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Example 3. Dif. Lon. 243 Depart. 138::R: cos Lat. ; and, by the tables, 2.385606: 2.139879::10.000000: 9.75 4273-55° 23′ 44′′.

Example 4. Dif. Lon. Depart. ::R: cos Lat.; that is, 60:30:1 cos Lat. ; and, by the tables, 1.778751 : 1.47 7121: 10.000000 ; 9.698970=60° the part of the earth where the degrees of longitude are half as long as at the equator.

MIDDLE LATITUDE SAILING.

Example 1. R Dif. Lon.::cos Mid. Lat. Depart. ; that is, 1: 72' cos 41° 10′ 30′′: Depart.; and, by the tables. 10.000000 1.857338::9.876666; 1.734004=54.21=the departure. By plane sailing, CASE VI, Dif. Lat. R::Depart. tan Course; that is, 13': 1:: 54.21; tan Course; and, by the tables, a. c. 8.886057: 10.000000::1.734079; 10.620136 76° 30′ 41′′-the course. The distance is found by plane sailing, CASE III. cos Course: Dif. Lat.::R: Dist.; that is, cos 76° 30′ 41′′ 13'::1: Dist. ; and, by the tables, 9.368600 1.113943: 10.000000 1.745343=55.63= the distance.

Example 2. By plane sailing, CASE IV, R: Dist::cos Course Dif. Lat; that is, 1; 67::cos 45°: Dif. Lat. ; and, by the tables, 10.000000:1.826075::9.849495 : 1.6755 60 47.38 the difference of latitude, and 40° 28′-47.38′= 39° 40.3' the required latitude; also 40° 28′— § 47.38'— 40' 04.6' the middle latitude. By Theorem II, cos. Mid. Lat. Dist.sin Course: Dif. Lon.; that is, cos 40° 04.3' 67::sin 45°: Dif. Lon. ; and, by the tables, a. c. 0.1161 75 1.826075::9.849485: 1.791735-61.9', and 74° 08′ — 61.9' 73° 06.1' the required longitude.

Example 3. By Art. 46, cos Course Dif. Lat. :: R: Dist.; that is, cos 39° 266'-(4° 26')::1 Dist.; and, by the tables, 9.890503 : 2.424882::10.000000 ; 2.534379= 342.28 the distance. Again, by Theorem II, cos Mid. Lat. Dist.sin Course Dif. Lon. ; that is, cos 47° 44': 342.28 sin 39°: Dif. Lon.; and, by the tables, 9.827745

=

2.534379::9.798872: 2.505506=320.3'-5° 20.3' which added to 5o 14'10° 34.3' the difference of longitude.

Example 4. By inverting the proportion in Art. 53, Dif. Lon. R. Depart. cos Mid. Lat.; that is, 18° 29′:1:: 789 cos Mid. Lat.; and, by the tables, 3.044932: 10.000 000::2.897077: 9.852145 = 44° 38′ 50′′ cos Mid. Lat. Then 49° 57'-44° 38′ 50′′ 5° 18' 10", which multiplied by 2=10° 36' 24" Dif. Lat. 637.5 miles. Again, Dif. Lat. R: Dep. : tan Course; that is, 637.5: 1:::789 : tan Course; and, by the tables, 2.9044802 10.0000000::2.89 70770 10.0925965=51° 04′ 13′′ the course. Finally, R: Dif. Lat. :: sec Course: Dist.; that is, 1: 637.5 sec 51° Dist.; and, by the tables, 10.0000000: 2.8044802:: 10.2016303.0062432=1014.5=the distance.

MERCATOR'S SAILING.

Example 1. See Figure 24. By Theorem II, Merid. Dif. Lat. Dif. Lon.::R: tan Course; that is, 17 : 1° 12': :1 : tan Course; and, by the tables, 1.230449: 1.85733 :: 10. 000000: 10.626884 = 76° 43′ the course. Again, by plane sailing, cos Course: Prop. Dif. Lat.::R: Dist. ; that is, cos 76° 43' 13'::1: Dist.; and, by the tables, a. c. 0.63 8713: 1.113943: 10.000000 : 1.752656 56.58 = the distance.

