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Equivalence requires that the area of certain spaces on the chart shall have the same ratio to one another as the corresponding spaces on the surface of the earth.

Equidistance requires that the distances from any two points to the center of the chart shall have the same ratio to each other as the corresponding distances on the earth.

33. The spherical representation of the earth (the globe) is, however, the only image that can satisfy all of these conditions. But a globe is a very inconvenient thing on which to find distances, bearings, and areas. Therefore, when representing some part of the earth's surface on a plane sheet of paper, the constructor will have to content himself with strictly fulfilling one of these conditions, while the others are satisfied as nearly as they possibly can be.

34. This may be accomplished in several ways by applying the rules of different methods of projection. Among the methods used for chart construction, the Polyconic, the Gnomonic, and Mercator's projection are prominent. Mercator's projection is the one usually adopted for navigators' charts, for on this alone is the ship's track, when steering a continuous course, represented as a straight line.

THE POLYCONIC CHART

35. Graphic Description.-When the extent of surface to be charted is limited, such as the plan of a harbor, island, small sections of the coast, etc., the polyconic projection is usually employed. Without touching on the mathematical principles underlying this projection, the characteristic features of the polyconic chart, briefly told, are as follows: The space to be laid down on the chart is considered to be divided by the latitude parallels into narrow zones, projected according to certain rules on a cone tangent to the zones' lower parallel (in north latitude the southernmost, in 'south latitude the northernmost). These zones aba'b', cdc'd', etc., Fig. 9, are then placed so that the middle meridian

mm', m' m', etc. of each constitutes a common meridian MM' for the whole chart; the zones therefore touch, or are tangent to one another only at points on or near this line and leave an open space between their ends, as shown

in the figure.

Now, in order to fill up these open spaces, and make the chart complete,

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the ends a b, c d, etc. of each zone are stretched, so to speak, So that the lower edge bm'b' of one zone will coincide with the upper edge cm'c' of the adjoining zone along the center line rm'r'. As a consequence of this stretching, it is evident that the parts of the chart near the vertical edges somewhat distorted. The latitude parallels are not parallel with one another, and the meridians ss', ss', etc. consist of curved lines converging toward the poles, with the exception of the middle meridian MM', which is a straight line.

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This projection is therefore advantageous for the representation of a coast line that lies in the direction of the meridian, and is for this reason extensively used by the United States Coast and Geodetic Survey in the preparation of working charts of the coast. The central part of this chart satisfies the conditions of conformity, equivalence, and equidistance.

36. The course, or line of bearing, between any two places on such a chart will necessarily be represented by a curved line, if properly laid out. However, if the scale of the chart is large, the bearing may without any practical error

be represented by a straight line, and especially so when the two places are situated in the central part of the chart.

37. If the places are widely separated in longitude, it should be borne in mind that when using a straight edge for finding the course between them, the ship following this course will at the end be nearer the elevated pole by a fraction that amounts to about 1 miles for every hundred on charts for parts of the ocean situated between latitude parallels 40° and 50° N and S. This error is greatly augmented on charts constructed on this projection where the scale is small.

THE MERCATORIAL CHART

38. Mercator's Systém.-The system on which Mercator's chart is constructed was invented in 1512 by Gerard Mercator, of Rupelmunde, Flanders. His system was not strictly a projection, but may be said to have resulted from operations illustrated in Fig. 10. By stripping the gores formed by the meridians of the globe, (a) representing the northern hemisphere, and placing them in regular order beside one another on a flat surface, a chart is formed similar to the one shown in (b). Owing to the openings, or vacant spaces, between the meridians, this chart is very defective; and, in order to remedy this defect, the upper parts of the gores are stretched so as to form the chart represented in (c). A glance at this chart, however, will reveal the fact that everything on it, except the equatorial parts, are distorted and that this distortion increases in the higher latitudes. Now, in order to restore a balance of orientation, or the relative position and direction of spaces, which are distributed horizontally (or in longitude), it is essential to distort the chart in an equal proportion vertically (or in latitude). When this has been done, the result is as shown in (d), which represents a complete map or chart of one-half the northern hemisphere according to Mercator's system.

On account of this distortion of the Mercatorial chart in the higher latitudes, due to the necessity of satisfying, as far as possible, the conditions of conformity, equivalence, and

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equidistance, it is evident that the areas and spaces in high latitudes on such a chart will appear much larger in comparison with equatorial parts than they are in reality.

39. Wright's Graphic Explanation. -Mercator did not demonstrate mathematically the principles of his system. Toward the close of the 16th century, however, Edward

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Wright, an English scientist, gave a mathematical demonstration of the law on which Mercator constructed his chart; at the same time he rendered a graphic explanation of the principles underlying its construction. He assumed that a cylinder contained a spherical globe representing the earth (a), Fig. 11, which was to swell like a bladder, equally in latitude and longitude, until it coincided with the concave surface of the cylinder; at the same time the meridians widen out until they are everywhere the same distance from one another, as on the equator. In this way the spherical surface is made to coincide with the cylindrical concave surface. Cutting the

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