Chapter II: Part 2

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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.

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:

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

Find algebraically the real difference of latitude, meridional difference of latitude and the difference of longitude. Reduce real difference of latitude and difference of longitude to minutes. Take log of the difference of longitude (Table 42) and add 10. From this log subtract the log of difference of meridional parts. The result will be the log tan of the True Course, which find in Table 44. On the same page find the log sec of true course. Add to this the log of the real difference of latitude, and if the result is more than 10, subtract 10. This result will be the log of the distance sailed. This method should be used only when steaming approximately a North and South course.

Note.--For detailed explanation of Tables 42 and 44 see Bowditch, pp. 271-276.

Assign for Night Reading Arts, in Bowditch: 183-184-185-186-187-188-189-194-259-260-261-262-263-264-265-266-267-268.

Also, one of the examples of Mercator sailing to be done by both the Inspection and Logarithmic method.

SATURDAY LECTURE

GREAT CIRCLE SAILING--THE CHRONOMETER

In Tuesday's Lecture of this week, I explained how a Great Circle track was laid down on one of the Great Circle Sailing Charts which are prepared by the Hydrographic Office.

Supposing, however, you do not have these charts on hand. There is an easy way to construct a great circle track yourself. Turn to Art. 194, page 82, in Bowditch. Here is a table with an explanation as to how to use it. Take, for instance, the same two points between which you just drew a line on the great circle track. Find the center of this line and the latitude of that point. At this point draw a line perpendicular to the course to be sailed, the other end of which must intersect the corresponding parallel of latitude given in the table. With this point as the center of a circle, sweep an arc which will intersect the point left and the point sought. This arc will be the great circle track to follow.

To find the courses to be sailed, get the difference between the course at starting and that at the middle of the circle, and find how many quarter points are contained in it. Now divide the distance from the starting point to the middle of the circle by the number of quarter points. That will give the number of miles to sail on each quarter point course. See this illustration:

Difference between ENE and E = 2 pts. = 8 quarter points. Say distance is 1600 miles measured by dividers or secured by Mercator Sailing Method. Divide 1600 by 8 = 200. Every 200 miles you should change your course 1/4 point East.

_The Chronometer_

The chronometer is nothing more than a very finely regulated clock. With it we ascertain Greenwich Mean Time, i.e., the mean time at Greenwich Observatory, England. Just what the words "Greenwich Mean Time" signify, will be explained in more detail later on. What you should remember here is that practically every method of finding your exact position at sea is dependent upon knowing Greenwich Mean Time, and the only way to find it is by means of the chronometer.

It is essential to keep the chronometer as quiet as possible. For that reason, when you take an observation you will probably note the time by your watch. Just before taking the observation, you will compare your watch with the chronometer to notice the exact difference between the two. When you take your observation, note the watch time, apply the difference between the chronometer and watch, and the result will be the CT.

For instance, suppose the chronometer read 3h 25m 10s, and your watch, at the same instant, read 1h 10m 5s. C--W would be:

3h -- 25m -- 10s
-- 1 -- 10 -- 05
----------------
2h -- 15m -- 05s

Now suppose you took an observation which, according to your watch, was at 2h 10m 05s. What would be the corresponding C T? It would be

WT 2h -- 10m -- 05s
C -- W 2 -- 15 -- 05
--------------------
CT 4h -- 25m -- 10s

If the chronometer time is less than the W T add 12 hours to the C T, so that it will always be the larger and so that the amount to be added to W T will always be +. For instance, CT 1h--25m--45s, WT 4h--13m--25s, what is the C-W?

CT 13h--25m--45s
WT 4 --13 --25
----------------
C--W 9h--12m--20s

Now, suppose an observation was taken at 6h 13m 25s according to watch time. What would be the corresponding CT?

WT 6h--13m--25s
C--W 9 --12 --20
----------------
15h--25m--45s
--12
----------------
CT 3h--25m--45s

Put in your Note-Book: CT = WT + C - W.

If, in finding C-W, C is less than W, add 12 hours to C, subtracting same after CT is secured.

