S

164. When the altitude of a celestial body s, above the visible horizon of a spectator is taken, it is obvious that, since the eye of the observer cannot be on the surface of the earth or water, but must be elevated above it as at A; and since the edge of the sea horizon is the circumference of a circle on which a cone, having a for its vertex, would be a tangent to the surface, the observed altitude SAH (the line AKH being a tangent to the earth's surface in a vertical plane passing through the observer and the celestial body, and the effects of refraction being neglected), will exceed the altitude SAh by the angle hAH, called the dip of the horizon, or by its equal ACK.

C

K

h

H

Hence a correction

must be made for this dip or depression, and it may be found from the triangle ACK, in which AB (the height of the spectator above the surface) being given, as well as the semidiameter of the earth, we have AC, CK, and the right angle at K; to find the angle ACK, or its equal HAh.

165. But the dip of the horizon is evidently affected by the refraction of the rays of light in passing from the edge of the sea to the eye of the spectator, in consequence of which, the sea line generally appears to be higher than it is in reality: and a like source of error exists in the observed angular elevations and depressions of objects on land. In order to determine the amount of refraction in these cases, the reciprocal elevations or depressions of two stations at the distance of several miles from each other are observed at the same instant (by signals or otherwise), and then a computation is made by means of a formula which is thus investigated.

H

Let A and B be the true, a and b the apparent places of two stations; let AH, BH' be tangents to spherical surfaces passing through A and B, and let the depressions H'вa, HAb be observed; then, c being the centre of the earth or of the spherical surfaces just mentioned, the angles CAH, CBH' are right angles; the angle AOB is the supplement of c which is subtended by the terrestrial arc between A and B, and is therefore supposed to be given. The sum of the angles OAB, OBA is consequently equal to the known angle at c; then, if from c, or from the sum of those angles, there be taken the sum of the two depressions H'вα, HAb,

the remainder will evidently be the sum of the two refractions, viz. bAB+ ABα ; and assuming that they are equal to each other, half that sum will be the refraction at each station A

or B.

If one of the apparent places, as a', had been above the line BH', drawn through the other, the sum of the refractions at the two stations would have been expressed by C + H'Ba' HAb, and half this quantity would have been the refraction at either station. Again, if one of the stations, as A, had been above the horizontal line BH' drawn through B, the line HA being produced would have met BH' in some point, and would have made with it an angle equal to c; and in this case also, the sum of the refractions would have been expressed by C+ H'вa'· HAB.

166. From numerous observations made during the progress of the trigonometrical survey of Great Britain, it has been determined, that the value of the refraction at each of two stations is equal to about one-twelfth of the angle ACB at the centre of the earth, between the radii drawn through the stations. Therefore, if D represent the dip of the horizon, or the value of the angle ACB (corresponding to ACK in the preceding figure), assuming that, by the effect of refraction, the elevation of the sea-line above K, or the angle KAL is equal to D, we have DD, or 11D for the apparent dip or depression hAL of the horizon; and this formula is used in the construction of the best tables of the dip, in treatises of navigation.

In order to obtain the value of D unaffected by refraction, let r represent the semidiameter of the earth, and h the height AB (in the preceding figure) of the station A above the surface of the sea then, in the right-angled triangle ACK, r+h: h:: rad. (= 1) : cos. ACK, or cos. hAK, or cos. D; therefore

h
r+h

COS. D

hence by trigonometry,

And developing the fraction, rejecting powers of h, above the

the value of D in seconds.

Assuming that the path of a refracted ray from one station to another, or from the edge of the sea to the eye of the observer, is an arc of a circle, Mr. Atkinson determined (Mem.

Astron. Soc., vol. iv. part 2.), that the apparent dip or the

angle hAL is expressed by the formula

1

where is the value of the refraction in terms of the angle

n

ACK (in general n = 12 as above stated). If n were equal to, or less than 2, it would follow from the formula that the dip then becomes zero or imaginary: and in these cases the surface of the sea is visible as far as objects on it can be distinguished, or the water and sky appear to be blended together so that no sea-line appears to separate them.

