I. MATHEMATICS IN INDIA: FROM VEDIC PERIOD TO MODERN TIMES

1. Introductory Overview

Mahāvīrācārya on the all-pervasiveness of Ganita. 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.

• 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 finding the cube root. Formula for the area of a triangle. Bhāskara I on altitude and area of a triangle. Numerical examples

Area of a circle, trapezium and other planar figures. Approximate value of π. Computation of tabular Rsines (geometric and difference equation methods). Approximate formula for Rsine (as given by Bhāskara I). Problems related to gnomonic shadow. Bhujā-koṭi-karṇa-nyāya, jyā-śara-nyāya and their applications. Arithmetic progressions. Finding sum of natural numbers, sum of sums, and so on.

Some algebraic identities. Rule of three. Problems on interest calculation. Ekavarṇa-samikaraṇa and anekavarṇa-samikaraṇa. The Kuṭṭaka problem (sāgra and niragra-kuṭṭaka). Illustrative examples.

• Brāhmasphuṭasiddhānta of Brahmagupta

Introduction. Twenty logistics. Cube root. Rule of Three, Five Seven, etc. Mixtures. Interest calculations,

etc. Progressions: Arithmetic and Geometric. Plane figures. Triangles, right triangles and quadrilaterals.

Diagonals of a cyclic quadrilateral. Rational triangles and quadrilaterals. Chords of a circle. Volumes with uniform and tapering cross-sections. Pyramids and frustum. Shadow problems.

Mathematical operations with plus, minus and zero. Rules in handling surds (karaṇī) Operations with unknowns (avyakta-ṣaḍvidha). Equations with single unknowns (ekavarṇa-samīkaraṇa). Equations with multiple unknowns (anekavarṇa-samīkaraṇa). Equations with products of unknowns (bhāvita). Brahmagupta on kuṭṭaka. The Second order indeterminate equation (Vargaprakṛti). Bhāvanā principle and its applications.

7. Bakṣālī Manuscript

The discovery of Bakṣālī Manuscript. Its antiquity and uniqueness. Use of symbols. Symbol for negative sign (kṣaya). Symbol for denoting unknown quantities (yāvatāvat). Solution of indeterminate equations. Formula for approximate value of surds. Some interesting problems involving simultaneous equations.

1. Gaṇitasārasaṅgraha of Mahāvīra

Introduction. Arithmetical operations, operations with zero. Squares, cubes, square roots, cube roots. Arithmetical and Geometric progressions, Citi (summation). Manipulations with fractions and solutions of equations. Mixed problems including interest calculations.

Vallīkāra-kuṭṭākara – linear indeterminate equations. Two and more simultaneous indeterminate equations. Other indeterminate equations. Vicitra-kuṭṭākara – Truthful and untruthful statements. Sums of progressions of various types. Variable velocity problem

Plane figures: Circle, Dīrghavṛtta, Annulus. Ratio of circumference and diameter. Segment of a circle. Janya operations: rational triangles, quadrilaterals. Excavations: Uniform and tapering cross-sections, volume of a sphere. Time to fill a cistern. Shadow problems.

8. Development of Combinatorics

Combinatorics in Āyurveda. Gandhayukti of Varāhamihira Mātrā-vṛttas or moric metres. Prastāra or enumeration of metres of n-mātrās in the form of an array. Saṅkhyā or the total number of metrical forms of given number of mātrās. The Virahāṅka sequence (so called Fibonacci sequence. Naṣṭa and Uddiṣṭa processes for finding the metrical form given the row-number and vice versa in a prastāra. Mātrā-meru to determine the number 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.

1. 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).

1. Bījagaṇita of Bhāskarācārya

Development of Bījagaṇita or Avyaktagaṇita (Algebra) and Bhāskara’s treatise on it. Understanding of negative quantities. Development of algebraic notation. The Vargaprakṛti equation X² – D Y² = K, and Brahmagupta’s bhāvanā process. The Cakravāla method of solution of Jayadeva and Bhāskara.

Bhāskara’s examples X² – 61Y² = 1, X² – 67Y² = 1. The equation X² – D Y² = -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 Ayyangar. History of the solution of the “Pell’s Equation” X² – D Y² = 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.

1. 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.

9. 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. 4×4 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.

10. 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^k + … n ^k 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.

11. 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.

12. 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^k + 2^k + … n ^k for large n. Yuktibhāṣā estimate of 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.