:

Example 2. By plane sailing, cos Course: Prop. Dif. Lat. ::R: Dist.; that is, cos 39° : 266'::1: Dist.; and, by the tables, a. c. 9.109497 2.424882::10.000000 : 2.534379= 342.28 = the distance. Again, by Theorem II, R: tan Course::Merid. Dif. Lat. : Dif. Lon.; that is, 1 : tan 39°:: 396: Dif. Lon.; and, by the tables, 10.000000 : 9.908369 ::2.597695: 2.506064=5° 20.7′, which added to 5° 14′= 10° 34.7' the west longitude of the ship.

:

Example 3. Prop. Dif. Lat. 49° 57′-39° 20′ =10° 37' =637', and Merid. Dif. Lat.=3470-2571-899. By The. orem I, (Fig. 24.) Prop. Dif. Lat. Merid. Dif. Lat.::De. part. : Dif. Lon.; that is, 637′: 899::789: Dif. Lon.; and, by the tables, a. c. 7.195861 : 2.953760::2.897077 : 3.04 6697 = 1113.4' 18° 33.4', which added to 5° 14′ (given longitude) = 23° 47.4' the required longitude. Again, Prop. Dif. Lat. R::Depart. : tan Course; that is, 637' 1:.:

:

789 tan Course; and, by the tables, a. c. 7.195861: 10. 0000000: 2.897077: 10.092938= 51° 05.3' the course. Again, R: Prop. Dif. Lat.::sec Course: Dist. ; that is, 1 : 637'::sec 51° 05.3′ : Dist. ; and, by the tables, 10.000000 : 2.804139: 10.201907 3.006046=1014.= the distance.

Example 4. Dif. Lon.=29' 26′-17° 33′ =11° 53′=713'. By Theorem II, tan Course : R:: Dif. Lon. : Merid. Dif. Lat.; that is, tan 28° 7′ 30′′: 1::713' : Merid. Dif. Lat; and, by the tables, 9.728008: 10.000000::2.853090 : 3.12 5082=1334=Merid. Dif. Lon. Again, Lat. of the port = 14° 45', Merid. parts=825, and 7° 18' 439 S. L. Lat. of the ship. Dif. of Lat.=14° 45′ +7° 18′=22° 03'=1323'. By plane sailing, cos Course: Dif. Lat.::R: Dist.; that is, cos 28° 7′30′′: 1323'::1 : Dist.; and, by the tables, 9.945430 : 3.121855::10.000000 : 3.176425=1501.2 the distance.

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Example 5. Dif. Lat. 41°-37° 4°, and Merid. Dif. Lat. 2707'-2393'=309. By plane sailing, Dist. : R::Dif, Lat. cos Course; that is, 300 1::240': cos Course; and, by the tables, 2.477121: 10.000000::2.380211 : 9.903090 =36° 52′ 30′′ the course. By Theorem II, R: tan Course ::Merid. Dif. Lat. : Dif. Lon.; that is, 1 : tan 36° 52′ 30′′:: 309 : Dif. Lon.; and, by the tables, 10.000000 : 9.875010 ::2.899560: 2.364968=231.72=3° 51′ 7′′—the difference of longitude.

Example 6. By plane sailing, sin Course: Depart.::R: Dist.; that is, sin 67° 30': 957::1: Dist.; and, by the tables, a. c. 0.034385: 2.980912::10.000000: 3.015297 = 1035.8 the Distance: Again, sin Course: Depart.::cos: Dif. Lat. ; that is, sin 67° 30': 957::cos 67° 30' : Dif. Lat.; and, by the tables, a. c. 0.034385: 2.980912::9.582840 : 2.598137 6° 36.4'-the difference of latitude. Again, by Theorem II, R: tan Course::Merid. Dif. Lat. : Dif. Lon.; that is, 1: tan 67° 30'::4157-3490 667 Dif. Lon. ; and, by the tables, 10.000000: 10.382776::2.824126: 3.2 06902-26° 50' the difference of longitude.

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Example 7. By plane sailing, R: Dist.:: cos Course : Dif. Lat.; that is, 1 : 231:: cos 27°: Dif. Lat.; and, by the tables, 10.000000: 2.363612::9.949881: 2.313493– 205.82'. 26° 13' 1631, and 22° 47.2' 1404, whose dif.

=

=227 Merid. Dif. Lat. By Theorem II, R: tan Course ::Merid. Dif. Lat. : Dif. Lon.; that is, 1 : tan 27°::227 : Dif. Lon.; and, by the tables, 10.000000 6026 2.063192 longitude.

