If the apparent diameter of a body be measured with a micrometer at any observed zenith distance, and the apparent and true zenith distances be obtained (80 and 91). the above proportion gives the horizontal diameter. For the moon, the difference between the apparent diameters in the horizon and zenith, amounts to about half a minute. For other bodies, the difference is nearly or quite insensible. = = 99. The sine of the equatorial parallax of a body, is to the sine of the apparent semidiameter in a constant ratio. For if R equatorial radius of the earth, R' radius of the body, and D = distance of the body from the earth, we have (93. E), R = D sin and (97), R' D sind. Hence sin sin :: RR. Therefore, since R and R' are constant quantities, the ratio of sin : sin ♪, is constant. For the moon this ratio is ascertained to be, sin : sin ♪ :: 1: 0.2725. Cor. From the proportion we have, R' 100. Apparent and real diameters of the sun and Moon. The apparent diameter of the sun at his mean distance from the earth is 32' 1".8. When least it is 31' 30".1 and when greatest, 32′ 34′′.6. The apparent diameter of the moon at her mean distance is 31' 34".4. When least it is about 29' 21" and when greatest 33' 31". Taking the sun's apparent semidiameter at his mean distance, and the corresponding parallax (96), we find (99. H), the sun's real diameter to be nearly 112 times the equatorial diameter of the earth, or more than 880 000 miles. His bulk is therefore more than a million times that of the earth. In like manner we find the moon's diameter to be about of the equatorial diameter of the earth, or 2160 miles. The moon's surface is therefore about of that of the earth, and her volume or bulk about, of the earth's volume. CHAPTER VII. POLAR DISTANCE OF A BODY-APPARENT DIURNAL MOTIONS OF THE FIXED STARS UNIFORM-MOTION OF THE EARTH 101. The polar distance of a body when on the meridian, is equal to the sum or difference of the complement of the latitude of the place and the zenith distance of the body, according as it culminates to the south or north of the zenith. Let M. Fig. 1, be the point at which a body is when on the meridian of the place A. Then PM PZ + ZM. But PM is the polar distance of the body, ZM its zenith distance, and PZ the complement of the latitude of the place. If the body be on the meridian at I, to the north of the zenith, we have PI PZ-IZ; if at F, we have PF =FZ-PZ. = 102. To find the polar distance or declination of a body. Let the meridian zenith distance of the body be observed at a place whose latitude is known, and be corrected for refraction and parallax. Then by the last article the polar distance becomes known. If the body is a fixed star, the zenith distance only requires correction for refraction, as the star has no sensible parallax. When the body has a sensible diameter, the apparent semidiameter added to or subtracted from, the observed zenith distance of the upper or lower limb, when corrected for refraction and parallax, gives the true zenith distance of the centre. The declination is evidently equal to the difference between the polar distance and 90°, and is north or south, according as the polar distance is less or greater than 90°. It therefore becomes known when the polar distance is known. The polar distances or declinations of the heavenly bodies, are found to vary more or less from day to day, except those of the fixed stars, which continue sensibly the same for several days in succession; but after a longer interval changes become also perceptible in them. 103. The apparent diurnal motion of a fixed star, is in u circle, of which the north pole of the heavens is a geometric pole, and it is uniform. As by the last article the polar distance PS, Fig. 1, of a fixed star S, does not sensibly change during the interval of a day, the apparent diurnal motion must be performed in a circle MSLU about the pole P. To prove that its motion is uniform, let the zenith distances of the star be observed, when it is on the meridian at M and when in other positions S, S', &c. and let the times of observation as shown by a good sidereal clock be also noted. The observed zenith distances when corrected for refraction, give the true zenith distances ZM, ZS, ZS' &c.; from the first of which the polar distance of the star becomes known. Then, since ZP is the complement of the latitude of the place, and PS, PS', &c. the polar distance of the star, we know all the sides in each of the triangles PZS, PZS', &c., and may compute the hour angles ZPS, ZPS', &c. The hour angles thus computed are found to be proportional to the intervals between the times the star has the positions M and S, M and S', &c. The hour angle ZPS therefore increases uniformly with the time, and consequently the apparent diurnal motion of the star is uniform. If one of the positions of the star be near the horizon as at S", a little discrepancy in the result is sometimes found. This depends on the uncertainty of the correction for refraction at these low latitudes; and observations of this kind serve to determine the proper amount of the correction. For, the rate of the clock having been ascertained, the true interval between the times at which the star has the positions M and S", becomes known, and thence the hour angle ZPS". Then in the triangle PZS", we have the sides ZP and PS" and the included angle ZPS", to find ZS". The difference between the observed and computed values of ZS" must be the refraction. 104. The apparent diurnal revolutions of the heavenly bodies from east to west, is produced by a real, uniform revolution of the earth on its axis from west to east, during a sidereal day. The apparent diurnal motion of a heavenly body, is not one that becomes at once perceptible to the view, like that of a meteor through the air. But in repeated observations at intervals of sufficient length, we see it at each succeeding observation, when it is to the east of the meridian, become more and more elevated and nearer the meridian, and when to the west, less and less elevated, and farther from the meridian; and not feeling conscious of any motion ourselves, we impute this continued change of position, to a westerly motion in the body. The change of position with regard to the horizon and meridian, and consequently the apparent motion of the body, must, however, be precisely the same, if instead of the body revolving round the earth from east to west, the earth itself revolves round its axis from west to east, making a complete revolution in a sidereal day. Thus, the hour angle MPS, Fig. 1, and therefore the apparent motion of a star S, will be exactly the same to an observer at A, whether we suppose the star to move westwardly from M to S, in any observed time, or suppose that in consequence of a rotation of the earth on its axis, the meridian PMP' of the place A, moves in the same time, eastwardly from the position PSP' to the position PMP'. As the appearance is therefore the same on either supposition, it is more reasonable to assume this rotation of the earth on its axis, than to suppose that all the heavenly bodies, situated at immense and various distances, should have motions so adjusted, as to revolve round it in the same or nearly the same time. This assumption of the earth's rotation on its axis, is confirmed by many astronomical facts. An experimental confirmation of the earth's diurnal motion may be mentioned here. Assuming this motion, the top of an elevated tower must, in consequence of its greater distance from the earth's axis, move eastwardly faster than the bottom. Hence a stone or other heavy body let fall from the top of the tower, and retaining by virtue of its inertia, the excess of the forward or eastwardly motion which it had at the top, must fall a little to the east of the vertical line through the point from which its fall commenced. Now, several experiments of this kind have been made, and the fall of the body has always been found to be in accordance with the assumed rotation of the earth. CHAPTER VIII. EARTH'S ANNUAL MOTION-THE SUN'S APPARENT PATHOBLIQUITY OF THE ECLIPTIC-POSITIONS OF THE FIXED STARS CONSTELLATIONS-CATALOGUES OF THE FIXED STARS. 105. Sun's apparent motion, meridian altitude and declination. When the sun is observed on any day to be on the meridian at the same instant with some fixed star, he is found on the next day to be a little distance to the east, when the star returns to the meridian, and comes to it about 4 minutes later than the star. On the third and succeeding days, he is still farther and farther to the east when the |