Example.-July 12th: lat. 33° 56′ S., long. 18° 29′ E.; equal altitudes of Antares (a Scorpii) were observed as follows. Required the error of the chronometer on mean time at Greenwich.

Example 1.-August 27th: in lat. 32° 3' S., long. 115° 46′ E.; the following times by chronometer were noted when the sun had equal altitudes—

Required the equation of equal altitudes; also the error of the chronometer on mean time at place, and on mean time at Greenwich.

Ans. Equation of equal altitudes-10-99s. Chronometer fast on M.T. at place 4h. 14m. 2s. Chronometer slow on M.T. Gr. 2m. 54s.

Example 2.-January 12th in lat. 20° 10' S., long. 57° 32′ E.; the following times by chronometer were noted when the sun had equal altitudes

Required the equation of equal altitudes; also the error of the chronometer on mean time at place, and on mean time at Greenwich.

Ans. No equation of equal altitudes. Chronometer slow on M.T. at place 3h. 58m. 55s. Chronometer slow on M.T. Gr. 8m. 475.

Example 3.-April 13th: in lat. 30° 25′ N., long. 81° 25′ W.; the following times by chronometer were noted when the sun had equal altitudes—

Required the equation of equal altitudes; also the error of the chronometer on mean time at place, and on mean time at Greenwich.

Ans. Equation of equal altitudes-75. Chronometer fast on M.T. at place 5h. 30m. 23s. Chronometer fast on M.T. Gr. 4m. 43s.

NOTE. Owing to the facility with which Greenwich mean time is obtained in the principal ports of the world, the method of determining the error of the chronometer by equal altitudes is almost obsolete in the Merchant Service; but it is still an essential problem in a surveying vessel.

GREAT CIRCLE SAILING

On the sphere the shortest distance between two places is the arc of a great circle intercepted between them, as may be readily tested experimentally by stretching a thread evenly between any two places selected at random on the surface of a terrestrial globe. Hence Great Circle Sailing is the method of finding what places a ship must go through, and what courses she must steer, that her track may be on the arc of a great circle (or nearly so) passing through the place sailed from and that bound to. This was well understood by the old navigators, who continually practised this method, and especially before the introduction of Mercator's Chart. Thus John Davis at the end of the sixteenth century, in "The Secrets of the Sea," says that "Great Circle navigation is the chiefest of all the sailings, in which all the others are contained, and by them this kind of sailing is performed, continuing a course by the shortest distance between places, not limited to any one course, but by it those courses are ordered to the full perfection of this rare practice, whose benefits in long voyages are to great purpose, disposing all traverses to a perfect conclusion." And again, the pilot "shall by this kind of sailing find a better and shorter course, so that without this knowledge I see not how courses may

be ordered to their best advantage.'

The fundamental theorem of what the old navigators usually called globular" sailing is therefore this, that the arc of a great circle joining two points on the surface of a sphere is the shortest distance between them; and on no other than on a great circle course does the ship steer for her port, heading towards it as if it were in sight.

Steamers being to a certain extent independent of winds and currents can take a great circle route which is impossible to sailing ships, but the latter may often shorten the distance, when adverse winds are encountered, by taking a course anywhere between the Great Circle and the Rhumb line. When the places are widely separated, as in high southern latitudes, a great circle course is impossible to steamer and sailing ship alike, but advantage may be taken of a composite route, formed by sailing partly on a great circle and partly on a parallel.

Windward Great Circle Sailing.—On this subject Towson's observations are trite and to the point.

When a ship cannot (on account of adverse winds) sail directly to her port, she obviously ought to be put on that tack by which she nears her port by the greatest proportion of the distance sailed. It is also evident that she must do this when her track deviates by the least amount from the direct line which connects her with her destination, or, in other words, when she is put on that tack which deviates less from the true course than the other tack. In adopting this rule, it must, however, be especially borne in mind that the true course alone can serve as a guide in choosing the tack; and that the Great Circle, and not the Rhumb, is the true course. But, since the mariner is more conversant with the Rhumb than the Great Circle, too much attention cannot be directed to the importance of making this

distinction between these two courses in connection with windward sailing. In crossing the Pacific, the Rhumb course frequently deviates four points from the true course; under such circumstances it is impossible that the mariner can navigate his vessel with advantage, if he fail to make himself acquainted with the Great Circle course.

