convenient. Choose a base, as XY, within the field, and from its ends measure the angles between it and the direction of each corner of the field, if the Theodo- r lite or Transit be used, or take the bearing of each, if the Compass be used. Consider first the triangles which have

XY for a base, and the corners of the field, A, B, C, &c., for vertices. In each of them one side and the angles will be known to find the other sides, XA, XB, &c. Then consider the field as made up of triangles which have their vertices at X. In each of them two sides and the included angle will be given to find its content, as in Art. (65). If Y be then taken for the common vertex, a test of the former work will be obtained.

The operation will be somewhat simplified by taking for the base line a diagonal of the field, or one of its sides.

(393) Inaccessible Areas. A field or farm may be surveyed, by this "Fourth Method," without entering

it. Choose a base line XY, from which all
the corners of the field can be seen. Take
their Bearings, or the angles between the
Base line and their directions. The dis-
tances from X and Y to each of them can
be calculated as in the last article. The
figure will then shew in what manner the
content of the field is the difference between X

Fig, 270.

Y

the contents of the triangles, having X (or Y) for a vertex, which lie outside of it, and those which lie partly within the field and partly outside of it. Their contents can be calculated as in the last article, and their difference will be the desired content. If the figure be regarded as generated by the revolution of a line one end of which is at X, while its other end passes along the boundaries of the field, shortening and lengthening accordingly, and if those triangles generated by its movement in one direction be called plus and those generated by the contrary movement be called minus, their algebraic sum will be the content.

In all the opera

(394) Inversion of the Fourth Method. tions which have been explained, the position of a point has been determined, as in Art. (8), by taking the angles, or bearings, of two lines passing from the two ends of a Base line to the unknown point. But the same determination may be effected inversely, by taking from the point the bearings, by compass, of the two ends of the Base line, or of any two known points. The unknown point will then be fixed by platting from the two known points the opposite bearings, for it will be at the intersection of the lines thus determined.

(395) Defects of the Method of Intersection. The determination of a point by the Fourth Method (enunciated in Art. (8), and developed in this Part) founded on the intersection of lines, has the serious defect that the point sighted to will be very indefinitely determined if the lines which fix it meet at a very acute or a very obtuse angle, which the relative positions of the points observed from and to, often render unavoidable. Intersections at right angles should therefore be sought for, so far as other considerations will permit.

PART VI.

TRILINEAR SURVEYING;

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 Fig. 271.

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 prob lem.

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' S'- B. Let BASU; Then we shall have (as will be shewn in Appendix B)

BCS = V.