Lectures in Navigation
by Ernest Gallaudet Draper
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[Transcriber's Note:- Words and phrases surrounded by underscores are italicised. There are four symbols which are denoted by :- (.) Circle surrounding a centred dot - Right Ascension of the Meridian; (+) Circle surrounding a cross - Earth's Central Progress; (_) Circle with centre dot and line under - Observed altitude of sun's lower limb; -(-)- Circle with line through - True Altitude. ]


Prepared for Use as a Text Book

at the



Lieutenant ERNEST G. DRAPER, U.S.N.R.F.

Head of the Department of Navigation

Officers' Material School, Naval Auxiliary Reserve



These Lectures have been compiled as speedily as possible to meet the demand for some quick but fairly comprehensive method whereby large bodies of men, divided into small classes, might learn the elements of Navigation and thus assume, without delay, their responsibilities as Junior Officers of the deck, Navigators and Assistant Navigators in the United States Naval Auxiliary Reserve.

I realize that the haste with which the book has been written is apparent in many places, and it is hoped that many evidences of this haste will disappear in case further editions are printed. Besides acknowledging the help and information which was secured from the list of navigational works, mentioned on another page, I wish to mention particularly Prof. Charles Lane Poor's book, entitled "Nautical Science," from which was secured practically all of the information in the Lecture on Planets and Stars (Tuesday—Week V); Commander W. C. P. Muir's book, "Navigation and Compass Deviations," and Lieutenant W. J. Henderson's book, "Elements of Navigation," the text of which was followed closely in discussing Variation and Deviation and Traverse Sailing.

I desire to express my gratitude to Lieutenant Commander R. T. Merrill, 2nd, U. S. N., for suggesting a detailed outline of the whole course; to Lieutenant Commander B. O. Wills, U. S. N., for his valuable criticisms and almost daily help during the preparation of these Lectures; to Lieutenant (j. g.) C. D. Draper, U. S. N. R. F.; Lieutenant (j. g.) R. Brush, U. S. N. R. F., and Lieutenant (j. g.) P. C. McPherson, U. S. N. R. F., for many criticisms and suggestions; and to Captain Huntington, Seamen's Church Institute, for suggesting helpful diagrams, particularly the one on page 44. This opportunity is also taken for thanking the many Instructors in the School for their opinions on various questions that have come up in connection with the course and for assistance in eliminating errors from the text.

E. G. D.














These lectures have been written with the idea of explaining, in as simple language as possible, the fundamental elements of Navigation as set forth in Bowditch's American Practical Navigator. They will be given you during the time at the Training School devoted to this subject. At present this time includes two morning periods of one and a half hours each, separated by a recess of fifteen minutes. In general the plan is to devote the first period to the lecture and the second period to practical work.

Not many examples for practical work have been included in this book, but one example, illustrating each new method, has been worked out. If you understand these examples you should be able to understand others similar to them.

Toward the end of the course a portion of each second period will be devoted to handling the sextant, work with charts, taking sights, etc. In short, every effort will be made to duplicate, as nearly as possible, navigating conditions on board a modern merchant ship.


Officers' Material School,

Naval Auxiliary Reserve



Tuesday—The Compass 1

Wednesday—Pelorus; Parallel Rulers; The Lead, Sounding Machine, Dividers and Log 6

Thursday—The Chart 10

Friday—The Protractor and Sextant 13

Saturday—Fixes, Angles by Bearings and Sextant 16


Tuesday—Latitude and Longitude 20

Wednesday—Useful Tables—Plane and Traverse Sailing 23

Thursday—Examples on Plane and Traverse Sailing (Continued) 27

Friday—Mercator Sailing 28

Saturday—Great Circle Sailing—The Chronometer 30


Tuesday—Celestial Co-ordinates, Equinoctial System, etc. 34

Wednesday—Time by the Sun—Mean Time, Solar Time, Conversion, etc. 36

Thursday—Sidereal Time—Right Ascension 43

Friday—The Nautical Almanac 47

Saturday—Correction of Observed Altitudes 52


Tuesday—The Line of Position 55

Wednesday—Latitude by Meridian Altitude 58

Thursday—Azimuths of the Sun 61

Friday—-Marc St. Hilaire Method by a Sun Sight 63

Saturday—Examples on Marc St. Hilaire Method by a Sun Sight 66


Tuesday—A Short Talk on the Planets and Stars—Identification of Stars—Time of Meridian Passage of a Star 66

Wednesday—Latitude by Meridian Altitude of a Star—Latitude by Polaris 73

Thursday—Marc St. Hilaire Method by a Star Sight 74

Friday—Examples: Latitude by Meridian Altitude of a Star; Latitude by Polaris; Marc St. Hilaire Method by a Star Sight 75

