fore, this angle PSE: 360°:: the time from inferior conjunction: T. The two angles SEP and PSE being known, the angle SPE is known, and hence SP relatively to SE; for sine angle SPE: sine ang. SEP:: SE: SP. Having thus obtained the distance of the planet from the Sun, we can, at any time, by help of the time T and the time of the preceding inferior conjunction, compute the angular distance of the planet from the Earth, as seen from the Sun, and thence, by help of the distances of the planet and Earth from the Sun, compute the planet's elongation from the Sun. Thus the planet being at O, and the Earth at E, we can compute the angle ESO; and having the sides SE and SO, we can, by trigonometry, compute the angle SEO, the elongation of the planet 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 inferior planets, Mercury and Venus, are explained by those planets moving in orbits nearly circular about the Sun in the centre.

Now, in order to find the greatest elongation of the inferior planets, upon the supposition of circular orbits, at their mean distances, we have this trigonometrical proportion; as ES:SG::radius: to sine angle SEG, because EG is a tangent to the orbit of the planet at the time of its greatest elongation, and the angle SGE, is therefore a right angle. Hence, the greatest elongation of an inferior planet is ex R. a

pressed by this formula; the sine of the angle SEG=-,

b

b being equal to the distance of the Earth from the Sun, a the distance of the planet, and R radius, or sine 900. Or, the greatest elongation may be expressed by this formula, sine ang. SEG=axradius, a being the relative distance of the planet from the Sun, that of the Earth being unity or 1. For instance, let us take Venus: in this case, b may be taken 69x radius

equal to 95, and a=69; then, sine angle SEG=

95

9.86113=46° 35. Again, b being taken equal to 1, a will be equal to .70526, and sine ang. SEG radius X.70526; therefore, the angle SEG is equal to 46° 35', the same as before.

The variations in the greatest elongations of the inferior planets, Mercury and Venus, is owing to the elliptical figure of their orbits, and that of the Earth, which also causes

variation in the stationary points, and in the conjunctions. The ancients observed the places of the fixed stars and planets with respect to the Sun, by the assistance of the Moon, or planet Venus. In the day time they very frequently could observe the situation of the Moon, with respect to the Sun. Venus also being occasionally visible to the naked eye in the day time, they used that planet for the same purpose. Now we can, owing to the convenience of our instruments, without the intervention of a third object, obtain the angular distance of a planet from the Sun, by observing the declinations of each, and the difference of their right ascensions. By which we have, in the triangle formed by the distances of each from the pole of the equator and from each other, two sides and the included angle, to find the third side, the angular distance of the planet from the sun.

3. The motion of an inferior planet is direct from its stationary point, before its superior conjunction, to its stationary point, after the same conjunction; and it appears retrograding from the stationary point, before its inferior conjunction, to the stationary point, after its inferior conjunction.

As the computed place of an inferior planet always agrees with the observed place, (see the preceding Art.) it necessarily follows that the retrograde, stationary appearances, and direct motions of the planets, Mercury and Venus, are explained, by assigning circular motions to them, in orbits which coincide with the plane of the ecliptic.

In order to demonstrate the retrograde and stationary appearances in a clear manner, it will be necessary to consider the effect of the motion of the spectator, arising from the motion of the Earth, in changing the apparent place of a distant body. The spectator not being conscious of his own motion, attributes the motion to the body, and conceives himself to be at rest.

Illustrations. 1. Let S be the Sun, ET the space described by the Earth in a small portion of time, which therefore may be considered as rectilinear; the motion being from E towards T.