13. 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

education  (c.1835). Survival of indigenous education system till 1880. Modern Scholarship on Indian Mathematics and Astronomy (1700-1900). Rediscovering the Tradition (1850-1900). Development of Higher Education and Modern Mathematics in India (1850-1910). Srinivasa Ramanujan (1887-1920). Brief outline of the life and mathematical career of Ramanujan. Hardy’s assessment of Ramanujan and his Mathematics (1922, 1940). Some highlights of the published work of Ramanujan and its impact. Selberg’s assessment of Ramanujan’s work (1988). The saga of Ramanujan’s Notebooks. Ongoing work on Ramanujan’s Notebooks. The enigma of Ramanujan’s Mathematics. Ramanujan not a Newton but a Mādhava.

Rediscovering the tradition (1900-1950). Rediscovering the tradition (1950-2010). Modern scholarship on Indian Mathematics (1900-2010). Development of modern mathematics in India (1910-1950). Development of modern mathematics in India (1950-2010). Development of higher education and scientific research in India (1900-1950). Development of higher education and scientific research in India (1950-2010). Comparison with global developments.

Suggested References

1. B. Datta and A. N. Singh, History of Hindu Mathematics, 2 Parts, Lahore, 1935, 1938; Reprint, Asia Publishing House, Bombay 1962; Reprint, Bharatiya Kala Prakashan, Delhi 2004.
2. C. N. Srinivasiengar, History of Indian Mathematics, The World Press, Calcutta, 1967.
3. T. A. Saraswati Amma, Geometry in Ancient and Medieval India, Motilal Banarsidass, Varanasi, 1979.
4. S. Balachandra Rao, Indian Mathematics and Astronomy: Some Landmarks, 3^rd Ed. Bhavan’s Gandhi Centre, Bangalore, 2004.
5. G. G. Emch, M. D. Srinivas and R. Sridharan, Eds., Contributions to the History of Mathematics in India, Hindustan Book Agency, Delhi, 2005.
6. C. S. Seshadri, Ed., Studies in History of Indian Mathematics, Hindustan Book Agency, Delhi, 2010.
7. G. G. Joseph, Indian Mathematics Engaging the World from Ancient to Modern Times, World

Scientific, London, 2016.

• P. P. Divakaran, The Mathematics of India Concepts Methods Connections, Hindustan Book Agency 2018. Rep Springer New York, 2018.
• Gaṇitayuktibhāṣā (c.1530) of Jyeṣṭhadeva (in Malayalam), Ed. with Tr. by K. V. Sarma with Explanatory Notes by K. Ramasubramanian, M. D. Srinivas and M. S. Sriram, 2 Volumes, Hindustan Book Agency, Delhi, 2008.

1.  BASICS OF INDIAN ASTRONOMY

1. Introduction

The science of Astronomy. Astronomy as one of earliest sciences; observational astronomy in the Vedic corpus. Emergence of Jyotiḥśāstra encompassing the three skandhas of Gaṇita (Astronomy), Horā (Horoscopic Astrology and Saṃhitā (Omens and Natural Phenomena). The purpose of Astronomy—as stated in the texts. Contents of a typical Indian astronomical Siddhānta text. Broad classification of the texts in Indian astronomy. Names of some of the prominent astronomers and their important contributions. Highlight the continuity of the Indian astronomical tradition (1400 BCE – 19th cent CE).

2. The different units of time discussed in the texts

Brief introduction to the concept of time (approach of physics and philosophy). Quote from Bhāskara I’s commentary at the beginning of Kālakriyā; also quote the verse in the famous text Sūryasiddhānta. Recount the currently used units of time—duration of year, month, week, etc. in the Gregorian calendrical system—subtly point out that they do not have any astronomical basis whatsoever. Introduce the different shorter units of time discussed in Indian astronomical texts year, month, fortnight, tithi, etc. Introduce larger units or time like yuga, mahāyuga, manvantara and kalpa.

3. Systems employed for representing numbers

Highlight the need for having different systems for representing numbers in those days. Explain the three systems adopted – Bhūtasaṅkhyā, Kaṭapayādi and Āryabhaṭīyapaddhati. With illustrative examples, bring out their beauty and ingenuity. Briefly discuss the advantages in each of these systems.

4.  Spherical trigonometry

Introduce the notion of shortest path on a non-Euclidean surface. Definition of great circle, small circle, spherical triangle, etc. Their illustration using the Earth as an example, which the students will be familiar with. Compare and contrast the properties of a spherical triangle with a planar triangle. Derive the cosine and sine formula from ‘first’ principles. Introduce the four-part formula. Work out a few illustrative problems (such as distance travelled by a flight along the great circle arc, small circle arc, etc) that would help visualise the circles on a sphere, as well as assimilate the application of the formulae. Demonstrate how to derive the sine formula simply using the planar triangles, and their projections inside the sphere.

