aryabhatiya

OF

ARYABHATA

(wikipedia info)

Critically edited with Introduction, English Translation. Notes, Comments and Indexes

By

KRIPA SHANKAR SHUKLA

Deptt. of Mathematics and Astronomy University of Lucknow

in collaboration with K. V. SARMA

V. V. B. Institute of Sanskrit and Indological Studies Panjab University

INDIAN NATIONAL SCIENCE ACADEMY NEW DELHI

Published for

THE NATIONAL COMMISSION FOR THE COMPILATION OF HISTORY OF SCIENCES IN INDIA

by

The Indian National Science Academy Bahadur Shah Zafar Marg, New Delhi— 1

©

Indian National Science Academy

1976

Rs. 21.50 (in India) $ 7.00 ; £ 2.75 (outside India)

EDITORIAL COMMITTEE

Chairman : F. C. Auluck

Secretary : B. V. Subbarayappa

Member : R. S. Sharma

Editors : K. S. Shukla and K. V. Sarma

Printed in India At the Vishveshvaranand Vedic Research Institute Press Sadhu Ashram, Hosbiarpur (Pb.)

CONTENTS

Page

FOREWORD iii

INTRODUCTION xvii

1. Aryabhata— The author xvii

2. His place xvii

1. Kusumapura xvii

2. Asmaka xix

3. His time xix

4. His pupils xxii

5. Aryabhata's works xxiii

6. The Aryabhatiya xxiii

1. Its contents xxiii

2. A collection of two compositions xxv

3. A work of the Brahma school xxvi

4. Its notable features xxvii

1. The alphabetical system of numeral notation xxvii

2. Circumference-diameter ratio, viz., tz xxviii . 3. The table of sine-differences xxviii

4. Formula for sin 0, when 6>rc/2 xxviii

5. Solution of indeterminate equations xxviii

6. Theory of the Earth's rotation xxix

7. The astronomical parameters xxix

8. Time and divisions of time xxix

9. Theory of planetary motion xxxi -

10. Innovations in planetary computation xxxiii

11. Celestial latitudes of the planets xxxiii

12. Use of the radian measure in minutes xxxiv

5. Its importance and popularity xxxiv

6. Commentaries on the Aryabhatiya xxxv

(a) Commentaries in Sanskrit

1. Bhaskara I's commentary xxxv

2. Prabhakara's commentary xxxvi

3. Somes vara' s commentary xxxvi

4. Suryadeva Yajva's commentary xxxvii

5. Paramesvara's commentary xl

6. Yallaya*s notes on Suryadeva 's commentary xl

1

li liii liv

ARYABHATIYA

Page

7. Mlakantfia Somayaji's commentary xhv

8. Raghunatha-ra-ja's commentary xlv j

9. Commentary of Madhava

10. Bhmivisnu's commentary

11. Ghatigopa's commentary

12. Kodandarama's commentary

(b) Commentaries in Telugu

13. Kodandarama's commentary lv

14. VirUpaksa's commentary lv

© Commentaries in Malay alam

15. Krsnadasa's commentary lv

16. Krsna's commentary lv * 17-18. Two commentaries by Ghatlgopa lvi

{d) Commentary in Marathi

19. Anonymous commentary in Marathi lviii

h Works based on the Aryabhatiya lvi »

1. The worksofBhaskaral lv »i

2. The Karana-ratna of Deva lvil1

3. The Gtaha-cara-nibandhana of Haridatta lix

4. The §isya-dhx-nddhida of Lalla l ix

5. The Karava-prakafa of Brahmadeva * x j

6. The Bhatatulya of Damodara ^ 1™

7. The Karava-paddhati of Putumana Somayaji lxii

8. The Aryabhata-siddhanta-tulya-karana of VIrasimha lxii 8. Transmission to Arab lxil

The Aryabhafa-Siddhanta lxni

1. The Aryabhata-siddhanta and the Aryabhatiya lxiii

2. The astronomical instruments and special methods lxv

3. Popularity of the Aryabhata-siddhanta and the Khanda-khadyaka lx ™

The present edition lxvlli

(a) Sanskrit text lxviii

1. Text manuscripts lxix

2. Text preserved in commentaries lxxl

3. Quotations from later commentators lxxu

4. Variations in reading lxxm

5. Selection of readings lxxiv

(b) English translation, notes etc. lxxiv

Acknowledgements lxxvl

CONTENTS

I. 1 HE GITIKA SECTION Invocation and Introduction Method of writing numbers Revolution-numbers and zero point Kalpa, Manu and beginning of Kali Planetary orbits, Earth's rotation Linear diameters

Obliquity of the ecliptic and inclinations of orbits

Ascending nodes and Apogees

Manda and Sighra epicycles

Rsine-differences

Aim of the Dasagitiku-Sutra

II. GANITA OR MATHEMATICS

Invocation and Introduction

The first ten notational places

Square and squaring

Cube and cubing

Square root

Cube root

Area of a triangle

Volume of right pyramids

Area of a circle

Volume of a sphere

Area of a trapezium

Area of plane figures

Chord of one-sixth circle

Circumference-diameter ratio

Computation of Rsine-table geometrically

Derivation of Rsine-differences

viii

Aryabhatiya

Construction of circle etc. and testing of level and verti- cally

Radius of the shadow-sphere

Gnomonic shadow due to a lamp-post

Tip of the gnomonic shadow from the lamp-post and height of the latter

Theorems on square of hypotenuse and on square of half- chord

Arrows of intercepted arcs of intersecting circles Sum (or partial sum) of a series in A.P. Number of terms in a series in A.P. Sum of the series l + (i+2)+(I+2+3)+ … to n terms.. Sum of the series 2 n 2 and 2n 3 Product of factors from their sum and squares Quantities from their difference and product Interest on principal Rule of three

Simplification of the quotients of fractions Reduction of two fractions to a common denominator Method of inversion

Unknown quantities from sums of all but one Unknown quantities from equal sums Meeting of two moving bodies Pulveriser

Residual pulveriser

Non-residual pulveriser

III. KAXAKRIYA OR THE RECKONING OF TIME Time divisions and circular divisions Conjunctions of two planets in a yuga Vyatipatas in a yuga Anomalistic and synodic revolutions

CONTENTS

Jovian years in a yuga Solar years and lunar, civil and sidereal days Intercalary months and omitted lunar days Days of men, manes and gods, and of Brahma Utsarpim, Apasarpini, Su$ama and Du$sama Date of Aryabhata I

Beginning of the Yuga, year, month and day Equality of the linear motion of the planets Consequence of equal linear motion of the planets Non-equality of the linear measures of the circular divisions

Relative positions of asterisms and planets Lords of the hours and days

Motion of the planets explained through eccentric circles Motion of planets explained through epicycles Motion of epicycles

Addition and subtraction of Mandaphala and Sighraphala

A special pre-correction for the superior planets

Procedure of Mandaphala and Sighraphala corrections for the superior planets

Mandaphala and Sighraphala corrections for inferior planets

Distance and velocity of a planet

IV. GOLA OR THE CELESTIAL SPHERE 1. Bhagola

Position of the ecliptic

Motion of thel nodes, the Sun and the Earth's shadow^

ARYABHATIYA

Motion of the Moon and the planets Visibility of the planets

Bright and dark sides of the Earth and the planets

Situation of the Earth, its constitution and shape

Earth compared with the kadamba flower

Increase and decrease in the size of the Earth

Apparent motion of the stars due to the Earth's rotation

Description of the Meru mountain

The Meru and the Badavamukha

The four cardinal cities

Positions of Lanka and Ujjayim

Visible and invisible portions of the Bhagola

Motion of the Bhagola from the north and south poles

Visibility of the Sun to the gods, manes and men

<* 2. Khagola

The prime vertical, meridian and horizon Equatorial horizon The observer in the Khagola The observer's Drhmandala and Drkksepavxtta The Automatic sphere (Gola-yantra)

3. Spherical astronomy (1. Diurnal motion)

The latitude-triangle

Radius of the day-circle

Right ascensions of Aries, Taurus and Gemim“

CONTENTS

Earthsine

Rising of the four quadrants and of the individual signs Rsine of the altitude Sahkvagra Sun's Agra

Rsine of the Sun's prime vertical altitude Sun's greatest guomon and the shadow thereof

(2. Parallax in a solar eclipse) Rsine of the zenith distance of the central ecliptic point Drggatijyas of the Sun and the Moon Parallax of the Sun and the Moon

(3, The visibility corrections)

Visibility correction Aksadrkkarma for the Moon Visibility correction Ayanadrkkarma of the Moon

(4. Eclipses of the Moon and the Sun)

Constitution of the Moon, Sun, Earth and Shadow, and the eclipsers of the Sun and the Moon

Occurrence of an eclipse

Length of the shadow

Earth's shadow at the Moon's distance

Half-duration of a lunar eclipse

Half-duration of the totality of the lunar eclipse

The part of the Moon not eclipsed

Measure of the eclipse at the given time

Aksavalana

Ayanavahna for the first contact Colour of the Moon during eclipse

gii ARYABHATIYA

Page

When the Sun's eclipse is not to be predicted «. 161

Planets determined from observation »• 162

Acknowledgement to BrahmS ••• 163

164

Conclusion

APPENDICES

I. Index of Half- verses and Key passages .» 165

II. Index-Glossary of Technical terms … 173

III. Subject Index — 182

IV. Select Bibliography on the Aryabhatlya and

allied texts

(a) Primary Sources ••• 1”

(b) Secondary Sources — 211

LIST OF ABBREVIATIONS

1. BOOKS

A

Aryabhatiya of Aryabhata I

ASi

A ryabhata-siddhftnia

BBi

Bhaskara H's Bljaga-nita

BM

Bakhshali Manuscript

BrJa

Brhat-jataka of Varahamihira

BrSam

Brhat-samhita of Varahamihira

BrSpSi

Br&hma-sphuta-siddhanta of Brahmagupta

GCN

Graha-cara-nibandhana of Haridatta

GSS

Ga'nita-sa.ra-sahgraha of Mahavira

GK

Ganita-Kaumudi of Narayana

GT

Ganyita-tilaka of Sripati

KK

Khanda-khadyaka of Brahmagupta

KR

Karava-ratna of Deva

L

Lllavati of Bhaskara II

L{ASS)

Wavati (Anandasrama Sanskrit Series)

LBh

Laghu-Bhaskarlya of Bhaskara I

LMa

Laghu-mtinasa of Manjula (Munjala)

MBh

Maha-Bhaskarlya of Bhaskara I

MSi

Maha-siddhanta of Aryabhata II

NBi

Narayaria's Bljagaxtita

PG

Patigapita of Sridhara

PSi

Panca-siddhantika of Varahamihira

PuSi

PulUa-siddhanta

RoSi

Romaka-siddhanta

&\DVr

$Uya-dhi-vrddhida of Lalla

SiSe

Siddhanta-iekhara of Sripati

Si$i

Siddhanta-siromavi of Bhaskara II

SiTV

Siddhanta-tattva-viveka of Kamalakara

SMT

Sumati-maha-tantra of Sumati

SuSi

Sundara-siddhanta of JHanaraja

SuSi

Surya-siddhanta

Tris

Trifotika of Sridhara

VSi

Vateivara-siddhanta

2. PERIODICALS ETC.

f

ABORT Annals of the Bhandarkar Oriental Research Institute AMM American Mathematical Monthly

LIST OF ABBREVIATIONS

An. SS RnandaSrama Sanskrit Series AR Asiatick Researches

BCMS Bulletin of the Calcutta Mathematical Society B\f Bibliotheca Mathematica

BNISI Bulletin of the National Institute of Sciences of India

Ep. Ind. Epigraphia Indica

IC Indian Culture

IJHS Indian Journal of History of Science

I HQ Indian Historical Quarterly

INS \ Indian National Science Academy

J A Journal Asiatique

J AOS Journal of the American Oriental Society

JBBRAS Journal of the Bombay Branch of the Royal Asiatic Society

JASB Journal of the {Royal) Asiatic Society of Bengal

JASGBI Journal of the Asiatic Society of Great Britain and Ireland

JBORS Journal of the Bihar and Orissa Research Society JBRS Journal of the Bihar Research Society

JDLjCV Journal of the Department of Letters of the Calcutta University

JIBS Journal of Indian and Buddhist Studies (Tokyo)

JIMS Journal of the Indian Mathematical Society

JORM { Journal of Oriental Research, Madras Math. Edu. Mathematics Education QJMS Quarterly Journal of the Mythic Society

TSS Trivandrum Sanskrit Series

ZDMG Zeitschrift fur Deutsche Morgenlandischen GeselU schaften

3. COMMENTATORS

Bh. Bhaskara I Ra. Raghunatha-raja

Br. Brahmagupta Sa. Sankaranarayana

Go. Govinda-svaml So. Somesvara

Kr. Krsnada-sa Sn. Sflryadeva

Ni. Nilakantha Ud. Udayadivakara

Pa. Paramesvara Ya - Yallaya

Pr. Prthndaka

ROMAN TRANSLITERATION OF DEVANAGARI

VOWELS

Short : 8T 5 3 «E H (and 5)

a i u r 1

Long : 3TT f 3; q art ^ aft

a 1 u e o ai au

AnusvSra : 1 Visarga : s Non-aspirant : '

Classified : ^ k

m

•»

c X

t P

Un-classed : if

y

Compound : sr

— m = h

= J

CONSONANTS

ll

•»

kh

8

gh

if

IT

ch

j

jh

*

(J

<Jh

th

d

dh

ph

b

bh

r

1

V

*

ti-

js

5

A 51

n

n 1

n m

if 3

XV

f

INTRODUCTION

The present volume, which forms Part I of our edition of the Aryabhatiya, contains a critically edited text of the Aryabhatiya and its English translation along with explanatory and critical notes and comments.

1. ARYABHATA— THE AUTHOR

The Aryabhafiya is a composition of Aryabhata. The author mentions his name at two places in the Aryabhatiya, first in the opening stanza of the first chapter (viz., Gitika-pada) and then in the opening stanza of the second chapter {viz., Gaijita-pada) . In the conclu- ding stanza, he calls the work Aryabhatiya ('A composition of Aryabhata') after his own name.

This Aryabhata is a different person from his namesake of the tenth century A.D., the author of the Maha-siddhnnta. To distinguish between the two, the author of the Aryabhatiya is called Aryabhata I, and the author of the Maha-siddhmta is called Aryabhata II.

It is Aryabhafa I, author of the Aryabhafiya, after whose name the first Indian satellite was designated 'Aryabhata' and put into orbit on April 19, 1975 and whose 1500th birth anniversary is being celebrated now.

2. HIS PLACE

2.1. Kusumapura

Aryabhata I does not expressly state the place to which he Belonged, but he mentions Kusumapura and there are reasons to believe that he lived at Kusumapura and wrote his Aryabhatiya there. In stanza 1 of chapter ii of the Aryabhatiya, he writes :

“Aryabhata sets forth here the knowledge honoured at Kusumapura.” 1

1. irotefa^ ffprefo f$<TT*PHrf%r STfru i (3, ii. 1)

xviii

INTRODUCTION

The commentator ParameSvara (A.D. 1431) interprets this statement as meaning :

*'Aryabhata sets forth in this country called Kusumapura, the

knowledge honoured at Kusumapura.“ The commentator RaghunStha-raja (A.D. 1597), too, interprets it in a similar way :

“Aryabhata, while living at Kusumapura, sets forth the know- ledge honoured at Kusumapura.” That Aryabhata I belonged to Kusumapura is substantiated by the following stanza which is quoted in connection with Aryabhata I :

“When the . methods of the five Siddhantas began to yield results conflicting with the observed phenomena such as the settings of the planets and the eclipses, etc., there appeared in the Kali age at Kusumapurl SQrya himself in the guise of Aryabhata, the Kulapa welt versed in astronomy/' 1 The Persian scholar Al-Blrum (A.D. 973-1048), too, has, on occasions more than one, called him 'Aryabhata of Kusumapura'. 2

Bhaskara I (A.D. 629), the earliest commentator of the Iryabhafiya, identifies Kusumapura with Pataliputra in ancient Magadha, and 'the knowledge honoured at Kusumapura' with the teachings of the Svayam- bhuva- or Br&hmarsiddhanta. He also informs us that at Magadha the year commenced on the first tithi of the dark half of the month Sravana and ended on the fifteenth tithi of the light half of the month Asadha. From the writings of the early Jaina scholars who belonged to Kusuma- pura we know that the astronomers of Pataliputra in Magadha were the followers of the Brahma school. We also know that in Magadha, since A.D. 593 down to the present day, the year, which is known as 'Sala' there, is taken to commence from the first tithi of the dark half of the month Sravana.

Hence we can conclude without any shadow of doubt that Arya- bhata I flourished at Kusumapura or Pataliputra in ancient Magadha, or modern Patna (long. 25° '37 N., lat. 85°' 13 E.) in Bihar State.

2. See for example, Al-BjrunVs India, translated by E.C. Sachau, Vol. I, London (1910), pp. 176, 246, 330 and 370.

XRYABHATA'S TIME

xix

Repeated homage to Brahma 1 (the promulgator of the Svayambhma- siddhnnta) and acknowledgement to 'the grace of Brahma' 2 in the Aryabhafiya, also point to the same conclusion.

2.2. Asmaka

Bh5skara I (629 A.D.), the commentator of the Aryabhafiya, refers to Aryabhata I as Asmaka, his Aryabhafiya by the names Asmaka- tantra and Asmakiya, and his followers by the designation Mmakiyah at several places in his writings in more than one context.

The use of the above-mentioned words shows that Aryabhata I was an Asmaka, i.e., his original homeland was ASmaka. According to the commentator Nllakantha (1500 A.D.), he was born in the ASmaka Janapada. 3 (For Asmaka, see vol. II, introduction, pp. xxvii- xxviii).

It seems that Aryabhata I was an Asmaka who lived at Pataliputra (modern Patna) in Magadha (modern Bihar) and wrote his Aryabhafiya there. Magadha in ancient times was a great centre of learning. The famous University of Nalanda” was situated in that state in the modern district of Patna. There was a special provision for the study of astronomy in this University. According to D.G. Apte, 4 an astronomical observatory was a special feature of this University, lm a passage quoted above, Aryabhata I has been designated as Kulapa (=Kulapati or Head of a University). It is quite likely that he was a Kulapati of the University of Nalanda which was in a flourishing state in the fifth and sixth centuries A.D. when Aryabhata I lived.

3. HIS TIME

The year of birth of Aryabhata I is known to us with precision. There is a verse in the Aryabhafiya which runs as follows : “When sixty times sixty years and three quarter-yugas had elapsed (of the

1. A, i. 1 ; A, ii. 1.

2. A, iv. 49.

3. See, opening lines of Nilakant,ha*s comm. on GanitapSda,

4. See Universities in Ancient India, p. 30,

xx INTRODUCTION

current yugth twenty-three years had then passed since my birth.*' 1 This shows that in the Kali year 3600 (elapsed), Aryabhata I was twenty- three years of age. Since the Kali year 3600 (elapsed) corresponds to A.D. 499, it follows that Aryabhata I was born in the year A.D. 476. The Gupta king Buddhagupta reigned at Pataliputra from A.D. 476 to A.D, 496. This shows that Aryabhata I was born in the same year in which Buddhagupta took over the reigns of government at Pataliputra.

To be more precise, 3600 years of the Kali era came to an end on Sunday, March 21, A.D. 499, at mean noon at Lanka or Ujjayini, at the time of Mean Sun's entrance into the sign Aries {madhyama- mesa'sahkranti) (See the table given below). So, the time of birth of Aryabhata I may be fixed at Me$a-sankranti on March 21, A. D. 476. Since at the end of the Kali year 3600 the precession of the equinoxes amounted to zero (see the next paragraph), the amount of the precession of the equinoxes 23 years before the time of Aryabhata's birth was negligible, Hence his birth may be taken to have occurred at nirayaria-me^a-sahkranti or at sayana-me$a- sankrVnti. The Bihar Research Society, Patna, celebrates the birth anni- versary of Aryabhata on April 13, the day on which the Sun now enters into the nirayana sign Aries (i. on the nirayana mesa-sahkranti day).

Mean positions of the Planets 2 at Kali 3600 elapsed, i.e., on Sunday, March 21, A.D. 499, mean noon

at Ujjayini.

Planet Aryabhatiya Aryabhata- Ptolemy Moderns

sfddhanta

Sun

0'

0'

0'

0”

357° 8'

16*

359° ■

42' 5”

Moon

280°

48'

0*

280°

48'

0'

278° 24'

58“

280°

24' 52”

Moon's apogee

35°

42'

0“

35°

42'

0”

32° 43'

42“

35°

24' 38”

Moon's asc. node

352°

12'

0'

352°

12'

0“

349° 25'

33”

352°

2' 26“

Mars

r

12'

0”

12'

0*

4° 20'

12“

52' 45”

Mercury

186°

00'

0“

180°

0'

0”

178° 0'

27“

183*

9' 51”

Jupiter

187°

12'

0“

186°

0'

0”

185° 20'

55“

187°

10' 47”

Venus

356°

24'

0“

356°

24'

0”

351° 4'

15“

356°

7' 51”

Saturn

49°

12'

0“

49°

12'

0*

45° 55'

39”

48°

21' 13“

stlfspPT firerfaxssTScr^ w ^ffts^fan: ll (A, iii. 10) 2. Taken from Siddhanta-Mhara of Srjpati, Part II, edited by

AfcYABHASA'S TIME

It may be asked: What consideration prompted Aryabhata I to mention the end of the Kali year 3600 which happened to occur on Sunday, March 21, A.D. 499, at mean noon at UjjayinI ? Or, does it denote the time of composition of the Aryabhafiya ? According to the commentators of the Aryabhafiya, the object of specifying the end of the Kali year 3600 was to show that at that time the precession of the equinoxes amounted to zero and the mean positions of the planets obtained from the astronomical parameters given in the Gitika-pada did not require any correction. The commentator SDryadeva (6. A.D. 1191), Paramesvara (A.D. 1431) and Nilakantha (AD 1500), however, are of opinion that this was also the time of composition of the Aryabhafiya. K. Ssmbaiiva Sastrt, W.E. Clark and Baladeva Misra, too, hold the same opinion. P.C. Sengupta once entertained this view but later discarded it.

The Kerala astronomer Haridatta (also called Haradatta) (c A D 683), the alleged author of the so-called Sakabda correction (with epoch at Saka 444), has, as remarked by the commentator Nilakantha 1 (rather in surprise), interpreted the above-mentioned verse of the Aryabhafiya (viz, iii. 10) in a different way : “When sixty times sixty years and three quarter-^ had elapsed (of the current yuga), twenty-three years of my age have passed since then.” No commen- tator of the Aryabhafiya, not even of Kerala, has' interpreted the above passage in this way. T. S. Kuppanna Sastri has called it a wrong interpretation. 2 Another Kerala astronomer (probably Jyes t hadeva), author of the Dfkkarana (A.D. 1603), an astronomical manual in Malayalam, has actually stated thatAryabhata I was born in A.D. 499 and that Aryabha^a I wrote the Aryabhafiya twenty-three

Babuaji Misra, Calcutta, 1947, introduction by P.C. Sengupta and N. C. Lahiri, p. xii.

1. See his commentary on A, iv. 48, p. 150 of the printed edition, Trivandrum Sanskrit Series, No. 185.

2. See Mahabhmkanyam, edited by T.S. Kuppanna Sastri, introduction, p. xvi.

xzii

INTRODUCTION

years later, in 522 A.D. 1 This, according to T.S. Kuppanna Sastrl, is a mistaken impression. 2

It must be noted that the translation of the verse in question as given earlier (on p. xix-xx) is in agreement with the interpretation of the commentators. This is also in conformity with what, according to Bhaskara I, Aryabhata I himself told his pupils while teaching the subject.

However, the duality of interpretation of the above verse has given rise to two epochs (called bhatabda, 'the year of Aryabhata') associated with Aryabhata I, viz. Saka 421 (= A.D. 499) and Saka 444 (=A.D. 522), and to two bija corrections, one taking the beginning of Saka 421 and the other the beginning of Saka 444 as the zero point.

4. HIS PUPILS

No more information regarding the life of Aryabhata 1 is now available to us. The Iryabhafiya does not throw light on such aspects as his parentage, his educational careeer, or other details of his personal life. From the writings of Bhaskara I (A.D. 629), it appears that Aryabhata I took up, as was expected of him, the profession of a teacher. BhSskara I mentions the names of Pandu- ranga-svami, Lstadeva and NiSanku amongst those who learnt astronomy at the feet of Aryabhata I. Of these pupils of Aryabhata I, Lata- deva is the most important and deserves special notice. He earned a name as a great scholar and teacher of astronomy. Bhaskara I t has called him Acarya ('Learned Teacher') and Sarva-siddhnnta-guru ('teacher of all systems of astronomy' or 'well versed in all systems of astronomy'). From the writings of VarShamihira (died A.D. 587) and Sripati (A.D. 1039), we learn that Latadeva was the author of at least two works on astronomy; in one, the day was measured from midnight at Lanka (lat. 0, long. 75°.43 E). Varahamihira has also

1. See Grahacaranibandhana, edited by K.V. Sarma, introduction, p. v. The same is stated in the Sadratnamalh of Sankaravarman (A.D. 1800-38). See A history of the Kerala school of Hindu astronomy, by K.V. Sarma, p. 8.

2. Ibid, p. xv.

iRYABHATA'S works

xxitf

ascribed to him the authorship of two commentaries, one on the Romaka-siddhanta and the other on the Paulisa-siddhQnta. According to the Persian scholar Al-Birum (A.D. 973 to A.D. 1048), Latadeva was the author of a Surya-siddhanta. Reference to 'Acarya Latadeva* has been made by Brahmagupta (A.D. 628) and his commentator Prthndaka (A.D. 860) too. PrthUdaka has also quoted a number of verses from some work of Latadeva. These verses are in arya metre and their language and style are similar to those of the Aryabhatiya.

5. ARYABHATA'S WORKS Aryabhata I wrote at least two works on astronomy :

1. Aryabhatiya

2. Aryabhata-siddhanta.

The former is well known ; the latter is known only through references to it in later works.

VarShamihira has distinguished the two works by the reckoning of the day adopted in them. “Aryabhata said,” writes he, 1 “that the day begins at midnight at Lanka ; the same (Aryabhata) again said that the day begins from sunrise at Lanka.” Other differences between the two works of Aryabhata I have been noted by Bhaskara I in this Mahn- Bhaskanya (vii. 2 1-35). 2 •

6 THE ARYABHATIYA

6.1. Its contents

The Aryabhatiya deals with both mathematics and astronomy. It contains 121 stanzas in all, and is marked for brevity and concise- ness of composition. At places its style is aphoristic and the case- endings are dispensed with. Like the Yoga-darsana of Patanjali, the subject matter of the Aryabhatiya is divided into 4 chapters, called Pada (or Section).

Pada 1 (viz., Gitika-pada), consisting of 13 stanzas (of which 10 are in gitika metre), sets forth the basic definitions and important astronomical parameters and tables. It gives the definitions of the larger units of time (Kalpa, Manu and yuga), the circular units (sign,

1. See PSi, xv. 20.

2, Se infra, Tables 1-5, under Sn, 8.1 below.

Introduction

degree and minute) and the linear units (yojana, nr t hasta and afigula) ; and states the number of rotations of the Earth and the revolutions of the Sun, Moon and the planets etc. in a period of 43,20,000 years, the time and place from which the planets are supposed to have started motion at the beginning of the current yuga as well as the time elapsed since the beginning of the current Kalpa up to the beginning of Kaliyuga, the positions of the apogees (or aphelia) and the ascending nodes of the planets in the time of the author, the orbits of Sun, Moon and the planets including the periphery of the so-called sky, the diameters of the Earth, Sun, Moon and the planets, the obliquity of the ecliptic, and and the inclinations (to the ecliptic) of the orbits of the Moon and the planets, the epicycles of the Sun, Moon and the planets, and a table of sine-differences.

Pada 2 (viz. Ganita-pada), consisting of 33 stanzas, deals with mathematics. The topics dealt with are the geometrical figures, their properties and mensuration ; problems on the shadow of the gnomon ; series ; interest ; and simple, simultaneous, quadratic and linear indeterminate equations. The arithmetical methods for extracting the square root and the cube root and rules meant for certain specific mathematical problems including the method of constructing the sine table are also given.

Pada 3 {viz* KalakriyH-pada), containing 25 stanzas, deals with the various units of time and the determination of the true positions of the Sun, Moon and the planets. It gives the divisions of the year (month, day, etc.) and those of the circle ; describes the various kinds of year, month and day ; defines the beginning of the time-cycle, the so-called circle of the sky, and the lords of hours and days ; explains the motion of the Sun, Moon and the planets by means of eccentric circles and the epicycles ; and gives the method for computing the true longitudes of the Sun, Moon and the planets.

Pada 4 {viz. Gola-pada), consisting of 50 stanzas, deals wtih the motion of the Sue, Moon and the planets on the celestial sphere. It describes the various circles of the celestial sphere and indicates the method of automatically rotating the sphere once in twenty-four hours ; explains the motion of the Earth, Sun, Moon and the planets ; des- cribes the motion of the celestial sphere as seen by those on the

THE ARYABHATlYA

xtv

equator and by those on the north and south poles ; and gives rules relating to the various problems of spherical astronomy. It also deals with the calculation and graphical representation of the eclipses and the visibility of the planets.

6.2. A collection of two compositions

The Aryabhatiya is generally supposed to be a collection of two compositions : (f) Dasagitika suira (Aphorisms in 10 gitika stanzas), which consists of PSda 1, stating the astronomical parameters in 10 stanzas in gitika metre, and (2) Aryatfasata (108 stanzas in arya metre) or Aryabhata-tantra (Aryabhaja's tontra), which consists of the second, third and fourth Padas, containing in all 108 stanzas in arya metre. It is noteworthy that the Da'sagitika-sntra and the Arya$ta'sata both begin with an invocatory stanza and end with a concluding stanza in praise of the work and look like two different works. The commentator Bhaskara I (A.D. 629) regards the two as two different works and designates his commentaries on them by the names Dasagitikn-sUtra-vySkhya and Aryabhata-tantra-bha^ya, respectively. He has also referred to the Daiagitika-sUtra as svatantrantara (author's own t ant ran tar a) in the Aryabhata-tantra-bhdfya. 1 Other commentators of the Aryabhatiya, too, hold the same opinion. The commentator Snryadeva (b. A.D. 1191) has called the Dasagitikd-sUtra and the Aryatfasata as two compositions (nibandhanadvaya). The commen- tator Raghunatha-raja (A.D. 1597) has also made similar statements. The commentators Yallaya (A.D. 1480) and Nllakantha (A.D. 1500) have commented upon the second, third and fourth chapters of the Arya- bhatiya only, which shows that they regarded these chapters as forming one complete work. The north Indian astronomer Brahmagupta (A.D. 628) has also referred to Pada 1 of the Aryabhatiya as Da'sagitika and the rest of the Aryabhatiya as Aryatfaiata.

It seems that the Da'sagitika-sutra, which begins with an invocatory stanza and ends with a concluding stanza in praise of it, was issued as a separate tract, like the multiplication tables of arithmetic, and was meant for the freshers who were expected to learn the astronomical parameters given therein by heart before embarking upon the study of

1. See Part II, p. 188. A. Bh. ir

INTRODUCTION

astronomy proper. The Aryatfasata was meant for those who had mastered the Dasagitika-sUtra and were qualified for the study of astronomy proper.

There is no doubt, however, that the Dasagitika-sUtra and the Aryastaiata, taken together, form the Aryabhatiya and that the Dasa- gitika-sUtra, the Ganita, the Kalakriya, and the Gola form the four chapters of the Aryabhatiya. This is quite clear from the following stanza of the Dasagitika-sUtra where the author proposes to deal in that work three topics, viz., ganita, kalakriya and gola :

“Having paid obeisance to Brahma — who is one and many, the real God, the Supreme Brahman — Aryabhafa sets forth the three, viz., mathematics (ganita), reckoning of time (kalakriya) and celestial sphere (gola)!'

Moreover, the four chapters are generally known as Gitika-pada, Gat^ita-pada, Kalakriya-pada, and Gola-pSda, respectively. The word pada means quarter or one fourth, and unless there are four chapters in a book its chapters cannot be rightly called Pndas or 'Quarters'.

It is noteworthy that Aryabhafa I himself has called Gitika-pada by the name Dasagitika-sUtra and the whole work by the name Arya- bhatiya. The names Aryastasata and Aryabhata-tantra were given to the second, third and fourth Pddas by later writers. The former occurs for the first tims in the Brahma-sphuta-siddhVnta of Brahmagupta ; the latter seems to be due to Bhaskara I.

6.3. A work of the Brahma school

From the obeisance to Brahma in the opening stanzas of the first and second Padas of the Aryabhatiya, it is evident that Aryabhafa I was a follower of the Brahma school of Hindu astronomy. Acknowledge- ment of His grace at the successful completion of the Aryabhatiya in one of the closing stanzas of this work shows how deeply was he devoted to Him. This devotion to God Brahma has led people to suppose that Aryabhata I acquired his knowledge of astronomy by performing penance in propitiation of God Brahma. In his commentary on the Aryabhatiya (i. 2), Bhaskara I writes :

“This is what one hears said : This Acarya worshipped God Brahma by severe penance. So, by His grace was revealed

THE XRYABHATIYA

xxvii

to him the true knowledge of the subjects pertaining to the true motion of the planets. It is said : '(Aryabbafa) who exactly followed into the footsteps of (Vyasa) the son of ParaSara, the ornament among men, who, by virtue of penance, acquired the knowledge of the subjects beyond the reach of the senses and the poetic eye capable of doing good to others'.”

Aryabhata's devotion to Brahma was indeed of a high order. For, in his view, the end of learning was the attainment of the Supreme Brahman and this could be easily achieved by the study of astronomy. In the closing stanza of the Daiagitika-sutra, he says :

“Knowing this Dasagitikd-sUtra, the motion of the Earth and the planets, on the celestial sphere, one attains the Supreme Brahman after piercing through the orbits of the planets and the stars.”

Aryabhata I's predilection for the Brahma school of astronomy may have been inspired by two main considerations. Firstly, the Brahma school was the most ancient school of Hindu astronomy promulgated by God Brahma himself. Secondly, the astronomers of Kusumapura, where Aryabhafa 1 lived and wrote his %yabhafiya, were the followers of that school. “The learned people of Kusumapura”, writes Bhaskara I, “held the Svayambhuva-siddhanta ifl the highest esteem, even though the Pauli'sa, the Romaka, the VHsistha and the Saurya SiddhQntas were (known) there.”

6.4. Its notable features

The following are the notable features of the Aryabhatiya :J 1. The alphabetical system of numeral notation, (i. 2)

The alphabetical system of numeral notation defined by Arya- bhata I is different from the so-called katapayadi system but much more effective in expressing number briefly in verse.

According to this notation —

Sfij denotes the number 43,20,000

afofaJ

» „ 1,58,22,37,500

For details see below, GTtika-pada, vs. 2, p. 3-$.

txtriii

INTRODUCTION

2. Circumference-diameter ratio, viz., *=3.1416. (ii. 10)

Aryabhata I states that

Circumference : diameter =62832 : 20000, which is equivalent to saying that » =3*1416.

This value of ir is correct to four decimal places and is better than the value 3' 14 1666 given by the Greek astronomer Ptolemy. 1 It does not occur in any earlier work on mathematics and constitutes a marvellous achievement of Aryabhata I.

It is noteworthy that Aryabhata I has called this value only 'approximate'.

3. The table of sine-differences* (i. 12)

Aryabhata I is probably the earliest astronomer to have given a table of sine -differences. He has also stated geometrical and theoretical methods for constructing sine-tables. For details see below Ganita-pada, vss. 11 and 12, pp. 45-54.

4. Formulae for sin 6, when d>ttj2, (iii. 22)

Aryabhata I uses the following formulae :

sin(77-/2 + 0)=shTjr/2 — versin 6

sin (^-1-0)=* sin t/2 — versm→r/2— sin 6

sin(3ir/2-(-0) =sinir/2— -versimr/2— sin tr j2-\- versin 9.

These formulae were later used by Brahmagupta also,* evidently under the influence of Aryabhata I.

5i Solution of indeterminate equations of the following types :

