SECTION VI.

TIE-LINES.

258. Tie-Lines. The external boundaries of a tract of land having more than three sides are not sufficient either for making a plat or calculating the area. In the methods heretofore laid down, diagonals were also used. In some cases, however, owing to obstructions, such as ponds, close woods, or buildings, it is difficult to run the diagonals. When this is the case, a line measured across one of the angles of a quadrilateral will determine the direction of two sides, and thus fix the relative position of all the lines of the tract. Such lines are called tie-lines. For example, suppose it were required to survey the tract represented in Fig. 113, the interior of which is filled with such thick woods that the diagonals cannot be measured: the external lines AB, BC, CD, and DA might be measured as before. Then on the lines adjacent to one angle, as C, measure carefully

D

A

Fig. 113.

B

C

E

E

CE and CF; also measure EF. These measures should be made with the greatest accuracy, as a slight error here will very materially affect the result. On the same account, the distances CE and CF should be taken as large as circumstances will allow.

If the tie-line cannot be run within the tract, the points may be taken at E and F in the sides produced.

To plat such a tract, commence with the triangle. This being formed, the direction of CB and CD is known.

259. To calculate the Area. First find in ECF the angle ECF, whence by trigonometry BD is found, and then the area of the triangles.

CF, EF will be the chord of the arc to the

EF

radius CE, whence the chord to radius 1

=

This

EC

quotient being found in the table of chords the corresponding arc will give the degrees and minutes of the angle ECF: or CE EF:: rad. sin. ECF.

:

260. Inaccessible Areas. By a combination of tie-lines and offsets, tracts that cannot be entered, such as a pond or a swamp, may be measured. For this purpose, surround the tract by a system of lines bound at the angles by tielines, and take offsets to the prominent points in the boundary of the tract.

261. Defects of this Method. Every system of measurement or drafting should commence with the longer lines and end with the shorter. By this means the errors that are unavoidable are diminished as we proceed. If, for example, a diagonal of thirty chains were measured, this would fix the distance of the ends to a degree of certainty precisely equal to that of the measurement; and if from this measurement the length of an inferior line joining two points in the sides were to be determined, the errors in the length of the diagonal would affect this length to a degree exactly proportional to its length, the error in a line of five chains long being one-sixth of that of the diagonal. Precisely the reverse is the case when the shorter line is measured: the error is magnified as we proceed. On this account, the method explained above should never be employed when it can be avoided. By the use of the compass, transit, or theodolite, this can always be done. The mode of using them for surveying purposes forms the subject of the next chapter.

CHAPTER V.

COMPASS SURVEYING.

SECTION I.

DEFINITIONS AND INSTRUMENTS.

262. IN chain surveying, the position of any point is determined either by directly measuring to it from other known points, or by determining its distance from such points by the indirect methods explained in last chapter. In the method about to be explained, its position is ascertained by angular measurements taken from known stations, or by its distance from a known point and the angle which it makes with the meridian.

All those methods, which have a direct reference to the meridian as the base of angular distance, are known under the head of compass surveying; whether the instrument used to determine the angle is a theodolite, a transit, or a

compass.

263. The Meridian. If the heavens are examined during a clear night, the stars to the north will be perceived to revolve around a star elevated about 40°. This is called the pole-star, and is very nearly in the point in which the axis of the earth if produced would meet the heavens. This point is called the north pole of the heavens. The north star is not exactly at the pole, but revolves around it in a small circle. If a transit or theodolite be levelled, and the telescope directed to the centre of this circle (see chap. ix.) it will point exactly north. Depress it, and run

160

out a line in the direction of the line of collimation. This will be a meridian line.

264. The Points of the Compass. If through any station a line be drawn perpendicular to the meridian it will run east and west. If we face the south, the west will be to the right hand and the east to the left. These four pointsnorth, east, south, west-are called the cardinal points of the compass, and are used as reference for all angular distances from the meridian.

For nautical purposes, each of the quadrants into which the horizon is divided is further divided into eight parts called points, and named as in Fig. 114, commencing at the north and going to the east.

North, N.; North by East, (N.6E.;) North Northeast, (N.N.E.;) Northeast by North, (N.E.6N.;) Northeast, (N.E. ;) Northeast by East, (N.E.6E.;) East Northeast, (E.N.E. ;) East by North, (E.6N.;) East, (E.) and so on, E.bS.; E.S.E.; S.E.6E.; S.E.; S.E.6S.; S.S.E.; S.6E.; S.

For land surveying only the cardinal points are mentioned, the direction being determined by the angular distance from the meridian.

265. Bearing. The bearing of a line is the angle which it makes with a meridian through one end. It is expressed either by naming the points, as N.¿E., S.S.E. 1 E., as is

done in navigation, or by mentioning the number of degrees in the angle accompanied by the cardinal points between which it runs. Thus, if a line runs between north and west and makes an angle of 37° 25' with the meridian, its bearing is N. 37° 25' W. It deflects 37° 25' from the north towards the west, and is therefore sometimes said to run from north towards the west. This expression, though convenient, is not strictly correct.

266. The Reverse Bearing. If the bearing of a line of moderate length is determined at one end, and then again at the other end, the latter is called the reverse bearing. It will be found to be of the same number of degrees as the bearing, but with the opposite points. Thus, if the bearing of a line be N. 271° E, its reverse bearing is S. 271° W.

If the line be long, there will be a continual variation from the initial course. Thus, if a line run N. 45° E. through its whole course, it will be found to deviate to the left from a straight line. A true east and west line in latitude 40° is a curve with a radius of about 4800 miles.

267. The Magnetic Needle. A magnetic needle is a light bar of magnetized steel suspended on a pivot, so that it may turn freely in a horizontal direction. Such a needle will always place itself in nearly the same direction, one end of it being northward and the other southward. The needle should move very freely on its pivot, so that it may always assume its proper position. The pivot should therefore be of very hard steel ground to a fine point. In the centre of the needle there should likewise be a cup of agate or some other hard material inserted for it to rest upon.

As the needle is generally balanced before being magnetized, the north end in northern latitudes will always "dip" after the magnetic force has been communicated to it. To restore the balance, a coil of fine brass wire is wrapped around the south end. This may be slipped along the bar so as perfectly to restore the balance. It serves also to distinguish the two ends of the needle.

A good needle will vibrate for a considerable time after