of reducing the Lunar Distances. 181 
ae = sin m, and a = sins; the moon’s altitude being 
=m, and that of the star =s. Join MV, SV, and produce 
them to meet TL in L and R, Then on account of the parallels 
LR:MS:: LV: MV, and PV: LT;: MV: LM;3_hence 
LRxPV:LT x MS;:LV:LM:: PK: MN, and LR: 
LT::MSxKP:PV x MN::sinm cos.d: sin s. 
Censequently with a given distance, R'T varies as sin am. cot 
d— sin s. cosec d. That is, as the cosine of the angle at the 
moon multified by the cosine of her altitude: a well known ex- 
pression for the principal effect of her parallax on the distance, 
supposing the horizontai parallax to be unity. 
This corzection is subtractive when R lies to the right of MT, 
otherwise it is additive. When m= oa, and the distance is ina 
vertical circle, RT becomes a maximum for the subtractive cor- 
rection, and represents the horizontal parallax; also if § =0, and 
the objects are in the same vertical quadrant, RT will show the 
greatest additive correction for parallax that the corresponding 
distance admits of. 
Corresponding to the sign of cos 
d, it is evident that LT will lie above T/ RL 
or below PQ, as also K will be on 
the right or left of P, according as 
the distance is less or greater than 
90°. However, since the paralleis 
of distanee or successive positions of 
LT, become somewhat crowded as 
the distance approaches 120°, it 
might perhaps be better after it ex- 
ceeds 90°, to transpose the altitudes, WS WN 
reckoning the moon’s on PQ, and 
that of the staron MN. In this case the parallel of distance 90° 
would coincide with MN, and the rest be continued upward from 
it till they reached the place of 120°. By this means the scale 
would be considerably enlarged, and the confusion of having PQ 
with its divisions of sines crowded in the midst of the other pa- 
rallels would also be avoided, 
There are different positions in which two lines of sines might 
be permanently placed to give a construction for solving the pro- 
blem. When these lines are parallel but lie in contrary direc- 
tions, the parallels of distances less than 90° fall between them, 
but are excesssively crowded and contracted as the distance ap- 
proaches 20°,—the least in common use. For distances greater 
than 90°, the parallels lie without these reversed lines of altitudes. 
Thus if M N were produced beyond M, and another equal line of 
sines 
make 
