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ing the elevated pole; and subtracted therefrom, when receding from that pole.

When the Sun is in the solstices, this correction vanishes; and it may be neglected in all cases, unless for some philosophical purposes.

EXAMPLE

Required the apparent time of the beginning and end of twilight, and its duration, at Buenos Ayres, in latitude 34° 35' S. and longitude 58° 32' W. October 17, 1804 ?

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If it be required to make an allowance for the change of declination between noon and the time of the beginning or end of twilight; the angle of position must be previously obtained, which being computed by either of the formulæ given in the note of page 130, vol. 1. will be found to be 48° 49', with which, the declination 9° 21', and its change in the above interval, 7' 22", the corresponding change in the hour angle will be 26" ; which being added to the approxiinate times of the beginning and end of twilight, because the Sun is approaching towards the elevated pole, will give to the nearest minute, 4h. 2' for the beginning, and 7h, 59' for the end of twilight.

Reduced declination
Latitude

9° 21' 34 35

tangent
tangent

1.21578
9.83849

83

33

2

Arch

429

co-sine 9.05427 Approx. time Sun rising 5 59 Appr. time Sun set. 6h. 26' 1" Equation Table A, + 16

16 Pro. part to hor. refr.

I

44 Apparent time Sun rising 5 31 App. time Sun set, Hence the time of duration of the morning twilight is lh. 29'', and that of the evening twilight 1h. 30'.

31

6

29

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sine

Secant

0114382 0.191481

Declin. 6

58

sine

9.083648

sine

PROBLEM VI. Given the Latitude of a Place, to find the Time of the shortest

Twilight and its Duration.

Rule. To the constant log. 9.199712* add the log. sine of the latitude, and the sum will be the log. sine of the declination, at the time of the shortest twilight, of a contrary name to the latitude ; the time corresponding to this declination will be that required.

Again, to the constant log. 9.194332+, add the lòg. secant of the latitude, the sum will be sine of an arch, which, multiplied by 8, will be the time of duration of the shortest twilight.

EXAMPLE. Required the time of the shortest twilight, and its duration, in the year 1804, in latitude 49 57 N.? Constant log.

9.199712

Constant log.
Latitude 49° 57'

9.883936
Arch 144

9.385913 Hence, the times of the shortest twilight in 1804, in latitude 49 57' N. are the evening of the 2d of March, and the morning of the 11th of October; and the duration of the twilight is (=14° 4'*8') =lh. 52.

PROBLEM VII. Given the Latitude of a Place, between the Limits of 48° 32' and

66° 32', to find when it ceases to be real Night, and the Time of its Commencement.

RULE. The complement of the sum of the given latitude and 18°, will be the declination of the Sun, of the same name as the latitude, at the time when it ceases to be real night, and also when real night commences.

EXAMPLE Required the period of time, in the year 1804, during which there will be no real night, in latitude 55° 3''N.? Given latitude

55° 3' N. Arch

18

73

3

16 57

Sum

Complement Now, the days answering to 16° 57' of north declination are, 7th May and 5th August. Upon the first of these days, therefore, real night ceases, and commences upon the last : during this interval, there is no real night, the Sun being less than 18° under the horizon.

* The log. tangent of half 18'.
of The log. sine of half 10°.

APPENDIX, No. II.

CONTAINING,

A TABLE of MERIDIONAL Parts, extended to one Decimul place,' and a GENERAL TRAVERSE TABLE.

TABLE I. - Meridional Parts. A Table of Meriodional parts extended to, at least, one decimal place, is useful in many cases, particularly when the course is near a parallel, or when one or both latitudes are given in degrees, minutes, or parts of a minute. The bearing and distance between any two places may be found as follows:

RULE.-From the logarithm of the difference of Longitude, its index being encreased by 10, subtract the log: of the Meridional difference of Latitude, the remainder will be the log. tangent of the course, the co-sine of which being subtracted from the log. of the proper difference of Latitude, its index being encreased by 10, the remainder will be the logarithm of the distance

EXAMPLE.-Required the bearing and distance of the Anholt lights, in Lat. 56° 44' 20" N. and Longitude 11° 40' E. from Aberdeen in Lat. 57° 9' N. and Longitude 2° 8' W. Lat. Anholt lights 56° 44' 20" M. parts 4154.0 Long. 11° 40' E. Lat. Aberdeen 57 90 M parts 4199.2 Long. 2 8 W.

Proper diff. Lat.