Example No. 1:

CT 3h--25m--10s
WT 1 --10 --05
----------------
C--W 2h--15m--05s

WT 2h--10m--05s
+ C--W 2 --15 --05
----------------
CT 4h--25m--10s

Example No. 2:

CT 1h--25m--45s
WT 4h--13m--25s

(+12 hrs.) CT 13h--25m--45s
WT 4 --13 --25
----------------
C-W 9h--12m--20s

WT 6h--13m--25s
+ C-W 9 --12 --20
----------------
15h--25m--45s
(-12 hrs.) 12
----------------
CT 3h--25m--45s

There is one more very important fact to know about the chronometer. It is physically impossible to keep it absolutely accurate over a long period of time. Instead of continually fussing with its adjustment and hands, the daily rate of error is ascertained, and from this the exact time for any given day. It is an invariable practice among good mariners to _leave the chronometer alone_. When you are in port, you can find out from a time ball or from some chronometer maker what your error is. With this in mind, you can apply the new correction from day to day. Here is an example (Put in your Note-Book):

On June 1st, CT 7h--20m--15s, CC 2m--40s fast. On June 16th, (same CT) CC 1m--30s fast. What was the corresponding G.M.T. on June 10th?

June 1st 2m--40s fast
16th 1m--30s fast
----------------
1m--10s
60
--
60
10
--
15) 70s (4.6 sec. Daily Rate of error losing

June 1st-10th, 9 days times 4.6 sec. = 41 sec. losing
June 1st 2m--40s fast
June 10th 41s losing
---------
June 10th 1m--59s fast

CT 7h--20m--15s
CC -- 1 --59
------------
G.M.T. 7h--18m--16s on June 10th

If CC is fast, subtract from CT
If CC is slow, add to CT

WEEK III--CELESTIAL NAVIGATION

TUESDAY LECTURE

CELESTIAL CO-ORDINATES, EQUINOCTIAL SYSTEM, ETC.

We have already discussed the way in which the earth is divided so as to aid us in finding our position at sea, i.e., with an equator, parallels of latitude, meridians of longitude starting at the Greenwich meridian, etc. We now take up the way in which the celestial sphere is correspondingly divided and also simple explanations of some of the more important terms used in Celestial Navigation.

As you stand on any point of the earth and look up, the heavenly bodies appear as though they were situated upon the surface of a vast hollow sphere, of which your eye is the center. Of course this apparent concave vault has no existence and we cannot accurately measure the distance of the heavenly bodies from us or from each other. We can, however, measure the direction of some of these bodies and that information is of tremendous value to us in helping us to fix our position.

Now we could use our eye as the center of the celestial sphere but more accurate than that is to use the center of the earth. Suppose we do use the center of the earth as the place from which to observe these celestial bodies and, in imagination, transfer our eye there. Then we will find projected on the celestial sphere not only the heavenly bodies but the imaginary points and circles of the earth's surface. Parallels of latitude, meridians of longitude, the equator, etc., will have the same imaginary position on the celestial sphere that they have on the earth. Your actual position on the earth will be projected in a point called your zenith, i.e., the point directly overhead.

From this we get the definition that the Zenith of an observer on the earth's surface is the point in the celestial sphere directly overhead.

It would be a simple matter to fix your position if your position never changed. But it is always changing with relation to these celestial bodies. First, the earth is revolving on its own axis. Second, the earth is moving in an elliptic track around the sun, and third, certain celestial bodies themselves are moving in a track of their own. The changes produced by the daily rotation of the earth on its axis are different for observers at different points on the earth and, therefore, depend upon the latitude and longitude of the observer. But the changes arising from the earth's motion in its orbit and the motion of various celestial bodies in their orbits, are true no matter on what point of the earth you happen to be. These changes, therefore, in their relation to the center of the earth, may be accurately gauged at any instant. To this end the facts necessary for any calculation have been collected and are available in the Nautical Almanac, which we will take up in more detail later.

