11. Given the base 1514, the verticle angle 75° 24′ 50′′, and the perpendicular 972.41; required the remaining sides and angles? Ans.-The sides are 1298 and 1172, and the angles are 56° 4′ 5′′ and 48° 31′ 5′′ respectively.

Navigation.

52. The various sailings in navigation are only the applications of trigonometry in particular circumstances.

The course is the angle formed between the meridian and the point on which the ship sails, indicated on the mariner's compass, the distance is the hypotenuse, and the difference of latitude and departure the legs of a right-angled triangle.

B

B

A

E

F

Thus let AB represent a portion of the meriD dian; then, if a ship sails north-easterly, the line AC is drawn to the right-hand, making an angle D BAC equal to the course, shown by the compass, and is generally in points and quarter-points, and AC represents the distance, AB the difference of latitude, and BC the departure. If she sails northwesterly, then BAD is supposed to be the angle of course, AD the distance, AB the difference of G E F latitude, and BD the departure. If she sails south-westerly, AG is the distance, AE the difference of latitude, GE the departure, and GAE the course. Lastly, if the ship sail south-easterly, AF is the distance, AE the difference of latitude, EF the departure, and FAE the course. If, however, AE' be the meridian difference of latitude, E'F' is the difference of longitude, E'AF' is the course, and AF is still the distance. Hence the course and distance between two places can be found, by this method, when their latitudes and longitudes are known. This is commonly called Mercator's sailing. Parallel, middle latitude, and oblique sailings, may readily be explained on similar principles.

Parallel Sailing.

If the earth be considered as a sphere, from which it does not deviate greatly, the difference of longitude between two places is the angle at the pole, formed by two meridians passing through these places, (Spher. Trig. prop. VI.) and measured by an arc of the equator intercepted between them. But since these meridians approach one another, as their distance from the pole diminishes, the meridian distance corresponding to the same difference of longitude varies according to the latitude.

B

P

A E

Let PCE be a section of one-fourth of the earth, PE an arc of the meridian from the equator, as E to the pole at P, and L any place on the earth's surface. Draw BL parallel to CE, AL parallel to PC, and join CL. Now CE or CL is to BL as radius is to cosine ECL; therefore, since circles, or any portion of them, are as their radii, any portion of a circle whose radius is CE, is to a similar portion of a circle whose radius is BL, as radius to cosine ECL. The latitude of L, of which BL is the radius of the parallel, is measured by the arc EL, supposing CE the radius of the equator, and PC the polar semiaxis, then the length of an arc on the equator is to the length of the corresponding arc in the latitude L, as radius to the cosine of the latitude. Hence, in any latitude, the radius of the parallel is equal to

D

the cosine of the latitude, the semidiameter of the equator being considered radius, and since similar arcs are as their radii, it follows, that the distance sailed on any parallel is to the difference of longitude as the cosine of the latitude to the radius. In this manner Table I. was computed.

Middle Latitude Sailing.

This is an easy approximate method of resolving problems in navigation, and is a combination of plane and parallel sailings, in which the difference of longitude is reckoned upon the middle parallel between the latitude sailed from and that come to, whence it derives its name.

Let ABC be a figure in plane sailing, in which AB is the difference of latitude, AC the distance, and BC the departure. Connect with this, DBC, a figure in parallel sailing, in which BC is still the departure, DCB an angle equal to the middle latitude, then will DC be the difference of longitude. Resolving these two triangles either separately, or in conjunction, as the case may require, the following rule will be obtained.

B

Diff. lat. diff. long. :: cos mid. lat. : tang. course. This analogy may be taken inversely, &c. according to the data.

EXAMPLES.

1. A ship from Bombay in latitude 18° 57′ N. sailed S. W. by S. 224 miles; required the latitude come to, and the departure? Ans.-The difference of latitude is 186.2, and the departure 124.4 Latitude of Bombay 18° 57' N.

Diff. of lat. 186 miles=

Latitude come to

3 6 S.

15 51 N.

