to the eastward.

In this way the diff. of lat. made good by the second course WNW 26 miles, is entered in the column for northing, and the departure in that for westing, agreeably to the nature of the course between N and W. The fifth run likewise being SW 13 miles, the diff. of lat. and departure are entered in the columns for southing and westing, the course lying between S and W.

When the diff. of lat. and departure for all the given courses and distances have been calculated and entered in their respective columns, each column is summed up and the amount entered at the bottom: thus in the preceding example the northing is 89.4 miles, the southing 12.1 miles, the easting 48.4, and the westing 71.5. Subtracting the southing from the northing, the remainder 77.3 miles or 1 degree 17.3 minutes is the total diff. of lat. made good upon the whole traverse; and it is northerly because the column of northing is the greatest. In the same way subtracting the easting from the westing the remainder 23.1 miles is the total departure on the whole traverse from the meridian of the port whence the ship sailed: and it is westerly, for the column of westing is the greatest.

We have now obtained a simple case of plane sailing in which the difference of latitude and the departure are given, and the course and the distance are required. By case 1st of plane sailing, if a ship from a port in N. lat. sail between N and W until her difference of latitude be 77.3 niles, and her departure 21.1, her distance will be found 20.7 miles, and her course will form an angle of 16° 38′ on the W side of the N part of the meridian of the place sailed from: hence it appears that after a vessel has run down the several courses and distances before given, amounting to not less than 173 miles, she will have advanced in fact only 80.7 miles in her voyage from the point of departure, and that after all the various directions in which she may have sailed the course made good on the whole will be N 16° 38 W,

nearly

nearly to the westward of N, or N by WW. If to the latitude of the place sailed from 33° 50' N we add the difference of latitude now discovered 1° 17.3', the ships course carrying her farther to the northward of the equator, we obtain 35° 07.3' N. lat. for her place at the end of the

traverse.

In complete treatises on navigation tables are found containing the difference of latitude and the departure, corresponding to any given distance from 1 mile to 100, 200, 300, miles according to the extent of the tables,and to every course from one quarter of a point to 74 points, as also to every degree of the quadrant. By the use of such tables much time and labour may be saved to the mariner: but as it is almost impossible to prevent errors from creeping into numerical tables through the inadvertency of the calculator or the printer, it is always desirable that the difference of latitude and the departure should be calculated, conformably to the rules laid down in the preceding cases of plane sailing.

PARALLEL SAILING.

Hitherto the earth has been considered as one vast extendded plane, having the degrees of latitude and longitude every-where of equal dimensions; this supposition however we know to be false, and in navigation it must be the source of many errors of the greatest importance to seamen. The earth being in fact spherical, the circles of latitude drawn round the poles and parallel to the equator, diminish in circumference in gradation as they recede from theequator; and consequently all these parallels being supposed to be divided into the same number of degrees of longitude, 360, each degree must occupy a space on the globe greater or smaller in a certain ratio to its distance from the equator or the poles. This ratio is that of the sine of the complement of the latitude to radius, so that knowing the latitude of

any

any place we can discover how much of the circumference of the earth on that parallel corresponds to any given portion of the eircumference at the equator. Thus for example, a degree of longitude on the equator containing 60 minutes or nautical and geographical miles, a degree of longitude at London, situated in N. lat. 51° 31', will contain only 87 of the same miles.

As radius

To S. comp. of lat. or of
So 60 m. 1 deg. at equator

To m. in 1 deg. at London

1

About 37 nautical miles therefore at London or any where else in 51° 31' N. lat. is equal in longitude to 60 of such miles at the equator.

By this example when the difference of longitude between any two places situated on the same parallel of latitude is given, we can discover their distance, and vice versa, from their distance we can discover their difference of longitude: thus if a ship sail from a port in N. lat. 45° 30' to another situated due W. from the former, or on the same parallel of latitude, but differing in longitude 35o 20', what is the real length of her voyage in nautical miles or minutes of a degree on the equator?

The difference of longitude 35° 20', reduced into minutes is 2120 minutes of the equator, which quantity employed as the 60 m. in the former example, will give the number of nautical miles of distance between the two ports.

As radius

To co-s. of lat. 45°. 30'

So diff of long. on the equator 2120
To diff of long. in miles on the parallel of the
given latitude, equal to the true distance 1485

A

Again if the distance between the two ports both in N. lat. 45° 30', be given 1485 nautical miles, by reversing the proportion we have this formula for discovering the diff. of longitude.

A ship from a port in N. latitude and W. longitude, sails due W. for 1485 nautical miles, and then finds by observation her difference of longitude to be 35° 20'; required the latitude of the parallel on which she sails.

To solve this case the following proportion formed on the foregoing is to be employed.

As given diff. of long. 35°. 20′ = 2120'

= 3.32634

MIDDLE LATITUDE SAILING.

This is a method of solving problems in navigation where the ship's course on the globe is neither on a meridian Nor S, nor on a parallel of latitude E or W, but in a direction oblique to both; and it is so named because in the calculations use is made not of the parallels sailed from or come to, but of that lying in the middle equidistant from both. This method is not however strictly correct, as it gives always a result somewhat less than the truth; bnt in short runs near

the

the equator, and on courses diverging less from the parallel of latitude than from the meridian of the place sailed from, the ship's place may by it be discovered with sufficient accuracy for ordinary purposes.

To conceive whence arises the inaccuracy of middle latitude sailing we must consider that the course performed by a ship sailing in any direction between the meridian and the parallel of latitude of the point of departure is very different from the circumference of a great circle of the globe passing through that point, or crossing the equator at an angle of inclination equal to that of the ship's course. Supposing there existed a globe in all respects similar to this earth, excepting that its surface entirely covered with water were freely permeable to shipping in every direction: supposing also that this globe were cut by the plane of a great circle passing through its center in any direction inclined to the equator and the meridian of the point of intersection, as for instance in a NE direction or in one equidistant between N and E, that is, making an angle of 45' with both the equator and the meridian of the point of intersection. Were the meridians all situated in one plane, and parallel the one to the other, the plane of this great circle would form with each an angle of 45o, equal to that formed with the first meridian, (Geom, vol. i. p. 353): but as this is not the case, the meridians converging to and centering in the pole, it is evident that the circumference of the given great circle must form with each successive meridian an angle more and more deviating towards E from the original inclination NE, until at the distance of 90° from the first meridian it cut the meridian at right angles or in a direction due E. Proceeding for another 90", the circumference will cross the equator to the S. at an angle of 45°, or in a direction SE; when arrived at the 3d quadrant it will again be at right angles to the southern meridian, and at last return to the original point of intersection, forming an angle with the equator of 45°, or in a

direction