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 intersection of the lines AE and BE will determine 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 deter mined, 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. The work done with the plane-table (Art. 28), is united to the work done with the theodolite, by simply placing the line AN on the paper of the plain-table, upon the line AN, drawn on the plot of the triangulation. SECOND METHOD OF PLOTTING. 41. Place the centre of the protractor near the centre of the paper, and draw a line through the points 0 and 180°. This line will have the same position with the cir cular protractor that the base line AB had with the limb Lay off then from the 0 point an arc equal to the direc tion 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 position with the circular protractor, as the corresponding 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 180°, 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 direc tions CE and ED, and the point of intersection will determine D. In a similar manner we may determine the stations F and G. METHOD OF CHORDS. 42. Let us first prove that the chord of a given arc equal to the sine of half the arc with double the radius. Let DAF be any given angle, and AH a line bisecting it. Let H K Since AF2AC we have, from similar triangles, HF2KC, but DC=2KC, hence EF = CD. TO LAY OFF AN ANGLE. 43. To avoid, as far as possible, the use of fractions, let us suppose the radius of the table of natural sines to be 1 ten, or 10 inches. A Take from a scale 5 equal parts, with which as a radius, from the centre A, describe an arc CD. Take from the table the natural sine of half the arc, and remove the decimal point one place to the right; the result will express the sine of half the arc to the radius 10, or the chord of the arc to the radius 5. From the same scale, take this sine in the dividers, and from C as a centre, describe an arc cutting CD in D; draw AD, and CAD will be the angle required. This is the most accurate of all the methods of laying off an angle, and it may also be applied advantageously to the second method of plotting, thus: 90° Draw a fine straight line, generally in the direction of the meridian or of the base line of the survey; and also a line exactly perpendicular to it. From the point of intersection, as a centre, with a radius of 5 equal parts of the scale, describe the circumference of a circle cutting the straight lines in the points marked 0 and 90°. 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 SECTION II. MARITIME SURVEYING. 44. When, in connection with a trigonometrical survey on shore, a harbor is to be surveyed 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 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. 45. 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 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 or the boat can be determined. 46. There is another method of determining the places 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 pro vided 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, page 62. 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. 47. 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 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 Boundings were made; and hence, the depth of water at those points becomes known. 48. If a person stationed on shore with a theodolite, takes the bearing of the boat, at every second or third sounding, |