Page 19 - Guidelines for Training/ Orientation of Faculty on IKS

Vallīkāra-kuṭṭākara – linear indeterminate equations. Two and more simultaneous indeterminate
equations. Other indeterminate equations. Vicitra-kuṭṭākara – Truthful and untruthful statements.
Sums of progressions of various types. Variable velocity problem

Plane figures: Circle, Dīrghavṛtta, Annulus. Ratio of circumference and diameter. Segment of a circle.
Janya operations: rational triangles, quadrilaterals. Excavations: Uniform and tapering cross-sections,
volume of a sphere. Time to fill a cistern. Shadow problems.

11. Development of Combinatorics

Combinatorics in Āyurveda. Gandhayukti of Varāhamihira Mātrā-vṛttas or moric metres. Prastāra or
enumeration of metres of n-mātrās in the form of an array. Saṅkhyā or the total number of metrical
forms of given number of mātrās. The Virahāṅka sequence (so called Fibonacci sequence. Naṣṭa and
Uddiṣṭa processes for finding the metrical form given the row-number and vice versa in a prastāra.
Mātrā-meru to determine the number of metrical forms with a given number of gurus. Representation
of any number as a sum of Virahāṅka numbers.

Saṅgīta-ratnākara of Śārṅgadeva (c.1225). Tāna-Prastāra or enumeration of permutations or tānas of
svaras. Prastāra, the rule of enumeration of permutations in the form of an array. Khaṇḍameru and the
processes of naṣṭa and uddiṣṭa. Factorial representation of Śārṅgadeva. Tāla-Prastāra: Enumeration
of tāla forms. The tālāṅgas: Druta, Laghu, Guru and Pluta and their values. Prastāra: Rule of
enumeration of all tāla-forms of a given value. Saṅkhyā and the Śārṅgadeva-sequence of numbers.
The processes of naṣṭa and uddiṣṭa. Representation of natural numbers as sums of Śārṅgadeva-
numbers. Laghu-Meru. The general relation between prastāra and representation of numbers.

12. Līlāvatī of Bhāskarācārya

Introduction. Importance of Līlāvatī. Arithmetical operations: Inversion method, rule of supposition.
Solution of quadratic equations. Mixtures. Combinations, progressions.

Plane figures: Right triangles, applications. Sūcī problems. Construction of a quadrilateral:
Discussion on earlier confusions. To find the second diagonal, given the four sides and a diagonal of
a quadrilateral. Cyclic quadrilaterals. Value of π, area of a circle, surface area of a sphere, volume of
a sphere.

Regular polygons inscribed in a circle. Expression for a chord in a circle. Excavations and contents
of solids. Shadow problems (advanced problems). Importance of rule of proportions. Combinations
(advanced problems).

13. Bījagaṇita of Bhāskarācārya
Development of Bījagaṇita or Avyaktagaṇita (Algebra) and Bhāskara’s treatise on it. Understanding

of negative quantities. Development of algebraic notation. The Vargaprakṛti equation X - D Y = K,
2
2
and Brahmagupta’s bhāvanā process. The Cakravāla method of solution of Jayadeva and Bhāskara.
Bhāskara’s examples X – 61Y = 1, X – 67Y = 1. The equation X - D Y = -1. Solution of general
2
2
2
2
2
2
quadratic indeterminate equations. Bhāskara’s solution of a bi-quadratic equation.

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Guidelines for Incorporating Indian Knowledge in Higher Education Curricula