Mr. Adams on [June, 



COS. H A I : A H ;. that is, rad. : tang. A C :: cos. BAG: tan. 

 A B, or rad. : tan. A C :: cot. A C : md. :; cos. B A C : tan. AB, 

 therefore, rad. cos. B A C = cot. A C tan. A B. 



9. In the right angled triangle G D B, we have rad. : tan. 

 B D G :: D B : B G ; that is. rad. : tan. BAG:: sin. A B : 

 tan. B C, or rad. : tan. BAG:: cot. BAG: rad. :: sin. A B : 

 tan. B G ; therefore, rad. sin. A B — tan. B G cot. BAG. 



10. In the similar right anded triangles O B D, O F E, we 

 have O B : O D :: O F : O E ; that is, rad. : cos. A B :: cos. 

 B G : cos. A G ; therefore, rad. cos. A G = cos. A B cos B G. 



In like manner, if the angle B G A enter into the process, we 

 shall have 



11.' Rad. sin. A B = sin. B G A sin. A G. 



12. Rad. COS. B G A ±= cot. A G tan. B G. 



13. Rad. sin. B G = cot. B G A tan. A B. 



, . T^ , o -n K r^ tan. A B cot. A C tan. A B cos. A C . 



14. By art. 8, cos, B.A G = = —-. — -— - , 



•^ ' rad. rad. sm. A C 



-IT • -r% r-i K rad. sin. A B rad. cos. A B ^, ^ , 



by art. 11, sm. B G A = — ■. — -77— = -: — --^ — nr^y therefore 



•/ * sin. A C sm. A C cot. A B' 



COS. E A C COS. A C COS. BC, /i.iA\ \ r^ 



—■ = = r — , because (art. 10) cos. A C = 



sin. B C A COS. A B rad. ' ^ ^ 



COS. A B COS. B c , ^^^^^^ ^^^ ^^^^ ^^^^ ^^g^ B A G = sin. B G A 



rad. 



cos. B G. 



15. Bvart.9, cot. BAG = '^^^^^^l-^; by art.13, cot.BGA 



•' ' tan. 15 U -' 



rad. sin. B C ^, r. .- T> n A ^^'^•'^ rad . tan. A B 



= tan. AB ' therefore tan. B G A = ^^^^-^jrj = -^^c~ > 



,, .„ tan. B C A tan. A B tan B C rad.- , r „ ^r 



^^^^^ ^^^1^ cot. BAG = s-nrAlTs-i^^nrC = cos-ABcosTbG' ^"* ^^ ^^*- 



10, cos. A^ COS. B G = rad. COS. A G ; therefore, ^^°' ^ ^ ^ = 



__^L_, from whence, rad. : tan. B G A :: cot. B G A : rad. :: 

 COS. A C 



COS. A G : cot. BAG; therefore, rad. cos. A C = cot. B G A 

 cot. BAG. 



16. The reverse of art. 14 will give rad. cos. B G A = sin. 

 BAG cos. A B. 



17. By comparing the articles 9 and 11, 7 and 13, 10 and 15, 

 8 and 14, 12 and Ifc), and making radius unity, we have the fol- 

 lovvino; equations : 



.^ /sin. BCAsin.AG \ 



^^ = \cot.BAG.tan. BG / 



. ^r^ r sin. BAG sin. AG \ 



sm. BC = ^^^^ BGAtan. AB / 



.^ fcos. AB COS. BG \ 



COS. AC = -^ ^^^ ^^c cot. BGAJ 



T,.^ (sin. BGA COS. BG \ 

 COS. BAG = 1^^^^ ^^ ^^^^ ^^ I 



T,^A rsin. BAG COS. AB \ 



COS. BGA = |cot.AGtan.BG / 



sm. 



