Circling the square – Amita Sharma

Nataraj & Sadhu

Who knows for certain?
Who shall here declare it?
Whence was it born, whence came creation?
The gods are later than this world’s formation;
Who then can know the origins of the world?
None knows whence creation arose;
And whether he has or has not made it;
He who surveys it from the lofty skies,
Only he knows — or perhaps he knows not.
– Rig Veda (X:129)

Amita SharmaThe Hymn of Creation is, perhaps, one of the profoundest critical gazes cast on the creator in the history of metaphysical thought, putting under erasure an a priori cognisant principle, manifesting a spirit free to doubt and question. This defines ancient Indian knowledge systems, cohabiting different realms of ‘realities’ generating at one level, the language of sophisticated argument, technical detail and codification, and at another, a language that is tentative, suggestive, pushing the borders of the known.

The genesis of ancient Indian knowledge systems is this coupling of intellectual enquiry with a sense of sublime wonder at the great mysteries of life. Ancient Indian knowledge forms were divided into two broad groups, namely para vidya and apara vidya. Para vidya or higher knowledge is knowledge by which the imperishable is known. Apara vidya encompassed worldly knowledge, like science, technology, arts, commerce and management. To the modern mind, thinking in exclusive categories, dichotomising knowledge forms, these two knowledge worlds are dualistically wedged. But the uniqueness of ancient Indian knowledge systems is the coexistence of the transcendental with the empirical, generating several distinctive features. The sacred and the secular flowed into each other.

There was no Inquisition to be feared, no imposed dogma. Savants built upon inherited knowledge forms while equally questioning them. We find Kautilya disagreeing with earlier thinkers of political science on issues of warfare, or Brahmagupta virulently rejecting Aryabhata’s theory of a rotating earth. A huge literature of commentaries, many of them sadly lost, is evidence of ceaseless and tireless scholarly discussion, debate and dissent

Technical knowledge often arose to serve religious practices. Over time, organised systems emerged dealing with language, philosophy, mathematics, astronomy, medicine, the arts, governance and administration, ethics and yoga and a host of lesser known knowledge systems related to agriculture, animal husbandry, water management, town-planning that were documented in numerous texts rarely studied now.

Today the enmeshing of ideas, of poetry with physics, maths with mantra, science with mysticism, might appear as pre-scientific and, therefore, mere objects of curiosity. Yet when submitted to rigorous tests, they have almost always proved their worth. A few examples illustrate how the complexity of Indian knowledge systems argues for analytical rigour, not a reductionist reading and uncritical rejection. It is not for nothing that 20th century physicists such as Erwin Schrödinger or Werner Heisenberg drew inspiration from Vedantic concepts.

Ancient Indian mathematics arose from the Vedangas — a reflection on the Vedas. The Shulba Sutras, India’s first texts of geometry, composed around 800 – 600 BCE, exemplify traditional epistemological forms and their contribution to modern knowledge. These texts addressed the theological requirement of constructing fire altars with different shapes such as a falcon in flight with curved wings or a tortoise with extended head and legs, but with the same surface area. The altars had to be constructed with five layers of burnt bricks, each layer consisting of 200 bricks and no two adjacent layers having congruent arrangements of bricks. One of the geometric constructions involved squaring the circle (and vice-versa) viz., geometrically constructing a square having the same area as a given circle.

Circling the SquareThe Baudha­yana Shulba Sutra says, a rope stretched along the length of the diagonal produces an area which the vertical and horizontal sides make together. This is a precise geometric expression of the Pythagorean theorem, which states that the sum of the squares of the two sides of a right-angle triangle equals the square of its hypotenuse. Were we more aware of the contribution of Indian geometricians, the Pythagorean theorem might today be equally known as the Shulba theorem.

The Vedic mantras are chanted till today after a ritual prayer. Have we ever noticed the incantation involves large numbers? For example, the mantra at the end of the annahoma (food-oblation rite) performed during the ashvamedha invokes powers of 10 from a hundred to a trillion.

Maths and verse enmeshed not just in style, but in substance. Pingala (300 – 200 BCE), author of the earliest known Sanskrit treatise on prosody, Chandaḥsāstra (the science of metres), gave elaborate rules for listing out all possible combinations of ‘heavy’ (long) and ‘light’ (short) syllables in Vedic metres. In the process, Pingala constructed prastāras or tables which noted the combinations in what would be called today a binary system of notation (for instance, ‘long-long-short … long-short-short’). The science of metres led to the Indian equivalent of the famous Fibonacci numbers (in which each number is the sum of the preceding two: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55…).

A Vedic hymn could have more than one meaning, embedding philosophical speculation with mathematical concepts. A famous shloka from Ishavasya Upanishad reads, purnamadah purnamidam purnat purnamudachyate purnasya purnamadaya purnameva vashishyate: “That is whole; this is whole. From the whole comes the whole; take away the whole from the whole, what remains is the whole.” Mathematically, this can be interpreted in terms of zero as well as infinity, both of which are meanings of purna.

While the zero (variously called shunya, purna, bindu …) as an empty place-holder in the place-value numeral system appears much earlier, algebraic definitions of the zero and its relationship to mathematical functions appear in the mathematical treatises of Brahmagupta in the 7th century whose Brahmasphutasiddhant had basic operations (including cube roots, fractions, ratios and proportions) as well as applied mathematics (including series, plane figures, stacking of bricks, sawing of timber, and piling of grain). His concept “divided by zero = infinity” is etymologically interesting: khachheda means divided by kha, where kha (space) stands for zero.

