PART THE NINTH.

SURVEYING BY THE THEODOLITE.

THERE are several cases in which the use of the theodolite is preferable, or absolutely necessary, both in limited and extensive surveys. First, in the survey of a wood or lake, by going round it, and taking the angles of its several bends.

Secondly, in that of a winding road or river, by taking the angles of the windings, as in the last case.

Thirdly, to determine the positions and distances of several points by taking angles at two stations.

Fourthly, in the survey of a town or city.

Fifthly, in that of extensive districts, as in railway surveys, &c., by taking angles to connect the main or base lines, and to determine the inaccessible parts, wherever they occur; also to range long lines over high summits, by fixing the theodolite in the line and turning the telescope to the back station-flags, and then reversing it, that the direction of the line may be accurately continued, by finding well-defined front marks at a great distance. This case frequently includes all the four preceding cases.

PRELIMINARY PROBLEMS.

By the following problems the areas of triangles, &c. may be expeditiously found, when required, without mapping them.

PROBLEM I.

To find the Area of a Triangle when two Sides and their included Angle are given.

To the logarithms of the two sides in links, add the log. sine of the included angle, and from the sum subtract 15.30103, and the remainder is the log. of the area in acres and decimals of an acre.

Ex. The two sides of a triangle are 960 and 576 links, and their included angle 53° 8'; required the area.

To find the Area of a Triangle when two Angles and their included Side are given.

From the sum of twice the log. of the given side in links and the log. sines of the given angles, subtract the log. sine of the sum of the given angles increased by 15.30103, and the remainder is the area in

acres.

Ex. Two angles of a triangle are 59° 46′ and 60° 14', and their included side 1000 links; required the area.

Log. 1000......3.00000

2

6.00000

Log. sin. 59° 46'......9.93665

60° 14'......9.93855

25.87520

Log. sin. (59° 46′ + 60° 14′) = 120°...9.93753

}

.25.23856

15.30103

PROBLEM III.

To find the Area of a Triangle when the three Sides are given. From the half sum of the three sides subtract each side separately, add together the logarithms of the half sum and the three remainders in links, and divide the sum by 2, diminishing the index by 5, and the result is the log. of the area in acres.

Ex. The three sides of a triangle are 4080, 5040, and 6100 links; required the area. Ans. 102a. Or. 16p.

PROBLEM IV.

To find the Area of a Trapezium when the two Diagonals and the Angle, made by their Intersection, are given.

The rule for the area is the same in this as in Prob. I., the diagonals being used as sides.

NOTE.

The investigations of the rules for solving these Problems are given by

various writers on Analytical Trigonometry.

CASE I.

To survey a Wood by the Theodolite.

The following figure represents a wood, the map and area of which are required.

Fix station-flags so that the lines compassing the wood may be as near it as possible, that the offsets may be conveniently taken, and that the stations, at the same time, may be on ground proper for fixing the theodolite. Let the stations be A, B, C, D, and E, the field-notes being as below.

Explanation of the Method of taking the Angles, &c.

From the preceding field-notes it will be seen that A B is first measured, as a base line for the plan. At B the angle A B C is then taken, which is 80° 31', thus giving the direction of the next line BC.

In taking this angle, the theodolite is fixed at B; the two zeros on the horizontal plates being clamped together, the telescope is directed to A, and the whole instrument clamped; the upper plate is then unclamped, and the telescope turned round to C, and the angle ABC found to be 80° 31'. In a similar way the angles B CD, CDE are found respectively 111° 41′ and 47° 48′, each of the three lines B C, CD, D E turning respectively to the left of the line that preceded it. But in taking the angle DEA, it is found to be 241° 40', which being greater than 180°, that is, greater than the semicircle adb by the arc bc; therefore, the line EA turns to the right. Lastly, the angle EA B is found 58° 20′, the line AB turning to the left.

It thus appears that the magnitude of the angle determines whether the new line turns to the right or the left of the old one, that is, the

new line turns to the left of the old one, if the angle is less than 180°, and to the right, if greater than 180°, provided the zero of the instrument be directed towards the beginning of the old line: therefore the bearings of the lines B C, CD, DE, AE, though put down by way of check, may be omitted in the field-notes.

The remainder of the field-notes are similar to those given in the preceding parts of this work, and therefore require no further explanation.

On plotting and proving the Work.

=

Draw the line A B in the given direction, and of the given length 1150 links, as a base line; place the centre of the protractor at B, with its straight edge close against A B, and from the end towards A mark off 8010, then through B and the protractor-mark draw BC= 1340 links. Proceed similarly with the lines CD, CE, and the angles C, D. At E the angle is 2413°: the next line EA must therefore turn to the right, and the angle to be marked off at Ę must be 360° - 2413° 1183°; or, if the protractor be a circular one, the whole angle may be marked off at once. At A, the commencement of the base line, the angle is 5830, which is a check angle to prove the accuracy of the previous work.

=

Moreover, since the sum of all the interior angles of any polygon is equal to twice as many right angles as the figure has sides, lessened by four; as the given figure has five sides, the sum of all its interior angles must be 2 × 54 6 right angles

=

80° 31'

111° 41'

= 6 x 90° = 540°, that is,

47° 48'

241° 40'

58° 20'

540° 0′ proof, as respects the angles.

The angles may, therefore, be considered as having been accurately taken. But the proof of the plotting being correct, is when the work is found to close, that is, when the line E A exactly reaches to the starting point A, or so very near to it, that the error is immaterial.

Having plotted the work, the area is found, by the methods already given in the preceding parts of this work, to be 9a. 1r. 22p.

A large Pond, Mere, or Lake is surveyed and mapped precisely in the saine manner, as in the case of the wood, just given.