Example 2. Let the apparent central distance between the moon and sun be 118:56:40", the sun's apparent altitude 16:40:10%, the moon's apparent altitude 9:39:50", and her horizontal parallax 59:19"; required the true central distance? Sun's apparent alt. = 16:40′10′′-Correc. 3: 0=true alt. = = 16:37:10? true alt, 10.32.53 Diff. of the app. alts. 7: 0:20" Diff. of the true alts. = . Apparent distance =118:56:40% N.V.S.=1.483961 = 6: 4:17 Diff. of appar. alts. 7. 0.20 N.V.S.= .007466 Log.diff.=9.998919 Diff. of the true alts.6: 4:17%Nat.V.S.= .005609 True central dist. =118:35 1Nat.V.S.=1.478434 METHOD IV. Of reducing the apparent to the true central Distance. RULE. Take the logarithmic difference from Table XXIV., and let it be corrected for the sun's, star's, or planet's apparent altitude, as directed in pages 49, 51, and 52. Find the sum of the apparent altitudes of the objects, and, also, the sum of their true altitudes; then, From the natural versed sine supplement of the sum of the apparent altitudes, subtract the natural versed sine of the apparent distance; to the logarithm of the remainder let the logarithmic difference be added, and the sum (abating 10 in 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 sun be 110:53:34", the sun's apparent altitude 38:11:59", the moon's apparent altitude 15:51:22", and her horizontal parallax 58:40"; required the true central distance? Sun's apparent alt. Sum of the ap. alts. = 38:11:59"- Correc. 1'15"-true alt.=38:10:54" 15.51.22 + Correc. 53. 7 =true alt.=16. 44. 29 54: 3:21 Sum of the true altitudes = 54:55:23′′ Sum of ap. alts.=54° 3′21′′Nat.V.S.sup.=1.586997 110.53.34 Nat. vers. S. 1.356620 Log.diff.=9.998150 Natural number = ..229398 Log. = 5.360589 Sum of true alts. 54:55:23 "Nat.V.S.sup.-1.574676 True cent. dis.=110:11:56"Nat. vers. S.= 1.345278 Note. See remarks, page 434, relative to the corrections of the apparent altitudes of the objects. Example 2. Let the apparent central distance between the moon and a fixed star be 41:11:7, the star's apparent altitude 43:10:20", the moon's apparent altitude 56:48 16", and her horizontal parallax 59.25"; required the true central distance? = Star's apparent alt. 43:10:20%- Correc. 1 1=true alt. 43: 9:19" Moon's appar. alt. 56. 48. 16+ Correc. 31.56 =true alt.=57.20.12 Sum of the app.alts.=99:58:36 Sum of the true altitudes 100:29:31" Sum of the app.alts.=99:58:36" N.V.S.sup.=826753 = App. central dist. 41. 11. 7 N. vers. S. 247416 Log.diff.=9.993895 Remainder 579337 Log. = 5.762931 571250 Log. = 5.756826 Natural number = sup.=817902 True cent. dist. = 41: 7: 8"Nat. vers. S. 246652 METHOD V. 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, Table XXIV., and the constant logarithm 6.301030: the sum of these four logarithms (rejecting 30 in the index,) will be the logarithm of a natural number; which, being added to the natural versed sine of the difference of the true altitudes, wiH give the natural versed sine of the true central distance. Example 1. Let the apparent distance between the moon and a fixed star be 37:56:43", the star's apparent altitude 19:32, the moon's apparent altitude 56:33, and her horizontal parallax 61:16"; required the true central distance? = Star's apparent alt. 19:32 0"-Correc. 2:40=true alt.=19°29′20′′ Moon's appar. alt. 56.33. 0 +Correc. 33. 9 =true alt.=57. 6. 9 Diff. of the app. alts. 37: 1 0 Diff. of the true altitudes 37:36:49% Diff, of the true alts. 37:36:49 Nat. vers. S.=207855 True central dist. = 38:31 1 Nat. vers. S.=217575 Example 2. Let the apparent central distance between the moon and sun be 106:22:48", the sun's apparent altitude 39°25', the moon's apparent altitude 19:56, and her horizontal parallax 58:0"; required the true central distance ? Sun's apparent alt. 39°25' 0"-Correc. 1 3 true alt. 39:23:57" Moon's apparent alt.=19.56. 0 + Correc. 51.56 =true alt.=20. 47.56 Diff. of the app. alts.=19:29′ 0′′ Diff, of the true altitudes=18:36′ 1′′ Diff. of the true alts.=18:36 1"Nat. vers. S.= 052234 True central dist.=105:41:24"Nat. vers. S.=1. 270430 METHOD VI. 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, Table XXIV., and the constant logarithm 6. 301030: the sum of these four logarithms (rejecting 30 in 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 distance. Example 1. Let the apparent central distance between the moon and a fixed star be 69:21:25", the star's apparent altitude 27:32:37", the moon's apparent altitude 22:28:56%, and her horizontal parallax 56:17"; required the true central distance? Star's apparent alt. 27:32:37"-Correc. 1:49"=true alt.=27:30'48" Moon's appar. alt. 22. 28. 56 +Correc. 49. 43 true alt. 23. 18.39 Sum of the ap. alts.=50: 1:33 Sum of the true altitudes 50:49:27′′ Half sum of ap.alts. 25: 0:46" Sum of true alts. 50:49:27" Nat.V. S. sup.-1.631703 True cent. dist. 69: 3:11"Nat. vers. S. = .642499 Example 2. Let the apparent central distance between the moon and Jupiter be 116:40:28", Jupiter's apparent altitude 10:40:20%, and his horizontal parallax 2", the moon's apparent altitude 15:10:30%, and her horizontal parallax 59′13′′; required the true central distance? = = Jupiter's appar. alt. 10:40:20"-Correc. 4.54" true alt. 10:35:26% Moon's appar. alt. 15. 10.30 +Correc. 53. 41 =true alt.=16. 4.11 = Sum of the ap. alts. 25:50:50 Sum of the true altitudes = 26:39:37" Sum of true alts.=26:39:37% Nat. V. S. sup.-1. 893683 True cent. dist.=116:23:26" Nat. vers. S. = 1.444489 METHOD VII. Of reducing the apparent to the true central Distance, RULE. To the apparent central distance add the apparent altitudes of the objects, and take half the sum; the difference between which and the apparent distance, call the remainder then, |