Now with these facts in mind, let us explain in simple words the meaning of some of the terms you will have to become acquainted with in Celestial Navigation.

In the illustration (Bowditch p. 88) the earth is supposed to be projected upon the celestial sphere N E S W. The Zenith of the observer is projected at Z and the pole of the earth which is above the horizon is projected at P. The other pole is not given.

The Celestial Equator is marked here E Q W and like all other points and lines previously mentioned, it is the projection of the Equator until it intersects the celestial sphere. Another name for the Celestial Equator is the Equinoctial.

All celestial meridians of longitude corresponding to longitude meridians on the earth are perpendicular to the equinoctial and likewise P S, the meridian of the observer, since it passes through the observer's zenith at Z, is formed by the extension of the earth's meridian of the observer and hence intersects the horizon at its N and S points. This makes clear again just what is the meridian of the observer. It is the meridian of longitude which passes through the N and S poles and the observer's zenith. In other words, when the sun or any other heavenly body is on your meridian, a line stretched due N and S, intersecting the N and S poles, will pass through your zenith and the center of the sun or other celestial body. To understand this is important, for no sight with the sextant is of value except with relation to your meridian.

The Declination of any point in the celestial sphere is its distance in arc, North or South of the celestial equator, i.e., N or S of the Equinoctial.

North declinations, i.e., declinations north of the equinoctial are always marked, +; those south of the equinoctial, -. For instance, in the Nautical Almanac, you will never see a declination of the sun or other celestial body marked, N 18 deg. 28' 30". It will always be marked +18 deg. 28' 30" and a south declination will be marked -18 deg. 28' 30". Another fact to remember is that Declination on the celestial sphere corresponds to latitude on the earth. If, for instance, the Sun's declination is +18 deg. 28' 30" at noon, Greenwich, then at that instant, i.e., noon at Greenwich, the sun will be directly overhead a point on earth which is in latitude N 18 deg. 28' 30".

The Polar Distance of any point is its distance in arc from either pole. It must, therefore, equal 90 deg. minus the declination, if measured from the pole of the same name as the declination or 90 deg. plus the declination if measured from the pole of the opposite name.

P M is the polar distance of M from P, or P B the polar distance of B from P.

The true altitude of a celestial body is its angular height from the true horizon.

The zenith distance of any point or celestial body is its angular distance from the zenith of the observer.

The Ecliptic is the great circle representing the path in which the sun appears to move in the celestial sphere. As a matter of fact, you know that the earth moves around the sun, but as you observe the sun from some spot on the earth, it appears to move around the earth. This apparent track is called the Ecliptic as stated before, and in the illustration the Ecliptic is represented by the curved line, C V T. The plane of the Ecliptic is inclined to that of the Equinoctial at an angle of 23 deg. 27-1/2', and this inclination is called the obliquity of the Ecliptic.

The Equinoxes are those points at which the Ecliptic and Equinoctial intersect, and when the sun occupies either of these two positions, the days and nights are of equal length. The Vernal Equinox is that one which the sun passes through or intersects in going from S to N declination, and the Autumnal Equinox that which it passes through or intersects in going from N to S declination. The Vernal Equinox (V in the illustration) is also designated as the First Point of Aries which is of use in reckoning star time and will be mentioned in more detail later.

The Solstitial Points, or Solstices, are points of the Ecliptic at a distance of 90 deg. from the Equinoxes, at which the sun attains its highest declination in each hemisphere. They are called the Summer and Winter Solstice according to the season in which the sun appears to pass these points in its path.

To sum up: The way to find any point on the earth is to find the distance of this point N or S of the equator (i.e., its Latitude) and its distance E or W of the meridian at Greenwich (i.e., its longitude). In the celestial sphere, the way to find the location of a point or celestial body such as the sun is to find its declination (i.e., distance in arc N or S of the equator) and its hour angle. By hour angle, I mean the distance in time from your meridian to the meridian of the point or celestial body in question.

Assign for Night reading, Arts, in Bowditch: 270-271-272-273-274-275-277-278-279-280-282-283-284.

