the gnomonic projection, is readily seen by inspecting Fig. 18. Take, for instance, the parallel mn, the latitude of which is the arc Em; the radius m P', or R, of that parallel when projected on the chart is equal to the tangent for the angle m O P', or, equal to the cotangent for the latitude of the parallel, which is measured by the angle E Om, or the arc Em, the radius of the globe being considered a unit in both cases. A gnomonic chart may therefore be readily drawn to any desired scale, the cotangent of each latitude parallel being taken from a table of natural cotangents. The chart represented in Fig. 19 is constructed between the limits of 40° and 75° of latitudes. As stated before, the main advantage of this chart is that the great-circle track between any two places within its limit may be represented by a straight line. Thus, the line AB is the great-circle track between A and B, and by simply inspecting the regions through which it passes it is evident that the navigator is enabled to see whether the track is practicable or not, or whether it passes through regions unfavorable to navigation. The position of the vertex V is obtained by inspection; its latitude may be found quite accurately by measuring its distance P V from the pole, this distance being equivalent to the cotangent of the latitude according to the given scale; the corresponding numbers of degrees and minutes in the table of natural cotangents will be the required latitude. Its longitude is best obtained by measuring, with a protractor, the angular distance between the nearest meridian and that of the vertex. In like manner,

the latitude and longitude of any desired point on the greatcircle track is determined and can be transferred to a Mercator's chart, whence the proper courses and distances to be sailed are readily found.

74. In this simple form of gnomonic chart the courses and distances along a great-circle track cannot be obtained directly, but on great-circle charts, devised by Mr. G. Herrle, and published by the United States Hydrographic Office, courses and distances are conveniently found by means of

diagrams attached to the chart. Full information regarding the use of these diagrams is printed at conspicuous places on each chart under the heading of "Explanation."

75. The gnomonic chart, though accurate in high latitudes, gives a very distorted representation of low latitudes; in such parts of the world, however, the advantage of great-circle sailing over that of Mercator's sailing is insignificant.

76. On a gnomonic chart of any part of the earth that does not include the pole, the meridians and other great circles are still represented by straight lines, but the latitude parallels become arcs of ellipses, the globe in Fig. 18 then being supposed to touch the paper at the middle of the portion represented. The great-circle chart of the North Atlantic Ocean is a representation of such a chart.

77. Airy's Method.-When gnomonic charts are not available, the following method for laying out a great-circle track on a Mercator's chart will be found satisfactory. It was originally communicated to the Royal Astronomical Society by Sir G. B. Airy, Astronomer Royal of Great Britain, in 1858.

The sweep of the track is accomplished by adhering to the direction in the following rules:

Rule.-I. Join the two places on a Mercator's chart by a straight line. Find its middle point. From this point draw a perpendicular toward the equator and, if necessary, continue it beyond the equator.

II.

With the middle latitude between the two places, enter the following table, and take out the corresponding parallel.

III.

The center of the required arc (which is drawn either with beam compasses or a pencil attached to a thread) will be the intersection of this parallel with the perpendicular, the radius being the distance from the center to either place.

TABLE FOR FINDING THE "CORRESPONDING PARALLEL” ACCORDING TO AIRY'S METHOD

NOTE.-The blank spaces arise from the fact that in relations otherwise than those contained in the table, great-circle sailing is of no particular advantage.

78. Sigsbee's Method.-Another method of determining graphically the data connected with great-circle sailing was devised by Commander C. D. Sigsbee, U. S. N., and published by the United States Hydrographic Office under the title "Graphical Method for Navigators." It contains a large stereographic projection of the sphere, exhibiting meridians and parallels at intervals of 1°. Full explanations and numerous examples accompany each diagram. No drawing is necessary except to plot points and occasionally to

trace parts of a curve from the diagram. The adjuncts needed are a piece of tracing paper about the size of the diagram, a lead pencil, and a rubber eraser. Besides being useful for solving problems in great-circle sailing, it can be advantageously used for the solution of spherical problems in general.

79. Comment on Graphical Methods.-Other graphical methods have been devised by which the great-circle track can, without any calculations whatever, be drawn on a Mercator's chart. Among these, the one devised by the late Richard A. Proctor, the astronomer, is most satisfactory in its result. His charts of the northern and southern hemispheres are based on the principles of the stereographic projection. Godfray's great-circle charts, and course and distance diagrams, also give very accurate results. Explanations and directions usually accompany these charts.

In simplicity, however, Airy's method is foremost. Its disadvantages are that the center of the arc to be described frequently falls beyond the limit of the chart to be used, and the track may occupy more than one chart, thus rendering it difficult to draw the required arc. On the other hand, Herrle's method as used on great-circle sailing charts, issued by the United States Hydrographic Office, is a most excellent one for all practical purposes. It combines simplicity and accuracy. By his method the course and distance run by a vessel can be measured independently of any great-circle track that may have been laid down, just as the rhumb course and distance to a desired place are measured on the Mercator's chart from the actual position in which the vessel is found to be.

COMPOSITE SAILING.

80. Explanation.-Whenever the great-circle track between two places passes through land or brings the ship into too high a latitude, or into a region frequented by ice and bad weather, or is met by other obstacles that would tend to either lengthen the passage or make it dangerous, a method of sailing called composite sailing is

resorted to. This sailing, which is simply a combination of great-circle sailing and parallel sailing, is described in the following article.

81. Assume a ship, in latitude 45° N and longitude. 15° W, or at A, Fig. 19, to be bound for a place B in latitude 42° N and longitude 123° E. Joining the two places with a straight line the master of the ship at once discovers that by following the great-circle track he will come as far north as 72°. Now, on account of the ice, fog, and cold weather that is likely to be met with at this latitude, and which would seriously interfere with the successful navigation of his vessel, he decides not to go farther north than 55°; in other words, 55° N will be his maximum latitude. A tangent from A to the 55th parallel is then drawn on the gnomonic chart, and likewise a tangent from B to the same parallel. The ship will now proceed first along the great-circle track Ax, then along the 55th parallel to y, and finally proceed along the great-circle track y B until her destination B is reached.

This in brief comprises what is known as composite sailing, and the track Axy B, composed of two separate great-circle tracks and an arc of a parallel, is called a composite track.

20 10 0 10 20 30 40 50 60 70 80 90 100 110 120 130 75

75

40

Rhumb Track

0 10 20 30 40 50 60 70 80 90 100 110 120 130

FIG. 20

82. Measuring roughly, we find that the first great-circle track Ax, Fig. 19, crosses the 50th parallel at about 3° W,