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 be tween 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. bb R (Art. 17.) : But, by similar triangles, we have bb' : pp' :: Ab' Ap', and consequently, proper lat. Ab' : mer. diff. of lat. Ap' :: 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 o all the elementary parts, on which it depends. If d be taken equal to 1', we shall have from the equation above D= 1' sec. l, D= sec. 1, it being understood that I expresses minutes or geographi cal miles. 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 cor responding 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 Mer. pts. of 2' = nåt. sec. 1' + nat. sec. 2'. Mer. pts. of 3' = nat. sec. 1 + nat. sec. 2 + nat. sec. 3'. Mer. pts. of 1'= Mer. pts. of 2' Mer. pts. of 3' = = 1.0000000+ 1.0000000 2.0000002 = 2.0000002 + 1.0000004 = 3.0000006 Mer. pts. of 4′ = 3.0000006 + 1.0000007 = 4.0000013, &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. 1, 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 lat itude, 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 meri dional 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 Now, let us suppose that we have. 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. B A B For the Distance. 6.982132 As cos A 51° 11' 0.202850 10.000000: AB' 745 3.112270: radius 2.872156 10.000000 3.075006 tang. A 51° 11' E. 10.094402: AB 1189 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 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 are 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. 23), 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 straight line. LINE OF MERIDIONAL PARTS ON GUNTER'S SCALE. 25. This scale corresponds exactly with the table of meridional parts, excepting, that in the table, the circle is divid ed 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 those scales, by using the scale of equal parts for the lines of longitude, |