The vertical position can be ascertained by means of a plumb-line. To see the spider's lines in the field of the telescope at the same time with the star, a faint light should be placed near the object-glass. When the plumb-line, the star Alioth, and the north star, fall on the vertical spider's line, the horizontal limb is firmly clamped, and the telescope brought down to the horizon; a light, seen through a small aperture in a board, and held at some distance by an as sistant, is then moved according to signals, until it is covered by the intersection of the spider's lines. A picket driven into the ground, under the light, serves to mark the meridian line for reference by day, when the angle formed by it and the ragnetic meridian may be measured. BOOK III. LE VELLING AND TOPOGRAPHICAL SURVEYING. SECTION I. OF LEVELLING. 1. LEVELLING is the art of determining the relative distances of points from the centre of the earth. 2. A line whose points are all equally distant from the centre of the earth, is called a line of true level, and a surface, all whose points are equally distant from the centre of the earth, as the surface of still water, is called a level surface. 3. One point is said to be above another, when it is farther from the centre of the earth; and this difference of distance from the centre, is called the difference of level between the two points. 4. A straight line drawn tangent to a line of true level at any point, is a horizontal line, and is called a line of apparent level. Thus (Pl. 4, Fig. 1), if C is the centre of the earth and AEF a line of true level, ABD is a line of apparent level. This is the line of level determined by an instrument. The difference between the apparent and true level at any distant station B, as determined from A, is BF, or the excess of the secant of the arc AE over the radius. 5. To find a general formula for computing this excess, we have (Geom. B. IV., Prop. XXX.) AB2 = BE (BE+2EC); but since the arc AE is very small in comparison with the 10 radius of the earth, the arc AE will not differ sensibly from the tangent AB; the diameter 2EC may, for the same reason, be taken for the secant (BE+2EC): hence, AE2=BEX2EC, or dividing by 2EC, AE 2EC (1). If we take the mean diameter of the earth to be 7919 miles, formula (1) gives BE= AE (2): hence, The departure of the apparent from the true level, starting from a given point, is equal to the square of the distance to the second point, divided by the diameter of the earth. If in formula (2) you give to AE, in succession, every value from 1 chain to any given number of chains, (say 100), and reduce at the same time both terms of the frac tion to inches, a table may be computed as below. Table showing the differences in inches between the true and ap parent level, for distances between 1 and 100 chains. Chains. Inches. Chains. Inches. Chains. Inches. Chains. Inches. Observing that for AE= 80 chains = 1 mile, BE is equal to 8.001 inches, or about two-thirds of a foot, and since the differences of level vary as the squares of the dis tances, we have the following easy rule for finding the cor rection in feet. The correction for curvature, in feet, is equal to two-third of the square of the distance in miles. INSTRUMENTS. 6. Before proceeding further in the discussion of the principles of levelling, we will describe some of the instruments used, and first, THE Y LEVEL. 7. A level is an instrument used to determine horizontal lines, and the difference of level of any two points on the surface of the earth. The part of the instrument shown in Pl. 4, Fig. 2, rests on a tripod, to which it is permanently attached at Z. HA is a horizontal brass plate, through which four levelling screws with milled heads are passed, and worked against a second horizontal plate GG. Two of these screws, K and I, are seen in the figure. S is a clamp-screw, which, being loosened, allows the upper part of the instrument to turn freely around its axis. Q is a tangent-screw, by means of which the upper part of the instrument is moved gently, after the clamp-screw S has been made fast. EE is a hori zontal bar, perpendicular to which are the wyes, designated Y's, that support the telescope LB. This telescope is confined in the Y's by the loops r, r, which are fastened by the pins p and p. The object-glass B, is adjusted to its focus by the screw ; the eye-glass L slides out and in freely. The screws f f work the slide which carries the horizontal hair; and two horizontal screws, only one of which, a, is seen, work the slide that carries the vertical hair. CD is an attached spirit-level. The screw N elevates and depresses the Y, nearest the eye-glass. In some instruments this Y is elevated and depressed, by Before using this level, it must be adjusted. The adjustment consists in bringing the different parts to their proper places. The line of collimation is the axis of the telescope. With this axis, the line drawn through the centre of the eyeglass and the intersection of the spider's lines, within the barrel of the telescope, ought to coincide. FIRST ADJUSTMENT.* To fix the intersection of the spider's lines in the axis of the telescope: Having screwed the tripod to the instrument, extend the legs, and place them firmly. Then loosen the clampscrew S, and direct the telescope to a small, well-defined, and distant object. Then slide the eye-glass till the spider's lines are seen distinctly; after which, with the screw X, adjust the object-glass to its proper focus, when the object and the spider's lines will be distinctly seen. Note now the precise point covered by the intersection of the spider's lines. Having done this, revolve the telescope in the Y's, half round, when the attached level CD will come to the upper side. See if, in this position, the horizontal hair appears above or below the point, and in either case, loosen the one, and tighten the other, of the two screws which work the horizontal hair, until it has been carried over half the space between its last position and the observed point. Carry the telescope back to its place; direct again, by the screws at M and R, the intersection of the spider's lines to the point, and repeat the operation, till the horizontal hair neither ascends nor descends while the telescope is revolv ed. A similar process will arrange the vertical hair, and the line of collimation is then adjusted. SECOND ADJUSTMENT. To make the axis of the attached level CD parallel to the line of collimation. Turn the levelling screws M and R, until the bubble * This, and some of the following adjustments, are so similar to those of the theo lolite, that they would not be here repeated, but that some may use the |