0 34 40 M.D. Lat. 45.2 Dif. Long. 13 48

In miles 828

Diff. Long;

828 log: +10 - 12.918030 Mer. Diff. Lat. 45,2

1.655138

Course S. 86° 52' 32" E. tang. 10.262899 co-sine 8.736453
Proper diff. of Lat.

24' 40" or 24'}" log 1.372111

Distance 452. 5

2.655658 Hence, the course per compass, from Aberdeen to the Anholt lights, the variation being 2 points westerly, is S. E. by E. J. E. nearly

REMARK. When the difference of Longitude is very considerable, with respect to the difference of Latitude, the distance may be found by adding the log. co-sine of the middle Latitude to the log. of the diff. of Longitude expressed in minutes. Thus, in the preceding Example, the middle Lat. is 56° 56' 40". Log. co-sine 9.736756 Diff. of Longitude 828

2.918030

Log.

Distance 451.6

2.654786 This, however, must' always be less than the true distance.

TABLE II.- GENERAL TRAVERSE TABLE. The second table is a concise, and at the same time an extensive traverse table. The difference of Latitude and departure may be found with the same ease, answering to any given course and distance.

MERIDIONAL PARTS.

Min

Min.

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0

2 3

3

6

8

8 9

10 11 12 13 14 15 16 17 18 19

10 11 12 13 14 15 16 17 18 19

20 21 22

23

LATITUDE. 2°

4° 5° / 6° 70 89 90 10° 11° 12° 0.0 00.0 120.0 (180.1240.2 300.4 360.7 +21.1451.6 542.2 603.1 664.1 725.3 1.0 61.0 121.0181.1 241.2 301.4 361.7 +22.1 452.6 543.3 604.1|665.1 726,3 2.0 62.0 122.0 182.1 2+2.2 302.4 362.7 123.1 483.6 544.3 605.1 (666.1727.4 3.0 63.0 (123.0183,1243.2 303.4 363.4 424.1 484.61545.3 606.21667.2 728.4 4.0 64.0 (124.01181.1244.2 304.4 301.7 1425.1 +85.6 546.3607.21668.2 729.4 5.0 65.0 125.01185.1 215.2305.4 365.7 +26.1 486.6 547.3608.2 669.2 730.5 6.0 66.0 126.0186.1 240.2 306.4 366.7 427.1 187.6 548.3 609.2 670.2 1731.5 7.0 67.0|127.0 187.1 2-7.2 307.4 307.7 1128.1 488.6 549.3610.2671.2 1732.5 8.0 68.0 128.0|188.1 (248.2 308.7 368.7 +29.1 489.6 550.3 611.2 1672,2 733.5 9.0) 69.0 129.0189.1 249.2 309.4 (369.7 +30.1 490.7 551.4 612.2 1673.3 1734.6 10.0 70.0 130.0 190.1|250.2 310.+ 370.7 +31.1 191.7 552.4 1613.2674.3 735.6 11.0 71.0131.0 191.1 251.2 311.4 371.7432.1 492.71553.4 614.2 1675.3 1736.6 12.0 72.0 132.0|192.1 252.2 312.4 372.7433.1 +93.7 554.41615.3 676.3 737.6 13.0 73.01133,0|193.1 |253.2 313.4 373.7 431.2 94.7 555.4 616,31677.3 738.6 14.0 74.0134.01194.1 254.2 314.4 374.7435.2 495.7 556.4 1017.31678.4 739.6 15.0 75.0 135.0 195.1 255.2 1315.4 375.7 436.2 496.7 557.4 618,3 1679.4 740.7 10.0 76.0:36.0 1196.1256.2 316.4 376.8 437,2 197.7 558.4 619.3 680.4 741.7 17.0 77.0 137.0 197.1 257.2 317.5 377,8 438.2 498.7 559.4 620.3 681.4 742.7 18 0 78.0 158.0 198.1 258.2 318.5 378.8 439.2 499.8 560.5 021.3 682.4 743,7 19.0 79.0 139.0 199.1 259.2 319.5 379.8 140.2 500.8 561.5 622.4 683.5 744.8 20.0 80.0 140.0 200,1260.2 320.5 380.8 1441.2 501.8 562.5623,4 684.5 745.8 21.0 81.01141.0 1201.1261.3 (321.5381.8 442.2 502.8 563.5 1624.4 685.5 746.8 22.0 82.01142.0 202.1 262,3392.5 382.8 443.2 503.8 564,5 1025.4 686.5 1747.8 23.0 83.) 