Saturday—Longitude by Chronometer Sight of the Sun 76


Tuesday—Longitude by Chronometer Sight of a Star 79

Wednesday—Examples on Longitude by Chronometer Sight of a Star 80

Thursday—Latitude by Ex-Meridian Altitude of the Sun 81

Friday—Examples: Latitude by Ex-Meridian Altitude of the Sun 83

Saturday—Finding the Watch Time of Local Apparent Noon 83


Tuesday—Compass Error by an Azimuth 88

Wednesday—Correcting Longitude by a Factor 89

Thursday—The Navigator's Routine—A Day's Work at Sea 91

Friday—Day's Work 105

Saturday—Day's Work 105


Monday to Thursday—Day's Work 107-108

Additional Lecture—Compass Adjustment 109




Everyone is supposed to know what a compass looks like. It is marked in two ways—the old way and the new way. Put in your Note-Book this diagram:

The new way marked on the outside of the diagram, starts at North with 0 deg., increases toward the right through East at 90 deg., South at 180 deg., West at 270 deg. and back to North again at 360 deg. or 0 deg..

The old way, marked on the inside of the diagram, starts at North with 0 deg., goes to the right to 90 deg. at East and to the left to 90 deg. at West. It also starts at South with 0 deg., goes to the right to East at 90 deg. and to the left to West at 90 deg..

A Compass Course can be named in degrees, according to either the new or old way. For instance, the new way is just 45 deg.. The old way for the same course is N 45 deg. E. New way—100 deg.. Old way for same course—S 80 deg. E.

There is another way to name a compass course. It is by using the name of the point toward which the ship is heading. On every ship the compass is placed with the lubber line (a vertical black line on the compass bowl) vertical and in the keel line of the ship. The lubber line, therefore, will always represent the bow of the ship, and the point on the compass card nearest the lubber line will be the point toward which the ship is heading.

The compass card of 360 deg. is divided into 32 points. Each point, therefore, represents 11-1/4 deg.. The four principal points are called cardinal points. They are—North, East, South, West. Each cardinal point is 90 deg. from the one immediately adjacent to it. It is also 8 points from the one adjacent to it, as 90 deg. is 8 points, i.e., 11-1/4 deg. (one point) times 8. Midway between the cardinal points are the inter-cardinal points. They are—N E, S E, S W, N W, and are 45 deg. or 4 points from the nearest cardinal point. Midway between each cardinal and inter-cardinal point—at an angular distance of 22-1/2 deg. or 2 points, is a point named by combining a cardinal point with an inter-cardinal point. For instance, NNE, ENE, ESE, SSE, SSW, WSW, WNW, NNW. Midway between the last points named and a cardinal or inter-cardinal point, at an angular distance of 11-1/4 deg., is a point which bears the name of that cardinal or inter-cardinal point joined by the word by to that of the cardinal point nearest to it. As, for instance, N by E, E by N, E by S, S by E, S by W, W by S, W by N, N by W. Also NE x N, NE x E, SE x E, SE x S, SW x S, SW x W, NW x W, NW x N. The angular distance between each and every whole point is divided into 4 parts called half and quarter points and each representing an angular measure of approximately 2 deg. 49'. In mentioning fractional points, the U. S. Navy regulations are to name each point from North and South toward East and West except that divisions adjacent to a cardinal or inter-cardinal point are always referred to that point: For instance, N 1/2 E, N x E 1/2 E, NE 1/2 N, NW 1/2 N, NW 1/4 W, NW 3/4 W, NW 1/4 N.

Boxing the compass is naming each point and quarter-point in rotation, i.e., starting at North and going around to the right back to North again. Every man should be able to identify and name any point or quarter-point on the compass card.

In changing a point course into a degree course, for either new or old compass, a guide is herewith furnished you. This should be pasted into the front of your Bowditch Epitome. It shows, from left to right, the name of the point course, its angular measure in the new compass and its angular measure in the old compass. It also shows at the bottom, the angular measure of each division of one point. In understanding this guide, remember that each course is expressed in degrees or degrees and minutes.

Put in your Note-Book:

In Navigation, each degree is written thus deg.. Each fraction of a degree is expressed in minutes and written thus '. There are 60' in each degree. Each fraction of a minute is expressed in seconds and is written ". There are 60" in each minute. Four degrees, ten minutes and thirty seconds would be written thus: 4 deg. 10' 30".

Although this guide just given you is given as an aid to quickly transfer a point course into a new or old compass course—or vice versa—you should learn to do this yourself, after awhile, without the guide.