5.  Celestial Sphere

The notion of celestial sphere and the need for its conception. The different coordinate systems (horizontal, equatorial and ecliptic) employed. The range of the coordinates in each of these systems. The advantages and disadvantages of one system over the other. Some illustrative examples for converting one set of coordinates into another. Indian names for the fundamental circles and the coordinates used in these systems.

• What is Pañcāṅga?

Division of the celestial sphere/ecliptic into 12 and 27 equal parts — rāśi and nakṣatra division. Explain their significance by pointing out their basis; that is, they are connected with the duration of 12 lunar months and the period of moon’s revolution around the earth, and not introduced arbitrarily. Explain the five elements that constitute Pañcāṅga – and also bring out their astronomical significance. Also point out that they are essentially different units of time. Illustrate with numerical examples the computation of these elements in a Pañcāṅga. Explain how to compute the average period of a lunar month; Bring out the need for the introduction of an adhikamāsa in the calendrical system. Outline the broad categories into which different calendars that are followed can be put into— namely solar, lunar and luni-solar.

7.  Key concepts pertaining to planetary computations

The revolution numbers of various planets, nodes, apogees, etc.; The count of the number of civil days, adhikamāsas, etc. in a mahāyuga. Introduce the concept of Ahargaṇa, and its significance; The basis for choice of epoch. Calculation of Ahargaṇa; Illustration with a few numerical examples choosing contemporary dates – using siddhāntic text (to begin with). Explain the computation of mean motion of planets, and how its computation along with the Ahargaṇa can help in finding the mean position of planets.

8. Computation of the true longitudes of planets

Provide an overview of the steps involved in the computation of the true longitudes. Explain manda- saṃskāra in detail using epicyclic model and eccentric model. Outline the nature of the resultant orbit, etc, and explain how this correction takes into account the eccentric nature of the planetary orbit. Emphasise and make the students appreciate the simplification achieved in computation by the ‘constraint’ r/R = r₀/R. Explain śīghra-saṃskāra in detail; Point out how this correction boils down to the transformation of the heliocentric coordinates to geocentric. Also indicate how this simple model takes care of the retrograde motion of the planets. Bring out the distinction between the inner and outer planets.

• Precession of equinoxes – sāyana and nirayaṇa longitude

Introduce the concept of precession of equinoxes. Explain solsticial and equinoctial points, and connect them to the concept of uttarāyaṇa and dakṣiṇāyana in the Indian calendrical system. Derive the formula for finding the declination of the sun on any day at any time, and also illustrate it with examples. Also highlight how crucial its accurate computation is for the computation of various other quantities precisely — including the problem of finding the direction and the latitude of the place — even if we choose to do them by experimental methods.

9. Finding the cardinal directions and the latitude of a place

Introduce śaṅku (the gnomon), and explain how it has to be prepared as described in the texts. Describe the experimental set up that has to be made meticulously for conducting experiments with śaṅku and doing shadow measurements. Explain how with a very simple experiment the directions at a given place can be easily and precisely determined. Also point out that this experimental method is very old—described even in the Śulbasūtras. Also outline the theoretical basis for the formula that has been given for correcting the points marked in connection with determination of the direction

using śaṅku. Bring out the versatility of this simple device śaṅku in determining a variety of physical quantities of interest including the latitude of the place. Explain the concept of parallax in general, and how it introduces an error – that is unavoidable in conducting this experiment for determining the latitude of the place. Outline the corrections that have been prescribed in the text that would take into account the above error, as well as the fact that sun is not a point source of light.

10. Determination of the variation of the duration of the day at a given location

Introduce the 6’o clock circle and its significance. Derive the formula for the hour angle at sunset, and explain how the latitude and the declination of the sun play a role in it. Explain the concept of cara as outlined in the Indian astronomical texts that captures the variation in the duration of the day at a given location. Present the formula for determining the local time using shadow measurements. Also outline how cara plays a role in determining this local time. Bring out the distinction between this local time and the standard time that we are generally familiar with and generally keep track of.

