It has been conjectured that zero is indicated by the word ks.udra(lit. petty, tri-
fling) in the Atharvaveda (19.22.6=19.23.21), and a negative number by anr.ca
(lit. without a hymn) (ibid.19.23.22). But this conjecture still awaits a proof.
It is probable that the Pythagorean Theorem (the sum of the areas of the
squares on the two sides of a right-angled triangle is equal to that on its
hypotenuse), which was explicitly stated and utilized by the authors of the
S ́ulbasu ̄tras, had already been known at latest in the times of the Bra ̄hman.as.
TheS ́atapathabra ̄hman.a(10.2.3.18) speaks of successive increases of the
areas of the fire altars (agni): the basic fire altar, called the “seven-fold agni,” has
the area of seven and one half square purus.as(onepurus.aor “man” is the height
of the sacrificer with his hands stretched upwards), and it is increased, in sub-
sequent rites performed by the same sacrificer, by one square purus.aeach time
up to the “one hundread and one-fold agni.” The augmentation of area was made
most probably by means of the Pythagorean Theorem.
The inverse case of the Pythagorean Theorem (if the sum of the areas of the
squares on two sides of a triangle is equal to that on the remaining side, then
the first two sides contain a right angle) is also possibly referred to in the
S ́atapathabra ̄hman.a.
TheVa ̄jasaneyisam.hita ̄(30.20) and the Taittirı ̄yabra ̄hman.a(3.4.15) include
the word gan.aka(“a calculator”) in their lists of people to be sacrificed on the
occasion of the purus.a-medha(“human-sacrifice rite”). The gan.akain the former
list, who is sacrificed for the divinity of a sea animal, is usually taken to be an
astrologer, but this interpretation seems not to be decisive at all since an
astrologer called naks.atradars ́a (“a star-gazer”), who is sacrificed for the divinity
of wisdom, is listed prior to it in the same list (30.10). The gan.akain the latter
list, who is sacrificed together with a vı ̄n.a ̄player for the divinity of songs, is pre-
sumably related with music.
I.2 S ́ulbasu ̄tras
S ́ulbasu ̄tras, which constitute part of, or appendices to, the S ́rautasu ̄tras, are basi-
cally practical manuals meant for the preparation of the sites of s ́rauta rituals,
the main topic being the construction of fire altars with burned bricks. A sac-
rificer (yajama ̄na) had to strictly follow the regulations affecting the sizes and
shapes of the sites and altars, because otherwise he was supposed to lose the
merit to be attained through the specific ritual he was performing. Hence follows
the connection between rituals and geometry. But it would be wrong to suppose
only practical geometry like the measurement of land in the S ́ulbasu ̄tras. The
geometry conceived by the s ́u1ba mathematicians already has theoretical
aspects as will be seen below.
The two oldest S ́ulbasu ̄tras, those of A ̄pastamba and Baudha ̄yana, can be each
divided into two parts. The first part treats geometry in general terms. The topics
dealt with in this geometry are certainly related to the construction of ritual
sites, but it contains few words which indicate rituals. The second part treats the
362 takao hayashi