Put in your Note-Book:

+ -+ + Ship's Head New Old By Point + -+ + NE 45 deg. N 45 deg. E NE 90 deg. 90 deg. N 90 deg. E EAST SE x E 123 deg. 45' S 56 deg. E SE x E S 20 deg. E 160 deg. S 20 deg. E S x E 3/4 E S 2 pts. E 157 deg. 30' S 22 deg. 30' E SSE NW 3/4 W 306 deg. 34' N 53 deg. W NW 3/4 W 289 deg. 41' 289 deg. 41' N 70 deg. W WNW 1/4 W + -+ +

I will show you just how each one of these courses is secured from the guide just given you.

Note to Instructor: After explaining these courses in detail, assign for reading in the class room the following articles in Bowditch: Arts. 25-26-27-28-29-30-31-32, 74-75-76-77-78-79-80-81-82.

Every compass, if correct, would have its needle point directly to the real or true North. But practically no compass with which you will become familiar will be correct. It will have an error in it due to the magnetism of the earth. This is called Variation. It will also have an error in it due to the magnetism of the iron in the ship. This is called Deviation. You are undoubtedly familiar with the fact that the earth is a huge magnet and that the magnets in a compass are affected thereby. In other words, the North and South magnetic poles, running through the center of the earth, do not point true North and South. They point at an angle either East or West of the North and South. The amount of this angle in any one spot on the earth is the amount of Variation at that spot. In navigating a ship you must take into account the amount of this Variation. The amount of allowance to be made and the direction (i.e. either East or West) in which it is to be applied are usually indicated on the chart. On large charts, such as those of the North Atlantic, will be found irregular lines running over the chart, and having beside them such notations as 10 deg. W, 15 deg. W, etc. Some lines are marked "No Variation." In such cases no allowance need be made. On harbor charts or other small charts, the Variation is shown by the compass-card printed on the chart. The North point of this card will be found slewed around from the point marking True North and in the compass card will be some such inscription as this: "Variation 9 deg. West in 1914. Increasing 6' per year."

Now let us see how we apply this Variation so that although our compass needle does not point to true North, we can make a correction which will give us our true course in spite of the compass reading. Note these diagrams:

The outer circle represents the sea horizon with the long arrow pointing to true North. The inner circle represents the compass card. In the diagram to the left, the compass needle is pointing three whole points to the left or West of True North. In other words, if your compass said you were heading NE x N, you would not actually be heading NE x N. You would be heading true North.

In other words, standing in the center of the compass and looking toward the circumference, you would find that every true course you sailed would be three points to the left of the compass course. That is called Westerly Variation.

Now look at the diagram to the right. The compass needle is pointing three whole points to the right or East of True North. In other words, standing in the center of the compass and looking toward the circumference, you would find that every true course you sailed would be three points to the right of the compass course. That is called Easterly Variation.

Hence we have these rules, which put in your Note-Book:

To convert a compass course into a true course

When the Variation is westerly, the true course will be as many points to the left of the compass course as there are points or degrees of Variation. When the Variation is easterly, the true course will be as many points or degrees to the right of the compass course.

To convert a true course into a compass course

The converse of the above rule is true. In other words, Variation westerly, compass to the right of true course; variation easterly, compass course to the left.


As stated before, Deviation causes an error in the Compass due to the magnetism of the iron in the ship. When a ship turns, the compass card does not turn, but the relation of the iron's magnetism to the magnets in the compass is altered. Hence, every change in course causes a new amount of Deviation which must be allowed for in correcting the compass reading. It is customary in merchant vessels to have the compasses adjusted while the ship is in port. The adjuster tries to counteract the Deviation all he can by magnets, and then gives the master of the ship a table of the Deviation errors remaining. These tables are not to be depended upon, as they are only accurate for a short time. Ways will be taught you to find the Deviation yourself, and those ways are the only ones you can depend upon.

Put in your Note-Book:

Westerly Deviation is applied exactly as westerly Variation. Easterly Deviation is applied exactly as easterly Variation.

The amount of Variation plus the amount of Deviation is called the Compass Error. For instance, a Variation of 10 deg. W plus a Deviation of 5 deg. W equals a compass error of 15 deg. W, or a Variation of 10 deg. W plus a Deviation of 5 deg. E leaves a net compass error of 5 deg. W.


Leeway is not an error of the compass, but it has to be compensated for in steaming any distance. Hence it is mentioned here. A ship steaming with a strong wind or current abeam, will slide off to the leeward more or less. Hence, her course will have to be corrected for Leeway as well as for Variation and Deviation.

Put in your Note-Book:

Leeway on the starboard tack is the same as westerly Variation. Leeway on the port tack is the same as easterly Variation. This is apparent from the following diagram:

As the wind, blowing from the North, hits the left hand ship, for instance, on her starboard side, it shoves the ship to the left of her true course by the number of points or degrees of leeway.

