(tiryan.ma ̄nı ̄)·ropesÒ produce separately.” What “the ropes produce” are not
explicitly mentioned, but are no doubt the square areas constructed on them.
Highlights of the s ́ulba mathematics are transformations of geometric figures
with their areas kept unchanged. This theme was originally related with the
practical requirement of drawing altars in various shapes with a given area, but
the S ́ulba makers seem to have taken a step forward and tried to treat the theme
with a theoretical perspective in mind.
Baudha ̄yana deals with seven transformations. Five out of them are con-
cerned with transformations from a square to a circle, a rectangle, an isosceles
trapezium, an isosceles triangle, and a rhombus, while the other two are those
from a rectangle and a circle to a square. All these, except the last one, can be
put into the following scheme for constructing various figures.
a rectangle Æa square Æa circle, etc.This scheme consists of three steps. Given an area A:
1 Construct a rectangle having the area A.
2 Transform it into a square having the area A.
3 Transform it into the desired figure (a circle, etc.) having the area A.
For example, in order to draw a circle with the area A, one first constructs a
rectangle with the two sides aandA/afor any rational number a, and then trans-
forms the rectangle into a square and the square obtained into a circle. In fact,
for the s ́ulba mathematicians, this was the most natural way of constructing a
circle with a given area, since they did not know the formula, A=pr^2 , from which
we would compute rto draw a circle with the radius r.
An apparent gap between s ́ulba mathematics and later Indian mathematics
used to puzzle scholars, and this made some suppose Western influence upon the
latter. But now we know some links between the two.
The word, karan.ı ̄, used in later mathematics to denote a square number and
the square root of a number, originated from its use in the S ́ulbasu ̄tras.
In order to draw a line measured by , Brahmagupta constructs an isosce-
les triangle, whose base and two sides are measured respectively by (n/m-m)
and by (n/m+m)/2, where mis any optional number. Its perpendicular is the
line to be obtained (Bra ̄hmasphut.asiddha ̄nta18.37). This is a generalization of
Ka ̄tya ̄yana’s rule (6.7), where m=1.
The square nature or an indeterminate equation of the second degree of the
type, Px^2 +t=y^2 , which was to be investigated in detail by Brahmagupta and
others, presumably has its origin in the s ́ulba mathematicians’ investigation of
the diagonal of a square or.
Two root-approximation formulas used in Jaina works and in the Bakhsha ̄lı ̄
Manuscript, seem to have been obtained by algebraically (or numerically) inter-
pretingS ́ulbasu ̄trasgeometric rules for the rectangle-squaring transformation.
2n364 takao hayashi