r 440 ] 
fore, the cafme of BDC, when D is the point of 
longeft afcenfion, is equal to the tangent of half the 
-complement of the angle, which the ecliptic makes 
with the horizon, when the folflitial point is amend- 
ing. 
But, the fine of the angle compofed of DAB, and 
twice ABD, muft be lefs than three times the fine 
-of the angle BAD. In the fpherical triangle ABD, 
the angles BAD, ABD together exceed the ex- 
ternal angle BDC. Therefore, in the third corol- 
lary of the lemma, let the angle BAN be equal to 
the fum of the fpherical angles BAD, ABD : but 
Iiere, AN is to A B as the cofine of the fpherical 
angle ABD to the cofine of BAD; and AN is alfo 
to A B as the fine of ABN to the fine of A N B, 
that is, as the cofine of B A P to the cofine of N A P ; 
confequently, fince the angle BAN is equal to the 
fum of the fpherical angles BAD, ABD, the angle 
NAP is equal to the fpherical angle BAD, and the 
angle BAP equal to the fpherical angle ABD ; but 
the fine of the angle compofed of NAP and twice 
PAB is lefs than three times the fine of NAP; 
therefore, the fine of the angle compofed of the 
fpherical angle BAD and 2 ABD will be lefs than 
three times the fine of the angle BAD; otherwife 
no fuch triangle DBA, as is here required, can take 
place, but the point A will be the point of longeft 
afcenfion. 
If the fine of the angle A be greater than one 
third of the radius, the point A can never be the 
point of longeft afcenfion ; but when the/ine of this 
angle is lefs, the angle compounded of BAD and- 
twice ABD, may be greater or lels than a quadrant ; 