Leave a space and put the following heading in your Note-Book:

I. Complete rule for converting a compass course into a true course: 1. Change the compass course into a new compass reading. 2. Apply Easterly Variation and Deviation +. 3. Apply Westerly Variation and Deviation -. 4. Apply port tack Leeway +. 5. Apply starboard tack Leeway -. II. Complete rule for converting a true course into a compass course: 1. Reverse the above signs in applying each correction.

I will now correct a few courses, and these are to be put into your Note-Book:

- - - - C Cos Wind Leeway Dev. Var. New Old - - - - N x E NW 1/2 pt. 5 deg. E 10 deg. W 12 deg. N 12 deg. E S 67 deg. E S 1 pt. 3 deg. W 5 deg. E 104 deg. S 76 deg. E E x N SE 1/2 pt. 5 deg. W 10 deg. E 78 deg. N 78 deg. E W x N NW 1-1/2 pts. 1 deg. E 15 deg. E 280 deg. N 80 deg. W - - - -

Assign for Night Work the following arts. in Bowditch: 36-8-10-13-14-15-16-17-18-19-20-21-22-23-24.



I. The Pelorus

This is an instrument for taking bearings of distant objects, and for taking bearings of celestial bodies such as the sun, stars, etc. It consists of a circular, flat metallic ring, mounted on gimbals, upon a vertical standard. The best point to mount it is in the bow or on the bridge of the ship, where a clear view for taking bearings can be had. The center line of the pelorus should also be directly over the keel line of the ship. The inner edge of the metallic ring is engraved in degrees—the 0 deg. or 360 deg. and the 180 deg. marks indicating a fore-and-aft line parallel to the keel of the ship. Within this ring a ground glass dial is pivoted. This ground glass dial has painted upon it a compass card divided into points and sub-divisions and into 360 deg.. This dial is capable of being moved around, but can also be clamped to the outside ring. Pivoted with the glass dial and flat ring is a horizontal bar carrying at both of its extremes a sight vane. This sight vane can be clamped in any position independently of the ground glass dial, which can be moved freely beneath it. An indicator showing the direction the sight vane points can be read upon the compass card on the glass dial. If the glass dial be revolved until the degree of demarcation, which is coincident with the right ahead marking on the flat ring, is the same as that which points to the lubber's line of the ship's compass, then all directions indicated by the glass dial will be parallel to the corresponding directions of the ship's compass, and all bearings taken will be compass bearings, i.e., as though taken from the compass itself. In other words, it is just as though you took the compass out of its place in the pilot house, or wherever it is regularly situated, put it down where the pelorus is, and took a bearing from it of any object desired.

In taking a bearing by pelorus, two facts must be kept in mind. First, that when the bearing is taken, the exact heading, as shown by the ship's compass, is the heading shown by the pelorus. In other words, if the ship is heading NW, the pelorus must be set with the NW point on the lubber line when the bearing is taken of any object. Second, it must be remembered that the bearing of any object obtained from the pelorus is the bearing by compass. To get the true bearing of the same object you must make the proper corrections for Variation and Deviation. This can be compensated for by setting the glass dial at a point to the right or left of the compass heading to correspond with the compass error; then the bearing of any object will be the true bearing. But naturally, you will not be able to make compensation for these errors unless you have immediately before found the correct amount of the compass error.

Parallel Rulers

The parallel rulers need no explanation except for the way in which they are used on a chart. Supposing, for instance, you wish to steam from Pelham Bay to the red buoy off the westerly end of Great Captain's Island. Take your chart, mark by a pencil point the place left and the place to go to and draw a straight line intersecting these two points. Now place the parallel rulers along that line and slide them over until the nearest edge intersects the center of the compass rose at the bottom or side of the chart. Look along the ruler's edge to find where it cuts the circumference of the compass rose. That point on the compass rose will be the true compass course, and can be expressed in either the new or old compass, as, for instance, 60 deg. or N 60 deg. E. Remember, however, that this is the true course. In order to change it into the compass course of your ship, you must make the proper corrections for the compass error, i.e., Variation and Deviation and for Leeway, if any.