Let V be a planet, supposed at rest, any where on the same side of the line of direction of the Earth's motion, as the Sun. Draw EP parallel to TV; then, while the Earth moves through ET, the planet supposed at rest, will appear to a spectator, unconscious of his own motion, to have moved by the angle VEP. which motion is direct, being the same way as the apparent motion of the Sun. And because the Earth appears at rest with respect to the fixed stars, the planet will appear to have moved forward among the fixed stars, by the angle VEP=EVT the motion of the Earth as seen from the planet supposed at rest. Thus the planet, being on the same side of the line of direction of the Earth's motion as the Sun, will appear, as far as the Earth's motion only is concerned, to move direct. Let M be a planet any where on the opposite side of the line of direction, then, the planet will appear to move retrograde by the angle MER. And therefore, as far as the motion of the Earth only is concerned, a planet, when the line of direction of the Earth's motion is between the Sun and planet, will appear retrograde.

;

2. To return to the apparent motion of the inferior planets. Let the Earth be at E, and draw two tangents GE and ED then, when the planet is at D or G, it is at its greatest elongation from the Sun S. It is clear that the planet being in the inferior part of its orbit between D and G, relatively to the Earth, and the Earth being supposed at rest, the planet will appear to move from left to right, that is, retrograde : and in the upper part of the orbit, from right to left, that is, direct. But the Earth not being at rest, we are to consider the effect of its motion. In the case of an inferior planet, the planet and the Sun are always on the same side of the line of direction of the Earth's motion; and therefore the effect of the Earth's motion is always to give an apparent direct motion to the planet.

Hence, in the upper part of the orbit between the greatest elongations, the planet's motion will appear direct, both on account of the Earth's motion and its own motion. In the interior part of the orbit, the planet's motion will only be direct between the greatest elongation and the points where the retrograde motion, arising from the planet's motion, be comes equal to the direct motion which arises from the Earth's motion. At these points the planet appears sta fionary; and between these points through inferior con junction, it appears retrograde. See Dr. Brinkley's Elements of Astronomy.

3. Or, the geocentric motions of the inferior planets may be explained in the following manner: Let S be the Sun, (fig. 1. 258) E the Earth, DPGON the orbit of one of the inferior planets, and AI the sphere of the fixed stars. Draw Ed, EC, EB, and EF through the several stations a N, O, G and D, of the inferior planets. The positions a and N are called conjunctions; the latter is the superior, and the former the inferior conjunction, they being then in a line, or the same vertical plane to the ecliptic, with the Sun. The lines EG and ED being tangents to the orbit at G and D; the planet, when in these points of its orbit, is at its greatest angular distance from the Sun, called its greatest elongation. Now, admitting the Earth to be stationary at E, and the planet to be moving in its orbit from a to b, and from b to G, &c.; it is obvious that when the planet is at a it must appear from the Earth among the fixed stars at d; when it is at b, it must appear at C; when at G, it must

appear at B; when at O, it will appear again at C; and when at N, it must appear at d; when at D, it will appear in the heavens at F; and when it returns to a, it must appear again at d. In this manner will an inferior planet, viewed from the Earth, seem to move backwards and forwards in the heavens from F to B, and from B to F. The points D and G would be the stationary points, if the Earth was at rest; but as the Earth moves in an orbit, the stationary points will not coincide, or be at the time of the greatest elongation, but some days after, when the planet approaches the inferior conjunction, and before the time it is approaching the superior conjunction. For instance, Mercury's greatest elongation at D, 1819, was on the 15th April; but that planet was not stationary until the 22d of the same month: Mercury was at its inferior conjunction at a, on the 3d of May, stationary on the 17th, and at its greatest elongation, on the 31st of the same month. See Squire's Astronomy.

4. A superior planet appears to move retrograde from its stationary point before opposition, to its stationary point after opposition; and direct, from its stationary point before conjunction, to its stationary point after conjunction, being retrograde through opposition, and direct while passing through conjunction.

The interval of time between two succeeding oppositions of a superior planet to the Sun can be observed, for it is known when a superior planet is in opposition, by observing when it is in the part of the zodiac opposite to the place of the Sun. Let T represent the time between two suc cessive oppositions; then viewing the planet from the Sun, the Earth will appear to have gained an entire revolution, or 3600 on the planet, in the time T; and the Earth and planet being supposed to move with uniform angular velocities about the Sun, the angle gained by the Earth will increase uniformly.