. A treatise on plane and spherical trigonometry, and its applications to astronomy and geodesy, with numerous examples . ertices of thetriangle; the three arcs are called the sides of the triangle. Any two points on the surface of a sphere can be joinedby two distinct arcs, which together make up a greatcircle passing through the points. Hence, when the pointsare not diametrically opposite, these arcs are unequal, oneof them being less, the other greater, than 180°. It is notnecessary to consider triangles in which a side is greaterthan 180°, since we may always replace such a side by therema
. A treatise on plane and spherical trigonometry, and its applications to astronomy and geodesy, with numerous examples . ertices of thetriangle; the three arcs are called the sides of the triangle. Any two points on the surface of a sphere can be joinedby two distinct arcs, which together make up a greatcircle passing through the points. Hence, when the pointsare not diametrically opposite, these arcs are unequal, oneof them being less, the other greater, than 180°. It is notnecessary to consider triangles in which a side is greaterthan 180°, since we may always replace such a side by theremaining arc of the great circle to which it belongs. 183. Geometric Principles. — It is shown in geometry(Art. 702), that if the vertex of a triedral angle is madethe centre of a sphere, then the planes which form thetriedral angle will cut the surface of the sphere in threearcs of great circles, forming a spherical triangle. Thus, let 0 be the vertex of a triedral angle, and AOB,BOC, COA its face-angles. We may construct a spherewith its centre at 0, and with any radius OA. Let AB, 267 268 SPHERICAL BC, CA be the arcs of great circles in which the planes of the face-angles AOB, BOC, COA cut the surface of this sphere; then ABC is a spherical triangle, and the arcs AB, BC, CA are its sides. Now it is shown in geometrythat the three face-angles AOB,BOC, COA are measured by the sides AB, BC, CA, re-spectively, of the spherical triangle, and that the diedralangles OA, OB, OC are equal to the angles A, B, C, respect-ively, of the spherical triangle ABC, and also that a diedralangle is measured by its plane angle. There is then a correspondence between the triedralangle O-ABC and the spherical triangle ABC: the sixparts of the triedral angle are represented by the corre-sponding six parts of the spherical triangle, and all therelations among the parts of the former are the same asthe relations among the corresponding parts of the latter. 184. Fundamental Definitions
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