Library. Of Califor LAND-SURVEYING PART I. GENERAL PRINCIPLES AND FUNDAMENTAL OPERATIONS. CHAPTER I. DEFINITIONS AND METHODS. (1) SURVEYING is the art of making such measurements as will determine the relative positions of any points on the surface of the earth; so that a Map of any portion of that surface may be drawn, and its Content calculated. (2) The position of a point is said to be determined, when it is known how far that point is from one or more given points, and in what direction there-from; or how far it is in front of them or behind them, and how far to their right or to their left, &c; so that the place of the first point, if lost, could be again found by repeating these measurements in the contrary direction. The "points" which are to be determined in Surveying, are not the mathematical points treated of in Geometry; but the corners of fences, boundary stones, trees, and the like, which are mere points in comparison with the extensive surfaces and areas which they are the means of determining. In strictness, their centres should be regarded as the points alluded to. (3) A straight Line is "determined," that is, has its length and its position known and fixed, when the points at its extremities are determined; and a plane Surface has its form and dimensions determined, when the lines which bound it are determined. Consequently, the determination of the relative positions of points is all that is necessary for the principal objects of Surveying; which are to make a map of any surface, such as a field, farm, state, &c., and to calculate its content in square feet, acres, or square miles. The former is an application of Drafting, the latter of Mensuration. (4) The position of a point may be determined by a variety of methods. Those most frequently employed in Surveying, are the following; all the points being supposed to be in the same plane. (5) First Method. By measuring the distances from the re quired point to two given points. Thus, in Fig. 1, the point S is "determined," if it is known to be one inch Fig. 1. SQ from A, and half an inch from B: for, its place, if lost, could be found by describing two arcs of circles, from A and B as centres, and with the given distances as radii. intersection of these arcs. A B The required point would be at the In applying this principle in surveying, S may represent any station, such as a corner of a field, an angle of a fence, a tree, a house, &c. If then one corner of a field be 100 feet from a second corner, and 50 feet from a third, the place of the first corner is known and determined with reference to the other two. There will be two points fulfilling this condition, one on each side of the given line, but it will always be known which of them is the one desired. In Geography, this principle is employed to indicate the posi tion of a town; as when we say that Buffalo is distant (in a straight line) 295 miles from New-York, and 390 from Cincinnati, and thus convey to a stranger acquainted with only the last two places correct idea of the position of the first. 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 dis tance 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 a third of a right angle, and 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 AAB, and measuring on it the given distance. Fig. 3. B In applying this principle in surveying, S, as before, may repro sent 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 direc tion 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 lino 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. 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. 66 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 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 deter mined 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. 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. (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, 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. |