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quantities of these angles may be determined by ob. servation in the following manner :
Let a graduated instrument, as DAE, (the larg er the better,) having a moveable Index with sightholes, be fixed in such a manner, that its plane surface may be parallel to the plane of the equator, and its edge AD in the plane of the meridian: so that when the Moon is in the equinoctial, and on the meridian ADE, she may be seen through the sight-holes when the edge of the moveable index cuts the beginning of the divisions at 0, on the graduated limb DE; and when she is so seen, let the precise time be noted. Now, as the Moon revolves about the Earth from the meridian to the meridian again in about 24 hours 48 minutes, she will go a fourth part round it in a fourth part of that time, viz. in six hours twelve minutes, as seen from C, that is, from the Earth's centre or pole. But as seen from A, the observer's place on the Earth's surface, the Moon will seem to have gone a quarter round the Earth when she comes to the sensible horizon at 0; for the index through the sights of which she is then viewed, will be at d, 90 degrees from D, where it was when she was seen at E. Now let the exact moment when the Moon is seen at 0 (which will be when she is in or near the sensible horizon) be carefully noted, that it may be known in what time she has gone from E to 0; which time subtracted from 6 hours 12 minutes (the times of her going from E to L) leaves the time of her going from 0 to L, and affords an easy method for finding the angle OAL, (called the Moon's horizontal parallax, which is equal to the angle ALC) by the following analo
*Here proper allowance must be made for the refraction, which being about 34 minutes of a degree in the horizon, will cause the moon's centre to appear 34 minutes above the horizon when her centre is really in it.
gy: As the time of the Moon's describing the arc EO is to 90 degrees, so is 6 hours 12 minutes to the degrees of the arc Dde, which measures the angle EAL; from which subtract 90 degrees, and there remains the angle OAL, equal to the angle ALC, under which the Earth's semi-diameter AC is seen from the Moon. Now, since all the angles of a right-lined triangle are together equal to 180 degrees, or to two right angles, and the sides of a triangle are always proportional to the sines of the opposite an- The gles, say by the Rule of Three, as the sine of the Moon's angle ALC, at the Moon L, is to its opposite side distance AČ, the Earth's semi-diameter, which is known to ed. be 3985 miles, so is radius, viz. the sine of 90 degrees, or of the right angle ALC, to its opposite side AD, which is the Moon's distance at L from the observer's place at A, on the Earth's surface; or, so is the sine of the angle CAL to its opposite side CL, which is the Moon's distance from the Earth's centre, and comes out at a mean rate to be 240,000 miles. The angle CAL is equal to what OAL wants of 90 degrees.
191. The Sun's distance from the Earth might The Sun's be found in the same way, though with more diffi- cannot be culty, if his horizontal parallax, or the angle OAS, yet so exequal to the angle ASC, were not so small, as to be termined hardly perceptible; being scarce 10 seconds of a as the minute, or the 360th part of a degree. But the Moon's horizontal parallax, or angle OAL, equal to the angle ALC, is very discernible, being 57' 18", or 3438" at its mean state; which is more than 340 times as great as the Sun's: and, therefore, the distances of the heavenly bodies being inversely as the tangents of their horizontal parallaxes, the Sun's distance from the Earth is at least 340 times as great as the Moon's: and is rather underrated at 81 millions of miles, when the Moon's distance is certainly known to be 240 thousand. But
because, according to some astronomers, the Sun's horizontal parallax, is 11 seconds, and according to others only 10, the former parallax making the Sun's distance to be about 75,000,000 of miles, and the latter 82,000,000; we may take it for granted that the Sun's distance is not less than as deduced from the former, nor more than as shewn by the latter: and every one, who is accustomed to make such ob. servations, knows how hard it is, if not impossible, to avoid an error of a second, especially on account of the inconstancy of horizontal refractions. And here the error of one second, in so small an angle, will make an error of 7 millions of miles in so great a distance as that of the Sun's. But Dr. HALLEY has shewn us how the Sun's distance from the Earth, and consequently the distances of all the planets How near from the Sun, may be known to within a 500th the truth part of the whole, by a transit of Venus over the Sun's disc, which will happen on the 6th of June, determin- in the year 1761; till which time we must content ourselves with allowing the Sun's distance to be about 81 millions of miles, as commonly stated by
it may soon be
192. The Sun and Moon appear much about the proved to same bulk; and every one who understands geombe much etry, knows how their true bulks may be deduced bigger than the from the apparent, when their real distances are known. Spheres are to one another as the cubes of their diameters; whence, if the Sun be 81 millions of miles from the Earth, to appear as big as the Moon, whose distance does not exceed 240 thousand miles, he must in solid bulk be 42 millions 875 thousand times as big as the Moon.
