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direct vision, holding the instrument for this purpose with its face downwards if the distant object be on the left hand. If both objects be near, the reflected image of a distant object, in a direct line with one of the objects, must be brought into coincidence with the direct image of the other object, and the parallax will thus be eliminated.

For the purposes of surveying, the horizontal angles between different objects are required, and the reduction of these angles from the actual oblique angles subtended by the objects, would be a troublesome and laborious process. If the angle subtended by two objects be large, and one be not much higher than the other, the actual angle observed will be, however, a sufficient approximation to the horizontal angle required; and, if the angle between the two objects be small, the horizontal angle will be obtained with sufficient accuracy by taking the difference of the angles observed between each of the objects, and a third object at a considerable angular distance from them With a little practice the eye will be able to select an object in the same direction as one of the objects, and nearly on a level with the other object, and the angle between this object and the object selected will be the horizontal angle required.

At sea the altitude of an object may be determined by observing the angle subtended by it and the verge of the horizon; but upon land a contrivance, called an artificial horizon, becomes necessary for correctly determining altitudes. The best kind of artificial horizon consists of an oblong trough, tt, filled with mercury and protected from the wind by a roof, rr, having in either slope a plate of glass with its two surfaces ground into perfectly parallel planes. The angle s Es' between the object and its reflected image seen in the mercury is double the angle of elevation s E H, and, the angle SE s' being observed, its half will consequently be the angle of elevation re

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quired. If the angle of elevation be greater than 60°, the angle s E s' will be greater than 120°, and cannot be observed with the sextant we have been describing.

The pocket sextant is a most convenient instrument for laying off offsets or perpendicular distances from a station line; for by setting the index at 90°, and walking along the station line, looking through the horizon glass directly at the further station staff, or any other remarkable object upon the station line, any object off the station line will be seen by reflection when the observer arrives at the point where the perpendicular from this object upon the station line falls, and the distance from this point to the object being measured, is its perpendicular distance from the station line.

For the mere purpose of measuring offsets an instrument called an optical square is now very generally employed, which consists of the two glasses of the sextant fixed permanently at an angle of 45°, so that any two objects seen in it, the one by direct vision, and the other by reflection, subtend at the place of the observer an angle of 90°.

THE THEODOLITE.

The theodolite is the most important instrument used by surveyors, and measures at the same time both the horizontal angles subtended by each two of the points observed with it, and the angles of elevation of these points from the point of observation

This instrument may be considered as consisting of three parts; the parallel plates with adjusting screws fitting on to the staff head, of exactly the same construction as already described for supporting the Y, and other, levels; the horizontal limb, for measuring the horizontal angles; and the vertical limb, for measuring the vertical angles, or angles of elevation.

The horizontal limb is composed of two circular plates, L and v, which fit accurately one upon the other. The lower plate projects beyond the other, and its projecting edge is sloped off, or chamfered, as it is called, and graduated at every half degree. The upper plate is called the vernier plate, and has portions of its edge chamfered off, so as to form with the chamfered edge of the lower plate continued portions of the same conical surface. These chamfered portions of the upper plate are graduated to form the verniers, by which the limb is subdivided to single minutes. The sixinch theodolite represented in our figure has two such ver

niers, 180° apart. The lower plate of the horizontal limb is attached to a conical axis passing through the upper parallel

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plate, and terminating in a ball fitting in a socket upon the lower parallel plate, exactly as the vertical axis of the Y level already described. This axis is, however, hollowed to receive a similar conical axis ground accurately to fit it, so that the axes of the two cones may be exactly coincident, or parallel.* To the internal axis the upper, or vernier, plate of the horizontal limb is attached, and thus, while the whole limb can

* Upon this depends, in a great measure, the perfection of the instrument, as far as the horizontal measurements are concerned; and, when we describe presently the adjustments of the instrument, we shall explain the method of detecting an inaccuracy in the grinding of the axes.