The Lead and Sounding Machine

The lead, as you know, is used to ascertain the depth of the water and, when necessary, the character of the bottom. There are two kinds of leads: the hand lead and deep-sea lead. The first weighs from 7 to 14 pounds and has markings to 25 fathoms. The second weighs from 30 to 100 pounds and is used in depths up to and over 100 fathoms. Put in your Note-Book:

Fathoms which correspond with the depths marked are called marks. All other depths are called deeps. The hand lead is marked as follows:

2 fathoms—2 strips of leather. 3 fathoms—3 strips of leather or blue rag. 5 fathoms—A white rag. 7 fathoms—A red rag. 10 fathoms—A piece of leather with one hole in it. 13 fathoms—Same as at 3. 15 fathoms—Same as at 5. 17 fathoms—Same as at 7. 20 fathoms—2 knots or piece of leather with 2 holes. 25 fathoms—1 knot. 30 fathoms—3 knots. 35 fathoms—1 knot. 40 fathoms—4 knots. And so on up to 100 fathoms.

The large hand leads are hollowed out on the lower end so that an "arming" of tallow can be put in. This will bring up a specimen of the bottom, which should be compared with the description found on the chart.

All up-to-date sea-going ships should be fitted with Sir William Thompson's Sounding Machine (see picture in B. J. Manual). This machine consists of a cylinder around which are wound about 300 fathoms of piano wire. To the end of this is attached a heavy lead. An index on the side of the instrument records the number of fathoms of wire paid out. Above the lead is a copper cylindrical case in which is placed a glass tube open only at the bottom and chemically colored inside. The pressure of the sea forces water up into this tube, as it goes down, a distance proportionate to the depth, and the color is removed. When hoisted, the tube is laid upon a prepared scale, and the height to which the water has been forced inside shows the depth in fathoms on the scale.


The dividers are nothing but an instrument for measuring distances, etc., on the chart.


There are two kinds of logs—the chip log, used for measuring the speed of the ship, and the patent log, used for measuring distance run.

The chip log consists of a reel, line, toggle and chip. Usually a second glass is used for measuring time. The chip is the triangular piece of wood ballasted with lead to ride point up. The toggle is a little wooden case into which a peg, joining the ends of the two lower lines of the bridle, is set in such a way that a jerk on the line will free it, causing the log to lie flat so that it can be hauled in. The first 10 or 15 fathoms of line from the log-chip are called "stray line," and the end of this is distinguished by a mark of red bunting. Its purpose is to let the chip get clear of the vessel's wake. The marks on the line (called knots) are pieces of fish line running through the strands of the reel line to the number of two, three, four, etc. A piece of white bunting marks every two-tenths of a knot. This is because the run of the ship is recorded in knots and tenths. The knots of fish line are 47 feet 4 inches from each other.

The log glass measures 28 seconds in time. For high rates of speed, a 14 second glass is used. Then the number of knots shown by the log line must be doubled. The principle of the chip log is that each division of the log line bears the same ratio to a nautical mile that the log glass does to the hour. In other words, if 10 knots or divisions of the log line run out while the 28 second glass empties itself, the ship's speed is 10 knots per hour. If ten knots or divisions run out while the 14 second glass empties itself, the ship's speed is 20 knots per hour.

The patent or towing log consists of a dial, line and rotator. The large circle of the dial records the knots and the small circle tenths of knots. When changing course, read the log and enter it in the log book. When changing course again, read the log again. The difference between the two readings will be the distance run.

Both logs are liable to error. A following sea makes them under-rate, a head sea over-rate. With both logs you must allow for currents. If a current is against you—and you know its rate—you must deduct its rate from that recorded in the log and vice versa. The reason for this is that your log measures your speed through the water. What you must find is your actual distance made good over the earth's surface.

Put in your Note-Book:

Between Sandy Hook and Fort Hamilton, bound due North, speed by chip-log was 10 knots, tidal current setting North 2 knots per hour; what did the ship make per hour? Answer: 12 knots.

At sea in North Sea ship heading S x W, patent log bet. 8 A.M. and 12 M. registered 32 miles, current running N x E 2 knots per hour; what was the actual distance made good? Answer: 24 miles.

Directions for allowing for a current setting diagonally across a ship's course will be given in the proper place.

Assign for Night Work the following articles in Bowditch: Arts. 161-162-163-164-165.



Aids to Navigation

A chart is a map of an ocean, bay, sound or other navigable water. It shows the character of the coast, heights of mountains, depths at low water, direction and velocity of tidal currents, location, character, height and radius of visibility of all beacon lights, location of rocks, shoals, buoys, and nature of the bottom wherever soundings can be obtained.

The top of the chart is North unless otherwise noted. When in doubt as to where North is, consult the compass card printed somewhere on the chart. On sea charts, such as those of the North Atlantic, only the true compass is printed, with the amount and direction of Variation indicated by lines on the chart.

Parallels of latitude are shown by straight lines running parallel to each other across the chart. The degrees and minutes of these parallels are given on the perpendicular border of the chart. Meridians of longitude are shown by straight lines running up and down, perpendicular to the parallels of latitude, and the degrees and minutes of these meridians are given on the horizontal border of the chart.

