34', which is rather greater than the apparent diameter of the sun or moon. Either of these bodies may therefore be wholly visible. when it is really below the horizon. 85. Oval form of the discs of the sun and moon when near the horizon. This is an effect of refraction. As R must be nearly equal to Z (82. D), and as the tangent of an angle increases rapidly when the angle approaches to 90°, it is evident from the expression for r (82. D), that the refraction must increase rapidly near the horizon. Hence the lower part of the disc, when in that situation, is considerably more elevated by refraction than the upper; and consequently the vertical diameter and chords parallel to it are shortened, while the horizontal diameter and its parallel chords are not sensibly affected. This necessarily causes the disc to assume an oval form. The apparent diminution of the vertical diameter amounts, at the horizon, to about of the whole diameter. 8 86. Apparent enlargement of the discs of the sun and moon when near the horizon. Although this is not an effect of refraction, it may properly be noticed here. It is an optical illusion of the same kind as that which makes a ball or other object appear larger when seen at a distance on the ground than when viewed, at the same distance from the eye, on the top of a high steeple. Our judgment of the magnitude of a distant object depends not only on the angle it subtends at the eye, but also on a concurring though sometimes very erroneous impression with regard to the distance; the same object, seen under the same angle, appearing larger as there is an impression of greater distance. Now in viewing the sun or moon when at or near the horizon, the various intervening objects near the line of sight, give the impression of its being more remote, than when seen in an elevated position. When the sun or moon is viewed through a smoked glass, which renders intervening objects invisible, the disc does not appear thus enlarged. 87. Twinkling of the Stars. From changes in the temperature, currents of air, and other causes, the atmosphere is continually more or less agitated. This agitation produces momentary condensations and dilatations in its constituent molecules, and thus occasions slight but sudden and continually repeated deviations in the directions of the rays of light which traverse it. As the stars appear merely as luminous points, presenting scarcely any visible discs, these irregularities in the directions of their rays of light give to them the apparent tremulous motion called the twinkling of the stars. The discs of the planets, though small, are much larger than those of the stars, as is shown by observations with the telescope. They are therefore less affected than the stars, and the twinkling is but little observable in them, except sometimes near the horizon, where the cause producing it usually acts with the greatest effect. 88. Twilight or Crepusculum. This depends on both reflections and refractions of the sun's rays in the atmosphere. When, in the evening, the sun has descended so far below the horizon as to cease to be visible by refraction (84), a portion of the lower part of the atmosphere ceases to receive his rays directly, and is only illumined by light diffused through it by reflection from the higher parts. As the sun continues to descend below the horizon, the part of the atmosphere that is not directly enlightened by his rays increases, and at the same time its illumination gradually diminishes, in consequence of the diminished portion of the atmosphere from which its light is received. This gradual diminution of the light continues till the sun has descended so far below the horizon as to cease to illuminate any sensible portion of the atmosphere above it. This takes place when he is about 18° below the horizon. The last appearance of twilight must evidently be in the western part of the heavens. In the morning the twilight commences, or the first dawn of day is perceived in the eastern part of the heavens, when the sun has arrived within about 18° of the eastern horizon; and the light then increases in the same gradual manner as it diminishes in the evening. CHAPTER VI. APPARENT AND TRUE PLACES OF A BODY-PARALLAX-METHOD OF FINDING THE PARALLAX OF A BODY-PARALLAXES AND DISTANCES OF THE MOON AND SUN-THEIR APPARENT AND REAL DIAMETERS. 89. Apparent and true places of a body. The place which a planet or any other of the heavenly bodies, except the fixed stars, appears to occupy in the celestial sphere varies with a change in the position of the observer. Astronomers,, therefore, in order to render their observations easily comparable, and for convenience in computations, reduce the place of a body as observed at any place on the surface of the earth, to that in which it would appear to be, if seen from the centre. The place in the celestial sphere in which a body would appear to be as seen from any point on the earth's surface, if there were no refraction, is called the apparent place of the body; and that in which it would appear to be if seen from the centre of the earth, is called the true place. Thus, if C, Fig. 15, be the centre of the earth, A a place on its surface, Z the zenith of this place, B the place of a body, and b and c the points in which AB and CB produced meet the celestial sphere, then is b the apparent place and e the true place of the body. 90. The Parallax or Parallax in altitude of a body is the angle contained between two straight lines conceived to be drawn from the centre of the body, one to the centre of the earth and the other to a place on its surface. Thus, for the place A, the angle ABC is the parallax of a body at B. The Horizontal Parallax is the parallax when the body is in the horizon, or, which is the same, when the apparent zenith distance is 90°. Thus, the angle AB'C is the horizontal parallax of the body. The Equatorial Parallax of a body is its horizontal parallax for a place at the equator. 91. The parallax of a body is equal to the difference between the apparent and true zenith distances of the body, or between the true and apparent altitudes. For as ZAB is an exterior angle of the triangle ABC, we have ang. ZCB+ ang. ABC ang. ZAB; or ABC = ZAB-ZCB. But ABC is the parallax, ZAB the apparent zenith distance, and ZCB the true zenith distance. As the altitudes are the complements of the zenith distances, the difference between them must be the same. Cor. It is evident that parallax increases the zenith distance, and consequently diminishes the altitude. Hence, to obtain the true zenith distance from the apparent, the parallax must be subtracted; and to obtain the true altitude from the apparent, it must be added. 92. The sine of the parallax at any altitude is equal to the product of the sine of the horizontal parallax by the sine of the apparent zenith distance. Since the angles ZAB and CAB are supplements of each other, their sines are equal, and we have from the triangles CAB and As the parallax is always a small angle, that of the moon, which is much the greatest, being only about a degree, we may frequently take the parallax itself instead of its sine (App. 51). We then have, p = P sin N.......... (C). When the spheroidal figure of the earth is taken into view, the zenith distance must be taken in reference to the geocentric zenith, and r must be the radius of the earth at the place of observation. 93. Distance of a body in terms of the horizontal parallax and (D) If R the equatorial radius of the earth and the equatorial parallax, then From these two expressions for D, we have the following relation between the equatorial parallax and the horizontal parallax at a place where r is the radius of the earth. R: r::x : P.......... (F) It also follows that the parallaxes of different bodies, or of the same body at different distances, are inversely as the distances. For, let D and D' be the distances of two bodies from the earth, and and 'the corresponding equatorial parallaxes. Then, 94. To find the equatorial parallax of a body. Let B, Fig. 16, be the body, and A and A' two places situated remote from each other on the same meridian. Let the meridian zenith distances ZAB and Z'A'B be observed at the same time by two observers at A and A', and let them be corrected for refraction. Also let the meridian zenith distances ZAS and Z'A'S of a star which passes the meridian at nearly the same time with the body, be observed and corrected for refraction. [The student should here be reminded that the lines AS and A'S are sensibly parallel (13)]. Then BAS, which is the difference of the corrected values of ZAS and ZAB, is known; and also BA'S, the difference of Z'A'B and Z'A'S. Now, ALA' being an exterior angle of the triangle ABL, ABA' ALA' - BAS, but ALA' BA'S, and and hence, ABA' = BA'S - BAS. If the zenith distance of the body is greater at each place than that of the star, as may sometimes occur when the zenith distances of the body and star are |