cylinder down a meridian, and opening it out into a flat surface (b), 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

• the same parallel of latitude; P, the adjacent pole; BD, the arc of the parallel of latitude through the two places, A being its center; and QE, the corresponding arc of the equator, its center being at C. Then, BD is the departure on the parallel, BCE the latitude of the parallel, and QE, the difference of longitude of the two places B and D. For convenience, let the radius AB of the parallel be denoted

Diff Long

by r, and the radius CE (=CB) be denoted by R. In the triangles ABC and BCE, the angle ABC is equal to BCE, because R and r are parallel. Since the angles BAD and ECO are equal we have, according to plane geometry, the following proportion:

In the right-angled triangle ABC, we have

(1)

and, since the angles ABC and BCE are equal,

But BCE is the latitude of the parallel BD;

or, the radius of a parallel of latitude is equal to the radius of the equator multiplied by the cosine of the latitude.

or

Substituting this value for in proportion (1), we get Dep. D.Long. = R cos Lat.: R,

whence,

and,

D. Long.

Dep. cos Lat.

=

Dep. sec Lat.

(2)

This establishes the relation between the arc of the equator QE(= D. Long.) and that of a latitude parallel (p) 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=p 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' + sec 3', and the distance from the equator to a certain parallel, the latitude of which is p is, therefore,

1'(sec 1'+ sec 2' + sec 3'+...+ sec p).

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=7,916.70447 X log tan (90+Z),

in which L is any latitude and M 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° x 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 length of the true difference of latitude, and that, since in such high latitudes the meridians rapidly approach one another, the meridional differences of latitude 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 lower margin of the paper, if the chart is to represent north latitude, and near the upper 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.

Illustration.—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 ab carefully erect the perpendiculars ac and bd 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