(/) N=ax-\-b=cy-\-d=eZ'\-f— …

() (ax ^c)jb=a whole number. 1. See Sir Thomas Heath, A History of Greek Mathematics, vol. 1, p. 233 ; and D. E. Smith, History of Mathematics, vol. 2, p. 308. 2, See firSpSi, ii. 15-1$. THE XRYABHAT1YA * xlx Aryabhata I is the earliest to have given the general solution of problems of the following types which reduce to the solution of the above equations : (i) Find the number which yields 5 as the remainder when divided by 8, 4 as the remainder when divided by 9, and 1 as the remainder when divided by 7. (ii) 16 is multiplied by a certain number, the product is diminished by 138, and the difference thus obtained being divided by 487 is found to be exactly divisible. Find the multiplier and the quotient. For Aryabhata Fs solution, see below, Ganita-pSda, vss. 32-33, p. 75 ff. 6. Theory of the Earth's rotation It was generally believed that that the Earth was stationary and lay at the centre of the universe and all heavenly bodies revolved round the Earth. But Aryabhata I differed from the other astronomers and held the view that the Earth rotates about its axis and the stars are fixed in space. The period of one sidereal rotation of the Earth according to Aryabhata I is 23 h 56 m 4*1. 1 The corresponding modern value is 23 h 56 m 4 B 091. 2 The accuracy of Aryabhata Fs value is remarkable. 7. The astronomical parameters The astronomical parameters given by Aryabhata I differ from those of the other astronomers and are based on his own observations. They are much better than those given by the earlier astronomers. The method used by Aryabhata I for their determination has been indicated by him in the Gola-pnda (vs. 48). For Aryabhata Fs astronomical parameters, see GltikSrpada. 8. Time and divisions of time Aryabhata I does not believe in the theory of creation and annihi- lation of the world. For him, time is a continuous process, without 1, See GWka-pada, vs. 3. 2. See W.M. Smart, Text-book on Spherical Astronomy, Cambridge, 1923, p, 492., INTRODUCTION beginning and end (anadi and ananta). The beginnings of the yuga. and Kalpa, according to him, have nothing to do with any terrestrial occurrence ; they are purely based on astronomical phenomena depending on the positions of the planets in the sky. In the Smrtis as also in the Snrya-siddhanta, we have the following pattern of time-division : 1 Kalpa— \A Manus 1 Mann =71 yugas 1 1^=43,20,000 years. In order to make the Kalpa equivalent to 1000 Yugas (in round number), every Manu is supposed to be preceded and followed by a period of 2/5 of a yuga, called twilight. A period of 3 95 yugas is further earmarked for the time spent in the creation of the world, so that when the world order starts all planets occupy the same place. A Kalpa is defined as a day of Brahma, 2 Kalpas as a nycthe- meron (day and night) of Brahma, 720 Kalpas as a year of Brahma”, and 100 years of Brahma (or 72,000 Kalpas) as the lifespan of Brahma. The age of Brahma, according to the Surya-siddhanta, at the beginning of the current Kalpa, was 50 years. The current Kalpa is the first day of the 51st year of Brahma's life, and 6 Manus with their twilights and 2 7i% yugas had elapsed at the beginning of Kaliyuga since the beginning of the current Kalpa. Moreover, a yuga is taken to be composed of 4 smaller yugas bearing the names Krta, Treta, DvSpara and Kali. The lengths of these smaller yugas are supposed to be 17,28,000 ; 12,96,000 ; 8,64,000 and 4,32,000 years, respectively. Aryabhata I rejects this highly artificial scheme of time-division, and replaces it by the following : 1 day of Brahms or Kalpa = 14 Manus or 1008 yugas 1 Manu⇒72 yugas I yuga =43,20,000 years. THE ARYABHATIYA Aryabhata I has dispensed with the periods of twilight and the time spent in creation, and has simplified the scheme enormously. Since 1008=0 (mod 7), every Kalpa under this scheme begins on the same day, which is an additional advantage. Under this scheme, 6 Mams and 27f yugas had elapsed at the beginning of the current Kaliyuga since the beginning of the current Kalpa. Aryabhata, too, divided a yuga into 4 smaller yugas, but he takes them to be of equal juration and calls them quarter-jwgas, the duration of each being 10,80,000 years. This is indeed a more scientific division, because in every quarter yuga the planets make an integral number of revolutions round the Earth. Although the time-divisions given in the Surya-siddhanta and by Aryabhata I differ so much, they have been so adjusted that the beginning of the current Kaliyuga according to both of them falls on the same day, viz., Friday, February, 18, 3102 B.C. Aryabhata I has also divided his yugas into 2 divisions, Utsarpini and ApasarpM; and further Utsarpini into Dussama and Susama, and Apasarpini into Susama and Dussama, respectively. 1 This is evidently under the influence of the Jaina scholars of Pataliputra where Aryabhata I lived. Pataliputra, was the original home, of the Jainas and a bulwark of Jaina saints and scholars in ancient times. 9. Theory of the planetary motion The computation of the planetary positions in the Aryabhatiya is based on the following hypotheses : Hypothesis 1. In the beginning of the current yuga, which occurred on Wednesday, 32,40,000 years before the commencement of the current quarter-yuga, all the planets together with the Moon's apogee and the Moon's ascending node were in conjunction at the first point of the asterism AsvinI Piscium). (Gitika-pada, vs. 4 (d) ). Hypothesis 2. The mean planets revolve in geocentric circular orbits. l s For details, see below, KalakriyS, vs. 9, pp. 93-94. xtxn Introduction The mean motions of the planets are given in terms of revolu- tions performed by the planets round the Earth in a period of 43,20,000 years. These revolutions, as already stated, are based on Aryabhata Fs own observations, and constitute the main distinguishing feature of Aryabhata Vs astronomy. For details see Gitika-pada, vss. 3-4. Hypothesis 3. The true planets move in eccentric circles or in epicycles. For details see Kalakriya-pada, vss. 17-21. The eccentric and the epicyclic theories of Aryabhata I have been explained in greater detail by BhSskara I in his Maha-Bhaskanya (ch. iv). Two things may, however, be mentioned here : (i) The manda epicycles are not the actual epicycles but the mean epicycles corresponding to the mean distances of the planets. (ii) The radius of the slgkra concentric (and therefore of .the sighra eccentric), according to Aryabhata I, is equal to the planet's distance called mcmdakarna. The Greek astronomer Ptolemy, too, explained the motion of the planets with the help of epicycles and eccentric circles, but the method used by Aryabhata I for explaining the planetary motion is quite different and much simpler than that used by Ptolemy. Bina Chatterji who has made a comparative study of the Greek and the Hindu epicycles and eccentric theories, concludes that “Aryabhata's epicyclic and eccentric methods are unaffected by Ptolemaic ideas. 1 It may be pointed out that whereas the epicycles of Ptolemy are of fixed dimensions, those of Aryabhata I vary in size from place to place. The variable (or pulsating) epicycles probably yielded better results. The later Hindu astronomers have followed Aryabhata I in taking variable epicycles. 1. See Bina Chatterjee, The Khaif4a-khadyaka of Brahmagupta, Vol. I, Appendix VII, p. 293. World Press, Calcutta, 1970. NOTABLE FEATURES OF THE ARYABHA^IY A xxxiii Hypothesis 4 All planets have equal linear motion in their respective orbits. (Kalakriya-pdda, vs. 12). 10. Innovations in planetary computation From the old Stirya-siddhanta, summarized by Varahamihira, and from the Sumati-mahatantra of Sumati, we learn that the earlier astronomers performed four corrections in the case of the superior planets (Mars, Jupiter and Saturn) and as many as five corrections in the case of the inferior planets (Mercury and Venus) in order to obtain their true positions. 1 The fifth correction applied to the inferior planets was again purely impirical in character and was artificially devised to get correct results. Aryabhata changed the old pattern of correction and, in the case of the inferior planets, reduced the number of corrections from five to three. In the the case of the superior planets, too, the corrections were the same, but a pre- correction (equal to half the equation of centre) was also prescribed. This innovation was an improvement and yielded more accurate results. In the case of finding planetary distances, the Surya-siddhanta* prescribed the formula mandakarna -\- sighrakarna ^ 2 but Aryabhata 3 changed it to mandakarna X sighrakarna R 11. Celestial latitudes of the planets In the time of Aryabhata, astronomers were helpless in finding the celestial latitudes of the planets. The methods given in the old Suryasiddhanta? summarised by Varahamihira, and even in the Siddhanta 1. See PSi, xvi. 17-22. Also see K. S. Shukla, The Pane* siddhantikd of Varahamihira (I), IJHS, vol. ix, no. 1, 1974, pp. 62-76. 2. vii. 14. 3. See A, iii. 25. 4. See PSi, xvi. 24-25. A. Bb.: v INTRODUCTION of Aryabhata himself, 1 (which was written earlier than the Aryabhatiya), are not correct. It is Aryabhata who in his Aryabhatiya* for the first time gave the correct method for finding the celestial latitude of planets, both superior and inferior. 12. Use of the radian measure in minutes Aryabhata is probably the earliest astronomer to use the radian measure of 3438' for the radius of his circles. His table of Rsine- differences is also given in the same measure. Measurement of the radius m minutes facilitates computation and most of the astronomers in India have followed Aryabhata in this respect. Brahmagupta (A.D. 628) who did not use this measure in his Rsine-table was criticised by Vatesvara (A.D. 904). 7*5. Its importance and popularity Brevity and conciseness of expression, superiority of astronomical constants, and innovations in astronomical methods rendered the Aryabhatiya an excellent text-book on astronomy. It gave birth to a new school of astronomy, the Aryabhata School, whose exponents called themselves 'disciples of Aryabhata'. These disciples of Aryabhata” deified Aryabhata I as 'Bhagavan' and 'Prabhu' and held the teachings of the Aryabhatiya in- the highest esteem, claiming greater accuracy for them. Bhaskara I, writing in the first half of the seventh century A. D., declares : “None except Aryabhata has been able to know the motion of the heavenly bodies : others merely move in the ocean of utter dark- ness of ignorance.” Bhaskara I was the most competent exponent of the Aryabhata school. He wrote a commentary on the Aryabhatiya and two other works on astronomy in illucidation of the teachings of Aryabhata I. He earned a great name as a teacher of astronomy and was well known as -guru'. The works of Bhaskara I throw a flood of light on the astronomical theories and methods of Aryabhata I and on the earlier followers of Aryabhata I. His commentary on the Aryabhatiya, which was utilized by most of the subsequent commentators, was recognized as a work of great scholarship and its author came to be designated as 'all-knowing commentator'. 1. See MBh, vii. 28(c-d)-33. 2. iv. 3, COMMENTARIES ON THE XfeYABHATlYA x«v The works of Bhaskara I provided a great stimulus to the study of the Aryabhafiya which became a popular work and continued to be studied at various centres of learning in South' India, especially in Kerala, till recent times. The extent of the popularity enjoyed by the Aryabhafiya can be easily estimated by the following facts : (l) There is hardly any work dealing with Hindu astronomy which does not refer to Aryabhafa or quote from the Aryabhafiya. (2) There exist a number of commentaries on the Aryabhafiya written in Sanskrit and other regional languages by authors hailing from far-flung places in South India. (3) There exist a number of independent astronomical works which are based on the Aryabhafiya. (4) Calendrical texts” and tables used in South India after the first half of the seventh century A.D. until the introduction of new works based on the western astronomical tables and the Nautical Almanac were based on the Aryabhafiya oi works based on it. In northern India, too, the Aryabhafiya continued to be studied at least up to the end of the tenth century A.D. Brahmagupta, who lived in the seventh century at Bhinmal in Rajasthan, made an intensive study of this work. He utilized this work in writing his Brahma-sphuta-siddhanta, and a number of passages in that work are strikingly similar to those found to occur in the Aryabhafiya. Prthtl- daka, who lived at Kannauj in Uttar Pradesh, quotes, in his commentary on the Brahma-sphuta-siddhanta written in A.D. 860, a number of passages from the Aryabhafiya. It is remarkable that for finding the volume of a sphere, Prthndaka prescribes exactly the same rule as found in the Aryabhafiya. He has evidently taken it from the Arya- bhafiya, since this rule is typically AryabhatVs and is not fcmnd tc occur in any other work on Indian astronomy. Several passages from the Aryabhafiya occur in the writings of the Kashmirian scholar Bhatfot- pala who wrote about A.D. 968. 7 6. Commentaries on the Aryabhafiya (a) Commentaries in Sanskrit 1. Bhaskara Ps commentary Bhaskara I's commentary on the Aryabhafiya has been critically edited in Part II of this Series. It is the earliest commentary on the Aryabhafiya that has come down to us. Written at ValabhJ in Sau- INTRODUCTION rSsjra (modern Kathiawar) in the year A.D. 629, it sets forth a compre- hensive exposition of the contents of the Aryabhatiya. “These who want to know everything written by Aryabhata”, writes Sankaranaiayana (A.D. 869), “should read the commentary on the Aryabhatiya and the Mahn-Bhaskanya (written by Bhaskara I).” 2. Prabhakara's commentary From two passages in Bhaskara I's commentary on the Arya- bhatiya it appears that Prabhakara was an earlier commentator of the Aryabhatiya. In both the places, Bhaskara I finds fault with the interpretations given by Prabhakara. Bhaskara I calls him 'Acarya Prabhakara', but says : “He is a teacher, bethinking thus I am not censuring him.” This Prabhakara may have been the same person as has been called 'a disciple (follower) of Aryabhata' by Bhaskara II (A.D. 1150) in his commentary on the §i$ya-dki-vrddhida of Lalla. Acarya Pra- bhakara has also been mentioned by SankaranSrayana (A.D. 869), Udaya-divakara (A.D. 1073), Snryadeva (b. A.D. 1191) and Nllakanjha (A.D. 1500). Prabhakara's commentary has not survived the ravages of time, nor has it been mentioned by any later writer. It is noteworthy that Snryadeva (b. A.D. 1191), in his commentary on the Laghu-manasa (iv. 2), refers to Prabhakara as 'Prabhakara- guru' and mentions his work Prabhakara-ganita. It is not known whether this Prabhakara was the same person as one criticised by Bhaskara I. 3. Somesvara's commentary A manuscript of Somesvara's commentary exists in the Bombay University Library, Bombay. 1 end of it are as follows : Beginning : «ft im*TR sm: fi«if^f *nhr m«iw< far* w*n\ i on the Aryabhatiya The beginning and 1, Catalogue No, 335, Bookshelf No. 272, Accession No. 2495. COMMENTARIES ON THE XRYAPHATlYA xxxvii srfaiqcS etc. End : ^T^^srffT'n^ ijfftrot snrtssrt ?rc*n: *ttt?pc fw^wj Tf%ff fn^r sr$*j u Colophon : fffT ^^^Tf^f%ff»TW'R^r«n«T fWWW I The contents of this commentary show that, as acknowledged by the author himself in the closing stanza, it is a summary of Bhaskara Fs commentary. Even the introductory lines given before the verses commented upon are sometimes almost exactly the same as found in Bhaskara I's commentary. In the commentary on the Ganitapada, however, Somesvara has set some new examples besides those taken from the commentary of Bhaskara I. Somesvara's commentary docs not throw any light on the life and works of its author. A commentary by Somesvara on the Khanda-khadyaka of Brahmagupta is mentioned by Amaraja in the opening^ stanza of his commentary on the same work. From the order in which Amaraja mentions the names of the earlier commentators of the Khanda- khadyaka, it appears that SomeSvara lived posterior to Bhaflotpala (A. D. 968). Since Amaraja lived about 1200 A.D., Somesvara must have lived sometime between A.D. 968 and A.D. 1200. 4. Snryadeva Yajva's commentary Snryadeva calls himself SUryadeva Yajva, Snryadeva Somasut and sometimes Snryadeva Dlksita. Snryadeva's commentary on the Aryabhatiya has been critically edited in Part III of this Series. It is usually known by the following names : Aryabhata-prakasa, Bhata-praka'sa, Prakasa, Aryabhata-praka- sika, Bhata-prakaiika and Prakasika. SOryadeva's commentary sets forth an excellent exposition of the Aryqbhaftyq. It has. been illucidated by further notes and. examples by INTRODUCTION Yallaya (A.D. 1480) and has been used as a source book by Paramesvara (A.D. 1431) in writing his own commentary on the Aryabhatiya* Snryadeva is the author of at least five commentaries, which he wrote in the following order : (1) An exposition of Govinda-svami's bhasya on the Maha - Bhaskanya of BhSskara I (A.D. 629). (2) Commentary on the Iryabhatiya (3) Commentary on the Maha-yMra of Varahamihira. (4) Commentary on the Laghu-manasa of Manjula (A.D. 932). (5) Commentary on the Jataka-paddhati of Sripati (A.D. 1039). From his commentary on the Laghu-mSnasa of Mafijula (A.D. 932), we learn that : (1) Suryadeva was born on Monday, 3rd tithi of the dark half of Magha, Saka 1113,(=A.D. 1191). The ahargana for that day, according to the Aryabhata-siddhUnta, was 15,68,004. 1 (2) He was a Brahmana of Nidhruva gotra.* (3) He belonged to the Cola country (which roughly comprised of Tanjore and Trichinopoly districts of Tamilnadu) and was resident of the town called by the names Gangapura, GangSpuri and Srtranga-gangapuri 3 which may be easily identified with Gangai-konda-Colapuram (lat. 11° 13' N., 1. Suryadeva writes : fa^raW 1113 *n% in*$««r-^tarat titmft tTTTOl qifzfesF clftrsts??T5^Tf^S|'FT: 1 5,68,004 . 2» See colophons at the ends of chs. i and ii. 3. Cf., srta^r '1^15* ( T?nu ) ^rffjTO^p^vnn^Pnfffir: (LMa, ii. 1 com.); sfm^T T^T^ ftrsr^grT: «f*r«r 48,38, 16 {LMn, ii. 1 com.); t^t^ ^cgorRR^r 24, 19, 8 t tftw pn wgtwt: Im? 254, w« 280, fop*»r 315 q^iTerrOr t^t^ imTfcsresrcregiW- fjnfre^RrfH ?refcr {LMa, ii. 2 com.) ; afe^SI^N“ Tff Ij^t SR^t* ' * *H^sir^toT5rwPT ^T^s^f mfc {LMa, iv. 4 com.); ^”te^T ^r?f SPT wwr {LMa, i ©. 3, com.). It may be mentioned that, out of the two manuscripts consulted, one has but the other does not have the word «fft^f prefixed to T^T$n? ; and »l^f in this last referettce. COMMENTARIES ON THE XRYABHATlYA xxxix long. 79° 30' E.), 1 for, according to SHryadeva, the equinoctial midday shadow at that place was 2\ angulas which corresponds to the latitude of ll°-3 N. This is also sub- stantiated by the ascensional differences and times of risings of the signs stated by Suryadeva for the said place. The distance of that place from the Hindu prime meridian is said to have been 1 1 yojanas eastwards. 1. Gangai-konda-CoJapuram ('the city of the Cola king, who conquered up to the Ganges'), also called Ganga-konda-puram, is a town and temple in Trichiriopoly district of Tamilnadu. It is located between Colaroon (a branch of the river Cauveri at its delta region) and the river Ve] lar flowing on its two sides and is situated about six miles from Jayamkonda-Solapuram. It is connected with Udaiyar- palaiyam by the Chidambaram road, and is one mile distant from the great Trunk Road running from Tanjore to South Arcot. Gangai-konda-Colapuram was founded by king RSjendra Cola I (A.D. 1012-44) who was called 'Gangai-konda-Cola' (lit. 'The Cola who conquered up to the Ganges') and who shifted his capital from^ Tanjore to this city, and was known after his name 'the city of the Cola king, who conquered up to the Ganges'. This city remained the#clpital of the Cola kings for many years to come. It has now lost its past glory and is no more than a village. Close to it stand the ruins of one of the most remarkable but least known temples in Southern India. The temple consists of one large enclosure, measuring 584 feet by 372 feet. The vimsna in the centre of the courtyard is a very conspicuous building and strikes the eye from a great distance. The pyramid surrounding it reaches a height of 174 feet. The ruins of six gopuras, or gate-pyramids, surmount different parts of the building. That over the eastern entrance to the main enclosure, was evidently once a very fine structure, being built entirely of stone except at the very top. All the lower part of the central building is covered with inscriptions. Gangai-kon<Ja-Colapuram was called GangSpuii in Sanskrit (Cf. Epigraphia Indica, xv, p. 49). The word &ri-ranga (literally meaning 'the stage of the goddess of prosperity') prefixed to the name Gangapuri in one of the manuscripts consulted seems to point to tho richness and magnificence of this city. ll Introduction (4) He wrote his commentary on the Laghu-manasa in A.D. 1248 (i.e., at the age of 57 years). This is inferred from the fact that Suryadeva has stated the dhruvakas (planetary positions) for Thursday noon, Caitradi, Saka year 1170. 5. Paramesvara's commentary Paramesvara's commentary on the Aryabhafiya was edited by H. Kern and printed at Leiden (Holland) in A.D. 1874. It was reprinted in A.D. 1906 by Udaya Narain Singh along with his Hindi translation of the Aryabhafiya. Paramesvara's commentary sets forth a brief but excellent exposition of the Aryabhafiya. In writing this commentary the author has utilized Suryadeva's commentary, and has quoted from the Sarya- siddhanta, the BrOhma-sphuta-siddhanta of Brahmagupta, the Brhat- samhita of Varahamihira, the Sisya-dhi-vrddhida of Lalla, the Trisatika of Srldhara, and the Lildvati of Bhaskara II. He has also referred to his Mahabhaskariya-bhasya-vyakhyn SiddhetntadipikS, which was written sometime after A.D. 1431. His commentary on the Aryabhafiya was evidently a later work. Paramesvara hails from Kerala. He lived in the village ASvattha (identified with modern Alattnr) situated on the north bank of the river Nila (Mai. B^aratappuzha) near the Arabian sea shore. His first composition was his commentary on the Laghu-Bhaskariya which he wrote in A.D. 1408 when he was still a student. If we presume that he was 28 years of age at that time, his date of birth may be fixed at 1380 A.D. His Drgganita was written in A.D. 1431 and his Goladipika in A.D. 1443. Paramesvara wrote a number of books on astronomy, astrology and allied subjects. See Part II, Scholiasts of Bhaskara I. 6. Yallaya's notes on SHryadeva's commentary * Yallaya has written notes on Ssryadeva's commentary dealing with the second, third and fourth Padas of the Aryabhafiya. Yallaya's commentary on each verse of the Aryabhafiya consists of SOryadeva's commentary followed by Yallaya's notes where necessary. In his notes Yallaya has sometimes illustrated the rules by giving suitable examples with solutions. A manuscript of this commentary exists in the Lucknow University Library. The colophon at the end of it runs as follows : COMMENTARIES on the aryabhatiya The scope of the commentary in the words of its author is as follows : “As the commentary written by Suryadeva Yajva, who had thorough knowledge of the science of words (i.e., grammar), is brief, so for the benefit of those astronomers who want to know the (detailed) meaning of the Ganita, Kalakriya and Gola Padas (of the Aryabhatiya) composed by Aryabhafa, I, learned Yallaya, son of Sridhararya, pupil of Stiryacarya son of Baladitya, well versed in many works on Pafiganita and proficient in the three branches of astronomy, and who has command over language by virtue of the boon acquired from God Siva, will first write those entire explanations of the Srya-sutras ('aphorisms in arya metre') which have been given by Snryadeva Yajva and then, wherever the explanations are brief, will supplement them by further explanations and alter- native illustrative examples.” 1 From the above passage we learn that Yallaya was a son of Sridhararya and a pupil of SnryacSrya. 2 This Sridhara was different from the author of the Patiganita and the Tri'satika* And this Suryacarya was the author of : (1) Ganaknnanda, (2) Daivajnabharana and (3) Daivajna-bhn$ana and was different from Suryadeva (b. A.D. 1191). Suryacarya's father Baladitya was also a famous astronomer. He was nrrensKi^ (em) 3gre^facrff5qfafc^^t*ffa3rr «fm^?5^aT- 2. In his commentary on the Surya-siddhanta (written in A.D. 1478) Yallaya has called his teacher by the name SHrya-Sun and has quoted a large number of passages from the astronomical work Daivajna- bharana written by him. A.Bb. Ti xlti IKTRODUCTIOK the author of a work called Bala-Bhaskariya which has been often quoted by the commentators hailing from the Andhra State. As regards the place where the present commentary was written, Yallaya himself writes : “This exposition was carefully composed by me in the town of Skandasomesvara which is situated towards the south-east of Srisaila.” 1 The equinoctial midday shadow and the equinoctial midday hypotenuse for Skandasomesvara are given to be 3f ahgulas and 12 T 7 3 angulas, respectively. From these figures the latitude of the place comes out to be 15° 30'N. approximately. The distance of Skandasomesvara from the Hindu prime meridian is stated to be 36 yojanas (according to the reckoning of the Sarya-siddhanta) or 23 x 7 3 yojanas (according to the reckoning of the Aryabhafiya),* which corresponds to 4° 5'. As the commentator applies the corresponding longitude correction to the planets negatively, it follows that Skandasomesvara was situated 4° 5' to the east of the Hindu prime meridian. 3 Skandasomesvara was not only the place where this commentary was written, but it was also the place to which Yallaya actually belonged. For in his commentary on the Smya-siddhanta (i. 57-58), Yallaya says : 1. tf^qT^fafOT^wtftftTO^ ^f%: *wt s^it fa-offenr \ (Comm. on A, iii. 6). Srisaila is a temple in the NandikotkQr taluk of Kurnool District, Andhra State, situated in latitude 16° 5' N. and longitude 78° 53' E. It lies in the midst of malarious jungles and rugged hills on the northernmost plateau of the Nallamalais, overlooking a deep gorge through which flows the Krishna river. The temple is 600 feet long by 510.feet broad. The Walls are elaborately sculptured with scenes from the RamUyana and the MahabhSrata. In the centre stands the shrine of Mallikarjuna, the name by which God Siva is worshipped there. 2. Comm. on A, iii. 6. 3. Our conclusions agree with the computations made in the ommentary. COMMENTARIES ON THE ARYABHATIYA xliii “My native country, however, is the town called SkandasomeS- vara which lies towards the south-east of Srisaila.” 1 The date of writing the commentary is 1480 A.D., which corres- ponds to the Kali year 4581 mentioned in the commentary. 2 Other dates mentioned in the commentary are A.D. 1456, AD. 1465, A.D. 1466 and A.D. 1469. In the present commentary there are a number of rules and examples which have been cited from the works of earlier writers. The sources are generally not mentioned. Some rules, however, can be easily traced to the TrisatikS of Srldhara. Several examples are borrowed from the commentary of Bhaskara I. Some rules and examples are Yallaya's own composition. At one place the commentator (Yallaya) refers to the people of Andhra and Karnafaka, saying that they call the number 10 l ° (arbuda) by the denomination htakofu The following tables given by Yallaya in the commentary will be useful to historians : 1. Table of linear measures 8 paramanus 1 trasarenu 8 trasarenus 1 ratharenu 8 rathareyus 1 ko'sa 8 ko'sas 1 tilabija 8 tilabijas 1 sar§apa 8 sar$apas 1 yava 8 yavas 1 angula 12 ahgulas 1 yitasti 2 vitastis 1 hasta 4 hastas 1 da$da 2000 dandas 1 kro'sa 4 krosas 1 yojanft 2, Comm. on 2., iii. 6. * ,iv INTRODUCTION 2. Table of grain measures 4 kudubas — \ prastha 4 prasthas =* l a dha 4 adhas = 1 drona 5 dronas = l khan 3. Table of gold or silver measures 4 vrihis = 1 gM H/a 2 gunjas = 1 m^afoj 2 masakas = I gumarta 10 gumartas = 1 j« rar na 1J suvarnas = 1 4 AttrSfitf as 1 ^0/ fl 4. Names of 29 notational places (1) eka, (2) dWfl, (3) sata, (4) *jtora, (5) (6) (7) proyMte, (8) koti, (9) tob/i, (10) ia/ofo*/, (11) arbuda, (12) njvrWa! (13) kharva, (14) mahu-kharva, (15) padma, (16) mahu-padma, (\1) iahkha, (18) maha-'sahkha, (19) *^f, (20) mahs-ksoni, (21) Jfcy/, (22) maha-ksiti, (23) ksobha, (24) maha-ksobha, (25) pardrdha, (26) sagara, (27) ananta (28) c/n/ya, and (29) Mflr/. 1 ' 7. Nilakan&a Somayaji's commentary This commentary bears the name Maha-bhasya and has been published in the Trivandrum Sanskrit Series, Nos. 101, 112 185 Nilakantha, like Yallaya, has commented only upon the Ganita, Kala- kriya and Gola Padas of the Aryabhatiya. From the colophon occurring at the end of the commentary on the Ganita-padd, we have the following information regarding the commentator (Nilakantha) : 1. His father was called Jstaveda, the same being the name of his maternal uncle also. 1. In place of parardha some people, says Yallaya, use the denomination sankrti. In the list given by Mahavira (A.D. 850), iafikha and maha-kahkha have been replaced by ksoni and maha-ksotf, res- pectively, and vice versa. See B. Datta and, A,N. Singh, History of Hindu Mathematics, Part I, p. 13, COMMENTARIES ON THE ARYABHATIYA xlv 2. His younger brother was named Sankara. 3. He was a Brahmana, follower of the AsvalSyana-sHtra, and belonged to the Gargya-gotra. 4. His teacher in astronomy was Damodara, son of the com- mentator ParameSvara (A.D. 1431) ; and his teacher in Vedanta was Ravi, who was probably the same person as the author of the AcSradipika — a commentary in verse on the Muhnrm$taka. 5. He was a native of the village Kunda, which has been identified with TrkkantiyHr in South Malabar, Kerala. In the commentary on verses 12-15 of the Knlakriya-puda t the commentator writes : “When 16,68,478 days had elapsed since the beginning of Kaliyuga, we observed a total eclipse of the Sun ; and when 16,81,272 days had elapsed, there occurred an annular eclipse in Ananta-ksetra.” The first epoch corresponds to A.D. 1467 and the latter to A.D. 1502. Thus it is evident that this commentary was written after A.D. 1502. From the commentator's SiddhGnta-darpana-vyakhya, we learn that he was born in December A.D. 1444. Hence at the time of writing the present commentary he was above 60 years of age. Nllakanfha's commentary on the Aryabhatiya is a valuable work as it incorporates the advances made in astronomy up to his time and contains a good deal of matter of historical interest. There are quotations from the works of Varahamihira, Prabhakara, Jaisnava or Jisnunandana (i.e., Brahmagupta, son of Jisnu), Bhaskara I, Munjaiaka (same as Munjala or Manjula), Srlpati, Bhaskara II, Madhava, resident of the village Sangama j 1 from the Sarya-stddhanta, the Siddhanta- dipika of Paramesvara, Snryadeva's commentary on the 2ryabhatiya t Govinda-svami's commentary on the Maha-Bhaskariya, and from his own works, the Golasara and the Tantra-sahgraha. Reference is also made to Vrddha-Garga, Garga, Paramarsi, Manittha, VySsa, Varttikakara, 1. Saftgama-grSma is identified with Irinjalakkuda or IrhmSla- kkuda, near Cochin, xlvi INTRODUCTION Pingala, Bhattapada, Haradatta (same as Haridatta), Damodara, Kausltaki NetranaTayana, and to the Vakyapadiya, the Vaijayantl, the Vyapti-nirnaya of Parthasarathi Misra, the Laghu-BhSskariya and Udaya- divakara's commentary on the Laghu-Bhaskariya. Passages from some of these works are also cited. At one place in the commentary, 1 Magadha and Baudhayana are reported to have stated in their works the amount of precession of the equinoxes for their times. The following hemistich is ascribed to the Garga-samhita ; 2 which means that if b, k, and h be the base, the upright and the hypo- tenuse of a right-angled triangle, then b l -\-k 2 =h i t 8. Raghunatha-raja's commentary A manuscript of this commentary is available in the Lucknow Uni- versity Library, Lucknow. The beginning and end of it are as follows : 1. Comm. on iii. 10. 2. Comm. on ii. 4. Garga, author of the Garga-sathhita (who was different from Vrddha-Garga), is said to have been born in the beginning of Kaliyuga. In support of this is adduced the evidence of ParaSara as also Garga's own assertion in the Garga-safhhitQ. Vide comrn. on iii. 10, COMMENTARIES ON THE ARYABHATlyA xlvU «??f: *i*q?rctfol. srfftfrf JSTOFI » End : 3* «ita<nitecg^rer: snwra: i m vrfmim sarftr I sroWi qarfr ^ fairer tut srwrarft i Colophon : ^^n^m^m^^^t^ tf*ra*n^^fircW«rc^ft- From the opening stanzas of the commentary we learn that Raghunatha-raja belonged to KarnSta (Karnatak or Mysore) and was a king. His mother's name was Lak§mi, and his genealogy was as follows : Vefikata I . Nagar&ja I Kondabhnpa l RaghunStha-raja xtviii INTRODUCTION The following stanza occurring in the commentary 1 throws light on the place where the commentator (Raghun3tha-raja) lived and wrote the commentary : ?fRNi$fkfl w*frsH?rf?T: ijFRsq^nw: u The last two lines mean : “Here at Ahobila, which is the abode of Parabrahma in the form of Ramanrsimha, the equinoctial midday shadow (of a gnomon of 12 angulas) is to be known as 3 (angulas) and 30 (vyahgulas) ; also for the very same place the distance from the (Hindu) prime meridian is equal to 22 yojanas.” 2 This equinoctial midday shadow corresponds to latitude 15° 50' approx. This conclusion is corroborated by the following facts : (i) In an example set in the commentary, 3 the commentator gives the Rsine of the local latitude as equal to 962' 38*. (ii) At another place in the commentary, 4 the commentator gives the times of rising of the signs for his local place as follows : Sign Time of rising in vimdikas Aries 243 Taurus 271 Gemini 311 Cancer 335 Leo 327 Virgo 313 1. under A, iii. 6 (c-d). 2. This translation agrees with Raghunatha-raja's interpretation. One vyahgula is one-sixtieth of an ahgula. 3. under A, iv. 26. 4. Comm. on A, iv. 27. COMMENTARIES ON THE XR. YABHATIYA xlix In the commentary on verse 6(c-d) of the Kalakriya-pSda, the local circumference of the Earth at Ahobila is given to be 3162$ yojanas, so that the distance 22 yojanas of that place from the (Hindu) prime meridian corresponds to 1° 40'. 1 Further, in an example solved in the commentary under verse 10 of the Kalakriyei-pada, the longitude correction for Ahobila has been applied negatively. It follows that Ahobila lay 1° 40' towards the east of the Hindu prime meridian. From what has been said above, we find that Ahobila, the native place of the commentator Raghunatha-raja, was situated approximately in latitude 15° 50' N and longitude 1° 40' E of the Hindu prime meridian. According to the verse quoted above, it was the seat of the Laksmi- nrsimha temple. So, it is the same Ahobila as is situated in Kurnool district, Andhra State. 2 1. It should be noted that the number 3 162 J yojanas accords to the jo/afla-reckoning of Aryabhata I, whereas the number 22 yojanas accords to the yojanareckomng of the Surya-siddhanta. The latter number, when reduced to the .yo/ami-reckoning of Aryabhata I, would become 14| yojanas. 2. Ahobila is a village and temple in the Kurnool District of Andhra State, situated in latitude 15° 8' N and longitude 78° 45' E on the Nallamalais. It is about 34 miles from Nandyal railway station, 22 miles from Nandyal to Allagadda by road and 12 miles from Allagadda to Ahobila by cart or on foot. The temple at Ahobila is the most sacred Vaisnava shrine in the District. It has three parts, namely : Diguva (lower) Ahobila temple at the foot of the hills, Yeguva (upper) Ahobila temple about four miles higher up, and a small shrine on the summit. The first is the most interesting as it contains beautiful reliefs of scenes from the Ramayana on its walls and on the two great stone porches which stand in front of it, supported by pillars 8 feet in circumference, hewn out of the rock. It is said that in ancient times Ahobila was the capital of the demon king Hiranyakasipu, whose son Prahlada was saved from the wrath of his father by God Nrsirhha at this very place. There are three hills at Ahobila, viz., GarudSdri, Vedadri and Acalacchaya-meru. The commentator Raghunatha-raja has remembered God Nrsimha, Lord of Garudari, at the commencement of his commen- tary and has sought His protection. Ahobila, as stated above, is now in Andhra State, but it appears from the commentary that in the time of Raghunatha-raja it formed part of Karnataka of which he was the king. A* Bh. vii 3KTR0DUCTION In the commentary on verse '6(c-d) of the Raktkriya-p&da, 1519 is mentioned as the current Saka year. This corresponds to A.D. 1597 and is evidently the time of writing the commentary. The same year is mentioned at other places 1 in the commentary also. The present commentary is based on those of Bhaskara I and Suryadeva. It goes deep into explanatory details and is, on the whole, a very valuable work. The number of quotations from anterior works is large but the commentator refers only to a few of them by name. Amongst these maybe mentioned the names of the Brahma'siddhantQ, the Soma-siddhSnta, the Surya-siddhanta, the Panca-siddhatttikd, the BhSskara-bhusya (i e. t Bhaskara I's commentary on the Aryabhafiyti), the Laghu-Bhaskanya, the Trisatika, the Utpala-parimala, the SiddhSnta- sekhara, the Lilavatl, the Siddhdnta-siromani, and MallikSrjuna Suri's commentary on the &isya-dhi-vrddhida of Lalla. Amongst the authors mentioned are Vrddha-Garga, Garga, Parasara, Virabhadra, Vasis^ha, Devala, Simharaja, Varahamihira, Brahmagupta, Lall5c2rya, and Manju- lacarya. Bhaskara I has been called Bhasyakara ('author of the Bha$ya'). The commentary contains a large number of solved examples. Forty-one of these examples have been taken from the commentary of Bhaskara I, fifteen from the commentary of SDryadeva, and some from the works of'Bhaskara II. 9. Commentary of Madhava, son of Virupaksa From the following passage occurring in the beginning of Madhava's commentary on the Brhajjntaka of Varahamihira, we learn that a commentary on the Aryabhatiya, giving rationales of the rules and illustrative examples, was also written by him : HTT^T fTrTWTT^ Hf^tTWT I 1. See comm. on A, iii. 10, iv. 4, and iv. 5. COMMENTARIES OH THE At* YABHATlY A H- ztm vmxwt&w *fik*M\iY<'&Mwui u etc. 1 The above passage shows that — (i) Madhava, the commentator, was a Brahmana of Atreya Gotra, and belonged to the family (anvaya) of Vantula. (ii) His father's name was VirUpaksa. (iii) His commentary on the Aryahkafiya was his earliest work. (iv) His other commentaries were on the Narada-samhita and on the Laghu-jataka and the Brhajjaiaka of Varahamihira. As the commentator himself says in the last two lines of the above passage, his commentary on the Brhajjataka was in Telugu. This shows that he belonged to the Andhra State. It is not known whether his commentary on the Aryabhatiya was composed in Sanskrit or in Telugu. 10. BhUtivisnu's commentary A manuscript of Bhutivisnu's commentary on the Aryabhatiya, entitled Bhatapradipa, exists in the Royal Library at Berlin. 2 The concluding verse of the commentary on the Gitika-pada (as recons- tructed from the corrupt reading in the manuscript) runs as follows :— g^ram*TEpp*r fast*??* ^fafawr: srewtafec'm 11 Bhfitivisnu is the author of a commentary on the SUrya-siddhcLnta also, of which an incomplete manuscript (containing a few pages in 1. A Descriptive Catalogue of the Sanskrit Manuscripts in the Government Oriental Manuscripts Library, Madras, 1918, Vol.24, Ms. No. 13835. 2. Catalogue No. 834, De Handschriften-verzeichnisse der Konig- lichen Bibliothek, Erster Band, Verzeichniss der Sanskrit- Handschrif ten, by Weber, Berlin, 1853, p. 232. There is also another manuscript of this commentary in the Anup Sanskrit Library, Bikaner. See Cat; No, 4447, . lu INTRODUCTION the beginning) exists in the Lucknow University Library, Lucknow. 1 It begins with : fe^TffFreriT sftifem^ — *Tmf?*RrfaT etc. “O Goddess— dedicated to bestowing favour on humble devotees, enable me to have firmer devotion towards God Visnu who resides at Kari-sikhari (i.e., the Elephant Mountain or Kaficl). “There was a learned and virtuous Devaraja, the crest-jewel of the lineage of, Garga, the son of Bhutivisnu and an ornament of the terrestrial world. His eldest son, Bhutivisnu, was regarded as the best amongst the wise and intelligent and was honoured with the epithet 'Srimat'. That Bhutivisnu, who I am, has the desire to comment on the Siddhanta acquired from SDrya (/.*., Surya-siddhanta). May God Visnu, the crest-jewel of the Hasti-giri (i.e., the Elephant Mountain), accomplish that desire of this ignorant self.” The above passage shows that Bhutivisnu belonged to the lineage of Garga and that he was the son of Devaraja and the grandson of his own namesake Bhutivisnu. This Devaraja was a different person from Devaraja, the author of the Kuttakara-iiromarti, as the former belonged to the lineage of Garga and the latter to the lineage of Atri. Bhntivisnu's commentary on the Aryabhattya was written earlier than his commentary on the Surya-siddhanta, which is evident from the 1. Accession No, 47070. COMMENTARIES ON THE XRYABHATlYA Uii reference of the former in the latter. For, commenting on SnSi, i. H-12, Bhutivisnu says : tun * ^H-srfft sremfawi— The initial few pages of Bhmivi§nu's commentary on the Surya- siddhanta, which are available in the Lucknow University Library, Lucknow, do not throw light on the time and place of Bhutivisnu. But they do contain numerous references to Srlpati and quotations from his SiddhSnta-'sekhara, which shows that he lived posterior to Srlpati (A.D. 1039). Similarly, his devotion to God Visnu residing on the Elephant Mountain (i.e., Kanci) suggests that he belonged to Tamilnadu, in South India. 11. Ghatigopa's commentary Two manuscripts of this commentary exist in the Kerala University Oriental Research Institute and Manuscripts Library, Trivandrum. 1 This commentary begins thus : It ends thus ; i. Mss. Nos. 13305- A and T. 736, »r INTRODUCTION 1% wnsn i The closing verses of the commentary sh ow that Ghatlgopa was a devotee of God Padmambha and a p UpiI of Paramesial” Parame.vara, however, was different from his namesake, the author o Dtgganita (A.D. 1431), for, according to K V Sarma Phi! JMJ T „er Malaya.am commentary on the^^ ' ' quotes from the PancabodHa of Putumana SomaySji, and Putumana Somay 5 , according to K.V. Sarma, lived between A.D l^nd ii « more two cemuries later to K.V Sarma is of the opinion that this Ghapgopa it tfc same- person as Pnnce Godavarma Koyit t amp„r S n (A.D. lilLo) TiZZ of the scholar* fami.y of Kilamanoor and a resident o TrivaTmT who bore the appel.ation ■ManikkSran' (=c,ockman) (in MaCZ)' equivalent to ■Ghapgopa' (in Sanskrit). Maiayalam), 12. Koiandarkma's (A.D. 1807-83) commentary A ' complete manuscript of Kodaodarama's commentary on tte- Madras^ ^Z^^T ^ Koda^Oarama is also the author of a work caller! J a*. K*. which was meant to be a seque, to the fST swam, sastri describes this work as follows: S. Kuppu- “A treatise i„ stanzas of the Srya metre dealing with tha H TtTT-'t ByK ° d ^ a - Ko.iku.aU L j He states that he adds a fourth Fata to the thr^^Tf Aryabhata, Ga , ita , K a!a and Gola> ^ '<> ca.culation is explained, and >, Hat in the S 1- Ms. No. R. 371 (0), COMMENT AfelBS'tiN 'tHfi iRYABHATlYA Pdda catted Ananda-pada, the nature of the Supreme Brahman lis explained.” 1 (b) Commentaries in Telugu 13. Kodandarama's commentary Kodandarama wrote a commentary in Telugu also. It is on the first three Padas {viz., Giiika, Ganita and KMakriya) only, and bears the name Sudhmarahga. This commentary has been edited by V. Lakshminarayana Sastri and published in Madras Government Oriental Series (No. CXXXIX) in 1956. 14. Vitflpaksa's commentary A manuscript of this commentary exists in the Oriental Manu- scripts Library, Mysore. 8 © Commentaries in Malayalam 15. Krsnadasa's commentary A, manuscript of Krsnadasa's commentary covering the GitikapAda occurs in the collection of K.V. Sarma. The beginning and end of it run as follows : . „ Beginning : sftpvft ^?%^> WRjwft qgqj : \ • fweifrfffevf 5T^ etc. End : srg *»z f *rrf?rfa£ aram *>wtortf»nn&f q«T ftramfa i Colophon: <5*i<fl(ti^<aHH*i ife&j ^F^l%^F^f?T I 1. T riennial Catalogue of the Govt. Oriental Manuscripts Library, Madras, Vol. Ill, Pa#t I, Sanskrit A, Madras, 1922. Ms. No. R 2156 (a). 2. Ms. No. B. 573. lvi Introduction Krsnadasa i s identified with Koccu-Krsnan Asan (A.D. 1756-1812), of the family of Nefumpayil in the Tiruvalla taluk of South Kerala, well known in Malayalam literary circles as the author of several poetical works. He is also the author of a number of astrological works in Malayalam. 1 16. Krsna's commentary A manuscript of Aryabhatiya-vynkhyd, a commentary in Malayalam, entitled BhUsayafn Krsna-flka, exists in the library of the India Office, London. 3 It begins with the words : More details regarding this commentary are not known and it • is difficult to say whether the author of this commentary was the same person as Krsnadasa or different from him. 17-18. Two commentaries by Ghatigopa In addition to his commentary in Sanskrit (already noticed), Ghatigopa wrote two commentaries in Malayalam, both on the Ganita, Kalakriys and Gola Padas only. The larger commentary extends to 1850 granthas (1 grantha = 32 letters), and the smaller one to 1200 granthas. Of tha, larger commentary, there exist two manuscripts (Nos. C. 2333- A and T. 157 B) in the Kerala University Oriental Research Institute and Manuscripts Library, Trivandrum. The beginning and end of it are as follows : Beginning : *k srro^nrrenN *tfiftfag*mfM*pKT srsrci WcftfeiwTf^- famirerN *ita<Tre^j ^rrcrfosr ^ktrwt %i<j 3nTfawp5| — s^TTOrfti etc. End : STSHTST usfq *“Tfe?m*tfft- fatfat * ^3?w?w: stater em. ^IlWt^f^ 3^3R|: HTfTHW: ft^FTW || 1. For details, see K. V. Sarma, A history of the Kerala school of Hindu astronomy, pp. 74-75, 2. Ms. No. 6273. COMMENTARIES ON THE ARYABHATJYA lvii Of the smaller commentary, there are three manuscripts (Nos. 11014, L. 1334 and T. lo7-A) in the Kerala University Oriental Research Institute and Manuscripts Library, Trivandrum, and one {No. 542-B) in the Government Sanskrit College Library, Tripunithura. This commentary begins thus : ^ E T5 — np^iftf etc. The commentary on the last two verses is in Sanskrit and not in Malayalam. It ends with the following sentence : which is exactly the same as in his Sanskrit commentary. The . colophon to Gariita-pada runs thus : The colophon to Gola-pnda runs thus : It is noteworthy that some of the verses occurring towards the end of all the three commentaries written by Ghaftgopa are exactly A. Ba. viii tt*TftO£)UCTlON the same and prove Beyond ddubf the common authorship of the three commentaries. (d) Commentary in Marathi 19 Anonymous commentary in Marathi A commentary (rather a translation) in Marathi exists in the Bombay University Library, Bombay. 1 The name of the author is not mentioned . 6.7 Works based on the Aryabhafiya Of the works written on the basis of the Aryabhatiya, mention may be made of the following : 1. The works of Bhaskara I For details see Introduction to Part II of this Series, pp. xxx ff. • 2. The Karapa-ratna of Deva (A.D. 689) son of Gojanma A manuscript of this work exists in the Kerala University Oriental Research Institute and Manuscripts Library, Trivandrum. 2 The Karana-ratna is a calendrical work in eight chapters, containing in all 183 verses. In the seeond opening stanza, the author says : * “Having taken a deep plunge into the entire ocean of the Aryabhata-sastra with the boat of intellect, I have acquired this jewel, the Karana-ratna, adorned by the rays of all the planets.” 3 This work, though essentially based on the teachings of the Aryabhatiya, is highly influenced by the Khan4a-kMdyaka of Brahma- gupta. It adopts a number of verses from the Laghu-Bhaskariya and the Khanda-khadyaka, 1. Ms. No. 334. 2. No. T.559. WORKS BASSO ON TKEE ARYABHATIYA Ut This is the earliest work of the Aryabhaja school that states the precession of the equinoxes and the so^cal!e4 &ak*M> Manuyuga and Kalpa corrections. 3. The Graha-cara-nibandhana of Haridatta (or Haradatta) This calendrical work was edited by K.V. Sarma and published by the Kuppuswami Sastri Research Institute, Mylapore, Madras, in 1954. The work is in three chapters and states simplified rules and tables for finding the true longitudes of the planets and therefrom the naksatra and tithi, two of the five elements of the Hindu calendar. This work does not prescribe any bija correction to the mean longitudes of the planets, although it is conjectured that Haridatta was the author of the so-called Sakabda correction. 4. The Sisya-dhuvrddhida of Lalla (or Ralla) The text of this work was published by Sudhakara Dvivedi at Benaras in A.D. 1886. In the opening stanzas, the author explains the scope of the work as follows : “That science of astronomy which, as told by Aryabhata, is diflicult to comprehend is being set forth by Lalla in such a way as to be easily understood by students. • “Although having mastered the sastra composed by Aryabha^a, his pupils (or followers) have written astronomical tantras, but they have not been able to describe the methods properly. I shall, therefore, state the procedures stated by him in proper sequence.” 1 In the penultimate stanza of the Grahaganita part of the same work, he again says : fit rftprin?; *w*j*w( u INTRODUCTION “Lalla … has composed this tantra which yields the same results as the Iryabhata-siddhanta (i.e., the Aryabhafiya).” Regarding his parentage, the author (Lalla) himself writes : This shows that he was a son of Samba, popularly known as Bhatta Trivikrama, and a grandson of the learned scholar Taladhvaja, and that he was a Brahmana. Chronologically, Lalla comes after BhSskara I and Brahmagupta, but in the absence of any definite evidence his date could not be fixed so far. On the basis of a passage (sake nakhabdhirahite etc.) generally ascribed to him, it is conjectured that he lived about A.D. 748. 1 But this date is doubtful, because the said passage does not lead to any definite conclusion. There is, however, no doubt that Lalla lived sometime between A.D. 665 and A.D. 904. The former is the date of the Khanda-khadyaka on which Lalla wrote a commentary, and the latter the date of Vatesvara who has utilized the Sisya-dhi-vrddhida of Lalla in writing his SiddhSnta. Lalla's places of birth and activity are also unknown. But the following example, which is the only example of this kind occurring in the Si?ya-dhi-vrddhida, 2 probably refers to the place where he lived : “Knowing that the sum of the Rsines of the latitude and the colatitude is 1308' and that the difference of the same is 538', say what are the Rsines of the latitude and the colatitude here.” 3 1. See, for example, introduction to P. C Sengupta's English translation of the Khanda-khadyaka, pp. xxvi-xxvii. 2, II, xii. 22. WORKS BASED ON THE XRYABHATIYA lxi If x and y denote the two quantities, then x =923'=Rsin (15° 34') approx. j, = 385'=Rsin (6° 26') approx. Thus the latitude of the place referred to in the above example is either 15° 34' or 6° 26'. The latter alternative is impossible as the circle of latitude 6° 26' does not cross the Indian continent. So we infer that Lalla lived in latitude 15° 34' N. There are also reasons to believe that Lalla belonged to LStadesa (i.e., Gujarat). For, in his Sifya-dhhvrddhida, Lalla has made a special reference to the ladies of the Lata country. He has compared the half-phased Moon with the forehead of a lady of the Lata country (lati). Although the Sisya-dht-vrddhida claims to set forth the teach- ings of the Aryabhattya, the impact of the teachings of Brahmagupta on this work is also visible. Two features of the Sisya-dhi-vrddhida deserve special notice : (i) arrangement of subject matter under two distinct heads— Grahaganita (dealing with astronomical calculations) and Goladhyaya (dealing with the celestial sphere, cosmogony, astronomical instruments, etc.), and (ii) language. This arrangement has been followed by Vat.e$vara (A.D. 904) in his SiddMnta, by Bhaskara II (A.D. 1150) in his SiddhQnta-siromani, by Jnanaraja (A.D. 1503) in his Siddhanta-sundara, and other later writers. The language used by Lalla is, at places, highly poetic and appealing. Some of his expressions and similies are so nice that posterior writers could not resist copying them. One can easily find a number of passages in the works of VateSvara, Srtpati and Bhaskara II which have been copied from the Si$ya-dhi-vrddhida of Lalla. 5. The Karaw-prakata of Brahmadeva (A.D. 1092) This calendrical work was edited by Sudhakara Dvivedi together with his own commentary. The epoch used in this work is A.D. 1092. This work holds an important place amongst the calendrical works. It makes use of the bija correction prescribed by Lalla, and tithis calculated from this work differ by about 2 to 3 ghatis, being in excess, from those calculated from the parameters of the Aryabhafiya. This work was in use in South India, particularly in Maharaja, INTRODUCTION amongst Vaisnavas, who preferred the 11th tithi calculated fwm this work. For details, see Diksita's Bhqrat\ya-Jyoiisa-sastra (Marathi), pp. 240-42. 