magnetic courses or bearings. The card we have described is by a particular contrivance so suspended, that it remains perfectly horizontal, notwithstanding the various irregular motions and concussions to which a ship at sea is liable. In the inside of the box containing the card, there are two black vertical lines, which lie in an imaginary straight line drawn through the ship, from head to stern, and which coincide respectively with the N and S points of the card or compass, when the ship's head is towards the magnetic north, or the north which that compass shows; so that when the ship's head is directed towards any other point, the points, or quarter points, which are then opposite to these vertical lines, show the points, or quarter points, of the horizon to which the head and stern of the ship are then directed; now as the circle of the compass card is divided into 32 equal parts, each point must contain 11° 15'; so that if the points marked on the card N b W and Sb E coincide with the vertical lines in the new position of the ship, (the Nb W with the vertical line towards the head, and the contrary,) the straight line drawn through the ship makes an angle of 11° 15' with the line drawn towards the magnetic north, or the magnetic meridian, as it is called: if the ship were now to set sail, and to sail with her head in this position, her apparent course would be Nb W, and her track or wake would make an angle of 11° 15' with the magnetic meridian ;* in this manner are ships steered by the compass :-the card of some compasses is divided into degrees; a compass of this description would show at once the angle which the line drawn through the ship's head makes with the magnetic meridian. (46.) In heaving the log at sea, the half minute is measured, not by a seconds watch, but by a half minute glass, which differs from a common hour-glass in nothing but the time the sand takes to run out. (47.) Besides the plain diagonal decimal scale, of which we have spoken, (Art. 7,) there are others, the most useful of which is that called Gunter's. But as these scales only enable the mariner to solve problems, in which no great accuracy is required, we shall do no more than mention them, and not stop to explain their utility. (48.) We have already described a plane chart, (Art. 19,) it remains to explain the principles on which a chart is constructed of much greater practical no leeway at the time. See Art. 55. This would be the case if the ship were making utility, called a Mercator's Chart.* This chart exhibits a most convenient and ingenious manner of representing the surface of the globe on a plane. The meridians of the globe, or secondaries to the equator, are drawn parallel to one another in this chart, as in the plane chart; and therefore the parts of the parallels of latitude, intercepted between any two meridians, must be equal to one another, and to the intercepted part of the equator in all latitudes; for these parallels of latitude cross the parallel meridians at right angles, and form a number of four-sided figures, which are all right angled parallelograms; (see note Art. 14.) the difference of longitude between any two places may therefore be measured on these parallels, as well as on the equator itself; and the distance between any two meridians is in all latitudes the same, and equal to their difference of longitude: the proportion, however, which subsists between the parts of meridians, and the same elementary parts of parallels, on the globe, is accurately preserved in a Mercator's Chart in all latitudes. If we look at fig. 7, we shall perceive, that though ae in that figure is the same part of the small circle a e b c d, that AE is of the circle AEB CD, or equator; yet that it is much shorter in length; and therefore one minute, for example, or any other elementary part or division of a circle, must be smaller on that circle than on the larger circle, (see Art. 28,) which represents the equator; in fact, the length of A E: length of a e: rad. of equator: rad. of parallel; that is, AE: ae:: AF : a G, for those lines are the radii of the equator and the parallel; but a G is the sine of the arc a P to radius A F, which are is the complement of the latitude of the place a, therefore a G is the cosine of the latitude of a; (Art 29 and 38;) also the length of a minute, on a meridian of the globe, is equal to the length of a minute on the equator, for all great circles of a sphere are equal. Hence we have, Length of 1' on meridian : length of 1' on parallel rad. : cos. lat. of parallel. Therefore in the globe the proportion between the length of an elementary part of a meridian and the length of the same elementary part of a parallel is, as rad. * Mercator never divulged the principles on which he constructed the charts which bear his name. It was to Mr. Edward Wright, that science was indebted for the first exposition of those principles. : cos. of the latitude in which that parallel is situated. It is this proportion of radius to the cosine of the latitude of the parallel, which is preserved in a Mercator's Chart. As on the globe the length of 1', taken on a parallel, becomes very small as