its place be again noted. Then the required distance will be equal to the difference of the readings on the rod, in feet, multiplied by the distance at which a foot was intercepted between the lines. One of the horizontal hairs may be made movable, and its distance from the other, when the space between them exactly covers an object of known height, can be very precisely measured by counting the number of turns and fractions of a turn, of a screw by which this movable hair is raised or lowered. A simple proportion will then give the distance. On sloping ground a double correction is necessary to reduce the slope to the horizon and to correct the oblique view of the rod. The horizontal distance is, in consequence, approximately equal to the observed distance multiplied by the square of the cosine of the slope of the ground. The latter of the above two corrections will be dispensed with by holding the rod perpendicular to the line of sight, with the aid of a right angled triangle, one side of which coincides with the rod at the height of the telescope, and the other side of which adjoining the right angle, is caused, by leaning the rod, to point to the telescope. Other contrivances have been used for the same object, such as a Binocular Telescope with two eye-pieces inclined at a certain angle; a Telescope with an object-glass cut into two movable parts; &c. (376) Ranging out lines. This is the converse of Surveying lines. The instrument is fixed over the first station with great precision, its telescope being very carefully adjusted to move in a vertical plane. A series of stakes, with nails driven in their tops, or otherwise well defined, are then set in the desired line as far as the power of the instrument extends. It is then taken forward to a stake three or four from the last one set, and is fixed over it, first by the plumb and then by sighting backward and forward to the first and last stake. The line is then continued as before. good object for a long sight is a board painted like a target, with black and white concentric rings, and made to slide in grooves cut in the tops of two stakes set in the ground about in the line. It is moved till the vertical hair bisects the circles (which the eye can determine with great precision) and a plumb-line dropped from their centre, gives the place of the stake. "Mason & Dixon's Line" was thus ranged. If a Transit be used for ranging, its "Second Adjustment" is most important to ensure the accuracy of the reversal of its Telescope. If a Theodolite be used, the line is continued by turning the vernier 180°, or by reversing the telescope in its Ys, as noticed in Arts. (325) and (362). (377) Farm Surveying, &c. A large farm can be most easily and accurately surveyed, by measuring the angles of its main boundaries (and a few main diagonals, if it be very large,) with a Theodolite or Transit, as in Arts. (366) or (371), and filling up the interior details, as fences, &c., with the Compass and Chain. Art. (366), will be the interior angles of the field, as noted in the figure. x The accuracy of the work will be proved, as alluded to in Art. (257), if the sum of all the interior angles be equal to the product of 180° by the number of sides of the figure less two. Thus in the figure, the sum of all the interior angles = 540° = 180° × (5-2). The sum of the exterior angles would of course equal 180° x (5+ 2) = 1260°. If the Transit be used, the farm should be kept on the right hand, and then the angles measured will be the supplements of the interior angles. If the angles to the right be called positive, and those to the left negative, their algebraic sum should equal 360°. If the boundary lines be surveyed by "Traversing," as in Art. (373), the reading, on getting back to the last station and looking back to the first line, should be 360°, or 0°. The content of any surface surveyed by "Traversing" with the Transit can be calculated by the Traverse Table, as in Chapter 'VI, of Part III, by the following modification. When the angle of deflection of any side from the first side, or Meridian, is less than 90°, call this angle the Bearing, find its Latitude and Departure, and call them both plus. When the angle is between 90° and 180°, call the difference between the angle and 180° the Bearing, and call its Latitude minus and its Departure plus. When the angle is between 180° and 270°, call its difference from 180° the Bearing, and call its Latitude minus and its Departure minus. When the angle is more than 270°, call its difference from 360° the Bearing, and call its Latitude plus and its Departure minus. Then use these as in getting the content of a Compass-survey. The signs of the Latitudes and Departures follow those of the cosines and sines in the successive quadrants. Town-Surveying would be performed as directed in Art. (261), substituting "angles" for "Bearings." "Traversing" is the best method in all these cases. Inaccessible areas would be surveyed nearly as in Art. (134), except that the angles of the lines enclosing the space would be measured with the instrument, instead of with the chain. (378) Platting. Any of these surveys can be platted by any of the methods explained and characterized in Chapter IV, of the preceding Part. A circular Protractor, Art. (264), may be regarded as a Theodolite placed on the paper. "Platting Bear ings," Art. (265), can be employed when the survey has been made by "Traversing." But the method of "Latitudes and Departures," Art. (285), is by far the most accurate. PART V. TRIANGULAR SURVEYING; OR By the Fourth Method. (379) TRIANGULAR SURVEYING is founded on the Fourth Method of determining the position of a point, by the intersection of two known lines, as given in Art. (8). By an extension of the principle, a field, a farm, or a country, can be surveyed by measuring only one line, and calculating all the other desired distances, which are made sides of a connected series of imaginary Triangles, whose angles are carefully measured. The district surveyed is covered with a sort of net-work of such triangles, whence the name given to this kind of Surveying. It is more commonly called "Trigonometrical Surveying;" and sometimes "Geodesic Surveying," but improperly, since it does not necessarily take into account the curvature of the earth, though always adopted in the great surveys in which that is considered. (380) Outline of operations. A base line, as long as possible, (5 or 10 miles in surveys of countries), is measured with extreme accuracy. From its extremities, angles are taken to the most distant objects visible, such as steeples, signals on mountain tops, &c. The distances to these and between these are then calculated by the rules of Trigonometry. The instrument is then placed at each of these new stations, and angles are taken from them to still more distant stations, the calculated lines being used as new base lines. This process is repeated and extended till the whole district is embraced by these "primary triangles" of as large sides as possible. One side of the last triangle is so located that its length can be obtained by measurement as well as by calculation, and the agreement of the two proves the accuracy of the whole work. Within these primary triangles, secondary or smaller triangles are formed, to fix the position of the minor local details, and to serve as starting points for common surveys with chain and compass, &c. Tertiary triangles may also be required. The larger triangles are first formed, and the smaller ones based on them, in accordance with the important principle in all surveying operations, always to work from the whole to the parts, and from greater to less. Each of these steps will now be considered in turn, in the following order: 1. The Base; articles (381), (382). 2. The Triangulation; articles (383) to (390). 3. Modifications of the method; articles (391) to (395). (381) Measuring a Base. Extreme accuracy in this is necessary, because any error in it will be multiplied in the subsequent work. The ground on which it is located must be smooth and nearly level, and its extremities must be in sight of the chief points in the neighborhood. Its point of beginning must be marked by a stone set in the ground with a bolt let into it. Over this a Theodolite or Transit is to be set, and the line "ranged out" as directed in Art. (376). The measurement may be made with chains, (which should be formed like that of a watch,) &c. but best with rods. We will notice in turn their Materials, Supports, Alinement, Levelling, and Contact. As to Materials, iron, brass and other metals have been used, but are greatly lengthened and shortened by changes of temperature. Wood is affected by moisture. Glass rods and tubes are preferable on both these accounts. But wood is the most convenient. Wooden rods should be straight-grained white pine, &c.; well seasoned, baked, soaked in boiling oil, painted and varnished. They may be trussed, or framed like a mason's plumb-line level, to prevent their bending. Ten or fifteen feet is a convenient length. Three are required, which may be of different colors, to prevent |