6. The Bhatatulya of Damodara The epoch used in this work is A.D. 1417. The authpr Damodara was a son of Padmanabha (c. A.D. 1400) and a grandson of Narmada (c. A.D. 1375). Use ofLalla's bija correction is made in this work also. A manuscript of this work exists in the Deccan College Library, Poena, The second stanza therein runs as follows : “I, Damodara, bowing to the lotus-like feet of my teacher Padmanabha, write, for the pleasure of the learned, this work, which will yield the same results as those of Aryabhata, by making use of the pratyabda-'suddhi method.* For details see Diksita, ibid., pp. 354-56. 7. • The Kara^a-paddhati of Putomana Somayaji (A.D. 1732) This work has been published in Trivandrum Sanskrit Series (No. 126), and the Madras Government Oriental Series (No. 98). The latter contains two Malayalam commentaries also. 8. The Aryabhata-siddhanta-tulyarkaraxia by Virasiriihaganaka sop of Kasiraja Three manuscripts of this work occur in Anup Sanskrit Library, Bikaner. 2 6.8. Transmission to Arab The Aryabhatiya was taken to Arab where it was translated into Arabic by Abul Hasan Ahwazi under the title Aryabhata (misread as Arajbahara or Arajbahaz). The Arabians misunderstood the exact significance of the title of the work and wrongly thought that it meant 'one thousandth part'. 3 5ic*r5^5mss*?”>rc*2r fast *tsj| ^tw ^Htfk n 2. Mss. Nos. 4448, 4449 and 4450. 3. See Arab aw BhSrat ke sgmbewlha, by Maulana Saiyad Suleiman Nadavl, translated into Hindi by Ram Chandra Varma, pub. by Hindustani Academy, Allahabad, 1930. p. H3, ARYABHA9A*6lDt)f3£NTA 7. THE ARYABHATA^SrDDHANTA From the writings of Varahamihira (died A.D. 587), Brahmagupta (A.D. 628), Bhaskara I (A.D. 629), Govinda-svami (ninth century), Mallikarjuna Snri (A. D. 1178), Ramakrsna Aradhya (1472 A. D.), Maithila Candesvara, Bhudhara (A.D. 1572) and Tamma Yajva (A.D. 1599), it is now established beyond doubt that Aryabhafa I, the author of the Aryabhafiya, wrote at least one more work on astronomy which was known as Aryabhata-siddhanta. Unlike the Aryabhafiya ifl which the day was measured from one sunrise to the next, this work reckoned the day from one midnight to the next as was done in the Surya- siddhanta. The astronomical parameters and methods given in the Iry&bhata-siddhanta differed in some cases from those of the Aryabhafiya. The important differences between the two works have been noted by Bhaskara I in Chapter VII of his Mahz-Bhaskariyat Some of the typical methods and the astronomical instruments des- cribed m the Aryabhata-shidharrttJ have been mentioned by MallikaYjmra: Snri, Tamma YajvS, RSmakrsna Aradhya and others in their commen- taries on the Surya-siddhartta. The astronomical parameters and methods ascribed to the Aryabhata-siddhanta are generally the same a4 those found in Varahamihira' s version of the Sarya-siddhSnta and tfa* Swtali-mahatantra of Acarya Sumati of Nepal, which was based on the Snrya-siddhanta. It appears that the Aryabhata-siddhWa was an independent work like the Aryabkatiya and that it bore the same relation to the earlie* Sarya-siddhanta as the Aryabhattya bore to the earlier Smyambhrna-siddhanta ; and that the Surya-siddhartta summarized by Var9fcamihira was the one anonymously revised by Latadeva fn the light of the Aryabhata-siddhanta. This is, perhaps, the reason why both Aryal&ata I and Latadeva are sometimes referred to as the author* of the SuryasiaMkanta. 7T. The Aryabhata-siddhanta and the Aryabhattya The following tables exhibit the main differences between the astronomical parameters of the Aryabhafiya and the Aryabhata-siddhanta according to Bhaskara I. i*tv Introduction Table 1. Diameters and distances of planets in yojanas Aryabhatiya Aryabhata-siddhanta Earth's diameter 1050 1600 Sun's diameter 4410 6480 Moon's diameter 315 480 Sun's distance 459585 689358 Moon's distance 34377 51566 circumference of the sky revolutions of the Moon 216000 324000 The numbers in the second and third columns are in the ratio 2:3, approximately. This is due to the fact that the measures of yojana employed in the two works are in the ratio 3 : 2. Table 2. Civil days, Omitted lunar days, and Revolutions of Sighrocca of Mercury and Jupiter in a period of 43,20,000 years Aryabhatiya Aryabhata-siddhanta Civil days 1,57,79,17,500 1,57,79,17,800 Omitted lunar days 2,50,82,580 2,50,82,280 Revolutions of Sighrocca of Mercury 1,79,37,020 1,79,37,000 Revolutions of Jupiter 3,64,224 3,64,220 «■ ■. It may be pointed out that the difference of 300 days between the civil days of the two works was so adjusted that both the works indi- cated Jhe same epoch at the end of the Kali year 3600 mentioned in the Aryabhatiya. Since 3600 years=l,57,79,l7,500/l200= 13,14,931-25 days according to the Aryabhatiya and= 1577917800/ 1200= 1314931*50 days according to the Aryabhata-siddhanta, the Kali year 3600 ended exactly on Sunday, March 21, A D. 499, at mean noon at Lanka or Ujjayinl, according to both the works of Aryabhata I. Table 3. Longitudes of the p lanets* apogees (or aphelia) in 499 A.D. Planet Aryabhatiya Aryabhata-siddhanta Sun 78° 80° Mars 118° 110° Mercury 210° 220° Jupiter 180° 160° Venus 90° 80° Saturn 236° 240° ARYABHATA-SIDDHXNTA lxv Table 4. Dimensions of planets' manda epicycles A ryabhatiya Aryabhata-siddhanta odd quadrant even quadrant Sun 13° 30' 13° 30' 14° Moon 31° 30' 31° 30' 31° Mars 63° 81° 70° Mercury 3l c 30' 22° 30' 28° Jupiter 31° 30' 36° 32° Venus 18° 9° 14° Saturn 40° 30' 58° 30' 60° The circumference of a planet's concentric or deferent (or mean orbit) is supposed to be of 360 units (called degrees) in length and the above dimensions are on the same scale. Table 5. Dimensions of planets' Ughra epicycles Aryabhafiya Aryabhata-siddhanta odd quadrant even quadrant Mars 238° 30' 229° 30' 234° Mercury 139° 30' 130° 30' 132° Jupiter 72° 00' 67° 30' '52° Venus 265° 30' 256° 30' 260 Q Saturn 40° 30' 36° 00' 40° 7 2. The astronomical instruments and special methods of the Aryabhata- siddhanta Ramakrsna ArSdhya (A.D. 1472) has quoted a set of 34 verses (composed in anustubh metre) from the chapter of the Aryabhata-siddhanta dealing with the astronomical instruments. The instruments described in these verses are : (1) Chaya-y antra ('the shadow instrument'), (2) Dhanuryantra ('the semi-circle'), (3) Yasti-yantra, (4) Cakra-yantra ('the whole circle'), (5) Chatra-yantra ('the umbrella'), (6) Water instruments, (7) Ghatika-yantra, (8) Kapala-yantra, and (9) the gnomon. 1 Of these instruments, some were indeed devised by Aryabha^a I. The 1. For details see K.S. Shukla, 'Aryabhaja I's astronomy with midnight day-reckoning', Gaqita, Vol. 18, No. 1, pp. 83-105, ABfa. iz livi INTRODUCTION gnomon (as described by Aryabhafa I) and the water instruments have been generally attributed to him. It is in connection with these instruments that the commentators of the Surya-siddhanta have remem- bered him. Mallikarjuna Suri (A.D. 1178), Ramakrsna Aradhya (A.D. 1472) and Tamma Yajva” (A.D. 1599) have referred also to some special methods of the Aryabhata-siddhanta. Of these methods J one relates to the approximate determination of time from the shadow of the gnomon. The method is interesting and also unique as it does not occur in any other known work on Indian astronomy. It may be briefly described as follows : When the Sun is in Scorpio, Capricorn or Aquarius, and it is within 2 ghatls from noon, set up a gnomon of 9 digits on the east-west line in such a way that the tip of its shadow may fall on the north- south line. Then the digits of the distance of the gnomon from the intersection of the east- west and north-south lines would approximately give the ghatls to elapse before noon or elapsed since noon (according as the observation is made before noon or after noon). 7-3. Popularity of the Aryabhata-siddhanta and the Kha^a-khadyaka of Brahmagupta The Aryabhata-siddhanta was a popular work and was studied throughout India. It was mentioned in the sixth century by Varahamihira of Kapitthaka (near Ujjayinl), in the seventh century by Brahmagupta of Bhinmal (in Rajasthan) and Bhaskara I of Valabhi (in Kathiawar), in the ninth century by Govinda-svami of Kerala, in the twelfth century by Mallik5rjuna-Surl of Andhra and Maithila-Candesvara of Banaras in Uttar Pradesh, in the fifteenth century by Ramakrsna-Aradhya of Andhra, and in the sixteenth century by Bhadhara of Kampilya (modern Kampil, twenty-eight miles north-east of Fatehpur in the Farrukhabad district, Uttar Pradesh) and Tamraa- Yajva of Andhra. There are reasons to believe that in the seventh century the popularity of this work in north India was at its highest peak and it was used not only as a text book of astronomy but also in everyday calculations such as those pertaining to marriage, nativity etc. The celebrated Brahmagupta who, in his youth, was a bitter critic of Aryabhafa I was so much impressed by its popularity that he ARYABHATA-SIDDHANTA livii cduld not resist the temptation of bringing out an abridged edition of this work under an attractive title, 'Food prepared with sugarcandy' {Khanda-khsdyaka). It was so much liked in some parts of India that it is in use even today. Brahmagupta was not in complete agreement with the teachings of Aryabhafa I. So he planned his Khaqda-khadyaka in two parts. In Part I he summarized the teachings of the Aryabhata-siddhanta without making any alteration, modification or addition (but rectifying one or two rules whose inaccuracy was obvious to him); and in Part II he stated the corrections and modifications which had to be applied to Part I in order to get accurate results. In the opening stanzas of the two parts, Brahmagupta himself says : “Having bowed in reverence to God Mahadeva, the cause of creation, maintenance and destruction of the world, I set forth the Khanda-khadyaka which yields the same results as the work of Aryabhafa. “As it is generally not possible to perform calculations pertaining to marriage, nativity, and so on, every day by the work of Aryabha^a, hence this smaller work giving the same results.” • “As the process of finding the true longitudes of {he planets as given by Aryabhafa does not make them agree with observation, so I shall speak of this process (now).” A sad consequence of the composition of the Khanda-khudyaka was that the original work of Aryabhat,a I on which it was based was lost. The Khanda-khadyaka t however, received, wide acclamation and, though it was a calendrical work, a large number of commentaries were written on it. Amongst the commentators of this work were Bala- bhadra, whose commentary (tika) has been mentioned by Al-BirnnI (A.D. 973-1048); Prthudaka (A.D. 860), whose commentary (vivarana) has been edited by P. C. Sengupta; Lalla, whose commentary {Khan4a- khndyaka-paddhati) has been mentioned by Amaraja {c. A.D. 1200) ; Bhaftotpala (A. D. 968), whose commentary {vivrti) has been edited by Bina Chatterjee ; Varuna (c. A. D. 1040), whose commentary {udaharand) is extant though not printed ; Someivara, whose commentary has been mentioned by Amaraja (c. A.D, 1200); Amaraja (c. A.D. 1200), {Part I) : {Part II) : lxviii INTRODUCTION W ^° S V?r ntary(V ^ n3& ^ fl) hasbeen edited b y Babuaji Misra; and Sridatfa, a manuscript of whose commentary exists in Nepal An anonymous commentary (udaharana) exists in the India Office Library London, and another written in Nepali in the Lucknow University Library, Lucknow. Snryadeva (b. A.D. 1191) proposed to write a commentary on the khanda-khadyaka* but it is not known whether he actually wrote it. The Khanda-khudyaka reached Arab where it was translated into Arabic under the title Zij-al-Arkand, and was widely used. 2 It was retranslated into Arabic under the title Az-Zij Kandakatik al-Arabi i^The Arabic Khanda-khadyaka) by the Persian scholar AI-Blrnnl (A.D. 973-1048), who has quoted some of the methods of this work in his other works. (E. gm , see Hisa'il, II, p. 150). From Arab, the Khanda-khadyaka reached Europe and had its impact on astronomy there. O. Neugebauer has shown that -Kepler's theory of parallax is identical with the theory of the Khanda-khudyaka.^ 8. THE PRESENT EDITION OF THE JlRYABHATIYA (a) Sanskrit Text The Sanskrit text incorporated in the present work is edited critically and all possible efforts have been made to reconstruct it as authentically as possible. It is based on: (i) original manuscripts of the text, (n) available commentaries and (iii) quotations from later astronomers. The said three sources, it might be seen are complementary and mutually corrective. Thus, while the text manu- scripts present the text as handed down by manuscript tradition the commentaries containing the meanings and derivations of the words in the text help in correcting scribal and other errors, besides indicating textual variants. Quotations in later works, cited either by way of 1. Vide his statement towards the end of his commentary on Snpati's Jataka-paddhati. y n i« 2 ' . See S - Kenned y» A survey of Islamic astronomical tables, p 138 According to Kennedy, it -was translated into Arabic, at or before the time of Ya'qub ibn Tanq, and was widely used”. p. 124, 3 ' °* Neusebauer ' Th * Gnomical tables of Al-Khmrizmi, PRESENT EDITION lzix approbation in establishing a point or by way of refutation by a critic, are particularly helpful in deciding upon the correct readings of the text. 1. Text manuscripts 1 Seven palmleaf manuscripts in Malayalam script, designated A to G (noticed below), have been collated towards fixing the text of the Aryabhaftya. A. Ker 2 . 475-A. Mai. (Malayalam script), PL (palmleaf), Cm. (complete) ; 17 cm.x5cm., 7ff., 7 lines per page with about 36 letters per line. The verses have been written continuously through the entire length of folios on both the pages. Old, damaged and brittle to the touch. Inked and revised. The writing is quite readable but not shapely. The date of the ms. is given in a post-colophonic Kali chronogram in the katapayadi notation which reads, sevyo dugdhabdhitalpah (16,99,817), and corresponds to A.D. 1552. The astronomical codex which contains A belonged originally to the reputed scholarly Namputiri family of Kntallur in South Malabar, and carries the undermentioned works, all on mathematics and astronomy : A, Aryabhatxya of Aryabhaja ; B. Maha-Bhaskariya and C Laghu-Bhaskanya of BhSskara I ; D. Siddhanta-darpana £nd E. Tantra-sahgraha of Nllakantha Somayaji ; F. Lilamti of Bhaskara II ; G. Pancabodhd, anon. ; H. Laghumanasa of Munjala ; L Candracchuya- gatfita of Nllakantha Somayaji; and J. Goladipikn and K. GrahartBftaka, both of Paramefivara (A.D. 1431). B. Ker. 5131-B. Mai., PI., Cm. ; 56 cm.x5 cm., 4 8 lines per page with about 75 letters per line. Old, damaged and brittle. Inked and revised. Readable writing. Generally correct text. Neither dated nor scribe mentioned. The other work contained in this codex is Bhagavata-Purana numbered as 513 1-A. The codex has been procured from Shri Vasudevan Namputiri of the village of Marappadi in Central Kerala. 1. The material of this section was supplied by K.V. Sarma. 2. Ker. stands for the Kerala University Or. Research Inst, and Mss t Library, Triyandrumt lxx INTRODUCTION C. Ker. 13300. Mai., PI., Cm., 40 cm.x5 cm.; 7 ff. with 7 lines a page and about 50 letters a line. Old, brittle and damaged, some of the folios being torn. Neither dated, nor scribe mentioned. The text preserved is fairly accurate. The works contained in this astronomical codex are : A. Aryabhattya, B. Surya-siddhanta, C. Laghumanasa of Munjala, D. Maht-Bhaskariya of Bhaskara I, and E. Drgganita of ParameSvara. The codex was procured by Shri Karuvelil Nilakantha Piilai of Karthikappalli (S. Kerala) from an unidentified source. D- Ker. 13305-B. Mai., PI., Icm„ 15 cm. X 5 cm.; 12ff. with 8 lines a page and about 24 letters a line. Late ms., in good preservation. Very legible writing. Inked and revised, the reviser's corrections being identifiable by their not being inked. The text preserved is accurate. The ms. is not dated; neither is any scribe mentioned. The codex contains also the Aryabhatiya-vyakhya in Sanskrit by Ghatfgopa, which is catalogued as No. 13305-A. The codex belonged originally to the family collection of Patififiaretattu Plsaram in Kitamiur in Central Kerala. E. Trip. 542- A, belonging to the Govt. Sanskrit College, Tripuni- thura, near, Cochin. Mai., PI., Cm., 20 cm.x3 cm. ; 11 ff, with 10 lines a page and about 24 letters a line. A comparatively late manuscript, written in shapely script. The text preserved is generally correct. No date is given, nor is any scribe named. The works contained in this astronomical codex are : A. Aryabhattya, B. Iryabhattya-vyakhya in Malayalam by Ghatfgopa and C. Venvaroha by Madhava with the Malayalam gloss of Acyuta Pisarati. F. Ker. 501-A. Mai., PI., fan., 20 cm.x3 cm. ; 12 ff., with 7 letters a page and about 25 letters a line. Old and damaged, with the corners worn out on account of constant use. Lacks the GitikZpUda. Inked and revised, the corrections being uninked. The text preserved is accurate. No date or scribe has been mentioned. The works contained in the codex are : A. Aryabhattya, B. Catuilokah (nanagranthoddhrtah), and C. MuhUrtapadavh The codex formed part of the famous mediaeval collect of the De$amangalara PrEsEnt Edition Variyam in North Kerala, as known from an uninked marginal statement on the first folio, which reads : Desamahgalattu Variyatte Aryabhatudi. G. Ker. C. 2475-B. Mai., PI., Icm., extending up to Ganita, verse 2 only, 25 cm. X 5 cm., 1 f., with 10 lines a page and about 37 letters a line. Carefully written in beautiful hand. Scrupulously revised. Not dated, nor any scribe mentioned. The text preserved is accurate. The other work contained in the codex is the Aryabhatiya- vyukhya by Suryadeva Yajva. The codex contains also folios with some miscellaneous matter inscribed thereon. The codex belonged originally to the library of the royal principality of EdappaUi in Central Kerala* All these manuscripts are completely independent of each other. Neither do they present any consistent common characteristic so as to enable them being grouped in any order or formulate any stemma codi- cum to portray their descent. 2. Text preserved in the commentaries Of the commentaries on the Aryabhatlya, those by Bhaskara I (A.D. 629), Suryadeva (b. A.D. 1191), Paramesvara (A.D*- 1431) and Nilakantha (A.D. 1500) are available in print. These commentaries have been referred to as Bh., So., Pa., and Nh, respectively, and the following editions have been used in the collation of the text : Bh. Edited by K.S. Shukla in Part II of the present series. Su. Edited by K.V. Sarma in Part III of the present series. Pa. Edited by H. Kern at Leiden in 1874. Ni. Edited by K. Sambafiiva Sastri (TSS, Nos. 101 and 110) in 1930, 1931 and by Suranad Kunjan Pillai (TSS , No. 185) in 1957. Commentaries by Somesvara, Yallaya (A.D. 1480), Raghunatha- raja (A.D. 1597), Krsnadasa (A.D. 1756-1812) and GhatTgopa (c. A.D. 1800-60) are available in manuscript form. The following manuscripts (designated as So., Ya., Ra., Kf., and Gh , respectively) have beed used in the collation of the text : INTRODUCTION So. Transcript. Accession No. 45886 of the Lucknow Uni- versity Library, Lucknow. (Transcribed from Bs. No. 272, Catalogue No. 335, Accession No 2495 of the Bombay University Library, Bombay. The original manuscript is complete but extremely defective and full of inaccuracies and omissions). Ya. Transcript in the collection of A. N. Singh. It contains Ganita-pada (up to vs. 28), Kalakriya-pada and Gola-pnda. Ra. Transcript. Accession Nos. 45771, 45772 and 45773 of the Lucknow University Library, Lucknow. Complete in four Padas. Kr. Transcript in the collection of K.V. Sarma. The original of this transcript, which contains the GWka-puda only, is available in the Government Sanskrit College Library, Tripunithura, Kerala. Gh. Transcript of the smaller version of Ghatfgopa's Malayalam commentary in the collection of K. V. Sarma. It is a copy from Ms. No. 542-B of the Government Sanskrit College Library, Tripunithura. See E, above. The variant readings noted or discussed in the commentaries have also been generally taken into consideration. 3. Quotations from later astronomers Extracts from the Aryabhatiya occur as quotations in the Brahma- sphuta-siddhSnta of Brahmagupta, Prthudaka's (A.D 860) commentary on the Brahma-sphuta-siddhanta, Govinda-sva”mfs commentary on the Maha-Bhnskariya, and Sankaranarayaija's (A. D. 869) and Udaya- divakara's (A.D. 1073) commentaries on the Laghu-Bhaskanya. These works have been referred to as Br., Pr., Go., &a. and Ud. respectively, and the following editions or manuscripts of them have been used : Br. Edited by Sudhakara Dvivedi, Benaras, 1902. Pr. Photostat copy of Ms. Egg. 2769 : No. 1304 of the India Office Library, London. Belonging to the Lucknow University Library, Lucknow, Accession No. 47047. PRESENT EDITICiN Go. Edited by T.S. Kuppanna Sastri and published in Madras Government Oriental Series, No. CXXX, 1957. Sa. Transcript in the collection of A.N. Singh. Complete. Ud. Transcript. Accession No. 46338 of the Lucknow University Library, Lucknow. Complete. 4. Variations in reading The collation of the manuscripts did not reveal many significant variations in the text. In the first Pada, the variations are mostly phonetic : (i) «?, n ( vs - 3) ^ Tfmi > (VS ' 4) (Hi)' *igTT:, *y*J\ (vs. 5) (iv) imW^W*!?, ( vs - 9 > (v) m, * (vs. 10) (vi) '*IT, (vs. 10) (vii) fas*, (vs. 12) Other variations in vs. 12 also seem 10 be inspired by phonetic require- ments. This Pada was generally learnt by heart and the students seem to have varied the readings to suit their pronunciations without affecting the meaning of the text. One significant variation in reading in this Pada is : m for W. (vs. 6) which seems to have been deliberately made under the pressure of VarShamihira's criticism of the theory of the Earth's rotation. Variations in Pada II are generally verbal and due to the scribes, The following variations, though not significant, are noteworthy : (i) ^srr^fgir: wk « V* I (Prthudaka)! ^ n) rsrran : ajtartan: (Others) J K&fr TO *T I (Others) id“ (vs. 25) (ii) fT^T *Rrafa 5FT^rT w?(5i«5w' i (Bbaskara and SomeSvara) jm *IHfrT ^TTrT sR<B«PT I (Others) _ In Pada III, there is one significant variation in reading, viz. : ^Tsrafasnfa for wrafc^Tfa (vs. 5) This, too, seems to have been made when bhuh was changed into bhaik. Other noteworthy, though not very significant, variations are : A.Bh. x l»iv INTRODUCTION q^Jfcr for s^cn^f *?%sq wfei <ji for «rare«r 3£*isr ?igfcT in vs. 22., both of these being mentioned by Bhaskara I. In Psda IV also, there is one significant reading-difference, viz. : M»^<ttlt$r for cTt^T^T (vs. 14) which is inspired by the teaching of Brahmagupta. Other notable variations are : n^rfora for fftgfcra (vs. 8) tmx or ctTTeWSTOeUT for f^§Wq^TlWT (vs. 8) mnitevrt for TOiteii 1 (vs. 13) «m ^q?taniff (Bh., So.), sttootukt a«nif (others) (vs. 16) for 3»*ta>tm: (vs. 27) IrBT for iffetTT (vs. 35) sn*nm staM for ^5^siT«n5t^?# (vs. 39) fc^tiN *m for ft«Tcq«f ?FR*i (vs. 42) f?*rare for fNfanesr (vs. 44) ^cTT for (vs. 49) GSIcT or m*i for fair (vs. 50) 5. Selection of readings In the selection of readings for the Sanskrit text, preference has been given to the most appropriate and, if possible, the oldest readings. Readings which were considered to be wrong or due to subsequent alteration in the text, or else, were less appropriate and unacceptable have been recorded in the footnotes. (b) English translation, notes etc. The English translation and explanatory and critical notes subjoined to the Sanskrit text as well as references to parallel passages given in the footnotes have been taken with necessary modi- 1. It is difficult to say whether it is yavakoti or yamakoti but most of the manuscripts give the former reading. PRESENT EDITION fications from my D. Litt. Thesis. Most of the matter in this intro- duction is also derived from the same source. The question of translating technical material written in Sanskrit into English presents considerable difficulty. It requires a thorough knowledge of both the languages, which few can claim. Effort has been directed towards giving, as far as possible, a literal version of the text in English. The portions of the English translation enclosed within brackets do not occur in the text and have been given in the translation to make it understandable and are, at places, explanatory. Without these portions, the translation, at these places, might appear meaningless to a reader who cannot consult the original for lack of knowledge of Sanskrit. Attempt has been made to keep the spirit of the original and as far as possible the sequence of the text has been kept unaltered. Sanskrit technical terms having no equivalents in English have been given as such in the translation. They have been explained m the subjoined notes and the reader can always refer to the glossary of the technical terms given in the end to find the meaning of such terms whenever the subjoined notes do not contain the explanations of the terms. Verses dealing with the same topic have been translated together and are prefixed by an introductory heading briefly summarizing their contents. This is in keeping with the practice followed by the commen- tators. For the convenience of the Sanskrit-knowing readers, the Sanskrit text of each passage translated has been given just before its English translation. The translation is followed by short notes and comments compri- sing : (1) elucidation of the text where necessary, (2) rationale of the rule given in the text, (3) illustrative solved examples, where necessary, (4) critical notes, and (5) other relevant matter, depending on the passage translated. In doing so, a vast literature has been consulted and parallel passages occurring elsewhere have been noted in_ the foot- notes. Practically all commentaries in Sanskrit on the Aryabhafiya, whether published or in manuscript, have been consulted. They have been of considerable help in translating the text ; without them quite a number of passages would have remained obscure. Advantage has also been taken of the interpretations and views of the earlier translators of the Aryabhafiya, such as P.C. Sengupta and W.E. Clark. l xxv i INTRODUCTION For the convenience of the reader, the chapter-name has been mentioned at 'the top on the left hand page and the subject matter under discussion at the top on the right hand page. The verse-number is also mentioned at the top. Four appendices have been given at the end : 1. Index of half- verses and key-passages. 2. v Index-glossary of technical terms. 3. Subject Index. 4. Bibliography. It is hoped that they would prove useful to the reader, 9. ACKNOWLEDGEMENT Parts I and II of the present Series were to appear >jis Part One (containing a general introduction to the works of Bhaskara I) and Pari Four (containing Bhaskara I's commentary and English translation of the Aryabhotiya) of the 'Bhaskara I and his works' series, of which Parts Two \ and Three were published by the Department of Mathematics and Astronomy, Lucknow University, Lucknow. However, in spite of pressing demand from interested readers, these could not be published so far. I am grateful to the Indian National Science Academy, New Delhi, for sponsoring the present publication. I am greatly indebted^ to Professor F.C. Auluck, Vice-President, National Commission for the Compilation of the History of Sciences in India, and President, Organizing Committee for the 1500th Birth Anniversary of Aryabha^a I, and to Dr B.V. Subbarayappa, Executive Secretary, Indian National Science Academy, New Delhi, and Secretary, Organizing Committee for the 1500th Birth Anniversary of Aryabhaja I, who have taken keen interest in the present work and have gladly offered all possible help and advice from time to time. Thanks are due also to Prof. B.P. Pal, President, Indian National Science Academy, New Delhi, for writing the foreword to the present series of works. Originally, the idea was to publish only the English translation of the Aryabhatiya along with notes and comments as was earlier done by P.C. Sengupta and W.E. Clark. The Sanskrit text was included at the suggestion of Shri K.V. Sarma, who took the responsibility of ACKNOWLEDGEMENT Ixzvii preparing a critically collated text on the basis of the manuscripts of the Aryabhafiya that existed in the Kerala University Oriental Research Institute and Manuscripts Library, Trivandrum, and in the Library of the Government Sanskrit College, Tripunithura, and were accessible to him. The four appendices occurring at the end of this volume were also prepared by him. These have enhanced the value and usefulness of the work. Shri Sarma also gave his wholehearted cooperation in the editing of the present series of books. I offer my sincere thanks to him. Thanks are due to the authorities of the Kerala University Oriental Research Institute and Manuscripts Library, Trivandrum, the Government Sanskrit College, Tripunithura, and the Lucknow University Library, Lucknow, whose manuscripts were utilized in the collation of the Sanskrit text, and to the many scholars whose works were consulted during the preparation of the present work. I wish to express my deep sense of gratitude to my teacher, the late Dr. A.N. Singh, and to the late Dr. Bibhutibhusan Datta, who, in 1954, had gone through the English translation and notes and had offered valuable suggestions for their improvement. An alternative interpretation of vs. 1 of the Citika-pnda, as suggested by the latter, is mentioned in his sacred memory. I am grateful to Dr R.P. Agarwal, Professor and Head of the Department of Mathematics and Astronomy, Lucknow University, Lucknow, also Hony. Librarian of the Lucknow University Library, Lucknow, for providing me all facilities in my work. My cordial thanks are due to my colleague and friend Pandit Markandeya Misra, Jyotishacharya, for his assistance in the present work. I must also express my thanks to the workers of the V.V.R.L Press, Hoshiarpur, for the excellent composing, printing and get-up of the book. K. S. SHUKLA CHAPTER I THE GITIKA SECTION ('TEN APHORISMS IN THE GITIKA STANZAS') [ In the 10 stanzas composed in the gitika metre, comprising the 10 aphorisms (sutra) of this chapter, Aryabhata sets out the parameters which are necessary for calculations in astronomy. A beginner in astronomy was supposed to learn them by heart so that he might not feel any difficulty while making calculations later on. For the convenience of the beginner, this chapter was written as an independent tract and issued under the name Dasagitika-smra (Ten aphorisms in the gitika stanzas') which is mentioned in the concluding stanza. When this DaSagitikn-smra is regarded as a chapter of the Aryabhafiya, it is called Gttika-pada {Gitika Section). ] INVOCATION AND INTRODUCTION 1. Having paid obeisance to God Brahma -who is one and many, the real God, the Supreme Brahman - Aryabhata sets forth the three, viz., mathematics (gayi to), reckoning of time (kalakriya) and celestial sphere {gold). Obeisance to God Brahma at the outset of the work points to the school to which the author Aryabhata I belongs. “Obeisance has been paid to Svayambhu (Brahma)”, writes the commentator 1. Abbreviations : Text mss. A to G. Text in later works and commentaries : Bh. (Bhaskara I), Br. (Brahmagupta), Go. (Govinda- svami), Kr. (Krsnadasa), Ni. (Nilakaijtha), Pa. (Paramesvara), Pr. (PfthH- daka), Ra. (Raghunatha-raja), £a. (SankaranSrayana), So. (Somesvara), SO. (SBryadeva), Ud. (Udayadivakara), Ya. (Yallaya). A. Bh. 1 2 GlTlKA SECTION £ GitikS Srt. Snryadeva (b. A.D. 1191), “because the science which is being set out was due to Him and the mysteries of that science were revealed to Aryabhata on worshipping Him.” Brahma is spoken of as one and many, because, as writes the commentator Bhaskara I (A.D. 629), when viewed as the unchangeable {nirvikara) and unstained (niranjana) God, He is one, but when taken to reside in the bodies of so many living beings, He is many ; or, in the beginning He was only one, but later He became twofold— man and woman -and created all living beings and became many; or, viewed as the omnipresent God (visvarnpa), He is unquestionably one and many. He is called the 'real god' (satya devata), because the other gods having been created by Him are not real gods. He is called the 'Supreme Brahman* (param brahma), because He is the root cause of the world. Bhaskara I thinks that the first half of the stanza may be inter- preted also as obeisance to the two Brahmans-tht Sabda-Brahman (satya devata) and the Para-Brahman ; or else, as obeisance to the triad, Hiranyagarbha (the Supreme Body), consisting of the subtle bodies of all living beings taken collectively), the Causative Power of the Supreme Body (satya devata), and the Master of that Power (Para- Brahma, the Supreme Brahman). For details, the reader is referred to Bhaskara I's commentary on the above stanza (in Vol. II). According to Bibhutibhushan Datta, ham in the text may be interpreted as anandakam (meaning 'supreme bliss'), satyQm as sat- svorupam (meaning 'really existent truth'), and devatUm as cit-svaruparh meaning 'pure intelligence'). The text should, then, be translated as : “Having paid obeisance to the Supreme Brahma who is one and also many, who is supreme bliss, really existent truth, and pure intelligence— Aryabhata sets forth the three, viz., mathematics (ganita), reckoning of time (halakriya), and celestial sphere (gola).” Vers* 2 ] WRITING OF NUMBERS METHOD OF WRITING NUMBERS 3 2. The varga letters (k to m) (should be written) in the varga places and the avarga letters (y to h) in the avarga places. (The varga letters take the numerical values 1, 2, 3, etc.) from fc onwards ; (the numerical value of the initial avarga letter) y is equal to « plus m {i.e., 5+25). In the places of the two nines of zeros (which are written to denote the notational places), the nine vowels should be written (one vowel in each pair of the varga and avarga places). In the varga (and avarga) places beyond (the places denoted by) the nine vowels too (assumed vowels or other symbols should be written, if necessary). In the Sanskrit alphabet the letters k to m have been classified into five vargas (classes) — ka-varga, ca-varga, ta-varga, ta-varga and pa-varga. These letters are therefore referred to above as varga letters. These are supposed to bear the numerical values 1 to 25 as shown in the following table i 1 Table 1. Varga letters and their numerical values Varga Letters and their numerical values ka-varga k= = 1, kh- = 2, g= ■■ 3, gh- = 4, h- = 5, ca-varga c— = 6, ch = = 7, J= 8, jh-. = 9, n~ = 10, , ta-varga = 11, th = = 12, 4= = 13, dh = 14, n = = 15, ta-varga r⇒16, d= = 18, dh = 19, n- =20, pa-varga P = = 21, ph = = 22, b= =23, bh– =24, m~ =25. The letters y to h are called avarga letters, because they are not classified into vargas (classes or groups). These letters bear the following numerical values : y=3Q, r=40, 1=50, v=60, i=70, $=80, 5 = 90, h = \00. 1. The word kat in the text is meant to show that in this system the vflrga v letters take the numerical values 1, 2, 3, … beginning with k and not with k, t, p and y as in the case of the katapayddl system and that n and n are not zero in this system. 4 GITIK1 SECTION [ G itikB Six. The values, of the said avarga letters are taken to increase by 10 because the avarga letters are written in the avarga places, and increase by 1 in the avarga place means increase by 10 in the varga place. On the analogy of the varga and avarga classification of the letters, the notational places are also divided into the varga and avarga places. The odd places denoting the units' place, the hundreds' place, the ten thousands' place and so on, are called the varga places (because 1, 100, 10000, etc. are perfect squares) ; and the even places denoting the tens' place, the thousands' place, and so on are called the avarga places (because 10, 1000, etc., are non-square numbers). The text says that the varga letters should be written down in the varga places and the avarga letters in the avarga places. But how ? This is explained below : The notational places are written first. The usual practice in India is to denote them by ciphers : 0000000000000 00000 Instead, it is suggested that they should be denoted by the nine vowels 1 {a, i, u, r, I, e, o 1 ai, au) in the following manner : an au ai ai o o e e 1 ] r r u u i i a a When a letter is joined with a vowel (for example, in gr the letter g is joined with the vowel r), the letter denotes a number and the vowel the place where that number is to be written down. Thus gr stands for the number g (~3) written in the varga place occupied by the vowel r in the varga place as below : (A= Avarga, V=Varga) AVAVAVAV r r u u i i a a g = 3000000 Thus gr= 3000000. g has been written in the varga place because g is a varga letter. 1. It is immaterial whether the short vowels a, i, «, etc. are used or the long vowels a, i, u, etc. Thus, in a number-chronogram, a letter joined with a short vowel means the same thing as the same letter joined with the same long vowel. Thus ka=^k$ = l, ki~kJ=\QQ t and so on. Vetse 2 ] WRITING OF NUMBERS 5 Similarly, nWbunl?khr (=*+*,! + /, H«. ?+/, $ + + r) denotes the number which is obtained by writing ri in the varga place and s in the avar^a place occupied by the vowel i ; b in the varga place occupied by the vowel u ; n in the avarga place occupied by the vowel / ; and $ in the avarga place and kh in the varja place occupied by the vowel r as follows : ] 1 r r u n i i a a n $ kh b 's h = 158223 7 500 Thus Mitbuntikht=\5%223n%to. The rule stated in the above stanza is meant essentially to provide a key to decipher the numerical values borne by the letter chronograms used by the author in the succeeding stanzas. The commentator Surya- deva (b. A.D. 1191), therefore, interprets the above stanza as follows : “The varga letters denoting numbers which occur in the GUi-sutras that follow should be written in the odd places, and the avarga letters should be written in the even places …” The instruction 'hmau yah* serves two purposes. Firstly, it gives the value of the letter y as equal to h plus m (=5+25=30) ; secondly, it suggests that the conjoint letter hm means h-\-m. The statement of “two nines of zeros” in the text refers to the Indian method of writing the notational places by means of zero's. In the present primary schools in India when a student is taught to write large numbers he is first made to write the notational places by means of zeros arranged horizontally as follows : 0000 0000 The teacher then points to the first zero on the right and says “units' place”, then to the next zero and says “tens' place” then to the next zero and says “hundreds' place”, and so on. This practice of writing the notational places is of immemorial antiquity in India. It has been mentioned by the commentator BhSskara I (A.D. 629), who says : “Writing down the places, we have 00000000 0.” 6 , GITIKA SECTION [ Gi tiki So. REVOLUTION-NUMBERS AND ZERO POINT ^t, f sr nsfai^j, wrcfkT: ii 3 || liter* irafNfolT, f^Tf^T^T^ SrfFTm II « II 3.4. In a yuga, the eastward revolutions of the Sun are 43,20,000 ; of the Moon, 5,77,53,336 ; of the Earth, 3 1,58,22,37,500 ; of Saturn, 1,46,564 ; of Jupiter, 3,64,224 • of Mars, 22,96,824 ; of Mercury and Venus, the same as those of the Sun ; of the Moon's apogee, 4,88,219; of (the ilghrocca of) Mercury, 1,79,37,020; of (the iighrocca of Venus, 70,22,388 ; of (the Sighroccas of) the other planets, the same as those of the Sun; of the moon's ascending node in the opposite direction (i.e., westward), 2,32,226. 4 These revolutions commenced at the beginning of the sign Aries on Wednesday at sunrise at Lanka (when it was the commencement of the current yugd). The 'Moon's apogee' is that point of the Moon's orbit which is at the remotest distance from the Earth, and ths 'Moon's ascending node' is that point of the ecliptic where the Moon crosses it in its northward motion. The sighroccas of Mercury and Venus are the imaginary bodies which are supposed to revolve around the Earth with the heliocentric mean angular velocities of Mercury and Venus, respectively, their directions from the Earth being always the same as those of the mean 1. Go. ^. 2. CD. Kr. Pa. Su. ^rNnr ; Bh. Ni. Pa. (alt.), Ra. So. ^fartr. 3. These are the rotations of the Earth, eastward. 4. These very revolutions, excepting those of the Earth, are stated in MBh } vii. 1-5 ; LBh % i. 9-14 ; and SiDVu Grahaganita, i. 3-6, Vetsea 3-4 3 ftEVOLUf tOMS AttD ZERO fOIrtt 1 positions of Mercury and Venus from the Sun. It will thus mean that the revolutions of Mars, the sighrocca of Mercury, Jupiter, the sighrocca of Venus, and Saturn, given above, are equal to the revolutions of Mars, Mercury, Jupiter, Venus and Saturn, respectively, round the Sun. The following table gives the revolutions of the Sun, the Moon and the planets along with their periods of one sidereal revolution. The sidereal periods according to the Greek astronomer Ptolemy (A.D. c. 100-c. 178) aod the modern astronomers are also given for the sake of comparison. Table 2. Mean motion of the planets Revolutions Sidereal period in Planet in terms of days 43,20,000 : years Aryabhata I Ptolemy 1 Moderns 2 Sun 43,20,000 365-25868 365-24666 365-25636 Moon 5,77,53,336 27-32167 27-32167 27-32166 Moon's apogee 4,88,219 3231.98708 3231-61655 3232-37543 Moon's asc. node 2,32,226 6794-74951 6796-45587 6793-39108 Mars 22,96,824 686-99974 686-94462 686*9797 Sighrocca of 87-9693 Mercury 1,79,37,020 87-96988 87*96935 Jupiter 3,64,224 4332-27217 4330-96064 4332-5887 Sighrocca of Venus 70,22,388 224-69814 224-69890 224-7008 Saturn 1,46,564 10766-06465 10749-94640 10759-201 The epoch of the planetary motion mentioned in the text marks the beginning of the current yuga and not the beginning 1. Taken from Bina Chatterjee, “The Khanda-khadyaka of Brahmagupta”, World Press, Calcutta, 1970, vol. I, Appendix VII, p. 281. 2. Taken from H.N. Russell, Dugan and J.Q. Stewart, Astrono- my, Part I : The Solar system, Revised edition, Ginn and Company, Boston, Appendix. Also, see ibid., pp. 150, 159. The sidereal periods of Moon's apogee and ascending node are taken from P.C. Sengupta and N.C. Lahiri's introduction (p. xiv) to Babuaji MiSra's edition of Sripati's Siddhanta-sekhara. 8 GlTIM SECT.ON £ . Guik , Sn of the current Kal„a as was supposed by P.C. Sengupta. The current accord.ug ,o Aryabhata I, started „„ Thu S rs P day , „ M ™ years or 7 24,26,41,32,500 days before the beginning o th'e ™ ma ; and ,98,61,20,000 years or 7,25,44,75,70,625 days Lfo l l 177 7 , ^ CUrrent KaliyUgll • , Tte C «™< K liyu^ LI If I I8 ' 3102 BC - a ' SUDrise *' ^ <a hypothXi Place on the equator where the meridian of Uj ja i„ intersects V w h ch SSST the ^'— — ftheiunar £3S F ar ,h' , thing “T deSefVeS Spedal notice is *• ~»t of the Earth s rotates. Aryabhata I is, perhaps, the earliest astronomic Indta who advanced the theory of the Earth's rotation and gaTe the number of rotattons that the Earth perform, in a period of 43 Z m years. The per.od of one siderea. rotation of the Earth acS ng S^ESS, Thf *”“v * Mrre « ™— value! 23 56 4 091. The accuracy of Aryabhata I's value is remarkable. V.r,v.°! °' her Indian astronomers wi ° “P^Id the theory of the Earth srotatton, mention may be made of Prthndaka (AD 8601 »*h Makkibhat.a (A D. ,377, In , he Skmia J m * 3^)* J ,ha,,h T p ,t ,mmentat0?S of ^ who hold the opinion hat th Earth » senary, think that Aryabhata I states the rotations of the Ear h because the asterisms, which revo.ve westward around the earth by the force of the provector wind, see that the Earth rotates eastwaT These commentators were indeed helpless because Aryabhata I's theory of the Earth's rotation received a severe blow a, the land ofVaraham.h.ra (d. A.D. 587) and Brahmagupta (A.D. 628) whos a~r: 8ain5t ' hiS ^ C<>Uld ^ — d * -J Aristo,” mr^rr, the Greek astronomer pt ° ,e ^ Anstotle (B.C. 384-322), beheved that the Earth was stationary and adduced arguments in support of his view. 1. Vide infra notes on verse 5. Camb^ge, S 1940 W p. V mart ' ~ T. Verse 5 ] KALPA, MANU AND BEGINNING OF KALI 9 KALPA, MANU AND BEGINNING OF KALI 5. A day of Brahma (or a Kalpa) is equal to (a period of) 14 Manns, and (the period of one) Manu is equal to 72 yugas. Since Thursday, the beginning of the current Kalpa, 6 M anus, 27 yugas and 3 quarter yugas had elapsed before the beginning of the current Kaliyuga (lit. before Bharata). Thus we have 1 Kalpa =14 Manus and 1 Manu =72 yugas, so that 1 Kalpa=1008 yugas or 4,35,45,60,000 years. Likewise, the time elapsed since the beginning of the current Kalpa up to the beginning of the current Kaliyuga ts:6 Manus +27f yugas *^(6x 72+271) yugas =2(432+271) x 4320000 years s£ 1986120000 years or 725447570625 days. It is interesting to note that Aryabhat.a I prefers to say “before Bharata” (bharatat pmvam) instead of saying “before the beginning pf Kaliyuga” which is the sense actually intended here. Regarding the interpretation of bharatzt purvam there is difference of opinion amongst the commentators. The commentator Spme&vara interprets it as meaning 'before the occurrence of the Bharata (battle)'. P. C. Sengupta (A.D. 1927) and W.E. Clark (A.D. 1930), too, 1. Bh. Sa. T^TT: ; all others *T3^T. 2. E. Bh. Sa. ^TT: ; all others A Bb. 2 10 G1TIKA SECTION [ cSltika Sn. have interpreted the word bh&rata as meaning 'the Bharata battle'. In the Mahabharata we are told that the Bharata battle occurred at the end of the Dvapara yuga and before the beginning of the Kali yuga : “The battle between the armies of the Kurus and the Pandavas occurred at Syamantapancaka (Kuruksetra) when it was the junction of Kali and Dvapara.” 1 So this interpretation of bharatat purvam ('before Bharata') is equivalent to kaliyugat purvam ('before Kaliyuga'), as it ought to be. The commentators Bhaskara I (A.D. 629), Suryadeva (b. A D. 1191) and others have interpreted bharatat purvam as meaning 'before Yudhisthira', i.e., 'before the time when Yudhisthira of the Bharata dynasty relinquished kingship and proceeded on the last journey (maha-prasthanay. 2 According to these commentators, this event 1. MafiSbharata, Adiparva, ch. 2, vs. 13. 2. According to the Bhagavata-Pwana {Skandha 1, ch. 15, vs. 36), Kaliyuga began on the day on which Lord Krsna left this earthly abode : And when Yudhisthira came, to know that Kaliyuga had commenced he made up his mind to proceed on the last journey {Skandha I ch. 15, vs. 37) : ' Other views are that Kaliyuga commenced the moment Lord Krsna left for heaven : *rft*r* $«suft for jirff^rff^^ { Bhagavata-Purana, Skandha 12, ch. 2, vs. 33 Or, when the Seven $sis {i.e., the seven stars of the constellation of Ursa Major) entered the asterism Magha : Bhagavata-PurSna, Skandha 12, ch, 2, vs. 31 Verse 5 ] KALPA, MANU AND BEGINNING OF KALI II took place on Thursday, the last day of the past Dvapara. But the basis of this assumption is not specified. The commentators simply say : “This is what is well known.” (iti prasiddhih) } According to these commentators, too, bharatat purvam ultimately means 'before the beginning of the current Kaliyuga'. Brahmagupta criticises Aryabhafa I for his teaching in the above stanza. Writes he : “Since the measures of a Manu, a (quarter) yuga and a Kalpa and the periods of time elapsed since the beginnings of Kalpa and Krtayuga (as taught by Aryabhafa) are not in conformity with those taught in the Smrtis, it follows that Aryabhata is not aware of the mean motions (of the planets).” 2 “Since Aryabhata stales that three quarter yugas had elapsed at the beginning of Kaliyuga, the beginning of the current yuga and the end of the past yuga (according to him) occurred in the midst of Krtayuga; so his yuga is not the true one.” 3 “Since the initial day on .which the Kalpa started according to (Aryabhaja's) sunrise system of astronomy is Thursday and not Sunday (as it ought to be), the very basis has become discordant.”* 1. It is, however, noteworthy that Indian astronomers of all schools are perfectly unanimous in taking Friday as the day on which the current Kaliyuga commenced. 2. T tfTT ^FnPPF^T: WTTfe*KT ^felTRf ^ I BrSpSi, xi. 10 3. srro^ qwsTFfta *rrerHT^ *rer \ BrSpSi, xi. 4 BrSpSi, xi. H GITIKA SECTION [ GitikS Sn. reply to this criticism, astronomer VajeSvara (A. D. 904) “If the yu?a stated by Brahmagupta conforms to the teachings of the Smrtis, how is it that the Moon (according to him) is not beyocd the Sun fas stated in the Smrtis). If that . is unacceptable because f that statement of the Smrtis is false, then, alas, the .yw^-bypothesis of the Smrtis, too is false.” 1 “Since a planet does not make complete revolutions during the quarter yugas acceptable to Brahmagupta, son of Jisnu, (whereas it does during the quarter yugas according to Aryabhata), it follows that the quarter yugas of Srlmad Aryabhata (and not those of Brahmagupta) are the correct ones.” 2 “If a Kalpa should begin with a Sunday, how is it that Brahmagupta's Kalpa does not end with a Saturday. Brahmagupta's Kalpa being thus contradictory to his own state- ment, it is a fabrication of his own mind (and is by no means authoritative).” 3 VaSi, Grahaganita, ch. 1, sec. 10, vs. 3 VaSi, Grahaganita, ch. 1, sec. 10, vs. 2 VaSi, Grahaganita, ch. I, sec. 10, vs. 10 VctM 6 ] ORBITS AND EARTH'S ROTATION 13 PLANETARY ORBITS, EARTH'S ROTATION 6. Reduce the Moon's revolutions (in a yuga) to signs, multiplying them by 12 (lit. using the fact that there are 12 signs in a circle or revolution). Those signs mutiplied successively by 30, 60 and 10 yield degrees, minutes and yojanas, respectively. (These yojanas give the length of the circumference of the sky). The Earth rotates through (an angle of) one minute of arc in one respiration (=4 sidereal seconds). The circumference of the sky divided by the revolutions of a planet in a yuga gives (the length of) the orbit on which the planet moves. 2 The orbit of the asterisms divided by 60 gives the orbit of the Sun. 3 Thus we have Orbit of the sky =57753336 X 12×30 X 60 X 10 yojanas -. ; 12474720576000 yojanas Orbit of the asterisms = 1 73260008 yojanas Orbit of the Sun=2887666| yojanas Orbit of the Moon =21 6000 yojanas 132027 Orbit of Mars = 5431291^!^;^fl»<tt 287103 373277 Orbit of (Stghrocca of) Mercury =695473^^^ yojanas 699 Orbit of Jupiter =34250 133 yojanas 255221 Orbit of (Sighrocca of) Venus =1776421^^- yojanas 5987 Orbit of Saturn =85 114493 ^± yojanas. 36641 1. Br. Pr. Ud. ; all others f. 2. Cf. Somesvara : u^sr^t ^frfa: I 3. The same rule, excepting the rate of the Earth's motion, occurs in MBh, viu 20, also. 14 GITIKA SECTION t Gitiki S>. These orbits are hypothetical and are based on the following two assumptions : 1. That all the planets have equal linear motion in their res- pective orbits. 1 2. That one minute of arc (1') of the Moon's orbit is equal to 10 yojanas in length. 2 From the second assumption, the length of the Moon's orbit comes out to be 216000 yojanas. Multiplying this by the Moon's revolution-number (viz. 57753336), we get 12474720576000 yojanas. This is the distance described by the Moon in a yuga. From the first assumption, this is also the distance described by any other planet in a yuga. Hence Orbit of a planet = dist ? Ilce descri bed by a planet in a yuga revolution-number of that planet This is how the lengths of the orbits of the various planets stated above have been obtained. In the case of the asterisms, it is assumed that their orbit is 60 times the orbit of the Sun. By saying that “the orbit of the asterisms divided by 60 gives the orbit of the Sun”, Aryabhafa I really means to say that “the orbit of the asterisms is 60 times the orbit of the Sun.” Indian astronomers, particularly the followers of Aryabhaja I, believe that the distance described by a planet in a yuga denotes the circumference of the space, supposed to be spherical, which is illumined by the Sun's rays. This space, they call 'the sky' and its circumference *the orbit of the sky'. Bhaskara I says : ”(The outer boundary of) that much of the sky as the Sun's rays illumine on all sides is called the circumference or orbit of the sky. Otherwise, the sky is beyond limit ; it is impossible to state its measure.” 3 For us the sky extends to as far as it is illumined by the rays of the Sun. Beyond that, the sky is immeasurable.“ 4 1. See A y iii, 12. 2. This is implied in the text under discussion. 3. See Bhaskara I's commentary on A, i. 6, in Vol. II. 4. See BhSskara I's commentary on A, iii. 12, in Vol. II. Verse 7 ] linear diameters 15 According to the Indian astronomers, therefore, Orbit of a planet = Orbit of the sky Planet's revolution-number The statement of the Earth's rotation through 1' in one respiration, 1 stated in the text, has been criticised by Brahmagupta, who says : “If the Earth moves (revolves) through one minute of arc in one respiration, from where does it start its motion and where does it go ? And, if it rotates (at the same place), why do tall lofty objects not fall down ?' The reading bham (in place of bhnh) adopted by the commentators is evidently incorrect. The correct reading is bhnh, which has been mentioned by Brahmagupta (A.D. 628), Prthndaka (A. D. 860) and Udayadivakara (A.D. 1073). 8 7. 8000 nr make a yojana. The diameter of the Earth is 1050 yojanas; of the Sun and the Moon, 4410 and 315 yojanas, (respectively) ;* of Meru, 1 yojana ; of Venus, Jupiter, Mercury, Saturn and Mars (at the Moon's mean distance), one-fifth, one-tenth, one-fifteenth, one -twentieth, and one-twentyfifth, (respectively), of the Moon's diameter. 5 The years (used in this work) are solar years. 1. 1 respiration =4 seconds of time. 2. srmfa ^*rt ^rfc aff fat *r%?r spin^m i LINEAR DI A MEIERS BrSpSi, xi. 17. 3. See his commentary on LBh, i. 32-33. The same values are given in MBh, v. 4 ; LBh, iv. 4, Cf, MBh, vi. 56. 4. 5. 16 GlTIKA SECTION f GitikS Slu Nr is a unit of length whose measure is equal to the height of a man. N? is also known as nam, pumsa, dhanu and danda, “Purusa, dhanu, danda and nam are synonyms”, says BhSskara I. The diameters of the Earth, the Sun, the Moon, and the Planets stated above may be exhibited in the tabular form as follows : Table 3. Linear diameters of the Earth etc. Linear diameter Linear diameter in yojanas irl yojanas (at the Moon's mean distance) Earth 1050 Sun 4410 Moon 315 Mars 12.60 Mercury 21.00 Jupiter 3150 Venus 63m Saturn 15 75 The following is a comparative table of the mean angular dia- meters of the planets : Table 4. Mean angular diameters of the Planets Planet Mean angular diameter according to Aryabha^a I Greek astronomers 1 Modern Moon 31' 30” 35' 20' (Ptolemy) Tycho Brahe 31' 8” Mars 1' 15*6 (1546-1631) 1' 40' Mercury 2' 6' 2' 10' Jupiter 3' 9' 2' 45* Venus 6' 18' 3' 15* Saturn 1' 34“5 1' 50' 1. See E. Burgess, Translation of the SUrya-siddhanta, Reprint Calcutta, 1935, p. 196 ; and introduction by P.C. Sengupta, p. xlvi. Verse 8 1 OBLIQUITY AMD ORBITAL INCLINATIONS 17 P. C. Sengupta translates the second half of the stanza as follows : 'The diameters of Venus, Jupiter, Mercury, Saturn and Mars are, respectively, 1/5, 1/10, 1/15, 1/20 and 1/25 of the diameter of the Moon, when taken at the mean distance of the Sun.” This is incorrect, because : 1. “When taken at the mean distance of the Sun” is not the correct translation of sam&rkasam&h. The correct translation is : “The years are solar years” as interpreted by Bhaskara Ij and Somes vara ; or “The years of a yuga are equal to the number of revolutions of the Sun in a yuga” 1 as translated by Clark and as interpreted by SHryadeva, Paramesvara and Raghunatha-raja. 2. The diameters of the planets stated in the stanza under consideration correspond to the mean distance of the Moon and not to the mean distance of the Sun as Sengupta has supposed, Sengupta's disagreement on this point from the commentator Paramesvara is unwarranted. All commentators agree with Paramesvara. OBLIQUITY OF ECLIPTIC AND INCLINATIONS OF ORBITS SS-f* % *TS*Jt iTT II c II 8. The greatest declination of the Sun is 24°. a The greatest celestial latitude (lit. deviation from the ecliptic) of the Moon is 4JV of Saturn, Jupiter and Mars, 2°, 1° and 1|° respectively; 2. The same value is given in MBh, iii. 6 ; LBh, ii. 16 • KK Part 1, iii. 7 ; KR, i. 50. 3. The same value occurs in MBh, v. 30 ; LBh, iv. 8 • KK Part 1, iv. 1 (c-d) ; KR, ii. 3 (a-b). A. Bb. 3 Id G1TIKA SECTION t GitilcS Sn. and of Mercury and Venus (each),! . 1 96 ahgul as or 4 cubits make a nr. The greatest declination of the Sun is the obliquity of the ecliptic. According to Aryabhafa I and other Indian astronomers, its value is 24° , a According to the modern astronomers its value is 23° 27' 8*.26— 46\84 T, where T is measured in Julian centuries from 1900 A.D. The value in common use is 23£°. The greatest celestial latitude of a planet is the inclination of the planet's orbit to the ecliptic. The values of the inclinations of the orbits of the Moon and the planets as given in the above stanza and those given by the Greek astronomer Ptolemy and the modern astronomers are being exhibited in the following table : Table 5. Inclinations of the Orbits Inclination of the orbit PauSt 4 and RoSi 6 • Planet Aryabhata I Ptolemy 3 of Modern 6 Varahamihira Moon 4° 30' 5° 4^ 40' 5° 9' Mars 1° 30' r i° 51' or Mercury 2° V 7° 00' 12' Jupiter 1° 1° 30' 1° 18' 28* Venus 2° 3° 30' 3° 23' 38“ Saturn 2° 2° 30' 2° 29' 20” 1. The same values occur in MBh, vii. 9 ; LBh, vii. 7 (a-b) ; KK, Part 1, viii. 1 (c-d) ; KR, vii. 8 (c-d). 2. According to the Greek astronomer Ptolemy, the obliquity of the ecliptic is 23° 51' 20“. See Great Books of the Western World, vol. 16 : The Almagest of Ptolemy, translated by R. Catesby Taliaferro, Book II, p 31. 3. See E. Burgess, Translation of the Surya-siddhanta, Reprint, Calcutta, 1935, p. 52. In the Almagest of Ptolemy, translated R. Catesby Taliaferro, the obliquities of the epicycles of Mercury and Venus are stated as 6° 15' and 2° 30' respectively. See pp. 435 and 433. 4. See PSi, iii. 31. 5. See PSi, viii. 11. 6. See H.N. Russell, R.S. Dugan and J.Q. Stewart, Astronomy, Part I, The Solar system, Revised edition, Ginn and Company, Boston, Appendix, Veise9 ] NODES AND APOGEES 19 In the case of Mercury and Venus, Aryabhata l' s values differ significantly from those of Ptolemy and modern astronomers because the values given by Aryabhata I are geocentric and those given by Ptolemy and modern astronomers are heliocentric. Combining the instruction in the last quarter of the above verse with that in the first quarter of verse 7, we have 24 angulas= 1 cubit (hasta) 4 cubits=l nr 8000 nr=l yojana. Since earth's (equatorial) diameter= 1050 yojanas (vs. 7), according to Aryabhata I, and =12757 km or 7927 miles, according to modern astronomy, it follows that Aryabhata Vs yojana is approximately equal to 12% km or 7£ miles. Likewise his nr = 152 cm. or 5 ft approx, and cubit= li ft. approx. The length of a cubit in common use is
