CONTOUR LEVELLING, OR CONTOURING.

WITHIN a comparatively recent period contour levelling, or contouring, has become an important part of the professional knowledge required both by the civil engineer and engineering surveyor. A plan of any city, town, or district, with the contour lines carefully laid down on it, enables the engineer to devise the best and most economic means of carrying out any system of sewage, drainage, irrigation, or waterworks, as well as railway or public road that may be required. By the contour lines the inequalities of the surface of any city, town, or district are correctly represented.

If we imagine a hill to be cut by any number of horizontal planes, and the outline of each cut, as seen from above, to be projected orthographically on the map or plan, the outlines of the cuts so projected are called contour lines. These lines are identical in position with the outlines that would be formed by the sea surrounding the hill, and rising to heights corresponding with those at which the horizontal planes just referred to would cut the hill. If a hill were similar in form to a right cone, its contour lines would be represented on the map or plan by a number of concentric circles; the apex of the cone being the centre, and the outermost circle the circumference of the base of the cone. A hill in shape like an oblique cone would be represented by eccentric circles.

On page 222 is a plan, on a scale of ten chains to the inch, of a part of the Borough of Liverpool, with the contour lines, or lines of equal altitude, represented thereon. The contour lines are shown at every four feet of altitude, as indicated by the numbers inserted on them.

The method generally adopted for determining the position of the contour lines is this: levels are taken along the most suitable streets or roads, or in a direct line between two or more points of the district to be contoured. The most suitable roads, streets, or lines to be levelled, are those which intersect the contour lines at right angles, or nearly so. Bench marks are made in the most favourable places, or stakes are driven in the ground at or near the points where the contour lines will run through; the altitude of each stake being carefully marked on it. Suppose, for instance, that the line from a to B on the plan (p. 222) had been correctly levelled, and that stakes had been driven down at 64 feet above the datum adopted by the Ordnance Survey authorities at 68 feet, at 72 feet, and so on. It is evident the contour lines of corresponding altitudes must pass through the positions of these stakes. Then, in order to determine the 64 feet contour, for example, the level is placed at a distance of 4 or 5 chains from the stake at that altitude, the back staff is placed on the stake, and read off by the level, after it has been carefully levelled;

the staff is then sent forward 4 or 5 chains in the direction the contour line seems to take, and placed on the ground; if the reading of it by the level agrees with what it had been on the stake, the position of the staff is in the contour line. If the reading of the staff should be more or less than that at the stake, the staff is to be moved to a higher or lower position, until the reading is the same as that at the stake. When the true position is found, the level is then moved forward, while the staff remains where it is until the back-sight or reading is taken, when the staff is again sent forward, and so on. Previously, however, to the staff being sent forward, some distinctive mark should always be placed where it stood, in order that the surveyor may be enabled to survey the contour lines, so as to fix their positions on the map or plan.

The marks usually placed at the positions of the staff on the contour lines, are twigs, or small cuttings off the branches of trees. These are stuck in the ground, with pieces of paper at the top, to render them more conspicuous. The surveyor then lays out his chain lines in the most suitable manner, and takes offsets to the twigs or cuttings, and notes them in his field-book. He is then able to plot the various courses of the contour lines as if they were fences, and to show, if necessary, the position on the map or plan of every point on the contour line where the staff stood. The dots on the contour lines, on the plan, p. 222, represent the positions of the staff. If the map or plan be correct, the chain lines for surveying the contours can generally be fixed by means of the fences and other details. The writer of these remarks prepared contour plans of the city of Londonderry, and the town of Preston, Lancashire, for their respective Corporations, preliminary to the carrying out of the sewerage and drainage of those places. The contour lines were shown at altitudes differing only five feet from each other.

On the Ordnance Survey of Great Britain, the lowest contour is run at 50 feet above the Ordnance datum, which datum is the height of mean tide at Liverpool, as determined by careful tidal observations taken in 1844. The next contour is run at 100 feet, then at 200, 300, and at intervals of 100 feet up to an altitude of 1,000 feet. Above this, no instrumental contours have been run, but sketched contours are shown at intervals of 250 feet.

