<< Previous Section Table of Contents The Polar Planimeter Page Numbers Next Section >>

CHAPTER VIII.

Quantities of Materials (Cont'd.)

III. Volumes from Original and Final Contours.

2. Grading Over Extended Areas.

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 A0 of Eq. 20, and giving each following base its proper subscript up to the dotted area 8.0, which is the An 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.

<< Previous Section Table of Contents The Polar Planimeter Page Numbers Next Section >>