reading both verniers "errors of eccentricity" arising from the plates and y and the vertical axis u, not being exactly concentric, are eliminated. If one of the verniers is adjusted to 360° or zero, the others should also be read before measuring the angle, as in most instruments the second vernier does not read exactly 180° when the first reads 360°, owing to errors of graduation and position of verniers, &c. Similarly when there are three verniers the second and third do not generally read exactly 120° and 240° when the first is set at zero. Fig. 71. To Measure an Angle with the Theodolite. Method of Repetition.* When great accuracy is required, errors of graduation may be reduced to any required extent by repetition. This process consists in repeating the observation of the angle any number of times, according to the degree of accuracy aimed at, the verniers not being read until the last observation. Thus if the angle is observed six times, the total angle, as read at the end of the six observations, divided by six, is the required angle. Be careful to count 360° for each complete revolution of the horizontal circle. By the process of repetition the errors of graduation are diminished, but errors of observation are as a rule accumulated, as an observer tends to make the same error of observation each time the telescope is directed upon the same object. Errors due to instability of clamping and tangent screws are also introduced (see also page 459, Chapter XII.). For the most accurate methods of measuring angles, see pages 460, 461, Chapter XII. Using both Faces of the Instrument. In making important observations both "faces" of the theodolite should be used in order to eliminate errors of adjustment of the instrument. For instance, errors arise from the horizontal axis of the telescope not being exactly level, either owing to the spirit levels v not being exactly parallel to the plate t, or owing to the bearings of the horizontal axis in the supports ll, I not being exactly level (adjustment 1, Chapter IV.). The telescope consequently does not move in a vertical plane from either of these causes, and the * See also page 459. This error accuracy of the observation is thereby affected. affects the observation, whether the telescope during the observation is revolved vertically through 180° or not. When the telescope is revolved vertically through 180°, another error arises, if the collimation line is not exactly perpendicular to the horizontal axis of the telescope (adjustment 2, Chapter IV.). All errors are eliminated by using both "faces" of the instrument.* This consists in turning the whole of the upper part of the instrument by hand through 180° and repeating the observation. The mean of the two observations will then be the correct result. By turning the whole of the instrument through 180°, the errors are reversed. For instance, suppose the telescope, instead of revolving in a vertical plane, revolves in a plane which slopes upwards to the left of the observer. If now the instrument is moved bodily through 180°, and the telescope again directed from the observer, it will be found that it now lies in a plane sloping upward the same amount to the right of the observer. Similarly if the error of the collimation line was to the left of the observer, it will now be the same amount to the right. The mean of the two observations is therefore the correct result. Of course the spirit levels v, and the bearings of the supports of the horizontal axis of the telescope, as well as the collimation line, may be adjusted by means of the screws provided for the purpose, as explained in Chapter IV., but as the instrument is always liable to get out of adjustment, in all important observations both "faces" should be used. A B Accurate Method of setting Instru ment in Line. Sometimes an error arises from the plumb bob not being exactly in the vertical axis of the instrument. In order to avoid this error, range in a point at a little distance back from the point over which the instrument is to be set up, and set the instrument exactly in line with the two back points by trial and error. If the cross hairs bisect the two back points, the instrument will be in the line. For instance, suppose the Fig. 72. Accurate Method of getting in Line. * With the exception of errors due to eccentricity, which are eliminated by reading both verniers, and errors of graduation, which are eliminated by measuring the angle several times on different parts of the graduated limb. instrument to be set exactly in the line at A, and it is required to set it up at B, Fig. 72. Line in the point c a little way back from B, and measure BC. The instrument is now to be set up at B, so that the cross hairs are directed exactly on both A and c, while the axis is at the measured distance BC from c. This is a more accurate way of getting into line than setting up by the plumb bob. Sometimes the operation is a little troublesome, but with the aid of adjusting screws (m',n', Fig. 51) on the instrument and by getting first approximately into line by the plumb bob it is usually easy enough. This method is of course not necessary for ordinary work, and is only used when great accuracy is required, as for instance in setting out tunnels, &c. Method of conducting an Ordinary Small Survey with the Theodolite.