The easiest method of setting out the parallel in the above case, is to fix the middle of the chain at the point C, and its ends on the lines CD, CE'; then carry the middle of the chain from C towards F, and mark the point to which it reaches; and repeat this on the other side of C, as shown by the finely dotted lines in the figure. INACCESSIBLE AREAS. Fig.142. (207) Triangles. In the case of a triangular field, in which one side cannot be measured, or determined by any of the methods just given, the two accessible sides may be prolonged to their full length, and an equal symmetrical triangle formed, all of whose sides. can be measured. Thus in Fig. 102, page 103, if CDE be the original triangle, of which the side EC is inaccessible, DFP will be equal to it. But if this also be impossible, portions of the sides may be measured, as AD, AE, B in the figure in the margin, and also DE, and the area of this triangle found by any of the methods which have been given. Then is the desired area of the triangle ABC= area of AB × AC D A E ADE × AD X AE In the case Fig. 143. (208) Quadrilaterals. of a four-sided field, whose sides cannot be measured, or prolonged, but whose diagonals can be measured, the area may be obtained thus. Measure the diagonals AC and BD; and also the portions AE, EC, into which one of them is divided by the other. Calcu B late the area of the triangle BCE, by the preceding method, or any of those heretofore given. Then the area of the quadrilateral (209) Polygons. Methods for obtaining the areas of inaccessible fields of more than four sides, have been given in Arts. (101,) &c. PART III. COMPASS SURVEYING; OR By the Third Method CHAPTER I. ANGULAR SURVEYING IN GENERAL. (210) Angular Surveying determines the relative positions of points, and therefore of lines, on the THIRD PRINCIPLE, as explained in Art. (7), which should now be referred to. (211) When the two lines which form an angle lie in the same horizontal or level plane, the angle is called a horizontal angle.* When these lines lie in a plane perpendicular to the former, the angle is called a vertical angle. When one of the lines is horizontal and the other line from the eye of the observer passes above the former, and in the same vertical plane, the angle is called an angle of elevation. When the latter line passes below the horizontal line, and in the same vertical plane, the angle is called an angle of depression. When the two lines which form an angle, lie in other planes which make oblique angles with each of the former planes, the angle is called an oblique angle. Horizontal angles are the only angles employed in common land surveying. A plane is said to be horizontal, or level, when it is parallel to the surface of standing water, or perpendicular to a plumb-line. A line is horizontal when it lies in a horizontal plane, (212) The angles between the directions of two lines, which it is necessary to measure, may be obtained by a great variety of instruments. All of them are in substance mere modifications of the very simple one which will now be described, and which any one. can make for himself. fix a needle, or sharp-pointed wire, and upon this fix a straight stick, or thin ruler placed edge-wise, (called an alidade), so that it may turn freely on this point and nearly touch the graduations. of the circle. Fasten the circle on a staff, pointed at the other end, and long enough to bring the alidade to the height of the eyes. The instrument is now complete. It may be called a Goniometer, or Angle-measurer. (214) Now let it be required to measure the angle between the lines AB and AC. Fix the staff in the ground, so that its centre shall be exactly over the intersection of the two lines. Turn the alidade, so that it points, (as determined by sighting along it) to a rod, or Fig. 145. -B 45° other mark at B, a point on one of the lines, and note what degree it covers, i. e. "The Reading." Then, without disturbing the circle, turn the alidade till it points to C, a point on the other line. Note the new reading. The difference of these readings, (in the figure, 45 degrees), is the difference in the directions of the two lines, or is the angle which one makes with the other. If the dis tance from A to C be now measured, the point C is "determined,' with respect to the points A and B, on the Third Principle. Any number of points may be thus determined. (215) Instead of the very simple and rude alidade, which has been supposed to be used, needles may be fixed on each end of the alidade; or sights may be added, such as those described in Art. (103); or a small straight tube may be used, one end being covered with a piece of pasteboard in which a very small eye hole is pierced, and threads, called "cross-hairs," being stretch- Fig. 146. ed across the other end of it, as in the figure; so that their intersection may give a more precise line for determining the direction of any point. (216) When a telescope is substituted for this tube, and supported in such a way that it can turn over, so as to look both backwards and forwards, the instrument (with various other additions, which however do not affect the principle), is called the Engineer's Transit. With the addition of a level, and a vertical circle, for measuring vertical angles, the instrument becomes a Theodolite; in which, however, the telescope does not usually admit of being turned over. Fig. 147. N (217) The Compass differs from the instruments which have been described, in the following respect. They all measure the angle which one line makes with another. The compass measures the angle which each of these lines makes with a third line, viz: that shown by the magnetic needle, which always points (approximately) in the same direction, i. e. North and South, in the Magnetic Meridian. Thus, in the figure, the line AB makes an angle of 30 degrees with the line AN, and the line AC makes an angle of 75 degrees with AN. The difference of these angles, or 45 degrees, is the angle which AC makes with AB, agreeing with the result obtained in Art. (214). 30 450 ---C (218) Surveying with the compass is, therefore, a less direct operation than surveying with the Transit or Theodolite. But as the use of the compass is much more rapid and easy (only one sight and reading at each station being necessary, instead of two, as in the former case), for this reason, as well as for its smaller cost, it is the instrument most commonly employed in land surveying in this country, in spite of its imperfections and inaccuracies. As many may wish to learn "Surveying with the Compass,' without being obliged to previously learn "Surveying with the Transit," (which properly, being more simple in principle, though less so in practice, should precede it, but which will be considered in Part IV), we will first take up COMPASS SURVEYING. (219) Angular Surveying in general, and therefore Compass Surveying, may employ either of the 3d, 4th and 5th methods of determining the position of a point, given in Part I; that is, any instrument which measures angles may be employed for Polar, Triangular, or Trilinear Surveying. The first of these, Polar Surveying, is the one most commonly adopted for the compass, and is therefore the one which will be specially explained in this part. The same method, as employed with the Transit and Theodolite, will be explained in the following part. The 4th and 5th methods will be explained in the next two parts. (220) The method of Polar Surveying embraces two minor methods. The most usual one consists in going around the field with the instrument, setting it at each corner and measuring there the angle which each side makes with its neighbor, as well as the length of each side. This method is called by the French the method of Cheminement. It has no special name in English, but may be called (from the American verb, To progress), the Method of Progression. The other system, the Method of Radiation, consists in setting the instrument at one point, and thence measuring the direction and distance of each corner of the field, or other object. The corresponding name of what we have called Triangular Surveying is the Method of Intersections; since it determines points by the intersections of straight lines. |