circumference of a circle cutting the straight lines in the points marked 0 and 90°.

To lay off an angle, as, for instance, the angle 14° 29′. The half of it is 7° 14′ 30′′, the natural sine of which is 0.126005, or 1.26 to the radius of 10 inches. Set off from 0 to b, as in the figure, this distance taken from the scale, and through the two points b, b, thus determined, draw a straight line. This line should pass through the centre, and will make with the line (0, 0) the angle 14° 29'; and any line on the paper drawn parallel to it, will make with the line (0, 0) the same angle. The further application is obvious.

MARITIME SURVEYING.

152. When, in connection with a trigonometrical survey on shore, a harbor is to be surveyed, as in the example, for the purpose of ascertaining the channels, their depth and width, the positions of shoals, and the depth of water thereon, other means must be used, and other examinations made, in addition to those already described.

Let buoys be anchored on the principal shoals and along the edges of the channel; and using any one of the lines already determined as a base, let the angles subtended by lines drawn from its extremities, to the buoys respectively, be measured with the theodolite. Then, there will be known, in each triangle, the base and angles at the base, from which the distances to the buoys are easily found; and hence, their positions become known.

Having made the soundings, and ascertained the exact depth of the water at each of the buoys, several points of the harbor are established, at which the precise depth of the water is known; and by increasing the number of the buoys, the depth of the water can be found at as many points as may be deemed necessary.

153. If a person with a theodolite, or with any other instrument adapted to the measurement of horizontal angles, be 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 the anchor, hoist the 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.

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

Having established, trigonometrically, three points which can be seen from all parts of the harbor, and having provided a sextant, let the sounding be made at any place in the harbor, 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, p. 93.

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 harbor.

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.

155. There is yet another method of finding the soundings, which, although not as accurate as those already explained,

truth. It is this:-Let a boat be rowed, with uniform speed, across the harbor, 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.

156. If a person stationed on shore with a theodolite, takes the bearing of the boat, at every second or third sounding, determined by hoisting a flag, it will fix the positions of the soundings with great accuracy. Soundings 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.

Sounding-lines should be made of strong cord, and divided into feet or fathoms, by different colored rags or other marks. The lead is shaped like the frustum of a cone, with the base B hollowed out, to hold some grease. The land or mud of the bottom adheres to the grease, and thus shows the nature of the bottom, which should be entered in the field-book, and laid down upon the map. As the cord is liable to change its length, it should be compared, from time to time, with some standard. In tide-waters, the exact time of each sounding is to be noticed, and an assistant should note the height of the tide at regular intervals, upon a tidegauge. The tide-gauge is permanently placed at some convenient point of the harbor, and its 0 point is referred, by

B

means of a spirit-level, to some fixed bench-mark, on a level with mean low-water mark, to which all the soundings must be reduced.

157. Having plotted the work done with the theodolite, as also the outline of the harbor traced with the compass, it remains to delineate the bottom of the harbor; and this is done by means of horizontal curves, hereafter explained, (Bk. III., Art. 29), which are used to represent broken or undulating ground.

Let the plane of reference be taken through low-water mark, or to coincide with the surface of the water, at low tide. The accuracy with which the bottom of the harbor 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 harbor, determined as in Book III., Art. 25. And similarly, for the other horizontal planes of section.

Having thus delineated the bottom of the harbor, 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 harbor, and also from the rocky shore across to the light-house.

BOOK III.

LEVELLING AND ITS APPLICATIONS.

SECTION I.

OF LEVELLING.

1. LEVELLING is the art of determining the relative distances of points from the centre of the earth.

2. A line whose points are all equally distant from the centre of the earth, is called a line of true level; and a surface, all whose points are equally distant from the centre of the earth, as the surface of still water, is called a level surface.

3. One point is said to be above another, when it is farther from the centre of the earth; and this difference of distance from the centre, is called the difference of level between the two points.

4. A straight line drawn tangent to a line of true level, at any point, is a horizontal line, and is called the line of apparent level. Thus, (Pl. 4, Fig. 1), if C is the centre of the earth, and AEF a line of true level, ABD is the line of apparent level. This is the line of level determined by an instrument. The difference between the apparent and true level of the points A and E, is BE, the excess of the secant CB, of the arc AE, over the radius CE.