Illustrations. 1. Let TEL represent the orbit of the Earth, IDOG that of a superior planet, N the place of the planet when the Earth is at E. Then in the triangle SNE, we have the angle SEN by observation, and the angle NSE by computation. For NSE is the angle at the Sun, which the Earth has gained on the planet since the preceding oppoT. The sition. This angle: 360°:: time since opposition: two angles NSE and SEN being known, the angle SNE is known, and therefore SN relatively to SE: for sine angle SNE: sin. angle SEN:: SE: SN. Having thus obtained the distance of a superior planet from the Sun, we can, at any time, by help of the time T', and time of the preceding opposition, compute the angular distance of the Earth from the planet, as seen from the Sun, and thence, by help of the Earth's distance, and planet's distance, from the Sun, planet's elongation from the Sun. Thus, the planet being at R and the Earth at E, we compute the angle RSE, and knowing the sides ES and SR, we can, (by can compute the we plane trig.) compute the angle RES, the elongation of the planet et from the Sun. This being compared with the observed angle, we always find them nearly agreeing, and thereby is shown that the motions of the superior planets are explained by those planets moving in orbits nearly circular about the Sun. As the computed place nearly agrees with the observed place, it necessarily follows that the retrograde and direct motions, and the stations of these planets are explained, by assigning to them these circular motions. And it is easy to demonstrate these appearances; for it is clear that the planet being in any part of its orbit, and the Earth being supposed at rest at any point E, the planet will appear to move from west to east, or direct. But the earth not being at rest, we are to consider the effect of its motion. The Earth being at E, draw the tangent DEG; then, if the planet is in the upper part of the orbit DIG, it is on the same side of the line of direction of the Earth's motion, as the sun; and therefore the effect of the Earth's motion is to give an apparent direct motion to the planet. The Earth being at E, and the planet at Dor G, the planet is is said to be in quadrature; consequently, from quadrature to conjunction, and from conjunction to quadrature, the planet appears to move direct, both on account of its own motion and the motion of the Earth. If the planet is in the lower part of the orbit DOG, the effect of the | Earth's motion is to give an apparent retrograde motion to the planet: consequently, from quadrature to opposition, and from opposition to quadrature, the planet moves direct or retrograde, according as the effect of the planet's motion exceeds, or is less, than the effect of the Earth's motion. Between quadrature and opposition their effects become equal, and the planet appears stationary; and afterwards, through opposition to the next station, retrograde. 2. These appearances may be also demonstrated in the following manner: Suppose S the Sun, e the Earth, e TEL the orbit of the Earth, IbDOGathe orbit of a superior planet, AF an arc of the heavens at the distance of the fixed stars. Through e and S draw the line OC, through Land a the line LD', through T and b the line TB, and through e and b the line nD'. Then, when the Earth is at e, and the planet at I, it is in opposition to the Sun; but when the planet is at L, it is in conjunction with the Sun, the latter body being in the line or vertical plane joining the Earth and planet. As the velocity of the Earth is greater than that of the superior planet, let us suppose that whilst it moves from L to e, the planet describes the small arcs al and Ib. Hence, when the Earth is at L, and the planet at a, it appears in the heavens at D'; when the earth is at e, the planet at I appears in the heavens at C; and when the Earth is at T, the planet at b appears in the heavens at B. So that whilst the Earth was moving through LeT according to the order of the signs, and the planet through alb, the latter when referred to the heavens, appears, to a spectator at the Earth, to have retrograded through the arc DCB. Suppose now, that when the Earth is at E, the planet is at I, or in conjunction; and whilst the Earth moves from E to n, the planet moves from I to b, then it must have appeared to have moved in the heavens from C to D', according to the order of the signs, or direct. To find the angle of elongation SLD' of any superior planet, when stationary, upon the supposition of circular orbits, at the mean distances of the planets from the Sun; we shall have this formula; sine of supplement of the angle SLD = a √a2+a+1' a being equal to the relative distance of the planet from the Sun, that of the Earth being unity or 1. For example, let us take Mars: in this case, a=1.5236925, a 1.5236925 43° 48′ therefore, Va2 a+12.2012111-.692207=sine 17"; so that the angle of elongation SLD' of Mars is 1360 -11' 43", when he is stationary upon the above supposition. We have supposed above that the orbits are accurately circular, that the planes of these orbits and that of the Earth coincide, and that