2, 6, 9, and 12 are taken by the theodolite, the tie-lines from 3 to 4, from 5 to 7, &c., will not be required.

Station-flags being set up at 1 and 2, and the line measured, a flag is then set up at 6, and the angle 2 taken, which being less than 180°, shows that the line 2 6 is on the left of 1 2. Having measured the line 2 6, the angle 2 6 9 is next taken, which, being 297° 13', or greater than 180°, shows that the line 6 9 turns to the right. Similar observations will apply to the remaining lines. But at station 12, besides the angle to determine the position of the line 12 14, a check angle is taken between an imaginary line, drawn from 12 to 1 and the line 12 14, which is 234° 20′; therefore, after the work is plotted, the protractor will show this angle if the work be right.

If the station-flag at 1 cannot be seen from the station at 12, a flag may be fixed in any other given point in the line 1 2, or in its prolongation, for the purpose of taking the check angle; and if the road is very long, and has several bends, a proportionate number of check angles ought to be taken.

The method of plotting the preceding work will be sufficiently obvious from the example given in Section II. of the survey of a wood.

2. If the figure just referred to, i.e. Problem V. Part IV., represent a river, it may be surveyed in the following way.

Measure a system of lines ac, cm, &c., to x, similar to those used in the last example, and take the angles between them, but omitting ab, nr, &c., which being tie-lines are not required.

If the breadth of the river is very unequal, a similar system of lines must be used on the other side of it: but if its width is everywhere nearly equal, its breadth may be found by any of the methods given in Prob. I., Heights and Distances, or by Prob. X. Part III.

The map of the river being thus obtained, the area, if required may be readily found by one or other of the methods already given.

SECTION IV.

THE PRACTICE OF TAKING BEARINGS AT TWO STATIONS.

Let F, G, H, E, D, C, be six points, or stations, the positions of which, and the distances FG, GH, HE, ED, DC, CF, are to be determined. Measure a base line AB on convenient and level ground, and at A and B angles being taken between AB and the points F, G, H, E, D, C, their positions and distances will be determined by the intersection of the lines that include these angles, as shown in the figure.

The lines FG, GH, &c., being thus determined, may form new base lines for other portions of the survey.

NOTE. It is not necessary that the stations A and B should be in the lines ER, CF, as in the figure, these positions being only assumed as being convenient for checking the work, which can be better done by taking angles at any given point o in AB.

The base AB, and the several angles taken at the stations A and B, are shown in the following field-notes, also the angles taken at 0, a station in AB, as check angles.

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Draw the base line AB in the given direction, and of the given length, 4323 links, making the station o thereon, for the check or proof angles. With the protractor, applied to the line AB, and its centre at A, lay off the angle BAF 16° 56', reckoning from B, Move its centre to B, and lay off the angle ABF = 360°-310° 56' = 49° 4′, reckoning from A, and the intersection of the lines AF, BF, will give the point F. The position of the point & may be determined in the same way, by the intersection of AG, BG; as also the points H, E, D, C; recollecting that when the angles exceed 180° to take them from 360°, and to plot the remainder from right to left, looking from the centre of the instrument along the base line.

If the work be correct, the lines HE, CF will respectively pass through the stations A, B ; this, however, does not check the points D and G: therefore join OD, and apply the protractor to AB with its centre at 0, and if it show the angle AOD to be 99° 59′, the point D is in its right position. In the same way, if the angle AOG = 360° -292° 18' 67° 42', the point G is in its correct position; and similarly with respect to the points E, C, F, H.

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AF and BF may be found. Again, in the triangle ABG are given AB, and the angles BAG = 48° 18', and ABG 360°-328° 38' = 31° 22′, from which AG may be found. Now in the triangle AFG, the two sides AF and AG, being determined by the two preceding operations, may be considered as given, and their included angle FAG is also given in the field-notes, being 48° 18' 16° 56′ 31° 22′, from which FG may be found. And similarly GH, HE, &c., may be found. By either of the preceding methods are found CD

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2576,

TWO EXAMPLES FOR PRACTICE BY TWO STATIONS.

It is required to find the positions and distances of the points. D, E, F and G, from the field-notes No. 1, also from the field-notes No. 2.

Let the following figure represent a portion of a town or city, the map of which is required.

Begin the survey at the meeting of three or more principal streets, and select a station, as A, where the longest prospects can be obtained along all the streets diverging from it, in order to get the longest main or station lines. Having fixed the theodolite at that station, take the direction of one of the principal streets as a base line, as AB,

which must be directed to some well-defined object, as to the right or left side of a door or window, or a projecting corner of a house, or to an angle, or upright post of railing, or any other mark that can be well remembered; a description of which should be also put down in the field-book.

Angles between this base line and the other lines, diverging from station A must then be taken, the objects to which these lines are directed being well defined, in the manner already explained with respect to the base line.

The lines thus set out must next be measured with the chain, taking offsets to all projecting corners of buildings, bends, or openings; and

to all public buildings, as churches, markets, &c., also to statues, obelisks, &c., leaving stations at the ends of the streets to the right and left of the line, and taking the angles of their several directions. The same process is to be repeated on the other lines till the whole is finished.

Thus, fix the theodolite at A, and take the bearings of all the streets meeting there; then measure AB, taking offsets to the openings, corners, &c., of buildings, leaving stations at m, n, o, opposite the directions of the streets on the right. If the survey is for a railway, for which some of the buildings are required to be taken down, the entries should be made at a sufficient distance