approaches the earth, towards the perpendicular direction; and as the spectator sees any object, not always in its true direction, but in that direction in which the light from it finally enters his eye, a celestial body appears higher than its true place. Thus, the ray SA, which proceeds from a star, is more and more bent towards the vertical line AZ as it approaches the surface, whereby the spectator sees the star in the direction A S', and therefore higher than its true position.

The ray A Z, which traverses the atmosphere perpendicularly, undergoes no refraction. Thus to the eye supposed at the centre all rays would proceed without any deviation; because lines drawn towards the centre of the sphere are perpendicular to its circumference, parallel to which the atmosphere is disposed.

433. This alteration in the apparent place of a celestial body, caused by the atmosphere, is called the ASTRONOMICAL REFRACTION.

The astronomical refraction is 0 at the zenith, and about 34' at the horizon; hence a celestial body, when really on the horizon, appears elevated 34' above it, and is seen on the horizon when really 34 below it. From the same cause all the celestial bodies rise earlier and set later than they would were there no atmosphere.

The refraction varies with the density or weight of the air, being greater when the barometer is high, or the air cold, and less when the barometer is low, or the air warm. The mean refraction, or that in the average state of the atmosphere, is given in Table 31, and corrections for different states of the air in Tables 32 and 33.

Since refraction causes the object to appear too high, it is to be subtracted from the apparent altitude in reducing it to the true altitude.

434. TWILIGHT is the effect of the illumination of the upper regions of the atmosphere by the sun, before he has risen or after he has set, at the place of the spectator. Twilight continues, generally, while the sun is less than 18° below the horizon.

435. PARALLAX IN ALTITUDE is the angular depression of a celestial body, in consequence of its being seen from the surface instead of the centre of the earth, thus:

The body S, which is vertical to the spectator (who always stands with his feet towards the centre) at B, in the line CS, appears at T, being seen in the direction CST; while to a spectator at A the same body appears below T at U, or in the direction ASU; the angle ASC, or TSU, which is equal to A S C, No. 116, is the parallax in altitude. (Tables 34 and 45.)

The spectator at B sees S in the same line as if he were at the centre; that is, a body in the zenith has no parallax. To a spectator at D, to whom S appears in the horizon, the depression, or parallax, is greater than at any other point.

The parallax at the horizon is called the HORIZONTAL PARALLAX. Since parallax makes the object appear too low, it is to be added to the apparent altitude, in reducing it to the true altitude.

436. It is evident, by the fig. No. 435, that the farther off a celestial body is, the less parallax it will have; and the nearer, the more. The sun has about 9′′ hor. par.: the moon has about 1°. Parallax is matter of actual observation, and determines definitively the distances of the sun, moon, and planets.

437. The parallax will obviously be less if the earth's radius is less. Now, the earth being shaped like an orange, the radius, or line from the centre to the surface, in any latitude, is less than at the equator; hence the moon's hor. par. in the Nautical Almanac, which is the equatoreal hor. par., is too great for any latitude. reduction is given in Table 41.

The

438. Since the apparent altitude is too great on account of refraction, and too small on account of parallax, the diff. between these quantities is the diff. between the true and apparent altitudes. This difference, or the combined effect of parallax and refraction, is called the Correction of Altitude.

The moon's Corr. of Alt. is given in Table 39; that of a star is merely its refraction.

439. The SEMI-DIAMETER of a celestial body is half the angle subtended by the diameter of the visible disc.

Thus to a spectator at S the semi-diameter of the body is half the angle subtended by the diameter D F, or contained between the lines SD, SF, supposed to be drawn from S to D and F; the half of this angle is DSC or CS F, and is called the semi-diameter.

It is evident that the semi-diameter will be greater as the body is nearer, and smaller as it is farther off. Thus the variations in the semi-diameter of the sun prove that the distance between the sun and the earth varies at different times of the year. (Table 34.)

