i^4 DEVELOP MENT of a 'cerhin 



'-' /;■'■•;, -0111?)'' -_^DE , .: fin 24> FG 



therefore, ^^.^^,^.^.,^^^;^^, - p]j. ^Bcl ^____,--^-^ _ ^^, 



hence it appears that ^ = ^> fo that the triangles FED, PGF, 



arefimilar; and',' as ',it' is^ obvious,' from what has been already 

 Ihewn, that the' arc lies' "Ai), BF, are el'tlier both lefs or both 

 greater thai^ quadrants, it follows that PD and PF lie in the 

 fame ftraight line. Join AD and BD, and draw BH parallel to 

 DF, meeting AD in H, then AP : PB : : AD : DH : : tan ABD : tan 

 DBH or tan BDF ; that is, i-j-e'-.i-^e'-.: tan p : tan 4^ j there- 

 fore, tan-y — "^~~' ^^^ f- But- if we denote the femi-conjugate 

 axis of the ellipfe by c, we have e' = ~^^ (Art 1 1.) ; and there- 

 fore c — - "* 7 • fo that we at laft derive from our formula, the 

 following very remarkable theorem, firfl obferved by, .CJ^unt 



Let ad be a quadrant of an ellipfe, of. which tli^ femi-tr^nf- 

 verfe axis = i *, and AK a quadrant of she-.cirpumfcribing; cir- 

 cle; let c = the femi-conjugate .axis,, and ^ip,tl)£^excen,t!ri,city.; 

 let 4- = AM, an arch of the circle,, iceckonfid fl'om.i^e es^treuns^- 

 ty of the tranfverfe, and <P = HK, an arch of the: cirde,-rec'kQiv 

 edfrom its interfedlion .with the.CQnj^^gaie a.fif ; ^^rjiw JV^L ^p^ 

 HG perpendicular t;o thetfanfverft, pjiSf,,,.jne^png.-tbf:7e^U|>% 

 N aridF; then^if tl\e arcT^9'4 au(|.? be fu<;h^tji^-t tW,'f.^f_,f:^Pi^ 

 the idifference of the ellifptic arch&^ ,^^jq )^, . ^._ \^qual;-^tp 



^V'"^-""^^ / ' '" ' ..;--i.^ - ;>0 bar 



r4. It would not^>e difficult to fliew, that by means of our 

 formula, other tTieorems fimikr to^that^of FAGN^Nijytijjg^ be 

 inveftigated; but. as ''ffi:fcH^'i^^arpe»'Bave.been pointed out by 

 other writers, ah^^aS'tlii^ f»aper ^asex\elic(€d(fc3 a greater length 

 than was at firff iiiteiide J, 1 ^afs jtl\em ovir for the prefent, and 

 t^ ^ ■ '"'• " ' •' "i" ■ - ^ 'i'lo'liB proceed, 



,31?rP;!e.irjg. page 279. 