ft. ASCENDING NODE5 AND APOGEES {APHEL1A) 9. The ascending nodes of Mercury, Venus, Mars, Jupiter and Saturn having moved to 20°, 60°, 40°, 80° and 100° respectively (from the beginning of the sign Aries) (occupy those positions); 3 and the apogees of the Sun and the same planets (viz., Mercury, Venus, Mars, Jupiter and Saturn) having moved to 78°, 210% 90°, 118°, 180° and 236° respectively (from the beginning of the sign Aries) (occupy those positions). 4 The following table gives the longitudes of the ascending nodes and the apogees of the planets for A. D. 499 as given by Aryabhata I and as calculated by modern methods. The corresponding longitudes for A.D. 150, as stated by Ptolemy are also given for comparison. 1. A.-G. sn. ; Bh. ; Pr. *tt^ 2. Kr. 3. The same values occur in MBh, vii. 10 ; LBh, vii. 6 (c-d) ; KK, Part 1, viii. 1 ; K.R, vii. 8 (a-b). 4. The same values occur in MBh, vii. U-12 (a-b) ; LBh, j. 22 {a-b), 18 ; KH, i. 10 {c-d) ; vii. 9 {c-d). 20 GITiKX SECTION [ GitikR Sn. Table 6. Longitudes of the Ascending Nodes for A.D. 499 Longitudes of the ascending nodes Planet Aryabhafa I Ptolemy 1 By modern (for A.D. 150) calculation 2 iviars 40° 25° 30' 37° 49' Mercury ^ AO 1 AO aa/ ts\o «% W 2M 10 00 30 35 Jupiter 80° 51° 00' 85° 13' Venus 60° 55° 00' 63° 16' Saturn 100° 183° 00' 100° 32' Table 7. Longitudes of the Apogees for A.D. 499 Longitudes of the apogees (aphelia) Planet Aryabhaja I Ptolemy 3 RoSi* By modern (for A.D. 150) of calculation* Varahmihira Sun 78° 65° 30' 75° 77° 15' Mars 118° 115° 30' 128° 28' Mercury 210° 190° 00' 234° 11' Jupiter 180° . 161° 00' 170° 22' Venus 90° 55° 00' 290° 4' Saturn 236° 233° 00' 243° 40' The word gatva (meaning 'having moved' or 'having moved to') is used in the text to show that the ascending nodes and the apogees 1. See E. Burgess, ibid. Appendix, p. 331. According to P.C. Sengupta and N.C. Lahiri. the longitudes of the ascending nodes of Mars, Jupiter and Saturn as given by Ptolemy are 30°, 70° and 90°, respectively. See their introduction (p. xiv) to the Siddhanta-sekhara of Sripati, Part II, edited by Babuaji MiSra, Calcutta, 1947. 2. See E. Burgess, ibid., Introduction by P.C. Sengupta, p. xlviii. 3. See E. Burgess, ibid., Appendix, p. 331 ; and Bina Chatterjee, The Khan4akhadyaka of Brahmagupta, with the commentary of Bhatfotpala, vol. I, p. 283, Verte 9 ] NODES AND APOGEES 21 of the planets are not stationary but have a motion. The commentator Bhaskara I says that by teaching their motion, Aryabha^a I has specified by implication their revolution-numbers in a yuga. The aged, who preserve the tradition, says he, remember those revolution-numbers by the continuity of tradition. The period of 35750224800 years, according to the tradition, is the common period of motion (yuga) of the ascending nodes of all the planets, in which the ascending nodes of Mars, Mercury, Jupiter, Venus and Saturn make 2, 1, 4, 3 and 5 revolutions, respectively. In the case of the apogees of the planets, the periods and the corresponding revolutions, as handed down to Bhaskara I by tradition, are shown in the following table : Table 8. Periods and Revolution-numbers of the Apogees Apogee of Period in years Revolution-number Sun 119167416000 13 Mars 357502248000 59 Mercury 23833483200 7 Jupiter 3972247200 1 Venus 7944494400 1 Saturn 178751124000 59 The commentators SUryadeva and Raghunatha-raja have also cited the above-mentioned periods and revolution-numbers to preserve (Footnotes of the last Page :) According to P.C. Sengupta, the longitudes of the apogees of Sun, Mars, Mercury, Jupiler, Venus and Saturn as given by Ptolemy are 65° 30', 106° 40', 181° 10', 152° 9', 46° 10' and 224° 10', respectively. See his introduction to E. Burgess' Translation of the Surya-siddhanta, pp. xlvi and xlvii. The same values are given also in Great Books of the Western World, vol. 16 : The Almagest by Ptolemy (translated by R. Catesby Taliaferro), Book XI, pp. 386-390. 4. See PSi, viii. 2. 5. See E. Burgess, ibid., introduction by P.C. Sengupta, p. x and xlvii. 22 GITIKA SECTION [ Gitiks Sn. the continuity of tradition of the school. It is remarkable that the first line of the stanza : which has been quoted in full by SBryadeva, has been cited by Bhaskara I too. This means that the passage was derived from some earlier source, and the tradition mentioned in the stanza is definitely older than Bhaskara I Whosoever might be the founder of the tradition, it is based on the misunderstanding that the ascending nodes and the apogees, after having started their motion from the first point of Aries at the beginning of the current .Kalpa, moved exactly through the degrees mentioned by Aryabhata I up to 499 A.D., the epoch mentioned by Aryabhata I. The motions of the nodes and the apogees of the planets ascribed to tradition by Bhaskara I and the other commentators are much less than their actual motions. For example, the node of Mercury, which is the slowest, actually requires about 166000 years to complete a revolution. 1 Similar is the case with the apogees. MAN DA AND &GHRA EPICYCLES (Odd quadrants) 10 The manda epicycles of the Moon, the Sun, Mercury, Venus, Mars, Jupiter and Saturn (in the first and third anomalistic quadrants) 1. See C.A. Young, A Text-book on General Astronomy, Revised edition, 1904, p. 337. 2. Bh. Pa. So. ^ ; others vfi. 3. Bh. G. T?rr ; all others
T. 4. Bh. and Go. *f ; all others ?5T, Vet 8 * 11 ] EPICYCLES 2S are, respectively, 7, 3, 7, 4, 14, 7 and 9 (degrees) each multiplied by 4\ {i.e., 31.5, 13.5, 31 5, 18, 63, 31.5 and 40.5 degrees, respectively) ; the slghra epicycles of Saturn, Jupiter, Mars, Venus and Mercury (in the first and third anomalistic quadrants) are, respectively, 9, 16, 53, 59 and 31 (degrees) each multiplied by 4| (i.e., 40.5, 72, 238.5, 265.5 and 139.5 degrees, respectively). (Even quadrants) *$tm? ftftssa f^Tg^w^T n ii 11. The manda epicycles of the retrograding planets (viz., Mercury, Venus, Mars, Jupiter and Saturn) in the second and fourth anomalistic quadrants are, respectively, 5, 2, 18, 8 and 13 (degrees) each multiplied by 4* (i.e., 22.5, 9, 81, 36 and 58.5 j degrees, respectively) ; and the Sighra epicycles of Saturn, Jupiter, Mars, Venus, and Mercury (in the second and fourth anomalistic quadrants) are, respectively, 8, 15, 51, 57 and 29 (degrees) each multiplied by 4H'.e > 36> 67.5, 229.5, 256.5 and 130.5 degrees, respectively). 1 3375 is the outermost circum- ference of the terrestrial wind. 2 The dimensions of the manda and sighra epicycles are stated in terms of degrees, where a degree stands for the 360th part of the circumference of the deferent (kaksyavrtta). Thus, when an epicycle is stated to be A°, it means that its periphery is A/360 of the circum- ference of the deferent. The following table gives the manda and slghra epicycles as stated above by Aryabhaja I and also those given by Ptolemy : 1. The same values occur in MBh, vii. 13-16; LBh, i. 19-22 ; KR, viii. 10-11. 2. The same value occurs in SiDV(, GoladhySya, v. 2. 24 GiTIKX SECTION [ Gitikft Sn. Table 9. Manda and Sighra epicycles of the planets Manda epicycles Sighra epicycles Planet Aryabhafa I Aryabhaja I Odd Even Ptolemy 1 Odd Even Ptolemy 2 quadrant quadrant quadrant quadrant Sun 13°.50 13°.50 15°.00 Moon 31°.50 31°.50 31°.40 Mars 63°.00 81°.00 72°.00 238°.50 229°.50 237° Mercury 31°.50 22°.50 18°,00 139°.50 130°.50 135° Jupiter 31°.50 36°.00 33°.00 72°. 00 67°.50 . 69° Venus 18°. 00 9°.00 15°.00 265°.50 256°.50 259° Saturn 40°.50 58°.50 41°.00 40°.50 36°,00 39° It is noteworthy that in stating the dimensions of the manda epicycles the planets have been mentioned in the order of decreasing velocities {manda-gati-krama), whereas in stating the dimensions of the sighra epicycles they have been mentioned in the order of increasing velocities (Hghra-gati-krama). It is perhaps done deliberately to emphasise this point to the reader. The use of ablative in yathoktebhyah is meant to indicate that in finding the manda anomaly the longitude of the apogee is to be subtracted from the longitude of the planet. Similarly, the use of the inverted forms uccasighrebhyah and uccacchighrat in place of sighroccebhyah and sighroccat, respectively, shows, as remarked by Bhaskara I and Somesvara, that, in finding the sighra anomaly, the longitude of the planet has to be subtracted from the longitude of the sighrocca. It may be pointed out that, according to Bhaskara I and Lalla, the manda and sighra epicycles stated above correspond to the beginnings of the respective anomalistic quadrants as is evident from the rules stated in MBh, iv. 38-39 (a-b), LBh, ii. 31-32 and SiDVr, I, iiii. 2 and explained in Bhaskara I's commentary on 1. See Bina Chatterjee, opxit., p. 284. In the Almagest of Ptolemy, translated by R. Catesby Taliaferro, however, the Moon's epicyclic radius is stated as 51 p 51' which yields 31°.50 as the value of the Moon's epicycle. See p. 151. 2. See Bina Chatterjee, op, cit. t p. 285, Vetse 11 ] EPrCYCLES 25 A, iii. 12. 1 The Kerala astronomer SankaranSrayana (A. D. 869) refers to some astronomers (without naming them) who said that there was also the view that the epicycles givei by Aryabhafa I corresponded to the end-points of the anomalistic quadrants. 2 The Kerala astronomer Govinda-svamI, who also refers to this controversy, is of the opinion that Hghra epicycles stated above correspond to the beginnings of the respective anomalistic quadrants, but the manda epicycles stated above correspond to the last points of the respective anomalistic quadrants,* and, consequently, he has replaced the rules referred to above by another rule (which has been quoted by Udayadivakara (1073 A.D.) in his commentary on LBh, ii. 31-32. This controversy is due to the fact that Aryabhata I himself does not specify whether the epicycles given by him correspond to the initial points or last points of the anomalistic quadrants. Since the epicycles stated in the text correspond to the beginnings of the odd and even anomalistic quadrants, their values at other positions of the planets are to be derived by the rule of three. Bhaskara I has prescribed the following rule .* Let a and p be the epicycles (manda or iighra) of a planet for the beginnings of the odd and even anomalistic quadrants, respectively. 1. The commentator SSryadeva, too, is of this view. See his comm. on A, iii, 24, p. 114. 2. See his commentary on LBh, ii. 32-33, where he w'rites : 3. Govinda-svamI has been led to this conclusion by the fact that in stating the dimensions of the manda epicycles Aryabhafa I has mentioned the planets in the order of decreasing velocities whereas in Stating the dimensions of the sighra epicycles he has mentioned the planets in the order of increasing velocities. See his comm. on MBh. iv. 38-39 (a-b). 4. See MBh, iv. 38-39 (a-b)) LBh, ii. 31-32. An equivalent rule is given in SiDVr, Graltaganita, iii. 2. A. Bh. 4 26 GlTIKA SECTION [ GitikS Sn. Then (i) If the planet be in the first anomalistic quadrant, say at P, and its anomaly be 0, epicycle at P=a+ fi»~ a) p Rsin ° , when «<p R (a- p) Rsin 6 . ^ n =a — ^ —L- , when a>p R and (ii) If the planet be in the second anomalistic quadrant, say at Q, and its anomaly be 90° -{-6, epicycle at Q=p- (P ~ g) * V — * when a<(5 R , («— P) Rvers . „^ fl =H- ^ , when«>p. Similarly in the third and fourth quadrants. The epicycles thus derived are called true epicycles (spatfa- or sphuta-paridhi). But the tabulated tnanda epicycles or the true manda epicycles derived from them are not the actual epicycles on which the true planet in the case of the Sun and Moon or the true mean planet in the case of the other planets is supposed to move. It is believed that they are the mean epicycles corresponding to the mean distances of the planets. In order to obtain the actual epicycles, one should either apply the formula : actual manda epicycle = tabulatedl or true “™*» epicycle 2 XH R where H is the planet's true distance in minutes obtained by the process of iteration {asakrtkalakarna or mandakarna), 3 or apply the process, of iteration.* In the case of the iighra epicycles, however, the actual epicycles are the same as the tabulated epicycles. 1 . In the case of the Sun and the Moon. 2. In the case of the other planets, Mars etc. 3. See MBh, iv. 9-12 ; LBh, ii. 6-7, 4. See &iDV(, Grahaga$ita, iii. 17. Vetse 11 3 EUCYCLES 27 Brahmagupta criticises Aryabhafa I for stating different epicycles for odd and even anomalistic quadrants. Writes he : “Since in (Aryabhaja's) sunrise system of astronomy, the epicycle 'which is the multipler of the Rsine of anomaly in the odd anomalistic quadrant is different from the epicycle which is the multiplier of the Rsine of anomaly in the even anomalistic quadrant, the {manda or sighra) correction for the end of an odd anomalistic quadrant is not equal to that for the beginning of the (next) even anomalistic quadrant (as it ought to be). This discrepancy shows that the differing epicycles (stated by Aryabhafa) are incorrect. “Since the epicycle which is the multiplier of the Rsine of anomaly in the odd anomalistic quadrant is different from the epicycle which is the multiplier of the Rversine of anomaly in the even anomalistic quadrant, the {manda or iighra) correction for the anomaly amounting to half a circle, does not vanish (as it ought to). This discrepancy, too, shows that differing epicycles (stated by Aryabhaja) are incorrect. 1 “Since the epicycles (stated by Aryabhat.a) correspond to odd and even anomalistic quadrants (and not to their first or last points), the (so-called true) epicycle which is obtained by multiplying the Rsine of anomaly by the difference of the epicycles (for the odd and even quadrants) and dividing by the radius and then subtracting the resulting quotient from or adding that to the epicycle for the odd quadrant, according as it is greater or less than the other, is not the correct epicycle. “If indeed there should be two different epicycles for the odd and even anomalistic quadrants, then, why have not two 1 . Let a and be the epicycles for the odd and even quadrants and let the anomaly be equal to 180°. Then, according to Aryabhata I (see A, iii. 22 (a-b)), the corresponding bhujaphala=«- X R sine 90°/360— x Rvers 90°/360 =« R/360— PR/360, which is not equal to zero, because 23 GIT1KA SECTION [ GitiiB Sn. different epicycles been stated in the case of the Sun and the Moon. It simply shows that the process of planetary correction stated in (Aryabhata's) Audayika-Tantra (i. e., Aryabhapya) does, in neither way, lead to a correct result” 1 Had Aryabhata I specified that the epicycles stated by him corresponded to the first or last points of the respective anomalistic quadrants, there would not have been any occasion for such a criticism. The number 3375, denoting the length of the outer boundary of the terrestrial wind, his reminded Bhaskara I of the following formula which also involves that number : . 4 (180°— &) 8. R Rsm 12×3375— (180°— W ' where B is in terms of degrees. BhSskara I thinks that the length of the outer boundary of the terrestrial wind has been stated simply to teach the method of finding the Rsine without the use of the Rsine Table which is implied in the above formula. Brahmagupta (A. D. 628) misreads giyihasa as giyigasa and unnecessarily criticises Aryabhata I for giving two different values of the Earth's diameter. Writes he : “The circumference being (stated as) 3393 yojanas, the Earth's diameter becomes equal to 1080 yojanas. By stating the same again as 1050 (yojanas) due to uncertainty of his mind, he (/.<?., Aryabhata I) has exposed his knowledge !” 8 1. s: qftfafira^ftsfiT: 3«r: l sirrah a> snf : qftfaft^rcnijer: ^^pr^r: I BrSpSi, xi. 18-21 2. mM n ^3FPTfr^: *rf3r %*fa*t\ sum \ ffrSpSi,. xi. 15 Vm l2 1 SINE- DIFFERENCES 29 RSINE-DIFFERENCES irfe <g% *sr% sftr sfa ^ ftw wfc ^ l s*rfo ftsr fa* 12. 225, 224, 222, 219, 215, 210, 205, 199, 191, 183, 174, 164, 154, 143, 131, 119, 106, 93, 79, 65, 51, 37, 22, and 7- tbese are the Rsine-differences (at intervals of 225 minutes of arc) in terms of minutes of arc. The following table gives the Rsines and the Rsine-differences at intervals of 225' (or 3° 45') according to Aryabha t a I and the corresponding modern values correct to three decimal places. Table 10. Rsines and Rsine-differences at the taterYars of 225' or 3*45' irvahhata I's values Modern Values Arc Rsine Rsine-differences Rsine Rsine-differences 225' 225' 225' 224\856 224'.856 450' 449' 224' 448'.749 223\893 675' 671' 222' 670'.720 221 '.971 900' 890' 219' 889'.820 219M00 1125' 1105' 215' 1105'. 109 215'.289 135tf 1315' 210' 1315'.666 210'.557 1575' 1520' 205' 1520'. 589 204\923 1800' 1719' 199' 1719'.000 198'.411 2025' 1910' 191' 1910' .050 191'.050 2250' 2093' 183' 2092' .922 182'.872 1. D.G. Sn. fararT ; others fktt. 2. A. %W *l fa fa Bh. Sa. ft* » E. ^ ^ Pa. Ra. Sis. *n^r ; So. fa* vrfo fiW 3. Bb,. TOT ; other* m- 30 GTTIKA SECTION [ GitikB Sn. Aryabhata Vs values Modern Values Arc Rsine Rsine-differences Rsine Rsine-differences 247S' 2267' 174' 2266'.831 173'. 909 2700' 2431' 164' 2 431 '.033 164'.202 2585' 154' 2584'. 825 153'.792 2728' 143' 2727'. 549 142' .724 117S' jj i j 2859' 131' 2858'.592 131'.043 2978' 119' 2977'.395 118'.803 3R25' 3084' 106' 3083'.448 106' .053 40S0' 3177' 93' 3176'.298 92'.850 427S' 3256' 79' 3255'.546 79'.248 3321' 65' 3320'.853 65'.307 4/23 3372' 51' 3371 '.940 51'.087 4950' 3409' 37' 3408'.588 36'.648 5175' 3431' 22' 3430'.639 22'.051 5400' 3438' 7' 3438\000 7.361 The twenty-four Rsines given in the Surya-siddhanta 1 are exactly the same as those in column 2 above. P.C. Sengupta is of the opinion that the author of the Surya-siddhunta has based his Rsines on the Rsine-differences given by Aryabhata I. 2 The 16th Rsine, viz,, 2978, was modified by Aryabhata II s (c. A.D. 950) who replaced it by the better value 2977. The table of Rsines given by Bhaskara II* (A.D. 1150) is the same as that of Aryabhata II (c.A.D. 950). Astronomer Sumati of Nepal, who lived anterior to Aryabhata II (c. A.D. 950), gives 5 the values of the 4th and 16th Rsines as 889' and 2977' respectively instead of 890' and 2078' given by Aryabhata I. Sumati's table contains ninety Rsines at the intervals of one degree. 1. ii. 17-22. 2. See P. C. Sengupta's introduction (p. xix) to E. Burgess' Translation of the Surya-siddhanta. 3. See MSi, iii. 4-6. 4. See SiSi, Grahaganita, ii. 3-6. 5. Both in SumatUmahmantra and Sumati-karana. Verse 13 1 AIM AIM OF THE DA&AG1TIKA-SUTRA 31 13. Knowing this DaSagltika-sntra, (giving) the motion of the Earth and the planets, on the Celestial Sphere (Sphere of asterisms or Bhagola), one attains the Supreme Brahman after piercing through the orbits of the planets and stars. This chapter is called Ten Aphorisms in Gitika Stanzas' (Da'sagitikasutra). But instead of 10 gitika stanzas there are 11 gitika stanzas (vss. 2-12) here. The question arises : Which of these are those 10 which contain the 10 aphorisms of this chapter ? This is indeed a controversial question. For, according to the commentators Bhaskara I (A.D. 629), Somesvara and Snryadeva (b. A.D. 1191), vss. 2-1 1 are the ten stanzas which contain the 10 aphorisms ; vs; 12, in their opinion, does not constitute an aphorism as it contains a table of Rsine-differences which is easily derivable. According to the commen- tator Paramesvara (A.D. 1431), however, vss, 3-12 are the 10 stanzas containing the 10 aphorisms; vs. 2, in his opinion, is a definition and not a mathematical aphorism. There is, however, another difficulty. Is vs. 12 composed in the gitika metre or in the arya metre ? According to Snryadeva, it is in the aryS metre and, according to Paramesvara, it is in the gitika metre. In fact, vs. 12 (in the form in which Suryadeva and Paramesvara state it) is, as pointed out by H. Kern,* metrically defective, as it contains 20 syllabic instants instead of 18, in the fourth quarter : 1. C.D.E. Kr. Ra. SS. 5*r*ftfr$Kr«? I 2. Kr. ?R[»it5PTft WT I 3. A. 5Wnftflr«jiT$'f STreir ; B.D. No colophon ; E. »ftf^ 4. See H. Kern, Iryabhafiyam, Leiden (1874), p. 17, footnote. 32 GlriKl SECTION I 1 I I I 1 1 t I I t 1 I 1 S 5 1 | S S | t S S Tf<sr *r% qsfa *ter ufar srfa, sfa gro wfa fa rofa ftre* i II SISISS S S 5 S S-||| tfrffa fa* W^T, ^ r WW 1ST qi <E ^ *?TWWIT: II It may be called a defective gitika. It is not a pure Qrya. How- ever, in the forms in which vs. 12 has been stated by Bhaskara I and Someivara, it is in the perfect gitika metre. Bhaskara I's reading I I I I I I I I 1 I IIMSS I 1 S S II SS *far nftr qsfa afar *rfa srfiar, sfa ^ wfr fawr wfo> fa i ii sin ii ii lis s siiiisss ssrfa fa*r $w afa fa*, ?it itw 5* fw «r q> *>?mrwT: i Somesvara's reading I I I I I I I 1 I I I I I I S S I I SSI t s s «rfa »rfar ufa afar *rfa *rfa, jpfa fa fa*ir rofa fare* i II S S II || II S S S S I I I I S 5 5 *?ifa fa* f* fan fa*, *r *w <rt <s ^ *$rrtff err; n We agree with Paramesvara in regarding vss. 3 to 12 as forming the 10 gitika stanzas containing the 10 aphorisms of this chapter. CHAPTER II GANITA OR MATHEMATICS INVOCATION AND INTRODUCTION 1. Having bowed with reverence to Brahma, Earth, Moon, Mercury, Venus, Sun, Mars, Jupiter, Saturn and the asterisms, Arya. bhata sets forth here the knowledge honoured at Kusumapura. Commenting on this stanza, Bhaskara I writes : “Kusumapura is Pataliputra. (Aryabhata) sets forth the knowledge honoured there. This is what one hears said : Indeed this Svayambhuva-siddhnnta was honoured by the learned people of Kusumapura (PSjaliputra), although the Paulisa-, Romaka-, Vasi$tha- and Saurya-Siddhantas were also (known) there. That is why (Aryabhaja) says— 'the knowledge honoured at Kusumapura'.” 2. Eka (units place), data (tens place), iata (hundreds place), sahasra (thousands place), ayuta (ten thousands place), niyuta (hundred thousands place), prayuta (millions place), koti (ten millions place), arbuda (hundred millions place), 1. C. 5TTf«W 2. Bh. adds ^ 3. A-G. Gh. NL Pa. Ra. So. Sa. ^r^rn^r A. Bh. 5 33 THE FIRST TEN NOTATIONAL PLACES 34 GA2SUTA SECTION [ Ganita Sn. and vrnda (thousand millions place) are, respectively, from place to place, each ten times the preceding. 1 The notational places are denoted by writing zeros as follows : 0000000000 The zero on the extreme right denotes the units place, the next one (on its left) denotes the tens place, the next one denotes the hundreds place, and so on. SQUARE AND SQUARING 3. (a-b) An equilateral quadrilateral with equal diagonals and also the area thereof are called 'square'. The product of two equal quantities is also 'square'. 2 The commentator Paramesvara explains the term samacaturasra as follows : “That four-sided figure whose four sides are equal to one another and whose two diagonals are also equal to each other is called a samacaturasra. 1 ' By defining a square as the product of two equal quantities the author has stated, by implication, the rule of squaring. That is, to find the square of a number, one should multiply that number by itself. The commentator Bhaskara I gives the terms varga, karani, krti, varganS? and yavakarana as synonyms, meaning 'square or squaring'. Of these terms, karani, vargana and yavakarana are unusual. The term yavakarana is derived from the fact that in Hindu algebra jc a is written t as yava (ya standing for yavat-tSvat, i.e., x, and va for varga, i.e., square).* 1. Cf. GSS, i. 63-68 ; PG, def. 7-8; GT, p. 1, vv. 2-3 ; L (ASS), Def. 10-11, pp. 11-12 ; GK, I, p. 1, vv. 2-3. 2. Cf. BrSpSi, xviii. 42 ; GSS, ii. 29 ; SiSe, xiii. 4 ; L (ASS), Rule 19 (a), p. 19. 3. Use of the term vargava in the sense of multiplication has been made by Bhaftotpala also. See his comm. on Br J a, vii. 13. 4. See vol. II, Introduction, p. lxxvii, sec. 4, V««e 3 ] CUBE AND CUBING 35 The terms for multiplication according to Bhaskara I are : saihvarga, ghata, gunana, hatih and udvartam. For the multiplication of equal quantities, Bhaskara I uses a special term, gata, meaning literally 'moved' >progressed>raised. “Gunana is the multiplication (abhyUso) of unequal quantities, and gata”, says he, “is the multiplication of equal quantities.” 1 The term dvigata, according to him, means square, trigata means 'cube'; and so on. The dvigata of 4 is the product of 4 and 4, i.e., 4 2 ; the trigata of 4 is the product of 4 and 4 and 4, i.e., 4 s ; and so on. According to this terminology, m” will be expressed by saying 'nth gata of m\ which corresponds to our present-day expression 'nth power of m\ Following the same terminology, the roots have been called gatartmla. Thus 4 is the gatamula of 4 2 , the trigatamlila of 4 8 , and so on. In general, m is the *«th gatamxda of m n \ This, too, corresponds to our present-day expression 'the nth root of m n \ It is interesting to note that Bhaskara I finds fault with the usual Hindu method 8 of squaring a number for the simple reason that it implies the use of the squares of the digits 1 to 9 but it neither states them nor gives the method for obtaining them. Aryabhafa i's method, according to him, is complete in itself. CUBE AND CUBING 3. (c-d) The continued product of three equals as also the (rectangular) solid having twelve (equal) edges is called a 'cube'. 4 The rule for cubing a number is implied as in the previous case. Here also, Bhaskara I finds fault with the usual Hindu method 6 of cubing a number for the reason that although it implies the use of 1. See vol. II, Bhaskara I's commentary, p. 43. 2. For example, see PG, Rule 23. 3. F. G. SRSnfsre ; Pa. Ra. SFRTT^eWT 4. Cf. BrSpSi, xviii. 42 ; GSS, ii. 43 ; Si$e, xiii. 4 ; L (ASS), Rule 24 (a), p. 23. 5. For example, see PG, Rule 27-28, 36 GA3JITA SECTION [ Gajdta Sa. the cubes of the digits 1 to 9, it neither states them nor tells how to find them out, SQUARE ROOT 4. (Having subtracted the greatest possible square from the last odd place and then having written down the square root of the number subtracted in the line of the square root) always divide 1 the even place (standing on the right) by twice the square root. Then, having subtracted the square (of the quotient) from the odd place (standing on the right), set down the quotient at the next place (i.e., on the right of the number already written in the line of the square root). This is the square root. (Repeat the process if there are still digits on the right). 8 The following example will illustrate the above rule. Example, Find the square root of 55,225. Let the odd and even places be denoted by o and e, respectively. The various steps are then as shown below : 235 e o e o line of square root 5 5 2 2 5 Subtract square 4 Divide by twice the root 4) l 5 (3 1 2 3 2 1. In dividing, the quotient should be taken as great as will allow the subtraction of its square from the next odd place. 2. Cf. GSS, ii. 36 ; PG, Rule 25-26 ; GT, p. 9, vs. 23 ; MSi, xv. 6 (c~d)-l ; Si$e, xiii. 5 ; L (ASS), p. 21, Rule 22 ; GK, I, p. 7, lines 2-9, Verse 5 ] CUBE ROOT 37 3 2 Subtract square of quotient 9 Divide by twice the root 46) 2 3 2 (5 2 3 2 5 2 5 Subtract square of quotient The process ends. The square root is 235. The remainder being zero, the square root is exact. G.R. Kaye's statement that Aryabhat-a l's method is algebraic in character and that it resembles the method given by Theon of Alexandria, are, as noted by W.E. Clark, B, Datta and A.N. Singh, 1 incorrect. CUBE ROOT 5. (Having subtracted the greatest possible cube from the last cube place and then having written down the cube root of the number subtracted in the line of the cube root), divide the second non-cube place (standing on the right of the last cube place) by thrice the square of the cube root (already obtained) ; (then) subtract from the first non.cube place (standing on the right of the second non-cube place) the square of the quotient multi- plied by thrice the previous (cube root) ; and (then subtract) the cube (of the quotient) from the cube place (standing on the right of the first non-cube place) (and write down the quotient on the right of the previous cube root in the line of the cube root, and treat this as the new cube root. Repeat the process if there are still digits on the right). 2 1 See Datta and Singh, History of Hindu mathematics, Part I, p. 171. For details see A.N. Singh, BCMS, 18 (1927). See also W. E. Clark, Aryabhatiya, pp. 23 f. 2. Cf. BrSpSi, xii. 7 ; GSS, ii. 53-54 ; PG t Rule 29-31 ; MSi, xv. 9-10 (a-b) ; GT, p. 13, lines 18-25 ; Site, xiii. 6-7 ; L (ASS), Rule 28-29, pp. 27-28 ; GK t I, pp. 8-9, vv. 24-25. 38 GAljlTA SECTION [ Ga^ita Sn. Beginning from the units place, the notational places are called cube place, first non-cube place, second non-cube place, cube place, first non-cube place, second non-cube place, cube place, and so on. Indicating the cube, first non-cube and second non-cube places by c, n and their positions may be shown as below : cnncnncn' ncn'nc 00 00 00 000000 The following solved example will explain the rule stated in the above stanza : Example, Find the cube root of 17,71,561 c n' n c ri n c 121 1 7 7 1 5 6 1 line of cube root Subtract I s 1 Divide by 3.1 a 3)~0~7~(2 J 1 7 Subtract 3.1.2 2 1 2 5 1 Subtract 2 1 a Divide by 3.12 2 432)T~3~~5H J 3 2 ~ 3 6 Subtract 3. 12.1 2 3 6 1 Subtract l 3 1 The process ends. The required cube root is 121. The remainder being zero, the root is exact. AREA OF A TRIANGLE 6. (a-b) The product of the perpendicular (dropped from the vertex on the base) and half the base gives the measure of the area of a triangle. Verse 6 ] VOLUME OF RIGHT PYRAMIDS 39 The term samadalakotJ means 'the perpendicular dropped from the vertex on the base of a triangle' , i.e., 'the altitude of a triangle'. BhSskara I criticises those who interpret it as meaning 'the upright which bisects the triangle into two equal parts', for, in that case, the above rule will be applicable only to equilateral and isosceles triangles. The word phala'sanra means, according to BhSskara I, phala- pramBna, i.e. t 'the measure or amount of the area'. The above rule is applicable when the base and the altitude of a triangle are known. When the three sides of a triangle are given but the altitude is not known, Bhaskara I gives the following formulae to get the segments of the base (called Qbadhn or abadhantara) and the altitude : A (3) p=*/c*-x* or s/&-y* It is remarkable that Bhaskara I does not mention the formula : Area of a triangle = y/s {s-a){s— b) (s -c) t 2s=a-\-b-\-c t although his contemporary Brahmagupta states it in his BrShma-sphuta- siddh&nta} VOLUME OF RIGHT PYRAMIDS ^W^ 1 ^ 2 3 W ISfaftft II $ II 6. (c-d) Half the product of that area (of the triangular base) and the height is the volume of a six-edged solid. 1. See BrSpSi, xii. 21. 2. A.B.F. 3*Tf$: 40 Capita section t Ga*it*Sa. This rule, which is based on speculation on the analogy of the area of a triangle, is inaccurate. The correct formula is found to occur in the Brahma-sphuta-siddhanta of Brahmagupta where it is stated as follows : “The volume of a uniform excavation divided by three is the volume of the needle-shaped solid.” 1 That is to say, Volume of a cone or pyramid = J (area of base) X (height). Bhaskara I seems to be unaware of this formula, for he has no comment to make on the rule of Aryabha^a I. Even the commentators SomeSvara and Suryadeva (b. A.D. 1191) have nothing to add. AREA OF A CIRCLE 7. (a-b) Half of the circumference, multiplied by the semi, diameter certainly gives the area of a circle. That is, area of a circle = £x circumference X radius. The same result in the form circumference X diameter area of a circle = 4 occurs earlier in the Tattvaithadhigama-sntra-bhusya* of UmSsvSti (1st century A.D.). It occurs in the Brhat-k$etra-samasa z of Jinabhadra Gani (A.D. 609) also. VOLUME OF A SPHERE 7. (c.d) That area (of the diametral section) multiplied by its own square root gives the exact volume of a sphere. 1. See BrSpSU xii. 44. 2. Comm. iii. 11. VclM'7 ] VOLUME OF A SPHERE 41 That is, if r be the radius of a sphere, then, according to Aryabhata I : Volume of a sphere = irr*<jTr^, This formula is based on speculation, and, as noted by Bhaskara I, is inaccurate, although called exact by Aryabhata f. The probable rationale of Aryabhata I's formula is as follows : The area of a circle of radius r = area of a square of side v/^. i^de vs. 9 a- b) On the analogy of this, Aryabhata I concludes that Volume of a sphere of radius r = volume of a cube of edge = irr 2 X Vtt?- Bhaskara I quotes the following formula from some earlier work, but he does not give it any credit and regards it as inferior to that given by Aryabhata I : Volume of a sphere of radius r = f r 3 . It is noteworthy that Bhaskara Fs contemporary Brahmagupta, who has criticised Aryabhata I even for his minutest errors, has not been able to make any improvement on Aryabhata's formula for the volume of a sphere. Still more noteworthy is the fact that mathe- maticians and astronomers in northern India, too, regarded Aryabhata I's formula as accurate and went on using it even in the second half of the ninth century A.D. Brahmagupta's commentator Prthudaka who wrote his commentary on the Brahmasphuta-siddhanta in 860 A.D. at Kannauj, has prescribed 1 Aryabhata I's rule for finding the volume of a sphere. The formulae given by other Indian mathematicians are : (1) Mahavlra's (850 A.D.) formula : a Volume of a sphere = jX T 5 r 8 . 1. In his comm. on BrSpSi, xi. 20. 2. See GSS, viii. 28£. A. Bh. 6 42 GA2SUTA SECTION [ Ga&ita Sn. (2) Sridhara's (c. 900 A.D.) formula r 1 Volume of a sphere =4(1 -I-1/18)/- 3 . The same formula is given by Aryabhafa II (c. 950 A.D.) a and Sripati (1039 A.D.). 8 All these formulae are approximate. The accurate formula was given by Bhaskara II (1150 A.D.). (3) Bhaskara IPs (1150 A.D.) accurate formula : 4 Volume of a sphere = | x surface X diameter. Bhaskara II also gave the following approximate formula, using 77=22/7 ; Volume of a sphere = J d ™ ete ^ (1 _j_ 1 12 \) or 4 (1-f- 1/21) r 3 approx. 5 AREA OF A TRAPEZIUM 8. (Severally) multiply the base and the face (of the trapezium) by the height, and divide (each product) by the sum of the base and the face : the results are the lengths of the perpendiculars on the base and the face (from the point of intersection of the diagonals). The results obtained by multiplying half the sum of the base and the face by the height is to be known as the area (of the trapezium). 1. See TrU, p. 39, Rule 56. 2. See MSi, xv. 108. 3. See Si$e, xiii. 46. 4. See L (Anandasrama), Rule 201 (c-d), p. 201. 5. See L (Anandasrama), Rule 203 (*?-/), p. 203. Verse 9 ] AREA OF PLANE FIGURES 43 Let a, b be the base and the face, p the height and c, d the lengths of the perpendiculars on the base and the face from the point where the diagonals intersect. Then b Fig. 2 area— J (a-\-b) p. The term ayama, meaning 'breadth', denotes the height of the trapezium. The term vistara, meaning 'length', denotes the base and face of the trapezium and so vistara yogClrdha means 'half the sum of the base and the face'. The term pnr'sve means, here, the two sides of a trapezium lying on the two sides of the height. Evidently, they are the base and the face. AREA OF PLANE FIGURES 9. (a-b) In the case of all the plane figures, one should determine the adjacent sides (of the rectangle into which that figure can be transformed) and find the area by taking their product. According to Bhaskara I, this rule is meant both for finding the area and for verifying the area of a plane figure. Writes he : Doubt : “Now, the word all means 'everything without exception'; so, here, all (plane) figures are included. The area of all (plane) figures being thus determined by this rule, the statement of tfce previously staled rules teccmes useless, 44 GASjUTA SECTION t Gtnita Sn. Answer : That is not useless. Both the verification and the calculation of the areas are taught by this rule. The areas of the previously stated figures have to be verified. The mathematiciam Maskarl, Purana and PQtana etc., prescribe the verification of all (plane) figures (by transforming them) into a rectangular figure. So has it been said : 'Having determined the area in accordance with the prescribed rule, verification should always be made by (transforming the plane figure into) a rectangle, because it is only of the rectangle that the area is obvious.' 'The determination of the area of the (plane) figures which have not been mentioned above is possible only by transforming them into rectangles.' The commentator Someivara, following Bhaskara I, is of the opinion that the above rule is meant for the verification of the plane figures. According to the commentators SOryadeva (b. 1191 A.D.), Paramesvara (1431 A.D.), Yallaya (1480 A.D,), and Raghunatha-raja (1597 A.D.), this rule is meant for finding the area of all plane figures including those already considered above. According to the commentator Nllakan^ha (c. 1500 A.D), however, this rule is meant only for finding the area of those plane figures that have not been considered heretofore. There is, however, no doubt that the above rule is based on the assumption that all plane figures can be transformed into a rectangle. In his commentary, Bhaskara I has shown how to find the area of a triangle, a quadrilateral, a drum-shaped figure, and a figure resembling the tusk of an elephant, by transforming them into rectangles. CHORD OF ONE-SIXTH CIRCLE 9. (c-d) The chord of one.sixtb of the circumference (of a circle) is equal to the radius. 1 h C/. ra, iv. 2 V*tM II ] COMPUTATION OF RSINE-TABLE 45 That is, chord 6G°=R, or Rsin30 p = R/2. ClKCUMFERENCE-DIAMETER RATIO ■qpftsR ^i^g^ srctewrr i 10. 100 plus 4, multiplied by 8, and added to 62,000 : this is the nearly approximate measure of the circumference of a circle whose diameter is 20,000. This gives circumference 62832 7r — -— =3 1416. diameter 20000 This value does not occur in any earlier work on mathematics, and forms an important contribution of Aryabhat,a I. It is noteworthy that Aryabhaja I has called the above value approximate. COMPUTATION OF RSIMETABLE GEOMETRICALLY airenniqrafft 3 ft*?*mir ^srft II ?? 11 11. Divide a quadrant of the circumferenc of a circle (into as many parts as desired). Then, from (right) triangles and quadri. laterals, one can find as many Rsines of equal arcs as one likes, for any given radius. Following BhSskara I, we explain the method implied in the above stanza by solving three examples. Example 1. Find six Rsines at intervals of 15° in a circle of radius 3438'. Let Fig. 3 represent a circle of radius R(=3438'). Divide its circumference into twelve equal parts by the points A, B, C, D, E, F, …,L. Join BL. This is equal to R and denotes chord 60°. Half of this, i.e., MB, is Rsin 30°, Thus Rsm 3Q°=R/2^1719\ This is the second Rsine. 46 GAljUTA SECTION [ Ganita Sn. A Fig 3 J D G Now, in the right-angled triangle OMB, OM= VR 2 -(R/2) a =^ R=2978\ This is the fourth Rsine, viz., Rsin 60°. Now, in the right-angled triangle AMB, AM=Rvers 30° and MB = Rsin 30°. /. AB = V / (Rsin 30°) a -j-(Rvers 30°)* This is chord 30°. Half of this (/.*., AN) is Rsin 15°. Thus Rsin 15°= i v^Rsin 30°) 2 + (Rvers 30°)» =890'. This is the first Rsine. Now, in the triangle ANO, AN=Rsin 15° and OA=R. .-.ON= VR^ fRsin 15°) a = 3321'. This is the fifth Rsine, i.e., Rsin 75°. Since this is the fifth Rsine, i.e., an odd Rsine, it would not yield any further Rsine. Thus, five Rsines have been obtained by using triangles. Now, we shall make use of the semi-square AOD, whose sides OA and OD are each equal to R. Therefore AD=V2 R. This is chor4 90°, Half V«tse 11 3 COMPUTATION OF RSINE-TABLE 4? of this (i.e., AP) is Rsin 45\ Thus, Rsin 45°=R/ v / 2 = 2431'. This is the third Rsine. Thus we get all the six Rsines, which are as follows : Analysis. Verse 9 (c-d) gives the second Rsine. This yields the first and the fourth Rsines. The first Rsine yields the fifth Rsine. The fourth and the fifth Rsines do not yield any other Rsines. This process ends here. Again, the radius is the sixth Rsine. it yields the third Rsine. The third Rsine being odd, does not yield and further Rsine. So this process also ends. Thus, from the second and the sixth Rsines, one gets all the six desired Rsines. Example 2. Find twelve Rsines at intervals of T 30' in the circle of radius R (=3438'). 9 Fig. 4 represents a circle of radius R (3438'). Join LB, as before. This is equal to R and denotes chord 60°. Half of this is Rsin 30*. Thus, Rsin 15° =890' ; Rsin 30° = 1719' ; Rsin 45° =2431' ; Rsin 60°=2978' ; Rsin 75° =3321' ; Rsin 90° = 3438'. r Fig. 4 ID Rsin 30° = R/2 = 1719'. This is the fourth Rsine. 4S G ANITA SECTION [ Ganit* Stt. Now, from the right-angled triangle OMB, as before, OM= V R a -(R/2) 2 =^-R— 2978'. This is Rsin 60°, / e., the eighth Rsine. Now, from the right-angled triangle AMB, AB= V(Rsin 30°)H(Rvers 30°) 8 = V / (1719')H(460') 1! = 1780'. This is chord 30°. Half of this, i.e., AN, is Rsin 15°. Thus, Rsin 15°= 890'. This is the second Rsine. Now from the right-angled triangle ANO ON= VCAO) 3 - (AN) 2 = y/ R 2 -(Rsn 15 ) 2 =3321'. Thisis Rsin 75°, le. t the tenth Rsine Now, from the right-angled triangle ANR, where R is the mid- point of the arc AB, we have AR=v/(AN) 2 + (NR) 2 = vXRsin 15°) a + (Rvers 15*) a - -s/(890') 2 -|-(117') 3 = 898'. This is chord 15°. Half of this (i.e., AS) is Rsin V 30', Thus, Rsin 7° 30'=449\ This is the first Rsine. Now, from the right-angled triangle ASO, OS = VR 2 -(Rsin 7°~30 7 ) 2 = 34 09'. This is Rsin (82° 30'), i.e., the eleventh Rsine. Now, Rvers 75°= R — Rsin 15°, so that chord 75°=v'(Rsin 75°) 2 + (Rvers 75°) a =4186'. Half of this is Rsin 37° 30\ This is the fifth Rsine. Now, Rsin 52° 30' = VR 2 -(Rsin 37° 30') 2 =2728'. This is the seventh Rsine, Thus, seven Rsines have been obtained by using triangles. Veil* 11 ] COMPUTATION OF R SINE -TABLE 49 Now, we make use of the semisquare AOD as before. Its side OA and OD are each equal to R. Therefore, AD = > /2 R=4862\ This is chord 90°. Half of this, /.*., AP, is Rsin 45°. Thus, Rsin 45° =2431'. This is the sixth Rsine. Now, from the right-angled triangle APT, AT= v^Rsin 45°) a +(Rvers 45T=2630'. This is chord 45°. Half of this is Rsin 22° 30'. This is the third Rsine. Hence, as before, Rsin 67° 30'= V R*- (Rsin 22° 30') a =3177'. This is the ninth Rsine. Thus, we get all the twelve Rsines, which might be set out as follows : Rsin 7° 30'=449' Rsin 37* 30'=2093' Rsin 67* 30'=3177' Rsin 15°= 890' Rsin 45° =2431' Rsin 75°= 3321' Rsin 22° 30' = 131 5' Rsin 52° 30' =2728' Rsin 82° 30'= 3409' Rsin 30°=1719' Rsin 60° =2978' Rsin 90° = 3438' Analysis. Stanza 9 (c-d) gives the fourth Rsine. This fourth Rsine yields the eighth and the second Rsines. The eighth Rsine does not yield any new Rsine. The second Rsine yields the tenth and the first Rsines. The first Rsine yields the eleventh Rsine, and the tenth Rsine yields the fifth and the seventh Rsines. These Rsines do not yield any new Rsines. So this process ends here. Again, the radius is the twelfth Rsine. This yields the sixth Rsine, and the sixth Rsine yields the third and the ninth Rsines. These do not yield any further Rsines. So the process ends here. Thus, from the fourth and the twelfth Rsines one gets all the twelve desired Rsines. Example 3. Find the twenty four Rsines at the equal intervals of 3° 45' in the circle of radius R ( = 3438'). A. Bb. 7 50 CAPITA SECTION [ Ganita Sn. From stanza 9 (c-d) 8th Rsine=R/2=1719'. This yields : 16th Rsine=VR2^(R72)^== 2978' 4th Rsine = } V~(^m^^^(R^r73(Ff = 890'. The 16th Rsine does not yield and new Rsine. The 4th Rsine yields : 20th Rsine = VR 2 — (890') 2 =3321' 2nd Rsine =449' The 20th Rsine yields : 10th Rsine =2093' The 2nd Rsine yields : 1st Rsine =225' 22nd Rsine = 3409' The 1st Rsine yields : 23rd Rsine = 3431' and the 22nd Rsine yields : 11th Rsine =2267'. The 23rd Rsine does not yield any new Rsine. The 11th Rsine yields : 13th Rsine=2585'. The 13th Rsine does not yield any new Rsine. The 10th Rsine yields : 5th Rsine=1105' 14th Rsine =2728'. The 5th Rsine yields : 19th Rsine =3256'. The 14th Rsine yields : 7th Rsine =1520'. The 7th Rsine yields : 17th Rsine =3084'. The 17th Rsine does not yield any new Rsine. Versa 12 ] DERIVATION OF RSINE-D1FFERENCES 51 Now, we start with : 24th Rsine =3438'. This yields : 12th Rsine =2431'. This 12th Rsine yields : 6th Rsine =1315'. The 6th Rsine yields : 3rd Rsine=671' 18th Rsine=3177'. The 3rd Rsine yields : 21st Rsine =3372' and the 18th Rsine yields : 9th Rsine =1910'. The 9th Rsine yields : 15th Rsine — 2859'. Thus, we get all the twentyfour Rsines. DERIVATION OF RSINE- DIFFERENCES 12. The first Rsine divided by itself and then diminished by the quotient gives the second Rsine-diffcrence. The same first Rsine diminished by the quotients obtained by dividing each of the preceding Rsines by the first Rsine gives the remaining Rsine-differences. Let R lf R 2 , …R a4 denote the twentyfour Rsines and §i (=Ri), 8 a» S 3i , S 24 denote the twentyfour Rsine-differences. Then, according to the above rule, S,_R, . (i) 52 G ANITA SECTION [ Ganlta Sn. The above translation is based on Prabhakara's interpretation of the text. The same interpretation is given by the commentators Somesvara, SHryadeva (b. 1191 A. D.), Yallaya (1480 A. D,) and Raghun2tha-raja (1597 A.D.). It is interesting to note that this interpre- tation is also in agreement with the rule stated in the Sarya-siddhanta (ii. 15-16), as interpreted by the commentator Ranganatha (1603 A.D.), viz.. W-R.+K. - *'+*« + -+*” Datta and Singh, following the commentator Paramesvara (1431 A.D.), have translated the text as follows : “The first Rsine divided by itself and then diminished by the quotient will give the second difference. For computing any other difference, (the sum of ) all the preceding differences is divided by the first Rsine and the quotient is subtracted from the preceding difference. Thus, all the remaining differences (can be calculated).” 1 That is Ri M-8 a -!-…+&„ Ri or K — R- (2) Ri ' J This is also how the commentator SomeSvara seems to have interpreted the text. One can easily see that (1) and (2) are equivalent. The commentator Nltakaniha (c. 1500 A.D.) interprets the text as follows : “The first Rsine divided by itself and then diminished by the quotient gives the second Rsine-difference. To obtain any other 1. History of Hindu mathematics, Part III, (unpublished). Also see A.N. Singh, 'Hindu Trigonometry', Proc, Benaras Math, Soc, vol. 1, N.S„ 1939, p. 88. Vm«* 12 ] DERIVATION OF RSIMI DIFFERENCES S 3 Rsine-differenc«, divide the preceding Rsine by the first Rsine and multiply the quotient by the difference between the first and second Rsine-differences and subtract the resulting product from the preceding Rsine-difference.” That is, R, 8 2 -Ri- Ri *„ + i=»»-(RJRi)(*i-*”)- ) (3) This is the accurate form of the formula, and reduces to the previous form because, according to Aryabhata I, §!— S 2 = 225-224=1. The following is the trigonometrical rationale of (3) : S n _S B+1 ={Rsin nh— Rsin (n-l)h}— {Rsin (« + l)/i-Rsin nh}, where /i=225' =s 2 Rsin nh— {Rsin (n+l)A+Rsin (n— \)h\ 2 Rsin nh. Rcos h = 2 Rsin nh — = 2 Rsin nh. = Rsin nh. R (R — Rcos h) R because Rsin h = (R»/Ri)(*i-«i)t The geometrical rationale as given by the commentator Nilakantha (c. A.D. 1500) is as follows : Let AOB be a quadrant of a circle, OA being horizontal and OB vertical. Let the arc AQ be equal to nh, where A=225' ; and let the arcs PQ and QR be each equal to h. Let L and M be the middle points of the arcs PQ and QR, so that the are LM is also equal to h. 54 GASTITA SECTION [ Ganita Sn. Let LU, QV, MW >nd RX be the perpendiculars on OA; PE, LF and QD perpendiculars on QV, MW and RX, respectively. Also let PQ, LM and QR be joined by straight lines, B Fig. 5 O X W V u Now, the triangles QEP and OUL are similar. Therefore QJE = (PQ/OL). OU. Similarly, RD — (QR/OM). OW = (PQ/OL). OW. Therefore, by subtraction, QE-RD=(PQ/OL)(OU— OW)= (PQ/OL). WU. (4) Again, since the triangles MFL and OVQ are similar, FL- (QV/OQ). LM, or WU=(PQ/OL). QV. (5) From (4) and (5), QE— RD= (PQ/OL) 8 . QV. In other words, K S B+1 ={(2 Rsin */2)/R} a . R n (6) In particular, S!