182 On reducing the Lunar Distances. 
sines laid off from M in the opposite direction, by using this for 
the moon’s altitude, we could read off the effect of parallax for 
a distance of 120° on the same parallel that is used for 60°; 
and in a similar way for those between 90° and 120°. But it is 
obvious that MN thus produced would be inconyeniently long. 
It is true, however, that M N being merely reversed, or rather 
another equal line of sines laid close to it but beginning from N, 
by using this for the moon’s altitude when the distance ex- 
ceeds 90°, we might still have the correction on the parallel of 
60°, &c.; but then neither the divisions nor numbers attached 
to them would suit well. The line LT, we may also observe; 
might have one permanent position, while the lines of altitude 
changed théir places or even magnitudes for each degree of di- 
stance; but this would likewise be attended with several incon- 
veniences. 
Preferring then the former arrangement, and that part of each 
parallel of distance lying to the right of M'T being divided into 
69 equal parts with the same divisions continued as far as neces- 
sary to the left, we shall obtain the effect of parallax in any given 
case, merely by applying a ruler to the two altitudes, and then 
the segment of the parallel of the distance intercepted between 
the ruler and MT will be the required correction in minutes, sup- 
posing a horizontal parallax of 60°. This number again being 
multiplied by the given horizontal parallax, and divided by 60, 
gives a quotient in minutes and a remainder in seconds corre- 
sponding to the given horizontal parallax. 
In all this we have assumed the effect of parallax as strictly 
proportional to the cosine of the angle at the moon: this how- 
ever is not in general quite correct. ‘The error is usually deno- 
minated the final correction, and is contained in the 15th of the 
Requisite Tables. It is nearly proportional-to the cotangent of 
the distance multiplied by the difference of the squares of the 
parallaxes in altitude and distance. 
In the expression, sin m cot d, if m be altered by a quantity 
m proportional to the par allax in altitude, the change in the cor-~ 
rection is as m* cot d, which is proportional to one term of the 
final correction. The other term may be allowed for, by every- 
where shifting the divisions of RT by a small quantity propor- 
tional to its square. If however this method of making out the 
final correction be used, it is evident we cannot transpose the al- 
titudes in the manner already proposed, because their lines of 
sines would not then be quite similarly divided. 
But the final correction may be effected with almost sufficient 
accuracy for a thing of this kind, by merely curving a little the 
parallels of distance and making their divisions somewhat un- 
equal, 