WEDNESDAY LECTURE

TIME BY THE SUN--MEAN TIME, SOLAR TIME, CONVERSION, ETC.

There is nothing more important in all Navigation than the subject of Time. Every calculation for determining the position of your ship at sea must take into consideration some kind of time. Put in your Note-Book:

There are three kinds of time:

1. Apparent or solar time, i.e., time by the sun.

2. Mean Time, i.e., clock time.

3. Sidereal Time, or time by the stars.

So far as this lecture is concerned, we will omit any mention of sidereal time, i.e., time by the stars. We will devote this morning to sun time, i.e., apparent time, and mean time.

Apparent or Solar Time is, as stated before, nothing more than sun time or time by the sun. The hour angle of the center of the sun is the measure of apparent or solar time. An apparent or solar day is the interval of time it takes for the earth to revolve completely around on its axis every 24 hours. It is apparent noon at the place where you are when the center of the sun is directly on your meridian, i.e., on the meridian of longitude which runs through the North and South poles and also intersects your zenith. This is the most natural and the most accurate measure of time for the navigator at sea and the unit of time adopted by the mariner is the apparent solar day. Apparent noon is the time when the latitude of your position can be most easily and most exactly determined and on the latitude by observation just secured we can get data which will be of great value to us for longitude sights taken later in the day.

Now it would be very easy for the mariner if he could measure apparent time directly so that his clock or other instrument would always tell him just what the sun time was. It is impossible, however, to do this because the earth does not revolve at a uniform rate of speed. Consequently the sun is sometimes a little ahead and sometimes a little behind any average time. You cannot manufacture a clock which will run that way because the hours of a clock must be all of exactly the same length and it must make noon at precisely 12 o'clock every day. Hence we distinguish clock time from sun time by calling clock time, mean (or average) time and sun time, apparent or solar time. From this explanation you are ready to understand such expressions as Local Mean Time, which, in untechnical language, signifies clock time at the place where you are; Greenwich Mean Time which signifies clock time at Greenwich; Local Apparent Time, which signifies sun time at the place where you are; Greenwich Apparent Time, which signifies sun time at Greenwich.

Now the difference between apparent time and mean time can be found for any minute of the day by reference to the Nautical Almanac which we will take up later in more detail. This difference is called the Equation of Time.

There is one more fact to remember in regard to apparent and mean time. It is the relation of the sun's hour angle to apparent time. In the first place, what is a definition of the sun's HA? It is the angle at the celestial pole between the meridian intersecting any given point and the meridian intersecting the center of the sun. It is measured by the arc of the celestial equator intersected between the meridian of any point and the meridian intersecting the center of the sun.

For instance, in the above diagram, suppose PG is the meridian at Greenwich, and PS the meridian intersecting the sun. Then the angle at the pole GPS, measured by the arc GS would be the Hour Angle of Greenwich, or the Greenwich Hour Angle. And now you notice that this angular measure is exactly the same as apparent time at Greenwich or Greenwich Apparent Time, for Greenwich Apparent Time is nothing more than the distance in time Greenwich, England, or the meridian at Greenwich is from the sun, i.e., the time it takes the earth to revolve from Greenwich to the sun; and that distance is exactly measured by the Greenwich Hour Angle or the arc on the celestial equator, GS.

The same is correspondingly true of Local Apparent Time and the ship's Hour Angle. Suppose, for instance, PL is the meridian intersecting the place where your ship is. Then your ship's hour angle would be the angle at the pole intersecting the meridian of your ship and the meridian of the sun or LPS and measured by the arc LS. And you will note that this distance is exactly the same as apparent time at the ship, for Apparent Time at ship is nothing more than the distance in time which the ship is from the sun. We can sum up all this information in a few simple rules, which put in your Note-Book:

Mean Time = Clock Time.

G.M.T. = Greenwich Mean Time.

L.M.T. = Local Mean Time.

Apparent Time = Actual or Sun Time.

G.A.T. (G.H.A.) = Greenwich Apparent Time or Greenwich Hour Angle.

