CHAP. VIII. SYNODIGAL REVOLUTIONS. 253 ring again to the figure of art. 401., 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 reproduce 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 115.877d, and 583-920d. (404.) In this interval, the planet will have described a whole revolution plus the arc a e, 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 C c 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 phenomena happen may easily be computed from a knowledge of the synodical periods and the radii of the orbits. (405.) 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 109400 miles per hour, of Venus 80060 and of the earth 68080. 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. (406.) 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; but when we regard them as otherwise than cir CHAP. VIII. PHASES OF MERCURY AND VENUS. 255 cles (which they really are), it becomes somewhat complex-too much so to be here entered upon. It will suffice to state the results which experience verifies, and which assigns the stationary points of Mercury at from 15° to 20° of elongation from the sun, according to circumstances; and of Venus, at an elongation never varying much from 29°. The former continues to retrograde during about 22 days; the latter, about 42. (407.) We have said that some of the planets exhibit phases like the moon. This is the case with both Mercury and Venus; and is readily explained by a consideration of their orbits, such as we have above supposed them. In fact, it requires little more than mere inspection of the figure annexed, to show, that to a spectator situated on the earth E, an inferior planet, illuminated by the sun, and therefore bright on the side next to him, and dark on that turned from him, will appear full at the superior conjunction A; gibbous (i. e. more than half full, like the moon between the first and second quarter,) between that point and the points B C of its greatest elongation; half-mooned at these points; and crescent-shaped, or horned, between these and the inferior conjunction D. As it approaches this point, the crescent ought to thin off till it vanishes altogether, rendering the planet invisible, unless in those cases where it transits the sun's disc, and appears on it as a black spot. All these phenomena are exactly conformable to observation; and, what is not a little satisfactory, they were predicted as necessary consequences of the Copernican theory before the invention of the telescope.* (408.) The variation in brightness of Venus in different parts of its apparent orbit is very remarkable. This arises from two causes: 1st, the varying proportion of its visible illuminated area to its whole disc; and, 2dly, the varying angular diameter, or whole apparent magnitude of the disc itself. As it approaches its inferior conjunction from its greater elongation, the half-moon becomes a crescent, which thins off; but this is more than compensated, for some time, by the increasing apparent magnitude, in consequence of its diminishing distance. Thus the total light received from it goes on increasing, till at length it attains a maximum, which takes place when the planet's elongation is about 40°. (409.) The transits of Venus are of very rare occurrence, taking place alternately at intervals of 8 and 113 years, or thereabouts. As astronomical phenomena, they are, however, extremely important; since they afford the best and most exact means we possess of ascertaining the sun's distance, or its parallax. Without going into the niceties of calculation of this problem, which, owing to the great multitude of circumstances to be attended to, are extremely intricate, we shall here explain its principle, which, in the abstract, is very simple and obvious. Let E be the earth, V Venus, and S the sun, and CD the portion of Venus's relative orbit which she describes while in the act of transiting the sun's disc. Suppose A B two spectators at opposite extremities of that diameter of the earth which is perpendicular to the ecliptic, and, to avoid complicating the case, let us lay *See ESSAY ON THE STUDY OF NATURAL PHILOSOPHY, Cab. Cyclo. Vol. XIV. p. 269. CHAP. VIII. TRANSIT OF VENUS. 257 out of consideration the earth's rotation, and suppose A, B, to retain that situation during the whole time of E the transit. Then, at any moment when the spectator at A sees the center of Venus projected at a on the sun's disc, he at B will see it projected at b. If then one or other spectator could suddenly transport himself from A to B, he would see Venus suddenly displaced on the disc from a to b; and if he had any means of noting accurately the place of the points on the disc, either by micrometrical measures from its edge, or by other means, he might ascertain the angular measure of a b as seen from the earth. Now, since AV a, BVb, are straight lines, and therefore make equal angles on each side V, ab will be to AB as the distance of Venus from the sun is to its distance from the earth, or as 68 to 27, or nearly as 2 to 1: ab, therefore, occupies on the sun's disc a space 2 times as great as the earth's diameter; and its angular measure is therefore equal to about 24 times the earth's apparent diameter at the distance of the sun, or (which is the same thing) to five times the sun's horizontal parallax (art. 298.). Any error, therefore, which may be committed in measuring a b, will entail only one fifth of that error on the horizontal parallax concluded from it. (410.) The thing to be ascertained, therefore, is, in fact, neither more nor less than the breadth of the zone P Q R S, p q r s, included between the extreme apparent paths of the center of Venus across the sun's disc, from its entry on one side to its quitting it on the other. The whole business of the observers at A, B, therefore, resolves itself into this;-to ascertain, with all S |