There are also references in the R.gveda to the basic unit-fractions 1/2 (ardha
or “a half ”), 1/4 (pa ̄daor “a foot,” from the 4 “feet” of a quadruped), 1/8 (s ́apha
or “a hoof,” from the 8 “hooves” of a cow), and 1/16 (kala ̄or “a digit,” from the
16 “digits” of the moon).
The Vedic people seem not to have felt much difficulty in performing the addi-
tion, subtraction, and, perhaps, multiplication of whole numbers, but division
appears to have troubled them when it left a remainder, for only Indra and Vis.n.u
are said to have succeeded in dividing a thousand cows into three. According to
theS ́atapathabra ̄hman.a(3.3.1.13, 4.5.8.1), they correctly obtained 333 for the
quotient with the remainder 1. This story is believed to be implicitly alluded to
already in R.gveda 6.69.8.
By the time of, at latest, the Yajurvedasam.hita ̄, the Indians had extended their
list of the names of powers of ten up to 10^12. In Va ̄jasaneyisam.hita ̄17.2, they are
used for counting the numbers of bricks. In Taittirı ̄yasam.hita ̄7.2.11–20, they
constitute part of the sacrificial formulas (mantras) to be uttered on the occasion
of an annahomaor “food-oblation rite,” which is performed at a certain stage in
theas ́vamedha or “horse-sacrifice rite.” In the annahoma, a priest makes a series
of oblations of ghee, honey, rice, barleycorns, etc. to the fire (agni) called
a ̄havanı ̄yathrough the night until sunrise, while uttering mantras in which a unit
formula, “the dative case of a numeral +sva ̄ha ̄(hail to),” is repeated.
The first mantrareads: “Hail to one, hail to two, hail to three,..., hail to nine-
teen, hail to twenty, hail to twenty-nine, hail to thirty-nine,..., hail to ninety-
nine, hail to one hundred, hail to two hundred, hail to all (sarva).” The numbers
that actually occur in this mantra are:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 29,
39, 49, 59, 69, 79, 89, 99, 100, 200;but this seems to be an abbreviation of a series of natural numbers from 1 to
200 or more. According to the traditional interpretation, “one” represents
Praja ̄pati (“lord of creatures” or creator), the rest being all things in the world
which have evolved from him. The mantras that follow consist of arithmetical
progressions such as odd numbers, even numbers, multiples of 4, etc., some of
which are accompanied by an additional number either at the end or at the
beginning. Then comes the last mantra, which contains the decimal names from
s ́ata(10^2 ) to para ̄rdha (10^12 ): “Hail to s ́ata, hail to sahasra, hail to ayuta,..., hail
topara ̄rdha, hail to the dawn (us.as), hail to the twilight (vyus.t.i), hail to the one
which is going to rise (udes.yat), hail to the one which is rising (udyat), hail to the
one which has just risen (udita), hail to the heaven (svarga), hail to the world
(loka), hail to all.” The four phrases beginning with “hail to the dawn” are uttered
immediately before the sunrise, and the four beginning with “hail to the one
which has just risen” immediately after that. Some regard the last seven
terms, us.astoloka, as the names of 10^13 to 10^19 , but there is no support for this
conjecture.
indian mathematics 361