EXAMPLE. What is the compass course corresponding to a magnetic course S. 71° W. ? By comparing the diagram with the deviation table in Art. 58 its construction will be readily seen. CORRECTING COURSES. Art. 60. In Fig. 20, NCS represents the meridian; N. and S. the North and South points of the horizon; CV the direction of the magnetic meridian, or the direction of the compass needle affected by terrestrial magnetism alone; CD the direction of the needle under the influence of the iron in its vicinity; then the angle NCV is the variation, east in this case; the angle VCD is the deviation west. If a ship is sailing on the rhumb CT, her true course is NCT; her compass course is DCT; hence, it will be seen that the true course and the compass course can be derived from each other by proper application of the variation and deviation. In reducing compass course, DCT, to the true course, NCT, the algebraic sum of the two corrections may be applied at once. But to find the compass course, DCT, from the true course, NCT, the two corrections must be separately applied, since tables of deviations are made for different directions of the ship's head, as indicated by compass; hence, the true course must first be corrected for variation to get VCT, the compass course nearly, and then take from the table the proper deviation for that direction of the ship's head.* One more correction must be considered in finding the course made good when the ship is under sail, and not running before the wind. A ship sailing on the rhumb CT, under the influence of the wind indicated by the arrow, is deflected from the course steered at a certain angle TCG, the value of which depends upon various circumstances; therefore, the angle TCG must be determined and applied to the right or left of the indicated compass course, according as the ship is on the port or starboard tack, to obtain NCG, the true course made good. In fact it may occur that in the case of a screw steamer under sail there are seven distinct causes to produce the course made good, all of which must be considered under certain circumstances. These are W N D FIG. 20. 5. Variation of the Compass. 6. Deviation of the Compass. G E EXAMPLES. Let it be required to correct for deviation the following courses steered, using the Table of Devi EXAMPLE. Given Compass course E. N. E. and variation 12 E., to find the true course: N. 5° 37′ W. 5 15 W. N. 10 52 W. S. 50 37 E. Deviation, Course, 19 30 E. 12 EXAMPLE. Given compass course N. by E. and the variation 12 W., to find the true course: EXAMPLE. Given Compass course S. W. by W. and the variation 20° E., to find the true course: In the above examples it will be readily seen that the variation and deviation may be combined, but the fol lowing examples to find the Compass courses will show, not only that these quantities must be separately applied, but also that when the deviations are large this will give only an approximation to the Compass course; then the corresponding deviation must be applied to the correct magnetic course, which will give a second approximation. The proper deviation to apply to the correct magnetic course will fall between the deviations corresponding to these two approximations. EXAMPLE. Suppose a true N. N. E. course is required where the variation is 2 points W. EXAMPLE. Suppose a true W. by N. course is required when the variation is 2 points W. Art. 61. A CHART is a representation upon paper of any portion of the earth's surface for the service of the Navigator; or, in other words, it is the Hydrographer's Map. According to the extent of surface to be represented or the manner used, depends the style of chart. Raper says: "As the surface of the globe is round, while that of the paper is flat, every chart exhibiting any extent of surface is necessarily an artificial construction, or, as it is called, projection of the real state of things." When the extent was limited, such as plans of harbors, islands, small sections of the coast, &c., the chart was formerly constructed on PLANE PROJECTION, but now on the POLYCONIC PROJECTION. For the general purposes of Navigation the MERCATOR'S PROJECTION is adopted, for on this alone is the ship's track represented as a straight line when steering the same course. For the purpose of representing areas of large or small extent along the coast line the Polyconic projection is used, in which each parallel of latitude is supposed to be developed upon its own cone, the vertex of which is on the axis at its intersection with the tangent to the meridian at the parallel. Each sheet is an independent one, and adjoining ones are connected by the triangulation points. For the convenience of representing the track to be followed in Great Circle Sailing as a straight line, special charts are constructed on the CENTRAL, or GNOMONIC PROJECTION. Upon this projection all great circles are represented as straight lines, thereby answering the same purpose for Great Circle Sailing that Mercator's Chart does for Rhumb Sailing. There are many other systems of projection but these are all that are required by the Navigator for purposes of Navigating and Marine Surveying. To construct a Chart on Plane Projection.* Art. 62. Divide the limits East and West by the number of miles of longitude required to be embraced Take out the Meridional Parts, Table 3, for any number of miles of latitude between the extremes of the she say 100 miles. *Mayne's Nautical Surveying. Then, as 100 miles: number of inches measured by the paper number of inches for 100 miles of latitude. = the meridional parts for 100 miles : the If the latitude and longitude of any two places at some distance apart are known, the Chart can be graduated thus: Find a course and distance between the two places, working to hundredths. Take out the meridional parts of 5, 10, or 20 miles between the latitudes of the given places. Take from the Chart the number of inches between the two places. Then, for Latitude Scale: As the distance in miles: the number of inches between the two places inches in 5, 10, or 20 miles of latitude. For Longitude Scale: As the number of miles of longitude (found by taking the difference of the meridional parts of 5, 10, or 20 miles): number of inches in 5, 10, or 20 miles of latitude = 5, 10, or 20 miles of longitude : number of inches in 5, 10, or 20 miles of longitude. Lay off the true North from both places with compasses, and measure off the miles of latitude and longitude, having first taken care that the two places are in their proper positions. These charts are now rarely used. In high latitudes they show truly no directions but North and South, East and West, and no distances but those on a meridian. Hence, the figure of every portion of surface, however small, is distorted. Art. 63. Mercator's projection.