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The Aryabhatiya of Aryabhata r ^ i I V* 1 1 1 M I U I i ■ I I 1 1 1 P^ 1 1 K \ 'i 1 1 1 on Mathematic and Astronomy English Translation by W. E, Clark THE ARYABHATIYA of ARYABHATA An Ancient Indian Work on Mathematics and Astronomy TRANSLATED WITH NOTES BY WALTER EUGENE CLARK Professor of Sanskrit in Harvard University THE UNIVERSITY OF CHICAGO PRESS CHICAGO, ILLINOIS THE TJNIVERSITT OF CHICAGO PKESS CHICAGO, ILLINOIS THE BAKER 4 TAYLOE COMPANY NEW YORK THE CAMBRIDGE UNIVEBSIIY PRESS LONDON THE MARUZEN-KABUSHIKI-KAISHA TOKYO, OSAKA, KYOTO, FUK0OKA, SENDAI THE COMMERCIAL PRESS, LIMITED SHANGHAI IXGHT 1930 B-r THE U J^JlVJflRSITY OS' CHICA.GO A.ZjXi RIGMT& XIX:S£1KVX3X>. FXTBI^ISBX3£> J'CTlj'Z' 1930 COMPOSSD AND PRINTED BY THE TTNIVEHSrT'ir OF CSTCAaO PKESS CBXCAOO, TT.T.T-Krr^Tp^ U.S.A. PREFACE In 1874 Kern published at Leiden a text called the Aryabhatiya which claims to be the work of Arya- bhata, and which gives (III, 10) the date of the birth of the author as 476 a.d. If these claims can be sub- stantiated, and if the whole work is genuine, the text is the earliest preserved Indian mathematical and astronomical text bearing the name of an individual author, the earliest Indian text to deal specifically with mathematics, and the earliest preserved astro- nomical text from the third or scientific period of Indian astronomy. The only other text which might dispute this last claim is the Suryasiddhdnta (trans- lated with elaborate notes by Burgess and ^Tiitney in the sixth volume of the Journal of the American Oriental Society) . The old Sitryasiddhdnta undoubt- edly preceded Aryabhata, but the abstracts from it given early in the sixth century by Varahamihira in his Pancasiddhdntikd show that the preserved text has undergone considerable revision and may be later than Aryabhata. Of the old Paidisa and Romaka Siddhdntas, and of the transitional Vdsistha Si- ddhdnta, nothing has been preserved except the short abstracts given by Varahamihira. ' The names of sev- eral astronomers who preceded Aryabhata, or who were his contemporaries, are known, but nothing has been preserved from their writings except a few brief fragments. The Aryabhatiya, therefore, is of the greatest im- vi PREFACE portance in the history of Indian mathematics and astronomy. The second section, which deals with mathematics (the Ganitapadd), has been translated by Rodet in the Journal asiatique (1879), I, 393- 434, and by Kaye in the Journal of the Asiatic Society of Bengal, 1908, pages 111-41. Of the rest of the work no translation has appeared, and only a few of the stan2as have been discussed. The aim of this work is to give a complete translation of the Aryahhatlya with references to some of the most important parallel passages which may be of assistance for fm-ther study. The edition of Kern m^kes no pretense of giving a really critical text of the Aryabhatiya. It gives merely the text which the sixteenth-century commentator Paramesvara had before him. There are several un- certainties about this text. Especially noteworthy is the considerable gap after IV, 44, which is discussed by Kern (pp. v-vi). The names of other commenta- tors have been noticed by Bibhutibhusan Datta in the Bulletin of the Calcutta Mathematical Society, XVIII (1927), 12. All available manuscripts of the text should be consulted, all the other commentators should be studied, and a careful comparison of the Aryahhatlya with the abstracts from the old si- ddhantas given by Varahamihira, with the Suryasi- ddhdnta, with the Sisyadhlvrddhida of Lalla, and with the Brahmasphutasiddhdnta and the Khandakhddyaka of Brahmagupta should be magle. All the later quota- tions from Aryabhata, especially those made by the commentators on Brahmagupta and Bhaskara, should be collected and verified. Some of those noted by Colebrooke do not seem to fit the pubKshed Arya- PREFACE vii bhatlya. If so, were they based on a_lost work of Aryabhata, on the work of another Aryabhata, or were they based on later texts composed by followers of Aryabhata rather than on a work by Aryabhata himself? Especially valuable would be a careful study of Prthudakasvamin or Caturvedacarya, the eleventh- century commentator on Brahmagupta, who, to judge from Sudhakara's use of him in his edition of the Brdhmasphutasiddhdnta, frequently disagrees with Brahmagupta and upholds Aryabhata against Brah- magupta's criticisms. The present translation, with its brief notes, makes no pretense at completeness. It is a prelimi- nary study based on inadequate material. Of several passages no translation has been given or only a ten- tative translation has been suggested. A year's work in India with unpublished manuscript material and the help of competent pundits would be required for the production of an adequate translation. I have thought it better to pubhsh the material as it is rather than to postpone publication for an indefinite period. The present translation will have served its purpose if it succeeds in attracting the attention of Indian scholars to the problem, arousing criticism, and en- couraging them to make available more adequate manuscript material. There has been much discussion as to whether the name of the author should be spelled Aryabhata or Aryabhatta.^ Bhata means "hirehng," "mercenary," ^ See especially Journal of the Royal Asiatic Society, 1865, pp. 392-93; Journal asiatique (1880), II, 473-81; Sudhakara Dvivedi, Gav^atarangiv>'i, p. 2. viii PREFACE "warrior," and hhatta means "learned man," "schol- ar." Aryabhatta is the spelling which would natural- ly be expected. However, all the metrical evidence seems to favor the spelling with one t It is claimed by some that the metrical evidence is inclusive, that hhata has been substituted for bhatta for purely metrical reasons, and does not prove that Arya- bhata is the correct spelling. It is pointed out that Kern gives the name of the commentator whom he edited as Paramadisvara. The name occurs in this form in a stanza at the beginning of the text and in another at the end, but in the prose colophons at the ends of the first three sections the name is given as Paramesvara, and this doubtless is the correct form. However, until more definite historical or n^etrical evidence favoring the spelling Aryabhatta is produced I prefer to keep the form Aryabh^ta. The Aryahhatvya is divided into four sections which contain in all only 123 stanzas. It is not a com- plete and detailed working manual of mathematics and astronomy. It seems rather to be a brief descrip- tive work intended to supplement matters and proc- esses which were generally known and agreed upon, to give only the most distinctive features of Arya- bhata's own system. Many commonplaces and many simple processes are taken for granted. For instance, there are no rules to indicate the method of calculat- ing the ahargana and of finding the mean places of the planets. But rules are given for calculating the true places from the mean places by applying certain cor- rections, although even here there is no statement of PREFACE ix the method by which the corrections themselves are to be calculated. It is a descriptive summary rather than a full working manual like the later karana- granthas or the Suryasiddhdnta in its present form. It is questionable whether Aryabhata himself com- posed another treatise, a haraiiagrantha which might serve directly as a basis for practical calculation, or whether his methods were confined to oral tradition handed down in a school. Brahmagupta^ implies knowledge of two works by Aryabhata, one giving three hundred sdvana days in a yuga more than the other, one beginning the yuga at sunrise, the other at midnight. He does not seem to treat these as works of two different Aryabhatas. This is corroborated by Paficasiddhantika, XV, 20: "Aryabhata maintains that the beginning of the day is to be reckoned from midnight at Lanka; and the same teacher [sa eva] again says that the day begins from sunrise at Lanka." Brahmagupta, however, names only the Dasagltika and the Arydstasata as the works of Aryabhata, and these constitute our Arya- hhatlya. But the word audayikatantra of Brahma- sphutasiddhdnta, XI, 21 and the words audayika and drdhardtrika of XI, 13-14 seem to imply that Brahma- gupta is distinguishing between two works of one Aryabhata. The pubhshed Aryabhatiya (I, 2) begins the yuga at sunrise. The other work may not have been named or criticized by Brahmagupta because of the fact that it followed orthodox tradition. Alberuni refers to two Aryabhatas. His later 1 Brahmasphutasiddhdnta, XI, 5 and 13-14. X PREFACE Aryabhata (of Kusumapura) cannot be the later Aryabhata who was the author of the Mahdsiddhdnta. The many quotations given by Alberuni prove con- clusively that his second Aryabhata was identical with the author of our Aryabhatlya (of Kusumapura as stated at II, 1). Either there was a still earlier Aryabhata or Alberuni mistakenly treats the author of our Aryabhatlya as two persons. If this author really composed two works which represented two slightly different points of view it is easy to explain Alberuni's mistake.^ The pubUshed text begins with 13 stanzas, 10 of ,. which give in a peculiar alphabetical notation and in a very condensed form the most important numerical elements of Aryabhata's system of astronomy. In ordinary language or in numerical words the material would have occupied at least four times as many stanzas. This section is named Dasagitikasutra in the concluding stanza of the section. This final stanza, which is a sort of colophon; the first stanza, which is an invocation and which states the name of the author; and a paribhdsd stanza, which explains the peculiar alphabetical notation which is to be em- ployed in the following 10 stanzas, are not counted. I see nothing suspicious in the discrepancy as Kaye does. There is no more reason for questioning the authenticity of the panhhdsa stanza than for ques- tioning that of the invocation and colophon. Kaye 1 For a discussion of the whole problem of the two or three Arya- bhatas see Kaye, BiUiotheca mathematica, X, 289, and Bibhutibhusan Datta, BvMetin of the Calcutta Mathematical Society, XVII (1926), 59. PREFACE xi would like to eliminate it since it seems to furnisli evidence for Aryabhata's knowledge of place-value. Nothing is gained by doing so since Lalla gives in numerical words the most important numerical ele- ments of Aryabhata without change, and even with- out this parihhdsd stanza the rationale of the alpha- betical notation in general could be worked out and just as satisfactory evidence of place-value furnished. Further, Brahmagupta {Brdhmasphutasiddhdnta, XI, 8) names the Dasagitika as the work of Aryabhata, gives direct quotations (XI, 5; I, 12 and XI, 4; XI, 17) of stanzas 1, 3, and 4 of our Dasagitika, and XI, 15 (although corrupt) almost certainly contains a quotation of stanza 5 of our Dasagitika. Other stanzas are clearly referred to but without direct quotations. Most of the Dasagitika as we have it can be proved to be earlier than Brahmagupta (628 a.d.). The second section in 33 stanzas deals with mathematics. The third section in 25 stanzas is called Kdlakriyd, or "The Reckoning of Time." The fourth section in 50 stanzas is called Gola, or "The Sphere." Together they contain 108 stanzas. The Brdhmasputasiddhdnta of Brahmagupta was composed in 628 a.d., just 129 years after the Arya- hhatiya, if we accept 499 a.d., the date given in III, 10, as being actually the date of composition of that work. The eleventh chapter of the Brdhmasphuta- siddhdnta, which is ^lled "Tantrapanksa," and is devoted to severe criticism of previous works on astronomy, is chiefly devoted to criticism of Arya- bhata. In this chapter, and in other parts of his work, xii PREFACE Brahmagupta refers to Aryabhata some sixty times. Most of these passages contain very general criticism of Aryabhata as departing from smrti or being igno- rant of astronomy, but for some 30 stanzas it can be shown that the identical stanzas or stanzas of iden- tical content were known to Brahmagupta and ascribed to Aryabhata. In XI, 8 Brahmagupta names the Arydstasata as the work of Aryabhata, and XI, 43, jandty ekam api yato ndryabhato ganitakdlago- landm, seems to refer to the three sections of our Arydstasata. These three sections contain exactly 108 stanzas. No stanza from the section on mathe- matics has been quoted or criticized by Brahma- gupta, but it is hazardous to deduce from that, as Kaye does,^ that this section on mathematics is spurious and is a much later addition.^ To satisfy the conditions demanded by Brahmagupta's name Aryd- stasata there must have been in the work of Arya- bhata known to him exactly 33 other stanzas forming a more primitive and less developed mathematics, or these 33 other stanzas must have been astronomical in character, either forming a separate chapter or scattered through the present third and fourth sec- tions. This seems to be most unlikely. I doubt the validity of Kaye's contention that the Ganitapdda was later than Brahmagupta. His suggestion that it is by the later Aryabhata who was the author of the MahdsiddJidnta (published in the "Benares Sanskrit 1 Op. dt., X, 291-92. *For criticism of Kaye see Bibhutibhusan Datta, op. dt., XVIII (1927), 5. PREFACE xiii Series" and to be ascribed to the tenth century or even later) is impossible, as a comparison of the two texts would have shown. I feel justified in assuming that the Aryahhatlya on the whole is genuine. It is, of course, possible that at a later period some few stanzas may have been changed in wording or even supplanted by other stanzas. Noteworthy is I, 4, of which the true reading hhul],, as preserved in a quotation of Brahmagupta, has been changed by Paramesvara or by some pre- ceding commentator to hham in order to eliminate Aryabhata's theory of the rotation of the Earth. Brahmagupta criticizes some astronomical mat- ters in which Aryabhata is wrong or in regard to which Aryabhata's method differs from his own, but his bitterest and most frequent criticisms are directed against points in which Aryabhata was an innovator and differed from smrti or tradition. Such criticism would not arise in regard to mathematical matters which had nothing to do with theological tradition. The silence of Brahmagupta here may merely indicate that he found nothing to criticize or thought criticism unnecessary. Noteworthy is the fact that Brahma- gupta does not give rules for the volume of a pyramid and for the volume of a sphere, which are both given incorrectly by Aryabhata (II, 6-7) . This is as likely to prove ignorance of the true values on Brahma- gupta's part as laten^s of the rules of Aryabhata. What other rules of the Ganitapdda could be open to adverse criticism? On the positive side may be pointed out the very close correspondence in termi- xiv PREFACE nology and expression between the fuller text of Brahmagupta, XVIII, 3-5 and the more enigmatical text of Aryabhatiya, II, 32-33, in their statements of the famous Indian method (kuttaka) of solving inde- terminate equations of the first degree. It seems prob- able to me that Brahmagupta had before him these two stanzas in their present form. It must be left to the mathematicians to decide which of the two rules is earlier. The only serious internal discrepancy which I have been able to discover in the Aryabhatiya is the follow- ing. Indian astronomy, in general, maintains that the Earth is stationary and that the heavenly bodies revolve about it, but there is evidence in the Arya- bhatiya itself and in the accounts of Aryabhata given by later writers to prove that Aryabhata maintained that the Earth, which is situated in the center of space, revolves on its axis, and that the asterisms are stationary. Later writers attack him bitterly on this point. Even most of his own followers, notably Lalla, refused to follow him in this matter and reverted to the common Indian tradition. Stanza IV, 9, in spite of Paramesvara, must be interpreted as maintaining that the asterisms are stationary and that the Earth revolves. And yet the very next stanza (IV, 10) seems to describe a stationary Earth around which the asterisms revolve. Quotations by Bhattotpala, the VS- sanavarttika, and the Marici iryiicate that this stanza was known in its present form from the eleventh cen- tury on. Is it capable of some different interpreta- tion? Is it intended merely as a statement of the PREFACE XV popular \dew? Has its wording been changed as has been done with I, 4? I see at present no satisfactory solution of the problem. Colebrooke^ gives caturvimsaty amsais coikram uhhayato gacchet as a quotation by Munisvara from the Arydstasata of Aryabhata. This would indicate a knowledge of a Hbration of the equinoxes. No such statement is found in our Arydstasata. The quotation should be verified in the unpublished text in order to determine whether Colebrooke was mistaken or whether we are faced by a real discrepancy. The words are not found in the part of the Marici which has already been published in the Pandit. The following problem also needs elucidation. Al- though Brahmagupta (XI, 43-44) janaty ekam api yato narj'abhato ganitakalagolanam | na maya proktani tatah ppthak prthag duganany e§am 1| aryabhatadu?ananairi samkhya vaktmh na sakyate yasmat | tasmad ayam uddeso buddliimatanyani yojyani || sums up his criticism of Aryabhata in the severest possible way, yet at the beginning of his Khanda- khddyaka, a karanagrantha which has recently been edited by Babua Misra Jyotishacharyya (University of Calcutta, 1925), we find the statement vaksydmi khandakhddyakam dcdrydryahhatatulyaphalam. It is curious that Brahmagupta in his Khandakhddyaka should use such respectful language and should foUow the authority of an author who was damned so un- mercifully by him in tlfe Tantraparlksd of his Brdhma- sphutasiddhdnta. Moreover, the elements of the Khan- ^ Miscellaneous Essays, II, 378. XVI PREFACE dakhadyaha seem to differ much from those of the Aryabhatlya} Is this to be taken as an indication that Brahmagupta here is following an older and a dif- fereirt Aryabhata? If so the Brdhmasphutasiddhdnta gives no clear indication of the fact. Or is he fol- lowing another work by the same Aryabhata? Ac- cording to Diksit,^ the Khandakhddyaka agrees in all essentials with the old form of the Suryasiddhdnta rather than with the Brdhmasphutasiddhdnta. Just as Brahmagupta composed two different works so Aryabhata may have composed two works which represented two different points of view. The second work may have been cast in a traditional mold, may have been based on the old Siiryasiddhdnta, or have formed a commentary upon it. The Mahdsiddhdnta of another Aryabhata who lived in the tenth century or later declares (XIII, 14) : vrddharyabliataproktat siddhantad yan mahakalat ] patiiair gatam ucchedam vise^itam tan maya svoktya | j But this Mahdsiddhdnta differs in so many particulars from the Aryabhatlya that it is difficult to believe that the author of the Aryabhatlya can be the one referred to as Vrddharyabhata unless he had composed an- other work which differed in many particulars from the Aryabhatlya. The matter needs careful investiga- tion.® 1 Cf. PancasiddhSntikd, p. xx, and BvUetin of tike Calcutta Maihe- matical Society, XVII (1926), 69. * As reported by THbaut, Astronor^e, Astrologie und Mathematik, pp. 55, 59. ' See Bulletin of the Calcutta Mathematical Society, XVII (1926), 66-67, for a brief discussion. PREFACE xvii This monograph is based upon work done with me at the University of Chicago some five years ago by Baidyanath Sastri for the degree of A.M. So much additional material has been added, so many changes have been made, and so many of the views expressed would be unacceptable to him that I have not felt justified in placing his name, too, upon the title-page as joint-author and thereby making him responsible for many things of which he might not approve, Hahvaed Univeesitt April, 1929 While reading the final page-proof I learned of the publication by Prabodh Chandra Sengupta of a translation of the Aryabhatiya in the Journal of the Department of Letters (Calcutta University), XVI (1927). Unfortunately it has not been possible to make use of it in the present publication. April, 1930 TABLE OF CONTENTS List of Abbreviations xxvii I. Dasagitika oe the Ten Giti Stanzas .... 