Put in your Note-Book:

A minute of latitude is always a mile, because parallels of latitude are equidistant at all places. A minute of longitude is a mile only on the equator, for the meridians are coming closer to each other as they converge toward either pole. They come together at the North and South poles, and here there is no longitude.

* * * * *

I can explain this very easily by reference to the following illustration:

As every parallel of latitude is a circle of 360 deg. the distance from A to B will be the same number of degrees, minutes and seconds whether measured upon parallel AA' or EE', but it will not be the same number of miles as the meridians of longitude are gradually converging toward the poles. On the other hand, the distances from A to C, C to D, D to E, etc., must be the same because the lines AA', CC', DD', EE' are all parallel. That is why the distance is always measured on the latitude scale (i.e. on the vertical border of the chart), and a minute of latitude is always a mile on the chart, no matter in what locality your ship happens to be.

You should be able to understand any kind of information given you on a chart. For instance, what are the various kinds of buoys and how are they marked?

Put in your Note-Book:

1. In coming from seaward, red buoys mark the starboard side of the channel, and black buoys the port side.

2. Dangers and obstructions which may be passed on either hand are marked by buoys with red and black horizontal stripes.

3. Buoys indicating the fairway are marked with black and white vertical stripes and should be passed close to.

4. Sunken wrecks are marked by red and black striped buoys described in No. 2. In foreign countries green buoys are frequently used to mark sunken wrecks.

5. Quarantine buoys are yellow.

6. As white buoys have no especial significance, they are frequently used for special purposes not connected with Navigation.

7. Starboard and port buoys are numbered from the seaward end of the channel, the black bearing the odd and red bearing the even numbers.

8. Perches with balls, cages, etc., will, when placed on buoys, be at turning points, the color and number indicating on which side they shall be passed.

9. Soundings in plain white are in fathoms; those on shaded parts are in feet. On large ocean charts fathom curves, showing the range of soundings of 10, 20, 30, 40, etc., fathoms are shown.

10. A light is indicated by a red and yellow spot. F. means fixed, Fl., flashing; Int., intermittent; Rev., revolving, etc.

11. An arrow indicates a current and its direction. The speed is always given.

12. Rocks just under water are shown by a cross surrounded by a dotted circle; rocks above water, by a dotted circle with dots inside it.

Practically all charts you will use will be called Mercator charts. Just how they are constructed is a difficult mathematical affair but, roughly, the idea of their construction is based upon the earth being a cylinder, instead of a sphere. Hence, the meridians of longitude, instead of converging at the poles, are parallel lines. This compels the parallels of latitude to be adjusted correspondingly. Although such a chart in any one locality is out of proportion compared with some distant part of the earth's surface, it is nevertheless in proportion for the distance you can travel in a day or possibly a week—and that is all you desire. The Hydrographic Office publishes blank Mercator charts for all latitudes in which they can be used for plotting your position. It makes no difference what longitude you are in for, on a Mercator chart, meridians of longitude are all marked parallel. It makes a great difference, however, what latitude you are in, as in each a mile is of different length on the chart. Hence, it will be impossible for you to correctly plot your course and distance sailed unless you have a chart which shows on it the degrees of latitude in which you are. For instance, if your Mercator chart shows parallels of latitude from 30 deg. to 40 deg. that chart must be used when you are in one of those latitudes. When you move into 41 deg. or 29 deg., you must be sure to change your plotting chart accordingly. In very high latitudes and near the North pole, the Mercator chart is worthless. How can you steer for the North pole when the meridians of your chart never come together at any pole? For the same reason, bearings of distant objects may be slightly off when laid down on this chart in a straight line. On the whole, however, the Mercator chart answers the mariner's needs so far as all practical purposes are concerned.

The instruments used in consulting a chart, i.e., parallel rulers, dividers, etc. have already been described. The only way to lay down a course and read it is by practice.

The one important thing to remember in laying down a course, is that what you lay down is a true course. To steam this course yourself, you must make the proper correction for your compass error.

Assign for Night Work in Bowditch, Arts. 9-239-240-241-243-244-245-246-247-248-249-251-252-253-254-255-256-257-258.

If any time in class room is left, spend it in laying down courses on the chart and reading them; also in answering such questions as these:

1. I desire to sail a true course of NE. My compass error is 2 points Westerly Variation and 1 point Easterly Deviation. What compass course shall I sail?

2. I desire to sail a true course of SW x W. My Variation is 11 deg. W, Deviation 2 pts. W and Leeway 1 pt. starboard. What compass course shall I sail?

3. I desire to sail a true course of 235 deg.. My compass error is 4 pts. E Variation, 27 deg. W Deviation, Leeway 1 pt. port. What compass course shall I sail?

