motion, and become again direct, they acquire sufficient speed to commence overtaking him at which moment they have their greatest western elongation; and thus is a kind of oscillatory movement kept up, while the general advance along the ecliptic goes on.

(468.) Suppose PQ to be the ecliptic, and ABD the orbit of one of these planets, (for instance, Mercury,) seen almost edgewise by an eye situated very nearly in its plane; S, the sun, its centre; and A, B, D, S, successive positions of the planet, of which B and S are in the nodes. If, then, the

sun S stood apparently still in the ecliptic, the planets would simply appear to oscillate backwards and forwards from A to D, alternately passing before and behind the sun; and, if the eye happened to lie exactly in the plane of the orbit, transiting his disc in the former case, and being covered by it in the latter. But as the sun is not so stationary, but apparently carried along the ecliptic PQ, let it be supposed to move over the spaces ST, TU, UV, while the planet in each case executes one quarter of its period. Then will its orbit be apparently carried along with the sun, into the successive positions represented in the figure; and while its real motion round the sun brings it into the respective points, B, D, S, A, its apparent movement in the heavens will seem to have been along the wavy or zigzag line ANHK. In this, its motion in longitude will have been direct in the parts A N, N H, and retrograde in the parts H n K; while at the turns of the zigzag, as at H, it will have been stationary.

(469.) The only two planets - Mercury and Venus whose evolutions are such as above described, are called inferior planets; their points of farthest recess from the sun are called (as above) their greatest eastern and western elongations; and their points of nearest approach to it, their inferior and superior conjunctions, the former when the

planet passes between the earth and the sun, the latter when behind the sun.

(470.) In art. 467. we have traced the apparent path of an inferior planet, by considering its orbit in section, or as viewed from a point in the plane of the ecliptic. Let us now contemplate it in plan, or as viewed from a station above that plane, and projected on it. Suppose then, S to represent the sun, a b c d the orbit of Mercury, and A B C D a part of that of the earth-the direction of the circulation being the same in both, viz. that of the arrow. When the planet stands at a, let the earth be situated at A, in the direction of a tangent, a A, to its orbit; then it is evident that it will appear at its greatest elongation from the sun, the angle a A S, which measures their apparent interval as seen from A, being then greater than in any other situation of a upon its own circle.

E

D

B

S

(471.) Now, this angle being known by observation, we are hereby furnished with a ready means of ascertaining, at least approximately, the distance of the planet from the sun, or the radius of its orbit, supposed a circle. For the triangle SA a is right-angled at a, and consequently we have Sa: SA: sin. SA a: radius, by which proportion the radii Sa, SA of the two orbits are directly compared. If the orbits were both exact circles, this would of course be a perfectly rigorous mode of proceeding: but (as is proved by the inequality of the resulting values of S a obtained at different times) this is not the case; and it becomes necessary to admit an excentricity of position, and a deviation from the exact circular form in both orbits, to account for this difference. Neglecting, however, at present this inequality, a mean or average value of Sa may, at least, be obtained from the frequent repetition of this process in all varieties of situation of the two bodies. The calculations being performed, it is concluded that the mean distance of Mercury from the sun is about 36000000 miles; and that of Venus, similarly derived, about 68000000; the radius of the earth's orbit being 95000000.

