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 P, thence to land h, and along to curve (2), and the point D: and on the other side to t and s. to 216. Thus far, we have said nothing of a plane of reference; it 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. 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 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 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 CHAPTER IX. OF SURVEYING HARBOURS. 222. THERE are two objects to be attained in the survey of a harbour. 1. To survey the shore along high or low-water mark,—to trace its windings, to note the points and inlets, and to ascertain and fix the places at which rivers and creeks discharge themselves. And, 2. To discover the channels, their direction, depth, and width, the position of shoals, the depth of water upon them, the nature of the bottom, and, in short, whatever may contribute to easy and safe navigation. 223. Having provided a boat and crew, row once or twice around the harbour-mark the more important and prominent points; at which, let station-staves with flags upon them be erected. Then, measure a base-line, and form a series of triangles, having their angles at the stations before chosen. Let the angles of these triangles be measured with the theodolite, and their sides calculated; after which, the high or low-water mark may be traced along the shore with the compass, and the work plotted as in Chapter VI. 224. When a harbour is surveyed for the second object, viz. for the purpose of ascertaining the channels, their depth and width, the positions of shoals, and the depth of water thereon, other means must be used, and other examinations made, in addition to those already referred to. 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 flag, 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. The sextant is not described, nor are its uses explained, in these Elements, because its construction combines many philosophical principles, with which the surveyor cannot be supposed conversant. 227. There is yet another method of finding the soundings, which, although not as accurate as those already explained, will, nevertheless, afford results approximating nearly to the truth. It is this:-Let a boat be rowed uniformly from one extremity to the other, of any of the lines determined trigonometrically. Let soundings be made continually, and let the precise time of making each be carefully noted. Then, knowing the length of the entire line, the time spent in passing over it, as also the time of making each of the soundings, we can easily find the points of the line at which the several soundings were made; and hence the depth of water at those points becomes known. Soundings may thus be made along any number of known lines, and a comparison of the depths found on different lines, at or near their points of intersection, will show with what degree of accuracy the work has been done. 228. If the soundings are made in tide-waters, the time of high tide must be carefully noticed, as also the precise time of making the sounding, so that the exact depth at high or low water may be known. It is considered preferable to reduce the soundings to high-water mark, and the number of feet which the tide rises and falls should be noted on the map. 229. Having plotted the work done with the theodolite, as also the outline of the harbour traced with the compass, (223) it remains to delineate the bottom of the harbour; and this is done by means of horizontal curves (Chap. VIII.) which have already been used to represent broken and undulating ground. Let the plane of reference be taken through high-water mark, or to coincide with the surface of the water at high tide. The accuracy with which the bottom of the harbour is to be delineated, will guide us in fixing the distance between the horizontal planes of section. The first horizontal plane is to be passed at a distance below the shallowest point that has been sounded, equal to the number |