Elements of geometry and trigonometry . l fall on each other, andthus the two triangles will exactly coincide : hence they areequal; and the surface JX^F-APC. Fora like reason, thesurface FQE:=CPB, and the surface DQE=:APB ; hence we • TJin circle which paiiwfi throuph tlio tliroo pointu A, R, (, or whicli cir-cumscribeN the triangln AHC, can only \w a Ninall riruh) of tho xphero ; for ifit were a great circle, thf thrcn aides AH, IK, AC, would lie in oiio piano, andihm trianglu AUC would b« reduced to ono of iti sidcii. 26 202 GEOMETRY. have DQF+FQE—DQE-APC + CPB^APB, or DFE=: ABC ; hence the
Elements of geometry and trigonometry . l fall on each other, andthus the two triangles will exactly coincide : hence they areequal; and the surface JX^F-APC. Fora like reason, thesurface FQE:=CPB, and the surface DQE=:APB ; hence we • TJin circle which paiiwfi throuph tlio tliroo pointu A, R, (, or whicli cir-cumscribeN the triangln AHC, can only \w a Ninall riruh) of tho xphero ; for ifit were a great circle, thf thrcn aides AH, IK, AC, would lie in oiio piano, andihm trianglu AUC would b« reduced to ono of iti sidcii. 26 202 GEOMETRY. have DQF+FQE—DQE-APC + CPB^APB, or DFE=: ABC ; hence the two symmetrical triangles ABC, DEF areequal in surface. Scholium, The poles P and Qmight lie within triangles ABC,DEF: in which case it would berequisite to add the three trianglesDQF, FQE, DQE, together, in or-der to make up the triangle DEF;and in like manner, to ?dd the threetriangles ÀPC, CPB, APB, together,in order to make up the triangleABC : in all other respects, the de-monstration and the result would still be the 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 circles.
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