we approach the pole, compared with its length on a meridian; it is evident that, as all the parallels in our chart are equal to the equator, in order to preserve this proportion, our meridians must be very considerably lengthened; and the more so as we approach nearer to the pole, in the neighbourhood of which the natural parallels are so very small. It is the mode of lengthening these meridians in proportion as the parallels on the chart are lengthened beyond their natural size, we have now to investigate. Now, cos. lat.: rad.:: rad. : sec. lat. (Art. 31,) and our proportion becomes, We began by observing that the parts of meridians on our chart were to have the same proportion to the same elementary parts of parallels, as they have on the globe: that proportion we have just expressed; and, therefore, in our chart also, 1' on parallel l' on meridian :: rad. : sec. lat. or, 1' on parallel rad. :: 1' on meridian: sec. lat. (See Algebra, Art. 127 and 128.) Let us commence, therefore, the construction of the chart by drawing a line to represent the equator, divide it into equal parts, which we will call 1' each; draw parallels, and through the equal divisions of the equator draw the meridians parallel to one another, and at right angles to the equator; and the parallels will likewise be divided into equal parts of l' each. Now if we suppose that the line which represents 1' of longitude on all the parallels, or on the equator of our chart, represents also the length of the radius of the natural trigonometrical tables, it is evident; looking at our proportion, that l'on a meridian should also represent the natural secant of the latitude in which the measure is taken; or the parallel of the latitude of one minute, for example, ought to intercept between itself and the equator a line equal to the secant of 1' to a radius equal to l' on a parallel; for the two first terms in our proportion are made equal, the other two therefore must be equal: in the same manner, when we draw the pa с rallel of the latitude of 2', we must make it intercept between itself and the parallel of 1' a line equal to the natural secant of 2'. Hence to find the distance of any parallel in the chart from the equator, or the number of lines representing elementary parts of parallels, which the part of the meridian intercepted between them contains, we have only to add the natural secants together, until we arrive at the secant of the given parallel: thus the part of the meridian intercepted between the equator and the parallel of 5', or the projected meridian, as it is called, = sec. 1' + sec. 2' + sec. 3′ + sec. 4'+ sec. 5'. This distance, which we have found for the parallel of 5', is called also the meridional parts of 5'; the meridional parts or distances are computed to every minute of the quadrant as far as 86°, and inserted in tables.* It is frequently necessary, in the solution of nautical problems, to refer to the number of these meridional parts intercepted between the parallels of two places, in lieu of their real or natural difference of latitude. This circumstance has introduced the terms, proper difference of latitude, and meridional difference of latitude; the latter term being used to express the number of meridional parts intercepted between the parallels of any two places on the earth's surface, when they are delineated on a Mercator's Chart. CHAPTER IV. Principles of Navigation.-Invention and Construction of the Four Triangles.-Proportions derived there from.-Examples. (49.) WHEN a ship sails due north or due south she does not alter her longitude, and when she sails due east or west she does not alter her latitude. These are the two most simple cases that can be proposed, in which it is necessary to take into account the spherical figure of the earth, and, therefore, with these cases we will commence. Let us suppose the ship (fig. 7) to sail on a meridian from the place ƒ to the place e, now the longitude of these two places is the same, (Art. 40,) and, consequently, the ship has not changed her In computing these tables, however, the addition of the natural secants is not the method resorted to, as they can be calculated more conveniently and correctly from the expression, log. merid. part=3,8981895+log. (10-log, tan. complement latitude.) See a paper by Dr. Halley, Trans. No. 219, longitude; the whole of her progress during the voyage, has been effective only in changing her latitude: the nautical distance also which she has made, is the arc of a great circle of the earth, and the number of miles in that distance, which is ascertained by the process of heaving the log, is the number of minutes in the arc fe, and represents the difference of latitude; so that by that process alone, the ship's place at e can be determined. In this instance only, and when she sails on the equator, is a ship, steered by a compass, able to trace out the arc of a great circle of the earth; and, consequently, these are the only two cases in which a ship, sailing on any given course from point to point, takes the shortest possible road between those points:-for the arc of a great circle is the shortest line which can connect two