1 A. Invocation 1 B. System of Expressing Numbers by Letters of Alphabet 2 L Revolutions of Sun, Moon, Earth, and Planets ina^t^a 9 2. Revolutions of Apsis of Moon, Conjunctions of Planets, and Node of Moon in a yuga; Time and Place from Which Revolutions Are To Be Calculated 9 3. Number of Manus in a kalpa; Number of yugas in Period of a Manu; Part of halpa Elapsed up to Bharata Battle 12 4. Divisions of Circle; Circumference of Sky and Orbits of Planets in yojanas; Earth Moves One kald in a prat^a; Orbit of Sun One-sixtieth That of Asterisms 13 5. Length of yojana; Diameters of Earth, Sun, Moon, Meru, and Planets; Number of Years in a yuga 15 6. Greatest Declination of Ecliptic; Greatest Deviation of Moon and Planets from Ecliptic; Measure of&nr 16 7. Positions of Ascending Nodes of Planets, and of Apsides of Sun and Planets 16 8-9. Dimensions of Epicycles of Apsides and Con- junctions of Planets; Circumference of Earth- Wind 18 10. Table of Sine-Dif erences 19 0. Colophon 20 xix XX TABLE OF CONTENTS II. Ganitapada or Mathematics 21 1. Invocation 21 2. Names and Values of Classes of Numbers Increas- ing by Powers of Ten 21 3. Definitions of Square (varga) and Cube (ghana) . 21 4. Square Root 22 5. Cube Root 24 6. Area of Triangle; Volume of Pyramid ... 26 7. Area of Circle; Volume of Sphere 27 8. Area of Trapezium; Length of Perpendiculars from Intersection of Diagonals to Parallel Sides . . 27 9. Area of Any Plane Figure; Chord of One-sixth Cir- cumference Equal to Radius 27 10. Relation of Circumference of Circle to Diameter 28 11. Method of Constructing Sines by Forming Tri- angles and Quadrilaterals in Quadrant of Circle . 28 12. Calculation of Table of Sine-Differences from First One 29 13. Construction of Circles, Triangles, and Quadri- laterals; Determination of Horizontal and Per- pendicular 30 14. Radius of hhavrtta (or svavrtta); Hypotenuse of Right-Angle Triangle Formed by Gnomon and Shadow 31 15-16. Shadow Problems 31-2 17. Hypotenuse of Right-Angle Triangle; Relation of Half-Chord to Segments of Diameter Which Bisects Chord 34 18. Calculation of sampatasaras When Two Circles Intersect 34 19-20. Arithmetical Progression 36-6 21. Sum of Series Formed by Taking Sums of Terms of an Arithmetical Progression 37 22. Sums of Series Formed by Taking Squares and Cubes of Terms of an Arithmetical Progression . 37 TABLE OF CONTENTS xxi 23. Product of Two Factors Half the Difference be- tween Square of Their Sum and Sum of Their Squares 38 24. To Find Two Factors When Product and Differ- ence Are Known 38 25. Interest ; . . 38 26. Rule of Three (Proportion) 39 27. Fractions • 40 28. Inverse Method 40 29. To Find Sum of Several Numbers When Results Obtained by Subtracting Each Number from Their Sum Are Known 40 30. To Find Value of Unknown When Two Equal Quantities Consist of EJiowns and Similar Un- knowns 41 31. To Calculate Their Past and Future Conjunctions from Distance between Two Planets .... 41 32-33. Indeterminate Equations of First Degree (kuttaka) 43 III, .Kalakeita or the Reckoning of Time .... 51 1-2. Divisions of Time; Divisions of Circle Corre- spond 51 3. Conjunctions and vyatlpdtas of Two Planets in a yuga ' . . 51 4. Number of Revolutions of Epicycles of Planets; Years of Jupiter 51 5. Definition of Solar Year, Limar Month, Civil Day, and Sidereal Day . . . ,. 62 6. Intercalary Months and Omitted Lunar Days . 52-3 7-8. Year of Men, Fathers, and Gods; yuga of All the Planets; Day of Brahman 53 9. Utsarpim, avasaTpinl, sxLsamQ, and dussama as Divisions of yuga ^ 53 10. Date of Writing of Aryabhatiya; Age of Author at Time 54 xxii TABLE OF CONTENTS 11. Yuga, Year, Month, and Day Began at First of Caitra; Endless Time Measured by Movements of Planets and Asterisms 55 12. Planets Move with Equal Speed; Time in Which They Traverse Distances Equal to Orbit of Aster- isms and Circumference of Sky 55 13. Periods of Revolution Differ because Orbits Differ in Size 66 14. For Same Reason Signs, Degrees, and Minutes Differ in Length 56 15. Order in Which Orbits of Planets Are Arranged (beneath the Asterisms) around Earth as Center 56 16. Planets as "Lords of Days" (of Week) ... 56 17. Planets Move with Their Mean Motion on Orbits and Eccentric Circles Eastward from Apsis and Westward from Conjunction 57 18-19. Eccentric Circle Equal in Size to Orbit; Its Center Distant from Center of Earth by Radius of Epicycle 58 20. Movement of Planet on Epicycle; When ahead of and When behind Its Mean Position .... 58 21. Movement of Epicycles; Mean Planet (on Its Orbit) at Center of Epicycle 59 22-24. Calculation of True Places of Planets from Mean Places 60 25. Calculation of True Distance between Planet and Earth 61 rV. GOLA OR THE SPHERE 63 1. Zodiacal Signs in Northern and Southern Halves of Ecliptic; Even Deviation of Echptic from Equator 63 2. Sun, Nodes of Moon and Planets, and Earth's Shadow Move along Ecliptiij, 63 3. Moon, Jupiter, Mars, and Saturn Cross Ecliptic at Their Nodes; Venus and Mercury at Their Conjunctions 63 TABLE OF CONTENTS xxiii 4. Distance from Stm at Which Moon and Planets Become Visible 63-4 5. Sim Illumines One Half of Earth, Planets, and Asterisms; Other Half Dark 64 6-7. Spherical Earth, Surroimded by Orbits of Planets and by Asterisms, Situated in Center of Space; Consists of Earth, Water, Fire, and Air . 64 8. Radius of Earth Increases and Decreases by a yojana during Day and Night of Brahman . . 64 9. At Equator Stationary Asterisms Seem To Move Straight Westward; Simile of Moving Boat and Objects on Shore 64 10. Asterisms and Planets, Driven by Provector Wind, Move Straight Westward at Equator — Hence Rising and Setting 66 11-12. Mount Meru and Vadavamukha (North and South Poles); Gods and Demons Think the Others beneath Them 68 13. Four Cities on Equator a Quadrant Apart; Sun- rise at First Is Midday, Sunset — Midnight at Others . . . i 68 14. Lanka (on Equator) 90° from Poles; Ujjain 22|° North of Lanka 68 15. From Level Place Half of Stellar Sphere minus Radius of Earth Is Visible ; Other Half plus Radius of Earth Is Cut Off by Earth 68-9 16. At Meru and Vadavamukha Northern and South- em Halves of Stellar Sphere Visible Moving from Left to Right or Vice Versa 69 17. At Poles the Sun, after It Rises, Visible for Half- Year; on Moon the Sun Visible for Half a Lunar Month 69 18. Definition of Prime Vertical, Meridian, and Horizon 69 19. East and West Hour-Circle Passing through Poles (unmandala) 69 xxiv TABLE OF CONTENTS 20. Prime Vertical, Meridian, and Perpendicular from Zenith to Nadir Intersect at Place Where Observer Is 70 21. Vertical Circle Passing through Planet and Place Where Observer Is {drhmandala); Vertical Circle Passing through Nonagesimal Point (drkk^epa- mandala) 70 22. Construction of Wooden Globe Caused To Re- volve So as To Keep Pace with Revolutions of Heavenly Bodies 70 23. Heavenly Bodies Depicted on This; Equinoctial Sine (Sine of Latitude) Is Base; Sine of Co- latitude {sahku at Midday of Equinoctial Day) Is koti (Perpendicular to Base) 70 24. Radius of Day-Circle 71 25. Right Ascension of Signs of Zodiac .... 71 26. Earth-Sine Which Measures Increase and De- crease of Day and Night 71 27. Obhque Ascension of Signs of Zodiac .... 72 28. Sahku of Sun (Sine of Altitude on Vertical Circle Passing through Sun) at Any Given Time . . 72 29. Base of sahku (Distance from Rising and Setting Line) . . ■ 73 ,30. Amplitude of Sun (agra) 73 31. Sine of Altitude of Sun When Crossing Prime Vertical 74 32. Midday sahku and Shadow 74 33. Sine of Ecliptic Zenith-Distance (drkk^epajya) . 74 34. Sine of Ecliptic Altitude (drggatijya) ; Parallax . 75 35-36. Drkkarman {ak§a and dyana) 76-7 37. Moon Causes EcUpse of Sun; Shadow of Earth Causes Eclipse of Moon 78 38. Time at Which Eclipses Occur 78 39. Length of Shadow of Earth <^ 78 40. Diameter of Earth's Shadow in Orbit of Moon . 79 41. Sthityardha (Half of Tinie from First to Last Contact) 79 TABLE OF CONTENTS xxv 42. Vimardardha (Half of Time of Total Obscuration) 79 43. Part of Moon \\Tiich Is Not Eclipsed .... 79 44. Amount of Obscuration at Any Given Time . . 79-80 45. Valana 80 46. Color of Moon at Diiferent Parts of Total Eclipse 81 47. Eclipse of Sun Not Perceptible if Less than One- eighth Obscured 81 48. Sun Calculated from Conjunction {yoga) of Earth and Sun, Moon from Conjunction of Sun and Moon, and Other Planets from Conjunction of Planet and Moon 81 49-50. Colophon 81 General Index 83 Sanskrit Index 89 LIST OF ABBREVIATIONS Alberuni Alberuni's India. Translated by E. C. Sachau. London, 1910. Barth {(Euvres) CEuvres de Auguste Barthe. 3 vols. Paris, 1917. BCMS Bulletin of the Calcutta Mathe- matical Society. BMskara, Ganitadhyaya Edited by Bapu Deva Sastri; re- vised by Ramachandra Gupta. Benares (no date). Bhaskara, Golddhy&ya Edited by Bapu Deva Sastri; re- vised by Ramacliandra Gupta. Benares (no date) . Edited by Girija Prasad Dvivedi. Lucknow: Newul Kishore Press, 1911. Bhattotpala. . . ; The Brhat Samhita by Varaha- miliira with the commentary of Bhattotpala. "Vizianagram Sanskrit Series," Vol. X. Be- nares, 1895-97. Bibl. math Bibliotkeca mathematica. Brahmagupta Refers to Brcthmasphuiasiddhanta. Brahmasphutasiddhanta Edited by Sudhakara Dvivedin in the PandU (N.S.), Vols. XXIII- XXrV. Benares, 1901-2. Bvennaiad, Hindu Astronomy.. 'W. Brennand, Hindu Astronomy. London, 1896. Brhat Sa'fiihita The Brhat Samhita by VaraJiami- hira with the commentary of Bhattotpala. "Vizianagram San- skrit Series," Vol. X. Benares, 1895-97. sxvii xxviii LIST OF ABBREVIATIONS ColebrookCj Algebra H. T. Colebrooke, Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bhdscara. London, 1817. Colebrooke, Essays Miscellaneous Essays (2d ed.), by H. T. Colebrooke. Madras, 1872. Hemacandra, AbhidhdTia- Edited by Bohtlingk and Rieu. cintdmani St. Petersburg, 1847. I A Indian Antiquary. IHQ Indian Historical Quarterly. J A Journal asiatique. JASB Journal and Proceedings of the Asiatic Society of Bengal. JBBRAS Journal of the Bombay Branch of the Royal Asiatic Society. JBORS Journal of the Bihar and Orissa Research Society. JIMS Journal of the Indian Mathematical Society. JRAS Journal of the Royal Asiatic Society. Kaye, Indian Mathematics . . . G. E,. Kaye, Indian Mathematics. Calcutta, 1915. Kaye, Hindu Astronomy "Memoirs of the Archaeological Survey of India," No. 18. Cal- cutta, 1924. Khandakhddyaha By Brahmagupta. Edited by Babua Misra Jyotishacharyya. University of Calcutta, 1925. Lalla The Sisyadhwrddhida of Lalla. Edited by Sudhakara Dvivedin. Benares (no date). Mahasiddhanta By Aryafbhata. Edited by Sudha- kara Dvivedin in the "Benares Sanskrit Series." 1910. LIST OF ABBREVIATIONS xxix Marici The Ganitadhydya of Bhaskara's SiddhantaHromani with Vasana- bha§ya, Vasanavarttika, and Marici. Pandit (N.S.), Vols. XXX-XXXI. Benares, 1908-9. Paneasiddhdntika G. Thibaut and Sudhakara Dvi- vedl, The Pancasiddhantika. The Astronomical Work of Varaha Mihira. Benares, 1889. Sudhakara, Oanakatarangim. .BenaTes, 1892. Suryasiddhdnta Edited by F. E. Hall and Bapu Deva Sastrin in the Bibliotheca indica. Calcutta, 1859. Translated by Burgess and Whit- ney, Journal of the American Oriental Society, Vol. VI. Vasanavarttika The Ganitadhydya of Bhaskara's Siddhdntasiromani with Vasa- nabha§ya, Vasanavarttika, and Marici. Pandit (N.S.), Vols. XXX-XXXI. Benares, 1908-9. ZDMG Zeitschrift der Deutschen Morgero- Idndischen Gesellschaft. I, II, III, and rV refer to the four sections of the Aryabhatzya. CHAPTER I DASAGlTIKA OR THE TEN GITI STANZAS A. Ha\ing paid reverence to Brahman, who is one (in causal- ity, as the creator of the universe, but) many (in his manifesta- tions), the true deity, the Supreme Spirit, Arj^abhata sets forth three things: mathematics [ganita], the reckoning of time [kalakriyd], and the sphere [gola]. Baidyanath suggests that satyd devatd may denote Sarasvati, the goddess of learning. For this I can find no support, and therefore follow the commen- tator Paramesvara in translating "the true deity," God in the highest sense of the word, as referring to Prajapati, Pitamaha, Svayambhu, the lower indi- viduahzed Brahman, who is so called as being the creator of the universe and above all the other gods. Then this lower Brahman is identified with the higher Brahman as being only an individuafized manifesta- tion of the latter. As Paramesvara remarks, the use of the word kam seems to indicate that Aryabhata based his work on the old Pitdmahasiddhdnta. Sup- port for this view is found in the concluding stanza of our text (IV, 50), drydbhatiyam ndmnd purvam svdyamhhuvam sadd sad yat However, as shown by Thibaut^ and Kharegat,^ there is a close connection between Aryabhata and the old Suryasiddhdnta. At 1 Pancasiddhdntikd, pp. sviii, xxvij. 2 JBBRAS, XIX, 129-31. 1 2 ARYABHATIYA present the evidence is too scanty to allow us to specify the sources from which Aryabhata drew. The stanza has been translated by Fleet.^ As pointed out first by Bhau Daji,^ a passage of Brahma- gupta (XII, 43), jdndty ekam api yato naryahhato ganitakalagoldnam, seems to refer to the Ganitapdda, the Kdlakriydpdda, and the Golapdda of our Arya-, bhatiya (see also Bibhutibhusan Datta).^ Since Brahmagupta (XI, 8) names the Dasagltika and the Arydstasata (108 stanzas) as works of Aryabhata, and since the three words of XI, 43 refer in order to the last three sections of the Aryabhatiya (which contain exactly 108 stanzas), their occurrence there in this order seems to be due to more than mere coincidence. As Fleet remarks,* Aryabhata here claims specifically as his work only three chapters. But Brahmagupta (628 A.D.) actually quotes at least three passages of our Dasagltika and ascribes it to Aryabhata. There is no good reason for refusing to accept it as part of Aryabhata's treatise. B. Beginning with, ka the varga letters (are to be used) in the varga places, and the avarga letters (are to be used) in the avarga places. Ya is equal to the sum of na and ma. The nine vowels (are to be used) in two nines of places varga and avarga. Navantya- varge vet. Aryabhata's system of expressing numbers by means of letters has been discussed by Whish,^ by 1 JRAS, 1911, pp. 114^15. » BCMS, XVIII (1927), 16. » Ibid., 1865, p. 403. ** JRAS, 1911, pp. 115, 125. 8 Transactions of the Literary Society of Madras, I (1827), 54, translated with additional notes by Jacquet, J A (1835), II, 118. THE TEN GITI STANZAS 3 Brockhaus,^ by Kern,^ by Barth,^ by Rodet,^ by Kaye,^ by Fleet,^ by Sarada Kanta Ganguly/ and by Sukumar Ranjan Das.* I have not had' access to the Prthimr Itihasa of Durgadas Lahiri.^ The words varga and avarga seem to refer to the Indian method of extracting the square root, which is described in detail by Rodet^" and by Avadhesh Narayan Singh.^^ I cannot agree with Kaye's state- ment^2 that the rules given by Aryabhata for the extraction of square and cube roots (II, 4-5) "are perfectly general (i.e., algebraical)" and apply to all arithmetical notations, nor with his criticism of the foregoing stanza: "Usually the texts give a verse explaining this notation, but this explanatory' verse is not Aryabhata's."!* Sufficient evidence has not been adduced by him to prove either assertion. The varga or "square" places are the first, third, fifth, etc., counting from the right. The avarga or "non-square" places are the second, fourth, sixth, etc., counting from the right. The words varga and avarga seem to be used in this sense in II, 4. There is no good reason for refusing to take them in the same sense here. As applied to the Sanskrit alphabet the varga letters referred to here are those from k to m, 1 Zeitschrift fur die Kunde des Morgenlandes, IV, 81. 2 JRAS, 1863, p. 380. » IHQ, III, 110. » CEiwres, III, 182. » III, 332 ff. < J A (1880), II, 440. " Op. at. (1879), I, 406-8. 5 JASB, 1907, p. 478. • " BCMS, XVIII (1927), 128 8 Op. di., 1911, p. 109. " Op. dt., 1908, p. 120. ' BCMS, XVII (1926), 195. « Ibid., p. 118. 4 ARYABHATIYA ■which are arranged in five groups of five letters each. The avarga letters are those from y to h, which are not so arranged in groups. The phrase "beginning with ka" is necessary because the vowels also are divided into vargas or "groups." Therefore the vowel a used in varga and avarga places with varga and avarga letters refers the varga letters fc to m to the first varga place, the unit place, multiplies them by 1. The vowel a used with the avarga letters y to h refers them to the first avarga place, the place of ten's, multiplies them by 10. In like manner the vowel i refers the letters A; to m to the second varga place, the place of hundred's, multi- plies them by 100. It refers the avarga letters y to h to the second avarga place, the place of thousand's, multiplies them by 1,000. And so on with the other seven vowels up to the ninth varga and avarga places. From Aryabhata's usage it is clear that the vowels to be employed are a, i, u, r, I, e, ai, o, and au. No distinction is made between long and short vowels. From Aryabhata's usage it is clear that the letters fc to m have the values of 1-25. The letters y to h would have the values of 3-10, but since a short a is regarded as inherent in a consonant when no other vowel sign is attached and when the virdma is not used, and since short a refers the avarga letters to the place of ten's, the signs ya, etc., really have the values of 30-100.^ The vowels themselves have no numerical values. They merely serve to" refer the consonants (which do have numerical values) to certain places. 1 See Sarada Kanta Ganguly, op, eit., XVII (1926), 202. THE TEN GITI STANZAS 5 The last clause, wMch has been left untranslated, offers great difficulty. The commentator Paramesvara takes it as affording a method of expressing stiU higher numbers by attaching anusvdra or visarga to the vowels and using them in nine further varga (and avarga) places. It is doubtful whether the word avarga can be so supplied in the compound. Fleet would translate "in the varga place after the nine" as giving directions for referring a consonant to the nineteenth place. In view of the fact that the plural subject must carry over into this clause Fleet's in- terpretation seems to be impossible. Fleet suggests as an alternate interpretation the emendation of vd to hau. But, as explained above, au refers h to the eighteenth place. It would run to nineteen places only when expressed in digits. There is no reason why such a statement should be made in the rule. Rodet translates (without rendering the word nava), "(s6p- arement) ou a un groupe termini par un varga." That is to say, the clause has nothing to do with the ex- pression of numbers beyond the eighteenth place, but merely states that the vowels may be attached to the consonants singly as gara or to a group of con- sonants as gra, in which latter case it is to be under- stood as applying to each consonant in the group. So giri or gri and guru or gru. Such, indeed, is Arya- bhata's usage, and such a statement is really nec- essary in order to avoid ambiguity, but the words do not seem to warrant the translation given by Rodet. If the words can mean ''at the end of a group," and if nava can be taken with what precedes, Rodet's in- 6 ARYABHATIYA terpretation is acceptable. However, I know no other passage which, would warrant such a translation of antyavarge. Sarada Kanta Ganguly translates, "'[Those] nine [vowels] [should be used] in higher places in a similar manner." It is possible for vd to have the sense of "beliebig," "fakultativ," and for nava to be sepa- rated from antyavarge, but the regular meaning of antya is "the last." It has the sense of ''the following" only at the end of a compound, and the dictionary gives only one example of that usage. If navdntyavarge is to be taken as a compound, the translation "in the group following the nine" is all right. But Ganguly's translation of antyavarge can be maintained only if he produces evidence to prove that antya at the begin- ning of a compound can mean "the following." If nava is to be separated from antyavarge it is possible to take it with what precedes and to trans- late, "The vowels (are to be used) in two nine's of places, nine in varga places and nine in avarga places," but antyavarge vd remains enigmatical. The translation must remain uncertain until further evidence bearing on the meaning of antya can be produced. Whatever the meaning may be, the passage is of no consequence for the numbers actually dealt with by Aryabhata in this treatise. The largest number used by Aryabhata himself (1, 1) runs to only ten places. Rodet, Barth, and some others would translate "in the two nine's of zero's," instead of "in the two nine's of places." That is to say, each vowel would serve to THE TEN GITI STANZAS 7 add two zero's to the niimerical value of the con- sonant. This, of course, will work from the vowel i on, but the vowel a does not add two zero's. It adds no zero's or one zero depending on whether it is used •with varga or avarga letters. The fact that khadvi- navake is amplified by varge 'varge is an added difficulty to the translation "zero." It seems to me, therefore, preferable to take the word kha in the sense of "space" or better ^ 'place. "^ Later the word kha is one of the commonest words for ''zero," but it is still disputed whether a sjrmbol for zero was actually in use in Aryabhata's time. It is possible that computation may have been made on a board ruled into columns. Only nine symbols may have been in use and a blank column may have served to represent zero. There is no evidence to indicate the way in which the actual calculations were made, but it seems cer- tain to me that Aryabhata could write a number in signs which had no absolutely fixed values in them- selves but which had value depending on the places occupied by them (mounting by powers of 10). Com- pare II, 2, where in giving the names of classes of numbers he uses the expression sthdndt sthanam dasagunam syat, "from place to place each is ten times the preceding." ■ There is nothing to prove that the actual calcula- tion was made by means of these letters. It is prob- able that Aryabhata was not inventing a numerical notation to be used in calculation but was devising a system by means of which he might express large, 1 Cf. Fleet, op. clL, 1911, p. 116. 