When they run from different points, towards the same point, add them together, and take the supplement of the sum.

When they run from different points, towards different points, subtract the less from the greater, and take the supplement of the remainder.

Note.-When the bearing of one of the lines is given towards the station, instead of from it, take the reverse bearing of such line; the angle may then be found by the above rule.

EXAMPLES.

1. Given the bearing of the line AB, Fig. 67, N. 34° E., and AD, N. 58° E.; required the angle A.

AD, N. 58° E.

AB, N. 34° E.

Angle A=24°

2. Given the bearing of BA, Fig. 57, S. 34° W., and BC, S. 35° E.; required the angle B. Ans. B=69°

3. Given the bearing of BC, Fig. CD, S. 87° W.; required the angle C.

67, S. 35° E., and

Ans. 58°.

4. Given the bearing of DC, Fig. 67, N. 87° E., and

DA, S. 58° W.; required the angle D. Ans. 151°

PROBLEM IX.

To change the bearings of the sides of a survey in a corresponding manner, so that any particular one of them may become a Meridian.

RULE.

Subtract the bearing of the side that is to be made a meridian, from those bearings that are between the same points that it is, and also from those that are between points directly opposite to them. If it is greater than either of the bearings from which it is to be subtracted, take the difference, and change E. to W., or W. to E.

Add the bearing of the side which is to be made a meridian, to those bearings which are neither between the same points that it is, nor between the points that are directly opposite to them. If either of the sums exceeds 90°, take the supplement and change N. to S., or S. to N.*

Note. When the bearings of some, or all, of the sides of a survey have been thus changed, and by calculation the changed bearing of another side or line has been

*The changing of the bearings so as to make a given side become a meridian, may be illustrated by means of a protracted survey. If a protracted survey or plot is held horizontally, with the meridian in a north and south direction, the north end being towards the north, the bearings of the sides of the plot will then correspond with the bearings of the sides of the survey. If then, keeping the paper horizontal, it be turned round till any particular side of the plot has a north and south direction, or becomes a meridian, the bearings of all the other sides of the plot will have been changed by a like quantity. But it is evident, that neither the relation of the different parts of the plot to one another, the area nor the lengths of the sides will have been altered by this change. It may be here observed, that some calculations in surveying are considerably shortened by changing the bearings so as to make a certain side become a meridian. The method was communicated to me by Robert Patterson, late Professor of Mathematics and Natural Philosophy

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. 51° W. Required the changed bearings, so that the 3th side may be a meridian.

2. Given the following bearings of the sides of a survey; 1st. S. 40° E.; 2d. N. 54° E.; 3d. N. 291° E.; 4th. N. 281° 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. 85° E.; 2d. North; 3d. N. 241° W., 4th. N. 25° W.; 5th. S. 69° W.; 6th. S. 7° E.

3. Given the bearings as in the 1st. example; viz. 1st. S. 45° W.; 2d. N. 50° W.; 3d. North; 4th. N. 85° E.; 5th. S. 47° E.; 6th. S. 20° W; 7th. N. 514° 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. 20° W.; 4th N. 64° E.; 5th. S. 67° E.; 6th. South; 7th. N. 713° W.

PROBLEM X.

Of the bearing, Distance, Difference of Latitude and Departure, 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 deAns. Diff. lat. 7.91 N.: dep. 10.63 E.

parture.