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K. Arvind 08/24/2017 Glimpse of Indian Mathematics दीर्घचतुरश्रस्य अक्ष्णयारज्जुः पार्श्वमानी तिर्यङ्मानी च यत् A rope stretched along the length of the diagonal produces (The socalled Pythagorean theorem (bhujaa koTi karNa nyaayaH) stated in Prof. K. Ramasubramanian from IIT, Mumbai, delivered a lecture titled “Glimpses of Indian Mathematics: Sutra Style to Paragon of Poetry” at MIT on Sunday, August 13, 2017. In this talk organized by Samskrita Bharati Boston and MIT Samskritam, Prof. K. Ramasubramanian offered an illuminating view of Indian Mathematics over the ages, its unique approach based on poetry rather than prose, the need for such an approach, and presented many surprising facts about mathematical results ranging from algebra and trigonometry through calculus, that were discovered in India much before the West. For example, Fibonacci numbers are described in Bharata’s Natya Shastra. Prof. Ramasubramanian mostly used the work of Pingala (prior to 300 BCE), Bhaskara (12^{th} century CE) and Nityananda (17^{th} century CE), mathematicians from three different periods of history to illustrate his ideas. It was fascinating as well as amusing to learn how complex ideas in Math were taught and transmitted over the millennia via the medium of delightful poetry, in effect giving a musical character to Mathematics! Here are highlights from his talk. Music in Mathematics In most of the world, technical literature is written in prose, while poetry is reserved for subjects involving fantasy and feeling. However, a significant corpus of scientific and mathematical literature in the Indian tradition have been composed in Sanskrit verses that can set to melodious music. This approach was driven by compulsion, because the Indian learning tradition was an oral tradition where ideas are captured and transmitted via sound rather than the written word. There was an element of choice also to this approach because structuring a concept as musical poetry makes it fun and easy to memorize. For example, this verse (link to recitation) captures the date of its composition in musical form (in a meter known as “shaardUla vikriiDita”). The chanted verse actually describes the following set of simultaneous equations: y=m^{2}, t = y/2, v= t x 3, b = v/2, that when solved will yield the exact date of the verse’s composition (April 25, 1629)! This musical method of conveying ideas in Math has been used for representing numbers, specifying the value of pi and representing it as an infinite series, specifying expressions for sums of series, and even computing the derivative of a quotient, among other things! The earliest shaastras were composed in sutra style, for example Panini’s grammar, and Bodhayana’s shulba (measurement) sUtras – see “Pythagorean” theorem above. A sutra is a pithy aphorism and its length is not uniformly regulated, and is not really constrained by any metrical discipline. The rules of prosody (chandas shaastra) were composed later in the 2^{nd} century BCE by Pingalanaga (though Prof. Ramasubramanian believes that Pingala may have predated Panini, based on the coarser style of the sUtras used in chandas shaastra), and consists of two classes of meters, one based on number of syllables (varNa vRRittaH), and another based on the number of beats (maatraa vRRittaH). The later shaastras were set to these poetic meters, and are more musical to hear than sutras. Use of a meter restricts the choice of words that can be used in a verse, and therefore created challenges in conveying mathematical ideas. This resulted in some innovations such as the “khyughRRi” representation invented by Aryabhata, and the number naming system of “bhUta sa~Nkhyaa”. The khyughRRi notation for instance used vowels to denote powers of 10 and consonants for other numbers (e.g., ka=1, ki=100, ku=100000), and created new (but difficult to pronounce) words such as “khyughRRi” (=4.32 million) for numbers used in astronomical calculations. The “bhUta sa~Nkhyaa” approach employs representative ideas to stand in for numbers that these ideas are typically associated with. For example, the word “eye” (netra) represents the number two (we have two eyes), “veda” represents the number four (there are four vedas), and “aakaasha” (space – is empty) the number zero. Further a composer can choose from any of the numerous synonyms of these words which may each have a different number of syllables to meet metrical requirements in a verse. For example, “kham”, “vyoma”, “aakaasha” are all synonyms for “sky” but with different number (1, 2 and 3 respectively) of syllables. An example of “bhUta sa~Nkhyaa” may be found here, where bhU” (earth) represents the number one, and “baaNa” (arrows of manmatha) represents