Jaina Formulas for the Arc of a Circular Segment
Summary
The article “Jaina Formulas for the Arc of a Circular Segment” examines how ancient Indian mathematicians, especially Jaina scholars, tried to find simple formulas for measuring the length of an arc of a circular segment when only the chord and the segment’s height were known. Let AB be a chord of a circle with centre O; N is the midpoint of the chord and M is the midpoint of the arc. The chord length is c, the arc length is s, the height MN is h and the radius OM is r. The exact relation between c, h and s is s = 2rθ, where θ is the angle subtended at the centre and c² = 4h(d – h), but calculating s via trigonometric functions requires tables that might not have been available. Greek mathematicians such as Heron of Alexandria provided an approximate formula: s ≈ √(c² + 4h²) + h/4. Early Indian mathematicians looked for formulas of the form s = √(c² + k h²) that would be correct in the case of a semicircle. Setting c = 2r and h = r makes s/r equal to π; choosing k = π² – 4 gives a value of π consistent with the approximation used. Jaina authors experimented with different values of π, including 3, √10 and 22/7, which produced formulas with k = 5, 6 and 288/49 respectively. Mahavira’s 9th century treatise Gaṇita-sāra-saṅgraha states in verse that the square root of the sum of five times the square of the segment height plus the square of the chord yields the arc, corresponding to k = 5. Another Jaina text, the Bhāṣya on the Tattvārthādhigama-sūtra, quotes an expression where six times the square of the height is added to the square of the chord. Similar rules appear in Nemicandra’s Tiloyasāra, Padmanandi’s Jambū-paññatti-saṅgaho and Aryabhata II’s Māhāsiddhānta. The author explains that these rules were derived from analogy with a semicircle rather than from rigorous trigonometric reasoning, and notes that Nilakantha Somayaji in the 16th century proposed a more accurate rule for small arcs, s = √(c² + (16/3)h²). The article concludes by showing how Mahavira applied the same technique to derive an approximate perimeter for an ellipse by treating half of the ellipse as a circular segment with c = 2a and h = b, giving s ≈ 2√((2a)² + 6b²). Through these examples, the article shows the ingenuity of Jaina mathematicians in developing practical approximations long before modern calculus.
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Conclusion
By surveying Greek, Vedanga and Jaina sources, the author demonstrates that Jaina mathematicians played a significant role in refining empirical formulas for circular arcs. Their search for a constant k that yields the correct semicircular arc shows an implicit awareness of π and a willingness to test different approximations. Although the rules lack the precision of later trigonometry, they provided usable methods for craftsmen and astronomers. The discussion underscores the interplay between observation and generalization in early Indian mathematics and highlights how Jaina scholars contributed to the wider mathematical tradition.