193. The horizontal parallaxes are best observed at the equator; 1. Because the heat is so nearly
equal every day, that the refractions are almost constantly the same. 2. Because the parallactic angle is greater there, as at A, (the distance from thence to the Earth's axis being greater,) than upon any parallel of latitude, as a or b.
194. The Earth's distance from the Sun being The rela determined, the distances of all the other planets from him are easily found by the following analogy, the plantheir periods round him being ascertained by obser- ets from vation. As the square of the Earth's period round are known the Sun, is to the cube of its distance from the Sun; to great precision, so is the square of the period of any other planet, to the cube of its distance in such parts or measures their real as the Earth's distance was taken; see 111. This are not proportion gives the relative mean distances of the well planets from the Sun to the greatest degree of exactness. They are as follows, having been deduced from their periodical times; according to the law just mentioned, which was discovered by KEPLER, and demonstrated by Sir ISAAC NEWTON.*
*All the following calculations except those in the two last lines before § 195, were printed in former editions of this work, before the year 1761. Since that time the said two lines (as found by the transit A. D. 1761) were added; and also § 195.
Periodical Revolutions to the same fixed Star, in
Venus The Earth Mars Jupiter Saturn 2246.176 365.2564 686.9785 4332.514 1079.275 Relative mean distances from the Sun.
From these numbers we deduce, that if the Sun's horizontal parallax be 10", the real mean distances of the planets from the Sun in English miles, are
[31,742,200 | 59,313,060 | 82,000,000 | 124,942,680 | 426,478,720 | 782,284,920 | 1,565,035,600 But if the Sun's parallax be 1" their distances are no more than 29,032,500 | 54,238,570 | 75,000,000 | 114,276,750 | 390,034,500 | 715,504,500 | 1,431,435,000
Errors in distance arising from the mistake of 1" in the Sun's
2,709,700 | 5,074,490 | 7,000,000 | 10,665,830 | 36,444,220 |__66,780,420 | 133,600,600 But, from the late transit of Venus, A. D. 1761, the Sun's parallax appears to be only 8'; and according to that,
their real distances in miles are
36,841,468 | 68,891,486 | 95,173,127 | 145,014,148 | 494,990,976 | 907,956,130 | 1,816,455,526
195. These numbers shew, that although we have the relative distances of the planets from the Sun, to the greatest nicety, yet the best observers could not ascertain their true distances until the late long-wished-for transit appeared, in 1761, which we must confess was embarrassed with several difficulties. But another transit of Venus over the Sun, has now been observed, on the third of June, 1769, much better suited to the resolution of this great problem than that in 1761 was; and the result of the observations does not differ materially from the result of those in 1761. No other transit will happen till the year 1874.
196. The Earth's axis produced to the stars, being carried parallel to itself during the Earth's annual revolution, describes a circle in the sphere of Why the the fixed stars equal to the orbit of the Earth. But celestial this orbit, though very large, would seem no bigger than a point, if it were viewed from the stars;
keep still in
By this is meant, that if a line be supposed to be drawn parallel to the Earth's axis in any part of its orbit, the axis keeps parallel to that line in every other part of its orbit: as in fig. I. of plate V. where abcdefgh represents the Earth's orbit in an oblique view, and Ns the Earth's axis keeping always parallel to the line MN.