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Example III. In a right angled spherical triangle given the hypothemuse = 64° 40', and an adjacent angle = 64° 35'; to find the rest.

Example IV. Given one leg = 42°12', and its opposite angle = 48°; to find the rest. Example V. Given a leg = 54° 48', and its adjacent angle = 48"; to find the rest, - Example VI, Given the two legs = 54° 13'; and 42°12', respectwely; to find the rest, . Example VII, Given the two oblique angles = 48° and 64° 35' respectively; to find the rest, Example VIII,

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Example IX, , * Given in an oblique angled spherical triangle, the former of which is the exterior angle ABD, the latter the angle B of the triangle.

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This arc p exceeds a, therefore the perpendicular Ap from the vertical angle falls on the base produced: hence the 2d expression becomes cos b cos (a -- ty cos 5 - Hence, to log cos 84° 14'29" .... 9-0014632 add cos 38° 35' 48". . . . 9.8929604

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from the sum..........1889,4236
take cos 82° 49'83" . . . . 9-0965132

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To find the remaining parts use the known proportion of the sines of sides to the sines of their opposite angles; thus

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2d Method, by Napier's Analogies.

Taking the 14th and 15th formulae at the end of sect. 4, of the preceding chapter, we have - sin #(* - a). sin (; Taj' cos # (b a) cos # (b. 4- a) The log, computation will therefore stand thus:

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From the sum.................... 204515085
Take log cos; (b+a) 64° 14' 7".... 9.6381663

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Hence 81° 15' 44” + 48° 49' 38” = 130° 5' 22" = B, and 81° 15' 44” – 48° 49' 38” = 32° 26' 6" = A; 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.

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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 aris 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 aris of the heavens ; its extremities are called the poles f 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: Jerence of longitude between any two places, is the dis

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