# Āryabhaṭīya of Āryabhaṭa

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THE ĀRYABHATĪYA
of
ĀRYABHATA
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
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COPYRIGHT 1930 BY THE THE UNIVERSITY OF CHICAGO
ALL RIGHTS RESERVED. PUBLISHED JULY 1930
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PREFACE
In 1874 Kern published at Leiden a text called the Āryabhatīya which claims to be the work of Āryabhata, and which gives (III, 10) the date of the birth of the author as 476 a.d. If these claims can be substantiated, 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 astronomical text from the third or scientific period of Indian astronomy. The only other text which might dispute this last claim is the Sūryasiddhānta (translated with elaborate notes by Burgess and Whitney in the sixth volume of the Journal of the American Oriental Society). The old Sūryasiddhānta undoubtedly preceded Āryabhata, but the abstracts from it given early in the sixth century by Varāhamihira in his Pañcasiddhāntikā show that the preserved text has undergone considerable revision and may be later than Āryabhata. Of the old Paulisa and Romaka Siddhāntās and of the transitional Vāsistha Siddhānta, nothing has been preserved except the short abstracts given by Varāhamihira. The names of several astronomers who preceded Āryabhata, or who were his contemporaries, are known, but nothing has been preserved from their writings except a few brief fragments.
The Āryabhatīya, therefore, is of the greatest im​portance in the history of Indian mathematics and astronomy. The second section, which deals with mathematics (the Ganitapāda), 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 stanzas have been discussed. The aim of this work is to give a complete translation of the Āryabhātīya with references to some of the most important parallel passages which may be of assistance for further study. The edition of Kern makes no pretense of giving a really critical text of the Āryabhātīya. It gives merely the text which the sixteenth-century commentator Parameśvara had before him. There are several uncertainties about this text. Especially noteworthy is the considerable gap after IV, 44, which is discussed by Kern (pp. v-vi). The names of other commentators 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 Āryabhātīya with the abstracts from the old siddhāntas given by Varāhamihira, with the Sūryasiddhānta, with the Śisyadhīvrddhida of Lalla, and with the Brāhmasphutasiddhānta and the Khandakhādyaka of Brahmagupta should be made. All the later quotations from Āryabhata, especially those made by the commentators on Brahmagupta and Bhāskara, should be collected and verified. Some of those noted by Colebrooke do not seem to fit the published Ārya​bhatīya. If so, were they based on a lost work of Āryabhata, on the work of another Āryabhata, or were they based on later texts composed by followers of Āryabhata rather than on a work by Āryabhata himself? Especially valuable would be a careful study of Prthūdakasvāmin or Caturvedācārya, the eleventh- century commentator on Brahmagupta, who, to judge from Sudhākara’s use of him in his edition of the Brāhmasphutasiddhānta, frequently disagrees with Brahmagupta and upholds Āryabhata against Brahmagupta’s criticisms.
The present translation, with its brief notes, makes no pretense at completeness. It is a preliminary study based on inadequate material. Of several passages no translation has been given or only a tentative 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 publish 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 encouraging them to make available more adequate manuscript material.
There has been much discussion as to whether the name of the author should be spelled Āryabhata or Āryabhatta. Bhata means “hireling,” “mercenary,” ​"warrior," and bhatta means "learned man," "scholar." Āryabhatta is the spelling which would naturally 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 bhata has been substituted for bhatta for purely metrical reasons, and does not prove that Āryabhata is the correct spelling. It is pointed out that Kern gives the name of the commentator whom he edited as Paramādīsvara. 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 Parameśvara, and this doubtless is the correct form. However, until more definite historical or metrical evidence favoring the spelling Āryabhatta is produced I prefer to keep the form Āryabhata.
