If both readings are on the arc, or both off the arc, half their sum is the index correction-subtractive when both on, additive when both off

One-fourth of the sum of the two readings should be equal to the sun's semi-diameter in the Nautical Almanac for the day; but if both readings be on or both off the are one-fourth their difference should be the sun's semi-diameter.

Thus, suppose the observations, in Example 1, to be made on September 26th, 1872, here one-fourth of the sum of the two readings is 16' o", agreeing with the semidiameter as giver in the Nautical Almanac for the given day.

This affords a test of the accuracy with which the observation has been made.

EXAMPLES FOR PRACTICE.

Ex. 1. 1872, April 17th, the reading on the arc 29′ 40′′, the reading off the arc 34' 10" required the index correction and semi-diameter.

Ex. 2.

1872, July 4th, the reading on 33' 10", off 29' 50": find index correction and semi-diameter.

Ex. 3. 1872, November 13th, on 4' 40", off 60' 10": find index correction and semi-diameter.

Ex. 4. 1872, July 10th, on 32′ 45′′, off 34′ 30′′: find index correction and semidiameter,

Ex. 5. 1872, March 21st, off 1° 10′ 0′′, off 6′ 40′′: find index correction and semidiameter.

Ex. 6. 1872, January 17th, on 67′ 40′′, on 2' 30': find index correction and semidiameter.

ON THE CHART.

A CHART is a map or plan of a sea or coast. It is constructed for the purpose of ascertaining the position of the ship with reference to the land, and of shaping a course to any place.

The use to be made of the chart in each case determines the method of projection, and the particulars to be inserted. (1.) The chart may be required for coasting purposes, for the use of the pilot, &c., and then only a very small portion of the surface of the globe being represented at once, no practical error results from considering that surface

a plane, and a “plane chart” is constructed in which the different headlands, lighthouses, &c., are laid down according to their bearings. The soundings on these charts are marked with great accuracy; the rocks, banks, and shoals, the channels, with their buoys, the local currents, and circumstances connected with the tides, are also noted. (2.) Again, for long sea passages the seaman requires a chart on which his course may be conveniently laid down. The track of a ship always steering the same course appears as a strait line (and can at once be drawn with a ruler) on the Mercator's chart. Hence the charts used in navigation are Mercator's charts. (3.) When great circle sailing is practicable, and of advantage, a chart on the "central projection," or gnomic, exhibits the track as a straight line, and is therefore convenient.*

ON MERCATOR'S CHARTS.

(See Norie, pages 126-131; or Raper's "Practice of Navigation," pages 120-127, on this subject.)

A CHART used at sea for marking down a ship's track and for other purposes, exhibits the surface of the globe on a plane on which the meridians are drawn parallel to each other, and therefore the parts, BH, CI, DK, &c. (fig. p. 64), arcs of parallels of latitude, are increased and become equal to the corresponding parts of the equator UV, VW, &c. Now, in order that every point of this plane may occupy the same relative position with respect to each other that the points corresponding to them do on the surface of the globe, the distance between any points A and O, and A and F must be increased in the same proportion as the distance FO has been increased. The true difference of latitude, AO, is thus projected on the chart into what is called the meridional difference of latitude (see p.p. 66, 67), and the departure BH + CI + DK, &c., into the difference of longitude, and the representation is called a Mercator's projection. It is evidently a true representation as to form of every particular small track, but varies greatly as to point of scale in

The method lately introduced by Hugh Godfray, Esq., M.A., St. John's College, Cambridge, deserves special mention, as its beauty and simplicity will ultimately lead to its general adoption. A chart on the central projection, as stated above, exhibits the great circle as a straight line, and thus it is seen at once, whether the track between two places is a practicable one; hence, also, we have by inspection the point of highest latitude. An accompanying diagram then gives the different courses and distances to be run on each, in order to keep within of a point to the great circle. This chart and diagram is fully described in the Transactions of the Cambridge Philosophical Society, vol. X, part II; and is published by J. D. Potter, Poultry. Mr. W. C. Bergen, of Blyth, Master in the Mercantile Marine, has also published Charts on the Gnomic Projection, and claims to share with Mr. Godfray the credit of proposing the use of this projection for charts in navigation,

its different regions, each portion being more and more enlarged as it lies farther from the equator, and thus giving an appearance of distortion.*

(1.) In charts generally, the upper part as the spectator holds it, is the North, and that towards his right hand the East, as on the compass card.

(2.) On Mercator's chart the parallel lines from North to South (from top to bottom) are termed meridians, and they are all perpendicular to the equator, the meridians on the extreme right and left are the graduated meridians-so called from showing the divisions for degrees and minutes. The latitude is measured on the graduated meridians, and also the distance.

(3.) The parallel lines from West to East (from left to right) are called parallels, and they are all parallel to the equator, the parallels at the top and bottom are graduated to degrees and minutes--and longitude is measured on the graduated parallel.

