ARYABHATIYA OF ARYABHATA 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 Studies V. V. B. Institute of Sanskrit and Indological Panjab University INDIAN NATIONAL SCIENCE ACADEMY NEW DELHI 1 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— © 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 table of sine-differences xxviii . 3. The 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. Somesvara' s commentary xxxvi 4. Suryadeva Yajva's commentary xxxvii 5. Paramesvara's commentary xl 6. Yallaya*s notes on Suryadeva 's commentary xl ARYABHATIYA Page xhv 7. Mlakantfia Somayaji's commentary xlv 8. Raghunatha-ra-ja's commentary j 1 9. Commentary of Madhava li 10. Bhmivisnu's commentary liii 11. Ghatigopa's commentary liv 12. Kodandarama's commentary (b) Commentaries in Telugu lv 13. Kodandarama's commentary lv 14. VirUpaksa's commentary (c) Commentaries in Malayalam lv 15. Krsnadasa's commentary lv * 16. Krsna's commentary lvi 17-18. Two commentaries by Ghatlgopa {d) Commentary in Marathi in Marathi lviii 19. Anonymous commentary lvi » h Works based on the Aryabhatiya lv»i 1. The worksofBhaskaral lvil1 2. The Karana-ratna of Deva of Haridatta lix 3. The Gtaha-cara-nibandhana Lalla lix 4. The §isya-dhx-nddhida of of Brahmadeva *x 5. The Karava-prakafa j Damodara 1™ 6. The Bhatatulya of ^ Somayaji lxii 7. The Karava-paddhati of Putumana of VIrasimha lxii 8. The Aryabhata-siddhanta-tulya-karana lxil 8. Transmission to Arab lxni The Aryabhafa-Siddhanta the Aryabhatiya lxiii 1. The Aryabhata-siddhanta and special methods lxv 2. The astronomical instruments and Aryabhata-siddhanta and the 3. Popularity of the Khanda-khadyaka lx™ lxvlli The present edition lxviii (a) Sanskrit text lxix 1. Text manuscripts lxxl 2. Text preserved in commentaries commentators lxxu 3. Quotations from later lxxm 4. Variations in reading lxxiv 5. Selection of readings lxxiv (b) English translation, notes etc. lxxvl Acknowledgements 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 2n2 and 2n3 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 planets Bright and dark sides of the Earth and the 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 quadrants and of the individual signs Rising of the four Rsine of the altitude Sahkvagra Sun's Agra altitude Rsine of the Sun's prime vertical the shadow thereof Sun's greatest guomon and (2. Parallax in a solar eclipse) point Rsine of the zenith distance of the central ecliptic 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 «. 161 When the Sun's eclipse is not to be predicted 162 Planets determined from observation »• 163 Acknowledgement to BrahmS ••• 164 Conclusion APPENDICES passages .» 165 I. Index of Half-verses and Key ... 173 II. Index-Glossary of Technical terms 182 III. Subject Index — IV. Select Bibliography on the Aryabhatlya and allied texts 1" (a) Primary Sources ••• 211 (b) Secondary Sources — 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.