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Problem III. 
tfhe iropic found , by Dr. Halley s method without 
any confederation of the parabola. 
The obfervations are fuppofed to give the pro- 
portions between the differences of the fines 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 ; therefore, the fame oblervations 
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 T and and its center F ; let 
B, C, D be three places of the Sun ; B G, Cl, 
D H the fines of thofe places refpedtively. 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 BD to OD 
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 CD to D O, in the tri- 
angle C D O, is given ; and confequently, the angle 
COD will be given : which angle is the diflance 
of the tropic from the middle point of the ecliptic 
between B and D : for, F P R being perpendicular 
to OC, and FQJ5 perpendicular to DB, the angle 
Q^F P is equal to QO P, the points O, P, Q, F, 
being in a circle. 
* Vide Philofophical Tranfaclions, N° 215. 
T HE 
