The following Table contains the elements of the sta tions and retrogradations of Venus.
All the elements contained in the two preceding Tables will vary according to the relative position of the greater axis of the orbits of Mercury and Venus.
When Mercury and Venus are examined with a good telescope, after their inferior conjunction with the sun, they appear like the new moon shining with a fine lumi nous crescent. As their elongation from the sun increases, the crescent increases in breadth ; and when they reach their point of greatest western elongation, nearly the half of their disc is illuminated by the sun. During their return to the sun, the enlightened part of their disc gra dually comes round to the earth, and their luminous side is completely turned to the earth at the time of their superior conjunction, when they would appear like the full moon if they were not then eclipsed by the rays of the sun. As they approach to their eastern elongation, the illuminated portion of their disc gradually diminishes till it is dichotomised, or half enlightened, at the point of greatest eastern elongation. In their return to the sun, the luminous disc continues to diminish, and the dark side is completely turned to the earth at the infe rior conjunction. These phenomena will be understood from Fig. 4, where E is the earth moving in its orbit EMN, and ABCDFGHI, the orbit of Venus, for exam ple. When the earth is at E, and Venus at A, her ob scure half is obviously turned to the earth, and she would be invisible even though she were not eclipsed by the solar rays. When Venus gets to B, a small portion of the eastern limb becomes visible like a fine crescent. When she reaches C, half of her luminous disc is turn ed to the earth. At D a greater portion of it is visible ; and the whole enlightened limb is turned to the earth when she is at F, in her superior conjunction with the sun. At G part of her enlightened disc is withdrawn from the earth, and part of her dark limb appears. When she reaches H, she is again half moon. At I she as sumes the crescent form ; and at A her enlightened disc is invisible. By moving an ivory ball, or a white ball of any kind round the flame of a candle, it will be illu minated in the same manner as Venus, and exhibit all the different phases of that planet.
By means of the following Table, the phases of Mer cury and Venus, or the proportion between the illumi nated and the obscure part of their disc, may be found at any given time : The Table serves also for finding the phases of the Moon.
The argument of the preceding Table, when applied to Venus or Mercury, is the angle formed at their cen tre by two lines drawn from Venus, or Mercury, to the sun, and to the earth. In order to find this angle, sup pose that another line is drawn joining the earth and sun. Then add the angle formed at the sun, or the ano maly of commutation, to the difference between the geo centric longitudes of the sun and Venus or Mercury, and this sum being subtracted from six signs, or 180 degrees, will leave the angle formed at Venus or Mer cury.
Let it be required, for example, to find the proportion between the enlightened and the dark portion of Venus' disc, on the 25th of December 1810. The sun's longitude being then 9: 3° 7'; the heliocentric longitude of Venus 3' 57'; and her geocentric longitude 9' 8° 59', as found from the Nautical Ahnanack. Then, since the anomaly of commutation is equal to the difference between the heliocentric longitude of the planet and the longitude of the earth, as seen from the sun, we have With this argument enter the Table, and you will find 0.059 answering to 8 signs, for the diameter of the en
lightened part of Venus. Her whole diameter being 12.000, the diameter of the dark part of her disc will be 11.941; so that she is nearly in conjunction with the sun.
When Venus and Mercury are in their inferior con junction with the sun, we should naturally expect to see them passing over the disc of that luminary like black spots. If the orbits of these planets were in the plane of the ecliptic, this phenomenon would happen at every in ferior conjunction, and the planets would pass exactly across the centre of the solar disc. As their orbits, however, arc inclined to the ecliptic, Venus and Mercu ry arc seldom seen upon the sun, excepting when the line of their nodes coincides nearly with a line joining the earth, the inferior planet, and the sun. At all other times, Venus and Mercury, at the time of their inferior conjunction, pass either above or below the disc of the sun. Thus let us suppose the ellipsis EE, Plate XXXIII. Fig. 2. to be the orbit of the earth moving round the sun, placed at 0 in the same plane, while Venus or Mer cury revolves in the orbit ADBC, inclined at a consider able angle to the ecliptic. The line DC joining the points where Venus has no latitude, or where she as cends above and descends below the ecliptic, will be the line of her nodes. If the earth be at F, and the longi tude of Venus' node is the same as the longitude of the sun, then when Venus comes to her inferior conjunction at C, she will be in the same line with the earth and sun, and will therefore seem to pass over the centre of his disc. If the earth be at G, then when Venus is in con junction she will be at the point v of her orbit ; but the point V is far above the plane of the ecliptic, and there fore she will pass above the sun's limb. In every other position of the line of Venus' nodes, with regard to the place of the earth at the time of Venus' inferior conjunc tion, she will pass either above or below the sun, unless the longitude of the nodes is nearly the same with that of the sun, which happens when the earth is about II and F, the line of the nodes having the position DC. This will be better understood from Plate XXXVIII. Fig. 5. where EE represents the ecliptic, and VV the orbit of Venus. When the sun is in the position C, or in the node of Venus' orbit, his longitude and that of the node will be exactly the same ; and therefore when Venus is in conjunction with the sun, or passes between him and the earth, she will appear to move over the sun's centre in the path ab. When the sun has the po sition D, and when CD is the difference between the sun's longitude and the longitude of Venus' node, then Venus will pass over his upper limb in the path cd. When the sun is at F, Venus will just touch his upper limb at M ; but when Venus is at 1\I, her latitude or dis tance from the ecliptic is MF, so that Mercury and Ve nus will always pass over the sun's disc, if, at the time of their inferior conjunction, their geocentric latitude 1\ IF is less than the semidiameter of the sun's disc. When the sun is at A and B, the same phenomena will happen upon his lower limb.