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PRACTICAL OBSERVATIONS ON SETTING OUT. 171
from this base, its position and measurement become most important objects, and whether such base be primary or secondary, as we shall presently explain, its position should be such not only to obtain in most cases a view from both ends of the most important points, but also such that the triangulation shall be as favourable as possible.
Thus much, which is almost self-evident, being admitted, the following rules may be adopted as maxims, except, however, when conditions are such as to oblige us to set them aside.
When practicable, the triangles should be as nearly as may be equiangular, as the nearer they approach to sixty degrees the more accurately they are likely to be all measured, and the more equally practical errors incident to observations will be probably divided amongst them. Generally, however, it would be very inconvenient thus to make the triangles equilateral, and practically the maxim means, that the angles should be neither very acute nor very obtuse; for instance as little as may be above 120°, and under 30° for the main triangles; this, however, should not prevent an observation being taken to a point so situated, but it should be carefully checked from other stations.
The greater the number of angles taken the greater the liability to error, and vice versa, therefore the triangles should be made to cover as large spaces as possible, which should be done also in order to work from large triangles to small ones, and not to make a main triangle, or certain points of an area which should be included in one, dependent on observations taken from the stations of a secondary triangle. This number of observations taken from the base must, however, be regulated by the number of important points to be fixed.
It is not absolutely necessary to confine the stations at the angles of the main triangles within the boundaries of the survey, if favourable stations for taking observations are to be found by doing otherwise. But the stations should not be so far distant from each other as to make it inconvenient for taking observations from them on to other points; on the contrary, they should be so fixed as to make it convenient to take angles from them on to the other stations on the main triangles, and also on to points which are to become stations on the secondary triangles, or a great part of their value will be sacrificed; these stations on the secondary triangles should, therefore, be carefully thought of and considered when determining the stations of the main triangles; it is very desirable, when possible, that when the angles of the main triangles are taken, observations should at the same time be made on to those points which are to become the stations of the secondary triangles.
Every station should be so fixed as to obtain the greatest view possible of the surrounding country; so that the theodolite may be planted over it, and in such a situation that angles may be conveniently observed therefrom.
We have already alluded to the terms primary and secondary base. This secondary base may or may not be dependent entirely on the observations made from the primary one, and it may or may not be actually chained; we have alluded also to the measurement of the base, and to its position relative to the remainder of the survey.
And first, as to the measurement; that which at first sight might appear to be one of the most simple operations on a survey is often rendered by natural circumstances one of the most troublesome, and under any conditions, more particularly in a survey of this character, where everything depends on it, if the instrumental observations are all correctly made, the measurement of the base must be most accurately taken; not only any error in this will be carried pro tanto throughout the survey, but it may become multiplied over and over again.
In geodetical operations great pains have been taken, and the most searching ingenuity has been resorted to in order to obtain the accurate length of a base of operations, and metal and deal rods laid end to end, adjusted with the utmost nicety in troughs set horizontally, have been used for this purpose, and the greatest attention given to allow for expansion and contraction due to temperature. For the purposes of the engineer these nice minutiæ of extreme accuracy are not required; it will be sufficient to employ a good ordinary chain, of which the length has been carefully tested by a standard chain, or by a standard measure set off precisely with ten-feet rods on some level piece of ground, or better, along the coping of a wall; we should, however, for this purpose decidedly prefer a chain that had been well used and stretched, still by all means testing it before commencing operations, and again at the end of each day's work during measurement of the base. The line to be measured as a base should be carefully boned out with the theodolite, by setting up rods or other marks at every five or ten chains, and every precaution taken to set out a truly straight line, with a sufficient number of marks along its length to ensure the chain's being laid exactly along the line each time. In such a case as this all hedges, or other interruptions, must be cut through, and the banks must all be levelled. The inclinations of all undulations of the ground must all be carefully measured, and allowed for on the spot, or by the calculations made after the day's work. Par
173 ticular care should be taken that there are no kinks in the chain in measuring the base line.
Where circumstances will allow of it, it is very desirable to have on the base from whence observations are taken, a third station, somewhere about the centre of the line, besides the two end stations, as such third station will become a station of verification. In measuring the base, it is well to have a third man to help to lay the chain straight and flat, more particularly if the 100 feet chain is used instead of the ordinary Gunter's chain.
Where a level piece of ground, or if not exactly level, then smooth and even, so as to ensure as much facility as possible for correct chainage, can be found situated in such a manner as to offer good positions for commanding stations, we have of course ready at hand all we can desire; such a situation will besides afford us probably an opportunity of making the length of the base some round number in chains, whereby calculations are shortened, and chances of error are diminished.
