I 440 ] 



-fore, the cofine 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 folftitial point is afcend- 

 -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 B D C, 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 

 here, 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 A B N to the fine of A N B, 

 that is, as the cofine of BAP to the cofine of NAP; 

 confequently, fince the angle B A N 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 fee greater than one 

 third of the radius, the point A can never be the 

 point of longeft afcenfion ; but when the fine of this 

 angle is lefs, the angle compounded of B A D and 

 twice A B Dj may be greater or lefs than a quadrant ; 



and 