L.A.T. (S.H.A.) = Local Apparent Time or Ship's Hour Angle.

Difference between apparent and mean time or mean and apparent time--Equation of Time.

Right under this in your Note-Book put the following diagram, which I will explain:

You will see from this diagram that civil time commences at midnight and runs through 12 hours to noon. It then commences again and runs through 12 hours to midnight. The Civil Day, then, is from midnight to midnight, divided into two periods of 12 hours each.

The astronomical day commences at noon of the civil day of the same date. It comprises 24 hours, reckoned from O to 24, from noon of one day to noon of the next. Astronomical time, either apparent or mean, is the hour angle of the true or mean sun respectively, measured to the westward throughout its entire daily circuit.

Since the civil day begins 12 hours before the astronomical day and ends 12 hours before it, A.M. of a new civil day is P.M. of the astronomical day preceding. For instance, 6 hours A.M., April 15th civil time is equivalent to 18 hours April 14th, astronomical time.

Now, all astronomical calculations in which time is a necessary fact to be known, must be expressed in astronomical time. As chronometers have their face marked only from 0 to 12 as in the case of an ordinary watch, it is necessary to transpose this watch or chronometer time into astronomical time. No transposing is necessary if the time is P.M., as you can see from the diagram that both civil and astronomical times up to 12 P.M. are the same. But in A.M. time, such transposing is necessary. Put in your Note-Book:

Whenever local or chronometer time is A.M., deduct 12 hours from such time to get the correct astronomical time:

CT 15d-- 9h--10m--30s A.M.
--12
------------------------
CT 14d--21h--10m--30s
------

L.M.T. 10d-- 4h--40m--16s A.M.
--12
------------------------
L.M.T. 9d--16h--40m--16s

Now we come to a very important application of time. You will remember that in one of the former lectures we stated that to find our latitude, we had to find how far North or South of the equator we were, and to find our longitude, we had to find how far East or West of the meridian at Greenwich we were. Never mind about latitude for the present. We can find our longitude exactly if we know our Greenwich time and our time at ship. For instance, in the accompanying diagram:

Suppose PG is the meridian at Greenwich, then anything to the west of PG is West longitude and anything to the East of PG is East longitude. Now suppose GPS is the H.A. of G. or G.A.T.--i.e., the distance in time G. is from the sun. And L P S is the H.A. of the ship or L.A.T.--i.e., the distance in time the ship is from the sun. Then the difference between G P S and L P S is G P L, measured by the arc L G, and that is the difference that the ship, represented by its meridian PL, is from the Greenwich meridian PG. In other words, that is the ship's longitude for, as mentioned before, longitude is the distance East or West of Greenwich that any point is, measured on the arc of the celestial equator. The longitude is West, for you can see LPG or the arc LG is west of the meridian PG.

Likewise if P E is the meridian of your ship, the Longitude in time is the S.H.A. or L.A.T., E P S (the distance your ship is from the sun) less the G.H.A. or G.A.T., G P S (the distance Greenwich is from the sun) which is the angle G P E measured by the arc G E. And this Longitude is East for you can see G P E, measured by G E, is east of the Greenwich meridian, P G.

In both these cases, however, the longitude is expressed in time, i.e., so many hours, minutes and seconds from the Greenwich meridian and we wish to express this distance in degrees, minutes and seconds of arc. The earth describes a circle of 360 deg. every 24 hours. Then if you are 1 hour from Greenwich, you are 1/24 of 360 deg. or 15 deg. from Greenwich and if you are 12 hours from Greenwich, you are 1/2 of 360 deg. or 180 deg. from Greenwich. By keeping this in mind, you should be able to transpose time into degrees, minutes and seconds of arc for any fraction of time. It is, however, all worked out in Table 7 of Bowditch which turn to. (Note to Instructor: Explain this table carefully). Put in your Note-Book:

89 deg. 24' 26" = (89 deg.) 5h--56m
(24') 1m--36s
(26") 1--44/60s
--------------------
5h--57m--37s 44/60s = 38s

4h--42m--26s
4h--40m = 70 deg.
2m--24s = 36'
2s = 30"
-----------
70 deg. 36' 30"

Also put in your Note-Book this diagram and these formulas: (For diagram use illustration on p. 40.)