-8 2 = {(2 Rsin hj2)lR}\ R, ( 7 ) From (6) and (7), Sn~V*=(R,/R 1 )(S } -S3). Verse 13 ] THE SHADOW-SPHERE 55 CONSTRUCTION OF CIRCLE, ETC., AND TESTING OF LEVEL AND VERTICAL1TY Itt m*i fam ^ ^ tRfcn^-i 13. A circle should be constructed by means of a pair of compasses ; a triangle and a quadrilateral by means of the two hypotenuses (karzia). The level of ground should be tested by means of water; and verticality by means of a plumb. 1 The two hypotenuses (karnas) in the case of a triangle are the two lateral sides above the base ; in the case of a rectangle, the two diagonals ; and in the case of a trapezium, the two lateral sides. 2 The reference is to the usual methods of constructing a triangle when the three sides {i.e., the base and the two lateral sides) are given ; a parallelogram, when one side and two diagonals called the hypotenuses are given ; and a trapezium, when the base, height and the two lateral sides (called hypotenuses) are given. As regards testing the level of the ground, Bha'skara I observes : “When there is no wind, place a jar (full) of water upon a tripod on the ground which has been made plane by means of eye or thread, and bore a (fine) hole (at the bottom of the jar) so that water may have continuous flow. Where the water falling on the ground spreads in a circle, there the ground is in perfect level ; where the water accumulates after departing from the circle of water, there it is low ; and where the water does not reach, there it is high.” 1. The same rule occurs in BrSpSi, xxii. 7 ; &iDVr, II, viii. 2 (c-d). Also see SnSi, iii. 1 ; Si$i, I, iii. 8. 2. See examples set by Bhaskara I and Snryadeva in their commentaries on A, ii. 6 (a-d). 56 G ANITA SECTION [ Ganita Sii. RADIUS OF THE SHADOW-SPHERE 14. Add the square of the height of the gnomon to the square of its shadow. The square root of that sum is the semi. diameter of the circle of shadow. 1 “The semi-diameter of the circle of shadow is taken here”, says the commentator Bhaskara I, “in order to accomplish the rule of three, viz. 'If these are the values of the gnomon and the shadow corresponding to the radius of the circle of shadow, what will correspond to the radius of the celestial sphere ?' Thus are obtained the Rsines of the Sun's altitude and zenith distance. At an equiuox, these are called the Rsines of colatitude and latitude (respectively).” Cf. LBh, iii. 2-3. As regards the shape of a gnomon, Bhaskara I informs us that the Hindu astronomers differed from one another. Some took a gnomon with one third at the bottom of the shape of a right prism on a square base, one third in the middle of the shape of a cylinder, and one third at the top of the shape of a cone. Others took a gaomon of the shape of a right prism on a square base. The followers of Aryabhafa T, writes Bhaskara I, preferred a cylindrical gnomon, made of excellent timber, free from holes, knots and scars, with targe diameter and height. In order to get a prominent tip of the shadow, a cylindrical needle (of height greater than the radius of the gnomon) made of timber or iron was fixed vertically at the top of the gnomon in the middle. Such a gnomon being large and massive was unaffected by the wind ; being cylindrical, it was easy to manufacture ; being surmounted by a needle of small diameter, the tip of the shadow was easily perceived. A gnomon was generally divided into 12 equal parts called aiigulas, but, according to Bhaskara I, there was no such hard and fast rule. A gaomon could bs of any length with any number of divisions. 1. Cf KK, I, iii. 10 ; MBh, iii. 4. Verses 15-16 ] GNOMONlC SHADOW 57 GNOMONlC SHADOW DUE TO A LAMP- POST 15. Multiply the distance between the gnomon and the lamp-post (the latter being regarded as base) by the height of the gnomon and divide (the product) by the difference between (the heights of) the lamp. post (base) and the gnomon. The quotient (thus obtained) should be known as the length of the shadow measured from the foot of the gnomon. 1 In Fig. 6, let AB be the lamp-post, CD the gnomon, and E the point where AC and BD produced meet. Then DE is the shadow cast by the gnomon due to light from the lamp at A. Let FC be parallel to BD. Then comparing the similar triangles CDE and AFC, we have A F Fig. 6 DE = FCXCD AF _ gBDxCD Hence the rule. AB — CD TtP OF THE GNOMONlC SHADOW FROM THE LAMP-POST AND HEIGHT OF THE LATTER 1. This rule occurs also in BrSpSi, xii. 53 ; GSS, ix. 40£ ; SiSe, xiii. 54 ; L (Anandairama), Rule 234, p. 243 ; GK, II, p. 208, Rule 14 (a-b). 2. Bh. TTrfof ; others TTTfaffT 3. A-G. Gh. ^tfe: 4. B. Tr. m ^tit A.Bh. 8 53 GANITA section [ Ganita So. 16. (When there are two gnomons of equal height in the same direction from the lamp-post), multiply the distance between the tips of the shadows (of the two gnomons) by the (larger or shorter) shadow and divide by the larger shadow diminished by the shorter one : the result is the upright (i.e., the distance of the tip of the larger or shorter shadow from the foot of the lamp- post). The upright multiplied by the height of the gnomon and divided by the (larger or shorter) shadow gives the base {i.e., the height of ithe lamp.post). 1 In Fig. 7, AB is the lamp-post (base), BC or BD is the upright, LM and PQ are the gnomons of equal height A Fig. 7 We have AB PQ BD (0 QD Since PQ==LM, therefore BD BC j\B LM~ BC MC CD QD MC QD— MC Hence from (iii), (i) and (ii), we have BD= CDX -Q D QD-MC (iii) BC= CDxMC QD-MC and AB _ BDxPQ BCxLM QD MC (1) (2) (3) 1. This rule reappears in BrSpSi, xii. 54 ; L (ASS), Rule 239, pp. 246-47 ; GK, vol. 2, p. 210, Rule 16. Verse 17 ] TWO THEOREMS 59 THEOREMS ON SQUARE OF HYPOTENUSE AND ON SQUARE OF HALF-CHORD 17 (In a right-angled triangle) the square of the base pins the square of the upright is the square of the hypotenuse. In a circle (when a chord divides it Into two arcs), the product of the arrows of toe two arcs is certainly equal to the square of half the chord Of the two theorems stated above, the first one is “the theorem of the Square of the Hypotenuse”, as Hankel has called it. This theorem has been known in India since very early times. Baudhayana (r. 800 B.C.), the author of the Baudhayana-sulba-stttra, has enunciated it thus : “The diagonal of a rectangle produces both (areas) which its length and breadth produce separately.” 2 This theorem is now universally associated with the name of the Greek Pythagoras (c. 540 B.C.), though “no really trustworthy evidence exists that it was actually discovered by him.” It were certainly the Hindus who enunciated the property of the right-angled triangle in its most general form. No other ancient nation is known to have made any attempt in this direction. The second theorem, states that if, in a circle, a chord CD and a diameter AB intersect each other at right-angles at E, then 1. Pr. ^STrerfVcT: tftzfcm ^ 2. Baudhayana-sulba-sUtra, i. 48. 3. Heath, Greek Mathematics, I, p. 144 f, 60 CAPITA SECTION [ Gagrita Sn. This result easily follows from comparison of the similar triangles CAE and CEB. This second theorem occurs earlier in the works of UmSsvati 1 (1st century A.D.). It has been mentioned by Jinabhadra Gani a (A.D. 609) and Brahmagupta (A.D. 628) also. 3 ARROWS OF INTERCEPTED ARCS OF INTERSECTING CIRCLES 18. (When one circle intersects another circle) multiply the diameters of the two circles each diminished by the erosion, by the erosion and divide (each result) by the sum of the diameters of the itwo circles after each has been diminished by the erosion : then are obtained the arrows of the arcs (of the two circles) intercepted in each other. 6 Let two circles intersect at P and Q and let ABCDE be the line passing through the centres of the two circles. Then BD is the erosion (gram), and BC, CD are the arrows of the intercepted arcs. 1. See Tatmrthadhigama-sutra, ' iii. 11 (comm.) md^JambU- dnpa-samSsa, ch. iv. 2. See Brhat-k^etra-samdsa, I 36. 3. See BrSpSi, xii. 41. 4. Gh. 5to 5. A. D. Pa. *r>1?T^ ; E. sta^ST 6. This rule occurs also in BrSpSi, m 42; GSS, vii, 23 H ; GK. vol. 2, p. 76 (Rule 68). Verse 19 1 SERIES ™ A ' ? ' ” (BE— BD). BD (2) and CD (AD—-BD) + (B&–BD) ”' These formulae may be derived as follows : Since AC X CD = BC X CE , therefore (AD— BD + BCXBD— BC) = BC(BE-BC) (AD— CD)CD = (BD— CD)(BE— BD + CD) Solving (3) for BC, we get (1), and solving (4) for CD, we get (2), (3) (4) SUM (OR PARTIAL SUM) OF A SERIES IN A.P. 19 Diminish the given number of terms by one then divide by two, then increase by the number of the precedmg terms (if any) then multiply by the common difference, an then the first term of the (whole) series : the result .. he arithmetic mean (of the given number of terms). ™» plied by the given number of terms is the sum of the given ^Alteritl^ly.-ol.iplytfce— of the first and las terms (of the series or partial series which is to be summed up) by half the number of terms. 3 Let an arithmetic series be a+0»+<0-Ka+2</)+ Then the rule says that (1) the arithmetic mean of the n terms (a+pd)M”+P+ V '>) + - +{«+0+”- 1 >''> =«+(-?=±+/> ) * 1. Bh. WW 2. C. 3 Cf. BrSpSi, xii. 17 ; GSS, vi. 290 j PC, Rule 85 ; MSi, >v. 47 ;SiSe. xiii. 20 ; I (ASS), Rule 121, P- U4. 62 GANIXA SECTION [ Ganita Sn. (2) the sum of the n terms In particular (when p~ 0) (3) the arithmetic mean of the series a + (a + d ) + . . . + { a + (» - 1 ) d} — « + d 2 (4) the sum of the series «+(a+rf)+…+{fl-f-(,i-l)4 Alternatively, the sum of n terms of an arithmetic series with A as the first term and L as the last term where i (A+L) is the arithmetic mean of the terms. The commentator Bhgskara I says : “Several formulae are severally set out here. They are obtained by suitable combination of the text as follows : Formula i. “Istam vyekam dalitam uttaragurtam samukham” iti madhya- dhanHnayanartham sutram. i.e., a-f-^i d is the formula for the arithmetic mean of n terms. Formula 2. “Madhyam i§tagunitanT iti istadhanam. i.e., ja -f ILj d^n gives the sum of n terms. Formula 3. “Istam vyekam sapurvam uttaragurtam samukham'* ityantyo- pantyadidhananayanartkam sutram. ie., l)}d is the formula for the *th tefm. V«f* 20 ] SERIES J# A. P. 63 Formula 4, “Iftam vyekam dalltafn sapUrvam uttaragunam samukham itfagunitam” ityavantarayathe^tapadasamkhyanayomrtham sBtram. i.e., nja + ^~2~~ p ) ^ } * s ^ e ^ ormu ^ a ^ or ^ e surn °f n terms beginning with the (/> + l)th term. Formula 5. “Adyantapadardhahatam” iti istadhanam. i.e., \n (A-f-L) is the sum of n terms, A and L being the first and last terms. Series in A.P. are found to occur in the Taittiriya Sdmhita (vii. 2. 12-17 ; iv. 3. 10), the Vajasaneya Samhita (xvii. 24. 25), the Pancavimsa Brahmana (xviii. 3) and other Vedic works. In the Brhaddevata* (500-400 B.C.), we have the result 2+3 +4+ +1000= 500499. Formal rules for finding the sum etc. of a series in A.P. occur in the Bakshali Manuscript (c. 200 A.D.) and other Indian works on mathematics written subsequently. NUMBER OF TERMS OF A SERIES IN A.P. 20. The number of terms (is obtained as follows) : Multiply (the sum of the series) by eight and by the common difference, increase that by the square of the difference between twice the first term and the common difference, and then take the square root ; then subtract twice the first term, then divide by the common difference, then add one (to the quotient), and then divide by two. 2 Let S be the sum of the series a-f- (« + </)+(a+2<0+ (a +3J) + to « terms. 1. The Brhaddevata has been edited in original Sanskrit with fcnghsh translation by Macdonell, Harvard, 1904. v ™ 2 ” Cf- BrSpSi,. -x\i.l%\ GSS, vi. 294 ; PG, Rule 87 ; MSi, *v. 50 ; Si$e, xiii. 24 ; L (ASS), Rule 128, p. 118. 64 GAlsriTA SECTION t Ganita Sn. Then V8^S + (2a^)2 —2a SUM OF THE SERIES l + (l+2) + (l+2+3)+ TO N TE&MS 21. Of the series {upaciti) which has one for the first term and one for the common difference, take three terms in continuation, of which the first is equal to the given number of terms, and find their continued product. That (product), or the number of terms plus one subtracted from the cube of that, divided by 6, gives the citighana. 1 The term upaciti or citi is used in the sense of a series in general. The series 1+2+3+ + n, which has one for the first term and one for the common difference is called ekottaradi-upaciti. The sum of this series is generally called sahkalita. Bhaskara X calls it sahkalanS. The term citighana is used in the sense of the sum of the series l + (l+2)+(l+2+3)+ (1) to any number of terms. This sum is generally called sahkalita-sahkalita. BhSskara I has called it sahkalana-sahkalanU. The above rule gives the sum to n terms of the series (I) in two forms : B («+1)(«+2) _ md („ + l)»-(„+l) 6 6 The term citighana literally means 'the solid contents of a pile (of balls) in the shape of a pyramid on a triangular base'. The pyramid is so constructed that there is 1 ball in the topmost layer, 1+2 balls in the next lower layer, 1 -[-2 4-3 balls in the further next lower layer, and so on. In the «th layer, which forms the base, there are l + 2+-3+…+« balls. 1. Cf. BrSpSi, xii. 19 ; PG, Rule 103 (c-d) ; Si&e, xiii, 21 ; L (ASS), Rule 118, p. 112. Ve«e 22 ] SERIES JgN* AND SN* 65 The number of balls in the solid pyramid, i.e., c*7zgfo™i=S 1 +S !s +… + S f .+….+S n , where S,= 1+2+3+ … +r. The base of the pyramid is called upaciti, so upaciti~\ +2 +3 + . , . +«. Bhaskara I illustrates the rule by the following example : Example. There are (three pyramidal) piles (of balls) having respectively 5, 8 and 14 layers which are triangular. Tell me the number of units (i.e., balls) (in each of them). The above citighana is a series of figurate numbers. The Hindus are known to have obtained the formula for the sum of the series of natural numbers as early as the fifth century B.C. It cannot be said with any certainty whether the Hindus in those times used the represen- tation of the sum by triangles or not. The subject of piles of shots and other things has been given great importance in the Hindu works, of which all contain a section dealing with citi ('piles'). It will not be a matter of surprise if the geometrical respresentation of figurate numbers is traced to Jlindu sources. SUM OF THE SERIES £N a AND 2N 3 22. The continued product of the three quantities, viz., the number of terms plus one, the same increased by the number of terms, and the number of terms, when divided by 6 gives the sum of the series of squares of natural numbers (vargaciiighana). The square of the sum of the series of natural numbers (citi) gives the sum of the series of cubes of natural numbers (ghanacitigh'ana). 1 1. Cf. BrSpSi, xii. 20 ; PG, Rule 102-3 (a-6), Si$e, xiii. 22 ; L(ASS), Rule 119, p. 113. A.Bb. 9 66 CAPITA SECTION t Sfr. The term vargacltighana is used in the sense of the sum of the series l a +2 a +3»+ … + n\ U., the sum of the series of squares of natural numbers ; and the term ghanacitighana is used in the sense of the sum of the series l'+2 3 +3 3 + … + »■, ; i.e., the sum of the series of cubes of natural numbers. Bhaskara I has i called these sums by the terms vargasahkalanZ and ghanasankalanH, respectively. Other mathematicians have called them vargasahkalita and ghanasahkalita, respectively. According to the above rule 1 , +2 . + 3« + „. +n ' = ^l ”+ 1 f ”+') , and l»+2 3 +3° + … +n'=(l + 2+3+ •■• +»)' The term vargacltighana literally means 'the solid contents of a pile (of balls) in the shape of a pyramid on a square base'. It is so constructed that there is 1 ball in the topmost layer, 2 a balls in the next lower layer, 3 a balls in the further next lower layer, and so on. In the nth layer from the top, which forms the base of the pile, there are n a balls. The term ghanacitighana similarly means 'the solid contents of a pile (of cuboidal bricks) in the shape of a pyramid having cuboidal layers'. It is so constructed that there is 1 brick in the topmost layer, 2 s bricks in the next lower layer, 3 3 bricks in the further next lower layer, and so on. In the nth layer (from the top), which forms the base of the pile, there are n 3 bricks, n bricks in each edge of the cuboidal base. Bhaskara I illustrates Aryabhafa's rule stated in the text by the following examples : Example 1. There are (three pyramidal) piles on square bases having 7, 8 and 17 layers which are also squares. Say the number of units therein (i.e., the number of balls or bricks, of unit size used in each of them). V«sm J PROPUCT AND FACTORS # Example 2. There are (three pyramidal) piles having 5, 4 and 9 cuboidal layers. They are constructed of cuboidal bricks (of unit dimensions) with one brick in the topmost layer. (Find the number of bricks used in each of them). PRODUCT OF FACTORS FROM THEIR SUM AND SQUARES f^n? «wrhotpH n 23. From the square of the sum of the two factors subtract the sum of their squares. One-half of that (difference) shouW be known as the product of the two factors. That is, AXB== (A+B)'-(V+B') QUANTITIES FROM THEIR DIFFERENCE AND PRODUCT *rrsw stf- ^ftra^ n w ii 24. Multiply the product by four, then add the square of the difference of the two (quantities), and then take the square root. (Set down this square root in two places). (In one place) increase it by the difference (of the two quantities), and (in the other place) decrease it by the same. The results thus obtained, when divided by two, give the two factors (of the given product). 1 That is, if x — y=a xy —b, then ~2 y 2 1. Cf. BrSpSU xviii. 99. 68 OAUITA SECTION [ Ga^ita Sb. . INTEREST ON PRINCIPAL 25. Multiply the interest on the principal plus the interest on that interest by the time and by the principal ; (then) add the square of half the principal ; (then) take the square root ; (then) subtract half the principal ; and (then) divide by the time : the result is the interest on the principal. 3 The problem envisaged is : A principal P is lent out at a certain rate of interest per month. At the expiry of one month, the interest T which accrues on P in one month is given on loan at the same rate of interest for T months. After T months T amounts to A. The problem is to find T when A is given. The solution to this problem is t _ VPTA+(P/2)'— (P/2) T as stated in the above rule. RULE OF THREE 26. In the rule of three, multiply the 'fruit' (phala) by the 'requisition* (iccha) and divide the resulting product by the 'argument' (pramana). Then is obtained the 'fruit corresponding to the requisition* (icchaphala)* 1. B. crossed out and tf^T substituted. 2. Bh. and So. read cP^T ^TTEffjf qpffi^i S^^IWT ; others read 3. Brahmagupta gives a more general rule. See BrSpSi, xii. 15. 4. Similar rules occur in BrSpSi, xii. 10 ; GSS, v. 2 (i) ; PG, Rule 43 ; MSi, xv 24-25 (a-b); GT, p. 68, vs. 86 ; Si$e, xiii. 14 ; L (ASS), p. 71, vs. 73 ; GK, I, p. 47, vs. 60. Vttu 27 1 RULE OF THR*B 69 Example 1. If A books cost P rupees, what will R books cost ? Here A is the 'argument', P the 'fruit' and R the 'requisition'. So the required answer is rupees. Example 2. If the interest on Rs. 100 for 2 months is Rs. 5, find the interest on Rs. 25 invested for 8 months. Here we have two arguments, viz, Rs. 100 and 2 months ; and two requisitions viz., Rs. 25 and 8 months. The fruit is Rs. 5. So the required answer is 25X8X5 or 5 rupees. 100×2 SIMPLIFICATION OF THE QUOTIENTS OF FRACTIONS 27. (a-b) The numerators and denominators of the multipliers and divisors should be multiplied by one another. For example, (0 -!= a J qd_ c be 1 00 a c ac ~b d_bd__ (ac)(fh) “e v g'~ eg (bd) (egY 7 ~h 7h This rule is a sequel to the previous rule of three, and relates to the case when the argument, fruit and requisition are each fractional. What is meant int his rule is that when the fractional fruit and the fractional requisition have been multiplied and a fractional product is obtained, then the product should be treated as the multiplier and the argument as the divisor. The numerator of the multiplier should then be multi- plied by the denominator of the divisor and the denominator of the GASWA SECTION multiplier by the numerator of the divisor. The commentator Snryadeva explains : “Here by the word gunakura (multiplier) are meant the fruit and the requisition, because, being the multiplicand and the multiplier, both of them are mutually multipliers. By the word bhagahara (divisor) is meant the argument. The (product of the) denominators of the fruit and the requisition should be multiplied by the (numerater of the) argument, and the product of (the numerators of) the fruit and the requisition should be multiplied by the denomi- nator of the argument.” REDUCTION OF TWO FRACTIONS TO A COMMON DENOMINATOR 27. (c-d) Multiply the numerator as also the denominator of each fraction by the denominator of the other fraction ; then the (given) fractions are reduced to a common denominator. 1 iKgri TOtf ^ II ^ u That is, bd bd be ad-\-bc a c ad be ad — be bd b d bd bd Example. Add h £ and J. Reducing £ and J to a common denominator and adding, we get Now adding \ and J, we get 1. Similar rules occur in BrSpSi, xii. 2 (a-b) ; PG, Rule 36; MSi, xv. 13 (c-d) ; GT, p. 30, line 16 ; Si$e, xiii. 11 (a-b) ; L (ASS), p. 28, lines 9 ; GK, I, p. 9, vs. 26 (a-b). Also see GSS, iii. 55 (a-b). EQUATIONS METHOD OF INVERSION 71 q: ^tsq^qtsq^q: ^q^ a frttlR* 8 II 3= II 28. In the method of inversion multipliers become divisors and divisors become multipliers, additive becomes subtractive and subtractive becomes additive. 4 Example. A number is multiplied by 2 ; then increased by I; then divided by 5 ; then multiplied by 3 ; then diminished by 2 ; and then divided by 7 ; the result (thus obtained) is 1. Say what is the initial number. Starting from the last number 1, in the reverse order, inverting the operations, the result is 1 X 7, + 2, -r 3, X 5, —1, -f- 2, /. e. 7. UNKNOWN QUANTITIES FROM SUMS OF ALL BUT ONE *rc$-i »t^sr fated I <pfa |<r *ri n rs. II 29. The sums of all (combinations of) the (unknown) quantities except one (which are given) separately should be added together ; and the sum should be written down separately and divided by the number of (unknown) quantities less one : the quotient thus obtained is certainly the total of all the (unknown) quantities. (This total severally diminished by the given sums gives the various unknown quantities). 6 1. Gh. Go. Ni. Pa. So. ^T^T * 2. Ra. £<T: *T 3. C. tff<rc^ W*ffa fa<T<ft 4. Similar rules occur in BrSpSi, xviii. 14 ; GSS, vi. 286 ; PG, Rule 73 ; MSi, xv. 23 ; GT, p. 65, vs. 83 ; Si$e, xiii. 13 ; L(ASS), p. 42, vs. 48; GK, I, p 46, lines 13-16. 5. Bh. 6. Cf. GSS, vi. 159 ; GK, I, p. 85, Rule 28. 12 GAtflTA SECTION t <Wita Sfi. That is if (tt+xa-f- … +x„)— x t =a x (xi+x 2 + … +*„)— X2=di (Xi+x a + … +x n )— x n =a n then … +*.= , so that n— 1 01 + 02-}- B3 - ' * 2 , n-1 • +«■ „ n-1 xi, x 2t x„ being the unknown quantities and ai, at an the given sums. UNKNOWN QUANTITIES FROM EQUAL SUMS S%$np? top ^fts^ 1 11 ^ 11 30. Divide the difference between the rftpakas with the two persons by the difference between their gulikas. The quotient is the value of one gulika, if the possessions of the two persons are of equal value. 8 1. A. B. E-G. *ref?r tre^m 2. The verse may also be translated as : “The difference of the known amounts (rupaka) relating to . the two persons should be divided by the difference of the coefficients of the unknowns (gulika) : The quotient will be the value of the unknown ; if their possessions be equal.” See B. Datta and A N. Singh, History of Hindu Mathematics, part II, p. 40. For similar rules see BrSpSi, xviii, 43; Si$e, xiv. IS ; BB (ASS), p. 113, Rule 89 ; NBi, II, Rule 5. Vetie 31 ] MEETING OF TWO MOVING BODIES Two persons are equally rich. Of them, one possesses a gulikas and b mpakas (coins), and the other possesses c gulikas and d rupakas. The rule tells how to find the value of one gulika in terms of mpakas. Algebraically, if ax-\-b=*cx~\-d, then d—b x= a — c ' The term gulika stands for 'a thing of unknown value'. “By the term gulika,” writes Bhaskara I (629 A.D.), “is expressed a thing of unknown value.” Gulika and yavattavat (commonly used in Hindu algebra for an unknown) are used as synonyms. Bhsskara I writes : “These very gulikas of unknown value are called yavattavat .” The term rupaka means a coin. “The mpaka”, writes Bhaskara I, “is (a coin such as) dtnSra etc.” MEETING OF TWO MOVING BODIES 31. Divide the distance between the two bodies moving in the opposite directions by the sum of their speeds, and the distance between the two bodies moving in the same direction by the difference of their speeds ; the two quotients will give the time elapsed since the two bodies met or to elapse before they will meet. 3 1. B. F. N. fori* St ; Go. Pa. So. flnft I 2. NI. m€t 3. Problems on meeting of travellers occur in Bakhshali Manuscript and later works. BM, III, Ai 3 , 3 recto, Rule 14 ; B a , 9 verso ; B 4 , 4 recto. ; GSS, vi. 326£ ; PG, Rule 65, Exs. 81-82. A. Bh. 10 GAtflTA SECTION [ Ga*it* Sn. The following cases may arise : Case 1. When the two bodies are moving in the opposite directions. If the bodies are facing each other, i.e., if they have not already met, the distance between them when divided by the sum of their velocities will give the time to elapse before they meet. If the bodies have already met and moving away from each other, the distance between them when divided by the sum of their velocities will give the time elapsed since they met each other. Case 2. When the two bodies are moving in the same direction. If the fast-moving body is behind, i.e., if they have not already met, the distance between them when divided by the difference of their velocities will give the time to elapse before they meet. If the slow-moving body is behind, i.e., if they have already met, the distance between them when divided by the difference of their velocities will give the time elapsed since they met each other. PULVERISER The rule stated in vss. 32-33 below is meant for solving a residual pulveriser (sngra-kutfakara), i.e., a problem of the following type : A number leaves 1 as the remainder when divided by 5, and 2 (as the remainder) when divided by 7. Calculate what that number is. RESIDUAL PULVERISER sr^f^f^^^*ra%*tf5it ^ i 1. A. No colophon ; B. D. sfh” TfoRPTR: ; C. 5% <lfrdm5: Verse* 32-33 ] RESIDUAL PULVERISER 75 32-33. Divide the divisor corresponding to the greater remainder by the divisor corresponding to the smaller remainder. (Discard the quotient). Divide the remainder obtained (and the divisor) by one another (until the number of quotients of the mutual division is even and the final remainder is small enough). Multiply the final remainder by an optional number and to the product obtained add the difference of the remainders (corresponding to the greater and smaller divisors ; then divide this sum by the last divisor of the mutual division. The optional number is to be so chosen that this division is exact* Nov* place the quotients of the mutual division one below the other in a column ; below them write the optional number and underneath it the quotient just obtained. Then reduce the chain of numbers which have been written down one below the other, as follows) : Multiply by the last but one number (in the bottom) the number just above it and then add the number just below it (and then discard the lower number). (Repeat this process until there are only two numbers in the chain). Divide (the upper number) by the divisor corresponding to the smaller remainder, then multiply the remainder obtained by the divisor corresponding to the greater remainder, and then add the greater remainder : the result is the dvicchedagra (i.e., the number answering to the two divisors). (This is also the remainder corresponding to the divisor equal to the product of the two divisors). 1 It may be pointed out that when the quotients of the mutual division are odd in number, the difference of the greater and smaller remainders is subtracted from the product of the last remainder of the mutual division and the optional number. To illustrate the above rule, we solve the following example. Example. Find the number which yields 5 as the remainder when divided by 8, 4 as the remainder when divided by 9, and 1 as the remainder when divided by 7. J. The same rule occurs in BrSpSi, xviii. 3-5, 76 GA1STITA SECTION [ Ga^ita Sn. (0 (“) (iii) Remainder 5 4 1 Divisor 8 9 7 To begin with, we apply the process of the pulveriser on the first two pairs of remainder and divisor, viz., (i) and (ii). Dividing 8 (the divisor corresponding to the greater remainder) by 9 (the divisor corresponding to the smaller remainder), we get 8 as the remainder and as the quotient* We discard the quotient and divide the remainder 8 and the divisor 9 mutually until there are even number of quotients and the final remainder is small : 8)9(1 8 T) 8 (8 8 T We choose 1 as the optional number and multiply the remainder by it and add 1 (the difference of the greater and smaller remainders) to it. The result is 1. Diving this 1 by 1 (the final divisor of the mutual division), the quotient obtained is 1. Now, we write the quotients of the mutual division, viz., 1 and 8 one below the other and below them the optional number 1 and then the quotient 1 just obtained. Thus we get 1 8 1 1 Reducing this chain, we successively get 1 1 10 8 9 9 1 1 1 Vewes 32-33 ] NON-RESIDUAL PULVERISER 77 Now dividing the upper number 10 by 9 (the divisor corresponding to the smaller remainder), we get 1 as the remainder. Multiplying this 1 by 8 (the divisor corresponding to the greater remainder), we get 8, Adding the greater remainder 5 to this 8, we get 13. This 13 is obviously the number which divided by 8 leaves 5 as remainder and divided by 9 leaves 4 as remainder. This number is] called dvicchedagra because this answers to two divisors. This 13 is also the remainder corresponding to the divisor 8×9 (i.e., 72). We now apply the process of the pulveriser on the following pairs of remainder and divisor t (0 (») Remainder 13 1 Divisor 72 7 Proceeding as above, we get 85. This is called triccheddgra because this answers to three divisors (viz., those in the example). One can easily see that 85 leaves 5 as remainder when divided by 8, 4 as remainder when divided by 9, and 1 as remainder when divided by 7. - This is the least integral solution of the problem. The general solution is 8x9x7 ro+85, i.e., 504 m+85, m=0, 1, 2, 3, … The commentators Bhaskara I, Suryadeva and others have also interpreted vss. 32-33 as a rule for solving a non-residual pulveriser (nlragra-kuttnkara), i.e., a problem of the following type : Problem: 11 is multiplied by a certain number, the product is dimi- nished by 3, and the difference thus obtained being divided by 23 is found to be exactly divisible. Find the multiplier and the quotient. NON-RESIDUAL PULVERISER Verses 32-33 may be translated also as follows : 32-33 Divide the greater number (denoting the divisor) by the smaller number (denoting the dividend) (and by the remainder 78 GAljnTA SECTION [ Ge 9 it» S». obtained the smaller number and so on. Dividing the greater and the smaller numbers by the last non-zero remainder of the mutual division, reduce them to their lowest terms. 1 ) Divide the resulting numbers mutually (until the number of quotients of the mutual division is even and the final remainder is small enough). Multiply the final remainder by an optional number and to the product obtained add the (given) additive (or subtract the subtractive). 2 (Divide this sum or difference by the last divisor of the mutual division. The optional number is so chosen that this division is exact. Now place the quotients of the mutual division one below the other in a column ; below them write the optional number and underneath it the quotient just obtained. Then reduce this chain of numbers as follows)* Multiply by the last but one number (in the bottom) the number just above it and then add the number just below it (and then discard the lower number). (Repeat this process until there are only two numbers in the chain). Divide (the upper number by the abraded greater number and the lower number) by the abraded smaller number. (The remainders thus obtained are the required values of the unknown multiplier and quotient). In the above translation, the word agra has been taken to mean 'number' and agrantara to mean 'the given additive or subtractive'. The operations mentioned in adhikagracchedagunam dvicchedigram adhikagrayutam are, as remarked by the commentator Swyadeva, not needed in the case of a non-residual pulveriser. Writes he : “Adhi- kagracchedagunam ityndi niragrakuttakiiresu nopayujyate” This rule might be illustrated as follows : Example. Solve 1. The additive or subtractive should also be reduced by dividing it by the last non-zero remainder. 2. By the additive and subtractive here is 'meant the reduced additive and reduced subtractive, V«f« 38-33 ] NON-RKSI0UAL PULVERISER ™ Here, the divisor=487, dividend = 16 and subtractive=138. Since 487 and 16 are already prime to each other, we proceed with their mutual division. The mutual division runs as follows : 16) 487 (30 480 7) 16 (2 J4 2×76—138= 14 (2 14 The cnain of the quotients oi the mutual division, the optional number and the final quotient is reduced as follows : 30 30 4696 2 154 154 76 76 2 Dividing 4696 by the divisor 487, the remainder is 313 : this is the value of x. Dividing 154 by the dividend 16, the remainder is 10 » this is the value of y. Hence x-313, j = 10. This is the least integral solution of the problem. The general solution is *=487 A+313 y= 16 A+ 10 where A=0, 1, 2, 3 … The commentator Somesvara, instead of interpreting the text in a different way in the case of a non-residual pulveriser, interprets a non-residual pulveriser itself as a residual pulveriser. 1 Thus, he 1. This is really the method of Brahmagupta. See BrSpSi, xviii. 7. 80 GAtflTA SECTION t Ga^iuSa. interprets the non-residual pulveriser ax — c — ft— =* as the residual pulveriser N=*7+c— ax+0, in which c is the adhikagra (i.e., the greater remainder), b is the adhikagrabhagahara (/.«., divisor corresponding to the greater remainder), is the unagra (i.e., the smaller remainder), and a is the (unagrabhagahara, i.e., the divisor corresponding to the smaller remainder). To illustrate this method, we solve the non-residual pulveriser of Example 2, viz. t I6x— 13 8 ^ 487 ~~ y by converting it into a residual pulveriser. This is equivalent to the residual pulveriser N = 16*-r-0=487y+138, where the greater remainder =138, the corresponding divisor 487, smaller remainder=0, and corresponding divisor = 16. To solve this, we proceed as in Example 1. The mutual division runs as follows : 16) 487 (30 480 7) 16 (2 14 2)7(3 6^ 1×2+138 optional number— 2 1 4 (70 1 40 Verses 32-33 ] NON-RESIDUAL PULVERISER 81 We discard the first quotient and write down the other quotients of the mutual division one below the other in a column, and underneath them the optional number and the final quotient ; and then reduce the chain of numbers obtained, as follows : 2 2 154 3 76 76 2 2 70 Dividing 154 by 16 (the divisor corresponding to the smaller remainder), the remainder is 10 ; this 10 multiplied by 487 (the divisor corresponding to the greater remainder) gives 4870 ; this increased by 138 (the greater remainder) gives 5008. This is the value of N (the dvicchedagra) The values of x and y are, therefore, the following : *=J™ =313 16 5008-138_ 10 y ~ 487 The rationale of Aryabhafa I's rule is as follows : Method 1. Since both residual and non-residual pulverisers reduce ultimately to an equation of the form ax+c — by, (1) it is sufficient to start with this equation. Let the mutual division of a and b (b>a) yield a) b {q x The mutual division may be continued to any number of even quotients. For convenience, we have stopped after obtaining the second quotient. A. Bb. 11 82 GANITA SECTION t Ganita Sa. The mutual division shows that the application of pulverisation to equation (1) gives rise to the equation ax-\rc—r x y (2) and then to r^x-^-c—riy. (3) Now if x~m, y=q z be solution of (3), then x—m y^mq t +q 3 is a solution of (2), and y=mq i + q 3 J *=('M?2 + 03)?1 + W J (4) is a solution of (li). 1 1. For, if jc« m, y~q* be a solution of (3), then or (a— nq2)m + c=r\q*. :. c = ri(mq2\qs)-am- Application of this value of c reduces (2) to ax+n (mqz + qa)—am=*ny, or a (x—m)=n[y—(mq2+ qs)], of which a solution is evidently y=*mqt + qz. Application of the same value of c reduces (1) to ax 4- n (mq 2 + q 3 )~am - 67, or + aqi)imqi+qs)-am=by, or <i[x— {(m? 8 4- ? 3 )?i + m}\ = &[»— (w? 2 + qj\, of which a solution evidently is y=mq 2 + q s X=(mq%+q*)q\+m, Verse* 32-33 ] NON-RESlDUAL PULVERISER 83 One can easily see that m is the so called 'optional number' (mati), q t the quotient obtained on dividing wr a + c by r lf and that the solution (4) is the same as obtained by reducing the chain m 03 the successive steps of reduction of the chain being q l q x (.mq 2 +g 3 )qi-^m q z mqi+qi rnq%-\-^ m m Method 2. Proceeding with the equation ax+c=by M and mutually dividing a and b (b>a) up to the second quotient as before, we get a) b fax h) a (q t n Since from the mutual division, b=aq l -\-n the equation (1) becomes ax + c=(aq!+Tjy,. or jcsnfty+Xi. where axj=r,y— c where x=qiy-\-x\. (2) (3) And since from th.e mutual division 84 GAiyiTA SECTION [ Ganiu Sn. the equation (2) becomes (riqt+r i )x 1 +c=r 1 y, or q2X 1 -{-y 1 =y where ly^r^+c. where y=q i x 1 +y 1 . (5) Now if x x = m {mati), y=q 3 be a trial solution of (4), then (5) gives y=qim+q 3 | and (3) gives x=q 1 (q % m-{ q z )-\-m. ^ Hence a solution of equation (1) is given by (6). Note. When the mutual division is carried to more than two quotients, the proof is similar. See Datta and Singh, History of Hindu Mathematics, Part II, pp. 95 ff. CHAPTER HI KALAKRIYA OR THE RECKONING OF TIME [ The aim of this section is to teach theoretical astronomy as far as the determination of true positions of the planets is concerned. ] TIME DIVISIONS AND CIRCULAR DIVISIONS qf^ife^ <ffe^ 3 f^^Tfe^r sn?ft n ? II 1. A year consists of 12 months. A month consists of 30 days. A day consists of 60 nails. A nodi consists of 60 vina&kas (or vina^is).^ 2. A sidereal vinafika is equal to (the time taken by a man in normal condition in pronouncing) 60 long syllables (with moderate flow of voice) or (in taking) 6 respirations {prSnas). This is the division of time. The division of a circle (lit. the ecliptic) proceeds in a similar manner from the revolution. 4 1. So. st^ptts: 2. Bh. forar. 3. so. Pa. qfcroj 4. See supra, i. 6, p. 13, above. 85 86 KAXAKRIYA SECTiON [ Kalakriya Sn. These definitions may be stated in tabular form as follows : Table 11. The Time divisions 1 year =12 months 1 month =30 days 1 day =60 ntidis (or rtSdikSs) 1 nndi (nadika) = 60 vinddikas 1 sidereal vinadiks=60 long syallables or = 6 respirations (pranas) Table 12. The Circular divisions 1 revolution = 12 signs 1 sign =30 degrees 1 degree =60 minutes (kalas) 1 minute =60 seconds (yikalas) 1 second =60 thirds (tatparSs). “The term ksetra means bhagola ('Sphere of the asterisms')”, writes Bhaskara I. More accurately, it means 'the circle of the asterisms* or the 'ecliptic'. The term ksetra is also used in the sense of 'a sign of the zodiac. 1 But in the present context it means a circle. CONJUNCTIONS OF TWO PLANETS IN A YVGA 3. (a.b) The difference between tha revolution .numbers of any two planets is the number of conjunctions of those planets in a yuga. VYATIPATAS in a yuga #*rftRWwrn afararc^ spirit: n 3 11 3. (c-d) The (combined) revolutions of the Sun and the Moon added to themselves is the number of Vyatipntas (in a yuga). The phenomenon called vyatlpata is of two types : (1) L&ta- vyattpata, and (2) Vaidhrta-vyatlpata. The former occurs when the 1 . Cf. Trftr-d'r-^i-w-'nfsT *wr 3*T*f*nR«rarr: I {JyotiscandrHrka by Rudradeva Sharma, N. K, Press, Lucknow, i. 165, gloss). Vtrie4 ] ANOMALISTIC AND SYNODIC REVOLUTIONS 87 sum of the (tropical) longitudes of the Sun and the Moon amounts to 180 degrees and the latter when that sum amounts to 360 degrees. 1 Thus, in one combined revolution of the Sun and the Moon there occur two vyatipatas. The conception of the phenomenon of vyatipata is very old. It occurs in the Vedahga-Jydutisa* (c. 1400 B.C.) which states the number of vyatipatas in the yuga of 5 years. It also occurs in the Jaina astronomical work Jyotiskaranda* (514 AD.) where the rule for finding the number of vyatipatas in a yuga of five years is formulated. ANOMALISTIC AND SYNODIC REVOLUTIONS 4 (a-b) The difference between the revolution-numbers of a planet and its ucca gives the revolutions of the planet's epicycle (in a yuga). What is meant is that the difference between the revolution- numbers of a planet and its matdocca (apogee) gives the anomalistic revolutions of that planet ; and that the difference between the revolution-numbers of a planet and its sighrocca gives the synodic revolutions of that planet. The number of the anomalistic revolutions of the Moon in a yuga, according to Aryabhaja is : = Revolution-number of the Moon minus Revolution-number of the Moon's apogee =5,77,53,336—4,88,219 = 5,72,65,117. The period of one anomalistic revolution of the Moon is, likewise, equal to 1,57,79,17,500/5,72,65,117, i e., 27*55459 days, or 27 rf 13* 18 m 36'.6 1. See Khan4a-khadyaka, i. 25. 2. vs. 19. The date of the Vedahga-Jyautisa may be derived from the position of the summer solstice (viz. the first point of the naksatra Dhani§tha) mentioned in that work. 3. Gathas 291-93. Eor details, see Vedahga-jyautisa, edited with English translation and Sanskrit commentary by R, Shamasastry, Mysore, 1936, vs. 19, notes and Sanskrit commentary. 4. G. SU, faqYforar: 88 KALAKRIYA SECTION [ KalakriyS Sn. approx. According to modern astronomers, it is equal to 27 4 13* 18”* 33U. The following table gives the synodic revolutions (in a yuga) and the synodic periods (mramsa-kdla) of the Moon and the planets according to Aryabhat,a I, Ptolemy and the modern astronomers. Table 13. Synodic revolutions and synodic periods Synodic revolutions Planet . according to Synodic period in days Aryabhaja I Aryabha^a I Ptolemy Modern Moon 53433336 29-53058 29-53059 29-53059 Mars 2023176 779-92125 779-9428 779-936 Sighrocca of Mercury 13617020 115-8783 115-8786 115-877 Jupiter 3955776 398-8895 398-8864 398-884 Sighrocca of Venus 2702388 583-8975 584-0000 583-921 Saturn 4173436 378*0859 378-0930 378-092 JOVIAN YEARS IN A YUGA 4. (c.d) The revolution-number of Jupiter multiplied by 12 gives the the number of Jovian years beginning with Aivayuk (in a yuga). A Jovian year is the time taken by Jupiter in passing through one sign of the zodiac. The following table gives the names by which the Jovian years are called when Jupiter passes through the various signs. Table 14. Names of the Jovian years Sign Jovian year Sign Jovian year 1 Aries Asvayuk 7 Libra Caitra 2 Taurus KSrtika 8 Scorpio Vaisakha 3 Gemini Margasirsa 9 Sagittarius Jyeftha 4 Cancer Pausa 10 Caricorn Asadha 5 Leo Magha 11 Aquarius SrSvana 6 Virgo Phalguna 12 Pisces Bhadrapada 1. B. ^tt# ^ siren; Bh. ^t!%3<tt to^sttst i Verse 4 ] JOVIAN YEARS 89 The Jovian years were named after the asterisms in which Jupiter rises heliacally in the various signs. The following table gives, according to Varahamihira (d. A.D. 587), 1 the asterisms in which Jupiter is normally seen to rise heliacally in the various Jovian years. Table 15. Asterisms in which Jupiter rises in the various Jovian years Jovian year Asterisms in which Jupiter rises Asvayuk Revaii, Asvini, Bharani Kartika Krttika, Rohini Margasirsa MrgaSira, Ardra Pausa Punarvasu, Pusya Magha ASlesa, Magha Phalguna PurvS Phalguni, Uttara Phalguni, Hasta Caitra Citra, Svail Vaisakha ViSakha, Anuradha Jyes^ha Jyesfha, Mula AsSdha PDrvasadha, UttarSsadba SrSvana Sravana, Dhani&ha Bhadrapada Satabhisak, Purva Bhadrapada, Uttara Bhadra- pada The above twelve-year cycle of Jupiter is taken to start at the beginning of the current yuga with A&vayuk, because in the beginning of the yuga Jupiter rose heliacally in the asterism A'sdrit* There is another cycle of Jupiter which consists of five 12-year cycles or 60 Jovian years. The sixty years of this cycle bear the following names : 8 1. Cf. Brhat-samhiM of Varahamihira, edited with Bhaftotpala's commentary by S. Dvivedi, Banaras (1895), viii. 2. 3. See Jyotiscandrdrka by Rudradeva Sharma, N. K. Press, Lucknow, pp. 27-28; Bfhijjyotihsara compiled and translated into A. Bb. 12 SO KALAKRIYA SECTION t KalakriyS S«. 1 Vijaya 31 Rudhirodgari 2 Jaya 32 Raktaksa 3 Manmatha 33 Krodhana 4 Durmukha 34 Ksaya 5 Hemalamba 35 Prabnava 6 Vilamba 36 Vibnava 7 Vikarl 37 Sukla 8 Sarvari 38 Pramoda 9 Plava 39 Prajapati 10 Subhakrt 40 Aftgira 11 Sobhana 41 Srimukha 12 Krodhi 42 Bhava 13 Visvavasu 43 Yuva 14 Parabhava 44 Dhata 15 Piavanga 45 Isvara 16 Kilaka 46 Bahudhanya 17 Saumya 47 Pramathi 18 Sadharana 48 Vikrama 19 Virodhakrt 49 Vrsa 20 Paridhavi 50 Citrabhanu 21 PramadI 51 buDnanu 22 Ananda 52 Tarana 23 Raksasa 53 Parthiva 24 Nala or Anala 54 Vyaya 25 Pingala 55 Sarvajit 26 Kalayukta 56 Sarvadharl 27 Siddhsrtha 57 Virodfa! 28 Raudra 58 Vikrta 29 Durmati 59 Khara 30 Dundubhi 60 Nandana Hindi by Surya Narayana Siddhanti, T. K. Press, Lucknow, 1972, pp. 2-3. Vecse 5-6. ] VARIOUS RECKONINGS OF TIME ' 9\ This cycle took a new round at the beginning of Kaliyuga. Likewise, the current Kaliyuga started with Vijaya, the first year of this cycle. SOLAR YEARS, AND LUNAR, CIVIL AND SIDEREAL DAYS 5. The revolutions of the Sun are solar years. The conjunctions of the Sun and the Moon are lunar months. The conjunctions of the Sun and Earth are (civil) days. The rotations of the Earth are sidereal days. Thus we have : Solar years in a yuga =43,20,000 Lunar months in a yuga = 5,34,33,336 Civil days in a yuga =1,57,79,17,500 Sidereal days in a yuga =1,58,22,37,500 The commentators have adopted the reading “ bhavartascapi naksatraK' in place of “kvavartascapi naksatrah.” BhSskara I (A.D. 629) and Raghunatha-raja (A.D. 1597) have, however, mentioned the latter as an alternative reading. The latter, evidently, is the correct reading as it agrees with Aryabhaja's theory of the Earth's rotation. The word yoga applied to the Sun and the Earth, as Clark notes, clearly indicates that Aryabhata I believed in the rotation of the Earth. Also see infra, ch. iv, vs. 48. INTERCALARY MONTHS AND OMITTED LUNAR DAYS 6. The lunar months (in a yuga) which are in excess of the solar months (in a yuga) are (known as) the intercalary months in ft yuga ; and the lunar days (in a yuga) diminished by the civil days (in a yuga) are known as the omitted lunar days (in a yuga). I, Bh. (quotes), Ra. (quotes) wraatwfa; others ?TR?lfo*rfif 92 KXLAKR1YA SECTION [ KBlakriyS f n . Thus, according to Aryabha^a I, Intercalary months in yi#a=5,34,33,336— 5 18,40,000=15,93,336 Omitted lunar days in a >wgfl=l 60,30,00,080— 1,57,79,17,500 7. A solar year is a year of men. Thirty times a year of men is a year of the Manes. Twelve times a year of the Manes is called a divine year (or a year of the gods). 8. 12000 divine years make a general planetary yuga. 1008 (general) planetary yugas make a day of Brahma. The statement that “thirty times a year of men is a year of the Manes” is inaccurate. The Manes are supposed to reside on the opposite side of the. Moon. Since the length of a day on the Moon is equal to one lunar month or 30 lunar days of men, a year of the Manes is equal to 30 times a lunar year of men, not 30 times a solar year. The Surya-siddhanta (xiv 14) makes the statement correctly : “Of thirty lunar days is composed a lunar month, which is declared to be a day and night of the Manes.” What Aryabhata means to say io the above stanza is that a yuga= 43,20,000 years a day of Brahma (or KaJpa) = 1,008 yugas or 4,35,45,60,000 years. This is in agreement with what he said in A t i. 5. (See p. 9, above) 1. B-E. Gh. Ni. Pa. SO. *nr?fe<S£JT 2. F. for 03 3. C. 5»f and F. 5”T for ffif =2,50,82,580 DAYS OF MEM, MrtNES GODS AND OF BRAHMA sretoc mil %st 3Tf5*TH!i(; n c n Verse 9 ] DIVISIONS OF YUGA 93 UTSARPINI, APASARP1NI, SUSAMA AND DUSSAMA 9. The (first) half of a yuga is Utsarpivi and the second half Apasarpivi. Susama occurs in the middle and Dussama in the beginning and end. (The time elapsed or to elapse is to be reckoned) from tbe position of the Moon's apogee 1 This terminology is in conformity with the teachings of the Jaina canons. The time-cycle is divided there into two halves : (1) the auspicious half called Utsarpim and (2) the inauspicious half called Apasarpinl (or Avasarpirii). The Utsarpim is subdivided into six divisions which occur in the following sequence : (1) Dussama-dussama\ (2) Dussama \ DUSSAMA (1) (3) Dussama-susama ) (4) Susama-dussamd (5) Susama } SUSAMA (2) } (6) Susama-susama The Apasarpinl is also similarly subdivided into six divisions which occur in the following succession : (7) Susama-susama \ (8) SusamS \ SUSAMA (3) (9) Susama- Dussama I (10) DussamS-susama 1 (11) Dussama > DUSSAMA (4) (12) DussamQ-dussama] Instead of dividing the Utsarpim and the Apasarpinl into six divisions each, Aryabhaja has divided each of them into two gross divisions, Utsarpim into Dussama and Susama, and Apasarpinl into Susama and Dussama. The two Susama divisions thus fall in the middle of a yuga and the two Dussama divisions in the beginning and end of a yuga. 1. Same is stated in VatesvQra-siddhanta, Grahaganita, ch, 1, sec. 2, vs. 6, 94 KSLAKRIYA SECTION [ KBlakriyS Sn. The time elapsed is reckoned from the position of the Moon's apogee, because in the case of the Moon's apogee there is no abraded yuga. The yuga of the Moon's apogee is the same as the general planetary yuga of 43,20,000 years (or 1,57,79,17,500 days), in the case of the other planets, there are abraded yugas of smaller durations which are not to be used. The idea is that in the determination of the elapsed or unelapsed portion of a yuga the general planetary yuga has to be taken into account and not the abraded yugas of the planets. Table 16. Abraded yugas and corresponding revolutions Planet Yuga Revolutions (in days) Sun 2,10,389 576 Moon 21,55,625 78,898 Moon's apogee 1,57,79,17,500 4,88,219 Moon's ascending node 78,89,58,750 1,16,113 Mars 13,14,93,125 1,91,402 Sighrocca of Mercury 7,88,95,875 8,96,851 Jupiter 13,14,93,125 30,352 Sighrocca of Venus 13,14,93,125 5,85,199 Saturn 39,44,79,375 36,641 In the Kasyapa-samhita, 1 a Hindu work on pediatrics, too, time is classified into two categories, auspicious time (subhakala) and inauspicious time {asubha-kala). The auspicious time is called Utsarpini and the inauspicious time Apasarpini. But, there, the Utsarpini is subdivided into three parts, viz. : (1) Adi-yuga (2) Deva-yuga (3) Krta-yuga The Apasarpini is also subdivided into three parts, viz, : (4) Tretn-yuga (5) Dvapara-yuga (6) Kali-yuga 1. See .Kasyapa-samhita, edited by Heroaraja Sarma. Nepal Sanskrit Series, No. 1, Bombay (1938), p. 44, Vetse 10 DATE OP ARYABHATA I 95 Kasyapa has evidently tried to establish an excellent compromise between the Jaina and the Hindu conceptions. According to the orthodox Hindu conception, a general planetary yuga is divided into 4 smaller yugas called Krta, Treffi, Dvapara and Kali. The lengths of these yugas and the measures of righteousness in them are supposed to be in the ratio of 4 : 3 : 2 : 1. The defect in the Hindu conception is that Kaliyuga, which is marked by one-quarter of righteousness, is abruptly followed by Krtayuga, which is marked by four quarters of righteousness. The commentators of the Aryabhatiya have taken full liberty in interpreting the text. The interpretations given by Bhaskara I (A.D. 629), Suryadeva {b. A.D. 1191) and Nilakantha (A.D. 1500) are quite arbitrary and different from one another. Two similar terms Avarohini and Arohini are found to occur in Hindu works on horoscopy. According to Gargi, so long as a planet moves from its apogee to perigee the Dasa or Antardasa of that planet is called Avarohini and so long as a planet moves from its perigee to its apogee, the Dasa or Antardasa of that planet is called Arohini. Varahamihira, too, says the same thing. (See Brhajjataka, viii. 6, and Bhaftotpala's commentary thereon.) DATE OF AR.Y ABriATA I 10. When sixty times sixty years and three quarter yugas (of the current yuga) had elapsed, twenty three years had then passed since my birth. This stanza mentions the epoch when 3600 years had elapsed since the beginning of the current Kaliyuga. Since 3600 years^ ^^qq 00 or 1314931 ' 25 ^ this epoch corresponds to mean noon at Ujjayini, Sunday, March 21, 499 A.D. At this time Aryabhaja I was exactly 23 years of age. Aryabhata I was therefore born on March 21, 476 A.D. % KALAKRlYA SECTION t KfilakriyH Sn. The object of specifying the year 3600 of the Kali era, according to the commentators of the Aryabhatiya, was to show that at that time the mean positions of the planets computed from the parameters given in the Da'sagttikH-sutra did not require any correction, and that to the mean longitudes computed for a subsequent date a bija correction was necessary. Such a correction is given by Lalla, who belongs to the school of Aryabhata I. Table 17. Bija corrections according to Lalla (Epoch of zero correction, though really Saka 421, is taken as Saka 420 for facility of computation) Planet Bija correcton to the mean longitudes of the planets per annum in terms of minutes of arc Moon — 25/250 Moon's apogee — 114/250 Moon's asc, node — 96/250 Mars + 48/250 Mercury +420/250 or +430/250 Jupiter — 47/250 Venus —153/250 Saturn + 20/250 The Kali year 3600 (or Saka year 421) was, according to SBryadeva (b. A.D. 1191), Raghunatha-raja (A.D. 1597), ViSvanStha 1 (A.D. 1629j, and also according to the author of the Vakya-karana (c. A.D. 1300), the time when the precession of the equinoxes was also zero. According to another view, A.D. 522 (corresponding to the Saka year 444) was the epoch of zero correction to the mean longitudes of the planets calculated from the parameters stated in the Dasagitikd-sutra. Astronomers holding this view have prescribed the following bija corrections. 