For the correction of areas we have this proportion, As the square of the length of the chain : the square of the length of the standard, : the area required. MEASUREMENT OF ANGLES. 20. We come now to the measurement of angles, and for this purpose several instruments are used. The one, however, which affords the most accurate results, and which indeed can alone be relied on for nice or extensive operations, is called a Theodolite. This instrument only will be described at present; others will be subsequently explained. OF THE THEODOLITE. Pl. 1. The theodolite is an instrument used to measure horizontal and vertical angles. It is usually placed on a tripod ABC, which enters by means of a screw the lower horizontal plate DE, and becomes firmly attached to the body of the instrument. Through the horizontal plate DE, four small hollow cylinders are inserted, which receive four screws with milled heads, that work against a second horizontal plate, FG. The upper side of the plate DE terminates in a curved surface, which encompasses a ball, that is nearly a semi-sphere, with the plane of its base horizontal. This ball, which is hollow, is firmly connected with the smaller base of a hollow conic frustum, that passes through the curved part of the plate DE, and screws firmly into the curved part of the second horizontal plate FG. A hollow conic spindle passes through the middle of the ball, and the hollow frustum with which it is connect ed. To this spindle, a third horizontal and circular plate HI, called the limb of the instrument, is permanently attached. Within this spindle, and concentric with it, there is a second spindle, called the inner, or solid spindle. To this which rests on the limb of the instrument, and supports the upper frame-work. The two spindles terminate at the base of the spherical ball, where a small screw enters the inner one, and presses a washer against the other, and the base of the ball. On the upper surface of the plate FG, rests a clamp which goes round the outer spindle, and which, being compressed by the clamp-screw K, is made fast to it. This clamp is thus connected with the plate FG. A small cylinder a, is fastened to the plate FG: through this cylinder a thumb-screw L passes, and works into a small cylinder b, connected with the clamp. The cylinders b and a, admit of a motion round their axes, to relieve the screw L of the pressure which would otherwise be occasioned by working it. Directly above the clamp, is the lower telescope MN. This telescope is connected with a hollow cylinder, which is worked freely round the outer spindle, by the thumbscrew P having a pinion working into a concealed cogwheel, that is permanently fastened to the limb of the instrument. By means of a clamp-screw Q, the telescope is made fast to the limb, when it will have a common motion with the limb and outer spindle. The circular edge of the limb is chamfered, and is gen erally made of silver, and on this circle the graduation for horizontal angles is made. In the instrument described, the circle is cut into degrees and half degrees; the degrees are numbered from 0 to 360. On the circular edge of the vernier plate, is a small plate of silver, called a vernier; this plate is divided into 30 equal parts, and numbered from the line marked 0 to the left. Two levels, at right angles to each other, are attached to the vernier plate by small adjusting screws; one of the levels is seen in the figure. The vernier plate turns freely around with the inner spindle. It is made fast to the limb of the instrument by the clamp-screw S; after which the smaller motions are made by the tangent-screw T There is a compass on the vernier plate, that is concentric with it, the use of which The frame-work which supports the horizontal axis of the vertical semicircle UV and the upper telescope, with its attached level, rests on the vernier plate, to which it is made fast by three adjusting screws, placed at the angu lar points of an equilateral triangle. The vertical semicircle UV, is called the vertical limb; its motions are gov erned by the thumb-screw Z, which has a pinion, that works with the teeth of the vertical limb. On the face of the vertical limb, opposite the thumb-screw Z, the limb is divided into degrees and half degrees: the degrees are numbered both ways from the line marked 0. There is a small plate resting against the graduated face of the verti cal limb, called the vernier; it is divided into 30 equal parts, and the middle line is designated by 0. On the other face of the vertical limb, are two ranges of divisions, commencing at the 0 point, and extending each way 45°. The one shows the vertical distance of any object to which the upper telescope is directed, above or below the place of the instrument, in 100th parts of the horizontal distance: the other, the difference between the hypothenusal and base lines: the hypothenuse being sup posed to be divided into one hundred equal parts: therefore, by mere