4. I desire to sail a true course of S 65 deg. W. My compass error is 10 deg. E Variation, 3 deg. E Deviation, Leeway 1/4 point starboard. What compass course shall I sail?



The protractor is an instrument used to shape long courses. There are many kinds. The simplest and the one most in use is merely a piece of transparent celluloid with a compass card printed on it and a string attached to the center of the compass card. To find your course by protractor, put the protractor down on the chart so that the North and South line on the compass card of the protractor will be immediately over a meridian of longitude on the chart, or be exactly parallel to one, and will intersect the point from which you intend to depart. Then stretch your string along the course you desire to steam. Where this string cuts the compass card, will be the direction of your course. Remember, however, that this will be the true course to sail. In order to convert this true course into your compass course, allow for Variation and Deviation according to the rules already given you.

In case you know the exact amount of Variation and Deviation at the time you lay down the course—and your course is not far—you can get your compass course in one operation by setting the North point of your protractor as far East or West of the meridian as the amount of your compass error is. By then proceeding as before, the course indicated on the compass card will be the compass course to sail. This method should not be used where your course in one direction is long or where your course is short but in two or more directions. The reason for this is that in both cases, either your Variation or Deviation may change and throw you off.

Practically all navigation in strange waters in sight of land and in all waters out of sight of land depends upon the determination of angles. The angle at which a lighthouse is seen from your ship will give you much information that may be absolutely necessary for your safety. The angular altitude of the sun, star or planet does the same. The very heart of Navigation is based upon dealing with angles of all kinds. The instrument, therefore, that measures these angles is the most important of any used in Navigation and you must become thoroughly familiar with it. It is the sextant or some member of the sextant family—such as the quadrant, octant, etc. The sextant is the one most in use and so will be described first.

Put in your Note-Book:

The sextant has the following parts: (Instructor points to each.)

1. Mirror 2. Telescope 3. Horizon Glass 4. Shade Glasses 5. Back Shade Glasses 6. Handle 7. Sliding Limb 8. Reading Glass 9. Tangent Screw 10. Arc

In getting angles of land-marks or buoys, the sextant is held by the handle No. 6 in a horizontal position. The vernier arrow in the sliding limb is set on zero. Now, suppose you wish to get the angular distance between two lighthouses as seen from the bridge of your ship. (Draw diagram.)

Look at one lighthouse through the line of sight and true horizon part of the horizon glass. Now, move the sliding limb along the arc gradually until you see the other lighthouse in the reflected horizon of the horizon glass. When one lighthouse in the true horizon is directly on top of the other lighthouse in the reflected horizon, clamp the sliding limb. If any additional adjustment must be made, make it with the tangent screw No. 9.

Now look through the reading glass No. 8. You should see that the arc is divided into degrees and sixths of degrees in the following manner:

11 deg. 10 deg. -+ - + + + + + + + + + + + + + + + + + +

Now, as every degree is divided into sixty minutes, one-sixth of a degree is 10 minutes. In other words, each of the divisions of a degree on this arc represents 10 minutes.

Now on the vernier in the sliding limb, directly under the arc, is the same kind of a division. But these divisions on the vernier represent minutes and sixths of a minute, or 10 seconds.

To read the angle, the zero point on the vernier is used as a starting point. If it exactly coincides with one of the lines on the scale of the arc, that line gives the measurement of the angle. In the following illustration the angle is 10-1/2 degrees or 10 deg. 30':

10 deg. 9 deg. + -+ - -+ -+ - -+ -+ - -+ -+ - -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ -+ ^ + -+ -+ 0

If however, you find the zero on the vernier has passed a line of the arc, your angle is more than 10 deg. 30' as in this:

11 deg. 10 deg. + -+ -+ -+ - -+ -+ - + -+ -+ -+ -+ -+ -+ ^ + -+ -+ -+ 0

You must then look along the vernier to the left until you find the point where the lines do coincide. Then add the number of minutes and sixths of a minute shown on the vernier between zero and the point where the lines coincide to the number of degrees and minutes shown on the arc at the line which the vernier zero has passed, and the sum will be the angle measured by the instrument.

Now in measuring the altitude of the sun or other celestial body, exactly the same process is gone through except that the sextant is held vertically instead of horizontally. You look through the telescope toward that part of the sea directly beneath the celestial body to be observed. You then move the sliding limb until the image of the celestial body appears in the horizon glass, and is made to "kiss" the horizon, i.e., its lowest point just touching the horizon. The sliding limb is then screwed down and the angle read. More about this will be mentioned when we come to Celestial Navigation.

