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without it, and let the ground between be more or less difficult for correct chain admeasurement. Select any length on the ground, as C D, from which A and B are visible, if not along the whole length, then from any three points; of course the length CD must be as suitable for chaining as it is possible to find it; measure CD accurately, making it some round number, and in doing so leave a station at E, also at a round number, having first, however, boned out the line with the theodolite. From C measure the angles ACB and BCD; from E measure AEC and AEB; and from D, CD A and ADB; now move the theodolite to A, and measure C AE, E AD, and DAB; and from B measure DBE, EBC, and CBA; from these we may calculate the length of A B, by means of CD, CE, and ED, as primary bases, when A B will be the secondary one.
If we take C D as base, and begin with the triangle BCD, we shall find
Sine B : sine D::CD: CB; then, with C B as base in triangle C AB,
Sine A : sine C::CB: A B. There must be the same final result if we begin with the triangle C A D instead of BCD.
Again, if we take E D as base, and beginning with the triangle EBD
Sine B : sine D::ED: EB, and in triangle E AB, with E B, last found, as base,
Sine A : sine E::EB : A B. The reader may readily accustom himself to these calculations, and the use of the tables by laying down a few triangles with a protractor and scale, and computing the sides, and afterwards practising in the field. The advantages to be obtained on extensive surveys are too manifest to be doubted; on many occasions it is indispensable, and the student cannot too soon become accustomed to the practice.
It will be hardly necessary to observe, that if we could not obtain a favourable line in the position of CD, we might select some other, as E F, FG, or G K, as a primary base, the last of which, GK, would be better than either of the other two, for though shorter than the first, the angles off it would be better proportioned than off either of the other two. This sufficiently instances how in many cases it may be practicable to lay down a short primary base to obtain a long secondary one, which, how
TRIANGULATION.-SELECTION OF BASE.
ever, once obtained, becomes the base of the survey. This working from short to long lines may sometimes be repeated three or four times, to obtain the length of the base required, and the operation is not to be confounded with that of laying down small triangles to obtain large ones over the area of survey, which would be an erroneous method, as we have already alluded to in this chapter. As an instance, take Fig. 108, where it is required to ascertain the length of base between two headlands on which are stations O Oʻ; set out and carefully measure the line A B, which must be very correctly done; plant the theodolite at A ; measure the angle O A O', and at the same time set up a station at C, making A C, judging by the eye, somewhat about half as long again as_AB; then measure the angle O'A E, and observe that A B E, being a straight line, is in fact an angle of 180°; then set up D, by making the angle E A D equal to O’AE. At B measure the angles C B A and A BD, which should be exactly equal if the work is correct; also D BE and E BC, equal to one another, observing that the four angles at C are equal to 360°; from C measure ACB and A ČE; from D, A Ď B and A DE, also equal each to each, and the three angles of each of the four triangles equal to 180, after having measured CEB and BED; from E measure also AEO, AEO, and CEO. From O, measure EO A and A O'0; also from 0, AO E and A 00. From the triangle
base A B, we obtain the length of A C, for the second base. From the triangle A C E, and base A C, we obtain the length of A E, for the third base. From the triangle A O’E, and base A E, we obtain the length of A O' for the fourth base; and from triangle O' A O, and base A 0', we obtain the length of O O', which was required.
Observe that from triangle A O'E and base A E, the side OE may be calculated, which may be made the base of the triangle O É O, also to calculate the length of 0 Oʻ, whereby all the former calculations are tested; observe also that A D, DE, E O', and O' A, form a four-sided figure of which all the angles together are equal to 360°; similarly in the quadrilateral A E, EO, 00', and 0 A, the sum of the angles is equal to 360°. Numerous other calculations may be made by means of these triangulations, but we will restrain ourselves to those above, as being all that are requisite for our purposes.