points on a sphere. (See the Treatise on Math. Geog. chap. 7.) (50.) Next let us suppose the ship sails due west from e to a; she evidently does not change her latitude, for she sails on a parallel of latitude all the points in which have the same latitude, (Årt. 39.) This second case, however, is not quite so simple as the former, unless indeed the ship sails on the equator itself; for here the heaving the log only gives the length of the arc of the parallel e a, whilst the mariner, in computing the place of his ship, must know the change of longitude,which so much nautical distance sailed on that particular parallel produces; that is, he must know the length of the arc E A on which the difference of longitude is measured; but we have seen (Art. 48) that, cos. lat. rad. ::ea: EA, and therefore E A=eux; rad. cos. lat. consequently, from this equation, E A may be found, and the place of the ship determined. This is called a case of parallel sailing. If we take, however, any right angled triangle BEF, (see fig. 11, page 22.) BE:EF:: rad. cos. BE F, or BE rad. = EFX (Art. 29.) Make cos. BEF therefore the angle BEF equal to any given latitude, and if E F represents the length of the part of the parallel which a ship has traversed in that latitude, or the arc ea, BE will represent the differSuppose that a ship sails twenty miles in ence of longitude made, or the arc E A. the parallel of London, then if BEF= rad. x 20 cos. lat. the latitude of London, BE the difference of longitude = about 32 miles. So that, in resolving cases of parallel sailing, we need not concern ourselves with the sphere, but may advert to this right angled triangle, by which we may see exhibited, and may measure, the comparative lengths of the corresponding parts of the parallel and equator. We shall now show that all cases of sailing whatever may be resolved through the medium of right angled triangles. (51.) We proceed to the third and by far the most difficult case, where the ship sails neither on a meridian, nor the equator, nor on a parallel; in which case she is continually changing both her latitude and longitude. In fig. 9, let P represent the north meridians, and AG, ML, and FE porpole of the earth, PF and PE parts of tions of parallels of latitude; let the ship sail from A to E. It is required to investigate rules by which, if the nautical distance sailed and course be given, the Fig. 9. differences of latitude and longitude between any two places, as A and E, may be found; and conversely, if the differences of latitude and longitude be given, the nautical distance and course may be found. It will be necessary, however, first to define some lines of which we shall have occasion to speak. The curve A E in the figure, which is the line the ship traces out in sailing from A to E, without altering her course by compass, is called a rhumb* line. A rhumb line may be defined to be the shortest line which can join two points on the globe, cutting all the meridians which it crosses at the same angle. From the nature of the compass it is evident, that a ship's way, so long as she steers the same course by a compass, must make the same angle with every meridian she crosses; she therefore must trace out the line we have defined; This name is derived from the Portuguese word rumbo, or rumo, which signifies a course. ↑ When there is no leeway, (see Art. 55.) and the angles c A b, def, &c. which her track makes with the meridians, or courses, must be all equal to one another. The angles a ship's way makes with the equator and parallels, when she sails on a meridian, or with meridians, when on the equator or parallels, are equal, since they are always right angles. The oblique rhumb line is called also the Loxodromic curve, from two Greek words signifying an oblique course, and is a line of a very peculiar nature; it is a spiral, and has the remarkable property of winding round and round the pole of the earth, constantly approaching, yet never reaching it: so that if a ship could sail on the same oblique course for ever, she would approach infinitely near, either to the north or south pole, but could never actually reach them. the nautical distance, and therefore the In the triangle AFE, AE represents portion of a rhumb line intercepted between any two places through which the rhumb line passes, is their nautical distance; A F also represents the difference of latitude, and the angle EAF the course. The meridian distance made, is the distance between the meridian arrived at and the meridian left, measured on the equator, or the parallel on which the ship is; thus, when a ship sails from A to E, FE is the meridian distance she has made; but when she sails from E to A, then GA is the meridian distance; and when she sails on a parallel, or the equator, the arc of the parallel or equator she describes is itself her meridian distance. In parallel sailing, therefore, the ship actually measures her meridian distance as she proceeds; but in sailing on an oblique rhumb line, it is the oblique rhumb line which she measures. the rhumb line AE divided into four (52.) Construct fig. 9 thus, suppose equal parts in the points c, d, and e; and