490 SURVEYING from a corner of each triangle to the opposite side serve to rectify the other measures of the triangle, and if perpendicular to the side afford a convenient means of calculating upon the ground the area of the triangle. Perpendicu- lars to any line are readily laid out with a chain, as carpenters and masons draw right angles by what they call the 6, 8, and 10 rule, the popular application of the principle of the square of the hypothenuse being equal to the sum of the squares of the two other sides. The method is to measure from the point where the perpendicular meets the line, either along this line or along the perpendicular, a dis- tance equal to six units of any kind, and then upon the other of these lines a distance of eight units. The two lines are perpendicular to each other when the two termini are just ten units apart. Convenient distances for this measurement might be 3, 4, and 5 rods or chains, or any similar multiples of these num- bers, as 21, 28, and 35. Other trigonometri- cal methods readily suggest themselves. A number of convenient instruments of simple form, known as the surveyor's cross, are in use for setting out perpendiculars by lines of sight, crossing each other at right angles ; and a temporary substitute for them is easily made by sticking a pin in each corner of a square piece of board, and sighting across these in the direction of the line and at right angles to it. Angles in the field are deter- mined by a chain, by measuring a "tie line" from a measured point on one side to another measured point on the other side. By this means the boundaries of a tract may be de- termined when it cannot be conveniently mea- sured off in triangles. A great variety of expedients are adopted for overcoming nat- ural obstacles and determining the extent and shape of inaccessible objects, systems of tri- angles being in such cases formed outside of and around such objects. Crooked lines are determined by means of perpendicular offsets measured from different points along a straight line run as nearly coincident to the crooked line as may be. In all the methods of survey- ing, the measurements, together with various incidental observations, are recorded, after some established system, in what are called field notes, and from these the results of the survey are afterward plotted to a convenient scale. A more common system of surveying is that in which instruments for taking angles are employed in connection with the chain. A graduated horizontal circle, with a straight edge called an alidade turning upon its cen- tral point, which may be conveniently sight- ed along, furnishes the means of ascertaining the angular distance of two lines, the instru- ment being set at their intersection, and the alidade pointed in the direction of one and then of the other. This involves the princi- ple of the engineer's transit, or of the the- odolite. (See THEODOLITE.) With these in- struments angles can be determined with great accuracy, especially when the observations are repeated by reversing the instrument and ta- king the mean, each including the reading of both verniers. With the transit and the chain for measuring distances, a tract of almost any dimensions is accurately surveyed by mea- suring the angles at its corners, and the cor- rectness of the work is proved when the sum of all the interior angles is found equal to the product of two right angles, or 180, by the number of sides of the tract less two ; or if the instrument be used by the method called traversing, or " surveying by the back angle " (which consists in noting the angle which each successive line makes, not with the preceding line, but with the first line observed, which is hence called the meridian of the survey), then the reading, on getting round to the last sta- tion, and looking back to the first line, should be 360, or 0. A compass and chain may be employed in filling up the interior details of a large survey with the transit ; and the compass may be used for determining the magnetic bear- ing of one of the lines, unless this be astronom- ically ascertained by observations of the north star or of the shadows before and after noon. The compass is the instrument in most common use in ordinary surveying. The magnetic nee- dle, wherever the instrument is set, establishes the meridian line, and from this, the sights of the instrument being turned to any other line, the angle of divergence is read on the gradua- ted circle around the compass box. This instru- ment has been described under its own name ; also the more perfect instrument, in which its inaccuracies are obviated, under the head of COMPASS, SOLAR. The details of surveys are variously modified according to the extent of the area, character of the ground, &c. With the transit or compass, the boundary lines may be all followed out, the angles they make with each other determined, and their lengths mea- sured by the chain ; the points of crossing of roads, brooks, fences, &c., measured, and the bearings of these objects taken ; and increased accuracy may be given to the work by run- ning diagonal or proof lines, as in chain sur- veying. Additional checks are furnished by taking at each station the bearings of some marked objects, which when the work is plot- ted should severally fall at the points of inter- section of the lines directed toward these ob- jects from the several stations. Sometinies a tract may be surveyed from a measured base line, either a line within or without it, or one of the boundary lines, by placing the compass successively at each end of this line and taking the bearings of each corner; or without a com- pass the work may be very conveniently per- formed with approximate correctness by the plane table method, provided no angles are taken less than 30 nor larger than 150. A drawing board covered with paper is set up at one end of a measured base line, and a ruler furnished with upright sights at each end, ex- actly over the drawing edge, is set with this