8 ARYABHATIYA unwieldy numbers in verse in a very brief form.^ The alphabetical notation is employed only in the Dasagltika. In other parts of the treatise, where only a few numbers of small size occur, the ordinary words which denote the numbers are employed. As an illustration of Aryabhata's alphabetical notation take the number of the revolutions of the Moon LQ a yuga (I, 1), which is expressed by the word . cayagiyihusuchlr. Taken syllable by syllable this gives the numbers 6 and 30 and 300 and 3,000 and 50,000 and 700,000 and 7,000,000 and 50,000,000. That is to say, 57,753,336. It happens here that the digits are given in order from right to left, but they may be given in reverse order or in any order which will make the syllables fit into the meter. It is hard to believe that such a descriptive alphabetical nota- tion was not based on a place-value notation. This stanza, as being a technical parihhdsa stanza which indicates the system of notation employed in the Dasagltika, is not counted. The invocation and the colophon are not counted. There is no good reason why the thirteen stanzas should not have been named Dasagltika (as they are named by Aryabhata himself in stanza C) from the ten central stanzas in Giti meter which give the astronomical elements of the system. The discrepancy offers no firm support to the contention of Kaye that this stanza is a later addition. The manuscript referred to by Kaye^ as containing fifteen instead of thirteen stanzas is doubtless com- » See J A (1880), II, 454, and BCMS, XVII (1926), 201. 2 Op. cii., 1908, p. 111. THE TEN GiTI STANZAS 9 parable to the one referred to by Bhau DajP as having two introductory stanzas "evidently an after-addi- tion, and not in the Arya metre." 1. In a yuga the revolutions of the Sun are 4,320,000, of the Moon 57,753,336, of the Earth eastward 1,582,237,500, of Saturn 146,564, of Jupiter 364,224, of IMars 2,296,824, of Mercury and Venus the same as those of the Sun. 2. of the apsis of the IMoon 488,219, of (the conjunction of) Mercurj' 17,937,020, of (the conjunction of) Venus 7,022,388, of (the conjunctions of) the others the same as those of the Sun, of the node of the Moon westward 232,226 starting at the beginning of Mesa at sunrise on Wednesday at Lanka. The so-called revolutions of the Earth seem to refer to the rotation of the Earth on its axis. The number given corresponds to the number of sidereal days usually reckoned in a yuga. Paramesvara, who follows the normal tradition of Indian astronomy and believes that the Earth is stationary, tries to prove that here and in IV, 9 (which he quotes) Aryabhata does not really mean to say that the Earth rotates. His effort to bring Aryabhata into agreement with the views of most other Indian astronomers seems to be misguided ingenuity. There is no warrant for treating the revolutions of the Earth given here as based on false knowledge (jnithyajnana) , which causes the Earth to seem to move eastward because of the actual westward movement of the planets (see note to I, 4). In stanza 1 the syllable su in the phrase which gives the revolutions of the Earth is a misprint for bu as given correctly in the commentary.^ 1 Ibid., 1865, p. 397. ^ See ibid., 1911, p. 122 n. 10 ARYABHATIYA Here and elsewhere in the Dasagltika words are used in their stem form without declensional endings. Lalla (Madhyamddhikdra, 3-6, 8) gives the same numbers for the revolutions of the planets, and differs only in giving "revolutions of the asterisms" instead of "revolutions of the Earth." The Suryasiddhanta (I, 29-34) shows slight varia- tions (see Pancasiddhdntikd, pp. xviii-xix, and Kharegat^ for the closer relationship of Aryabhata to the old Suryasiddhanta). Bibhutibhusan Datta,^ in criticism of the number of revolutions of the planets reported by Alberuni (II, 16-19), remarks that the numbers given for the revolutions of Venus and Mercury really refer to the revolutions of their apsides. It would be more accu- rate to say "conjunctions." Alberuni (I, 370, 377) quotes from a book of Brahmagupta's which he calls Critical Research on the Basis of the Canons a number for the civil days accord- ing to Aryabhata. This corresponds to the number of sidereal days given above (cf . the number of sidereal days given by Brahmagupta [I, 22]). Compare the figures for the number of revolutions of the planets given by Brahmagupta (1, 15-21) which differ in detail and include figures for the revolutions of the apsides and nodes. Brahmagupta (I, 61) akrtaryabhatat sighragam induccam patam alpagam svagate^i | tithyantagrahananam gh.unak§aram tasya samvadah (| criticizes the numbers given by Aryabhata for the revolutions of the apsis and node of the Moon.^ 1 JBBRAS, XVIII, 129-31. « BCMS, XVII (1926), 71. * See further Bragmagupta (V, 25) and Alberuni (I, 376). THE TEN GITI STANZAS 11 Brahmagupta (II, 46^7) remarks that according to Aryabhata all tlie planets were not at the first point of IMesa at the beginning of the yuga. I do not know on what evidence this criticism is based.^ Brahmagupta (XI, 8) remarks that according to the Arydstasata the nodes move while according to the Dasagitika the nodes (excepting that of the Moon) are fixed: arya§tasate pata bhramanti dasagitike stlurah pata^ | muktvendupatam apamandale bhramanti sthira nata^i. (| This refers to I, 2 and IV, 2. Aryabhata (I, 7) gives the location, at the time his work was composed, of the apsides and nodes of all the planets, and (I, 7 and IV, 2) implies a knowledge of their motion. But he gives figures only for the apsis and node of the Moon. This may be due to the fact that the numbers are so small that he thought them negligible for his purpose. Brahmagupta (XI, 5) quotes stanza 1 of our text : yugaravibhaganah khynghriti yat proktam tat tayor yugam spa^tam | trisati ravyudayanam tadantaram hetuna kena. ||'' 1 See Suryasiddhanta, pp. 27-28, and JRAS, 1911, p. 494. 2 Cf. JRAS, 1865, p. 401. This_ implies, as Sudhakara says, that Brahmagupta knew two works by Aryabhata each giving the revolu- tions of the Sun as 4,320,000 but one reckoning 300 savana days more than the other. Cf. Kharegat (op. ciL, XIX, 130). Is the reference to another book by the author of our treatise or was there another earlier Aryabhata? Brahmagupta (XI, 13-14) further implies that he knew two works by an author named Aryabhata in one of which the yuga began at sunrise, in the other at midnight (see JRAS, 1863, p. 384; JBBRAS, XIX, 130-31; JRAS, 1911, p. 494;.7HQ, IV, 506). At any rate, Brahmagupta does not imply knowledge of a second Aryabhata. For the whole problem of the two or three Arya- bhatas see Kaye {Bihl. math., X, 289) and Bibhutibhusan Datta 12 ARYABHATIYA 3. There are 14 Manus in a day of Brahman [a kalpa], and 72 y-ugas constitute the period of a Manu. Since the beginning of this kalpa up to the Thursday of the Bharata battle 6 Manus, 27 yugas, and 3 yugapddas have elapsed. Tiie word yugapada seems to indicate that Arya- bhata divided the yuga into four equal quarters. There is no direct statement to this effect, but also there is no reference to the traditional method of dividing the yuga into four parts in the proportion of 4, 3, 2, and 1. Brahmagupta and later tradition ascribes to Aryabhata the division of the yuga into four equal parts. For the traditional division see Suryasiddhdnta (I, 18-20, 22-23) and Brahmagupta (I, 7-8). For discussion of this and the supposed divisions of Aryabhata see Fleet.^ Compare III, 10, which gives data for the calculation of the date of the composition of Aryabhata' s treatise. It is clear that the fixed point was the beginning of Aryabhata' s fourth yugapada (the later Kaliyuga) at the time of the great Bharata battle in 3102 b.c. Compare Brahmagupta (I, 9) yugapadan aryabhatas catvari samSni kftasmgadini [ yad abhihitavan na te§ani smrtyuktasamanam ekam api |[ and XI, 4 aryabhato yugapadarfis trin yatan aha kaliyugadau yat | tasya krtantar yasmat svayngadyantau na tat tasmat || (op. dt., XVII [1926], 60-74). The PancasiddhantiM also (XV, 20), "Aryabhata maintains that the beginning of the day is to be reckoned from midnight at Lanka; and the same 'teacher again says that the day begins from sunrise at Lanka," ascribes the two theories to one Aryabhata. 1 Op. at., 1911, pp. Ill, 486. THE TEN GiTI STANZAS 13 with the commentary of Sudhakara. Brahmagupta (I, 12) quotes stanza I, 3, manusandhiiii jiigam iccliaty an-abhatas tanmanur yatati skhajTigah \ kalpas caturjoiganaih sahasram a§tadliikam tasj'a. ||^ Brahmagupta (I, 28) refers to the same matter, adhikat smrtj-uktamanor aryabhatoktas catunnagena manul;i | adhikam -v-imsamsajTitais tribhir joigais tasya kalpagatam. 1| Brahmagupta (XI, 11) criticizes Aryabhata for be- ginning the Kaliyuga with Thursday (see the com- mentary of Sudhakara). Bhau Daji^ first pointed out the parallels in Brahmagupta I, 9 and XI, 4 and XI, 11.® 4. The revolutions of the Moon (in a yiiga) multiplied by 12 are signs [rasi].* The signs multiphed by 30 are degrees. The degrees multiplied by 60 are minutes. The miautes multiplied by 10 are yojanas (of the circumference of the sky). The Earth moves one minute in a prdna.^ The circumference of the sky (in yojanas) di^-ided by the. revolutions of a planet in a yuga gives the yojanas of the planet's orbit. The orbit of the Sun is a sixtieth part of the circle of the asterisms. In translating the words sasirasayas tha cakram I have followed Paramesvara's interpretation sasinas cakram hhagand dvddasagunitd rdsayah. The Sanskrit construction is a harsh one, but there is no other way of making sense. Sasi (without declensional ending) is to be separated. Paramesvara explains the word grahajavo as fol- 1 Cf. Ill, S. . 2 Op. ciL, 1865, pp. 400-401. ' Cf . Alberuni, I, 370, 373-74. * A rail is a sign of the zodiac or one-twelfth of a circle. ^ For jrrana see III, 2. 14 ARYABHATIYA " lows : ekapanvrttau grahasya javo gatimdnam yojana- tmakam bhavati. The word yojanani must be taken as given a figure in yojanas for the circumference of the sky (akasa^ kaksya). It works out as 12,474,720,576,000, which is the exact figure given by Lalla {Madhyamadhikdra 13) who was a follower of Aryabhata. Compare Suryasiddkdnta, XII, 80-82; Brahmagupta, XXI, 11-12; Bhaskara, Golddhydya, Bhuvanakosa, 67-69 and Ganitddhydya, Kaksddhydya, 1-5. The statement of Alberuni (I, 225) with regard to the followers of Aryabhata, It is sufficient for us to know the space which is reached by the solar rays. We do not want the space which is not reached by the solar rays, though it be in itself of an enormous extent. That which is not reached by the rays is not reached by the per- ception of the senses, and that which is not reached by per- ception is not knowable, may be based ultimately upon this passage. The reading bham of our text must be incorrect. It is a reading adopted by Paramesvara who was de- termined to prove that Aryabhata did not teach the rotation of the Earth. This passage could not be ex- plained away by recourse to false knowledge (mith- ydjndna) as could I, 1 and IV, 9 and therefore was changed. The true reading is bhuh, as is proved con- clusively by the quotation of Brahmagupta (XI, 17) : pra,nenaiti kalarh bhur yadi tarhi kuto vrajet kam adhvanam | avarttanam urvyas cen na patanti samucchraya^i kasmat. || Compare Brahmagupta (XXI, 59) and Alberuni (I, 276-77, 280). THE TEN GITI STANZAS 15 5. A yojana consists of 8,000 times a nr [the height of a man]. The diameter of the Earth is 1,050 yojanas. The diameter of the Sun is 4,410 yojanas. The diameter of the Moon is 315 yojanas. Meru is one yojana. The diameters of Venus, Jupiter, Mercury, Saturn, and Mars are one-fifth, one-tenth, one-fifteenth, one- twentieth, and one-twenty-fifth of the diameter of the Moon. The years of a yuga are equal to the number of revolutions of the Sun in a yuga. As pointed out by Bhau Dajl,^ Brahmagupta (XI, 15-16) seems to quote from this stanza in his criticism of the diameter of the Earth given by Aryabhata §odasagaviyojana paridhim pratibhuvyasam pulavadata | atmajnanaih khyapitam aniscayas tanikrtakanyat f[ bhuvyasasyajnanad vyartharh desantaram tadajnanat | sphutatithyantajnanaih tithinasad grahanayor nasalj. j| The text of Brahmagupta is corrupt and must be emended. See the commentary of Sudhakara, who suggests for the j5rst stanza nf§iyojanabhuparidhim prati bhm^asam punar nila vadata | atmajnanam khyapitam aniscayas tatkf tavyasati. 1 1 Lalla (Madhyamddhikara, 56 and Candragrahand- dhikdra, 6) gives the same diameters for the Earth and the Sun but gives 320 as the diameter of the Moon, and {Grahayutyadhikara, 2) gives for the planets the same fractions of the diameter of the Moon.2 Alberuni (I, 168) quotes from Brahmagupta Aryabhata's diameter of the Earth, and a confused 1 JRAS, 1865, p. 402. 2Cf. Suryasiddhanta, I, 59; IV, 1; VII, 13-14; Brahmagupta, XXI, 32; Kharegat {op. cit, XIX, 132-34, discussing Suryasiddhanta, IX, 15-16). 16 ARYABHATlYA passage (I, 244-46) quotes Balabhadra on Arya- bhata's conception of Meru. Its height is said to be a yojana. The context of the foregoing stanza seems to imply that its diameter is a yojana, as Paramesvara takes it. It is probable that its height is to be taken as the same. If Paramesvara is correct in interpreting samarka- samdh as yugasamd yugdrkahhaganasamd, the nomi- native plural samdh has been contracted after sandhi. 6. The greatest declination of the ecliptic is 24 degrees. The greatest deviation of the Moon from the ecliptic is 4| degrees, of Saturn 2 degrees, of Jupiter 1 degree, of Mars 1| degrees, of Mercury and Venus 2 degrees. Niaety-six angulas or 4 hastas make 1 nr. Paramesvara explains the words hhdpakramo grahdmsdh as follows: grahdndm bha amsds catur- vimsatihhdgd apakramah. paramdpakrama ity arthah. The construction is as strange as that of stanza 4 above.^ 7. The ascending nodes of Mercury, Venus, Mars, Jupiter, and Saturn having moved (are situated) at 20, 60, 40, 80, and 100 degrees from the beginning of Me§a. The apsides of the Sun and of the above-mentioned planets (in the same order) (are situated) at 78, 210, 90, 118, 180, and 236 degrees from the beginning of Mesa. I have followed Paramesvara's explanation of gatvdmsakdn as uktdn etdn evdrhsakdn mesddito gatvd vyavasthitdh. In view of IV, 2, "the Sun and the nodes of the planets and of the Moon move constantly along the ^ Cf . SUryasiddhanta, I, 68-70 and II, 28; Brahmagupta, IX, 1 and XXI, 52. THE TEN GITI STANZAS 17 ecliptic," and of I, 2, which gives the number of revo- lutions of the node of the Moon in a yuga, the word gatva ("having gone") seems to imply, as Parame- svara says, a knowledge of the revolution of the nodes of the planets and to indicate that Aryabhata in- tended merely to give their positions at the time his treatise was composed. The force of gatvd continues into the second line and indicates a knowledge of the revolutions of the apsides. Aryabhata gives figures for the revolutions of the apsis and node of the Moon. Other siddhdntas give figures for the revolutions of the nodes and apsides of all the planets. These seem to be based on theory rather than on observation since their motion (except in the case of the Moon) is so slow that it would take several thousand years for them to move so far that their motion could easily be detected by ordinary methods of observation.^ Aryabhata may have re- frained from giving figures for the revolutions of nodes and apsides (except in the case of the Moon) because he distrusted the figures given in earher books as based on theory rather than upon accurate observa- tion. Brahmagupta XI, 8 (quoted above to stanza 2) remarks in criticism of Aryabhata that in the Dasa- gltika the nodes are stationary while in the Aryd- stasata they move. This refers to I, 2 and IV, 2. In the Dasagitika only the revolutions of the nodes of the Moon are given; in the Arydsiasata the nodes and apsides are said expHcitly to move along the ecliptic. In the present stanza the word gatvd seems clearly to ^ Cf. SUryasiddhdnta, pp. 27-28. 18 ARYABHATIYA indicate a knowledge of the motion of the nodes and apsides of the other planets too. If Aryabhata had intended to say merely that the nodes and apsides are situated at such-and-such places the word gatvd is superfluous. In a text of such studied brevity every word is used with a very definite purpose. It is true that Aryabhata regarded the movement of the nodes and apsides of the other planets as negligible for pur- poses of calculation, but Brahmagupta's criticism seems to be captious and unjustified (see also Bra- hmagupta, XI, 6-7, and the commentary of Sudha- kara to XI, 8). Earth's criticism^ is too severe. Lalla {Spastddhikdra, 9 and 28) gives the same positions for the apsides of the Sun and five planets (see also Pancasiddhdntikd, XVII, 2) . For the revolutions of the nodes and apsides see Brahmagupta, 1, 19-21, and Suryasiddhdnta, 1, 41-44, and note to I, 44. 8. Divided by 4| the epicycles of the apsides of the Moon, the Sun, Mercury, Venus, Mars, Jupiter, and Saturn (in the first and third quadrants) are 7, 3, 7, 4, 14, 7, 9; the epicycles of the conjunctions of Saturn, Jupiter, Mars, Venus, and Mercury (in the first and third quadrants) are 9, 16, 53, 59, 31; 9. the epicycles of the apsides of the planets Mercury, Venus, Mars, Jupiter, and Saturn in the second and fourth quadrants are 5, 2, 18, 8, 13; the epicycles of the conjunctions of the planets Saturn, Jupiter, Mars, Venus, and Mercury in the second and fourth quadrants are 8, 15, 51, 57, 29. The circumference within which the Earth-wind blows is 3,375 yojanas. The criticism of these stanzas made by Brahma-' gupta (II, 33 and XI, 18-21) is, as pointed out by 1 Op. cit., Ill, 154. THE TEN GITI STANZAS 19 Sudhakara, not justifiable. For the dimensions of Brahmagupta's epicycles see II, 34-39). Lalla (Spastadhikdra, 28) agrees closely with stanza 8 and (Grahabhramana, 2) gives the same figure for the Earth-wind. Compare also Suryasiddhdnta, II, 34-37 and note, and Pancasiddhdntikd, XVII, 1, 3. 10. The (twenty-four) sines reckoned in minutes of arc are 225, 224, 222, 219, 215, 210, 205, 199, 191, 183, 174, 164, 154, 143, 131, 119, 106, 93, 79, 65, 51, 37, 22, 7. In Indian mathematics the "half-chord" takes the place of our "sine." The sines are given in minutes (of which the radius contains 3,438) at intervals of 225 minutes. The numbers given here are in reality not the values of the sines themselves but the differences between the sines. Compare Suryasiddhanta (II, 15-27) and Lalla {Spastadhikdra, 1-8) and Brahmagupta (II, 2-9). Bhaskara (Ganitddhydya, Spastadhikdra, Vdsanabhd- §ya to 3-9) refers to the Suryasiddhdnta and to Aryabhata as furnishing a precedent for the use of twenty-four sines.^ Krishnaswami Ayyangar^ furnishes a plausible explanation of the discrepancy between certain of the values given in the foregoing stanza and the values as calculated by II, 12.