**Dr B. Lakshmi**

Researcher, Department of Mathematics, Prayoga, Bengaluru

**Acharya Pingala -** an ancient Indian mathematician was the first inventor of the Binary number system and the Fibonacci sequence authorized in his book **Chandahsastra**.

“I loved Indian Mathematicians to such an extent above all others that I completely devoted myself to them’" - By Leonardo Pisano Bigollo (Fibonacci)

We all know that the Fibonacci sequence are the mysterious collection of numbers stand for 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233… generally represented by a mathematical equation where n ≥ 3.

This sequence of numbers are found in many biological environments, including the arrangement of petals of flowers, branching of trees, arrangement of leaves on stems, arrangement of sunflower seeds, spiral geometries in sea creatures and also branching proportions of the human body. These sequences of numbers are believed to be first discovered by the Italian Mathematician Leonardo Pisano Bigollo also called Fibonacci (1170-1250). But it's fascinating to note that this was well known to Ancient Hindus even before Leonardo’s time. The first expert on metrical sciences in India whose writings demonstrate the understanding of the Fibonacci sequence is **Acharya Pingala**, who referred this sequence of numbers as **Maathra Meru.**

**About Acharya Pingala**

*Fig.1: **Pingala** - the author of Chandaḥśāstra (Chandaḥsūtra)*

Acharya Pingala was an ancient Indian poet and Mathematician who lived around 300 BCE. Sadgurasisya in his commentary on Rganu Kramani (1187 A.D) refers to Pingala as a young brother of Palini. It is speculated that he might have been born on the west coast of India. The phrase in Panchatantra **[Ref 6]** indirectly supports the claim that he resided close to the coast and it is also stated in one of the phrases of Panchatantra that,

*‘Chandognananidhim jaghana makaro velatate Pingalam’*

Means Pingala, the repository of the knowledge of meters was killed by a crocodile on the sea shore.

**About Pingala’s work**

**Acharya Pingala **was the first authority on the metrical sciences whose book named **Chandahsastra **indicates the method of formation of Pascal’s triangle, Binary number system, and the Fibonacci sequence. Chandahsastra (Pingala sutra) means the science of meters or chandah, it is the timing for pronouncing syllables. The book contains **315 sutras** distributed over eight chapters. The Sanskrit prosody consists of two basic units: **Laghu** (single matra) and **Guru** (two matras). The systematic enumeration of chandas with predetermined arrangements of Laghu (short syllables) and Guru (long syllables) syllables are described in Chandahsastra as the first instance of a binary number system.

The prosodist and mathematician **Virahankara** (who lived between A.D 600 and 800) was the first to describe Pingala’s sutras. He established the Fibonacci numbers and the method for their formation. Later **Halayudha**, a mathematician who lived in the 10th century, wrote the commentary on Pingala’s Chandahsastra. In his comprehensible commentary, he describes the formation of Pingala’s Meru Prastara (Means Mountain with peak with numbers). A selected portion of Halayudha’s commentary on Pingala’s Chandahsastra is given below **[Ref 1]**. Following each step of commentary results in the formation of Meru Prastara or Pascal’s triangle which intern also gives Maatra Meru or Fibonacci sequence.

**Sutra -1**

अतोऽनेकद्वित्रिलघुक्रियासिद्यर्थयावदभिमतंप्रथमप्रस्तारवन्मेरुप्रस्तारंदर्शयति- परेपूर्णमिति॥८।३५॥

उपरिष्टादेकंचतुरस्रकोष्ठंलिखित्वातस्याधस्तादुभयतोऽर्धनिष्क्रान्तंकोष्ठकद्वयंलिखेत्।

तस्याप्यधस्तात्रयंतस्याप्यधस्ताच्चतुष्टयंयावदभिमतंस्थानमितिमेरुप्रस्तारः॥

**Meaning**: Two squares are drawn below the top of one (Prathama prasthava) square, with half of each square extending on either side. Three squares are drawn below it, followed by four squares, and so on until the required pyramid is achieved. This is shown in **Fig.2**

*Fig.2: **Development of Pingala’s Meru prastara*

**Sutra -2**

तस्यप्रथमेकोष्ठेएकसंख्यांव्यवस्थाप्यलक्षणमिदंप्रवर्तयेत्।तत्रपरेकोष्ठेयद्वृत्तसंख्याजातंतत्पूर्वकोष्टयोःपूर्णंनिवेशयेत्।

तत्रोभयोःकोष्ठकयोरेकैकमंगंदद्यात्, मध्येकोष्ठेतुपरकोष्टद्वयांकमेकीकृत्यपूर्णनिवेशयेदितिपूर्णशब्दार्थः।

**Meaning**: 'One' symbol needs to be written in the first square. Then, one has to be positioned in each of the two squares of the second line figure. One is then to be positioned on each of the two extreme squares in the third line. One is then to be positioned on each of the two extreme squares in the fourth line.

