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No III
History of Mathematics in India from Jaina Sources
65
b and height h, and performing construction and deformation one
obtains the body in fig. 3.
In the figure,
A A BB - 2ab BD B'D'
a-b
BC B'C' AD=A'D'=h
This body is then cut into three parts by vertical planes passing through A and A'. We get the prism ABDD B' A', and two equal tetrohedrons ABDC and A'B'D'C'. The volume of ABDD B'A', which is a prism of height 2 h on a triangular baɛe ABD, is
!BDxADx2 s =(a-b)xhx2b
=bh (a - b)............. (1)
The volume of the two tetrahedrons together is 2 } } BDxBCxAD
= ! (a-b'xa-b) x h =! a ( hixh............
The volume of the frustum is, therefore,
*bh
-
"
(a-b)
..(ii).
bh (a−b)+! * (a−b)' h h3b+Jab-3b+a+b2 - 2abl ha+b+2 abi,
which is the well known formula.
AN INFINITE PROCESS.
The volume of the two tetrahedrons has been directly found out. Each tetrahedron is cut into three parts by drawing horizontal and vertical planes through the middle point G(G) of AB A B (see fig. 4). The bodies BDHEGIF and B'DHGTF being placed one on the other produce a parallelopiped of height & on a rectangular base
with sides
LD= (a - b)
and BF = (-b)
Let K denote the volume of this parallelopiped, i. e..
K= * (a-b).
h
7
= (a - b). h
W
Now cat the four tetrahedons produced in the above construction, each into three parts. by horizontal and vertical planes drawn as