440. When the body S is in the zenith, it is nearer to the spectator by half the earth's diameter, CB, than when it is on the horizon; hence it appears larger when in the zenith. This increase of apparent dimensions due to increase of altitude is sensible in the case of the moon only, and is called her AUGMENTATION.* This is given in Table 42.

*The apparent increase of the magnitudes of the sun and moon when near the horizon is a mere optical illusion, whatever explanation may be given of it; for the instruments by

441. The DECLINATION of a celestial body is the portion of the meridian between the equator and the body; it is reckoned from the equator, and is either north or south. Thus, A B, fig. 2, p. 144, is the Declin. of A, and is north.

Since the declination is measured on the celestial meridians, these are called also declination circles.

442. Parallels of Declination are circles parallel to the equator, as the dotted line through A, in both figures, p. 144.

Thus declination is reckoned from the celestial equator as latitude on the surface of the earth is reckoned from the terrestrial equator; and as both these circles are in one and the same plane, declination and terrestrial latitude correspond: that is, a star in 28° N. Decl. passes every day vertically over all places in 28° N. Lat.

443. POLAR DISTANCE is the arc of the celestial meridian between a celestial body and the pole, or the angular distance of a body from the pole. When the Lat. and Decl. are of the same name, the pol. dist. is the compl. of the Decl. to 90°, because the distance from the pole to the equator is 90°; when the lat. and decl. are of different names, the pol. dist. is the sum of the decl. and 90°. Thus the pol. dist. of A is PA; that of A' in S. decl., fig. 2, is P A', which is the sum of 90° and A'B.

444. The AZIMUTH of a celestial body is the angle at the zenith contained between the meridian of the place of the spectator and the circle of altitude passing through the body. It is reckoned to begin from that part of the meridian which is on the polar side of the zenith, that is, from the N. in north latitude; thus, the angle PZA is the azimuth of A.

The angle MZA is the supplement of the azimuth to 180°. This is often used for convenience; thus, instead of N. 132° E., we say S. 48° E.

445. The angle NZA or PZA is the same thing as an angle NCH on the horizontal plane, contained between the north and south line CN, and a line from the eye at C to the foot of the circle of altitude H, which is the " point of the compass" on which A is seen. Now the angle NCH is measured by the arc NH; the azimuth, accordingly, is measured by the arc of the horizon between the meridian of the place and the circle of altitude of the body. The ship's course is the azimuth of the ship's head; so, also, the bearing of an object is its azimuth; and difference of bearing is difference of azimuth.

When a body is on the prime vertical, its azimuth is 90°.

Since refraction and parallax take place vertically, they do not affect the azimuth of a body.

446. The AMPLITUDE is the arc of the horizon between a celestial body at rising or setting and the E. or W. point, and is the com

which the angles subtended by the discs are measured discover no change of magnitude. The constellations, as the Great Bear, Orion, &c., appear in like manner, when near the horizon, to occupy a vast space in the heavens, but when near the zenith much less.

*This cannot be distinctly represented to the eye by figs. 1 and 2, because in fig. 1 the points Z and C coincide, and in fig. 2 the horizon NW SE appears as a straight line.

plement of the azimuth; thus E H is the amplitude of a body rising at H. Amplitude is reckoned from the E. or W.; thus, if EH is 27°, the amplitude of H is E. 27° S.

(1.) The great refraction at the horizon affects sensibly the apparent amplitude. Thus, suppose the spectator in north lat. facing the east, EQ part of the equator, EZ part of the prime vertical, A' a star having north decl. then EA' is the apparent amplitude at the instant of rising; but the star is known to be raised, that is, brought into view, in this case, by refraction, and therefore has not yet, in its revolution, arrived at the horizon; A'is consequently to the left of the place A, where it would rise were there no atmosphere. Hence the arc A'A is applied to the right of the compass-bearing on which A' is observed, in order to correct the apparent place of the star for the effect of refraction. This quantity is given in Table 59 A.

In facing the west the line EQ (which would become W Q) would

E

lie on the other side of the prime vertical, and the star would be seen to set to the right of its true place.

In south lat. the figure drawn above answers to setting, putting W. for E.

