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and the precision with which the directions of the meridians have been previously determined.

It would be proper to observe at three or more known stations the angles contained between the lines imagined to join them, and other lines supposed to be drawn from them to each of the objects whose positions are required; in order that, by the concurrence, in one point, of the intersections of all the lines tending towards each object, the correctness of the operations might be proved. Such concurrence is, however, scarcely to be expected, and the mean point among the intersections must be assumed as the true place of the object.

424. It occurs frequently, in the secondary operations of a survey, particularly in those which take place on a sea-coast, that the position of an object, or the position and length of a line joining two remarkable objects, are to be determined when, from local impediments, it would be inconvenient to convey the instrument to any known stations from whence the objects might be visible; and, in order to meet such cases, the following propositions are introduced. As a graphical construction alone may sometimes suffice, there are given, with the formulæ for the computations, the processes of determining the positions and distances by a scale.

PROB. I.

To determine the positions of two objects, and the distance between them, when there have been observed at those objects, the angles contained between the line joining them, and lines imagined to be drawn from them to two stations whose distance from each other is known.

Let P and Q be the two objects whose positions are required, A and B the stations whose

distance A B from each other is known; then QPB, QPA, PQA, PQB will be the observed angles.

A'

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b

B'

On paper draw any line as pq, and at p and q lay down with a protractor, A or otherwise, angles equal to those which were observed at P and Q; then the intersections of the lines containing the angles will determine the points A' and b. Produce the line a'b if necessary, and, with any convenient scale, make A'B' on that line equal to the given distance from A to B: from B' draw B'Q' parallel to bg till it meets A'q, produced, if necessary, in Q';

and from Q' draw Q'P' parallel to pq till it meets A'p, produced, if necessary, in P'. The figures A'pqb and A'P'Q'B' are (Euc. 18. 6.) similar to the figure APQB formed by lines imagined to join the objects on the ground: therefore P'Q', A'P', &c., being measured on the scale from whence A'B' was taken, will give the values of the corresponding distances between the objects.

The processes to be employed in computing the distances are almost obvious. Thus, let any number, as 10 or 100, represent pq. Then, in the triangle A'pq there are known the angles A'pq (=APQ), A'qp (= AQP), with the side pq; from whence (Pl. Trigon., art. 57.) A'p may be found. In the triangle pqb there are known the angles bpq (=BPQ), pqb (PQB), with pq; to find pb. In the triangle a'pb there are known the angle A'pb (=APB) and the sides A'p, pb; from which a'b may be found. Again, from the similarity of the figures,

A'b: A'B' (=AB) :: pq : P'Q' (=PQ);

and by like proportions any other of the required distances may be found.

The construction and formulæ of computation will, manifestly, be similar to those which have been stated, whatever be the positions of the stations P and Q with respect to a and B.

PROB. II.

425. To determine the position of an object, when there have been observed the angles contained between lines imagined to be drawn from it to three stations whose distances from each other have been previously determined.

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Let A, B, C be the three given stations, and P the object whose place is to be determined; then APC, CPB, or APB will represent the observed angles.

With the three given distances AB, AC, BC, lay on paper, by any convenient scale, the triangle A'B'C', and on the side A'B' make the angles B'A'D, A'B'D, respectively, equal to the

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observed angles CPB, APC; through D and C' draw an indefinite line, and from A' a line to meet it, suppose in P', making the angle B'A'P' equal to B'DC': the point P' will be the station. For if the circumference of a circle be imagined to pass through A, D, and B', since the angle B'A'P' is by construction equal to B'DP', those angles will be in the same segment (Euc. 21. 3.), and the circumference will also pass through P'; therefore the angle B'P'C' will be equal to B ́A ́D, and the angle A'P'C' to A'B'D. But the angles B'A'D, A'B'D are by construction equal to BPC, APC; therefore the latter angles are, respectively, equal to B'P'C' and A'P'c': thus, the angles at P', on the paper, are equal to the corresponding angles which were observed at P; and P' represents the object P: therefore the lines A'P', C'P', B'P', measured on the scale, will give the required distances in numbers.

The like construction might be used if A, C, and в were in one straight line. If the point D should coincide with c' the case would evidently fail; and the determination of p' will be less accurate as D falls nearer to c'.

The formula for computation may be briefly stated thus: In the triangle A'B'C' the three sides are given; therefore one of the angles, as B'A'C', may be found (Pl. Trigon., art. 57.). In the triangle B'A'D all the angles are known, and the side A'B'; therefore the side A'D may be computed. In the triangle DA'C', the sides A'D, A'c', and the angle C'A'D are known; therefore the angles A'DC', A'C'D may be found. In the triangle A'C'P' all the angles and the side A'C' are known; therefore the sides A'P', C'P' may be computed. Lastly, in the triangle A'B'P' there are known the side A'B' and all the angles (for A'B'P' is equal to the computed angle A'DC'); therefore P'B' may be obtained.

