Example X. In an oblique angled spherical triangle ABC, given the side c = 114° 30′, side a = 56° 40′, and the angle c opposite the first side = 125° 20′: to find the rest. Ans. a = 48° 30′, û = 62° 54′, b = 83° 12′. Example XI. Given A= 48° 30′, c = 125° 20′, c = 114° 30′; to find the rest. Example XII. Given a = 56° 40′, c = 114° 30′, B = 62° 54; to find the rest. Example XIII. Given A 48° 30′, c = 125° 20′, b = 83° 12′; to find the rest. Example XIV. Given a = 56° 40, b = 83° 12′, c = 114° 30'; to find the rest. Example XV. Given A 48° 30′, B = 62° 54′, c = 125° 20′; to find the rest. * For more examples see chap. x. * CHAPTER VIII. On Projections of the Sphere. SECTION I. Astronomical Definitions. I. SINCE the figure of the earth differs but little from that of a sphere, it is usual in the greater part of the inquiries and computations of astronomers, to proceed as though it were a sphere in reality; and since, to an observer on the earth, the heavens appear as a very large concave sphere, every part of which is equidistant from him, it has been found expedient to imagine various lines and circles to be described upon the earth, and the planes of several of them to be extended every way until they mark other similar lines and circles upon the imaginary concave sphere of the heavens. Some of these it now becomes necessary to explain. 2. The axis of the earth is an imaginary right line passing through the centre, about which line it is supposed to turn uniformly once in a natural day. 3. The extremities of this axis are called the poles of the earth. 4. That great circle of the earth, the poles of which are the poles of the earth, is called the equator. 5. If the axis of the earth be supposed produced both ways to the concave heavens, it is then called the axis of the heavens; its extremities are called the poles of the heavens; and the circumference formed by extending the plane of the equator to the celestial concavity is called the celestial equator, or the equinoctial. 6. A secondary to the equator drawn through any place on the earth, and passing through the poles, is called the meridian of that place. 7. The latitude of any place upon the surface of the earth, is its distance from the equator measured on an arc of the meridian passing through it. A less circle passing through any place parallel to the equator is called a parallel of latitude. Places that lie between the equator and the north pole have north latitude; if they lie between the equator and the south pole they have south latitude. 8. All places that lie under the same meridian have the same longitude; and those places which lie under different meridians have different longitudes. The dif ference of longitude between any two places, is the dis tance of their meridians measured in degrees, &c. upon the equator. 9. The sensible horizon is a circle, the plane of which is supposed to touch the spherical surface of the earth, in the place of the spectator whose horizon it is. The rational horizon is a circle whose plane passes through the centre of the earth, parallel to the plane of the sensible horizon. The radius of the earth being exceedingly minute compared with that of the celestial sphere, the sensible and rational horizon may, in many astronomical inquiries, be supposed, without error, to coincide. 10. Great circles which are drawn as secondaries to the rational horizon, are called vertical circles; they serve to measure the altitude or the depression of any celestial object. 11. The two points in which all the vertical circles that can be drawn to any rational horizon meet, are called, the one above the spectator the zenith, and that which is below him the nadir.. 12. Almucantars, or parallels of altitude, are circles parallel to the horizon, or whose poles are the zenith and nadir. All the points of any one almucantar are at equal altitudes above the horizon. 13. The real motion of the earth about the sun once in a year, gives rise to an apparent motion of the sun about the earth in the same interval of time. The circle in which the sun appears to move is called the ecliptic; the angle in which it crosses the equinoctial the obliquity of the ecliptic;* and the two points where it intersects that circle, the equinoxes. *The obliquity of the ecliptic is a variable quantity, oscillating between certain limits which it never passes. According to the profound investigations of Laplace in physical astronomy, the obliquity may always be determined very nearly by this formula, viz. 23° 28' 23" 05 1191" 2184 [1 cos (t 13"-94645)] where denotes the number of years run over from 1750; it is 14. The distance of the sun, or of any of the heavenly bodies, from the equator, measured on an arc of the meridian, is called the declination; north or south, according as the body is situated north or south of the equator. 15. Secondaries to the celestial equator are called circles of declination; of these, twenty-four, which divide the equator into equal parts of 150 each, are called hour circles; because the sun in his apparent diurnal motion passing over 360° of a circle parallel to the equator, goes throughth of them, or 15°, in an hour. 16. The right ascension of a celestial body is an arc of the equinoctial, intercepted between one of the equinoxes, and a declination circle passing through the body; it is measured according to the order of the sun's apparent motion through the twelve signs. 17. The longitude of a heavenly body is an arc of the ecliptic, contained between the 1st point of Aries (that is, one of the equinoctial points), and a secondary to the ecliptic, or a circle of latitude passing through the body. 18. The latitude of a body is its distance from the ecliptic measured upon a secondary to that circle. And the angle formed at the body by two great circles, one passing through the pole of the equator, the other through the pole of the ecliptic, is called the angle of position. 19. The tropics are two circles parallel to the equinoctial, and touching the ecliptic at the two points where it is most remote from the equator; that is to say, the first points of Cancer and of Capricorn; the former is denominated the tropic of Cancer, the latter the tropic of Capricorn. negative before, positive after, that epoch. This theorem is found to answer very well up to the time of Pytheas, 350 years before the Christian era. The obliquity at the beginning of 1816 is 23° 27'49"-2. 20. The points where the tropics touch the ecliptic are called solstices, because the sun when in either of them appears to be at a stand with regard to his declination. 21. Colures are two secondaries to the equinoctial: one, passing through the equinoctial points, is called the equinoctial colure; the other, passing through the solstices, is called the solstitial colure. 22. Small circles drawn at the distance of 23° 28′ (or correctly 23° 27′ 49′′) from the north and south poles of the equator, are called polar circles; the former the arctic, the latter the antarctic circle. 23. That vertical circle which intersects the meridian of any place at right angles, is called the prime vertical: the points where it cuts the horizon are the east and west points; at the distance of 90° from each of these on the horizon are the north and south; all four being called cardinal points. 24. The distance on the horizon of a vertical circle that passes through any body from the north or south points is the azimuth of that body; the distance of the same circle from the east or west points is the amplitude. 25. In order to represent on a plane the celestial' sphere with all its circles great and small, the ancients invented two kinds of projection. The first, named by them Analemma, has since received the name of Orthographic Projection. The second, originally denoted by the generic term Planisphere, received from the Jesuit Aguilon the name of Stereographic Projection. The adjective orthographic is given to the former, because it is produced by lines which fall at right angles upon the plane which represents the sphere. That of stereographic was given to the other, because it results from the intersection of two solids, a sphere and a cone. G |