inspection, we can ascertain the number of links, which must be subtracted from every chain of an oblique line, to reduce it to a true horizontal distance. The supports of the upper telescope are called the wyes, and designated Y's. Two loops, turning on hinges, pass over the telescope, and are made fast by the pins c and d; these loops confine the telescope in the Y's. By withdrawing the pins, and turning the loops on their hinges, the telescope may be removed for the purpose of being reversed in position; and in both situations, the telescope can be revolved in the Y's about its axis. In the telescopes attached to the theodolite, are two principal lenses, one at each end. The one at the end where the eye is placed, is called the eye-glass, the other the object-glass In order that the axis of the telescope may be directed hairs, are fixed at right angles to each other, and placed within the barrel of the telescope, and at the focus of the eye-glass. The vertical hair is moved by two small horizontal screws, one of which, f, is seen in the figure; and the horizontal hair, by two vertical screws, g and h. Before using the instrument it must be adjusted, that is, the parts must be brought to their proper relative positions: there are four principal adjustments. FIRST ADJUSTMENT.-To fix the intersection of the spider's lines in the line of collimation or axis of the telescope. Having screwed the tripod to the instrument, extend the legs, and place them firmly. Then loosen the clampscrew S of the vernier plate, and direct the telescope to a small, well-defined, and distant object. By means of a small pin i, on the under side of the telescope, slide the eye-glass till the spider's lines are seen distinctly; then with the thumb-screw X, which forces out and draws in, the object-glass, adjust this glass to its proper focus, when the object, as well as the spider's lines, will be distinctly seen: after which, by the tangent-screw T and the thumb-screw Z, bring the intersection of the spider's lines exactly upon a well-defined point of the object. Having done this, revolve the telescope in the Y's half round, when the attached level mn, will come to the upper side. See, in this position, if the horizontal hair appears above or below the point, and in either case, loosen one, and tighten the other, of the two screws that work the horizontal hair, till the horizontal hair has been carried over half the space between its last position and the ob served point. Carry the telescope back to its place; direct again 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 revolved. similar process will arrange the vertical hair, and the line of collimation is then adjusted. A SECOND ADJUSTMENT.-To make the axis of the attached level of the upper telescope, parallel to the line of collimation. over two of the levelling screws, between the plates DE and FG. Turn these screws contrary ways, keeping them firm against the plate FG, till the bubble of the level mn, stands at the middle of the tube. Then, open the loops, and reverse the telescope. If the bubble still stands in the middle of the tube, the axis of the tube is horizontal; but if not, it is inclined, the bubble being at the elevated end. In that case, by means of the small vertical screws m and n, at the ends of the level, raise the depressed end, or depress the elevated one, half the inclination; and then, with the levelling screws, bring the level into a horizontal position. Reverse the telescope in the Y's, and make the same correction again; and so on, until the bubble stands in the middle of the tube, in both positions of the telescope: the axis of the level is then horizontal. telescope be now revolved in the Y's. If the bubble continue in the middle of the tube, the axis of the level is not only horizontal, but also parallel to the line of collimation. If, however, the bubble recede from its centre, the axis of the level is inclined to the line of collimation, and must be made parallel to it by means of two small antagonistic screws, (one of which is seen at p,) which work horizontally. By loosening one of them, and tightening the other, the level is soon brought parallel to the line of collimation, and then, if the telescope be revolved in the Y's, the bubble will continue in the middle of the tube. Let the It is difficult to make the first part of this adjustment, while the axis of the level is considerably inclined to the line of collimation; for, if the level were truly horizontal in one position of the telescope, when the telescope is reversed, the bubble would not stand in the middle of the tube, except in one position of the level. This suggests the necessity of making the first part of the adjustment with tolerable accuracy; then, having made the second with care, let the first be examined, and proceed thus tili the adjustment is completed. THIRD ADJUSTMENT. To make the axes of the levels on the limb perpendicular to the axis of the instrument. |