sitle a = 44° 13′ 45′′, b = 84° 14′ 29′′, and their included angle c = 36° 45′ 28′′; to find the rest. This example corresponds with case 2, A prob. 2, of oblique angled spherical triangles; and may first be solved by means of the subsidiary arc, in the manner there explained. Thus, first find o, so that tan cos c tan b. This arc 9.9037261 tan 84° 14′ 29′′....10.9963395 tan tan 82° 49′ 33′′.... 10.9000656 exceeds a, therefore the perpendicular AD from the vertical angle falls on the base produced: hence the 2d expression becomes Hence, to log cos 84° 14′ 29′′ cos 38° 35′ 48′′. cos b cos (a) COS C= .... add 9.0014632 from the sum 18.8944236 take cos 82° 49′ 33′′, 9.0965132 Rem. cos c = cos 51° 6' 11”........ 97979104 To find the remaining parts use the known propor tion of the sines of sides to the sines of their opposite angles; thus As sin c. 51° 6' 11"....9-8911340 To sin c So is sin a.. To sin A 32° 26' 7"....9.7294447 And so is sin b.. 84° 14′ 29′′ ....9.9978028 To sin B ..130° 5′ 21′′.... .9.8836846 Here the logarithmic sine 9.8836846 answers either to 49° 54′ 39′′ or to its supplement 130° 5′ 21′′; the former of which is the exterior angle ABD, the latter the angle B of the triangle. 2d Method, by Napier's Analogies. Taking the 14th and 15th formulæ at the end of sect. 4, of the preceding chapter, we have cos (b -a (b + and tan (B+ A) = cot c cos The log. computation will therefore stand thus: From the sum .... 9.5341789 20-0127184 Take log sin (b + a) 64° 14′ 7′′.... 9′9545255 Rem.logtan (B—A) 48° 49′ 38′′".... 10-0581929 Also, to log cot c.. 18° 22' 44"....10-4785395 Add log cos (b— a) 20° 0′ 22′′. ..... 9.9729690 20.4515085 Take log cos(b+a) 64° 14′ 7′′.... 9-6381663 Rem.logtan (B+ A) 81° 15′ 44"....10-8133422 Hence 81° 15′ 44′′ + 48° 49′ 38′′ 130° 5′ 22′′ = B, 32° 26′ 6′′ = a; and 81° 15′ 44′′ 48° 49′ 38′′ = agreeing nearly with the result of the former computation. Then to find c, use the proportion, as sin a: sin a :: sin c: sin c sin 51° 6' 12′′. Here it would seem, from a comparison of the methods, that the first is rather quickest in operation, while the last is probably the easiest to remember, and provides best against the occasions of ambiguity. 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′, в = 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′, в = 62° 54; to find the rest. Given A find the rest. Example XIII. 48° 30′, c = 125° 20′, b = 83° 12′; to 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. 1. 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 t denotes the number of years run over from 1750; it is |