Goniomelric Problems. 131 



= /, ASB (/, CSD + /, ESD) ; but by Cagnoli 89, /, CSD x 



*,ASFc,CSDc, ESD = /, ASB ^,CSE r, ASBc, FSB; but 

 i, X c, = s; therefore s, CSD s, ASF c, ESD = 5, ASB 

 .9, CSE c, FSB ; whence 



.,ASF =.,CSL (.-^^yir.lLSD)' 

 which gives the relation between the six angles ASF, CSE, 

 ASB, FSB, CSD, and ESD. 



Cor. ist. Since /, BSD : /, ASC : : c, ASB : rad. and 

 /,FSE : /,BSD :: rad. : r, BSFj 

 therefore t, FSE : t, ASC : : c, ASB : c, BSF. 

 Cor. 2d, /, ASB : /, CSD : : rad. : c, BSD and 

 /, FSB : f, ESD : : rad. : c, BSD, also 

 /, BSD : /, ASC : : rad. : c, ASB and 

 t, BSD : t, FSE : : rad. : r, FSB, whence 

 i, ASC : c, ASB : : /, FSE : c, FSB and 

 t, CSD : /, ASB : : /,ESD : /, FSB ; therefore 

 bv mukiplving the two last proportions together, we have 

 /,ASC/, CSdI ^ ASB c,ASB : : ^FSE^P:SD : /, FSB c, FSB 

 or, /, ASC /, CSD : s., ASB : : (, FSE t, ESD : 5, FSB. 

 Cor, 3d, From preceding cor. 



t, FSB t, CSD = /, ESD i, ASB ; therefore 

 /, FSB : /, ASB : : /, ESD : t, CSD. 

 But when the plane SBD, that is perpendicular to AE 

 passing through S, falls without the figure, we have (fig. 14.) 

 /, CSD - t, ESD : /, ASB - t, FSB : : c, BSD : radius ; 

 but by Cagnoli, art, 91, • 



/, CSD - /, ESD = -'t7~^^^ ; therefore 



;, (CSD - FRD) s,(ASV, - FSB) 

 £, CbD c, ESD c, ASB c, l SB 



AQF - -^ CSE ^, ASB r, FSB 



.9, ASi- _ -^cSDf,KSDf,BSlV 



Svippo?c now the line EF to move up to BD and to 



coincide with it, then the equation becomes ,v, ASB = 



i.CSDr, ASB .r,ASB „^,^ x,CSD , , .sine 



f, CSDo BbD' f, ASB f, CSD' tus. 



I 2 = tau>r. ; 