^ Some of the discrepancies may be due to bad readings of the manuscripts. Kern ^ For discussion of the stanza see Bartb, ihid., Ill, 150 n., and JRAS, 1911, pp. 123-24. 2 JIMS, XV (1923-24), 121-26. * See also Naraharayya, "Note on the Hindu Table of Sines," ihU., pp. 105-13 of "Notes and Questions." 20 ARYABHATlYA in a footnote to the stanza and Ayyangar (p. 125 n.) point out that the text-reading for the sixteenth and seventeenth sines \dolates the meter. This, however, may be remedied easily without changing the values.^ C. Whoever knows this Dasagltika Sutra which describes the movements of the Earth and the planets in the sphere of the asterisms passes through the paths of the planets and asterisms and goes to the higher Brahman. 1 Cf. JRAS, 1910, pp. 752, 754, and lA, XX, 228. CHAPTER II GANITAPADA OR MATHEMATICS 1. Having paid reverence to Brahman, the Earth, the Moon, Mercury, Venus, the Sun, Mars, Jupiter, Saturn, and the aster- isms, Aryabhata sets forth here [in this work] the science which is honored at Kusumapura.^ The translation "here at Kusumapura the revered science" is possible. At any rate, Aryabhata states the school to which he belongs. Kusumapura may or may not have been the place of his birth. 2. The numbers eka [one], dasa [ten], sata [hundred], sahasra [thousand], ayuta [ten thousand], niyuta [hundred thousand], prayuta [million], koti [ten million], arbuda [hundred million], and vrnda [thousand million] are from place to place each ten times the preceding.^ The names for classes of numbers are given only to ten places, although I, B describes a notation which reaches at least to the eighteenth place. The highest number actually used by Aryabhata himseK runs to ten places. 3. A square, the area of a square, and the product of two equal quantities are called varga. The product of three equal quantities, and a soUd which has twelve edges are called ghana} 1 Translated by Ffeet, JRAS, 1911, p. 110. See Kern's Preface to his edition of the Brhat Samhita, p. 57, and BCMS, XVIII (1927), 7. 2 See JRAS, 1911, p. 116; IHQ, III, 112; BCMS, XVII (1926), 198. For the quotation in Alberuni (1, 176), which differs in the last two names, see the criticism in BCMS, XVII (1926), 71. 5 Read dvdda§a§Tas with ParameSvara. For oiro in the sense of "edge" see Colebrooke, Algebra, pp. 2 n. and 280 n. The translations given by Rodet and Kaye are inaccurate. 21 22 ARYABHATIYA 4. One should always divide the avarga by twice the (square) root of the (preceding) varga. After subtracting the square (of the quotient) from the varga the quotient will be the square root to the next place. Counting from right to left, the odd places axe called varga and the even places are called avarga. According to Paramesvara, the nearest square root to the number in the last odd place on the left is set down in a place apart, and after this are set down the successive quotients of the division performed. The number subtracted is the square of that figure in the root represented by the quotient of the preceding division. The divisor is the square of that part of the root which has already been found. If the last sub- traction leaves no remainder the square root is exact. "Always" indicates that if the divisor is larger than the number to be divided a zero is to be placed in the line (or a blank space left there). Sthandntare ("in an- other place") is equivalent to the pahkti ("line") of the later books. This process seems to be substantially correct, but there are several difficulties. Sthandntare may mean simply "to another place," that is to say, each division performed gives another figure of the root. Nityam ("always") may merely indicate that such is the regular way of performing the operation. All the translators except Saradakanta Ganguly translate vargdd varge suddhe with what precedes. I think he is correct in taking it with what follows. In that case the parallelism with the following rule is exact. Otherwise the first rule would give the opera- GANITAPADA OR MATHEMATICS 23 tion for the varga place and then that for the avarga place while the second rule would give first the opera- tions for the aghana places and then that for the ghana place. However, for purposes of description, it makes no difference whether the operations are given in one or the other of these orders. Parallelism with ghanasya mulavargena of the following rule seems to indicate that vargamulena is not to be translated "square root" but "root of the (preceding) varga." If the root is to contain more than two figures the varga of vargamulena is to be interpreted as applying to all the preceding figures up to and including the varga place which is being worked with. That is to say, the word mula would refer to the whole of that part of the root which had already been found.^ For discussion see Kaye,^ Avadhesh Narayan Singh,^ Saradakanta Ganguly.* I cannot agree with Ganguly's discussion of the words hhdgam hared avargdt. I see no reason to question the use of hhdgam harati with the ablative in the sense of "divide." Brahmagupta (XII, 7) in his description of the process of extracting the cube root has chedo 'ghandd dvitiyat, which means "the divisor of the second aghana." Kaye^ insists that this rule and the next are per- fectly general (i.e., algebraical) and apply to all arithmetical notations. He offers no proof and gives 1 See Colebrooke, op. cit., p. 280 n. * JASB, 1907, pp. 493-94. ■« JBORS, XII, 78. » BCMS, XVIII (1927), 124. « Op. cit., 1908, p. 120. 24 ARYABHATIYA no example of the working of the rule according to his interpretation. To what do the words "square" and "non-square" of his translation refer? The words of Aryabhata exactly fit the method employed in later Indian mathematics. Although Brahmagupta does not give a rule for square root, his method for cube root is that described below, although the wording of his rule is different from that of Aryabhata's. I fail to see any similarity to the rule and method of Theon of Alexandria. In the following example the sign ° indicates the varga places, and the sign - indicates the avarga places. 16129 (root =1 Square of the root 1 Twice the root 2)05(2= quotient (or next digit of root) (2X1) 4 11 Square of the quotient 4 Twice the root 24)72(3 = quotient (.or next digit of root) (2X12) 72 09 Square of the quotient 9 Square root is 1 2 3 5. One should divide the second aghana by three times the square of the (cube) root of the (preceding) ghana. The square (of the quotient) multiplied by three times the purva (that part of the cube root already found) is to be subtracted from the first GANITAPADA OR MATHEMATICS 25 aghana, and the cube (of the quotient of the above di\dsion) is to be subtracted from the gliana. The translation given by Avadhesh Narayan Singh^ as a "correct literal rendering" is inaccurate. There is nothing in the Sanskrit which corresponds to "after ha\dng subtracted the cube (of the quo- tient) from the ghana place" or to "the quotient placed at the next place gives the root." The latter thought, of course, does carry over into this rule from the preceding rule. In the same article (p. 132) the Sanskrit of the rule is inaccurately printed with trighanasya for trigunena ghanasya.^ Kaye^ remarks that this rule is given by Brahma- gupta "word for word." As a matter of fact, the Sanskrit of the two rules is very different, although the content is exactly the same. Counting from right to left, the first, fourth, etc., places are named ghana (cubic); the second, fifth, etc., places are called the first aghana (non-cubic) places; and the third, sixth, etc., places are called the second aghana (non-cubic) places. The nearest cube root to the number in (or up to and including) the last ghana place on the left is the first figure of the cube root. After it are placed the quotients of the succes- sive divisions. If the last subtraction leaves no remainder the cube root is exact. 1 BCMS, XVIII (1927), 134. 2 The rule has been discussed in JBORS, XII, 80. Cf . Brah- magupta (XII, 7) and the translation and note of Colebrooke (op. cit, p. 280). ' Op. cit, 1908, p. 119. 26 ARYABHATIYA In the following example the sign ° indicates the ghana places and the sign - indicates the aghana places. 1860867 (root=l Cube of root 1 Three times square of root 3) 08 (2 = quotient (or next digit of (3XP) 6 root) 26 Square of quotient multiplied 12 by three times the purva (2^X3X1) 140 Cube of quotient 8 Three times square of root 432) 1328(3 = quotient (or next digit (3X122) 1296 of root) 326 Square of quotient multiplied 324 by three times the pilwa (3*X3X12) 27 Cube of quotient 27 Cube root is 1 2 3 6. The area of a triangle is the product of the perpendicular and half the base. Half the product of this area and the height is the volume of a solid which has six edges (pyramid). If samadalakoti can denote, as Paramesvara says, a perpendicular which is common to two triangles the rule refers to all triangles. If samadalakoti refers to a perpendicular which bisects the base it refers only to isosceles triangles.^ 1 For asra or asri in the sense of "edge" see note to stanza II, 3. See JBORS, XII, 84-85, for discussion of the inaccurate value given in the second part of the rule. GANITAPADA OR MATHEMATICS 27 7. Half of the circumference multipKed by half the diameter is the area of a circle. This area multiplied by its own square root is the exact volume of a sphere.'- 8. The two sides (separately) multiplied by the perpendicu- lar and di\'ided by their sum will give the perpendiculars (from the point where the two diagonals intersect) to the parallel sides. The area is to be kno-wn by multipl5ing half the sum of the two sides by the perpendicular. aXc a-i-6 bXc a+b =d, =e, Area = cia+b) The rule applies to any four-sided plane figure of which two sides are parallel, i.e., trapezium. The word translated "sides" refers to the two parallel sides. The perpendicular is the perpendicular be- tween the two parallel sides. In the example given above a and b are the parallel sides, c is the perpendicular between them, and d and e are the perpendiculars from the point "of intersection of the two diagonals to the sides a and b, respectively. 9. The area of any plane figure is found by determining two sides and then multiplsdng them together. The chord of the sixth part of the circumference is equal to the radius. 1 See ibid, and Bibl. maih., IX, 196, for discussion of the inac- curate value given in the second part of the rule. For a possible reference to this passage by Bhaskara, Goladhydya, Bhuvanakosa, stanza 61 {Vdsanabhd^a) (not stanza 52 as stated), see BCMS, XVm (1927), 10. 28 ARYABHATIYA The very general rule given in the first half of this stanza seems to mean, as Paramesvara explains in some detail, that the mathematician is to use his in- genuity in determining two sides which will represent the average length and the average breadth of the figure. Their product will be the area. Methods to be employed with various kinds of figures were doubt- less handed down by oral tradition. Rodet thinks that the rule directs that the figure be broken up into a number of trapeziums. It is doubtful whether the words can bear that interpretation. 10. Add 4 to 100, multiply by 8, and add 62,000. The result is approximately the circumference of a circle of which the diameter is 20,000. The circumference is 62,832. The diameter is 20,000. By this rule the relation of circumference to diameter is 3.1416.^ Bhaskara, Golddhydya, Bhuvanakosa (stanza 52), Vasandbkdsya, refers to this rule of Aryabhata. 11.. One should divide a quarter of the circumference of a circle (into as many equal parts as are desired). From the tri- angles and quadrilaterals (which are formed) one wiU have on the radius as many sines of equal arcs as are desired.'^ The exact method of working out the table is not known. It is uncertain what is intended by the triangle and the quadrilateral constructed from each point marked on the quadrant.^ 1 See JBORS, XII, 82; JRAS, 1910, pp. 752, 754. * See the table given in I, 10 of the differences between the sines. Twenty-four sines taken at intervals of 225 miautes of arc are regu- larly given in the Indian tables. * Note the methods suggested by Kaye and Rodet and of. JIMS, XV (1923-24), 122 and 108-9 of "Notes and Questions." GANITAPADA OR MATHEMATICS 29 12. By what number the second sine is less than the first sine, and by the quotient obtained by di\dding the sum of the preceding sines by the first sine, by the sum of these two quanti- ties the following sines are less than the first sine. The last phrase may be translated "the sine- differences are less than the first sine."^ This rule describes how the table of sine-differ- ences given in I, 10 may be calculated from the first one (225). The first sine means always this first sine 225. The second sine means any particular sine with which one is working in order to calculate the follow- ing sine. Subtract 225 from 225 and the remainder is 0. Di- \'ide 225 by 225 and the quotient is 1 . The sum of and 1 is subtracted from 225 to obtain the second sine 224. Subtract 224 from 225 and the remainder is 1. Divide 225 plus 224 by 225 and the nearest quotient is 2. Add 2 and 1 and subtract from 225. The third sine will be 222. Proceed in like manner for the fol- lowing sines. If this method is followed strictly there results several shght divergences from the values given in I, 10. It is possible to reconcile most of these by assum- ing, as Klrishnaswami Ayyangar does, that from time to time the neglected fractions were distributed among the sines. But of this there is no indication in the rule as given. 1 For discussion of the Indian sines see the notes of Rodet and Kaye; Pancasiddkaniikd,, chap, iv; SuryasiddMMa, II, 15-27; Lalla, p. 12; Brahmagupta, II, 2-10; JRAS, 1910, pp. 752, 754; lA, XX, 228; Brennand, Hindu Astronomy, pp. 210-13; JIMS, XV (1923-24), 121-26, with attempted explanation of the variation of several of the values given in the table from the values calculated by means of this rule, and ifnd., pp. 106-13 of "Notes and Questions." 30 ARYABHATIYA How Kaye gets "If the first and second be bisected in succession the sine of the half -chord is obtained" is a puzzle to me. It is impossible as a translation of the Sanskrit. 13. The circle is made by turning, and the triangle and the quadrilateral by means of a harm; the horizontal is determined by water, and the perpendicular by the plumb-line. Tnhhuja denotes triangle in general and catur- bhuja denotes quadrilateral in general. The word ,karna regularly denotes the hjrpotenuse of a right- angle triangle and the diagonal of a square or rec- tangle. I am not sure whether the restricted sense of karria limits trihhuja and caturhhuja to the right-angle triangle and to the square and rectangle or whether the general sense of tribhuja and caturhhuja general- izes the meaning of kariia to that of one chosen side of a triangle and to that of the diagonal of any quadri- lateral. At any rate, the context shows that the rule deals with the actual construction of plane figures. Paramesvara interprets it as referring to the con- struction of a triangle of which the three sides are known and of a quadrilateral of which the four sides and one diagonal are known. One side of the triangle is taken as the karna. Two sticks of the length of the other two sides, one touching one end and the other the other end of the karna, are brought to such a posi- tion that their tips join. The quadrilateral is made by constructing two triangles, one on each side of the diagonal. GANITAPADA OR MATHEMATICS 31 The circle is made by the turning of the karkata or compass.^ 14. Add the square of the height of the gnomon to the square of its shadow. The square root of this sum is the radius of the khavrtta. The text reads khavrtta ("sky-circle"). Para- mesvara reads svavrtta ("its circle")- I do not know which is correct. Kaye remarks that in order "to mark out the hour angles on an ordinary sun-dial, it is necessary to , describe two circles, one of which has its radius equal to the vertical gnomon and the other with radius equal to the hypotenuse of the triangle formed by the equinoctial shadow and the gnomon." It may be that this second circle is the one referred to here. Para- mesvara has chayagramadhyam sankusirahprdpi yan mandalam urdhvadhahsthitam tat svavrttam ity ucyate, "the circle which has its centre at the extremity of the shadow and which touches the top of the gnomon is called the svavrtta.'' As Eodet remarks, it is diffi- cult to see for what purpose such a circle could serve. 15. Multiply the length of the gnomon by the distance be- tween the gnomon and the bhuja and divide by the difference between the length of the gnomon and the length of the bhuja. The quotient will be the length of the shadow measured from the base of the gnomon.^ 1 For parallels to the stanza see Lalla {Yantradhyaya, 2) and Brahmagupta, XXII, 7. See BCMS, XVIII (1927), 68-69, which is too emphatic in its assertion that karoa must mean "diagonal" and not "hjrpotenuse." 2 See Brahmagupta, XII, 53; Colebrooke, op. dt., p. 317; Brennand, op. dt., p. 166. 32 ARYABHATIYA Because of the use of the word kotl in the following rule Rodet is inclined to think that the gnomon and the hhujd were not perpendicular but projected hori- zontally from a waU. Bhujd denotes any side of a triangle, but kotl usually refers to an upright. It is possible, however, for kotl to denote any perpendicu- lar to the bhuja whether horizontal or upright. A BA is the bhuja which holds the light, DE is the gnomon, DEXBD DC=- AF 16. The distance between the ends of the two shadows multi- plied by the length of the shadow and divided by the difference in length of the two shadows gives the kotl. The kotl multiplied by the length of the gnomon and divided by the length of the shadow gives the length of the bhuja. The literal translation of chdydgunitam chdyd- gravivaram unena bhdjitd kotl seems to be "The dis- tance between the ends of the two shadows multiplied by the length of the shadow is equal to the kotl divided by the difference in length of the two shad- ows." This is equivalent to the translation given above. GANITAPADA OR MATHEMATICS 33 AB is the bkujd, AE is the koii, CD is the gnomon in its first position, CD' is the gnomon in its second position, CE and C'E' are the first and second shadows, CEXEE' AE = AB = C'E'-CE' AEXCD CE The length of the hhujd which holds the light and the distance between the end of the shadow and the base of the hhujd are unknown. In order to find them the gnomon is placed in another position so as to give a second shadow. The length of the shadow is its length when the gnomon is in its first position. The koti is the dis- tance between the end of the shadow when the gno- mon is in its first position and the base of the hhujd. The word koti means perpendicular (or upright) and the rule might be interpreted, as Rodet takes it, as meaning that the hhujd and the gnomon extend horizontally from a perpendicular wall. But the words hhujd and kopi also refer to the sides of a right- angle triangle without much regard as to which is horizontal and which is upright. 34 ARYABHATlYA Or the first position of the gnomon may be CD' and the second CD. To find AE' and AB} 17. The square of the bhuja plus the square of the hotl is the square of the karna. In a circle the product of two saras is the square of the half- chord of the two arcs. The hhujd and koti are the sides of a right-angle triangle. The karna is the hypotenuse. The saras or ''arrows'^ are the segments of a diameter which bisects any chord.^ where c is the half-chord. 18. (The diameters of) two circles (separately) minus the grdsa, multiplied by the grasa, and divided separately by the sum of (the diameters of) the two circles after the grasa has been sub- tracted from each, will give respectively the sampatasaras of the two circles. When two circles intersect the word grdsa ("the bite") denotes that part of the common diameter of the two circles which is cut off by the intersecting chords of the two circles. ^ See Brahmagupta, XII, 54; Colebrooke, op. cit., p. 318; Bren- nand, op. cit., p. 166. 