The Āryabhatīya is divided into four sections which contain in all only 123 stanzas. It is not a complete and detailed working manual of mathematics and astronomy. It seems rather to be a brief descriptive work intended to supplement matters and processes which were generally known and agreed upon, to give only the most distinctive features of Āryabhata's own system. Many commonplaces and many simple processes are taken for granted. For instance, there are no rules to indicate the method of calculating 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 corrections, although even here there is no statement of ​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 karanagranthas or the Sūryasiddhānta in its present form. It is questionable whether Āryabhata himself composed another treatise, a karanagrantha 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 Āryabhata, one giving three hundred sāvana 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 Āryabhatas. This is corroborated by Pañcasiddhāntikā, XV, 20: "Āryabhata maintains that the beginning of the day is to be reckoned from midnight at Lankā; and the same teacher [sa eva] again says that the day begins from sunrise at Lankā." Brahmagupta, however, names only the Daśagītika and the Āryāstaśata as the works of Āryabhata, and these constitute our Āryabhatīya. But the word audayikatantra of Brāhmasphutasiddhānta, XI, 21 and the words audayika and ārdharātrika of XI, 13-14 seem to imply that Brahmagupta is distinguishing between two works of one Aryabhata. The published Āryabhatīya (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 Āryabhatas. His later ​Āryabhata (of Kusumapura) cannot be the later Āryabhata who was the author of the Mahāsiddhānta. The many quotations given by Alberuni prove conclusively that his second Āryabhata was identical with the author of our Āryabhatīya (of Kusumapura as stated at II, 1). Either there was a still earlier Āryabhata or Alberuni mistakenly treats the author of our Āryabhatīya 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 published 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 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 Daśagītikasūtra 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 paribhāsā stanza, which explains the peculiar alphabetical notation which is to be employed 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 paribhāsā stanza than for questioning that of the invocation and colophon. Kaye ​would like to eliminate it since it seems to furnish evidence for Āryabhata's knowledge of place-value. Nothing is gained by doing so since Lalla gives in numerical words the most important numerical elements of Āryabhata without change, and even without this paribhāsā stanza the rationale of the alphabetical notation in general could be worked out and just as satisfactory evidence of place-value furnished. Further, Brahmagupta (Brāhmasphutasiddhānta, XI, 8) names the Dasagītika as the work of Āryabhata, gives direct quotations (XI, 5; I, 12 and XI, 4; XI, 17) of stanzas 1, 3, and 4 of our Dasagītika, and XI, 15 (although corrupt) almost certainly contains a quotation of stanza 5 of our Dasagītika. Other stanzas are clearly referred to but without direct quotations. Most of the Dasagītika 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 Kālakriyā, or "The Reckoning of Time." The fourth section in 50 stanzas is called Gola, or "The Sphere." Together they contain 108 stanzas.
The Brāhmasphutasiddhānta of Brahmagupta was composed in 628 A.D., just 129 years after the Āryabhatīya, 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 Brāhmasphutasiddhānta, which is called "Tantraparīksā," and is devoted to severe criticism of previous works on astronomy, is chiefly devoted to criticism of Āryabhata. In this chapter, and in other parts of his work, ​Brahmagupta refers to Āryabhaṭa some sixty times. Most of these passages contain very general criticism of Āryabhaṭa as departing from smṛti or being ignorant of astronomy, but for some 30 stanzas it can be shown that the identical stanzas or stanzas of identical content were known to Brahmagupta and ascribed to Āryabhaṭa. In XI, 8 Brahmagupta names the Aryāṣṭaśata as the work of Āryabhaṭa, and XI, 43, jānāty ekam api yato nāryabhaṭo gaṇitakālagolānām, seems to refer to the three sections of our Aryāṣṭaśata. These three sections contain exactly 108 stanzas. No stanza from the section on mathematics has been quoted or criticized by Brahmagupta, 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 Aryāṣṭaśata there must have been in the work of Āryabhaṭa 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 sections. This seems to be most unlikely. I doubt the validity of Kaye's contention that the Gaṇitapāda was later than Brahmagupta. His suggestion that it is by the later Āryabhaṭa who was the author of the Mahāsiddhānta (published in the "Benares Sanskrit ​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 Āryahhaṭīya 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 bhūḥ, as preserved in a quotation of Brahmagupta, has been changed by Parameśvara or by some preceding commentator to bham in order to eliminate Āryabhaṭa's theory of the rotation of the Earth.
