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The departure is less than VQ (fig. 1), the intercepted arc of the equator, or than the intercepted arc of the parallel through A; but it is greater than FG. But since the arc of the parallel gradually decreases from A to F, there is some point intermediate in position between A and F (y), the intercepted arc of the parallel of which will be exactly equal to the departure. The exact determination of this point is not very easy. Various methods have been proposed to determine this latitude nearly, with as little trouble as possible: first, by taking the arithmetical mean of the two latitudes for that of the mean latitude; secondly, by using the arithmetical mean of the cosines of the latitudes; thirdly, by using the geometrical mean of these cosines; and, lastly, by employing the latitude deduced from the mean of the meridional parts of the two latitudes. The first of these methods is the one usually employed. It has the merit of great simplicity; and as all the rules in navigation are approximate only, it may perhaps be depended on as much as any of these. Hence y may be considered as the middle point between A and F; and dep. = xyz.
But xyz=VQ cos lat. of y;
or dep. diff. long. x cos y, where y is the arithmetical mean of the latitudes of A and F= (1+l'), if = latitude of A, l' latitude of F.
Mercator's Chart-Meridional Parts.
The chart used at sea for tracing the ship's track exhibits the surface of the earth on a plane, in which the meridians are parallel, and consequently the distance between them throughout their length equal to the equatorial distance, instead of gradually decreasing as the latitude increases. In other words, FG is increased so as to become equal to VQ. Now, in order that on this chart all points may occupy the same relative position with respect to each other that the points corresponding to them do on the surface of the globe, the distance AF and the distance AG must be increased in the same proportion that BH + CI+DK+ &c., i.e., the departure has been increased.
ing arc of the equator, they have all been increased in Prelimin the ratio of sec lat. to 1.
Construction of Table of Meridional Parts.
This table may be constructed approximately, by dividing the whole meridian from 0° to 90° into intervals of 1', and supposing the increase of the arc of the parallel of latitude, and consequently that of the arc of latitude, to take place at the end of the successive minutes.
Now we have seen that an arc of any parallel = corresponding arc of equator x cos lat.; and since the arcs of the successive parallels have all become equal to the correspond
Hence, if the length of l' of the meridian be 1, and
c, d, the corresponding increased lengths between 0 and 1', between l' and 2′, 2′ and 3′, &c.,
a=1 x sec l'
And a+b+c, &c.=sec l'+ sec 2+ sec 3' + &c.; or the meridional parts in any arc of the meridian is equal to the sum of the secants of all the successive angles, differing by 1', from 1' up to the given latitude.
The true investigation is as follows:-Let m = the cir cular measure of the angle subtended at the centre by tl and let / become 7+ 81, and let 8m be the corresponding inmeridional parts in the arc between the latitudes 0 and ?; crease of m. Then dm is proportional to the secant of l+dl;
And ultimately taking the limit.
cos 45+ 2 1,
(1 + tan31⁄2) dl al =S
= sec 1;
=, or 90-1=4;
The distance AG so increased is called the meridional difference of latitude, or mer. diff. lat.; and the chart constructed on this principle is called Mercator's Chart. If for any latitude the meridional difference of latitude between this point and the equator be expressed in miles or minutes, the number of miles so expressed is called the meridional parts for that latitude. A table of meridional parts for .. meridional parts for lat. every minute of latitude from 0° up to 90°, is given in every collection of nautical tables.
= 2.3025851 logo cot 4. arc (in minutes) Now m= radius radius (in miles) meridional parts for lat. 57.29577 × 60
If, then, in AB produced (fig. 4) AB' be taken equal to mer. difference of latitude, and B'C' be drawn parallel to BC, meeting AC produced in C'; the sides of the triangle AB'C' will be the meridional difference of latitude, difference of longitude, and increased distance.
Hence, in equations (1.), (II.), (111.), (IV.), (v.), (vI.), (vII.), and (VIII.), we may substitute meridional difference of latitude for true difference of latitude, difference of longitude for departure, and increased distance for distance; and the equations will still hold.
