two courses run in the ordinary way, as we go around the field, the bearing of one of them must be reversed before the calculation for the angle is made.

1. The bearings of two courses, from the same point, are N 37° E, and S 85° W: what is the angle included between them?

Ans. 132°.

2. The bearings of two adjacent courses, in going round a piece of land, are N 39° W, and S 48° W: what is the angle included between them?

Ans. 87°.

3. The bearings of two adjacent courses, in going round a piece of land, are S 85° W, and N 69° W: what is the angle included between them?

Ans. 154°.

4. The bearings of two adjacent courses, in going round a piece of land, are N 55° 30' E, and S 69° 20′ E: what is the angle included between them?

OF DIVIDING LAND.

Ans. 124° 50'.

that it is difficult

It is by practice branch of survey

38. Fields are so variously shaped to give rules that will apply to all cases. alone that facility is obtained in that ing relating to the division of estates. We shall add only a few examples that may serve as general guides in the application of the principles of Plane Geometry to such cases as may arise.

I. To run a line from the vertex of a triangular field which shall divide it into two parts, having to each other the ratio of M to T.

39. Let ABC be any triangular field.

Divide the side BC into two

parts, such that (Geom., Bk. IV., Prob. 1.)

BD : DC :: m : and draw the line AD:

n;

B

For, the two triangles ABD, ADC having the same altitude are to each other as their bases (Geom., Bk. IV., P. 6, C.): hence, the triangle is divided into parts having the ratio of m to n.

II. To run a line parallel to one side of a triangular field, that shall form with the parts of the two other sides a

m

triangle equivalent to the part of the field.

n

40. Let CBA represent a triangular field and CA the side parallel to which the dividing line is to be drawn.

On the side BC describe a semicircle then divide BC

G

B

1

at D, so that

BD : BC :: m : n.

E

A

At D, erect the perpendicular DG to the diameter BC, and draw BG. Then, with B as a centre, and BG as a radius, describe the arc of a circle cutting BC at E. Through E draw EF parallel to CA, and it will divide the triangle in the required ratio.

For, (Geom., Bk. IV., P. 23.)

or,

2

BE

2

BD

Boi whence,

: BC :: BD : BC :: m : n.

But, since the triangles BEF, BCA are similar,

REMARK. The points E and F may easily be found

by computation.

m

For, since BE=BCX BD, and BD=

× BC,

Let it be required to divide the triangular field CAB, in which AC= 9 ch. AB= 11 ch. and CB=7 ch. into two such parts that ADE shall be one-fourth of the whole field.

In this case, we have

hence,

AE-4 ch. 50 1. and AD=5 ch. 50 1.

III. To run a line from a given point in the boundary of a piece of land, so as to cut off, on either side of the line, a given portion of the field.

41. Make a complete survey of the field, by the rules already given. Let us take, as an example, the field whose area is computed at page 118. That field contains 1044 1R 16P, and the following is a plot of it.

Let it now be required to run a line from station A, in such a manner as to cut off on the left any part of the

field; say,

26A 2R 31P.

9

It is seen, by examining the field, that the division line will probably terminate on the course CD. a line from A to C, which we will call line.

Therefore, draw the first closing

The bearings and lengths of the courses AB, BC, are always known; and in the present example are found in the table on page 118: hence, the bearing and distance from C to A, can be calculated by Art. 35: they are in this example,

Bearing S 9° 28' E: Course 22.8 ch.

Having calculated the bearing and length of the closing line, find, by the general method, the area which it cuts off: that area, in the present case, is.

13A 3R 3P.

It is now evident that the division line must fall on the right of the closing line AC, and must cut off an area ACH, equal to the difference between that already cut off, and the given area: that is, an area equal

to

26A 2R 31P given area,

13A 3R 3P area already cut off,

12A 3R 28P.

Since the bearing of the next course CD, and the bearing of the closing line AC are known, the angle ACD which they form with each other, can be calculated, and is in this example 80° 32°. Hence, knowing the hypothenuse AC, and the angle ACG at the base, the length AG of the perpendicular let fall on the course CD, can be found, and is 22.49 chains.

Since the area of a triangle is equal to its base multiplied by half its altitude, it follows, that the base is equal to the area divided by half the altitude. Therefore, if the

rea

12A 3R 28P

be reduced to square chains, and divided by 11.244 chains, which is half the perpendicular AG, the quotient, which is 11.58 chains, will be the base CH. Hence, if we lay off

then run the line AH, it will cut off from the land the required area.

REMARK I. If the part cut off by the first closing line, should exceed the given area, the division line will fall on the left of AC.

REMARK II. If the difference between the given area and the first area cut off, divided by half the perpendicular AG, gives a quotient larger than the course CD; then, draw a line from A to D, and consider it as the first closing line, and let fall a perpendicular on DE.

REMARK III. When the point from which the division line is to be drawn, falls between the extremities of a course, dividing the course into two parts, consider one of the parts as an entire course, and the other as forming a new course, having the same bearing. The manner of making the calculation will then be the same as before.

SECTION IV.

PUBLIC LANDS-VARIATION OF THE NEEDLE.

1. Soon after the organization of the present government, several of the states ceded to the United States large tracts of wild land, and these, together with the lands since acquired by treaty and purchase, constitute what is called the public lands, or public domain. Previous to the year 1802, these lands were parcelled out without reference to any general plan, in consequence of which the titles often. conflicted with each other, and in many cases, several grants covered the same premises.

In the year 1802, the following method of surveying the public lands, was adopted by Colonel Jared Mansfield, then surveyor-general of the North-Western Territory.

2. The country to be surveyed is first divided by