CASE VII. One latitude, distance sailed, and departure given, to find the course, difference of latitude, and difference of longitude. A ship in the latitude of 49° 30' N. and the longitude of 250 W. sails southeasterly 645 miles, making 500 miles departure; required the course steered, and the latitude and longitude in? BY PROJECTION. Draw the meridian ABC, and on A any point of it draw BD perpendicular thereto, and make it equal to the departure 500 miles; with an extent equal to the distance 645 miles in your compasses, and one foot on D as a centre describe an arch to cut AB in A, join AD;" then will AB be the proper difference of latitude 407.5 miles, and the angle BAD will be the course 50° 49'; hence we have the other latitude, and the meridional difference of latitude, to which C make AC equal; and draw CE parallel B Dist 645 to BD, meeting AD produced in E; then will CE be the difference of longitude, 722.6 miles. 1st. The extent from the distance 645 to the departure 500 on the line of numbers, will reach from the radius 90° to the course 50° 49' on the line of sines. 2dly. The extent from radius 90° to the complement of the course 39° 11' on the line of sines, will reach from the distance 645 to the difference of latitude 407.5 on the line of numbers. 3dly. The extent from the radius 450 to the course 50° 49′ on the line of tangents, will reach from the mer. diff. of lat. 589 to the difference of longitude 722.6 on the line of numbers. Or, the extent from the proper difference of latitude 407.5 to the departure 500, will reach from the meridionaldifference of latitude 589 to the difference of longitude 722.7 on the line of numbers. BY INSPECTION. Find the course and difference of latitude, as in Case V. Plane Sailing, by seeking in Tab. II. till the distance and departure are found to correspond in their respective columns, adjoining to which, in the column of latitude, will This log, was found above-it differs a little from the log, of 407.5, be found the true difference of latitude, which if greater than the departure the course will be found at the top; but if less, the course will be found at the bottom: with this course seek the meridional difference of latitude in the latitude column, adjoining to which in the departure column will be found the difference of longitude. Thus one-third of the distance 215, and one-third of the departure 166,7 are found nearly to correspond to a course of 51 degrees, and a difference of latitude of 135.3, which multiplied by 3, gives the true difference of latitude 406 nearly. Then one-fourth of the meridional difference of latitude 147, in the latitude column, is found nearly to correspond to the departure 181.9; this multiplied by 4, gives 727.6 the difference of longitude nearly. Having explained the method of calculating single courses by Middle Latitude and Mercator's Sailing, it now remains to explain the method of calculating compound courses. To do this, you must construct a Traverse Table, and find the difference of latitude and departure for each course and distance, as in Traverse Sailing, and from thence the whole difference of latitude, departure, and latitude in, with the departure and latitudes, find the difference of longitude and longitude in, as in Case II. of Middle Latitude or Mercator's Sailing. This method is exact enough for working any single day's work at sea, except in high latitudes, where it will be a little erroneous; in this case the difference of longitude and longitude in, may be calculated for every single course and short distance; but in general this nicety in calculation may be neglected. To illustrate the method of working compound courses, we shall here work an example, by Middle Latitude and Mercator's Sailing. EXAMPLE. TRAVERSE TABLE. A ship from Cape Henlopen, in the latitude of 380 47' N. longitude 75° 17' Courses. W. sails the following true courses, viz. E. by S. 20 miles, E. N. E. 15 miles, S. E. 26 miles, South 16 miles, W. S. W. 6 miles, N. W. 10 miles, and East 30 miles: required her latitude and longitude? By constructing the Traverse Table with these courses and distances, it appears that the ship has made 27.8 miles of southing, and 69.3 miles of easting; and by subtracting the southing from the latitude of Cape Henlopen there remains the latitude in 38° 19' N. E. by S. 20 S. E. 26 W. 30 Meridional parts 2528 36 By inspection of Table II. it appears that the difference of latitude 27.8. and departure 69.3 correspond to a course of 680 nearly, and a distance of 75 miles; and in the same page of the Table opposite to the meridional difference of latitude, found in the column of latitude, stands the difference of longitude 89 miles in the departure column; this subtracted from the longitude of Cape Henlopen 75° 17' W leaves the longitude in 73° 38' W. by Mercator's Sailing. Or, with the Middle Latitude 38° 33′ to 390 as a course, find the departure 69.3 in the latitude column, opposite to which is 89 in the distance column, which is the difference of longitude by Middle Latitude Sailing; consequently the longitude in is 73° 38' W. as above. Thus we see that such examples are performed as in Traverse Sailing and Case II. of Mercator's or Middle Latitude Sailing, either by Inspection, as above, or by the scale of logarithms. Having gone through the necessary problems in Mercator's Sailing, we shall now show how Mercator's Chart may be constructed by means of the Table of Meridional Parts. To construct a Mercator's Chart to commence at the Equator. Suppose it was required to construct the Chart in the Plate prefixed to this work which begins at the equator, and reaches to the parallel of 50 degrees, and contains 95 degrees of longitude west from the meridian