Page 20 - Guidelines for Training/ Orientation of Faculty on IKS

Review of the Cakravāla method. Analysis of the Cakravāla method by Krishnaswami Ayyangar.
History of the solution of the “Pell’s Equation” X - D Y = 1. Solution of “Pell’s equation” by
2
2
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 (paṅkti) and tabular figures (meru) used in
combinatorics. Enumeration (prastāra) of permutations with repetitions. Enumeration (prastāra) of
combinations.

15. Magic Squares

The earliest textual references and references in inscriptions. The sarvatobhadra square of
Varāhamihira. Nārāyaṇa’s classification of magic squares into samagarbha (doubly-even numbers of
the form 4m), viṣamagarbha (singly-even or numbersof the form 4m + 2) and viṣama (odd). Use of
Kuṭṭaka to find the arithmetic sequences to be used in magic squares. 4x4 Pandiagonal magic squares
of Nārāyaṇa.

Ancient method for the construction of odd magic squares and doubly even squares. The folding
method (sampuṭīkaraṇa) of Nārāyaṇa for samagarbha squares. The folding method for Viṣama
squares. Illustrative examples.

16. Kerala School of Astronomy and Development of Calculus
Background to the Development of Calculus (c.500-1350). The notions of zero and infinity. Irrationals
and iterative approximations. Second order differences and interpolation in computation of Rsines.
Summation of infinite geometric series. Instantaneous velocity (tātkālika-gati). Surface area and

volume of a sphere. Summations and Repeated summations (saṅkalita and vārasaṅkalita). The Kerala
School of Astronomy and the Development of Calculus. Mādhava (c. 1340-1420) and his successors
to Acyuta Piśāraṭi (c. 1550-1621). Nīlakaṇṭha (c.1450-1550) on the irrationality of π. Nīlakaṇṭha and
the notion of the sum of infinite geometric series. Binomial series expansion. Estimating the sum 1 +
k
2 + ... n for large n.
k
k

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