2'^ ON THE FIGURE OF THE SARTIt^ 



Solution of the meridian PAO be one of the extremities of the meafured chorcf, 

 problem for ^^^ g ;„ the meridian P B the other extremity ; let AD, BF 

 figure of the be drawn perpendicular to CP, B E perpendicular to the 

 earth from the plane CPAO, and let AB, AE, FE be joined. Then will 



]:r^in;to''"' AE^ be=(CD-CF)^ + (^^^ 



knownplaces, FE' -2 C D X CF-2 AD x^FE, and AB*=AE^ + BE* = 



*'• CA^+CB^-2CDxCF-2ADxFE. 



Now let A, <p be the latitudes of A and B expreHTed in deci- 

 mals of the radius 1, w the difference of longitude or the angle 

 BFE, and D the length of the meafured chordizAB. Then 

 from the properties of the ellipfis we have 



CA^=-- ;. , o r • =q'— 2 ac Sm X* 



a2 cof. X^-\-b^ fm. X* 



AD a'-coLx — _ ^^^^j. ^^^ ^^c ^^^^ ^2 



v/{a*coi.A^.f6*i5n.x2) 



^(a* cot. x*+^* fin. ^* j 

 (2 — fin. ?i.*), neglecting the powers of eingher than the firfi, 

 becaufe c is very rnall in comparifon of a. Whence by fub- 

 flitution, and putting ^^ = 2 a' (i — fin. fin. X— cof. (^(cof, 

 Ti. cof. w), we obtain the following equation; 



r-tY4.a(fin.X-fin. <?)*-— (fin. x* + fin. (p*) )=D*, 



and by extrafting the fquare root of each fide, and reje6ling 

 the fquar". cube, &c. of c, there refults, 



S-c\~{(in. X-fin. (f))8--(fin. X* + fin. (P*)|=sD, 



or a+c I f (fin. x2 -f fin. ci:)^)~?^(fin. x-fin.^)2 j =5JL 



This equation may be otherwife exprefied thus ; let a fphe- 

 rical triangle be conftruded, having two fides equal to the 

 polar diftances of A, B, and contained angle = their difference 

 of longitude; whence find the third fide, which put = 9. 

 Then will fin. (p fin. x -j- cof. (fcof. x x cof. w = cof. S, and 

 1 -fin. <P fin. .\ - cof. <P cof. x cof. w = 1 -cof. ^ = 2 fin. |S*; 



therefore S = 2 a fin. l^, and D = 2 a fin. p + c | fin. p 



(fin.x-xfin.^-)- <^"-^^-j^^-'^^' } . _ Thevalueof^ 



is manifeftly equal to the length of a ftraight line joining two 

 places, whofe latitudes are X, <p, and difference of longitude 

 ojf on a fphere, whofe radius is a. 



From 