the angular motions were uniform; but if the planes of the orbits coincided, if the orbits were accurately circular, and were uniformly described, the planets would always appear in the ecliptic, and would always be found exactly in the places which the computation on the circular hypothesis points out; but none of these things take place exactly. The deviation however can be explained, by showing that the planes of the orbits of the planets, except that of Pallas, are inclined to the plane of the Earth's orbit at small angles, and that the orbits are not circles, but only nearly circles, being ellipses, not differing much from circles, as has already been observed. Every phenomenon, even the most minute, can be deduced from such an arrangement; no doubt therefore would remain of the motions of the planets, in such orbits, round the Sun, even had we not the evidence derived from physical astronomy. 5. The apparent velocities of the planets, whether direct or retrograde, are accelerated from one of the stationary points, to the midway between that and the following stationary point; from thence they are retarded till the next station. 6. Their greatest direct velocity is in their conjunction, and their greatest retrograde velocity is in the opposition of the superior planets, and in the lower conjunction of the inferior planets. The greatest apparent motion of a planet when in opposition or conjunction with the Sun, is owing to the parallel motion of the Earth and planet in these points of its orbit. The lower conjunction of an inferior planet, is the same as the inferior conjunction. 7. The shorter the periodic time of an inferior planet, the more frequent are its stations and retrogradations, the shorter time they continue, and the less they are in quantity. This is well known to be the case, both from observation and calculation. For instance, in the year 1819, Mercury was stationary no less than six times, and retrograded four times, whilst Venus was stationary only once, and retrograde only once. The mean arc of Mercury's retrogradation is about 13° 30′, and its mean duration about 23 days; whilst that of Venus is about 16° 12', and its mean duration about 42 days. 8. The longer the periodic time of a superior planet, the more frequent are its stations and retrogradations, but they are less in quantity, yet continue a longer time. The greater the relative motion of the Earth and a superior planet is, the more frequent will a given situation of the two bodies occur; and the less it is, the longer time it will be before similar situations of the two bodies take place. The mean arc of Mars, is 16° 12, and its mean duration about 73 days; whilst the mean arc of retrogradation of Jupiter is only 9° 54', but its mean duration is about 121 day 4 9. When the planets are in their syzygies, their longitude, seen from the Earth is the same as their longitude seen from the Sun, except in the lower conjunction of an inferior planet, when its longitude seen from the Earth, differs 180 degrees from. its longitude as seen from the Sun. Obs. 1. That the superior planets have the same longitude as seen from the Earth and Sun, when in conjunction or opposition, will readily appear, (see fig. 3, page 264,) for when the planet is at I, and the Earth at e, in opposition, it will have the same longitude as seen from the Earth or Sun, the three bodies being in the same vertical plane, or right line directed to the same part of the heavens. The like will be the case when the Earth is at E, or the planet is in conjunction at I. 2. When an inferior planet is in its superior conjunction at N, (fig. 1, page 258) it will have the same longitude, whether observed from the Sun or Earth; but when the planet is in its inferior conjunction at a, it will appear from the Sun to be in the opposite part of the heavens, or 180 degrees from its place, as seen from the Earth, the planet being at the time between the Earth and the Sun. 3. To find the geocentric latitude of a planet, we have the proportion; as the sine of the difference of longitudes of the Earth and planet: the sin. of elongation in longitude :: to tang. of the heliocentric latitude: the tang. of the geocentric latitude. For example, to find the geocentric latitude of Mars, December 1st, at noon, 1819. Sun's long. 8s. 8° 30′ 7", hence the Earth's place, 2s. 8° 30′ 7"; heliocentric long. of planet, 3s. 4° 7', geocentric longitude, 4s. 50 29', heliocentric latitude of the planet 10 20. Then, 2s. 8° 30′ 7" subtracted from 3s. 4° 7', gives 25° 36′ 53′′ diff. long. of Earth and planet. Again, 4s. 50 29' taken from 8s. 8° 30′ 7", leaves 123° 1' 7", elongation in longitude. Hence, sin. 25° 36′ 35′′: sin. 123° 1'7":: tang. 10 20': tang. 2° 35' 5" the geocentric latitude, as required. 4. It may not be improper to observe, that by knowing the longitude of the Earth, its distance from the Sun, the heliocentric long. of the planet, and its distance from the Sun when referred to the ecliptic, there are given two sides of a plane triangle, and the included angle, to find the angle at Farth, or elongation in longitude: so that by knowing |