represented upon a Mercator's chart by the rhumb, or straight line, but rather by that represented on a globe by a fine line of silk thread stretched from point to point upon its surface.

Or, more correctly, by means of a terrestrial globe, especially one of 12 inches diameter, the shortest distance, and at the same time all the other elements of the great circle track would be found by bringing both places to coincide with the upper edge of the wooden horizon, which itself represents a great circle of the sphere.

The elements which would then appear would be

1. The distance between the two places in degrees and minutes of arc. 2. The angles of position at the two places.

3. The latitude of the highest point of the circle, called the vertex.

4. The longitude of the meridian of vertex.

5. A succession of points on the arc, few or many, according to the choice of the navigator.

6. The course and distance from point to point, successively.

VIII. The angle of position is the angle at the place included between the plane of the great circle and the plane of the meridian, and shows the angular position of that place from any other place through which the great circle passes. Two such angles are found, and they are equivalent to the initial and terminal courses at the two places.

IX. The distance is the arc of the great circle between the two places; and being the shortest distance is the track on which the ship should be steered so as to head directly towards her port.

X. The vertex is that point in a great circle which is farthest from the equator. There are two such vertices in every great circle, one in the northern and one in the southern hemisphere; they mark in each hemisphere the points of greatest separation from the equator, which bisects every other great circle on the earth's surface. The arc intercepted between the vertex and the equator is called the latitude of vertex; the meridian that passes through the vertex is the meridian of vertex; and the arc of the equator contained between the meridian of vertex and the meridian of any place on the great circle is named the longitude from vertex.

The vertex may or may not fall between the two places; but if the two places are on a parallel of latitude, the vertex will be midway between them. The meridian of the vertex always intersects the great circle at right angles, and, with the equator, divides a great circle into quadrants; and in each of these quadrants the elements are the same; that is, the latitudes, courses, and distances, corresponding to each degree of longitude from the vertex in one quadrant, truly represent those for the corresponding degree in each quadrant belonging to the same great circle.

XI. The angle at which the great circle crosses successive meridians is constantly altering, therefore, it becomes necessary to calculate, at recurring intervals, the approximate course and distance from point to point along the great circle.

There are several methods of determining the various parts of a spherical triangle, in which the fundamental data are the two sides and the included angle; in the method here given the following order is adopted as most direct arising out of the two co-latitudes and difference of longitude.

1. The angles of position at the two places, or the initial and terminal

courses;

2. The distance between the two places;

3. The position of the vertex, in latitude and longitude; and

4. A succession of points on the arc of the great circle, with the course and distance from point to point.

The following diagrams will illustrate the various elements and definitions. In Fig. 1, let A P B be a spherical triangle in the northern hemisphere, the latitudes of A and B are known, A being in the higher latitude; the longitudes of A and B are also known. Let P A be the arc of the meridian passing through A, then P A is the polar distance, or co-latitude of A (or 90° Latitude of A); and if P B be the arc of the meridian passing through B, then P B is the polar distance, or co-latitude of B (or 90° Latitude of B); these are the two sides of the triangle; and the angle A P B, equal to the difference of longitude between A and B, is the angle included between the two sides P A and P B, and sometimes called the polar angle.

The third side of the triangle, which has to be found, is A B; it is the arc of the great circle passing through A and B, and is also the distance.

PA B is the angle of position at A, and initial course from A to B.
PBA is the angle of position at B, and initial course from
B to A.

M is the vertex, or highest latitude on the great circle of which
A B is an arc.

P M is the arc of the meridian extending through M, and is hence
the meridian of vertex, reckoned from Greenwich-in this case
falling between A and B.

A P M is the difference of longitude between A and M, and is hence the longitude of A from vertex.

In the lower part of Fig. I all the elements and definitions just noted are given, with the same letters dashed, but M' is outside the arc of the great circle A' and B'.

In Fig. 2, A being a place in N. latitude, and B in S. latitude, A's

+

co-latitude will be 90° latitude of A, and B's co-latitude = 90° latitude of B, reckoned from the pole of higher latitude.

When the two places lie on the same meridian their difference of latitude will be the arc of the meridian between them, and the position from one place to the other will be directly north or south.

When the two places are on the equator the distance between them is equal to their difference of longitude, and the position (or course) of one from the other will be due east or west.

General Rule to find the Angles of Position, or Course from A to B

and from B to A

Reckon both co-latitudes from the pole NEARER to A.-Find half the sum, and half the difference of the co-latitudes; also find half the difference of longitude between A and B.

To find half sum of the angles at A and B.-Add together the L cotangent of half the difference of longitude, the L secant of half the sum of co-latitudes, and the L cosine of half the difference of co-latitudes; the sum of the three logarithms (rejecting index 20) will be the L tangent of half the sum of the angles A and B.

To find half the difference of the angles A and B.-Add together the L co-tangent of half the difference of longitude, the L co-secant of half the sum of co-latitudes, and the L sine of half the difference of co-latitudes; the sum of the three logarithms (rejecting index 20) will be the L tangent of half the difference of the angles A and B.

and

The co-latitude of B being greater than that of A, the sum of the two angles A+ B A B will be the first great circle course from A towards B, and their difference the first great circle course from B towards A.

2

2

The course is reckoned from N. if A is in N. latitude, but from S. if in S.

Add together the L cosine of half the sum of co-latitudes, the L secant of (sum of angles A and B), and the L sine of half the difference of longitude; the sum of the three logarithms (rejecting index 20) will be the L cosine of half the distance in arc, which take out, multiply by 2, and convert into nautical miles.

Formula- Cos.

= COS.

X sec.

X sin.

To know whether the Vertex falls within or without the triangle

If the angles A and B are both greater or both less than 90°, the meridian of vertex falls within the triangle, but if one angle is greater and the other less than 90° the meridian of vertex falls without the triangle.

The remaining parts of the "Great Circle" problem are solved by means of Napier's rules for "Circular parts" as follows

To find Latitude of Vertex.

In the spherical triangle B PM right-angled at M (Fig. 3) given B and side B P to find P M, and its complement equal to the lat. of M—

To find Longitude of Vertex from B

In the same triangle, given B P, and P M, to find P