and setting on every fifth day during the same period. The time for any intermediate day must be found by proportion, as already explained. TABLE Of the Sun's Rising and Setting for every fifth Day. January 1, Sun rises 5 m. after 8. Sets 55 m. after 3 1 6, 8 7 7 7 7 7 59 3 9 Equation of Time. 30 3 The following table shows what must be added to apparent time, or that as shown by a good sundial, to obtain mean time for every fifth day of the present month. The correction for any intermediate day must be found by proportion, as already directed. Phases of the Moon. First Quarter 3d, at 40 m. after 8 morning. Full Moon 11th, 36 Last Quarter 19th, Moon's Passage over the Meridian. The Moon's centre will pass the first meridian of this country, or that on which the Royal Observatory is situated, at the following times during the present month; and which will therefore af ford convenient times of observation to such as are not very distant from that meridian. The time of her passing any other meridian, or of her being exactly south of any observer, must be found by means of his longitude and her horary motion. TABLE Of the Moon's Meridional Passage. 3d, 4th, - 53 6 6 7 21 8 9 9 10 5 in the morning. 58 9th, 48 19th, 36 20th, - 23 21st, 16 22d, Our astronomical readers are already aware that the phases of this beautiful planet vary like those of the Moon, but are subject to a much longer period in their entire revolution. It is our intention, therefore, to insert the appearance of Venus in this respect, on the first of every month, as calculated from the rule given in the following pages, which will enable our readers to ascertain these phases at any intermediate periods at pleasure; and it deserves to be remarked, that such computations afford excellent exercises for our youthful studies. January 1st, disc of Venus (Enlightened part 0:15144 digits. Dark part 11.84856 Venus is therefore near the point of her inferior conjunction; and by comparing her appearance this month with that stated under the Occurrences in February, it will be seen that the illuminated part of her disc is increasing, and that she has consequently passed her inferior conjunction. Eclipses of Jupiter's Satellites. None of the eclipses of Jupiter's satellites are visible this month, on account of his nearness to the Sun. Form of Saturn's Ring. As the plane of Saturn's ring is sometimes in the plane of the Earth's orbit, and at others considerably inclined to that orbit, it appears more elliptical or open at one time than another. The change in this elliptical appearance, however, is very slow, and we have therefore only inserted it for every third month. The method of calculation is explained under the head of next month. January 1. {Conjugate axis Transverse axis 0.058. 1.000. The ring will therefore appear at this time as a straight line. Other Phenomena. Mercury will be in his inferior conjunction at 45 m. after midnight on the 7th of this month, he will be stationary on the 19th, and attain his greatest elongation on the 31st. Venus will be stationary on the 16th. Jupiter will be in his inferior conjunction at half past 5 in the evening of the 15th. The Moon will be in conjunction with the star ẞ in Taurus at 12 m. after 11 at night on the 8th; and with the star marked a in Scorpio, at 29 m. past 4 in the morning of the 22d. On the PHASES of MERCURY and VENUS. We have, in the former volumes of Time's Telescope, already explained the appearances of the inferior planets; and such of our readers as have attended to these explanations, will not be at any loss to comprehend the subsequent remarks relative to the phases of Mercury and Venus. When these planets are examined with a good telescope, after their inferior conjunctions with the Sun (see T. T. for 1815, p. 22), they appear like the new Moon, shining with a fine luminous crescent. As their elongation from the Sun increases, this crescent augments in breadth; and when they reach their point of greatest western elongation, nearly half of their illuminated disc becomes visible. During their return to the Sun, the enlightened part of their disc is gradually turned towards the Earth, till their luminous side is opposite the Earth at the time of their superior conjunction; when they would appear like the full Moon, if they were not at that time rendered invisible by the intensity of the solar light. As they proceed from this point to their greatest eastern elongation, the enlightened part of their disc gradually diminishes till they become apparently about half illuminated, when they attain that elongation. In their return to the Sun, the enlightened part now conti*nues to diminish, as the dark side of the planet is turned towards the Earth, which is completely the case, at the moment of their inferior conjunctions. From these remarks, it is evident that the phases of the inferior planets are subject to constant variation like those of the Moon; and the following method will enable the reader to calculate these phases, or to find the greatest breadth of the enlightened part, at any time; the whole diameter of the disc being supposed to be divided into twelve equal parts or digits. In order to accomplish this, let S, E, and V, in the following Figure, be the respective positions of the Sun, the Earth, and Venus, at any given time; then that side of Venus which is turned towards the Sun will be completely illuminated by his rays; but only a part of that enlightened face can evidently be seen from the Earth. Draw EvVv from the Earth through the centre of Venus, and join VS and ES; and draw rs perpendicular to EC and po perpendicular to rs; then pq will be the circle of illumination. Now as pV, a part of that circle, will be seen obliquely, it will, according to the laws of vision, and the immense distance of the planet compared with its magnitude, be projected into an elliptic arc on the plane of projection rs, and the point p will, to a spectator at E, appear at o; so that ro will be the versed sine of the arc rp, and consequently so the versed sine of the arc sp. Fig. 1. E Now the angle pVs is equal the angle v'VS; for pVv is equal vVq, and vVs equal to qVS, being both right angles; and therefore, by addition, the sum of the two angles pVv, vVs, is equal to the sum of the two vVq, qVS; but pVv + vVs = pVs, and v'Vq + qVS = v'VS; and consequently os, which is the versed sine of the former pVs, is also the versed sine of the latter v'VS. As this proof is not limited by any particular position of Venus, it follows as a necessary consequence, |