horizontal hair of the telescope will cut the vane, and drive here another stake, marked 6; and so or, until a sufficient nuinber 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 ines may be found in a similar manner.

31. We will add another example for determining the contour of an undulating piece of ground (Pl. 4, Fig. 7,) by means of horizontal sections. Let rows of stakes DA, HE, IF, &c., be driven at intervals, depending upon the required accuracy of the survey, and let f, g, h, &c., be stakes driven along the lines, at such points as will best show the accidents of ground. Determine as before the difference of level between each stake, and some fixed. point, and then determine where the contour lines cut the lines AD, EH, &c., by the rules already laid down.

After the stakes are all placed, and the distances meas ured, let the differences of level of all the points so desig nated be found. In the present example, the results of the measurements are,

d above D 4 h below H3

below 13 peow L 4t below C5

The heights of the points are here compared with each

clearly their relative Leights, we must assume some one point, and compare all the others with it. Let the point A be taken. The height of

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 D. 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 D. Arranging them in their order, we have

In this example, the plane of reference is assumed through D, the lowest point of the ground; and the secant planes are taken 3 feet apart.

32. 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 ex plained. These shading lines are drawn close together, when the slope is abrupt, and further apart, as it grows

33. 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 th other, negative. The curves having a negative reference are distinguished by placing the minus sign before the number; thus-( ).

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 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 Topographi cal Bureau. It is, therefore, recommended, that the conventional signs given in Plates 5 and 6, be carefully

1. WHEN a large extent of territory, or a long line of sea-coast is to be surveyed, it becomes necessary to consider the curvature of the earth's surface; this branch of surveying is called Geodesic surveying.

2. Extensive geodesic operations prove that the earth is an oblate spheroid, the shortest diameter of which coincides with the terrestrial axis, and all of whose meridians, are equal ellipses. The meridian lines, however, differ so little from the circumferences of circles, that they may be taken for them, except when great accuracy is required. The earth, will, therefore, in the following pages, be regarded as a perfect sphere.

3. The operations necessary to the successful execution of a Geodesic Survey, require the minutest attention, and when performed, numerous corrections are to be applied to the measured lines and angles, on account of the various causes of error incident to such operations.

To investigate those causes of error, and to deduce rules for correcting the errors, in all cases, would far exceed the limits of an elementary treatise. We shall, therefore,

of a trigonometric survey, with the application of some of the more important corrections.

4. It may be observed that most of the operations described in this section, are equally applicable, whether we regard the area surveyed as plane or spherical: in either case, the basis of an accurate survey, is an extensive system of triangulation.

5. After having made a preliminary examination or reconnaissance of the territory to be surveyed, suitable stations are selected at the most prominent points, and these points are marked by staves or signals.

A base line is then measured. The length of the base will, in general, depend upon the magnitude of the survey, and each extremity is marked by a signal.

The next step is the triangulation. At each extremity of the base, the angles between the base, and the lines drawn to each of the visible signals, are carefully measured by means of a theodolite. The sides of the triangles thus obtained, serve as new bases upon which other triangles may be formed, and so on, until the entire area is covered by a net-work of triangles.

6. This system of triangles is called the primary system, and the operation of forming them is called the primary triangulation. Within the primary triangles, and depending upon them, a system of smaller triangles is formed in the same manner, called the secondary system; and if the extent or importance of the work should demand it, the secondary may be sub-divided into tertiary triangles.

Having completed the triangulation, the characteristics of the surface, such as roads, streams, villages, boundaries, &c., are filled in by means of the compass, plain table, or some of the methods already explained.

After the field work is completed, the triangles, when regarded as spherical, are reduced by applying the formula for spherical excess, hereafter explained, and other necessary corrections, and thus the whole work is plotted upon