182 PHILOSOPHICAL TRANSACTIONS. [aNN0 1771. 



measure of the angle dih, or of its complement to 4 right angles : and the 

 square of the radius is to the rectangle under the sines di, ih, as the square of 

 the sine of half the angle dih, or of half its complement to 4 right angles, to 

 the rectangle under the radius, and half the excess of the cosine of the difterence 

 between di and ih, above the cosine of dh, or the sine of df. 



In the next place, the arch ad being drawn, in the rectangular triangle aed, 

 the radius is to the cosine of de, as the cosine of ae to the cosine of ad ; and 

 in the rectangular triangle afd, the cosine of df is to the radius, as the cosine 

 of AD to the cosine of af ; therefore, by equality, the cosine of dp is to the 

 cosine of de as the cosine of ae to the cosine of af ;* the arch af, counted ac- 

 cording to the order of the signs, being to be taken similar in species to ae : for 

 when AE is less than a quadrant, as in fig. 1 , af will be less than a quadrant ; 

 and when ae shall be greater than 1, 2, or 3 quadrants, af, counted according 

 to the order of the signs, shall exceed the same number of quadrants. For, 

 since de and df are each less than quadrants, when ae in the triangle dea is 

 also less than a quadrant, the hypothenuse ad is less than a quadrant, when in 

 the triangle dfa the legs df, fa are similar, that is, fa will be less than a qua- 

 drant ; as in fig. ] : but if ae be greater than a quadrant, as in fig. 2, that is, 

 dissimilar to de, the hypothenuse da will be greater than a quadrant, and the 

 arches df, fa likewise dissimilar, and af greater than a quadrant ; also in fig. 3 

 and 4, the arches ae, af counted from a, in consequence, will be the comple- 

 ments to a circle of the arches ae, af in the triangles ade, adf. 



For an example, let the case be taken in Dr. Halley's astronomical tables, 

 where an occultalion of the moon with a fixed star is prpposed to be computed, 

 the latitude of the place being 65° 50' 50'', and the point of the equinoctial cul- 

 minating 25° 36' 24", from the first point of Aries. This case relates to fig. 1, 

 and the computation will stand thus : 



For the distance of the nonagesime degree from the zenith, 



Distance of e in consequence from a, the equinoctial point 25° 36' 24'' 



Add 90 



Gives the angle hid . . : 115 36 24 L. Sines 



9.92749 



Half HID 57 48 12 o.gZj^g 



HI, the obliquity of the ecliptic used by Dr. Halley 23 29 



ID, the complement of the latitude 24 9 10 



Natural number corresponding 0.1 1676 



Its double, to be deducted from the nat. cosine of id a> in (0" 40' 10") 0.99.993 

 leaves the nat. cosine of h d (39 58 0) 0.7664 1 



Therefore df is 50 2 



For the arch at. 



Cosine of df, or sine of hd (co. arith 



Cosine of the latitude, or sine of id 



Cosine of a e 



Cosine of the long, of the 90th deg. (54° 56' 24") 9-75924 



9.60041 

 9.61190 



9.06729 

 Sum thrice 

 d. deduct 



0.1 92-^3 

 .9.61191 

 9-95510 



