EXPLANATION From the logarithm of the departure 214'.o subtract the logarithm of the distance 428'.0, which gives the logarithm sine of the course S. 30° E. To the logarithm cosine of the course S. 30° E. add the logarithm of the distance 428'.o, which gives the logarithm of the difference of latitude 370'.7 and is subtracted from the latitude of the point left because the ship has sailed south. Add together the latitude of the point left and that of the ship. Then divide by 2, which will give the middle latitude. To the logarithm of the departure 214'.o add the logarithm secant of the middle latitude 17° og' 39" and the result will be the logarithm of the difference of longitude 224.0, which is subtracted from the longitude of the point left because the ship has sailed east and in doing so has decreased the longitude. EXAMPLE NUMBER 8 Both latitudes (one north and the other south) and both longitudes to find the course and distance:, A ship is in latitude 4° 32' 45" N. and longitude 25° 35' 45" W. Find the course and distance to a point in latitude 1° 30' 45" S. and longitude 24° 05' 15" W. Find the course and distance. Lat. ship 4° 32' 45" Tv. Long. ship 25° 35' 45" W. Lat. point +1 30 45 S. Long. point - 24 05 15 W. Dif. 363.5=6 03 30 S. Dif. 90'.5 = I 30 30 E. 90'.5 1.95665 cos. + 9.99966 90'.43 = 1.95631 363.5 2.56050 tang. 9.39581 sec. .01303 363.5 + 2.56050 374.6 = 2.57353 This method will satisfy the ordinary demands of practice; but should not be used when the latitudes of the places under consideration lie on opposite sides of the equator. In such cases the course, distance, etc., should be found by Mercator sailing. EXPLANATION Find the difference of latitude by adding the latitude of the point to that of the ship, convert it into miles and name it south because the point is south of the latitude of the ship. When one latitude is north and the other is south, the middle latitude can best be found by taking half the greater latitude, which has been done in this case, and gives the middle latitude 2° 16' 22". Find the difference of longitude by subtracting the longitude of the point from that of the ship, convert it into miles and name it east because the longitude of the point is east of the longitude of the ship. To the logarithm of the difference of longitude 90'.5 add the logarithm cosine of the middle latitude 2° 16' 22", which gives the logarithm of the departure 90'.43 and from that logarithm subtract the logarithm of the difference of latitude 363'.5 and the result will be the logarithm tangent of the course S. 13° 58' E. To the logarithm secant of the course S. 13° 58' E. add the logarithm of the difference of latitude 363'.5, which gives the. logarithm of the distance 374'.6. When the places under consideration lie so near the equator, it is not necessary to use middle latitude sailing unless requested to do so at an examination, as plain sailing, in which the degrees of latitude and longitude are supposed to be equal, will give a result practically the same. EXAMPLE NUMBER 9 Find the course and distance from Sandy Hook light in latitude 40° 27' 42" N. and longitude 74° oo' 09" W. to Cape of Good Hope light in latitude 34° 21' 12" S. and 18° 29' 26" E. — COS. COS. sec. Lat. Sandy Hook 40° 27' 42" N. Mer. parts 2644.2 Lat. Cape G. H. + 34 21 12 S. Mer. parts + 2183.7 Dif. 4488.9=74 48 54 S. Dif. 4827.9 S. Long. Sandy Hook 74° 00' 09" W. Long. Cape Good Hope + 18 29 26 E. Dif. 5549'.6=92 29 35 E. Prop. dif. lat. 4488.9 3.65215 Mer. dif. 4827.9 3.68377 Middle lat. 21° 36' 9.96838 Dif. long. 5549.6 3.74429 Mid. lat. 21° 36' + 9.96838 3.71267 Prop. dif. lat. 4488'.9 3.65215 Co. S. 48° 58' E. tang. 0.00052 Co. S. 48° 59' E. .18291 Prop. dif. lat. 4488'.9 + 3.65215 Dist. 6840.0 = 3.83506 EXPLANATION Find the difference of latitude by adding the latitude of Cape of Good Hope to that of Sandy Hook, convert it into miles and name it south because Cape of Good Hope is south of the latitude of Sandy Hook. From Table 3 take the meridional parts for the latitudes to the nearest mile and find their difference by adding, which is the meridional difference of latitude and takes the same name as the proper difference of latitude. Find the difference of longitude by adding the longitude of Cape of Good Hope to that of Sandy Hook, convert it into miles and name it east because Cape of Good Hope is east of the longitude of Sandy Hook. When the places under consideration are on opposite sides of the equator, the middle latitude should not be found in the usual manner, if an accurate result is required. From the logarithm of the proper difference of latitude 4488.9 subtract the logarithm of the meridional difference of latitude 4827'.9, which gives the logarithm cosine of the middle latitude 21° 36'. By using the middle latitude found in this manner, the course and distance found by middle latitude sailing will be exactly the same as by Mercator sailing. To the logarithm of the difference of longitude 5549'.6 add the logarithm cosine of the middle latitude 21° 36' and from the resulting logarithm subtract the logarithm of the proper difference of latitude 4488'.9, which gives the logarithm tangent of the course S. 48° 59' E. To the logarithm secant of the course S. 48° 59' E. add the logarithm of the proper difference of latitude 4488'.9, which gives the logarithm of the distance 6840'.O. EXAMPLE NUMBER 10 Find the course and distance from Cape Horn in latitude 55° 58' 41" S. and longitude 67° 16' 15" W. to Cape of Good Hope in latitude 34° 21' 12" S. and longitude 18° 29' 26". E. Co. N. 70° 02' E. sec. .46664 1297'.5 + 3.11311 3799.7 = 3.57975 EXPLANATION Find the difference of latitude by subtracting the latitude of Cape of Good Hope from that of Cape Horn, convert it into miles and name it north because Cape of Good Hope is north of Cape Horn. From Table 3 take the meridional parts for the latitudes to the nearest mile and find their difference, which is the meridional difference of latitude and takes the same name as the proper difference of latitude. Find the difference of longitude by adding the longitude of Cape of Good Hope to that of Cape Horn, convert it into miles and name it east because Cape of Good Hope is east of the longitude of Cape Horn. The middle latitude should not be found in the usual manner, if an accurate result is required. From the logarithm of the proper difference of latitude 1297'.5 subtract the logarithm of the meridional difference of latitude 1869'.0, which gives the logarithm cosine of the middle latitude 46° 02'. To the logarithm of the difference of longitude 5145'.7 add the logarithm cosine of the middle latitude 46° 02' and from the resulting logarithm subtract the logarithm of the proper difference of latitude 1297.5, which gives the logarithm tangent of the course N. 70° 02' E. The logarithm secant of the course N. 70° 02' E. add the logarithm of the proper difference of latitude 1297'.5 and the result will be the logarithm of the distance 3799'.7. |