1. See his comm. on Makaranda-sSrani, Bombay (1935), p. 84. Verie 10 1 DATE OF XfcYABHAfA W Table 18. Blja correction according to Hartdatta 1 and Deva 1 (689 A.D.) {Epoch of zero correction being tiaka 444 or A.D. 522) Planet Bija correction to the mean longitudes of the planets per annum in terms of minutes of arc Moon — 25/235 Moon's apogee —114/235 Moon's asc. node — 96/235 Mars + 45/235 Mercury +420/235 Jupiter — 47/235 Venus —153/235 Saturn -f 20/235 Table 19. Another blja 3 correction with epoch at Saka 444 (A.D. 522) Planet Blja correction per annum in terms of minutes of arc Moon — 25/235 Moon's apogee —114/235 Moon's asc node — 96)235 Mars + 50/235 Mercury +430/235 Jupiter — 50/235 Venus —160/235 Saturn + 21/235 1. See GrahacVra-nibandhana of Haridatta, ed. K.V. Sarma, p. 25, vss. 17-18. 2. KR, i. 16-18. 3. See Grakacara-nibandhana, pp. 25-26, vss. 19-22. Snryadeva, in his comm. on LMn, i. 1-2, calls this correction 'traditional' (sampradaya-siddha) ; but for Venus he gives — 180/235 mins. in place of —160/235 mins. A. Bh. 13 98 KALAKRIYA SECTION t K«IaktiyS Sn. Bfahmadeva <A.D. 1092), who wrote his caiendrieal work Karana-prekQsa on the basis of Lalla's &i§ya-dhi-vrddhida, Bhoja (A D. 1042), who wrote the caiendrieal work Raja-mrganka, Ganesa Daivajna (A.D. 1520), the author of the caiendrieal work Grahalaghava, Manjula (A.D. 932), the author of the Laghu-manasa, and some followers of the Khan4a-khadyaka have regarded Saka 444 (A.D. 522) as the epoch when the precession of the equinoxes was zero. Some astronomers of Kerala have associated both Saka 421 and Saka 444 with the life of Aryabhaja and have called them Bhatabda ('the years associated with Aryabhata'). The Kerala astronomer Haridatta (c. A.D. 683), the alleged author of the Sakdbda correction, has, as remarked by Nllakantba in his commentary on the Aryabhatiya (rather in surprise^ interpreted the above stanza in a different way, viz. : “When sixty times sixty years and three quarter yugas had elapsed, twentythree, years of my age have passed since then.” This means that Aryabhata was born in Saka 421 and wrote the Aryabhatiya in Saka 444. But no commentator of the Aryabhatiya has interpreted the above stanza in this way, and T.S. Kuppanna Sastri has rightly called it a “wrong interpretation”. Another Kerala astronomer (probably Jyesfhadeva), the author of the Drkkarana (A.D. 1603), an astronomical manual in Matayalam, has actually stated that Aryabhata was born in Saka 421 and that he wrote the Aryabhatiya in Saka 444. This is, according to T.S. Kuppanna Sastri, a “mistaken impression”. According to the commentators Snryadeva (b. A.D. 1191) Paramesvara (A. D. 1431) and Nilakantha (A. D. 1500), the Kali year 3600 (corresponding to Saka 421), besides being the epoch of zero correction, indicates the time of composition of the Aryabhatiya. K. SambaSiva Sastri, W.E. Clark, and Baladeva Misra, too, hold the same opinion. BEGINNING OF THE YUGA, YEAR, MONTH. AND DAY l. D. sr^mr Verse 11 ] BEGINNING OF YOGA ETC. 99 11. The yuga t the year, the month, and the day commenced Simultaneously at the beginning of the light half of Caitra. 1 This time, which is without beginning and end, is measured with the help of the planets and the asterisms on the Celestial Sphere. What is meant is that time is endless and has no beginning or end, but for practical purposes it is measured by means of the yuga, the year, the month, and the day etc., which are defined on the basis of the positions of the planets and the asterisms in the sky, in the same way as length is measured by means of the units of length. The commentator Bhaskara I provides an alternative inter- pretation for grahabhair anumiyate. “Others”, writes he, “interpret grahabhair anumiyate in a different way, viz., the beginning or end of time is defined with the help of the planets and the asterisms.” The commentator Somesvara, too, interprets the above stanza in the same way. Writes he : “The yuga, the year, the month, and the day started simultaneously at the beginning of Caitra, i.e., at the beginning of the light fortnight of Caitra when half the Sun had risen (above the horizon at Lanka). Then it would mean that time has a beginning ; to contradict this, Aryabhafa says : 'This time is without beginning or end', i.e., the time which we have referred to as yuga etc. and which started at the beginning of the light half of Caitra, has neither a beginning nor an end. It is only for the use of the people that its beginning and end are defined. How are its beginning and end defined for use ? Aryabhata says : 'These are defined by (the positions of ) the planets and the asterisms'. As for example, the beginning of the yuga is defined as the time when all the planets are simultaneously on the horizon (at Lanka) at the first point of Aries/' Since the calculation of the positions of the planets is ultimately aimed at the determination of time, the nomenclature 'the reckoning of time* given to the present chapter is highly significant. 1, Cf. BrSpSi, i. 4 ; MSi, i. 5 ; Si&t, i. 10 ; Sify I. i. 15. KALAKR1YA SECTION [ KtUkriyi Sn. EQUALITY OF THE LINEAR MOTION OF THE PLANETS The planets moving with equal linear velocity in their own orbits 3 complete (a distance equal to) the circumference of the sphere of the asterisms in a period of 60 solar years, and (a distance equal to) the circumference of the sphere of the sky in a yuga.* That is, a planet moves through a distance of 17,32,60,008 yojanas in 60 solar years and a distance of 1,24,74,72,05,76,000 yojanas in 43,20,000 solar years. Since there are 1,57,79,17,500 days in 43,20,000 solar years, it follows that the mean daily motion of a planet, according to Aryabhafa, is 12474720576000 , -^7-977500— y0jaMS or 7905 '8 yojanas approx. For further details, the reader is referred to our notes on 2. i.6 (pp. 13-15, above). CONSEQUENCE OF EQUAL LINEAR MOTION OF THE PLANETS 3<#sTq; srcfaf STC^ft II ?3 I) 13. The Moon completes its lowest and smallest orbit in the shortest time ; Saturn completes its highest and largest orbit in the longest time. 4 For the lengths of the orbits of the planets, see supra, i. 6 (p. 13, above). 1. C. <Tfon«W 2. Cf. PSi, xiii. 39 (c-d). 3. Cf t BrSpSi, xxi. 12. 4. Cf. PSi $ xiii. 41 ; SgSi, xii. 76-77 ; Si$r t I, i. 5. 27 (c-d). 100 12. Verse 14 ] circular Divisions 101 NON-EQUALITY OF THE LINEAR MEASURES OF THE CIRCULAR DIVISIONS w s^rca^ 1 ^img^n: II ?w 11 14. (The linear measures of) the signs are to be known to be small in small orbits and large in large orbits ; 2 so also are (the linear measures of) the degrees, minutes, etc. The circular division is however, the same in the orbits of the various planets. The first part of the above statement would be self-evident from the following table which gives the lengths in yojanas of one sign, one degree and one minute in the orbits of the various planets. The planets have been arranged, for the sake of convenience, in the order of increasing orbits. Table 20. Lengths in yojanas of the circular divisions of the orbits Orbit of Lengths in yojanas of 1 sign 1 degree 1 minute Moon 18000 600 10 Sighrocca of Mercury 57956 1932 32 Sighrocca of Venus 148035 4935 82 Sun 240639 8021 134 Mars 452608 15087 251 Jupiter 2854178 95139 1586 Saturn 7092874 236429 3940 1. C. E. SU. erf* ; C. Sn. add *r here. 2. Cf. PSi, xiii.40; SuSi, xii. 75 (c-d) ; Si&, I, i. 5. 27 (b). 102 KALAKRIY3 SECTION [ KBlikxiy* Sn. The second part of the statement means that in the orbits of all the planets one sign is equal to 1/12 of the orbit, one degree is equal to 1/30 of a sign, one minute is equal to 1/60 of a degree, and so on. The non-equality of the linear measures of the circular divisions in the orbits of the various planets implies that although the planets have equal linear velocity, their angular velocities are different* The following table gives the mean angular velocities of the planets according to Aryabhaja 1 1 Table 21. Mean angular velocities of the planets Planet Mean angular velocity per day Sun 59' 8* Moon 790' 35* Moon's apogee 6' 41' Moon's ascending node 3' 11* Mars 31' 26' &ighrocca of Mercury 4° 5' 32* Jupiter 4' 59' or 5' 00' approx. Sighrocca of Venus 1° 36' 8' Saturn 2' 0' RELATIVE POSITIONS OF ASTERISMS AND PLANETS 2 ^rc^ ^Bw^it \\\t
15. (The asterisms are the outermost). Beneath the asterisms lie (the planets) Saturn, Jupiter, Mars, the Sun, Venus, Mercury, 1. See SiDVr, I, i. 40-41. 2. All others than Bh., %m for x^tj Vers* 16 1 LORDS OF HOURS AND DAYS 103 and the Moon (one below the other) ; and beneath them all lies the Earth like the hitching peg in the midst of space. 1 LORDS OF THE HOURS AND DAYS wife wtentf f^w n \$ ii 16. The (above-mentioned) seven planets beginning with Saturn, which are arranged in the order of increasing velocity, are the lords of the successive hours. The planets occurring fourth in the order of increasing velocity are the lords of the successive days, which are reckoned from sunrise (at Lanka). 3 That is to say, the lords of the twenty-four hours (the hours being reckoned from sunrise at Lanka) are : Saturn, Jupiter, Mars, Sun, Venus, Mercury, Moon, Saturn, Jupiter, Mars, Sun, Venus, Mercury, Moon, Saturn, Jupiter, Mars, Sun, Venus, Mercury, Moon, Saturn, Jupiter, and Mars, respectively ; and the lords of the seven days are : Saturn, Sun, Moon, Mars, Mercury, Jupiter, and Venus, respectively. The lord of a day is the lord of the first hour of that day, the day being measured from sunrise at Lanka. It is to be noted that the lords of the hours and the days are to be reckoned from sunrise at Lanka (and not from sunrise at the local place). Since Aryabhata I mentions, in the above rule, sunrise without specifying that it refers to Lanka, Brahmagupta finds occasion to criticise him. 1. Cf. PSl, xiii. 39 ; BrSpSi, xxi. 2 ; Si&e, xv. 70 ; Si&i, II. iii. 2. 2. D. 3. Cf. PSh xiii- 42 \ SnSi, xii. 78-79. 104 KALAKRIYA SECTION [ KSIakriyB Sn Writes he : “The statement of Aryabhafa, v/z., 'Reckoned from sunrise, the planets occurring fourth (in the order of increasing velocity) are the lords of the successive days' is not true, because he has himself declared sunset at Siddhapura when it is sunrise at Lanka.” 1 On this criticism, Brahmagupta's commentator Prthndaka comments : “This is a phantom of a defect, for, in the Dasagitika, Aryabhafa has (already) said — 'from sunrise at Lanka'. 1 ' As regards the first day of the week cycle, it is perhaps implied in the above rule that it was Saturday. Vajesvara (A.D. 904) is the only Hindu mathematician who supposed that the world-order commenced on Saturday. He has criticised Brahmagupta for starting the Kalpd on Sunday : “The lords of the hours, days, months and years have been stated by Brahma to succeed one another in the order of increasing velocity beginning with Saturn and not with the Sun. Even the order of the planets are not known to him.”* MOTION OF THE PLANETS EXPLAINED THROUGH ECCENTRIC CIRCLES (BrSpSt, xi. 12) VaSi, Grahagantia, ch. 1, sec. 10, vs. 9 Verset 17-20 ] MOTION OF PLANETS 105 17. (The mean planets move on their orbits and the true planets on their eccentric circles). All the planets, whether moving on their orbits (Jcak&OrnmialaY or on the eccentric circles (prati-mavdala), move with their own (mean) motion, anticlock- wise from their apogees and clockwise from their Sighroccas. 18. The eccentric circle of each of these planets is equal to its own orbit, but the centre of the eccentric circle lies at a distance from the centre of the solid Earth. 19 (a-b) The distance between the centre of the Earth and the centre of the eccentric circle is (equal to) the semi.diameter of the epicycle (of the planet). MOTION OF PLANETS EXPLAINED THROUGH EPICYCLES 19. (c-d) All the planets undoubtedly move with mean motion on the circumference of the epicycles. 20. A planet when faster than its ucca moves clockwise on the circumference of its epicycle and when slower than its ucca moves anticlockwise on its epicycle. 4 1. B. D. E Bh. Gh. So. SO. HwnRTO^ ; others 2. Bh. WP?^«f*l 3. B-E, Gh. NI. Pa. SO. ST*fa 4. Cf. BrSpSi, xxi. 25-26 ; &iDV(, II. i. 12 (a-b) ; $t$e, xvi. 5 ; Sl&t, II.' v. 30. A. Bh. 14 106 KALAKR1YA SECTION [ KalaktiyH Sn. What is meant is that a planet moves clockwise on its manda epicycle and anticlockwise on its Hghra epicycle. According to the commentator Bhaskara I, verse 20 relates to the determination of the true daily motion, retrograde or direct. He has interpreted this verse as follows : “When the Hghragatiphala (iig/w-a-motion-correction) is negative but numerically greater than the true-mean motion, their difference gives the retrograde motion ; and when Hghragati- phala (ijgArd-motion-correction) is negative but numerically less than the true-mean motion, their difference gives the direct motion. This latter motion when less than mean motion is called slow motion (mandagati)” Following Bhaskara I, the commentator Somesvara, too, interprets the verse in the same way. Evidently, both Bhaskara I and Somesvara have misunderstood the text. MOTION OF EPICYCLES 21- The epicycles move anticlockwise from the apogees and clock- wise from the ilghroccas. The mean planet lies at the centre of its epicycle, which is situated on the (planet's) orbit. What is meant is that the manda ep icycles move anticlockwise from the apogees and the itghra epicycles move clockwise from the ilghroccas. ADDITION AND SUBTRACTION OF MANDAPHALA AND &IGHRAPHALA 1. E. F. Ni. Pa. 2. «WiMi*«|i*4 var. recorded by Bh. (in his com, on this verse) Verse 22 ] APPLICATION OF COHRICTIONS 107 22 (a.b) The corrections from ike apogee (for the four anomalistic quadrants) are respectively minus, pins, plus, and minus. Those from the ilghrocca are just the reverse. 1 In the time of Aryabhata I, the Rsines of the arcs (> 90”) were obtained by the application of the following formulae : Rsin (90°-M)=Rsin 90°— Rvers 9 Rsin (1 8O°+0)=Rsin 90°— Rvers 90°— Rsin 9 Rsin (270°-H)=Rsin 90°— Rvers 90°— Rsin 90°+Rvers 0, where 9<90*. Suppose that a planet lies in the fourth manda anomalistic quadrant and that the manda anomaly is 270° Then Rsin (27O°4-<0=Rsin 90°— Rvers 90'— Rsin 90°+Rvers 6, so that Rsin (90°— *)=— Rsin 90° + Rvers 90°+Rsm 90°- Rvers 9, or Mandakendrabhujajyli=—Ksm 90° 4 Rvers 90°+Rsin 90° —Rvers 9. (1) Now, multiplying both sides of (1) by the planet's manda epicycle and dividing by 360, we get Correction from the apogee (mandaphala) = —correction for the first quadrant+correction for the second quadrant -f correction for the third quadrant— correction for the fourth quadrant, whence it is clear that the corrections for the first, second, third, and fourth quadrants are — , +, +, and — , respectively. The same can be seen to be true when the planet is in the other anomalistic quadrants. In the case of the sighraphala, the correction for the four quadrants are of the contrary signs, because the mandakendra and the Hghrakendra are defined contrarily : m — 1. Cf. BrSpSi, ii. 16 (a-b). 108 KALAKRIYS SECTION [ K «lakri y « St*. mandakendra «= longitude of planet-longitude of planet's apogee. tighrakendra^ghudc of planet's Hghrocca- longitude of planet. f The law of addition and subtraction of the mandaphala and sighraphala in the four quadrants is mentioned also by Bhaskara I (A.D. 629), Brahmagupta (A.D. 628) and Sripati (c, A.D. 1039), but it was more convenient to apply the mandaphala as obtained by the formula : mandnphnln^ Rsin 6 * ™nda ep icycle 360 ' (0 being the planet's mandakendra reduced to Mh» negatively or positively, according as the mandakendra was less than or greater than 180°, and the sigh.-a-phala as obtained by the formula &if>hrnphnl*= Rsia e XSighra epicycle R 360 X H' (6 being the planet's tighrakendra reduced to M w y a and H the planet's sighrakarna) positively or negatively, according as the nghrakendra was less than or greater than 180°. And so the Hindu astronomers have generally adopted these latter rules. A SPECIAL PRE-CORRECTtON FOR THE SUPERIOR PLANETS 22. (c-d) In the case of (the superior planets) Saturn, Jupiter and Mars, first apply the mandaphala negatively or positively (as the case may be). Here the following rule is implied : In the case of Saturn, Jupiter and Mars, first apply half the mandaphala to the mean longitude of the planet negatively or positively, according as the mandakendra is Jess than or greater than 180°. comm. 1 ' ^ ^ » var - recorded by Bhaskara I in his Vent 23 } ALTERNATIVE METHOD 109 This pre-correction is meant only for the superior planets — Mars, Jupiter and Saturn. It should not be applied to the inferior planets, Mercury and Venus. (See full rules given below). PROCEDURE OF MANDAPHALA AND SlGHRAPHALA CORRECTIONS FOR SUPERIOR PLANETS *$#T*IT: ^sfr?^ S$ST f*Ti: II R\ || 23. Apply half the mandaphala and half the Slghraphala to the planet and to the planet's apogee negatively or positively (as the case may be). The mean planet (then) corrected for the mandaphala (calculated afresh from the new mandakendra) is called the true-mean planet and that (true-mean planet) corrected for the ixghraphala (calculated afresh) is known as the true planet. 1 This rule may be stated fully as follows : Apply half the mandaphala to the mean longitude of the planet negatively or positively, according as the mandakendra is less than or greater than 180° and to the longitude of the planet's apogee reversely. Then apply half the nghraphala to the corrected longitude of the planet's apogee negatively or positively, according as the sighrakendra is less than or greater than 180°, Then calculate the mandaphala afresh and apply the whole of it to the (original) mean longitude of the planet negatively or positively, according as the mandakendra is less than or greater than 180° : this would give the true-mean longitude of the planet. Then calculate the sighraphala again and apply the whole of it to the true-mean longitude of the planet positively or negatively, according as the sighrakendra is less than or greater than 180° : this would give the true longitude of the planet. -m 1. Cf. MBh, iv. 40-43 ; LBh, ii. 33-37 (a-b) ; SiDVr, I, iii. 4-7. 110 KALAKR1YA SECTION t KalakriyB Sn. MANDAPHALA AND &GHRAPHALA CORRECTIONS FOR INFERIOR PLANETS 24. (In the case of Mercury and Venus) apply half the Sighraphala negatively or positively to the longitude of the planet's apogee (according as the fighrakendra is less than or greater than 180°). From the corrected longitude of the planet's apogee (calculate the mandaphala afresh and apply it to the mean longitude of the planet ; then) are obtained the true-mean longitudes of Mercury and Venus. (The sighraphala, calculated afresh, being applied to them), they become true (longitudes) 2 The old Snrya-siddhanta applied the mandaphala and sighraphala corrections in the following order : (a) For obtaining the true longitude of the planet's apogee : 1. Half sighraphala to the longitude of the planet's apogee (reversely). 2. Half mandaphala to the corrected longitude of the planet's apogee (reversely). (b) For obtaining the true longitude of the planet : 3. Entire mandaphala (calculated from the corrected longitude of the planet's apogee) to the mean longitude of the planet. 4. Entire sighraphala to the corrected mean longitude (called true-mean longitude) of the planet. But instead of applying a pre-correction in the case of the superior planets (as done by Aryabhata), it prescribed an impirical correction (called the fifth correction) in the case of the inferior planets.* 1. A. $sp^ 2. Cf. MBh, iv. 44; LBh, ii. 37(c-d)-39; SiDVt, I, iii. 8. 3. See PSt, xvi. 17-22. Also see K.S. Shukla, The PaHca* siddhantika of VarShamihira (I), IJHS, vol. 9, no. 1, pp. 69-71. Verse 25 3 DISTANCE AND VELOCITY 111 It seems that the procedure used by the author of the old Surya- siddhanta did not lead to accurate results and that Aryabhafa's method was an improvement. DISTANCE AND VELOCITY OF A PLANET >5pwsi$fM ^jranJpK 1 ^isNpt: i 25. The product of the mandakarryi and the Sighrakarna when divided by the radius gives the distance between the Earth and the planet. 3 The velocity of the (true) planet moving on the (slghra) epicycle is the same as the velocity of the (true-mean) planet moving in its orbit (of radius equal to the mandakaryta). Aryabhata and his followers take the distance between the Earth and a planet as equal to mandakarna X sighrakarna R the mandakarna and the sighrakarna being obviously the karnas obtained in the last two operations. The Sarya-siddhllnta* takes the distance between the Earth and a planet as equal to mandakarna -f- sighrakarna 2 ” • 1. D. E. F. sjmnsf^f 2. A. 5% vrsrftwrrre: ; (E. om. ^fa) ; F. sfa *mOwi httct: 3. The same rule occurs in MBh, vi. 48 ; LBh, vii, 8. 4. SnSi t vii. 14. 112 K5LAKRIYA SECTION [ CtltkriyC Sn. Aryabhata takes the orbit of the true-mean planet as equal to the mandakarw. Hence the rule in the second half of the stanza. The commentator Snryadeva interprets the second half of the verse as meaning : “The velocity of the (true) planet in the (sighra) epicycle is the same as the velocity of the planet in the orbit constructed with radius equal to the distance of the planet from the Earth.” What he means to say is that the velocity of the (true) planet moving on the (sighra) epicycle is equal to the true-mean velocity. CHAPTER IV GOLA OR THE CELESTIAL SPHERE [ In order to demonstrate the motion of the heavenly bodies, the Hindu astronomers make use of spheres constructed by means of circles made of flexible wooden sticks or bamboo strips. These are called Gola and correspond to the Celestial Sphere of modern astronomy. The Gola which is supposed to be centred at the Earth's centre is called Bhagola ('Sphere of the asterisms')- It is used to demonstrate the motion of the Sun, the Moon and the planets in their orbits. The principal circles of this sphere are : (1) the celestial equator, (2) the ecliptic, (3) the orbits of the Moon and the planets, and (4) the day-circles, etc. The Gola which is supposed to be centred at the observer is called Khagola ('Sphere of the sky'). It is fixed in position and is used to demonstrate the diurnal motion of the heavenly bodies ; the principal circles of this sphere are : (1) the horizon, (2) the meridian, (3) the prime vertical, and (4) the six o'clock circle, etc. In the present Section, Aryabhafa aims at teaching spherical astronomy. He begins by giving a brief description of the Bhagola and the Khagola and then, with their help, demonstrates the motion of the heavenly bodies. ] 1. Bhagola POSITION OF THE ECLIPTIC 1. One half of the ecliptic, running from the beginning of the sign Aries to the end of the sign Virgo, lies obliquely inclined (to the equator) northwards. The remaining half (of the ecliptic) running from the beginning of the sign Libra to the end of the sign Pisces, lies (equally inclined to the equator) southwards. 2 U D. 5T^: 2. Cf. BrSpSi, xxi. 52 ; SiDVr, II, ii. 7 ; VSi, Gola, iv. 7 ; % Si&e, xvi. 32 ; Si&i, II, vi.12 ; SuSi, II, iv. 6 (a-b). A. Bh. 15 113 114 GOLA SECTION t Gola Sn. Reference to the equator without defining it shows that its position was supposed to be well known and that it was already shown on the Bhagola. The word eva, says the commentator Bhaskara I, is superfluous and is meant to complete the arya verse. In case the alternative reading evam is adopted, the word 'similarly' will have to be added in the beginning of the second sentence (in the translation above). Bhaskara I thinks that the word sama is intended to suggest that the signs of the ecliptic are of equal measure, i.e., each of 30°. MOTION OF THE NODES, THE SUN AND THE EARTH'S SHADOW 2. The nodes of the star-planets (Mars, Mercury, Jupiter, Venus and Saturn) and of the Moon incessently move on the ecliptic. So also does the Sun. From the Sun, at a distance of half a circle, moves thereon the Shadow of the Earth. 2 The nodes of a planet are the two points where the orbit of the planet intersects the ecliptic. The point where the planet crosses the ecliptic in its northerly course in called the 'ascending node' and the point where the planet crosses the ecliptic in its southerly course is called the 'descending node'. MOTION OF THE MOON AND THE PLANETS ^ TOT? 3[f%W I 3. The Moon moves to the north and to the south of the ecliptic (respectively) from its (ascending and descending) nodes. So 1. B. ^RTfcTS (wr.) 2. Cf. BrSpSi, xxi. 53 ; * &DV t , II, ii. 8 ; VSi, Gola, iv. 8 ; Si$e, xvi. 33 ; S/&,II,vi. 11. 3. B. C. F. Pa. 5^pr Vetse 3 ] MOTION OF MOON AND PLANETS 115 also do the planets Mars, Jupiter and Saturn. Similar is also the motion of the iighroccas of Mercury and Venus. 1 With regard to the last statement, Prthndaka (A. D. 860) says : “As much is the (celestial) latitude of Mercury or Venus at its iighrocca, so much is its (celestial) latitude at the place occupied by it.” 2 This is so, writes Bhaskara II (A.D. 1150), because the revolution-number of the node (in the case of Mercury and Venus) is the sum of the revolution-numbers of the planet's node and the planet's txghra anomaly (i.e., ixghrocca minus planet.). 8 The correct explanation, however, is that Mercury and Venus revolve round the Sun with the velocity of their sighroccas and so the (celestial) latitudes of Mercury and Venus are really the latitudes of their Ughroccas. The following rules are implied in the instructions of the text : 1. In the case of the Moon „. „ . - v Rsin (M — °-)xRsin / Rsin (latitude) = » H where M and Q. are the true longitudes of the Moon and its ascending node, i the inclination of the Moon's orbit to the ecliptic, and H the Moon's true distance (called mandakarva).* 2. In the case of Mars, Jupiter and Saturn _ . . j v Rsin (P— Q)xRsin / Rsin (latitude) — - jj , where P and Q are the true longitudes of the planet and its ascending node, i the inclination of the planet's orbit to the ecliptic, and D the distance of the planet from the Earth (as defined in Kalakriya, 25). 5 3. In the case of Mercury and Venus „. „ . x , v Rsin (S—Q)x Rsin / Rsin (latitude) = p > where S and ft are longitudes of the planet's Hghrocca and ascending 1. Cf. BrSpSi, xxi. 54 ; SiDVr, II, ii. 9 ; VSi, Gola, iv. 9 ; SiSe, xvi, 34-35 ; Si&i, II, vi. 14. 2, 3. See Si&i, II, Golabandha, 23-25 (a-b) ; and Bhaskara IPs comm. on it. % 4. Cf. LBh, iv. 8. 5. Cf. MBh % vi. 52-53 ; LBh, vii. 6-9 (a-b). 116 GOLA SECTION [ GoU Sn. node, i the inclination of the planet's orbit to the ecliptic, and D the distance of the planet from the Earth (as defined in Kdakriya. 25). > These formulae are not accurate but, according to Bhaskara I, they conform to the teachings of Aryabhaja I. The correct formula for the celestial latitude of a planet is : Rsin (l,ti^.W ^ Rsin (II-Q>^Rsin/ where II is the heliocentric longitude of the planet. 2 VISIBILITY OF THE PLANETS 4. When the Moon has no latitude it is visible when situated at a distance of 12 degrees (of time) from the Sun. Venus is visible when 9 degrees (of time) distant from the Sun. The other planets taken in the order of decreasing sizes (viz., Jupiter, Mercury, Saturn, and Mars) 5 are visible when they are 9 degrees (of time) increased by two-s (i.e., when they are 11, 13, 15 and 17 degrees of time) distant from the Sun. 6 One degree of time is equivalent to 4 minutes. Thus the Moon, when ahead of the Sun, is visible towards the west if the arc of the ecliptic joining the Sun and the Moon, takes at least 12×4 minutes in setting below the horizon ; and when behind the Sun, it is visible towards the east if the arc of the ecliptic joining the Sun and the Moon takes at least 12×4 minutes in rising above the horizon. In other words, the Moon will be visible at a place if the time-interval between sunrise and moonrise, or between sunset and moonset, amounts to 12×4 minutes or more. But this is the case when the Moon has no latitude. “When, however, the Moon has some latitude,” comments Bhaskara I, “it is visible earlier or later than when it is two ghatikas 1. Cf. MBh, vi. 52-53 ; LBh, vii. 6-9 (a-b). 2. See BrSpSi, ix. 9 ; SiDVr, I, xi. 6, 9 etc. 3. All except Bh. and So., fer5&3: 5. See supra, i. 7. 6. Cf PSi, xvj\ 23 ; MBh, vi. 4 (c-d)-5 (a-b), 44-45 ; 44 ; IBh, vi. 5, vii. 1. Verse 5 ] BRIGHT AND DARK SIDES OF PLANETS 117 (i.e., 12 degrees of time) distant from the Sun. For, when it has north latitude, the (Moon's) sphere being elevated towards the north, it is visible earlier than when it is two ghatikas distant from the Sun ; and when it has south latitude, the (Moon's) sphere being depressed towards the south, it is visible later than when it is two ghatikas distant from the Sun. That is why it is said— 'When the Moon has no latitude'. There- fore, the distance of the planet from the Sun should be taken after the visibility correction has been applied to the longitude of the planet.” The degrees of time for the heliacal visibility of the planets as given by the old Snrya-siddhanta, 1 are the same as those given above. Those given by the Vasktha-siddhanta summarised by Varahamihira are : 12° for the Moon, 14° for Mars, 12° for Mercury, 15° for Jupiter, 8° for Venus and 15° for Saturn. 2 According to the Greek astronomer Ptolemy (c. A. D. 100-178) the distances of the planets, when in the beginning of the sign Cancer (f. e., when the equator and ecliptic are nearly parallel), from the true Sun, at which they become heliacally visible, are : for Saturn, 14°; for Jupiter 12 p 45'; for Mars, 14° 30'; and for Venus and Mercury, in the west, 5° 40' and 11° 30', respectively. See The Almagest, xiii. 7. BRIGHT AND DARK SIDES OF THE EARTH AND THE PLANETS *ps*n5ri *rfam?ft *ra5R*n fawffa f 5. Halves of the globes of the Earth, the planets and the stars are dark due to their own shadows ; the other halves facing the Sun are bright in proportion to their sizes. 3 The Hindu astronomers believed that the Sun was the only source of light in the universe and all other celestial bodies, which were spherical in shape, received their light from the Sun. Their conception that the stars too received light from the Sun and were half-luminous and half-dark is indeed wrong. The next eight stanzas give a description of the Earth which occupies the centre of the Bhagola. 1. See PSi, xvi. 23. 2. See PSi, xvii. 58. 3. Cf. PSi, xiii, 35 ; SiDV(, II, iii. 40 ; SiSe, xviii. 14. 118 GOLA SECTION [ Gola Sn. SITUATION OF THE EARTH, ITS CONSTITUTION AND SHAPE 6. The globe of the Earth stands (supportless) in space at the centre of the circular frame of the asterisms (i.e., at the centre of the Bhagola) surrounded by the orbits (of the planets); it is made up of water, earth, fire and air and is spherical (lit. circular on all sides). 1 The commentator Somesvara's statement that “the Earth, mother of all beings, stands 'motionless' in space” is against the teachings of Aryabhata. It is remarkable that Aryabhata, unlike the other astronomers, takes the Earth as made up of four elements, viz., earth, water, fire and air, only. The other astronomers take it as made up of five elements, viz., earth, water, fire, air and ether. EARTH COMPARED WITH THE KADAMBA FLOWER 7. Just as the bulb of a Kadamba flower is covered all around by blossoms, just so is the globe of the Earth surrounded by all creatures, terrestrial as well as aquatic. 2 INCREASE AND DECREASE IN THE SIZE OF THE EARTH srarfwto* *ptorffal ^ 1 f^^% 6 *r5qr g|q^rirqr^% 6 sift: H * II 8. During a day of Brahma, the size of the Earth increases externally by one yojana ; and during a night of Brahma, which is as long as a day, this growth of the earth is destroyed. 7 1. Cf PSi, xiii. 1 ; BrSpSi, xxi. 2 ; SiDVr, If, iv, 1 ; SiSe, xv. 22-23 ; SiSi, II, iii. 2 (a-b) ; Golasara, ii. 1. 2. Cf PSi, xiii. 2 ; SiDVr, II, iv. 6. 3. A. E. Gh. NI. Pa. sTT^lfe^vf 4. B. D. E. 5T^r 5. F. NI. Pa. *h- 6. So. *r*far for grf^ 7. The same statement occurs in SiDVr, II, v, 20 ; SiSi, II, iii. 62, Verses 9-10 ] APPARENT MOTION OF THE STARS Modern astronomers, too, believe in the growth of the Earth's size, but this growth, according to them is extremely insignificant. C. A. Young, in his Text Book on Astronomy, writes : “Since the earth is continually receiving meteoric matter, and sending nothing away from it, it must be constantly growing larger: but this growth is extremely insignificant It would take about 1000000000 years to accumulate a layer one inch thick over the earth's surface.” According to modern geologists, the rate of uplift of the earth varies from place to place and time to time. The minimum rate of uplift of the Himalayas is about 6 in. per century, 1 whereas the present rate of uplift of the earth in Greenland is 3 mm. per year. 2 APPARENT MOTION OF THE STARS DUE TO THE EARTH'S ROTATION ft^ 5 1 9. Just as a man in a boat moving forward sees the stationary objects (on either side of the river) as moving backward, just so are the stationary stars seen by people at Lanka (on the equator), as moving exactly towards the west. 10. (It so appears as if) the entire structure of the asterisms together with the planets were moving exactly towards the west of Lanka, being constantly driven by the provector wind, to cause their rising and setting. The theory of the Earth's rotation underlying the above passage was against the view generally held by the people and was severely criticised by Varahamihira (d. A.D. 587) and Brahmagupta (628 A.D.) The followers of Aryabhata I, who were unable to refute the criticism against the theory, fell in line with Varahamihira and others of his ilk and have misinterpreted the above verses as conveying the contrary 1. See D.N. Wadia, Geology of India, Macmillan and Company, London, 1949, p. 300 fn. 2. See Richard Foster Flint, Glacial and Pleistocene Geology, John Wiley and Sons, Inc., 1963, p. 256. GC-LA SECTION t Gota Sa. sense. See how the commentator Somesvara interprets the above verses : “Just as one seated on a boat sees the stationary objects such as trees etc. standing on the two sides of the river or sea moving in the contrary direction, in the same way those situated on the Earth rotating eastwards see the stationary stars located in the sky as moving in the opposite direction towards the west. Likewise, those living in Lanka see the stars as moving towards the west. Lanka is only a token, others also see in the same way. So, it is the Earth that moves towards the east; the stars are fixed. And that part of the circle of the asterisms which lies (at the moment) towards the east appears to rise, that which lies in the middle of the sky appears to culminate, and that which lies towards the west appears to set. Otherwise, the rising and setting of the stars is impossible.” After saying all this he adds : “This is the false view. For, if the Earth had a motion, the world would have been inundated by the oceans, the tops of the trees and castles would have disappeared, having been blown away by the storm caused by the velocity of the Earth, and the birds etc. flying in the sky would never have returned to their nests. So, there exists not a single trace of the Earth's motion. Hence this stanza must be interpreted in another way (as follows) : “Just as a man seated on a boat moving forward sees the stationary objects moving in the contrary direction, in the same way the asterisms driven by the provector wind, due to their own motion, see the objects at Lanka as moving in the opposite direction, i.e., they see the stationary Earth lying below as if it were rotating. Apparently also the asterisms rise in the east and move towards the west.” PrthUdaka (860 A.D.) in his commentary on the Brtihma-sphuta- siddhanta, supports Aryabhaja Fs theory of the Earth's rotation. The followers of Aryabha^a I, who misinterpreted Aryabhaja I, were, according to him, afraid of the public opinion which was against the motion of the Earth. It is noteworthy that the Greek astronomor Ptolemy (c. A. D. 100-178) holds that the Earth is stationary and does not move in any way locally. 1 1. See The Almagest, translated by R.C. Taliaferro, pp. 10-12. THE MERU MOUNTAIN 121 DESCRIPTION OF THE MERU MOUNTAIN Jfafarcro: 1 vwwt ft^T qftf^: I 11. The Meru (mountain) is exactly one yojana (in height). It is light-producing, surrounded by the Himayat mountain, situated in the middle of the N and ana forest, made of jewels, and cylin- drical in shape. The height of the Meru mountain taught here is quite different from the teachings of the Pura^as. It is also different from the teachings of the Buddhists and the Jaiuas. According to the Puranas, the Meru mountain is 84,000 yojanas high, of which 16,000 yojanas lie inside the Earth. 2 According to the Buddhists, it is 1,60,000 yojanas high, of which 80,000 yojanas lie sub- merged in water and 80,000 yojanas above the Earth. 3 According to the Jainas, it is 1,00,000 yojanas high, of which 1000 yojana* lie inside the Earth and 99,000 yojanas outside the Earth. 4 The commentator Nilakantha thinks that the above stanza is meant to refute the enormous size of the Meru advocated in the Purdnas and elsewhere. The commentators Bhaskara I, Somesvara and Raghunatharaja, however, reconcile the two views by interpreting the word Meru as meaning “the highest peak of the Meru mountain”. It seems that, according to the instruction of verse 8 above, the maximum uplift of the earth cannot exceed one yojana and so the height of any mountain cannot be greater than one yojana. This is perhaps the reason that Aryabhafa takes the height of the Meru mountain as one yojana only and not more. Combining the instructions given in A, i. 7 with those given above, we see that, according to Aryabhafa I, the Meru mountain is cylindrical in shape, with its diameter and height each equal to one yojana. 1. D. E. Ttf 2. See \ ayu-purana, ch. 34, gathn 1-45 ; ch. 35, gatha 11-32 ; Visnupurana, Am'sa 2, ch. 2, gStha 5-19 ; Markandeya-purSna, ch. 54, gmha 5-19 ' ; Afatsya-purana, ch. 113, gatha 4-40. ▼3. See Abhidharmakosa of Vasubandhu. 4. See LokaprakUsa, 18.15-16. A, fib. 16 122 GOLA SECTION I Sn - In calling the Meru mountain as 'light-producing* Aryabha^a I probably has in mind the 'northern lights' or the 'aurora', which Robert H. Baker describes in the following words : “Characteristic of many displays of the 'northern lights' of our atmosphere is a luminous arch across the northern sky, having its apex in the direction of the geomagnetic pole. Rays like searchlight beams reach upward from the arch, while bright draperies may spread to other parts of the sky, altogether often increasing its brightness from 10 to 100 times that of the ordinary night sky “The light of the aurora is believed to be produced by the streams of protons and electrons, which emerge from solar upheavals and are trapped by the Earth's magnetic field “Most of the light of an auroral display is produced in the colors green, red and blue by the combining of electrons with oxygen atoms and nitrogen molecules. . . . * n The Meru mountain is supposed to be made up of jewels of different colours because the light of an auroral display is of various colours. THE MERU AND THE BADAVAMUKHA 12. The heaven and the Meru mountain are at the centre of the land (i.e., at the north pole); the hell and the Badavamukha are at the centre of the water {i.e., at the south pole). 5 The gods (residing at the Meru mountain) and the demons (residing at the Badava- 1. See Robert H. Baker, Astronomy, East- West Student Edition, New Delhi, 1965, pp. 312-14. 2. E. Go. Pa. 3*: ; F. \® rev. to I Pr reads ^fe TOT- 3. F. «g*g*r; Pr. wro^ti 4. Ni. So. Ya. fe*RTT 5. C/. SiDVu H, iv s 4 ; VSt, Gola, vii. 11. Verse 13 ] THE FOUR CARDINAL CITIES 123 mnkha) consider themselves positively and permanently below each other. 1 The above statement is based on the conception that half of the Earth lying north of the equator is land and half of the Earth lying south of the equator is water. THE FOUR CARDINAL CITIES 13. When it is sunrise at Lanka, it is sunset at Siddhapura, midday at Yavakoti, and midnight at Romaka. 6 The time-distance relation v explained here with the help of four cities supposed to lie on the equator separated by one-quarter of the Earth's circumference. Lanka is supposed to be at the place where the meridian of Ujjayim (long. 75°.43 E., lat 23°.09 N) intersects the equator, Yavakoti 90° to the east of Lanka, Romaka 90” to the west of Lanka, and Siddhapura diametrically opposite to Lanka. POSITIONS OF LANK5. AND UJJAYINI 14. From the centres of the land and the water, at a distance of one- quarter of the Earth's circumference, lies Lanka ; and from Lanka, at a distance of one-fourth thereof, exactly northwards, lies Ujjayini. 8 1. Cf. PSi, xiii. 3 ; MSi, xvi. 7 (a-b). 2. F. ^« 3. So. ippflCTt 4. D. *Tt*W 5. B. fa^foffci ; Ni. So. fWwftW 6. Cf. PSi, x\^23. 7. Ni. SH. qs^IT%; Gh. Ni. Pa. Ra. SB. note both readings 8. The same statement is made in KR, u 33 (a-b); SiDVf, II, Bhuvfinakosp, 40 (c-d) ; SiSi, II, Bhuvgnako^a, 15 (a-b). 124 GOLA SECTION E QtAn ^ The positions of Laaks and Uifayim” have been give n because the Hindu prime meridian is supposed to pass through them. By stating the positions of Lanka and Ujjayinl, Aryabha t a has, by implication defined the position of the prime meridian. The distance of Ujjayinl from Lanka as stated in the above passage is one-sixteenth of the Earth's circumference. This makes the latitude of Uijayinl equal to 22° 30' N. This is in agreement with the teachings of the earlier followers of Aryabhata, such as Bhaskara P (A.D. 629), Deva* (A. D. 689), and Lalla* and the interpretations of the commentators Somesvara, Sttryadeva (b. A.D. 1191) and Paramesvara (A.D. 1431).* Even the celebrated Bhaskara IP (A.D. 1150) has chosen to adopt it. But Brahmagupta (A.D. 628) differed from this view. He takes Ujjayinl at a distance of one-fifteenth of the Earth's circumference from Lanka 6 , and likewise the latitude of Ujjayinl as equal to 24° N. Some of the commentators of Aryabhatiya, who favoured Brahmagup.ta's view, changed the reading taccaturafnie into pancadasamte. The commentator SOryadeva, who first interprets the original reading taccaturamie, later remarks : Ujjayinl tahkayah pdficadaiaMe samottaratah \ (/. e., Ujjayinl is at a distance of one-fifteenth of the Earth's circum- ference to the exact north of Lanka) is the proper reading because Brahmagupta writes : Lahkottarato 'vanti bhaparidheh pancadaiabhuge \ (i.e., Avantlis to the north of LankS at a distance of one-fifteenth of the Earth's circumference).” In defence of the reading pancadatemie, Suryadeva again says : 1. See his comm, on A, i. 7, where he gives the distance between Laftka and Ujjayinl as approximately equal to 200 yojanas. 2. See KR, i. 33 (a-b). 3. See SiDVr, II, Bhuvanakosa, 40 (c-d). 4. Paramesvara notes the other reading trs^srt^r also. 5. See SiSi, II S Bhuvanakosa , 15 (a-b). 6. See BrSpSi, xxi. 9 (c-d). Vetw 14 ] LANKA AND UJJAYINI 125 “24° to the north of Lanka lies Ujjayini. So, when the Sun is situated at the end of Gemini, then, due to its greatest decimation of 24°, it causes midday when it is exactly overhead at Ujjayini. In a place to the north of Ujjayinl, the Sun is never exactly overhead. To the south (of Ujjayini), it is exactly overhead when the Sun's north declination becomes equal to the latitude of the place. There- after it gets depressed towards the north. So the instruction of Ujjayinl for the knowledge of a place having a latitude equal to the Sun's greatest declination is appropriate. We Fdo not see any use in the instruction of Ujjayini lying at a distance of one-sixteenth of the Earth's circumference (to the north of Lanka), for its latitude being 22°30' N.. it is of no use anywhere So we have rightly said : Ujjayinl lahkayah pancadatttmse samoitaratah” The commentator Nilakant.ha (1500 A.D.) mentions the reading taccaturamie but adopts the reading pancadasamie taking it to be correct. Writes he : “Some read taccaturamie. According to them the word tat means one-fourth of the Earth's circumference, one-fourth of one-fourth is indeed one- sixteenth. So there is difference of meaning between the two. (However,) between facts there can be no option. So only one of the two readings is correct. Which of the two is correct can be decided upon from the equinoctial midday shadow at Ujjayini. That the janapada of Ujjayini lies at a distance of one-fifteenth of the Earth's circumference is well known from other works on astronomy. For the son of Jisnu (i.e., Brahma gupta) writes : 'Avanti is to the north of Lanka at a distance of one-fifteenth of the Earth's circumference*. So also writes Varahamihira, who belonged to Avanti : 'When the Sun is at the end of Gemini, it revolves 24° above the horizon of the gods ; and at Avanti it is then exactly overhead (at midday)'. This shows thalthe latitude there is 24°. Now 24° is one- fifteenth of aHircle and not one-sixteenth, because there are 360° in the whole circle and 24° is one-fifteenth of 360°. So the reading paftcadaiamte is the correct reading.” 126 GOLA SECTION [ Gola Sn. But he adds : “However, that janapada being large and the latitude being different at different places, somewhere (in that janapada) a latitude of 24° is also possible. Whether it occurs at Ujjayinl or not, can be decided (only) by the people there. Varaha- mihira has shown it to be 24° in respect of his village. Following him the son of Jisnu, too, has said the same. But Ujjayinl is to the south of that (village). There a latitude of 22£ degrees is also possible. In that case the other reading (taccaturamse) would be correct, for latitude has been stated (here) for Ujjayinl (and not for the village of Varahamihira).” The commentator Raghunatha-raja (1597 A.D.) adopts the reading pancadatamie. He interprets the reading taccaturamse also, but he prefers the other reading on the same grounds as given by Suryadeva. The majority of the Hindu astronomers, however, favours Brahmagupta's view and takes the latitude of Ujjayinl as 24° N. But there is no doubt that according to Aryabhaja I it is 22* 30' N. VISIBLE AND INVISIBLE PORTION OF THE BHAGOLA 15. One half of the Bhagola as diminished by the Earth's semi, diameter is visible from a level place (free from any obstructions). The other one. half as increased by the Earth's semi-diameter remains hidden by the Earth. 1 What is meant is that that portion of the Bhagola is visible at a place O on the Earth's surface which lies above the sensible horizon at O, i.e., which lies above the tangent plane to the Earth's surface at O, and that portion of the Bhagola which lies below the sensible horizon at O is invisible at O. 1. Cf. §WVr II, vi. 35, Veree 16 3 MOTION Of THE BHAGOLA 127 From this we easily deduce that according to Aryabhata I Sun's mean horizontal parallax = 3' 56* Moon's mean horizontal parallax=52' 30”, the corresponding modern values being 8“'794 and 57' 2'-7, respectively. MOTION OF THE BHAGOLA FROM THE NORTH AND SOUTH POLES 16. The gods living in the north at the Meru mountain (i.e., at the north pole) see one half of the Bhagola as revolving from left to right (or clockwise) ; the demons living in the south at the Badavamukha (i.e., at the south pole), on the other hand, see the other half as revolving from right to left (or anti- clokwise). 2 VISIBILITY OF THE SUN TO THE GODS, MANES AND MEN jiftmrari forc Jiftron f grafts » » 17. The gods see the Sun, after it has risen, for half a solar year ; so is done by the demons too. 3 The manes living on (the other side of ) the Moon see the Sun for half a lunar month ; 4 the men here see it for half a civil day. 5 This verse stating how long do the gods (living at the north pole), the demons (living at the south pole), the manes (living on the other side of the Moon) and men see the Sun after it has: once risen. 1. All others except So. trraapf ?r*rra 2. Cf. PSi, xiii. 9. 3. Cf PSi, xiii. 27. 4. Cf. PSi, xiii. 38. 5. Cf. PSi* xv. 14. 128 GOLA SECTION t GolaSn. 2. Khagola THE PRIME VERTICAL, MERIDIAN AND HORIZON 18. The vertical circle which passes through the east and west points is the prime vertical, and the vertical circle passing through the north and south points is the • meridian. The circle which goes by the side of the above circles (like a girdle) and on which the stars rise and set is the horizon. 1 P.C. Sengupta's remark that “here we have the rational horizon and not the apparent horizon” is incorrect. Since the centre of the Khagola is at the observer lying on the surface of the Earth, the horizon is evidently the apparent or sensible horizon and not the rational horizon. EQUATORIAL HORIZON 19. The circle which passes through the east and west points and meets (the meridian above the north point and below the south point) at distances equal to the latitude (of the place) from the horizon is the equatorial horizon (or six o' clock circle) on which the decrease and increase of the day and night are measured. 2 THE OBSERVER IN THE KHAGOLA 1. Cf. SiDVr, II, ii. 2 ; VSi, Gola, iv 2. ; SiSe, xvi. 29 (d) ; Si&t, II, vi. 3 (c-d) ; vii. 2 (o-d). 2. Cf BrSpSi, xxi. 50 ; SiDVr. II, ii. 3 ; VSi, Gola, iv. 3 ; SiSe, xvi. 30 ; SiSi, II, vi. 4 ; SuSi, II, iv, 4. 3. F. «rf, rev. to ^ Ver.se 22 ] THE AUTOMATIC SPHERE 129 20. The east. west line, the nadir-zenith line, and the north-south line intersect where the observer is. 1 What Aryabhata I means to say is that the centre of the Khagola lies at the observer, or that (the position of ) the observer forms the centre of the Khagola. THE OBSERVER'S DgtiMAlSfpALA AND DJZKKSEPAVRTTA ^ fiRR^ II R\ II 21. The great circle which is vertical in relation to the observer and passes through the planet is the drhmaniala (i. e., the vertical circle through the planet). The vertical circle which passes through that point of the ecliptic which is three signs behind the rising point of the ecliptic is the drkk$epavrtta. THE AUTOMATIC SPHERE {GOLA-YANTRA) toe* *Ni*gM src^ ^iw ^ w^m^ n w ii 22. The Sphere (Gola-yantra) which is made of wood, perfectly spherical, uniformly dense all round but light (in weight) should be made to rotate keeping pace with time with the help of mercury, oil and water by the application of one's own intellect. The Gola-yantra is the representation of the Bhagola. The method used by Aryabhata for rotating the Sphere {Bhagola) at the rate of one rotation per twentyfour hours may be briefly described in the words of the commentator SHryadeva as follows : “Having set up two pillars on the ground, one towards the south and the other towards the north, mount on them the 1. Cf. SiDVr, II, vi. 33-34. 2. E. Ni. Pa. So. <rn*r A, Bh. 17 30 GOL* SECTION t Gola Sn . ends of the iron needle (rod) (which forms the axis of rotation of the Sphere). In the holes of the Sphere, at the south and north poles, pour some oil, so that the sphere may rotate smoothly. Then, underneath the west point of the Sphere, dig a pit and put into it a cylindrical jar with a hole in the bottom and as deep as the circumference of the Sphere. Fill it with water. Then having fixed a nail at the west point of the Sphere, and having fastened one end of a string to it, carry the string downwards along the equator towards the east point, then stretch it upwards and carry it to the west point (again), and then fasten to it a dry hollow gourd (appropriately) filled with mercury and place it on the surface of water inside the cylindrical jar underneath, which is already filled with water. Then open the hole at the bottom of the jar so that with the outflow of water, the water inside the jar goes down. Consequently, the gourd which, due to the weight of mercury within it, does not leave the water, pulls the Sphere westwards. The outflow of water should be manipulated in such a way that in 30 ghatis (=12 hours) half the water of the jar flows out and the Sphere makes one-half of a rotation, and similarly, in the next 30 ghatis the entire water of the jar flows out, the gourd reaches the bottom of the jar and the Sphere performs one complete rotation. This is how one should, by using one's intellect, rotate the Sphere keeping pace with time.” 3. Spherical Astronomy (1) Diurnal motion THE LATITUDE-TRIANGLE 23. Divide half of the Bhagola lying in the visible half of the Khagola by means of Rsines (so as to form latitude-triangles). The Rsine of the latitude is the base of a latitude-triangle. The Rsine of the colatitude is the upright of the same (triangle). The statement “half of the Bhagola lying in the visible half of the Khagola,” implies that the radius of the Earth is disregarded 1. A. SRFHT *r*WM*: Verse 23 ] THE LATITUDE TRIANGLE 131 here and the centre of the Khagola is supposed to be coincident with the centre of the Bhagola. What is meant is the standard Khagola, i.e., Khagola for the centre of the Earth. A right-angled plane triangle whose sides are proportional to Rsin 6, Rcos 8 and R, where R (=3438') is the radius of the Bhagola, is called a latitude-triangle (aksa-k$etra). The right-angled plane triangle whose sides are equal to Rsin 6, Rcos 6 and R is the main latitude-triangle, defined above. The latitude-triangles play an important role in the solution of the spherical triangles in Indian astronomy. For, a number of results in astronomy are obtained simply by comparing two latitude- triangles. Because of this importance of the latitude-triangles, Arya- bha?a II (c. 950 A. D.) and Bhaskara II (1150 A.D.) have given a list of such triangles in their works. “It is only he who is versed in the latitude- triangles,” adds Bhaskara II, “that enjoys respect, fortune, fame, and happiness”. 