The datum plane adopted for the whole of the Ordnance Survey of Ireland was the level of low water spring tide observed at Poolbeg Light-house in Dublin Bay on the 8th April, 1837, which level was found by a combined system of tidal observation and spirit levelling to be 8.094 feet below the mean sea level.

Seven counties in Ireland have been instrumentally contoured by the Ordnance Survey, but the vertical interval between the contours varies in different localities.

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ADDENDA TO THE METHODS OF LAYING OUT

RAILWAY CURVES.

PROBLEM I.

To join the straight portions A B, C D of a railway by a serpentine curve FOIN M, meeting A B at F between A and B, and CD at M between C and D; the distance BC and the 8 ABC, BCD being given, to find the common radius GI = IH.

Take the cotangents of half the angles ABC, BCD to rad. = 1; then as the sum of these two cotangents is to the cotangent of half A B C to rad. = 1, so is BC to BI; hence BC-BI IC.

=

In the triangle BCI; as sin. BGI is to sin. ▲ G BI, so is BI to GI = I H, the common radius.

EXAMPLE.

Let the angle ABC = 71° 40′, the angle B C D = 129° 15', and the distance BC 95 chains: required the length of the radius G I = I H of the serpentine curve for uniting A B to C D.

Here the cotangents of half the angles A B C, B C D to rad. = 1 are respectively 1.3848 and 4743, their sum being 1.8591. Then as 1.8591: 1.3848 :: 95: 70·76 chains = B I. Whence BC-BI= 95 — 70·76: 24.24 chains = I C.

GI

Again, as sin. BGI is to sin. G BI, so is 70.63 to 51.09 chains I H, which is the common radius of the required serpentine curve.

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PROBLEM II.

To lay out a railway curve by means of tangental angles.

This method, already noticed in the note to Problem IV., p.169, is preferred by many eminent engineers, where the ground on which the curve is to be laid out is level and free from numerous obstructions, except such as do not fall on the curve itself; I shall therefore give it more in detail than in the note referred to.

If from any point B, in a straight line A D, we lay off any number of equal angles, as DBS, SBT, TBU, &c., making the chords B S, ST, TU, &c., equal to each other; then the points B, S, T, U, &c., will be situated in the circum

ference of a circle, which is tangental to the line AD at the point B.

Fix the theodolite at B, and lay off the tangental angle DBS, the chain being extended from B to S to meet the visual line B S; in the same manner lay off the other angles SB T, TB U, &c., cill the fourth point V is reached. If any obstruction, as H, should prevent our seeing from B further than to V, the curve may be continued by removing the instrument to U, the point preceding V; thence sighting first on V, continue to lay off additional tangental angles V UW, WUX, &c., as before. Otherwise, moving the instrument to V instead of U, sight back to U, and lay off first the exterior angle P V W equal to double the tangental angle (P being in U V prolonged), and afterwards continue the tangental angles W V X, XV Y, &c., as before, to the end of the curve.

Finally, in order to pass to the end of the curve at Y to a tangent Y Z, place the instrument at Y, and sighting back to X. lay off the tangental angle X Y O; then O Y continued towards Z will be the required tangent to the curve.

The method of finding the quantity of the tangentil angles is given in the note, p. 166

ON THE DIVISION OF LAND.

Three straight fences WX, X Z, ZY are given in position; there is a well at W, and it is required to substitute a new straight fence W Y instead of X Z, that the ground on both sides of the divisional fence may have the use of the well; the value of the land on the side WXZ is to that on the side X Z Y as m to n per acre : required the position of the new fence W Y, that the proprietors on each side of it may neither gain nor lose by the change of boundary.

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Let the required straight fence W Y cut XZ in P; draw W Q parallel to Y Z, meeting X Z in Q; then Q is a given point, and the area W QX is also given, which put A, also put ZQ a, QW= b, QP x, and the sine of the angle Z 0; then from the nature of the problem we readily obtain the following quadratic equation:

sine of the angle ZQ W

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