If therefore, in the annexed figure, ABC and ADE be two right angled plane triangles, and AD be the difference of latitude, the common angle A the course, and AD the meridional difference of latitude, then BC will be the departure, or the sum of the elementary meridional distances, AC the nautical distance, and DE the difference of longitude, all exactly. Hence from such parts of these triangles as may be given, the others may in any case be determined, either by computation, inspection, or otherwise.

The following obvious proportions will be found useful: — 1st. AB: BC::AD: DE; 2d. AD: DE:: rad. : tan. A. That is, diff. lat.: dep. mer. diff. lat. diff. long.; and mer. diff. lat. diff. long. :: rad. : tan. course.

:

Erample.-1. Required the course and distance from lat. 28° 30′ N., long. 30° 15′ W., to Cape Clear in Ireland, lat. 51° 25′ N., long. 9° 29′ W.?

DE

By construction.-Draw the meridian AD, fig. 20, and make D E perpendicular to A; make AD 1824, the meridional difference of latitude, 1246 the difference of longitude, and A B 1375 the difference of latitude. Join AE and draw CB parallel to DE, then AC the distance will be found 1665, and angle A the course N. 341° E.

By inspection.-With 1824 in latitude coluinn, and 1246 in dep. column, the angle A, the course, will be found 341°; with this course and 1375 in latitude column, AC will be found in the distance column 1665 miles.

By Gunter's scale.-Extend on the line of numbers from 1824 to 1246, and that extent will reach from radius to 344°, the course on the line of tangents. Extend again from radius to 1665, the complement of the course on the line of sines, and the extent applied to the line of numbers will reach from 1375, the difference of latitude, to the distance.

S. W. by W. W. 197 miles; required her latitude and longitude in?

By construction.-Draw the meridian AD, and draw AC making the angle A= 51 points, the given course, and make AC the distance 197. From C let fall the perpendicular CB on AD; then AB will be the difference of latitude, and BC the departure; and AB, being measured, will be found = 101 1° 41', which, added to 34° 29', the latitude left, gives 36° 10' S. the latitude in. Then take the difference of the meridional parts of these two latitudes, and lay it off from A to D, and draw D E parallel to BC, and DE will be the difference of longitude, 205 W. 3° 25', which subtracted from 18° 23′, the east longitude left, leaves 14° 58′ E., for the longitude arrived at.

By inspection. With the given course and distance the difference of latitude is found = 101.4, and the latitude in as above. And with the same course and the meridional difference of latitude 123 in the latitude column, the difference of longitude ED is found in the departure column= 205, whence the longitude in is obtained also as above.

By Gunter's scale.-Extend from radius to 2 points, the complement of the course, on the line of sines, and the extent applied to the line of numbers will reach from the distance 197, to the difference of latitude 101. Again, extend from radius to 51 points, the course on the line of tangents, and the extent will reach from 123, the meridional difference of latitude to 205, the difference of longitude on the line of numbers.

Mer. diff. lat.

Long. left
Diff. long.

Long. in

36 10 S.

2207 2330

123

18° 23' E. 3 25 W.

14 58 E.

In Mercator's sailing, though the principles are perfectly correct, they do not apply to the case in which two places are in the same parallel of latitude; in that case the principles of parallel sailing must be resorted to. And in all cases when the difference of latitude is small, and the course large, a trifling error in the difference of latitude may produce a very important error

= or Rm C, or C =

CL

Then L:: C:

LR CL

Hence

RI

m

the

== d =

for C. We have

cation of this method. It is used in the following manner:-Take half the sum of the latitudes for the mean middle latitude, and take also the difference of the latitudes. Then enter the table with the nearest degree of mean middle latitude at the side, and the nearest degree of difference of latitude at the top, and in the angle of meeting is the correction to be added to the mean middle latitude, to obtain what in this case may be called the true middle latitude.

Example.-What is the true middle latitude between 40° 17' N. and 55° 51' N.?

The subjoined Table will facilitate the appli

48 37 true mid. lat.

Method of finding the difference of longitude when a ship sails on a traverse.

Rule.-Resolve the traverse by means of the traverse table, as shown in traverse sailing, and find the latitude in the middle latitude between that left and that arrived at, and the meridional difference of latitude between the same two latitudes. Then with the middle latitude as a course, and the departure made on the traverse, in the latitude column of a traverse table, the difference of longitude will be found in the distance column. Or, with the difference of latitude and departure made good on the traverse, find the course and distance; and with the same course, and the meridional difference of latitude in the latitude column, the difference of longitude will be found in the departure column.