The relation principally required is that which connects the meridional difference of latitude, difference of longitude, and course;
or, diff. long. =mer. diff. lat. x tan course.
dist. diff. long. x cosine lat.,
from which, any two of the quantities being given, the third may be found. Also, if d and d be distances corresponding to the same difference of longitude in parallels and l, we have d = diff. long, cos / d' diff. long. cos l'; cos / or d=d. cos l';
which enables us to find the distance on one parallel corresponding to a given distance on another, the difference of longitude being the same.
Since departure diff. of longitude x cos. mid. latitude, if in equations (II.), (IV.), (VI.), (VIII.), we substitute this value for departure, we shall obtain the following equations:
diff. long. x cos mid. lat. dist. x sin course
diff. long. x cos mid. lat.=true diff. lat. x tan course. true diff. lat. diff. long. x cos mid. lat. x cot course sin course = diff. long. x cos mid. lat. dist. .
from which all the rules for middle-latitude sailing may be derived.
A table in which the true difference of latitude and departure, corresponding to certain distances for every course, expressed in points and degrees, are laid down, is called a traverse table. It is very useful to enable the seaman to solve the several problems which occur in navigation by simple inspection. It must of course be calculated on some of the principles laid down in this chapter.
It is evident that as this table contains the relations of the sides and angles of a right-angled triangle, the solution of any right-angled triangle, whose sides represent any other quantities, will be given by it, by making the requisite changes.
Thus, the course remaining the same, if, for difference of latitude we look out in the table the meridional difference of latitude, the corresponding departure will be the difference of longitude; also, the difference of longitude can be found from the departure by these tables, by looking out the middle latitude as a course, and the departure as a true difference of latitude; then the corresponding distance is the difference of longitude.
SECT. II.-LONGITUDE AND LATITUDE.
PROB. I.-Given latitude from, and latitude in; to find the true difference of latitude.
Rule. Under latitude in, with its proper name, i.e., N. or S., place latitude from. Subtract the greater from the less, if of the same name, and reduce to minutes. The result is the true difference of latitude required, and is of the same or different name with latitude in, according as latitude in is greater or less than latitude from. If they are of different names, add and affect with the name of latitude in.
True diff. latitude..........
True diff. latitude..........
Ex. 2.-A vessel sails from New York, Lat. 40° 42′ N., to Liverpool, Lat. 53° 25' N.; find the true diff. of latitude.
53° 25' N.
40 42 N.
12 43 N. 763 miles N.
F. 3.-A ship sailed from Funchal, Lat. 32° 38′ N., to the Ca of Good Hope, Lat. 34° 29' S.; what is the true diff. of latitude? Latitude in................... Latitude from......
34° 29' S.
32 38 N.
True diff. latitude... 67 7 S. 4027 miles S. PROB. 2.-Given the latitude from, and true difference of latitude; to find latitude in.
Rule. If the latitude from and true difference be of the turned, if necessary, into degrees and minutes); the sum is same name, add them (the true difference of latitude being the latitude in of the same name. If of unlike names, under
latitude from place the true difference of latitude, subtract the less from the greater; the result, with the name of the greater, is the latitude in.
Prelimin. ary prin ciples.
PROB. 4.—To find the difference of longitude; having given longitude in, and longitude from.
Rule. Under longitude in place longitude from; subtract, if of like name; reduce the result to minutes, and call it E. or W., according as longitude in is E. or W. of longitude from. Add, if of unlike names, and attach the name E. or W., according as the longitude in is E. or W. of longitude from. If the longitude found by this rule exceed 180° it must be subtracted from 360°, and affected with the
Mer. parts.......... 499 S. Mer. parts....... 335 N.
Mer. diff. lat.... 834 S.
of an hour. Hence the proper length of a knot is =51 feet nearly. But although the line and glass be at any time perfectly adjusted to each other, yet as the line shrinks after being wet, and as the weather has a considerable effect on the glass, it will be necessary to examine them from time to time; and the distance given by them must be corrected accordingly. The distance sailed, therefore, may be affected by an error in the glass or in the line, or in both. The true distance may, however, be found as follows::
PROB. 1.-The distance sailed by the log, and the seconds run by the glass, being given; to find the true distance run, the line being supposed right.