1. Lagna and its computation

Introduce the concept of lagna, and how non-trivial a problem it is to determine it precisely. Also point out how this is deeply connected with fixing times for various social and religious functions such as marriage, etc. Bring out the connection between its computation and the computation of declination, cara etc. that would have been discussed before. Explain how this can be determined ‘reasonably’ accurately using interpolation. Also outline more precise formulations that have been given by later astronomers by introducing the notion of kālalagna.

11. Eclipses and their computation

Briefly explain the phenomenon of lunar and solar eclipses, and the crucial role played by the position of the lunar nodes in their computation. Also bring out how difficult it is to precisely determine the position of the nodes—as they are not physical objects available for observation. Explain how the latitude of the moon is computed, and then outline the procedure for the determination of the semi- diameters of the eclipsing and the eclipsed bodies. Derive the simple formula for determining the duration of eclipses as well as the obscuration. Also mention that iterative procedures are followed to improve accuracy. Point out the role of parallax in the determination of solar eclipses.

Suggested References:

1. S. N. Sen and K. S. Shukla, History of Astronomy in India, 2^nd Ed., INSA, Delhi, 2001.
2. S. Balachandra Rao, Indian Astronomy An Introduction, Universities Press, Hyderabad, 2000
3. History of   Astronomy: A Handbook, Edited   by K. Ramasubramanian, Aniket Sule and Mayank Vahia, SandHI, IIT Bombay, and T.I.F.R. Mumbai, 2016.
4. B.V. Subbarayappa and K.V. Sarma, Indian Astronomy: A Source Book, Nehru Centre, Bombay, 1985.
5. Tantrasaṅgraha of Nīlakaṇṭha Somayājī, Translation and Notes, K. Ramasubramanian and

M. S. Sriram, Hindustan Book Agency, New Delhi, 2011.

I. INTRODUCTION TO INDIAN ASTRONOMY

1. Preliminaries

Sky viewed as the inside of a hemisphere. Cardinal directions, zenith, horizon, pole star at any location. Daily motion of celestial objects (Sun, Moon, planets, stars) and diurnal circles. Motion in the stellar background. Ecliptic. Basic time units: Day, Month and Year. Celestial coordinates and elementary spherical trigonometry. Cosine and Sine formulae. Horizontal (z,A), Equatorial (δ, α and H), and Ecliptic (λ, β) systems. cos z = sin φ sin δ + cos φ cosδ cos H, and other relations. Planetary positions.

2. Developments from the Vedic period up to the Siddhāntic period

Vedic Astronomy: Astronomical concepts in Vedic literature regarding Sun, Moon, Stars, Earth. Months, seasons, year. 27 nakṣatras. Ecliptic and ayana. Planets, Comets etc. Pole star in an earlier era. Nakṣatra division of the ecliptic and motion of the Sun along it in Vedāṅga Jyotiṣa (VJ) and other texts. VJ calendar. VJ computations. Duration of a day. Better value for an year in Vedic literature.

Siddhāntic astronomy: Earlier Siddhāntas and Pañcasiddhāntikā. Introduction of trigonometry, Indian jyā–astronomy. Āryabhaṭīya . Mahāyuga. (Kalpa etc., and also smaller units of time can be introduced at this stage). Revolution numbers of planets. Ahargaṇa and Mean longitudes, Examples. Obtaining the true longitudes by applying corrections to mean longitudes.

Epicycle models: Manda correction (Equation of centre) in detail. Its significance. Latitude of Moon.

Śīghra correction to planets and its significance: Essential features only with the aid of diagrams and final formulae. Latitudes of planets.

Precession of equinoxes— Nirayana and Sāyana longitudes.

Nature and organisation of texts. Sūtra (algorithmic) format. Siddhānta, Tantra, Karaṇa and Vākya

texts. Sāraṇis or Tables.

3. Indian Calendar

Pañcāṅga. Adhikamāsas. Solar and Luni-Solar systems.

4. Solar and Lunar Eclipses

Angular diameters of the Sun, Moon and Earth’s shadow. Possibility of eclipses. Finding the middle of an eclipse by iteration. Amount of obscuration at any time.

5. Tripraśna Topics (Diurnal problems)

Description of the celestial spheres and various circles. Similarity to modern description. Determination of the East-West directions. Derivation of the expression for the declination in terms of the longitude. Shadow of a gnomon. Equinoctial day when the locus of the tip of the shadow is a straight line. Finding the latitude. Mid-day shadow. Finding the declination. Relation between the time and the shadow at an arbitrary instant (no derivation).

6. Planetary longitudes and latitudes  and  Nīlakaṇṭha Somayājī’s revised planetary

model

True longitudes of planets: Manda and Śīghra corrections in detail. Geometrical description. Comparison with Kepler’s model. Latitudes of planets.