"The term Windward Great Circle Sailing,' is employed with special reference to these facts. This new form of describing the application of the true course is rendered necessary on account of the prevalent erroneous opinions that 'to a sailing vessel, Great Circle Sailing is of comparatively little value;' and that steamers, being in a measure independent of the winds, could, more readily than sailing vessels, avail themselves of the advantages of Great Circle Sailing.' The reverse is the fact to a sailing vessel the advantage of being guided by the true course, when contending with adverse winds, is fourfold as great as that which is conferred on a steamer. Thus, for example, the increase of distance arising from the direct track being diverted two points is only one mile in 12; but, if a ship that sails six points from the wind deviate two points further from the angle of the true position of her port on account of the wrong course being employed, she cannot in the least degree near her port, whilst, under the same circumstances, the knowledge of the true course would enable the mariner so to choose his track as to make good 8 miles by a run of 12 miles.

"The rule for Windward Great Circle Sailing is as follows:-Ascertain the Great Circle course, and put the ship on that tack which is the nearer to the Great Circle course."

Composite Great Circle Sailing.—When the track shows that the Great Circle would take the ship into too high a latitude, a maximum latitude is selected, and two great circles are drawn touching the parallel of the maximum latitude chosen, the one through the point of departure, the other through the point of destination. The ship first sails on the great circle from the point of departure until she reaches the selected parallel of maximum latitude, then she sails along this parallel until she reaches the point where the second great circle drawn from the point of destination touches the maximum latitude, and finally along the second great circle to their destination.

All the computations in Great Circle Sailing are effected through the methods of spherical trigonometry. Into these it is presently intended to enter, but it may be well to indicate a few of the principal terms connected therewith. The equator, which is a great circle, bisects every other great circle on the earth's surface, and there must necessarily be two points in every such circle equidistant from the equator and at the same time farthest removed from it; in Great Circle Sailing each of these points is called the Vertex; and the Latitude of Vertex, which is the highest latitude attained in sailing on a great circle, is the nearest approach to the elevated pole. The meridian cutting the Great Circle at the vertex is the Meridian of Vertex; and the Longitude from Vertex is the arc of the equator intercepted between the meridian of any place and the meridian of vertex.

The Great Circle and the Rhumb line differ most widely from each other in high latitudes and between places on nearly the same parallel. When the two places are on opposite sides of the equator, the Great Circle and the Rhumb line intersect each other, and the difference between them is not so perceptible.

Bear in mind that every point of a Great Circle course lies in a higher latitude than any point having the same longitude on the Rhumb line; and a course taken anywhere between the Great Circle Course and the Rhumb line will be attended with some saving of distance as compared with the Rhumb.

Mercator's Sailing answers every purpose for short voyages, or within the tropical regions; but in the present day of long voyages and great competition, much time and distance can be saved by resorting to the Great Circle track, or to a compromise between it and the Rhumb track, called a Composite track.

The great obstacle which once existed against the practice of Great Circle Sailing, viz., the determination of the longitude, a necessary element in the calculations, no longer exists. Again, the great labour of determining various points and pricking them off on a Mercator's Chart, drawing through them a freehand curve, is obviated by the use of charts upon which all great circles are represented as straight lines; hence, as with a Rhumb line on the Mercator's Chart, the entire track may be seen and obstacles thereby avoided.

Hence, there are two advantages in favour of the Great Circle track over the Rhumb track: the difference of distance and the great ratio in which it increases in high latitudes and between places separated by many degrees of longitude; and the difference of time saved on a voyage by a proper application of the principles of the great circle.

DEFINITIONS CONNECTED WITH GREAT CIRCLE SAILING

I. Circles on the sphere or globe are of two kinds, great and small. II. A great circle divides the sphere into two equal parts or hemispheres, its plane passing through the centre of the earth. The equator and meridians are great circles, but an infinite number of such circles can be projected on the sphere.

III. A small circle is one which divides the sphere into two unequal parts, its plane not passing through the centre of the earth, such as a parallel of latitude.

IV. A spherical arc is any portion of the circumference of a great circle. V. A spherical angle is formed by the intersection of two great circles, and is measured by the plane angle which measures the inclination of the planes of the containing arcs.

VI. A spherical triangle is a portion of the sphere's surface included by three arcs of different great circles; and each of these arcs is less than a semi-circumference, or semi-circle.

The data required in the computation are the latitudes and longitudes of the two places, and the selection of one of the poles of the earth; preferentially, and for convenience, take the pole nearest to the place in the higher latitude. The spherical triangle is formed by projecting the meridians passing through the two places, and then joining the two places by a great circle: thus the co-latitudes (or two polar distances) form the two sides, and the difference of longitude the included (or polar) angle.

VII. Any two points upon the surface of a sphere must be situated upon the arc of a great circle; and this arc is the shortest distance between the two places.

The shortest distance between two places would therefore not be that