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cluded within the rectangle ABCD. On the line AB, measure the horizontal distances AE, EF, FG, and GB; and on the line DC, the distances DH, HI, IL, and LC, respectively equal to the distances on AB: that is, DH=AE, HI=EF, &c. The distances AE, EF, &c. are regulated by the inequalities of the ground, being less if the changes in the surface are considerable, and greater if the changes are nearly uniform. In the present example, they are 100 feet each, which, upon ordinary ground, would render the work tolerably accurate.
Let stakes be driven at A, E, F, G, B, C, L, I, H, and D. Measure now the line AD, and place stakes at convenient distances, as a, b, c, and d: place stakes also along the other lines EH, FI, GL, and BC, at suitable points, and measure the respective distances Ef, fg, &c. It is best to use the telescope of the theodolite or level, in order to run the lines and place the stakes truly. In placing the stakes, it should be borne in mind, that the difference of level of either two that follow each other, ought not to be very great; and also, that they ought not to be on the same horizontal plane.
After the stakes are all placed, and the distances measured, let the differences of level of all the points so designated be found. In the present example, the results of the measure
B below G 2
d above D 4 h below H 3 below I 3 p below L 4 t below C 5
The heights of the points are here compared with each. other, two and two. Before, however, we can conceive
and compare all the others with it. Let the point A be taken.
This being done, a mere inspection shows us the highest and lowest points, as also the relative heights of the others, reckoning upwards or downwards. Let them be now written in the order of their heights above the lowest point, which is The difference of level between A and D being 20 feet, if the difference of level of each of the points below A, be taken from 20 feet, the remainder will be the height above Arranging them in their order, we have
c above D 2 H above Dp above D 9
B above D 12
A above D, 20 feet.
Let the surface be now intersected by a system of horizontal planes at 3 feet from each other,—the first plane being 3 feet above the point D. The point b being 9 feet above D, and the point c, 2 feet, the first plane will intersect the line AD between b and c : let the proportional distance be found, as in the last example, and one point u, of the first curve, will be known. The point II being 7 feet above D, the plane will cut the line DC between II and D, and finding the proportional distance as before, a second pc'at, v, of the first urve, is determined. Now, in drawing this curve, it will be borne in mind, that the point h is but 4 feet above D, and consequently but 1 foot above the first curve, so that the curve must run from u towards h, and then turn around to the point v. The curve is maked (3), which is the number of
tance in feet above D.
other curves of the figure; their number showing their disAround the point d, there is a small curve, also marked (3). By inspecting the table, it will be seen that d is 4 feet above D, and that the ground descends from d towards D and c: d is therefore a small knowl, the top of which is cut off by the first plane. To show that the ground descends from d, even below the first curve (3), a plane is passed 1 foot below the first plane, or 2 feet above D; the curve of section is marked (2).
The second of the system of curves, or the one marked (6), must cut the line AD between b and c, the line EH between f and g, the line FI between k and 1, and also between l and I; it also cuts EH again between h and II, and the line DC between II and D.
The third curve, or the one passing 9 feet above D, passes through b, cuts the line EH between E and f, the line FI between i and k; thence it passes to p, and thence to the line DC, crossing it between I and L. There is also another curve determined by this plane, since it passes through the points C and q, leaving the points t and s below it. This curve runs from C to p, and from p to q, as drawn in the figure.
The fourth curve, marked (12), intersects the line AD between a and b, EH between E and f, FI at i, GL at m, and BC at B. There is also another curve lying around the point L: for the plane cuts GL between p and L, the line DC between C and L, and again between I and L. The fifth curve, marked (15), cuts AD at a, EH between E and f, and AB at F. The sixth curve, marked (18), cuts AD between A and a, and AB between A and E. The proportional distances in all these cases are found as in the first example.
In looking on the little map that has been made, it is clearly indicated by the curves and shading, that the ground slopes from A to c, thence rises to d, and then slopes to D. It also slopes from A along the line AB; from E in the directions f and i, from F in the directions i and m, from G in the directions m and B, and from B in the direction Bqs. The ground also slopes from L to p, thence to 7 and h, and along to curve (2), and the point D: and on the other side
187. Thus far, we have said nothing of a plane of reference, which is any horizontal plane to which the levels of all the points are referred. In the first example, the plane of reference was assumed through the point A (Pl. 4, Fig. 6), and tangent to the surface of the hill in the second example, it was taken through D, the lowest point of the work.
188. After having compared all the levels with any one point, the highest and the lowest points are at once discovered, and the plane of reference may be assumed through either of them. As, however, in comparing the heights of objects, the mind most readily refers the higher to the lower, it is considered preferable to take the plane of reference through the lowest point. We say, for example, that the summit of a hill is 200 feet above a given plain, and not that the plain is 200 feet below the summit of the hill; so we say that a plain is at a given distance above a river, and not that the river is below the plain. This habit of the mind of referring the higher to the lower objects, suggests the propriety of taking the plane of reference through the lowest point, where there is no other circumstance to influence its selection. If, however, there are fixed and permanent objects, to which, as points of comparison, the mind readily refers all others, such as the court-house or church of a village, the market-house of a town, or any public building or monument, it is best to assume the plane of reference through some such point; for, it must be kept in mind, that the ends proposed in the construction of maps, are, to present an accurate view of the ground, its form, its accidents, and the relative position of objects upon it.
189. When the plane of reference is so chosen that the points of the work fall on different sides of it, all the references on one side are called positive, and those on the other, negative. The curves having a negative reference are distinguished by placing the minus sign before the number; thus - ( ).
190. In topographical surveys, great care should be taken to leave some permanent marks, with their levels written on them in a durable manner. For example, if there are any rocks, let one or more of them be smoothed, and the vertical distance from the plane of reference marked thereon: or let
the vertical distance of a point on some prominent building, be ascertained and marked permanently on the building. Such points should also be noted on the map, so that a person, although unacquainted with the ground, could by means of the map, go upon it, and trace out all the points, together with their differences of level.
191. The manner of shading the map, so as to indicate the hills and slopes, consists in drawing the lines of shading perpendicular to the horizontal curves, as already explained.
192. In making topographical surveys, the great point is, to determine the curves which result from the intersection of the surface by horizontal planes.
Besides the methods of diverging and parallel sections, we may assume a point on the surface of a hill, place the level there, and run a line of level round the hill, measuring the angles at every turn or change of direction: such a line will be a horizontal curve. Then, levelling up or down the hill, a distance equal to the vertical distance between the horizontal curves, let a second curve be traced; and similarly for as many curves as may be necessary.
This method, however, is not as good as the methods before explained.
193. Besides representing the contour of the ground, it is often necessary to make a map which shall indicate the cultivated field, the woodland, the marsh, and the winding river. For this, certain characters, or conventional signs, have been agreed upon, as the representatives of things, and when these are once fixed in the mind, they readily suggest the objects for which they stand. Those which are given in Plates 5 and 6, have been adopted by the Engineer Department, and are used in all plans and maps made by the United States Engineers.
It is very desirable that a uniform method of delineation should be adopted, and none would seem to be of higher authority than that established by the Topographical Beaureau. It is, therefore, recommended, that the conventional signs given in Plates 5 and 6, be carefully studied and closely fol