ADD together the reserved logarithm (found as directed, page 111 and 112 of the Appendix to the third edition of the Requisite Tables) the log. sines of half the sum, and half the difference of the apparent distance, and difference of apparent altitudes, and 0.3010300, the log. of 2. Then, to the natural number corresponding to the sum of these four logarithms, add the natural verse sine of the difference of true altitudes, and the sum will be the natural verse sine of the true distance.
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″ =12sum of app. dist. and diff. app. altitudes | 9.8368895 | |
| Log. sin. 6° 45′ 36″ =12diff. 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 × sin12(D + d) × sin12(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 Scos 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 Scos M cos S = 1−cos D′−sin M′ sin Scos M′ cos S′; that is, cos M cos S+sin M sin S−cos Dcos 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 Dcos 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 sin212D − 2 sin212d); that is, cos D′ = cos d′−2 cos M′ cos S′cos M cos S × sin12(D + d) × sin12(D−d); whence also vers D′ = vers d′ + 2 cos M′ cos S′cos M cos S × sin 12(D + d) × sin12(D−d).
It may be observed, that Requisite Tables 9–11, answer logarithmically to cos M′ cos S′cos M cos S; and the verse sines, and the cosines can be very readily taken out of the tables in the Appendix. Also no ambiguity can arise from the application of the rule before given: for all the arcs concerned in the operation will always be (each of them) less than a quadrant, except the resulting true distance, which cannot cause any ambiguity; and the verse sines are given in the Appendix, to 126°.