[ 445 ] 



Problem III. 



tfhe tropic found, by Dr. Halleys method *, without 

 any confideration of the parabola. 



The obfervations are fuppofed to give the pro- 

 portions between the differences of the lines of three 

 declinations of the Sun near the tropic ; but the fine 

 of the Sun's place is in a given proportion to the fine 

 of the declination j therefore, the fame obfervations 

 give equally the proportion between the differences 

 of the fines of the Sun's place, in each obfervation. 



Now (Fig. 9.), let ACE be the ecliptic, AE its 

 diameter between ~f and £}, and its center F ; let 

 B, C, D be three places of the Sun j B G, CI, 

 D H the fines of thofe places refpectively. Draw 

 C K, B L parallel to A E, which may meet HD, 

 in N and M. Then, by the obfervations, the ratio 

 of DM to DN is given. Therefore, if BD be 

 drawn to meet KL in O, the ratio of B D to O D 

 is given ; and the ratio of B D to D C is alfo given, 

 they being the chords of the given angles B F D, 

 CFD: hence the ratio of C D to D O, in the tri- 

 angle C D O, is given 5 and confequently, the angle 

 COD will be given : which angle is the difl ance 

 of the tropic from the middle point of the ecliptic 

 between B and D : for, F P R being perpendicular 

 to OC, and FQ^S perpendicular to DB, the angle 

 QFP is equal to QOP, the points O, P, Q, F ? 

 being in a circle. 



* Vide Pfailofophical Tranfaclions, N° 215. 



The 



