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Jyotihshastra: Astral and Mathematical Literature c5: Chapter III. Mathematics

Jyotiḥśāstra: Astral and Mathematical Literature c5

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English

Contents

Chapter III. Mathematics

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Mathematics¹ is useful to a number of *śāstras* in India as elsewhere—the *śulbasūtras*, astronomy, and astrology, which are discussed in this volume, as well as such subjects as alchemy, medicine, prosody, and philosophy (particularly Jaina). But there also exists a literature directly concerned with mathematics, in particular with arithmetic (especially its commercial and other practical applications), algebra, and geometry. There was never in India a *jāti* of mathematicians, and rarely anything that could be called a school; most mathematicians were *jyotiṣīs* (astronomers or astrologers). Therefore, the mathematical literature consists either in the form of chapters of astronomical *siddhāntas* or of treatises composed, with a few exceptions, by scholars who were also authors of astronomical texts. Moreover, several important branches of Indian mathematics—analemmata and trigonometry, for instance—rarely if ever are treated outside of the astronomical context that endowed them with significance.

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¹ Among general works on Indian mathematics, the most useful are: B. DATTA and A. N. SINGH, History of Hindu Mathematics, 2 vols., Lahore 1935–1938, reprinted Bombay 1962 (not always reliable); C. N. SRINIVASIENGAR, The History of Ancient Indian Mathematics, Calcutta 1967; T. A. SARASVATI AMMA, Geometry in Ancient and Medieval India, Delhi 1979; and A. K. BAG, Mathematics in Ancient and Medieval India, Varanasi 1979.

Gaṇita

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The earliest text of this genre that we have is the first chapter of Āryabhaṭa’s A entitled Gaṇitapāda.² In thirty-two verses he deals succinctly with a number of selected topics: the decimal place-value system of numbers; squares and cubes; square-roots and cube-roots; the areas of triangles, rectangles, trapezia, and circles, and the volume of a pyramid;³ chords and sines of arcs of a circle; π;

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² The most recent treatment of the Gaṇitapāda is K. ELFERING, Die Mathematik des Āryabhaṭa, München 1975; see also R. C. GUPTA, “On Some Mathematical Rules from the Āryabhaṭīya,” IJHS 12 (1977), 200–206; and A. VOLODARSKY, “Mathematical Achievements of Āryabhaṭa,” ibid., 167–172.

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³ K. ELFERING, “The Area of a Triangle and the Volume of a Pyramid as well as the Area of a Circle and the Surface of the Hemisphere in the Mathematics of Āryabhaṭa I,” IJHS 12 (1977), 232–236.

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shadow-problems; intersecting circles; sums of various series;⁴ solutions to quadratic equations; proportions; and the solution of indeterminate equations of the first degree.⁵ Thus he includes in this chapter on *gaṇita* not only arithmetic, geometry, and algebra, but several topics of primary interest to astronomers—i.e., trigonometry, intersecting circles (for eclipse calculations), and indeterminate equations (for deriving the numbers of rotations of the planets in a Mahāyuga). The solution to the indeterminate equations utilizes continued fractions; continued fractions had been investigated by Euclid, though not applied to the problem of indeterminate equations, so far as we know, by the Greeks. Many useful illustrations to Āryabhaṭa’s rules were given by his early commentator, Bhāskara.⁶

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⁴ On the history of series in India see also G. CHAKRAVARTI, “Growth and Development of Progressive Series in India,” JDL/UC 24 (1934), art. 6; and T. A. SARASVATI, “Śreḍhikṣetras or Diagrammatical Representations of Mathematical Series,” JOR Madras 28 (1958–59), 74–85, and “The Development of Mathematical Series in India after Bhāskara II,” BNISI 21 (1963), 320–343.

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⁵ A. K. BAG, “The Method of Integral Solution of Indeterminate Equations of the Type: BY = AX ± C in Ancient and Medieval India,” IJHS 12 (1977), 1–16.

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⁶ K. S. SHUKLA, “Hindu Mathematics in the Seventh Century as Found in Bhāskara I’s Commentary on the Āryabhaṭīya, Gaṇita 22 (1971), 1, 115–130, and 2, 61–78; and 23 (1972), 1, 57–79, and 2, 41–50.