:

=

115.66'

9.707166::2.35

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1° 55.66' =

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Example 8. By plane sailing, Dist.: R: Depart. : sin Course; that is, 260: 1::173: sin 41° 42.5'; and, by the tables, a. c. 7.585027: 10.000000::2.238046 : 9.823073= 41° 42′ 40′′=sin Course. Again, by Art. 47, R: Dist.::cos Course Dif. Lat.; that is, 1: 260 cos 41° 42.5': Dif. Lat.; and, by the tables, 10.000000 2.414973: 9.8731 10: 2.288083-194.1=3° 14.13' Dif. Lat. The Merid. parts of 14° 10.50, and of 17° 14′ 848, whose dif.=202. By Theorem II, R: tan Course: Merid. Dif. Lat. Dif. Lon.; that is, 1 : tan 41° 42.5': :202 : Dif. Lon.; and, by the tables, 10.000000 9.949989::2.305351 : 2.255340= 180.3=32 00.3' the difference of longitude.

TRAVERSE SAILING.

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Note. In the solution of this and subsequent examples, also in some that have preceded, the frequent references to the several cases in Plane Sailing are so obvious, that particular mention of them has not in all cases been considered necessary. The two problems which follow may be solved by rules already given in Plane Sailing; but if the student has a Traverse Table at hand, much labor may be saved by taking the Dif. Lat. and Depart. from that. A valuable one may be found in Davies' Tables, also one in Gummer's Surveying, the latter of which has been used in this work. In the first example, it will be observed, that the several steps are not tabulated after the manner of Day.

Example 2. Art, 77. By CASE I. Art. 44, making the distance radius.

1. S. 13° E. 21

2. S. 18° W. 16

3. N. 84° E. 9

4. S. 67° E. 12

5. N. 78° E. 14

{

{

::

:

SR Dist. sin Course: Dep.-4.72 E.
R: Dist.cos Course: D. Lat. 20.46 S.
R Dist.sin Course: Dep.-4.94 W.
R: Dist.::cos Course: D. Lat.-15.22 S.
R Dist.sin Course: Dep.-8.95 E.
R: Dist.::cos Course: D. Lat.=0.94 N.
R: Dist.sin Course: Dep. 11.05 E.
R: Dist.::cos Course : D. Lat.=4.69 S.
R: Dist. sin Course: Dep. 13.69 E.
R: Dist. cos Course: D. Lat.=2.91 N.

6. S. 12° W. 35

R: Dist.sin Course: Dep.-7.28 W. R: Dist.::cos Course : D. Lat.=34.24 S. By adding, we find the sum of the northings 3.85, the sum of the southings 74.61, the sum of the eastings 38.41, and the sum of the westings 12.22. By subtracting, we have the difference of latitude 70.761 S. and the departure 26.19 E. By CASE IV, (Art. 47.) Dif. Lat.: R::Depart.

tan Course; that is, 70.76: 1:26.19 tan Course; and, by the tables, 1.8497878: 10.0000000::1.4181355: 9.568 3477-20° 18′ 38"+the direct course. Again, by Art. 47, R: Dif. Lat.::sec Course: Dist.; that is, 1 : 70.76:: sec 29° 18′ 38′′+ : Dist. ; and, by the tables, 10.0000000 : 1.8497878 ::10.0279263: 1.8777141 = = 75.46 the dis. tance. By Mercator's sailing, (Art 72.) R: tan Course:: Merid. Dif. Lat. : Dif. Lon. ; that is, 1: tan 20° 18′ 38′′+ ::95.5: Dif. Lon.; and, by the tables, 10.0000000 : 9.5683 477: 1.9754318: 1.5457750-34.977-Dif. Lon.

The difference of latitude=70.76'.

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Length of a degree of longitude for each degree of latitude.

D. L. Miles.

D. L. Miles. 1 59.99 19 56.73 37 2 59.96 20 56.38 38 3 59.92 21 56.01 39 4 59.85 22 55.63 40 5 59.77 23 55.23 41 6 59.67 24 54.81 42 7 59.55 25 54.38 43 8 59.42 26 53.93 44 9 59.26 27 53.46 45 10 59.08 28 52.97 46 11 58.89 29 52.47 47 12 58.68 30 51.96 48 13 58.46 31 51.43 49 14 58.22 32 50.88 50 15 57.95 33 50.32 51 16 57.67 34 49.74 52 17 57.38 35 49.15 53 18 57.06 36 48.54 54

D. L.

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