each at his own station, this path,—where it enters, where it quits, and what segment of the sun's disc it cuts off. Now, one of the most exact ways in which (conjoined with careful micrometric measures) this can be done, is by noting the time occupied in the whole transit: for the relative angular motion of Venus being, in fact, very precisely known from the tables of her motion, and the apparent path being very nearly a straight line, these times give us a measure (on a very enlarged scale) of the lengths of the chords of the segments cut off; and the

U

sun's diameter being known also with great precision, their versed sines, and therefore their difference, or the breadth of the zone required, becomes known. To obtain these times correctly, each observer must ascertain the instants of ingress and egress of the center. To do this, he must note, 1st, the instant when the first visible impression or notch on the edge of the disc at P is produced, or the first external contact; 2dly, when the planet is just wholly immersed, and the broken edge of the disc just closes again at Q, or the first internal contact; and, lastly, he must make the same observations at the egress at R, S. The mean of the internal and external contacts, corrected for the curvature of the sun's limb in the intervals of the respective points of contact, internal and external, gives the entry and egress of the planet's

center.

(481.) The modifications introduced into this process by the earth's rotation on its axis, and by other geographical stations of the observers thereon than here supposed, are similar in their principles to those which enter into the calculation of a solar eclipse, or the occultation of a star by the moon, only more refined. Any consideration of them, however, here, would lead us too far; but in the view we have taken of the subject, it affords an admirable example of the way in which minute elements in astronomy may become magnified in their effects, and, by being made subject to measurement on a greatly enlarged scale, or by substituting the measure of time for space, may be ascertained with a degree of precision adequate to every purpose, by only watching favourable opportunities, and taking advantage of nicely adjusted combinations of circumstance. So important has this observation appeared to astronomers, that at the last transit of Venus, in 1769, expeditions were fitted out, on the most efficient scale, by the British, French, Russian, and other governments, to the remotest corners of the globe, for the express purpose of performing it. The celebrated expedition of Captain Cook to Otaheite was one of them. The general result of all the observations made on this most memorable occasion gives 85776 for the sun's horizontal parallax. The two next

occurrences of this phænomenon will happen on Dec. 8. 1874 and Dec. 6. 1882.

(482.) The orbit of Mercury is very elliptical, the excentricity being nearly one fourth of the mean distance. This appears from the inequality of the greatest elongations from the sun, as observed at different times, and which vary between the limits 16° 12′ and 28° 48', and, from exact measures of such elongations, it is not difficult to show that the orbit of Venus also is slightly excentric, and that both these planets, in fact, describe ellipses, having the sun in their commom focus.

(483.) Transits of Mercury over the sun's disc occasionally occur, as in the case of Venus, but more frequently; those at the ascending node in November, at the descending in May. The intervals (considering each node separately) are usually either 13 or 7 years, and in the order 13, 13, 13, 7, &c.; but owing to the considerable inclination of the orbit of Mercury to the ecliptic, this cannot be taken as an exact expression of the said recurrence, and it requires a period of at least 217 years to bring round the transits in regular order. One will occur in the present year (1848), the next in 1861. They are of much less astronomical importance than that of Venus, on account of the proximity of Mercury to the sun, which affords a much less favourable combination for the determination of the sun's parallax.

(484.) Let us now consider the superior planets, or those whose orbits enclose on all sides that of the earth. That they do so is proved by several circumstances:- 1st, They are not, like the inferior planets, confined to certain limits. of elongation from the sun, but appear at all distances from it, even in the opposite quarter of the heavens, or, as it is called, in opposition; which could not happen, did not the earth at such times place itself between them and the sun : 2dly, They never appear horned, like Venus or Mercury, nor even semilunar. Those, on the contrary, which, from the minuteness of their parallax, we conclude to be the most distant from us, viz. Jupiter, Saturn, Uranus, and Neptune, never appear otherwise than round; a sufficient proof, of