NOTE 3.-When the point from which the division line is to be drawn falls between the extremities of a course, divide the course into two parts, at this point. Then 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.

PROBLEM VI.

109. To cut off from a field, a given area, by a line running in a given direction.

In this case, as in the previous one, a complete and correct survey is first necessary. Then, when the whole area is known, the position of the line may be approximately determined by the inspection of a correct map of the whole.

We will take, for illustration, Example 4, page 138, of which the plot is on the next page.

Let it be required to cut off from this area, 50 acres, by a line whose bearing shall be S 60° E, or N 60° W.

We will make a trial of a line starting at 25 chains from station 6, on the 6th course. We will call this station A,

and the trial line AB.

In order to determine if the area cut off is equal to the required area, we must first determine the length of AB and of B5. These cannot be determined by the method of supplying lost notes.

We must first calculate the length of a line, starting at the proposed point, and running to the station nearest to the other extremity of the closing line. In this example, from A to 5. This is easily found to be 36.406 chains, and its bearing

Now, in the triangle AB5 we have one side and the angles, to find the remaining parts. AB is found to be 28.88

It now remains to move this line northerly, so that the area contained between its present position and the new one shall be equal to 8.5029 acres.

Suppose the lines A6 and B5 be prolonged till they meet at some point, as V.

Calculate AV and BV, also the area ABV. AV is found to be 92.19 chains and BV 88.18 chains. The area of the triangle ABV, is 127.29 acres. Let MN represent the line sought. Then, we have two similar triangles, with all the sides of the one, and the areas of each, known; for, VMN must contain. 8.5029 acres less than AVB.

V

M

N

Then,

A

B

AM and BN are easily determined.

The complete notes of the area to be cut off, are

NOTE.-Fields are so variously shaped that it is difficult to give rules that will apply to all cases. It is by practice alone that facility is obtained in that branch of surveying relating to the division of estates. We have given 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.

PUBLIC LANDS.

110. 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

general plan, in consequence of which the titles often conflicted with each other, and in many cases, several grants covered the same area.

In the year 1802, the following method of surveying the public lands, was adopted by Colonel Jared Mansfield, then Surveyor-General of the Northwestern Territory.

111. The country to be surveyed is first divided, by meridians, six miles distant from each other; and then again, by a system of east and west lines, also six miles from each other. The country is thus divided into equal squares, which are called townships. Hence, each township is a square, six miles on a side, and contains thirty-six square miles.

112. For the purpose of illustration, we have obtained from the general land-office the accompanying map, which represents a considerable portion of the State of Arkansas.

The principal meridian in this survey is called the 5th meridian, and passes through the point of junction of the White river with the Mississippi. The principal base-line, running east and west, intersects this meridian a little to the east of White river; and from the meridian and baseline, reckoned from this point of intersection, all the ranges of townships are laid off.

For example, 1 North, will apply to all the townships lying in the first row north of the base-line: 1 South, will apply to all the townships in the first row south of the base-line. Range 1 East, will apply to all the townships lying in the first row, east of the 5th meridian: and Range 1 West, will apply to all lying in the first row to the west of it. The small figures designate the rows of townships, reckoned north and south from the base-line, and the ranges reckoned east and west from the 5th meri