Three orders of triangulation are recognized, viz.: Primary, in which the sides are from 20 to 150 miles in length. Secondary, in which the sides are from 5 to 40 miles in length, and which connect the primary with the tertiary. Tertiary, in which the sides are seldom over 5 miles in length, and which bring the survey down to such dimensions as to admit of the minor details being filled in by the compass and plane-table. § 28. THE MEASUREMENT OF BASE LINES. Base lines should be measured with a degree of accuracy corresponding to their importance. Suitable ground must be selected and cleared of all obstructions. Each extremity of the line may be marked by cross lines on the head of a copper tack driven into a stub which is sunk to the surface of the ground. Poles are set up in line about half a mile apart, the alignment being controlled by a transit placed over one end of the line. The preliminary measurement may be made with an iron wire about one-eighth of an inch in diameter and 60m in length. In measuring, the wire is brought into line by means of a transit set up in line not more than one-fourth of a mile in the rear. The end of each 60m is marked with pencil lines. on a wooden bench whose legs are thrust into the ground after its position has been approximately determined. If the last measurement exceeds or falls short of the extremity of the line, the difference may be measured with the 20m chain. The final measurement is made with the base apparatus, which consists of bars 6m long, which are supported upon trestles when in use. These bars are placed end to end, and brought to a horizontal position, if this can be quickly accomplished; if not, the angle of inclination is taken by a sector, or a vertical offset is measured with the aid of a transit, so that the exact horizontal distance can be computed. A thermometer is attached to each bar, so that the temperature of the bar may be noted and a correction for temperature applied. The method of measuring lines varies according to the required degree of accuracy in any particular case, but the brief description given above will give the student a general idea of the methods employed. § 29. THE MEASUREMENT OF ANGLES. Angles are measured by the transit with much greater accuracy than by the compass, since the reading of the plates of the transit is taken to minutes, and by means of microscopes to seconds, while the reading of the needle of the compass is to quarter or half-quarter degrees. In order to eliminate errors of observation, and errors arising from imperfect graduation of the circles, a large number of readings is made and their mean taken. methods are in use; viz., repetition and series. Two The method of repetition consists, essentially, in measuring the angles about a point singly, then taking two adjacent angles as a single angle, then three, etc.; thus "closing the horizon," or measuring the whole angular magnitude about a point in several different ways. The method of series consists, essentially, in taking the readings of an angle with the circle or limb of the transit in one position, then turning the circle through an arc and taking the readings of the same circle again, etc.; thus reading the angle from successive portions of the graduated circle. To On account of the curvature of the earth, the sum of the three angles of a triangle upon its surface exceeds 180°. This spherical excess, as it is called, becomes appreciable only when the sides of the triangle are about 5 miles in length. determine the angles of the rectilinear triangle having the same vertices, one-third of the spherical excess is deducted from each spherical angle. CHAPTER IV. LEVELLING. § 30. DEFINITIONS, CURVATURE, AND REFRACTION. A Level Surface is a surface parallel with the surface of still water; and is, therefore, slightly curved, owing to the spheroidal shape of the earth. A Level Line is a line in a level surface. Levelling is the process of finding the difference of level of two places, or the distance of one place above or below a level line through another place. The Line of Apparent Level of a place is a tangent to the level line at that place. Hence, the line of apparent level is perpendicular to the plumb-line. The Correction for Curvature is the deviation of the line of apparent level from the level line for any distance. Let t (Fig. 41) represent the line of apparent level of the α P FIG. 41. approximately, since c is very small compared with d, and ta without appreciable error. Since'd is constant (=7920 miles, nearly), the correction for curvature varies as the square of the distance. EXAMPLE. What is the correction for curvature for 1 mile ? By substituting in the formula deduced above, Hence, the correction for curvature for any distance may be found in inches, approximately, by multiplying 8 by the square of the distance expressed in miles. NOTE. The effect of curvature is to make an object appear lower than it really is; and the effect of refraction of light, caused by the greater density of the atmosphere near the surface of the earth, is to make an object appear higher than it really is. When both effects are taken into a2 account c is more correctly expressed by c = 1⁄2 of § 31. THE Y LEVEL. This instrument is shown on page 61. d The telescope is about 20 inches in length, and rests on supports called Y's, from their shape. The spirit level is underneath the telescope, and attached to it. The levelling-head and tripod are similar to the same parts of the transit. § 32. THE LEVELLING ROD. The two ends of the Philadelphia levelling rod are shown in Fig. 42. The rod is made of two pieces of wood, sliding upon each other, and held together in any position by a clamp. 62 The front surface of the rod is graduated to hundredths of a foot up to 7 feet. If a greater height than 7 feet is desired, the back part of the rod is moved up until the target is at the required height. When the rod is extended to full length, the front surface of the rear half reads from 7 to 13 feet, so that the rod becomes a self-reading rod 13 feet long. FIG. 42. The target slides along the front of the rod, and is held in place by two springs which press upon the sides of the rod. It has a square opening at the centre, through which the division line of the rod opposite to the horizontal line of the target may be seen. It carries a vernier by which heights may be read to thousandths of a foot (§ 7). § 33. DIFFERENCE OF LEVEL. To find the difference of level between two places visible from an intermediate place, when the difference of level does not exceed 13 feet. Let A and B (Fig. 43) represent the two places. Set the Y level at a station equally distant, or nearly so, from A and A' FIG. 43. Β' B B, but not necessarily on the line AB. Place the legs of the tripod firmly in the ground, and level over each opposite pair of levelling screws, successively. Let the rodman hold the levelling rod vertically at A. Bring the telescope to bear upon the rod (§ 8), and by signal direct the rodman to move the target until its horizontal line is in the line of apparent level of the telescope. Let the rodman now record the height ДA' of the target. In like manner find BB'. The difference between AA' and BB' will be the difference of level required. If the instrument be equally distant from A and B, or nearly so, the curvature and the refraction on the two sides of the instrument balance, and no correction for curvature or refraction will be necessary. |