1. To find the quantity of earth required for the parapet, and banquette of a field work, &c.
Divide the parapet, and banquette into trapezoids, and triangles; compute the contents of each separately (by the rules in Mensuration of Planes) and the sum of them will be the superficial content of a section of the parapet, and banquette. Multiply this by the length of the periphery of the redoubt, battery, &c., for the solid content of the parapet, and banquette.
2. To find, rapidly, the quantity of earth required for a parapet, and banquette.
Multiply the height of the crest of the parapet, into the sum of the bases of the superior, and exterior slopes; which will give the superficial content, very nearly.
3. To compute the superficial content of the ditch.
Multiply the depth into the breadth at bottom, to which product add the areas of the escarp, and counterscarp, for the content required.
4. To find the breadth of the ditch, of the usual form.
Divide the area of the section of the parapet by the intended depth of the ditch, and the quotient will be the mean breadth of the ditch; to this add half the sum of the bases of the slopes of the escarp, and counterscarp, for the breadth at top, and deduct the same for the breadth at bottom.
5. To find the breadth of the ditch, having a triangular section.
Divide the area of the section of the parapet by half the given depth of the ditch, and the quotient will be the required breadth at the top.