By this proposition the Observatory of the Royal Military College at Sandhurst was connected with three stations whose positions are given in the account of the Trigonometrical Survey of England. The stations are Norris's Obelisk, which may be represented by A; Yately Church Steeple, represented by B, and the middle of a Tumulus near Hertford Bridge represented by c; P representing the centre of the dome in the Observatory.

With an altitude and azimuth instrument whose circles are, each, twenty inches in diameter, the following angles were taken :

APC 134° 32′ 4′′, APB = 171° 37′ 16′′; whence
CPB = 37° 5' 12" ;

and from data furnished by the Trigonometrical Survey, there

were obtained AB 20252 feet, AC 18086 feet, and BC 8243.66 feet: with these data and the observed angles there were found by computation, as above, AP 6985.2 feet, CP 12488 feet, and BP 13315.3 feet. There was,

at the same time, observed the bearing of Norris's Obelisk from the meridian of the Observatory, which was found to be S. 86° 11′ 6′′ E., with which, and the computed distance AP, it was found by the formulæ in art. 410. that the difference between the latitudes of the Obelisk and the centre of the dome is 8".19, and the difference, in time, between the longitudes is 7".28; the Observatory being northward and westward of the Obelisk. Hence, from the latitude and longitude of the latter in the Trigonometrical Survey, it is ascertained that the latitude of the Observatory is 51° 20′ 32".99, and its longitude, in time, is 3' 3".78 westward of Greenwich.

If the object whose position is required were, as at p, within the triangle formed by the three given stations A, B and C, the observed angles being then Apc, CpB or AрB, the construction would be similar to that which has been given, except that the angles B'A'D', A'B'D' must in that case be made, respectively, equal to the supplements of Bpc and Apc, and the angle B'A'p' equal to B'D'C'; for then, as in the former, the circumference of a circle supposed to pass through A', B' and D' would also pass through p', and this point would represent the object. The formulæ of computation would be precisely the same as in the other case.

When many points are to be determined in circumstances corresponding to those which are stated in this proposition, it is found convenient to obtain their positions mechanically by means of an instrument called a station pointer. This consists of a graduated circle about the centre of which turn three arms extending beyond the circumference, and having a chamfered edge of each in the direction of a line drawn through the centre: by means of the graduations these arms can be set so as to make with one another angles equal to those which have been observed, as APC and CPB; and then, moving the whole instrument on the paper till the chamfered edges of the arms pass through the three points A', B′ and c', the centre of the instrument will coincide with P', and consequently will indicate on the paper the required position of the object.

The position of an object, as P, with respect to two given stations as A and B, may be found by the method of crossbearings in art. 364.

PROB. III.

426. To determine the positions of two objects with respect to three stations whose mutual distances are known, when there have been observed, at the place of each object, the angles contained between a line supposed to join the objects and other lines imagined to be drawn from them to two of the three given stations, some one of these being invisible from each object.

Let P and Q be the two objects whose positions are required; A, B and C the three stations, and let APB,

APQ or BPQ, PQB, PQC or BQC be the four angles which have been observed.

With any scale lay on paper a triangle A'B'C' having sides equal to the given distances AB, BC,

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AC, then on A'B' describe (Euc. 33. 3.) a segment A'P'B' of a circle, which may contain an angle equal to APB, and on B'C' describe a segment B'Q'C' which may contain an angle equal to BQC: it is manifest that the representations of P and Q will be somewhere on the circumferences of those segments.

Now, in order to discover readily what should be the next step in the process, imagine P and Q to be the places of the two objects, and imagine lines to be drawn as in the figure: then wherever P' and 'be situated, the angles A'P'B' and B'Q'C' will be equal to the corresponding angles APB and BQC; also the angle B'A'x will be equal to B'P'Q', and B'C'Y to B'Q'P'. Therefore, if the angles B'A'X, B'C'Y be made respectively equal to BPQ and BQP, a line drawn through x and Y, and produced if necessary (as in the figure), will cut the circumferences of the circles in P' and Q'; and these points will represent the required places of the two objects, since it is manifest that the angles at P' and Q' will be equal to the observed angles at P and Q. The lines P'Q', A'P', &c., being measured on the scale, will give the required distances in numbers.

The points P and Q may be on opposite sides of the triangle ABC, as at p and 9, or one of them may be within and the other on the exterior of the triangle, but the construction will be similar to that which has been given. It is evident that

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