on these lines drive stakes at equal distances, a1, a2, a 3, com mencing from the points B and C. If represents the radius of the circle, and d the distance between the points a,, a,, ɑ 39 &c., then (Art. 172), √r2+d2-r a will be the distance which must be set off from the first point a,, in the direction a, E, to obtain a point of the circular arc. In like manner, A √r2+(2d) *—: E b, b2 63 64 65 64 bз b2 b, D will be the distance to be set off from the point a,, in the direction a,E; and, generally, √ r2+(nd)3· will be the distance to be set off at the nth points from B and C. For example, let r be one mile, or 5280 feet, and d equal to 100 feet; then, √52802+1002-5280-.94 feet, will be the distance a,b,. In a similar manner, we find at a2, or 200 feet from B, the offset will be 3.79 feet. 8.52 66 (180.) Second Method.-When the center of the circle can not be seen from every part of the curve, the offsets may be set off perpendicularly to the tangent BF, in which case they must be computed from the formula a,G=BH=BE-HE=r-√r3-ď. 5 64 F If r=5280 feet, we shall find the offsets at intervals of 100 For small distances, the offsets will be given with sufficient accuracy by the formula d 2r' see Art. 172. 1 2 2 It is very common for surveyors, after they have found the first point, b,, of the curve, to join the points B, b,, and produce the line Bb, to the distance d, and from the end of this line set off an offset to determine the point b,; then, producing the line b12 set off a third offset to determine the point b,, and The objection to this method is, that any error committed in setting out one of the points of the curve will occasion an error in every succeeding one. Whenever this method, therefore, is employed, it should be checked by determining the position of every fourth or fifth point by independent computation and measurement. so on. (181.) Third Method.-Where the radius of the curve is small, place a theodolite at B, and point its telescope toward C. Place another theodolite at C, and point its telescope toward E, the point of intersection of the lines AB, CD produced. Then, if the former be moved through any number of degrees toward a1, and the latter the same number of degrees toward the point a, will be a point of the curve, for the angle Ba, C will be equal to BCD (Geom., Prop. 16, B. III.). In the same manner, ɑ„, α ̧, &c., any number of points of the curve, may be determined. It will be most convenient to move the a a a I theodolites each time through an even number of degrees, for example, an arc of two degrees, and a stake must be driven at each of the points of intersection a,, a, a,, &c. The accuracy of this method is independent of any undulations in the surface of the ground, so that in a hilly country this method may be preferable to any other. When the position of one end of the curve is not absolutely determined, the engineer may proceed more rapidly. Suppose it is required to trace an arc of a circle having a curvature of two degrees for a hundred feet. Place a theodolite at C, the point where the curve commences, and lay off from the line CE, toward B, an angle of two degrees, and in the direction of the axis of the instrument set off a distance of 100 feet, which will give the first point a, of the curve. Next lay off from CE an angle of four degrees, and from a, set off a distance of 100 feet, and the point where this line cuts the axis of the instrument produced will be the second point a2. In the same manner, lay off from CE an angle of six degrees, and from a, set off a distance of 100 feet, and the point where it cuts the axis of the instrument produced will be the third point a ̧. All the points a,, a, a,, etc., thus determined lie in the circumference of a circle (Geom., Prop. 15, B. III.). Circles thus drawn are generally made with a curvature of one or two degrees, or some convenient fraction of a degree, for every hundred feet. This method is very ex tensively practiced in the United States. 3 Surveying Harbors. (182.) In surveying a harbor, it is necessary to determine the position of the most conspicuous objects, to trace the outline of the shore, and discover the depth of water in the neighborhood of the channel. A smooth, level piece of ground is G chosen, on which a base line of considerable length is measured, and station staves are fixed at its extremities. We also erect station staves on all the prominent points to be surveyed, forming a series of triangles covering the entire surface of the harbor. The angles of these triangles are now measured with a theodolite, and their sides computed. After the principal points have been determined, subordinate points may be ascertained by the compass or plane table. Let the following figure be a map of a harbor to be survey ed. We select the most favorable position for a base line, which is found to be on the right of the harbor, from A to B. We also erect station flags at the points C, D, E, F, and G. Having carefully measured the base line AB, we measure the three angles of the triangle ABC, which enables us to compute the remaining sides. We then measure the three angles of the triangle ACD, and by means of the side AC, just computed, we are enabled to compute AD and CD. We then measure the three angles of the triangle CDF, and by means of the side CD, just found, we are enabled to compute CF and DF. Proceeding in the same manner with the triangles CEF, DFG, we are enabled, after measuring the angles, to compute the sides. (183.) Having determined the main points of the harbor, we may proceed to a more detailed survey by means of the chain and compass. If it is required to trace the shore, HCK, wo commence at H, and observe the bearings with the compass, and measure the distances with the chain. Where the shore is undulating, it is most convenient to run a straight line for a considerable distance, and at frequent intervals measure offsets to the shore. When a great many objects are to be represented upon a map, the most convenient instrument is The Plane Table. (184.) The plane table is a board about sixteen B subdivided as minutely as the size of the table will admit. On one side of the board there is usually a diagonal scale of equal parts. A compass box is sometimes attached, which renders the plane table capable of answering the purpose of a surveyor's compass. The ruler, A, is made of brass, as long as the diagonal of the table, and about two inches broad. A perpendicular sightvane, B, B, is fixed to each extremity of the ruler, and the eye looking through one of them, the vertical thread in the other is made to bisect any required distant object. To the under side of the table, a center is attached with a ball and socket, or parallel plate screws, like those of the theodolite, by which it can be placed upon a staff-head; and the table may be made horizontal by means of a detached spirit level |