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This same Bhāskara, in his MB (1, 41), appears to be the first to name the continued fraction method of solving indeterminate equations the “pulverizer” (*kuṭṭakāra*); he applies the “pulverizer” to problems of the planets’ mean motions in MB 1, 41–52. In MB 8, 7–25 he sets the student various mathematical problems that arise in astronomy, most of which are intended to exercise his facility in applying the “pulverizer.”

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Bhāskara’s contemporary, Brahmagupta, devotes two *adhyāyas* of his BSS to mathematics. Chapter 12⁷ is the earliest extant formal treatise on arithmetic in Sanskrit. In it he gives succinct rules for twenty arithmetical operations (*parikarma*) and eight determinations (*vyavahāra*). These are listed by Pṛthūdakasvāmin; the *parikarmāṇi* are: addition (*saṅkalita*), subtraction (*vyavakalita*), multiplication (*pratyutpanna*), division (*bhāgahāra*), square (*varga*), square-root (*vargamūla*), cube (*ghana*), cube-root (*ghanamūla*), five⁸ rules for operating with fractions (*pañcajātayaḥ*), the rule of three (*trairāśika*), the inverse rule of three (*vyastatrairāśika*), the rules of five, seven, nine, and eleven terms, and barter (*bhāṇḍapratibhāṇḍa*), while the *vyavahārāḥ* are: mixtures (*miśraka*), series (*śreḍhī*), plane geometry (*kṣetra*), (solid geometry, subdivided into) excavations (*khāta*), stacks (*citi*), sawn lumber (*krākacika*), and mounds of grain (*rāśi*), and shadow-problems (*chāyā*). Of special interest is his exhaustive investigation of triangles and quadrilaterals—mostly right triangles and cyclic quadrilaterals (BSS 12, 21—39). Chapter 18 is devoted to algebra, here denominated kuṭṭaka after the pulverizer;⁹ it is illustrated by exemplary problems (udāharaṇa). Among the topics discussed, in addition to the pulverizer itself, are the mathematics of zero and of surds, quadratic equations, equations with several unknowns, and indeterminate equations of the second degree (vargaprakṛti). Brahmagupta’s method for solving vargaprakṛti equations is not universally applicable. There is no convincing evidence to support the theory that Brahmagupta had a Greek source.¹⁰

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⁷ Translated with excerpts from Pṛthūdakasvāmin’s commentary by H. T. COLEBROOKE, Algebra with Arithmetic and Mensuration from the Sanscrit, London 1817, repr. Walluf bei Wiesbaden 1973, pp. 277–324.

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⁸ Five only are given by Brahmagupta; a sixth is said by Pṛthūdaka to have been described by Skandasena.

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⁹ A rearrangement of this chapter with excerpts from an anonymous commentary as found in an India Office Library manuscript, 596 A, was translated by COLEBROOKE, ibid., pp. 325—378.

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¹⁰ Most recently supported by B. L. VAN DER WAERDEN, "Pell's Equation in Greek and Hindu Mathematics," Russian Mathematical Surveys 31: 5 (1976), 210—225.

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Śrīdhara evidently wrote his four works on mathematics between the time of Brahmagupta, whom he gently criticizes without naming,¹¹ and before Govindasvāmin, who quotes a verse from Śrīdhara’s Triśatikā in the Prakaṭārthadīpikā that he wrote in ca. 850.¹² Thus Śrīdhara flourished most probably in the eighth century. Though his work on algebra that was known to Bhāskara¹³ is now lost, there still survive three of his writings on arithmetic; all seem to be epitomes of a longer, lost work entitled Pāṭīgaṇita, Bṛhatpāṭī, or Navaśatī.¹⁴ The first of the three works is the incomplete Pāṭīgaṇita,¹⁵ of which we have some 251 verses (118 are sūtras in āryā meter, and 133 are udāharaṇāni mostly in āryā). Following BSS 12 it divides arithmetic into parikarmāṇi (Śrīdhara adds nine to Brahmagupta’s twenty) and vyavahārāḥ (Śrīdhara adds the mathematics of zero, which was treated in BSS 18, to Brahmagupta’s eight). The work begins with verses on mensuration (paribhāṣā). Unfortunately, the unique manuscript breaks off in the middle of the third vyavahāra (kṣetra). Sharing many verses with this Pāṭīgaṇita is the Pāṭīgaṇitasāra or Triśatikā.¹⁶ Despite its second title, this work contains only some 180 āryā verses, of which 73 are sūtras and 107 udāharaṇāni. It follows the order of the Pāṭīgaṇita and thereby provides some idea of the contents of vyavahārāḥ 4–8; but it fails to discuss the mathematics of zero. The third of Śrīdhara's works is entitled Gaṇitapañcaviṃśī, of which a fragment of two folia has recently been discovered.¹⁷ Of this work, written almost exclusively in ślokas, there survive 53 verses, of which 20 are udāharaṇāni; the section between the inverse rule of three (vyastatrairāśika) and the discussion of quadrilaterals in the section on plane geometry (kṣetra) is missing.

