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Jyesthadeva (c. 1500 – c. 1575 CE)

Overview

Jyesthadeva was a seminal figure of the Kerala School of Mathematics and Astronomy, renowned for authoring Yuktibhāṣā, the first known text in the world to present calculus in a logically rigorous form. Writing in Malayalam rather than Sanskrit, Jyesthadeva bridged the scholarly and vernacular traditions, ensuring the transmission of advanced mathematics to a broader intellectual audience.

Birth and Background

• Born: Circa 1500 CE
• Place: Possibly Thrikkandiyur, Kerala, India
• Tradition: Nambudiri Brahmin and a student in the Madhava–Nilakantha lineage

Jyesthadeva was trained under Nilakantha Somayaji, and he became the intellectual vehicle through which the accumulated mathematical advances of the Kerala School were compiled, systematized, and logically defended.

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Major Contributions

Yuktibhāṣā (c. 1530–1550 CE)

• A foundational text of Indian mathematics, written in Malayalam prose (a rarity in scientific literature of the time).
• “Yukti” means reasoning, and “Bhāṣā” means language—the book systematically explains the logic and derivations behind mathematical and astronomical rules.
• Presented full derivations for:
  • Trigonometric series (sine, cosine, arctangent)
  • π series
  • Geometric summations
  • Calculations of volumes and surface areas
• Discussed concepts akin to:
  • Limits
  • Continuity
  • Infinitesimal increments
  • Integration by parts

This makes Yuktibhāṣā the earliest known text to embody the principles of integral calculus, predating Newton and Leibniz by over a century.

Jyesthadeva – Key Equations and Mathematical Content

In his Yuktibhāṣā, Jyesthadeva elaborates on and rigorously justifies the infinite series discovered by Mādhava. Here are some of the central equations:

Infinite Series Expansions

Sine Series:
\[ \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \]

Cosine Series:
\[ \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots \]

Arctangent Series:
\[ \arctan x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots \quad (|x| \leq 1) \]

Setting \( x=1 \) gives the Madhava–Leibniz series for \( \pi \):
\[ \frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots \]

Geometric Series Summation

For a geometric series:
\[ S = a + ar + ar^2 + \cdots + ar^{n-1} = \frac{a(1 - r^n)}{1 - r} \quad \text{if } r \neq 1 \]

Jyesthadeva also explored early differential concepts, like approximating small changes:
“If a quantity increases by a small increment, then the increment of the square is approximately twice the original multiplied by the increment.”

This foreshadows the derivative of \( x^2 \):
\[ \frac{d}{dx} (x^2) = 2x \]

Drk Karanam

• A practical manual based on astronomical tables.
• Emphasized computational methods to determine planetary positions and eclipses.
• Linked theoretical astronomy with calendar-making and navigation.

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Key Features of Yuktibhāṣā

• Language: Written in Malayalam, making complex mathematics accessible beyond Sanskrit scholars.
• Pedagogy: Emphasized step-by-step reasoning, rather than rote formulas.
• Structure: Contains two major parts—mathematics (Gaṇita) and astronomy (Jyotiṣa).

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Legacy and Importance

• Mathematical rigor: His explanations of infinite series are logically complete, a feat rare even in European works of his time.
• Transmission: Preserved and expanded Mādhava’s calculus concepts, ensuring they were not lost.
• Global recognition: Today, Yuktibhāṣā is cited as proof that calculus developed independently in India, supported by textual and historical analysis.

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Philosophical and Cultural Impact

Jyesthadeva embodied the Kerala School’s emphasis on rationality (yukti) over mere tradition. His bold decision to write in Malayalam, combined with his methodical reasoning, represents a turning point in Indian scientific literacy—from elite scholasticism to broader intellectual democratization.