In Analytical Geometry, the lines AS and BS are known as "Focal Co-ordinates," the general name "co-ordinates" being applied to the lines or angles which determine the position of a point.

(6) Second Method. By measuring the perpendicular distance from the required point to a given line, and the distance thence along the line to a given point.

Thus, in Fig. 2, if the perpendicular distance SC be half an inch, and CA be one inch, the point S is "determined": for, its place could be again found by measuring one inch from A to C, and half an inch from C, A at right angles to AC, which would fix the point S.

Fig. 2.

The Public Lands of the United States are laid out by this method, as will be explained in Part XII.

In Geography, this principle is employed under the name of Latitude and Longitude.

Thus, Philadelphia is one degree and fifty-two minutes of longitude east of Washington, and one degree and three minutes of latitude north of it.

In Analytical Geometry, the lines AC and CS are known as "Rectangular Co-ordinates." The point is there regarded as determined by the intersection of two lines, drawn parallel to two fixed lines, or "Axes," and at a given distance from them. These Axes, in the present figure, would be the line AC, and another line,perpendicular to it and passing through A, as the origin.

(7) Third Method. By measuring the angle between a given line and a line drawn from any given point of it to the required point; and also the length of this latter line.

Thus, in Fig. 3, if we know the angle BAS to be half a right angle, and the line AS to be one inch, the point S is determined; for, its place could be found by drawing from A, a line making the given angle with AB, and measuring on it the given distance.

Fig. 3.
SØ

A

B

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 relative 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 re-
spectively angles of a half and of a third
of a right angle with the line AB, which A

B

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 determining the same point S, by each of the four preceding methods. In the First Method, we measure the distances AS and A

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 arcs 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

A

Fig. 6.

B

C

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.

S

Fig. 7.

D

(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 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. It is said to be employed

B

by smugglers to fix the spot where they have sunk contraband goods.

(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 "Triangular 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.