(4.) The numerals in harbours, bays, channels, &c., indicate soundings reduced to low water spring tides.

* It is plain from the principles of Mercator's projection, and from the diagram (page 101) which connects the enlarged meridian with the difference of longitude, that if a ship set out from any point on the globe, and sail on the same oblique rhumb towards the pole, it can reach it only after an infinite number of revolutions round it. For from any point to the pole, the projected meridian is infinite in length, and so, therefore, is the difference of longitude due to this advance in latitude upon an oblique course. Consequently, this latitude can be reached only after the ship has circulated round the pole an infinite number of times.

These endless revolutions, however, are all performed in a finite time, the entire track of the ship being of limited extent. This, however paradoxical it may appear, is necessarily true from the principles of plane sailing, which shows that any finite advance in latitude is always connected with a finite length of track, this length being

diff. lat.

cos. course.

The apparent paradox of the infinite number of revolutions about the pole being performed in a finite time, becomes explicable when we consider that, whatever be the progressive rate of the ship along its undeviating course, the times of performing the successive revolutions continually diminish as the ship approaches the pole, both the extent of circuit and the time of tracing it tending to zero, the limit actually attained at the pole itself; hence there must ultimately be an infinite number of such circuits to occupy a finite time.

When the pole is reached the direction all along preserved may still be continued; and a descending path will be described similar to that just considered, and which will conduct the ship to the opposite pole, after an infinite number of revolutions round it, as in the former case. In receding from this pole the track described will at length unite with that at first traced, the point of junction being that from which the ship originally departed. But for the strict mathematical proof of these latter circumstances the student may consult Professor Davies' curious and instructive papers on Spherical Co-ordinates in the Edinburgh Transactions, vol. XII.

(5.) When the true course between two places is known, it must be remembered that Westerly variation is allowed to the RIGHT, and Easterly to the LEFT hand of the true course, in order to obtain the

COMPASS COURSE.

(6.) With respect to the method of determining the ship's position by cross bearings, it may be observed that this is the most complete of all methods when the difference of bearings is near 90°; but if the difference is small-as, for example, less than 10° or 20°, or near 180°-the ship's position will be uncertain, because a small error in the bearing will cause a great error in the distance.-(Raper, page 120, No. 367.)

EXERCISES ON THE CHART.

FOR ONLY MATE, FIRST MATE, AND MASTER.

(1.) Latitude 55° 5' N.

Longitude o 5 E.

North Sea.

(2.)

Latitude 57° 30′ N.

Required the course and distance to Hartlepool.

(3.) Latitude 53° 35' N. Longitude o 55 E.

Required the course and distance to

the Dudgeon Light.

(5.) Latitude 60° 21' N.

Longitude o 35 E.

Required the course and distance to Udsire,

(7.) Latitude 55° 28' N.

Longitude o 30 W.

Required the course and distance to Tynemouth Light.

(9.) Required the true and magnetic Bearing and Distance between Whitby and the Naze of Norway.

Longitude o 40 E.

Required the course and distance to Tynemouth Light.

(4.) Latitude 55° 10' N.

Longitude o 35 E.

Required the course and distance to Flambro' head.

(11.) A ship from Kinnaird's Head, in Scotland, sailed S.E. by E. (true) 186 miles required the latitude and longitude she is come to, and the direct course and distance she must sail, in order to arrive at Heligoland.

(12.) A ship from Heligoland sailed on a direct course between the North and West 197 miles, and spoke a ship which had run 170 miles on a direct course from Hartlepool: required the latitude and longitude of the place of meeting; also the course steered by each ship.

(13.)

Sunderland Light, bearing by compass S.W. S.
Coquet Island,

N.W.

Required the latitude and longitude of ship; also the course and distance to Hartlepool Light.

N N

(14.)

Buchaness Light, N. by W. W., by compass.

Girdleness Light, West,

Required the latitude and longitude of ship; also the course (by compass) and distance to the Staples.

Required the latitude and longitude in; also the compass course and distance to Peterhead.

Required the latitude and longitude in; also the compass course and distance to Outer Dowsings

(17.)

Farn Lights, S.W, by S. by compass

Berwick Lights, W. by N.

Required the latitude and longitude; also the distance from each light.

(18.)

The Dudgeon Light, W. by N. by compass.
Hasbro' Sand-end Light, S.S.W.,

Required the latitude and longitude of ship; also the compass course and distance to Flambro' Head.

(19.) Scarbro' light was observed to bear S.W. by compass, then sailed E.S.E. II miles, and the light then bore West: required the latitude and longitude of the ship at each station, and her distance from the light.

(20.) Coasting along shore, I observed Tynemouth light to bear W. by S. by compass; I then sailed S. by W. 16 miles, and the light bore N.W. by N.: required the latitude and longitude of the ship, and her distance from the light.

English and Bristol Channels, and South Coast of Ireland.