**Sutra -3**

चतुर्थ्यांपंक्तावपिपर्यन्तकोष्ठयोरेकैकमेवस्थापयेत्।मध्यमकोष्ठयोस्तुपरकोष्ठद्वयांकमेकीकृत्यपूर्णंत्रिसंख्यारूपंस्थापयेत्।

उत्तरत्राप्ययमेवन्यासः।तत्रद्विकोष्ठायांपंक्तौएकाक्षरस्यविन्यासः।तत्रैकगुर्वेकलघुवृत्तंभवति।तृतीयायांपंक्तौद्वयक्षरस्यप्रस्तारः।

**Meaning:** In the middle square of the third line the sum of the figures in the two squares immediately above is to be placed. In each of the two middle squares immediately above, three is placed. Subsequent squares are filled in this way. Thus, the second line gives the expansion of combinations of syllables: the third line the same for two syllables, the fourth line for three syllables and so on.

Following these steps, finally establishes the **Pingala’s Meeru Prastara** which is known as **Pascal’s triangle** as shown in Fig.3

*Fig.3: **Pingala’s Meeru Prastara (Pascal’s triangle)*

**Sutra-4**

तत्रैकंसर्वगुरु,द्वेएकलघुनी, एकंसर्वलघ्वितिकोष्ठक्रमेणवृत्तानिभवन्ति॥चतुर्थ्यांपंक्तौत्यक्षरस्यप्रस्तारः।

तत्रैकंसर्वगुरुत्रीण्येकलघूनित्रीणिद्विलघूनिएकंसर्वलघु॥तथापंचमादिपंक्तावपिसर्वगुर्वादिसर्वलघ्वन्तमेकद्वयादिलघुद्रष्टव्यमिति॥

**Meaning:** The above lines of the Halayudha commentary clearly states that the second line of the Meru Prastara (Fig.2) is the expansion of a meter with one syllable. The third line of a meter is expansion of two syllabus and the fourth line of the meter is the expansion of three syllabus. Similarly the fifth line of the meter is the expansion of four syllabus and so on as shown in the Fig.4. So the ways of arranging all patterns of short and long syllables in a line with “n” syllables led to the formation of Pascal triangle, Binomial coefficients.

*Fig.4 **Pingala’s Meeru Prastara leading to Binomial expansion [Ref.1]*

The Maathra Meru or Fibonacci numbers emerged while finding out the number of sequences with a certain total length if long syllables are assigned with length 2 and short syllables are assigned with length 1. That is when we consider the sum of the numbers along the diagonal of the Meeru Prastara gives Maathra Meru or Fibonacci sequence {1, 1, 2, 3, 5, 8, 13….} (Fig 5)

*Fig.5 **Shallow diagonals of the Meru Prastara sum to the Fibonacci series [Ref.5]*

Pingala’s Chandahsastra sutra not only describes Pascal’s triangle, Binomial expansion and Fibonacci sequence. It also explores numerous recursive algorithms, computation of Binomial coefficients, the application of repeated partial sums of sequences and the formula for summing a geometric series and many more.

**References**

Binomial Theorem in Ancient India. Amulya Kumar Bag, History of Science, Ancient Period Unit II, No.1, Park Street, Calcutta-16(1966).

Paramananda Singh. The so called Fibonacci Numbers in Ancient and Medieval India. Historian Mathematics, 12(1985), 229-244.

Paramananda Singh, Acharya Hemachandra and the Fibonacci numbers, math.Ed.siwan, 20(10):28-30, 1986, IISN0047-6269.

Madhusudhana, V.(ed.)1981.Pingala Naga Viracitam Chandahsastram (with commentary by Halayudha).Parimala Publications,33/17. Delhi-110007:Shakti Nagar.

Sanskrit Prosody, Pingala sutras and Binary Arithmetic by R. Shridharan, Chennai Mathematical Institute.

Panchatantram of Visnusarma (edited with Hindi commentary by Shyamacharana Pandeya), Motilal Banarsi Das, Delhi, 2002.

Subhash Kak ”The Golden mean and the Physics of Aesthetics”, Archive of Physics: Physics/0411195, 2004.

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