Star's apparent alt. 10: 8' 6"-Correc. 5:11-true alt. Diff. of the app. alts. 2: 6'41" 10: 2.55% 8. 52. 23 Diff. of the true altitudes = 1:10:32" Half diff. of true altitudes = 0:35.16% Log.diff. 19.999162 = 28:35:47"Log. sine= 9.680006 Sum = Half sum = Half diff, of true alts. 0:35:16"Log.sine 8.011083 Example 2. Let the apparent central distance between the moon and sun be 91:26:8", the sun's apparent altitude 14:45:41", the moon's apparent altitude 53:41:1", and her horizontal parallax 58′29′′; required the true central distance? Sun's apparent alt. 14:45'41"-Correc. 3:26%=true alt.=14:42:15% 39:32:42" 19:27:40" Half diff. of the true alts. 19:46:21" 45.43. 4. --Log.diff.-19.994220 65:10:44"Log.sine = 9.957905 = 26. 15.24 Log.sine 9. 645809 39.597934 Half sum = Half diff. of true álts. 19:46:21" Log.sine 61:44:43"Log.tan. 10. 269682 Log. sine 9. 944902 45:36:38 Log, sine 9. 854065 91:13:17" METHOD XI. Of reducing the apparent to the true central Distance. RULE. To the logarithmic difference (its index being increased by 10,) add the logarithmic co-sines of the sum and the difference of half the apparent distance and half the sum of the apparent altitudes; from half the sum of these three logarithms subtract the logarithmic co-sine of half the sum of the true altitudes, and the remainder will be the logarithmic sine of an arch; the logarithmic tangent 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 68:52:40%, the star's apparent altitude 10:52:17", the moon's apparent altitude 6:39:28", and her horizontal parallax 58:31"; required the true central distance? Star's apparent alt. 10:52:17%-Correc. 4'50= true alt.=10:47.27" Moon's appar. alt. 6.39. 28 +Correc. 50. 26 true alt. 7, 29.54 = 17:31:45 Sum of the true altitudes = 18:17:21" 8:45'524" Half sum of the true alts. 9:8:40" Half sum of true alts. 9: 8:40 Log.co-si. 9. 994445 55 7 4 Log.sine=9.913988Log.T.=10. 156675 Arch=. Example 2. Let the apparent central distance between the moon and sun be 120:10:40%, the sun's apparent altitude 13:30:0", the moon's apparent altitude 6:10:0", and her horizontal parallax 61:12"; required the true central distance ? 1 Sun's apparent alt, 13:30, 0%-Correc. 3:45" true alt. 13:26:15% Sum of the true altitudes = 20:28:51" Half sum of the true alts. 10:14.25" = Log, diff. 19.999345 69:55:20% Log. co-sine 9.535668 50, 15.20 Log. co-sine 9. 805749 19.670381 Half sum of true alts. 10:14:25" Log, co-sine 9. 993026 Of reducing the apparent to the true central Distance. RULE. From the natural versed sine supplement of the sum of the apparent altitudes, subtract the natural versed sine of their difference, and call the remainder arch first. Proceed in a similar manner with the true altitudes, and call the remainder arch second; and from the natural versed sine supplement of the sum of the apparent altitudes, subtract the natural versed sine of the apparent distance, and call the remainder arch third. Now, to the arithmetical complement of the logarithm of arch first add the logarithms of arches second and third, and the sum (rejecting 10 from the index,) will be the logarithm of a natural number; which, 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. Example 1. Let the apparent central distance between the moon and a fixed star be 83:15:19, the star's apparent altitude 7:39:4", the moon's apparent altitude 10:57:36", and her horizontal parallax 58:55"; required the true central distance? 'sap.alt. 7:39 4-Cor. 6:45"-True alt. 7:32:19% D's ap.alt. 10.57.36 +Cor.53. 3 True alt.11.50.39 Sum ofap.alts. Arch third = 18:36:4031.947707 Sum = 19:22:58"N.V.S1.943322 3. 18. 32N.V.S.. 001668 Diff. = 4. 18. 20N.V.S. .002822 Arch second = Arch second = 1.940500 18:36:40" N.V.S. sup. = 1.947707 Arch first= Natural number = .882554 Sum of true alts. 19:22′58′′ N.V.S. sup. = 1.943322 True cent. dist. 83:10:28 Nat. vers. sine = 881153 Example 2. Let the apparent distance between the moon and sun be 111:27:17, the sun's apparent altitude 24:40:16%, the moon's apparent altitude 16:52.31%, and her horizontal parallax 54'56"; required the true central distance? = = O's ap.alt. 24:40:16-Cor. 1:56" True alt.24:38:20? 41:32:471. 748419 Sum = 42:20:19"}1.739177 7.47.45N.V.S.. 009242 Diff. = 6.56. 21N.V.S..007325 Arch first 1.739177 = Arch second = 1.731852 Sum of ap.alts. 41:32:47"N.V.S.sup.-1.748419 Sum of true alts.42:20:19% N.V.S.sup.-1.739177 True central dist. 110:58:56 N.V.S.=1.358080 METHOD XIII. To the apparent distance add the apparent altitudes of the objects; take half the sum, and call the difference between it and the apparent distance. the remainder. Then, To the logarithmic difference (its index being augmented by 10,) add the logarithmic co-sines of the half sum and the remainder; from half the sum of these three logarithms subtract the logarithmic co-sine of half the sum of the true altitudes, and the remainder will be the logarithmic sine of an arch. Now, the logarithmic co-sine of this arch, being added to the logarithmic co-sine of half the sum of the true altitudes (rejecting 10 from the index), will give the logarithmic sine of half the true central distance. Example 1. Let the apparent central distance between the moon and Spica Virginis be 37:12:40, the star's apparent altitude 11:27:50%, the moon's apparent altitude 40:55.15%, and her horizontal parallax 54'10"; required the true central distance? Star's apparent alt. -11°27'50%-Correc. 4:35"-true alt. 11:23:15 Moon's appar. alt.=40.55.15 +Correc. 39.51 =true alt. 41.35. 6 Appar. cent. dist. = 37. 12. 40 Half sum of true alts. 26:29:10"Log.co-sin. 9. 951844 68:48:45 Log. sine=9. 969604Log.co-si.9. 558014 Arch= Half the true distance = True central distance = 18:52 261 Log. sine=9. 509858 37:44:53" Example 2. Let the apparent central distance between the moon and sun be 117:42:28", the sun's apparent altitude 10:19:19", the moon's apparent altitude 42:551", and her horizontal parallax 60:2"; required the true central distance? |