tance, as also the difference of longitude between the two places? Lat. from 51° 18′ N. Sum of latitudes 37° Lat. to Diff. lat. As distance 1024 : radius :: diff. lat. 858 cos course 33° 5' 6.989700 Cos mid lat 44° 9' ar c 0.144167 10.000000 tang course 33° 5' 9.813899 2.933487:: diff. lat. 858 9.923187 diff. long. 2.933487 779 2.891553 In this operation the middle latitude has not been corrected, so that the difference of longitude here determined is not without error. To find the proper correction, look for the given middle latitude, viz., 44° 9', in the table of corrections, the nearest to which we find to be 45°; against this and under 14° diff. of lat. we find 27'; and also, under 15° we find 31', the difference between the two being 4'; hence, corresponding to 14° 18' the correction will be about 28'. Hence, the corrected middle latitude is 44° 37', therefore, Cos corrected mid. lat. 44° 37' ar. comp. 0.147629 therefore, the error in the former result is about 6 miles. X 2. A ship sails in the N. W. quarter, 248 miles, till her departure is 135 miles, and her difference of longitude 310 miles required her course, the latitude left, and the lat itude come to. Ans.} Course N. 32° 59' W.; Lat. left 62° 27' N.; lat. in 65° 55' N. 3. A ship, from latitude 37° N., longitude 9° 2′ W., naving sailed between the N. and W., 1027 miles, reckons that she has made 564 miles of departure: what was her direct course, and the latitude and longitude reached? 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 table. Ans. Course N. 51° 11′ E.; dist. 1189 miles. MERCATOR'S SAILING. 22. 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 of 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 B P b PP B and departure. Each of these elements is supposed to be taken so small, as to form on the surface of the sphere a series of triangles, differing insensibly from plane triangles. Let ABB' be a triangle, in which the angle A represents the course, AB the difference of latitude, B'B the departure, and AB the distance run. Produce the side AB' to C', until CC' shall be equal to the difference of longitude of the two extremities of the course the sake of distinction, we call AB' the proper difference of latitude, AC' the meridional difference of latitude, = then, for and we are now to explain the manner of constructing table, called a table of meridional parts, which will furnish the meridional differences of latitude when the proper differ ences are known. Let Abb represent one of the 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 equator containing an equal number of degrees, as the cosine of its latitude is to radius (Art. 17). This similar portion of the equator, is the difference of longitude between b' and b. Suppose, now, that Ab' is prolonged to p', making p'p equal to the difference of longitude between b and b': then bb': pp' :: cos lat. b'b : R (Art. 17.) But, by similar triangles, we have bb' pp' :: Ab' Ap', : : and consequently, cos lat. bb': 1. Denoting the proper difference of latitude by d, the meridional difference of latitude by D, the latitude of b'b by l, and the radius by 1, which is, indeed, the radius of the table of natural sines, we shall have If then, we know the latitude 7 of the beginning of a course, and the proper difference of latitude d of the extremity of the course, we can easily find the meridional latitude D corresponding to that course. The determination of AC' which represents the meridional difference of latitude, involves the determination of all the elementary parts, on which it depends. If d be taken equal to 1', we shall have from the equation above D = D = sec. 1, it being understood that I expresses minutes or geographi cal miles. 1' sec. 1, or 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 proper 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 actually formed by Mr. Wright, the proposer of this valu able 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'. Mer. pts. of 4′ = nat. sec. 1' + nat. sec. 2′ + nat. sec. 3' + &c. Hence, by means of a table of natural secants we have Mer. pts. of 1'= Nat. Secs. Mer. Pts. 1.000000 = 1.0000000 = 2.0000002 3.0000006 Mer. pts. of 2' = 1.0000000 + 1.0000000 Mer. pts. of 3' 2.0000002 + 1.0000004 : Mer. pts. of 4'3.0000006 + 1.00000074.0000013, &c. 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 D=1' sec. l, 1' is a geographical mile. 23. Having thus formed the table of meridional parts, if we find from it, 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 denoted by C'C may then be found by the fol lowing proportion. 1. 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 proper difference of latitude is to the departure, so is the meridional difference of latitude to the longitude. 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 have Now, let us suppose that we sailed from A to B: we shall then know AB' equal proper diff. lat. = 745 miles; AC' meridional diff. of lat. = = 1042; and O'C= the difference of longitude equal to 1295 miles. It is required to find the course B'AB, and the distance AB. 3.075006 tang. A 51° 11' E. 10.094402: AB 1189 X 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. 24. 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 |