L.M.T. + West Lo. = G.M.T. L.A.T. + West Lo. = G.A.T. L.M.T. - East Lo. = G.M.T. L.A.T. - East Lo. = G.A.T.

G.M.T. - West Lo. = L.M.T. G.A.T. - West Lo. = L.A.T. G.M.T. + East Lo. = L.M.T. G.A.T. + East Lo. = L.A.T.

If G.M.T. or G.A.T. is greater than L.M.T. or L.A.T. respectively, Lo. is West.

If G.M.T. or G.A.T. is less than L.M.T. or L.A.T. respectively, Lo. is East.

Example:

In longitude 81 deg. 15' W, L.M.T. is April 15d--10h--17m--30s A.M. What is G.M.T.?

L.M.T. 15d--10h--17m--30s A.M.
--12
------------------
L.M.T. 14d--22h--17m--30s
5 --25 W +
------------------
G.M.T. 15d-- 3h--42m--30s
--------

G.M.T. April 15d-- 3h--42m--30s
L.M.T. April 15d--10h--17m--30s A.M.

In what Lo. is ship?

G.M.T. 15d 3h--42m--30s
L.M.T. 14d 22h--17m--30s
------------------
Lo. in T 5h--25m--00s W

Lo. = 81 deg. 15'W

Assign also for Night Work reading the following articles in Bowditch: 276-278-279-226-228-286-287-288-290-291-294 (omitting everything on page 114.)

THURSDAY LECTURE

SIDEREAL TIME--RIGHT ASCENSION

Our last lecture was devoted to a discussion of sun time. Today we are going to talk about star time, or, using the more common words, sidereal time.

Now, just one word of review. You remember that we have learned that astronomical time is reckoned from noon of one day to noon of the next and hence the astronomical day corresponds to the 24 hours of a ship's run. The hours are counted from 0 to 24, so that 10 o'clock in the morning of October 25th is astronomically October 24th, 22 hours or 22 o'clock of October 24th.

Now Right Ascension is different from both astronomical and civil time. Right Ascension is practically celestial longitude. For instance, the position of a place on the earth is fixed by its latitude and longitude; the position of a heavenly body is fixed by its declination and right ascension. But Right Ascension is not measured in degrees and minutes nor is it measured East and West. It is reckoned in hours and minutes all the way around the sky, eastward from a certain point, through the approximate 24 hours. The point from which this celestial longitude begins is not at Greenwich, but the point where the celestial equator intersects the ecliptic in the spring of the year, i.e., the point where the sun, coming North in the Spring, crosses the celestial equator. This point is called the First Point of Aries. You will frequently hear me speak of a star having, for instance, a Right Ascension of 5h 16m 32s. I mean by that, that starting at the celestial meridian, i.e., the meridian passing through the First Point of Aries, it will take a spot on the earth 5h 16m 32s to travel until it reaches the meridian of the star in question.

Roughly speaking then, just as Greenwich Apparent Time means the distance East or West the Greenwich meridian is from the sun and Local Apparent Time means the distance East or West your ship is from the sun, so R.A.M.G. means the distance in time the Meridian of Greenwich is from the First Point of Aries, measured eastward in a circle. And this distance is the same as Greenwich Sidereal Time, i.e., Sidereal Time at Greenwich or the distance in time the meridian of Greenwich is from the First Point of Aries.

Now, what is the star time that corresponds to local time? It is called the Right Ascension of the Meridian, which means the R. A. of the meridian which intersects your zenith. Just as L.A.T. is the distance in time your meridian is from the sun, so Local Sidereal Time is the R. A. of your meridian, i.e., the distance in time your meridian is from the First Point of Aries. Put in your Note-Book:

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Lectures in NavigationChapter II: Part 2

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