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103. The apparent diurnal motion of a fixed star is in a 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.

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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 altitudes; 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.






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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 four minutes later than the star. On the third and succeeding days, he is still farther and farther to the east when the star returns to the meridian. He has therefore an apparent motion among the stars from west to east, as has been already stated (7).

The meridian altitude of the sun changes from day to day. On the 20th of March and 22d of September, it is the same or nearly the same, as that of the point of the equator which is on the meridian. He is therefore, at these times, in or nearly in the equator. From the 20th of March to the 22d of September, his meridian altitude is greater than that of the meridian point of the equator; and consequently, during this period, he is to the north of the equator and his declination is north. From the 22d of September to the 20th of March, his meridian altitude is less than that of the meridian point of the equator; he is therefore to the south of the equator and his declination is south. The greatest meridian altitude and north declination occur on the 21st of June ; and the least meridian altitude and greatest south declination on the 21st of December. The greatest north and south declinations. are found to be equal, each being about 23° 28'.

It follows, from the preceding observations, that about one half of the sun's apparent path among the fixed stars lies to the north of the equator, and the other to the south.

106. The sun's apparent annual path is a great circle of the celestial sphere.

On several consecutive days about the 20th of March, let the

sun's polar distance, when on the meridian, be obtained (102), and also the interval of time, as shown by a well regulated sidereal clock, between the time of the passage of the sun's centre over the meridian and that of some fixed star. It will commonly be found that, on the first of some two consecutive days, the sun's polar distance is greater than 90°, and on the second, less than 90°. The sun must, therefore, in the intermediate time, have passed from the south to the north side of the equator.

Let EQFB, Fig. 18, be the equator, P and P' its poles, and ECFD the sun's apparent path. Let a and b be the places of the sun in his apparent path, when on the meridian at the two noons preceding, and following his passage from the south to the north side of the equator, S, the star whose passages over the meridian were observed, and Pa'a, Pbb', and PSG, arcs of declination circles. The intervals of time between the passages of the sun over the meridian and those of the star give, when converted into degrees (63), the angles GPa' and GPb', or the arcs Ga' and Gb', which are their measures. The difference between Ga' and Gb' gives a'b'. Then, the changes in the sun's polar distances and in the intervals of time being very nearly uniform, as will appear from examination of their values on several preceding and following days, we have, Pa- Pb: aa' : : a'b' : a'E. The arc a'E taken from Ga' leaves GE, the distance of the point E from the declination circle through the star.*

Let e and d be the places of the sun in his apparent path when on the meridian at any subsequent times, and let the declinations cc' and dd' and the arcs Ge' and Gd' be obtained from observations as above. From the values of the latter and of GE, we know Ec' GE-Ge', and Ed' GE Gd. Then, whatever be the sun's places c and d, it is found that the values of the quantities Ec', Ed', cc', and dd' are such that the proportion, sin Ec': sin Ed' : : tang ce' tang dd', is always true. But assuming ECFD to be a great circle, we have, from the right angled spherical triangles Ec'c and Ed'd (App. 48), tang ce' = tang E sin Ec', and tang dd' = tang E sin Ed'; which gives the same proportion. Hence the


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* From the observations of several consecutive days, the arc GE may be found with great precision, by a method of computation called interpolation. Some cases of this method will be found in the appendix.

sun's apparent path ECFD is a great circle, cutting the equator in two opposite points E and F.

107. The apparent motion of the sun around the earth, is produced by a real annual motion of the earth round the sun.

Let S and E, Fig. 19, be the situations of the sun and earth respectively, at any instant of time, fg, a part of the sun's apparent path in the celestial sphere, a, the apparent place of the sun, and s, a fixed star, supposed to be situated in the apparent path. Then will sEa be the angular distance of the sun from the star. If we suppose the sun to move from S to S' in any given interval of time, his angular distance from the star will become sEb. But if, instead of supposing the sun to move, we suppose the earth to move, in the same interval of time, through the same angular distance from E to E', the sun's angular distance from the star will then become sE'c. As the angles E'SE and SES' are equal, E'S and ES' are parallel, and the angle sEb = sFc=sE'c + EsE'. Hence the angular distance of the sun from the star, at the end of the interval, differs, on the two suppositions, by the angle EsE'; and consequently the sun's apparent motion, during the interval, differs by the same quantity.

If we assume the distance Es of the star to be so great that the distance from E to E', whatever be their situations, is extremely small in comparison with it, the angle EsE' will also be extremely small. Consequently, on this assumption, the sun's apparent motion will be sensibly the same, whether we suppose the sun to revolve round the earth, or the earth to revolve round the sun. But, as the bulk of the sun is more than a million times that of the earth (100), it seems highly improbable that the former revolves round the latter as the central body. The reasonable conclusion therefore is, that the earth revolves round the sun in the course of a year, in the same plane in which the sun appears to move, and thus produces the sun's apparent motion. This conclusion is confirmed by various astronomical facts; some of which will be noticed in their proper places. But although the earth's annual motion is fully established, astronomers frequently find it convenient to speak of the sun's motion; always, however, meaning the apparent motion.

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