VOL. LXII.] PHILOSOPHICAL TRANSACTIONS. -^ 353 



than the complement of half bcd, another situation of the horizon, as klm, 

 may be found, toward the other equinoctial point a, wherein the arc of the 

 ecliptic CK will be less than the arc of the equator, and their difference be greater 

 than in any other situation. But if the angle cBAbe not greater than the com- 

 plement of half BCD, the arc of the ecliptic, between c and the horizon, will 

 never be less than the arc of the equator, between the same point c and the 

 horizon. In the two situations of the horizon, the angles chb and kma are equal. 



Schol. 1 . To find the point in the ecliptic, where the arc of the ecliptic most 

 exceeds the right ascension, is a known problem: that point is, where the 

 cosine of the declination is a mean proportional between the radius and the 

 cosine of the greatest declination. 



In the preceding figure, supposing the angle cbd to be right, then, because 

 when CD most exceeds cb, the cosine of bd is to the radius as the sine of cb to 

 the sine of cd, and, in the triangle cbd, the sine of cb is to the sine of cd as 

 the sine of the angle cdb to the radius, also the sine of cdb is to the radius as 

 the cosine of bcd to the cosine of bd; therefore the cosine of bd is to the radius 

 as the cosine of the angle bcd to the cosine of the same bd, and the cosine of bd 

 is a mean proportional between the radius and the cosine of bcd. 



Schol. 2. In any given declination of the sun, to find when the azimuth 

 most exceeds the angle which measures the time from noon, is a problem 

 analogous to the preceding. 



Prob. 3. — The Tropic found, by Dr. Halley^s method,* without any considera- 

 tion of the parabola. 



The observations are supposed to give the proportions between the differences 

 of the sines of 3 declinations of the sun near the tropic; but the sine of the 

 sun's place is in a given proportion to the sine of the declination ; therefore the 

 same observations give equally the proportion between the differences of the sines 

 of the sun's place, in each observation. 



Now, (fig. 12), let ACE be the ecliptic, ae its diameter between 7" and ^iz, and 

 its centre p; let b, c, d be 3 places of the sun; bg, ci, dh the sines of those 

 places respectively. Draw ck, bl parallel to ae, which may meet hd in n and m. 

 Then, by the observations, the ratio of dm to dn is given. Therefore, if bd be 

 drawn to meet kl in o, the ratio of bd to od is given ; and the ratio of bd to dc 

 is also given, these being the chords of the given angles bfd, cfd: hence the 

 ratio of cd to do, in the triangle cdo, is given; and consequently the angle cod 

 will be given: which angle is the distance of the tropic from the middle point of 

 the ecliptic between b and d : for, ppr being perpendicular to oc, and fqs 

 perpendicular to db, the angle qfp is equal to qop, the points o, p, a, f, being 

 in a circle. 



« Vide Phil. Trans. N" 215. 



VOL. XIII. Z z 



