which is taken as the elementary part of the meridian, so much the more exactly will the length of the parts of it in the projection be obtained. Instead of taking l' as the elementary part, we might have taken 1", and the method of computation would have been precisely the same. The calculation of course would have been length, ened; but even in the latitude of 70° the length of the projected meridian from the equator would not have differed much more than half a minute from that by the calculation having 1' for its base. There are other methods by which the computation of the length of the projected meridian may be more expeditiously made; but the above will give the student a distinct conception of the nature of the artifice by which the proportion between the meridians and parallels is preserved, and of the principles of one method by which he may examine for himself the accuracy of the numbers contained in his table, Now, as in this projection the meridians are parallel straight lines, the rhumb lines which cut the meridians all at the same angle on the globe will all be straight lines cutting them at the same angles in the projection that they do upon the globe; and the distance between any two places on the globe will be to the projected distance, as the difference of latitude of the places on the globe is to the projected or meridional difference of latitude ; and the difference of latitude on the globe will be to the departure, as the projected or meridional difference of latitude, is to the difference of longitude. For let the annexed figure be the Mercator's projection of that part of the figure, p. 148, which is marked by like letters without dashes ; then all the elementary triangles A b B, Bc C, &c. in that F E figure are respectively similar to each other ; ak and, by the nature of the projection, they are also similar to the corresponding triangles A'W B’, B Ć C', &c. in this figure; and as the angles and the sides of the elementary trian H н K'L M'NO gles A b B, B cC, &c. may be collectively represented by the angles and sides of a similar right angled plane triangle, whose perpendicular is equal to A F, hypothenuse to A. E, and base to 6 B + c C, &c.; therefore the sides and angles of the projected triangles Al V B', B' dC', &c. may be represented also by plane triangle, similar to that by which those on the globe are represented, having A F', the projected difference of latitude, for its perpendicular, F'E' or B' b' + Ç¢', &c. the difference of longitude for its base, A' E' the projected distance for its hypothenuse, and the angle F' A' E' equal to the course. B D C angled triangles, and A B be the difference of latitude, E the common angle A the course, and A D the meridional difference of latitude ; then B C will be the departure, or the sum of the elementary meridian distances, A C the nautical distance, and D E the difference of longitude, all exactly. Hence from such parts of these triangles as may be given in any case proposed for solution, the others may be obtained by computation, inspection, or otherwise. The following obvious proportions will be found useful in practice. 1. AB:BC::AD:DE 2. AD:DE: : rad : tan 2 A That is diff lat : dep : : mer diff lat : diff long. And mer diff lat : diff long :: rad : tan . From the preceding elementary view of the principles of navigation, it will be seen that the whole business of practical computation resolves itself into the solution of plane triangles ; and that therefore navigation may be considered as only a particular application of plane trigonometry, course. ON THE INSTRUMENTS BY WHICH A SHIP'S COURSE AND DISTANCE ARE DETERMINED AT SEA. The course of a ship, or the angle which the rhumb line on which she sails makes with the meridian, is determined by an instrument called TøE COMPASS ; which is merely a circular card suspended horizontally on a point, and having a magnetised bar of hardened steel, called THE NEEDLE, for one of its diameters. The circumference of the card is generally divided into thirtytwo equal parts, called points, and each of those divisions is again subdivided into four parts, called quarter points. A point of the compass being therefore the 32d part of the circumference of a circle, is equal to 11° 15'. But in some compasses, for delicate observations, particularly those called azimuth compasses, the rim of the card is divided into degrees. The magnetised needle has the peculiar property of pointing always in a particular direction, generally not far from the direction of the meridian. That point of the card which coincides with the northerly end of the needle is called the magnetic north, and the opposite point the magnetic south ; and, looking towards the north end of the needle, the middle point on the right, between the north and south, is called the east, and the opposite point the west. These four are called cardinal points, and the others are named according to their situation with respect to these cardinal points, as in the annexed figure. The situation of the needle with respect to the meridian is not the same at every place, nor is it always the same at the same place. At present at London the north end of the needle points about 241° towards the west of the true north point of the horizon, but at the North Cape it points only about 1° towards the west, while in some parts of Davis's Straits its direction is more than 64 points towards the west, and near Cape Horn it points about 22° towards the east of the true north. Again in the year 1580, the direction of the needle, at London, was about one point towards the east of the north, while, as has been already observed, it at present points about 241° towards the west. But in the West Indies, for a very long period, the deviation of the needle has undergone but a very trifling variation. Delicate observations appear to indicate that it is again at London retrograding towards the east, and Mr. Barlow, in his valuable “ Essay on Magnetic Attractions," observes that all the phenomena attending the progressive change of the needle's deviation from the meridian, may be accounted for by conceiving the magnetic pole to revolve in a parallel of