Hindu Geometry - Part 1

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Aniruddha Avanipal

Hindu Geometry - Part 1

 

Introduction

In ancient India, geometry was known as 'Shulba' or 'Rajju'. The

word 'shulba' itself is derived from the Sanskrit root 'shulb'

meaning 'to measure' and hence its etymological significance

is 'measuring' or 'act of measurement'. The word 'Rajju' literally

means a 'rope' or an instrument of measuring. The rules of geometry

itself were known as 'Shulba-Sutra', the Sanskrit word 'sutra'

meaning 'aphorism' or a short rule.

 

Shulba-Sutras or the rules of geometry were instrumental for the

Vedic Hindus to construct altars. Those altars used to vary in their

shapes and sizes and acted as symbolic representations for higher

spiritual conceptions invented by Vedic Sages. From Shulba-sutras we

get a glimpse of advanced knowledge of geometry that the Vedic Hindus

developed.

 

The root of the Shulba-Sutras

The Vedas which were composed between 4000 BCE to 3100 BCE are not

only ancient repository of spiritual Hindu knowledge but also contain

in-depth analysis of physical sciences such as Mathematics, Physics,

Astronomy, Architecture etc. The Vedas also have six supplementary

texts known as the 'Vedangas' or limbs of the Vedas. One of these

is 'Kalpa-Sutra-Vedanga' or set of rules related to rituals or

ceremonies. The Kalpa-sutras are broadly divided in two classes,

the 'Grhya-sutras' or the rules for ceremonies relating to family or

domesti affairs such as marriage, birth etc and the 'Shrauta-sutras'

or the rules for ceremonies ordained by Vedas. The shulba-sutras

belong to the latter class.

 

Patanjali, the great writer of 'Yoga Sutra', stated that there are as

many as 1131 to 1137 different schools of the Vedas. In his own

words, "There were 21 different schools of the Rig-Veda; 101 schools

of the Yajur-Veda; 1000 of the Sama-Veda; and 9 or 15 of the Atharva

Veda". Each school of the Veda had its own 'Shrauta-sutra' and hence

probably its own Shulba. It appears there were numerous manuals of

geometry in ancient India. But many of them are lost and at present

we know of only seven distinct Shulba-sutras compiled by Baudhayana,

Apastamba, Katyayana, Manava, Varaha and Vadhula.

 

Dating of Hindu Geometry

Before providing the actual dates of Shulba-sutra, it is crucial to

point out that many of the important development in India have been

dated by the western historians based on the foundation laid by the

erroneous and mythical "Aryan Invasion Theory". Max Müller (1823-

1903), who was largely responsible for this theory based it on pure

conjectures rather than on scientific proofs. He concluded that

northern India was invaded and conquered by nomadic, light-skinned

RACE of a people called 'Aryans' who descended from Central Asia (or

some unknown land ?) around 1500 BC, and destroyed an earlier and

more advanced civilization of the people habitated in the Indus

Valley and imposed upon them their culture and language. According to

this erroneous theory, Rig Veda was composed around 1200 BCE.

 

However, Aryan Invasion theory has been proved to be entirely

mythical and baseless by modern historians like David Frawley, N S

Rajaram, N Jha, Dinesh Agrawal etc. A complete description on the

myth of Aryan Invasion Theory is outside the scope of this article.

Please refer to the 'reference section' of this article for a

comprehensive description of the scientific, archaeological and

logical proofs disproving the Aryan Invasion theory.

 

Through their ground-breaking work, historians like Frawley and

Rajaram not only proved the baslessness of Aryan Invasion Theory, but

also established correct dating of many important developments of

Hindu civilization.