2. A ship from latitude 47° 30' N. sails S. W. by S. 98 miles; what latitude is she in, and what departure has she made?

Ans.-Difference of latitude 81.48, departure 54.45 miles, and the latitude come to 46° 9' N

3. A ship from latitude 48° 32′ N. sails between north and west till her departure is 54 miles, and then finds herself in latitude 49° 54' N.; what course did she steer, and what distance did she run? Ans. Course 33° 22′ N. W., and distance 98.18 miles.

4. Required the course and distance from Cape Wrath in Scotland, in latitude 58° 36′ N. longitude 5° 20′ W., to New York in North America, in latitude 40° 28′ N. longitude 74° 2′ W.?

Ans. Course 67° 32′ or W. S. W. nearly, and distance 2847 miles. 5. A ship from latitude 60° 24′ N. and longitude 43° W. sails between south and east till she is in latitude 56° 30′ N., and has made 226 miles of departure; required her course, distance, and longitude? Ans-Course S. 44° E. or S. E. nearly, distance 325.4 miles, and the longitude of the ship 35° 47′ W.

6. Required the course and distance between the Isle of May, in latitude 56° 12' N. longitude 2° 33′ W., and Heligoland, in latitude 54° 12' N. longitude 7° 53′ E.?

Ans. Course S. 71° 27′ E. and dist. 377 miles.

7. A ship from Inchkeith, in latitude 56° 3′ N., and longitude 3° 10 W. sailed E. S. by compass, or, since the variation is about 24

points West, E. N. E. true, 3h 20m, at the rate of 5.8 knots an hour, what was her situation with regard to the isle of May in latitude 56° 11′ N., and longitude 2° 33′ W.?

Ans. Latitude in 56° 10′ N. and longitude 2° 47′ W. or about 1 mile south of the isle of May light, and 14′ of longitude west of it.

8. Required the true and magnetic course and distance between Ayr light, in latitude 55° 25′ N. and longitude 4° 26′ W., and the Mull of Cantire, in latitude 55° 18′ N. and longitude 5° 40′ W., the variation of the compass being 23 points W.?

Ans. The true course is S. 80° 47′ W. or W. by S. nearly, the course by compass W. N. W., and the distance 44 miles nearly.

9. Coasting along shore I saw a cape bearing N. E. by N. After standing N. W. 20 miles, the same cape bore E. N. E.; required the distance of the ship at each station?

Ans. From the first station 33.26, and from the second 35.31 miles.

10. From a ship at sea, I observed a point of land to bear E. by S., and after sailing 12 miles N. E., it bore S. E. by E.; required the distance of the last place of observation from the point of land? Ans.-26 miles.

11. Sailing N.N.W. at the rate of 6 knots an hour, at 8h. P. M. I discovered two light-houses, the northernmost of which bore N. N. E. and the other E. by N., and at 10h. 30m. the northernmost light bore E. N. E., and the other E. S. E.; the bearing and distance of the lights from each other are required.

Calculation.In the triangle ACD are given the side AC equal to 15 miles, the angle ADC 3 points, the interval between E by N. and E. S. E. and the angle CAD 4 points, the distance between S.S. E. the opposite point to N. N. W., and E. S. E.; to find CD= 19.09. Again, in the triangle ABC are given AC as before equal to 15 miles, the angle ABC equal to 4 points, the interval between N. N. E. and E. N. E. and the angle ACB also 4 points, the interval between the N. N. W. and N. N. E. points; hence the angle CAB is a right angle; consequently we get BC-21.21.

Lastly, in the triangle BCD are given the sides CB, CD, equal to 21.21 and 19.09 respectively, and the included angle BCD 5 points, the interval between N. N. E. and E. by N.; to find the angles CDB =67° 30′, CBD=56° 15′-5 points, CBF=BCN=2 points, and the distance BD=19.09, and bearing S. E by S.

Most of these problems may be very expeditiously performed by Mercator's chart, of so much utility for laying down a ship's place at sea, and particularly at noon.

