Dist. at 3h. (Naut. A.) 66 24 23 Proportional Log. of diff. 2537 And the error of the chronometer is 52s. fast on Greenwich mean time. A base line is selected as level as can be found, and as long as possible, this is lined, leveled, and measured with rods of Norway pine, with platt inum plates and points to serve as indices to connect the rods. They are daily examined by a standard measure, reference being had to the change of temperature. (See p. 165.) At each extremity stones are buried, and at the trig. points are put discs of copper or brass, with a centre poinFrom these extreme points angles are taken to points selected on high places, thus dividing the country into large triangles, and their sides calculated. in them. These are again subdivided into smaller triangles, whose sides may range from one mile to two miles. These lines are chained, horizontally, by the chain and plumb-line; or, as on the ordnance survey of Ireland, the lines of slopes are measured, and the angles of elevation and depression taken. Spires of churches, angles of towers and of public buildings are observed. On the main lines of the triangles, the heights of places are calculated from the field book, and marked on the lines. When inaccessible points are observed from other points, we must take a station near the inaccessible one, and reduce it to the centre by (sec. 244.) On the second or third pages of the field book, we sketch a diagram of the main triangle, and all chain lines, with their numbers written on the respective lines, in the direction in which the lines were run. The main triangle may be subdivided in any manner that the locality will allow. See Fig. 64 is the best. Here we have three check-lines, D F, D E, and F E, on the main triangle, and having the angles at A, B, and C, with the distances, A D, D C, C E, B E, B F, and F D, we can calculate F D, D E, and F E, insuring perfect accuracy. We chain as stated in Section 211. In keeping our field book we prefer the ordnance system of beginning at the bottom, and enter toward the top the offsets and inlets, stating at what line and distance we began, and on what; we note every fence and object that we pass over or near; leave a mark at every 10 chains, or 500 feet, and a small peg, numbered as in the field book. 398. See the diagram (figure 65). Here we began 114 feet farther on line 1 than where we met our picket and peg at 3500 feet, and closed on line 3 at 870, where we had a peg and a long Isoceles' triangle dug out of the ground. We write the bearings of lines as on line 3, and also take the angles, and mark them as above. When there are Woods. Poles are fastened to trees, and made to project over the tops of all the surrounding ones. The position of these are observed or Trigged. The roads, walks, lakes, etc., in these woods can be surveyed by traversing, closing, from time to time, on the principal stations or Trig. points, but we require one line running to one of the forest poles, on which to begin our traverse, and continue, closing occasionally on the main lines and Trig. points. 399. Traverse Surveying. See Secs. 216, 217, 255. The bearing of the most westerly station is taken. At Sec. 216 is given a good example where we begin at the W. line of the estate, making its bearing 0, and the land is kept on the right. There we began with zero and closed with 180, showing the work to close on the assumed bearing. 400. To Protract these Angles at Sec. 216. Draw the line A B through the sheet; let A be S, and B, N. On this lay of other lines parallel to A B, according to the number of bearings, size of protractor and scale. We lay down A B, then from B set off four, five, or more angles, L, K, I, and H. Lay the parallel ruler from A to L, draw a line and mark the distance A L of the second line on it. Lay the ruler from A to K, move one edge to pass through L, draw a line, mark the third line L K on it. Lay the ruler on A I, move the other edge to pass through K, draw the line K I, equal to the fourth line. Lay the ruler on A to H, make the other edge pass through I, and mark the fifth line, I H. Now, we suppose that we are getting too far from our first meridian, A B. We now remove the protractor to the next meridian, and select a point opposite H, and then lay off the bearings, G, F, E, D, etc. Now, from this new station, which we will call X, we lay the parallel ruler to F and make the other edge pass through H, and set off the sixth line H G. Lay the parallel ruler from X to F, and move the other edge through G, and mark the seventh line, G F, and so proceed. We have used a heavy circular protractor made by Troughton & Simms, of London, it is 12 inches diameter, with an arm of 10 inches, this, with a parallel ruler 4 feet long, enabled us to lay down lines and angles with facility and extreme accuracy. 