Nīlakaṇṭha Somayājī’s revision of the planetary model: Nīlakaṇṭha’s analysis of the motion of the interior planets (Mercury and Venus). His geometrical model which is geometrically similar to the Tycho Brahe model (planets moving around the Sun which itself orbits the Earth), but computationally approximates the Kepler model.

7.  Rates of motion of planets

Idea of derivative in finding the Mandagatiphala (manda-correction to the mean rate of motion). The correct formula due to Nīlakaṇṭha. True rates of motion of planets: Correct expression due to Bhāskara. Application to calculate retrograde motion of planets.

• Tripraśna topics

Latitudinal triangles (of Bhāskara) and applications. Agrajyā or the distance between rising-setting line and the east-west line. Correction to the east-west line due to change in Sun’s declination. Zenith distance in terms of the declination, hour angle and latitude (cos z = sin φ sin δ + cos φ cos δ cos H). Derivation of this formula as in Siddhāntaśiromaṇi. Relation among Śaṅkutala (Śaṅkvagra), Bhujā, Agrajyā and its applications.

• Rising times of Rāśis and finding Lagna

Relation between the right ascension and longitude and rising times of rāśis at the equator. Rising times at an arbitrary latitude. Finding the Lagna at any instant after Sunrise (approximate).

10. Eclipse calculations

Details of calculations of the middle of a lunar eclipse and half-durations iteratively, using the correct expression for the rate of motion of the Moon. Parallax and the calculation of the middle of a solar eclipse.

1. The Vākya system

Longitude of the Sun from the ‘subtractive minutes’ at any time (‘Bhūpajña etc. vākyas). Vākyas for zodiacal transit times (‘Śrīrguṇamitra’ etc.). Longitude of the Moon using the Candravākyas (‘gīrnaśreyaḥ’ etc). More accurate values due to Mādhava.

12. Astronomical Instruments

Gnomon. Cakra and Dhanur yantras for measuring the zenith distance of the Sun. Approximate and exact times from a ‘yaṣṭi’. Phalakayantra to measure the hour angle. Equatorial sundial to measure time. Clepsydra for measuring time. Celestial globe and Armillary sphere for explaining celestial coordinates and various circles.

13. Indian Astronomy in the 18^th and 19^th centiries

Astronomical endeavours of Savai Jayasiṃha. Samrat-yantra and other instruments in the observatories of Jayasiṃha. European observers on the simplicity and accuracy of Indian eclipse computations. The work of Śa◻karavarman and Candraśekhara Sāmanta. Efforts to update the Indian calendar.

Suggested References

1. M. S. Sriram, Man and the Universe- An elementary account of Indian Astronomy, (Unpublished 1993).
2. M. S. Sriram, Elements of Indian astronomy- 5 Lectures, Instructional Course on Indian Sciences, Prof. K.V. Sarma Research Foundation, 2019.
3. (Videos             available             at             https://www.youtube.com/watch?v=Qzam3vEnD- 8&list=PLF72fmBZVDxlkv0Ih_aSHnax5S5-wug8v)
4. S. N. Sen and K. S. Shukla, Eds., History of Astronomy in India, 2^nd Ed., INSA, New Delhi, 2001.
5. S. Balachandra Rao, Indian Astronomy-Concepts and Procedures, M.P. Birla Institute of Management, Bengaluru, 2014.
6. K. Ramasubramanian, A. Sule and M. Vahia, Eds. History of Astronomy: A Handbook, SandHI, I.I.T Bombay and T.I.F.R., Mumbai, 2016.
7. Āryabhaṭīya of Āryabhaṭa, Edited with translation and notes, K. S. Shukla and

K. V. Sarma, Indian National Science Academy, New Delhi, New Delhi, 1976.

• B.V. Subbarayappa and K.V. Sarma, Indian Astronomy: A Source Book, Nehru Centre, Bombay, 1985.
• Tantrasaṅgraha of Nīlakaṇṭha Somayājī, Translation and Notes, K. Ramasubramanian and M.S. Sriram, Hindustan Book Agency, New Delhi 2011 (Rep. Springer, New York 2011).

Karaṇapaddhati of Putumana Somayājī, Translation and Notes, R. Venkateswara Pai, K. Ramasubramanian, M.S. Sriram and M. D. Srinivas, Hindustan Book Agency, New Delhi, 2018 (Rep. Springer, New York 2018)