Or, by computation,

rad. dep. sect. mid. lat. diff. long.
diff. lat. dep.:: mer. diff. lat.; diff. long.

The difference of longitude, being always of the same denomination as the departure, must be added to, or subtracted from, the longitude left, according as they are of the same or different names, to obtain the longitude in.

If the traverse is performed in a very high latitude, it will be advisable to compute the difference of longitude separately on each course and distance, which may easily be done thus: -Find the latitude in at the end of each course, and the middle latitude between it, and that at the end of each preceding course, and the meridional difference of latitude between each pair of successive latitudes. Then with the middle latitude for each course, and the departure made on that course, find, as above, the difference of longitude; or with the meridional difference of latitude on each course, and the course made good, find, as above, the difference of longitude.

Enter the different longitudes thus found in two columns, headed E. and W., according as the course has been easterly or westerly; then the difference between the sum of the numbers in these columns will be the whole difference of longitude made on the traverse, and of the same denomination with the greater sum.

Note. When the course is north or south, no difference of longitude will be made; and, when it is east or west, the difference of longitude must be found by parallel sailing. In practice, the difference of longitude will generally be determined with sufficient exactness by inspection; and it may be a useful precaution generally to compare it both by middle latitude and Mercator's sailing. If however the course is small, and any little discordance should appear in the results by the two methods, that by Mercator's ought to be preferred; and the contrary when the course is large.

Example.-If a ship sail from Cape Farewell, in Greenland, lat. 59° 42′ N., long. 45° 16′ W., S. E. by E. 38, E. N. 26, S. by W. W. 40, N. E. by N. 33, S. E. 16, and S. W. W. 20 miles, required her latitude and longitude in; and course and distance to the Butt of Lewis, lat. 58° 29′ N., long. 6° 12′ W.?

Long. in 43 19 W.

Mer. diff. lat. 104 With the difference of latitude 53° 5' S., and departure 60-0 E. the course is found S. 48° 30' E., and distance 85; and with this course and meridional difference of latitude 104 in latitude column, the difference of longitude is found 117 E. = 1° 57′ E. in departure column.

Or, with middle latitude 59° 15' as a course, and departure 60 in latitude column, the difference of longitude is found 115 in distance column.

We shall now find the difference of longitude on each separate course.

I.-By MIDDLE LATITUDE SAILING.

sails, a piece of wood cut out in the form of the sector of a circle is cast into the sea, and it is then considered as a fixed point, the distance from which is measured by means of a string attached to the log, and unwound from a reel, as the vessel moves through the water.

The curved part of the log is loaded with lead so as to sink the whole just below the surface of the water, and at the same time keep it upright. One string is securely fixed to the central point of the log, and a second is attached, by means of a peg, to the middle of the curve. These two strings, at a short distance, are united in one; and when stretched keep the broad side of the log towards the ship, thereby causing it to oppose the most powerful resistance to motion in that direction.

Thus prepared the log is cast into the sea, and about ten fathoms of the line, called stray line, is suffered to run off the reel, for the purpose of stretching it, and to allow the log to get out of the wake of the vessel. Afterwards the line run off in a certain number of seconds is measured, whence the rate of sailing at the time is deduced.

The person who heaves the log knows when the stray line is run off, by a particular mark on the line, and the length afterwards run out by means of knots which are at distances from each other bearing the same proportion to a nautical mile that the time run by the sand-glass used in marking the time does to an hour. If one of these knots pass in the interval, the ship's rate of sailing is one mile per hour; if two knots pass, the rate of sailing is two miles per hour; and so on.

In the navy a sand-glass running twenty-eight seconds is generally used, and the length of a knot is forty-seven feet three inches; which bears the same proportion to 6075-5 feet, the length of a mean nautical mile, that twenty-eight seconds does to an hour. In the merchant service, however, the sand-glass generally runs thirty seconds, and the knots are about fifty-one feet apart; rather more than the just proportion, but the excess is allowed as a compensation for the dragging of the log by the weight of the line and

the friction of the reel.

In the navy the knot is generally subdivided into eight parts, denominated fathoms; in East Indiamen, and the better class of merchantmen in general, the knot is subdivided into ten parts, also called fathoms; and this mode of subdivision is decidedly the most convenient. ships in which little nicety is pretended to in the navigation the knots are subdivided into four parts, and sometimes they are simply halved.