143.0 23.1 263.3|323.5 383.8 414.2 504.8 565.5 626.4 687.5 748.9 24 0 84.0 144.0204.1|264.3 324.5|384.8 445.2 503.8 566.61627.4 688.6 749.9 25.083.01145.0205.11265.3 325.5 385.8 +46.3 506.8 567.6|628.5 (689.6 750.9 26.0 86.0 146.0 206.1266.3|326.5 386.8 147.3 507.8 568.61629.5 1690.6 751.9 27.0 87.0 147.0 207.1 267.3 327.5 387.8 +48.3 508.9 569.6630.5 691.6 753.0 28.0 88.0 148.0 208.1 268.3328.5 388.8 449.3 509.9570.6 631.5 692.6 754.0 29.0 | 89.9149.0 209.1|209.3 329.5 389.8 150.3 510.9 571.6632.5 693.6 755.0 30.0 90.01150.0210.1|270.3 330.5 390.8 151.3 511.91572.61633.5 694.7 756.0 31.0 91.01151.! 211.1|271.3331.5 391.8 +52.3 512.9573.7 1034.6 1695.7757,1 32.0 92.0152.1|212.1 272.3 332.5 392.9 453.3 513.9 574.7 635.6 696,7 1758.1 33.0 93.0 153.1213.1 279.3 333.5 393.9 454.3 514.9 1575.7636.6 1697.7 17 59.1 34.0 94.0 154.1 214.127.1.3 334.5 394.9455.3 1515.9 576.7637.6 1698.77 60.1 35.0 95.0 155.1|215.1275.3335.5 395.. 456.3 516.9 577.7638.6 699.8 761.2 36.0 96.0156.1|216.1 |276.3 336.5 396.9 457.31518.4 578.7|639.6 700.8 762.2 37.0 97.0 157.1 217.1 27.3 337.5 397.9 58.4 519.0 579.7 640.7 701.8 763.2 38.0 98.0 158.1 1218.2 278.3 338.5 398.9|159.4 520.0 580.81641.7 702.8 1764.2 39.0 99.0 (159.1 219.2 279.3 339.6 399 9 460.4 521.0 581.81642.7 703.8 1765.2 40.0 100.0 160.1 220.2 280.3 340.6 404) 9161.41522.01582.8 1643,7 1704.9 766.3 41.0 101.0 161.1 221.2 281.3 (341.6 401.9 462.4 523.0 583.8 (644.7 705.9 767.3 42.0102.01162.1 222.2 282.3342.6 402.9 463.4 524.01584.8 645.8 706.9768.3 43.0 103.0 (163.1|223.2 283.3.43.6 403.9 464:4 525.0 1585.8 646.8 707.9 769.3 44.0 104.0 164.1224.2 284.3344.6 404.9 465.4 526.0 586.8 647.8709.0 770.4 45.0 105.0 1165.1 225.2 285.3345.6 405.9 166.4 527.1 587.91648.8 710.0 1771.4 46.0 106.0160.1226.2 286.3 346.6 407.0 167.4 528.1 588.9 1649.8 711.0 772.4 47.0 167.0 167.1 227.2 -87.3|347.6 408.0468.4 529.1 589.9 650.8 712.0 773.4 48.3 108.0 168.1 228.2 288.3 348.6 409.0 469.5 530.1 590.9051.91713.1 774.5 49.0 109.01169.1 229.2 289.3349.6410.0 +70.5 531.1591.9 1652.91714.1 775.5 50.0 110.0 170.1 230.2 290.3350.6 411.0 171.5 $532.1 592,9653.91715.1 776.5 51.0 111.01171.1 231.2 291.3 351.6 412.0 |172.5 533.1 593.9 654.91716.11777.5 52.0 112.0172.1 232.2 292.4352.6 413.0 473.5 534.1 (595.0 1655.9 717.1 778.6 53.0 13.01173.1 233.2 293.4353.6 414.0 1474.5 1535.1 596.0 1657.0 718.2 779.6 54.0 114.0 174.1 234.2294.4 354.6 415.0 475.5 536.2 (597.0658.0719.2 780.6 55.0 115.0 175.1235.2 295.4 355.6 416.0 476.5 537.2 598.0 659.01720.2 781.7 56.0 116.0 176.1 236.2 296.4 356.6 417.0 477.5 538.2 599.0 660.0 721.2 782.7 57.0|117.0177.1 237.2297.4 1357.6418.0 478.5 539.2 600.0 661.07722.3 1783.7 58.0 118.0 178.1238.2 298.4358.6 419.01479.5 540.2 601.01662.1 723.3 784.7 59.0 119.0 179.1 289.2 299.4 359.7 420.0 480.5 541.2 602. i 1663.1 1724.3 785.8 1° 2° 30 5° 6° 7°

10 11°

20 21 22 23 24 25 26 27 28 29

25 26 27 28 29

30 31 32 33 34 35 36 37 38 39

30 31 32 33 34 35 36 37 38 39

40

40 41 49 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59

48 49 50 51 39 53 54

55 56 57 58

59 12° Min.

