tonnages is to calculate
the volume and value of each block separately. Under the law of
averages, the multiplicity of these blocks tends in proportion
to their number to compensate the percentage of error which might
arise in the sampling or estimating of any particular one. The
shapes of these blocks, on longitudinal section, are often not
regular geometrical figures. As a matter of practice, however, they
can be subdivided into such figures that the total will approximate
the whole with sufficient closeness for calculations of their areas.

The average width of the ore in any particular block is the arithmetical
mean of the width of the sample sections in it,[*] if the samples be
an equal distance apart. If they are not equidistant, the average
width is the sum of the areas between samples, divided by the total
length sampled. The cubic foot contents of a particular block is
obviously the width multiplied by the area of its longitudinal
section.

[Footnote *: This is not strictly true unless the sum of the widths
of the two end-sections be divided by two and the result incorporated
in calculating the means. In a long series that error is of little
importance.]

The ratio of cubic feet to tons depends on the specific gravity
of the ore, its porosity, and moisture. The variability of ores
throughout the mine in all these particulars renders any method
of calculation simply an approximation in the end. The factors
which must remain unknown necessarily lead the engineer to the
provision of a margin of safety, which makes mathematical refinement
and algebraic formulae ridiculous.

There are in general three methods of determination of the specific
volume of ores:--

_First_, by finding the true specific gravity of a sufficient number
of representative specimens; this, however, would not account for
the larger voids in the ore-body and in any event, to be anything
like accurate, would be as expensive as sampling and is therefore
of little more than academic interest.

_Second_, by determining the weight of quantities broken from measured
spaces. This also would require several tests from different portions
of the mine, and, in examinations, is usually inconvenient and
difficult. Yet it is necessary in cases of unusual materials, such
as leached gossans, and it is desirable to have it done sooner
or later in going mines, as a check.

_Third_, by an approximation based upon a calculation from the
specific graviti