144 Ivory on a fiew Method of deducing 
hypothenuse, and [i — i°) the angle opposite to tt; it is plain 
that we shall have 
sin. 7 t = sin. (z — /°) sin. h° 
sin. n = sin. i sin. [e° — n). 
Therefore, by substitution, the last equation will become 
O sin. 7T ^ T30 
^ ^ sin. n 2 * Rosj 
This equation is equivalent to the second of the equations 
(lo): and we infer from it that r° will be less, or greater, than 
R°, according as is positive or negative ; that is, accord- 
ing as the arcs tt and n are on the same side, or on different 
sides, of the great circle to which they are both perpendicular. 
Hence if we mark, on the surface of a celestial globe, three 
geocentric places of a comet, and likewise the places in the 
ecliptic occupied by the earth at the middle observation ; then 
the comet's distance from the sun will be greater than the 
earth's distance, W'hen the great circle drawn through the two 
extreme places of the comet passes between the earth's place, 
and the remaining place of the comet ; but when these two 
places are both on the same side of the great circle, the comet's 
distance from the sun will be less than the earth's distance. 
When all the three places of the comet are in one great circle 
of the heavens, then tt = o, and = o : and in this case, 
the comet and the earth will be equally distant from the sun. 
From these rules we must how'ever except the cases in which 
sin. n = 0 : for, since cannot become infinite, it is neces- 
sary that the numerator vanish together with the denominator, 
so that ^ 4 ^ will become indeterminate. This will happen, as 
