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found, its true bearing will be obtained by applying to the changed bearing, the bearing of the side which was made a meridian, in a contrary manner to what is directed in the rule ; that is, by adding in the case in which the rule directs to subtract, and by subtracting in the case in which it directs to add.

EXAMPLES.

1. Given the bearings of the sides of a survey as follow ; 1st. S. 45° W.; 2d. N. 50° W., 3d. North ; 4th. N. 85° E.; 5th. S. 47° E.; 6th. S. 201° W.; and 7th. N. olto W. Required the changed bearings, so that the 3th side may be a meridian.

1st. S. 45° W.

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2. Given the following bearings of the sides of a survey ; 1st. S. 404° E.; 2d. N. 54° E. ; 3d. N. 291° E.; 4th. N. 28° E.; 5th. N. 57° W.; and 6th. S. 47° W.; to find the changed bearings so that the 2d. side may be a meridian. Ans. 1st. N. 854° E.; 2d. North; 3d, N. 24° W., 4th. N. 251° W.; 5th. S. 69° W.; 6th. S. 7° E.

3. Given the bearings as in the 1st. example ; viz. 1st. S. 451° W.; 2d, N. 50° W.; 3d. North; 4th, N. 85° E.; 5th. S. 47° E. ; 6th. S. 20° W ; 7th. N. 51to W.; to find the changed bearings so that the 6th side may be a meridian. Ans. 1st. S. 25° W.; 2d. N. 70° W.; 3d. N. 201° W.; 4th N. 641° E.; 5th. S. 673° E. ; 6th. South ; 7th. N. 713° W.

PROBLEM X.

Of the bearing, Distance, Difference of Latitude and De

parture, any two being given, to find the other two.

RULE.

When the bearing and distance are given. As Rad. : cos. of bearing :: distance : dif. of latitude.

When the bearing and difference of latitude are given,

As Rad. : sec. of bearing : : diff. lat. : distance.

Rad. : tang. of bearing :: diff. lat. : departure.

When the bearing and departure are given.

As Rad. : cosec. of bearing :: departure : distance.

Rad. : cotang. of bearing : : departure : diff. lat.

When the difference of latitude and the departure are given.

As diff. lat. : departure : : rad. : tang. of bearing.

Rad. : sec. of bearing : : diff. lat. : distance.

When the distance and difference of latitude are given.

As Diff. lat. : distance : : rad. : sec. of bearing.

Rad. : tang. of bearing :: diff. lat. : departure.

When the distance and departure are given.

As Distance : departure :: rad. : sin. of bearing.

Rad. : cos. of bearing :: distance : diff. lat.

Note.It is evident the above proportions are the solutions of a right-angled triangle, having for its sides the distance, difference of latitude, and departure.

EXAMPLES.

1. Given the bearing of a line, N. 53° 20' E., distance 13.25 ch.; to find the difference of latitude and the departure. Ans. Diff. lat. 7.91 N.: dep. 10.63 E.

2. Given the bearing of a line, S. 32° 30' E., and the departure 10.96 ch. to find the distance and difference of latitude. Ans. Dist. 20.40 ch.; diff. lat. 17.20 S.

3. Given the distance of a line, running between the north and east, 44 ch. and its difference of latitude 34.43 ch.; to find the bearing and departure.

Ans. Bearing, N. 38° 30 E.; dep. 27.39 ch. E.

4. The bearing of a line S. 32° 30' E., and the difference of latitude 17.21 ch. being given, to find the distance and departure. Ans. Dist. 20.41 ch.; dep. 10.96 E.

5. Given the difference of latitude of a line 27.92 N., and the departure 5.32 E.; to find the bearing and distance. Ans. Bearing, N. 10° 47' E.; dist. 28.42.

6. The distance of a line, running between the north and west, is 35.35 ch., and its departure 15.08 ch., required the bearing and difference of latitude.

Ans. Bearing N. 25° 15' W.; diff. lat. 31.97 N.

PROBLEM XI.

To find the difference of latitude and departure correspond

ing to any given bearing and distance, by means of the Traverse Table.

RULE.

When the distance is any number of whole chains or

perches, not exceeding 100. Find the given bearing at the top or bottom of the table, the given distance, found in the column of distances at the side of the table, and under or over the given bearing, is the difference of latitude and departure; which must be taken as marked at the top of the table, when the bearing is at the top; but as marked at the bottom, when the bearing is at the bottom.

When the distance is a number of whole chains or perches,

exceeding 100.

Separate the distance into parts that shall not exceed 100 each; and find, as before, the difference of latitude and departure, corresponding to the given bearing and to each of those parts; the sums of these will be the difference of latitude and departure required.

When the distance is expressed by chains or perches and

the decimal of a chain or perch.

Find, as above, the difference of latitude and departure corresponding to the given bearing and to the whole chain or perches, Then considering the decimal part as a whole number, find the difference of latitude and departure corresponding to it, and remove the decimal point in each of them, two figures to the left hand if there are two decimal figures in the distance, or one figure to the left if there is but one; then these added to the former will give the difference of latitude and departure required.

Note. When the number of whole chains or perches is less than 10, and the second decimal figure is a cipher, the difference of latitude and departure may be taken out at once, by considering the mixed number, rejecting

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