##
CHAPTER VI.

##
Quantities of Materials.

###
I. Volumes from Cross Sections.

###
2. Volumes of Single Prismoids.

###
Method of Plotting the Cross Sections.

It is evident from the definition given above
that, since the area ABCPD is the A_{0},
and the area ABHNG is the A_{1} of our Equations, if we bisect
the distances CH, PN, DG on our diagram and draw the lines FM and ME, the
resulting area ABFME will be the A_{m} of the Equations, since
by so doing we have drawn an area the lengths of whose sides are the means
of the lengths of the corresponding sides of the two end sections A_{0}
and A_{1}.
If, now, we suppose the sum of the three areas A_{0}, 4 A_{m}
and A_{1}, to be say 1 Sq. Inch, and substitute in Eq. 5 we have:

V Cu. Yds. = 39.5061 × (1) ... (6)
or
V = 39.5061 Cu. Yds. ... (7)
Eq. 7 then shows that when the sections of any given prismoid are plotted
to a scale of 1 in. = 8 ft., each sq. in. of actual area represented by
the term (A_{0} + 4 A_{m} + A_{1}) will represent
a volume of 39.5061 Cu. Yds., the prismoid being 100 ft. in length.

If, then, we adjust the Planimeter to that Setting which will cause
the Instrument to record 3950.6 Vernier Units when tracing an actual area
of 1 Sq. inch, and with the instrument so adjusted we trace continuously
and successively the plotted sections as indicated in the expression (A_{0}
+ 4 A_{m} + A_{1})— that is, tracing the area ABCPD once,
the area ABFME four times and the area ABHNG once— it is evident that the
number of Vernier Units recorded for that tracing when multiplied by .01,
the value of the Relative Vernier Unit, will be the number of Cu. Yds.
contained in the given prismoid.