-The chief advantage of using a theodolite is that the measurement of tie lines is unnecessary, and also the measurement of those lines which in a chain survey are measured only to enable the work to be plotted. With the theodolite a sufficient number of angles is taken to enable the work to be plotted and also to give a check, in the same way that the tie lines act as checks in a chain survey. D Fig. 73. B Small Theodolite Survey. For example, in Fig. 73, if the four sides ABCD are measured, and also the angle DAB, the survey may be plotted. Thus we may lay down the line AB to scale on the paper, and then AD, making the angle BAD equal to the observed angle. By now taking the lengths CD and BC in the compasses, and sweeping out arcs from D and B as centres, we get by their intersection the point c. As a check on the work one at least of the other angles must be measured, say the angle at c, and it is preferable to measure all the angles. Suppose the angle at c to be measured, then if there is any error either in the measurement of any of the sides or in the measurement of the angle at A, the angle at c when measured on the paper with the protractor will not agree with the angle observed at c on the ground. The advantage of measuring all four angles at A, B, C, and D is that we can check the accuracy of the angular measurements before leaving the ground, as "the sum of the interior angles of any rectilinear figure is equal to twice as many right angles as the figure has sides, less four right angles." In the case of Fig. 73, as the figure is four-sided the sum of the interior angles is equal to 2 x 4 × 90° - 4 × 90° = 360°. If then the sum of the four interior angles as measured on the ground is equal to 360°, or, as perfect accuracy is impossible, to 360° plus or minus a fair allowable amount of error, we are satisfied before leaving the ground that the angular measurements at all events are correct. The accuracy of the work is then proved by testing the angles with the protractor after plotting. If now the same figure be surveyed with the chain only, in order to plot the work it is necessary to measure one of the diagonals AC or DB, say AC, while the two tie lines aв and bD or the other diagonal BCD are necessary to check the two triangles ABC, ACD. By the use of the theodolite, therefore, the measurement of the dotted lines shown in Fig. 73 is rendered unnecessary. This is a not inconsiderable saving of labour, as it is much easier to measure a few angles with the theodolite than to chain several lines. At the same time it must be admitted that in many cases in surveying with the chain only it is possible to select the main and subsidiary chain lines, so that those lines which must in any case be chained to take up interior fences and other details, themselves act as tie lines. Thus in a chain survey, lines which are run for no other purpose than to act as ties or to enable the work to be plotted, are avoided where possible. For example, referring to the chain survey in Fig. 24, it will be seen that all the Fig. 74. Entering Angles in Field Book. chain lines shown are required to take up the fences, &c. On the other hand, however, if all the interior angles of the boundary lines at A, B, C, D, E, and F had been measured, by a rearrangement of the chain lines, the measurement of some of the lines OA, OB, OC, OD, OE, and of might have been avoided. Field Book.-The field book is kept in the same manner as the field book of the chain survey (Fig. 24, pages 21 to 39), and the angles may either be entered in the field book at the beginning of each line, or marked on the rough sketch of the survey which is made before commencing to chain the lines. If preferred, the angles may be entered all together on separate pages of the field book, as in Fig. 74. As a rule it is better to take all the angles last after the chaining and offsets are finished. They may then be entered in their proper places in the field book. Reconnaissance of Ground. The same remarks as to reconnaissance of ground as made on page 15 apply to theodolite surveys. Practical Hints.-See that the instrument is firmly planted in the ground by pressing in the legs, and get it as nearly level as possible by means of the legs before proceeding to level it up with the levelling screws. In sighting on to a point on which a ranging rod is being held, always take the 'iron-shod point at the lower end of the ranging rod, or, if that is not visible, see that the ranging rod is carefully plumbed, and sight on to the lowest visible part of the ranging rod.* Obstacles to Measuring. In passing an obstacle by "squaring off," as in Fig. 12, Chapter I., the operation is much facilitated by having the theodolite to set out the right angles at a, C, D, and b, and greater accuracy is attained. a Fig. 75. Obstacle. Fig. 76. Distance across a River. An easier method when a theodolite is at hand is to triangulate round, as shown in Fig. 75. If the angles at a, b, c are each made 60°, the triangle is equilateral, and ac = ab or be. If, owing to the See also page 202. |