136 Astronomical and Nautical Collection, 



Example. 



(From page 1 12, Appendix to Requisite Tables.) 



Reserved log. from Tables (Req.) 9th and 11th . . 9.9938860 

 Jjog. sin. 43° 23' 5" = i sum of app. dist. and diff. 



app. altitudes . 9.8368895 



Lo^. sin. 6° 45' 36" = I diff. ditto ditto 9.0708157 



Log. of 2 0.3010300 



Nat. num. to sum of 4 logarithms . . .1594488 9.2026212 

 Nat.vers.37° 13' 12"=diff. true altitudes .2036812 



Nat. vers. 50° 26' 28" = true distance .3631300 



. Or, Nat. cos. 37° 13' 12" =r diff. true altitudes .7963188 

 Nat. number found above 1594488 



Nat. cosin. 50° 26' 28" = true distance . .6368700 



Demonstration of the Rule. 



Let M\ S\ D', d' and ikf, Sy D, c?, respectively denote the true 

 and apparent altitudes, distances, and differences of true and appa- 

 rent altitudes of the moon and sun (or a star) ; then will the theorem 

 answering to the above rule be expressed by 



vers. D' = ^ ^Q^^^^Q^ -^f' sin J- (D+£^)sinl(D-cZ)+Yers.cZ', 

 cosMcos -S 2 2^- ^ 



By Bonnycastle's Trig. p. 175, the cosine of the angle contained 



, ,, ,^., J . cos D—sin ikf sin 5> cos D' — sin M' sin S' 



bytheco-altitudesis. = ; 



cos M cos ^ cos M' cos S' 



consequently the verse sine of the same angle 



_ cos D — sin ilf sin *S , cos D'— sin JW'sin -S ,, . . 

 Et: 1— =1-- ; that is, 



cos M cos -S cos ikf' cos -S' 



cosMcosS+sinMsin-S— cosD__cos7kf'cos8'+sinM'sin-S'— cosD' 

 cos M cos S cos M' cos S' 



Substituting cos d and cos d' for cos M cos S + sin M sin S and 

 cos M' cos S' + sin M' sin S'. (Bon. Trig. p. 282), we have 

 cos J -cos J) ^ 2^:Z^2lE; whence 

 cos M cos S cosM' cos. S' 



T\i J' cos M' cos 8' X , rk\ u* I,- ii 



cos U = cosa (cos a — cosD); or, which is the same, 



cos M cos -S' 



cos D'= cos d— (versD— verse?); or,(Bon.Trig.p.286.) 



cos M cos <S 



