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

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

Upapattis or proofs in Indian mathematical tradition. Early European scholars of Indian Mathematics
were aware of upapattis. Some important commentaries which present upapattis. Bhāskarācārya II
on the nature and purpose of upapatti. Upapatti of bhujā-koṭi-karṇa-nyāya (Baudhayana-Pythagoras
theorem). Upapatti of kuṭṭaka process. Restricted use of tarka (proof by contradiction) in Indian
Mathematics. The Contents of Gaṇita-yukti-bhāṣā. Yukti-bhāṣā demonstration of bhujā-koṭi-karṇa-
nyāya. Estimating the circumference by successive doubling of circumscribing polygon.

Expression for abādhās, area and circum-radius of a triangle. Theorem on the sum of the product of
chords (jyāvargāntara-nyāya). Theorem on the difference of the squares of the chords (jyāvargāntara-
nyāya). From jyāsaṃvarga-nyāya to jyotipatti (generation of tabular sines). The cyclic quadrilateral.
Expression for the diagonals in terms of the sides. Expression for the area in terms of the diagonals.
Expression for the area and circum-radius in terms of the sides.

Yuktibhāṣā estimate of the samaghāta saṅkalita 1 + 2 + ... n for large n. Yuktibhāṣā estimate of
k
k
k
Vārasaṅkalita. Yuktibhāṣā derivation of Mādhava Series for π. Yuktibhāṣā derivation of end-correction
terms. Yuktibhāṣā derivation of Mādhava Rsine and Rcosine Series. Upapatti and “Proof”. Lessons
from history.

19. Mathematics in Modern India

Continuing tradition of Indian Astronomy and Mathematics (1770-1870). Surveys of indigenous
education in India (1825-1835). The Orientalist-Anglicist debate shaping the British policy on

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Guidelines for Incorporating Indian Knowledge in Higher Education Curricula