CHAP. VIII. The value of a is 59' 8", and then a solution of these several equations gives the mean angular motion, per day, of the several planets, as follows: Mars. 31' 27" Times of Dividing the whole circle 360° by the mean daily motion revolution of each planet, will give their respective times of revolution, derived from 687 days. 4331 days. Saturn. 10840 days. Uranus. 28610 days. (106.) For the inferior planets, Mercury and Venus, we have the same principle, only making a greater than a, and Mean an gular motion of the inferior tion round the sun. These diurnal angular motions correspond to 89 days for the revolution of Mercury, and 224.8 days for the revolution planets, and of Venus. All these results are, of course, understood as their revolu- first approximations, and accuracy here is not attempted. We are only showing principles; and it will be noticed, that the times here taken in these considerations, are only to the nearest days, and not fractions of a day, as would be necessary for accurate results. By this method accuracy is never attempted, on account of the eccentricity of the orbits. No two synodical revolutions are exactly alike; and therefore it is very difficult to decide what the real mean values are. (107.) To obtain accuracy, in astronomy, observations must be carried through a long series of years. The following is an example; and it will explain how accuracy can be attained in relation to any other planet. On the 7th of November, 1631, M. Cassini observed Mercury passing over the sun; and from his observations then taken, deduced the time of conjunction to be at 7 h. 50 m., mean time, at Paris, and the true longitude of Mercury 44° 41′ 35′′. Comparing this occultation with that which took place in tions carried 1723, the true time of conjunction was November 9th, at 5 h. long course 29 m., P. M., and Mercury's longitude was 46° 47′ 20′′. Observa through & The elapsed time was 92 years, 2 days, 9 h. 39 m. Twenty- CHAP. VIII. two of these years were bissextile; therefore the elapsed time was (92×365) days, plus 24 d. 9 h. 39 m. of years, to secure accu In this interval, Mercury made 382 revolutions, and 2° 5' racy. 45" over. That is, in 33604.402 days, Mercury described 137522.095826 degrees; and therefore, by division, we find that in one day it would describe 40.0923, at a mean rate. Thus, knowing the mean daily rate to great accuracy, the mean revolution, in time, must be expressed by the fraction 360 4.0923; or, 87.9701 days, or 87 days 23 h. 15 m. 57 s. (108.) The following is another method of observing the periodical times of the planets, to which we call the student's special attention. Another method of observing the periodical re. The orbits of all the planets are a little inclined to the volutions of the planets. plane of the ecliptic. The planes of all the planetary orbits pass through the center of the sun; the plane of the ecliptic is one of them, and therefore the plane of the ecliptic and the plane of any other planet must intersect each other by some line passing through the center of the sun. The intersection of two planes is always a straight line. (See Geometry.) The reader must also recognize and acknowledge the following principle: That a body cannot appear to be in the plane of an observer, unless it really is in that plane. For example: an observer is always in the plane of his meridian, and no body can appear to be in that plane unless it really is in that plane; it cannot be projected in or out of that plane, by parallax or refraction. Hence, when any one of the planets appears to be in the plane of the ecliptic, it actually is in that plane; and let the time be recorded when such a thing takes place. The planet will immediately pass out of the plane, because the two planes do not coincide. Passing the plane of the ecliptic is called passing the node. Keep track of the planet until it comes into the same plane; that is, crosses the other node in this interval of time the planet has described just What is by meant other, as seen CHAP. VIII. 180°, as seen from the sun (unless the nodes themselves are Two nodes in motion, which in fact they are; but such motion is not 180 degrees sensible for one or two revolutions of Venus or Mars). from each Continue observations on the same planet. until it comes from the sun. into the ecliptic the second time after the first observation, or to the same node again; and the time elapsed, is the time of a revolution of that planet round the sun. From such observations the periodical time of Venus became well known to astronomers, long before they had opportunities to decide it by comparing its transits across the sun's disc; and by thus knowing its periodical time and motion, they were enabled to calculate the times and circumstances of the transits which happened in 1761, and in 1769; save those resulting from parallax alone. First idea of the perigee of the plan ots. Final results. (109.) On comparing the time that a planet remains on each side of the ecliptic, we can form some idea of the position of its apogee and perigee. If it is observed to be on each side of the ecliptic the same length of time, then it is evident that the orbit of the planet is circular, or that its longer axis coincides with its nodes. If it is observed to be a shorter time north of the plane of the ecliptic than south of it, then it is evident that its perigee is north of the ecliptic; but nothing more definite can be drawn from this circumstance. (110.) Finally. By the combination of the different methods, explained in articles (98), (100), (101), (105), (107), and (108), and extending the observations through a long course of years, and from age to age, the times of revolution, the mean relative distances of the planets from the sun, were approximated to, step by step, until a great degree of exactness was attained, and the following were the results: (111.) By inspecting the preceding table, we find that the CHAP. VIII. greater the distance from the sun, the greater the time of Times of revrevolution; but the ratio for the time is greater than the ratio olution and corresponding to distance; yet we cannot doubt that some compared connection exists between these ratios. For instance, let us compare the Earth with Jupiter. The ratio between their times of revolution, is near 12. The ratio between their relative distances from the sun, as we perceive, is nearly 5.2. The square of 12 is 144; the cube of 5.2 is near 141. But 12 is a little greater than the real ratio between the times of revolution, and 5.2 is not quite large enough for the ratio of distance; and by taking the correct ratios, they seem to bear the relation of square to cube. Without a very rigid or close examination, we perceive that five revolutions of Jupiter are nearly equal to two revolutions of Saturn; that is, is nearly the ratio between their times of revolution. By inspecting the column of distances, we perceive that the ratio of the distances of these two planets, is nearly; and if we square the first ratio, and cube the second, we shall have nearly the same ratio. distances Now let us compare two other planets, say Venus and Result dis. Mars, more exactly. covered. Log. of the square of the ratio of time, 0.970694 Log. of the cube of the ratio of distance, 0.970680 Thus we perceive that the squares of the times of revolution, are to each other as the cubes of the mean distances of CHAP. VIII. the planets from the sun,* and this is called Kepler's third Kepler's law; and it was by such numerical comparisons that Kepler discovered the law.† laws. We may now recapitulate the three laws of the solar system, called Kepler's laws, as they were discovered by that philosopher. 1st. The orbits of the planets are ellipses, of which the sun occupies one of the foci. 2d. The radius vector in each case describes areas about the focus, which are proportional to the times. 3d. The squares of the times of revolution are to each other as the cubes of the mean distances from the sun. * For a concise mathematical view of this subject, we give the following: Let d and D represent mean distances from the sun, and t and T the times of revolution. T D t d Then =m; n and m taken to represent the ratios. Square the 1st equation and cube the 2d. Then + It appears that Kepler did not compare ratios, as we have done ; but took the more ponderous method of comparing the elements of the ratios (the numbers themselves); for, says the historian:- It was on the 8th of March, 1618, that it first came into Kepler's mind to compare the powers of the numbers which express their revolutions and distances; and by chance he compared the squares of the times with the cubes of the distances; but from too great anxiety and impatience, he made such errors in computation, that he rejected the hypothesis as false and useless; but on examining almost every other relation in vain, he returned to the same hypothesis, and on the 15th of May, of the same year, he renewed his calculation with complete success, and established this law, which has rendered his name immortal |