the two places and then the upper part of the ruler is pushed forward to the center c of the nearest compass diagram and the bearing read off immediately.

In some cases it may be necessary to take several "steps" with the ruler in order to reach the diagram, but the operation in itself is so simple as to need no further explanation. The main point, however, is to preserve the parallelism of the last step with the first.

81. The dividers, shown in Fig. 29, are used to lay off and measure distances. They should be made of well-tempered steel; their points should be sharp but not too fine. When using the dividers to lay off a distance into a number of equal parts, hold them at the top between the thumb and forefinger and step off the spaces, turning the instrument alternately to the right and left. When laying off a distance in this manner, great care must be exercised not to press the divider points into the chart; they should be turned lightly and the final point indicated by a pencil mark, lightly applied.

FIG. 29

82. The course protractor is shown in Fig. 30. The outer edge is a semicircle, with a center at O, and is divided into 360 parts. Each division is one-half of 1°, and, for convenience, the degrees are numbered from 0° to 180° from both A and B. The protractor is used for laying off or measuring courses, or bearings, and should be made of transparent horn or celluloid, and with a radius of at least 3 inches. Protractors are often made of metal, in which case the central part is cut away so as to make the chart under it visible.

83. When using the protractor it must be placed so that the line O B, Fig. 30, will coincide with the line forming one

side of the angle to be laid off, or measured, and the center O must be at the vertex of the angle. For example, let it be required to find the bearing between the two places

a and b, Fig. 31. Proceed then as follows: Lay any straight edge, for instance, that of a ruler, between the two

places; then place the protractor along the ruler with its center on one of the meridians, as shown in figure, and read off the bearing, as indicated by the number of degrees of the arc de reckoned from the meridian I in this case

N 75° W.

84. To Lay Down a Position on the Chart. - When the latitude and longitude of a ship or a place are given,

Place the

its position on the chart is laid down as follows: edge of a parallel ruler along the latitude parallel nearest the given latitude; then move it until one of its edges passes accurately through the given latitude on the graduated meridian, and keep it there. This done, take, with the

dividers, from the upper or lower graduated margin the distance of the given longitude from the nearest meridian, and lay off this distance along the edge of the ruler from the same meridian. The position thus found is the position of the ship or place required, and should be lightly marked on the chart by a small cross.

For example, the latitude and longitude of a place is 11° 45' N and 61° 15' W, respectively; mark down its position on the chart, Fig. 32. The parallel ruler is first placed along the twelfth parallel and then moved down until its edge is exactly at 11° 45′ on the graduated meridian, as shown in figure, whence the distance c (= 15' of longitude) is taken by the dividers from either the upper or lower longitude scale and laid off along the edge of the ruler from the sixty-first meridian. The position thus found (indicated by a cross) is the position of the given place according to its latitude and longitude.

85. This position could have been just as easily found by first placing the ruler in the position shown in the figure and then moving it along the sixty-first meridian to 61° 15', when the point of intersection between the two lines formed by its edge would be identical with the point indicated by the cross.

86. To Find the Latitude and Longitude of a Place or Position on the Chart. - First Method. - Place the edge of a ruler through the given point, parallel to the nearest latitude parallel, and read off the degree and minute at which the edge of the ruler cuts the graduated meridian. This will give the required latitude. Similarly, place the ruler parallel to the nearest meridian through the given point and note the degree and minute at which the edge of the ruler cuts the graduated parallel, or longitude scale. This will give the required longitude.

87. Second Method. -The latitude and longitude of the same position may also be conveniently found by means of the dividers as follows: Measure the perpendicular distance of the given point from the nearest parallel; refer this distance.

to one of the graduated meridians, or latitude scales, and place one leg of the divider on the same meridian; the other leg should then be at the required latitude. For the longitude, proceed in precisely the same manner, but use a meridian and the longitude scale at the top or bottom of the chart.

For example, if it be required to find the latitude and longitude of the point x, Fig. 32, measure with the dividers the distance of the point from the eleventh parallel and place one leg at d; the other leg will then be at e and the latitude read off at once-in this case 11° 30' N. For the longitude, measure the distance a from the point to the nearest meridian, and lay off this on the longitude scale either at the top or bottom from the same meridian. This will give the longitude of the point x-in this case 62° 15′ W.

88. To Find the Distance Between Any Two Points on the Chart. If the two points lie on the same meridian, find on the chart their respective latitudes according to the preceding article; the difference or sum of these, according as the given points are situated on the same or on different sides of the equator, expressed in minutes, will give the required distance.

89. If the two points are situated on the same parallel, half the distance should be measured on the graduated meridian, or latitude scale, on each side of the parallel; the total length thus measured expressed in minutes is very nearly the required distance. For example, if it be required to find the distance between two places a and b, Fig. 33, situated on the same parallel, take half the distance, or a c, and lay it off on the latitude scale from the parallel to a'; then lay off the other half cb from the same parallel to b'. The sum of these lengths, or the distance d, will give very nearly the required distance a b.

90. If the two points differ both in latitude and longitude, take the distance between them by a pair of dividers and lay