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But, when D is the point of longeft afcenfion, the 
ratio of BE to DF is the greateft that can be; there- 
fore, then the ratio of the redangle under the tangent 
of BDH and the fine of B D C to the given redr- 
angle under the cotangent and fine of the given angle 
ABD muft be the greateft that can be ; and confe- 
quently, the redangle under the tangent of B D IB 
and the fine of B D C, muft be the greateft that 
can be. 
In the triangle BDA, the fine of BDH is to the 
line of H DA, as the cofme of ABD to the cofine of 
B AD. Now, in the preceding lemma, let the angle 
BAG of the triangle AGB be equal to the fpherical 
angle BDC : then will the fum of the angles ABG, 
AGB be equal to the fpherical angle BDA. And, 
if AG in the triangle AGB, be to AB as the cofine 
of the fpherical angle DBA to the cofine of DAB, 
that is, as the fine of BDH to the fine 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 fkft corollary of the lemma, that the 
redangle 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 
eofine is to the radius, as AG to AB, in the triangle, 
or as the cofine of the fpherical angle ABD to the 
cofine of the fpherical angle BAD. 
If IK be the fituation of the horizon, when the 
folftitial point is afcending, in the quadrantal triangle 
AIK, the cofine of KIC is to the radius as the co- 
fine of IK A (==3 DBA) to the cofine of JAR. There- 
fore, 
