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

Guidelines for Training/ Orientation of Faculty on IKS

Course 5: Mathematics in India: from Vedic period to modern times

1. Introductory Overview
 Mahāvīrācārya on the all-pervasiveness of Gaṇita. The algorithmic approach of Indian
Mathematics. Overview of development of Mathematics in India during the ancient and
early classical Period (till 500 CE), later classical period (500-1250) medieval period
(1250-1750) and the modern periods (1750- present). Proofs in Indian Mathematics. The
genius of Srinivasa Ramanujan (1887-1920).Lessons from History.
2. Mathematics in the Vedas and Śulva Sūtras
 Mathematical references in Vedas. The extant Śulbasūtra texts & their commentaries.
The meaning of the word Śulbasūtra. Qualities of a Śulbakāra. Finding the cardinal
directions. Methods for obtaining perpendicular bisector. Bodhāyana's method of
constructing a square. The Bodhāyana Theorem (so called Pythagoras Theorem)
 Applications of Bodhāyana Theorem. Constructing a square that is the difference of two
squares. Transforming a rectangle into a square. To construct a square that is n times a
given square. Transforming a square into a circle (approximately measure preserving).
Rational approximation for √2. Construction of Citis. Details of fabrication of bricks,
etc.
3. Pāṇini's Aṣṭādhyāyī
 Development of Vyākaraṇa or Śabadaśāstra. Pāṇini and Euclid. Method of Pāṇini's
Aṣṭādhyāyī. Śiva-sūtras and Pratyāhāras. Context-sensitive rules and other techniques
of Aṣṭādhyāyī. Pāṇini and zero. Patañjali on the method of Aṣṭādhyāyī. Vākyapadīya on
Aṣṭādhyāyī as an upāya.
4. Piṅgala's Chandaḥśāstra
 Development of Prosody or Chandaḥśāstra. Long (guru) and short (laghu) syllables.
Scanning of Varṇavṛtta and the eight Gaṇas. Pratyayas in Piṅgala's Chandaḥśāstra.
Prastāra or enumeration in the form of an array. Saṅkhyā or the total number of metrical
forms of n syllables. Naṣṭa and Uddiṣṭa (the association between a metrical form and the
row-number in the prastāra through binary expansion). Lagakriyā or the number of
metrical forms in the prastāra with a given number of Laghus. Varṇameru and the so
called "Pascal Triangle".
5. Mathematics in the Jaina Texts
 Place of Mathematics in Jaina literature. Important Jaina mathematical works. Jaina
geometry. Circumference of a circle. Area of a circle. Relation between chord, śara
(arrow) and diameter, etc. Approximation for the value of π. Notion of different types of
infinity. The law of indices. Permutations and Combinations.
6. Development of Place Value System
 Earliest evidence of the use of place value system. Numerals found in the inscriptions
(Brāhmi & Kharosṭhi). Use of Zero as a symbol in Piṅgala's Chandaḥśāstra. References
to use of decimal place value system in the commentary Vyāsabhāṣya on Yogasūtra and
in Southeast Asian Inscriptions. Different systems of numeration employing place value
system. Bhūtasaṅkhyā system. Āryabhaṭan system. Kaṭapayādi system. Algorithms for
arithmetical operations based on decimal place value system.
7. Āryabhaṭīya of Āryabhaṭa
 Āryabhaṭa, his period and his work Āryabhaṭīya. Names of the notational places. Square
and Squaring. Algorithm for finding the square root. Cube and cubing. Algorithm for

17