(2.) As the elevation of the observer depresses the sea-horizon while it does not affect the place of the star, it produces a further effect of the same kind as that of refraction.

In the case of the moon, as her parallax exceeds the refraction, the opposite effect is produced; that is, when she appears to rise, she has already, to an eye at the centre, passed the rational horizon thus A would be the apparent place of the moon at rising, to the right of the true place A'.

447. The latitude, or distance of the observer from the equator, is measured, on the celestial sphere, by the distance of his zenith from the celestial equator; or ZM is the measure of the latitude, figs. p. 144.

Suppose now D, a star of N. decl., on the meridian at D., then MD is its decl. and ZD its zenith distance; here Z M, the Lat., is the sum of the decl. and zen. dist.

If D' be a star of S. decl., Z M is the diff. of Z D' and M D'.

If a star d be between Z and P, the lat. Z M is the difference of Md and Z d.

448. When the object is to the south of the observer, that is, when his zenith is to the north of the body, the zen. dist. is commonly called N.; when his zenith is to the south of the body, the zen. dist. is called S. In fig. 2, Z D and Z D′ are therefore called North; Zd is called South.

It appears, hence, that when the Decl. and Zen. Dist. are of the same name, their sum is the latitude; when of different names, their difference is the latitude.

But when the star is below the pole, as at d', the Lat. Z M is

the Diff. of Md' and Zd', and Md' is the sum of MP and Pď, or of 90°, and the compl. of the decl.

449. MZ being the lat., PZ is the Colat., since PM is 90o. Also Z N being 90°, PN is the compl. of PZ, and therefore equal to MZ; or the elevation of the pole is equal to the lat. of the place.

450. The altitude of the uppermost point of the equator on the meridian, or MS, is equal to the colatitude, because ZS is 90°. By noting this, and also that the equator passes through the E. and W. points, it is easy, in looking towards the heavens, to figure in the mind, roughly, the position of this circle. This is often useful.

451. In high latitudes, P in the figure falls near Z; in low latitudes, P falls near N. On the equator, Z and M coincide, the celestial equator there passing over the spectator's head.

In S. Lat. the letters N and S in the figures are changed; also the direction of the celestial motions (which we in N. lat. consider from left to right) is there reversed, because in S. lat., in looking towards the equator, the E. is on the right hand.

452. By the help of the preceding considerations (No. 447 and following) it is easy to construct a figure, in any case, to exhibit at once the manner in which the latitude is obtained from the meridian altitude and the declination.

Ex. 1. The Mer. Alt. of the sun, observed to the southward, is 58°; his Decl. 14°N. Fig. 1. Draw a quadrant Z CS by means of the chord of 60° (No. 107). Lay off, by the scale of chords, the Alt. SO, 58°, or the zen. dist. Z O, 32°. Lay off the Decl. 14° to the southward of the sun, as O M, since he is to the northward of the equator; then M is on the equator, and Z M is the LAT. north, and measures 44°.

Ex. 2. The Mer. Alt. of the sun, south of the observer, is 29°; his Decl. 18° S.

Fig. 2. Lay off SO, 29°, and M, 18° to the N. of the sun; then M is the place of the equator, and Z M, the LAT. north, measures 43°.

Ex. 3. The Mer. Alt. of the sun, north of the observer, is 38°; his Decl. 14° N. Fig. 3. Lay off No, the Mer. Alt. 38°, and M the Decl. 14° to the S. of ; then ZM is the LAT. South, and measures 38°.

These figures, which are varieties of fig. 2, p. 144, are of the simplest kind. The point Z being marked on the quadrant, the place of the sun at O, north or south of the observer, is given by the observation; his declination gives M the place where the equator cuts the meridian; whence it is at once seen whether Z is north or south of M, that is, whether the Lat. is N. or S.*

*After a little practice the observer will perceive, at the time of observation, how to deduce the latitude from the mer. alt. and decl. independently of the distinctions of names above (No. 448), which are adopted for the purpose of forming a general rule.