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¹¹ K. S. SHUKLA, The Patiganita of Sridharacarya, Lucknow 1959, pp. xix—xx.

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¹² On uttarabhāga 14, 97 of Parāśara’s Horāśāstra.

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¹³ Bījagaṇita 116 and granthasamāpti 2; in the latter verse Bhāskara also names as his predecessors Brahma(gupta) and Padmanābha.

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¹⁴ In Śrīdhara’s works occur the earliest instances of the technical term pāṭī for arithmetic. Various authors have claimed that it refers to the calculating board (see, e.g., B. DATTA, "The Science of Calculation by the Board," AMM 35 (1928), 520—529), though others, with good reason, have questioned this meaning (e.g., R. VENKATACHALAM IYER, "Pāṭīgaṇita and the Hindu Abacus," MS 18 (1950), 79—82).

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¹⁵ Edited with an anonymous commentary and an English translation by K. S. SHUKLA, Lucknow 1959. There is a Russian translation and commentary by A. VOLODARSKII (and O. F. VOLKOVOI) in Fiziko-matematičeskie Nauki v Stranakh Vostoka, Moskva 1966, pp. 141—246.

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¹⁶ Edited by S. DVIVEDIN, Benares 1899; partial English translation by N. RAMANUJACHARIA and G. R. KAYE, BM 3, 13 (1912—13), 203—217.

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¹⁷ I have used ff. 1 and 3 of G. 111f. in the Wellcome Institute for the History of Medicine.

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Perhaps contemporary with Śrīdhara is the work on mathematics preserved in the Bakhshālī manuscript;¹⁸ though some scholars would date it as early as 200 A.D. on the basis of its symbols, its language, its meter, and its references, none of their arguments is very persuasive. The work is earlier than the manuscript, which is written in "early" Śāradā and may therefore have been copied in the eighth, ninth, or tenth century. Arguments based on its relation to extant Sanskrit texts on gaṇita appear to me rather unreliable since we have so little from an early period to compare it with, and since it is clearly not in the main tradition of these texts. However, the fact that it is composed of sūtrāṇi and udāharaṇāni in verse with symbolic statements and prose solutions of the examples as are the works of Śrīdhara makes me believe that it follows the same model as does he. Aside from Āryabhaṭa and Brahmagupta, who did not follow this model, the only mathematicians earlier than Śrīdhara that we know of are Maskari, Pūraṇa, Pūtana, and Mudgala whom the earlier Bhāskara mentions,¹⁹ and Nārmada whom Śrīdhara himself mentions,²⁰ while Lalla was probably contemporary with Śrīdhara.²¹ The tradition, then, of works on gaṇita based on the classification into parikarmāṇi and vyavahārāḥ (the latter are specifically ascribed to Maskari, Pūraṇa, and Mudgala by Bhāskara) was not a very long one; the treatment of rules and examples offered by Śrīdhara and the Bakhshālī manuscript was probably even shorter, though no certain terminus post quem can be attached to it. The subjects treated in the Bakhshālī manuscript include the rule of three, series, square-roots, mixtures (miśraka), simultaneous equations, and quadratic equations; presumably the rest of the traditional contents of a work on gaṇita once were present when the manuscript was complete.

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¹⁸ Edited by G. R. KAYE as ASI, NIS 43, 2 vols., Calcutta 1927–1933. See also, against KAYE, B. DATTA, "The Bakhshālī Mathematics," BCMS 21 (1929), 1–60, and A. A. KRISHNASWAMI AYYANGAR, "The Bakhshālī Manuscript," MS 7 (1939), 1–16.

