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The necessity for the above correction is not of common occurrence,

as in the principal triangles stations

are generally selected from whence observations can be made; and in those of the secondary order, the measurement of the third angle is not considered imperative.

In observing the angles for triangulation, too much care cannot be bestowed upon the adjustments of the instrument. These are briefly as follows for the 5 or 7-inch theodolites used in fixing points in the interior, and for traversing. The large theodolite, 3 feet in diameter, known by the name of its maker, Ramsden†, is fully described in the "Trigonometrical Survey; and the peculiarities in the construction and management of the

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* Instead of deducing the angle at the station on which the instrument cannot be set up from that observed at any spot convenient to it, it is often found more expeditious, particularly if there are many observations made, to correct the other angles of the triangles; this latter method is generally now practised on the Ordnance Survey.

+ An instrument of the same size has since been made by Messrs. Troughton and Simms for the survey of India, as also another for the Ordnance Survey. A theodolite of 18 inches diameter upon a repeating stand was constructed by General Mudge, with an idea of its superseding the larger theodolite, the weight and size of which rendered its carriage an affair of difficulty; but the advantage of repetition (so desirable in single observations) possessed by moderate sized instruments does not appear to compensate for the diminished size of the circumference of the horizontal circle. Theodolites of 24, 18, 12, 10, 9, and 8 inches diameter are also used on the Ordnance Survey, as well as those of smaller dimensions, of 7 and 5 inches.

other large instruments with which the angles of the principal and secondary triangles are observed, are soon understood by any officer conversant with the adjustment of the smaller class, which he most generally has to work with, and which is therefore the one selected for description.

The first adjustment is for the line of collimation, and consists in making the cross wires* in the diaphragm of the telescope coincide with the axis of the supports in which the telescope rests; the proof of which is their intersection remaining constantly fixed upon some minute, well-defined, distant point, during an entire revolution of the telescope upon its own axis in the Ys, which are left open for the purpose. When this intersection on the contrary forms a circle round the object, the wires require adjusting. They are generally placed crossing each other, at an angle inclined to the horizon of about 45°, and the operation is facilitated by first turning the telescope partly round, till they appear horizontal and vertical; half the divergence of each of these lines from the point is then corrected by the screws near the eye-piece, working in the diaphragm, loosening one screw as that opposite to it is tightened. One or two trials will perhaps be required, the diaphragm being moved in the contrary direction to that which in the inverting eye-piece it appears to require.

The second adjustment is for the purpose of setting the level

* Platinum wire is the best adapted for the purpose, though cobwebs are generally used by surveyors; and as they are liable to break from the slightest touch, it is necessary that every person using a theodolite should be able to replace them himself. They must be stretched tight across the diaphragm, and confined in their places (indicated by faint notches on the metal) by gum, or varnish, the latter of which is to be preferred on account of its not being affected by the humidity of the atmosphere. The following simple and ingenious mode of fixing these cobwebs, which to a novice is often a difficult and tedious operation, was mentioned to me by Mr. Simms, who constructs all the mathematical and astronomical instruments for the Ordnance Survey. A piece of wire is bent into a shape something like a fork, the opening a b being rather larger than the diameter of the diaphragm. A cobweb being selected, at the extremity of which a spider is suspended, it is wound round the fork in the manner represented in the sketch, the weight of the insect keeping it constantly tight. The web is thus kept stretched ready for use; and when it is required to fix on a new hair, it is merely necessary to put a little gum or varnish over the notches on the diaphragm, and adjust one of the threads to its proper position.



attached to the telescope parallel to the optical axis, and to the surface of the cylindrical rings on which it is supported; this is done by simply levelling the telescope by means of the tangent screw to the vertical arc, and then reversing it end for end in the Ys. If the air-bubble does not remain in the centre of the tube after this reversion, it must be corrected, one half of the error by the screw attached to one end of the level, and the remainder by the vertical arc. A few trials will be necessary to obtain this adjustment perfectly; and the level should be at the same time adjusted laterally, so as to be in the same vertical plane as the line of collimation, if it should be found, on moving the telescope slightly on either side, that the bubble becomes deranged from its central position.