When the latitude and longitude are known, the ship's place may be laid down on the chart by laying a ruler first over the latitude on the sides of the chart, and drawing a pencil-line, then over the longitude on the top and bottom, and drawing another line intersecting the former; this point of intersection will be the ship's place. To facilitate the operation of finding the bearing and distance of one place from another, several compasses are drawn on convenient parts of the chart. By laying a ruler over the two places, and extending the points of the compasses from the centre of one of the cards on the chart to the nearest point of the ruler, then sliding both feet of the compasses forward and backward, keeping the one foot close to the ruler, the other will point out the course. The extent between the places being applied to the graduated scale of

latitude on the side of the chart, so that one-half may be above the mean latitude, and the other below it, will give the distance in degrees and minutes, which may be readily reduced to miles. If the distance be too great for the compasses, some aliquot part may be taken, such as a half, a third, &c. whence, by doubling, tripling, &c. the whole distance becomes known.

12. A ship from Aberdeen, in latitude 57° 9' N. and longitude 2° 8′ W. sailed on the following true courses; required her situation ?

56

7

Now to 56° 7 as a course, Table VIII., the distance 640', answering to the departure 357.6, found in a latitude-column, will be the difference of longitude=10° 40′E.

Long. left,

Long. made,

Long. in

2 8 W.

10 40 E.

8 32 E.

Hence the ship is close on the coast of Jutland opposite Hallam, between the islands Sylt and Romoe.

By computing the courses and run by a ship in this manner daily, from noon to noon, her place will be found each day at twelve, which may be laid down on a chart. Such other matters occurring on board as may be thought worthy of notice are also recorded in a book kept for that purpose, constituting what is called the logbook or journal.

JOURNAL.

The journal is a register of the various transactions which take place on board a ship both in harbour, and more especially at sea. At sea the day begins at noon, 12h before the civil day, and 24h before the astronomical day. The first 12 from noon till midnight are marked P. M., and the next 12h from midnight to noon, A. M. In ships of the Royal Navy and East India Company's service, the log is hove every hour, but in many trading vessels only once in two hours. To correct the course steered for variation and leeway, the observer is supposed to be placed at the centre of the card of the compass. Now the variation must be allowed to the right hand of the course steered, if it is easterly, but to the left if westerly, and the leeway must be allowed to the right or left, according as the ship is on the larboard or starboard tack.

When the true course is known, that by compass will be obtained by allowing the variation to the right if westerly, but to the left if easterly. When the ship is lying to, the middle point between that to which her head comes up, and that to which it falls off, is taken as the course, and the drift is assumed as the rate per hour in the direction in which the ship is carried. The latitude and longitude obtained by observation is reduced to noon, if necessary, by means of the log-book, showing the run of the ship in the interval. The latitude and longitude should be ascertained by observation daily, if possible. If lunars cannot be conveniently taken, the longitude by chronometer should not be neglected.

Rules for Working a Day's Work.

Having corrected the courses according to the foregoing directions, place these and the distances run in a table, as in example 12, and find the difference of latitute and departure. From the latitude left and that come to, the middle latitude may be found, being equal to half the sum of the latitudes if they are of the same name, and half the difference if of contrary names. Now, in a traverse table, to middle latitude as a course, the distance answering to the departure found in a latitude-column, is the difference of longitude, which being properly applied to the longitude left, according to its name, will be the longitude of the ship by dead reckoning.

The log-board is usually divided into seven columns: the first column on the left hand contains the hours from noon to noon; the second and third contain the knots and tenths of a knot at which the ship is sailing; the fourth contains the courses steered; the fifth the winds; the sixth the leeway; and the seventh the various remarks thought necessary, such as the state of the weather, the management of the sails, the observations for ascertaining the ship's place, the variation of the compass, &c. The log-board is transcribed every day at noon into the log-book, which is ruled and diIvided in the same manner. The nautical mile is 6076 feet, and since the sand-glass runs out in half a minute, or 10 part of an hour, the knot is 6076=50.6 feet, or say 50 feet, and the tenth part of a knot, in the following journal marked F, or fathom, must be five feet. Hence the number of knots and fathoms run off the reel in half a minute shows the miles and tenths which the ship is sailing per hour.

120