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¹⁹ Maskari, Pūraṇa, and Mudgala on A Daśagītikā 1 (pp. 6–7 SHUKLA); and Maskari, Pūraṇa, and Pūtana on A Gaṇita 9 (p. 67 SHUKLA).

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²⁰ Gaṇitapañcaviṃśī, sūtra 23.

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²¹ SHUKLA, Pāṭīgaṇita, p. ix.

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The earliest influence of Indian mathematics upon Islam presumably was contemporaneous with the earliest translations of Sanskrit astronomical texts into Arabic in the eighth century. The first concrete evidence that survives for that transmission is the work on performing arithmetical operations with the ten Indian number symbols written by al-Khwārizmī, probably at Baghdād, in the early ninth century; this is preserved in several recensions of a Latin translation of the early twelfth century.²² A large group of Arabic arithmetics characterized by the use of the Indian number symbols (for both integers and fractions) and the dust board (takht) were denominated al-ḥisāb al-hindī ("Indian calculation") by their authors. The oldest of these is the Kitāb al-fuṣūl fī al-ḥisāb al-hindī, composed by al-Uqlīdisī at Damascus in 952/3.²³ The precise relationship of such mathematical works to Indian texts on gaṇita is not as yet established.

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²² A facsimile of the unique manuscript of the earliest of these versions with an inaccurate transcription is given by K. VOGEL, Mohammed ibn Musa Alchwarizmi's Algorismus, Aalen 1963.

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²³ A. S. SAIDAN, The Arithmetic of al-Uqlīdisī, Dordrecht–Boston 1978. Further information on the Indian influence on Islamic arithmetic will be found in H. HERMELINK, "Arabic Recreational Mathematics as a Mirror of Age-old Cultural Relations between Eastern and Western Civilizations," Proceedings of the First International Symposium for the History of Arabic Science, vol. 2, Aleppo 1978, pp. 44–54.

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Mahāvīra,²⁴ the most famous of the Jaina mathematicians, wrote his Gaṇitasārasaṅgraha²⁵ during the reign of the Rāṣṭrakūṭa Amoghavarṣa, who ruled parts of Karṇāṭaka and Mahārāṣṭra between ca. 814/15 and ca. 880. In this work he has divided the traditional contents of gaṇitaśāstra into an opening saṃjñādhikāra (paribhāṣā, mathematics of zero, and addition and subtraction) and eight vyavahāras. The parikarma is concerned with multiplication, division, squares, square-roots, cube-roots, and series; the kalāsavarṇa and prakīrṇaka with fractions; the trairāśika with the rules of three, five, seven, and nine; the miśraka with mixtures, the pulverizer, and sums of series; the kṣetragaṇita with plane geometry;²⁶ the khāta with solid geometry; and the chāyā with shadow problems. The GSS was commented on in Kannaḍa by Daivajñavallabha²⁷ (Bālabodha) and in Sanskrit by Varadarāja. Furthermore, it was translated into Telugu by Pāvulūri Mallana²⁸ in the eleventh century and into Rājasthānī by Amīcandra²⁹ in about 1850.

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²⁴ CESS A 4.

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²⁵ Edited with an English translation by M. RAṄGĀCĀRYA, Madras 1912; and with a Hindi version by L. JAINA as JJG 12, Solāpura 1963.

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²⁶ B. DATTA, "On Mahāvīra's Solution of Rational Triangles and Quadrilaterals," BCMS 20 (1930), 267–294; E. T. BELL, "Mahavira's Diophantine System," BCMS 38 (1946), 121–122; and R. C. GUPTA, "Mahāvīrācārya on the Perimeter and Area of an Ellipse," ME 8, 1 (1974), B 17–19, and "Mahāvīrācārya's Rule for the Surface-area of a Spherical Segment," Tulasī prajñā 2 (1975), 63–66.

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²⁷ CESS A 4.

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²⁸ CESS A 4.

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²⁹ CESS A 1, 45b–46a; A 2, 13b; and A 4.

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The second Āryabhaṭa included two chapters on mathematics in the MS that he wrote in about 950–1000. Adhyāya 15, the pāṭyadhyāya, deals with the traditional topics of arithmetic and geometry in 120 verses, while adhyāya 18, the kuṭṭakādhyāya, discusses the pulverizer, of whose use it gives many examples, in 70 verses. This treatment is clearly modelled on the BSS.

