IKS Series : Book-6 

Mathematics in Ancient India

Chapter 1: Geometry in Śulbasūtras - Expressions for Surds

This chapter explores the expressions for surds (square roots of non-square numbers) as described in the Śulbasūtra texts, highlighting their role in ancient geometric constructions and approximations used in ritual altar designs.

Chapter 2: Methods for Obtaining Perpendicular Bisectors in Śulbasūtras

Focusing on techniques to find perpendicular bisectors, this chapter discusses general methods outlined in Śulbasūtras, essential for symmetric constructions in Vedic geometry.

Chapter 3: Construction of Perpendicular Bisectors - Cord-Folding Method

Detailed examination of the cord-folding method for constructing perpendicular bisectors, a practical approach using strings in ancient Indian geometry for accurate divisions.

Chapter 4: Bodhāyana's Method of Constructing a Square

This chapter covers Bodhāyana's specific technique for constructing squares, including step-by-step processes and their applications in larger geometric figures.

Chapter 5: The Bodhāyana-Pythagorean Theorem

An in-depth look at the Bodhāyana statement resembling the Pythagorean theorem, its proof, and significance in pre-Euclidean geometry within Śulbasūtras.

Chapter 6: Decimal Place Value System and Numbers in the Vedas

Exploring the origins of the decimal place value system in Vedic literature, including representations of numbers and their evolution in ancient texts.

Chapter 7: Enumeration of Large Numbers in Traditional Literature

This chapter details the naming and enumeration of large numbers, from koti to mahaaugh, akshauhini, and other numerals in epic and mathematical traditions.

Chapter 8: Numeration Systems - Āryabhaṭan and Bhūtasaṅkhyā

Comparative analysis of the Āryabhaṭan system (alphabetic notation for numbers) and Bhūtasaṅkhyā system (using objects to represent digits), their structures, and uses.

Chapter 9: The Kaṭapayādi Numeration System

Dedicated to the Kaṭapayādi system, where consonants represent digits, this chapter explains its mnemonic applications in astronomy and mathematics.

Chapter 10: Arithmetic in Āryabhaṭīya - Square Roots and Cube Roots

Covering Āryabhaṭa's methods for calculating square roots and cube roots, including algorithms and their accuracy in computational contexts.

Chapter 11: Trigonometry in Āryabhaṭīya - Sine Table and Modifications

This chapter discusses Āryabhaṭa’s sine table, its construction, and later modifications in works like Tantrasaṅgraha for improved precision.

Chapter 12: Mādhava Series for Sine and Cosine Functions

Exploration of Mādhava's infinite series expansions for sine and cosine, marking early developments in calculus-like methods in Indian trigonometry.

Chapter 13: Methods for Obtaining Sine Values in Karaṇapaddhati

Review of various techniques from Karaṇapaddhati for computing sine values, including iterative and approximation methods for astronomical calculations.

Chapter 14: Brahmagupta's Mathematics - Positive, Negative, Zero, and Equations

This chapter covers Brahmagupta's treatment of positive, negative numbers, zero, and solutions to linear and quadratic equations, with examples.

Chapter 15: Brahmagupta and Cyclic Quadrilaterals; Introduction to Līlāvatī's Arithmetical Operations

Combining Brahmagupta's rules for cyclic quadrilaterals (area formula) with Līlāvatī's arithmetical methods like inversion and rule of supposition.

Chapter 16: Advanced Topics in Līlāvatī, Kuṭṭaka, Continued Fractions, and Kerala School Contributions

Encompassing Līlāvatī's quadratic equations, mixtures, combinations, progressions, plane figures (right triangles, Sūcī problems), quadrilateral constructions, cyclic quadrilaterals, π value, circle area, sphere surface and volume; Kuṭṭaka methods by Āryabhaṭa and Brahmagupta, Vallyupasaṃhāra (Methods I/II from Karaṇapaddhati), nearest-integer continued fractions in Dṛkkaraṇa; and Kerala School's Mādhava π series, end-correction, fast convergent series, Putumana-Somayājī series, Nilakantha's irrationality of π, and infinite geometric series sum.

Professional Note

Authors may submit chapters other than those listed above, provided the proposed chapter aligns with the overall theme and objectives of the book.