Of the MOON from the SUN. 199 



ther, without any fenfible error j and, by this means, r the calcu- 

 lator may avail himfelf of that convenient and ufeful table of 

 the proportional logarithms, and fave a great deal of trouble in 

 making proportions for feconds. When the arcs themfelves 

 are taken inftead of the tangents of the arcs, it will fhorten 

 the operation, and the refult will be the fame, if half the rec- 

 tangle under the fum and difference of LM and aM is fubfti- 

 tuted for twice the rectangle under half the fum and half the 

 difference of LM and aM ; each of them is equal to half the 

 fquare of La, or twice the fquare of half La, and half the fquare 

 of La, divided by the tangent of half the fum of SL, and Sa is 

 equal to ab. 



WHAT is called the rectangle logarithm is the proportional 

 logarithm of the fquare of La ; its half is the proportional lo- 

 garithm of La, and La, divided by the fine of LS, is equal to 

 the angle LSa, which being fubtracted from ZSM, leaves 

 ZSL, equal to the oppofite angle dSR, fig. 2. ; or, when the an- 

 gle at the Star is obtufe, as in fig. i . the angle pSi, equal to the 

 oppofite angle LSa, being added to the fupplement of the angle 

 at the Star, makes the angle ZSi equal to the oppofite angle 

 LSR or dSR, whence the remaining corrections are found in 

 parts of the arc Ld. 



THE third correction Sc is equal to SR, multiplied by the co- 

 fine of cSR. 



THE fourth correction cd is the difference between Lc and LR, 

 and Re being perpendicular to Ld, makes the triangles ScR and 

 Lc R right-angled at c, fo that cd is found by the fame propor- 

 tion as ab. 



VIII. 



