This difference of longitude is westerly, because the column of westerly departure is greater than that for easterly departure. It is, however, more accurate to calculate the difference of longitude on each separate run than as here to compute it in all the runs made in one day.

N. B. Maps or charts constructed on Mercator's, or to speak correctly on Wright's principles, are called by the French Cartes réduites, and by the Spaniards Cartas esfericus or reducidas, terms signifying reduced or spherical charts and maps.

OBLIQUE SAILING.

Under the head Practical Geometry (vol i. p. 425) were given rules for the application of trigonometry, both right angled and oblique angled, in the measurement of distances accessible and inaccessible: the same rules are equally applicable to certain parts of navigation, as in running along a coast to determine the ship's distance from it by observing the bearings and distance between objects on shore, in making the survey of a bay or other portion of the coast, &c. as in the following examples:

1st. A ship in passing down the English Channel observed at 10 o'clock A. M. that the Start Point, in Devonshire, bore W. by N. and at two P. M. the ship in the mean time running S. W. by W. four knots (that is miles) per hour, the same point was again observed to bear NNE: required the distance of the Start from the ship at each place of observation.

At the first station the point bore from the ship W. by

N. or one point to the northward of W.; she then steered SW by W. that is three points to the southward of W. the angle thus formed at the first station by the bearing of the point and the ship's course must have been four points or 45°. Again at the second station the Start bore NNE. or two points to the eastward of N. while the course sailed, reversed that is NE by E. from the same second station was five points from N. the difference between these two bearings three points or 33° 45' is the angle formed at the second station.

Having thus in a plane triangle two angles given 45° and 33° 45', their sum 71° 45' subtracted from two right angles 180° will leave 101° 15' for the angle formed at the Start by lines drawn to the ship at both stations: the side representing the ship's course is also given, for she sailed four miles an hour for four hours, in all 16 miles between the places of observation.

It was shown in Prop. 2 of Trignometry (vol. i. p. 405) that in all plane triangles the sides are to each other in the proportion of the sines of the angles respectively opposite to each. In the present case we have all the angles and one side, if then we state the proportion, as the sine of the angle at the Start opposite to the given side or ship's course 101° 15' to the sine of the angle at second station 33° 45′ so is the given side 16 miles to the ship's distance from the Start at the first place of observation, which will be found a little more than nine miles. Working in a similar way for the remaining side of the triangle, as the sine of the angle at the Start 101° 15' to the sine of the angle at the first station 45°, so is the given side or ship's course 16 miles to the ship's distance from the Start at the second station, which comes out to be a little more than 114 miles (See vol. i. page 430.)

ed. A ship coming in with the land in the night observed two light-houses, the first N. 20° 15′ E. and the

second

second N. 53° 40′ W. the ship stood on in a due W. course for 12 miles, and again observed the first bearing N. 56° 09′ E. and the second N. 9° 28' W: required the distance of the ship from both light-houses at each station, also the bearing and distance of the one light-house from the other.

In this example we have two triangles, of which are given all the angles and one side common to both. The other sides may therefore readily be found by employing the proportion that the sides are to each other as the sines of the angles opposite to cach respectively; and the manner of performing the whole problem may be seen in Example 5th of Pract. Geom. page 431 of vol i.

The first thing to be done is to determine the quantity of the angles formed at the ship at both stations, by lines drawn to each light-house, and by the line of her course: thus at the first station the first light bore from the ship 20° 15' to the eastward of N. while the second light bore 53° 40' to the westward of N. these two quantities therefore added together will give the angle formed at the first station by lines from both lights, equal to 73° 55'. Again at the second station, the first light was observed to bear 56° 09' E. of N. while the second light bore 9° 28' W. of N. the sum of these two quantities 65° 37' is the angle formed at the second station by lines drawn to the two lights.

Further, by subtracting from the angle of the ship's course, due W or 90° the bearing of the second light from the first station 59° 40', we obtain the angle formed at the first station by the bearing of the second light, and the ship's course 36° 20′; and subtracting the bearing of the first light from the second station 56° 09′ from 90°, we obtain the angle formed by that bearing and the ship's course 33° 51'

Lastly, by adding together the angles formed at the two stations by the ship's course, and lines drawn to the first light, we obtain 85° 54'; for the angle formed at the first

light by lines drawn to the ship at both stations: in the same way the angle formed at the second light by lines drawn to each station comes out 44° 12'.

Employing now these angles and the given side or ship's course, 12 miles, we find the distances of the ship at both stations from each light-house to be the following:

We are next to calculate the distance and bearing between the two light-houses themselves, which may be done from either station; thus having discovered the distance from the ship at the first station to each light, and the angle formed at that station by their bearings being given, we discover the remaining side opposite to the given angle to be 17.6 miles (see Case 3d of oblique-angled Trigon. vol. I. p. 420.)

Then in the triangle formed by lines from the first station to both lights, and by that from the one light to the other, knowing all the sides the angles are readily discovered: by this means the angle formed at the second light by lines to the first light, and the first station is found to be 38° 24′, which added the bearing of the second station from the second light, S. 9° 28′ E, (N 9° 28′ W. reversed) will give the bearing of the first light-house from the second S. 47 52 E. The two light-houses, therefore, bear the one from the other, N 47° 52′ W, and S. 47° 52' E, distance 17.6 miles, the things required to be known.

This problem is the foundation of the practice in making a survey of a bay or other tract of coast, where it is inconvenient

venient or impracticable to perform the operations on shore; for by carefully observing the true bearings of objects on the land, and measuring the courses and distances run by the ship from one place of observation to another a series of triangles may be formed, of which all the angles are either given or may readily be calculated, and one side is given, that is the ship's course from station to station.

WINDWARD SAILING.

Were a ship or other inert body to be placed in and acted upon by one substance or medium only it would remain at rest or move precisely in the direction and with the velocity of the surrounding medium: but if, as is the case. with a ship, one part of the body, the hull, is immersed in one medium, the water, while another part, the rigging, is surrounded by a different medium, the air or wind, the motion impressed on the ship will be compounded of the effects of the resistance of the different fluids in which she is inclosed; her motion through the water will, therefore, be much slower than that of the wind, in proportion to the greater density and resistance of the water above those of the air; and the pressure of the wind on the ship must amount to a certain quantity exceeding the opposite pressure or resistance of the water before the ship can move at all.

From the structure of the hull of a ship, narrowing gradually to a sharp edge, and also from the arrangement of the sails, by which the pressure of the wind is made to act in a direction oblique to its own, a ship under the impulse of the air, instead of moving forward in the course of the wind, is pressed on with the sharp end forwards, in a course inclined to the direction of the wind, varying according to the circumstances of her construction and management. Suppose the wind to blow from the N off a shore stretch