Why you should care
Sometimes the proof is in a little-known language.
Long before Sir Isaac Newton, Pierre de Fermat, Gottfried Leibniz or the rest of the crew credited with the development of calculus, an astronomer and mathematician named Jyesthadeva put ink to palm leaves to record the mathematics of his teachers and possibly some of his own.
In a small town in southern India in the 1500s, Jyesthadeva penned concepts important to developing a calculus system, and he did so in complete proofs that demonstrated infinite series expansions of trigonometric functions and gave precise approximations for complex calculations. “Calculus and everything derived from it depends to some extent on these concepts of infinitesimals and infinite series,” says Kim Plofker, author of Mathematics in India. By way of comparison, it wasn’t until the 1660s in Europe that a Westerner named James Gregory was able to independently do the same proof.
I have ascertained beyond a doubt that the invention of infinite series of these forms has originated in Malabar.
Charles M. Whish
The text, called the Yuktibhasa, is broken into 15 chapters and spans hundreds of pages of proofs and commentary. It was a compilation of a century-plus of Indian mathematics developed by the Kerala school, led by mathematician Madhava of Sangamagrama in the 14th century. Most of Madhava’s work would have been lost if not for the writings of pupils like Jyesthadeva, who recorded everything in Dravidian, the hyper-localized dialect of Malayalam. One theory is that Jyesthadeva wasn’t fluent in Sanskrit or he was helping others who weren’t.
By the 16th century, the school was on the wane, which might have been Jyesthadeva’s impetus for writing down the proofs that had been passed from pupil to pupil orally for 200 years — the Yuktibhasa may have been his way of preserving that information.
There are a few theories as to why the school faded. Perhaps a dynasty change led to funding cuts. Or perhaps it was because the practical use of mathematics was primarily for astronomy, and once the tables had been made accurate to the 11th decimal place, there was no more need for mathematicians. But whatever the cause, “by the 1700s, almost no one is reading or copying [the Kerala school] texts anymore,” says Homer White, professor of mathematics at Georgetown College.
While some historians have speculated that Jesuits traveling between India and Europe brought the Yuktibhasa back to Europe and that it served to inspire European calculus, most aren’t convinced. “There is no reason to believe that our use of these ideas was directly descended from or influenced by the Kerala school,” Plofker says. Located between the Western Ghats mountain chain and the Arabian Sea, Kerala was perfectly situated to have its own culture. It wasn’t completely isolated — Kerala was a hub for pepper production and export — but the school was “quite removed from that trading nexus,” Plofker says, which suggests that ideas from the Yuktibhasa were unlikely to have spread across the ocean.
After the school fizzled out, it took more than 100 years before the work was studied by a Western audience. British colonists in India began studying the culture in the 1700s; in the 1830s, Charles M. Whish published a paper about the Yuktibhasa in the journal Transactions of the Royal Asiatic Society of Great Britain and Ireland. Whish was a busy fellow, serving in the East India Company in South Malabar, and then, after a few years, as a criminal judge. But he found time to study Indian texts on the side, and before he died at the age of 38, Whish shared with a European audience how the Yuktibhasa had complete proofs.
This was an important “discovery.” Prior to Whish’s translation, Europeans commenting on Indian mathematics denied that the subcontinent had invented its own concepts. John Warren, another East India Company employee, wrote that “a Native Astronomer who was a perfect stranger to European Geometry” could demonstrate the infinite series, but the astronomer could not explain how he knew it to be true — the proof, in other words, was missing. “The Hindus never invented the series; it was communicated with many others, by Europeans, to some learned Natives in modern times,” Warren wrote, quoting George Hyne, also of the East India Company. Whish disagreed, but the prevailing notion, as Hyne had written to Warren, was that “the pretensions of the Hindus to such a knowledge of geometry, is too ridiculous to deserve refutation.”
The Yuktibhasa was the key to proving Warren wrong. It revealed that the Kerala mathematicians had not “taken” the logic but had found it themselves and derived their solution, and had done so far earlier than any European. Whish wanted to correct the misconception, noting, “I have ascertained beyond a doubt that the invention of infinite series of these forms has originated in Malabar” — further proof that history did not have to be written in English to be true.