Having the five points: the two places, the intersection with the equator, and the two points of maximum separation in latitude, the great circle may be traced. EXAMPLE. Given one place in Lat. 40° S., Long. 20° E., and another in Lat. 40° S., Long. 80° E., to project the great-circle track.

CHAPTER IV.

NAVIGATING THE SHIP.

TAKING THE DEPARTURE.

Art. 144. Upon clearing the land, the first step previous to shaping the course and commencing the log is to fix the ship's position, or, in other words, taking the departure.

The position of the ship may be referred to some well-known point of land, or conspicuous object, whose latitude and longitude are tabulated, either by a compass bearing, or an astronomical bearing, and an estimated or calculated distance; then the reverse of the bearing and the distance entered in the log-book as the first course and distance run.

Art. 145. 1. BY A SINGLE BEARING AND DISTANCE. The bearing of the point of land or conspicuous object being observed by the Azimuth Compass, the reverse of the reading entered in the log-book as the first course subject to the same compass corrections of variation and deB viation as all other compass courses-using the deviation corresponding to the course of the ship at the time the bearing was taken.

The distance to be estimated by the Navigator and entered as distance run. This estimation is liable to error, and may be considered sufficiently accurate when the distance is small; but, when great, the error becomes too large, and resort must be had to a more accurate method.

Art. 146. 2. BY TWO BEARINGS OF THE SAME OBJECT.If the ship's track lay along the land, 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 to change the bearing several points, note the same angle again.

Then, in Fig. 37, there are known the angles CBA and CAB; consequently ACB, and the side AB, being the distance run in the interval, to find the distance AC or BC.

EXAMPLE. A point of land, C, bore from the ship's first position E. N. E.; after running 25 miles N. by W. it bore S. E.; required, the distance of the ship from the point of land at the time of the last bearing.

Then the departure will be entered as a course N. W., and distance run 26.5 miles.

Art. 147. Should either angle be found equal to a right angle, or the bearing at right angles to the course, the distance may be found by direct inspection of the Traverse Tables; using the angle at the other observation as a Course and the Distance run as a Diff. Lat., the corresponding Dep. will be the distance of the object at the 90° bearing. As has been explained before, the Traverse Table is but a Table of Right Triangles, so that all similar problems may thus be solved by direct inspection.

The errors liable to become involved in this method will arise only from observation of the compass bearings, and the most favorable application will be when the triangle is equilateral.*

Solution by Inspection of Table 5.

Art. 148. The solution of this problem may be facilitated by the use of Table 54 and 5B. The course and distance must be first corrected for leeway and current, and then if the bearings are in points use Table 54; if in degrees, Table 5B.

RULE.

Multiply the distance run in the interval between taking the bearings by the two numbers found in the table under the "Difference between the course and first bearing," and opposite the "Difference between the course and second bearing;" the greater product is the distance of the object when the second bearing was taken; the other is its distance when abeam.

In the foregoing example the "Difference between the course and first bearing" being 7 points, "Difference *If the ship continues on the course until the 2d bearing the 1st bearing + the Diff. between the course and the 1st bearing; in other words, until the angle between the two bearings the angle between the course and 1st bearing: the triangle ACB (Fig. 37) becomes an isosceles triangle and BC = AB, or the distance from the position of the second bearing = the dis tance run between the two bearings.

between the course and second bearing" 13 points, the corresponding numbers in Table 5A are found to be 1.06 and 0.59, which, multiplied by 25, the distance run, will give 26.5 and 14.75, the distances, respectively, at times of taking second bearing and when the object was abeam.

Art. 149. 3. BY CROSS BEARINGS.-When the ship is so located that bearings may be taken of two prominent objects (located on the chart) differing nearly 90°, this method is sufficiently accurate to define the position of the ship.

Observe the true bearings of two points of land, drawing lines through them on the chart corresponding to the bearings, and the place of the ship will be at the point of intersection.

If the angle denoting the difference of bearings varies much from a right angle, the position will be uncertain, for the reason that a small error in the bearing will produce a large one in the distance.

Art. 150. 4. BY BEARINGS AND DISTANCES OF THREE OBJECTS.-When it becomes necessary to define the ship's position with great accuracy and there can be observed three prominent and known points on shore, resort must be had to this method.

Let A, B, and C, in Fig. 38, be three fixed objects on shore, and from the ship at D suppose the angles CDB and ADB were found: 400 and 60°.

