BOOK IV. SPHERICAL GEOMETRY. IOt

when the point O falls on that part ofthe circumference of DE which is con-cave to the point V.— (PI. 4. fig. 8. 9.)

Draw any other arch CPQj of equal curva-ture with A O B, (and alike concave or convexto G) to touch DE on the fame side of itscenter G with A B, and intersecting V D in C,VE in i, and AB in P. Also, from Bdraw BF of equal radius, and convexity withAB touching DE in F, (P. IX. B. I.) andcutting C i produced in I.

Then A 0-0 B--BF, (C. IV. T. XXII.B. I.) and P 0 = 0Q. Now, QJ — IF beingless than B F or A O, we have A P = A 0 +O P, greater than P Qj+- CM, and thereforemuch greater than P Qjb Qj —Pi.

Again, lay off m P = P i, L P = P B, and frontm draw the arch mL of equal curvature, &c.with B P and B i. Then the triangle B P i= LmP (C. II. T. VI. B. II.) is less than thetriangle A C P. Po each of these add the tra-pezium C V B P, (PI. 4. fig. 9.) and we havethe triangle A V B greater than C V i. Alsoto each of the foregoing unequals add thetrapezium A V i P, (PI. 4. fig. 8.) and WOhave the triangle AVB less than the triangleCVi.