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to the eye. This angle is very nearly equal to the angle subtended by the object and its image at the place of the eye, differing from it only by the small angle subtended at the object by the place of the eye and the center of the index glass. This small angle is called the parallax of the instrument, and is scarcely perceptible at the distance of a quarter of a mile, while for distances greater than that it is so small that it may be considered to vanish. It also varies with the amount of deviation, and vanishes altogether whenever the center of the index glass is in a direct line between the object and the eye*.

To see if the instrument be in perfect adjustment, place the dark glass before the eye-end of the telescope, and looking at the sun, and moving the index backwards and forwards a little distance on either side of zero, the sun's reflected image will be seen to pass over the disc as seen directly through the horizon glass, and if in its passage the reflected image completely covers the direct image, so that but one perfect orb is seen, the horizon glass is perpendicular to the plane of the instrument; but, if not, the screw at a must be turned by the key, k, till such is the case. The key, k, fits the square heads of both the screws seen at a and b, and fits into a spare part of the face of the instrument, so as to be at hand when wanted This adjustment being perfected, bring the reflected image of the sun's lower limb in exact contact with the direct image of his upper limb, and note the reading of the vernier; then move the index back beyond the zero division of the limb, till the reflected image of the sun's upper limb is in exact contact with the direct image of his lower limb, and, if the zero of the vernier be now exactly as far behind the zero of the limb as it was at the former reading in front of it, so that the reading now on the part of the limb called the arc of excess, behind its zero divisiont, be the same as the former reading,

*We have seen a method given for what is called correcting the parallax, when an observation is made at a short distance, by finding the deviation at this distance, when the angle between the object and its image is equal to zero; this deviation being given by the reading of the instrument, when the reflected image of the object observed exactly coincides with the object itself, seen through the unsilvered part of the horizon glass. This deviation, however, is not the parallax, even for a small angle between the object and its image, and, if the angle be not very small, the error introduced by the method will be greater than the parallax itself.

In reading an angle upon the arc of excess, the division to read on the limb is that next in front of the zero of the vernier, or between the zero of the vernier and the zero of the limb, and the divisions of the vernier itself are to be read from the end division, marked 30, and not, as usually, from

the instrument is in perfect adjustment; but, if not, half the difference of the two readings is the amount of the error, and is called the index error, being a constant error, for all angles observed by the instrument, of excess, if the first reading be the greatest, and of defect, if the second reading on the arc of excess be the greatest.

In the former case, then, the true angle will be found by subtracting the index error from, and in the latter by adding it to, the reading of the instrument at every observation.

This method of correcting for the index error is to be used with the larger instruments, hereafter to be described under the head of Astronomical Instruments; but in the box sextant this error should be removed by applying the key, k, to the screw at b, and turning it gently till both readings are alike, each being made equal to half the sum of the two readings first obtained. When this adjustment is perfected, if the zeros of the vernier and limb are made exactly to coincide, the reflected and direct image of the sun will exactly coincide, so as to form but one perfect orb, and the reflected and direct image of any ine, sufficiently distant not to be affected by parallax, as the distant horizon, or the top or end of a wall more than half a mile off, will coincide so as to form one unbroken line.

To obtain the angle subtended by two objects situated nearly or quite in the same vertical plane, hold the instrument in the right hand, and bring down the reflected image of the upper object by turning the milled head B, till it exactly coincides with the direct image of the lower object, and the reading of the instrument will give the angle between the two objects.

To obtain the angle subtended by two objects nearly in the same horizontal plane, hold the sextant in the left hand, and bring the reflected image of the right-hand object into coin cidence with the direct image of the left-hand object.

It will be seldom that the surveyor need pay any attention to the small error arising from parallax; but, should great accuracy be desirable, and one of the objects be distant while the other is near, the parallax will be eliminated by observing the distant object by reflection, and the near one by

the zero division: thus, if the zero division of the vernier were a little further from the zero division of the limb, then the first division on the arc of excess; and if the twenty-seventh division on the vernier, or the third from the end division, marked 30, coincided with a division upon the limb, then the reading would be 33'.

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 witn 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

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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 E s' between the object and its reflected image seen in the mercury is double the angle of elevation s E H, and, the angle sEs' 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 fiveinch theodolite represented in our figure has two such ver

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

Upon this depends, in a great measure, the perfection of the instrument, as far as the horizontal measurements are concerned; and, when we

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