Two angular pieces of brass, each having a small and sharp steel pin at its extremity, are fastened to the index, and revolve freely around the lines ab and cd. The small screws, a, b, c, and d, move them in the directions of the lines ab, cd, for the purpose of bringing the steel pins exactly into the line which passes through the 0 of the index and the centre of the protractor.

To adjust them to their places, place the centre of the protractor over a marked point, and the o of the index to the o of the limb. Then mark the place of the index by the pins : after which, turn the index 180o, and see if the pins will mark the same points as before. If they do, the index is adjusted; if they do not, correct the error with the screws a, b, c, and d

To lay off an angle with the Protractor.

202. Let its centre be placed over the angular point, and the diameter passing through 0 and 180o, on the given line. Turn the screw that works the index, until the 0 of the vernier coincides with the division corresponding to the given angle; then let the angular brass pieces be turned down; the points dotted by the steel pins will show the direction of the required line.

If this line does not pass through the angular point, the pins are out of place, and must be adjusted.

First Method of Plotting.

203. Suppose it were required to make the plan of the harbour on a scale of 450 yards to an inch.

Divide the length of the base line AB, which we will suppose equal to 1140 yards, by 450, and the quotient 2.53 will express the length which is to represent the base line on the paper (Art. 33).

Draw an indefinite line AB, to represent the base, and having chosen any point, as A, for the first station, lay off 2.53 inches to B. The other extremity of the base line will thus be determined.

Then, place the circular protractor at A, and lay off the angle BAE, and then the angle EAG. Next, place the protractor at B, and lay off the angles ABE and EBC.

the station E. Let the protractor be then placed at this point, and all the angles of station E, laid down.

The point G, where EG intersects AG, and the point C, where EC intersects BC, will then be found.

By placing the protractor at C and G, we can determine the points D and F, when the place, on the paper, of all the stations will be known.

To unite the work done with the compass, spread the compass-notes before you, and draw through A a line to represent the meridian. This line makes an angle of 12° with the course AE.

Then, lay off from the scale the distances Aa, Ab, Aq, Ac, Ad, Ae, and at the several points erect perpendiculars to AE. Lay off on these perpendiculars the lengths of the offsets, and the curve traced through the points so determined, will be the margin of the lake.

At E, draw a parallel to the meridian through A, and lay down the course EH, which makes an angle of 50° with the meridian. Then, lay down the several distances to the offsets, and draw the offsets and lay off their lengths. Do the same for the course HI, and all the compass-work will be plotted.

Had there been work done with the plain-table, it could easily be united to that done with the theodolite.

Second Method of Plotting.

204. Place the centre of the protractor near the centre of the paper, and draw a line through the points 0 and 180o. This line will have the same position with the circular protractor that the base line AB had with the limb of the theodolite.

Lay off then from the 0 point an arc equal to the direction from A to E, also an arc equal to the direction AG, and through the centre point, and the points so determined, draw lines. Lay off in succession, in a similar manner, the directions taken at all the stations; and through the centre point, and the points so determined, draw lines, and designate each by the letters of the direction to which it corresponds.

Now, since all the lines drawn on the paper have the same

lines on the ground have with the limb of the theodolite, it follows that each direction will be parallel to its corresponding line upon the ground.

Hence, any line may be drawn parallel to that passing through 0 and 180o, to represent the base line AB. Having drawn such a line, and marked a point for the station A, lay off the length of the base, and the extremity will be the station B.

Through A and B, so determined, draw parallels respectively to the lines corresponding to the directions AE and BE, and the point of intersection will determine station E. Through B and E draw parallels to the lines which correspond to the directions BC, CE, and their point of intersection will determine station C. Through C and E draw lines parallel to the lines corresponding to the directions CE and ED, and the point of intersection will determine D. In a similar manner we may determine the stations F and G. Of surveying a harbour for the purpose of determining the depth of water, &c.

205. When a harbour is surveyed for the second object, viz., 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 referred to.

Let buoys be anchored on the principal shoals and along the edges of the channel, and using any 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 harbour 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.

206. If a person with a theodolite, or with any other in

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. similarly, for the other horizontal planes of section.

And

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 from the rocky shore across to the light-house.