In 14th century Kerala, the astronomer and mathematician Madhava of Sangamagrama (1349-1425) can lay claim to the title of the man who knew infinity. The first known proofs of an infinite series expansion were devised here, in what came to be known as the Kerala school of astronomy and mathematics. It’s a discovery that serves as a precursor to calculus.
More than 200 years after these proofs were first made, Western mathematicians such as Isaac Newton and James Gregory ‘discovered‘ them anew, calling the proof the Gregory-Leibniz series). Madhava operated near modern day Irinjalakuda, near Cochin – a prominent trading port in his time. Writing on the forgotten history of Kerala’s mathematicians, George Gheverghese Joseph calls Madhava a pioneer in his field:
We may consider Madhava to have been the founder of mathematical analysis. Some of his discoveries in this field show him to have possessed extraordinary intuition.
Though little is known about Madhava’s life, George writes that he belonged to a priestly class called Emprantiri, comprised of Brahmins who integrated into Kerala after arriving from coastal Karnataka. He hailed from a house called ‘Bakuladhistitaviharam’ (also known as Ilaininnapalli in Malayaam), from the village of Sangamagrama.
Madhava was the first in a line of teacher-student relationships that preserved his work for future generations; his pupil was Vatasseri Paramesvara, who wrote many commentaries on mathematical and astronomical treaties. Like the series Madhava discovered, his pupils created a chain of mathematical researchers stretching up to the mid-17th century:Madhava (1350-1425), Parameswara (1360-1455), Damodara (1410-1510), Neelkantha (1500-1610), Jyesthadeva (1444-1545), Achyuta (1550-1621) and Melpathur (1569-1632).
Tragically, only few of Madhava’s works survive, mostly to do with astronomy. Though much of Madhava’s original texts were lost, his students and their own students preserved his findings in their own commentaries, attributing Madhava with phrasing like “Thus Madhava said…”.
Then, mathematics became a tool to create better astronomical calculations. Madhava revised the Chandra vakyas creating a method to calculate the exact positions of the moon for every 36 minutes of the day. Earlier methods were correct up to the minute; Madhava’s introduced accuracy to the level of a second.
Madhava’s math survived due to his pupils. This raises the problem of whether the works were his or that of his students, who were bound to attribute their master as the original founder. Whoever it was, these methods enabled the most accurate calculation of the value of pi at the time. He provided a helpful analogy for his finding:
33 Gods, 2 eyes, 8 elephants, 8 snakes, 3 fires, 3 qualities, 4 vedas, 27 naksatras, 8 elephants, and 2 arms – the wise say that this is he measure of the circumference when the diameter of a circle is 900,000,000,000.
Dividing the quantities mentioned by 900,000,000,000 gives a value of pi accurate to 11 digits – 3.141592653592222. The anonymously-authored Mahajyanayana-prakara (Method for computing the great sines) gives the power-series method that was later claimed by Gregory.
Madhava’s work allowed the calculation of accurate values of sin and cosine in any angle, building on Aryabhata’s sine tables using series-expansion. His work on these series is considered as a prelude to modern calculus. But why did the credit for this discovery go to Europe?
While the trade between Kerala and the world has been covered, it’s always been from the perspective of the colonizer extracting resources. Thus, we have the fallacy that Vasco de Gama ‘discovered’ India. Dennis F. Almeida and George Geverghese Joseph argue that the lack of scholarship on whether Indian maths made its way to Europe alongside its spices, suggests that the history of math too suffers from an overabundance of Eurocentrism.
In contrast, the Islamic world keenly recognized Indian prowess in this field. In circa 1068 A.D., Said Al-Andalusi wrote:
[The Indians] have acquired immense information and reached the zenith in their knowledge of the movements of the stars [astronomy] and the secrets of the skies [astrology] as well as other mathematical studies. After all that, they have surpassed all the other peoples in their knowledge of medical science and the strengths of various drugs, the characteristics of compounds, and the peculiarities of substances.
Madhava’s students were not just mathematicians and astronomers. Some, even managed poetry on the side. The final scholar of the Kerala school, Melpathur, composed the famous Sanskrit text Narayaniyam, studying the poetic form of the Bhagavata Purana – reappearing in print more than 250 years after it was first published in 1586 A.D.
Indian mathematics stems from millennia-old traditions, from the insight of Aryabhata to the genius of S. Ramananujan. There is a need for a de-colonized history of math; one that can explain how the ‘language of God’ was studied in ‘God’s own country’.
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