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CHAPTER II.

On supplying omissions in the dimensions of a survey.

When the bearings and distances of all the sides of a survey are known, except one bearing and one distance, or two bearings, or two distances, these can be obtained by calculation, provided those that are known can be depended on, as sufficiently accurate. This may sometimes be necessary when there are obstacles in the way of obtaining one or two of the bearings or distances; or when, after they have all been taken on the ground, the notes of one or two of them happen to be obliterated. As, however a bearing, or distance thus obtained, must be affected by any error or errors that may have been made in taking the others, it is better, when practicable, to have the bearings and distances of all the sides, as taken on the ground.

PROBLEM I.

The bearings and distances of all the sides of a tract of

land, except the bearing and distance of one side, being given, to find these.

RULE.

Find by prob. 11, of the preceding chapter, the differences of latitude and the departures for the sides whose bearings and distances are given, and place them in their proper columns in a table ruled for the purpose : Add up the northings and southings, and taking the dif

ference of their sums, place it opposite the unknown side, in the column whose sum is the least. The sums of the two columns will then be equal. This is called balancing the latitudes. Do the same with the eastings and westings. The two numbers inserted to make the latitudes and the departures balance, will be the difference of latitude and the departure of the unknown side ; with which its bearing and distance may be found, by prob. 10, of the preceding chapter.

Note 1.-By the application of this rule, the bearing and distance of a line joining two corners or stations, may be found, when there are obstacles in the way which prevent our going directly from one corner to the other, or when one cannot be seen from the other. To do this, let one or two, or more stations, if necessary, be taken out of the line, and take the bearing and distance from the first corner to the first assumed station ; from this station to the second ; and so on, to the second corner. Then considering these bearings and distances, as the bearings and distances of the sides of a survey, the required bearing and distance of the line may be found by the above rule. The bearing thus found must be reversed, in order to have the bearing from the first corner to the sccond

2. In the same way the bearing and distance of a straight road to run between two given places, may be found, by taking the several bearings and distances of the old road if there is one; or of lines joining assumed stations and extending from one of the places to the other.

EXAMPLES.

1. The bearings and distances of the side of a tract of

are not known, are as in the following field-notes; re quired the unknown bearing and distance.

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6 S. 200 W.23.80

22.29

8.33

7 N. 511 W.26.47 16.56

20.65

52.70 52.70 56.72 156.72

As diff. of lat. : dep. .

rad.

19.79 S.
21.20 E.

Ar. Co. 8.70355

1.32674 10.00000

: tang. bear. S. 47° E.

- 10.03029

As rad.
: sec. bearing
: . diff. lat.

47°
19.79

10.00000
10.16622
1.29645

: dist.

29.02

1.46267

Ans. S. 47° E. 29.02 ch.

1

2. Given the bearings and distances of the sides of a tract of land, as follow : 1st. N. 15% W. 9.40 ch.; 2d N. 634 E. 10.43 ch.; 3d. S. 49° E. 8.12 ch.; 4th. S. 13E. 8.45 ch.; 5th. S. 16% E. 6.44 ch.; 6th. Unknown; 7th. N. 60° W. 9.72 ch.; 8th. N. 174 E. 7.65 ch.; required the bearing and distance of the 6th. side.

Ans. S. 60° 8' W. 12.27 ch. 3. One side of a tract of land of which a survey is to be taken, passes through a pond. Two stations are therefore taken on one side of the pond as represented in Fig. 80. The bearings and distances from the first end of the side to the first station, from that to the second, and thence to the other end of the side are; 1st. S. 52° W. 10.70 ch.; 2d. S. 74° W. 13.92 ch.; and 3d. S. 34; E. 9 ch. Required the bearing and distance of the side.

Ans. S. 10° 33' W. 28.31 ch. 4. Given the bearings and distances of an old road, running between two places, as follow ; 1st. S. 10° E. 92.20 ch.; 2d. S. 15° W. 120.50 ch. ; 3d. S. 181 W. 205. ch. ; 4th. S. 711 E. 68 ch. Required the bearing and distance of a straight road, that shall connect the two places.

Ans. S. 2° 8' W. 423.47 ch.

PROBLEM II.

Given all the bearings and distances of the sides of a sur

vey, except the distances of two sides, to find these.

RULE.

By prob. 9, of the preceding chapter, change all the given bearings, in a corresponding manner, so that one of the sides whose bearings only are given, may become a meridian. With the changed bearings and given distances find the corresponding differences of latitude, and the departures. Add up the eastings and westings,

departure of that unknown side, which is not made a meridian. With this departure and the changed bearing, find by prob. 10, of the preceding chapter, the distance and difference of latitude of this side, which place in their proper

columns. Now add up the northings and southings, and take their difference, which will be the distance of the side made a meridian.*

EXAMPLES.

Given the following bearings and distances of the sides of a survey; 1st. S. 454 W. 15.16 ch. ; 2d. N. 50° W. 22.10 ch.; 3d. North 18.83 ch.; 4th. N. 85° E. 35.65 ch.; 5th. S. 47° E. dist. unknown; 6th. S. 20: W. dist. unknown; 7th. N. 51+ W. 26.47 ch. to the place of beginning. Required the unknown distances.

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* The reason of the rule is obvious. For as the side made a meridian has no departure, the difference of the sums of the departures, must be the departure of the other unknown side. And when the difference of latitude of this side has been found and placed in its proper situation, the difference of the sums of the latitudes must evidently be the difference of latitude of the side made a meridian; or which, in this case, is the same thing, its distance.

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