the number five. Mathematics in Music Meters based on syllables and beats lead to innovations that are today known by the names of scientists in the West who also discovered them much later. For example, Pascal’s triangle, which captures binomial coefficients in a triangular array of numbers, is described by the concept of “meru prastaara” (arrangement in the form of a mountain), the construction of which is described by a Sanskrit verse composed by Halaayudha. Prof. Ramasubramanian presents this verse and its translation here. Each level of the “meru prastaara” captures the number and types of meters that are possible for a given number of syllables. When the number of beats is fixed instead of the number of syllables, and a rhythm is constructed either using a set of ‘laghu’ (1 beat long) or ‘guru’ (2 beats long) units, determining the number of rhythms possible leads to the discovery of what is known as the Fibonacci sequence in the west. Prof. Manju Bhargava from Princeton University, the young winner of the 2014 Fields Medal (considered by many as the Nobel Prize equivalent in Mathematics), who as a tabla player encountered this problem of number of rhythms given a fixed number of beats, popularized the fact that this problem has been solved before Fibonacci (1202 CE) by an Indian Mathematician known as Hemachandra (1150 CE). Fibonacci numbers are now also referred to as “Hemachandra numbers” by many. Prof. Ramasubramanian pointed out that there are mathematicians even before Hemachandra, such as Virahanka (600 CE), Pingala (200 CE), and Bharata (100 CE) who described the same idea. In fact, these numbers are described in Bharata’s “naaTya shaastra”. To add to the fun, the Hemachandra numbers can be derived from the “meru prastaara” by adding along the diagonals. Musical Mathematics Prof. Ramasubramanian also presented verses from Bhaskaracharya’s Lilavati, that has problems at the level of high school algebra expressed as poetry. One verse dealt with a problem involving a single variable linear equation. This verse expresses what would now be called a “word problem”, but with a poetic touch and describes a collection of bees split divided into various fractions groups. Here is a link to the recitation of the verse. Another verse described a “word problem” involving a quadratic equation. The problem uses a setting in the “Mahabharata” war and deals with the number of arrows that Arjuna needs to discharge to kill Karna on the battlefield (link to recitation). A more interesting verse (link to recitation) specifies not only the sum (sankalita) of the first ‘n’ natural numbers (1+2+3+…+n = n(n+1)/2), but also the sum (sankalitaikya) of such sums: (1 + (1+2) + (1+2+3) + … (1+2+3+…n) = n(n+1)(n+2)/1.2.3). Trigonometry was important for astronomical calculations. Prof. Ramasubramanian described the work of Nityananda, who was a brilliant astronomer in the court of Mughal Emperor Shaj Jahan, and the author of the monumental treatise “Sarvasiddhaantaraaja” (1639 CE). In addition to original contributions, Nityananda also absorbed ideas from various sources including Arabic Astronomy and Mathematics, and incorporated them into his Sanskrit works. Nityananda devoted 65 verses to trigonometric sine formulae, provided a summary of Arab mathematician AlKashi’s method for determining the sine of 1 degree, and an original procedure to solve the cubic equation that arises in AlKashi’s method, all in verse form. The sine formulae that he describes includes the wellknown expansion for sin(a+b) = sin(a)cos(b) + cos(a)sin(b), but in verse form. Prof. Ramasubramanian also mentioned the work of other earlier Indian mathematicians who came up with methods of computing sines, such as Aryabhata’s recurrence relation and Bhaskara’s approximation function, He also pointed out how infinite series expansions for trigonometric functions and pi  the ratio of the circumference to the diameter of a circle, and the idea of a derivative defined in calculus appear in the work of Indian mathematicians a few centuries before Newton and Leibniz. The Speaker Prof. K. Ramasubramanian is at IIT Mumbai in the Cell for Indian Science and Technology in Sanskrit, Department of Humanities and Social Sciences. He holds a doctorate in Theoretical Physics, Bachelors in Engineering, and Masters in Samskritam. His research interests include Indian Science and Technology and other disciplines such as Indian Logic and Philosophy. Links: a video recording of this fascinating talk can be found at this link. Slides from the talk may be viewed at this link.
चतुरधिकं शतमष्टगुणं द्विषष्टिस्तथा सहस्राणां  (Aryabhata’s (499 CE) verse describing the approximation correct to 4 decimal places of pi
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