•40 Mr. Bugge’s Determination of 
of the node from one of the longitudes or places may be 
found. Let DE be the ecliptic, AF the orbit of the planet, 
N the node, DE the difference between the two obferved he- 
liocentric longitudes = a, EF the fouthern latitudes /3, AD 
the northern latitude = b , NE the diftance of the node from the 
heliocentric place at E, and correfponding to the fouthern 
latitude = In the fpherical triangles ADN and FEN, 
im. (a—x) _ G0t> n — , By placing the value of fin. (rf - x) 
tang, b 
in the equation 
tang. £ 
fin. m . cof. x— fin. x . col', a fin. x 
tang, b 
~ tang. (3* 
fin. a . tang. 0 
By refolving 
this equation ^L^=tang.^ = — — 7 , • 
T- col.* o tang, £-j-coi. a . tang, p 
If <?, b, and /3, are very fmall arcs, which commonly is 
the cafe with the planets, then fin. a = a, tang. j3 = j3, tang. 
b—b , and cof. a=\. Hence the fpherical formula will be 
This formula belongs to 
plane geometry, and may befides be thus demonflrated. DN : NE 
= AD : EF. Hence DN + NE : NE=: 
transformed into another x = -^L 
4 + £ 
AD + EF : EF ; and NE 
DEx EF 
If 
ad + eF 
the difference of the longitudes do not 
•exceed one degree, and the latitudes are not greater than ten 
minutes, the fpherical and the redtilineal formula will agree to 
very few feconds. Small faults in the longitude will not very 
much alter the true place of the node ; but very fmall errors 
in the latitude are of great confequence. Let the error in the 
fouthern heliocentric latitude be FG = 
4 - d. The error in the northern latitude 
AH= - d. Hence DH : D«=GE : En , 
and Es=^~^. By fubtratting EN 
