To the logarithmic difference, Table XXIV., add the logarithmic co-sines of the above half sum and remainder: the sum of these three logarithms (rejecting 20 in the index,) will be the logarithm of a natural number. Now, twice this natural number being subtracted from the natural versed sine supplement of the sum of the true altitudes, will leave the natural versed sine of the true central distance. Remarks.—If the remaining index of the three logarithms (after 20 is rejected) be 9, the natural number is to be taken out to six places of figures; if 8, to five places of figures; if 7, to four places of figures; if 6, to three places of figures,-and so on. The logarithmic difference is to be corrected for the sun's, star's, or planet's apparent altitude, as directed in pages 49, 51, and 52;—this, it is presumed, need not be again repeated. Example 1. Let the apparent distance between the moon and a fixed star be 48:20:21", the star's apparent altitude 11:33:29", the moon's apparent altitude 11:10:35", and her horizontal parallax 55'32"; required the true central distance? Star's apparent alt. = 11:33:29"-Correc. 4:33"=true alt.=11:28:56′′ Moon's appar. alt. = 11.10.35 +Correc. 49. 46 =true alt.=12. 0.21 Appar. central dist.= 48. 20. 21 Sum of the true altitudes-23:29:17" Sum of true alts.=23:29:17" Nat.V. S. sup.-1.917143 True cent. dist. 48:16:17" Nat. vers. S. = .334399 Example 2. Let the apparent distance between the moon and sun bé 108:42′ 3′′, the sun's apparent altitude 6:28, the moon's apparent altitude 54:12, and her horizontal parallax 55.19"; required the true central distance? Twice the natural number = Sum of true alts. 619 3:55% Nat.V. S. sup.-1.483813 = True cent. dist.=108:27:43% Nat. vers. S. 1.316677 83568 Log. 8.922038 . 167136 METHOD VIII. Of reducing the apparent to the true central Distance. RULE. To the logarithmic sines of the sum and the difference of half the apparent distance and half the difference of the apparent altitudes, add the logarithmic difference: half the sum of these three logarithms (10 being previously rejected from the index,) will be the logarithmic sine of an arch. Now, half the sum of the logarithmic co-sines of the sum and the difference of this arch and half the difference of the true altitudes, will be the logarithmic co-sine of half the true central distance. Example 1. Let the apparent central distance between the moon and a fixed star be 41:24:22, the star's apparent altitude 12: 4'27", the moon's apparent altitude 7:47′47′′, and her horizontal parallax 57:24" ; required the true central distance? 12: 4.27%-Correc. 4:22" true alt. 12: 0:5% 7.47.47 +Correc. 50. 13 true alt. 8.38.0 Diff. of the app. alts. 4:16:40 Diff. of the true altitudes 3:22′5′′ Half diff. of app. alts. 2: 8:20 Half diff. of the true alts. 1:41:2" Half the true dist. 20:38:15" Log. co-sine = True central dist. 41:16:30" Example 2. Let the apparent central distance between the moon and Saturn be 110:14:34", Saturn's apparent altitude 9:40:48", and his horizontal parallax 1, the moon's apparent altitude 15°:40:6%, and her horizontal parallax 58.43; required the true central distance? = Saturn's apparent alt. 9:40:48"-Correc. 5'24"=true alt.=9:35:24" Moon's apparent alt.=15.40. 6+ Correc. 53. 11=true alt. 16, 33. 17 = Sum Half the true dist. = 55: 0:30 Log, co-sine = 9.758499 True central dist. 110: 1:12 METHOD IX. Of reducing the apparent to the true central Distance. RULE. To the logarithmic co-sines of the sum and the difference of half the apparent distance and half the sum of the apparent altitudes, add the logarithmic difference: half the sum of these three logarithms (10 being previously rejected from the index,) will be the logarithmic co-sine of an arch. Now, half the sum of the logarithmic sines of the sum and difference of this arch and half the sum of the true altitudes, will be the logarithmic sine of half the true central distance. Example 1. Let the apparent central distance between the moon and a fixed star be 41:29:58, the star's apparent altitude 11:31:2", the moon's apparent altitude 8:44:35", and her horizontal parallax 57'24"; required the true central distance? = Star's apparent alt. 11:31 2"-Correc. 4:34"-true alt. 11:26:28: Moon's appar. alt. 8. 44. 35+ Correc. 50. 46 =true alt. Sum of the app.alts.=20:15:37 Half sum of ap.alts. 10: 7:48" 9.35.21 Let the apparent central distance between the moon and sun be 101:54:51", the sun's apparent altitude 39:34:35", the moon's apparent altitude 29:23:2", and her horizontal parallax 58:53"; required the true central distance? Sum of the app. alts.=68:57:37" Sum of the true altitudes 69:46′12′′ Of reducing the apparent to the true central Distance. RULE. To the logarithmic sines of the sum and the difference of half the apparent distance, and half the difference of the apparent altitudes, add the logarithmic difference, its index being increased by 10: from half the sum of these three logarithms subtract the logarithmic sine of half the difference of the true altitudes, and the remainder will be the logarithmic tangent of an arch; the logarithmic sine of which, being subtracted from the half sum of the three logarithms, will leave the logarithmic sine of half the true central distance. Example 1. Let the apparent central distance between the moon and a fixed star be 55:4:53, the star's apparent altitude 10:8:6", the moon's apparent altitude 8:1:25", and her horizontal parallax 58:17; required the true central distance? |