from Table 42, Bowditch. Now from the log of the difference of longitude subtract the log of this meridional difference of latitude, and the result is the log tangent of the course. Look up the angle corresponding to this log tangent in Table 43; this gives the course. Also in Table 43 find the log secant of the course, and add this to the log of the true difference of latitude, and the number corresponding to this sum is the distance. For a more complete account see Bowditch, pages 49-53. The following example may help make the problem clear:

Find by Mercator's sailing the course and distance from Cape Cod lighthouse, in latitude 42° 03′ N, longitude 70° 04' W, to the Island of St. Mary, one of the Western Islands, in latitude 26° 59' N and longitude 25° 10' W.

Table 42 log. tan. 0.83623 = 81° 42′ = Course

Diff. Lat. 304' log. 2.48287

log. sec. 81° 42′ = 0.84056

log. 3.32343 = 2,106 miles dist.

True Course, S 81° 42′ E.

Distance, 2,106 miles.

Formulæ by which all possible cases of Mercator's sailing may be solved are tabulated on page 50, Bowditch.

Great Circle Sailing. The shortest distance between two points on a sphere is along the great circle passing through the points. The simplest way of finding the Great Circle Course is to stretch a string from one point to the other on a globe, note the latitude of the points at which the string cuts the different meridians, and mark upon the chart the positions thus found. Join these points, and you have the course. It is necessary to steer from point to point in order to come nearest to the most direct path between places. On long voyages, such as a trans

Atlantic or trans-Pacific trip, a great circle track is considerably shorter and is usually taken by steamers. Sailing vessels may find great circle sailing very useful, not only in indicating the shortest distance between two places, but also as showing which is the best tack if the course cannot be laid before the wind. The tack which will keep the vessel nearest to the great circle will be the gaining tack; it may happen that what is apparently a losing tack is in reality a gaining one. Great circle sailing is applicable mainly to steam vessels, for they can generally keep any course desired.

Current or Oblique Sailing. A current is a body of water flowing steadily in a certain direction. The set of a current is the direction in which it moves; the drift is the rate at which it moves in a specified length of time, as, for example, so many knots per hour. When a ship sails with a current her motion is increased by an amount equal to the drift of the current; when sailing against a cur

rent her speed is diminished by an amount equal to the drift. When sailing obliquely to a current a ship's motion is increased if the direction of the current is from abaft the beam, and decreased if from forward of the beam.

The true course of a vessel sailing in a current is the diagonal of a parallelogram, in which the drift and set of the current in a certain time, and the course steered and speed in the same time, form the adjacent sides. However, in case the set of the current and the course steered are in the same straight line, the current will not affect the course. A simple illustration will show how to find the course if the current is across the path of the ship. Let A be a ship's position, A C the direction of the course steered, and also let A C be the distance in miles the vessel will travel in an hour; further, let A B be the drift and set of a current. Complete the parallelogram A B D C, and the diagonal A D gives the course made good and the distance sailed in one hour.

This is never used when in deep water. The reverse of the

above problem is worked at noon each day when in open ocean, to find the set and drift of the ocean currents. However, in coasting, and where known currents exist, the above may be useful.

Navigating the Ship. Taking the Departure. Upon clearing the land, the first step before shaping the course and commencing the log is to fix the ship's position or to take the departure. The position of the ship may be referred to some known point of land, lighthouse, or any prominence whose latitude and longitude are tabulated. The latitudes and longitudes of such places all over the world are tabulated in Table 49, Bowditch.

Departure by a Single Bearing and Distance. Get the compass bearing of the point or lighthouse, and reverse the bearing; this reversed bearing is entered or set down as the first course, sub

B

C

ject to the same compass corrections as all
compass courses, as previously explained.
The distance of the ship from the object,
to be estimated by the navigator, is taken
as the distance run; this latter is of course
somewhat inaccurate, but ordinarily will
suffice. The bearing of the object is re-
versed in order to get the course of the
ship, for suppose a light was found to
bear by compass NW, then the ship is in a
SE direction from the lighthouse.
more accurate method is to take two
bearings of the same object; for example,
if a ship's track lay so that the object will

A

remain in view for a time, take the bearing of the object and note the angle between it and the direction of the ship's head. After the ship has run on the same course far enough to change the bearing of the object a few points, note again the angle between the bearing of the object and the direction of the ship's head. In the following figure let C be a point of land, and let A be the first position of the ship and B the second; the angle C A B and CBA is obtained as noted above, and the side A B of the triangle is the distance the ship has run between the times of taking the bearings. We may find the necessary distance A C or B C by plane trigonometry, as follows:

For example, when the ship was at A it was running ENE; after running 25 miles N by W it bore SE: required the distance of the ship from the point of land C at the time of the last bearing at B.

For further examples see Bowditch, pages 61 and 62.

Shaping the course. Having taken the departure, the course must then be shaped, whether by rhumb or by great circle. The true or magnetic course may be taken from the chart; if the magnetic course be taken, allowance must be made for the deviation for that direction of the ship's head to obtain the compass course. If sailing in a known current, proper allowance for it should be made. Should the wind be ahead, the tack must be chosen upon which the greatest distance will be made good toward the point of destination; this latter must be altered, however, should there be any outlying shoals.

The day's work is the operation of computing from the data expressed in the log-book the ship's run for twenty-four hours preceding each noon. The term is usually restricted to the dead reckoning, and the data given are the latitude and longitude at the preceding noon (by observation), the compass courses, the distance run on each course (given by the log), compass variation, compass deviation for each direction of the ship's head, the leeway, the set and drifts of currents if they are known, the force and direction of the wind, state of sea, sail carried, etc. This data is recorded to find the latitude and longitude by dead reckoning, the true course made good, and the compass bearing and distance

of the port of destination, or that point toward which the ship is to be directed during the coming twenty-four hours.

Strictly speaking, the day's work includes all the computations the navigator must make each day, the results of which are entered in the log-book. In sailing vessels the mate keeps the log-book and the position of the ship by dead reckoning. Dead reckoning is merely the application of the problems already laid down; for instance, if the ship's position is known at noon on a certain day, and during the following twenty-four hours it has run a certain number of miles on a certain compass course, the position at noon on the next day is found as follows: First correct the compass course for variation, deviation, leeway and known currents, to find the true course made good; then in Table I or II, (depending upon whether the course is given in points or degrees) under the distance run, find the latitude made good and the departure. We may change this departure into difference of longitude, as indicated by example in parallel sailing. By applying our difference of latitude and difference of longitude to our position on the preceding day, we get our new position. This method of finding the ship's position is called dead reckoning. On making the land the navigator is aided in his dead reckoning by the lead line, which gives another clue to the ship's position.

Making the Land. Every precaution and forethought should be observed as the ship draws near the land, and the careful navigator will be on the lookout for the many evidences of the proximity of the land, such as the shoaling of the water, the change of color of the water, the change of temperature of the surface water, the presence of land birds, etc. As soon as land is made, the ship's position should be marked off on the chart by reckoning and verified by bearings of well-defined points, if any are in view. An easy and satisfactory method of finding position on making land is to take sextant angles of any well-defined prominent points, which, when used in connection with Lecky's "Off Shore Danger Angles," gives the distance in miles from the point of land to the ship, and by also observing the bearing of the point the exact position of the ship may readily be established.

It is always a good plan to make land, if it can be done without going too far out of one's way, to get a line on the chronome