of Greenwich? Draw the line AD representing the equator, then take from any scale of equal parts the number of minutes contained in 95 degrees, viz. 5700, which set off from A to D; subdivide this line into 95 equal parts representing degrees of longitude. Through A and D draw the lines AB, DC perpendicular to AD, and make each of them equal to 3474 which are the meridional parts, corresponding to 50 degrees. Join BC which must be subdivided in the same manner as the line AD; and through the corresponding points of the lines AD, BC must be drawn (at the distance of 100 or 200) the lines parallel to AB, representing meridians of the earth; these lines must be numbered 0, 10, 20, &c. beginning at the line AB which represents the meridian of Greenwich. Set off in like manner upon the meridians AB, DC, (beginning from the equator AD) the meridional parts corresponding to each degree of latitude from 0° to 50°; and through the corresponding points (at the distance of 100 or 200) draw lines parallel to the equator AD, to represent the parallels of latitude. Then the upper part of the chart will represent the north, the lower the south, the right hand the east, and the left hand the west (which is generally supposed in charts, unless the contrary is expressly mentioned.) If the Chart does not commence at the equator, but is to serve for a certain portion of the globe contained between two parallels of latitude on the same side of the equator, you must draw the meridians as directed in the last example; then subtract the meridional parts of the least latitude of the chart from the meridional parts of the other latitudes, and set off these differences on the extreme meridians, draw lines through the corresponding points, and they will be the parallels of latitude on the chart. If the chart is to be bounded by parallels of latitude on different sides of the equator, you must draw a line representing the equator, and perpendicu lar to it draw the lines to represent the meridians, continuing them on both sides of the equator; then set off the parallels of latitude on both sides of the equator, in the same manner as in the first example. Take from the Table of latitudes and longitudes of places the latitude and longitude of each particular place contained within the bounds of the chart, and lay a rule over its latitude and another crossing that over its longitude; the point where these meet will represent the proposed place upon the chart. The most remarkable point of a sea coast being thus laid down, lines may be drawn from point to point which will form the outlines of the sea coast, islands, &c. to which may be annexed the depths of water expressed in common Arabian numbers, the time of high water on the full and change days expressed in Roman numbers: the setting of the tide expressed by an arrow; and whatever else may be thought convenient for the chart to contain. This chart is not to be considered as a just representation of the earth's surface, for the figures of islands and countries are distorted towards the poles, as is evident from the construction; but the degrees of latitude and longitude are increased in the same proportion, so that the bearings between places will be the same on the chart as on the globe; and as the meridians are right lines, it follows, that the rhumbs, which form equal angles with the meridians, will be straight lines, which render this projection of the earth's surface much more easy and proper for the mariner's use than any other. Having the latitude and longitude of a ship or place, to find the corresponding point on the chart. RULE. Lay a ruler across the chart in the given parallel of latitude: take in your compasses the nearest distance between the given longitude and the nearest meridian drawn across the chart; put one foot of the compasses in the point of intersection of the ruler and meridian, and extend the other along the edge of the ruler on the same side of the meridian as the place lies, and that point will represent the place of the ship. If the longitude on the chart be counted from a different meridian from that you reckon from, you must reduce the given longitude to the longitude of the chart, by adding or subtracting the difference of longitude of those meridians, and then mark off the ship's place as before directed. Or, you may draw a meridian line through the place you reckon your longitude from; then measure off the ship's longitude on the equator, and apply it to the edge of the ruler, from this meridian, and you will obtain the ship's place. To find the bearing of any place from the ship. RULE. Lay a ruler across the given place and the place of the ship; set one foot of the compasses in the centre of some compass near the ruler, and take the nearest distance to the edge of the ruler; slide one foot of the compasses along that edge keeping the other extended to the greatest distance from the ruler, and observe what point of the compass it comes nearest to, for that will be the bearing required. To find the distance of any place from the ship. RULE. Take the distance between the ship and given place in your compasses and apply it to the side of the chart or graduated meridian, setting one foot as much above one place as the other is below the other place, the number of degrees between the points of the compasses will be the distance nearly. When the places bear north and south of each other this rule is accurate; but when they bear nearly east and west, and the distance is large, it will err considerably; but in general it is exact enough for common purposes; if greater accuracy is required, it is best to find the distance by calculation. If any one wishes to estimate the distance accurately by the chart, he must proceed in the following manner: 1. If the place be in the same longitude that the ship is in, then the preceding rule is accurate. 