2' &J. DEVELOPMENT of a certain 



. -fin i<p DE , fin 2-4, FG 



therefore, > .,,,,.,,. r >, — ^?tj a* 10 - > , ,, , r . -, = sf> 



hence it appears that p=r zr -=r, fo that the triangles PED, PGF, 



are fimilar ; and, as it is obvious, from what has been already 

 ihewn, that the' arches AD, BF, are either both lefs or both 

 greater than 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 

 PBH or tan BDF j that is, 1 + e \ 1 — e : : tan <? : tan -^ \ there- 



fore, tan -y = ~[~T~ e ' tan ?• But *fi we denote the femi-conjugate 



T c 



axis of the ellipfe by c, we have f — ^r € (Art 1 1.) j and there- 



1 — e' 



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

 following very remarkable theorem, firft obferved by. Qaunt 



Fagnani. 



Let AD be a quadrant of an ellipfe, of which the femi-tranf- 

 verfe axis — 1 *-, and AK a quadrant of the circumfcribing- cir- 

 cle ; let c — the femi-conjugate axis,, and e-, ^=. ,trfe excentricity ; 

 let 4/ -AM, an arch- of the circle, reckoned from the extremi- 

 ty of the tranfverfe, and <p = HK, an arch of the circle, reckon- 

 ed from its interferon with the conjugate axis ; .draw IVJL and 

 HG perpendicular t;o the tranfverfe axis-,, meeting .the ; ellipfe. in 

 N and F ; then^ if the arcrTejs'-l and f be fuch, thai tan -t. ™ ^ taUj- %, 

 the difference of the elliptic arches ^ANn £$> .is. ^Cjual -tp 



■ =: . / . ]. ••} — DO bar 



t/i — e 1 fin V / 



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



mveitigated ; out as tneie. 1 Relieve* .nayevDeen pointed, out Dy 



other writers, anc! as this paper has extehqecKr.3 a greater length 



than was at firft intended, I pafs them oyer for the prefent, and 



' • {■ II' proceed. 



1 =:J 



[$fy$ KJg-page 279- 