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MERIDIONAL PARTS.

22°

23°

16

LATITUDE. Min., 13° 14° 150 16° 17° 18° 19° 20°

27°

24°

Min. 0 786.81 848.5) 910.5) 972.7|1035.31098.2 1161.5 1225.1|1289.2 1353.7 1418.6 1484.1 0

787.8 849.5| 911.5| 973.8 1036.31099.31162.5 1226.2|1290.3 1354.8 1419.7|1485.2 1 788.8 850.6) 912.6 974.8) 1037.41100.31163.6 1227.31291.3|1355.814?0.5 1486.3 2

789.9) 351.6 913.6 975.9 1038.411101.41164.7|1228.3 1292.4 1356.91421.9 1487.3 3 4 790.9) 852.6914.6 976.9|1039.5|1102.4 1165.7 1229.4|1293.5 1358.0 1423.01488.4 4

791.9 853.5| 915.7 978.0 1040.5|1103.5 1166.8 1230.4 1294.5/i359.0 142+.1|1489.5 5 792.9 857.7 916.7979.0/1041.61104.5/1167.8|1231.5 1295.6|1960.11425.11490.6 6

794.0 855.7) 917.7 980.011042.6 1105.6 1168.31232.6|1296.7|1361.211426.2 1491.7 7 8 795.0 856.7 918.8) 981.1 1043.7|1106.6 1170.0|1233.61297.8|1362.3|1427.3|1492.8

796.01 857.8 919.8 982.1 1044.7/1108,7 1171.0|1234.71298.8 1363.4.1428.4/1493.9 9 10 797.1 858.81 920.8 983.2 1045.8 1108.711172.1 1235.81299.91364.5 1429.5 1495.0) 10 11 798.1 859,8 921 9 984,2 1046.81109.8|1173.1 1236.8 1201.0 1365.6 1430.61496.1 11 12 799.11 960.9 922.9 985.2 1047.91110.8 1174.2 1237.9 1302.0/1366.6 1431.711497.2 12 13 800.11 861.9 923.9 986.3 1048.91111.91175.2 1239,0 1303,1 1367.7 1432.31498.3 13

801.21 862.9 925.0 987.3 1049.9|1112.9|1176.3 1240.0 1304.2(1368.81433.9 1499.4 14 15 802.2 864.0 926.0 988.4/1051.0 1114.0 1177.4/1241.1 1305.31369.91434.9 1500.5 15 16 803.2 865.0 927.0989.4 1052.0/1115.0 1178,4 1242.2 1.306.3|1370.9 1436.01501.6 17 804. 866.0 929.1 990.411053.1 1116.1 1179.5 1244.2|1307.411372.0 1437.1 1502.7 17 18 805.3 867.1 929.1 991.5 1054.1 1117.1 1180.5|1244.3|1908.5 1373.1 1438.2/1503,81 18 19 806.3 868.1 930.11 992.5 1055.2 1118.21181.6 1245.4 1309.61374.2 1439.3 1504.9 19 20 807.3 869.11 931.21 993.6 1056.21119.2 1182.71246.4 1310.6 1375.3 1440,41506.0 20 21

803.4 870.11 932.2 994.61057.31120.3 1183.7|1217.5 1311.71376.4 1441,511507.1 21 22

