1112
THEORY ANI> PRACTICE OF ENGINEERING.
Book II.
Considering the complement II Q O M V of the frustum as composed of an infinity ofrectangles A R C D, which have for their base the double or-dinate AD, and for their height the corresponding element EF ofthe right-angled triangle X Y R, we shall find their sums bysubtracting the frustum, which makes the difference from thedemi-cylinder Q O M N RVQ. Supposing then XY=YR=
a,and the half circumference QR V = 6, we shall have for
the solidity of the demi-cylinder; consequently--^——or
Saab — 4a * ^ va ] ue 0 f the complement of the frustum.
6 6
Thus the ratio of these two solids will be -
4a 3
4a
or •
36 —4a 36— 2 a’
3aa6 —4a 3 'which shows that the frustum is to its
2
complement, as the diameter of the circle, is to the difference between the semi-diameterand three-fourths of the circumference. It follows that if we could find the exact value ofthe complement of the frustum, we should have the quadrature of the circle.
To have in figures the ratio of the frustum to its complement, supposing a = 7 and 6 = 22,
consequently ^ ^ ° = which shows that the frustum is to its complement as 14 to 19.
2
Considering also this solid as composed of an infinity ofplanes E, F, G, II, comprised between the double ordinateEH, and the element I K of the triangle PBR, we shallhave the sum of all these planes by multiplying the circleP V Q R by the mean element DC, serving as the axis of thecylinder PL NR, because the frustums OLRM andROMN being equal, the solid we are speaking of will beequal to the cylinder. Cutting this same solid into twoparts by the plane QOMV, which passes through the axisDC, and whose base QV is perpendicular to the diameterP R, the large piece O Q P B M V will be to the lesser as47 to 19.
Supposing Ii P = P R, we shall have B L = L D, and D C= CR; consequently if the frustum O LB Mis expressedby 14, its complement OQRVMwill be so by 19; and asthese two solids together equal the half cylinder OQPLMV,it may be expressed by the sum of the preceding twonumbers, to which adding that of the frustum, we shallOQPRMV, and 19 for the lesser OQRVM.
Fig. 1697. Fig 1698. Fig. 1699.
To have the sum of all the planes C F H L comprised by the double ordinate C L, andthe element C I of the trapezium A B R D, we must, as in the preceding case, multiply thesuperficies of the circle A D by the mean element X Y, serving as the axis of the cylinderA L N D, since this cylinder is equal to the solid of which we are speaking.