Ex. At what time will the sun rise and set at New Haven, Lat. 41° 18′ on the 10th of July? PROBLEMS ON THE CELESTIAL GLOBE. 75. To find the Declination and Right Ascension of a heavenly body: Bring the place of the body (whether the sun or a star) to the meridian. Then the degree and minute standing over it will show its declination, and the point of the equinoctial under the meridian will give its right ascension. It will be remarked, that the declination and right ascension are found in the same manner as latitude and longitude on the terrestrial globe. Right ascension is expressed either in degrees or in hours; both being reckoned from the vernal equinox, (Art. 37.) Ex. What is the declination and right ascension of the bright star Lyra?—also of the sun on the 5th of June? 76. To represent the appearance of the heavens at any time: Rectify the globe for the latitude, bring the sun's place in the ecliptic to the meridian, and set the hour index to XII; then turn the globe westward until the index points to the given hour, and the constellations would then have the same appearance to an eye situated at the center of the globe, as they have at that moment in the sky. Ex. Required the aspect of the stars at New Haven, Lat. 41° 18', at 10 o'clock, on the evening of December 5th. 77. To find the altitude and azimuth of any star : Rectify the globe for the latitude, and let the quadrant of altitude be screwed to the zenith, and be made to pass through the star. The arc on the quadrant, from the horizon to the star, will denote its altitude, and the arc of the horizon from the meridian to the quadrant, will be its azimuth. Ex. What is the altitude and azimuth of Sirius (the brightest of the fixed stars) on the 25th of December at 10 o'clock in the evening, in Lat. 41°? 78. To find the angular distance of two stars from each other: Apply the zero mark of the quadrant of altitude to one of the stars, and the point of the quadrant which falls on the other star, will show the angular distance between the two. Ex. What is the distance between the two largest stars of the Great Bear.* 79. To find the sun's meridian altitude, the latitude and day of the month being given: Having rectified the globe for the latitude, (Art. 66,) bring the sun's place in the ecliptic to the meridian, and count the number of degrees and minutes between that point of the meridian and the zenith. The complement of this arc will be the sun's meridian altitude. Ex. What is the sun's meridian altitude at noon on the 1st of August, in Lat. 41° 18'? CHAPTER III. OF PARALLAX, REFRACTION, AND TWILIGHT. 80. PARALLAX is the apparent change of place which bodies undergo by being viewed from different points, Thus in figure 6, let A represent the earth, CH' the horizon. H'Z a quadrant of Fig. 6. * These two stars are sometimes called "the Pointers," from the line which passes through them being always nearly in the direction of the north star. The angular distance between them is about 50, and may be learned as a standard for reference in estimating by the eye, the distance between any two points on the celestial vault. a great circle of the heavens, extending from the horizon to the zenith; and let E, F, G, H, be successive positions of the moon at different elevations, from the horizon to the meridian. Now a spectator on the surface of the earth at A, would refer the place of E to h, whereas, if seen from the center of the earth, it would appear at H'. The arc H'h is called the parallactic arc, and the angle H'Eh, or its equal AEC, is the angle of parallax. The same is true of the angles at F, G, and H, respectively. 81. Since then a heavenly body is liable to be referred to different points on the celestial vault, when seen from different parts of the earth, and thus some confusion occasioned in the determination of points on the celestial sphere, astronomers have agreed to consider the true place of a celestial object to be that where it would appear if seen from the center of the earth. The doctrine of parallax teaches how to reduce observations made at any place on the surface of the earth, to such as they would be if made from the center. 82. The angle AEC is called the horizontal parallax, which may be thus defined. Horizontal Parallax, is the change of position which a celestial body, appearing in the horizon as seen from the surface of the earth, would assume if viewed from the earth's center. It is the angle subtended by the semi-diameter of the earth, as viewed from the body itself. If we consider any one of the triangles represented in the figure, ACG for example, Sin. AGC: Sin. GAZ::AC: CG... Hence the sine of the angle of parallax, or (since the angle of parallax is always very small*) the parallax itself varies as the sine of the zenith distance of the body directly, and the distance of the body from the center of the earth inversely. Parallax, therefore, increases as a body approaches the horizon, (but increasing * The moon, on account of its nearness to the earth, has the greatest horizontal parallax of any of the heavenly bodies; yet this is less than 1° (being 57') while the greatest parallax of any of the planets does not exceed 30". The difference in an arc of 10, between the length of the arc and the sine, is only 0.18. with the sines, it increases much slower than in the simple ratio of the distance from the zenith,) and diminishes, as the distance from the spectator increases. Again, since the parallax AGC is as the sine of the zenith distance, let P represent the horizontal allax, and P' the parallax at any altitude; then, P' par P: P::sin. zenith dist. : sin. 90°..P= sin. zen. dist. Hence, the horizontal parallax of a body equals its parallax at any altitude, divided by the sine of its distance from the zenith. 83. From observations, therefore, on the parallax of a body at any elevation, we are enabled to find the angle subtended by the semi-diameter of the earth as seen from the body. Or if the horizontal parallax is given, the parallax at any altitude may be found, for Hence, in the zenith the parallax is nothing, and is at its maximum in the horizon. 84. It is evident from the figure, that the effect of parallax upon the place of a celestial body is to depress it. Thus, in consequence of parallax, E is depressed by the arc H'h; F by the arc Pp; G by the arc Rr; while H sustains no change. Hence, in all calculations respecting the altitude of the sun, moon, or planets, the amount of parallax is to be subtracted: the stars, as ad we shall see hereafter, have no sensible parallax. As the depression which arises from parallax is in the direction of a vertical circle, when the body is on the meridian, the body has only a parallax in declination; but in other situations, there is at the same time a parallax in declination and right ascension; for its direction being oblique to the equinoctial, it can be resolved into two parts, one of which (the declination) is perpendicular, and the other (the right ascension) is parallel to the equinoctial. 85. The mode of determining the horizontal parallax, is as follows: Let O, O', (Fig. 7,) be two places on the earth, situated under the same meridian, at a great distance from each other; one place, for example, at the Cape of Good Hope, and the other in the north Let of Europe. The latitude of each Fig. 7. M M and the two radii, CO, CO', whence the side CM may be easily found. But, CM: CO:: sin. ZOM: sin. CMO=sine of the angle of parallax; or (since the arc is very small) equals the parallax P'. But when M as seen from O is in the horizon, ZOM becomes a right angle, and its sine equal to radius. Then, CO CM: CO::1 : P=horizontal parallax=CM On this principle, the horizontal parallax of the moon was determined by La Caille and La Lande, two French astronomers, one stationed at the Cape of Good Hope, the other at Berlin; and and in a similar way the parallax of Mars was ascertained, by observations made simultaneously at the Cape of Good Hope and at Stockholm. 86. On account of the great distance of the sun, his horizontal parallax, which is only 8".6, cannot be accurately ascertained by this method. It can, however, be determined by means of the transits of Venus, a process to be described hereafter. 87. The determination of the horizontal parallax of a celestial body is an element of great importance, since it furnishes the means of estimating the distance of the body from the center of the earth. Thus, if the angle AEC (Fig. 6,) be found, the radius of the earth AC being known, we have in the right angled triangle AEC, the side AC and all the angles, to find the hypo |