Or, having obtained the natural number, as directed above, subtract it from the natural cosine of the difference of the true altitudes, and the remainder will be the natural cosine of the true distance. [p136] EXAMPLE. (_From page 112, Appendix to Requisite Tables_.) Reserved log. from Tables (Req.) 9th and 11th 9.9938860 Log. sin. 43° 23′ 5″ = 1/2 sum of app. dist. and diff. app. altitudes 9.8368895 Log. sin. 6° 45′ 36″ = 1/2 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″ = 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 M, S, D, d, respectively denote the true and apparent altitudes, distances, and differences of true and apparent altitudes of the moon and sun (or a star); then will the theorem answering to the above rule be expressed by vers. D′ = ((2 cos M′ cos S′)/(cos M cos S)) sin 1/2(D + d) × sin 1/2(D−d) + vers. d′. By Bonnycastle’s Trig. p. 175, the cosine of the angle contained by the co-altitudes is (cos D − sin M sin S)/(cos M cos S) = (cos D′ − sin M′ sin S′)/(cos M′ cos S′); consequently the verse sine of the same angle = 1−(cos D − sin M sin S)/(cos M cos S) = 1−(cos D′ − sin M′ sin S)/(cos M′ cos S′); that is, (cos M cos S + sin M sin S − cos D)/(cos M cos S) = (cos M′ cos S′ + sin M′ sin S′ − cos D′)/(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 d−cos D)/(cos M cos S) = (cos d′−cos D′)/(cos M′ cos S′); whence cos D′ = cos d′−((cos M′ cos S′)/(cos M cos S)) (cos d−cos D); or, which is the same, cos D′ = cos d′−((cos M′ cos S′)/(cos M cos S)) (vers D−vers d); or, (Bon. Trig. p. 286.) [p137] cos D′ = cos d′−((cos M′ cos S′)/(cos M cos S))· (2 sin^2 ((1/2)D) − 2 sin^2 ((1/2)d)); that is, cos D′ = cos d′−((2 cos M′ cos S′)/(cos M cos S)) sin ((1/2)(D + d))· sin 1/2(D−d); whence also vers D′ = vers d′ + ((2 cos M′ cos S′)/(cos M cos S)) · sin ((1/2)(D + d)) sin ((1/2)(D−d)).