2 Cf. Brahmagupta, XII, 41. See BCMS, XVIII (1927), 11, 71, with discussion of the quotation given by Colebrooke, op. cit., p. 309, from Prthudakasvaml's commentary to Brahmagupta. GANITAPADA OR MATHEMATICS 35 AE = AB{d-AB) EB = AB is the grasa, AE and BE are the sampaiaiaras. AB(D-AB) D+d-2AB' D+d-2AB ' where d and D are the diameters of the two circles. The sampdtasaras are the two distances (within the grdsa), on the common diameter, from the cir- cumferences of the two circles to the point of inter- section of this common diameter with the chord con- necting the two points where the circumferences intersect. 1 19. The desired number of terms minus one, halved, plus the number of terms which precedes, multiplied by the common difference between the terms, plus the first term, is the middle term. This multiplied by the number of terms desired is the sum of the desired number of terms. Or the sum of the first and last terms is multiplied by half the number of terms. This rule tells how to find the sum of any desired number of terms taken anywhere within an arith- metical progression. Let n be the number of terms extending from the (p-l-l)th to the {p-\-n)tla. terms in an arithmetical progression, let d be the common difference between the terms, let a be the first term of the progression, and I the last term. ^ Cf . Brahmagupta, XII, 43; Colebrooke, op. cU., p. 311. 36 ARYABHATIYA The second part of the rule applies only to the sum of the whole progression beginning with the first term. S As Paramesvara says, samukhamadhyam must be taken as equivalent to samukham madhyam. Whether Paramesvara is correct in his statement hahusutrdrtha'pradarsakam etat sutram. ato hahudhd yojana karyd and subsequent exposition seems very doubtful. Brahmagupta, XII, 17 has only the second part of the rule.^ 20. Multiply the sum of the progression by eight times the common difference, add the square of the difference between twice the first term and the common difference, take the square root of this, subtract twice the first term, divide by the common differ- ence, add one, divide by two. The result will be the number of terms. ^_l[ V8dS+(d-2ay-2a , ,-\ , ''-2L d """^J • As Rodet says, the development of this formula from the one in the preceding rule seems to indicate knowledge of the solution of quadratic equations in the form ax^-\-hx+c = 0.^ 1 Cf . Colebrooke, op. dt., p. 290. 2 See Brahmagupta, XII, 18; Colebrooke, op, dt., p. 291. ' See also J A (1878), I, 28, 77, and JBORS, XII, 86-87. GANITAPADA OR MATHEMATICS 37 21. In the case of an wpaciti which has one for the first term and one for the common difference between the terms the product of three terms ha\'ing the number of terms for the first term and one as the common difference, di\dded by six, is the dtighana. Or the cube of the number of terms plus one, minus the cube root of this cube, di\-ided bj' six. Form an arithmetical progression 12 3 4 5, etc. Form the series 1 3 6 10 15, etc., by taking for the terms the sum of the terms of the first series. The rule gives the smn of this series. It also gives the cubic contents of a pile of balls which has a triangular base. The wording of the rule would seem to imply that it was intended especially for this second case. Citighana means ''cubic con- tents of the pile," and wpaciti ("pile") refers to the base (or one side) of the pile, i.e., 12 3 4 5, etc.^ As Rodet remarks, it is curious that in the face of this rule the rule given above (stanza 6) for the volume of a pyramid is incorrect. n{n+l){n+2) {n+\f-{n+l) 6 6 • 22. The sixth part of the product of three quantities con- sisting of the number of terms, the number of terms plus one, and twice the number of terms plus one is the sum of the squares. The square of the sum of the (original) series is the sum of the cubes. From the series 12 3 4, etc., form the series 1 4 9 16, etc., and 1 8 27 64, etc., consisting of the 1 Cf. Brahmagupta, XII, 19; Colebrooke, op. cit., pp. 292-93. Brahmagupta (XII, 20) directs that the summation of certain series be illustrated by means of piles of roimd balls. 38 ARYABHATIYA squares and cubes of the terms of the first series. The rule tells how to find the sums of the second and third series.^ The rule for finding the sum of the first series was given above in stanza 19. The sum of the squares is w(n+l)(2n+l) 6 23, One should subtract the sum of the squares of two factors from the square of their simi. Half the result is the product of the two factors. ab= . 24. Multiply the product (of two factors) by the square of two (4), add the square of the difference between the two factors, take the square root, add and subtract the difference between the two factors, and divide the result by two. The results wiU be the two factors. l/4ab+(a-6)^±(a-6) .,, . ,, ^^ — ^ ^^ will give a and o. 25. Multiply the sum of the interest on the principal and the interest on this interest by the time and by the principal. Add to this result the square of half the principal. Take the square root of this. Subtract half the principal and divide the remainder by the time. The result will be the interest on the principal.^ 1 Cf. Brahmagupta, XII, 20; Colebrooke, op. cit., p. 293; BCMS, XVIII (1927), 70. ' Cf. the somewhat similar problem in Brahmagupta, XII, 15; Colebrooke, op. cit, pp. 287-28, and see the discussion of Kaye. GANITAPADA OR MATHEMATICS 39 The fonnula involves the solution of a quadratic equation in the form of ax^+2bx = c.^ A sum of money is loaned. After a certain unit of time the interest received is loaned for a known num- ber of units of time at the same interest. What is known is the amount of the interest on the principal plus the interest on this interest. Call this B. Let the principal be A. Let t be the time. x = ^BxAXt+{jy-j t The following example is given by Paramesvara. The sum of 100 is loaned for one month. Then the interest received is loaned for six months. At that time the original interest plus the interest on this interest amounts to 16. f = 100 "t^l6Xl0QX6+25Q0-50 __^Q The interest received on 100 in one month was 10. 26. In the rule of three multiply the fruit by the desire and divide by the measure. The result will be the fruit of the desire. The rule of three corresponds to proportion. In the proportion a is to b as c is to a; the measure is a, the fruit is 6, the desire is c, and the fruit of the desire is x. he a 1 See JBOBS, XII, 87. * See Brahmagupta, XII, 10 and Colebrooke, op. dt., p. 283. x=- 2 40 ARYABHATIYA 27. The denominators of multipliers and divisors are multi- plied together. Multiply numerators and denominators by the other denominators in order to reduce fractions to a common denominator. For the first part of the rule I have given what seems to be the most likely hteral translation. The exact sense is uncertain. Kaye (agreeing with Rodet) translates, "The denominators are multiplied by one another in multiplication and division." If that is the correct translation the genitive plural is curious. Paramesvara explains gunakara as gunaguriyayor ahatir air a gunakdrasahdena vivaksitd. hdrya ity arthah and then seems to take bhdgahdra as referring to a fractional divisor of this product. Can the words bear that construction? In either case the inversion of numerator and denominator of the divisor would be taken for granted. It is tempting to take gunakdrahhdgahdra as mean- ing "fraction" and to translate, "The denominators of fractions are multiplied together." But for that interpretation I can find no authority. 28. Multipliers become divisors and divisors become multi- pliers, addition becomes subtraction and subtraction becomes addition in the inverse method. The inverse method consists in beginning at the end and working backward. As, for instance, in the question, "What number multiphed by 3, divided by 5, plus 6, minus 1, will give 5?" 29. If you know the results obtained by subtracting suc- cessively from a sum of quantities each one of these quantities set these results down separately. Add them all together and divide by the number of terms less one. The result will be the sum of all the quantities. GANITAPADA OR MATHEMATICS 41 The translation given by Kaye is incorrect. The revised translation given in his Indian Mathematics, page 47, is not an improvement.^ According to the rule 3a+3&+3c+3(i X— d=a+6+c , , , J since 4x=4a+4&+4c+4d. x—c = a+o+a 30. One should di\-ide the difference between the pieces of money possessed by two men by the difference between the ob- jects possessed by them. The quotient ■ndll be the value of one of the objects if the wealth of the two men is equal. Two men possess each a certain number of pieces of money (such as rupees) and a certain number of objects of merchandise (such as cows). Let a and 6 be the number of rupees possessed by two men, and let m and p be the number of cows possessed by them. h—a . , , , x= smce mx-\-a = v^+o . m—p If one man has 100 rupees and 6 cows and the other man has 60 rupees and 8 cows the value of a cow is 20 rupees provided the wealth of the two men is equal. 31. The two distances between two planets moving in 6ppo- site directions is divided by the sum of their daily motions. The two distances between two planets moving in the same direction is divided by the difference of their daily motions. The two results (in each case) wiU give the time of meeting of the two in the past and in the future. 1 Cf. JBORS, XII, 88-90. 42 ARYABHATIYA In each case there will be two distances between the planets, namely, that between the one which is behind and the one which is ahead, and, measiu-ing in the same direction, the distance between the one which is ahead and the one which is behind. This seems to be the only adequate interpretation of the word dve. The translations of Rodet and Kaye fail to do full justice to the word dve.^ The next two stanzas give a method for the solu- tion of indeterminate equations of the first degree; but no help for the interpretation of the process in- tended, which is only sketchily presented in Arya- bhata, is to be found in Mahavira, Bhaskara, or the second Aryabhata. The closest parallel is foimd in Brahmagupta, XVIII, 3-5.2 The verbal expression is very similar to that of Aryabhata, but with one im- portant exception. In place of the enigmatic state- ment matigunam agrdntare ksiptam, ''(The last re- mainder) is multipUed by an assumed number and added to the difference between the agras," Brahma- gupta has, "The residue (of thfe reciprocal division) is multiplied by an assumed number such that the product having added to it the difference of the remainders may be exactly divisible (by the residue's divisor). That multipher is to be set down (under- neath) and the quotient last." It is possible that this same process is to be understood in Aryabhata. ^ Cf. Paramesvara, dve Hi vacanam antarasya dvaividhyat. iighragalihino mandagatir antaram bhavaii. numdagatihlnas slghragatis cdntaram hhavati. Hi dvaividhyam and his further interpretation of the results. Cf. Brahmagupta, IX, 5-6 and Bhaskara, Gav-itSLdhydya, Gra- hayviyadhikara, 3-4, and Vasanabha^ya, and see J A (1878), 1, 28. * Colebrooke, op. cit., p. 325. GANITAPADA OR MATHEMATICS 43 First I shall explain the stanza on the basis of Paramjesvara's interpretation and of Brahmagupta's method : 32-33. Divide the dmsor which gives the greater agra by the divisor -which gives the smaller agra. The remainder is reciprocal- ly di^'ided (that is to say, the remainder becomes the di\isor of the original di\'isor, and the remainder of this second di\-ision becomes the di-\dsor of the second di-visor, etc.)- (The quotients are placed below each other in the so-called chain.) (The last remainder) is multiplied by an assumed number and added to the difference between the agras. Multiply the penultimate number by the number above it and add the number which is below it. (Continue this process to the top of the chain.) Divide (the top number) by the di\'isor which gives the smaller agra. Multiply the remainder by the di\'isor which gives the greater agra. Add this product to the greater agra. The result is the number which will satisfy both divisors and both agras. In this the sentence, "(The last remainder) is multiplied by an assumed number and added to the difference between the agras," is to be understood as equivalent to the quotation from Brahmagupta given above. The word agra denotes the remainders which con- stitute the provisional values of x, that is to say, values one of which will satisfy one condition, one of which will satisfy the second condition of the prob- lem. The word dviccheddgra denotes the value of x which will satisfy both conditions. I cannot agree with the translation given by Kaye (and followed by Mazumdar, BCMS, III, 11) or accept the method given by Kaye. Kaye's transla- tion of matigunam agrdntare ksiptam, "An assum.ed number together with the original difference is thrown in," is an impossible translation, and any method 44 ARYABHATIYA based on that translation is bound to be incorrect. It omits altogether the important word gunam (".multi- plied"). Since the preceding phrase dealt with the remainders of the reciprocal division, the natural word to supply with matigunam seems to be sesam ("remainder"). Something has to be supplied, and Brahmagupta's method offers a possible interpreta- tion. A second possible interpretation, which will be given below, would supply "quotient" instead of "remainder." The following example is given by Paramesvara. Sa; 17x -TTT gives a remainder of 4 -rz- gives a remainder of 7. These are equivalent to ^^ =y and — 7^ = 2 or 8a;— 29 45 292/ =4 and l7x-45«=7 where y and 2 are the quotients of the division (y and 2 to be whole numbers). 1. First process. To find a value of x which will satisfy the first equation: 8)29(3 24 5)8(1 5 3)5(1 3 2)3(1 Take an assumed number such that multiplied by 1 (the last remainder of the reciprocal division) and GANITAPADA OR ^MATHEMATICS 45 plus or minus 4 (the original remainder) it Tvill be exactly di\'isible by 2 (the last divisor of the recip- rocal division). 6 is taken because —^ — 1. Therefore 6 and 1 are to be added to the quotients to form the chain. 3 73 29)73(2 1 20 58 1 13 15 1 7 This remainder 15 is the agra, that is to say, a value of x 6 which win satisfy the equation. 1 2. Second process. To find a value of x which ^sill satisfy the second equation: 17)45(2 34 11)17(1 11 6)11(1 6 5)6(1 5 Take an assumed number such that multiplied by 1 (the last remainder of the reciprocal division) and plus or minus 7 (the original remainder) it will be exactly divisible by 5 (the last divisor of the reciprocal division) . 3 is taken because —zr- =2. 46 ARYABHATIYA Therefore 3 and 2 are to be added to the quotients to form the chain. 2 34 34)45(1 1 13 34 1 8 — 15 11 3 This remainder 11 is the agra, that is to say, a Talue of x 2 which mil satisfy the equation. These numbers 15 and 11 are the agras mentioned at the beginning of the rule. The corresponding divisors are 29 and 45. The difference between the agras is 4, i.e., 15 — 11. 3. Third process. To find a value of x which wiQ satisfy both equations: 29)45(1 29 16)29(1 16 13)16(1 13 3)13(4 12 Take an assumed number such that multiphed by 1 (the last remainder of the reciprocal division) and plus or minus 4 (the difference between the agras) it will be exactly divisible by 3 (the last divisor of the reciprocal division). 2 is taken because -~— =2. GANITAPADA OR IMATHEMATICS 47 Therefore 2 and 2 are to be added to the quotients to form the chain. 1 34 45)34(0 1 22 1 12 — 4 10 34 2 Therefore 34 is the remainder, 2 Then in accordance with the rule 34X29 = 986 and 986+15 = 1001 This number 1001 is the smallest number which will satisfy both equations. Strictly speaking, the rule applies only to the third process given above. The solution of the single inde- terminate equation is taken for granted and is not given in full. There is nothing to indicate how far the reciprocal division was to be carried. Must it be carried to the point where the last remainder is 1? Must the number of quotients taken to make the chain be even in number? On page 50 of Kern's edition a 1 has been omitted by mistake (twice) as the fourth member of the chains given. The following method was partially worked out by Mazimadar/ who was misled in some details by fol- lowing Kaye's translation, and by Sen Gupta,^ and fully worked out by Sarada Kanta Ganguly.^ 1 BCMS, III, 11-19. * Journal of the Department of Letters (Calcutta University), XVI, 27-30. 3 BCMS, XIX (1928), 170-76. 48 ARYABHATIYA According to Gangulj^'s interpretation, the trans- lation would be : 32-33. Di\-icle the di\isor corresponding to the greater re- mamder by the di\-isor corresponding to the smaller remainder. The remainder (and the divisor) are reciprocally divided. (This process is continued until the last remainder is 0.) (The quotients are placed below each other in the so-called chain.) Multiply any assumed number by the last quotient of the reciprocal division and add it to the difference between the two remainders. (Interpreted as meaning that this product and this difference are placed in the chain beneath the quotients.) Multiply the penulti- mate number by the number above it and add the number which is below it. (Continue this process to the top of the chaia.) Divide (the lower of the two top numbers) by the di-visor cor- responding to the smaller remainder. Multiply the remainder by the di-\dsor corresponding to the greater remainder. Add the product to the greater remainder. The result is the (least number) which will satisfy the two divisors and the two re- mainders. He remarks : The impUcation is that the least number satisfying the given conditions can also be obtained by multiplying the remainder, obtained as the result of division of the upper number by the divisor corresponding to the greater given remainder, by the di-visor corresponding to the smaller given remainder and then adding the smaller remainder to the product. From this point of view the problem would be that of finding a number which will leave given remainders when divided by given positive integers. For example, to take a simple case: What num- ber divided by 3 and 7 will leave as remainders 2 and 1? Following the rule literally, even though a smaller GANITAPADA OR MATHEMATICS 49 number has to be divided by a larger number we get the following: 7)3(0 3)7(2 6 1)3(3 3 Multiply the last quotient (3) by an assumed number (for instance, 3) and set this product and the difference between the remainders 2 and 1, i.e., (1) down below the quotients to form the chain. 28 7)65(9 2 65 63 3 28 — 9 2 1 Then 2X3 = 6 and 6+2 = 8. or 3)28(9 27 1 Then 1X7=7 and 7+1=8. Therefore 8 is the number desired. The two methods attach different significations to the word agra and supply different words with bhajite in the third line ("remainder," ia one case; "quo- tient," in the other). They differ fundamentally in their interpretations of the words matigunam agra- 50 ARYABHATIYA ntare k^iptam. In the first method it is necessary to supply much to fill out the meaning, but the transla- tion of these words themselves is a more natural one. In the second method it is not necessary to supply anything except "quotient" with matigunam (in the first method it is necessary to supply "remainder"). But if the intention was that of stating that the product of the quotient and an assumed number, and the difference between the remainders, are to be added below the quotients to form a chain the thought is expressed in a very curious way. Gangtdy finds justification for this interpretation (p. 172) in his formulas, but I cannot help feeling that the San- skrit is stretched in order to make it fit the formula. The general method of solution by reciprocal division and formation of a chain is clear, but some of the details are uncertain and we do not know to what sort of problems Aryabhata appHed it. CHAPTER III KALAKRIYA OR THE RECKONING OF TIME 1. A year consists of twelve months. A month consists of thirtj' days. A day consists of sixty nad'is. A m<fi consists of sixty vinddikas} 2. Sixty long letters or six prdnas make a sidereal vinadikd. This is the division of time. In like manner the di\ision of space beginning with a revolution.^ 3. The difference between the number of revolutions of two planets in a yuga is the number of their conjunctions. Twice the sum of the revolutions of the Sun and Moon is the number of vyatipdtas.^ This is a yoga of the Sun and Moon when they are in different ayanas, have the same dechnation, and the sum of their longitudes is 180 degrees. 