be moved through any horizontal angle desired, the upper plate only can also be moved through any desired angle, when the lower plate is fixed by means of the clamping screw, C, which tightens the collar, D. T is a slow-motion screw, which moves the whole limb through a small space, to adjust it more perfectly, after tightening the collar, D, by the clamping screw, c. There is also a clamping screw, c, for fixing the upper, or vernier, plate to the lower plate, and a tangent screw, t, for giving the vernier plate a slow motion upon the lower plate, when so clamped. Two spirits levels, B, B, are placed upon the horizontal limb, at right angles to each other, and a compass, G, is also placed upon it in the center, between the supports, F, F, for the vertical limb.

The vertical limb, N N, is divided upon one side at every 30 minutes, each way from 0° to 90°, and subdivided by the vernier, which is fixed to the compass box, to single minutes. Upon the other side are marked the number of links to be deducted from each chain, for various angles of inclination, in order to reduce the distances, as measured along ground rising or falling at these angles, to the corresponding horizontal distances. The axis of this limb must rest, in a position truly parallel to the horizontal limb, upon the supports, F F, so as to be horizontal when the horizontal limb is set truly level, and the plane of the limb, N N, should be accurately perpendicular to its axis. To the top of the vertical limb, N N, is attached a bar which carries two Ys for supporting the telescope, which is of the same construction as that before described for the Y spirit level, and underneath the telescope is a spirit level, s s, attached to it at one end by a joint, and at the other end by a capstan-headed screw as in the Y level. The horizontal axis can be fixed by a clamping screw, and the vertical limb can then be moved through a small space by a slow-motion screw, i.

Before commencing observations with this instrument, the following adjustments must be attended to :

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2. Adjustment of the horizontal limb: viz., to set the levels on the horizontal limb to indicate the verticality of the azimuthal axis.

3. Adjustment of the vertical limb: viz., to set the level

beneath the telescope to indicate the horizontality of the line of collimation.

1.-Parallax and Collimation.-These adjustments have already been described (p. 100) under the head of the Y level. 2.-Adjustment of the Horizontal Limb.-Set the instrument up as accurately as you can by the eye, by moving the legs of the stand. Tighten the collar, D, by the clamping screw, c, and, unclamping the vernier plate, turn it round till the telescope is over two of the parallel plate-screws. Bring the bubble of the level, s s, beneath the telescope to the center of its run by turning the tangent screw, i. Turn the vernier

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plate half round, bringing the telescope again over the same pair of the parallel plate screws; and, if the bubble of the level be not still in the center of its run, bring it back to the center, half way by turning the parallel plate screws over which it is placed, and half way by turning the tangent screw, i. Repeat this operation till the bubble remains accurately in the center of its run in both positions of the telescope; and, then turning the vernier plate round till the telescope is over the other pair of parallel plate screws, bring the bubble again to the center of its run by turning these screws. bubble will now retain its position while the vernier plate is turned completely round, showing that the internal azimuthal axis about which it turns is truly vertical. The bubbles of the levels on the vernier plate, being now, therefore, brought to the centers of their tubes, will be adjusted to show the verticality of the internal azimuthal axis. Now, having clamped the vernier plate, loosen the collar, D, by turning back the screw, c, and move the whole instrument slowly round upon the external azimuthal axis, and if the bubble of the level s s, beneath the telescope, maintains its position during a complete revolution, the external azimuthal axis is truly parallel with the internal, and both are vertical at the same time; but, if the bubble does not maintain its position, it shows that the two parts of the axis have been inaccurately ground, and the fault can only be remedied by the instrumentmaker.

3. Adjustment of the Vertical Limb.-The bubble of the level, s s, being in the center of its run, reverse the telescope end for end in the Ys, and, if the bubble does not remain in the same position, correct for one-half the error by the capstanheaded adjusting screw at one end of the level, and for the other half by the vertical tangent screw, i. Repeat the opera

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