In applying this principle in surveying, S, as before, may represent any station, and the line AB may be a fence, or any other real or imaginary line.

In "Compass Surveying," it is a north and south line, the direction of which is given by the magnetic needle of the compass.

In Geography, this principle is employed to determine the rela tive positions of places, by "Bearings and distances"; as when we say that San Francisco is 1750 miles nearly due west from St. Louis; the word "west" indicating the direction, or angle which the line joining the two places makes with a north and south line, and the number of miles giving the length of that line.

In Analytical Geometry, the line AS, and the angle BAS, are called "Polar Co-ordinates."

(8) Fourth Method. By measuring the angles made with a given line by two other lines starting from given points upon it, and passing through the required point.

Thus, in Fig. 4, the point S is determined by being in the intersection of the two lines AS and BS, which make respectively angles of a half and of a third of a right angle with the line AB, which A

Fig. 4.

S

is one inch long; for, the place of the point could be found, if lost, by drawing from A and B lines making with AB the known angles. In Geography, we might thus fix the position of St. Louis, by saying it lay nearly due north from New-Orleans, and due west from Washington.

In Analytical Geometry, these two angles would be called "Angular Co-ordinates."

(9) In Fig. 5, are shown together all the measurements necessary for determinng the same point S, by each of the four preceding methods. In the First Method, we measure the distances AS and A

Fig. 5.

B

C

BS; in the Second Method, the distances AC and CS, the latter at right angles to the former; in the Third Method, the distance

AS, and the angle SAB; and in the Fourth Method, the angles SAB and SBA. In all these methods the point is really determined by the intersection of two lines, either straight lines or ares of circles. Thus, in the First Method, it is determined by the intersection of two circles; in the Second, by the intersection of two straight lines; in the Third, by the intersection of a straight line and a circle; and in the Fourth, by the intersection of two straight lines.

(10) Fifth Method. By measuring the angles made with each other by three lines of sight passing from the required point to three points whose positions are known.

Thus, in Fig. 6, the point S is determined by the angles, ASB and BSC, made by the three lines SA, SB and SC.

Geographically, the position of Chicago would be determined by three straight lines passing from it to Washington, Cincinnati, and Mobile, and mak

Fig. 6.

A

B

ing known angles with each other; that of the first and second lines being about one-third, and that of the second and third lines, about one-half of a right angle.

From the three lines employed, this may be named the Method of Trilinear co-ordinates.

Fig. 7.

(11) The position of a point is sometimes determined by the intersection of two lines, which are themselves determined by their extremities being given. Thus, in Fig. 7, the point S is determined by its being situated in the intersection of AB and CD. This method is sometimes employed to fix the position of a Station on a Rail-Road

line, &c., when it occurs in a place where

B

a stake cannot be driven, such as in a pond; and in a few other cases; but is not used frequently enough to require that it should be called a sixth principle of Surveying.

(12) These five methods of determining the positions of points, produce five corresponding systems of Surveying, which may be named as follows:

I. DIAGONAL SURVEYING.

II. PERPENDICULAR SURVEYING.
III. POLAR SURVEYING

IV. TRIANGULAR SURVEYING.

V. TRILINEAR SURVEYING.

(13) The above division of Surveying has been made in harmony with the principles involved and the methods employed.

The subject is, however, sometimes divided with reference to the instruments employed; as the chain, either alone or with crossstaff; the compass; the transit or theodolite; the sextant; the plane table, &c.

(14) Surveying may also be divided according to its objects.

In Land Surveying, the content, in acres, &c., of the tract surveyed, is usually the principal object of the survey. A map, showing the shape of the property, may also be required. Certain signs on it may indicate the different kinds of culture, &c. This land may also be required to be divided up in certain proportions; and the lines of division may also be required to be set out on the ground. One or all of these objects may be demanded in Land Surveying.

In Topographical Surveying, the measurement and graphical representation of the inequalities of the ground, or its "relief," i. e. its hills and hollows, as determined by the art of "Levelling," is the leading object.

In Maritime or Hydrographical Surveying, the positions of rocks, shoals and channels are the chief subjects of examination. In Mining Surveying, the directions and dimensions of the subterranean passages of mines are to be determined.

(15) Surveying may also be divided according to the extent of the district surveyed, into Plane and Geodesic. Geodesy takes into account the curvature of the earth, and employs Spherical Trigonometry. Plane Surveying disregards this curvature, as a needless refinement except in very extensive surveys, such as those of a State, and considers the surface of the earth as plane, which may safely be done in surveys of moderate extent.

(16) Land Surveying is the principal subject of this volume; the surface surveyed being regarded as plane; and each of the five Methods being in turn employed. For the purposes of instruction, the subject will be best divided, partly with reference to the Methods employed, and partly to the Instruments used. Accordingly, the First and Second Methods (Diagonal and Perpendicular Surveying) will be treated of under the title "Chain Surveying," in Part II. The Third Method (Polar Surveying) will be explained under the titles "Compass Surveying," Part III, and "Transit and Theodolite Surveying," Part IV. The Fourth and Fifth Methods will be found under their own names of "Triangu lar Surveying," and "Trilinear Surveying," in Parts V and VI.

(17) In all the methods of Land Surveying, there are three stages of operation:

1° Measuring certain lines and angles, and recording them; 20 Drawing them on paper to some suitable scale;

30 Calculating the content of the surface surveyed.

The three following chapters will treat of each of these topics in their turn.

CHAPTER II.

MAKING THE MEASUREMENTS.

(18) THE Measurements which are required in Surveying, may be of lines or of angles, or of both; according to the Method employed. Each will be successively considered.

MEASURING STRAIGHT LINES.

(19) The lines, or distances, which are to be measured, may be either actual or visual.

Actual lines are such as really exist on the surface of the land to be surveyed, either bounding it, or crossing it; such as fences, ditches, roads, streams, &c.

Visual lines are imaginary lines of sight, either temporarily measured on the ground, such as those joining opposite corners of a field; or simply indicated by stakes at their extremities or otherwise. If long, they are "ranged out" by methods hereafter to be described. Lines are usually measured with chains, tapes or rods, divided into yards, feet, links, or some other unit of measurement.

(20) Gunter's Chain. used in Land surveying. such chains make one mile.

This is the measure most commonly
It is 66 feet, or 4 rods long.* Eighty

Fig. 8.

It is composed of one hundred pieces of iron wire, or links, each bent at the end into a ring, and connected with the ring at the end of the next piece by another ring. are placed between the links.

Sometimes two or three rings The chain is then less liable to

This length was chosen (by Mr. Edward Gunter) because 10 square chains of 66 feet make one acre, (as will be shown in Chapter IV,) and the computation of areas is thus greatly facilitated. For other Surveying purposes, particularly for Rail-road work, a chain of 100 feet is preferable. On the United States Coast Survey, the unit of measurement (which at some future time will be the universal one) is the French Metre, equal to 3.281 feet, nearly.