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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 removed from the centre 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 EM, 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 SME, and therefore of the proportion of the distance SM, 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} times that of the earth’s. The phases of Jupiter, Saturn, Uranus, and Neptune, being imperceptible, it follows that their orbits must include not only that of the earth, but of Mars also.
(485.) 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, of that planet than of Uranus, and of Uranus again than Neptune. The angular velocity with which a planet appears to re
trograde 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 periodical 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, me 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 SEMX. Then the angles E Se and MSm are given. Now, if em be joined and prolonged to meet SM continued in X, the angle e XE, 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 Xe, being given in the right-angled triangle Ee X, the side E X is easily calculated, and thus S X becomes known. Consequently, in the triangle Sm 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.
(486.) 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 passages 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 ascer
tainable, 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 180°. 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 and by the process of calculation, indicated (art. 418.) 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.
(487.) 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 correspondence. The greater the distance, or the larger the orbit, evidently the longer the period. The order of the planets, beginning from the sun, is the same, whether we arrange them according to their distances, or to the time they occupy in completing their revolutions; and is as follows:- Mercury, Venus, Earth, Mars, — the ultra-zodiacal planets, or, as they are sometimes also called, Asteroids, – Jupiter, Saturn,
Uranus, and Neptune. Nevertheless, when we come to examine the numbers expressing them, we find that the relation between the two series is not that of simple proportional increase. The periods increase more than in proportion to the distances. Thus, the period of Mercury is about 88 days, and that of the Earth 365 — being in proportion as 1 to 4:15, while their distances are in the less proportion of 1 to 2.56; and a similar remark holds good in every instance. Still, the ratio of increase of the times is not so rapid as that of the squares of the distances. The square of 2.56 is 6.5536, which is considerably greater than 4:15. An intermediate rate of increase, between the simple proportion of the distances and that of their squares is therefore clearly pointed out by the sequence of the numbers; but it required no ordinary penetration in the illustrious Kepler, backed by uncommon perseverance and industry, at a period when the data themselves were involved in obscurity, and when the processes of trigonometry and of numerical calculation were encumbered with difficulties, of which the more recent invention of logarithmic tables has happily left us no conception, to perceive and demonstrate the real law of their connection. This connection is expressed in the following proposition:— “ The squares of the periodic times of any two planets are to each other, in the same proportion as the cubes of their mean distances from the sun.” Take, for example, the Earth and Mars *, whose periods are in the proportion of 3652564 to 6869796, and whose distance from the sun is that of 100000 to 152369; and it will be found, by any one who will take the trouble to go through the calculation, that
(3652564)2 : (6869796)2::(100000)3 : (152369). (488.) Of all the laws to which induction from pure observation has ever conducted man, this third law (as it is called) of Kepler may justly be regarded as the most remark
• The expression of this law of Kepler requires a slight modification when we come to the extreme nicety of numerical calculation, for the greater planets, due to the influence of their masses. This correction is imperceptible for the Earth and Mars.
able, and the most pregnant with important consequences. When we contemplate the constituents of the planetary system from the point of view which this relation affords us, it is no longer mere analogy which strikes us — no longer a general resemblance among them, as individuals independent of each other, and circulating about the sun, each according to its own peculiar nature, and connected with it by its own peculiar tie. The resemblance is now perceived to be a true family likeness; they are bound up in one chain — interwoven in one web of mutual relation and harmonious agreement — subjected to one pervading influence, which extends from the centre to the farthest limits of that great system, of which all of them, the earth included, must henceforth be regarded as members.
(489.) The laws of elliptic motion about the sun as a focus, and of the equable description of areas by lines joining the sun and planets, were originally established by Kepler, from a consideration of the observed motions of Mars; and were by him extended, analogically, to all the other planets. However precarious such an extension might then have appeared, modern astronomy has completely verified it as a matter of fact, by the general coincidence of its results with entire series of observations of the apparent places of the planets. These are found to accord satisfactorily with the assumption of a particular ellipse for each planet, whose magnitude, degree of excentricity, and situation in space, are numerically assigned in the synoptic table before referred to. It is true, that when observations are carried to a high degree of precision, and when each planet is traced through many successive revolutions, and its history carried back, by the aid of calculations founded on these data, for many centuries, we learn to regard the laws of Kepler as only first approximations to the much more complicated ones which actually prevail; and that to bring remote observations into rigorous and mathematical accordance with each other, and at the same time to retain the extremely convenient nomenclature and relations of the ELLIPTIC SYSTEM, it becomes necessary to modify, to a certain extent, our verbal