254 LUNAR' CALCULATION^. 



angle : the square of radhis : : the rectangle of the sines 

 of the differences between thie half-sura of the three sides, 

 and .each of those lirst-mentioned sides : the square of 

 the sine of half the contained angle; and : : the differ- 

 ence between the versed sines, (or the sum or difference 

 of the co-si)ies.) of the base and of the difference of the 

 sides : the versed sine, (or sum or difference of radius 

 and co-sine,) of the same angle. 



RULE I*. 



Arstrule. j. If the apparent distance be greater than 90°, take 



the sum, otherwise take the difference,, of its natural 

 co-sine and the natural co-sine of the difference of th« 

 apparent altitudes, and call it A. 



2. Add together the arithmetical complements of the 

 logarithmic co-sines of the observed altitudes, the logarith- 

 mic co-sines of the true altitudes, and the logarithm of A; 

 reject 20 from the index, and find the correspondent na- 

 tural number, which call B. 



3. The difference between this number and the natural 

 co-sine of the difference of the true altitudes is the natur^,! 

 co-sine of the observed distance, which v?ill be greater op-. 

 less than 90<» accordingly as B js greater or less than the 

 last-n?entiQned co-sine, 



RULE II. 



Second rule, 1, Add together the two apparent co-altitudies and the 



use of ^ at r 1 ^PP^^*^"*^ distance, and take the difference between their 

 niimberg. half-snm and each of the apparent co-altitudes sepa- 



rately. 



2. Add together the arithmetical complements of the 



4 logarithmic co-sines of the two apparent altitudes, the 



logarithmic sines of the two before-mentioned differences, 



and the logarithmic co-sines of the true altitudes, and 



halve the sum, 



* Since writing the above I find that there is a method perfectly 

 analogous to this in Mr. Thos. Keith's Trigonometry, published in 

 , and differing only in hjs using secants in one part of the 

 process instead of co-sines. As I think it, however, a very useful 

 mode of reduction, I have not struck it out, but shall content my- 

 self with thus resigning to that gentleman the merit of the first 

 publicatipn of it, Q^ 



3. Subtract 



