The angle which this line makes with the meridian of Lo. intersecting any point in question is the Course, and the length of the line between any two places is called the distance between them. Example: T or T'.

_Chart Projections_

The earth is projected, so to speak, upon a chart in three different ways--the Mercator Projection, the Polyconic Projection and the Gnomonic Projection.

_The Mercator Projection_

You already know something about the Mercator Projection and a Mercator chart. As explained before, it is constructed on the theory that the earth is a cylinder instead of a sphere. The meridians of longitude, therefore, run parallel instead of converging, and the parallels of lat.i.tude are lengthened out to correspond to the widening out of the Lo.

meridians. Just how this Mercator chart is constructed is explained in detail in the Arts. in Bowditch you were given to read last night. You do not have to actually construct such a chart, as the Government has for sale blank Mercator charts for every parallel of lat.i.tude in which they can be used. It is well to remember, however, that since a mile or minute of lat.i.tude has a different value in every lat.i.tude, there is an appearance of distortion in every Mercator chart which covers any large extent of surface. For instance, an island near the pole, will be represented as being much larger than one of the same size near the equator, due to the different scale used to preserve the accurate character of the projection.

_The Polyconic Projection_

The theory of the Polyconic Projection is based upon conceiving the earth's surface as a series of cones, each one having the parallel as its base and its vertex in the point where a tangent to the earth at that lat.i.tude intersects the earth's axis. The degrees of lat.i.tude and longitude on this chart are projected in their true length and the general distortion of the earth's surface is less than in any other method of projection.

[Ill.u.s.tration]

A straight line on the polyconic chart represents a near approach to a great circle, making a slightly different angle with each meridian of longitude as they converge toward the poles. The parallels of lat.i.tude are also shown as curved lines, this being apparent on all but large scale charts. The Polyconic Projection is especially adapted to surveying, but is also employed to some extent in charts of the U. S.

Coast & Geodetic Survey.

_Gnomonic Projection_

The theory of this projection is to make a curved line appear and be a straight line on the chart, i.e., as though you were at the center of the earth and looking out toward the circ.u.mference. The Gnomonic Projection is of particular value in sailing long distance courses where following a curved line over the earth's surface is the shortest distance between two points that are widely separated. This is called Great Circle Sailing and will be talked about in more detail later on.

The point to remember here is that the Hydrographic Office prints Great Circle Sailing Charts covering all the navigable waters of the globe.

Since all these charts are constructed on the Gnomonic Projection, it is only necessary to join any two points by a straight line to get the _curved_ line or great circle track which your ship is to follow. The courses to sail and the distance between each course are easily ascertained from the information on the chart. This is the way it is done:

(Note to Instructor: Provide yourself with a chart and explain from the chart explanation just how these courses are laid down.)

Spend the rest of the time in having pupils lay down courses on the different kinds of charts. If these charts are not available a.s.sign for night work the following articles in Bowditch, part of which reading can be done immediately in the cla.s.s room--so that as much time as possible can be given to the reading on Dead Reckoning: 167-168-169-172-173-174-175-176--first two sentences 178-202-203-204-205-206-207-208.

Note to pupils: In reading articles 167-178, disregard the formulae and the examples worked out by logarithms. Just try to get a clear idea of the different sailings mentioned and the theory of Dead Reckoning in Arts. 202-209.

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 lat.i.tude and longitude of which you know, and sailed so many miles in a certain direction, in what lat.i.tude and longitude have you arrived?

[Ill.u.s.tration]

If you sail due North or South, the problem is merely one of arithmetic.

Suppose your position at noon today is Lat.i.tude 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 lat.i.tude is always equal to a nautical mile. Hence, you have sailed 300 minutes of lat.i.tude or 5. This 5 is called difference of lat.i.tude, and as you are in North lat.i.tude and going North, the difference of lat.i.tude, 5, should be added to the lat.i.tude 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 lat.i.tude. 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 Lat.i.tude 40 30' N, Longitude 70 25' W, sails SSW 50 miles. What is the new lat.i.tude and longitude she arrives in? She sails a course like this:

[Ill.u.s.tration]

Now suppose we draw a perpendicular line to represent a meridian of longitude and a horizontal one to represent a parallel of lat.i.tude. 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 lat.i.tude, 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 compa.s.s reading. Look down the distance column to the left for the distance sailed, i.e., 50 miles. Opposite this you find the difference of lat.i.tude 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 lat.i.tude of 46' 24" (46.4), and as we went South--toward the equator--we should subtract this 46' 24" from our lat.i.tude left to give us our lat.i.tude 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 lat.i.tude in which you have been sailing. Do this by adding the lat.i.tude left and the lat.i.tude 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 lat.i.tude_ 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 Lo. left 70 25' W Diff. Lat. 46 24 Diff. Lo. 24 ------------- ---------- Lat. in 39 43' 36" N Lo. in 70 49' W

There is one more fact to explain. When the course is 45 or less (old compa.s.s reading) you read from the top of the page of Table 2 down. When the course is more than 45 (old compa.s.s 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 Lat.i.tude 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 compa.s.s reading). Take out the difference of Lat.i.tude and Departure and mark the difference of Lat.i.tude minutes ('). When the Lat.i.tude left and the difference of Lat.i.tude 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 Lat.i.tude, known as the Lat.i.tude in.

To find the new or Lo. in: Find the middle lat.i.tude by adding the lat.i.tude left to the lat.i.tude in and dividing by 2. With this middle lat.i.tude, enter Table 2. Seek for the departure in the difference of lat.i.tude 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 Lat.i.tude to the Lat.i.tude 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 compa.s.s NNW, 15 knots away. Ship heading South with a Deviation of 4 W. She sailed on the following courses:

--------+----+-------+---------+--------+------------------------------ Course |Wind| Leeway|Deviation|Distance| Remarks --------+----+-------+---------+--------+------------------------------ SE 3/4 E| NE | 1 pt. | 3 E | 30 |Variation throughout day 8 W.

S 11 W | NE | 0 | 6 E | 55 | A current set NE magnetic NNW | NE | 0 | 2 W | 14 | 1/2 mi. per hr. for the day.

S 87E | NE | 0 | 3 E | 50 | Required Lat. and Lo. in | | | | | and course and distance | | | | | made good.

----------------------------------------------------------------------------- C. Cos. |Wind|Leeway| Dev.| Var.| NEW | OLD |Dist.|Diff. Lat. |Departure | | | | |T. Cos.|T Cos. | +-----+-----+-----+---- | | | | | | | | N | S | E | W --------+----+------+-----+-----+-------+-------+-----+-----+-----+-----+---- SSE | .. | .. | 4 W| 8 W| 145 | S 35E| 15 | .. |12.3 | 8.6 | ..

SE 3/4 E| NE | 1 pt.| 3 E| 8 W| 133 | S 47E| 30 | .. |20.5 |21.9 | ..

S 11 W | NE | 0 | 6 E| 8 W| 189 | S 9W| 55 | .. |54.3 | .. | 8.6 NNW | NE | 0 | 2 W| 8 W| 327 | N 33W| 14 |11.7 |.. | .. | 7.6 S 87 E | NE | 0 | 3 E| 8 W| 88 | N 88E| 50 | 1.7 |.. |50 | ..

NE | .. | .. | mg | 8 W| 3 | N 3 E| 12 | 9.6 |.. | 7.2 | ..

--------+----+------+-----+-----+-------+-------+-----+-----+-----------+---- 23.0 |87.1 |87.7 |16.2 .. |23.0 |16.2 |..

.. |64.1 |71.5 |..

+-----+-----+---- .. S E