From the results of long experience, however, engineers have concluded that computations sufficiently exact for practical purposes can be made upon the hypothesis that _for common mines and those approximating closely to them in form_, the volumes of the craters are directly proportional to the charges used. =3.= In order to apply this rule in practice the volumes of craters formed by known charges must be measured; but since the soil in the immediate vicinity of the crater is more or less disintegrated, and the crater itself is partly filled up by the material which falls back into it, the outlines of the original crater cannot usually be recognized or its exact geometrical figure be determined. Besides, the craters formed under circumstances seemingly identical differ more or less among themselves. For convenience in computation, however, several simple geometrical figures have been assumed as giving with sufficient accuracy the form of the crater of a common mine. See Pl. XI, Fig. 1. Among these Vauban assumed a cone, _ACD_, with its vertex at the centre of the charge; Valière a paraboloid of revolution, _AHD_, with its focus at the centre of the charge; Müller truncated this paraboloid by a horizontal plane through its focus; while Gumpertz and Lebrun adopted the form in common use at their time, and which has been generally accepted since, viz., a frustum of a cone, _AEFD_, the smaller base of which passes through the centre of the charge and has a radius, _EC_, equal to one-half the crater radius, _AB_ (or one-half L. L. R., _CB_). The volumes of these figures are as follows: Vauban’s cone 1.05 (L. L. R.)^3, Valière’s paraboloid 1.90 (L. L. R.)^3, Müller’s truncated paraboloid 1.84 (L. L. R.)^3, The frustum of a cone 1.83 (L. L. R.)^3 = nearly (11/6)(L. L R.)^3. The cone of Vauban (lately assumed also by Höfer) was abandoned as unsatisfactory, because it did not conform to the craters produced, and, as treated by Höfer, because the charges computed by its use were found to be too small (an error in the wrong direction). The paraboloid of Valière or Müller would seem to conform more nearly to the actual shape assumed by the crater; but it will be observed that the volume of the latter is sensibly the same as that of the truncated cone, and as the volume of earth thrown out is the quantity to be considered, the truncated cone will be assumed as the measure for it. =4.= The principle that “the volumes of the craters are proportional to the charges used” is the general statement of the _miner’s rule_. Assume _C_ and _C´_ to represent the charges of two mines whose volumes are _V_ and _V´_, lines of least resistance _l_ and _l´_, and crater radii _r_ and _r´_. Assume also that the craters are frustums of cones, the radii of whose larger bases are twice those of the smaller. Then _C_ : _C´_ :: _V_ : _V´_ :: (11/6)(_lr_^2) : (11/6)(_l´r_´^2), or