the ecliptic will be obtained. The instant of the equinoxes is determined in the same manner, only that in the equation between the time and the declination, the declination is made zero, for in these points the sun is in the plane of the ecliptic. The length of the year is determined by comparing together the time of the sun's being in either equinox, or in either tropic, with the time of his being in the same point for another year distant from the former by a long period; the interval reckoned in days and parts of a day, divided by the num ber of years elapsed, will give the true length of the year; and the greater the interval, the more correct will it be. The length of the year however, like all astronomical data, was determined by successive approximations, but it was very early known to be 365.25 days. The Julian year being known, if the synodic revolutions of the planets be known, their mean motions for any given interval may be found. 610. The longitude of an inferior planet in conjunction, or of a superior planet in opposition, is the same as if viewed from the centre of the sun. The synodical revolution of the planet, which is the interval between two conjunctions, or two oppositions, may be ascertained by observation, and from thence its periodic time. Let T be the synodic revolution of a planet, P its periodic time, then P: 365.25 :: 360° : 360° ± «, But these the angle described by the planet in 365.25 days. If it be an inferior planet, its angular motion will be greater than that of the earth; hence the angle described in 365.25 days is equal to 360° plus the angle gained by the planet on the earth, or 360° + «. But if it be a superior planet, its angular velocity being less than that of the carth, the angle described in a Julian year is 360° - α. angles are as the times in which they are described, therefore 360° 360° ±a :: T: 365.25 ±T; P: 365.25:T: 365.25 ±T, 365.25 x T 365.25T P = hence and As the synodic revolutions are known, the sidereal revolutions of the planets are as follow: Mercury The Earth Mars Vesta. Juno Ceres. Pallas Jupiter Saturn Uranus Mercury Whence it will be found by simple proportion that the mean sidereal motions of the planets in a Julian year of 365.2564 days, or the values of n, n', &c., are Days. 87.9705 224.7 365.2564 686.99 1592.69 1331. 1681.42 1686.56 4332.65 10759.4 30687.5 5381034".99 2106644".82 1295977".74 689051".63 355681".17 297216".21 281531".00 280672".32 109256".78 43996".13 15425".64 These have been determined by approximation, continually corrected by a long series of observations on the oppositions and conjunctions of the planets. Mean Distances of the Planets, or Values of a, a', a', &c. 611. The mean distances are obtained from the mean motions of the planets: for, assuming the mean distance of the earth from the sun as the unit, Kepler's law of the squares of the periodic times being as the cubes of the mean distances, gives the following values of the mean distances of the planets from the sun. 1820 Ratio of the Eccentricities to the Mean Distances, or Values of e, e', &c., for 1801. 612. The eccentricity of an orbit is found by ascertaining that heliocentric longitude of the planet at which it is moving with its mean angular velocity, for there the increments of the true and mean anomaly are equal to one another, and the equation of the centre, or difference between the mean and true anomaly is a maximum, and equal to half the eccentricity. By repeating this process for a series of years, the effects of the secular variations will become sensible, and may be determined; and when they are known, the eccentricity may be determined for any given period. The values of e, e', e'', &c., for 1801, are 0.3870981 0.7233316 1.0000000 1.5236923 2.3678700 2.6690090 2.7672450 2.7728860 5.2011524 9.5379564 19.1823927 0.20551494 0.00686074 0.01685318 0.09330700 0.08913000 0.25784800 0.07843900 0.24164800 0.04816210 0.05615050 Inclinations of the Orbits on the Plane of the Ecliptic, in 1801. 613. When the earth is in the line of a planet's nodes, if the planet's elongation from the sun and its geocentric latitude be observed, the inclination of the orbit may be found; for the sine of the elongation is to the radius, as the tangent of the geocentric latitude to the tangent of the inclination. If the planet be 90° distant from the sun, the latitude observed is just equal to the inclination. By this method Kepler determined the inclination of the orbit of Mars. The secular inequalities become sensible after a course of years. The values of p, p', p', &c. were in 1801 Longitudes of the Perihelia. 614. The angular velocity of a body is least in aphelion, and greatest in perihelion; consequently, if its longitude be observed when the increments of the angular velocity are greatest or least, these points will be in the extremities of the major axis: if these be really the two observed longitudes, the interval between them will be exactly half the time of a revolution, a property belonging to no other diameter in the ellipse. As it is very improbable that the observations should differ by 180°, they require a small correction to reduce them to the true times and longitudes. On this principle the longitudes of the perihelia may be determined, and if the observations be continued for a series of years, their secular motions will be obtained, whence their places may be computed for any epoch. The longitude of the perihelion is the distance of the perihelion from the ascending node estimated on the orbit, plus the longitude of the node. In the beginning of 1801, the values of w, w', w", &c., were, 1820 1820 Mercury Venus Mars Vesta Juno Ceres Pallas. Jupiter Saturn Uranus Longitudes of the Ascending Nodes. 615. When a planet is in its nodes, it is in the plane of the ecliptic; its longitude is then the same with the longitude of its node, and its latitude is zero. The place of the nodes may therefore be found by a series of observations, and if they be continued long enough, their secular motions will be obtained; whence their positions at any time may be computed. In the beginning of 1801 the values of 0, e', 0", &c., were, Mercury. Venus. The Earth O 1 "1 74 21 46.8 128 43 53.0 99 30 4.8 332 23 56.4 249 33 24.2 53 33 46.0 147 7 31.1 121 7 4.3 11 8 34.4 89 9 29.5 167 30 23.7 O • "I 45 57 30.9 74 54 12.9 48 0 3.5 103 13 18.2 171 7 40.4 80 41 24.0 172 39 26.8 98 26 18.9 111 56 37.3 72 59 35.4 616. Mean longitudes of the planets on the 1st January, 1801, at midnight, or values of e, e', e", &c. O / "I 163 56 26.9 10 44 21.6 100 9 12.9 |