send the flag-man ahead to the next corner; note the bearing of his pole; and so proceed with the sides, in succession, taking a back sight at each station.

If the end of the line cannot be seen from the beginning, let the flag-man erect his pole, in the line, at a point as distant from the beginning as possible. Sight to the pole, as before; then, going forward, set the compass by sighting to the last station. The flag-man should now be placed, exactly in line, at another station. So proceed until the end of the line has been reached.

289. Random Line. If the first position of the flagstaff were not exactly in line, the course run will deviate to the right or left of the corner. Where such is the case, measure the perpendicular distance to the corner, and determine the correction by the following rule :

As the length of the line is to the deviation found as above, so is 57.3 degrees, or 3438 minutes, to the correction in the bearing.*

In running through woods, it is very frequently necessary to correct the bearing in this manner. In all cases, however, where back sights are taken, the compass should be allowed to stand at the last station on the random line, since the local attraction often varies very considerably in a short distance. If it is desired to run the next line precisely on its location, the corner should be sighted to from the end of the random line, and a back sight taken.

* This rule is founded on the ordinary rule for the solution of right-angled triangles,—the length being the hypothenuse, and the deviation the perpendicular, an arc of 57.3 degrees being equal in length to the radius.

Thus, supposing, in running a line N. 85° 30′ E. 27.53 chains, the corner is found 35 links to the right hand: the calculation would be

27.53: 35:: 57.3°: 0° 43′.

The proper bearing would therefore be N. 36° 13′ E.

290. When the far end of the line cannot be seen, it I will sometimes be found convenient to run to a station as near the middle of the line as possible, if one can be found from which both ends can be seen. Then, instead of continuing on in the same course, sight to the corner. The chain-men should note the distance to the assumed station. A very obtuse-angled triangle will thus be formed, and the correction in bearing may be readily calculated.

Thus, supposing the line were AB, (Fig. 117,) Fig. 117. passing over an elevation at C. At A the bearing of AC was found to be N. 43° W., distance 10.50 chains. At C, CB was N. 43° W., distance 7.36 chains.

We have

AC BC: sin. B: sin. A; or, as the angles are small, AC: BC :: B: A; AC+BC: BC :: B+A : A.

whence

That is, 17.86: 7.36 :: 45′ : A=19', the required correction. The true bearing of AB is therefore N. 4310 W.

B

Where the deviation from the correct line is not much greater than in the example given, AB is sensibly equal to AC+ CB. Where the deviation is considerable, the angles and side should be calculated by Trigonometry.

The above rule may be expressed thus:

As the sum of the distances is to the last distance, so is the whole deviation to the correction to be applied at the first station.

291. Proof Bearings. In the course of the survey, bearings or angles should be taken to prominent objects. These form a test of the accuracy of the work. Three bearings are necessary to each object: two of these, being required to fix its position, will afford no check on the intermediate measurements; but their coincidence with a third will determine the probable correctness of all, and of the connecting measurements. Diagonal bearings and distances may likewise be taken as proof lines.

292. Angles of Deflection. In surveying with the transit or theodolite, it is most convenient to record the angles of deflection; that is, the angle by which the new course deviates to the right or to the left from that of the last line. This is always done in surveying roads, rivers, &c. From the angles of deflection the bearings are very readily deduced, by rules to be given hereafter. As checks to the work, the bearings of some of the lines may likewise be taken.

In a closed survey the whole deflection must equal 360°. To determine whether it is so, arrange the deflections to the left in one column, and those to the right in another. Sum the numbers in each column: the difference of these sums should equal 360°.

In practice this will rarely occur; though in open ground, where the angles can readily be taken, the error should not exceed four or five minutes in a tract of ten or twelve sides, provided a good transit or theodolite is employed.

EXAMPLE.

The following are the notes of a survey taken by the author:-1. S. 53° 10′ W.; 2. Deflect 97° 3′ to the right; 3. 97° 45' to the right; 4. 81° 14′ to the right; 5. 30° 12' to the left; 6. 12° 14' to the left; 7. 27° 48′ to the right. Whence the first line deflects 98° 34' to the right.

Where the difference amounts to several minutes, it is best to distribute it among the angles.

The rule which is sometimes given: to determine the angles from the bearings, and ascertain whether the sum of the internal angles is equal to twice as many right angles as the figure has sides, less four right angles-proves nothing in regard to the correctness of the field work. Any set of bearings will prove in this way.

SECTION III.

OBSTACLES IN COMPASS SURVEYING.*

A.-PROBLEMS IN RUNNING LINES.

293. MANY of the obstacles that occur in angular surveying have already been alluded to. These, and all others which the operator will meet with, may be overcome by the principles of Trigonometry. As, however, there is frequently a choice in the means to be used, the following methods are given, as being perhaps the most simple:

294. Problem 1.--To run a line making a given angle with a given line from a given point within it.

Place the instrument at the point, and sight along the line. Turn the plate the required number of degrees, and the sights or telescope will be in the required line.

Many more such methods may be found in Gillespie's "Land Surveying."

295. Problem 2.-To run a line making a given angle with a given inaccessible line at a given point in that line.

Run CE, making ACE = CAB: CE will then be parallel to AB. Now, if we suppose AE to be drawn, we shall have in the triangle ACE all the angles and side AC to find CE. Lay off this distance from C to E, and run the line EF towards A.

If A cannot be seen from E, calculate CEF, and run the line from E, making the proper angle with CE.

Problem 3.-From a given point out of a line, to run a line making a given angle with that line.

296. Where the line is accessible.

If the compass is used. Take the bearing of the given line. Then place the compass at the given point, and set it to same bearing. Deflect the compass the number of degrees required, and run the line.

A), and CB or CB' will be the line required.

In all cases, unless the line is to be a perpendicular, there will be two lines that will answer the conditions.