construction of ritual sites including various altars. It is mostly devoted to the
description of various fire-altars (agni) to be employed in the agni-cayana (“fire-
altar-construction rite”). For the construction of a standard agni, the following
requirements had to be satisfied: (1) an agni consists of five layers of 200 bricks
each with the total height equal to the knee-height of the sacrificer; (2) the odd-
numbered and the even-numbered layers each have one and the same arrange-
ment of bricks, and no brick should coincide with the one above or below it; and
(3) an agni should occupy an area of seven and one half square purus.as.The
area of an agni was increased by one square purus.aeach time the same rite was
repeated by the same sacrificer.
The topics dealt with in the geometric portion of the S ́ulbasu ̄tras of A ̄pastamba
(abbr. A) and Baudha ̄yana (B) are as follows.
Linear measures (B 1.3–21; A provides them in later chapters at need).
Construction of geometric figures (A 1.2–3, 1.7–2.3; B 1.22–44, 46–7).
Relationships between the diagonal and the side of a rectangle (oblong and
square).
Pythagorean Theorem (A 1.4–5, B 1.45, 48–9).
Computation of the diagonal of a square (A 1.6, B 1.61–2).Sum and difference of two squares (A 2.4–6, B 1.50–1).
Equi-area transformation of geometric figures (A 2.7, 3.1–3, cf. 12.5, 12.9,
15.9; B 1.52–60).
Relationship between the area and the side of a square (A 3.4–10).
The tools used for drawing figures in the S ́ulbasu ̄tras are a rope called rajju or
s ́ulba and pegs or posts called s ́an.ku. A bamboo rod is sometimes used instead of
a rope. By means of these tools, one can draw a straight line, cut out a line
segment having any desired length, and draw a circle or an arc with any desired
radius.
No specific rules are given for the drawing of a line, a circle, and an arc; these
can be easily obtained by a rope and pegs. The main problem for the s ́ulba
geometers, who were required to construct geometric figures like a square, a rec-
tangle, and a trapezium, was how to draw a line orthogonal to, or parallel to, a
given line.
The most important mathematical motif of the S ́ulbasu ̄tras is the area. The
core of the S ́ulba mathematics is concerned with the religious requirement that
one should construct altars in varoius shapes with a given area, a requirement
which seems to have originated from agriculture where a harvest greatly
depends on the area of the land (bhu ̄orbhu ̄mi).
The S ́ulbasu ̄tras contain the earliest extant verbal expression of the
Pythagorean Theorem in the world, although it had already been known to
the Old Babylonians. It is stated in exactly the same words by A ̄pastamba (1.4),
Baudha ̄yana (1.48) and Ka ̄tya ̄yana (2.7): “The diagonal rope (aks.n.aya ̄-rajju)
of an oblong produces both which the flank (pa ̄rs ́vama ̄nı ̄) and the horizontal
indian mathematics 363