[439 ] 



Bu*, when D is the point of longeft afcenfion, the 

 ratio of BE to DF is the greateft'that can be j there- 

 fore, then the ratio of the re&angle under the tangent 

 of B D H and the fine of B D C to the given rect- 

 angle under the cotangent and fine of the given angle 

 ABD mufr. be the greateft that can be ; and confe- 

 quently, the rectangle under the tangent of B D H, 

 and the fine of BDC ? mull be the greateft that 

 can be. 



In the triangle BDA, the line of- BDH is to the 

 fine of H DA, as the coiine of ABD to the cofine of 

 BAD. Now, in thepreceding lemma, let the angle 

 BAG of the triangle AGB be equal to the fpherica! 

 angle BDC : then will the fum of. the angles ABG 5 

 AGB be equal to the fpherical angle BDA. And, 

 if AG in the triangle AGB, be to A B as the cofine 

 of the fpherical angle DBA to the cofine of DAB, 

 that is, as the fine of BDH to the line of HDA, 

 the angle ABG, in the triangle, will be equal to the 

 fpherical angle BDH; and the angle AGB, in the 

 triangle, equal to the fpherical angle HDA. There- 

 fore, by the fir ft corollary of the lemma, that the 

 rectangle under the tangent of the fpherical angle 

 BDH and the fine of BDC be the greateft that 

 can be, the cofine of BDC muft be equal to the 

 tangent of half the complement of the angle, whofe 

 cofine is to the radius, as AG to AB, in the triangle 3 

 or as the cofine of the fpherical angle ABD to the 

 coline of the fpherical angle BAD. 



If I K be the fituation of the horizon, when the 

 folftitial point is afcending, in the quadrantal triangle 

 A IK, the cofine of KIC is to the radius as the co- 

 fine of IK A (^ DBA) to the cofine of IAK, There- 

 fore 



