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 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 gives the entry and egress of the planet's center.

(411.) 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 8" 5776 for the sun's horizontal parallax.

(412.) 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 common focus.

(413.) 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, and Uranus, never appear otherwise than round; a sufficient proof, of itself, that we see them always in a direction not very remote from that in which the sun's rays illuminate them; and that, therefore, we occupy a station which is never very widely

S

E

removed from the center of their orbits, or, in other words, that the earth's orbit is entirely enclosed within theirs, and of comparatively small diameter. One only of them, Mars, exhibits any perceptible phase, and in its deficiency from a circular outline, never surpasses a moderately gibbous appearance,- -the enlightened portion of the disc being never less than seven-eighths of the whole. To understand this, we need only cast our eyes on the annexed figure, in which E is the earth, at its apparent greatest elongation from the sun S, as seen from Mars, M. In this position, the angle S M E, included between the lines S M and E M, is at its maximum; and, therefore, in this state of things, a spectator on the earth is enabled to see a greater portion of the dark hemisphere of Mars than in any other situation. The extent of the phase, then, or greatest observable degree of gibbosity, affords a measure-a sure, although a coarse and rude one-of the angle SM E, and therefore of the proportion of the distance S M, of Mars, to SE, that of the earth from the sun, by which it appears that the diameter of the orbit of Mars cannot be less than 1 that of the earth's. The phases of Jupiter, Saturn, and Uranus being imperceptible, it follows that their orbits must include not only that of the earth, but of Mars also.

M

(414.) All the superior planets are retrograde in their apparent motions when in opposition, and for some time before and after; but they differ greatly from each other, both in the extent of their arc of retrogradation, in the duration of their retrograde movement, and in its rapidity when swiftest. It is more extensive and rapid in the case of Mars than of Jupiter, of Jupiter than of Saturn, and of that planet than Uranus. The angular velocity with which a planet appears to retro

grade is easily ascertained by observing its apparent place in the heavens from day to day; and from such observations, made about the time of opposition, it is easy to conclude the relative magnitudes of their orbits as compared with the earth's, supposing their periodic times known. For, from these, their mean angular velocities are known also, being inversely as the times. Suppose, then, Ee to be a very small portion of the earth's orbit, and M m a corresponding portion of that

of a superior planet, described on the day of opposition, about the sun S, on which day the three bodies lie in one straight line SEM X. Then the angles E Se and MS m are given. Now, if e m be joined and prolonged to meet SM continued in X, the angle e X E, which is equal to the alternate angle X e y, is evidently the retrogradation of Mars on that day, and is, therefore, also given. Ee, therefore, and the angle E X e, being given in the right-angled triangle E e X, the side EX is easily calculated, and thus S X becomes known. Consequently, in the triangle S m X, we have given the side S X and the two angles m S X and m X S, whence the other sides, Sm, m X, are easily determined. Now, Sm is no other than the radius of the orbit of the superior planet required, which in this calculation is supposed circular as well as that of the earth; a supposition not exact, but sufficiently so to afford a satisfactory approximation to the dimensions of its orbit, and which, if the process be often repeated, in every variety of situation at which the opposition can occur, will ultimately afford an average or mean value of its diameter fully to be depended upon.

(415.) To apply this principle, however, to practice, it is necessary to know the periodic times of the several planets. These may be obtained directly, as has been already stated, by observing the intervals of their pas

sages through the ecliptic; but, owing to the very small inclination of the orbits of some of them to its plane, they cross it so obliquely that the precise moment of their arrival on it is not ascertainable, unless by very nice observations. A better method consists in determining, from the observations of several successive days, the exact moments of their arriving in opposition with the sun, the criterion of which is a difference of longitudes between the sun and planet of exactly 180o. The interval between successive oppositions thus obtained is nearly one synodical period; and would be exactly so, were the planet's orbit and that of the earth both circles, and uniformly described; but as that is found not to be the case (and the criterion is, the inequality of successive synodical revolutions so observed), the average of a great number, taken in all varieties of situation in which the oppositions occur, will be freed from the elliptic inequality, and may be taken as a mean synodical period. From this, by the considerations employed in art. 353., and by the process of calculation indicated in the note to that article, the sidereal periods are readily obtained. The accuracy of this determination will, of course, be greatly increased by embracing a long interval between the extreme observations employed. In point of fact, that interval extends to nearly 2000 years in the cases of the planets known to the ancients, who have recorded their observations of them in a manner sufficiently careful to be made use of. Their periods may, therefore, be regarded as ascertained with the utmost exactness. Their numerical values will be found stated, as well as the mean distances, and all the other elements of the planetary orbits, in the synoptic table at the end of the volume, to which (to avoid repetition) the reader is once for all referred.

(416.) In casting our eyes down the list of the planetary distances, and comparing them with the periodic times, we cannot but be struck with a certain correspondThe greater the distance, or the larger the orbit,

ence.