Brahmagupta criticizes some astronomical matters in which Āryabhaṭa is wrong or in regard to which Āryabhaṭa's method differs from his own, but his bitterest and most frequent criticisms are directed against points in which Āryabhaṭa was an innovator and differed from smṛti 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 Brahmagupta does not give rules for the volume of a pyramid and for the volume of a sphere, which are both given incorrectly by Āryabhaṭa (II, 6-7) . This is as likely to prove ignorance of the true values on Brahmagupta's part as lateness of the rules of Āryabhaṭa. What other rules of theGaṇitapāda could be open to adverse criticism? On the positive side may be pointed out the very close correspondence in termi​nology and expression between the fuller text of Brahmagupta, XVIII, 3-5 and the more enigmatical text of Āryabhaṭīya, II, 32-33, in their statements of the famous Indian method (kuttaka) of solving indeterminate equations of the first degree. It seems probable 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 Āryabhaṭīya is the following. Indian astronomy, in general, maintains that the Earth is stationary and that the heavenly bodies revolve about it, but there is evidence in the Āryabhaṭīya itself and in the accounts of Āryabhaṭa given by later writers to prove that Āryabhaṭa 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 Parameśvara, 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 Bhaṭṭotpala, the Vāsanāvārttika, and the Marīci indicate that this stanza was known in its present form from the eleventh century on. Is it capable of some different interpretation? Is it intended merely as a statement of the ​popular view? Has its wording been changed as has been done with I, 4? I see at present no satisfactory solution of the problem.
Colebrooke gives caturviṁṡaty aṁṡaiṡ cakram ubhayato gacchet as a quotation by Munīśvara from the Āryāṣṭaśata of Āryabhaṭa. This would indicate a knowledge of a libration of the equinoxes. No such statement is found in our Āryāṣṭaśata . 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 Marīci which has already been published in the Pandit.
The following problem also needs elucidation. Although Brahmagupta (XI, 43-44)
jānāty ekam api yato nāryabhaṭo gaṇitakālagolānām |
na mayā proktāni tataḥ pṛthak pṛthag dūṣaṇāny eṣām ||
āryabhaṭadūṣaṇānām saṁkhyā vaktuṁ na śakyate yasmāt |
tasmād ayam uddeśo buddhimatānyāni yojyāni ||
sums up his criticism of Aryabhata in the severest possible way, yet at the beginning of his Khaṇḍakhādyaka, a karaṇagrantha which has recently been edited by Babua Misra Jyotishacharyya (University of Calcutta, 1925), we find the statement vakṣyāmi Khaṇḍakhādyakam ācāryāryabhaṭatulyaphalam. It is curious that Brahmagupta in his Khaṇḍakhādyaka should use such respectful language and should follow the authority of an author who was damned so unmercifully by him in Tantraparīkṣā of his Brāhmasphuṭasiddhānta. Moreover, the elements of the Khaṇ​ḍakhādyaka seem to differ much from those of the Āryabhaṭīya. Is this to be taken as an indication that Brahmagupta here is following an older and a different Āryabhaṭa? If so the Brāhmasphuaṭasiddhānta gives no clear indication of the fact. Or is he following another work by the same Āryabhaṭa? According to Dīkṣit, the Khaṇḍakhādyaka agrees in all essentials with the old form of the Sūryasiddhānta rather than with the Brāhmasphuaṭasiddhānta. Just as Brahmagupta composed two different works so Āryabhaṭa 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 Sūryasiddhānta, or have formed a commentary upon it.
The Mahāsiddhānta of another Āryabhaṭa who lived in the tenth century or later declares (XIII, 14):
vṛddhāryabhaṭaproktāt siddhāntad yan mahākālāt | pāṭhair gatam ucchedaṁ viśeṣitam tan mayā svoktyā ||
But this Mahāsiddhānta differs in so many particulars from the Āryabhaṭīyathat it is difficult to believe that the author of the Āryabhaṭīya can be the one referred to as Vṛddhāryabhaṭa unless he had composed another work which differed in many particulars from the Āryabhaṭīya. The matter needs careful investigation.
​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,
Harvard University
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
CHAPTER I
DASAGlTIKA OR THE TEN
GITI STANZAS
A. Having paid reverence to Brahman, who is one (in causality, as the creator of the universe, but) many (in his manifestations), the true deity, the Supreme Spirit, Aryabhata sets forth three things: mathematics [ganita], the reckoning of time [kalakriyaa], and the sphere [gola].
Baidyanath suggests that satya devata may denote Sarasvati, the goddess of learning. For this I can find no support, and therefore follow the commentator Paramesvara in translating "the true deity," God in the highest sense of the word, as referring to Prajapati, Pitamaha, Svayambhu, the lower individuahzed 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 manifestation of the latter. As Paramesvara remarks, the use of the word kam seems to indicate that Aryabhata based his work on the old Pitamahasiddhanta. Support for this view is found in the concluding stanza of our text (IV, 50), dryabhatiyam namna purvam svayambhuvam sada sad yat. However, as shown by Thibaut and Kharegat, there is a close connection between Aryabhata and the old Suryasiddhanta. At ​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 Brahmagupta (XII, 43), janaty ekam api yato naryabhato ganitakalagolanam, seems to refer to the Ganitapada, the Kalakriyapada, and the Golapada of our Aryabhatiya (see also Bibhutibhusan Datta). Since Brahmagupta (XI, 8) names the Dasagitika and the Aryastasata (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 Dasagitika 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. Navantyavarge va.
