Where the rise or fall of the ground is considerable, the operation will be most conveniently and accurately performed by the use of a theodolite ; for this purpose pickets should be set up in the ground, in the direction of the line to be measured, at every place where a change occurs in the inclination of the ground to the horizon, and marks made on them at heights above the ground equal to that of the telescope belonging to the theodolite ; then, while the chairmen are employed in measuring the length of the line on the grouud, the surveyor takes the angular elevations or depressions of the marks on the pickets, with respect to the horizon. From the data thus obtained the horizontal distances between points of ground, and the positions of the points above or below any assumed horizontal plane, can be computed. In order to save the trouble of making trigonometrical computations, the vertical arch of the theodolite usually carries two series of graduations, from which, by inspection, when the telescope is directed to an object, the portion of the measured line which should be subtracted from it in order to reduce it to the corresponding horizontal length may be found ; and also the portion of that horizontal length to which the vertical height or depression is equal.
This method may be conveniently put in practice when it is required to exhibit sections of the ground, for the purpose of guiding the civil engineer in the choice of a line for a road or canal ; the great accuracy with which the section might be determined by a spirit-level not being requisite. It is now the practice to represent on a plan of the ground a vertical section in the direction of a proposed line of road, for the purpose of showing the depths to which the excavations are to be carried, and the heights to which the embankments are to be raised ; a strong line, as d', representing the surface of the proposed road : on one side of this line, as at a', d", are shown the profiles of the requisite excavation ; and on the other side, as at L", are shown the profiles of the embankments : both the heights and depths being deter mined with relation to the surface of the road. This method was first proposed by Mr. Macneil.
The principal and secondary station-lines constitute a triangulation on the plan of the ground ; and when the lengths of these lines have been ascertained by admeasurement, the superfieies of the whole track may be found by the rules of mensuration. The area of each triangle should be calculated separately from the measured lengths of the lines, and the several results added together, if all the triangles lie within the given bonndaries of the tract ; should any of them lie on the exterior of the boundary, the areas must of course be subtracted.
But as the boundaries of the several fields, &c., seldom coincide exactly with the station-lines, offsets must have been measured from every such line to each remarkable bend in the nearest boundary; and between the station-line, the boundary, and every two offsets from the former, there exists a small trapezoid, whose area must be computed separately, and either subtracted from or added to the areas of the triangles formed by the measured station-lines, according as it lies within or on the exterior of these triangles.
The accurate method just described is not always put in practice by surveyors. When the boundaries of a field or tract of ground have numerous small bends, a straight line is sometimes drawn through portions of the boundary in such a manner that the small areas on the exterior of the line shall be equal to those which fall in the interior, this equality being estimated by the eye : the complex figure of the contour line is thus reduced to one more simple ; and the area of the field or tract is then computed. For this purpose either the plan is divided into two or more triangles, or by a geometrical construction the whole irregular figure is reduced to one triangle of equal magnitude, and in either ease the lengths of the sides are measured by the scale of the plan.
When a road, river, or any boundary-line is surveyed with the theodolite and chain, the successive operations are registered in a book according to a particular form, by which a person without any know ledge of the ground may be enabled with facility to lay the work down on paper. This is called the Field-Book,' and the manner of entering in it the series of operations will be best explained by means of an example. Let a, Q, R, D, be the principal bends hi the direction of a road, and the stations at which, in succession, the theodolite is placed for the purpose of observing the bearings of the several lines G Q, Q it, and R D, from the magnetic meridian passing through the first station a.
At a let the bearing of the object, or mark set up at Q, be observed ; let the line a Q be measured with the chain, and let offsets be measured perpendicularly to that line up to any remarkable points near it. At q let the bearing of a staff at at be observed ; also let the length of Q a, and of several offsets from it at remarkable points towards the right and left hand along that part of the road, be measured. Again at a let, the bearing of the staff at n be observed ; let also the length of a n, and of various offsets along [hat line, be measured ; and let it be supposed that the like prucesa is continued as far as may be required.