Let the number of seconds in which the glass runs out be n, and let d and d' be the true distance and distance by log respectively. Then evidently the longer the time the glass is running, the less is the distance by log compared with the true distance, and conversely. Hence we have the proportion
120 ary principles.
d: d' :: 30" : "";
or true dist. dist. by log :: 30": number of seconds the glass is running; d' x 30 or d=
Rule.-Multiply the distance given by log by 30, and divide by the number of seconds the glass is running; the
result is the true distance run.
Ex. 1.-The hourly rate of sailing by the log is 9 knots, and the glass is found to run out in 35"; required the true rate of sailing.
35)270(7.7=true rate of sailing.
Ex. 2. The distance sailed by the log is 73 miles, and the glass runs out in 26"; required the true distance.
PROB. 3.-Given the length of a knot, the number of seconds run by the glass in half a minute, and the distance
159 420 3180 636
SECT. IV.-ON COURSES AND CORRECTIONS OF COURSES.
A ship is enabled to keep her course at sea by means of an instrument called the mariner's compass. It consists of a magnetic steel bar attached to the under side of a card, divided into points and quarter points, and supported by a fine pin, on which it turns freely within a box covered with glass. By reason of the directive property of the magnet, the north point, which is commonly denoted by a fleur de lis, is readily known. The circumference of the card is generally divided into thirty-two points, which, in the best compasses, are again subdivided into half points and quarters. These are reckoned sufficient for nautical purposes. On the inside of the box is drawn a dark vertical line called lubber's point. This point, or rather line, and the pin on which the card turns, are in the same line or plane with the keel of the ship; and hence the point on the circumference of the card opposite to lubber's point shows the angle which the ship's course makes with the magnetic meridian, called the course of the ship. The annexed diagram (fig. 5) gives a general view of
Compass course Variation .....
Compass course Variation
True course Variation
effects of local attraction, arising from the effects of the iron, guns, &c., in the vessel itself.
course on account of three causes:-1. The variation o The compass course generally differs from the true the compass; 2. The deviation of the compass; 3. The leeway. We shall now explain how these errors are to be applied.
1. The Variation of the Compass.-This is fully explained under the article MAGNETISM, which see. The mode of ascertaining its amount will be given hereafter.
PROB. I.-To find the true course, having given compass course.
Rule 1.-Allow easterly variation to the right, and allow westerly variation to the left.
Pts. qrs. 4 2
Ex. 1.-The compass course is W.N.W., and variation 3 pts. W.; find the true course.
3 right of S.
2 right of S.
Ex. 2.-The compass course is S.W.W., and variation 2 E.; find the course.
O left of N.
1 left of N.
1 left of N., or W. by S.1S.
True course......... 7 1 right of S., or W. by S. W. Ex. 3.-The compass course is N. W., the variation is 3 E.; required the true course.
Compass course........... 4 O left of N.
1 right of N.
3 left of N., or NW.
PROB. II.-Given the true course, to find the compass course.
Rule 2.-Allow easterly variation to the left, and westerly to the right.
Ex. 1.-The true course is N.N.E.E., and variation 1 W.; find the compass course.
2 right of N.
2 right of N.
O right of N., or N.E.
Ex. 2.-The true course is N.E., the variation 31 E.; required the compass course.
2. The Deviation of the Compass.-This error arises from the effects of local attraction, and varies with every different position of the ship's head. Several methods are employed in order to ascertain its amount. most commonly adopted is to place a compass on shore, out of reach of the ship's attraction, and to take the bearing of the ship's compass, or some other object in the same direction with it; while at the same time the bearing of the compass on shore is taken on board. If now 180° be added to the bearing of the shore compass, so as to bring it round to the opposite point, the difference between this augmented bearing and the bearing at the ship's compass will be the amount of deviation for that position of the ship's head.