4. The difference between the number of revolutions of a planet and the number of revolutions of its ucca is the number of revolutions of its epicycle. The number of revolutions of Jupiter multipUed by 12 are the years of Jupiter beginning with Asvayuja.* 1 Cf. SuryasiddMnta, I, 11-13; Albenmi, I, 335; Bhattotpala, p. 24. 2 Cf. Suryasiddharda, I, 11, 28; Bhattotpala, p. 24; Pancasi- ddhdntika, XIV, 32, for the first part, of 2; Brahmagupta, I, 5-6, and Blmskara, Ganitadhaya, Kdlamanddhydya, 16-18, for both stanzas. ^ See Lalla, Madhyamddhik6ra, 11; Brahmagupta, XIII, 42, for the first part. For vyatlpata see Suryasiddhdnta, XI, 2; Pancad- ddhantika, III, 22; Lalla, Mahapatddhikdra, 1; Brahmagupta, XIV, 37, 39. *For the first part see Lalla, Madhyamddhikara, 11; Brahma- gupta, XIII, 42; Bhaskara, Gai^itadhyaya, Bhaganadhyaya, 14. For the second part see JRAS, 1863, p. 378; iUd., 1865, p. 404; Suryasiddharda, I, 55; Bhattotpala, p. 182. 51 52 AEYABHATIYA The vrord ucca refers both to mandocca ("apsis") and sighrocca ("conjunction"). Paramesvara explains that the number of revolu- tions of the epicycle of the apsis of the Moon is equal to the difference between the number of revolutions of the Moon and the revolutions of its apsis; that since the apsides of the six others are stationary, the number of revolutions of the epicycles of their apsides is equal to the number of revolutions of the planets; and that the number of revolutions of the epicycles of the conjunctions of Mercury, Venus, Mars, Jupiter, and Saturn is equal to the difference between the revolutions of the planets and the revolutions of their conjunctions. As pointed out in the note to I, 7, the apsides were not regarded by Aryabhata as being stationary in the absolute sense. They were regarded by him as sta- tionary for purposes of calculation at the time when his treatise was composed since their movements were very slow. 5. The revolutions of the Sun are solar years. The conjunc- tions of the Sun and Moon are lunar months. The conjunctions of the Sun and the Earth are [civil] days. The revolutions of the asterisms are sidereal days. The word yoga applied to the Sun and the Earth (instead of bhagana or avarta) seems clearly to indi- cate that Aryabhata believed in a rotation of the Earth (see IV, 48). Paramesvara's explanation, ravi- bhuyogasahdena raver bhuparibhramaiiam abhihitam, seems to be impossible. 6. Subtract the solar months in a yitga from the lunar months in a yxiga. The result will be the number of intercalary months in KALAKRIYA OR THE RECKONING OF TIME 53 a yuga. Subtract the natural [chil] days in a yiiga from the lunar days in a yu^a. The result ■will be the number of omitted lunar days in a yuga.^ 7. A solar year is a year of men. Thirty of these make a year of the Fathers. Twelve years of the Fathers make a year of the gods. S. Twelve thousand years of the gods make a yuga of all the planets. A thousand and eight yiigas of the planets make a day of Brahman.'*- 9. Thefirsthalf of a jTiga is called wfsorpim [ascending]. The latter half is called avasarpinl [descending]. The middle part of a yuga is called summa. The beginning and the end are called dus^ama. Because of the apsis of the Moon. Alberuni (I, 370-71) remarks: Aryabhata of Kusumapura, who belongs to the school of the elder Aryabhata, says in a small book of his on Al-nij (?), that "1,008 caturyugas are one day of Brahman. The first half of 504 caiuryugas is called utsarpim, during which the sun is ascending, and the second half is called avasarpinl, during which the sun is descending. The midst of this period is called sama, i.e., equality, for it is the midst of the day, and the two ends are called durtama (?)• "This is so far correct, as the comparison between day and kalpa goes, but the remark about the sun's ascending and descending is not correct. If he meant the sun who makes our day, it was his duty to explain of what kind that ascending and descending of the sim is; but if he meant a sun who specially be- longs to the day of Brahman, it was his duty to show or to describe him to us. I almost think that the author meant by these two expressions the progressive, increasing development of things during the first half of this period, and the retrograde, decreasing development in the second haK." 1 Cf. Suryasiddhdnta, I, 35-36; Lalla, MadhyamMhikdra, 10; Brahmagupta, I, 24 and XIII, 26. »Cf. Suryasiddhanta, I, 13-15; I, 20. Brahmagupta (I, 12) criticizes Aryabhata's figure of 1,008 yugas instead of 1,000 yugas. Cf . JBAS, 1866, p. 400. Cf. also I, 3 and see JBAS, 1911, p. 486. 54 ARYABHATlYA The reference is to the foregoing stanza. The mid- dle of the yuga seems to be called susamd (''even") because good and bad are evenly mixed. The begin- ning and the end are called dussamd, ("uneven") be- cause in one case goodness and in the other case badness predominates. Paramesvara remarks that the vydkhyakara has given no explanation. Then he quotes from the Bhataprakdsikd a statement to the effect that our text refers to the increase and decrease of men's Hves in the course of a yuga and a criticism (asydrtho 'bhiyuktair nirupya vaktavyah) of the last phrase of the stanza. He then continues by saying that he does not see what meaning can be intended by the word induccat, and adds that the word has nothing to do with the matter under discussion, has no significance for the calculation of the places of the planets. Then he adds two forced explanations. The meaning of induccdt is quite uncertain. Sudhakara (Ganakatarangini, p. 7) suggests the emendation to agnyamsdt. The terminology is distinctively Jaina.^ 10. When three yugapddas and sixty times sixty years had elapsed (from the beginning of the ytiga) then twenty-three years of my hfe had passed." If Aryabhata began the KaHynga at 3102 b.c. as later astronomers did, and if his fourth yugapdda ^See Hemacandra, Abhidhdnacintamazd, 128-35; Glasenapp, Der Jainismus, pp. 244-45; Kirfel, Kosmographie, p. 339; ZDMG, LX, 320-21 ; Stevenson, The Heart of Jainism, pp. 272 ff . See also Hardy, Manual of Buddhism, p. 7. 2 See JRAS, 1863, p. 387; ibid., 1865, p. 405; Kern, Brhat Samhita, Preface, p. 57; JRAS, 1911, pp. 111-12. KALAKRIYA OR THE RECKONING OF TIME 55 began mth the beginning of the Kaliyuga, we arrive at the date 499 a.d. It is natural to take this as the date of composition of the treatise. Paramesvara quotes the Prakdsikakdra to the effect that this is to be taken as the date at which the calculations of the true places of the planets made by it would be correct, and that for later times a correction would have to be made. The word iha may mean "here" or "now." Paramesvara takes it as referring to this present twenty-eighth caturyuga. 11. The yuga, the year, the month, and the day began all together at the beginning of the bright fortnight of Caitra. Time, which has no beginning and no end, is measured by (the move- ments of) the planets and the asterisms on the sphere. Bhau Daji^ first pointed out the criticism made of this stanza by Brahmagupta (XI, 6) : jTigavarsadin vadata caitrasitadeh samam pra\'rttan yat| tad asad yata^^ sphutayugam tat sthairyan mandapatanam.|| Compare Brahmagupta, I, 4, and Bhaskara, Ganitd- dhydya, Kdlamdnddhydya, 15, who refers to an earlier commentary in which time is called endless.^ 12. The planets moving equally (traversing the same distance in yojanas each day) in their orbits complete the circle of the asterisms in sixty solar years, and the circle of the sky in a divine age [caturyv^a]. In sixty years they move a distance in yojanas equal to the circle of the asterisms. In a caturyuga they move a distance in yojanas equal to the cirexun- ference of the sky (akdsakaksyd) (cf. I, 4). 1 JBAS, 1865, p. 401. s For discussion of the stanza see Fleet, ibid., 1911, pp. 489-90; cf. I, 2. 56 ARYABHATIYA The planets really all move at the same speed. The nearer ones seem to move more rapidly than the more distant ones because their orbits are smaller. ^ 13. The Moon, being below, completes its small orbit in a short time. Saturn, being above all the others, completes its large orbit in a long time.^ 14. The zodiacal signs (a twelfth of the circle) are to be known as small in a small circle and large in a large circle. Like- \sise the degrees and minutes are the same in number in the various orbits.^ 15. Beneath the asterisms are Saturn, Jupiter, Mars, the Sun, Venus, Mercury, and the Moon, and beneath these is the Earth situated in the center of space like a hitching-post.* 16. These seven lords of the hours, Saturn and the others, are in order swifter than the preceding one, and coimting suc- cessively the fourth in the order of their swiftness they become the Lords of the days from sumise. They are called "swifter than the preceding" be- cause their orbits being successively smaller they complete their revolutions in less time (traverse a given number of degrees in less time). The order of the planets is Saturn, Jupiter, Mars, the Sun, Venus, Mercury, and the Moon. Therefore they become rulers of the days of the week as follows: 1 Cf. Suryasiddhanta, I, 27 and note; Brahmagupta, XXI, 12; Pancasiddhantika, XIII, 39; Bhaskara, Goladhydya, Bhuvanakosa, 69; JBAS, 1911, p. 112. » Cf. JRAS, 1863, p. 375; Suryasiddhanta, XII, 76-77; Pancasi- ddhantika, XIII, 41; Brahmagupta, XXI, 14; Bhattotpala, p. 45. » Cf. JRAS, 1863, p. 375; Pancasiddhantika, XIII, 40; Suryasi- ddhanta, XII, 75; Brahmagupta, XXI, 14; Bhattotpala, p. 45. * Cf. JRAS, 1863, p. 375; Pancasiddhantika, XIII, 39; Lalla, MadhyarmdhikSra, 12; Brahmagupta, XXI, 2; Bhattotpala, p. 44. KALAKRIYA OR THE RECKONING OF TIME 57 Saturday — Saturn Wednesday — Mercury Sunday — Sun Thursday — ^Jupiter Monday — Moon Friday — ^\'enus Tuesday — jSIars For the first part see Brahmagnpta, XXI, 13; Suryasiddhdnta, XII, 78.^ Bhau Dajl^ first pointed out tlie criticism of this stanza made by Brahmagupta (XI, 12) : stiryadayas caturtlia dinavara yad uvaca tad asad arj-abhatahj lankodaye yato 'rkasyastamayam praha siddhapure. || As Sudhakara shows, the criticism is a futile one. 17. AH the planets move by their (mean) motion on their orbits and their eccentric circles from the apsis eastward and from the conjunction westward.^ The mean planet moves with, its mean motion on its orbit the center of which is the center of the Earth. The true planet moves with its (mean) motion on an eccentric circle the center of which does not coincide with the center of the Earth. Kaksyd in this passage stands for kaJcsydmandala, the orbit on which the mean planet moves. The pra- timandala is the eccentric circle on which the true planet moves. Because of the eccentricity of this second circle the planet .is sometimes seen ahead of and sometimes back of its mean place.'* 1 See Barth {CEuvres, III, 151) concerning this as the only refer- ence to astrology in Aryabhata's treatise. The reference to vyailpata (III, 5) should be added. 2 JRAS, 1865, p. 401. = See Lalla, ChedyaJcadhilcara, 12-13; Brahmagupta, XIV, 11 and XXI, 24. *See Brennand, Hindu Astronomy, pp. 224 fl.; Surijasiddh&nUi, p. 64. 58 ARYABHATIYA 18. The eccentric circle of each planet is equal to its hah^ya- mai),4o.la [the orbit on -vrhich the mean planet moves]. The center of the eccentric circle is outside the center of the solid Earth. The kaksydmandala is determined by I, 4. 19. The distance between the center of the Earth and the center of the eccentric circle is equal to the radius of the epicycle. The planets move -n-ith their mean motions on their epicycles.^ Brahmagupta, XI, 52, has nIcocca\Tttamadhyasya golabahyena nama krtam uccam | tatstho na bhavati ucco yatas tato vetti noccam api. || If this really refers to Aryabhata the criticism is futile since Aryabhata does not call the center of the epi- cycle ucca. As Caturvedacarya says, vdgbalam etat 20. The planet in its swift motion from its ucca has a pra- tiloma motion on its epicycle. In its slow motion from its iicca it has an anuloma motion on its epicycle. The exact meaning of this is not clear to me. It can hardly mean that the planet moves on its epicycle pratiloma from its slghrocca and anuloma from its mandocca. On the epicycle of the apsis the motion should be exactly the reverse of these.^ Anulomajnesuas "eastward" or "ahead" ; Pratiloma means ''westward" or "behind." Paramesvara remarks that anuloma and pratiloma refer to the planet's position with reference to the mean planet as ahead of it or behind it. He also re- marks that the planet is sighragati in the six signs ' Cf. Lalla, Chedyakadhikara, 8-9; Brahmagupta, XIV, 10 and XXI, 24r-26. ' See Brahmagupta, XXI, 25-26 and Suryasiddhanta, pp. 63-64, 67-68. KALAKRIYA OR THE RECKONING OF TIME 59 which are above, and mandagati in the six signs which are below the ucca. When pratiloma the true planet is below the mean planet. When anuloma the true planet is above the mean planet.^ madhyamakaksa^Ttte madhyama3'a gacchati graho gatya | uparisthat tallaghvya tadadMkagatya tv adhahsthah syat. jj Paramesvara sums up the content of the stanza with madhyamdt sphutasya pratilomdnulomagatitvam uktam. The meaning of the stanza seems to be that during half of its revolution on its epicycle the planet is ahead of the mean planet and during half of its revolution is behind the mean planet. 21. The epicycles move eastward from the apsis and west- ward from the conjunctions. The mean planet, situated on its orbit, appears at the center of its epicycle.^ The next three stanzas state briefly the method of calculating the true places of the planets from their mean places. Paramesvara explains the method as follows: For the Sun and Moon only one process of cor- rection is required, that for the apsis. For Mars, Jupiter, and Saturn four processes are necessary: (1) From the mean place the mandaphala is calculated and (half of it is) applied to the mean place. (2) From this corrected place the sighraphala is calculated and half of it is applied to the corrected place. (3) From this result the mandaphala is again calculated and applied to the mean place. (4) From this result the sighraphala is again calculated and applied to the place obtained in the third process. 1 Cf . Lalla, Bhuvanakosa, 38. » Cf . Brahmagupta, XXI, 25. 60 ARYABHATlYA For Venus and Mercury three processes are nec- essary: (1) From the mean place the sighraphala is calculated and half of it is applied vyastam (in reverse order) to the mandocca (apsis). (2) This corrected mandocca is subtracted from the mean place, the mandaphala is calculated from this and applied to the mean place. (3) From this corrected place the sighra- phala is calculated and applied to the place obtained in the second process. 22-23. (The corrections) from the apsis are minus, plus, plus, minus (in the four quadrants). (The corrections) from the con- junctions are just the reverse. In the case of Saturn, Jupiter, and Mars in the first process half of the mandaphala obtained from the apsis is minus and plus to the mean planet. Half (the correction) from the conjunction is minus and plus to the manda planets. (By applying the correc- tion) from the apsis they become sphutamadhya. (By applying the correction) from the conjunction they become sphuta. 24. Half (the correction) from the conjunction is to be ap- plied minus and plus to the apsis. (By applying the correction) from the manda [apsis] thus obtained Venus and Mercury become sphutamadhya. They become sphuta (by applying the correction from the conjunction). The first half of stanza 22 gives the general rule as to whether the equations of anomaly and of com- mutation (mandaphala and sighraphala) are to be added or subtracted in each of the four quadrants. The equation from the apsis is minus in the half of the orbit beginning with Mesa, plus in the half of the orbit beginning with Tula. The equation from the conjunction is plus in the half of the orbit beginning with Mesa, minus in the half of the orbit beginning with Tula. KALAKRIYA OR THE RECKONING OF TIME 61 The planet is called manda after the first correc- tion from the apsis has been apphed to the mean place. Sphuta means "true." In stanza 24 Paramesvara gives no explanation of the two last words, sphutau hhavatah. It would be natural to take these words as summing up what precedes and to understand that only two processes are involved. But Paramesvara's detailed description of the process in his commentary to stanza 21 indi- cates that three processes are involved, that sphutau hhavatah indicates a further application of the equa- tion from the conjunction. The commentary to stanza 24 gives in detail the process of calculating the equations for apsis and conjunction.^ Brahmagupta (II, 19, 33, 46-47) criticizes Arya- bhata for the inaccuracy of his method of calculating the true places. 25. The product of its hypotenuses di\ided by the radius wHl give the distance between the planet and the Earth. The planet has the same speed on its epicycle that it has on its orbit. Paramesvara explains that the karnas referred to are the slghrakarv^ and the mandaharna employed in the last and the next to the last processes for calculating the true places of the planets. The second half of the stanza is uncertain. This same statement was made in unmistakable terms in ^See Pancasiddkantikd, XVII, 4-10; SHryasiddhanta, II, -43-45; Brahmagupta, II, 34r-40; Lalla, Spa§iadhiMra, 31-36; Bhaskara, Gavitadhyaya, Spa^tadhik&ra, 34-36 and GolSdhyaya, Chedyaka- dhikara, 10 ff.; JRAS, 1863, pp. 353-59; Brennand, op. dt., pp. 214- 28; Kaye, Hindu Astronomy, pp. 87-89. 62 ARYABHATlYA III, 19. Paramesvara quotes the author of the earlier Prakasika, hhutaragrahavivaravyasdrdhaviracAtaydrh kdksyayarh yo grahasya javas sa mandanlcocce bhavati. tdvatpramandydm kaksydydm graho mandasphutagatyd gacchatlty arthah. ity aha. asmdn kirn tv etan nopa- pannam iti pratibhdti. Then he explains that the meaning may be that the radius of the epicycle is equal to the greatest distance by which the mean orbit lies inside or outside of the eccentric circle. Grahavegah is reminiscent of grahajavah in I, 4 but the meaning can hardly be the same. Karna ("hypotenuse") is the distance between the center of the Earth and the planet.^ 1 Cf. Brahmagupta, XXI, 31; Bhaskara, Gariitadhyaya, Candra- graharicLdhikara, 4—5; SUryasiddhanta, p. 69. CK^^TER IV GOLA OR THE SPHERE 1. From the beginning of Me§a to the end of Kanya is the northern half of the ecUptic. The other half from the beginning of Taulya to the end of Mina is the southern half of the echptic. Both deviate equally from the Equator. Therefore the greatest declinations north and south are equal, and the declinations of the first three signs in each half are equal to the declinations of the last three signs taken in reverse order.^ 2. The Sun, the nodes of the planets, and the node of the Moon move constantly along the echptic. The shadow of the Earth moves along the echptic at a distance of 180 degrees from the Sun. Bhau Daji'- first pointed out the reference to this passage made by Brahmagupta, XI, 8.^ Barth^ questions the stanza, but without good reason. 3. The Moon, from its nodes, moves northward and south- ward of the ecliptic. Likewise Jupiter, Mars, and Saturn. Venus and Mercury do the same from their conjunctions.* 4. When the Moon has no dechnation it is visible when 12 degrees from the Sun. Venus when 9 degrees. The other planets 1 Cf. JBAS, 1863, p. 374; Bhattotpala, p. 45. 2Ji2A/S, 1865,p. 401. » Cf. I, 7 and note; Brahmagupta, XXI, 53; SuryasiddhdrUa, IV, 6. * (Euvres, III, 154. 5 Cf . SHrya^ddhanta, I, 68-69 and AryabhaUya, I, 6. 63 64 ARYABHATIYA in succession according to their decreasing sizes when at 9 degrees increased by two's. Compare Brahmagupta, VI, 6; Suryasiddhanta, IX, 6-9 and X, 1; Paficasiddhantikd, XVII, 12 and XVIII, 58. Bhau Dajl^ first pointed out the criticism of this stanza made by Brahmagupta, VI, 12 : aryabhatah ksetramsair dfsyadrsyan yad nktavams tad asat | drggaioitavisamvadad drgganitaikyam svakalamsaih. || 5. Half of the spheres of the Earth, the planets, and the asterisms is darkened by their shadows, and haK, being turned toward the Sun, is light (being small or large) according to their size.2 6. The sphere of the Earth, being quite round, situated in the center of space, in the middle of the circle of asterisms, surrounded by the orbits of the planets, consists of water, earth, fire, and air.' 7. Just as a ball formed by a Kadamba flower is surrounded on all sides by blossoms just so the Earth is surrounded on all sides by all creatures terrestrial and aquatic* 8. During a day of Brahman the sphere of the Earth increases a yojana in size all around. During a night of Brahman, which is equal in length to a day of Brahman, there is a decrease by the same amount of the Earth which has been increased by Earth.