Elements of geometry and trigonometry . PROPOSITION XIX. THEOREM. If the circumferences of two great circles intersect each other onthe surface of a hemisphere, the sum of the opposite trianglesthus formed, is equivalent to the surface of a lune whose angleis equal to the angle formed by the Let the circumferences AOB, COD,intersect on the hemisphere OACBD ;then will the opposite triangles AOC,BOD, be equal to the lune whose an-gle is BOD. For, producing the arcs OB, OD, onthe other hemisphere, till they meet inN, the arc OBN will be a semi-circum-ference, and AOB one also ; and taki
Elements of geometry and trigonometry . PROPOSITION XIX. THEOREM. If the circumferences of two great circles intersect each other onthe surface of a hemisphere, the sum of the opposite trianglesthus formed, is equivalent to the surface of a lune whose angleis equal to the angle formed by the Let the circumferences AOB, COD,intersect on the hemisphere OACBD ;then will the opposite triangles AOC,BOD, be equal to the lune whose an-gle is BOD. For, producing the arcs OB, OD, onthe other hemisphere, till they meet inN, the arc OBN will be a semi-circum-ference, and AOB one also ; and takingOB from each, we shall have BN= a like reason, we have DN=CO, and BD=AC. Hence,the two triangles AOC, BDN, have their three sides respect-ively equal ; they are therefore symmetrical ; hence they areequal in surface (Prop. XVIII.) : but the sum of the trianglesBDN, BOD, is equivalent to the lune OBNDO, whose angle isBOD: hence, AOC + BOD is equivalent to the lune whoseangle is BOD. Scholium. It is likewise evident that the two spherical pyra-mids, which have the triangles AOC, BOD, for bases, are toge-ther equivalent to the spherical ungula whose angle is BOD.
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Keywords: ., bo, bookcentury1800, booksubjectgeometry, booksubjecttrigonometry
