272 On reducing the Lunar Distance. 



that time I have made some comparison among tliese different 

 forms, with the view of ascertaining which of them is on the 

 whole Ukely to be the most convenient in practice. This has 

 at length inclined me rather to prefer the one in which the 

 line for correction has only one position, whilst the lines of al- 

 titude chance their places for each degree of distance. In it 

 the parts ot the various Imes are better proportioned to suit 

 the several distances, it is more applicable to the solution of 

 other problems, and it also admits of a convenient mode of 

 reading off the correction in terms of any horizontal parallax. 

 I shall therefore describe more particularly that construction 

 in which the same line of correction answers alike for every 

 distance, and in which there is only one line for the moon's 

 altitude, whilst the star's altitude takes a different line for each 

 degree of distance. 



Let MN as formerly be a line of 

 sines whose zero is at M, and parallel 

 to it draw I T equal the horizontal pa- 

 rallax ; divide I T into a Ime of sines 

 beginning from T, and produce it to 

 contain another equal line of sines 

 T I', also beginning at T, but in the 

 opposite direction. 



Now to find the position of the line 

 of star's altitude for a given distance: 

 Lay a ruler from the sine of the distance 

 taken on M N to its cosine reckoned on 



T r ; then through the point P where the ruler cuts M T 

 draw P Q parallel to M N, and it is the position required. 



Next to find the divisions on PQ for the sine of each de- 

 gree of the star's altitude : Imagine the corresponding distance 

 to be in a veitical circle, and lay a ruler from the sine of the 

 moon's altitude on M N to the cosine of her altitude on I' T 

 if the effect of parallax is additive, or if the distance exceed 

 90°, but on I T when that effect is subtractive and the distance 

 also less than 90° : the ruler will then cut P Q in the point that 

 is to be marked witn the intended degree of the star's altitude. 



This latter precept obviously comprehends the preceding 

 one, and the investigation formerly given is applicable to both ; 

 for by this construction, when N T is joined, it cuts any line 



of star's altitude P Q in a point K, such that r^— = cos d. So 



that, retaining the former notation, if S V M V be drawn and 

 produced to meet 1 1' in R and L, we have just the same state 

 of things as before. In this arrangement, the lines of star's 

 altitude for distances greater than 90' evidently lie to the left 



of 



