[438 



Problem I. 

 'To fold in the ecliptic the point of longed ajcenjion. 



Analysis. 



Let (Fig. 4\ ABC hQ the equator, ADC the 

 ecliptic, BD the fituation of the horizon, when D 

 is the point of longed afcenfion. Let EFG be an- 

 other fituation of the horizon. Then the ratio of 

 the fine of EB to the fine of FD is compounded of 

 the ratio of the fine of BG to the fine of GD, and 

 of the ratio of the fine of AE to the fine of AF 5 

 but the angles B and E being equal, the arcs EG, 

 G B together make a femicircle ; and, by the ap- 

 proach of EG towards GB, the ultimate magnitude 

 of BG will be a quadrant, and the ultimate ratio of 

 EB to FD will be compounded of the ratio of the 

 radius to the fine of DG (that is, the cofine of BD) 

 and of the ratio of the fine of A B to the fine of AD. 

 Draw the arc DH perpendicular to AB. Then, in the 

 triangle BDII, the radius is to the cofine of BD, as 

 the tangent of the angle BDH tothecotangentof HBD. 

 Alfo, in the triangle BD A, the fine of AB is to the 

 fine of AD as the fine of the angle B D A (or BDC) 

 to the fine of ABDj therefore, the ultimate ratio 

 of BE to DF is compounded of the ratio of the 

 itangent of BDH to the cotangent of ABD, and 

 of the ratio of the fine of BDC to the fine of 

 .ABD; which two ratios compound that of the 

 redangle under the tangent of BDH and the fine of 

 BDC to the rectangle under the cotangent and the 

 fine of the given angle ABD. 



4 But 



