If you sail due North or South, the problem is merely one of arithmetic. Suppose your position at noon today is Latitude 39° 15' N, Longitude 40° W, and up to noon tomorrow you steam due North 300 miles. Now you have already learned that a minute of latitude is always equal to a nautical mile. Hence, you have sailed 300 minutes of latitude or 5°. This 5° is called difference of latitude, and as you are in North latitude and going North, the difference of latitude, 5°, should be added to the latitude left, making your new position 44° 15' N and your Longitude the same 40° W, since you have not changed your longitude at all.
In sailing East or West, however, your problem is more difficult. Only on the equator is a minute of longitude and a nautical mile of the same length. As the meridians of longitude converge toward the poles, the lengths between each lessen. We now have to rely on tables to tell us the number of miles in a degree of longitude at every distance North or South of the equator, i.e., in every latitude. Longitude, then, is reckoned in miles. The number of miles a ship makes East or West is called Departure, and it must be converted into degrees, minutes and seconds to find the difference of longitude.
A ship, however, seldom goes due North or South or due East or West. She usually steams a diagonal course. Suppose, for instance, a vessel in Latitude 40° 30' N, Longitude 70° 25' W, sails SSW 50 miles. What is the new latitude and longitude she arrives in? She sails a course like this:
Now suppose we draw a perpendicular line to represent a meridian of longitude and a horizontal one to represent a parallel of latitude. Then we have a right-angled triangle in which the line AC represents the course and distance sailed, and the angle at A is the angle of the course with a meridian of longitude. If we can ascertain the length of AB, or the distance South the ship has sailed, we shall have the difference of latitude, and if we can get the length of the line BC, we shall have the Departure and from it the difference of longitude. This is a simple problem in trigonometry, i.e., knowing the angle and the length of one side of a right triangle, what is the length of the other two sides? But you do not have to use trigonometry. The whole problem is worked out for you in Table 2 of Bowditch. Find the angle of the course SSW, i.e., S 22° W in the old or 202° in the new compass reading. Look down the distance column to the left for the distance sailed, i.e., 50 miles. Opposite this you find the difference of latitude 46-4/10 (46.4) and the departure 18-7/10 (18.7). Now the position we were in at the start was Lat. 40° 30' N, Longitude 70° 25' W. In sailing SSW 50 miles, we made a difference of latitude of 46' 24" (46.4), and as we went South - toward the equator - we should subtract this 46' 24" from our latitude left to give us our latitude in.
Now we must find our difference of longitude and from it the new or Longitude in. The first thing to do is to find the average or middle latitude in which you have been sailing. Do this by adding the latitude left and the latitude in and dividing by 2.
| 40° 30'
00" 39 43 36 ————— 2) 80 13 36 ————— 40° 06' 48" Mid. Lat. |
Take the nearest degree, i.e., 40°, as your answer. With this 40° enter the same Table 2 and look for your departure, i.e., 18.7 in the difference of latitude column. 18.4 is the nearest to it. Now look to the left in the distance column opposite 18.4 and you will find 24, which means that in Lat. 40° a departure of 18.7 miles is equivalent to 24' of difference of Longitude. We were in 70° 25' West Longitude and we sailed South and West, so this difference of Longitude should be added to the Longitude left to get the Longitude in:
| Lo. left
70° 25' W Diff. Lo. 24 ————— Lo. in 70° 49' W |