The Rule of Three of Acarya Nemicandra

1. Introduction and Historical Context Ācārya Nemicandra (c. 981 C.E.), a Jain saint-scholar and contemporary of Camuṇḍarāya of the Ganga Empire, was a witness to the first Mahamastakābhiṣeka of the Bahubali statue at Śravaṇabeḷagoḷa. He composed important mathematical texts in Prakrit like Trilokasāra and Gommatasāra, where he used the Rule of Three—a mathematical principle combining multiplication and division to solve proportional problems. 2. Rule of Three in Indian Tradition The Rule of Three—called trairāśika—dates back to ancient Indian mathematicians like Āryabhaṭa, Brahmagupta and Bhāskara I, and was later passed to Arabs and Europe (as the "Golden Rule"). The formula is: Desired Result = Consequent×Desire / Standard ​This structure is applied throughout Jain cosmological and mathematical texts to derive proportions. 3. Applications in Nemicandra’s Works A. In Trilokasāra Used to calculate the diameter of Bharata and areas in Jain cosmology. Example: To compute the diameter of Bharata from that of Jambūdvīpa by proportioning. B. In Gommatasāra Applies logarithmic interpretation of the Rule of Three using ardhaccheda (base-2 logarithms). Introduces a formula for cases involving exponential growth or multiplication (e.g., computing 64 2’s multiplied together from knowledge of 16 2’s multiplied). The Jaina School of Indian Mathematics is divided into two classes: canonical and exclusive. 4. Advanced Geometrical Concepts Nemicandra further expands into: A. Mensuration of Circles Gives formulas for: i) Circumference: C=3d ii) Differentiates between gross and subtle values. iii) Compares with known approximations like Chinese and Greek values of π. B. Segments of Circles i) Gives precise rules to find the area of a segment using arc length, chord, diameter, and height. ii) Relates segment area to semi-circles via analogies—some rules derived from earlier works, others appear original. C. Elements of Segments Presents six different rules to compute diameter, arc, chord, and height of a circular segment using geometric and algebraic relations. 5. Circular Annulus in Jain Cosmology i) Applies Rule of Three and other geometrical formulas to calculate properties of concentric land-sea rings in the Jambūdvīpa. ii) Gives formulas for: Inner/outer diameters. Breadths. Gross/subtle areas of annular regions. 6. Comparative Analysis Tables (e.g., on pages 50–54) compare Nemicandra’s formulas with: Chinese works (e.g., Chou Pei, Chiu-Chang Suan Shu), Greek authors like Heron, Other Jain and Hindu mathematicians like Srīdhara and Bhāskara. Shows that many formulas had Chinese parallels, suggesting either mutual influences or common ancestry.

Summary reviewed by a volunteer editor.