is to 10000000, so is 1° 16' 34" to 47' 18", which is the Moon's horizontal parallax. If the two places of observation be not exactly under the same meridian, their difference of longitude must be accurately taken, that proper al. lowance may be made for the Moon's change of declination while she is passing from the meridian of the one to the meridian of the other. 6. The Earth's diameter, as seen from the Moon, subtends an angle of double the Moon's horizontal parallax; which being supposed (as above) to be 57' 18", or 3438", the Earth's diameter must be 1° 54' 36", or 6876". When the Moon's horizontsl parallax (which is variable on account of the eccentricity of her orbit) is 57' 18", her diameter subtends an angle of 31' 2", or 1862": therefore the Earth's diameter is to the Moon's diameter, as 6876 is to 1862; that is, as 3.69 is to 1. And since the relative bulks of spherical bodies are as the cubes of their diameters, the Earth's bulk is to the Moon's bulk, as 49.4 is to 1. 7. The parallax, and consequently the distance and bulk of any primary planet, might be found in the above manner, if the planet were ncar enough to the Earth, to make the difference of its two apparent places sufficiently sensible: but the nearest planet is too remote for the accuracy required. In order therefore to determine the distances and relative bulks of the planets with any tolerable degree of precision, we must have recourse to a method less liable to error: and this the approaching transit of Venus over the Sun's disc will afford us. 8. From the time of any inferior conjunction of the Sun and Venus to the next, is 583 days 22 hours 7 minutes. And if the plane of Venus's orbit were coincident with the plane of the ecliptic, she would pass directly between the Earth and the Sun at each inferior conjunction, and would then appear like a dark round spot on the Sun for about 7 hours and 3 quarters. But Venus's orbit (like the Moon's) only intersects the ecliptic in two opposite points called its nodes. And therefore one half of it is on the north side of the ecliptic, and the other on the south : on which account Venus can never be seen on the Sun, but at those inferior conjunctions which happen in or near the nodes of her orbit. At all the other conjunctions, she either passes above or below the Sun; and her dark side being then toward the Earth, she is invisible. The last time when this planet was seen like a spot on the Sun, was on the 24th of November, old style, in the year 1639. ARTICLE II. Shewing how to find the horizontal parallax of Ve. nus by observation, and from thence, by analogy, the parallax and distance of the Sun, and of all the planets from him. 9. In Fig. 4. of Plate XIV. let DBA be the Earth, V Venus, and TSR the eastern limb of the Sun. To an observer at B the point t of that limb will be on the meridian, its place referred to the heaven will be at E, and Venus will appear just within it at S. But, at the same instant, to an observer at A, Venus is east of the Sun, in the right line AVF; the point t of the Sun's limb appears at e in the heavens, and if Venus were then visible, she would appear at F. The angle CV A is the horizontal parallax of Venus, which we seek; and is equal to the opposite angle FVE, whose measure is the arc FE. ASC' is the Sun's horizontal parallax, equal to the opposite angle eSE, whose measure is the arc e£: and FAe (the same as V Av) is Venus's horizontal parallax from the Sun, which may be found, by observing how much later in absolute time her total ingress on the Sun is, as seen from A, than as seen from B, which is the time she takes to move from V to u in her orbit OVU, 10. It appears by the tables of Venus's motion and the Sun's, that at the time of her ensuing tran. sit, she will move 4' of a degree on the Sun's disc in 60 minutes of time; and therefore she will move 4." of a degree in one minute of time. Now let us suppose, that A is 90° west of B, so that when it is noon at B, it will be VI in the morning at A; that the total ingress as seen from B is at 1 minute past XII, but that as seen from A it is at 7 minutes 30 seconds past VI: deduct 6 hours for the difference of meridians of A and B, and the remainder will be 6 minutes 30 seconds for the time by which the total ingress of Venus on the Sun at S is later as seen from A than as seen from B: which time being converted into parts of a degree is 26", or the arc Fe of Venus's horizontal parallax from the Sun: for, as 1 minute of time is to 4 seconds of a 4 degree, so is 64 minutes of time to 26 seconds of a degree. 