Aryabhata's system of expressing numbers by means of letters has been discussed by Whish, by ​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 Prthivir 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 statement 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, ​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 k 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 k to m to the second varga place, the place of hundred's, multiplies 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 k 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 virama 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.
​The last clause, which has been left untranslated, offers great difficulty. The commentator Paramesvara takes it as affording a method of expressing still higher numbers by attaching anusvara 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 interpretation seems to be impossible. Fleet suggests as an alternate interpretation the emendation of va 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), "(separement) ou a un groupe termini par un varga." That is to say, the clause has nothing to do with the expression 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 consonants as gra, in which latter case it is to be understood as applying to each consonant in the group. So giri or gri and guru or gru. Such, indeed, is Aryabhata's usage, and such a statement is really necessary 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 ​interpretation 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 va to have the sense of "beliebig," "fakultativ," and for nava to be separated 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 navantyavarge 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 beginning 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 translate, "The vowels (are to be used) in two nine's of places, nine in varga places and nine in avarga places," but antyavarge va 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 ​add two zero's to the numerical value of the consonant. 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 khadvinavake 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 symbol 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 certain to me that Aryabhata could write a number in signs which had no absolutely fixed values in themselves but which had value depending on the places occupied by them (mounting by powers of 10). Compare II, 2, where in giving the names of classes of numbers he uses the expression sthanat sthanam dasagunam syat, "from place to place each is ten times the preceding."
There is nothing to prove that the actual calculation was made by means of these letters. It is probable 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, ​unwieldy numbers in verse in a very brief form. The alphabetical notation is employed only in the Dasagitika. 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 in a yuga (I, 1), which is expressed by the word. cayagiyinusuchlr. 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 notation was not based on a place-value notation.
This stanza, as being a technical paribhasa stanza which indicates the system of notation employed in the Dasagitika, 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 Dasagitika (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 ​comparable to the one referred to by Bhau Daji as having two introductory stanzas "evidently an after-addition, and not in the Arya metre."
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 (mithyajnana) , 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.
​Here and elsewhere in the Daśagītika words are used in their stem form without declensional endings.
Lalla (Madhyamādhikāra, 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 Sūryasiddhānta (I, 29-34) shows slight variations (see Pañcasiddhāntikā, pp. xviii-xix, and Kharegat for the closer relationship of Āryabhaṭa to the old Sūryasiddhānta).
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 accurate 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 according to Āryabhaṭa. 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)
akṛtāryabhaṭaḥ śīghragam indūccaṁ pātam alpagaṁ svagateḥ । tithyantagrahaṇānāṁ ghuṇākṣaraṁ tasya saṁvādaḥ ॥
criticizes the numbers given by Āryabhaṭa for the revolutions of the apsis and node of the Moon.
​Brahmagupta (II, 467) remarks that according to Aryabhata all the planets were not at the first point of Meṣa 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 :
āryāṣṭaśate pātā bhramanti daśagītike sthirāḥ pātāḥ । muktvendupātam apamaṇḍale bhramanti sthirā nātaḥ. ॥
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:
yugaravibhagaṇāḥ khyughriti yat proktam tat tayor yugaṁ spaśṭaṁ । triśatī ravyudāyānaṁ tadantaraṁ hetunā kena. ॥
​3. There are 14 Manus in a day of Brahman [a kalpa], and 72 yugas constitute the period of a Manu. Since the beginning of this kalpa up to the Thursday of the Bhārata battle 6 Manus, 27 yugas, and 3 yugapādas have elapsed.
The word yugapāda seems to indicate that Āryabhața 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 Āryabhața the division of the yuga into four equal parts. For the traditional division see Sūryasiddhānta (I, 18–20, 22–23) and Brahmagupta (I, 7–8). For discussion of this and the supposed divisions of Āryabhața see Fleet. Compare III, 10, which gives data for the calculation of the date of the composition of Āryabhața's treatise. It is clear that the fixed point was the beginning of Āryabhața's fourth yugapāda (the later Kaliyuga) at the time of the great Bharata battle in 3102 b.c.