It will be unnecessary to observe that the selection of a base line is a most important operation, requiring a careful examination of the ground before it is finally determined on, to see how the triangulation will be laid down, and what advantages may be gained by selecting one station instead of another, which may
often be the case by moving a station even 40 or 50 feet one way or another, though permanent objects are very desirable to select as stations when conveniently situated; it is also to be remembered that the object of the angles taken at the base is to fix stations, which are to be the centres of further observations, or for determining correctly the position of lines which may be useful on the survey, that is, in the operations of engineering field-work, which differ practically from those merely geodætic, when it is only sought to fix the stations of triangles independent of chaining. It is therefore necessary to determine how we are going to work on or from the stations determined upon, unless they are merely fixed as instrumental stations for angular observations. The examination thus required does not by any means result in lost time, as the engineer thereby acquires a mental bird's-eye view of his survey which never fails to be of great service throughout the remainder of his operations. It has already been shown that besides the stations at the ends, there may be other stations on the base, to which observations should be taken when the base is being fixed, so as not to rely merely on measurement only, for portions of the base between intermediate stations, any more than for the whole of the line.
Very little reflection will now point out the advantages to be gained in many localities by pursuing the system of triangulation above-mentioned, and how many points may thus be accurately fixed, which by any ordinary method it would scarcely be possible to effect even with considerable labour. Even when surveying by traverse, which is the best ordinary system, we must depend on the accuracy with which a multitude of angles is taken, which will not merely depend on the observer, but also on the instrument and the diameter of it; we must also rely on the chainage of every line. By the above system each triangle is fixed by three angles, two of which are taken off one measured line. When the stations on the triangles have been thus fixed, the remainder of the survey may be completed by traversing lines run from station to station, as circumstances may require, and however numerous the sides may be, the amount of practical error will be ascertained ; neither will it be carried beyond the terminal stations on the lines of triangulation.
Station Poles.--As it is often necessary to distinguish the triangulation station poles several miles off, and as it should be done readily and clearly, it will follow that they should be conspicuous; straight clean natural spars will serve the purpose very well; they should be from 15 to 20 feet long; the longer ones should be placed of course where the shorter ones would not be visible, or would not so readily catch the eye. To the
TRIANGULATION STATION POLES.
top of these are fastened flags of white or coloured buntin about 3 feet long, and about 2 feet wide at one end and pointed at the other; or the flag may be red and white, which will be found to attract the attention better, the one colour coming out where the other would not be perceived ; dark blue and white flags are also very useful, more particularly on the survey of an estuary or a navigable river, keeping the red and white on one side and the blue and white on the other, the advantage of which is sometimes very great, more particularly in situations where the banks are invisible. The most secure way of fastening the flag to the pole is to lay a stout piece of cord along one end of the flag, and to nail through cord and flag together up to the pole, leaving the cord outermost to lay against the heads of the nails. When the flag and the observer lay with the wind, there is often considerable inconvenience in catching sight of the flag, though we may be often assisted in doing so by means of a pocket-glass, which will always be found very useful on such work. A brush of twigs fastened up to the pole is also very useful. An iron ring may also be fastened up to within a foot of the top of the pole, and above a collar to play freely round the pole; round this collar should be fastened, at equal distances from each other, tin vanes about 9 inches long, painted black on one side and left bright on the other, so that there will always be one bright and one black side presented to the eye of the observer; ingenuity may devise many other plans equally good, but generally, station poles at long distances from the observer should have some such conspicuous mark, for it will often be found very troublesome to distinguish a mere spar a long way off. Station poles should be sunk some four or five feet into the ground ; they should be fixed perfectly vertical, by means of a line and plummet carried quite round the poles, and let it be remembered that unless they keep perfectly vertical throughout the survey, they are better out of the way altogether; the filling in should be thoroughly well rammed or punned; this demands much more careful attention than might at first be supposed. It has already been observed that where convenient permanent objects can be selected it should always be done. When station poles require to be removed for the purpose of planting the theodolite over the centre, this spot should be previously marked by driving in four stumps round the pole at a few feet off, so that lines strained across them may show the centre by their intersection. In some situations it will be found necessary to adopt three poles with a wide-spread, triangle fashion, and a fourth fastened above, a plummet fastened from the bottom of which will denote the centre of the station.
The details previously given, in conjunction with the observations made in this chapter, will sufficiently guide the reader as to the application of this system of surveying on particular occasions, and we will now turn our attention to another part of the subject.
There can be no possible doubt that when it comes to be well understood, the student will see how much time may be saved, and greater accuracy ensured, than by common systems which are sufficient on many occasions merely to show the relative positions of a few hedges. Indeed, if a surveyor begins by making a few observations of this kind, any intelligent man who has proved a good chainman, and can read and write, is quite competent to do a great deal of the filling in.