CHAP. VI.

DETERMINATION OF THE EQUINOCTIAL POINTS AND THE OBLIQUITY OF THE ECLIPTIC BY OBSERVATION.

167. WHEN the observer has succeeded in obtaining the latitude of his station, he is prepared with an astronomical circle, to ascertain the apparent declinations of fixed stars, of the sun, the moon, and the planets. For, supposing the observer to be in the northern hemisphere, it will be evident from an inspection of the figure in art. 150., in which HZO represents half the meridian, z the zenith, P the pole, and Q the place where the equator cuts the meridian, that if any celestial body culminate south of the zenith, as at M, its observed distance Zм from the zenith at the time of culmination being subtracted from zQ, the geocentric latitude of the station will give MQ the required declination; and this will be either north or south of the equator, according as the remainder is positive or negative. If it culminate north of the zenith and above the pole, as at M', the declination will be equal to the sum of the observed zenith distance and the latitude of the station; and lastly, if it culminate north of the zenith and below the pole, as at M", the declination will be equal to the supplement of such sum.

168. The declinations of those which are called fixed stars are not always the same; for independent of certain proper motions in the stars themselves, the plane of the equator, from which the declinations are reckoned, changes its position in the celestial sphere by the effects of planetary attraction on the terrestrial equator; but the declinations of the sun, moon, and planets are subject to considerable variations.

169. If the celestial body be the sun, the observations of the declinations may be considered as the first step in the determination of the elements of its apparent motions; for let it be imagined that, by means of the zenith distances observed during a whole year, beginning, for example, at midwinter, and continuing to the next succeeding midwinter, the declinations of the sun are obtained every time that the luminary arrives at the meridian of the station; the corrections on account of the errors of the instrument, and the effects of refraction and parallax being applied, on comparing such declinations one with another it will be found, that at midwinter

K

the observer being in the northern hemisphere, they have their greatest value on the southern side of the equator, that they afterwards gradually diminish, at first slowly, then more rapidly till the 21st of March, when the declination is exactly or nearly zero; and that they afterwards increase, the sun appearing to be on the northern side of the equator till June 21. From that time till the next midwinter, the changes of declination indicate that the sun descends towards the south, the differences of declination following the same law, nearly, as they followed while the sun was ascending from south to north.

170. The greatest observed northern and southern declinations of the sun may be considered as constituting approximate values of the angle at which the plane of the ecliptic, or of the earth's orbit, and the plane of the equator intersect each other; and the two times at which the declinations are nearly zero, are the approximate times when the sun, in ascending and descending, crosses the plane of the earth's equator produced; but as the observations are only made at the moments of apparent noon at the station, it is scarcely possible that the maxima or the zeros of declination should take place precisely at such moments for any one station; and therefore computation must be made by the rules of trigonometry, in order to obtain those elements with sufficient correctness. Before such computations can be made it will be necessary to explain the method of obtaining by observation the daily movements of the sun in right ascension, or from west to east.

171. The transit telescope and the sidereal clock are used for this purpose. And, supposing the former to be duly adjusted so as to move in the plane of the meridian, also the clock to be regulated, so that the hour hand will perform exactly one revolution (twenty-four hours) in the time of the earth's rotation on its axis; the times of the transits of the sun's centre must be obtained, from the daily observed times of the transits of the limbs: the daily transit of some remarkable fixed star must moreover be observed, the telescope being supposed capable of rendering the star visible should it pass the meridian during the daylight. The times of these transits may be observed on the sidereal clock, though the latter may not indicate the absolute value of sidereal time, which, in fact, cannot be known from observation till the precise moment of the sun's arrival in the plane of the equator has been determined.

The differences between the daily times of the sun's transit are evidently equal to the apparent daily movements of the