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
[Illustration]
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.
[Illustration]
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?
[Illustration]
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:
[Illustration]
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.
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.
THURSDAY LECTURE
EXAMPLES ON PLANE AND TRAVERSE SAILING (_Continued_)
1. Departure taken from Cape Horn. Lat. 55 deg. 58' 41" S, Lo. 67 deg. 16' 15"
W, bearing by compass SSW 20 knots. Ship heading SW x S, Deviation 4 deg. E,
steamed the following courses:
---------+----------+------------+-----------+----------
C. Cos. | Wind | Leeway | Deviation | Distance
---------+----------+------------+-----------+----------
SW x S | SE | 1 pt. | 4 deg. E | 40
WNW | N | 2 pts. | 5 deg. E | 25
S 40 deg. E | NE | 2 pts. | 4 deg. W | 20
---------+----------+------------+-----------+----------
_Remarks_
Variation 18 deg. E throughout. Current set NW magnetic 30 mi. for the day.
Required Latitude and Longitude in and course and distance made good.
2. Departure taken from St. Agnes Lighthouse, Scilly Islands, Lat. 49 deg.
53' S, Lo. 6 deg. 20' W, bearing by compass E x S, distance 18 knots,
Deviation 10 deg. W, Variation 23 deg. W. Ship headed N steamed on the following
courses:
-------+------+-------+-------+------+------------------------------
C. Cos.| Wind |Leeway |Devia- |Dis- | Remarks
| | |tion |tance |
-------+------+-------+-------+------+------------------------------
N | | .. | 10 deg. W | 60 |Variation 23 deg. W. Current set
S 1/2 E| W | 3 pts.| 10 deg. E | 40 |SE mg 1-1/2 miles for 24 hrs.
NNE | NNW | 2 pts.| 8 deg. W | 45 |Req. Lat. and Lo. in and
| | | | |course and distance made
| | | | |good,
-------+------+-------+-------+------+------------------------------
Assign for Night Work the following articles in Bowditch:
179-180-181-182. Also additional problems in Dead Reckoning.
FRIDAY LECTURE
MERCATOR SAILING
This is a method to find the true course and distance between two
points. The method can be used in two ways, i.e., by the use of Tables 2
and 3 (called the inspection method) and by the use of logarithms. The
first method is the quicker and will do for short distances. The second
method, however, is more accurate in all cases, and particularly where
the distances are great. The inspection method is as follows (Put in
your Note-Book):
Find the algebraic difference between the meridional parts corresponding
to the Lat. in and Lat. sought by Table 3. Call this Meridional
difference of Latitude. Find the algebraic difference between Longitude
in and Longitude sought and call this difference of Longitude. With the
Meridional difference of Latitude and the difference of Longitude, find
the course by searching in Table 2 for the page where they stand
opposite each other in the latitude and departure columns. Now find the
real difference of latitude. Under the course just found and opposite
the _real_ difference of Latitude, will be found the distance sailed in
the distance column. Example:
What is the course and distance from Lat. 40 deg. 28' N, Lo. 73 deg. 50' W, to
Lat. 39 deg. 51' N, Lo. 72 deg. 45' W?
Lat. in 40 deg. 28' N Meridional pts. 2644.2
Lat. sought 39 51 N Meridional pts. 2596.0
--------- ------
0 deg. 37' Mer. diff. Lat. 48.2
Lo. in 73 deg. 50' W
Lo. sought 72 45 W
---------
1 deg. 05' = 65'
On page 604 Bowditch you will find 48.7 and 64.7 opposite each other,
and as 48.7 is in the Lat. column only when you read from the bottom,
the course is S 53 deg. E. The real difference of Lat. under this course is
opposite 62 in the distance column. Hence the distance to be sailed is
62 miles.
If distances are too great, divide meridional difference of Lat., real
difference of Latitude and difference of Longitude by 10 or any other
number to bring them within the scope of the distances in Table 2. When
distance to be sailed is found, it must be multiplied by the same
number. For instance, if the difference of Lat., difference of Lo.,
etc., are divided by 10 to bring them in the scope of Table 2, and with
these figures 219 is the distance found, the real distance would be 10
times 219 or 2190.
Now let us work out the same problem by logarithms. This will acquaint
us with two new Tables, i.e., Tables 42 and 44. Put this in your
Note-Book:
Lat. in 40 deg. 28' N Mer. pts. 2644.2 Lo. in 73 deg. 50'
Lat. sought 39 51 Mer. pts. 2596.0 Lo. sought 72 45
--------- ------ -------
Real diff. 0 deg. 37' 48.2 1 deg. 05'
60
--
60
5
--
(Table 42) log (+ 10) 65 = 11.81291
Log 48.2 = 1.68305
--------
Log tan TC (Table 44) 10.12986
TC = S 53 deg. 26' E
Log sec TC (53 deg. 26') = 10.22493
Log real diff. Lat. = 1.56820 +
--------
11.79313
- 10.
--------
1.79313
Distance (Table 42) = 62.11 miles