Chapter 3
The ship at A finds the light bearing NNW 2 points off her bow. At B, when the light bears NW and 4 points off, the log registers the distance from A to B 9 miles. 9 miles, then, will be the distance from the light itself when the ship is at B. The mathematical reason for this is that the distance run is one side of an isosceles triangle. Such triangles have their two sides of equal length. For that reason, the distance run is the distance off. Now the same fact holds true in running from B, which is 4 points off the bow, to C, which is 8 points off the bow, or directly abeam. The log shows the distance run between B and C is 6.3 miles. Hence, the ship is 6.3 miles from the light when directly abeam of it. This last 4 and 8 point bearing is what is known as the "bow and beam" bearing, and is the standard method used in coastwise navigation. Any one of these methods is of great value in fixing your position with relation to the land, when you are about to go to sea.
4. Sextant angles between three known objects.
This method is the most accurate of all. Because of its precision it is the one used by the Government in placing buoys, etc. Take three known objects such as A, B and C which are from 30 deg. to 60 deg. from each other.
With a sextant, read the angle from A to B and from B to C. Place a piece of transparent paper over the compass card and draw three lines from the center of the compass card to the circumference in such a way that the angles secured by the sextant will be formed by the three lines drawn. Now take this paper with the angles on it and fit it on the chart so that the three objects of which angles were taken will be intersected by the three lines on the paper. Where the point S is (in my diagram) will be the point of the ship's position at the time of sight. To secure greater accuracy the two angles should be taken at the same time by two observers.
5. Using a compass, log and lead when you are in a fog or unfamiliar waters.
Supposing that you are near land and want to fix your position but have no landmarks which you can recognize. Here is a method to help you out:
Take a piece of tracing paper and rule a vertical line on it. This will represent a meridian of longitude. Take casts of the lead at regular intervals, noting the time at which each is taken, and the distance logged between each two. The compass corrected for Variation and Deviation will show your course. Rule a line on the tracing paper in the direction of your course, using the vertical line as a N and S meridian. Measure off on the course line by the scale of miles in your chart, the distance run between casts and opposite each one note the time, depth ascertained and, if possible, nature of the bottom. Now lay this paper down on the chart which can be seen under it, in about the position you believe yourself in when you made the first cast. If your chain of soundings agrees with those on the chart, you are all right. If not, move the paper about, keeping the vertical line due N and S, till you find the place on the chart that does agree with you. That is your line of position. You will never find in that locality any other place where the chain of soundings are the same on the same course you are steaming. This is the only method by soundings that you can use in thick weather and it is an invaluable one.
Put in your Note-Book this diagram:
10 \ 8.30A.M. | 12 \ 9.00A.M. | 13 \ 10A.M. | 13-1/2 \ 10.30A.M. | 14 \ 11A.M. | 14-3/4 \ 11.30A.M. |
Assign for Night Work, Review for Weekly Examination to be held on Monday.
Add an explanation of the Deviation Card in Bowditch, page 41.
Put in your Note-Book:
Entering New York Harbor, ship heading W 3/4 N, Variation 9 deg. W. Observed by pelorus the following objects:
Buoy No. 1--ENE 1/4 E " " 2--E 1/2 N " " 3--NE 1/4 E " " 4--NW 1/4 N
Required true bearings of objects observed.
Answer:
From Deviation Card in Bowditch, p. 41, Deviation on W 3/4 N course is 5 deg. E. Hence, Compass Error is 5 deg. E (Dev.) + 9 deg. W (Var.) = 4 deg. W.
C. B. C. E. T. B. ENE 1/4 E 70 deg. 4 deg. W 66 deg. E 1/2 N 84 deg. 4 deg. W 80 deg. NE 1/4 E 48 deg. 4 deg. W 44 deg. NW 1/4 N 318 deg. 4 deg. W 314 deg.
WEEK II--DEAD RECKONING
TUESDAY LECTURE
LATITUDE AND LONGITUDE
We have been using the words Latitude and Longitude a good deal since this course began. Let us see just what the words mean. Before doing that, there are a few facts to keep in mind about the earth itself. The earth is a spheroid slightly flattened at the poles. The axis of the earth is a line running through the center of the earth and intersecting the surface of the earth at the poles. The equator is the great circle, formed by the intersection of the earth's surface with a plane perpendicular to the earth's axis and equidistant from the poles. Every point on the equator is, therefore, 90 deg. from each pole.
Meridians are great circles formed by the intersection with the earth's surface of planes perpendicular to the equator.
Parallels of latitude are small circles parallel to the equator.
The Latitude of a place on the surface of the earth is the arc of the meridian intercepted between the equator and that place. It is measured by the angle running from the equator to the center of the earth and back through the place in question. Latitude is reckoned from the equator (0 deg.) to the North Pole (90 deg.) and from the equator (0 deg.) to the South Pole (90 deg.). The difference of Latitude between any two places is the arc of the meridian intercepted between the parallels of Latitude of the places and is marked N or S according to the direction in which you steam (T n').
The Longitude of a place on the surface of the earth is the arc of the equator intercepted between the meridian of the place and the meridian at Greenwich, England, called the Prime Meridian. Longitude is reckoned East or West through 180 deg. from the Meridian at Greenwich. Difference of Longitude between any two places is the arc of the equator intercepted between their meridians, and is called East or West according to direction. Example: Diff. Lo. T and T' = E' M, and E or W according as to which way you go.