In

The number of seconds run by the glass, and the length of the knots on the log-line should be frequently examined, and, if necessary, altered. Or, what would answer better than shifting the marks on the log-line so frequently as would thus be required, the error of the several knots might he determined at least once a-day immediately after heaving the log, and a correction applied to the distances from a table constructed for the purpose.

If the knots have a uniform error, and the glass is also erroneous, the true distance may be obtained by the following

The circumference of the card is generally divided into thirty-two equal parts, called points, and each division is subdivided into four parts, called quarter points. A point is therefore equal to 11° 15'; but in some compasses the rim of the card is divided also into degrees.

From the manner in which the card is suspended, the polarity of the needle causes its plane, in every situation, to assume a particular position, the point of the card which coincides with the north pole of the needle is marked north, and the opposite point south. The compass, placed in a box and secured by a glass cover from the wind and sea, is placed on the binnacle near the steersman, who is said to steer

on any particular point of the compass, when he brings the fore and aft line of the ship in the same direction with the line on the card which indicates that point.

The middle points between the north and south are called east and west, and the other points are named according to their situation with reference to the cardinal points, north, south, east, and west, as in Plate II. fig. 1.

There are few situations in which the needle points due north and south, and even in the same situation its direction undergoes a slow and gradual alteration. At present, at London, it deviates from the meridian about 24° 30′ towards the left or the west; but at the North Cape it points only about 1° towards the west, while in some parts of Davis's Straits it points upwards of six points and a half towards the west, and near Cape Horn it points about 22° towards the east. In the year 1580 the needle pointed about 11 or 12° towards the east, though now, as has just been stated, it points about 24° 30′ towards the west.

But in the West Indies the deviation has scarcely undergone any perceptible alteration for a very long period.

This deviation is called the variation of the compass, and the method of finding it by celestial observations will be given in a subsequent part of this article. It is denominated easterly or westerly variation, according as the needle points to the east or west of the true north.

Having computed the true course from one place to another by the appropriate rules, which have already been given, it is necessary, before that course can be steered by the compass, to VOL. XV.

make a proper allowance for the variation; that is, to find what point of the card corresponds with the course determined by calculation.

To explain the manner of doing this, we shall first suppose that the ship lies on the true computed course, and that the needle has no variation; in this case it is evident that no correction would be required. When the ship and compass are thus situated, let the north point of the needle be turned a little towards the west; then every point of the card will be carried to the left of its former place, and the course of the ship, though in reality the same, will appear by the compass to be the right of what it was before, or is the right of the true course. If we conceive the needle to deviate towards the east, a contrary effect will be produced, the apparent course by the compass appearing to the left of the true

course.

Hence in finding the compass course when the true course is known westerly variation is allowed to the right, and easterly to the left of the true course; and, in finding the true course from a given course by the compass, the variation if west is allowed to the left, and if right to the east of the course, as shown by the compass.

The following considerations will aid the student in recollecting the points of the compass. The middle point between

N. and E. is N. E. N. and W. is N. W. S. and E. is S. E.

S. and W. is S. W.

ther the letters of the cardinal points between The names being got simply by putting togewhich they lie, and from which they are distant four points. The middle point between

N. and N. E. is N. N. E.
E. and N. E. is E. N. E.

S. and S. W. is S.S.W., &c.

So that the second set of middle points are also got by putting together the letters of the points already determined, and from which they are distant two points. The remaining points, which may be considered as the third set of middle points, receive their names from the side on which they lie of the first middle point. Thus the point to the right of N. is called N. by E., and that to the left N. by W.; the point to the right of S.W. is called S.W. by W., and that to the left S.W. by S.

ON LEEWAY.

The action of the wind on the sails, rigging, and sides of a ship, when it lies near the point from which the wind blows, not only tends to impel her in the direction of a fore and aft line, Iut it also presses her sideways, and thus, upon the whole, the real track made in the sea is to leeward of the apparent line on which the ship is steered, and makes an angle with it, which angle is called the leeway.

When the wind is on the right hand side of a ship, she is said to be on the starboard tack; and when on the left hand side she is said to be on the larboard tack; and when she sails as near the wind as she will lie, she is said to be close hauled. Large vessels will seldom lie within 2 L