Wednesday Lecture
Useful Tables—Plane And Traverse Sailing
The whole subject of Navigation is divided into two parts, i.e., finding your position by what is called Dead Reckoning and finding your position by observation of celestial bodies such as the sun, stars, planets, etc.
To find your position by dead reckoning, you go on the theory that small sections of the earth are flat. The whole affair then simply resolves itself into solving the length of right-angled triangles except, of course, when you are going due East and West or due North and South. For instance, any courses you sail like these will be the hypotenuses of a series of right-angled triangles. The problem you have to solve is, having left a point on land, the latitude and longitude of which you know, and sailed so many miles in a certain direction, in what latitude and longitude have you arrived?
Dead Reckoning Illustration
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:
Course SSW
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
The whole problem therefore would look like this:
Lat. left 40° 30' N
Diff. Lat. 46 24
——————
Lat. in 39° 43' 36" N Lo. left 70° 25' W
Diff. Lo. 24
————
Lo. in 70° 49' W
There is one more fact to explain. When the course is 45° or less (old compass reading) you read from the top of the page of Table 2 down. When the course is more than 45° (old compass reading) you read from the bottom of the page up. The distance is taken out in exactly the same way in both cases, but the difference of Latitude and the Departure, you will notice, are reversed. (Instructor: Read a few courses to thoroughly explain this.) From all this explanation we get the following rules, which put in your Note-Book:
To find the new or Lat. in: Enter Table 2 with the true course at the top or bottom of the page according as to whether it is less or greater than 45° (old compass reading). Take out the difference of Latitude and Departure and mark the difference of Latitude minutes ('). When the Latitude left and the difference of Latitude are both North or both South, add them. When one is North and the other South, subtract the less from the greater and the remainder, named North or South after the greater, will be the new Latitude, known as the Latitude in.
To find the new or Lo. in: Find the middle latitude by adding the latitude left to the latitude in and dividing by 2. With this middle latitude, enter Table 2. Seek for the departure in the difference of latitude column. Opposite to it in the distance column will be the figures indicating the number of minutes in the difference of longitude. With this difference of Longitude, apply it in the same way to the Longitude left as you applied the difference of Latitude to the Latitude left. The result will be the new or Longitude in.
Now if a ship steamed a whole day on the same course, you would be able to get her Dead Reckoning position without any further work, but a ship does not usually sail the same course 24 hours straight. She usually changes her course several times, and as a ship's position by D.R. is only computed once a day - at noon - it becomes necessary to have a method of obtaining the result after several courses have been sailed. This is called working a traverse and sailing on various courses in this fashion is called Traverse Sailing.
Put in your Note-Book the following example and the way in which it is worked:
Departure taken from Barnegat Light in Lat. 39° 46' N, Lo. 74° 06' W, bearing by compass NNW, 15 knots away. Ship heading South with a Deviation of 4° W. She sailed on the following courses:
Courses Sailed
Calculations
The rule covering all these operations is as follows:
1. Write out the various courses with their corrections for Leeway, Deviation, Variation and the distance run on each.
2. In four adjoining columns headed N, S, E, W respectively, put down the Difference of Latitude and Departure for each course.
3. Add together all the northings, all the southings, all the eastings and all the westings. Subtract to find the difference between northings and southings and you will get the whole difference of Latitude. The difference between the eastings and westings will be the whole departure.
4. Find the latitude in, as already explained.
5. Find the Lo. in, as already explained.
6. With the whole difference of Latitude and whole Departure, seek in Table 2 for the page where the nearest agreement of Difference of Latitude and Departure can be found. The number of degrees at the top or bottom of the page (according as to whether the Diff. of Lat. or Dep. is greater) will give you the true course made good, and the number in the distance column opposite the proper Difference of Latitude and Departure will give you the distance made.
It is often convenient to use the reverse of the above method, i.e., being given the latitude and longitude of the position left and the latitude and longitude of the position arrived in, to find the course and distance between them by Middle Latitude Sailing. The full rule is as follows:
1. Find the algebraic difference between the latitudes and longitudes respectively.
2. Using the middle (or average) latitude as a course, find in Table 2 of Bowditch the Diff. of Lo. in the distance column. Opposite, in the Diff. of Lat. column, will be the correct Departure.
3. With the Diff. of Lat. between the position left and the position arrived in, and the Departure, just secured, seek in Table 2 for the page where the nearest agreement to these values can be found. On this page will be secured the true course and distance made, as explained in the preceding method.
4. Use this method only when steaming approximately an East and West course.
For an example of this method, see Bowditch, p. 77, example 3.