LAVLESH - Geometry In Ancient India

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Geometry in Ancient India

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 M�ller (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.