Compare Brahmagupta (I, 9)
yugapādān āryabhatas 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 ।।
​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.
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.
^ For jrrana see III, 2. ​Page:Aryabhatiya of Aryabhata, English translation.djvu/44 ​Page:Aryabhatiya of Aryabhata, English translation.djvu/45 ​Page:Aryabhatiya of Aryabhata, English translation.djvu/46 ​Page:Aryabhatiya of Aryabhata, English translation.djvu/47 ​Page:Aryabhatiya of Aryabhata, English translation.djvu/48 ​Page:Aryabhatiya of Aryabhata, English translation.djvu/49 ​Page:Aryabhatiya of Aryabhata, English translation.djvu/50 ​Page:Aryabhatiya of Aryabhata, English translation.djvu/51 ​Page:Aryabhatiya of Aryabhata, English translation.djvu/52 ​Page:Aryabhatiya of Aryabhata, English translation.djvu/53 ​Page:Aryabhatiya of Aryabhata, English translation.djvu/54 ​Page:Aryabhatiya of Aryabhata, English translation.djvu/55 ​Page:Aryabhatiya of Aryabhata, English translation.djvu/56 ​Page:Aryabhatiya of Aryabhata, English translation.djvu/57 ​Page:Aryabhatiya of Aryabhata, English translation.djvu/58 ​Page:Aryabhatiya of Aryabhata, English translation.djvu/59 ​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.
Tribhuja denotes triangle in general and caturbhuja denotes quadrilateral in general. The word karna regularly denotes the hypotenuse of a right angle triangle and the diagonal of a square or rectangle. I am not sure whether the restricted sense of karna limits tribhuja and caturbhuja to the right-angle triangle and to the square and rectangle or whether the general sense of tribhuja and caturbhuja generalizes the meaning of karna to that of one chosen side of a triangle and to that of the diagonal of any quadrilateral. At any rate, the context shows that the rule deals with the actual construction of plane figures.
Paramesvara interprets it as referring to the construction 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 position that their tips join. The quadrilateral is made by constructing two triangles, one on each side of the diagonal.
​Page:Aryabhatiya of Aryabhata, English translation.djvu/61 ​Page:Aryabhatiya of Aryabhata, English translation.djvu/62 ​Page:Aryabhatiya of Aryabhata, English translation.djvu/63 ​Page:Aryabhatiya of Aryabhata, English translation.djvu/64 ​Page:Aryabhatiya of Aryabhata, English translation.djvu/65 ​Page:Aryabhatiya of Aryabhata, English translation.djvu/66 ​Page:Aryabhatiya of Aryabhata, English translation.djvu/67 ​Page:Aryabhatiya of Aryabhata, English translation.djvu/68 ​Page:Aryabhatiya of Aryabhata, English translation.djvu/69 ​Page:Aryabhatiya of Aryabhata, English translation.djvu/70 ​Page:Aryabhatiya of Aryabhata, English translation.djvu/71 ​Page:Aryabhatiya of Aryabhata, English translation.djvu/72 ​Page:Aryabhatiya of Aryabhata, English translation.djvu/73 ​Page:Aryabhatiya of Aryabhata, English translation.djvu/74 ​Page:Aryabhatiya of Aryabhata, English translation.djvu/75 ​Page:Aryabhatiya of Aryabhata, English translation.djvu/76 ​Page:Aryabhatiya of Aryabhata, English translation.djvu/77 ​Page:Aryabhatiya of Aryabhata, English translation.djvu/78 ​Page:Aryabhatiya of Aryabhata, English translation.djvu/79 ​agrantare ksiptam. In the first method it is necessary to supply much to fill out the meaning, but the translation 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. Ganguly finds justification for this interpretation (p. 172) in his formulas, but I cannot help feeling that the Sanskrit 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 applied it.
CHAPTER III
KALAKRIYA OR THE RECKONING
OF TIME
This is a yoga of the Sun and Moon when they are in different ayanas, have the same declination, and the sum of their longitudes is 180 degrees.
The number of revolutions of Jupiter multiplied by 12 are the years of Jupiter beginning with Asvayuja.
​The word ucca refers both to mandocca ("apsis") and sighrocca ("conjunction").
Parameśvara explains that the number of revolutions 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 Āryabhața as being stationary in the absolute sense. They were regarded by him as stationary for purposes of calculation at the time when his treatise was composed since their movements were very slow.
