LtrjfAR CALCULATIONS. 255 



S. Subtract from this half-sum the logarithmic sine of 

 half the difference of the true altitudes, and the remainder 

 will be a logarithmic tangent. 



4. Find the correspondent logarithmic sine, subtract 

 it from the before-mentioned half-sum, and the remainder 

 will be the logarithmic sine of half the true distance. 



RULE III. 



1. Add together the arithmetical complemetit of the Third rule, 

 logarithmic sine of half the apparent distance and the ^j^^ without 

 logarithmic sine of half the difference of the apparent ^^^al numbers. 

 altitudes, and their sum will be the logarithmic co-sine 



of an arc, which call A. 



2. Add together half the sum of the logarithmic co- 

 sines of the true altitudes, the logarithmic sine of half the 

 difference of the apparent altitudes, and the logarithmic 

 tangent of A. 



3. Add together also half the sum of the logarithmic 

 co-sines of the apparent altitudes and the logarithmic sine 

 of half the difference of the true altitudes* 



4. The difference between these sums is a logarithmic 

 tangent of an arc, which call B. 



5. To the logarithmic sine of half the difference of the 

 true altitudes add the arithmetical complement of the 

 logarithmic co-sine of B, and their sum will be the loga- 

 rithmic sine of half the true distance. 



RULE IV. 



1. Add together the arithmetical complement of the Fourth rule, 

 logarithmic sine of half the apparent distance and the analogous to 

 logarithmic sine of half the difference of the apparent ^ ^ 

 altitudes, and their sum will be the logarithmic co-sine of 



an arc, which call A. 



2. Find the logarithmic sine of A ; subtract from it 

 the before-mentioned arithmetical complement, and dou- 

 ble the remainder. 



3. Add to this doubled remainder the arithmetical 

 complements of the logarithmic co-sines of the apparent 

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

 tudes, and halve the sum. 



4. From this half-sum smbtract the logarithmic sine of 



LI 2 half 



