stationed at each extremity of the base line, it will not be necessary to establish buoys. A boat, provided with an anchor, a sounding line, and a signal flag, has only to throw its anchor, hoist its signal flag, and make the sounding, while the persons at the extremities of the base line measure the angles;-from these data, the precise place of the boat can be determined,

207. There is also another method of determining the places at which the soundings are made, that admits of great despatch, and which, if the observations be made with care, affords results sufficiently accurate.

Having established, trigonometrically, three points which can be seen from all parts of the harbour, and having provided a sextant, let the sounding be made at any place in the harbour, and at the same time the three angles subtended by lines drawn to the three fixed points, measured with the sextant.

The problem, to find from these data the place of the boat at the time of the sounding, is the same as example 6, page 74.

It is only necessary to measure two of the angles, but it is safest to measure the third also, as it affords a verification of the work.

The great rapidity with which angles can be measured with the sextant, by one skilled in its use, renders this a most expeditious method of sounding and surveying a harbour.

The sextant is not described, nor are its uses explained in these Elements, because its construction combines many philosophical principles, with which the surveyor cannot be supposed conversant.

208. There is yet another method of finding the soundings, which, although not as accurate as those already explained, will, nevertheless, afford results approximating nearly to the truth. It is this-Let a boat be rowed uniformly across the harbour, from one extremity to the other of any of the lines determined trigonometrically. Let soundings be made continually, and let the precise time of making each be carefully noted. Then, knowing the length of the entire line, the time spent in passing over it, as also the time of making each of the soundings, we can easily find the points of the line at which the several soundings were made; and hence, the depth of water at those points becomes known. Sound

ings may thus be made along any number of known lines, and a comparison of the depths found on different lines, at or near their points of intersection, will show with what degree of accuracy the work has been done.

209. If the soundings are made in tide-waters, the time of high tide must be carefully noted, as also the precise time of making the sounding, so that the exact depth at high or low water may be known. It is considered preferable to reduce the soundings to high-water mark, and the number of feet which the tide rises and falls should be noted on the map.

210. Having plotted the work done with the theodolite, as also the outline of the harbour traced with the compass, it remains to delineate the bottom of the harbour; and this is done by means of horizontal curves (Chap. VI), which have already been used to represent broken or undulating ground.

Let the plane of reference be taken through high-water mark, or to coincide with the surface of the water at high tide. The accuracy with which the bottom of the harbour is to be delineated, will guide us in fixing the distance between the horizontal planes of section.

The first horizontal plane should be passed at a distance below the shallowest point that has been sounded, equal to the number of feet fixed upon for the distance between the planes of section; and the curve, in which it intersects the bottom of the harbour determined as in Chapter VI. And similarly, for the other horizontal planes of section.

Having thus delineated the bottom of the harbour, and noted on the map the distance of each intersecting plane below the plane of reference, let such lines be drawn as will indicate the channels, shoals, sunken rocks, and direction of the current.

In the example given in plate 6, soundings have been made in three directions from the sand-bar in the harbour, and also

CHAPTER VIII.

Of Navigation.

1. We have given, in the preceeding chapters of this work, various applications of Trigonometry. We propose, in the following chapter to explain the best methods of determining the place of a ship at sea. This application constitutes the science of Navigation.

There are two methods of determining the place of a ship at

sea.

1st. When a ship departs on her voyage, if we note her courses and the distance sailed, we may, at any time, by means of Plane Trigonometry, determine her place very nearly.

2nd. By means of observations on the heavenly bodies and the aid of Spherical Trigonometry, we may determine with great accuracy, the exact place of the ship. This method is called Nautical Astronomy.

The first part of Navigation, viz. the cases which can be solved without the aid of observations on the heavenly bodies, will be alone treated of in this chapter.

2. The earth is nearly spherical. For the purposes of Navigation it may be considered as perfectly so. It revolves round one of its diameters, called the axis, in about twenty-four hours.

3. The great circle, whose poles are the extremities of the axis, is called the equator. The poles of the equator are called the poles of the earth-the one is called the north pole, and the other the south pole.

