Page 22 - Guidelines for Training/ Orientation of Faculty on IKS
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Guidelines for Training/ Orientation of Faculty on IKS
(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.
k
k
k
Binomial series expansion. Estimating the sum 1 + 2 + ... n for large n.
Mādhava Series for π. End-correction terms and Mādhava continued fraction.
Transformed series for π which are rapidly convergent. History of Approximations to π.
Nīlakaṇṭha's derivation of the Āryabhaṭa relation for second-order Rsine differences.
Mādhava series for Rsine and Rcosine. Nīlakaṇṭha and Acyuta formulae for
instantaneous velocity.
Āryabhaṭa's sine table (makhi, bhaki, phaki…). Āryabhaṭa 's recursion relation and the
approximation involved in it. Attempts to improve the sine values by Lalla,
Govindasvāmi, Vaṭeśvara, etc. Bhāskara's formula for sin(A + B) and its application. The
refined recursion relation in Taṅtrasangraha and its commentary. Mādhava's sine series
and the use of mnemonics vidvān, tunnabala etc. Mādhava's sine table. Comparison of
sine-tables of Āryabhaṭa, Govindasvāmi, Vaṭeśvara and Mādhava.
17. Trigonometry and Spherical Trigonometry
Crucial role of trigonometry in astronomy problems. Indian sines, cosines : Bhujājyā,
Koṭijyā, sine tables. Interpolation formulae. Determination of the exact values of 24
sines. Bhāskara's Jyotpatti sin (18º), sin (36º).
Sine of difference of two angles. Sines at the interval of 3º, 1.5º. Jīve-paraspara-nyāya.
Sines at the interval of 1º. Trigonometry in later texts such as Siddhāntatattvaviveka of
Kamalākara
Spherical trigonometry in astronomy: Tripraśna problems. Applications to specific
diurnal problems: Duration of day (carajyā), Time from shadow. Systematic treatment
of spherical trigonometry problems in Nīlakaṇṭha's Tantrasaṅgraha. Proofs of
Tantrasaṅgraha results in Yuktibhāṣā.
18. Proofs in Indian Mathematics
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