Departure is the actual linear distance measured on a parallel of Latitude between two meridians. Difference of Latitude is reckoned in minutes because miles and minutes of Latitude are always the same. Departure, however, is only reckoned in _miles_, because while a mile is equal to 1' of longitude on the equator, it is equal to more than 1' as the latitude increases; the reason being, of course, that the meridians of Lo. converge toward the pole, and the distance between the same two meridians grows less and less as you leave the equator and go toward either pole. Example: TN, N'n'. 10 mi. departure on the equator = 10' difference in Lo. 10 mi. departure in Lat. 55 deg. equals something like 18' difference in Lo.
The curved line which joins any two places on the earth's surface, cutting all the meridians at the same angle, is called the Rhumb Line. 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 latitude 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 latitude in which they can be used. It is well to remember, however, that since a mile or minute of latitude has a different value in every latitude, 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 latitude intersects the earth's axis. The degrees of latitude 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.
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 latitude 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 circumference. 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 assign for night work the following articles in Bowditch, part of which reading can be done immediately in the class 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 latitude and longitude of which you know, and sailed so many miles in a certain direction, in what latitude and longitude have you arrived?
If you sail due North or South, the problem is merely one of arithmetic. Suppose your position at noon today is Latitude 39 deg. 15' N, Longitude 40 deg. 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 deg.. This 5 deg. is called difference of latitude, and as you are in North latitude and going North, the difference of latitude, 5 deg., should be added to the latitude left, making your new position 44 deg. 15' N and your Longitude the same 40 deg. 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 deg. 30' N, Longitude 70 deg. 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 deg. W in the old or 202 deg. 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 deg. 30' N, Longitude 70 deg. 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 deg. 30' 00" 39 43 36 ----------- 2)80 13 36 ----------- 40 deg. 06' 48" Mid. Lat.
Take the nearest degree, i.e., 40 deg., as your answer. With this 40 deg. 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 deg. a departure of 18.7 miles is equivalent to 24' of difference of Longitude. We were in 70 deg. 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 deg. 25' W Diff. Lo. 24 ----------- Lo. in 70 deg. 49' W
The whole problem therefore would look like this:
Lat. left 40 deg. 30' N Lo. left 70 deg. 25' W Diff. Lat. 46 24 Diff. Lo. 24 ------------- ---------- Lat. in 39 deg. 43' 36" N Lo. in 70 deg. 49' W
There is one more fact to explain. When the course is 45 deg. or less (old compass reading) you read from the top of the page of Table 2 down. When the course is more than 45 deg. (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 deg. (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 deg. 46' N, Lo. 74 deg. 06' W, bearing by compass NNW, 15 knots away. Ship heading South with a Deviation of 4 deg. W. She sailed on the following courses:
+----+-------+---------+--------+------------------------------ Course |Wind| Leeway|Deviation|Distance| Remarks --------+----+-------+---------+--------+------------------------------ SE 3/4 E| NE | 1 pt. | 3 deg. E | 30 |Variation throughout day 8 deg. W. S 11 deg. W | NE | 0 | 6 deg. E | 55 | A current set NE magnetic NNW | NE | 0 | 2 deg. W | 14 | 1/2 mi. per hr. for the day. S 87 deg.E | NE | 0 | 3 deg. 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 deg. W| 8 deg. W| 145 deg. | S 35 deg.E| 15 | .. |12.3 | 8.6 | .. SE 3/4 E| NE | 1 pt.| 3 deg. E| 8 deg. W| 133 deg. | S 47 deg.E| 30 | .. |20.5 |21.9 | .. S 11 deg. W | NE | 0 | 6 deg. E| 8 deg. W| 189 deg. | S 9 deg.W| 55 | .. |54.3 | .. | 8.6 NNW | NE | 0 | 2 deg. W| 8 deg. W| 327 deg. | N 33 deg.W| 14 |11.7 |.. | .. | 7.6 S 87 deg. E | NE | 0 | 3 deg. E| 8 deg. W| 88 deg. | N 88 deg.E| 50 | 1.7 |.. |50 | .. NE | .. | .. | mg | 8 deg. W| 3 deg. | N 3 deg. E| 12 | 9.6 |.. | 7.2 | .. --------+----+------+-----+-----+-------+-------+-----+-----+-----------+---- 23.0 |87.1 |87.7 |16.2 .. |23.0 |16.2 |.. +-----+-----+---- .. |.. |.. |.. .. |64.1 |71.5 |.. +-----+-----+---- .. S E
Lat. left 39 deg.-46'-00" N Mid. Lat. 39 deg. Diff. Lat. 1 -04 -06 S Dep. 71.5 ---------- Lat. in. 38 -41 -54 N Table 2--Under 39 deg. Dep. in 39 -46 -00 Diff. Lat. col. = 92' = 1 deg. 32' Diff. Lo. ---------- 2)78 -27 -54 ---------- Mid. Lat. 39 -13 -57
Lo. left 74 deg.-06'-00" W Diff. Lo. 1 -32 -00 E -------------- Lo. in. 72 deg.-34'-00" W
Table 2--Diff. Lat. 64.1, Dep. 71.5. Course S 48 deg. E--Distance 96 miles.
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.