THE ELEMENTS OP

gS

Produce B D to E in A C. Then (by T. IX.B. I.) A B + A E are greater than B E. Toeach of these add E C; then B A+AC, aregreater than BE + EC. (A. 8.) Again, CE+ DE, are greater than DC# add DBj there-fore, BE+EC, are greater than BD+DCj(Ibid.) much more then will BA+AC, begreater than BD + DC.

THEOREM III*

If from two given points A, B, on theIphere, lying on the fame side of a greatcircle P Q , two great circles A E, B E,be drawn to meet on and make equalangles A E Q , BE I, with the said circle.The arches so drawn, taken together,-shall be less than any other two, AG,BO, drawn from the same points, tomeet on the same line P CT.—(PI. 4. 3 )

For let the arch BIV be perpendicular top 1 C>, and let A E be produced to meet it inY. Also let V G be drawn.

Because the angle V EI = A E G* (T. XIII.B. I.) V EI = B E I, (A. 1 .) and since angleB I E = V I E, the arch YE--BE. (C. V.