Eternal India Encyclopedia

Eternal India

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VEDIC MATHEMATICS

Ancient Concepts, Sciences & Systems

Baudhayana's method of designing spoked wheels by successive

circling of concentric squares. Any number of spokes may be placed

by dividing one of the outer squares into the desired number of parts

before circling them. It is one of the most interesting mathematical

examples from antiquity.

BACKGROUND

Vedic mathematics as currently used is often an imprecise term,

referring to Indian mathematics from the hoariest antiquity to rela-

tively modem works that claim the Vedas for their inspiration. But

properly speaking, the term can only apply to the mathematics found

in the

Vedas

themselves and to the technical works that followed

immediately upon the redaction of the

Vedas

into the four-fold divi-

sion, the form in which they are currently found. These early post-

Vedic works, known as the

Sulbasutras

(or the Sulbas) contain

mathematical formulas and geometrical constructions necessary for

Vedic rituals. Thus as the American mathematician and historian

Seidenberg has noted, the Sulbas preserve the religious origins of

mathematics. The present article is concerned primarily with the

mathematics of the Sulbas which may be taken as being synony-

mous with Vedic mathematics. Vedic astronomy which also pre-

supposes considerable knowledge of mathematics is not discussed.

ORIGINS

The evidence for the existence of mathematics as part of Vedic

rituals is both direct and indirect. Until recently, the wholly unten-

able belief that the sites of the so-called Indus Valley Civilisation

(c.2700-1800 BCE) must have belonged to a pre-Vedic society has

been an artificial barrier to a proper appreciation of ancient Indian

mathematics. In addition, a strong tendency on the part of nine-

teenth century historians to trace all technical knowledge to Py-

thagorean Greece has led to irreconcilable contradictions in the

history and chronology of ancient mathematics. The very existence

of elaborately planned cities like Harappa, Mohenjo-Daro and

many other sites going back to the third millennium before Christ is

evidence of considerable knowledge of geometry at least two thou-

sand years before Pythagoras. But perhaps more interestingly, the

so-called Harappan sites from Baluchistan to Lothal in Gujarat to

Eastern Uttar Pradesh have yielded

Yajnashalas,

or sacrificial

altars of the kind prescribed in the Vedic literature, and whose

method of construction with the necessary mathematical details are

found given in the Sulbasutras. Thus, from its very beginnings,

mathematics arose from the needs of the religion and ritual, gradu-

ally giving rise to secular applications like architecture and town

planning. This is precisely how it is described in Indian tradition

also.

The Vedas being religious literature have little directly to say

about mathematics but there is enough even in the ancient

Rig Veda

suggesting that the mathematical knowledge of its composers was

more than rudimentary. In the

Rig Veda

(ii. 18.4-6) we find the

series 2, 4, 6, 8, 20, 30, 40, 50, 60, 70, 80, 90, 100. There are

numerous examples of seasons and the year being expressed in

terms of numbers of days and half days. Interestingly, both the

word

Aditi

(infinite) and

Kham

(zero) are found in the Vedic

literature. There is also evidence of the use of a limited decimal

system consisting of multiples of ten. Terms like

dasa

(10),

sata

(100),

sahasra

(1,000),

ayuta

(10,000) and others are known. Most

remarkably the number paradha equal to 10

or a trillion is also

known. The very first verse of the

Atharvaveda

refers to

trisaptah,

or all types of combinations of threes and sevens like: 3+7=10,

3x7=21,3+5+7=15 and many others. But of the famous Indian con-

tribution, the use of the zero for the place value notation, there

seems to be no signs.

Thus even going back to Vedic times, the knowledge of numbers

and arithmetic is found to be of no mean order. There are terms

describing different aspects of various fire altars that presuppose

basic knowledge of geometry also. The spoked wheel described in

minute detail in the

Rig Veda

(1.164. 11-15) shows knowledge of

circles, radii and methods of drawing them. This is also confirmed

by devices from drawing circles dating back to at least 2500 BC

found and described by Mackay. Thus the germs of both arithmetic

and geometry taken to such heights in the Sulba works of Baudhay-

ana and his schools are already found in the Vedas themselves.

THE SULBASUTRAS

With the Sulbasutras, or the Sulbas as they are more commonly

known, we reach a level of comprehensive mathematical exposition

not to be attained until the time of the Greeks. The word Sulba in

Sanskrit means a rope or cord, and may be derived from the root sulb

or sulv meaning to measure. The Sulbas are concerned primarily

with the mathematical details involved in the construction of sacri-

ficial altars as prescribed in the Vedas and the Brahmanas. They

always appear as appendixes to the Srauta or the ritual part of the

Kalpasutras —

religious works. Nevertheless, the Sulbas are

enormously interesting mathematical works and the earliest com-

prehensive treatises of their kind found anywhere. The Sulba of

Baudhayana

in particular is a work of great perfection.

In style and content, the Sulbas may be described as texts of

geometric algebra: a problem is stated in geometric terms, but its

solution is given in a form that combines geometry and algebra.

Two problems dominate the Sulbas: the square on the diagonal

theorem and the problem of equivalence of area. The square on the

diagonal theorem, which is more often known in its triangular

formulation as the ‘Theorem of Pythagoras’ is derived by