- INDIA 263
the square root of the product is the area. In our terms this rule says that the area
of a quadrilateral of sides a. b, c, and d is v(s — a)(s — b)(s — c)(s — d), where s is
half of the sum of the lengths of the sides. The case when d = 0, which is a triangle,
is known as Heron's formula. Brahmagupta did not mention the restriction that
the quadrilateral must be a cyclic quadrilateral, that is, it must be inscribed in a
circle.
Like Aryabhata, Brahmagupta knew that what we are calling one- and two-
dimensional ð were the same number. In Stanza 40 he says that when the diameter
and the square of the radius respectively are multiplied by 3, the results are the
"practical" circumference and area. In other words, ð = 3 is a "practical" value.
He also gives the "neat" ("exact") value as \/ºè· Since = 3.1623, this value
is not an improvement on Aryabhata's 3.1416 in terms of accuracy. If one had to
work with ð^2 , however, it might be more convenient. But ð^2 occurs in very few
contexts in mathematics, and none at all in elementary mathematics.
Section 5 of Chapter 12 of the Brahmasphutasiddhanta gives a rule for finding
the volume of a frustum of a rectangular pyramid. In keeping with his approach of
giving approximate rules, Brahmagupta says to take the product of the averages of
the sides of the top and bottom in the two directions, then multiply by the depth.
He calls this result the "practical measure" of the volume, and he knew that this
simple rule gave a volume that was too small.
For his second approximation, which he called the "rough" volume, he took the
average of the areas of the top and bottom and multiplied by the depth.^15 He also
knew that this procedure gave a volume that was too large. The actual volume
lies between the "practical" volume and the "rough" volume, but where? We know
that the actual volume is obtained as a mixture of two parts "practical" and one
part "rough", and so did Brahmagupta. His corrective procedure to give the "neat"
(exact) volume was: Subtract the practical from the rough, divide the difference
by three, then add the quotient to the practical value.
The phrasing of this result cries out for speculation on its origin. Why use the
"practical" volume twice? Why not simply say, "The exact volume is two-thirds
of the practical volume plus one-third of the rough volume"? Surely Brahmagupta
could do this computation as well as we can and could have used this simpler
language. Perhaps his roundabout way of expressing the result reveals the analysis
by which he discovered it. Let us investigate what happens when we subtract the
"practical" volume from the "rough" volume. First of all, since each is merely an
area times the height of the frustum, we are really just subtracting the average
area of two rectangles from the area of the rectangle formed by the averages of
their parallel sides. Let us simplify by taking the case of two squares of sides a and
b. What we are getting, then, is the average of the squares minus the square of the
average:
a^2 + b^2 sa + b\^2
2 ~^2~)
Figure 15 shows immediately that this difference is just the square on side
(a — b)/2. In that figure, half of the squares of sides a and b are set down with
their diagonals in a straight line. The two isosceles right triangles below and to
the right of the dashed lines fit together to form a square of side (a + b)/2. If the
(^15) This is the same procedure followed in the cuneiform tablet BM 85 194, discussed above in
Subsection 2.3.