Every sextant is liable to be in error. To detect this error there are four adjustments to be made. These adjustments do not need to be learned by heart, but I will mention them:

1. The mirror must be perpendicular to the plane of the arc. To prove whether it is or not, set the vernier on about 60 deg., and look slantingly through the mirror. If the true and reflected images of the arc coincide, no adjustment is necessary. If not, the glass must be straightened by turning the screws at the back.

2. The horizon glass must be perpendicular to the plane of the arc. Set the vernier on zero and look slantingly through the horizon glass. If the true and reflected horizons show one unbroken line, no adjustment is necessary. If not, turn the screw at the back until they do.

3. Horizon glass and mirror must be parallel. Set the vernier on zero. Hold the instrument vertically and look through the line of sight and horizon glass. If the true and reflected horizons coincide, no adjustment is necessary. If they do not, adjust the horizon glass.

4. The line of sight (telescope) must be parallel to the plane of the arc. This adjustment is verified by observing two stars in a certain way and then performing other operations that are described in Bowditch, Art. 247.

Do not try to adjust your sextant yourself. Have it adjusted by an expert on shore. Then, if there is any error, allow for it. An error after adjustment is called the Index Error.

Put in your Note-Book:

How to find and apply the IE (Index Error):

Set the sliding limb at zero on the arc, hold the instrument perpendicularly and look at the horizon. Move the sliding limb forward or backward slowly until the true horizon and reflected horizon form one unbroken line. Clamp the limb and read the angle. This is the IE. If the vernier zero is to the left of the zero on the arc, the IE is minus and it is to be subtracted from any angle you read, to get the correct angle. If the vernier zero is to the right of the zero on the arc, the IE is plus and is to be added to any angle you read to get the correct angle. Index error is expressed thus: IE + 2' 30" or IE - 2' 30".

Quadrants, octants and quintants work on exactly the same principles as the sextant, except that the divisions on the arc and the vernier differ in number from the sixth divisions on the arc and vernier of the sextant.

If any time is left, spend it in marking courses with the protractor and handling the sextant.

Assign for Night Work the following Arts. in Bowditch: 134-135-136-138-142-144-145-151-152-157-158-159-160-161-162-163.



There are five good ways of fixing your position (obtaining a "fix," as it is called) providing you are within sight of landmarks which you can identify or in comparatively shoal water.

Put in your Note-Book:

1. Cross bearings of two known objects.

2. Bearing and distance of a known object, the height of which is known.

3. Two bearings of a known object separated by an interval of time, with a run during that interval.

4. Sextant angles between three known objects.

5. Using the compass, log and lead in a fog or in unfamiliar waters.

1. Cross bearings of two known objects.

Select two objects marked on the chart, so far apart that each will bear about 45 deg. off your bow but in opposite directions. These bearings will be secured in the best way by the use of your pelorus. Correct each bearing for Variation and Deviation so that it will be a true bearing. Then with the parallel rulers carry the bearing of one object from the chart compass card until you can intersect the object itself and draw a line through it. Do exactly the same with the other object. Where the two lines intersect, will be the position of the ship at the time the bearings were taken.

Now supposing you wish to find the latitude and longitude of that position of the ship. For the latitude, measure the distance of the place from the nearest parallel with the dividers. Take the dividers to the latitude scale at the side of the chart and put one point of them on the same parallel. Where the other point touches on the latitude scale, will be the latitude desired. For the longitude, do exactly the same thing, but use a meridian of longitude instead of a parallel of latitude and read from the longitude scale at the top or bottom of the chart instead of from the side.

2. Bearing and distance of a known object, the height of which is known.

Take a bearing of, say, a lighthouse the height of which is known. The height of all lighthouses on the Atlantic Coast can be found in a book published by the U.S. Dept. of Commerce. Correct the bearing, as mentioned in case No. 1. Now read the angle of the height of that light by using your sextant. Do this by putting the vernier 0 on the arc 0, sliding the limb slowly forward until the top of the lighthouse in the reflected horizon just touches the bottom of the lighthouse in the true horizon. With this angle and the known height of the light, enter Table 33 in Bowditch. At the left of the Table will be found the distance off in knots. This method can be used with any fairly perpendicular object, the height of which is known and which is not more than 5 knots away, as Table 33 is not made out for greater distances.

3. Two bearings of the same object, separated by an interval of time and with a run during that interval.

Take a compass bearing of some prominent object when it is either 2, 3 or 4 points off the bow. Take another bearing of the same object when it is either 4, 6 or 8 points off the bow. The distance run by the ship between the two bearings will be her distance from the observed object at the second bearing. "The distance run is the distance off."

A diagram will show clearly just why this is so:

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.


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.




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.



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.



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 - -


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.



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.



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




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.



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.

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