4. Required the course and distance from the east point of St. Michael's, lat. 37° 48′ N., long. 25° 13′ W., to the Start Point, lat. 50° 13′ N., long. 3° 38′ W.; the middle latitude being corrected by Workman's tables. Ans. Course N. 57° 11′ E; dist. 1189 miles. Mercator's Sailing. 18. It has already been observed, that when a ship sails on an oblique rhumb, the departure, the difference of latitude, and the distance run, are truly represented by the sides of a right angled triangle. Thus, if a ship sails from A to B, the departure B'B will represent the sum oi all the very small meridian distances, or elementary departures, b'b, p'p, &c.; the difference of latitude AB' will represent, in like manner, the small differences of latitude Ab, b'p', &c; and the hypothenuse AB, will express the sum of the distances corresponding to these several differences of latitude and departure. Each of these elements is supposed to be taken so small, as to form on the sur B P b P B face of the sphere a series of triangles, differing insensibly from plane triangles. Let Abb represent one of these elementary triangles; b'b will then be one of the elements of departure; and Ab' the corresponding difference of latitude. Now, as b'b is a small arc of a parallel of latitude, it will be to a portion of the equator or of a meridian containing an equal number of degrees, as the cosine of its latitude is to radius (Art. 16). This similar portion of the equator, or of the meridian, will be the difference of longitude between b' and b. Let us now suppose Ab to be prolonged until the perpendicular p'p shall become equal to the difference of longitude between b' and b: then, : bb' will be to p'p, as the cosine of the latitude of b'b, to radius. But, b'b : p'p :: Ab' Ap': hence, Ab Ap : :: cos. lat. of b'b : radius; that is, if the latitude be so increased that p'p shall become the true difference of longitude, then, true diff. lat. Ab' increased lat. Ap' cos. lat. : radius. The increased latitude Ap' is called the meridional difference by d, the increased or meridional difference of latitude by D, the latitude of bb by l, and the radius by 1, which is, indeed, the radius of the tables of natural sines, and we shall have =sec. 7. cos. l D=d secant l, since It then, we know the latitude 7 of the beginning of a course, and the true difference of latitude d of the extremity of the course, we can easily find the meridional latitude D corresponding to that course. Conceiving each elementary distance to be increased in this manner, giving the meridional differences of latitude on the line AC', the sum of all the corresponding elements will be the entire meridional departure during the course. To represent, therefore, the difference of longitude due to any departure, as BB, and to its corresponding difference of latitude AB', we must produce AB' till AC' is equal to the meridional difference of latitude; the perpendicular CC will then be the difference of longitude actually made in sailing from A to B. The determination of AC' requires the determination of all its elementary parts. If d be taken equal to 1', we shall have from the equation above D=1' secl. or D= sec. l, it being understood that l expresses minutes or geographical miles. From this equation, the value of D, corresponding to every minute of 1, from the equator to the pole, may be calculated; and from the continued addition of these there may be obtained, in succession, the meridional parts corresponding to 1, 2, 3',4', &c. of true latitude, and when registered in a table, they form a table of meridional parts, given in all books on Navigation. The following may serve as a specimen of the manner in which such a table may be constructed, and, indeed, of the manner in which the first table of meridional parts was actually formed by Mr. Wright, the proposer of this valuable method. Mer. pts. of 1'=nat. sec. 1'. Mer. pts. of 2=nat. sec. 1'+nat. sec. 2′. Mer. pts. of 3'=nat. sec. 1'+nat. sec. 2′+nat. sec. 3'. Nat. Secs. 1.000000 Mer. Pts. 1.0000000 Mer. pts. of 1'= There are other methods of construction, but this is the most simple and obvious. The meridional parts thus determined, are all expressed in geographical miles, because in the general expression 1' is a geographical mile. D=1' sec. l. Having thus formed the table of meridional parts, if we enter it, and find the meridional parts corresponding to the latitudes of the place left and the place arrived at, their difference will be the meridional difference of latitude, or the line AC' in the diagram. The difference of longitude C'C may then be found by the following proportion. I. As radius is to the tangent of the course, so is the meridional difference of latitude to the difference of longitude. But if the departure be given instead of the course, then, II. As the true difference of latitude, is to the departure, so is the meridional difference of latitude to the tangent of the course. Other proportions may also be deduced from the diagram. EXAMPLES. As an example of Mercator's or rather Wright's, sailing, let us take the following: 1. Required the course and distance from the east point of St. Michael's to the Start point: the latitudes being 37° 48′ N., and 50° 13' N., and the longitudes 25° 13′ W., and 3° 38' W. = Now, let us suppose that we have sailed from A to B: we shall then know AB' equal true diff. lat. 745 miles; AC' = meridional diff. of lat.=1042; and C'C= the difference of longitude equal to 1295 miles. It is required to find the course For the Course. For the Distance. 3.017868 As cos. A. 51° 11' 9.797150 10.000000: AB' 745 1295 3.112270: radius . : tangt. A 51° 11′ E. 16.094402: AB 1189 2.872156 10.000000 3.075006 2. A ship sails from latitude 37° N. longitude 22° 56′ W., on the course N: 33° 19′ E: till she arrives at 51° 18' N.: required the distance sailed, and the longitude arrived at. Ans. Dis. 1027 miles; long. 9° 45′ W. Mercator's Chart. MERCATOR'S CHART is a Map constructed for the use of Navigators. In this chart all the meridians are represented by straight lines drawn parallel to each other, and the parallels of latitude are also represented by parallel straight lines drawn at right angles to the meridians. The chart may be thus constructed. Draw on the lower part of the paper a horizontal line to represent the parallel of latitude which is to bound the southern portion of the chart. From a scale of equal parts, corresponding in size to the extent of the map to be made, lay off, on this line, any number of equal distances and through the points draw a series of parallels to represent the meridians. Then draw a line on the side of the map, and for the second parallel of latitude, find from the table of meridional parts the meridional difference of latitude corresponding to the degrees between the first and second parallel, and lay off this distance for the interval between the two parallels. Then find the meridional difference between the second and third, and lay it off in the same way for the third parallel, and so on, for the fourth, fifth, &c. A place whose latitude and longitude is known, may be laid down in the same manner; for it will always be determined by the intersection of the meridian and parallel of latitude. If the chart is constructed on a small scale the divisions on the graduated lines, may be degrees instead of minutes; and the meridians and parallels may be drawn only for every fifth or tenth degree. We have already seen (Art. 18.), that the meridional difference of latitude bears a constant ratio to the difference of longitude, so long as the course remains unchanged: and hence we see that on Mercator's chart, every rhumb will be represented by a Line of Meridional Parts on Gunter's Scale. This scale corresponds exactly with the table of meridional parts, excepting, that in the table the circle is divided to minutes, while the scale is divided only to degrees. A scale of equal parts is placed directly beneath the scale of meridional parts; if the former corresponds to divisions of longitude, the latter will represent those of latitude. Hence, a chart may be constructed from these scales by using the scale of equal parts for the lines of longitude, and the scale of meridional parts for those of latitude. THE END. |