A glimpse of Geometry of Certain Figures
Summary
This scholarly article surveys the significant developments in Indian geometry from the 9th to 15th century CE, focusing on how ancient Indian mathematicians—particularly Jain scholars—formulated rules to calculate dimensions and areas of various geometric figures such as: A) Regular Polygons: i) Bhāskara II’s Līlāvatī provides coefficients for side lengths of polygons inscribed in a circle. ii) Mahāvīra's Gaṇitasārasaṅgraha and Utpala’s commentary on Bṛhatsaṃhitā give approximate area formulas. iii) Errors in traditional formulas are compared with modern trigonometric values to assess their accuracy. B) Ellipses: i) Early Jain and Buddhist texts (e.g., Sūryaprajñapti, Bhagavatīsūtra) refer to ellipses. ii) Mahāvīra's formula gives perimeter and area approximations, with tables showing percentage errors in comparison to modern formulas. iii) Ramanujan’s corrections are referenced for higher precision. C) Circular Segments: i) Various Jain texts, including the Tiloyapaṇṇatti and Bṛhatkṣetrasamāsa, provide formulas for arcs, sectors, and segments. ii) Several historical formulas are presented, tested, and error-analyzed using modern trigonometry. D) Spherical Segments: i) The Gaṇitasārasaṅgraha describes surface area estimation using fractional circumferences and chords. ii) Errors are calculated to determine ranges of practical applicability. iii) The paper uses mathematical derivations, diagrams (see page 4, 5, 8, and 11), and comparison tables to demonstrate the ingenuity and limitations of classical Indian geometry.
Summary reviewed by a volunteer editor.
Conclusion
This paper highlights the remarkable but approximate mathematical insights of ancient Indian scholars, especially in Jain literature. Though not always accurate by modern standards, their rules for calculating areas and lengths reveal a deep intuitive grasp of geometry. Their work laid a foundational base for applied geometry in architecture, astronomy, and cosmograph, especially through the Jain cosmological models. The study confirms that many formulas were impressively accurate within small error margins, particularly for simpler shapes and smaller angles.