where the lines at right angles, cut fences, roads, streams, &c., we can, by joining the points, obtain a plot of the ground.

30. The contour lines may be traced on the ground, as follows: Set up the level at a, and observe that the back-sight, to the stake, placed at (E3), gave a reading of 11.432. Depress the vane equal to the distance between the horizontal secant planes, that is, 6 feet, which is done by placing it at the reading 5.432. Then direct the rodman, by signals, up or down the hill, till the horizontal hair of the telescope coincides with the horizontal line of the vane. The foot of the staff is then 6 feet above the first point. Let a stake, marked 6, be driven here, and direct the rodman around the hill, until a second position shall be found, when the horizontal hair of the telescope will cut the vane, and drive there another stake, marked 6; and so on, until a sufficient number of stakes have been driven to determine the curve (6).' Then, let the line of stakes, marked 6, be surveyed with the compass and chain, and plotted. Other contour lines may be found in a similar manner.

31. When the plane of reference is so chosen that 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 ( ).

SHADING AND DELINEATION.

32. Figure 7 (Pl. 4), represents a piece of ground sloping towards D, which is the lowest point; and through this point the plane of reference is supposed to pass. The following table indicates the heights of the several points above the plane of reference.

Ft.

Ft.

Ft.

Ft.

c above D, 2 H above D, 7p above D, 9 B above D, 12

The first horizontal plane is passed 2 feet above D, and the curve of intersection with the surface passes through c. The second secant plane is passed at 3 feet above D, and intersects the surface, in the curve uv, and also near d, which is one foot above the curve. All the other secant planes are passed at three feet from each other; and, comparing the height of each point above D, with the curves lying nearest, on either side, the positions of all the points, with respect to the curves, and with respect to each other, are easily seen.

33. The manner of shading the map, so as to indicate the

pendicular to the horizontal curves, as already explained. These shading lines are drawn close together, when the slope is abrupt, and further apart, as it grows more gentle. Fig. 7 indicates the method of shading.

34. 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.

35. 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 ir 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 Bureau. It is, therefore, recommended, that the conventional signs given in Plates 5 and 6, be carefully studied and uniformly followed.

SECTION III.

RAILWAY CURVES.

36. The preliminary survey of a railroad establishes a succession of straight lines, of greater or less length, according to the obstacles to be avoided or the advantages to be gained, arising from the nature and the contour of the ground.

The angle formed, at each change in the direction of the route, is carefully measured and recorded.

In the final survey or location, these angles are replaced by curves; and in order that the change in direction shall be as gradual as practicable, the straight lines of direction. are made tangents to the curves at their point of meeting.

The preliminary survey is termed, by the engineer, “running out tangents."

37. We will proceed to describe the method of locating curves, first giving the mathematical principles applicable to the subject.

Let AD and DB (next fig.) be two tangents, to the arc of a circle, AB. Draw the radii AC, BC, and the secant CD.

The following relations are easily deduced. The tangents AD and DB are equal, (Leg., Bk. III., Prob. 14). The angles A and B are right angles (Leg., Bk. III., Prop. 9), consequently the angles C and D, of the quadrilateral ADBC, must be supplements of each other. The angle TDB, therefore, must be equal to the angle ACB. The right-angled triangles ADC and BDC are equal (Leg., Bk. I., Prop. 17); hence, the angle DCA

Let the radius AC be represented by r; the distance AD by d, and the angle TDB by a. Then will (Trig., Art. 66), d = r tanga (1)

The angle TDB is the angle formed by two straight lines of the preliminary survey, and is carefully measured by the engineer, in locating tangents.

From formula (1), we can determine the value of d, for any given values of a and r; and hence we can determine, at what point on the tangent, laid off from D, the curve of any given radius must commence.

It is evident, both from the diagram and the formula, that for any given angle between the tangents, the greater the radius of the curve, the greater will be the distance cut off between the intersection of the two tangents and the point of tangency.

It is sometimes necessary to give a particular value to d. In such case, we use the formula,