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By the Fifth Method.

(396) TRILINEAR SURVEYING is founded on the Fifth Method of determining the position of a point, by measuring the angles betwen three lines conceived to pass from the required point to three known points, as illustrated in Art. (10).

To fix the place of the point from these data is much more difficult than in the preceding methods, and is known as the "Problem of the three points." It will be here solved Geometrically, Instrumentally and Analytically.

(397) Geometrical Solution. Let A, B and C be the known

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objects observed from S, the angles ASB and BSC being there measured. To fix this point, S, on the plat containing A, B and C, draw lines from A and B, making angles with AB each equal

to 90°-ASB. The intersection of these lines at O will be the centre of a circle passing through A and B, in the circumference of which the point S will be situated.* Describe this circle. Also, draw lines from B and C, making angles with BC, each equal to 90°-BSC. Their intersection, O', will be the centre of a circle passing through B and C. The point S will lie somewhere in its circumference, and therefore in its intersection with the former circumference. The point is thus determined.

In the figure the observed angles, ASB and BSC, are supposed to have been respectively 40° and 60°. The angles set off are therefore 50° and 30°. The central angles are consequently 80° and 120°, twice the observed angles.

The dotted lines refer to the checks explained in the latter part of this article.

When one of the angles is obtuse, set off its difference from 90° on the opposite side of the line joining the two objects to that on which the point of observation lies.

When the angle ABC is equal to the supplement of the sum of the observed angles, the position of the point will be indeterminate; for the two centres obtained will coincide, and the circle described from this common centre will pass through the three points, and any point of the circumference will fulfil the conditions of the problem.

A third angle, between one of the three points and a fourth point, should always be observed if possible, and used like the others, to serve as a check.

Many tests of the correctness of the position of the point determined may be employed. The simplest one is that the centres of the circles, O and O', should lie in the perpendiculars drawn through the middle points of the lines AB and BC.

Another is that the line BS should be bisected perpendicularly by the line 00'.

A third check is obtained by drawing at A and C perpendiculars to AB and CB, and producing them to meet BO and BO' produced,

For, the arc AB measures the angle AOB at the centre, which angle = 180° -2 (90°-ASB) = 2 ASB. Therefore, any angle inscribed in the circumfer ence and measured by the same arc is equal to ASB

in D and E. The line DE should pass through S; for, the angles BSD and BSE being right angles, the lines DS and SE form one straight line.

The figure shews these three checks by its dotted lines.

(398) Instrumental Solution. The preceding process is tedious where many stations are to be determined. They can be more readily found by an instrument called a Station-pointer, or Chorograph. It consists of three arms, or straight-edges, turning about a common centre, and capable of being set so as to make with each. other any angles desired. This is effected by means of graduated arcs carried on their ends, or by taking off with their points (as with a pair of dividers) the proper distance from a scale of chords (see Art. (274)) constructed to a radius of their length. Being thus set so as to make the two observed angles, the instrument is laid on a map containing the three given points, and is turned about till the three edges pass through these points. Then their centre is at the place of the station, for the three points there subtend on the paper the angles observed in the field.

A simple and useful substitute is a piece of transparent paper, cr ground glass, on which three lines may be drawn at the proper angles and moved about on the paper as before.

(399) Analytical Solution. The distances of the required point from each of the known points may be obtained analytically. Let AB = c; BC= a; ABC B; ASB = S; BSC S'. Also, make T = 360° — S — S′ — B. Let BAS U; BCS = V. Then we shall have (as will be shewn in Appendix B)

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c. sin. S'

a. sin. S. cos. T



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Attention must be given to the algebraic signs of the trigonometrical functions.

Example. ASB=33° 45'; BSC= 22° 30′; AB=600 feet; BC= 400 feet; AC800 feet. Required the distances and directions of the point S from each of the stations.

In the triangle ABC, the three sides being known, the angle ABC is found to be 104° 28′ 39". The formula then gives the angle BASU 105° 8' 10"; whence BCS is found to be 94° 8'11"; and SB = 1042.51; SA=710.193; and SC 934.291.



(400) Maritime Surveying. The chief application of the Trilinear Method is to Maritime or Hydrographical Surveying, the object of which is to fix the positions of the deep and shallow points in harbors, rivers, &c., and thus to discover and record the shoals, rocks, channels and other important features of the locality. To effect this, a series of signals are established on the neighboring shore, any three of which may be represented by our points A, B, C. They are observed to from a boat, by means of a sextant, and the position of the boat is thus fixed as just shewn. The boat is then rowed in any desired direction, and soundings are taken at regular intervals, till it is found convenient to fix the new position of the boat as before. The precise point where each sounding was taken can now be platted on the map or chart. A repetition of this process will determine the depths and the places of each point of the bottom.



(401) THE obstacles, such as trees, houses, hills, vallies, rivers, &c., which prevent the direct alinement or measurement of any desired course, can be overcome much more easily and precisely with any angular instrument than with the chain, methods for using which were explained in Part II, Chapter V. They will however be taken up in the same order. As before, the given and measured lines are drawn with fine full lines; the visual lines with broken lines; and the lines of the result with heavy full lines.



(402) Erecting Perpendiculars. To erect a perpendicular to a line at a given point, set the instrument at the given point, and, if it be a Compass, direct its sights on the line, and then turn them till the new Bearing differs 90° from the original one, as explained in Art. (243). A convenient approximation is to file notches in the Compass-plate, at the 90° points, and stretch over them a thread, sighting across which will give a perpendicular to the direction of the sights.

The Transit or Theodolite being set as above, note the reading of the vernier and then turn it till the new reading is 90° more or less than the former one.

The Demonstrations of the Problems which require them, and from which they can conveniently be separated, will be found in Appe x B.

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