Other matters demanding consideration when it is proposed to make a voyage on a great circle, such as deducing the place of the ship, and shaping the course afresh from time to time, are treated in the division of the work appropriated to Navigating the Ship.

The elements of several tracks on a great circle are collected together in the following Table for reference.

CHAPTER IV.

TAKING DEPARTURES.

I. BY A SINGLE BEARING AND DISTANCE. II. DETERMINATION OF DISTANCE. III. METHODS BY THE CHART.

347. DETERMINING the place of the ship with reference to a point of land, or other position of known latitude and longitude, is called Taking a Departure.

348. The position of the ship with respect to a point of land or other fixed and conspicuous object is defined by the direction in which she lies, and her distance from it.

The direction or bearing of the ship from the land, being the opposite of the bearing of the land from the ship, is furnished at once by the compass, or it may be found by observation of an Astronomical Bearing; but the distance from the point, when it cannot be estimated or guessed with sufficient precision, must be deduced by means of some further observation, taken at the same time as the bearing, or after an interval.

When a former position of the ship herself is adopted as a point of departure, the direction (or course) and the distance are deduced from the reckoning.

I. BY A SINGLE BEARING AND DISTANCE.

349. The object being set by the compass, its distance is estimated by the eye.

This, which is the common method of taking departures, is near enough when the distance is small; but the error or uncertainty in the estimation of the distance, which, perhaps, may be stated generally at one-fifth of the whole, becomes considerable when the distance is great. Distances thus estimated are generally overrated.

II. DETERMINATION OF DISTANCE.

1. By two Bearings of the same Object.

350. When the ship's path lies across the line of direction of the object, the distance can be obtained by two bearings and the distance run by the ship in the interval of time between them.

Take the bearing of the object, and note the number of points contained between it and the ship's head. After the bearing has altered not less than two or three points, note the number of points in the same angle again.

NOTE. The ship is supposed to keep the same course; if not, the course made good must be employed, and the local deviation, if considerable, allowed for, as it will affect the different courses differently.

(1.) To find the distance when the last bearing was taken.

Enter Table 7 with the first number of points at the top and the second number of points at the side; take out the number corresponding, and multiply it by the number of miles made good by the ship: the result is the dist. in miles at the time the last bearing was taken.*

Ex. The Eddystone bore N.W. by W.; after running W. by S. 8 miles, it bore N.N.E.: required its Dist. at this last bearing.

The number of points between N.W. by W. and W. by S. is 4; that between N.N.E. and W. by S. is 11; under 4 at the top and against 11 at the side stands 0.72, which multiplied by 8 (miles), gives 5.8 miles, the Dist. required.

The student can easily supply a figure.

(2.) To find the distance when the first bearing was taken.

Enter the Table with the supplement (or difference from 16 points) of the second number of points at the top, and the supplement of the first number of points at the side; take out the multiplier, and proceed as above directed.

Ex. Find the Distance of the Eddystone at the time the first bearing (or N.W. by W. above) was taken.

The second number of points is 11, the supplement of which is 5; the first number is 4 points, the supplement of which is 12; then 5 at the top and 12 at the side give the number 085, which multiplied by 8 gives 6-8 miles, the DIST. required.

When the number of points between the object and the ship's head at either observation is 8, that is, when the bearing is at right angles to the course, the distance may be found by the Traverse Table (Table 1 or 2), by entering the table with the number of points at the other observation as a course, and the distance run as D. Lat.; the corresponding Dep. is the distance of the object when observed at 90° from the course.

351. Degree of Dependence. As both the bearings and the distance run are in all cases more or less liable to error, the resulting distance cannot be considered as exactly determined.

The degree of dependence may be estimated by altering the bearings and distance by small quantities equal in amount to the probable errors, and repeating the work. (See p. 52.)

Ex. In the example above, if the first bearing be taken as N.W. W. instead of N.W. by W., the resulting distance will be 65. If, on the other hand, the second bearing be taken as N. by E. E. instead of N.N.E., the resulting dist. will be 5'9. Since thus an error of a point in either bearing produces at most an error of 6.5-5.8, or o'7 of a mile, the case is a good one.

(1.) In general it will be enough to suppose an error in that bearing which differs the most from 8 points, or 90°.

This Table was constructed at the suggestion of Sir F. Beaufort, and first appeared in the Nautical Magazine, vol. i. p. 208.

(2.) The error of the required distance produced by an error in the dist. run, is a matter of simple proportion. For example, if the dist. run be of itself in error, the distance required will also be of itself in error. Hence the dist. run should not be much less

than the distance required.

