麻豆淫院

Add zero and one to get one, one and one to get two, one and two to get three, two and three to get five. Most of us know this鈥攖hat each successive number is the sum of the two numbers that came before it鈥攁s the Fibonacci sequence, named after a 12th-century Italian mathematician. But as early as 200 BCE, an Indian poet and mathematician named Acharya Pingala used that sequential concept to analyze poetry, and 7th-century scholar Virahanka later described it in more detail.

In fact, the use of math on the Indian subcontinent stretches back more than 3,000 years, and curiosity about this ancient and understudied history is at the center of Priya Nambrath's research. As a fifth-year doctoral candidate in the Department of South Asia Studies, Nambrath is studying the applied practice of mathematics during medieval and premodern times in what is now Kerala, a state in southwestern India.

It's "a deeply grounded and long-lasting mathematical tradition," she says, one in which people drew on local religious and metaphysical themes, as well as the rhythm and structure of Sanskrit poetry. In the process, they uncovered many ideas and approaches long before Europeans did鈥攄iscoveries that go largely underrecognized: "For the most part," Nambrath says, "even students in India are not taught this aspect of cultural and intellectual history."

Initially, Nambrath planned to dig into the topic independently. Ultimately, however, she realized she needed more academic support, "not just in the methodologies of Indian mathematics, but also in the literary and social histories of the region," she says. With that mission in mind, she came to Penn, where her studies have led her down many surprising routes, including a recently completed nine-month Fulbright Research Fellowship.

That fellowship gave Nambrath the opportunity to travel to the Indian cities of Thiruvananthapuram, Chennai, and Pune, where she visited manuscript archives to examine medieval mathematical and astronomical texts written on palm leaf. Nambrath also accessed late 19th-century print editions of Sanskrit and vernacular mathematical works, among many others.

Since returning to Philadelphia, she's been hard at work translating everything she saw into her dissertation鈥攁 "dream project" that the fellowship helped make a reality.

Different approaches

Nambrath's decision to apply for the Fulbright was driven by her curiosity around a range of issues, like how the subcontinent's many religions, cultures, and languages shaped its inhabitants' approach to math, or how Europeans reacted when they first encountered Indian mathematical texts. To understand this history firsthand, she decided she needed to be on the ground in India.

"This research involved a lot of time spent in several different archives and dealing with different categories of archival material," she explains. From December 2023 to September 2024, Nambrath visited manuscript libraries in India, where she identified a few mathematical texts that had not been previously studied or translated. Those texts provided insights into "a medieval system of pedagogy," Nambrath says, one that incorporated local approaches to mathematics.

She also found that European colonial scholars struggled to completely understand Indian math. One stumbling block, she observed, was cultural prejudice and a sense of mathematical superiority. But Nambrath surmises they may also have been flummoxed by how different it was from anything they'd encountered, something she ran into herself.

"My STEM background had encouraged me to think of mathematics as a kind of universal language, not susceptible to cultural and historical nuance like art, music, and literature," she says. "But what I was seeing in Indian mathematical texts convinced me otherwise."

Besides the close links with poetry, mathematical progress was sometimes driven by the precise requirements of ritual practice, and advancements in astronomy were often motivated by the needs of astrology. These efforts resulted in unique modes of mathematical expression, according to Nambrath.

One example is the ku峁�峁�膩k膩ra method, which Nambrath says translates to "the pulverizer," or the idea of reducing or grinding something down. The method is actually an algorithm that helps to solve what we now call linear Diophantine equations. Those take the form ax + by = c, with x and y representing unknown quantities, and the other letters representing known quantities. Through the ku峁�峁�膩k膩ra method, coefficients in this type of equation are broken up into smaller numbers to make it easier to find a solution.

The ku峁�峁�膩k膩ra method has some similarities with modern computational algorithms, but it first appeared in a 5th-century text, the 膧ryabha峁�墨yam, with many other Indian mathematicians building on it over the years. The text is a treatise written in Sanskrit verses, using what Nambrath describes as an obscure system of word-numerals鈥攖hat is, consonants representing digits, vowels denoting place value.

"We think of sciences and the humanities as embodying some kind of essential disciplinary binary," she says. "But here I was, encountering mathematical ideas and techniques encased in metrically precise and linguistically lush poetry."

Ups and downs

Like with any scientific inquiry, hurdles punctuated these breakthrough moments for Nambrath. During the summer of 2024, for example, she endured a blazing heat wave in Pune, where air conditioners are uncommon due to the typically mild climate.

To avoid the heat, Nambrath headed to one of the city's many cafes, making headway on her work before a glass of freshly squeezed sugarcane juice gave her food poisoning. Though "absolutely delicious," that drink also confined her to a steaming apartment in 105-degree temperatures amid power outages. Even after recovering, she found work challenging for several months, grappling with heat and exhaustion.

Still, Nambrath pushed on, and ultimately used the downtime to finalize papers that she eventually submitted for publication. Looking back, it's now clear to her that the Fulbright鈥攁nd her broader time at Penn鈥攈ave opened many opportunities and paths for her studies. "I am so grateful to the supportive academic community within Penn and the incredible resources that I have been able to access through the program," she says.

Nambrath, who is aiming to graduate next year, is now deep into writing her dissertation, along with developing a module for the Penn Museum that links artifacts in their Egyptian, Babylonian, and Greek galleries with the mathematics practiced by those cultures. Museum visitors should be able to see the result this fall.

And though that activity is a side project, Nambrath says it's bringing her research full circle. "It gives me a much more holistic view of how humans across time and geography have wrestled with mathematical problems," she says. "These approaches can be unique, but they are always logical, and it is fascinating to see how grounded they are in culture and custom."