1 The latitude-triangles (Aryabhata IPs list) Base (1) Rsin0 (2) equinoctial midday shadow (3) earthsine (4) unmav.$ala$ahku (5) other part of agm (6) agra (7) Rsin 8 Upright Rcos 8 gnomon (=12) Rsin 8 first part of agra unmandalaiahku samasahku Hypotenuse R hypotenuse of equinoc- tial midday shadow agm Rsin 8 earthsine taddhrti upper part of taddhrti samahaiiku (taddhrti— earthsine) (Bhaskara II' s additional triangle) (8) first part of agra upper part of upper part of samaiahku taddhrti SiSi f Grahaga^ita^ iii. 13 (c-d) 132 GOLA SECTION [ GcJa Sn. Explanation : When a heavenly body is on the six o* clock circle, the perpendicular dropped from it on the plane of the horizon is called unma^alaiahku ; the distance of the foot of the perpendicular from the east-west line is called the first part of agra ; the distance of the heavenly body from the rising-setting line is called the earthsine. When the heavenly body is on the prime vertical, the perpendicular dropped from it on the east-west line is called soma- Satiku ; the perpendicular dropped from it on the rising-setting line is called taddhrti ; the distance between the east-west line and the rising-setting line is called agra. When a perpendicular is dropped from the foot of the samafahku on the taddhrti, the latter is divided into two parts called upper and lower ; when a perpendicular is dropped from the foot of this perpendicular on the samasahku, the latter is divided into two parts called upper and lower; when from the same foot a perpendicular is dropped on the agm, the latter is divided into two parts called the 'first part of agra' and the 'other part of agra\ KADIUS OF THE DAY-CIRCLE 24. Subtract the square of the Rsine of the given declination from the square of the radius, and take the square root of the difference. The result is the radius of the day circle, whether the heavenly body is towards the north or towards the south of the equator. 2 That is, day radius = VR 2 — (Rsin S) 2 , (1) Rsin AxRsin 24° Rsin 8= “R” ~~ ' (2) “h and 8 being, respectively, the Sun's tropical longitude and declination of the heavenly body. Aryabhata does not state formula (2) for finding Rsin 8, because it can be easily derived by applying the rule of three as follows : “When the Rsine of the Sun's tropical longitude is equal to R, the Rsine of the Sun's declination is equal to Rsin 24°; what then 1. B. C. E. Sa. *T5T 2. Cf. MBh, iii. 6, V«m 25 ] RIGHT-ASCENSIONS OF THE SIGNS 133 will be the value of the Rsine of the Sun's declination when the Rsine of the Sun's tropical longitude has the value Rsin A ? The result is Rsin 8.” RIGHT ASCENSIONS OF ARIES, TAURUS AND GEMINI 25. Multiply the day radius corresponding to the greatest declination (on the ecliptic) by the desired Rsine (of one, two or three signs) and divide by the corresponding day radius : the result is the Rsine of the right ascension (of one, two or three signs), measured from the first point of Aries along the equator. 4 Let a, )3 and f denote the right ascensions of one sign, two signs and three signs, respectively, and Si, 8 2 and 8 3 the declinations at the last points of the signs Aries, Taurus and Gemini, respectively. Then _ . „ Rsin 30° X Rcos 24° Rsin «■= Rcos (1) _ . a Rsin 60° X Rcos 24° Rsin 8= — Rsin y=- Rcos 8, (-2) Rsin 90° X Rcos 24° Rcos 8 3 (3) Now, Rcos 8i -=3366', Rcos S 2 =3218' and Rcos 8,-3141'. Hence substituting these values and simplifying, we get a=1670' p«=3465' and y-5400'. Consequently, right ascension of Aries =<*= 1670 respirations right ascension of Taurus = p— * = 1 795 respirations right ascension of Gemini =y—P= 1935 respirations. 1. So. 2. Ra. SpTRT for wrstrt 3. Ra. So. srFKTT: 4. Cf. MBh, iii. 9. 134 GO LA SECTION [ Gola Sn. The right ascensions of Aries, Taurus and Gemini in the reverse order are the right ascensions of Cancer, Leo and Virgo ; and the right ascensions of the first six signs, Aries etc. in the reverse order are the right ascensions of the last six signs, Libra etc. Table 22. Right ascensions of the signs of the ecliptic Sign Right ascension in respirations Sign 1 Aries 1670 12 Pisces 2 Taurus 1795 11 Aquarius 3 Gemini 1935 10 Capricorn 4 Cancer 1935 9 Sagittarius 5 Leo 1795 8 Scorpio 6 Virgo 1670 7 Libra The Indian method for deriving formula (1) is as follows : Consider the Celestial Sphere for [a place on the equator. Let the first point of Aries coincide with the east point of the horizon ; and let A be the last point of the sign Aries, AB the perpendicular from A on the eastwest line, and AC the perpendicular from A on the plane of the horizon. Also let G be the last point of the sign Gemini, GO the perpendicular from G on the east-west line and GM the perpendicular from G on the plane of the horizon. Then comparing the triangles ABC and GOM, which are evidently similar, we have ABxGM _ Rsin 30°xRcos 24° ~ R Now Rsin a : R : : AC : Rcos 8. Therefore ACXR _ R sin 30° X R cos 24 ' Rsin<X R~co78 Rcos 8 The rationales of formulae (2) and (3) are similar, Verte 26 ] EARTHSINE EARTHS INE 26. The Rsine of latitude multiplied by the Rsine of the given decli- nation and divided by the Rsine of colatitude gives the earthsine, lying in the plane of the day circle. This is also equal to the Rsine of half the excess or defect of the day or night (in the plane of the day circle). 2 That is, - . Rsin 8 X Rsin <f> earthsine = „ 7 — Rcos This result may be easily obtained by comparing the following latitude- triangles ; Base Upright Hypotenuse (1) earthsine Rsin S agrn (2) Rsin <f> Rcos j> R By the 'excess or defect of the day or night' is meant the amount by which the day or night at the local place is greater or less than 30 ghatis (or 12 hours). The earthsine, as the text says, is the Rsine of half the excess or defect of the day or night in the plane of the day circle. Since the time is measured on the equator, one should first find the corresponding Rsine in the plane of the equator and then reduce that to the arc of the equator. The Rsine of half the excess or defect of the day or night in the plane of the equator is called carardhajya and is obtained by the following formula : „ . earthsine xR ctrardhajyn^- day radius The corresponding arc of the equator is called carclrdha and gives the amount by which the semi-duration of the day or night at the local place is greater or less than 15 ghatis. 1. F. isTcfiT for f^T 2. Cf. PSi, iv. 34 ; MBh,,iiu 6. 136 GOLA SECTION t (Sola Sft. The cqrardha is also equal to the difference between the oblique and right ascensions and so it is called the 'ascensional difference'. The oblique ascension is the time of rising of an arc of the ecliptic at the local place and the right ascension is the time of rising of an arc of the ecliptic at the equator. RISING OF THE FOUR QUADRANTS AND OF THE INDIVIDUAL SIGNS 27. The first as well as the last quadrant of the ecliptic rises (above the local horizon) in one quarter of a sidereal day diminished by (the ghafis of) the ascensional difference. The other two (viz. the second and third quadrants) rise in one quarter of a sidereal day as increased by the same (i.e. the ghatls of the ascensional difference). The times of rising of the individual signs (Aries, Taurus and Gemini) in the first quadrant are obtained by subtracting their ascensional differences from their right ascensions in the serial order ; in the second quadrant by adding the ascensional differences of the same signs to the corresponding right ascensions in the reverse order. The times of risings of the six signs in the first and second quadrants (Aries, etc.) taken in the reverse order give the risings of the six signs in the third and fourth quadrants (Libra, etc.). 4 Let Fig. 10 represent the Celestial Sphere (Khagola) for the local place. SEN is the horizon, RET the equator, UEV the ecliptic and PEQ the equatorial horizon. The small circle WBV is the day circle through V (the end of the first quadrant of the ecliptic). EV is the first quadrant of the ecliptic. At the moment the first point of Aries coincides with E. With the motion of the Celestial Sphere E will move along the equator and V along the diurnal circle X E. af|cft 3. A. E. Gh. Go. NI. Pa. ^Ht?^«TcT: 4. Cf. PSi, iv. 31 ; LBh, ifi. 6. Vene 27 ] RISINGS 'OF THE SIGNS 137 Fig. 10 in the direction of the arrowhead. When V reaches A, the whole of the first quadrant of the ecliptic, which is at the moment on the point of rising above the horizon, will be above the local horizon. So the time of rising of the first quadrant of the ecliptic at the local place is the time taken by V in moving from V to A, or, what is the same thing, the time taken by T in moving from T to C. This time is given by the arc TC or ET-EC of the equator (because time is measured on the equator). Since ET is one-fourth of the equator, it corresponds to one-quarter of a sidereal day. So the time of rising of the first quadrant of the ecliptic =one quarter of a sidereal day— ghafls corresponding to arc EC -=15 ghatfs—ghapis of ascensional difference. (1) Also, the first quadrant of the ecliptic is, at the moment, at the point of rising above the equatorial horizon QEP. When the point V reaches the point B, the first quadrant of the ecliptic will be completely above the equatorial horizon. The time taken by V to reach B is equal to the time taken by T in reaching E. So the time of rising of the first quadrant of the ecliptic above the equatorial horizon is equal to the arc ET of the equator, which has been just shown to correspond to one quarter of a sidereal day or 15 ghapls. This differs from (1) by the time given by the arc EC of the equator. EC therefore gives the difference between the times of rising of the first quadrant at the local and equatorial places. EC is, therefore, called the 'ascensional difference' of the first quadrant (or the ascensional difference of the last point of the first quadrant)* A. Bh. 18 38 GOLA SECTION [ Gola Sn. Hence, from (1), we have (1) Time of rising of the first quadrant at the local place ■=15 ghatis—ghatis of the ascensional difference. When the first point of Aries is at E, the first point of Libra is at the west point W. The first point of Libra will reach the point E exactly after 30 ghofis and then the second quadrant of the ecliptic will be completely above the local horizon. Hence we have (2) Time of rising of the second quadrant of the ecliptic at the local place =30 ghatis -(15 ghatis—ghatis of asc. diff.) = 15 ghatis+ghafis of asc. diff. Similarly, we can show that (3) Time of rising of the third quadrant of the ecliptic at the local place = 15 ghapls+ghatts of asc. diff. (4) Time of rising of the fourth quadrant of the ecliptic at the local places 1 5 ghatis - ghafis of asc. diff. Proceeding exactly in the same manner, we can show that Time of rising of the sign Aries at the local place = Time of rising of the sign Aries at the equator— asc. diff. of tha last point of Aries. (2) Time of rising of the signs Aries and Taurus at the local place =Time of rising of the signs Aries and Taurus at the equator— asc. diff. of the last point of Taurus. (3) Time of rising of the signs Aries, Taurus and Gemini at the local place — Tune of rising of the signs Aries, Taurus and Gemini at the equator— asc. diff. of the last point of Gemini. (4) Diminishing (2), (3), (4), each by the preceding (if any), we have Time of rising of the sign Aries at the local place ==Time of rising of the sign Aries at the equator— asc. diff. of Aries. Time of rising of the sign Taurus at the local place Time of rising of the sign Taurus at the equator— (asc. diff. of the last point of Taurus— asc. diff. of the last point of Aries) =0!ime o£ rising: of the sign Taurus at the equator— asc, diff. of Taurus. Vetse 28 ] RSINE OF THE ALTITUDE 139 Time of rising of the sign Gemini at the local place = Time of rising of the sign Gemini at the equator— (asc. diff. of the last point of Gemini— asc. diff. of the last point of Taurus) =*Time of rising of the sign Gemini at the equator— asc. diff. of Gemini. Let A, B, C be the times of rising of the signs Aries, Taurus, and Gemini at the equator and a, b, c the ascensional differences of the same signs in their respective order. Then the times of rising of the signs at the local place are as shown in the following table. Table 23. Times of rising of the signs at the local place Sign Time of rising Sign I Aries A- a 12 Pisces 2 Taurus 11 Aquarius 3 Gemini C-c 10 Capricorn 4 Cancer C+c 9 Sagittarius 5 Leo B+£ 8 Scorpio 6 Virgo A+a 7 Libra RSINE OF THE ALTITUDE 28. Find the Rsine of the arc of the flay circle from the horizon (up to the point occupied by the heavenly body) at the given time ; multiply that by the Rsine of the colat itufle and divide by the radius : the result is the Rsine of the altitude (of the heavenly body) at the given time elapsed since sunrise in the forenoon or to elapse before sunset in the afternoon. By 'the Rsine of the arc of the day circle from the horizon up to the point occupied by a heavenly body', is meant the distance of the heavenly body from the rising-setting line, irhich i& known as itfuhfti . 140 GOLA SECTION I Gola Sn, Thus the formula in the text may be stated as Rsin (Sun's altitude) — R This formula may be obtained by comparing the following latitude-triangles : Base (1) fahkvagra or iahkutala (2) Rsin^ The method intended by Ayabhaja I may be fully explained in the case of the Sun as follows : “With the help of the Sun's declination and the local latitude calculate the Sun's ascensional difference. Subtract the Sun's ascensional difference from or add that to the given time reduced to asus (1 ghati=360 asus), according as the Sun is in the northern or southern hemisphere. By the Rsine of that difference or sum multiply the day radius and divide by the radius. If the Sun is in the northern hemisphere, add the earthsine to the result obtained ; if the Sun is in the southern hemisphere, subtract the earthsine from the result obtained : the result is the istahrti. Multiply that by the Rsine of the colatitude and divide by the radius: the result is the Rsine of the Sun's altitude.” 'When the Sun is the northern hemisphere and the given time reduced to asus is less than the Sun's ascensional difference reduced to minutes of arc, one should proceed as follows : 'Subtract the asus of the given time from the minutes of the Sun's ascensional difference ; multiply the difference by the day radius and divide by the radius. Subtract whatever is obtained from the earthsine : the result is the istahrti. Multiply that by the Rsine of colatitude and divide by the radius ; the result is the Rsine of the Sun's altitude as before'.” 1 Upright Hypotenuse Rsin (Sun's altitude) istahrti Rcos tj> R %. See MBh> \\\. 13-20, 25, Also see PSi, iv, 41-43. Vetse 29-30 ] 8ASrKVAGRA AND AGRA Ml 4 AN KV AGRA 29. Multiply the Rsine of the Sun's altitude for the given time by the Rsine of latitude and divide by the Rsine of colatitude : the result is the Sun's r $ahkvagra 9 which is always to the south of the Sun's rising.setting line. 8 The Sun's iahkvagra is the distance of the Sun's projection on the plane of the observer's horizon from the Sun's rising-setting line. Or, it is the projection of the itfahrti on the plane of the observer's horizon. The formula stated in the text is „ , _ Rsin (Sun's altitude) X R sin ^ Sun s sankvagra— Rcos ^ — , which can be easily derived by comparing the following latitude- triangles : Base Upright Hypotenuse (1) Sun's iahkvagra Rsine (Sun's altitude) iftahrti (2) Rsin<£ Rcos^ R Although the rule is stated for the Sun, it is applicable to any heavenly body whatsoever. SUN'S AGRA 30. Multiply the Rsine of the (Sun's tropical) longitude for the given time by the Rsine of the Sun's greatest declination and then divide by the Rsine of colatitude : the resulting Rsine is the Sun's agra on the eastern or western horizon. 3 1. Gh. ft|3T: for gPrs: 2. Cf. MBh, iii. 54 ; LBh, iii. 16. 3. Cf. MBh, iii. 37 ; LBh, iii. 21. 142 GOLA SECTION [ Gok Sn. The Sun's agra is the distance of the rising or setting Sun from the east-west line. The formula stated in the text is Rsin AxRsin 24° Sun's agra* Rcos $ where A is the Sun's tropical longitude, and ^ the latitude of the place. This formula may be obtained as follows : Comparing the latititde-triangles : Base Upright Hypotenuse (1) earthsine Rsin S agra (2) Rsin^ Rcos^ R we get « > , Rsin SxR Sun's agr9= — — . Rcos <f> But Rsin A x Rsin 24° Rsin 8 = K Therefore , _ Rsin AxRsin 24° Sun's agra=- R cos tj> RSINE OF THE SUN'S PRIME VERTICAL ALTITUDE ftp33q*n &*raaT <$ik II « ii 31. When that (agra) is less than the Rsine of the latitude and the Son is in the northern hemisphere, multiply that (Sun's agra) by the Rsine of colatitude and divide by the Rsine of latitude : the result is the Rsine of the Sun's altitude when the Sun is on the prime vertical. 1 U Cf. MBh, ifr 37 (c-d)-3$. V«« 31 ] SUN'S PRIME VERTICAL ALTITUDE 143 That is, . Sun's agm X Rcos Rsm a= % . . Rsm <f> where a is the Sun's prime vertical altitude: This formula may be easily derived by comparing the following latitude-triangles : Base Upright Hypotenuse (1) agra Rsin a itfahrti (2) Rsin <j> Rcos <j> R The conditions necessary for the existence of the prime vertical altitude of the Sun are : (1) that the Sun should be in the northern hemisphere, and (2) that the Sun's declination should be less than the latitude of the place. The condition given by Aryabhata that the Sim's agra should be less than the Rsine of the latitude is incorrect. Brahmagupta (A.D. 628) has therefore rightly criticised Aryabhata on this account : “The statement (of Aryabhata) that the Sun, in the northern hemisphere, enters the prime vertical when the (Sun's) agra is less than the Rsine of the latitude is incorrect, because, this happens when the Rsine of the (Sun's) declination satisfies this condition (and not the Sun's agra)” 1 It is interesting to note that the commentator Bhaskara I (A.D. 629), committed the same error in his Maha-Bhaskanya 2 , but he has corrected himself in his Laghu-Bhaskarxya? Snryadeva (b. A.D. 1191), SomeSvara,, and other commentators, however, have interpreted the word sa as referring to the Sun's declination and mot to the Sun's agra. Although the rule in vss. 30-31 is stated for the Sun, it is appli- cable to any heavenly body whatsoever. 1. BrSpSi, xi. 22, 2. MBh, iii. 37. 3. LBh, iii. 22, 144 GOLA SECTION t Gola Sn. SUN'S GREATEST GNOMON AND THE SHADOW THEREOF ftfosns mwjH\ *n wit m <rct irt^agp i 32. The Rsine of the degrees of the (Sun's) altitude above the horizon (at midday when the Sun is on the meridian) is the greatest gnomon (on that day). The Rsine of the (Sun's) zenith distance (at that time) is the shadow of the same gnomon. The Sun's zenith distance at midday =^~S or <f>±S, according as the Sun is in the northern or southern hemisphere. Consequently, the greatest gnomon or the Rsine of the Sun's altitude at midday =Rcos(^~8) or Rcos and the shadow of the greatest gnomon or the Rsine of the Sun's zenith distance at midday = Rsin(^-S) or Rsin f S), according as the Sun is in the northern or southern hemisphere. (2) Parallax in a solar eclipse RSINE OF THE ZENITH DISTANCE OF THE CENTRAL ECLIPTIC POINT ^^mf^ft^^ w^?: MM 33. Divide the product of the madhyajya and the udayajya by the radius. The square root of the difference between the squares of that (result) and the madhyajya is the (Sun's or Moon's) own dfkksepa? 1. F. STf^te^ 2. Cf. PSi, ix, 19-20 ; MBh, v. 19 s Verse 33 ] D$KKSEPA 145 The Sun's madhyajya is the Rsine of the zenith distance of the meridian ecliptic point. The Sun's udayajya is the Rsine of the amplitude of the rising point of the ecliptic. The Sun's drkk$epa( jya) is the Rsine of the zenith distance of that point of the ecliptic which is at the shortest distance from the zenith. The Moon's madhyajya, is the Rsine of the zenith distance of that point of the Moon's orbit which lies on the observer's meridian. The Moon's udayajya is the Rsine of the amplitude of that point of the Moon's orbit which lies on the eastern horizon of the observer. The Moon's drkksepa(jya) is the Rsine of that point of the Moon's orbit which is at the shortest distance from the zenith. Let Z be the zenith, M the meridian ecliptic point and C that point of the ecliptic which is at shortest distance from the zenith. Then in the triangle ZCM Rsin (arc ZM) = Sun's madhyajya, Z ZCM =90°, and Rsin (MZC) = Sun's udayajya* Therefore Rsin (arc ZM) x Rsin (MZC) Rsin (arc MC) = ^ — Sun's madhyajya x Sun's udayajya ~~ R { The final result, viz. Sun's drkk§epajya — V (Sun's madhyajya) 2 — (Rsin MC) a is obtained by treating the triangle formed by the Rsines of the sides of the triangle ZCM as a plane right-angled triangle (which assumption is however incorrect). i, The Moon's drkk$epajya has been similarly obtained by taking i the Moon's orbit in place of the ecliptic. ! Brahmagupta has rightly criticised the above rule for being ' inaccurate. 1 1. See BrSpSi, xi. 29-30, A. Bh, 19 146 GOLA SECTION [ Gola Sn. DQGGATIJY1S OF THE SUN AND THE MOON 34. (i) The square Toot of the difference between the squares of (i) the Rsiue of the zenith distance (of the Sun or Moon) and (ii) the drkkfepajya, is the (Sun's or Moon's) own drggatijya. 1 The Sun's drggatijya is the Rsine of the arcual distance of the zenith from the secondary to the ecliptic passing through the Sun. The Moon's drggatijya is the Rsine of the arcual distance of the zenith from the secondary to the Moon's orbit passing through the Moon. The formula for the (Sun's or Moon's) drggatijya stated in the text is drggatijya= v^Rsm (z.d.)] 2 — {drkk$epajya)* . This formula is correct and can be proved as follows : Let CS be the ecliptic and K its pole ; S the Sun and Z the zenith ; KZC and KS the secondaries to the ecliptic ; and ZA the perpendicular to KS. Since the arcs ZC and ZA are perpendicular to CS and AS respectively, (Rsin ZA) a =(Rsin ZS) a -(Rsin ZC) a , i.e., (Sun's drggatijya)^ (Sun's drgjyaf— (Sun's drkk§epajya)\ Similarly, (Moon's drggatijya)* -(Moon's drgjya)*- (Moon's drkfyepajya)*. According to Brahmagupta (A.D. 628), this is wrong. Says he : “Dxkksepajya is the base and drgjya the hypotenuse ; the square root of the difference between their squares is the drnnatijya (= drggatijya). This configuration is also improper.” 2 Brahmagupta's criticism is valid if the drggati means “the arc of the ecliptic lying between the central ecliptic point and the Sun or Moon” as explained by the commentator SOryadeva. 1. Cf. MBh, v. 23 ; LBh, v, 7(c-d)-8 (a-b). J. BrSpSi t J&27* Vetse 34 ] PARALLAX PARALIAX OF THE SUN AND THE MOON 347 f%ft§r ^mi 1 y*mA *qta*n^ n ^ n 34. (ii) On account of (the sphericity of) the Earth, parallex increases from zero at the zenith to the maximum value equal to the Earth's semi-diameter (as measured in the spheres of the Sun and the Moon) at the horizon. The word drkchaya in the text means parallax. The instruction of the text implies, according to the commen- tators, the following formulae : „ . , , A , Earth's semi-diameter Xdrggatijya parallax in longitude— yojanas in the sphere of the planet concerned Earth's semi-diameter X drggatijya . = — i it- : tt-7 . . — minutes. planet s true distance in yojanas Earth's semi-diameter xdrkksepajya parallax in latitude = R - yojanas _ Earth's semi-diameter x drkksepajya m | flUtes — planet's true distance in yojanas 1 e * On the use of the word svadrkksepa, Bhaskara I observes : “The orbits of the Sun and the Moon being different, the (five) Rsines (viz., udayajya, madhyajya, drkksepajya t drgjyS and drggatijya) for them are said to differ. This difference is indicated by the words 'svadfkksepa' etc. of the Master (Aryabhata I).” a 1. E. msjj&wn 2. MBh, v. 12, 148 GOLA SECTION t Gcla Sn. 2. The visibility corrections VISIBILITY CORRECTION AKSAD&KKARMA FOR THE MOON 35. Multiply the Rsine of the latitude of the local place by the Moon's latitude and divide (the resulting product) by the Rsiue of the colatitude : (the result is the ak$adrkkarma) for the Moon). When the Moon is to the north (of the ecliptic), it should be subtracted from the Moon's longitude in the case of the rising of the Moon and added to the Moon's longitude in the case of the setting of the Moon ; when the Moon is to the south (of the ecliptic), it should be added to the Moon's longitude (in the case of the rising of the Moon) and subtracted from the Moon's longitude (in the the case of the setting of the Moon). 3 That is aksadrkkarma Rsin ^ X Moon's latitude Fig. 11 1. A.B.C.E.F. NI. Ra. SH. vfkm; D. fa^T 2. D. ^*^ff 3. The same rule occurs in PSi (Paulisa), [v, 8 ; KK, I, vi. 3 ; MSh, vi. l-2(a-b) ; IBh, yi. 1-2 ; Verse 36 ] VISIBILITY CORRECTIONS 149 Let the figure represent the Celestial Sphere (Khagola) for the local place in latitude SEN is the eastern horizon and Z the zenith : YE is the equator and P its north pole ; yT is the ecliptic and K its north pole. Suppose that the Moon is rising at the point M' on the horizon. Let M be the point where the secondary to the ecliptic drawn through M' meets the ecliptic, L the point where the hour circle through M' meets the ecliptic and T the point where the horizon intersects the ecliptic. Then the arc TL of the ecliptic is called the ak$adrkkarma and the arc LM of the ecliptic is called the ayanadrkkarma. Let A be the point where the diurnal circle through M intersects the hour circle through M' and B the point where the diurnal circle through M intersects the horizon. Then, since MM' is small, regarding the triangle M'AB as plane, we have _ a -rj Rsin (BM'A)xM'A _ v arc AB = approx. Rsin (M BA) F _ Rsin (BM'A) X M'M Rsin (M'BA) Rsin ^X Moon's latitude Rcos <f> ' Assuming the ak$adrkkarma as roughly equal to arc AB, Aryabhafa gives Rsin 6 X Moon's latitude ak$adrkkarma— Rcos <f> This rule is generally used when the celestial latitude of the body concerned is small. When the celestial latitude is large, a more accurate rule is prescribed. 1 VISIBILITY CORRECTION AYANADRKKARMA OF THE MOON 36. Multiply the Rversed sine of the Moon's (tropical) longitude (as increased by three signs) by the Moon's latitude and also by the (Rsine of the Son's) greatest declination and divide (the resulting 1. See BrSpSi, x. 18-19; &DV r , /, xi. 12-13 ; and Si&i I, vii. 6. Bhaskara II gives a slightly modified formula for small celestial latitude also. See SiSi, I, vii, 7. The most accurate formula occurs in SiTV, vii, 103-104, ISO GOLA 8ECTION [ Gola Sn. product) by the square of the radius. When the Moon's latitude is north, it should be subtracted frcm or added to the Moon's longitude, according as the Moon's ayana is north or south (i.e., according as the Moon is in the six signs beginning with the tropical sign Capricorn or in those beginning with the tropical sign Cancer) ; when the Moon's latitude is south, it should be added or subtracted, (respectively). 1 That is Rvers (M +90°) X Moo n's latitude x Rsin 24° R a where M is the Moon's tropical longitude. ayanadtkkarma— a The rationale of the formula is as follows : From triangle M'MA (See Fig. 11, p. 148), we have ' _ . JA Rsin (MM'A)x Rsin (arc MM') arc MA=— — — — - ^ approx. K ayanavalanax Moon's latitude = ^ approx. But (vide infra vs. 45), we have Rvers (M-f90°)xR sin 24° ayanavalana— .'.arc MA R Rvers (M f 90°) x Moon's latitude X Rsin 24° R 2 Assuming the arc LM of the ecliptic (which denotes the ayana- drkkarma) as approximately equal to arc MA, we have R vers (M+90°) x Moon's latitude X Rsin 24° ayanadrkkarma= - R 2 When the ayanadrkkarma and ak$adrkkarma are applied to the rising or setting Moon, we get the longitude of that point of the ecliptic which rises or sets with the Moon. There is difference of opinion regarding the interpretation of the word utkramanam. The commentator Somesvara interprets it as meaning “The Rversed sine of the Moon's longitude as increased by three signs”, whereas the commentators Bhaskara I, SOryadeva and Paramesvara interpret it as meaning “The Rversed sine of the 1. The same rule occurs in KK, /, vi. 2 ; MBh, vi. 2 (c-d)-3 ; LBh, vi. 3-4 ; KR, v. 3. More accurate formulae occur in Si&' f I, vii. 4, 5 an4 in $iTV, vii. 77-80, Verse 38 ] OCCURRENCE OF AN ECLIPSE 151 Moon's longitude as diminished by three signs.” 1 The commentator Raghunatha-raja interprets it as meaning Rvers (M+90°) or Rvers (M— 90°), according as the desired ayana commences with Capricorn or with Cancer. We have followed Somesvara's interpetation, because it agrees with the teachings of Aryabhata in stanza 45 below and also because it agrees with the teachings in his midnight system. 2 Brahmagupta has modified this rule by replacing the Rversed sine of the Moon's longitude as increased by three signs by the Rsine of the same. The commentator Nllakantha, however, interprets the word utkramanam itself as meaning “the Rsine of the complement of the Moon's longitude”. (4) Eclipses of the Moon and the Sun CONSTITUTION OF THE MOON, SUN, EARTH AND SHADOW AND THE ECLIPSERS OF THE SUN AND MOON 37. The Moon is water, the Sun is fire, the Earth is earth, and what is called Shadow is darkness (caused by the Earth's Shadow). The Moan eclipses the Sun and the great Shadow of the Earth eclipses the Moon. The statement that the Moon is water has proved false. OCCURRENCE OF AN ECLIPSE q^T5% ^Tf^#^ jpr^ ii 3= n 38. When at the end of a lunar month, the Moon, lying near a node (of the Moon), enters the Son, or, at the end of a lunar fortnight, enters the Earth's Shadow, it is more or less the middle of an eclipse, (solar eclipse in the former case and lunar eclipse in the latter case). 1. Govinda-sv2mi, too, says the same thing. Writes he : «r?r vfi [wsst^t] ^rerf^nRfre^g^iT i See his comm. on MBh, vi. 3. 2. See KK, I, vi. 2. 3» So* ^^«w*nfr 152 GOLA SECTION t Gola Sn. Aryabhata evidently takes the time of conjunction of the Sun and Moon as the middle of a solar eclipse, and the time of opposition of the Sun and Moon as the middle of a lunar eclipse. This is only approximately true. The phrase “more or less”, according to the commentators, is indicative of the fact that, on account of parallax, the time of apparent conjunction is not exactly the same as that of geocentric conjuction. LENGTH OF THE SHADOW Multiply the distance of the Sun from the Earth by the diameter of the Earth and divide (the product) by the difference between the diameters of the Sun and the Earth : the result is the length of the Shadow of the Earth (i.e. the distance of the vertex of the Earth's shadow) from the diameter of the Earth (i.e. from the centre of the Earth). 8 39. That is, length of Earth's Shadow = Su n's distance X Earth's diameter Sun'slliameter— Earth's diameter The Hindu method for deriving this formula, called “The lamp and Shadow method” (pradipacchaya-karma), is as follows : Consider the figure below. S is the centre of the sun and E that of the Earth. SA and EC are drawn perpendicular to SE and denote the semi-diameters of the Sun and the Earth, respectively. BC is parallel to SE. V is the point where SE and AC produced meet each other. 1. C. E. NI. S3. ^fcfcr 2. Cf. BrSpSi, xxiii 8 ; MBh, v. 71 i LBh, iv. 6, Vette 40 ] EARTH'S SHADOW 153 Hindu astronomers compare SA with a lamp post, EC with a gnomon, and EV with the shadow cast by the gnomon due to the light of the lamp. Consequently, they call EV 'the length of the Earth's shadow from the diameter of the Earth'. The triangles CEV and ABC are similar ; therefore EV = BC SE EC AB~ SA— EC ™ r SE X EC SEX 2 EC EV— — SA— EC 2 S A- 2 EC ■ . , 4t fD .., . . „ Sun,'s distance X Earth's diameter /.*?., length of Earth's shadow= _ , „ * ° Sun s diameter— Earth's diameter * EARTH'S SHADOW AT THE MOON'S DISTANCE ft mtf ft^ aw ^Prs
^ 2 ii a ° ii 40. Multiply the difference between the length of the Earth's shadow and the distance of the Moon by the Earth's diameter and divide (the product) by the length of the Earth's shadow : the result is the diameter of the Tamas (i. the diameter of the Earth's shadow at the Moon's distance). 3 That is Diameter of Tamas (length of Ea rth's shadow— Moon's distance) X Earth's diameter length of Earth's shadow * The rationale of this formula is as follows : See the previous figure (Fig. 12). M is the position of the Moon when it is just on the point of entering the Earth's Shadow. MD is perpendicular to SE ; so it denotes the semi-diameter of the Earth's shadow at the Moon's distance, i.e., the semi-diameter of the Tamas. 1. B. D. ^TfTWTW 2. Pa. Frfirarwr: 3. C/. BrSpSi, xxiii. 8-9 ; MBh, v. 72 ; LBh, iv. 7. A.Bh. 20 154 GOL A SECTION [ Gola Sn; The triangles MDV and CEV are similar ; therefore MP „ CE DV EV ™ (EV— ED)xCE or MD=A — EV ^ . xjrr , (EV — ED) X 2 CE (EV - EM) X 2 CE or 2 MD= * £y approx. i.e., Diameter of Tamas _ (le ngth of Earth's shadow— Moon's distance) X Earth's dameter . ” length of Earth's shadow approx. HALF-DURATION OF A LUNAR ECLIPSE 41. From the square of half the sum of the diameters of that (Tamas) and the Moon, subtract the square of the Moon's latitude, and (then) take the square root of the difference ; the result is known as half the duration of the eclipse (in terms of minutes of arc). The corresponding time (in ghatxs etc.) is obtained With the help of the daily motions of the Sun and the Moon. 3 That is, Half the duration of a lunar eclipse = */a*— pa minutes of arc „ 60XVV-P', gkaris a where a=sum of the semi-diameters of Tamas and Moon p= Moon's latitude tf=(Moon's daily motion-Sun's daily motion) in minutes of arc. 1. F. Pa. $ h 2. Gh. So. fa£<reFTcT ; others M«TW *F»Rf *3. A. B. D. NI. ^fwnt^r 4. A. D. Ni.' farwf 3^ 5. Cf, MBh, v. 74-76 (a-b) ; LBh, iy. 10-12 ; KK, I, iv. 4, Verse 42 ] LUNAR ECLIPSE 155 This gives only an approximate value of the semi-duration of the eclipse. To obtain the best approximation, the process should be iterated until the value of the semi-duration is fixed. For details, see M Bh, v. 75-76. HALF-DURATION OF TOTALITY OF A LUNAR ECLIPSE 42. Subtract the semi-diameter of the Moon from the semi-diameter of that Tamas and find the square of that difference. Diminish that by the square of the (Moon's) latitude and then take the square root of that : the square root (thus obtained) is half the duration of totality of the eclipse. 2 That is, half the duration of totality = vV— minutes of arc = -j= — — ghatis, where s*= semi-diameter of Tamas— semi-diameter of Moon p= Moon's latitude (Moon's daily motion— Sun's daily motion) in minutes of arc. This also gives only a rough approximation. To obtain the best approximation, the process should be iterated until the semi-duration of totality is fixed. THE PART OF THE MOON NOT ECLIPSED fNNTO^} i§Rt g^Tf^f 3 ii h n 43. Subtract the Moon's semi-diameter from the semi-diameter of the Tamas ; then subtract whatever is obtained from the 1. F. transposes this verse to after 44. 2. Cf. PSi, x. 7 ; MBh, v. 76(c-d) ; LBh, iv. 14 ; KK, I, iv. 4. 3. NT. The text in the TSS edn. adds ig^suTTT, which is not warranted by the Com. of Ni. Moreover, it is metrically superfluous. 56 GOLA SECTION [ Qota Sn. Moon's latitude : the result is the part of the Moon not eclipsed (by the Tamas). 1 That is, the length of the Moon's diameter which is not eclipsed -=Mbon's latitude— (semi-diameter of Tamas— Moon's semi-diameter). It is easy to see that the obscured part of the Moon's diameter (at the time of opposition of the Sun and Moon in a partial lunar eclipse) =4 (diameter of ra/mw+diameter of Moon)— Moon*s latitude, and hence the unobscured part of the Moon's diameter at that time —Moon's diameter — {^(diameter of Tamas-]- diameter of Moon) — Moon's latitude} —Moon's latitude— (semi-diameter of Tamas- semi-diameter of Moon). As stated earlier, Aryabhaja I does not make any distinction between the time of opposition and the time of the middle of the eclipse. Hence the above rule, MEASURE OF THE ECLIPSE AT THE GIVEN TIME 44. Subtract the ispa from the semi-duration of the eclipse ; to (the square of) that (difference) add the square of the Moon's latitude (at the given time) ; and take the square root of this sum. Subtract that (square root) from the sum of the semi, diameters of the Tamas and the Moon: the remainder (thus obtained) is the measure of the eclipse at the given time. The term ista denotes, says the commentator Sflryadeva, the Moon's motion (in longitude) relative to Tamas corresponding to the time elapsed at the given time, since the first contact, or to elapse at the given time before the last contact. Let AB be the ecliptic, the circle centred at T the Tamas at the time of opposition of the Sun and Moon, CD the Moon's orbit 1, The same rule occurs in LBh, iv. 9. 2. A. B. C. D. F. Nl. Pa. S*. Vmt 44 1 MEASURE OF THE ECLIPSE 137 relative to the Tamos at T. Mi the position of the Moon at the time of the first contact, and M» the position of the Moon at the given time. MiP and MiQ are perpendiculars dropped on AB. Then PT denotes the semi-duration of the eclipse, PQ denotes the ifta, M 2 Q denotes the Moon's latitude at the given time, and LM denotes the eclipsed portion of the Moon's diameter at the given time. where Fig. 13 It is evident from the figure that LM — (LT + MaM) — MiT, M|T v / M 2 Q a + QT“ = VM a Q 3 + (PT-PQ)». Hence the above. The i?pa is generally given in terms of time (in ghapls) elapsed since the first contact or to elapse before the last contact, and the above rule-is stated as follows : “Subtract the i$a (ghatis) from the ghatis of the semi- duration of the eclipse. Multiply that by the difference between the true daily motions of the Sun and Moon and divide by 60. Add the square of that to the square of the Moon's lati- tude for the given time, and take the square root (of that sum). This subtracted from the sum of the semi-diameters of the Tamas and the Moon gives the measure of the Moon's diameter eclipsed at the given time/' 158 GOLA SECTION i G°la Sn. The portion sthitimadhyadi&avarjitanmalam of the text is defective as it does not convey the correct sense intended here. A correct reading would have been vistasthityardhavargitanmulam. AKSAVALANA 45. (a-b) Multiply the Rversed sine of the hour angle (east or west) by (the Rsine of) the latitude, and divide by the radius : the result is the ak?avalana. Its direction (towards the east of the body in the afternoon and towards the west of the body in the forenoon) is south. (In the contrary case, it is north). 1 That is, Rvers H X Rsin j> ak$avalana—^ ^ — » where H is the hour angle of the eclipsed body and $ the latitude of local place. The ak$avalana is the deflection of the equator from the prime vertical on the horizon of the eclipsed body. The above formula is incorrect. Brahmagupta (A.D. 628) modified it by replacing Rvers H by Rsin H. 2 Better and accurate formulae were given by Bhaskara II (A.D. 1150). 3 The word dik in the text means valana. The PaulUa-siddhanta summarised by Varahamihira gives the following rule : 4 ak$avalana—~yQ- 1. The same rule occurs in PSi, xi. 2 ; MBh, v. 42-44 ; LBh, iv. 15-16 ; KK, I, iv. 7 (as interpreted by Bhattotpala) ; SMT, lunar eclipse. 2. See BrSpSi, iv. 16. 3. See Si&, h v. 20-21(a-b) ; , viii. 68 ; and , viii. 66(c-d)-67. 4. See PSi, vi. 8. Veise 45 ] V ALAN A 159 The formula of the old Surya-siddhanta summarised by Varahamihira is i 1 . RsmHxRsin^ ak§avaiana= — ^ , which is the same as given by Brahmagupta. AY AN AVAL AN A FOR THE FIRST CONTACT 45. (c-d) Making use of the semi-duration of the eclipse, calculate the longitude of the Sun or Moon (whichever is eclipsed) for the time of the first contact. Increase that longitude by three sign 9 and (multiplying the Rversed sine thereof by the Rsine of the Son's greatest declination and dividing by the radius) calculate the Rsine of the corresponding declination : this is the ayanavalana (or krftntivalana) for the time of the first contact. (Its direction in the eastern side of the eclipsed body is the same as that of the ayana of the eclipsed body ; in the western side it is contrary to that). 4 That is, Rvers (M + 90°)xRsin 24° ayanavalana— ^ » where M is the longitude of the eclipsed body, the Sun or Moon. 1. See PSi, xi. 2. 2. A. E. transpose this verse to after 46. 3. Govindasvami writes : faTTfsiraf^fa STS^f tifctfirfta% I hiitos^t ^hufit: i ^^^55^15^% i ^t&t^wmFr Sci*t: i So also writes Paramesvara : ftrcifiraf^aCT ^ftcW^W^it ^3^*^*1^ VRRfte'W I See Govindasvami's comm. on MBh, v. 46-47 and Parame- svara's supercommentary Sidhnnta-dipiket on. vs. 45. 4. The same rule occurs in PSi, xi. 3 ; MBh, v. 45 ; KK, 1, iv. 7. Also see LBh, iv. 17 and SMT, ch. on lunar eclipse, where M+90* is replaced by M— 90°. KR, iii. 14(c-d)-15(a-b) gives a mixed rule. 160 GOLA SECTION t Gola So. The ayanavalana is the deflection of the ecliptic from the equator on the horizon of the eclipsed body. It is defined as above for the first contact because in the .case of a lunar eclipse in the eastern side (which is first eclipsed) the direction of the ayanavalana is the same as that of the Moon's ayana. The first contact is only a token ; for the middle of the eclipse or for the last contact, it is obtained similarly. The above formula for the ayanavalana is incorrect. It was modified by Brahmagupta, who replaced Rvers (M+90°) in the formula by Rsin (M+90 ). 1 An accurate expression for the ayanavalana was given by Bhaskara II (1 1 50).* The akfavalana the ayanavalana, and the vikfepavalana are required in the graphic representation of an eclipse. For the details of graphic representation of an eclipse according to Bhaskara I, the reader is referred to MBh, v. 46-47 and LBh, iv. 19-32. The commentators of the Aryabhafiya are of the opinion that the word sthityardhacca refers to the vik^epavalana. “Since the Moon's latitude is obtained from the sthityardha”, writes the commentator Somesvara, “the Moon's latitude is meant by the word sthityardha” '. So also says the commentator Parameivara : “By the word sthityardha are meant the moon's latitude corrected for parallax (in the case of a solar eclipse) and the Moon's latitude (in the case of a lunar eclipse) which are based on that”. Thus, according to the interpretation of the commentators, the verse 45 (c-d) should be translated as follows : “With the help of the semi-duration of the eclipse, calculate (the Moon's latitude for the first contact, reverse its direction in the case of a lanar eclipse, and treat it as the yikfepa)valana. Also calculate the longitude of the Sun or Moon (which- eTer is eclipsed) for the time of the first contact and increase it by three signs and (multiplying the Rversed sine thereof by the Rsine of the Sun's greatest decli- 1. See BrSpSi, iv. 17. 2. See Si&t, I. v. 21(c-d)-22(a-b). Verse 46 ] COLOUR OF ECLIPSED MOON 161 nation and dividing by the radios) calculate the Rsine of the corresponding declination : this is the ayanavalana (or krantivalana) for the time of the first contact. (Its direction in the eastern half of the eclipsed body is the same as that of the ayana of the eclipsed body ; in the western side it is contrary to that.)” The word sparse in the text means 'at the time of the first contact', which seems to suggest that the calculation is to be made for the first contact. But this is not the case. “The time of the first contact is only a token”, writes the commentator SUryadeva. “The calculation of the valana should be made for the time of the first contact, the time of last contact, the time of the middle of the eclipse, and for any other desired time.” COLOUR OF THE MOON DURING ECLIPSE mm^i mznm unfit ^ft w i srcm§ $ft^: «i^iTOR^in% ii ii 46. At the beginning and end of its eclipse, the Moon {i.e., the obscured part of the Moon) is smoky ; when half obscured, it is black ; when (just) totally obscured, (i e., at immersion or emersion), it is tawny ; when far inside the Shadow, it is copper-colon red with blackish tinge. 1 In the case of a solar eclipse, the obscured part of the Sun looks black at every phase of the eclipse. WHEN THE SUN'S ECLIPSE IS NOT TO BE PREDICTED 47. When the discs of the Sun and the Moon come into contact, a solar eclipse should not be predicted when it amounts to 1. Cf. PSi (Paulisa), vi. 9(c-</)-10. 2. Pa. gives a variant, 3. So. UTOWTRl 4. Ni, ?g^35PR^PB*r A* Bb. 23 162 GOLA SECTION t Gola Sn. one-eighth of the Sun's diameter (or less) (as it may not be visible to the naked eye) on account of the brilliancy of the Son and the transparency of the Moon. 1 PLANETS DETERMINED FROM OBSERVATION 48. The Sun has been determined from the conjuction of the Earth and the Sun, the Mooo from the conjunction of the Sun and the Moon, and all the other planets from the conjunctions of the planets and the Moon. What is meant is that the revolution numbers etc. of the Sun, Moon and the planets stated in the first chapter of the present work have been determined by observing the conjunctions as stated above. According to the commentator Somesvara, the method of finding the revolutions of the Sun, Moon and the planets in a yuga by observing the conjunctions is as follows : (1) Revolutions of the Sun Let the number of conjunctions of the Sun and the Earth, i.e., the number of civil days (including the fraction of a civil day) in one (sidereal) solar year be c. Then the number of revolutions of the Sun in a yuga^CJc, where C denotes the number of civil days in a yuga. (2) Revolutions of the Moon Let the number of conjunctions (including the fraction of a conjunction) of the Sun and the moon in one solar year be c. Then the number of revolutions of the Moon in a yuga={c+ 1)S, where S denotes the number of solar years in a yuga. 1. Cf. MBh, v. 41. 2. Ra, srarfacre^*:; So. srarr^ Verse 49 ] ACKNOWLEDGEMENT TO BRAHMA 163 Alternative method. Let the number of risings of the Moon (including the fraction also) in one solar year be m. Then the number of revolutions of the Moon in a yuga—E— mS, where E denotes the number of rotations of the Earth and S the number of solar years in a yuga. (3) Revolutions of the planets (Mars, Mercury, Jupiter, Venus and Saturn) round the Earth Let the number of conjunctions of a planet and the Moon (including the fraction also) in one solar year be c. Then the number of revolutions of the planet in a yuga=M— cS, where M denotes the number of revolutions of the Moon and S the number of solar years in a yuga. (4) Revolutions of the fighroccas of Mercury and Venus Let C denote the number of civil days in a yuga, and d the number of days (including the fraction of a day) between ? two consecutive ; inferior or superior conjunctions of the planet and the Sun. Then C/d-= revolution-number of the planet's itghrocca— revolution- number of the Sun. .*. number of revolutions of the planet's sxghrocca in a yuga =C/d-{- Sun's revolutions in a yuga. For other methods see Somesvara's Commentary, in Part II. ACKNOWLEDGEMENT TO BRAHMA H535^R*i§^ *rcn^T V w^m^i *im few ^fircjsrr 11 ^ it 49. By the grace of Brahma, the precious jewel of excellent knowledge (of astronomy) has been brought out by me by means of the boat of my intellect from the sea of true and false knowledge by diving deep into it. I. AU others except Bh. and So. read ^TT for 5^”T; 164 CONCLUSION 50. This work, Aryabhapiya by name, is the same as the ancient Smyambhuva (which was revealed by Syayambhu) and as such it is true for all times. One who imitates it or finds fault with it shall lose his good deeds and longevity. [ % fcf litaTO: SWcr: ] 1. B. srefff for STT 2. Ni. *R*PT ; Pa. eiRT ; Ra. So. fa&m 3. C. $f»cn^ ; F. g$ <tt$<t: ry^f^fer^^ $f*pfaw (in Mai., meaning 16,99,817 is the Kali day of the completion of the transcription). B. ^TTsr*rcto JTTT Tfl?PPMf^n i Tt«T- snrfsr: 1 C. No formal col. D. ^JT^^tffaTfwT *\t*NJ*: WW^: I E. iftarcre: *pn*er: i ^tt4^?W qftyf stt^ i F. sfa » APPENDIX I INDEX OF HALF. VERSES AND KEY PASSAGES swtt: qreres'fa 14 c) 101 •raw *§r* fkertara (nfar, 5 a) 37 areHTfr *rfa tisa (nter, 9 c) 119 (ff*RT, 33 a) 74 (nftnr, 13 d) 55 5lftl*1U'o^^yi (lf*Tcf, 33 c) 74 srftupunrm^rc (*ffara, 32 a) 74 wftnrwwt g*t ^ (vto, 6 a) 91 a^taiferefor: (nt^r, 9 a) 119 *gritanfH$ (m*f, 20 c) 105 argeitamfa h?3Tct (war, 21 a) 106 (*lf^T?f, 24 c) 67 btospt: £<rsra fair^ (ifto, 28 d) 71 STOTO5r*iT ^sr: 3 a) 114 am^TT (*fta, 12 c) 122 (if^cT, 10 c) 45 (qfc?r, 30 d) 141 srofcer T'^tTsr (<for, 2 c) 1 14 srf^tf^on (iftfwi, 7 b) 15 W^fsfa: (*iter, 37 a) 151 (nf*nr, 17 d) 59 arc? ?*<rowm?i (*fta, 16 c) 127 (fl?r, 23 b) 109 (*T5T, 22 d) 108 srar 'tjlH'^H 15 c) 126 sroffa u*rre 5 c) 117 sr^ % iro&seqT: («bt*t, 14 a) 101 (mfe^T, 2 b) 3 anr^gsriraT gfft«*r: (sftsI, 4 d) 88 snssttnc w^r 8 c) 92 srcairoteii$«n5 (nto, 29 c) 141 WRT^f |Ml^r^-orHd 9 d) 93 aTFUT^ (nPtrf, 8 a) 42 4i4<H<it4H*r ('ftfH^»T, l c) l 3THT*rata *n*n (*it5r, 50 a) 164 3mT*ft sjrTTft^: («TfacT, 10 d)45 wfc! (nfafl, 19 a) 61 ^facTfaGn-R (irfacT, 19 c) 61 STOimgfara (nta, 25 a) 133 smmwgfr r am 26 a) 135 366 rssmi^f (nta, 24 a) gysrfajft st^WT: (*fta, 14 c 3c*fo>ft puf {vm, 9 a) 3TOSR*S»ro% ('fta, 36 c) ^cmfa f$ WWTC (*ffa, 27 a) 136 3*HTCc!*nifafaTf 10 a) 119 ^ tJ?TOf^ 35 c) ssift jj> *r^f rot (nt^r, 13 a swni3?i ^%fTct (nta, 19 c) ^qfelfl S^faf (^»TH, 13 c) 3e^TrT^*TM (vfrm, 6 c) 39 33^*3*31* SFE: 21 a) 129 sfcHlSRSFrcRT: («PT^, 22 a) 123 93 149 148 123 128 100 106 qsp g * ?Tci ^ (Tff^cl, 2 a) 33 ^treraqfafc: (irfaer, 21 a) 64 W qj (»Tta, 20 c) 128 ^ *f!!*lf?nn*T: 2 c) 85 3*msre<sr %Tm (nxvt, 15 c) 102 » TOTTOrfaiTOim: 17 a) 104 *»scrm«3H§5*i 18 a) 105 *>sirm<»2srer«T- (vra, 21 c) ^surci (*>m, 25 c) (iftfs^T, 1 b) * irtt: (iftfiwr, 7 b) isemfsm: (»ftf?r*T, 12 d) ^H^*NT3T n (»ftf*W>T, 5 c) 9 13 b) 100 106 110 1 15 29 (*T?T, 11c) 98 *n*5*ni H^Rri (»fta, 22 a) 129 V^t HWt 3 (HtfeWfT, 5 a) 9 f fef*Tf>5S<*| sn* (nHewT, 3 b) 6 j3i fRfasrgi ('ftf^Tj 3d) 6 $9grc>m<H (Hta, 3 c) 114 f f^mf^ *g*T: 17 d) 127 (nfas, 1 d) 33 jfevfi * (»rfa?r, 2 c) 33 srniPTOircTm: {*™, 22 a) 106 M?m *re<n*ft*i 13 c) 128 fefHSTTf^tTWTm^T 32 a) 144 fafe* *rc$nn (*fta, 34 c) 147 (nta 48 a) 162 (*T*r, 2 d) 85 46 b) 161 *sfegfc ^TT (*&fa*T, 2 c) 3 ngmSr <^3r> (tffaiflT, 6 d) 13 (»ftf?W*, 10 b) (if^RT, 20 a) (ifflaW, 1 d) itm^f * (ntffwr, 5 b) ncM^A”! (ifal, 31 c) 22 63 1 9 73 INDEX OF HALF-VERSES 167 19 (iftfa*T, 9 b) (iftf?WT, 11 d) 23 qmm tr^tt: (nftra, 28 a) 71 (^rcwn, 3 c) 6 (tff?WT, 5 d) 9 gsSTOfa (^TH, 2 a) 85 gfcwi?^ fe*%?T (Tf^, 30 a) 72 ^HiR«rfT^«f (tffcT^T, 13 c) 31 ^TOtiTHSSft (ifaa, 18 c) 60 *mrft I ^ (ifa&i 18 a) 60 (nfacT,7d) 40 TOOT sresnfa: ^ (qfqcT, 3 d) 35 ^ *T (*fiWT, 8 d) 17 ^ few 12 c) 29 ■sj^rfsrar 5Rm^gnr (nfaci, 10 a)45 raamrrafa (*frtf, 42 a) 155 ratal sfafis (ntfa^T, 4 a) 6 rat 3fsre*?sfrT: (ifhf, 37 a) 151 Vffewtfwrffc 4 a) 116 3* (*tor. 37 c) 151 fjTOigf^ W*™- (t^RI. 16 a) 57 qtmrenrfarc (nfcs, 40 a) 153 |}*g«i W&fi 27 c) 70 §i*T: q*R7§m: (nfia, 27 a) 69 gn-vr-^-^-iRtan^B)' strt ^ (*ftf?WT, 11 c) 23 (ifta. 23 b) 130 em 531%* ?t«ot (»fta, 30 c) 141 (tffa^T, 10 c) 22 sngffa J*tT (tffcWT, 10 a) 22 fa*n yatmt (ntf^PT, 7 a) 15 !RTO$ *W>tTOT: 14 d) 123 *RBjfilW»wfti$ar: (ft*, 41 a) 154 (Kst'Efresratfr (irfatfT, 12 c) 51 m^T: 33 48 d) 162 . («W,10d) 95 af^ST^PT ^ (tf^cT, 7 c) 40 HSUWWIlSWt: 33 c) 144 cf?f?T gSTTSlW (*tf«ia, 25 c) 68 W. fo*wini 40 d) 153 fa*P*Bra («fta» 43 a) 155 aRraH'“™ (»far, 2 a) 114 (ntfcror, 6 b) 13 afem%^f^ (nta, 1 c) 113 fa*if^a> to* a 1 b) 85 168 APPENDIX I foiJSTCT %*WTti (irfinT, 6a) 38 “TTT^^nrcrm (*iH 22 c) 129 fas* ffrorgfiRf (*m, 7 c) 92 (lf<T?T, 13 b) 55 SpfofaiNrT- (*TT*T, 20 a) 128 5*rfsra>T EhiBk*?: (vtw, 10 c ) 95 3*fqTfcT5T*f (iffa, 19 a) 128 #TTfor*q»5!TTr*r (»rf«r?l, 26 a) 68 JjafaTWSB* (iftsr, 18 a) 128 s*rotf?wij*fo*Ti (ifrfNn, 13 a) 31 SNT^F^ arST: (ifta, 46 a) 161 118 Sir«IimwMT (*ftfi!^T, 1 a) 1 fes?T 8 a) 92 srfN^rorW (wwx, 19 a) 105 fcsfcr wrqftfsr (*t*r, 12 c) 100 Srf?m«3?IFT qwj (^TTfT, 18 c) 105 ^PsqwsnrfiT (nter, 21 c) 129 ?«itanfoqT& (*fta, 23 a) 130 (W5T t 17 d) 104 si«wr3^TqwJiTsrfcT (^fora, 12 a ) 51 (»ft5I, 34 a) 146 g«njteqiHigrreft (ifta, 27 c) 136 U^lftrpf (*Tta, 21 b)l29 sn*|«r «npT ftr<r: (ifta, 10 b) 119 (*fta, 16 a) 127 (ft^TcT, 9 b) 43 ffCST *jfc*nT *T%3 Iff (»fta, 20 d) 128 («it?r, 2id) 129 ffT-oTfa-HT-^ST-^lT-fo^T Sin»mf5I ^5TT 11: (*ltffWT, 6 c) 13 Mrql^^ (iftfH*l, 9 d) 19 rgfftfgiTTcr («FfwRT, 24 a) 67 <?> 3 vht^jt: (tftfinrr, 12 d) 29 f5*A«T*MWcfldwfl (lf«RT, 31 d) 73 TT^rg^^ (liter, 11 c) (iPfcT, 3 b) 34 121 (nta, 12 b) 122 (^fcTfTT, 9 a) 19 ?T^f«n|3^»ft^: (iffcr, 4 c) 116 3* ggfsm* (»itr?fvT, 4 b) 6 factf sra|*r ^rgjfi feeff: (nta, 10 b) 119 (»ftf?WT, 4 d) 6 •jfa *IT*FT (*irf?W>T, 7 a) 15 $fa>*W «TT5Tf^fr*¥T (*itf?W»T, 4 c) 6 <re*Tq*R3ftaf (nte, 30 a) 141 *^5«r?iw^n^r- (nfoifr, l a )33 qfat; SIWTRT (ifrRT, 9 c) 44 fgifutft ij^: (nta, 8 a) IIS INDEX OF (<FT3T, 8 d) 92 •n^ fa*ttorrV^ (nfnr?T, 31a) 73 FftgftWf 3 a) 86 iw^sct: tragi wrfir (liter, 10 d) 119 TOftrs^: (iftfef^T, 6d) 13 (TpUfT, 28 b) 71 Wf g^irff^ (irfoi?T, 4 a) 36 SR^T (*TH, 15 a) 102 TRT Ta*te*T?fWGft (liter, 18 d) 128 TRt'rtgTmWTcT (ifor, 47 c) 161 WTOift 7^f7TT: (ifrf<T*T, 8 a) 17 TOafararfa TrcTer?: (*>r*r, 5 d) 91 filter: trial ^?r: (liter, 6 d) 118 (ifrffrerr, 13 b) 31 TOgJTTCT iftengffo (liter, 5 a) 117 •J^ftJiwi fwa (liter, 40 c) 153 ^BGnHTiNff (qter, 39 c ) 152 <T?fTT^ (qta, 38 c) 151 TOTRnrgfaST *nm$- (*T?r, 25 a) 110 (*vx, 6 d) 91 iTTfafasR fa*%<r (liter, 39 a) 152 wrmrafjfN ?