These three proportions comprise the theory of middle latitude sailing, and when to the middle latitude the proper correction, taken from Mr. Workman's table, is added, these theorems will be rendered strictly accurate. This is Table XXIX; the middle latitude is to be found in the first column to the left; in a horizontal line with which, and under the given difference of latitude, is inserted the proper correction to be added to the middle latitude to obtain the latitude in which the meridian distance is accurately equal to the departure. The formula for constructing this table is obtained as follows.* 1. A ship, in latitude 51° 18' N., longitude 22° 6' W., has sailed S. 33° 5' E., required her latitude and longitude. The required latitude is found by plane sailing, as follows: The investigation of this formula should be postponed until after reading the next article, and may be omitted entirely. 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', 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. log. 0.14763 therefore, the error in the former result is about 6 miles. 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 latitude come to. Course N. 32° 59′ W.; lat. left 62° 27' N.; lat. in 65° 52′ N. 3. A ship, from latitude 37° N., longitude 9° 2' W., having 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? Course N. 33° 19' W. or N.W. by N. nearly; lat. 51° 18' N.; long. 22° 8 W. 4. Required the course and distance from a point in lat. 37° 48' N., long. 25° 13′ W., to a point in lat. 50° 13′ N., long. 3° 38′ W., the middle latitude being corrected by Workman's Table. Course N. 51° 11' E.; distance 1189 miles. MERCATOR'S SAILING. This is for the determination of difference of longitude when a ship sails on any oblique rhumb. B 101. It has already been seen that when a ship sails on any oblique rhumb, the difference of latitude, the departure, and the distance run, are truly represented by the sides of a right-angled plane triangle. Let AB'B in the annexed diagram be this triangle, A representing the course, AB the diff. of lat., and B в the departure. Let Ac' be a sufficiently greater difference of latitude to make the corresponding departure cc' equal to the difference of longitude required. This increased difference of latitude AC' is called the meridional difference of latitude, AB' being called the proper difference of latitude, by way of distinction. The solution of the triangle AC'c then will serve to determine the difference of longitude c c. In this triangle we know the course A, and we shall now show how to construct a table for finding the side AC, the meridional difference of latitude. The departure BB represents the sum of all the very small meridian distances, or elementary departures, b'b, c'c, &c., in the diagram at Art. 100, the difference of latitude AB' represents the sum of all the corresponding small differences in the figure referred to, and the distance AB the sum of all the corresponding distances ab, bc, cd, &c., and each of these elements is supposed to be taken so exceedingly small as to form on the sphere a series of triangles, differing insensibly from plane triangles. 門 b P Let ab'b in the annexed diagram represent one of these elementary triangles, b'b will be one of the elements of the departure, and ab', the cor responding difference of latitude; and as b'b is a small portion of a parallel of latitude, it will be to a similar portion of the equator, or of the meridian, as the cosine of its latitude to radius (Art. 99), this similar portion of the equator, or of the meridian, being the difference of longitude between band b. Suppose now the distance ab prolonged to p, till the departure pp is equal to the difference of longitude of 5', and b, then b'b will be to p'p as the cosine of the latitude of bb to the radius ; but bb: pp:: Ab': Ap'; hence the proper difference of Ab' is to the increased difference Ap as the cosine of the latitude of b'b to the radius. Calling, therefore, the proper difference of latitude d, the increased difference of latitude D, the latitude of b'b, l, and the radius 1, which it is in the table of natural sines, this proportion will be in symbols, The ship, therefore, having made the small departure b'b, and the difference of latitude ab, must continue her course till the difference of latitude becomes D, in order that her departure may become equal to the dif ference of longitude corresponding to b'b. Conceiving all the elementary distances to be in this manner increased, the sum of all the corresponding increased departures will necessarily be the whole difference of longitude made by the ship during the course. The determination of Ac' requires the previous determination of all its elementary parts; if d be taken equal to 1', each of these parts will be expressed by D = 1' sec l, or D = sec 1, that is, sec expresses the meridional difference of latitude corresponding to a proper difference of latitude of 1' at the latitude 7, giving / successively the values 1', 2, 3', &c., up to 90°, and adding the result of the second substitution to that of the first, and so on, we shall have in succession, the values of the increased latitude corresponding to 1', 2, 3, &c. of proper latitude; these values are called the meridional parts, corresponding to the several proper latitudes, and when registered in a table, form a table of meridional parts, given in all books on Navigation.* The following scheme may serve as a specimen of the manner in which such a table may be constructed, and, indeed, of the manner in which the * In other words, a table of meridional parts is a table of differences of latitude expressed in geographic miles, each difference of latitude being enough greater than its corresponding proper difference of latitude, to make the departure equal to the difference of longitude. The table gives the meridional difference between the equator and any given latitude. first table of meridional parts was actually formed by Mr. WRIGHT, the proposer of this ingenious and valuable method. Mer. parts of 1'= nat. sec l'. Mer. parts of 2': nat. sec l'+ nat. sec 2'.* Mer. parts of 3'= nat. sec l'+ nat. sec 2′+nat. sec 3'. Mer. parts of 4' &c., &c. nat. sec l'+ nat. sec 2'+nat. sec 3′+ nat. sec 4', Hence, by means of a table of natural secants, we have Mer. parts of 1'= Nat. secs. 1.0000000 Mer. parts. 1.10000000 = Mer. parts of 2' 1·0000000 + 1·0000002 = 2⚫00000002 Mer. parts of 3'= &c. 2′0000002 + 1.0000004 = 3.00000006 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, l' is a geographical mile. Having thus formed a table of meridional parts (Table III. at the end), if we enter it with the latitudes sailed from, and reached, and take the difference of the corresponding parts in the table, the remainder will be the meridional difference of latitude, or the line Ac' in the preceding diagram. The angle A is the given course, so that there are known in a right angled triangle Ac'c, an angle and the side adjacent, to find the side opposite. The following is the rule. 1. The tangent of the course meridional difference of latitude the difference of longitude; or if the departure be given instead of the course, then from the similar triangles AB'B, AC'c, the proportion will be 2. As the proper difference of latitude is to the departure, so is the meridional difference of latitude to the difference of longitude. Other proportions immediately suggest themselves from the preceding figure. * Observe that the meridional parts, or meridional diff. of latitude from the equator to 2' of latitude will be the sum of the meridional parts from the equator to 1', plus the meridional parts from 1' to 2', which latter is nat. sec. 2'. Again, that the meridional parts from the equator to 3' is the sum of the meridional parts from the equator to 2', and the meridional parts from 2' to 3', which latter is the nat. sec. 3'. |