401. By a table of tangents we lay off on one of the lines, A B, the distance, 20 inches, on a scale of 20 parts to the inch. Then find the nat. tangent to the required angle, and multiply it by 400 divisions of the scale, it will give the perp., B C, at the end of the base. Join A and C, and on A C lay off the given distance, and so proceed. By this means we can, without a protractor, lay off any required angle. There is always a content plat or plan made, which is lettered and numbered, and the Register Sheet made to correspond with it. 3 403. Computation by Scale. drawn on a large scale, of 2 or by bringing the edges together. Draw a line about an inch from the margin; on this line mark off every inch, and dot through; now open the sheet and draw corresponding lines through these dots; make a small circle around every fifth one, and number them in pencil mark. Where the plats or maps for content are chains to the inch, we double up the sheet Lines are now drawn through the part to be computed. Where every pair of lines meet the boundaries, the outlines are then equated with a piece of thin glass having a perpendicular line cut on it, or, better, with a piece of transparent horn. When all the outlines of the figure are thus equated, we measure the length in chains, which, multiplied by the chains to one inch, will give the content in square chains. This gives an excellent check on the contents found by triangulation or traversing. It will be very convenient to have a strip of long drawing paper, on the edge of which a scale of inches is made. We apply zero to the left-hand side of the first parallel, and make a mark, a, at the other end; then bring mark a to the left side of the second parallelogram, and make a mark, b, at the other end, and so continue to the end. Then apply the required scale to the fractional part, to find the total distance. The English surveyors compute by triangulation on paper, and sometimes by parallels having a long scale, with a movable vernier and cross-hairs, to ་ equate the boundaries. We do not wish to be understood as favoring computation from paper. The Irish surveyors always draw the parallel lines on the content plat or map, and mark the scale at three or four places, to test the expansion or contraction of the sheet during the construction or calculation. We prefer, when possible, 3 chains, or 200 feet, to an inch for estates in the country, and 40 feet for city property. (Fig. 66.) triangle A C B, by a line parallel to one of its sides. AE 2 B (AD. Sin. A --- draw a line, We find the area B NOTE. We prefer this to any other complicated formula, in cutting off a given area from a quadrilateral or triangular field. 406. When the area B or A is to be cut off by the line D E, (Fig. 66,) making a given angle, C, with the line A B, let area Let the angle at A = b, the required side. S. that at D = c, and that at E d, and AD, From the value of x we find A E and DE from above. = Having A D and A E from these formulas, let us assume A D 10 chains, and having found the value of A E by substituting 10 chains for x. Multiply the numerical value of A E by 10 chains, and again by 1⁄2 the natural sine of the angle D A B, let its area = s, L, Then s S: A D2: the required A E 2, This useful problem was proposed to us in Dublin, at our examination for Certified Land Surveyor, September, 1835, by W. Longfield, Esq., Civil Engineer and Surveyor. NOTE. When the given area is to be cut off by the shortest line, D E, in the triangle A D E, (Fig. 66.) then A D D E. 407. When the area B is to be cut off by the line D E, starting from the point D. (Fig. 66.) 2 B 2 B 40S. From the quadrilateral, (Fig. 67,) A B C D, to cut off the area A by the line F E, parallel to the side B C. Produce the lines B A and C D to meet at G. Take the angles at Measure G D and AB, and of the triBy Sec. 404 we find 2 the line A For G E. For triangle G C B : triangle GFE :: G B 2 : GF or GC2: GE 2. 2 :: By taking the square roots we find G F and G E. 409. To divide any quadrilateral figure into any number of equal parts, by lines dividing one of the sides into equal parts. Let A B C D be the required figure, (see Fig. 70,) whose angles, sides, and areas are given, produce the the sides CD and BA to meet in E. As the angles at A and D are given, we find the angle E, and consequently the sides A E and D E, and area B of the triangle A E D. We have the distances E A, E F, and E G, EF K, and B + 2 A and areas B + A triangle EG II and by Sec. 29. triangle 410. If, in the last problem, it were required to have the sides BA and CD proportionally divided so as to give equal areas, Let B A a, C D = n a, A E b, D E = c, and 1⁄2 sin. E S, and |