-This is not strictly a projection, but may be said to result from the development upon a plane of a circumscribing cylinder tangent to the earth along the equator; the various points of the earth's surface having been projected upon the cylinder in such manner as to satisfy the following condition: That the Rhumb, or ship's track on the surface of the sea under a constant bearing, shall appear on the development as a right line, preserving the same angle of bearing with respect to the meridians it intersects as that of the ship's track. In order to realize the foregoing condition, the equator, being the circumference of a right section of the cylinder, will appear as a straight line on the development, while different right-lined elements of the cylinder, corresponding to the meridians, will appear as a system of equidistant straight lines, parallel to each other and perpendicular to the development of the equator, maintaining the same distances apart and same relative positions on that equator as the primitive meridians have on the terrestrial spheroid. Moreover, the series of parallels of latitude will also appear as a system of right lines parallel to each other and to the equator, and will so intersect the meridians as to form a system of rectangles, whose successive widths must be variable, increasing from the equator toward the poles in such manner that the required equality of angles shall be maintained, for cor responding elements of the ship's track, on the spheroid and on the chart representation. 3, Fig. 21 will illustrate this. Let pp', Pip', Pep′2, Pap's, &c., be arcs of equidistant parallels of latitude intercepted between the two meridians PM, PM', and there fore decreasing in length as they recede from the equator; the figures pp'p'iPi, Pip'iP'2 P2, PzP'ap'3ps, &c., are supposed to be indefinitely small on the projection; the corresponding arcs of the par-allels pp', þ1p' \, pap'2, pap's, &c., are all equal to MM', and therefore have been unduly lengthened, and more and more so as they recede from the equator; hence the increments of the meridian (i. e., the small portions successively added to the length under consideration), which are equal on the globe (ppi, Pip2, Papa, P3P4, &c.), must be magnified in the projection (1, pipa, pips, pap4, &c.), and more and more so as they recede from the equator. If they are made to lengthen in the same proportion as the arcs of the parallels of latitude (Pip'i, P2P2, P3p'3, &c.) were lengthened in the projection (Pip, pala, pap's, &c.), then the small areas of the projection pp'p'ipi, pip'ı p'atz, Pat'e p'sts, &c., will be depicted similar to the corresponding areas of the sphere pp'p'iP1, P1 P'iP'2P2, P2P 2P 3P3, &c., and the representation is a Mercator's projection.* Art. 64. A second system may well be inserted here as a more simple and graphic representation of Mercator's projection. Qa In Fig. 22, C is the centre of the globe; P1P2, two points near one another on the same meridian; AQ1Q2 is a line touching the meridian at the equator. Hence, PICA is the latitude P1, which represent by , and let P1CP1⁄2 the increment of latitude of P2, be dl. Hence, QQ, i. e., the projection of the arc P,P2 on the circumscribing cylinder Now, if dl be taken sufficiently small, tan 7 tan dễ may be neglected in comparison with unity, and for tan đ may be written dl. sec 1. Hence, QQ = CA sec2. l. dl : PP2 sec, but in Mercator's projection the corresponding projection is P1P2 Hence, QQ is greater than the arc of Mercator's projection in the ratio of sec / to I, and as increases this ratio increases, until toward the poles it is indefinitely great. Now, from P2 let fall the perpendicular P2M on the line CA, cutting CQ1, in S1. or, PS represents the arc of Mercator's projection corresponding to the arc of the meridian P2P1. This construction may be stated in words as follows: To represent the proper length of an arc of a meridian on Mercator's projection, draw a tangent to the merid ian at the equator; draw two radii through the extremities of the arc (supposed very small), and from the higher of the two extremities draw a line parallel to the tangent already drawn. The length of the line intercepted between the upper extremity of the arc and the radius drawn to the lower extremity, is the corresponding distance between these points on Mercator's projection.. Art. 65. The following construction shows how the chart itself may be drawn: In Fig. 23 let the arc of the meridian be divided into a sufficient number of equal small portions, AP1, PIP2, P2P3, P3P4, &c. Draw the corresponding radii and the perpendiculars P1M1, P2M2, P3M3 on CA, and let MP1 be prolonged indefinitely; then PM, P2S1, P3S2 are the distances on Mercator's projection corresponding to the arcs AP, PP2, P2P3. Let CA be produced as the base of the projection, and let equal distances N, N2, N2N3, N3N4, &c., be taken on this line to represent equal arcs, say of 1° of longitude. Draw PPPPP," parallel to CA; then this line is the parallel of latitude of the point P1. Draw PR2 parallel to CP1, meeting MP produced in R; draw RP PP parallel to CA; this represents the parallel of the point P ̧. Join SR. Draw PR parallel to SR, meeting MP1 produced in R3, and draw R3P3 P3 P3", &c., parallel to CA; then this is the paralfel of P3; and so, by a similar construction for any number of points. Art. 66. MERIDIONAL PARTS.-At the equator a degree of longitude is equal to a degree of latitude, but approaching the poles, while (supposing the earth to be a perfect sphere) the degrees of latitude remain the same, the degrees of longitude become less and less. In Mercator's projection the degrees of longitude are made everywhere of the same length, and, therefore, to preserve the proportion that exists at different parts of the earth's surface between the degrees of latitude and longitude, the former must be increased from their natural lengths more and more as we recede from the equator. The lengths of small portions of the meridian thus increased, expressed in minutes of the equator, are called Meridional Parts, and the meridional parts for any latitude is the fine, expressed in minutes of the equator, into which the latitude is thus expanded. The meridional parts computed for every minute of latitude from 0 to 90° form the Table of Meridional Parts, which is chiefly used for finding the meridional difference of latitude, in solving problems in Mercator's Sailing, and for constructing charts on the Mercator projection. Art. 67. Mercator's Chart.—The feature which makes this, the Navigator's Chart, chiefly valuable is, that on it the track of a ship always steering the same course appears as a straight line. It represents with perfect accuracy the relative positions of places, but not their relative distances. |