MEASURING ANGLES. (33) The angle made by any two lines, that is, the difference of their directions, is measured by various instruments, consisting essentially of a circle divided into equal parts, with plain sights, or telescopes, to indicate the directions of the two lines. As the measurement of angles is not required for "Chain Surveying," which is the first Method to be discussed, the consideration of this kind of measurement will be postponed to Part III. NOTING THE MEASUREMENTS. (31) The measurements which have been made, whether of lines, or of angles, require to be very carefully noted and recorded. Clearness and brevity are the points desired. Different methods of notation are required for each of the systems of surveying which are to be explained, and will therefore be given in their appropriate places. CHAPTER III. DRAWING THE MAP. (35) A MAP of a survey represents the lines which bound the surface surveyed, and the objects upon it, such as fences, roads, rivers, houses, woods, hills, &c., in their true relative dimensions and positions. It is a miniature copy of the field, farm, &c., as it would be seen by an eye moving over it; or as it would appear, if from every point of its irregular surface, plumb lines were dropped to a level surface under it, forming what is called in geometrical language, its horizontal projection. (36) Platting. A plat of a survey is a skeleton, or outline map. It is a figure "similar" to the original, having all its angles equal, and its sides proportional. Every inch on it represents a foot, a yard, a rod, a mile, or some other length, on the ground; all the measured distances being diminished in exactly the same ratio. PLATTING is repeating on paper, to a smaller scale, the mea surements which have been made on the ground. Its various operations may therefore be reduced, in accordance with the principles established in the Fig. 12 first chapter, to two, viz: drawing a straight line in a given direction and of a given length; and describing an arc of a circle with a radius whose length is also given. The only instruments absolutely necessary for this, are a straight ruler, and a pair of " dividers," or "compasses." Others, however, are often convenient, and will be now briefly noticed. (37) Straight Lines. These are usually drawn by the aid of a straight-edged ruler. But to obtain a very long straight line upon paper, stretch a fine silk thread between any two distant points, and mark in its line various points, near enough together to be afterwards connected by a common ruler. The thread may also be blackened with burnt cork, and snapped on the paper, as a carpenter snaps his chalk line; but this is liable to inaccuracies, from not raising the line vertically. (38) Arcs. The arcs of circles used in fixing the position of a point on paper, are usually described with compasses, one leg of which carries a pencil point. A convenient substitute is a strip of pasteboard, through one end of which a fine needle is thrust into the given centre, and through a hole in which, at the desired dis tance, a pencil point is passed, and can thus describe a circle about the centre, the pasteboard keeping it always at the proper distance. A string is a still readier, but less accurate, instrument. (39) Parallels. The readiest mode of drawing parallel lines is by the aid of a triangular piece of wood and a ruler. Let AB be the line to which a parallel is to σ Fig 13. B slide the triangle along it till the side of the triangle which had coincided with the given line, passes through the given point. This side will then be parallel to that given line, and a line drawn by it will be the line required. Fig. 14. Another easy method of drawing parallels, is by means of a T square, an instrument very valuable for many other purposes. It is nothing but a ruler let into a thicker piece of wood, very truly at right angles to it. For this use of it, one side of the cross-piece must be even, or "flush," with the ruler. To use it, lay it on the paper so that one edge of the ruler coincides with the given line AB. Place another ruler against the cross-piece, hold it firm, and slide the T square along, till its edge passes through the given point C, as shown by the lower part of the figure. Then draw by this edge the desired line parallel to the given line. A C Fig. 15. -B, (40) Perpendiculars. These may be drawn by the various problems given in Geometry, but more readily by a triangle which has one right angle. Place the longest side of the triangle on the given line, and place a ruler against a second side of the triangle. Hold the ruler fast, and turn the triangle so as to bring its third side against the ruler. Then will the long side be perpendicular to the given line. By sliding the triangle along the ruler, it may be used to draw a perpendicular from any point of the line, or from any point to the line. (41) Angles. These are most easily set out with an instrument called a Protractor, usually a semi-circle of brass. But the description of its use, and cf the other and more accurate modes of laying off angles, will be postponed till they are needed in Part III, Chapter IV. (42) Drawing to Scale. The operation of drawing on paper lines whose length shall be a half, a quarter, a tenth, or any other fraction, of the lines measured on the ground, is called "Drawing to Scale." To set off on a line any given distance to any required scale, determine the number of chains or links which each division of the scale of equal parts shall represent. Divide the given distance by this number. The quotient will be the number of equal parts to be taken in the dividers and to be set off. For example, suppose the scale of equal parts to be a common carpenter's rule, divided into inches and eighths. Let the given distance be twelve chains, which is to be drawn to a scale of two chains to an inch. Ther. six inches will be the distance to be set off. If the given distance had been twelve chains and seventy five links, the distance to be set off would have been six inches and three-eighths, since each eighth of an inch represents 25 links. If the desired scale were three chains to an inch, each eighth of an inch would represent 37 links; and the distance of 1275 links would be represented by thirty-four eighths of an inch, or 4 inches. A similar process will give the correct length to be set off for any distance to any scale. If the scale used had been divided into inches and tenths, as is much the most convenient, the above distances would have become on the former scale 6,375 inches, or nearly 64 inches; and on the latter scale 4,25 inches, coming midway between the 2d and 3d tenth of an inch. (43) Conversely, to find the real length of a line drawn on paper to any known scale, reverse the preceding operation. Take the length of the line in the dividers, apply it to the scale, and count how many equal parts it includes. Multiply their number by the number of chains or links which each represents, and the product will be the desired length of the line on the ground. This operation and the preceding one are greatly facilitated by the use of the scales to be described in Art. (48) (44) Scales. The choice of the scale to which a plat should be drawn, that is, how many times smaller its lines shall be than those which have been measured on the ground, is determined by several considerations. The chief one is, that it shall be just large enough to express clearly all the details which it is desirable to know. A Farm Survey would require its plat to show every field and building. A State Survey would show only the towns, rivers, and leading roads. The size of the paper at hand will also limit the scale to be adopted. If the content is to be calculated from the plat, that will forbid it to be less than 3 chains to 1 inch. Scales are named in various ways. They should always be expressed fractionally; i. e. they should be so named as to indicate what fractional part of the real line measured on the ground, the representative line drawn on the paper, actually is. When custom requires a different way of naming the scale, both should be given. It would be still better, if the denominator could always be some power of 10, or at least some multiple of 2 and 5, such as 3001 1000, 2000, 2500, &c. For convenience in printing, these may be written thus: 1:500, 1:1000, 1:2000, 1: 2500, &c. 1 Plats of Farm Surveys are usually named as being so many chains to an inch. Maps of Surveys of States are generally named as being made to a scale of so many miles to an inch. Maps of Rail-road Surveys are said to be so many feet to an inch, or so many inches to a mile. (45) Farm Surveys. If these are of small extent, two chains |