The case is most favourable when the triangle is equilateral.

2. By Found.

352. An excellent mode of determining the distance is obtained by noting the number of seconds elapsed between seeing the flash of a gun and hearing the report. Sound travels, in a calm, about 1130 feet in one second; hence it is easy to deduce the following approximate rule.

Divide the seconds elapsed by 5, and subtract from the quotient of itself; the result is the Dist. in miles very nearly.

Ex. The mean of the intervals given by 4 guns fired from C. Shilling was 141: required the Dist. of the ship.

This method is capable of much precision when the gun and the ear are at the same temperature and at the same height.* A moderate breeze in the direction of the sound causes a variation of about 20 feet a second in the velocity; a strong breeze more.

3. By the Altitude of High Land.

[1] When the Object is seen on the Sea-Horizon.

353. The distance of the visible horizon from the spectator is equal to the true depression or dip of the eye in Table 8, increased by about of itself. Thus, if the eye be twenty feet above the sea, the horizon is distant five miles and about half a mile more.

When, therefore, the sea-horizon is seen beyond the object, the distance of the latter is less than the depression.

354. When the summit, or any other point of known height of an object situated beyond the sea-horizon is seen on this line, its distance is at once known; for since the eye, the horizon, and the object are in the same straight line, the same horizon corresponds to both the height of the eye and that of the object; the distance, therefore, between these two points is, by No. 205, the sum of the depressions corresponding to the two heights.

Ex. From the mast-head, 87 feet above the sea, the Lizard Light, the height of which is 223 feet above low-water mark, is seen on the horizon: required its distance.

The dip (Table 8) to 87 feet is 10', that to 223 is 16′; the sum 26 increased by † of 26, or z', is 28 miles, the DIST. required.

*The uncertainty to which this method is liable (though not worth notice in navigation) may, when precision is required, be removed, in the ordinary state of the atmosphere, by firing a gun at each extremity of the line, and taking the mean of the observed intervals.

In this and the following rules is used instead of (see No. 207), because 12 is an easier divisor than 14. The difference is not worth notice.

This method will often be useful, but from the great uncertainty of terrestrial refraction it is impossible to assign with precision the degree of dependance.

[2.] When the Object is seen above the Sea-Horizon.

355. Case I. When the height of the summit, or other point of high land, is known, its distance is found by means of the altitude observed above the sea-horizon with a quadrant or sextant.*

356. The Observation. Observe the altitude of the summit, and estimate its distance in miles.

When the altitude exceeds 3° see No. 359. 357. The Computation. Alt. under 3°. (1.) index error (No. 496), and subtract from it distance; the remainder is the true alt.

Correct the alt. for of the estimated

When the height of the eye exceeds 30 feet, add of the corresponding Depression; the sum is the true altitude.

(2.) From the true alt. subtract the true Depression to the height of the eye, Table 8: note the remainder.

To the square of the Depression corresponding to the height of the summit add the square of the remainder (which is found at once in the column headed "Square," against the remainder as a Depression). Look for the sum in the column headed "Square," and take out the Depression corresponding; from this take the remainder: the result is the distance of the summit in miles.+

Ex. 1. The alt. of a hill 2000 feet high is observed 56'; corr. for index error, -3'; the height of the eye, zo feet; estimated Dist. 8 leagues, or 24 miles: required its Distance. Deducting of 24, or 2', and 3′ error, leaves true alt. 51′.

True alt.

51'

True Depr. to 20 ft. — 5

Rem. 46

Square of Depr. to 2000 ft. 2304
Ditto of Rem. 46'
+2116

Depr. 67' Square 4420
Rem. -46

DIST. required 21′ or miles.

Ex. 2. April 19th, 1829, Mr. Fisher observed from the poop of H.M.S. Spartiate, 74, the alt. of Mount Etna, 1° 26' 30"; index corr. + 1' 30"; height of eye, 30 feet; estimated dist. 20 leagues: required its Distance. Height of Etna, 109co feet.

358. When the distance is too great for estimation, and the altitude low, the computation must be repeated.

Ex. Captain Beechey observed from H.M.S. Sulphur, the Peak of Teneriffe clearly defined against the setting sun; mean of 3 alts. on the arc, 19' 32"; off the arc, 19′ 50′′; the

*In this instance, reference is necessarily made to the use of instruments which belong principally to Nautical Astronomy, and are, therefore, described in that subject, Chap. II.

When the height of the eye exceeds 30 feet, subtract from the sum of the two squares (above) the square of the corresponding Depression. From the nature of the observation, it is enough to work to minutes only.