through those points draw three meridians; then draw the parts of parallels bc, fd, and ge; the result is the formation of the four small triangles Abc, cfd, dge, and eh E; but if instead of four small triangles, constructed in the manner above mentioned, we were to imagine many thousand, in short, an indefinite number so constructed, by a continual subdivision of the rhumb line A E, and a continual drawing of meridians through the points of division, and to consider the triangl A be one of such infinitely small triangles; then the bases bc, &c. of these numerous triangles added together, would represent what is termed the departure made in sailing from A to E; and as an arc of a parallel of latitude, or of the equator, between any two places, may be supposed to consist of an infinite number of such small lines, it represents the departure between the two places by which it is intercepted. That this departure answers to the line which we have called the departure in the first Part, will appear from the following considerations: as the triangles A bc, &c, are indefinitely small, the lines composing them may be assumed to be straight lines; suppose, therefore, the ship to sail at first only from A to c, an indefinitely small distance; in this case the line b c represents the line we have hitherto called the departure, for it is a straight line perpendicular to a meridian; so when the ship continues her vogage and arrives at E, she has sailed over an indefinite number of lines, all equal to A c; now the very small right angled triangles Abc, &c. have their hypotenuses equal, and all the angles of the one equal to all the angles of the others, each to each, therefore the triangles are equal and similar;* therefore the ship has also made an indefinite number of small departures all equal to and identical with bc; therefore their sum bc + &c. must be the departure to the sum Ac+ &c. or A c taken as many times as b c, or the nautical distance A E. As the several small elementary. triangles Abc, &c. are equal, whatever number of times the line AE contains the line A c, the same number of times will the sum Ab+ &c. contain Ab, and the sum bc + &c. contain bc; hence we have sum of all the A. c's: sum = or of all the Ab's: Ac: Ab; and sum of all Ab's sum of all be's: Ab: bc. But the sum Ab+ &c. AF the difference of latitude; (Art. 42;) and the sum bc + &c. = the departure. Let there be a plane triangle A" E" F", having the angle at its vertex A", E" A" F", equal to the angles c A b, &c. or the course, and having the side A" E" equal to the number of nautical miles in the spiral line A E, and the side A" F equal to the number of nautical miles in the difference of latitude A F; in other words, let A"E" A E, and A" F" AF: we have already proved = Simson's Euclid, book 1. prop. 26. and book 6. prop. 4. and (Art. 5.) The triangle A" É" F" is not drawn. that, sum of all the A c's: sum of all the A b's Ac: A b, that is, Ac: Ab:: A" E": A" F". Now it is demonstrated in treatises on geometry,* that if two triangles have one angle of the one equal to one angle of the other, and the sides about the equal angles proportionals, the triangles are equiangular, and consequently similar; we have seen that the two triangles A bc, A" F"E" are so circumstanced, and therefore they are equiangular, but Abc is a right angled triangle, therefore A" F" E" is also a right angled triangle; and Ab: bc:: A" F": F"E"; and we have shown, A b : bc:: sum of all the Ab's: sum of all the bc's, but the sum of all the Ab's= A" F", and the sum of all the be's the departure; therefore the last proportion becomes, Ab: bc:: A" F": departure, A" F" xbc Ab hence departure = A" F" xbc. Ab = but F"E" hence F"E" = the departure. So that we have arrived at this conclusion, that when a ship sails on an oblique rhumb line, and the nautical distance made forms the hypotenuse, and the course forms the angle at the vertex of a right angled plane triangle; the perpendicular of that triangle will represent the difference of latitude, and the base the departure made; if any two therefore of these four parts be given, the others can be found; the triangle being a right angled triangle. It appears, therefore, that the departure made by a ship, may be defined to be, the sum of all the successive elementary meridian distances, when the nautical distance is assumed to be divided into an indefinite number of equal parts. The departure therefore is a species of imaginary quantity, the result of an hypothesis made for the purpose of obtaining a straight line to represent it. We shall presently discover, however, that it is of great utility as a connecting link; for it joins and adapts the rule for the solution of cases of parallel sailing to cases of sailing on an oblique rhumb line. Let the meridians in fig. 10 be the representation on a Mercator's Chart of the meridians in fig. 9, marked with the same letters without the dashes; then a straight line A' E' will represent the rhumb line AE on such a chart; for that line has been defined to be the shortest line that, in connecting two places, as A' and E', cuts all the meridians it Simson's Euclid, book 6. prop, 6. |