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a serious difficulty occurs-land has to be purchased for the purpose of digging earth to supply the deficiency, which is usually called side cutting.

Suppose in the cut below the upper figure to represent the section of an old line of road, and that it were required, by cutting and embankment, to reduce it from its present hilly surface to one uniform rate of inclination from the point A to the point B. The lower

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extremity A is 10 feet above the datum line of the section, and the higher point B 46 feet above the datum ; consequently, 46-1036 feet, the rise from A to B, and the distance 4356 feet, which, divided by the rise (36), will give 1 in 121 for the rate of inclination the road may be brought to.

Upon the section draw the straight line AB, which will show the extent of cutting and embanking to be made. The number of cubic yards of earth to be removed in the cutting between the points B and C, and the cubical contents, in yards, of embankment between C and A, may then be computed in the following man

ner:

Divide the quantities of cuttings and embankments as shown upon the longitudinal section, into triangles

and trapeziums, determined by the undulations of the surface lines, as shown in the above engraving, where, in the cuttings, a and c are triangles, b a trapezium ; and in the embankments d and ƒ are triangles, e a trapezium. The form of the excavation and embankment is shown by the transverse or cross sections. Let the width of the roadway (or base of the cutting, and top of the embankment) be 50 feet, including the footpath, &c., on each side; the slope of the cutting to be 11⁄2 to 1, that is, 1 horizontal to 1 perpendicular; consequently, where the depth is 20 feet, the width of the slope at the surface will be 30 feet; the slope of the embankment to be 2 to 1, that is, for 19 feet perpendicular, the base is to be 38 feet. With these data, the cubical quantities, as computed by the valuable Tables of Sir John Macneill,* are as follows:

Excavation........ 81517 yards.

Embankment...... 57081

24436 surplus cutting.

We have an excess of 24436 cubic yards of excavation, which is a quantity far too great. In order, therefore, to make the quantity of cutting and embankment more nearly balance each other, it would be necessary to continue the embankment beyond the point A, which would lengthen the inclination, as shown by the dotted

*

"Tables for calculating the cubic quantity of earthwork in the cuttings and embankments of canals, railways, and turnpike roads." By Sir John Macneill, Civil Engineer, F.R.A.S., &c.

line drawn from the point B to a "; this dotted line would now represent the proposed surface of the road. By such means we diminish the quantity of cutting, and, at the same time, increase that of the embankments; and also by lengthening the inclination, we The alteration of the proposed

reduce its steepness. surface line must be so made, that the cubical quantities of excavation and embankment are nearly equal; leaving, however, a preponderance in favour of the latter of about 10 per cent. to supply the deficiency occasioned by the consolidation and shrinking of the earth; and if any portion of the excess be then remaining, it may be disposed of in flattening the slopes of the embankments, when no more convenient mode presents itself.

The quantities of earthwork on a given section depend upon the arrangement and disposition of the gradients, or proposed surface lines; and there is no practical consideration of more consequence to the engineer, in laying out a proposed line of surface upon a section, especially if it be of any great extent (as the present projected lines of railway), than the most judicious distribution of the cuttings and embankments, which should not only be nearly equal to each other in quantity, but the circumstances must be considered under which the various embankments have to be supplied, it not being alone sufficient that for every hollow on the section there should be a corresponding protuberance, but that such protuberances be advanta

geously situated for filling the hollows; for otherwise. the work assumes a character of difficulty, in consequence of the great additional expense of removing the earth to a considerable distance; and if, in addition, the material has to be conveyed up an ascent, it will be more tedious in the execution.

Knowing the value of practical examples in elementary books, we shall here give the calculations of the above results in full, both by the common method; viz., The Prismoidal Formula, and Sir John Macneill's Tables, by which the saving of labour by the use of the Tables will be made apparent.

Prismoidal Formula.-The area of each end added to four times the middle area, and the sum multiplied by the length divided by 6, will give the solid content. If the measures used in the calculation are yards, the result will be the content in cubic yards; but if they are feet, the result must be divided by 27, to obtain the corresponding number of yards.

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