^r (iftsr, 15 a) 126 TO^f>*ftm: (ifrrfr^, 7 c) 15 *?l swfsrtiB (iftffwr, 4 b) 6 TO * ( , nf?WN f . 8 d) 17 qgittftti: (ifrfa^, 3 d ) 6 4. Bh. 24 HALF- VERSES i 6 g ffa Iffa <ET% (iftfiWT, 12 a) 29 ^^dH^THERrTTrT (f>m, 13 a) 100 TfenTTOTF?r% ftrenr (TNtr, 32 d) 74 T«iw*it3*?sftaT- (»iter, 33 a) 144 sRsnF*tHnnn (»rter, 32 c) 144 TOT^t ( *ft5T, 13 c ) 123 wr^pqiTrqat (»iter, 45 a) 158 (trm, 9 c) 93 t33»n: ^ (iftfewT, 5 b) 9 igi^n: (ntfer^T, 5 a) 9 *J«ERT S^-s-gr-ST (•ftfe^T, 11 a) 23 23 a) 109 *rafc*Tcr ?^6Hi:(^T?r, 23 c) 109 V^T^Stter (spTST, 17 c) 104 (um, 13 d) 100 $fci i^Rr^r *m<it a<n: (to, 14 b) 101 g*T1»5r (irf«l?T, 25 a) 68 fff^ra^ (qfim, 20 c) 63 ^^fafa*ig*rit (liter, 6 c) lis S: (liter, 37 b) 151 irat^m itrt^tt (^m, 15 d) 102 SMfainna: (liter, 11 a) 121 *r*r%: *J*iTiti (liter, 1 a) 113 it: WT: ^s<mJT> (irforcT, 28 c) 71 1 (to, 20 a) 105 170 WfW (lf*KI, 23 c) WWW Vl^ni (lf”KI, 14 c) 55 ngc^>^wgwre^: (nw, 7 a) 118 «t spm (iftlfl, 15 c) 57 ^srwil: (»rf*RT, 17 a) 59 gnrfawn: sg* (ntfswT, 3 a) 6 gnw*»mrfTOn: («Ft, 11 a) 98 xfawm mm (wra, 5 a) 91 *faij*i>n fwn: (wm, 5 c) 91 (ww, 6 b) 91 TfsRTC «ftfsa (wm, 7 a) 92 Tfagirfsf t*T: (ifm, 17 a) 127 *ftnfiM*WW i: (wh, 3 c) 86 (WW, 5 b) 91 (nm, 48 b) 162 TOf* Ttt^f (ifas, 29 a) 71 (ifw, 13 d) 123 «^TTO«rfaro*ft 10 c) 119 fqftanravirc (*fw, 25 d) 133 *m'i&*W* (nf«T?r, 30 c) 72 319 5PTl«wf3Rr (*if*<ti, 26 c) 68 (qm?f, 4 d) 36 (nmnn, 11 b) 23 VWt iqwjw: (“ifcw, 3 a) 34 APPENDIX I 67 (*ITOtf , 22 c) 65 «nf fta$#gfai?i: (*rf«rci, 5 c) 37 Wff«ffTTW W«f (»flfkWT, 2 a) 3 Wnf? 1*T 33 (ffliRf, 4 c) 36 rfspf «WK* re < T WT (nVffT^T 2 d) 3 85 155 148 156 155 149 130 ffra^T mm: (ww, 1 a) froqffafafjfci (*iw, 42 c) faw«ig«n«FiTT (*TW, 35 a) Jw«q«l^r^ie (m®, 44 a) firaroMJK (nra, 43 c) fa#ll<re»Hg<'T- (“ita, 36 a) fWf^afRT^STT (l”w, 23 c) fwgWSsNigfacT: (nw, 29 a) 141 frg^wOTT fasRBT 31 c) 142 fa|?rfSf««t?T: (iftH, 24 c) 132 fTOwngfww^ (m*t, 28 c) 139 fw^*TSR m 5^TT (*rf«l?T, 9 d)44 frorTiit«n^n (ifrrcr, 8 c) 42 «*qfoft (ww, 19 c) 105 Wm »« KH M > (#>sr, 6 a) 118 WtT ^1 *n*i (ifa?!, 13 a) 55 (irflw, 17 c) 59 q|?f ^ (ifaa, 29 c) 71 (irf*I?l, 15 a) 57 w^fgnn wte* *n (^fara, 16 c) 57 Jl^t: SRMVI (iflcr, 14 a) 56 (ifffimr, 8 c) 17 INDEX OF flPr-^J- *pr-*jij-f «ffaft 5? (iftfiwiT, 10 d) 22 (*m, 22 c) 108 fifr ®pf^r (“ftfir^T, 3 c) 6 (tffasiT, 7 d) 15 (»iH?R>T, 3 a) 6 irfaduiv^Mm (nta, 48 c) 162 ^ftlfotil fafTTT: 6 c) 91 *rfonnraf (tfferarT, 10 b) 22 (*fter, 37 d) 151 (nHl7c) -127 irfilTTOUtS 'TO (»ft^WT, 6 a) 13 (*ftf?PfrT, 8 b) 17 *ftsri&us3rg*rf: (*rrer t 16 c) 103 (*T5T, 21 b) 106 (“PFT, 23 d) 109 jftSTfaETTOrft (TO*r, 24 a) 110 (nil*, 3d) 114 $nrqT*<TC*rcci (tPhh, 32 c) 74 44 d) 156 JP*VERSES 171 (»rf*RI, 21 c) 64 ifesrfwt («PTH ; 1 c) 85 (*Wr, 1 d) 85 qWTOTTt 10 a) 95 TO*n ^T«4Ml («PW, 12 a) 100 (ntec, 46 d) 161 «tan*flfl*Uc<H (»ft?T, 49 c) 163 ti^^H^^it m 49 a) 163 STOdJ^H'il (f f'Tcf, 3 c) 35 W&t jftwi: (*T?T, 16 a) 103 CTT3TOv3T£rfft (tpT?!, 11 c) 45 (*rf«W, 6 b) 38 ^Hqf^II^FUTtf (nfaff, 7 a) 40 HW^qfrftWR (if^Rf, 11a) 45 tiqrfwn: (»ftf?TOT, 7 d) 15 («CT5T, 11 b) 98 12 d) loo (nfticr, 23 a) 67 *F<refaf^«a* (»it?T, 44 c) 156 (iftfcrn, 13 d) 31 ashnfc wfl^r: (nt*T, 46 c) 161 fTflfat ST^TTT (»lf*RT, 9 a) 43 172 (fifim, 9 c) 19 m&n st^r *w%: (Tfarcr, 13 c) 55 (tfm, 31 a) g^rig^: 5n»mt (»fta 50 c) 5 d) ^if^qfrftflft't ('fin, 47 a) *r*^lT>sji?T?ri (ifaret, 22 a) (ifHNi, 8d) fTOUCpraneff^T (*fta, 14 a) 123 (Tf^W, 2 d) 33 forcusw (nta, 41 c) 154 APPENDIX I (WW«T*HW*ftl*. 45 c) 159 (*T*f, 24 c) 110 pratffwmn^ (»iter, 38 a) 151 ^nrfk: frmcf (*faT, 34 b) 146 f^fe mm wft (*ft?T, 12 a) 122 21 d) 106 (*ifcc f 50 b) 164 tm^ftrara^f 25 c) 133 rei k is fafcrm («ft5T, 26 c) 135 ^TT^^^iTT 28 a) 139 (*m, 4 a) 87 142 164 117 161 65 17 APPENDIX II INDEX -GLOSSARY OF TECHNICAL TERMS USED BY ARYABHATA [ Note : In each reference, the initial figures I, 2, 3 and 4 refer, respectively, to the GitikS, Gariita, Kalakriya and Gola p&das and the further figures to verse numbers in the respective padas. ) 3WI (=nnr) (1. degree) 1. 6 ; 3.14 ; (2. part), 4.14 «W (1. latitude), 4.19; (2. Latitude- triangle), 4.23 Wffsm (Rsine latitude), 4.26, 35 am (1. tip or end), 4.19; (2. resi- due or remainder), 2.32, 33 «nn (amplitude at rising or the Rsine thereof), 4.30 JNTRiT (residue-difference), 4.32 8ltJf (non-cube), 2.5 WFgsr (unit of length, l/24th of a cubit), 1.8 8WT (sign of Aries), 4.25 sneraTO (greater remainder) 2.32, 33 air t WIV U HI^K (divisor correspon- ding to greater remainder), 4.32, 33 arftrem* (=3rfsnmc), (intercalary month), 3.6 gpn%*ros«r (eclipse not to be predicted), 4.47 3T|^T (direct or anticlockwise), 3.17, 20, 21 angsffaiT (having direct motion), 3.21 aig5l>?»Tfa (having direct motion), 3.20; 4.9 3Tf srtafr^T (distance between two planets in direct motion), 2.31 (difference between two quantities), 2,24 3R*TC* (last term in a series), 2.19 STOW (l. greatest declination), 1.8 ; (2. declination), 4.24, 26 aitRsr (subtractive), 2.28 3C«m<is5T (=3WW^W) (ecliptic), 1.8 ; 4.1, 2, 3 3T*m*i (anticlockwise), 4.16 tnnfaft (a designation of the second half of the yuga), 3.9 173 174 APPENDIX II ■rwim (multiplication), 2.9 WW (northward or southward motion of a planet), 4.36 snfifaT (Sun's amplitude at rising, or the Rsine thereof), 4.30 SNpfau (sunrise), 1.4 arcfcm (=^n), (Rsine), 2.17 «fr (=10 8 ), 2.2 8WT (not belonging to the varga-s which as ka-varga, ca-varga etc), 1.2 WF*gflF (month of Asvina), 3.4 STCcRU (setting, diurnal or helia- cal), 4.13 aflRWfteinja (rising-setting line), 4.29 «T^n^T^fil«V«I ( day-radius ), 4.24, 25 anfa or srtfasR (first term), 2.19, 20 (length or breadth), 2.8 TOwa (Trapezium), 2.8 WW? (sidereal), 3.2 WTtfff (approximate), 2.10 see $wfc4r<nfw ^SBjI'FtT (fruit corresponding to iccha), 2.26 5^ejl?lfs[T (requisition, one of the three quantities in the rule of three) , 2.26 (ascending node of the Moon), 4.2 (desired or given number), 2.19 [mandocca or Sighrocca), 1.4 ; 3.4, 20 ^^*ft^rf^ ( anomalistic or synodic revolutions), 3.4 353T*Tta^T (epicycle), 3.19 33qrfq?ft ( City of Ujjayini, modern Ujjain in Madhya Bharat), 4.14 (yc*WvWl) (Rversed sine), 4.36 ^tTT (common difference), 2.19, 20 m<l*H (stTOW), (Sun's north- ward journey from winter solstice to summer solstice), 4.36 3?flfq'ift (a designation of the first half of the Yuga), 3. 9 (^*flR*n) (Rsine ampli- tude of the rising point of the ecliptic), 4.33 34UltdW? (heliacal rising and setting), 4.10 ^RrHtFI (degrees of altitude), 4.32 xJ*H<MM (equatorial horizon), 4.19 ^qfeffl (a series in general), 2.21 3?TPBr-§j? or “fnn^R (divisor corresponding to smaller remainder), 2.32, 33 ^gqw sn (altitude or vertical side), *2.6 INDEX OF TECHNICAL TERMS (negative or minus quantity), 3.22 WT (orbit), 3.14 *WT«*W (mean orbit, deferent or concentric), 3. 17, 18, 21 *ran (sign Virgo), 4.1 jtTTSl (hemisphere), 4.23 V«f (hypotenuse, lateral side), 2.13.17 ; 3.25 HiSfT (minute of arc), 1.6 ; 3.14 *>5fTO«in: (the 24 Rsine-differences in terms of minutes), 1.12 (a period of 1008 yugas), 1.5 ( time in problems of princi- pal and interest), 2.25 qrofw ( computations using divisions of time ), 1.1 *mftrinT (division of time), 3.2 flRTjf (a day of Brahma known as kalpa), 1.5 (terrestrial. wind), 1. 11 ffe (square), 2.24 *ftfs, VtSt (i. vertical side of a right-angled triangle ), 2.16; (2. complement of the bhuja), 2.17; (3. crore), 2.2 (Saturn), 2.1 WSTTO (rotations of the Earth), 3.5 qfU (minus, decrease, negative), 3.22 175 fafa'BBjTOT (Earth's shadow), 4.2 fafasr (horizon), 4.18, 19, 28, 30, 32 fsrffl«n (=feffa«9T) (earthsine), 4.26 fa (Bhagola, sphere of asterisms), 3.11 Sf*f%*TT»T ( division of space ), 3.2 fa^i (area), 2.8 #T (1. additive quantity) 2.28 ; (2. celestial latitude), see under fasSTI <S (sky), 3.15 ; 4.6, 20 (sphere of the sky), 4.20 *a*3rtf jpr (partial eclipse), 4.46 srsjt (middle of the sky), 3.15 ; 4.6 (number of terms), 2. 20, 21, 22 irfafT (mathematics), 1.1 ifir (motion), 2.31 (motion-difference), 2.31 (multiplier), 2.27, 28 (long syllable) 3.2 15*5? (Jovian year), 3.4 gfa^T (a thing of unknown value), 2.30 iffar (1. circle ; 2. celestial sphere; 3. sphere), 1.1 ; 4.5 »ftH (*ra*), (automatic Sphere, model of the Bhagola), 4.22 176 APPENDIX II Vg (planet), 1.13 ; 3.8, 17, 20, 21, 23, 25 Vim (eclipse), 4.37 ff. M^lHW (middle of the eclipse), 4.38 ?TTO (l. measure of eclipse), 4.35 ; (2. erosion by over- lapping), 2.18 *R (1. cube of a number ), 2. 3 ; (2. solid cube), 2.5 crc»ita (solid sphere), 2.7 qifftfr^ (volume of a solid sphere), 2.7 tmfafira* (sum of the series of cubes of natural numbers), 2.22 SRI!?! (volume), 2.7 sptwts* (Earth's centre), 3.18 q>T3tr(cube root), 2.5 TO (circle), 1.6 ; 4.2 ^gT«r (quadrilateral), 2.3 <3§*fa (quadrilateral), 2.11, 13 m?g (lunar), 3.6 ■TON* (Moon's apogee), 1.4 m<ivi (ascensional difference), 4.27 «rTrjram (lunar month), 3.6 W?(arc), 2.11 snqwra (— *m), 2.12 fefir (sum of a series of natural numbers), 2.21 fafotPt (sum of a series SS«), zai, 22 fafiWT (square of the sum of a series of natural numbers), 2.22 3RTT (shadow) ; 2. 14, 15, 16; 4.5. 37, 38, 39, 40 BJPnsHr* (length of the Earth's shadow), 4 39 (denominator), 2.27 sfon (Rsine), 4.23 (sign Libra), 4.1 em (Rsine), 1.12; 2.9 «*ra (=nn, Rsine), 1.12 ; 2.11, 12 ?Pf (l. Section of Earth's shadow- cone at the Moon's dis- tance), 4. 42 ; (2. Earth's Shadow), 4.37 ?l*ftiro (section of Earth's shadow- cone at Moon's distance), 4.42 3tftftc$r*? (diameter of Shadow, i.e., diameter of the Earth's shadow cone at Moon's distance), 4.40, 43 mrws (Star planets, /. e., the planets Mars, Mercury, Jupiter, Venus and Saturn), 4.2 f?rf«T (lunar day), 3.6 fkfasrsrc (=3raw, i.e., omitted tithi), 3.6 «jtTT (sign Libra), 4.1 fan5* (triangle), 2.6, 11, 13 INDEX OF TECHNICAL TERMS (rule of three), 2.26 tfaTUPT (Sun's southward motion from summer solstice to winter solstice), 4.36 TO (half), 2.24 fop (=3unB), (deflection due to latitude), 4.45 fan (=fs*H, day), 4.8 foW (lord of the day), 3.16 fom (=f*T, day), 3.1, 5 fit*TOf (divine year, equal to 360 years of men), 3.7, 8 I^crt (designation of the first and the last quarters of a yuga), 3.9 (ecliptic-zenith distance or its Rsine), 4.33 5<HHMM««SI (vertical circle through the central-ecliptic point), 4.21 Miqm (parallax), 4.34 ipirffr (arc of the ecliptic between the Sun or Moon and the central ecliptic point or its Rsine), 4.34 ^wrfil(^n) (Rsine of drggati), 4.34 $nfta (visible celestial sphere),. 4.23 f^qviM (vertical circle), 4.21 177 tnwnfa ( twelve-edged solid, particularly a cube ), 2.3 fff«t$*nr (a number which yields the given remainders when divided by the two given divisors), 2.32-33 (additive, positive) ; (sum), 2.19; 2.3, 6 ; 3.22, 23, 24 (arc), 2.17 *wra*TT (=Wl) (revolutions- number), 3.3 ftiwn (Rsine of zenith distance), 4.32 sreTTTOSiTT (=TtT5tn) (Rsine of zenith distance), 4.32 TOT (hell, south pole), 4.12 HTSTsrfiwr (sidereal day), 3.5 ?TT (a unit of linear measure equal to four cubits), 1.8 JUST* (sidereal), 3.5 ftp ( = 10 B ), 2.2 fiTCrtro (exact), 2.7 mtf (mfcm, tife^i), 3.1 ?rfaf (perigee or perihelion), 3.4 «W (1. square root) ; (2. quad- rant of a circle), 1.11 ; (3. term of a series), 2.19, 21, 22, 29 “WlW (greatest declination ; obliquity of the ecliptic), 4.30 A. Bh. 23 178 APPENDIX II iremTOfsfan (°WT) (Rsine of the greatest declination), 4.30 greatest altitude, i.e., Rsine of meridian altitude), 4.32 iftUT? (periphery, circumference), 2.7, 10 qftfa (circumference), 2.9, 11 qft^ (revolution), 3.4 qTcT (ascending node) , 2.4, 9; 4. 2,3 qt^f (lateral or adjacent side), 2.8, 9 fq^ij (year of the manes), 3.7 yrfaT (east-west), 4,31 STO^T (first contact in an eclipse), 4.46 srfaTO5r (eccentric circle of a planet), 3.17, 18, 19 srfa?ita (retrograde, clockwise), 3.17, 20, 21 5Pnt (argument in the rule of three), 2.26 3Tp (=10«), 2.2 Wfgm (provector wind), 4.10 (=WH) (rising point of the ecliptic), 4.21 STPwinvTW (right ascension), 4.25 jrt«! (a unit of time equal to four sidereal seconds or one-sixth of a vms<*a), 1.6 ; 3.2 q??f (interest on principal), 2.25 qmrrfa (fruit, one of the three quantities in the rule of three), 2.26 l?*rig& (south pole), 4.12, 16 S^rfacia (a day of BrahmS, a Kalpa), 3.8 * (=*) (asterism), 1.6; 3. 11; 4.9, 15 q*m (revolutions, number of revolutions performed by a planet in a yuga), 1. 3 ; 3.2, 3, 4 wfc* (sphere of asterisms, with its centre at the Earth's centre), 4.15, 16, 23 xrwK. (*W?), (circle of the asterisms), 4.10 infamy (circumference of the circle of asterisms), 3.12 tTPT (degree),. 4.32 *nn37«I (division), 2.4 (*tT^)> (divisor), 2. 27, 28, 32 nan (lateral side of a right angled triangle), 2.15, 16, 17 ^pfta (sphere of the Earth), 4.6, 7, 8 *£Tfafa«* (diameter of the Earth), 4.39, 40 ^3pm (Earth's shadow), 4. 38, 39,40 ijflCTtf (terrestrial day, or civil day), 3.6 tftiT (motion, daily motion), 4.41 (compasses), 2.13 q<H3T (circle, revolution), 3.13, 14,17; 4.1,8 qfa (optional number), 2.32 x&n (1. centre, middle) 3.9, 15, 18,21 ; (2. mean), 3. 21 ; (3. middle term in a series), 2.19 qsmr^ (mean planet), 3.21 *e*wn (meridian sine, i.e., Rsine of the zenith distance of the meridian ecliptic point), 4.33 xwmp (or FGCTS*) (true-mean position of a planet), 3. 23,24 ng (a period of time equal to 72 yugas), 1.5 (slow, apex of slow motion, apogee), 1.5 TO*! (hypotenuse associated with mandocca), 3.25 vfrTOrT (manda epicycle), 1. 10, * 11 ; 3.21 cr^t^r (apogee or aphelion), 1.9; 3.17, 22, 23 WTOTO (year of men), 3.7 *rra (month), 3,1 *fta (sign Pisces), 4,1 INDEX OF TECHNICAL TERMS g?I (first term in a series), 2.19 «pr (1. square root), 2. 4, 24 ; (2. principal), 2.25 (interest), 2.25 (mountain at north pole), 1.7; 4.11,12, 16 ift (sign Aries), 4.1 giT (a period of 43,20,000 years), 1.3, 5 ; 3.3, 6, 8 sn^TT* (quarter yuga), 2.10 H>T (conjunction of two planets), 3.3 iftsrc (a unit of linear measure equal to 8000×4 cubits), 1.6, 7 tfama (solar month), 3.6 Tfaw (T3S3*, ^pr?s^) (solar year), 3.5, 7, 12 Tiftr (1. sign), 3.14 ; ( 2. quan- tity ), 2.29; (3. twelve), 3.4 *q (one), 2.20 *k<W> (a coin), 2.30 5T5*;t (a hypothetical city on the equator where the meridian of Ujjain intersects it), 1.4; 4.9, 10, 13, 14 ?rr ; *fa*j (times of rising of the signs at LankS, i.e., right ascensions of the signs), 4.25 *T¥<GfrroT'5m*ifc6 (right ascen- sion), 4,25 ' . ” 180 (1. plumb), 2. 13 ; (2. Rcosine of latitude), 4. 26, 29, 30, 35 qfe (a planet in retrograde motion), 1.1 in! (1. square), 2.3, 4, 5, 14, 22 ; (2. odd places indicated by letters of the vargas viz., ka-varga, ca-varga etc.), 1.2 toWhstc (sum of a series of squares of natural numbers), 2.22 JR^T (square root), 2.4, 14 *i (year), 3.1 tWI (celestial latitude), 1. 8 ; 4.35, 42, 43, 44 fonfavi (f*qfe*a), (one-sixtieth of a ghatika), 3.1,2 faTOftr (half-duration of the totality of an eclipse), 4.42 fasftT (retrograde), 1.4 fojftafw (difference of two planets, one direct and the other retrograde), 2.31 ftrtT (difference), 2.31 ft (difference), 2.15, 20; 4.33, 34, 39 fcgqrota ( = 8WOTT), (Rsine of latitude), 4.29, 31 ftgq* (equator), 4.24 fl|CVTO (diameter), 2. 7, 9, 11 ; 4,40 APPENDIX II fa^PRTO (semi-diameter, radius), 2.7, 9, 11, 14 fcmx (=M> (length), 2.8 ^tT (1. circle), 2. 10, 11, 13, 17 ; (2. epicycle), 3.19,20,21 wwrftrt^ (circumference of a circle), 2.10 ^1!?f (area of a circle), 2.7 wmfMs (circumference of a circle), 3.19 (=10»), 2.2 *T (velocity), 3.25 WifllRT, 3.3 wiTO (diameter), 1.7 snKTO (semi-diameter, radius), 3.19, 25 ; 4.24 Hf| (gnomon), 2.14, 15 ; 4. 28, 29, 31 V^m (distance of the planet's projection on the plane of the horizon from the rising- setting line), 4.29 f[X (arrow, Rversed sine), 2. 17, 18 trfcrfara (lunar day), 3.6 frftrtrm (lunar month), 3.5 jftsw* (Ughra epicycle), 1. 10, *11;3.21 tffitw*, (stghrocca), 3.17, 22, 23, 24 ; 4.3 (a solid with six edges, a triangular pyramid), 2.6 - INDEX OF TECHNICAL TERMS iftnf (multiplication), 2.3, 6, 17, 22, 23 ; 3.25 HU^^TST (square), 2.3 ttroritf? (altitude of a triangle), 2.6 *nr<?fW|? (circumference of a circle), 2.7 fTW (circle), 2.11 IWT: (year), 1.28 (sum), 2.23 gwnsftl (half the sum of the diameters of the eclipsed and eclipsing bodies), 4.44 «*rra*TT (arrows of intercepted arcs), 2.18 CPU (clockwise), 4.16 trim* (total eclipse), 4.46 *pfcm (sum of a series), 2.29 (reduction to common denominator), 2.27 181 ?pm (designation of the second iSnd third quarters of a yuga), 3.9 flirfs* (solar year), 3.12 tfT (solar), 1.3 f$&m (half the duration of an eclipse), 4.41, 44 Wlf (first contact of an eclipse), 4.45 ( true P^et), 3.23 C^sto, ( true-mean planet), 3.23, 24 **qi?l^ffT ( perpendicular drawn on the base or the face of a trapezium from the point of intersection of the dia- gonals), 2. 8 |3( (cubit, measure of length), 1.8 ^1 (lord of the hour), 3.16 APPENDIX HI SUBJECT INDEX [ Note : — The numbers refer to the relevant pages. ] A.P. Series, — mentioned in Vedic texts, 63 ; — number of terms in, 63-64 ; —sum of different types of series, 64-66 ; — sum or partial sum, 61-63 abadha (segment of the base by the altitude), 39 Abhidharmakosa, on the Meru, 12 In AHoratra-vrt/a (day-circle), — radius of, 132-33 Aksa-drkkarma of the Moon, 148- 49 Aksa-ksetra (latitude triangle), 130- 32, 135, 140-43 Aksavalana, 158-59, 160 Almagest of Ptolemy, Tr., 18n, 21n, 120n Amsa (bhaga), (degree), — measure of, 86, 116 angula, unit of length, 19 Anomalistic quadrants, beginnings of, 24-25 Apasarpini, 92-93, 94 Aphelia : See under Apogees. Apogees of planets, 19-22, 87 ; — motion of, 20-21 Ardharatrika (midnight) system of Aryabhata I, 151 Area of, —a circle, 40 ; — a plane figure, 42-44 ; —a trapezium, 42-43 ; —a triangle, 38-39 Arithmetic progression : See under A.P. Series. Arkagra, 14 Iff. Arkagrahana : See under Solar eclipse. Armillary sphere, (gola-y antra), 113, 129-30 Aryabhata I,— date of birth, 95 ; — epoch of, blja corrections therefrom, 95-96 ; — midnight system of, 151 ; — named as the author in the text, 1, 33. For his views expressed in the Aryabhatiya, see under the rele- vant subjects in this Index, Aryabhata II, on : — latitude-tri- angles, 131; — modified Rsinc- difference, 30 ; — volume of a sphere, 42. See also Maha- siddhania. Aryabhatiya, stated to be the title of the work in the text itself, 164 SUBJECT 2ryabhaftya-vyakhya by the differ- ent commentators : See under the respective commentators. Ascending nodes of planets, 19-22 ; — motion on the ecliptic, 20-21, 114 Ascensional difference (carardha), 136 ff. Asterisms, — orbit of, 13, 14, 15 ; — position in relation to the planets, 102-3 Astronomy, I. The solar system, by H.N. Russell etc., 18n Audaylka-tantra (—Aryabhatiya), 28 Automatic sphere {Gola-y antra), 129-30 Avarga letters denoting numbers, 3, 4, 5 Avarga places for numbers, 3, 4, 5 Avantl (Ujjayinl) : See under Ujjayinl. AyOma-ksetra (trapezium), area of, 42-43 Ayana-drkkarma for the Moon, 149-51 Ayana-vaJana, 159-61 Babuaji Misra, 7n, 20n BadavSmukha, located at the south pole, 122-23 Baker, Robert H,, on 'northern lights, or aurora' vis a vis the Meru mountain, 122 INDEX ^ Bakhshali manuscript, on : — A.P, Series, 63 ; — problems on travellers, 73n Baladeva Misra, takes Kali 3600 as the time of composition of the Aryabhatiya, 98 Baudhnyana-sulbasntra, on square of hypotenuse, (Pythagoras' theorem), 59 Bhaga (=amsa) (degree), measure of, 86 BhSgavata-Purana, on the com- mencement of Kaliyuga, lOn Bhagola (sphere of asterisms)— motion of, as seen from the poles, 127;— nature of and com- putations based on, 113-27 ; —principal circles of, 113 ; — visible and invisible portions at any place, 126 Bharata (battle), denoting the com- mencement of Kaliyuga, 9, 10 Bhaskara I, on : — A, hi. 5, 91 ; — A, iv.l, 114 ; — A.P. Series, [A, ii.19), 62-63; — aksavalana, 160 ; — apogees, revolution- numbers of, 21-22 ; — area of plane figures (A, ii. 9), 43-44 ; — area of a triangle (A, ii.6), 39 ; —beginning of Kali, 10 ; — Brahman, 2 ; — composition of A, 33 ; — cube and cubing, 35-36 ; — daily motion of planets, 106; — drkksepa, 147 ; — ghanacttighana, 66-67 ; — 184 APPENDIX III gnomon, 56 \-gulika (Z, iUO), 73 ; — identification of the ten gltikas, 31 ; —kuttakaro, 77 ; — kuvHyu, 28 ; —reads kvUvarta for bhavarta in A, iii.5, 91 ; — location of Ujjayini, 124 ; — manda and sighra epicycles, 24-26 ; — measuring of time, 99 ; — Meru mountain, 121 ; — motion of ascending nodes and apogees, 21 ; — nr (no), 16; —reading of A, i.13, 32 ; — Rsines, 45-51 ; —sankalana (sum of a series), 64 ; — squar- ing (il, ii. 3), 34-35 ; — Sun's prime vertical altitude, 143 ; — testing of level ground {A, ii.13), 55 ; — triangles, A, ii.6, 55 ; — viksepavalana, 160 ; — volume of a sphere, 41 ; — the word utkramana in 2., iv.36, 150-51 ; — writing down of numbers, 5 Bhaskara II, on : — aksavalana, 158 ; —ayanavalana, 160 ; — latitude triangles, 131 ; — location of Ujjayini, 124 ; — motion of the moon and the planets, 115 ; — Rsine diffe- rences, 30 ; — Sighrocca of Mercury and Venus, 115 ; — volume of the sphere, 42. See also Bijaganita, Lilavatl, Sid- dhanto-slromani. Bhaiabda, 98 Bhatfotpala, 20n, 34n, 89n ; — on aksavalana, 158n Bhoja, 98 Bijaganita (BBi) of Bhaskara II, 72n. See also Bhaskara II. Brahma{n), 1, 2 Brahma, God, originator of the science of astronomy, 1,2,33, 163, 164 ; —day of, 9, 118 Brahmadeva, author of Karana- prakasa, 98 Brahmagupta, on : — aksavalana, 158-59 ; —ayanavalana, 160; — location of Ujjayini, 124-26 ; — volume of a pyramid, 40 ; — wrong reading of A, i.ll, 28 Brahmagupta's criticism of Arya- bhar.a I on : ii, iii.16, 103-4 ;— beginning of Kallyuga, 11-12 ; — drggatijya, 146 ; — Earth's revolution, 15 ; —Earth's ro- tation, 119 ; — parallax, 145 ; — Sighra epicycles, 27-28 ; — Sun's prime vertical altitude, 143. Brahma-sphuta-siddhanta (BrSpSi), lln, 15n, 27-28, 34-35n, 37n, 39n, 40, 41, 55n, 57-58n, 60- 61n, 63-65n, 67-68n, 70-72n, 74n, 79n, 99-100n, 103-106n, 113-16n, 118n, 124n, 128n, 143n, 145-46n, 149n, 152-53n, 158n, 160n. See also Brahma- gupta. Brhaddevata, on A.P. Series, 63 Brhat-jataka {BrJa), of Varaha- mihira, 34n. See also VarSha- mihira. brhat-ksetra-samdsa, 60n Brhat-samhlta (BrSam), of Varaha- mihira, 89n. See also Varaha- mibira. Burgess, E., 16n, 20n, 2In, 30n Candrocca (Moon's apogee) : See under Moon. Carllrdhd (ascensional difference), 135ff. Cardinal cities, 123 Catesby, R., 24n Chatterjee, Bina, 7n, 20n Chord of one-sixth circle (60°) , 44-45 Circle (vrtfa), - area of, 40 ; — arrows of intercepted arcs of inter-secting circles, 60-61 ; — chord of one-sixth-circle (i.e., 60°), 44-45 ; — circumference- diameter ratio, 45 ; — con- struction of, 55 ; — divisions of, 85-86; —eccentric, designed for the explanation of planetary motion, 104 ff.; — square on half-chord, theorem on, 59-60 Circumference-diameter ratio, 45 Citi, — citighana, 64-65 ; — ghana- citlghana, 65-67 ; — varga- citlghana, 65-66 Clark, W.E.. Translation of Arya- bhatiya, 37, 98 Conjunctions of planets, in a yuga, 86 Gtibe and cubing, 35-36 A. Bb. 24 iNDiX 115 Cube root (ghanamnld), 37-38 DasagitikasUtra 1-32 ; ■ — identi- fication of the gitikas, 31-32 Datta, Bibhutibhushan , —on Brahma and Brahman, 2 Datta and Singh, History of Hindu Mathematics, 37, 52n, 72n, 84 Day(s) (divasa, dina), —civil, in a yuga, 91 ; —commencement of the first day of the yuga, 99 ; — measure of, 85-86 ; — of Brahma, 92 ; — omitted lunar days in a yuga, 91 ; - sidereal, in a yuga, 91 Day-circle (ahoratra-vrtta), radius of, 132-33 Declination of the Sun, 17-19 Degree {amsa, bhaga), circular mea- sure, 85 Demons, residing at Badavamukha, 122-23 Deva, —Bija correction for Saka 444, 97 ; — on the location of UjjayinI, 124. See under Kara- naratna also. Dina, divasa (day) : See under Day. Diurnal motion, 130-31 Drggatijya, of the Sun and the Moon, 146-47 Drkkarana, probably by Jyesfha- deva, 98 Drkksepa, 144-45 Drkksepa-jya, 144-45 Drkksepa-vrtta, 129 186 APPENDIX III D(hman4ala, 129 Dugan, R.S., 18n DussomH, 92-93 Dvapara-yuga, 10, 11,95 Dvigata (square), 35 Dvivcdi, Sudhakara, 89n Earth, — at the centre of the Bhagola, 1 1 8ff. ; —bright and dark sides of, 117 ; — consti- tution of, 118, 151 ; — increase and decrease in size of, 118-19; — linear diameter of, 15, 16, 17 ; — occupation by living beings all round, 118 ; — rate of growth of, 119 } — shape of, 118 ; — situation of, 118 Earth's revolution, — number of revolutions in a yuga, 6, 7 ; — refutation by Brahmagupta, 15 Earth's rotation, 8, 13-15, 91, 1 19-20 : — according to com- mentators of A, 8 ; — according to Makkibhatfa, 8 ; — acc. to Prthndaka, 8 ; — mention in Skanda-purVna,% ; —objection to the theory, 8 ; — refutation by Varahamihira, 8 Earth's shadow (Tamas), — at the Moon's distance, 153-54 ; — calculation of the length of, 152-53 ; — motion on the ecliptic, 114 ; — nature of, 151 Earth-sine (k&ijy*), 135-36 Eccentric circles, — planetary motion explained through, 104-5 Eclipses of the Sun and the Moon, 151-62 ; — condition for occur- rence, 151-52. See also under Lunar eclipse, Solar eclipse. Ecliptic, — motion of planets etc. on, 114 ff. ; — obliquity of, 17-18 ; —position of, 113-14 ; — right ascensions of, 133-34 Epicycles, —manda and tighrt, of the planets, 22-28 ; — plane- tary motion explained through, 105 ff. Epoch of time-reckoning, 6, 7, 8 Flint, Richard Foster, Glacial and Pleistocene Geology, 119 Fractions, — reduction of, 70 ; — inversion of, 71 ; — simpli- fication of, 69-70 Ganesa Daivajna, 98 Ganita, 1, 2 Ganita-kaumudi (GK) of NffrSyana, 34n, 37n, 57-58n, 60n, 68n, 70n, 71n Ganita-sUra-sahgraha (GSS) of Mahavira, 34-37n, 4 In, 59n, 60-61n, 63n, 68n, 70-71n, 73. See also Mahavira. Ganita-tilaka (GT) of Srlpati, 34n, 36-37n, 68n, 70-71n Geology of India, by D.N. Wadia, 119n SUBJECT Ghana (cube), 35-36 Ghana-gold (sphere), — volume of, 40-42 Ghana-mula (cube root), 37-38 Ghana- sahkalana, 66 GhQi a (multiplication), 35 GttikH-pada, 1-32 Glacial and Pleistocene Geology, by Richard Foster Flint, 119, Gnomon, — computations based on, 56-58 ; — description of, 56 Gods (dtvas), —residing at Meru, 122-23 ; —visibility of the Sun for, 127 ; —year of, 92 Gola, 1, 2 Gola-pada, 113-64 Gola-y antra (armillary sphere), 113, 129-30 Govinda-svSmi, on ; — aksavalana, 159 ; — ayanadrkkarma, 15 In ; manda and slghra epicycles, 25. See also Laghu-Bhaskariya and Maha-Bhaskariya. Graha-cnra-nibandhana (GCN) of Haridatta, 97n. See also Haridatta. Grahalaghavd of Ganesa Daivajna, 98 Greek astronomers, — planetary diameters (angular), according to, 16 Greek Mathematics, by Sir Thomas Heath, 59n INDEX 187 Gulikn (unknown quantity), — cal- culation from given data, 72-73 Gunana (multiplication), 35 Gurvabda (Jovian years), 88-89 Gurvaksara (time for pronouncing a long syllable), 85, 86 Hankel, 59 Haridatta, 97n ; on : — A, hi. 10, 98 ; — Bija correction for Saka 444, 97. See also Graha-cdra- nibandhana. Hast a, measure of length, cubit, 19 Hati (multiplication), 35 Heath, Sir Thomas, 59n Hemaraja Sarma, 94n Himalayas, rate of uplifting of, 119 History of Hindu Mathematics, by B. Datta and A. N. Singh, 37, 52n, 84 Hora (hour), — horeia-s, 103-4 Horizon (ksitlja), 128 Hour (hora), — lords of the hours (horesa-s), 103-4 Hypotenuse {karnd), — basis for the triangle and the rectangle, 55 ; —theorems based on, 59-60 Iccha (requisition), in rule of three, 68-69 Intercalary months in a yuga, 91 Interest on principal, 68 Jaina texts, on the Meru, 121 198 APPENDIX IU Jumbvdvipa-samasa, 60 n Jinabhadra Gani, 60 Jovian years (gurvabda), 88-91 ; -^asterisms in which Jupiter rises, 89 ; —names of, 88 Jupiter, — ascending nodes and apo- gees of, 19-22 ; — inclination of orbit, 17-18 ; —linear dia- meter of, 15, 16, 17 ; — manda and slghra epicycles, 22-28 ; — measure of orbit, 13 ; — revo- lutions in a yuga, 6,1 ; — iighrocca of, 6, 7, See also under Planets. Jyotisakarandaka, on vyatipata, 87 Jyotiicandrika by Rudradeva Sharma. 86n, 89n Kaksy*man4ala t °vrUa, 23 ; — motion of planets along, 104-6 Kala (minute of arc), measure of, 86 Kalakriya, 1, 2 KalakriyH-pada (section on reckon- ing of time), 85-112; — signi- ficance of the term, 99 Kalardhajyah (Rsine-differences, in minutes), 29-30 Kaliyuga, 95 ; — commencement of, 8-11 ; — started with the Jovian year Vijaya, 91 ; — the measure of, 8, 9-12 Kalpa, commencement of current, according to Aryabha^a I, 8 Karana-ratna (KR), of Deva, 17-19n, 23n, 97n, 123-24n, 150n. See also Deva. Karanl (square), 34 Karna : See under Hypotenuse. Kusyapa-samhita, on the division of the yuga, 94n Katapay&di system of writing numbers, 3n Kaye, G.R., 37 Kern, H., on A, i.13, 31 Khagola (sphere of the sky), 128-30; — observer at the centre of, 128-29 ; — principal circles of, 113 Khanda-khndyaka (KK) of Brahma- gupta, 7n, 17-20n, 56n, 87n, 98, 150-5 In, 155n, 158-59. See also Brahmagupta. KrUntivalana (= ayanavalana)^ See Ayanavalana. Kr?na, Lord, — leaving the Earth at the advent of Kali, lOn Krtayuga 11, 95 Krti (square), 34 Ksetra, — in the sense of bhctgola, 86 ; — in the sense of a sign of the zodiac, 86 Ksetraphala (area) : See under Area. KsitijyB (earthsine), 135-36 Kuruksetra, 10 Kurus, 10 Kusumapura (Pafaliputra), 33 SUBJECT KuUSkSra (pulveriser), — ntragra (non-residual), 77-84 ; — sQgra (residual), 74-77 Kuvdyu (terrestrial wind), 23-28 Laghu-Bhaskanya(LBh), of Bhaskara I, 15n, 17-19n, 23n, 24, 25-26n, 109n, 11 In, 115-1611,13611, 141n, 143n, 146n, 148n, 150n, 153 56n, 158-59n. See also under Bhaskara I. Laghu-Bhaskariya-vyakhya, — by Sankaranarayana, 25 ; — by Udayadivakara, 25 Laghu-imnasa (LMa) of Manjula, 98 Lahiri, N.G, 7n, 20 Lalla, 98 ; — on manda and Mghra epicycles, 24; —on the location of UjjayinI, 124, See also §isya-dhi-vrddhida. Lamp-post, — problems based on, 57-58 Lanka, hypothetical city on the equator where the meridian of Ujjain intersects it, 119,123-25; — commencement of the revo- lutions of the planets at, 6, 7, 8 ; — location of, 8 ; —planets at the beginning of the yuga, 99 Lankodaya (right ascension), 133-34 Lata, a type of vyatipMa, 87 Latitude- triangles (aksa-ksetra), 130-32, 143 INDEX 199 Level ground, testing of, 55 Lllavati (£) of Bhaskara II, 34n, 36-37n, 42n, 57-58n, 61n, 63? 65n, 68n, 70-71n. See also under Bhaskara II. Loka-prakasa, on the Meru, 12I&. Lunar eclipse, — Aksavalana, 15&- 59, 160 — ayanavalana, 159- 61 ; — colour of the Moon, 161 ; — eclipsed part at any time, 156-58 ; — half-duration (sthttyardha), of, 154-55, 160 ; uneclipsed part, 155-56 Macdonell, A.A., 63n. Mahabharata (battle), — denoting the commencement of the Kaliyuga, 9, 10 Maha-Bhaskarlya (MBh), of BhSs- kara I, 13n, 15n, 17-19n, 23n] 24, 25-26n, 56n, 109n, llln, 115-16n, 132-33n, I35n, 140- 44n, 146-48n, 150n, 153-55n, 158-59n, 160, 162n. See also BhSskara I. Maha-Bhaskarlya-vyakhya by Go- vinda-svami, 25n, 121n, 151n Mahasiddh&nta of Aryabha^a II> 30n, 36n, 42n, 61n, 63n, 68n, 70n, 7 In, 99n, 123n, 156n Mahavlra, — on the volume of a sphere, 41. See also Ganita-sara- sahgraha. Makaranda-saranl, 96n ld0 APPENDIX Makkibhatfa, on the rotation of the Earth, 8 Manda and slghra epicycles of planets, 22-28 Mandaphala of planets, 106-11 Mandocca : See under Apogees. Manes, —visibility of the Sun for, 127; —year of, 92, 127 Manjula (Munjala), 98. See also his work Laghumanasa. Manu (yuga), measure of, 9-12 Markandeya-purana, on the Meru, 121n Mars, — ascending nodes and apogees, 19-22 ; —inclination of orbit, 17-18 ; —linear dia- meter of, 15, 16, 17 ; —manda and fighra epicycles, 22-28 ; — measure of orbit, 13 ; ■ — revolutions in a yuga, 6,7 ; slghrocca of, 6,7. See also under Planets. Masa (month), 85-86 Mali (optional number), (in KuWknra),VS. Matsya-purSna, on the Meru, 121n Mean planets, computation of, 104- 11 Men, visibility of the Sun by, 127 Mercury, —ascending nodes and apogees, 19-22: —inclination of orbit, 17-19 ; —linear dia- meter, 15, 16, 17; — ifMiHfeand 'slghra epicycle, 22-28 ; — revo- > lutions in a yuga, 6,7; —speci- alities in computing. 114 flf; — sighrocca of, measure of orbit, 13. See also under Planets. Meridian of the Khagola, 128 Meru mountain, —description of, 121-23 ; — linear diameter of, 15 Minute (kald), circular division, measure of, 86 Months (mnsa), 85-86 ; —com- mencement of the first month of the yuga, 99 ; —intercalary, in a yuga, 91 ; —lunar, in a yuga, 91 Modern astronomers, on :— angular diameters of the planets, 16 ; — apogees, 20 ; — ascending nodes, 20 ; —inclination of orbits, 18, 19 ; —sidereal period of planets in terms of days, 7 ; — synodic revolutions and periods, 88 Modern geologists, rate of uplifting of the Himalayas, 119 Modern values, —of Rsine-diffe- rences, 30 Moon, —colour during eclipse, 161 ; — constitution of, 151; drggatljya, 146-47 ; —heliacal visibility of, 116-17 ; —incli- nation of orbit, 17-18 ; — linear diameter of, 15, 16, 17 ; —manda epicycles of, 22-28 ; — measure of orbit, 13 ; — motion of, 114-16, 162, 163 motion of node, 114 ; — parallax, 147 ff.; — revolutions in a yuga, 6, 7 ; — revolutions of apogee (candrocca) in a yuga, 6, 7 ; — time of revolution, smallest, ICO ; —visibility at sunrise and sunset, 116-17; — visibility of the Sun for the manes who live on the Moon, 127. See also under Planets. Mountains, rate of growth of, 119 Moving bodies, time for meeting of, 73-74 Multiplication, 34-35 Mula (Principal), 68 Na (N(), unit of length, 15, 16, 19 Nadi (ghatt), measure of, 85, 86 Narayana : See under Ganita- kaumudi. Nilakantha, on : A, ii.9, 44 ; — Kali 3600 as epoch of zero correction, 98 ; — location of Ujjayinl, 125-26 ; — Meru mountain, 121 ; —plane figures, (A, ii. 9), 44 Niragra-kuttakara ( non-residual pulveriser ), 77-84 Notation by letters of the alphabet, 3-5 Notational places, 33-34 subject index idi Nr (no), unit of length, 15, 16, 19 Numbers, — manipulations of, 67 ff.; — method of writing, 3-5 Omitted lunar days in a yuga, 91 Orbit of planets : See under 'Planets' and also under the individual planets* Pi (t), value of, 45 Panca-siddhantika (PSi), of VarSha- mihira, 18n, 21n, 44n, 100-ln, 103n, 116n, 117, 118n, 123n, 127n, 135-36n, 140-4 In, 144n, 155n, 158-59n, 161n. See also Varahamihira. PSncJavas, 10 Para-Brahman^ 2 Parallax of the Sun and the Moon, 147ff. ParameSvara, on : A, iv. 38, on visibility correction of planets, 150-51 ; — aksavalana, 159; — identification of the ten gitlkas t 31, 32; —Kali '3600 as epoch of zero correction, 98 ; — location of Ujjayinl, 124 ; — plane figures {A, ii9), 44 ; — square (A, ii. 3), 34 ; — sthityardha, 160 ; word 'utkramana* in A, iv.38, 150-51 Para-'sahku (Sun's greatest gno- mon), 144 Pdta (nodes) of planets, —motion on the ecliptic, 114 PStaliputra (Kusumapura), 33 APPENDIX lit Patlganfta (P8) f of Srtdhara, 34- ^n, 61n, 63-65n, 68n, 70-71n, 73n. See also Sridhara. Pauliia-siddhanta, 33; on : — ak$a- drkkarma, 148n ; — inclination of the orbits, 18 ; — method for akfavahna, 158 Phala ('fruit'), in rule of three, 68-69 Phala (interest), 68 Plane figures in general, area of, 43-44. See also Circle, Square, Trapezium, Triangle. Planets, —angular diameters, 16 ; — anomalistic revolutions, 87- 88 ; — ascending nodes and apogees, 19-22 ; — bright and dark sides, 117 ; —computation of mean positions, 104-11 ; — computation of true positions, 104-11 ; — conjunctions of, in a yuga, 86 ; daily motion of, 106 ; —determination of the motion of, 162-63 ; —distance of, from the Earth, 111; — equality in the linear motion of, 100 ; — heliacal visibility of, 117; — inclination of the orbits, 17-19 ; — lengths in yojanas of the circular divisions in the orbits, 101-2; — linear diameters of, 15-17; — manda and Sighra epicycles, 22-28 ; mandophala, 106-11 ; — mean angular velocities, 102; mean motion acc. to Aryabhata, 7 ; — mean motion acc. to mo- derns, 7 ; — motion explained through eccentric circles, 104- 5 ; — motion explained thro- ugh epicycles, 105-6; -motion of, 114-16; — motion of nodes of, 114 ; —non-equality of the linear measures of the circular motions, 101-2 ; — orbits, mea- sure of, 13-15 ; —orbits, theory of, 14-15 ; — orbits, inclination of, 17-19; — position in relation to the asterisms, 102-3 ; — revolutions of, in a yuga, 6-8 ; Hghraphala, 106-11 ; —sighro- ccas of, 6, 7 ; — synodic revo- lutions of, 87-88; — velocity of, 111-12 ; —visibility of, 116-17. See also under the individual planets. Plumb-test for vertically, 55 Prabhavadi Jovian years, 90 Pramana (argument) in the rule of three, 68-69 Prana (time for one respiration), 85, 86 Pratimandala (eccentric circle), motion of planets on the, 104-5 Prawha-vayu (-anila), (provector wind), 119 Prime vertical, 128 Principal and interest, 68 Prthndaka, on : —correct reading bhah for bham in A, i.6, 15 ; SUBJECT INDEX 193 —justification of A, iii.16, 104 ; — motion of the sighroccas of Mercury and Venus, 115 ; — supporting the theory of the rotation of the Earth, 8, 120 ; — volume of a sphere, 41 Ptolemy, — angular diameters of planets, 16 ; — apogees of planets, 20; — ascending nodes of planets, 22 ; -—distances of the planets from the Sun when heliacally visible, 117 ; —'Earth is stationary', 8, 120 ; — inclination of planetary orbits, 18, 19 ; — manda and sighra epicycles, 23-24 ; — mean motion of planets, table for, 7 ; — synodic revolutions and periods, 88 Pulveriser (kmtakdra), 73-84 ; — non-residual (niragra), 77-84 ; —residual (sagra), 74-77 Puranas, on the Meru, 121 Pyramid ($adasri), —sum of pile of balls in, 64-65 ; —volume of, 39-40 Pythagoras' theorem, 59 Raghunatha-raja, on : — apogees, revolution of, 21-22 ; —ayana- drkkarma, 151 ; —Kali year 3600, 96; — reads kvavarta for bhavarta in A, ii.5, 91 ; — location of UjjayinI, 126 ; —Meru mountain, 121; -plane figures (A, ii.9), 44 Rnja-Mrganka of Bhoja, 98 Hasi (sign), measure of, 85 Rectangle, construction of, 55 Revolution numbers of planets, 6-8 Right-ascension (Lahkodaya), 133- 34 Romaka, hypothetical city on the equator, 123 Romaka-siddhanta, 33; —Apogees, 20 ; —inclination of the orbits, 18 Rotation of the Earth : See under Earth's rotation. Rsines, geometrical derivation of 45-51 Rsine-differences, 29-30 : —deri- vation of, 51-54 Rudradeva Sharma, 86n, 89n Rule of three (trairasika), 68-69 Russel, H.N., 7n, 18n Sabda-Brahma, 2 Sadasri (pyramid) : See under Pyra- mid. Sdgra-kumkara, (residual pulveri- ser), 74-77 Saka 444, — Bija corrections for, 97 ; — year of zero precession, 98 A. Bh. 25 194 APPENDIX III Sak&bda correction, 98 Sambasiva Sastri, K., — Kali 3600 taken as epoch of zero correc- tion, 38 Samvarga (multiplication), 34, 35 SahkalanS, 64ff. — ghana-sankalana, 66*67 ; — varga-sank'alana, 66 Sahkalita-sahkalita, sum of the series %2,n, 64-65 Sankaranarayana, on : — end points of the anamolistic quadrants, 25 ; — manda and slghra epi- cycles, 25 Sahku (gnomon), — computations, based on, 56-58 ; — description of, 56 Sahku (Rsine of altitude), 139ff. Sankvagra of the Sun, 14 Iff. Sara (arrows) of intercepted arcs of intersecting circles, 60-61 Sastri, T.S. Kuppanna, 98 Saturn, —ascending nodes and apogees of, 19-22; — inclination of orbit, 17-19 ; —linear dia- meter of, 15, 16, 17 ; —manda and slghra epicycles of, 22-28 ; — measure of orbit, 13 ; — revolutions in a yuga, 6, 7 ; — sighrocca of, 6,7 ; — time of revolution, longest, 100. See also under Planets. Sengupta, P.C, 7n, 8, 16n, 20, 21n, 30 Series : See A.P. Series, Shadow-sphere, radius of, 56 Shamasastry, R., 87n Siddhanta-dtpika, of Paramesvara, 159n Siddhunta-sekhara (SiSe) of Srlpati, 7n, 20n, 34-37n, 42n, 57n, 6 In, 63-65n, 68n, 70-72n, 99n, 103n, 105n, 113-15n, 117-19n, 128n, 156n. See also Sripati. Siddhanta-siromani (SiSi) of Bhas- karall, 30n, 55n, 99n, 103n, 105n, 113-15n, 118n, 123-24n, 128n, 131n, 149-50n, 158n, 160n. See also Bhaskara II. Siddhapura, hypothetical city on the equator, 123 Sidereal division of time, 85-86 Slghra epicycles of planets, 22-28 Slghraphala of planets, 106-11 Sighroccas, 8 ; — determination of the motions of, 163 Sign (roii), circular division, mean- ing of, 86 Sisya-dhi-vrddhida (SiDVr), of Lalla, 23 n, 24, 25-26n, 30n, 55n, 98, 100-102n, 105n, 109n, 11 In, 113-19n, 122-24n,126n,128-29n, 149n, 156n. See also Lalla. Sixty-year cycle of Jovian years, 89-91 Skanda-Purana, supports the ro- tation of the earth, 8 Sky, orbit of, 13, 14, 15 SUBJECT Solar eclipse, 151-62 ; —parallax in, 144-47 ; —when not to be predicted, 161-62 Solar system, 7n Solar year : See under Year. Somesvara, on: — A, i.13, 32 ; — 2, iv.4, 116; A, iv.31, 143 r-beg. of Kali, 10 ; — daily motion of planets, 106 ; — determi- nation of the planets from observation, 162-63; — identi- fication of the ten gitikas, 31 ; —Kuttaktlra, 79; —location of Ujjayini, 124 : — Meru moun- tain, 121; -opposing the theory of the rotation of the Earth, 120; — plane figures (A, ii.9), 44 ; —planetary orbits, 13n ; — sthityardha, 160 ; — word utkramana in A, iv.36, 150-51 Smart, W.M., 8n Sphere (ghanagola), volume of, 40-42 Spherical astronomy, 130-51 Spherical astronomy, Text-book on, 8n Square and squaring, 34-35 Square root (vargamula), 36-37 Srldhara, on the volume of a sphere, 42 Sripati, — on the volume of a sphere, 42. See also Ganita- tilaka, Siddhanta-sekhara. Stars, apparent motion of, U9-20 INDEX 195 Stewart, J.Q., 7n, 18n Sthityardha : See under Lunar eclipse. Sumati, 30 Sumatikarana of Sumati, 30n Sumati-maha-t antra (SMT), 158n, 159n. Sumatitantra of Sumati, 30n Sun, —Arkagra, 141 ff. ; —consti- tution of, 151 ; — declination of, 17-19 ; — determination of the motion of, 162 ; — drggati- Jya, 146-47 ; —eclipse of, see under Solar eclipse ; — linear diameter of, 15, 16, 17 ; — manda epicycles of, 22-28 V — measure of orbit of, 13 j — motion along the ecliptic, 114 ; —para-sahku (greatest gnomon), 144 ; —parallax of, 147AF.; — revolutions in a yuga, 6, 7 ; — visibility for the gods, manes and men, 127. See also under Planets. Snryadeva, on : A, ii.9, 91 ; —A t iii. 22-23, on the computation of planets, 110; 2, iv.31,143 ; — apogees, revolution of, 21, 22 ; — beginnings of the anomalistic quadrants, 25n ; — drggatijya, 146 ; — examples on triangle, A } ii.6, 55 j — God Brahma 1-2 ; ^-golayantra, 129-30 ; — identification of the ten gitikas, 31 ; — Kali year 360Q, 96 j — - 196 APPENDIX III Kali 3600 as epoch of zero correction, 98; —ku\\akara, 11\ — location of Ujjayini, 124- 25 ; -—plane figures, (A, 11 9), 44; —valana, 161; — word utkramana in A, iv.38, 150-51 j Surya Narayana Siddhanti, 90n Surya-siddhanta (SnSi), 30, 55n, 92, 100-ln, 111, 113n, 128n ; — Tr. by E. Burgess, 20n, 2 In, 30n Surya-siddhanta (Old), 33 ; —on aksavalana, 159 Susama, 92-93 Svuyambhuva-siddhanta, (BrShma- siddhanta\ 33, 164 Syamantapancaka, 10 Taittinya Samhita, —on A.P, Series, 63 Taliaferro, R.C., 18n, 20n, 21n, 24n, 120n Tatpara (Third of arc), measure of, 86 Tamas (Earth's shadow) : See under Earth's shadow. Tantra-sahgraha (TS) of NUakantha, 156n. See also Nllakantha. TattvUrthadhigama-sutra, 60n Text-book on Astronomy, by C.A. Young, 119 Text-book of General Astronomy, by C.A, Young, 22n Texi'book on spherical astronomy, by W.M, Smart, 8n. Theon of Alexandria, 37 Third (tatpara), in circular division, measure of, 86 Three, rule of, (trairSsika), 68-69 Time, — divisions of, 85-86 ; — measuring of, 99 ; — reckoning of, 85-1 12 ; — without begin- ning or end, 99 Tithipralaya (omitted lunar days), 91 Traira'sika (rule of three), 68-69 Trapezium, area of, 42-43 Treta-yuga, 95 Triangle (tribhuja), —area of, 38- 39 ; — construction of, 55 Tribhuja (triangle) : See Triangle. “Trlsatika (Tri's), of Sridhara, 42n. See also Sridhara. True planets, computation of, 104-11 Twelve-year cycle of Jupiter, 88-89 Tycho Brahe, angular diameters of planets, 16 Udayadivakara, on -.—beginning of the anomalistic quadrants, 25 ; — correct reading bhuh for bham in A, i.6, 15 ; —manda and slghra epicycles, 25 Udvartana (multiplication), 35 SUBJECT INDEX 197 Ujjain, (Avanti), —meridian at, 8 j I — situation of, 123-26 UmSsvSti, 60. 1 1. Umasvati is reputed to be one of the greatest metaphysicians of India and is held in high esti- mation by the two main sections of the Jainas. Unfortunately, his time and place of birth have not been settled definitely. According to the tradition of the Svetambara Jainas, Umasvati was born in the now forgotten city of Nyagrodhika. His name is said to have been a combination of the names of his parents, the father Svati and the mother Uma\ He was the disciple | of the saint Ghosanandl. He I lived about 150 B.C. His disciple Syamarya or SySmScarya, the author of the Prajnapana-sutra, is said to have died 376 years after Sri Vlra, that is, in 92 B.C. and his earliest (commentator is said to have been Siddhasena Gani, or Divakara who lived c. 56 B.C. The Digambara tradition, on the other hand, sometimes, even changes his name and thinks it to be Umasvami, not Umasvati. Accord- ing to it, he lived from 135 A.D. to 219 A.D. Satish Chandra Vidya- bhushan is of the opinion that he flourished in the first century A.D. All are, however, agreed on one point, viz. f that Umasvati resided in the city of Kusumapura (modern Patna). Unknown quantities, — from equal sums, 72-73; —from sum of all but one, 71-72 Utkramana, word in A, iv.36, inter- pretation of, 150-51 Vtsarpinh 92-93, 94 i Vaidhrti, a type of vyatipata, 87 Vaiasaneya Samhita, on A. P. Series, 63 Varahamihira, 18, 20, 117 ; —on akfavalana, 158, 159; —on the location of Ujjaymi, 125-26 ; — refutation of the rotation of the Earth, 8, 119. See also Brhat-jStaka, Brhat-samhita, Panca-siddhSntika. Varga (square), 34-35 Farga -letters denoting numbers, 3-5 Varga-places for numbers, 3, 4, 5 Vargamula (square-root), 36-37 Vargana (square), 34 Varga-sahkalann, 66 Varsa (year) : See under Year. Vasitfha-siddhanta, 33, 117 Vasubandhu, 121 n Vatesvara, — justification of A, iii. 16 against Brahmagupta, 104 ; —reply to Brahmagupta on the concept of the yugas, 12 Vateivara-siddhanta, 12n, 113-15n, 122n, 128n VJlyu-purStfa, on the Meru, 121n Vedanga-Jyautifa, 87n 198 appen: Venus, — ascending nodes and apogees, 19-22 ; —inclination of orbit, 17-19 ; —linear dia- meter, 15, 16, 17 j — manda and sighra epicycles, 22-28; — revolutions in a yuga, 6, 7 j — revolutions of sighrocca in a yuga, 6, 7 ; — sighrocca of, measure of orbit, 13 ; — specialities in the computation of, 114 ff. See also under Planets, Vertically, testing of, 55 Vijayadi-varsa (sixty-year cycle of Jupiter), 89-91 Vikala (second of arc), measure of, 86 Vina&ka (vighati), measure of, 85, 86 Visnu-purtoa, on the Meru, 121n Visvanatha, on Kali year 3600, 96 Vowels, denoting numbers, 3, 4, 5 Vrtta (circle) : See under Circle. Vyatipata, 86-87 ; — in a yuga t 86-87 Wadia, D.N., Geology of India, 119. Water-test for level ground, 55 [X III YSvakaraiia (squaring), 34 Yavakotf, hypothetical city on the equator, 123 Year {abda, varsa), 85, 86 ; — commencement of the first year of the yuga, 99; — measure of, 15 ; —of gods (divine), 92 ; — of manes, 92 ; —of men, 92 ; — solar, in a yuga, 91 Yojana, 19 Young, C.A., — on the motion of Mercury, 22n; —on thfc growth of the Earth, 119 Yudhisthira, lOn Yuga, — abraded, with correspon- ding revolutions, 94 ; — ano- malistic revolutions in, 87-88 ; — commencement of, 99 ; — commencement of the current, 6, 7, 8 ; — divisions of, 92-94, 95 ; — Jovian years in, 88 ; — length of, 92 ; — revolutions of planets in, 6-8 ; — synodic revolutions in, 87-88. See also under the individual yugas. Yuga of five years, vyatlpatas in, 87 Zero-point of time reckoning, 6 APPENDIX IV SELECT BIBLIOGRAPHY ON ARYABHATIYA AND ALLIED LITERATURE A. Primary Sources Apastamba-iulba-sntra 1. Ed. with the corns, of Kapardisvamin, Karavinda and Sundararaja, Oriental Res. Inst., Mysore, 1931, (BibL Sanskrita, No. 73). 2. Ed. with German Tr. by A. Biirk, ZDMG, 55 (1901) 543-91 ; 56 (1902) 327-91. 3 Ed. and Tr. into English, by Satya Prakash and R.S. Sharma, New Delhi, 1968. Aryabhapiya of Aryabhata (b. A.D. 476) 1. Cr. ed. with Intro., Eng. Tr., Notes, Comments and Indexes, by K.S. Shukla and K.V. Sarma, Indian National Sc. Academy, New Delhi, 1976. 2. Cr. ed. with the commentary of Bhaskara I, with Intro, and Appendices, by K.S. Shukla, Indian National Sc. Academy, New Delhi, 1976. 3. Cr. ed. with the commentary of Suryadeva Yajva, with Intro, and Appendices, by K.V. Sarma, Indian National Sc. Academy, New Delhi, 1976. 4. Cr. ed. with the com. Bhatadipika of Paramadisvara (*=Paramesvara) by H. Kern. Leiden, 1874 ; Reprint, 1972 ; Together with Hindi tr. by Udaya Narain Singh, Madhurapur (Muzaffarpur), 1906. 5. Pt. I, Gaiyitapnda and Pt. II, Kalakriyapada, ed. with the com. entitled Mahabh^ya of Nilakantha Somasutvan, by K. Sambasiva SSstri, Trivandrum, 1930, 1931 ; Pt. Ill, ed. by Snranad Kufijan Pi]lai, 1957, (TSS, 101, 110, 185). 6. Ed. with Hindi Tr., Notes and Intro., by R. N. Rai, Indian National Sc. Academy, New Delhi, 1976. 199 200 APPENDIX IV 7. Ed. with Sanskrit and Hindi com., by B. Mishra, Bihar Res. Soc, Patna, 1966. 8. Ed.withMal.com. by P. S. Purushottaman Namputiri, S.T. Reddiar and Sons, Quilon, 1113 M.E. 9. (Gltika-pada) : Ed. with Mai. com. and Intro., by Kolatheri Sankara Menon, Trivandrum, 1927, (Sri Vanchi Setu Lakshmi Series, No. 7). 10. With the Telugu com. entitled Sudhataranga of K. Kodan- darama Siddhanti, ed. by V. Lakshmlnarayana £astn, Madras, 1956, (Madras Gov. Or. Ser., No. 139). 11. (Tr. into Eng. of Gawtapsda) : 'Notes on Indian mathematics : 2. Aryabhata', by G.R. Kaye, JASB, (NS), 4 (1908) 111-41. 12. Tr. into Eng. with Notes, by W.E. Clark, Univ. of Chicago Press, Chicago, 1930. 13. Tr. into Eng. by P.C. Sengupta, J of the Deptt of Letters, Calcutta Univ., 16 (1927) 1-56. 14. The Aryabhatiya translated into English, by B. R. Shastry, Chicago, 1925 (unpublished thesis). 15. (Tr. into German) : Die Mathematik des Aryabhata, Text, Ubersetzung aus dem Sanskrit und Kommentar, von Kurt Elfering. Wilhehm Fink Verlag, Nikolaistrasse 2, 8 Munchen- 40, 1975. 16. (Tr. into French of the Gamtapada) : 'Lecons de calcul d' Aryabhata', par M. Leon Rodet, J A, ser. 7, tome 13 (1879) 393-434. Atharvana-Jyotisa 1. Ed. by Bhagavad Datta, Lahore, 1924. , 2. With Hindi tr. by C. L. Sarma and O. N. Dvivedi, Datya (M. P.). Bakhshedl Manuscript Ed. by G. R. Kaye, Archaeological Survey of India, New Imperial Series, No. 43, Pts. I and II, Calcutta, 1927 ; Pt. Ill, Delhi, 1933. Baudhayana-iulba-sUtra 1. Ed. by W. Caland in vol. Ill of his edn. of Baud hay ana-irauta- sv.tr a in 3 vols., Calcutta, 1913- BIBLIOGRAPHY : PRIMARY SOURCES 201 2. Ed. with Dvarakanatha Yajvan's com. and Tr. into Eng. by G. Thibaut, Pandit (OS), 9-10 (1874-75); (NS) 1 (1876-77), Benares. 3. With Thibaut's Tr., ed. by Satya Prakash and R.S. Sharma, New Delhi, 1968. Bijagavita of Bhaskara II (b. A.D. 1114) 1. Ed. by Sudhakara Dvivedi, with expository notes, Benares, Rev. edn., 1927, (Benares Skt. Ser., No. 154). 2. Ed. with the com. Navahkura of Krsna Daivajna, by V.G. Apte, Poona, 1930, (Anandairama Skt. Ser., No. 99). 3. With com. Bijapallava of Krsna Daivajna by T. V. Radha- krishana Sastri, Tanjore, 1958. (Tanjore Sarasvati Mahal Ser., No. 78). 4. Ed. with Hindi Tr. and Notes by Durga Prasad Dvivedi, N.K. Press, Lucknow, 1917. 5. Ed. with a Skt. com. and tr. and notes in Hindi, by Visuddhanand Gaur, Benares, 1943. 6. (Tr. of Bljaganitd) : Algebra with Arithmetic and Mensuration. Translated from the Sanskrit of Brahmegupta and Bhascara, by H.T. Colebrooke, London, 1817 ; Rep. 1972. 7. With Notes (in Mai.), by P.K. Koru, Mangalodayam, Trichur, 1960. Brahma-sphuta-siddhanta of Brahmagupta (A.D. 628) 1. Ed. with his own com. by Sudhakara Dvivedi, Benares, 1902. 2. Ed. with Sudhakara's com. and his own com. Vasana and Hindi Tr. by Ram Swarup Sharma, with Introduction by Satya Prakash, 4 vols., New Delhi, 1966. 3. (Tr. of chs. 12 and 18) : Colebrooke, H.T., Algebra with Arith- metic and Mensuration from the Sanskrit of Brahmegupta and Bhascara, London, 1817 ; Rep. 1972. 4. With the com. of Prthudaka (A.D. 860), Ms. in India Office Library, London ; Transcript in Lucknow Univ. Library, Lucknow. Bijagayiitavatamsa (incomplete) of Narayana Pandita Ed. by K. S. Shukla, Akhila Bharatiya Sanskrit Parishad, Lucknow, 1970. A. Bb. 25 202 APPENDIX IV Brhajjataka of Varahamihira (d. A D. 587). 1. Ed. with com. Vivrti of Bhajtotpala, by Sita Ram Jha, Benares, 1922. 2. HoraSastra with Vivara^a of Rudra, Trivandrum, 1958, (Trivandrum Skt. Ser., No. 91). 3. Ed. with Eng. Tr. by V. Subrahmanya Sastri, Mysore, 1929. 4. Tr. with Notes in Hindi, by Ram Ratna Awasthi, 5th edn., T. K. Press, Lucknow, 1972. Brhajjyotih-sara Compiled and Tr. by Surya Narayana Siddhanti, T.K. Press, Lucknow, 1972. Brhatsamhita of Varahamihira (d. A.D. 587) 1. Ed. with the com. of BhattotpaJa, by Sudhakara Dvivedi, 2 vols., Benares, 1895-97, (Vizianagaram Skt. Ser., No. 10) ; by Avadh Bihari Tripathi, 1968, (Sarasvati Bhavana Grantha- maJa, No. 97). 2. (Tr. in Eng.) : B or Complete system of Natural Astrology, by H. Kern, Verspriede Geschriften (Collected Works), 's-Graven- hage, 1913. 3. Ed. with Eng. Tr. by V. Subrahmanya Sastri and M. Rama- krishna Bhat, 2 vols., Bangalore, 1947. 4. Hindi Tr. by Baldev Prasad Misra, Bombay, 1940. 5. Ed. with Mai. Tr. by P.S. Purushottaman Namputiri, Sri Rama Vilasam, Quilon, 2nd edn., 1955. Candraprajnapti Ed. with Hindi Tr. by Amolakarisi, Hyderabad, Vlrasarhvat 2445. Dhlkotida-karaw of Sripati (A.D. 1039) Cr. ed. with Intro., Tr. and Notes by K.S. Shukla, Lucknow, 1969. Dhi-vrddhida-tantra of Lalla (A.D. 8th cent.) See $isya-dhi-vrddhida. 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