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 B 5. 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 N 81° 13' E.

FIG. 113.

Now, in the triangle AB5 we have one side and the angles, AB is found to be 28.88 chains We have now the complete field

to find the remaining parts. and B5 to be 22.81 chains.

notes of the area cut off.

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 A 6 and B5 be prolonged till they meet at some point, as V, Fig. 114.

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. Then, AM and BN are easily

determined.

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

M

A

FIG. 114.

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.

SECTION II.

PUBLIC LANDS OF THE UNITED STATES.

225. Soon after the organization of the 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 the public lands, or public domain. These lands were at first parceled 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 area. Through many years of labor and experiment, the present admirable system has been wrought out.

226. This system is briefly that the territory to be surveyed shall be divided by true north and south, and east and west lines, into tracts, each of six miles square and containing as near as may be 23,040 acres, called Townships; each township into thirty-six tracts, each of one mile square and containing as near as may be 640 acres, called Sections; and each section into halves, quarters, and smaller portions, as may be deemed expedient.

227. In the survey, all primary lines north and south are

run on meridians of longitude, and all primary lines east and west are laid out on perpendiculars to the meridians.

It is of the first importance, then, that meridians of longitude should be accurately determined, and that lines perpendicular to them should be determined with equal precision.

The ordinary Surveyor's Compass is, for several reasons, not sufficiently accurate for the work of running out the standard and township lines, and such lines must be run by Burt's Solar Compass," or other instrument of equal utility."

228. The Solar Compass for determining a true meridian was invented by William A. Burt, of Michigan, and patented by him in 1836. It has been improved, by him and others, from time to time since, and in its present state is represented and described in Appendix A.

229. In commencing the division of the public lands in unsurveyed territory, an initial point is selected, with reference to its convenience in making the survey, perpetuated by a substantial monument suitably marked, and its true position in latitude and longitude determined.

230. From the initial point the principal base line is run out due cast and west with the solar compass, and permanently marked at each 40 chains, or half mile, with a quarter-section corner, and at each 80 chains, or mile, with a section corner.

231. The principal meridian is then run out due north and south from the initial point, and also marked with monuments at intervals, like the base line.

232. As meridians of longitude converge toward the poles, the distance between two such meridians decreases as the surveyor goes north. To counteract the error that would otherwise result from the convergency of meridians, and also to arrest

error arising from inaccuracies in measurements on meridian lines, standard parallels, or correction lines, are run east and west from the principal meridian and at stated intervals. On the north of the principal base-line, about latitude 35° north, these standard parallels are, in general, run at distances of every four townships, or twenty-four miles, and south of the principal base, at distances of every five townships, or thirty miles. Each of these standards is run out and marked in the same way as the principal base, and forms the base for laying out the townships north to the next standard parallel. The standards are numbered according to their position with respect to the principal base-line, as 1st Standard Parallel South, 2d Standard Parallel South, 1st Standard Parallel North, &c.

233. The principal meridian, the base-line, and the standard parallels having been first run, measured, and marked, and the corner boundaries thereon established, the exterior lines of townships are then run, measured, and marked.

The townships, consisting of a series of townships lying along a parallel, are numbered north or south of the principal base; the first series north of the base being Township 1 North, the second series north being Township 2 North, &c.; and the first series south being Township 1 South, &c.; these are designated T. 1 N., T. 2 N., T. 1 S., &c.

The ranges, consisting of tiers of townships, are numbered from the principal meridian both ways; the first tier west of the meridian being Range 1 West, the first tier east being Range 1 East, &c.; designated R. 1 W., R. 1 E., &c.

234. The accompanying map, from the U. S. General Land Office, representing a considerable portion of the State of Arkansas, will serve for illustration.

The principal meridian in this survey is called the 5th