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CHAPTER VIII.

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Quantities of Materials (Cont'd.)

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III. Volumes from Original and Final Contours.

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2. Grading Over Extended Areas.

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Method of Measurement.

Beginning then with the area
included within the dotted line marked 0.0 and the sides of the lot
and making this the A_{0} of Eq. 20, and
giving each following base its proper subscript up to the dotted area 8.0,
which is the A_{n} of the formula in this case; the successive
A's are traced in the same manner as in all the other cases of the measurement
of Continuous prismoids previously
explained and the final and total desired volume is obtained.
It is to be remembered that, as stated above,
the various areas thus traced are not the *actual* areas cut out by
the parallel cutting planes, since in this case the cutting planes are
not horizontal, but are parallel to the final surface of the lot, but they
are the *horizontal projections* of those areas, and the required
volumes are equal to the *horizontal projections* of those areas thus
cut out multiplied by the *vertical* distance between the cutting
planes, which in this particular case is evidently
one foot.

To simplify the explanation of the method of measurement just described
a simple case was taken as an illustration of
the method, the final surface being an inclined plane, but the extension
of the method to much more complex final forms will be found to present
no more serious difficulties, while the method itself will be found to
relieve all similar forms of calculation which may be treated in like manner
of a very large portion of the time and labor inseparably connected with
such calculation when made by any other of the usual methods.

It is of course readily seen that Table
9 contains all the factors for this form of operation the only correction
necessary in any case being that required for any
value of L other than one foot as the length between sections which
is the value assumed throughout in the calculation of the Table.
The method of making such correction is too evident to need further description.