* 9. As a man in a boat going forward sees a stationary object moving backward just so at Lanka a man sees the stationary asterisms moving backward (westward) in a straight line. The natural interpretation of this stanza seems 1 JRAS, 1865, p. 401. 2 Cf. Lalla, MadhyagaUvasamM, 40-41; Bhattotpala, p. 100; PaTV- casiddhantika, XIII, 35, for the Moon. 'Cf. Ill, 15. Cf. Lalla, BhUgoladhyaya, 1; PancasiddhantiM, XIII, 1; Bhattotpala, p. 58 (and see JRAS, 1863, pp. 373-74); Alberuni, I, 268. * Cf. Lalla, BhUgoladhyaya, 6; Bhattotpala, p. 58 (and see JRAS, 1863, pp. 373-74); Bhaskara, Goladhyaya, BhuvanakoSa, 3. ^Cf. Lalla, GrahabhramasaThsthadhydya, 20; Bhaskaxa, Gola- dhyaya, Bhuvanako§a, 62. GOLA OR THE SPHERE 65 to be that an observer at the Equator of the Earth, which , rotates toward the East, sees the stationary celestial objects as though moving westward. But Paramesvara explains that whereas the Earth does not really move, it appears to move tow^ard the east because of the westward movement of the asterisms. He is forced to take the w^ords anuloma and viloma, which regularly mean "ahead," "eastward," and "backward," "westward," in exactly the opposite senses. He explains that persons on the asterisms, which move toward the west, would seem to see sta- tionary objects on the Earth moving eastward. As Barth^ points out, this explanation is quite unac- ceptable. It seems that Paramesvara completely mis- represents the opinion of Aryabhata, as clearly stated in several places in the text, and as described by Brahmagupta and other critics of Aryabhata. There is nothing to indicate that this stanza repre- sents a state of affairs caused by mithydjndna ("false knowledge)." Bhattotpala (pp. 58-59) quotes this stanza and then refutes it by quoting the Pancadddhantikd, XIII, 6-8, Pauhsa, Brahmagupta, and strangely enough Aryabhata himself (the following stanza, IV, 10). It is curious that Aryabhata should be quoted against himself, and that Bhattotpala should not indicate clearly which view really represents Arya- bhata's own opinion. It looks as though Bhattotpala regarded the first stanza as containing a purvapaksa or erroneous view.^ 1 Op. cit, III, 158 n. 2 cf . jrj^js^ iges, pp. 375-77. 66 ARYABHATIYA For criticisms of the rotation of the Earth see Alberuni (I, 276-77, 280); Lalla, MithyajMnd- dhydya, 42-43; Sripati as reported in the Lucknow edition of Bhaskara's Golddhyaya, page 83; see also Barth.i Colebrooke^ quotes Prthtidaka the commentator on Brahmagupta as follows : bhapafijarati sthiro bhur evavftyavrtya pratidaivasikau | udayastamayau sampadayati nak|atragrahanam. ||' The Vdsandvdrttika to Bhaskara's Grahaganita, page 113,* quotes the foregoing stanza and remarks that according to Aryabhata the planets move toward the east, the asterisms are stationary, and the Earth rotates eastward. 10. The cause of their rising and setting is due to the fact that the circle of the asterisms, together with the planets, driven by the provector wind, constantly moves straight westward at Lanka.^ Bhattotpala (p. 59) quotes this stanza to disprove the preceding stanza which he quoted on page 58 (cf . JRAS, 1863, p. 377). 6 The Marici (p. 43) to Bhaskara's Grahagardta^ quotes this stanza. 1 Op. cit., Ill, 158. * Essays, II, 392. 3 See JRAS, 1865, pp. 403-4; IHQ, I, 666 (the words given as a direct quotation from Aryabhata are incorrect) ; BCMS, XVII (1926), 175. The author of the last article remarks that it is not clear whether Aryabhata had in mind the geocentric or the heliocentric motion of the Earth. The latter is out of the question. Cf . Ill, 15, bkumir medhlbhuta khamadhyastfia, and IV, 6, khamadhyagaiah,. * Pandit, Vol. XXXI. *Cf. Suryasiddhanta, II, 3: LaUa, Madhyamadhikdra, 12. 8 See Earth, op. dt., Ill, 158. ' Pandit, Vol. XXX. GOLA OR THE SPHERE 67 The Vdsandvarttika to Bhaskara's Grahaganita (p. 118)^ quotes this stanza apparently without seeing in it anything contradictory to the preceding stanza whichjvas quoted on page 113, and with the remark that Aryabhata is here following the opinion of Vrddhavasi§tha. If the readings of our text are correct it is difficult to see how the two stanzas can be brought into agree- ment. The ninth stanza states unequivocally that the asterisms are stationary and implies the rotation of the Earth. The tenth stanza seems to state that the asterisms, together with the planets, are driven by the pro vector wind. This would imply the ordinary- point of view of most Indian astronomers that the Earth was stationary. Paramesvara avoids the diffi- culty by assuming that stanza 9 describes a state of mind brought about by mithydjndna ("false knowl- edge"). But since several other stanzas (I, 1; I, 4; III, 5 ; IV, 48) and the testimony of later writers who quote Aryabhata prove that Aryabhata believed in the rotation of the Earth, it is impossible to follow Paramesvara. We might understand in stanza 10 the phrase "they seem to move" as stating a purvapaksa (the erroneous view), but in the absence of any word to suggest this interpretation it is a doubtful expedi- ent. Stanza 10 cannot be regarded as an interpolation (unless one stanza has been dropped out in order to make room for it) because the last three sections of Aryabhata's work were known to Brahmagupta as "The Hundred and Eight Stanzas" (and our text con- tains 108 stanzas). 1 Ibid., Vol. XXXI. 68 ARYABHATIYA 11. In the center of the Nandana forest is Mount Mem, a yojana in measure (diameter and height), shining, surrounded by the Himavat Moimtains, made of jewels, quite round.^ 12. Heaven and Meru are at the center of the land, Hell and Vadavamukha are at the center of the water. The gods and the dwellers in Hell both think constantly that the others are beneath them.2 Quoted by Bhattotpala, page 58.' 13. Sunrise at Laiika is sunset at Siddhapura, midday at Yavakoti, and midnight at Romaka.* Brahmagupta's criticism (XI, 12) suryadayas caturtha dinavara yad uvaca tad asad aryabhatat | lankodaye yato 'rkasyastamayam praha siddhapure || is incorrect, as pointed out by Sudhakara in his com- mentary. 14. Lanka is 90 degrees from the centers of the land and water [north and south poles]. Ujjain is straight north of Laiika by 22§ degrees.^ 15. From a level place half of the sphere of the asterisms 1 Cf. I, 5. Cf. also Suryasiddhdnta, XII, 34; Lalla, Bhuvanako&a, 18-19; Bhaskara, Goladhydya, Bhuvanakoia, 31; Alberuni (I, 244, 246). Quoted by Bhattotpala, p. 58 (cf. JRAS, 1863, p. 373). 2 Cf. Paficasiddhaniika, XIII, 2-3; SuryasiMhmta, XII, 35-36, 53; Brahmagupta, XXI, 3; Lalla, BhugolSdhyaya, 3-4; Bhaskara, Goladhyaya, Bhuvanako&a, 17-20, 31. ' Cf. JRAS, 1863, p. 373. * Cf. Kem, Brhat Samhita, Preface, p. 57; Suryasiddhanta, XII, 38r-41; Panca^iddhdntika, XV, 23; Lalla, Bhugoladhyaya, 12; Bhas- kara, Goladhyaya, Bhmanako&a, 17, 44; Alberuni I, 267-68; JRAS, 1865, p. 402. * Cf. SUryasiddMnta, I, 62; PancasiddhdnHka, XIII, 17; Lalla, MadhyamOdhikdra, 55, and Bhuvanakoia, 41; Brahmagupta, XXI, 9; Bhaskara, Goladhydya, Bhuvanakoia, 50 {Vdsandhhd^a), and Madhyagati, 24; Alberuni, I, 316 (cf. BCMS, XVII [1926], 71). GOLA OR THE SPHERE 69 decreased by the radius of the Earth is \'isible. The other half, plus the radius of the Earth, is cut off by the Earth.^ 16. The gods, who dwell in the north on Mem, see the northern half of the sphere of the asterisms mo\'ing from left to right. The Pretas, who dwell in the south at Vadavamukha, see the southern half of the sphere of the asterisms moving from right to left.2 Quoted by Bhattotpala, page 324.' 17. The gods and the Pretas see the Sun after it has risen for half a solar year. The Fathers who dwell in the Moon see it for half a Ixinar month. Here men see it for half a natural [ci'S'il] day.* Referred to by Albemni, I, 330. 18. There is a circle east and west (the prime vertical) and another north and south (the meridian) both passing through zenith and nadir. There is a horizontal circle, the horizon, on which the heavenly bodies rise and set.* 19. The circle which intersects the east and west points and two points on the meridian which are above and below the horizon by the amount of the observer's latitude is called the unmar),4ala. On it the increase and decrease of day and night are measured. The unmandala is the east and west hour-circle which passes through the poles. It is also called "the horizon of Lanka."^ 1 Cf. LaUa, Bhuvanahosa, 36; Brahmagupta, XXI, 64; Bhaskara, GolSdhyaya, Tripra&nwasana, 38. * Cf. Suryasiddhdnta, XII, 55; Pancasiddhantika, XIII, 9; Brahmagupta, XXI, 6-7; Lalla, GrahabhramasaThsthadhySya, 3-5; Bhaskara, GolSdhyaya, Bhuvanako§a, 51. » Cf. JRAS, 1863, p. 378. * Cf. Suryadddhanta, XII, 74 and XIV, 14; Lalla, Gralvahhror masamsthadyaya, 14; Brahmagupta, XXI, 8; Pancasiddh&ntika, XIII, 27, 38. * Cf. Lalla, GolabandkSdhikdra, 1-2; Brahmagupta, XXI, 49. 6 Cf. Lalla, Golabandkadhikara, 3; Brahmagupta, XXI, 50. 70 ARYABHATlYA 20. The east and west line and the north and south line and the perpendicular from zenith to nadir intersect in the place where the observer is. 21. The vertical circle which passes through the place where the observer is and the planet is the drhmandala. There is also the drkk§epamandala which passes through the nonagesimal point.^ The nonagesimal or central-ecliptic point is the point on the ecliptic which is 90 degrees from the point of the ecliptic which is on the horizon. These two circles are used in calculating the parallax in longitude in eclipses. 22. A light wooden sphere should be made, round, and of equal weight in every part. By ingenuity one should cause it to revolve so as to keep pace with the progress of time by means of quicksilver, oil, or water.^ Sukumar Ranjan Das^ remarks that two instru- ments are named in this stanza (the gola and the cakra). I can see no reference to the cakra. 23. On the visible half of the sphere one should depict half of the sphere of the asterisms by means of sines. The equinoctial sine is the sine of latitude. The sine of co- latitude is its kolti. The sine of the distance between the Sun and the zenith at midday of the equinoctial day is the equi- noctial sine. This is the same as the equinoctial shadow and equals the sine of latitude. It is the base. ^ Cf. SuryasiddharUa, V, 6-7 n.; Kaye, Hindu Astronomy, p. 76. 2 Cf. Suryasiddhanta, XIII, 3 ff.; Lalla, Yantradhyaya, 1 ff.; IHQ, IV, 265 ff . 8/HO, IV, 259, 262. GOLA OR THE SPHERE 71 The sine of co-latitude is the koti (the side perpen- dicular to the base) or sanku (gnomon).^ 24. Subtract the square of the sine of the given declination from the square of the radius. The square root of the remainder ■will be the radius of the day-circle north or south of the Equator. The day-circle is the diurnal circle of revolution described by a planet at any given declination from the Equator. So these day-circles are small circles parallel to the Equator.^ • 25. Multiply the day-radius of the circle of greatest declina- tion (24 degrees) by the sine of the desired sign of the zodiac and divide by the radius of the day-circle of the desired sign of the zodiac. The result -will be the equivalent in right ascension of the desired sign beginning with Me§a. To determine the right ascension of the signs of the zodiac, that is to say, the time which each sign of the ecliptic will take to rise above the horizon at the Equator.3 26. The sine of latitude multiplied by the sine of the given declination and divided by the sine of co-latitude is the earth- sine, which, being situated in the plane of one's day-circle, is the sine of the increase of day and night. The earth-sine is the distance in the plane of the day-circle between the observer's horizon and the 1 Cf. Brahmagupta, III, 7-8; Lalla, SamanyagolabandJm, 9-10; BhSskara, Ganitadhydya, Triprasnadhikara, 12-13. 2 Cf . Lalla, Spa^iadhikara, 18; Pancasiddhantiha, IV, 23; Surya- siddhanta, II, 60; Brahmagupta, II, 56; Bhaskara, Gav-itSdhyaya, Spa§tadhikara, 48 {Vasanabha^a); Kaye, op. cii., p. 73. 3 Cf. Lalla, Triprainadhikara, 8; Brahmagupta, II, 57-58; Suryasiddhanta, III, 42-43 and note; PaficasiddhantiM, IV, 29-30; Bhaskara, Gamiadhyaya, Spa§tadhikara, 57; Kaye, op. cit., pp. 79-80. 72 ARYABHATIYA horizon of Lanka (the unmandala). When trans- formed to the plane of a great circle it becorrfes the ascensional difference.^ 27. The first and fourth quadrants of the ecliptic rise in a quarter of a day (15 ghatikds) minus the ascensional difference; The second and third quadrants rise in a quarter of a day plus the ascensional difference, with regular increase and decrease. The last phrase means that the values for signs 1, 2, 3 are equal, respectively, to those of signs 6, 5, 4 and that the values of 7, 8, 9 are equal, respectively, to those of 12, 11, 10. They increase in the first quadrant, decrease in the second, increase in the third, and decrease in the fourth. There are, there- fore, only three numerical values involved, those cal- culated for the first three signs. See the table given in Siiryasiddhdnta, III, 42-45 n.^ 28. The sine of the Sun at any given point from the horizon on its day-circle multiplied by the sine of co-latitude and divided by the radius is the sahku when any given part of the day has elapsed or remains. The sanku is the sine of the altitude of the Sun at any time on the vertical circle from the zenith pas- sing through the Sun. Cf. Brahmagupta, XXI, 63, drgmandale natdrhsajya drgjya sankur unnatarhsajyd, * Cf. Suryasiddhanla, 11, 61-63; Lalla, Spa^tadhikdra, 17, and Samanyagolabandha, 4; Brahmagupta, II, 57-60; Paricasiddhantika, IV, 26 and note; Bhaskara, Qaiyitadhydya, Spa§tadhikara, 48; Kaye, op. dt., p. 73. ^ Cf. Lalla, Madhyagativasand, 15; Bhaskara, Gardiddhydya, Spa^tddhikdra, 65 {Vdsan&bha^ya) who names Aryabhata in connec- tion with this rule. GOLA OR THE SPHERE 73 and Bhaskara, Golddhydya, Tri'prasnavdsand, 36, sankuv unnatalavajyakd hhavet} Paramesvara remarks: uttaragole gatagantavyd- suhhyas caradaldsun visodhya jivdm dddya svdhord- trdrdhena nihatya trijyayd vibhajya lahdhe hhujydm praksipet. sd ksitijdd utpannd svdhoratrestajyd bhavati. This corresponds to the so-called cheda of Brahma- gupta. 29. Multiply the given sine of altitude of the Sun by the sine of latitude (the equinoctial sine) and di%"ide by the sine of co- latitude. The result mil be the base of the sahku of the Sun south of the rising and setting line. Sankvagra is the same as sankutala {"the base of the sanku") and denotes the distance of the base of the sanku from the rising and setting line.^ 30. The sine of the greatest decUnation multiplied by the given basersine„pf„the»Sun and divided by the sine of co-latitude is the Sun's agra on the east and west horizons. The agrd is the Sun's amplitude or the sine of the degrees of difference between the day-circle and the east and west points on the horizon.^ The proportions employed are those given in Suryasiddhdnta, V, 3 n. 1 Cf. SHryasiddhanta, III, 35-39 and note; Brahmagupta, III, 25-26; BCMS, XVIII (1927), 25. 2 Cf. Brahmagupta, III, 65 and XXI, 63; Bhaskara, Goladhyaya, Triprainavasana, 40-42 (and Vdsandbhdsya) and Ganitadhyaya, Tri- pra§nMhikara, 73 (and Vasanabha§ya) ; Lalla, Triprainadhikara, 49. *See S-uryasiddhanta, III, 7 n.; Brahmagupta, XXI, 61; Bhas- kara, Goladhy&ya, Tri-praknavdsana, 39 and GavitadhySya, Tripraina- dhikara, 17 (yssanabhd§ya). 74 ARYABHATIYA 31. The measure of the Sun's amplitude north of the Equator [i.e., when the Sun is in the Northern hemisphere], if less than the sine of latitude, multipUed by the sine of co-latitude and divided by the sine of latitude gives the sine of the altitude of the Sun on the prime vertical.^ Bhau Dajl^ first pointed out that Brahmagupta (XI, 22) contains a criticism of stanzas 30-31. uttaragole 'grayam vi§uvajjyato yad uktam unayam | samamandalagas tad asat krantijyayam yato bhavati || Paramesvara remarks: visuvajjyond cet. visuvaj- jyonayd krdntya sddhitd ced ity arthah. visuvaj- jyonakrdntisiddhd sodaggatdrkdgrd. 32. The sine of the degrees by which the Sim at midday has risen above the horizon will be the sine of altitude of the Sim at midday. The stae of the degrees by which the Swa is below the zenith at midday will be the midday shadow. 33. Multiply the meridian-sine by the orient-sine and divide by the radius. The square root of the difference between the squares of this result and of the meridian-sine will be the sine of the ecliptic zenith-distance. The madhyajyd or "meridian-sine" is the sine of the zenith-distance of the meridian ecliptic point. The udayajyd or "orient-sine" is the sine of the amplitude of that point of the ecliptic which is on the horizon. The sine of the ecliptic zenith-distance of that point of the ecliptic which has the greatest altitude (nonagesimal point) is called the drkksepajyd.^ ^Cf. Suryasiddhanta, III, 25-26 n.; Brahmagupta, III, 52; Pancasiddhdntika, IV, 32-3, 35 n. 2 JRAS, 1865, p. 402. ^ Cf . Suryasiddhanta, V, 4-6; Pancasiddhdntika, IX, 19-20 and note; Lalla, Suryagrahanadhik&ra, 5-6; Kaye, op. cit., pp. 76-77; BCMS, XIX (1928), 36. , GOLA OR THE SPHERE 75 Brahmagupta (XI, 29-30) criticizes this stanza as follows : vitriblialagne dfkk^epamandalam tadapamandalaj-utau jya | madhya dykk^epajya naxj^abhatoktanaya tulya \\ drkk§epajyato 'sat taimasad avanater nasalj I avanatinasad grasasyonadhikata ra\igraha;ie. |i 34. The square root of the difference of the squares of the sines of the ecliptic zenith-distance and ofjthe zenith-distance is the sine of the echptic-altitude. *^2-'>* ^1 ^7 A kuvasat k§itije sva drk chaya bhuvj'asardham nabhomadhyat. The sine of the altitude of the nonagesimal point of the ecliptic is called the drggatijyd. Drk is equivalent to drgjyd the sine of the zenith- distance of any planet.^ This stanza is criticized by Brahmagupta (XI, drkk§epajya bahur drgjya l/arno 'nayoh krti%-isesat | miilam drgnatijiva samsthanam ayuktam etad api. 1| The construction of the second part of the stanza and the exact meaning of drk and chdyd are not clear to me. It seems to mean that when the sine of the zenith-distance is equal to the radius the greatest parallax (horizontal parallax) is equal to the radius of the Earth. Kuvasat (''because of the Earth") seems to indicate that parallax is due to the fact that we are situated on its surface and not at its center, and that parallax, therefore, is the difference between the positions of an object as seen from the center and from the surface of the Earth. 1 Cf. Suryasiddhanta, V, 6; Lalla, Suryagrahapa, 6; Paficasi- ddhardika, IX, 21 and p. 60; Bhaskara, Ga^iMdhyaya, Suryagrahaxia, II, 5-6 (and Vasanabha$ya); BCMS, XIX (1928), 36-37. 76 ARYABHATIYA Paramesvara's explanation is as follows: Dfgbhedahetubhuta svacchaya drgjya va svadrggatijya va drkk§epajya vety arthah. sa yadi ksitije bhavati nabhomadhyat k?itijanta bhavati vyasardhatulya bhavatity arthas tada kuvasad bbumivasan nifpanno drgbhedo vyasardham bhavati bhuvyasardhatulyam dj-gbhedayojanam ity arthajt. antarale 'nupatat kalpyam. Sukumar Ranjan Das^ states that there is no reference to parallax in Aryabhata. If Paramesvara is correct in interpreting the second part of the rule as giving yojanas of drghheda (parallax), we must ascribe to Aryabhata the knowledge of parallax, even though no rules are given for its calculation at intermediate points. It is hard to see what else the "radius of the Earth" can refer to when given immediately after rules for finding the drkksepajyd and the drggati- jya (cf. Brahmagupta, XXI, 64-65, and Bhaskara, Golddhydya, Grahanavdsand, 11-17), especially since parallax was well known to the old Suryasiddhdnta which antedated Aryabhata.^ It seems to me that the passage is probably to be interpreted in the light of Brahmagupta, XXI, 64-65 : drsyadfsyam drggolardham bhuvyasadalavihinayutam [ dra§ta bhugolopari yatas tato lambanavanati || k§itije bhudalaliptah kak§ayam drnnatir nabhomadhyat | avanatilipta yamyottarS ravigrahavad anyatra. | p 35. The sine of latitude multiplied by the sine of celestial latitude and divided by the sine of co-latitude is minus and plus to the Moon when it is north of the ecliptic depending on 1 "ParaUax in Hindu Astronomy," BCMS, XIX (1928), 29-42. * Cf. Pancasiddkantika, p. 60. ' See also LaUa, Madhyagativasana, 23-28. GOLA OR THE SPHERE 77 whether it is in the Eastern or Western hemisphere, plus and minus when it is south of the ecliptic under the same circum- stances. This stanza and the next give the calculation called drkkarman, an operation for determining the point on the ecliptic to which a planet ha\'ing a given latitude will be referred by a secondary to the prime vertical It has been called "operation for apparent longitude" and falls into two parts, namely, the "operation for latitude" (ahsadrkkannan) treated in this stanza and the "operation for ecliptic-de^dation" (dyanadrkkarman) treated in the following stanza.