Consider the way the decimal story unravels. The present system of decimal numbers needed two fundamental discoveries: the concept of zero and the principle of place value. Both were developed in India between the 1st and the 6th centuries CE. The first inscription with a decimal place-value notation is from Sankheda in Gujarat, dated 346 in the Chhedi Era, or 594 CE, where ‘3’ stands for hundreds, ‘4’ for 10s and ‘6’ for units. But five centuries earlier, the Buddhist philosopher Vasumitra, discussing the counting pits of merchants, had remarked, “When [the same] clay counting-piece is in the place of units, it is denoted as one, when in hundreds, one hundred.”

Decimal representation was also employed in a verse composition technique, later labelled bhuta-sankhya (literally, object numbers) used in technical books. Since those were composed in verse, numbers were often represented by objects. The number 4, for example, could be represented by the word Veda (there are four Vedas), the number 32 by the word teeth (a full set consists of 32), and the number 1 by moon, sun or atman, all of which are unique. So, “Veda-teeth-moon” would correspond to 4-32-1 or our decimal numeral 1324, as the convention for numbers then was to enumerate their digits from right to left.

This rich tradition of mathematics flowered in some of the greatest mathematicians of their times who contributed towards discovering and formalising mathematical principles. Aryabhata I (born 476 CE) described fundamental principles of mathematics and astronomy in 121 verses of Aryabhatiya which deal with quadratic equations, trigonometry or the value of π, correct to 4 decimal places. His calculation of the circumference of earth was within 12 per cent of the actual value, his table of planetary positions and his lengths of the sidereal and solar years remarkably precise. Brahmagupta (born 598 CE) studied many geometrical figures, introduced negative numbers and defined mathematical infinity as “that which is divided by zero”.

Bhaskara IIBhaskara II (born 1114), author of Lilavati and Bijaganita, proposed solutions to cubic and biquadratic equations, worked out an efficient algorithm for some types of second-degree indeterminate equations and laid down some of the foundations of calculus.

As in the case of mathematics, early astronomical insights were embedded in sacred texts often veiled in allegorical or poetical forms. The Rig Veda refers to a wheel consisting of 360 spokes, clearly the days of the year; some of its verses have been interpreted in terms of eclipses and meteor showers.

The Aitareya Brahmana (3.44) declares, “The sun never really sets or rises. … Having reached the end of the day, he inverts himself; thus he makes evening below, day above…. Having reached the end of the night he inverts himself; thus he makes day below, night above; He never sets; indeed he never sets.” This seems to reflect an awareness of the sphericity of the earth.

The famous scholar Sayana (c. 1315-1387) commented thus on a hymn to the sun from the Rig Veda (1.50.4): “Thus it is remembered, O Surya, you who traverse 2,202 yojanas in half a nimesha.” With a yojana of about 13.6 km and a nimesha of 16/75th of a second, this amounts to 280,755 km/sec — just 6 per cent from the speed of light (299,792 km/sec) — a coincidence worth noting.

The Puranas describe time units from the infinitesimal truti, lasting (according to Bhaskara II) one 2,916,000,000th of a day or about 30 microseconds, to a mahamantavara of 311 trillion years. Time is seen as cyclical, an endless procession of creation, preservation and dissolution. The end of each kalpa brought about by Shiva’s dance is also the beginning of the next. Rebirth follows destruction. Each Brahma day and each Brahma night lasted a kalpa or 4.32 billion years, adding up to 8.43 billion years which is not too far from the current 13.7 billion years for the age of the universe (another coincidence, which the US astronomer Carl Sagan noted).

In contrast, till the 19th century, much of Europe was convinced that the universe was no more than 6,000 years old. And while we are on coincidences, let us mention that Jain texts state that there are 8.4 million species on earth, which compares well with the figure of 8.7 million arrived at in a 2011 research paper.

Chitrakut

Sacred geographies enfolded astronomical observations. According to the German archaeologist Holger Wanzke, the east-west alignment of the main streets of Mohenjodaro’s citadel (or acropolis) was probably based on the Pleiades star cluster (Krittika), which rose due east at the time; it no longer does because of the precession of the equinoxes. Ujjain, associated with the legendary king Vikramaditya, is located on the Tropic of Cancer; it was a centre of astronomical observations. Chitrakoot is associated with Rama, who is often represented symbolically as an arrow. Mapped with GPS, ashrams and other holy sites there form arrows that point to the sunrise and sunset on the summer solstice. Varanasi’s 14 Aditya shrines precisely track the sun’s path through the year, embedding time in the ancient city’s map.

Surya Shrines Varanasi

The purpose of alluding to diverse forms of cultural codification of knowledge is that even while ambiguity may en-wrap them, they have a steady stream of scientific insights that merit research. The significance of culturally embedded knowledge is supported by our material inheritance.

For the growth of a truly scientific spirit, it is necessary that we critically evaluate our intellectual inheritance. Bhaskara II said, “It is necessary to speak out the truth accurately before those who have implicit faith in tradition. It will be impossible to believe in whatever is said earlier unless every erroneous statement is criticised and condemned.”

In ignoring our own knowledge legacy, or rejecting it as myth, are we guilty of creating an uncritical modernism that stultifies its own growth?

Fields Medal winner, Manjul Bhargava has said that he was inspired not only from ancient Indian mathematicians, but also from his practice of tabla and his knowledge of Sanskrit. The statue of Nataraja, a symbol of the dance of subatomic particles, which adorns the CERN where the hunt for the ‘ultimate’ particles goes on, is a reminder that India’s wisdom offers much to the world in its tireless research into the mysteries of life in its infinite complexity. – Financial Chronicle, 13 April 2015

» Amita Sharma is former additional secretary in the Ministry of Human Resources Development. 

» With inputs from the French-born Indian author Michel Danino, currently guest professor at IIT Gandhinagar.  

» Maps courtesy Rana P.B. Singh & J. McKim Malville.

The SandHI Series | Indian Knowledge Series

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