Suppose the ship's head is N., and that the reading off at the shore compass is S. 17° 15′ W., and that the reading off at the ship's compass is N. 20° E. Adding 180' to the bearing of the shore compass, we get S. 197° 15′ W., or N. 17° 15′ E.; and subtracting this from the bearing of the ship's compass, N. 20° E., we get the deviation equal to 2° 45′ E., when the ship's head is N. The ship is now turned round, so that the head points successively to every point of the compass, and the deviation for each position found as before.
A table is then made, showing the deviation for every point of the compass. The deviation so found is treated exactly as the variation,―i.e., in correcting the compass course to find the true course, easterly deviation is allowed to the right, and westerly deviation to the left; and conversely, to find the compass course from the true course, easterly deviation is allowed to the left, and westerly deviation to the right. Hence it appears, that when both variation and deviation are given, we may consider the latter as a correction of the former-to be added to it if of the same name, and to be subtracted from it if of the opposite
Ex. 1.-The compass course is S.W.W., variation 1 E., and leviation W.; what is the true course?
Pts. qrs. Variation...... 2 2 left Deviation...... 0 3 left
1 3 right
3 right of N.
1 left of N.
2 left of N., or N.N.W.W.
Pt. qrs. Variation............ 1 3 left. Deviation...... 0 1 right.
Ex. 2.-The compass course is W.N., the variation 2 W., and deviation W.; what is the true course?
3 right of S.
True course 3 left of N., or S.W.by W.W. Ex. 3. The true course is N.N.W.W., variation 1 E., and deviation W.; required the compass course.
O right of S., or W.S.W.
2 left of N.
Pts. qrs. 2
1 left of N., or N.W W. The following table is taken from the monthly examination-papers at the Royal Naval College, Portsmouth, and will serve as a specimen of the tables which ought to be made for all ships :—
3 left of N.
3. Leeway. The effect of the action of the wind upon the sails and hull of a ship is sometimes to produce a motion of apparent course, as well as in this latter direction. The true the ship in a direction at right angles to that of the head or which is due to the composition of the two velocities of the course, therefore, is not that given by the compass, but that ship, viz., that in the direction of its head, and that at right angles to this direction. To obtain the true course from angle of leeway, which is the angle between the compass the compass course, therefore, we must add or subtract the
course and the true course.
who is looking towards the head, the real course is then eviIf the wind be on the right of a person on board ship dently to the left of the direction of the ship's head,—i.e., of the apparent course. If the wind be on his left hand, the true course is to the right. In the former case the ship is said to be on the starboard tack, and in the latter on the true course from the compass course. port tack. Whence is derived the rule for obtaining the
Rule.-If the ship is on the starboard tack, allow leeway to the left; if on the port tack, allow leeway to the right. Conversely, to obtain the compass course from the true course on the starboard tack, allow leeway to the right; if on the port tack, allow it to the left.
There are many circumstances which prevent laying down accurate rules for the allowance of leeway. The construction of different vessels, their trim with regard to the nature and quantity of cargo, the position and area of sail set, the velocity of the ship and the swell of the sea, are all susceptible of great variation, and very much affect the leeway.
The following rules are usually given for the purpose:1. When a ship is close-hauled under all sail, the water smooth, and with a light breeze, allow no leeway.
2. When the top-gallant sails are handed, allow one point. 3. Under close-reefed topsails allow two points.
4. When one topsail is handed, allow two points and a half.
5. When both topsails are handed, allow three points. 6. When the fore-course is handed, allow four points. 7. When under mainsail only, allow five points.
8. Under balanced mizen, allow six points.
9. Under bare poles, allow seven points. These rules, however, are not much to be depended upA very good method of estimating the leeway is to observe the bearing of the ship's wake as frequently as may be judged necessary, which may be conveniently enough done by drawing a small semicircle on the tafferel, with its diameter at right angles to the ship's length, and dividing its circumference into points and quarters. The angle contained between the semi-diameter which points right a^.