With the complement of CDB = 50°, draw the lines BE and CE; the point of intersection will be the centre of a circle, on some point of whose circumference the ship must be. Then, with the complement of the angle ADB = 30°, draw the lines AF and BF, meeting at F, which point will be the centre of another circle, on some point of whose circumference the ship must be. D, the point of intersec

tion of the circumferences of the two circles, will be the position of the ship. Should either of the angles exceed 90°, the excess of the angle over 90° must be laid off on the side of the line joining the two stations opposite to that on which the ship lies.

It is evident that this method will be defective when the centres E and F are near each other and the two circles nearly coincide.

Art. 151. 5. BY SOUNDINGS.-Under some circumstances, where the depth of water is not great and varies gradually with the distance from the land, the ship's position may be approximately determined by soundings. Art. 152. 6. ASTRONOMICALLY.-When a Sumner line may be observed nearly at right angles to the bearing of a prominent object on shore, the intersection of the two will give accurately the position of the ship (see Chap. IX, Part II).

SHAPING THE COURSE.

Art. 153. Having taken the departure, the course must then be shaped, and from that moment the run by log must be determined. Having decided upon the track, whether by rhumb or by great circle, the true or the magnetic course may be taken from the chart; if the latter, 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 there should be deduced and allowed a proper amount for it. Should the wind be ahead and the ship under sail, that tack must be chosen upon which the greatest distance will be made good toward the port of destination; and, if sailing on the great circle, that track must be chosen which lies nearest the track.

THE DAY'S WORK.

Art. 154. 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, the variation of the compass, the deviation for each direction of the ship's head, the leeway, the set and drift of currents if any accurately known, the force and direction of the wind, state of sea, sail carried, &c.; to find the latitude and longitude by dead reckoning, the true course and distance made good, and the compass bearing and distance of the port of destination, or that point toward which the ship is to be directed dur ing 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, but that portion to be described here will be limited to the dead reckoning. Art. 155. The method of keeping the log-book has been already described, and to illustrate the method of computing the day's work the run shown in the columns of Form Z, Art. 98, Chap. II, Part I, will be taken.

Rules for working a day's work.

1. Correct the several courses sailed* for variation and leeway, and enter them in a traverse table, and opposite to each course place the distance run on that course, found by summing up the knots and fathoms sailed by the ship on that course. Find in Table 1 or 2 the difference of latitude and departure corresponding to each course and distance, and set them in their respective columns; then the difference between the sums of the northings and southings will be the difference of latitude made good, of the same name with the greater; and the difference between the sums of the eastings and westings will be the departure made good, of the same name with the greater quantity.

2. Seek in Table 1 or 2 until the above difference of latitude and the departure are found together in their respective columns; opposite to these will be the distance made good, and at the top or bottom of the page, according as the departure is less or greater than the difference of latitude, will be found the course.

*The set and drift of a current (if there be any) is to be reckoned as a course and distance, and on the first day after losing sight of the land the bearing and distance of it are to be taken into account.

3. If the latitude from which the ship's departure is taken, or yesterday's latitude, be of the same name as the difference of latitude, add them together; but if of different names, take their difference; the sum or remainder will be the present latitude of the same name as the greater.

4. Find the middle latitude between the latitude of yesterday and this day, which take as a course in Table 2, and seek for the departure in the column of difference of latitude; then will the distance corresponding be the difference of longitude, of the same name as the departure.

5. If the longitude in yesterday be of the same name as the difference of longitude, add them together; but if of different names take their difference; the sum or remainder will be the longitude in, of the same name as the greater.

EXAMPLE. At noon, January 22d (see Form Z, Chap. II, Part I), in Lat. 35° 10' N., Long. 134° 01′ W.; variation, 15° E.; deviation from E. to N. E, 1⁄2 point E.: to determine Lat. and Long. by D. R. at noon of January 23d.

By inspection of the columns it will be seen that the ship was on the port tack during the entire day, and, owing to wind and sea, making an average leeway of 14 points easterly. Hence, the total correction to be applied to the compass courses to obtain the true courses will be 3 points easterly.

Then form the following abstract:

From this abstract arrange the Traverse Table, and in Table 1 find the corresponding differences of latitudes and the departures.

Having obtained the Lat. and Long. by observation, the difference will show the set and drift of current, being careful to eliminate the combined errors of steering, estimating leeway, and logging.

In this case the Lat. and Long., by observation, are

From which the bearing and distance between positions by Obs. and by D. R. are found to be→