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cylinder down a meridian, and opening it out into a flat surface (5), Fig. 11, we shall find on it a representation of a Mercator's chart. For this reason, all the meridians are parallel straight lines, and the degrees of longitude are all equidistant; the latitude parallels are everywhere at right angles to the meridians; and hence the loxodromic line, or track of a ship that steers a straight continuous course, or the relative direction or bearing between two places, can be represented on a Mercator's chart as a straight line. Consequently, this chart satisfies the first condition—conformity—the other conditions being satisfied to a great degree of accuracy. The degrees of latitude are all unequal, being increased in length from the equator to the pole in the same proportion as the degrees of longitude decrease on the globe.
40. Meridional Parts.-At the equator the degrees of latitude are equal to the degrees of longitude, but upon approaching the poles the latter decrease, becoming smaller and smaller until the pole is reached, when their value is zero, while the degrees of latitude remain the same. Thus, in the latitude of 60°, a degree of longitude is only one-half what it is at the equator, and, therefore, all dimensions east and west would be twice as great as they should be if drawn to the same scale as those north and south. To correct this distortion, therefore, the distance between the latitude parallels is increased precisely in the same proportion as that between the meridians; hence, on a Mercator's chart the relative magnitude of a degree of latitude and longitude is everywhere truly preserved.
The lengths of the small portions of the meridians thus increased expressed in minutes of the equator are called meridional parts, and the meridional parts for any latitude is the line expressed in minutes of the equator into which the latitude is thus expanded.
41. The meridional difference of latitude is the difference between the meridional parts for any two latitudes; or the length of the line on a Mercator's chart that represents the difference of latitude.
42. How Meridional Parts for Any Latitude Are Determined.-In Fig. 12, let B and D be two places on
P • the same parallel of latitude; /’, the adjacent pole; B /), the arc of the parallel of latitude through the two places, A being its center; and Q so, the correspond
A. r B ing arc of the equator, its i center being at C. Then, so ep. A /) is the departure on
the parallel, B C E the latitude of the parallel, and O AE, the difference of longitude of the two places Q ...} A and /). For convenience, let the radius A B of the parallel be denoted by r, and the radius C/E (= C /3) be denoted by R. In the triangles A /3 C and /8 C /2, the angle .1/3 C is equal to B C /3, because R and r are parallel. Since the angles B A D and A. CO are equal we have, according to plane geometry, the following proportion: /3 D: /E O = A /3: C/E; Or Dep. : D. Long. = r. A’. (1) In the right-angled triangle A /3 C, we have - A /3 cos A B C = C/3' and, since the angles A / C and B C /2 are equal, A /? cos /3 C /2 = C /?' But B C E is the latitude of the parallel B D; A B_ r. C/3 TA" whence, r= A' cos Lat.;
hence, cos Lat. =
or, the radius of a parallel of latitude is equal to the radius of the equator multiplied by the cosine of the latitude.
Substituting this value for r in proportion (1), we get Dep. : D. Long. = R cos Lat. : R,
Or Dep.: D. Long. = cos Lat. : 1;
This establishes the relation between the arc of the equator Q E(= D. Long.) and that of a latitude parallel (/) intercepted by the same meridians. But 1' of the meridian m is equal to 1' of the equator, or D. Long. = m. Substituting this in equation (2), we get m =/ sec Lat.
43. Each degree of latitude on a Mercator's chart must, therefore, be increased as it recedes from the equator, as the secant of the latitude increases; also, all latitude parallels on a Mercator's chart are larger than on the globe in the proportion sec Lat. : 1. Hence, the method of computing a table of meridional parts is as follows: Commencing from the equator and taking a minute of that great circle as a unit, the length of the first minute of latitude is the natural secant of 1', the length of the next is the natural secant of 2', and so on. Hence, the distance from the equator to latitude 3' is equal to sec 1/+ sec 2'-H sec 3', and the distance from the equator to a certain parallel, the latitude of which is 0 is, therefore,
44. The result obtained in this way is, however, only approximate, and the process is somewhat tedious. Other methods of computation have been devised that are more accurate, but the one just given shows more plainly the nature of the meridional parts. For accurate determination of meridional parts, a formula obtained by means of the Integral Calculus is required, wherein the fact that the earth is not a perfect sphere is taken into account. This formula is as follows:
M=7,916.70447X log tan (90+/.), in which L is any latitude and J/ the corresponding meridional part.
45. Tables of Meridional Parts.-Tables known as Tables of Meridional Parts have been constructed showing the lengths in nautical miles into which the true latitude must be expanded when constructing a Mercatorial chart. They are to be found in the collection of Nautical Tables.
On reference to these tables, it will be seen that the length of the true latitude or distance from the equator in miles in latitude 30° (30°2' 60 = 1,800) must be expanded to 1,876.9, and in latitude 60° each mile of true latitude is expanded to twice its natural length, since in that latitude the meridians passing through two places are only one-half the distance apart that they are at the equator. Also notice that the
meridional difference of latitude between the equator and
latitude 83° 55' is twice the /eng//, of the true difference of satitude, and that, since in such high latitudes the meridians rapidly approach one another, the meridional differences of 1atitude also rapidly increase, and the mile of latitude between 89° 59' and 90° becomes infinite in length. In high latitudes, therefore, charts are not constructed on the Mercator's, but on the Gnomonic, projection.
Tables of meridional parts are principally used in solving problems in Mercator's Sailing and in the construction of Mercator's charts.
46. To Construct a Mercator's Chart.-First, determine the limits of the proposed chart—in other words, the number of degrees and minutes it is to contain, both of latitude and of longitude. Then draw a straight line near the Wozo'er margin of the paper, if the chart is to represent north latitude, and near the u//er margin, if it is to represent south latitude; or, at a suitable position toward the center, if both north and south latitude are to be represented. Divide this line into as many equal parts as the number of degrees of longitude required; for instance, if the chart is to contain 15 degrees of longitude, divide the line into 15 equal parts. At each extremity, erect lines perpendicular to it. Take out from Table of Meridional Parts the meridional part for each convenient degree of latitude for the limits between which the chart is to be drawn, and take the difference between each successive pair, thus obtaining the meridional difference of latitude. Reduce these meridional differences to degrees, by dividing them by 60; the result will be the lengths on the longitude scale between the chosen degrees of latitude. Lay off these lengths successively on the perpendicular lines, and through the points thus obtained draw straight lines parallel to the original line, to represent latitude parallels. At convenient intervals, or through each division on the base line, draw lines parallel to the perpendiculars to represent meridians. The accuracy of the frame of the chart thus completed should be tested by measuring the two diagonals of the rectangle formed; if they are of the same length, the frame is perfect. The principal points in the chart are now laid down according to their respective latitudes and longitudes, and whatever formations and contours of water or land are required, together with other useful items, are drawn in freehand. A compass diagram may also be inserted, but it should be remembered that the direction of the meridians indicates true north and south.
47. Inustration.-In order to illustrate the foregoing, let it be required to construct a chart on the Mercator's projection extending from 59° to 75° west longitude, and from 30° to 42° north latitude. Proceed as follows: Draw the bottom line a b, Plate I, to represent the thirtieth parallel of latitude, and divide it into 75–59 = 16 equal parts, subdividing one of the parts in divisions of, say, 5' each. At the extremities of a b carefully erect the perpendiculars a c and b d to represent meridians. Then consult the tables of meridional parts and take out the values corresponding to every second degree of latitude (which is quite sufficient in