(472.) The sidereal periods of the planets may be obtained (as before observed), with a considerable approach to accuracy, by observing their passages through the nodes of their orbits; and indeed, when a certain very minute motion of these nodes and the apsides of their orbits (similar to that of the moon's nodes and apsides, but incomparably slower) is allowed for, with a precision only limited by the imperfection of the appropriate observations. By such observation, so corrected, it appears that the sidereal period of Mercury is 87d 23h 15m 43.9; and that of Venus, 224d 16h 49m 8.0. These periods, however, are widely different from the intervals at which the successive appearances of the two planets at their eastern and western elongations from the sun are observed to happen. Mercury is seen at its greatest splendour as an evening star, at average intervals of about 116, and Venus at intervals of about 584 days. The difference between the sidereal and synodical revolutions (art. 418.) accounts for this. Referring again to the figure of art. 470., if the earth stood still at A, while the planet advanced in its orbit, the lapse of a sidereal period, which should bring it round again to a, would also produce a similar elongation from the sun. But, meanwhile, the earth has advanced in its orbit in the same direction towards E, and therefore the next greatest elongation on the same side of the sun will happen-not in the position a A of the two bodies, but in some more advanced position, e E. The determination of this position depends on a calculation exactly similar to what has been explained in the article referred to; and we need, therefore, only here state the resulting synodical revolutions of the two planets, which come out respectively 115.8774, and 583.920.

(473.) In this interval, the planet will have described a whole revolution plus the arc a ce, and the earth only the arc A CE of its orbit. During its lapse, the inferior conjunction will happen when the earth has a certain intermediate situation, B, and the planet has reached b, a point between the sun and earth. The greatest elongation on the opposite side of the sun will happen when the earth has come to C, and the planet to c, where the line of junction Ce is a tangent

to the interior circle on the opposite side from M. Lastly, the superior conjunction will happen when the earth arrives at D, and the planet at d in the same line prolonged on the other side of the sun. The intervals at which these phænomena happen may easily be computed from a knowledge of the synodical periods and the radii of the orbits.

(474.) The circumferences of circles are in the proportion of their radii. If, then, we calculate the circumferences of the orbits of Mercury and Venus, and the earth, and compare them with the times in which their revolutions are performed, we shall find that the actual velocities with which they move in their orbits differ greatly; that of Mercury being about 109360 miles per hour, of Venus 80000, and of the earth 68040. From this it follows, that at the inferior conjunction, or at b, either planet is moving in the same direction as the earth, but with a greater velocity; it will, therefore, leave the earth behind it; and the apparent motion of the planet viewed from the earth, will be as if the planet stood still, and the earth moved in a contrary direction from what it really does. In this situation, then, the apparent motion of the planet must be contrary to the apparent motion of the sun; and, therefore, retrograde. On the other hand, at the superior conjunction, the real motion of the planet being in the opposite direction to that of the earth, the relative motion will be the same as if the planet stood still, and the earth advanced with their united velocities in its own proper direction. In this situation, then, the apparent motion will be direct. Both these results are in accordance with observed fact.

(475.) The stationary points may be determined by the following consideration. At a or c, the points of greatest elongation, the motion of the planet is directly to or from the earth, or along their line of junction, while that of the earth is nearly perpendicular to it. Here, then, the apparent motion must be direct. At b, the inferior conjunction, we have seen that it must be retrograde, owing to the planet's motion (which is there, as well as the earth's, perpendicular to the line of junction) surpassing the earth's. Hence, the

stationary points ought to lie, as it is found by observation they do, between a and b, or c and b, viz. in such a position that the obliquity of the planet's motion with respect to the line of junction shall just compensate for the excess of its velocity, and cause an equal advance of each extremity of that line, by the motion of the planet at one end, and of the earth at the other: so that, for an instant of time, the whole line shall move parallel to itself. The question thus proposed is purely geometrical, and its solution on the supposition of circular orbits is easy. Let E e and P p represent small

arcs of the orbits of the earth and planet described contemporaneously, at the moment when the latter appears stationary, about S, the sun. Produce p P and e E, tangents at P and E, to meet at R, and prolong EP backwards to Q, join ep. Then since P E, pe are parallel we have by similar triangles P p Ee:: PR: RE, and since, putting v and V for the respective velocities of the planet and the earth, Pp: Eev: V; therefore

v : V :: PR: RE :: sin. PER : sin. EPR

:: cos. SEP: cos. SPQ

:: cos. SEP: cos. (SEP+ESP)

because the angles SER and SPR are right angles. More