The longer the base the greater the number of chains that have to be laid in order to measure it, and the greater therefore the chances of error arising from this source, more particularly over any portion of the base running along rough or broken ground. It is therefore sometimes advisable to lengthen the base by observation in the following manner :
Let AB, Fig. 106, be a portion of a base line which it is desired to produce as towards C; if there does not happen to be any object in that direction in line with A B, send forward a pole to be set up anywhere on the hill, and set it in line by planting the theodolite at D, which is made any convenient number of chains, ten, if possible, from B; and, before removing the instrument, set off DE any convenient angle, BDE, and next set off the angle BDF, equal to 60°. In both cases read both verniers, and repeat both angles. Keep the vernier plate clamped, and move to B, where plant the instrument very correctly over the centre of the station, and set off D B F, also equal to 60°, and E B F, equal to 60°. B F and DF should both measure exactly equal to BD; move the instrument to F, when the angle B FD should measure 60° also. If the nature of the ground is such as to render an angle of 60° at B and D unfavourable, then select any other that may be convenient, but 60° is better. Whilst the instrument is at F, measure the angle CF B, and at the same time set off EB F, which is B F produced, and make it intersect D E, previously laid out. Remove the instrument to E, and measure the angle CEF and FED, the latter of which-180°-EFD+EDF; measure B E, and add it to
BF. Leave poles at E B F and D, under charge of an assistant, and remove the instrument to C; bring the telescope to bear on B, then let the pole B be made to lean a little on one side, when the line of sight through the telescope should also bear on D, and then, putting the pole at D out of the way, it will also bear on any distant station left on the base beyond A. If this is all correct, then the centre of the instrument is on the base produced, and so therefore is the station C. Now measure the angles EC B, and also ECF, the latter being equal to ECB+ ECF. We have now ample means of calculating the length of CB. Supposing the ground such at B that we have been obliged to make BF or EF rather short, the work may be tested by making the angle BCG = DBF and HCB= its supplement = EBD, and measuring the angles EH C, and H CE, and BG F ; this, however, can be but rarely necessary, as the former operations have been closely tested, there being six triangles, in all of which the three angles of each must be equal to 180°.
Should the nature of the ground be such on the hill B as to preclude the practicability of laying out such lines as E F and DF, we may make the angle BDF more acute, or we may carry out the operation at C, the object being to obtain a base, EF, or H G, by means of which we may calculate the length of BC; therefore this base should not be so small a proportion of the line required as to make some of the angles of the triangles too obtuse, and the others too acute.
To compute B C by means of the above triangles and the base BD, in triangle BDF, the angle B = 62° 30′, D = 54° 30′, F = 63° 0', base BD = 20.00 chains.
Sine F sine B: BD: FD;
then sine 63°: sine 62° 30':: 20.00; F D. Sine 63° = .89100;
sine 62° 30′ = '88701;
then, 89100 88701: 20.00 19.607 F D.
the angle E = 39° 40'; F = 63; D =
In triangle FDE,
77° 20′, and base F D
19.607, as above.
Sine E sine D:: FD: EF;
then sine 39° 40′ : sine 77° 20′ :: 19·607: EF; sine 39° 40′′ =
63832; sine 77° 20′ = '97566;
then 63832 97566: 19-607: 29.969 EF.
In triangle E CF, the angle C=19° 10′; F = 52° 20′; E = 108° 30′; EF = 29.969;
Sine C sine E:: EF: CF;
here the angle E being above 90°, we shall have to take the angle of the supplement to 108° 30′ = 71° 30′, and we shall find 94832 :: EF: CF:: 29.969: 86-563 C F.
In triangle BCF, C= 11° 10′, F = 52° 20′, B = 116·30, C F = 86-563; then
Sine B sine F: CF: CB,
89493 79158 :: 86-563: 76.555
This may be checked by calculating in the first triangle B D F the side B F, instead of F D, for
Sine F sine D :: BD: BF, and in triangle BC F,
Sine C sine F: BF: CB.
Should the length of base now obtained be found equal to an odd number of chains and links, it may easily be reduced to a round number by shortening.
It may occur to the reader that the number of observations to be made renders this method troublesome; to a great extent this will depend on his ability to handle the theodolite; if he is familiar with the use of it, a couple of hours will do all that is required, and that much more correctly than by ordinary means; besides this, the number of inclinations to be measured by observation in such a case will be found quite as troublesome as, if not more so than, the horizontal angles to be taken; to this is to be added the difficulty and uncertainty of chaining a straight line over hilly ground. In truth, the only practical objection lays in the use of the trigonometrical tables by the most ordinary arithmetical means, and this is really too futile not to be at once set aside, when it is observed how insignificant it is, the object of such a base being to fix with accuracy throughout an extensive survey a given number of points, which it would be difficult and laborious, if not impracticable, to do by ordinary means and chain triangles. To this we may add, that in marine surveying this method is indispensable, and therefore the sooner practical proficiency in carrying it out is obtained the better. Where a theodolite is not at hand, a little study of the ground, and a small amount of ingenuity, will often enable us to effect our purposes with
As another instance, let A and B, Fig. 107, be two positions on high ground, from which we can obtain a view of the surrounding district, and let them be far apart or near to each other, within reasonable limits, and within the survey or