THE COMPASS CHRONICLES. The square root of evil

In 1881, the oldest extant manuscript in Indian mathematics was discovered 50 miles from Peshawar in the village of Bakhshali. We speculate that this copy was created between 700-1200CE, even though some of the ideas could be as old as 200BCE. The author declares, “This has been written by the son of Chajaka, a brâhmana and king of mathematicians, for the sake of Hasika, son of Vasistha, in order that it may be used by his descendants.”

The above testimonial provides the enlightened view of a culture in which mathematics is highly respected, compelling us to examine its relationship with society. The invention of zero in India made possible the positional (or ‘place value’) number system, a huge event in world history. In such a system, the same digit in a different position, like one in 100 or 110000001, means different quantities. Subsequent thinkers tried to place this discovery in a social context; a good example of which is the Yoga Sûtra 3:13 of Patañjali (5th century): “Just as a line in the hundreds place means a hundred, in the tens place ten, and one in the ones place, so one and the same woman is called mother, daughter, and sister by different people.”

The Bakhshali manuscript contains an early method to compute square roots. Much earlier, in Greece around 600BCE the pursuit of a square’s diagonal length may have led the cult of Pythagoras to encounter the square root of two, which is an irrational number. Legend has it that irrational numbers went against their religious beliefs, leading to outrage and murder. The inability to comprehend mathematical ideas leads to confusion, and a great confusion often leads to catastrophe.

At the start of the first millennium CE, the science of Greek antiquity was squeezed through various political tumults that followed the rise of Byzantine and Islam, constantly reshaping Europe and Asia. Arabic translations of the Greek classics, and their own scientific temperament played a pivotal role in saving that knowledge. Most world religions contain some form or the other of mathematical mysticism; but no other religion embodies geometry more than Islam. Since it couldn’t use human images to depict celestial forces, Islam began to visualise the crystalline fabric of space time itself. As Greek knowledge passed through the sieve of Islamic thought, it was transformed.

The writings of Plato, in particular, were interpreted widely in the Arab world by a secret society called Ikhwãn al-safã, or The Brethren of Purity. Written in Basra (Iraq) in the 10th century CE, this rasa’il (epistle) from a scientific encyclopaedia compiled by them states their intention clearly: “Know, oh brother… that the study of sensible geometry leads to skill in all the practical arts, while the study of intelligible geometry leads to skill in the intellectual arts because this science is one of the gates through which we move to the knowledge of the essence of the soul, and that is the root of all knowledge...”

The Brethren may have been very secretive because their outlook was very radical for that age, or even revolutionary and futuristic. They define a perfect man as — “of East Persian derivation, of Arabic faith, Iraqi in education, Hebrew in astuteness, a disciple of Christ in conduct, as pious as a Syrian monk, a Greek in natural sciences, an Indian in the interpretation of mysteries and, above all, a Sufi or a mystic in his whole spiritual outlook.” There is a lot of mystery surrounding who the Ikhwãn al-safã really were because they wrote their texts anonymously.

A similar situation exists with Patañjali, who is often confused with Panini and Pingala, the author of Chhanda Shastra, a mathematical theory of rhythm in poetry. While Islam sought the root of knowledge in geometry, the Indians sought it in language and grammar. Professor of mathematics Jayant Shah writes, “A remarkable example of the mathematical spirit of Pingala’s work is his computation of the powers of two. He provides an efficient recursive algorithm based on what computer scientists now call the divide-and-conquer strategy.”

Pingala’s treatise also contains an idea known as Matra Meru (The Mountain of Cadence), which is taught to us as Pascal’s triangle. Certain diagonals of this triangle sum to the numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181… better known as the Fibonacci sequence, which ascends rather rapidly like a mountain would.

If each person in the world were to demand an eye for an eye, the pile of eyeballs on the ground would increase in this sequence. In merely the 49th mass extraction, the pile’s count would exceed the population of the world, and in the next round… the entire world would turn blind.

(This column is dedicated to the children who died in the terrorist attack in Peshawar on December 16, 2014.)

Rohit Gupta explores the history of science as Compasswallah

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