Sulba Sutras: Ancient Indian Origins of Pythagorean Theorem and Irrational Approximations — E8 Intelligence Research

FINDING: The Sulba Sutras (c. 800–500 BCE) contain the earliest known explicit statement of the Pythagorean theorem, geometric constructions for squaring the circle, and irrational approximations — predating Greek formalization. | MATH: Baudhayana's theorem: *dīrghasyākṣṇayā rajjuḥ pārśvamānī, tiryaḍmānī, yatpṛthagbhūte kurutastadubhayaṅ karoti* — "The diagonal of a rectangle produces both [areas] which the sides separately create" (i.e., a² + b² = c²). Explicit √2 approximation: *saviśeṣaḥ* — 1 + 1/3 + 1/(3·4) − 1/(3·4·34) = 577/408 ≈ 1.414215686 (error ~2×10⁻⁶). Also 1.41421356 (true √2). Circle-squaring construction: side of square = 1/3 of diameter + 1/3 of (1/3 of diameter) + 1/3 of (1/3 of 1/3) − 1/3 of (1/3 of 1/3 of 1/3) — equivalent to π ≈ 3.088 (crude) but refined elsewhere to π ≈ 3.1416 via *śulba* altars. | CONNECTION: The √2 approximation 577/408 is a convergent of the continued fraction [1;2,2,2,...] — directly linked to the silver ratio (1+√2 ≈ 2.414) and to the octagona Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com