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Suppose it is desired to excavate and remove the dirt or rock until the surface of the lot shall be brought to a given grade or surface which shall be level parallel to the front line, and shall have a regular up grade, so that the surface at the rear line of the lot shall be say 8 feet above the surface of the front line— the front line to be at the same elevation as the datum or zero line of the contours. As the lot is taken as being 80 feet square, the rise in grade of the final or finished surface is very evidently one foot in ten.

The lot being 80 ft. deep, it is evident that if we divide the depth of the lot into 8 equal parts, and through each part draw straight lines of the final surface when graded in the manner described, and that the elevation of each final contour above the zero level will be given by the figures in the margin at the ends of each such contour line on the diagram.

The two sets of lines just described are evidently then simply the contours of the original and final surfaces respectively, of the lot, or the lines of intersection of the parallel cutting planes with the original and desired ground surfaces, observing, however, that in this particular case the contour interval of the final contours is 1 ft., while for the original contours it is 2 ft.

It is evident that at each point of intersection of the two sets of contours, since one contour gives the height of the original surface, and the other contour gives the height of the final surface of the ground at that particular point, if we subtract the height of one contour from the heights of the other intersecting contours, the difference will be the depth of the material which must be excavated at that point in order to bring the surface from the original height to the final height.

If now, we make this subtraction at each such point of intersection, and write at each such point the number of feet given by such subtraction, by connecting all points of equal difference by dotted curved lines these dotted lines will evidently be lines of equal cutting, and at every point of each such dotted line the depth of cutting necessary to bring the original surface to the final surface will be the same.

A little consideration of the diagrams as thus
constructed will show that these dotted lines or lines of equal cutting
are also contours of the original surface, which may be considered as the
horizontal projection of the lines of intersection of a series of equidistant
parallel planes passed through the original surface and *parallel to
the designed final surface*— these planes being spaced *one foot apart
vertically*.

It is also seen that these planes have divided up the quantity of material to be excavated necessary to bring the original to the final surface into a number of continuous prismoids. The bases of the various prismoids are evidently the horizontal projections of the areas included within the various dotted lines and the sides of the lot, the excavation being vertical on all the sides of the lot.

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