 

The correct dates of some of those land-mark incidents are listed

below:

 

 

4000 BCE: Rig Veda

3700 BCE: Battle of Ten Kings (referred to in the Rig Veda) Beginning

of Puranic dynastic lists: Agastya, the messenger of Vedic religion

in the Dravida country. Vasistha, his younger brother, author of

Vedic works. Rama and Ramayana

3600 BCE: Yajur-, Sama-, Atharvaveda: Completion of Vedic Canon. 3100

BCE: Age of Krishna and Vyasa. Mahabharata War. Early Mahabharata.

3000 BCE: Shatapathabrahmana, Shulbasutras, Yajnavalkyasutra, Panini,

author of the Ashtadhyayi, Yaska, author of the Nirukta

 

As we can see, literature like Shulba sutras and Shatapatha Brahmana

date back to as far as 3000 BCE. Hence Shulba of Hindus which

includes both 2-dimensional and 3-dimensional geometry is almost 5000

years old.

 

Influence of Hindu Geometry on Greeks

In his monumental work, The origin of mathematics, Archive for

History of Exact Sciences. vol. 18, 301-342, A. Seidenberg remarks:

 

"By examining the evidence in the Shatapatha Brahmana, we now know

that Indian geometry predates Greek geometry by centuries. For

example, the earth was represented by a circular altar and the

heavens were represented by a square altar and the ritual consisted

of converting the circle into a square of an identical area. There we

see the beginnings of geometry!

 

Two aspects of the 'Pythagoras' theorem are described in the Vedic

literature. One aspect is purely algebraic that presents numbers

a,b,c for which the sum of the squares of the first two equals the

square of the third. The second is the geometric, according to which

the sum of the areas of two square areas of different size is equal

to another square. The Babylonians knew the algebraic aspect of this

theorem as early as 1700 BCE, but they did not seem to know the

geometric aspect. The Shatapatha Brahmana, which precedes the age of

Pythagoras, knows both aspects. Therefore, the Indians could not have

learnt it from the Old-Babylonians or the Greeks, who claim to have

rediscovered the result only with Pythagoras. India is thus the

cradle of the knowledge of geometry and mathematics."

 

So, contrary to the European belief that Hindus were influenced by

the Greek geometry, the facts prove that it is the other way round.

Most of the aspects of planar geometry described by Euclid and other

Greek mathematicians were already known to Indians at least 2500

years before the Greeks. In fact, there are proofs which hint towards

the fact Greeks were influenced by the ancient Hindu Mathematics and

Geometry. Bibhuti Bhushan Datta in his book "Ancient Hindu Geometry"

states:

 

"...One who was well versed in that science was called in ancient

India as samkhyajna (the expert of numbers), parimanajna (the expert

in measuring), sama-sutra-niranchaka (Uinform-rope-stretcher), Shulba-

vid (the expert in Shulba) and Shulba-pariprcchaka (the inquirer into

the Shulba).

 

Of these term, viz, 'sama-sutra-niranchaka' perhaps deserves more

particular notice. For we find an almost identical

term, 'harpedonaptae' (rope-stretcher) appearing in the writings of

the Greek Democritos (c. 440 BC). It seems to be an instance of Hindu

influence on Greek geometry. For the idea in that Greek term is

neither of the Greeks nor of their acknowledged teachers in the

science of geometry, the Egyptians, but it is characteristically of

Hindu origin."

 

I will conclude this part of the article by pointing out the fact

that the English word 'Geometry' has a Greek root which itself is

derived from the Sanskrit word 'Jyamiti'. In Sanskrit 'Jya' means an

arc or curve and 'Miti' means correct perception or measurement.

 

(To be continued...)

 

References:

 

The Myth of Aryan Invasion of India by Dr. David Frawley

Aryan Invasion (published in Hindustan Times) by N S Rajaram

Demise of the Aryan Invasion Theory by Dr. Dinesh Agarwal

Questioning the Aryan Invasion Theory and Revising Ancient Indian

History by Klaus Klostermaier

World's oldest writing (Part 1): Vedic origins by N S Rajaram

World's Oldest Writing (Part 2): Challenge to Orthodoxy N S Rajaram