11. The times in which the planets perform their annual revolutions about the Sun, are already known by observation. From these times, and the universal power of gravity by which the planets are retained in their orbits, it is demonstrable, that if the Earth's mean distance from the Sun be divided into 100000 equal parts, Mercury's mean distance from the Sun must be equal to 38710 of these parts—Venus's mean distance from the Sun, to 72333—Mars's mean distance, 152369—Jupiter's 520096—and Saturn's, 954006. Therefore, when the number of miles contained in the mean distance of any planet from the Sun is known, we can, by these proportions, find the mean distance in miles of all the rest. 12. At the time of the ensuing transit, the Earth's distance from the Sun will be 1015 (the mean distance being here considered as 1000), and Venus's distance from the Sun will be 720 (the mean distance being considered as 723), which differences from the mean distances arise from the elliptical figure of the planets' orbits-- Subtract 726 parts from 1015, and there will remain 289 parts for Venus's distance from the earth at that time. 13. Now, since the horizontal parallaxes of the planets are* inversely as their distances from the Earth's centre, it is plain, that as Venus will be between the Earth and the Sun on the day of her transit, and consequently her parallax will be then greater than the Sun's, if her horizontal parallax can be on that day ascertained by observation, the Sun's horizontal parallax may be found, and consequently his distance from the Earth.—Thus, suppose Venus's horizontal parallax should be found to be 36".3480; then, As the Sun's distance 1015 is to Venus's distance 289, so is Venus's horizontal parallax 36". 3480 to the Sun's horizontal parallax 10.3493, on the day of her transit. And the difference of these two parallaxes, viz. 25”.9987 (which may be esteemed 26'') will be the quantity of Venus's horizontal paral. lax from the Sun; which is one of the elements for prejecting or delineating her transit over the Sun's disc, as will appear further on. To find the Sun's horizontal parallax at the time of his mean distance from the Earth, say, As 1000 parts, the Sun's mean distance from the Earth's centre, is to 1015, his distance from it on the To prove this, let S be the Sun (Fig. 3.) V Venus, AB the Earth, Cits centre, and AC its semidiameter. The angle AVG is the hori. zontal parallax of Venus, and ASC the horizontal parallax of the Sun. But by the property of plane triangles, as the sine of AVC (or of SVA its supplement to 180) is to the sine of AVC, so is As to AV, and so is CS to CV.-N. B In all angles less than a minute of a degree, the sines, tangents, and arcs, are so nearly equal, that they may, without error be used for one another. And here we make use of Gardiner's logarithmic tables, because they have the sines to every second of a degree. day of the transit, so is 10".3493, his horizontal parallax on that day, to 10".5045, his horizontal parallax at the time of his mean distance from the Earth's centre. 14. The Sun's parallax being thus (or any other way supposed to be) found, at the time of his mean distance from the Earth, we may find his true distance from it in semidiameters of the Earth, by the following analogy. As the sine (or tangent of so small an arc as that) of the Sun's parallax 10".5045 is to radius, so is unity, or the Earth's semidiameter, to the number of semidiameters of the Earth that the Sun is distant from its centre, which number, being multiplied by 3985, the number of miles contained in the Earth's semidiameter, will give the number of miles which the Sun is distant from the Earth's centre. Then, by § 11, As 100000, the Earth's mean distance from the Sun in parts, is to 38710, Mercury's mean distance from the Sun in parts, so is the Earth's mean distance from the Sun in miles to Mercury's mean distance from the Sun in miles.-And, As 100000 is to 72333, so is the Earth's mean distance from the Sun in miles to Venus's mean distance from the Sun in miles.-Likewise, As 100000 is to 152369, so is the Earth's mean distance from the Sun in miles to Mars's mean distance from the Sun in miles.-Again, As 100000 is to 520096, so is the Earth's mean distance from the Sun in miles to Jupiter's mean distance from the Sun in miles.-Lastly, As 100000 is to 954006, so is the Earth's mean distance from the Sun in miles to Saturn's mean distance from the Sun in miles. And thus, by having found the distance of any one of the planets from the Sun, we have sufficient data for finding the distances of all the rest.-And then from their apparent diameters at these known |