CHAPTER XII. On determining, by Obsercation, the Magnitudes, Situations, &c. of the Orbits of the Planets. ART. 326. IN the determination of the various particulars relative to the orbits of the planets, different authours have adopted different methods, each of which has probably its peculiar advantages in this work the necessary elements will be determined by a process nearly similar to that which we had recourse to in the preceding chapters. We fhall pur sue the enquiry in what appears to be the natural order, by ascertaining, 1st, The times of the periodic revolutions of each planet. 2dly, The position of the line of the apsides, and an epocha of the passage of the planet through that line. 3dly, The excentricity, and thence the proportion of the axes of the orbit. 4thly, The position of the line of the nodes. 5thly, The inclination of the orbit to the ecliptic." 6thly, The relation of the principal axis of each orbit to that of the earth's, or the relation of their mean distances, and thence the absolute dimensions of each orbit. These particulars are determined, according to the best methods now known, in the problems which immediately follow. 327. PROB. I. To determine the time of a planet's periodical revolution. The periodic time of a planet may be determined either by means of observations upon it when in one of its nodes; or by knowing the place and time when it is in oppo ition to the sun, if it be a superiour planet, or in conjunction, if it be inferiour. As the methods depending upon the oppositions and conjunctions are best in practice, it becomes requisite to shew how to determine when a planet is in either of these situations. 32. The latitude and longitude of a planet may be found by observing its situation with respect to known fixed stars (Art. 190. 191.): and by comparing the longitudes deduced from such observations, with the longitude of the sun, either determined by the right ascensions and declinations (Art. 66.), or taken from the Nautical Almanac, or other Ephe mers, the time when the longitudes are the same, or when they differ 180°, may be discovered. But if there are opportunities of finding the longitude and latitude, by the passage of the planet over the meridian, such opportunities furnish us with a method which is more accurate. The time of the planet's culminating being carefully taken, its longitude may be known by finding the longitude of the culminating point of the ecliptic (Art. 160.) at that instant: for the longitude of the planet, and of that point of the ecliptic which culminates with it, are manifestly the same. Then, if it be a superiour planet, to ascertain the time of opposition, choose a day when the difference of the longitudes of the sun and planet is nearly 180°, and on that day, knowing the sun's longitude at noon, find by proportion, or by observation, the longitude at the exact instant (i) when the planet was on the meridian. Now in opposition, the planet's apparent motion is retrograde (Art. 254.); and consequently, the difference between the longi tudes of the planet and sun increases by the sum of their motions. Hence, subtract the sun's longitude from that of the planet at the instant (i), to get the difference (d) of longitudes, estimated according to the order of the signs, and say, as the sum of the daily motions in longitude: the difference between 180° and d 24 hours: the interval between the the time i and the time of opposition. This interval must either be added to, or subtracted from, 7, according as d is greater or less than 180°, and the time of opposition is obtained. As to the conjunction of inferiour planets, it must be found by means of neighbouring fixed stars, as first explained, unless their elongation at the time of conjunction be sufficiently great to render them visible. 329. Thus much being premised, we may find the periodic time of a superiour planet in this manner : observe when the planet is in opposition to the sun, also when it is next in opposition, and note its place at both times; for in these situations the heliocentric and geocentric longitudes are the same. (Art. 260.) Then will be known the arch the planet describes in the interval between the two observations; and it will be, as that arch; the whole circumference: the time elapsed the periodic time nearly; but this can only be exact, on the supposition that the orbits are circular, 350. For an inferiour planet, two observations might be made when the planet was stationary: and in the interval it would have described its whole orbit, together with an arch or angle, equal to that which the earth has described in the same interval, Then it will be, as the whole circumference + that angle; the time elapsed; 360°; the periodic time, nearly. 351, The methods given in the two last articles will give us the periodic time only so far correct aş the motion in the orbit is equable; but this approxi pration will be of considerable service, as it will enable us to tell nearly the number of a planet's revolutions in a given interval. Then to know the periodic time correctly, two observations must be chosen, which were made at the most distant times possible, when the planet was found in the same position relatively to some fixed stars, and at the same time in opposition or conjunction with the sun, according as it is a superiour or inferiour planet. The intervening time between these observations, being divided by the number of the planet's revolutions, will give the time of the periodic revolution respectively to the fixed stars. And as the motions of the planets are usually reckoned from the first point of aries, the periodic revolution with respect to the stars may be reduced to the revolution relatively to that point, by saying, as 360°: the revolution found: the precession of the equinoxes during the revolution: a fourth term which must be deducted, to obtain the revolution. with respect to aries. If the revolution, as thus determined, is not thought sufficiently accurate, it may be corrected after the other elements of the planets are determined, by allowing for the difference between the equations of the centre, at the two observations; but this will very seldom make any material differ ence. 332. For an example, we shall find the revolution of the planet Mars. On April 21, 1715, at 11", this planet was in opposition in m 1° 9′ 30′′. On June 11, 1717, the opposition happened at 9h 11min 20° 17' 15". Now in this time, which was two years (one a bissextile) and 20d 22h 11m, mars had made one revolution, and 49° 27′ 45′′ over; hence from these observations we get an approximation to a revolution, by saying, 360° + 49° 27′ 45′′ : 360° : : 781 22h 11m: 687d 11h 15m, the time of a revolution. From the observations of Ptolemy, it ap pears, that mars was in opposition on December 13, at 11h 48' at Paris, 130 years after the christian æra, in 21° 22′ 50′′. In 1709, January 4, at 5 48", mars was in opposition in 14° 18′ 25′′, The interval between these observations was 15789 11d 18; and allowing for the bissextiles, &c. we get 6864 22h 16m, for the time of a tropical revolu. tion. To obtain the sidereal revolution, say, as the mean daily motion of the planet: the precession of the equinoxes in the time of a tropical revolution :: one day: a number of days, which added to the tro, pical revolution, will give the sidereal, 333. After nearly similar methods, the tropical and sidereal revolutions of all the planets have been ascertained, and are very nearly as below: From a comparison of the most ancient, with the modern observations, there is some reason to con clude, that the time of jupiter's revolution is decreas ing, and that of saturn increasing, each very slowly, 334. Before we quit this problem, it may be necessary to say a few words, respecting what are called the annual and secular motions of the planets. If N (fig. 4, Pl. IV.) be the mean place of a planet in its orbit, at any given instant, and at the end of a year |