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 H being 7 feet above D, the plane will cut the line DC between H and D, and finding the proportional distance as before, a second point, v, of the first curve, is determined.

V, Now, in drawing this curve, it will be borne in mind, that the point h is but 4 feet above D, and consequently, but I foot above the first curve, so that the curve must run from u near to h, and then turn around to the point v. The curve is marked (3), which is the number of feet that it is above the lowest point, and similarly for the other curves of the figure; their number showing their distance in feet above D. Around 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, a plane is passed I 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 I, and also between 1 and I; it also cuts EH again between h and H, and the line DC between H 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 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, 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 l and h, and along to curve (2), and the point D: and on the other side to t and s.

216. Thus far, we have said nothing of a plane of reference ; it is any hwrizontal 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.9, 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.

217. 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, to refer the higher to the lower objects, suggests the propriety of taking he 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

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 a clear view of the ground, its form, its accidents, and the relative positions of objects upon it.

218. 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. Let the curves having a negative reference be distinguished by placing the minus sign before the number, thus -( ).

219. In these topographical surveys, great care should be taken to leave some permanent marks with their levels written upon 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 be marked thereon: or, let the vertical distance of a point on some prominent building be ascertained and marked permanently upon the building with paint. Such points must also be noted on the map, so that a person, though 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.

220. The manner of shading the map, so as to indicate the eminences and slopes, although highly important, belongs to the department of drawing, rather than to that of practical mathematics.

221. 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 is a horizontal curve. Then, levelling up or down the hill, a distance equal to the required distance between the horizontal curves, let a second horizontal curve be traced, passing through this point, and similarly for as many as may be necessary.

This method, however, is not as good as the

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OF SURVEYING HARBOURS.

Harin

> Trans are iwo copects to be attained in the survey of

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* wants and inlets, and to ascertain
en tomane creeks discharge then-

Jer direction depth, and om at of water upon them,

wprswhatever may contri

edges of the channels, and using any of the lines already determined as a base, let the angles subtended by lines drawn from its extremities, to the buoys respectively, be measured with the theodolite. Then there will be known in each triangle the base and angles at the base, from which the distances to the buoys are easily found ; and hence, their positions become known.

Having made the soundings, and ascertained the exact depth of the water at each of the buoys, several points of the harbour are established, at which the precise depth of the water is known; and by increasing the number of the buoys, the depth of the water can be found at as many points as may be

deemed necessary.

225. If a person with a theodolite, or with any other instrument adapted to the measurement of horizontal angles, be stationed at each extremity of the base line, it will not be necessary to establish buoys. A boat, provided with an anchor, a sounding line, and a signal flag, has only to throw its anchor, hoist its signal flug, and make the sounding, while the persons at the extremities of the base line measure the angles ;-from these data, the precise place of the boat can be determined.

226. There is also another method of determining the places at which the soundings are made, that admits of great despatch, and which, if the observations be made with care, affords results sufficiently accurate.

Having established, trigonometrically, three points which can be seen from all parts of the harbour, and having provided a sextant, let the sounding be made at any place in the harbour, and at the same time, the three angles subtended by lines drawn to the three fixed points, measured with the sextant.

The Problem, to find, from these data, the place of the boat at the time of the sounding, is the same as Example 6, p. 57.

It is only necessary to measure two of the angles, but it is safest to measure the third also, as it affords a verification of the work.

The great rapidity with which angles can be measured with the sextant, by one skilled in its use, renders this a most expeditious method of sounding and surveying a harbour.