^ The stanza is criticized by Brahmagupta (XI, 34) : vikiepagunaksajya lambakabhakta grahe dhanam rnam yat | uktam udayastamayayor na pratighatikam yatas tad asat. |i Brahmagupta, X, 13-14 gives a general criticism of Aryabhata's drkkarman, followed by an exposition of his own method. 36. Multiply the versed sine (of the Moon) by the celestial latitude and by the (greatest) declination, and divide by the square of the radius. The result is minus or plus to the Moon when it is in the northern ayana depending on whether its celestial latitude is north or south, and plus or minus when it is in the southern ayana under the same conditions. Paramesvara explains utkramanam by kotyd utkramajyd. The ayanas are the northern and southern paths of the Sun from solstice to solstice.^ ^ Cf. Suryasiddhanta, VII, S-9 n.; Kaye, op. cU., pp. 78-79. * Cf. Suryasiddhdnta, VII, 10 n.; Lalla, MadhyagativasanS,, 47-48. 78 ARYABHATlYA Criticized by Brahmagupta (XI, 35) : trijyakftibhakta vik^epapakramagunotkramajyendoti | ayanante yad rnadhanam tat tasyadau tato 'sat tat. 1 1 37. The Moon consists of water, the Sun of fire, the Earth of earth, and the Earth's shadow of darkness. The Moon ob- scures the Sun and the great shadow of the Earth obscures the Moon.^ Brahmagupta (XI, 9) remarks: aryabhato janati grahagtagatim yad uktavaihs tad asat | rahukrtam na grahanam tatpato nagtamo rahuh. 1 1 There is no such statement in our text and Brahma- gupta himself (XXI, 43-48) ascribes eclipses to Rahu. 38. When at the end of the true lunar month the Moon, being near the node, enters the Sun, or when at the end of the half- month the Moon enters the shadow of the Earth that is the middle of the eclipse which occurs sometimes before and some- times after the exact end of the lunar month or half-month. Paremesvara remarks, sphutasasimdsdnte lamha- nasaihskrte 'mdvdsydntakdle. He also takes the words adhikonam as meaning "middle of the eclipse which lasts for a longer or shorter time," but gives as an alternate explanation offered by some the foregoing translation.^ 39. Multiply the distance between the Earth and the Sun by the diameter of the Earth and divide by the difference between the diameters of the Earth and the Sun. The result will be the length of the shadow of the Earth (measured) from the diameter of the Earth. * Cf. SuryaMddhanta, IV, 9; Lalla, Madhyagativasana, 29, 34. * Cf. Suryasiddhanta, TV, 6, 16; Lalla, Candragrahatia, 10; Albe- runi, II, 111. GOLA OR THE SPHERE 79 The last clause seems to indicate that the measure- ment is to be reckoned from the center of the Earth.^ 40. The difference between the length of the Earth's shadow- and the distance of the Moon from the Earth multiplied by the diameter of the Earth and di^^ded by the length of the Earth's shadow is the diameter of the Earth's shadow (in the orbit of the Moon) .=2 41. Subtract the square of the celestial latitude of the iloon from the square of half the sum (of the diameters of the Sun and Moon or of the Moon and the shadow). The square root of the remainder is known as the sthityardha. From this the time is calculated by means of the daily motions of the Sun and ^Nloon. The sthityardha is half of the time from first to last contact.^ 42. Subtract the radius of the Moon from the radius of the Earth's shadow. Subtract from the square of the remainder the square of the celestial latitude. The square root of this remainder will be the vimarddrdha. The vimarddrdha denotes half of the time of total obscuration.^ 43. Subtract the radius of the Moon from the radius of the Earth's shadow. Subtract tliis remainder from the celestial latitude. The remaiader is the part of the Moon which is not ecUpsed. 44. Subtract the given time from half of the duration of the obscuration. Add this to the square of the celestial latitude. Take 1 Cf . Brahmagupta, XXIII, 8. 2 Cf. Brahmagupta, XXIII, 9. ' Cf. Pancaaiddhdntikd, VI, 3 and X, 26-3; Swryasiddhanta, W, 12-13; Brahmagupta, IV, 8. *Cf. SuryasMdhanta, IV, 13; Pancasiddhantikd, X, 7; Brah- magupta, IV, 8. 80 ARYABHATIYA the square root. Subtract tliis from half the sum of the diameters. The remainder will be the obscuration at the given time.^ The first sentence ought to be: "Subtract the koti of the given time from the koti of the sthityardha. Square this." 45. The sine of the latitude multiplied by the sine of the hour-angle and divided by the radius is the deflection due to latitude. It is south. sthityardhac carkendos trirasisahitayanat sparse. For the difficulty of the stanza and the gap in the commentary of Paramesvara see the Preface to Kern's edition (pp. v-vi) with the references to Bhaskara. "Hour-angle" is expressed by madhydhnat krama (gunitah). "Deflection due to latitude" seems to be the meaning of dik. The first part deals with the dksavalana or "deflec- tion due to latitude." According to Paramesvara, it is south in the Eastern and north in the Western hemisphere. The other books give just the opposite. Paramesvara remarks, etad aksavalanam sthitya- rdhdc ca. sihityardhasabdena tanmulahhuto viksepa ucyate. Paramesvara also remarks, ayanasahdendpakrama ucyate. trirdsisahitdd arkdc candrdc ca nispanno 'pakramo 'pi tayor arkendor valanam bhavati. Paramesvara explains sparse as sparsa iti grahana ity evdrthatah. However the second part of the stanza is to be ^ Cf. Suryasiddhanta, IV, 18-20; Pancasiddhantika, X, 5-6; Brahmagupta, IV, 11-12. GOLA OR THE SPHERE 81 translated it must deal with the so-called dyanavalana or "deflection due to the deviation of the ecliptic from the equator." Both valanas ("deflection of the ecliptic") were employed in the projection of eclipses.^ 46. At the beginning of an eclipse the Moon is dhumra, when half obscured it is kr§na, when completely obscured it is kapila, at the middle of an eclipse it is hrsnatamra."^ 47. When the Moon eclipses the Sun even though an eighth part of the Sun is covered this is not preceptible because of the brightness of the Sun and the transparency of the Moon.' 48. The Sun has been calculated from the conjunction of the Earth and the Sun, the Moon from the conjunction of the Sun and Moon, and all the other planets from the conjunctions of the planets and the Moon.* 49. By the grace of God the precious sunken jewel of true knowledge has been rescued by me, by means of the boat of my own knowledge, from the ocean which consists of true and false knowledge.' 50. He who disparages this universally true science of astronomy, which formerly was revealed by Svayambhu, and is now described by me in this Aryabhatiya, loses his good deeds and his long life.^ Read pratikuncuko. 1 Cf. Bralunagupta, IV, 16-17 and XXI, 66; Lalla, Candroffraha- Xtadhikara, 23, 25; Suryasiddhdnta, IV, 24-25: "From the position of the eclipsed body increased by three signs calculate the degrees of declination." See Brennand, Hindu Astronomy, pp. 280-83; Kaye, Hindu Astronomy, pp. 77-78. * Cf. SHryasiddhanta, VI, 23; Lalla, Candragrahavadhikara, 36; Brahmagupta, IV, 19. ' Cf . Suryasiddhanta, VI, 13. * Cf. BCMS, XII (1920-21), 183. s Cf. iUd., p. 187. « a. JBAS, 1911, p. 114. GENERAL INDEX [Including the most important Sanskrit proper names] Alberuni, 10, 14, 15, 53, 69 Alphabet, letters of, used with numerical value, 2-9 Altitude of Sun, sine of, 72, 73 at midday, 74 on prime vertical, 74 Amplitude, of Sun, 73, 74 Anomaly, equation of, 60 Apparent longitude, 77 Apsides of planets, epicycles, 18 motion, 16-18, 52 position, 16 Apsis of Moon, 53, 54 epicycle, 18 revolutions, 9 Apsis of Sun, epicycle, 18 motion, 16-18, 52 position, 16 Area, any plane figure, 27 circle, 27 square, 21 trapezium, 27 triangle, 26 Ascensional difference, 72 Asterisms, 55, 56 half-dark, half-light, 64 revolutions, 52 stationary, xiv, 64-67 Asterisms, circle of, 55 driven by provector wind, 66 sixty times orbit of Sun, 13 surrounds Earth as center, 64 Asterisms, sphere of, half, de- picted on a sphere, 70 half, minus radius of Earth visible to men, 68-69 \-isible to Gods, half to Pretas, 69 Balabhadra, 16 Balls, pile of, with triangular base, 37 Base of sanku, 73 Base of triangle, 26, 33, 70 Base-sine of Sun, 73 Bharata battle, 12 Bhaskara, 14, 19, 27 n., 28, 55, 66, 67, 73, 76, 80 Bhattotpala, 65, 68, 69 Brahmagupta, 2, 10, 11, 12, 13, 14, 15, 17, 18, 55, 57, 58, 64, 68, 74, 75, 76, 78 Brahman, day of. See Day night of, 64 Central ecHptic-point. See Non- agesimal point Chain, in indeterminate equa- tions, 43, 45-50 Circle, area, 27 chord of one-sixth circumfer- ence, 27 construction, 30 quadrant, in constructing sines, 28 relation of circumference to diameter, 28 saras, 33 sampatasaras, 34r-35 avil day, 52, 69 Co-latitude, sme of, 70, 71, 72, 73, 74, 76 Commutation, equation of, 60 Compass, 31 83 84 ARYABHATIYA Conjunctions, • Earth and Sun, 52, SI !Moon and Sim, 52, 81 ■\Ioon and planets, 81 planets with one another number in a yuga, 51 past and future, calculated from distance apart, 41-42 Conjunctions, of planets, epi- cycles, 18 revolutions, 9 Venus and Mercury cross ecliptic at, 63 Cube, defined, 21 Cube root, 24-26 Day, 51 civil, 52, 69 sidereal, 52 Day-circle, 71, 72 radius, 71 Day of Brahman, 12 increase of Earth during, 64 measurement of, 53 part which has elapsed, 12 Day-radius, 71 Declination, 71 Declination, greatest : of ecliptic, 16 day-radius, 71 sine of, 73, 77 Deflection, in eclipses, 80-81 Degrees, 13, 56 Deviation, of ecliptic from Equa- tor, 63 of Moon and planets from ecliptic, 16 Diameter, of circle, relation to circumference, 28 of Earth, Sun, Moon, and planets, 15 Earth, compared to round Ka- damba flower, 64 conjunction with Sun, 52, 81 constitution, 64, 78 diameter, 15 half dark, half light, 64 increase and decrease in size, 64 located in center of space, 56, 64 moves one minute in a prdna, 13 revolutions eastward, 9 rotation, 9, 14, 65-66 shadow. See Shadow simile of man moving in a boat, 64 Earth-sine, 71 Earth--wind, 18 East and west hour-circle, 69 Eccentric circle, equal in size to orbit, 58 location of center, 58 movement of planet on, 57 Eclipses, 78-81 causes and time, 78 color of Moon, 81 deflection, 80-81 half-duration, 79 middle, 78 obscuration at given time, 79-80 of Sun not perceptible if less than one-eighth obsciured, 81 part of Moon not eclipsed, 79 shadow of Earth, 78-79 total obscuration, 79 Ecliptic, deviates equally from Equator, 63 deviation of Moon and plan- ets from, 16 greatest declination, 16 northern and southern halves, 63 quadrants, 72 Sun, nodes of Moon and plan- ets, and shadow of Earth move along, 63 GENERAL INDEX 85 Ecliptic-altitude, sine of, 75 EcUptic-deviation, 77 Ecliptic zenith-distance, sine of, 74,75 Epicycles, dimensions, 18, 58 mean planets at centers, 59 movement of, 59 planets on, 58, 59, 61 number of revolutions, 51 Equator, celestial, 63 Equator, terrestrial. See Lanka Equinoctial shadow, 70 Equinoctial sine, 70, 73 Factors, problems relating to two, 38 Fathers, dwell in Moon, 69 see Sun for half a month, 69 year of, 53 Fractions, 40 Gnomon, 31-33, 71 Gods, dwell on Meru, 69 see northern half of stellar sphere, 69 see Sun for half a year, 69 think dwellers in Hell are be- neath them, 68 year of, 53 Half-duration, of eclipse, 79 Heaven, at center of land, 68 Hell, at center of ocean, 68 Himavat Mountains, 68 Horizon, 69 Horizon of Lanka, 69, 72 Horizontal, how determined, 30 Hour-angle, 80 Hypotenxise, of a planet, 61 relation to side of right-angle triangle, 31, 33 Indeterminate equations of first degree Qeuttaha), xiv, 42-60 Intercalarj' months, 52 Interest, 3S-39 Inverse method, 40 Jupiter. (See Planets years of, 51 Kusumapura, 21 Lalla, 10, 14, 15, IS, 19, 59, 66 Lanka (terrestrial Equator), 9. 64, 66, 6S horizon of, 69, 72 90 degrees from poles, 68 Latitude, celestial, 76, 77, 79 Latitude, sine of, 70, 71, 73, 74, 76,80 Long letter, as measure of time, 51 Longitude, apparent, 77 Lunar days, omitted, 53 Lunar month, 52, 69 Mahasiddhanta, xvi Manu, period of, 12 Marici, 66 Mars. See Planets Mean planet, 57, 58, 59, 60 Mean motion, of planets, 57, 58 Mercury. See Planets Meridian, 69 Meridian-sine, 74 Meru, at center of land, 68 description, 68 dimensions, 15, 68 home of the gods, 68 Midday shadow, 74 Midnight school, 11 n. Minutes, 13, 56 Month, 51 intercalary, 52 lunar, 52, 69 Moon, calculated from conjunc- tion of Sun and Moon, 81 86 ARYABHATIYA causes eclipse of Sun, 78 crosses ecliptic at node, 63 diameter, 15 distance from Sun at which \'isible, 63 eclipse. See Eclipses epicycle of apsis, 18 greatest deviation from eclip- tic, 16 half dark, half light, 64 home of the Fathers, 69 Lord of Monday, 56-57 made of water, 78 nearest to Earth, 56 node moves along ecliptic, 63 revolutions, 9 of apsis, 9 of node, 9 true place, calculation of, 59 Munisvara, xv Nandana forest, 68 Night of Brahman, decrease of Earth during, 64 Node of Moon, moves along ecliptic, 63 revolutions, 9 Nodes of planets Jupiter, Mars, and Saturn cross ecliptic at nodes, 63 move along ecliptic, 16-18, 63 position of ascending nodes, 16 Nonagesimal point, 70 altitude, 75 zenith-distanee, 74 Numbers, classes of, enumerated by powers of ten, 21 Oblique ascension, equivalents in, of quadrants of ecliptic, 72 Omitted lunar days, 53 Orbits, of planets, in yojanas, 13 movement of planets on, 57 surroimd Earth as center, 56, 64 Orient-siae, 74 Pancasiddhantika, 10, 12 n 18 19, 64, 65 Parallax, 75-76 Perpendicular, how determined 30 Perpendiculars, from intersection of diagonals of trapezium, 27 Plane figure, area of any, 27 Planets, as lords of the days. 56-57 calculated from conjunctions with the Moon, 81 conjunctions past and future calculated from distance apart, 41 number in a yuga, 51 cross ecliptic at nodes or con- junctions, 63 diameters, 15 distance from Earth at which visible, 63-64 driven by provector wind, 66 epicycles, 18, 51, 58-59, 61 greatest deviation from eclip- tic, 16 half dark, half light, 64 mean, 57, 58, 59, 60 motion, 57, 68 move with equal speed, 55 movement of apsides, 16-18, 52 on orbits and eccentric cir- cles, 57 nodes move along ecliptic, 63 orbits of, in yojanas, 13 periods of revolution differ because orbits differ in size, 56 positions of apsides and as- cending nodes, 16 relative position with refer- ence to Earth as center, 56 revolutions, 9 of conjunctions, 9 time in which they traverse distance equal to circle of asterisms and of sky, 55 true distance from Earth, 61 GENERAL INDEX 87 true places, calculation of, 59-61 yugdoi, 53 Plumb-line, 30 Pretas, dwell at Vadavamukha, 69 see southern half of stellar sphere, 69 see Sun for half a year, 69 Prime vertical, 69 altitude of Sun on, 74 Progression, arithmetical, num- ber of terms, 36 sum, 35-36 of any number of terms taken any^vhere within, 35-36 of series made by taking sum of terms, 37 sums of series made by taking squares and cubes of terms, 37-38 Proportion, 39 Provector wind, 66 Prthudaka, quoted, 66 Pjoramid, volume, 26 Quadrilateral, construction, 30 formed in quadrant of circle, 28 Radius, 61, 72, 74, 80 equals chord of one-sixth cir- cmaoference, 27 square of, 71, 77 Reciprocal division, 42-46, 48- 49 Revolutions, of apsis of Moon, 9 of asterisms are sidereal days, 52 of conjunctions of planets, 9 of Earth eastward, 9 of epicycles, 51 of node of Moon, 9 of Sun, are solar years, 52 Moon, and planets, 9 in Tjvga equal years of yuga, 15 time and place from which calculated, 9 Right ascension, equivalents in, of signs of zodiac, 71 Romaka, 68 Rule of three, 39 Saturn. See Planets farthest from Earth, 56 Series, made by taking sum of terms of arithmetical progres- sion, 37 made by taking squares and cubes of terms of arith- metical progression, 37-38 Shadow, midday, 74 Shadow of Earth, causes eclipse of Moon, 78 diameter of, in orbit of Moon, 79 length, 78 made of darkness, 78 moves along ecliptic, 63 Shadow of gnomon, 31-33 Sidereal day, 52 Sidereal mid4ika, 51 Siddhapura, 68 Signs of zodiac, 13, 56 day-circle of, 71 right ascension, equivalents in, 71 sine of, 71 Sine of Sun, 72 Sines, construction of, on radius of quadrant, 28 table of sine-differences, 19 calculation of, 29 Sky, circumference of, 13, 14, 55 Solid with twelve edges, 21 Solai year, 52, 53, 69 Space, measurement of, 51 Sphere, volume, 27 88 ARYABHATIYA wooden, made to revolve, 70 half of stellar sphere de- picted on, 70 Square, defined, 21 Square root, 22-24 Sripati, quoted, 66 Sun, amplitude, 73, 74 apsis, position of, 16 base-sme, 73 calculated from conjunction of Earth and Sun, 81 diameter, 15 eclipse. iSee Eclipses epicycle of apsis, 18 Lord of Sunday, 56—67 made of fire, 78 moves along ecUiptic, 63 orbit is one-sixtieth circle of asterisms, 13 relative position among plan- ets, 56 revolutions, 9, 52 in yiiga equal years of yuga, 15 sine of, 72 altitude. See Altitude true place, calculation of, 59 visible to gods and Pretas for half a year, 69 to Fathers for half a month, 69 to men for haK a day, 69 Sunrise school, 11 n. Suryasiddh&nta, 10, 12, 14, 18, 19, 57, 64, 72, 73 Three, nile of, 39 Time, beginningless and end- less, 55 measurement, 51 Total obscuration, in eclipse, 79 Trapezium, area, 27 perpendicular from intersec- tion of diagonals, 27 Triangle, area, 26 construction, 30 formed in quadrant of circle, 28 hypotenuse of right-angle, 31, 33 True places of planets, 59-61 Ujjain, 22^ degrees north of Lanka, 68 Vadavamukha, 68, 69 Vdsandvarttika, 66, 67 Venus. See Planets Versed sine, of Moon, 77 Volume, pyramid, 26 solid with twelve edges, 21 sphere, 27 Yavakoti, 68 Year, 51 of gods, 53 of the Fathers, 53 of men, 53 solar, 53, 69 Zenith-distance, 75 Zero, 6-7 Zodiacal signs. See Signs SANSKRIT INDEX Agra, 43, 45-46, 49 Agra, 73 Aghana, 24-26 Angula, 16 Adhikonam, 79 Anuloma, 58, 65 Antyavarga, 2, 6 Ay ana, 51, 77 Avarga, letters and places, 2-5 in square root, 22-24 Avasarjiixfl,, 53 ^sra, 21 n., 26 n. Akvayuja, 51 Akasakaksya, 14, 55 Ak§adrkkarman, 77 Ak§avalana, 80 Ayqnadrkkarman, 77 Ayanavalana, 81 Induccdt, 54 CTcco, 51, 52, 58 Utkramav,am, 77 Utsarpizil, 53 Udayajyd, 74 Unmai^dala, 69, 72 Upaciti, 37 Kak$yS, 57 Kak§yamari4ala, 57, 58 Kanya, 63 Kam, 1 Karkata, 31 iTarna, 30, 33, 61, 62 ITdZpa, 12 Kuiiaka, xiv Kuvaiat, 75 iToii, 70, 71 iCofi, 32, 33 iiT/ia. 7 Kkadvinavaka, 7 Khavrtta, 31 Gunakarabhdgahdra, 40 GoZa, 70 Grahajava, 13, 62 Grahavega, 62 Grdsa, 34 Ghatikd, 72 Ghana, defioed, 21 in cube root, 24-26 Cakro, 70 Caiurbhuja, 30 Caturyuga, 55 Citighana, 37 Caitra, 55 Chat/a, 75 C/ierfa, 73 Tuia, 60 Taulya, 63 Tribhuja, 30 Difc, 80 Du$§aina, 53-54 DrA;, 75 Drkkarman, 77 Dfkkqepajya, 74, 76 DTrkk^epamav-^la, 70 Drggatijya, 75, 76 Drgjyd, 75 Drgbheda, 76 Drhmazidala, 70 Dvdda§d^a, 21 Dvicckedagra, 43 89 90 ARYABHATlYA A'oi^I, 51 iVr, measurement of, 16 number in a yojana, 15 Pankti, 22 ParibhcLia, S Purva, 24, 26 Purvapak§a, 65, 67 PTaiimai}.iala, 57 Pratiloma, 58 Prar}.a, 13 number in a vina^ikd, 51 Bham, 14 Bhuja, 31-33 Madhyajya, 74 Madhyahnat krama, 80 Mando, 60, 61 Mandakarzia, 61 Mandagati, 59 Mandaphala, 59-60 Mandocca, 52, 58, 60 Mithydjnana, 9, 14, 65, 67 Mma, 63 Mesa, 9, 11, 16, 60, 63, 71 -3^*,"Feginmng of, 9, 55 at midnight, 11 n. at sunrise, 9, 11 n. division into four equal parts, 12 measurement of, 53 names for parts of, 53 number in period of a Manu, 12 revolutions of Earthj Sim, Moon, and planets m, 9 years of, equal revolutions of Sun, 15 Yitgapada, 12, 54 Yoga, of Sun and Earth, 52, 81 Yojana, measure of increase and PKIKTED IN U-SA- decrease of Earth, 64 measiurement of, 15 Yojanas, in drcumferencfe of sky, in planet's orbit, 13 of parallax, 76 same number traversed by each planet in a day, 55 RaH, 13 Varga, defined, 21 in square root, 22-24 letters and places, 2-7 Valana, 80-81 Vina,4ika, 51 Vimardardha, 79 Viloma, 65 Vyatipata, 51 Vyastam, 60 Sanhu, 71, 72, 73 Sankutala, 73 Sankvagra, 73 Sara, 33 iSlghrakarria, 61 Slghragaii, 58 Sighraphala, 59-60 Sighrocca, 52, 58 Satya devaia, 1 Samadalakotl, 26 Sampdta&ara, 34-35 Svsama, 53-54 Sth&nantare, 22 Sthityardka, 79 Sphuta, 60, 61 Sphutamadhya, 60 Svayamhhu, 1, 81 Svavrtta, 31 Hasta, 16 I