809.41 871.2 933.3 995.6 1058.3 1121.31184.8/1248.6 1312.8 1377.4 1442,6 1508.2 22 23 810.4) 872.2/ 937.3 996.7 1059.41122,4 1185.8 1243.0 1313.8|1378.5/1443.711509.3 23 24 811.4 873.2 935.3 997.71060.41123.4 1186.9/1250.7|1314.9 1379.6 1444.8 1510.4 24 25 812.5 874.31 936.3 998.8 1061.4 1124.51188.0 1251.8 1316.0 1380.7 1445.8 1511.5 25 26 813.5 875.3937.41 999.8|1062.5 1125.5|1189.0 1252.8/1317.1 1381.8|1446.9|1512.6 26 27 .814.5 876.3 938.4 1000.8 1063.5 1126.6 1190.11253.9|1318.1 1382.8 1448.0 1513.7 27 28 815.5 877.4 939.5 1001.9 1064.6 1127.61191.1 1255.0 1319.2 1333.9 1449.1 1514.8 28 29 816.6 878.4) 940.5 1002.9/1065.6 1128.7|1192.2 1256.1 1320.3 1385.0 1450.21515.9 29 30 817.6 879.4 941.6 1004.01066.7/1199.7 1193.21257.1 1321.4 1386.1 1451.3 1517.0 30 31 818.6 880.5 942.6 1005.0 1067.7|1130.8 1194.31258.21322.5 1387.2 1452.4 1518.1 31 32 819.6 881.5| 943.6 1006.1 1068.8 1131.8/1195.41259.2 1323.5 1388.31453.5 1519.2 32 33 820.71 882.5 944.7/1007.1 1069.8|1132.911196.4 1260.3 1324.6 1389.4 145.4.6 1520.3 34 821.71 883.6945.7 1008.1/1070.9 1134.01197.5 1261.4 1325.711390,4 1455.7 1521.4 34 35 822.7 884.6 946.711009.2 1078.101135.1 1198.5/1262.4 1326.71391.5 1456.71522.5 S5 36 823.8 885,6 947.8 1010.2 1073.01136.1|1199.611263.5 1327.8 1392.6 1457.8|1523.6 36 37 824.8| 886.7 948.8 1011.31074.1/1137.21200.7 1264.61328.911393.71458.9|1521.7 37 38 825.8) 887.7 949.9/1012.51075.11138.2 1261.7|1265.6 1330.0 1394.9 1460.0 1525.8 38 39 826.91 888.7) 950.9 1013.4 1076.2 1139.3 1202.8|1266.711331.1|1395.8/1451.1 1526.9 39 40 827.9 889.8 951.91014.4 1077.2 1140.3 1203.9 1267.811332.11396.9 1462.2|1528.0 40 41 828.9 890.8\ 953.0 1015.4 1078.3/1141.4 1204.911268.8/1333.211398.0 1463.3 1529.1 41 42 829.9| 891.8 954.0 1016.5/1079.3 1142.4 1206.0|1269.9 1334,3 1399.11464.4/1530.2 42 43 831.0 892.91 955.0 1017.5 1080.4 1143.5 1207.1 1271,011335.31:400.2 1465.5|1531.3 43 44 832.0 893.9 956.1 1018.6 1081.4 1144.6 1208.1 1272.1(1336.4/1401.31466.6 1532.4 44 45 833.01 894.9 957.1 1019.6/1082.5 1145.6 1209.21273.1 1337.5|1402.41467.7 1533.5 45 46 834.1 896.0 958.1 1020.61083.5 1146.71210.21274.2 1338.6 1403.4 1469.8/1534.6 46 47 835.1} 897.0 959.2 1021.71084.6|1147.71211.31275.3 1339.7 1404.5 1469.8 1535.7 47 48 836.1 898.1) 960.2 1022.7/1085.6 1148.8/1212.4 1276.31310.71405.6 1470.9 1536.8 48 49 837.2 899.1 961.3 1023.8/1086.7 1149.9/1213.4 1277.4/1341.8 1406.71 1472.41537.9 49 50 338.2 900.1 962.3|1024.8|1087.7 1150.91914.5 1278.5 1342.9|1407.811473.1 1539.0 50 51 839,2 901.2 963.4 1025.9 1088.8/1152.0/1215.5 1279.5/1344.0 1409.8/1474.2 1540.1 51 52 840.2) 902.2 964.4/1026.9 1089.8 1153.0|1216.6 1280.6 1345.1 1409.9 1475.3 1541.2 52 53 841.3 903.2 965.5|1028.01090.9|1154.1 1217.7|1281.71346.1 1411.011476.4 1542.3 53 54 842.3 904.3 966.5/1029.01091.9 1155.1 1218.7 1282.8 1347.21412.111477,41543.5 54 55 843,51 905.3 967.5 1030.1 1093.0 1156.2 1219.8 1283.8 1348.3 1413.2 1478.6/1544.5 55 56 814.41906.3 968.61031.11094.01157.2 1220.9 1284.91349.41414.31479.7 1545.6 56 57 845.41 907.41 969.6 1032.2 1095.1 1158.3 1221.91286.0 1350.41415.4 1480.8|1546.7 57 58 846.4) 908.4) 970.7|1033.2 1096.1 1159.4 1223.011287.0 1351.5 1416.5 1481.91547.8 58 59 847.5| 909.4) 971.711031.3 1097.2 1160.4 1224.11288.1/1352.6 1417.5 1483.0 1548.9 59 Min. 13° 14° 15° 16° 170 18° 190 200 21° 22° 23° 24° Min

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LATITUDE.

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