2. If the place be in the same latitude as the ship, or bear east or west, the distance cannot be obtained without calculating it by Case I. of Parallel Sailing. 3. If the place be neither in the same latitude, nor in the same longitude as the ship, the distance must be found in the following manner: Lay a ruler over both places, and draw through one of them a parallel to the equator: take the difference of latitude between both places in your compasses from the equator; slide one foot on that parallel, keeping the other extended so that both points shall be on the same meridian, and note the point of the ruler which is touched by the other foot of the compasses, take the distance from this point to the given place through which the parallel was drawn and apply it to the equator, and you will have the sought distance. The bearing and distance of any two places from each other may be found in the same manner as the bearing and distance of any place from the ship. EXAMPLE. Required the bearing and distance between the east end of Long-Island and the north part of Bermudas? A ruler being laid over both places as directed in the preceding rule, it will be found to lay parallel to the N. W. by N. and S. E. by S. line; and the distance between the two places being taken in the compasses, and applied to the graduated meridian, will measure about 10 degrees or 600 miles; therefore these places bear from each other N. W. by N. and S. E. by S. and their distance is 600 miles nearly. OF THE LOG-LINE & HALF MINUTE GLASS. VARIOUS methods have been proposed for measuring the rate at which a ship sails, but that most in use is by the Log and Half-Minute Glass. The Log is a flat piece of thin board, of a sectoral or quadrantal form, (see Plate VI. Fig. 3) loaded on the circular side with lead sufficient to make it swim upright in the water: to this is fastened a line about 150 fathoms long, called the Log-line, which is divided into certain spaces called knots, and is wound on a reel (see Plate VI. Fig. 4) which turns very easily. The Half-Minute Glass is of the same form as an Hour Glass, (see Plate VI. Fig. 2) and contains such a quantity of sand as will run through the hole in its neck in half a minute of time. The making of the experiment to find the velocity of the ship is called heaving the log, which is thus performed. One man holds the reel, and another the half-minute glass; an officer of the watch throws the log over the ship's stern, on the lee side, and when he observes the stray line is run off (which is about ten fathoms, this distance being usually allowed to carry the log out of the eddy of the ship's wake) and the first mark (which is generally a red rag) is going off, he cries turn! the glass holder answers done! who watching the glass, the moment it is run out says stop! the reel being immediately stopt, the last mark run off shows the number of knots, and the distance of that mark from the reel is estimated in fathoms. Then the knots and fathoms together, show the distance the ship has run the preceding hour, if the wind has been constant. But if the gale has not been the same during the whole hour, or time between heaving the log, or if there has been more sail set or handed, a proper allowance must be made. Sometimes when the ship is before the wind, and a great sea setting after her, it will bring home the log; in such cases it is customary to allow one mile in 10, and less in proportion if the sea be not so great; allowance ought also to be made if there be a head sea. This practice of measuring a ship's rate of sailing is founded upon the following principle: That the length of each knot is the same part of a sea mile, as half a minute is of an hour. Therefore the length of a knot ought to be of a sea mile; but by various admeasurements it has been found that the length of a sea mile is about 6120 feet; hence the length of a sea knot should be 51 feet: each of these knots is divided into 10 fathoms of about 5 feet each. If the glass be only 28 seconds in running out, the length of the knot ought to be 47 feet and 6 tenths. These are the length generally recommended in books of navigation, but it may be observed, that in many trials it has been found, that a ship will generally over-run her reckoning with a log-line thus marked; and since it is best to err on the safe side, it has been generally recommended to shorten the above measures by 3 or 4 feet, making the length of a knot about 74 fathoms of 6 feet each, to correspond with a glass that runs 28 seconds. In heaving the log you must be careful to veer out the line as fast as the log will take it; for if the log is left to turn the reel itself, the log will come home and deceive you in your reckoning. You must also be careful to measure the log-line pretty often, lest it stretch and deceive you in the distance. Like regard must be had that the half-minute glass be just 30 seconds, otherwise no accurate account of the ship's way can be kept. The glass is much influenced by the weather, running slower in damp weather than in dry. The half-minute glass may be examined by a watch with a second hand, or by the following method-Fasten a plummet on a line and hang it on a nail, observing that the distance between the nail and middle of the plummet be 39 inches, then swing the plummet and notice how often it swings while the glass is running out, and that will be the number of seconds measured by the glass. To correct the distance when the log-line and half-minute glass are faulty. If there be any error in the log-line or glass, the measured distance müst |