264 9. MEASUREMENTrectangle that is shaded dark, which lies inside these two isosceles triangles but
outside the squares of sides b and a, is moved inside the square of side á so as to
cover the rectangle that is shaded light, we see that the two isosceles triangles cover
all of the two half-squares except for a square of side (a — b)/2. Since this figure
is a very simple one, it seems likely that Brahmagupta would have known that the
difference between his two estimates of the volume of a square frustum amounted
to the volume of a prism of square base (a - 6)/2 and height h.
But how did he know that he needed to take one-third of this prism, that is,
the volume of a pyramid of the same base and height, and add it to the practical
volume? To answer that question, consider a slight variant of the dissection shown
in Fig. 3. First remove the four pyramids in the corners, each of which has volume
(/é/3)((á-6)/2)^2 , which is exactly one-third of the difference between the gross and
practical volumes. Doing so leaves a square platform with four "ramps" running
down its sides. In our previous dissection we sliced off two of these ramps on
opposite sides and glued them upside down on the other two ramps to make a
"slab" of dimensions á ÷ 6 x h. This time we slice off the outer half of all four ramps
and bend them up to cover their upper halves. The result, shown in Fig. 16, is
the cross-shaped prism of height h whose base is a square of side (a + b)/2 having
a square indentation of side^9 ~ at each corner. Filling in these square prisms
produces the volume that Brahmagupta called the practical measure. The volume
2 2
needed to do so is 4/é((á - b)/4) = h((a - b)/2). Now three of the four pyramids
removed from the corners, taken together, have exactly this much volume. If we
use these three to fill in the practical volume, we have one pyramid left over, and
its volume is one-third of the difference between the rough and practical volumes.
A person who followed the dissection outlined above would then very naturally
describe the volume of the pyramid as the practical volume plus one-third of the
difference between the gross and practical volumes. That would be natural, but it
would be rash to infer that Brahmagupta did imagine this dissection; all we have
shown is that he might have done that.Questions and problems
9.1. Show how it is possible to square the circle using ruler and compass given the
assumption that ð = (16\/2)/7.
9.2. Prove that the implied Egyptian formula for the volume of a frustum of a
square pyramid is correct. If the sides of the upper and lower squares are a and b
and the height is h, the implied formula is:
V = ^(a^2 + ab + b^2 ).9.3. Looking at the Egyptian pyramids, with their layers of brick revealed, now that
most of the marble facing that was originally present has been removed, one can
see that the total number of bricks must be 1 + 4 + 9 Ç h n^2 if the slope (seked)
is constant. Assuming that the Egyptian engineers had the kind of numerical
knowledge that would enable them to find this sum as \n(n + l)(2n + 1), can you
conjecture how they may have arrived at the formula for the volume of a frustum?
Is it significant that in the only example we have for this computation, the height
is 6 units?