The object of the third adjustment is to ensure the verticality of the axis of the instrument, and consequently the horizontal position of the azimuth circle, which is instrumentally at right angles to it. The level of the telescope already adjusted furnishes the means of effecting this. The instrument being placed approximately level, and the lower plate clamped, the upper plate is moved till the axis of the telescope is nearly over two of the opposite plate screws; the bubble of the telescope level is then adjusted by the vertical arc, and the upper plate turned round 180°; if the level is not in adjustment, half the error is to be corrected by the plate screws, and half by the tangent screw of the vertical arc. The same operation must be repeated with the telescope over the other pair of plate screws; and when, after several trials, the air-bubble of the level attached to the telescope remains constantly in the centre of the tube, in whatever position it is turned, it is only necessary to adjust the two small levels on the upper plate to correspond, and they will serve to indicate when the axis of the instrument is vertical, care being taken to verify their adjustment from time to time.

The vernier of the vertical arc is the last adjustment; it should indicate zero when all the above corrections have been made. If it differs from this point, it can be set to zero by releasing the screws by which the arc is held; but if the difference is small, it is better to note it as an index error +, or —, than to make the alteration.

A better plan of obtaining the index error of the vertical arc with

accuracy is by observing reciprocal angles of depression and elevation from two stations, about four hundred or five hundred yards distant. If none exists, the angles will correspond; otherwise the errors will be equal, but in an opposite direction; and half their difference is the index error.

If the distance selected be too long, it becomes necessary to take into account the corrections for refraction and the curvature of the earth, depending upon the arc of distance, which subjects will be explained hereafter; but for the purpose of ascertaining the index error of the vertical arc of a theodolite, the distance named is quite sufficient.

The mean of all the verniers should invariably be taken*, and each angle repeated six or eight times. The errors of eccentricity, and graduation of the instrument, are thus almost annihilated; and those of observation of course much diminished. The repetition of angles is also the only means by which they can be measured with any degree of minuteness by small instruments.

It is frequently necessary to refer to trigonometrical stations long after the angles have been observed; either for the purpose of fixing intermediate points, or of rectifying errors that may have crept into the work. Large marked stones should therefore be always buried under the principal stations which are not otherwise identified by permanent erections, and a clear description of the relative position of these marks with reference to objects in their vicinity should be always recorded. If, however, any station should be lost, and its site required to be ascertained for ulterior observations, the following method, which has been adopted by General Colby, will


* On the azimuth circle of the large theodolite used on the triangulation of the Ordnance Survey, the original verniers were only at the two opposite points A and B, the mean of the readings at which were, of course, always taken. Subsequently, the verniers at C and D were added, each of them equidistant 120° from A, and also from each other. It has since been sometimes the custom, first to take the mean of A and B, and afterwards the mean of A C and D, and to consider the mean between these two valuations as the true reading of the angle; this method has, however, been objected to as being incorrect in principle, an undue importance being given to the reading of the vernier A, and also in a smaller degree to B. The influence assigned to each vernier is, in fact, as follows:-A. 5; B. 3; C and D, 2 each.



be found to answer the purpose with very little trouble and with perfect accuracy.

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Let D be the lost station, the position of which is required. Assume T as near as possible to the supposed site of the point in question (in the figure the distance is much exaggerated, to render the process intelligible), and take the angles AT B, BTC; A, B, and C being corresponding stations which have been previously fixed, and the distances of which from D are known. If the angle ATB be less than the original angle A D B, the point T is evidently without the circle in the segment of which the stations A and B are situated; if the angle be greater, it is of course within the segment. The same holds good with respect to the angles BTC and BDC.

Recompute the triangle ABD, assuming the angle at D to have been so altered as to have become equal to the angle at T, and that the angle at A is the one affected thereby.

Again, recompute the triangle, supposing the angle at B the one affected. In like manner in the triangle BDC recompute the triangle, supposing the angles at B and C to be alternately affected

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