Elements of geometry and trigonometry . ides AB, AC, tillthe V meet again in D. The arcs ABD,ACD, will be semicircumferences,since two great circles always bisecteach other (Book VIII. Prop. 2.). But in the triangle BCD, wehave the side BC<BD-fCD (Prop1.); add AB+ACto both; we shallhave AB + AC + BC<ABD + ACD,thatistosay,lesst!ianacircumference. PROPOSITION IV. THEOREM 27ie sum of all the sides of any spherical polygon is less than thecircumference of a great circle. Take the pentagon ABCDE, forexample. Produce the sides AB, DC,till they meet in F; then since BC isless than BF+C
Elements of geometry and trigonometry . ides AB, AC, tillthe V meet again in D. The arcs ABD,ACD, will be semicircumferences,since two great circles always bisecteach other (Book VIII. Prop. 2.). But in the triangle BCD, wehave the side BC<BD-fCD (Prop1.); add AB+ACto both; we shallhave AB + AC + BC<ABD + ACD,thatistosay,lesst!ianacircumference. PROPOSITION IV. THEOREM 27ie sum of all the sides of any spherical polygon is less than thecircumference of a great circle. Take the pentagon ABCDE, forexample. Produce the sides AB, DC,till they meet in F; then since BC isless than BF+CF, the perimeter ofthe pentagon ABCDti will be less than that of the quadrilateral AEDF. Again, produce the sides AE, ^ A they meet in G ; we shall have ED<EG + DG; hence the pe-rimeter of the quadrilateral AEDF is less than that of the tri-angle AFG ; which last is itself less than the circumference ofa great circle ; hence, for a still stronger reason, the perimeterof the polygon ABCDE is less than this same
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