The word yoga applied to the Sun and the Earth (instead of bhagaņa or āvarta) seems clearly to indicate that Āryabhața believed in a rotation of the Earth (see IV, 48). Parameśvara's explanation, ravibhūyogaśahdena raver bhūparibhramaņam abhihitam, seems to be impossible.
Quoted by Bhattotpala, page 324.
Referred to by Alberuni, I, 330.
The unmandala is the east and west hour-circle which passes through the poles. It is also called "the horizon of Lanka."
​ 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.
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.
Sukumar Ranjan Das remarks that two instruments are named in this stanza (the gola and the cakra). I can see no reference to the cakra.
The sine of the distance between the Sun and the zenith at midday of the equinoctial day is the equinoctial sine. This is the same as the equinoctial shadow and equals the sine of latitude. It is the base.
​The sine of co-latitude is the koți (the side perpendicular 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 dadting: 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 Mega.
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.
   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 ​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,
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.
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. ​Page:Aryabhatiya of Aryabhata, English translation.djvu/106 ​Page:Aryabhatiya of Aryabhata, English translation.djvu/107 ​Page:Aryabhatiya of Aryabhata, English translation.djvu/108 ​Page:Aryabhatiya of Aryabhata, English translation.djvu/109 ​Page:Aryabhatiya of Aryabhata, English translation.djvu/110 ​translated it must deal with the so-called ayanavalana 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.
Read pratikuncuko.
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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
* ↑ See especially Journal of the Royal Asiatic Society, 1865, pp. 392-93; Journal asiatique (1880), II, 473-81; Sudhākara Dvivedī, Ganakataranginī, p. 2.
* ↑ Brāhmasphutasiddhānta, XI, 5 and 13-14.
* ↑ For a discussion of the whole problem of the two or three Āryabhatas see Kaye, Bibliotheca mathematica, X, 289, and Bibhutibhusan Datta, Bulletin of the Calcutta Mathematical Society, XVII (1926), 59
* ↑ Op. cit., X, 291-92
* ↑ For criticism of Kaye see Bibhutibhusan Datta, op. cit., XVIII (1927), 5.
* ↑ Miscellaneous Essays, II, 378.
* ↑ Cf. Pañcasiddhāntikā, p. xx, and Bulletin of the Calcutta Mathematical Society, XVII (1926), 69.
* ↑ As reported by Thibaut, Astronomie, Astrologie und Mathematik, pp. 55, 59.
* ↑ See Bulletin of the Calcutta Mathematical Society, XVII (1926), 66-67, for a brief discussion.
* ↑ Pancasiddhantika, pp. sviii, xxvii.
* ↑ JBBRAS, XIX, 129-31.
* ↑ JRAS, 1911, pp. 114-15.
* ↑ BCMS, XVIII (1927), 16.
* ↑ Ibid., 1865, p. 403.
* ↑ JRAS, 1911, pp. 115, 125.
* ↑ Transactions of the Literary Society of Madras, I (1827), 54, translated with additional notes by Jacquet, JA (1835), II, 118.
* ↑ Zeitschrift fur die Kunde des Morgenlandes, IV, 81.
* ↑ JRAS, 1863, p. 380.
* ↑ CEuvres, III, 182.
* ↑ JA (1880), II, 440.
* ↑ JASB, 1907, p. 478.
* ↑ Op. cit., 1911, p. 109.
* ↑ BCMS, XVII (1926), 195.
* ↑ IHQ, III, 110.
* ↑ III, 332 ff.
* ↑ Op. cit. (1879), I, 406-8.
* ↑ BCMS, XVIII (1927), 128
* ↑ Op. cit., 1908, p. 120.
* ↑ Ibid., p. 118.
* ↑ See Sarada Kanta Ganguly, op, cit., XVII (1926), 202.
* ↑ Cf. Fleet, op. cit, 1911, p. 116.
* ↑ See JA (1880), II, 454, and BCMS, XVII (1926), 201.
* ↑ Op. cit., 1908, p. 111.
* ↑ Ibid., 1865, p. 397.
* ↑ See ibid., 1911, p. 122 n.
* ↑ JBBRAS, XVIII, 129-31.
* ↑ BCMS, XVII (1926), 71.
* ↑ See further Bragmagupta (V, 25) and Alberuni (I, 376).
* ↑ See Sūryasiddhānta, pp. 27-28, and JRAS, 1911, p. 494.
