Pingala

Pingala is the name attributed to an ancient Indian mathematician and music theorist, believed to have lived sometime around the 3rd century BCE or possibly later, up to the 2nd century CE. He is best known for his work Chandahshastra (also spelled Chandaḥśāstra or Chandaḥ-shāstra), which translates to "The Art of Prosody." This text is a foundational treatise on Sanskrit prosody, the study of poetic meters and rhythms.

The Chandahshastra contains the earliest known description of binary numeral systems. Pingala employed a system of representing meters and rhythms using light and heavy syllables, which can be interpreted as 0 and 1, respectively. He used this binary system to classify and count the different possible patterns of syllables within a given meter.

Crucially, Pingala's work also implicitly contains an understanding of what are now known as Fibonacci numbers. He uses a recursive method to calculate the number of meters with a specific length, and the results of this calculation correspond to Fibonacci numbers. Though not explicitly stated as a mathematical concept in its own right, the sequence emerges from his analysis of metrical patterns.

Furthermore, the Chandahshastra describes a method for finding the number of combinations of n things taken r at a time, a concept central to combinatorics. His method is essentially equivalent to the binomial coefficient, although expressed in the context of prosody.

While little is known about Pingala's life beyond the association with the Chandahshastra, his contributions to the understanding of prosody, binary numbers, Fibonacci sequences (implicitly), and combinatorial analysis are significant in the history of mathematics and music theory, especially within the Indian tradition. His work provides a window into the advanced mathematical and linguistic thinking of ancient India.