4. Every great circle which passes through the poles cuts the equator at right angles, and is called a meridian circle. Every place on the surface of the earth has its own meridian; but for the purposes of Geography and Navigation, all these meridians are reckoned from a particular meridian, which is called the first meridian. The English have fixed on the meridian of Greenwich Observatory for the first meridian.

5. The longitude of any place is the arc of the equator intercepted between the meridian of that place and the first meridian, and is east or west, according as the place lies east or west of the first meridian.

6. The difference of longitude of two places is the arc of the equator included between their meridians; this arc is equal to the difference of longitudes when they are of the same name, and to their sum, when they are of different names.

measured on the meridian of the place, and is north or south according as the place lies north or south of the equator.

8. The small circles drawn parallel to the equator, are called parallels of latitude. The arc of any meridian intercepted between the parallels passing through any two places, measures the difference of latitude of those places; this difference is found by subtracting their latitudes when they are of the same name, and by adding them when they are of different names.

9. The sensible horizon of any place is an imaginary plane, supposed to touch the earth at that place, and to be extended to the heavens. A plane passing through the centre of the earth, and parallel to the sensible horizon, is called the rational horizon. The north and south line, is the intersection of the plane of the meridian circle with the sensible horizon, and the line which is drawn perpendicular to this, is called the east and west line.

10. The course of a ship, at any point, is the angle which her track makes with the meridian. So long as the course is unchanged, the ship would sail in a straight line, provided the meridians were truly parallel; but as the meridians bend constantly toward the pole, the direction of her path is continually changing, and she moves in a curve called the rhumb line. The course of a ship is indicated by the mariner's compass.

11. The mariner's compass consists of a circular card, whose circumference is divided into thirty-two equal parts called points; each point being subdivided into four equal parts called quarter points.

To the under side of this card a slender bar of magnetized steel, called a needle, is permanently attached. The direction of the needle corresponds to the diameter NS. The

diameter EW, at right angles to NS, is intended to indicate the east and west points. The points of the compass are thus read:

by east, north north east, north east and by north, north east; and so on, round the compass, as indicated by the letters.

The card being permitted to turn freely on the pin, on which it is poised as a centre, the line NS will always indicate the true magnetic meridian, but this, as we have seen it Art. 153, page 127, is not the true meridian, and hence, the variation must always be allowed for.

On the interior of the compass box, in which the card swings, are two marks, a and b, which lie in a line passing through the centre of the card, and the compass box is so placed that this line shall be parallel to the keel of the ship. Consequently, if a be placed towards the bow of the vessel, the point which it marks on the card will show the compass course, for the line NS is always north and south, and EW east and west. The course is generally read to quarter points, and as a quadrant contains eight points, each point will be equal to 90°÷8=11° 15'; and a quarter point=11° 15′÷4=2° 48′ 45′′. The table of Rhumbs, after the Traverse Table, shows the degrees of each course to quarter points.

12. A ship's rate of sailing is determined by means of an instruments, called the log, and an attached line called the log line. The log is a piece of wood in the form of a sector of a circle, the rim of which is loaded with lead, so that when it is heaved into the sea it assumes a vertical position. The log line is so attached as to draw the log square against the water, that it may not be drawn along after the ship as the line unwinds from the reel, by the ship's forward motion.

The time in which the log line unwinds from the reel, is noted by a sand-glass, through which the sand passes in half a minute; that is, in the one hundred and twentieth part of an

hour.

For convenience, the log line is divided into equal parts, marked by knots, and each part is equal to the one hundred and twentieth part of a nautical or geographical mile.*

Now, since half a minute is the one hundred and twentieth part of an hour, and each knot measures the one hundred and twentieth part of a mile, it follows that the number of knots reeled off while the half minute glass runs out, will indicate how fast the ship sails per hour.

A geographical mile is one minute, or one-sixtieth of a degree, measured on the equator. Taking the diameter at 7916 English miles, the geographical mile will be about 6079 feet; that is, about one-sixth greater than the English mile, which is 5280 feet.