* ↑ Cf. JRAS, 1865, p. 401. This implies, as Sudhakara says, that Brahmagupta knew two works by Aryabhata each giving the revolutions of the Sun as 4,320,000 but one reckoning 300 savana days more than the other. Cf. Kharegat (op. cit., 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 ; IHQ, IV, 506). At any rate, Brahmagupta does not imply knowledge of a second Aryabhata. For the whole problem of the two or three Aryabhatas see Kaye (Bibl. math., X, 289) and Bibhutibhusan Datta (op. cit. 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.
* ↑ Op. cit., 1911, pp. 111, 486.
* ↑ Cf. Suryasiddhanta, I, 11-13; Albenmi, I, 335; Bhattotpala, p. 24.
* ↑ Cf. Suryasiddhanta, I, 11, 28; Bhattotpala, p. 24; Pancasiddhantika, XIV, 32, for the first part, of 2; Brahmagupta, I, 5-6, and Bhaskara, Ganitadhaya, Kalamanadhyaya, 16-18, for both stanzas.
* ↑ See Lalla, Madhyamadhikara, 11; Brahmagupta, XIII, 42, for the first part. For vyatipata see Suryasiddhanta, XI, 2; Pancasiddhantika, III, 22; Lalla, Mahapatadhikara, 1; Brahmagupta, XIV, 37, 39.
* ↑ For the first part see Lalla, Madhyamadhikara, 11; Brahmagupta, XIII, 42; Bhaskara, Gainitadhyaya, Bhaganadhyaya, 14. For the second part see JRAS, 1863, p. 378; ibid., 1865, p. 404; Suryasiddhanta, I, 55; Bhattotpala, p. 182.
* ↑ Cf. Lalla, Bhuvanakosa, 36; Brahmagupta, XXI, 64; Bhaskara, Goladhyaya, Triprasnwasana, 38.
* ↑ Cf. Suryasiddhanta, XII, 55; Pancasiddhantika, XIII, 9; Brahmagupta, XXI, 6-7; Lalla, GrahabhramasaThathadhyaya, 3-5; Bhaskara, Goladhyaya, Bhuvanakosa, 51.
* ↑ Cf. JRAS, 1863, p. 378.
* ↑ Cf. Suryasiddhanta, XII, 74 and XIV, 14; Lalla, Grahabhramasamsthadyaya, 14; Brahmagupta, XXI, 8; Pancasiddhantika, XIII, 27, 38.
* ↑ Cf. Lalla, Golabandhadhikara, 1-2; Brahmagupta, XXI, 49.
* ↑ Cf. Lalla, Golabandkadhikara, 3; Brahmagupta, XXI, 50.
* ↑ Cf. Suryasiddhanta, V, 6-7 n.; Kaye, Hindu Astronomy, p. 76.
* ↑ Cf. Suryasiddhanta, XIII, 3 ff.; Lalla, Yantradhyaya, 1 ff.; IHQ, IV, 265 ff .
* ↑ IHQ, IV, 259, 262.
* ↑ Cf. Brahmagupta, III, 7-8; Lalla. Samanyagolabandha, 9-10; Bhiaskara. Garitadhyaya. Triprasnadhikara, 12-13.
* ↑ Cf. Lalla, Spastadhikara, 18; Paficasiddhantika, IV, 23; Suiryasiddhanta, II, 60; Brahmagupta, II, 56; Bhaskara, Ganitadhyaya, Spastadhikara, 48 (Vasanabhasya); Kaye, op, cit., p. 73.
* ↑ Cf. Lalla, Triprasnddhikara, 8; Brahmagupta, U, 57-58; Suryasiddhanta, Y, 42-43 and note; Paficasiddhantikd, IV, 29-30; Bhiskara, Ganitadhydya, Spastadhikara, 57; Kaye, op. cit., pp. 79-80.
* ↑ Cf. Brahmagupta, IV, 16-17 and XXI, 66; Lalla, Candragrahanadhikara, 23, 25; Suryasiddhanta, 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. Suryasiddhanta, VI, 23; Lalla, Candragrahanadhikara, 36; Brahmagupta, IV, 19.
* ↑ Cf . Suryasiddhanta, VI, 13.
* ↑ Cf. BCMS, XII (1920-21), 183.
* ↑ Cf. ibid., p. 187.
* ↑ Cf. JRAS, 1911, p. 114.