latitude, from east to west ; but that every place appears to have its individual pole. The direction of the needle, or as it is called, the variation of the compass, may however be determined at any time, (as will afterwards be shewn) by astronomical observations, the points of the horizon, which correspond to the several points of the compass, may therefore easily be found by allowing for the variation, when it is known. Thus if it be found that the north end of the needle points to the NN W point of the horizon, the compass. is then said to have two coincide with the meridian, the east point of the compass with the EN E point of the horizon, &c. But if the north point of the compass points to the N E by N point of the horizon, the compass is said to have three points easterly variation, and the N W by N point of the compass will coincide with the meridian, the east point of the compass with S E by E point of the horizon, &c. If therefore a ship is steered N W by a compass which has two points westerly variation, the angle which her way makes with the true meridian will be six points, or the ship's true course will be W NW. Hence when the compass course is given to determine the true course, allow the variation, if it be westerly, to the left of the compass course, and when easterly to the right of the compass course. On the contrary, when the true course is known, and the corresponding course is required by a compass whose variation is given, allow the variation, when it is west, to the right of the true course, and when east to the left of the true course, and the point thus determined will be the required compass course. When a needle which is balanced horizontally on a point is magnetised, it not only acquires the property of pointing in a particular horizontal direction, but it loses its balance, or becomes inclined to the horizon; and it requires an additional weight to be applied to the elevated end of the needle to restore it to its horizontal position. This inclination of the needle to the horizon is called the dip, and as it is different in different situations, a magnetised needle which is horizontal in one place, will not be horizontal in another. The weight therefore which is a counterpoise to the dip in one place will not be so in another; and on this account needles properly fitted up for mariners' compasses have a sliding weight applied to them, which acts with greater or less force, according as it is at a greater or less distance from the centre of the needle. The needle, with its apparatus, is generally placed in a brass case, which being slung in gimbals, the card is always at liberty to assume a horizontal position ; and in the inside of the case there are two black vertical lines, which with respect to the card are diametrically opposite to each other. The imaginary horizontal line joining these two vertical ones ought to be exactly in the vertical plane cutting the ship from stem to stern, and the point of the card which coincides with the vertical line towards the stem of the vessel will indicate the direction of the ship's head, or shew her apparent course by the compass. No iron whatever should be allowed to be near the compass. Indeed it has been lately noticed, that the whole mass of iron in a ship often exerts a perceptible influence on the direction of the needle, which varies according to the situation of the ship's head with respect to the magnetic meridian. We shall afterwards advert more particularly to this local attraction, when we consider the methods by which the variation of the compass may be found astronomically. A ship’s rate of sailing is estimated by heaving into the sea a piece of wood called The Loo, so loaded with lead that it will just swim. The log is then conceived to remain stationary in the water, and a line is attached to it, called the Log LINE, which at its other end is wound round a reel. The reel being turned, the part of the line that is withdrawn from it by the log in a given time will be the distance which the ship has run from the log during that interval ; and hence, by proportion, her distance for any other time may be obtained, while her rate of sailing continues the same. The log is made in the form of a sector of a circle, and the lead with which it is loaded is applied to the arc; the central point will therefore be vertical. The line is so attached to it that the flat side of the log is kept 'towards the ship, that the resistance of the water against the face of the log may prevent it, as much as possible, from being dragged after the ship by the weight of the line or the friction of the reel. The time which is usually occupied in determining a ship's rate is half a minute, and the experiment for the purpose is generally made at the end of every hour, but in common merchantmen at the end of every second hour. As the time of operating is half a minute, or the hundred and twentieth part of an hour, if the line were divided into 120ths of a nautical mile, whatever number of those parts a ship might run in half a minute she would, at the same rate of sailing, run exactly a like number of miles in an hour. The 120th part of a mile is by seamen called a knot, and the knot is generally subdivided into smaller parts, called fathoms. Sometimes (and it is the most convenient method of division) the knot is divided into ten parts, more frequently perhaps into eight; but in either case the subdivision is called a fathom. In ships where no great accuracy in navigation is attempted, the knot is subdivided into four parts, and sometimes only into two. We shall however consider a fathom as the tenth part of a knot, and as a nautical mile (p. 148) is 6079 feet, the 120th part of this, or the length of a knot, will be 50:66 feet, or nearly 50 feet 8 inches. Hence a fathom ought to be 5 feet and eight tenths of an inch nearly. In practice however 50 feet is generally considered as sufficient for the length of a knot, for the log is always in some degree drawn towards the ship, and therefore the distance given by a correct line is always less than the true distance. The operation for estimating the rate is called by seamen heaving the log. 1 |