Page 21 - Guidelines for Training/ Orientation of Faculty on IKS

Guidelines for Training/ Orientation of Faculty on IKS

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
2
2
Vargaprakṛti equation X - D Y = K, and Brahmagupta's bhāvanā process. The
Cakravāla method of solution of Jayadeva and Bhāskara.
2
2
2
2
2
2
 Bhāskara's examples X – 61Y = 1, X – 67Y = 1. The equation X - D Y = -1. Solution
of general quadratic indeterminate equations. Bhāskara's solution of a bi-quadratic
equation.
 Review of the Cakravāla method. Analysis of the Cakravāla method by Krishnaswami
2
2
Ayyangar. History of the solution of the "Pell's Equation" X - D Y = 1. Solution of
"Pell's equation" by expansion of √D into a simple continued fraction. Bhāskara semi-
regular continued fraction expansion of √D. Optimality of the Cakravāla method.
14. Gaṇitakaumudī of Nārāyaṇa Paṇḍita
 Importance of Gaṇitakaumudī. Solutions of quadratic equations. Double equations of
second and higher degree – rational solutions. Determinations pertaining to the mixture
of things. Interest calculations – payment in installments
 Meeting of travelers. Progressions. Vārasaṅkalita: Sum of sums. The kth sum. The kth
sum of a series in A.P. The Cow problem. Diagonals of a cyclic quadrilateral – Third
diagonal, area of a cyclic quadrilateral. Construction of rational triangles with rational
sides, perpendiculars, and segments whose sides differ by unity. Generalisation of
binomial coefficients and generalized Fibonacci numbers.
 Vargaprakṛti. Nārāyaṇa's variant of Cakravāla algorithm. Solutions of Vargaprakṛti and
approximation of square roots. Bhāgadāna: Nārāyaṇa's method of factorisation of
numbers. Aṅkapāśa (Combinatorics). Enumeration (prastāra) of generalised mātrā-vṛttas
(moric metres with more syllabic units in addition to Laghu and Guru). Some sequences

19