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More closely based on the same prototype, though of shorter extent, are the two adhyāyas devoted to mathematics in the SŚŚ composed by Śrīpati in about 1040: adhyāya 13, vyaktagaṇita, on pāṭī in 55 verses, and adhyāya 14, avyaktagaṇita, on algebra in 37 verses. Many of the verses of SŚŚ 13 appear as sūtras in Śrīpati's Gaṇitatilaka³⁰ to which many examples (some taken from Śrīdhara) have been added. Unfortunately, however, the Gaṇitatilaka does not survive in its entirety. After a paribhāṣā modelled on Mahāvīra's, it proceeds in the normal order, though inserting the mathematics of zero in verse 45, in the middle of its treatment of fractions; this mathematics of zero differs from that in SŚŚ 14, 6. The text breaks off in the middle of the miśrakavyavahāra. A commentary on the Gaṇitatilaka was written by Siṃhatilaka Sūri, the pupil of Vibudhacandra; this Siṃhatilaka wrote a vṛtti on the Bhuvanadīpaka of Padmaprabha Sūri at Vijāpura in 1269.

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³⁰ Edited with the commentary of Siṃhatilaka Sūri from a single, incomplete manuscript by H. R. KAPADIA as GOS 78, Baroda 1937. The following table illustrates the relationship between SŚŚ 13 and GT:

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Perhaps contemporary by Śrīpati was the mathematician Jayadeva.³¹ Though none of his works are now extant, a quotation in the Sundari composed by Udayadivākara in 1073 proves that he knew the cyclic solution (cakravāla) of Pell's equation—that is, of indeterminate equations of the second degree (vargaprakṛti) for which Brahmagupta possessed a partial solution. Jayadeva's solution was independently discovered in the West in the seventeenth century.

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³¹ CESS A 3, 60b, and A 4.

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The second Bhāskara, to whom this cakravāla has sometimes been erroneously attributed, wrote two works on mathematics in the middle of the twelfth century. The Līlāvatī,³² on pāṭī, after the usual paribhāṣā section on measure-

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³² The Līlāvatī was published at Calcutta in 1832; edited by TĀRĀNĀTHA ŚARMAN, Kalikātā 1846; published at Calcutta in 1846, 2nd ed. Calcutta 1876; published at Calcutta in 1852; edited with the vivaraṇa of Mahīdhara and the Telugu ṭīkā of TAḌAKAMALLA VEṄKAṬA KṚṢṆARĀVA by V. RĀMASVĀMIN SĀSTRIN, Madras 1863; edited by JĪVĀNANDA VIDYĀSĀGARA, Calcutta 1876, 4th ed. Calcutta 1909; edited by SUDHĀKARA DVIVEDIN, Benares 1878; edited with his own Sanskrit ṭīkā by BĀPŪDEVA ŚĀSTRIN, Benares 1883; edited by BHUVANACANDRA BASAK, Calcutta 1885; as an appendix to BANERJI'S edition of COLEBROOK'S translation, Calcutta

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| SŚŚ 13 | GT | SŚŚ 13 | GT | SŚŚ | GT | SŚŚ | GT |
| 2 | 15 | 7 | 30 | 14 | 86 | 17 | 111 |
| 5 | 23 | 9 | 38 | 15 | 97 | 18 | 113 |
| 6 | 29 | 13 | 83 | 16 | 108 | | |

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ment, contains the usual parikarmas (adding the mathematics of zero after the discussion of fractions) and vyavahāras (adding at the end a section on the pulverizer and one on aṅkapāśa or combinations and permutations). This became the standard textbook on arithmetic in India, as is easily seen from the hundreds of manuscript copies, many editions, various translations, and numerous commentaries. For it was translated into Kannaḍa by Rājāditya,³³ who is said to have been a Jaina from Pāvinabāge at the court at Dorasamudra of the Hoysaḷa Ballāla—presumably Ballāla II (1173–1220); into Persian by Abū al-Fayḍ Fayḍī³⁴ for the Mughal Akbar (1556–1605) in 1587, by Medinīmalla³⁵ in 1663/4, and by Muḥammad Amīn,³⁶ who flourished between 1661 and 1678; and into Hindī by Amīcandra at Jayapura in 1842. Commentaries on the Līlāvatī were composed by Parameśvara at Aśvatthagrāma, Kerala, before 1432 (vivaraṇa); by Gaṅgādhara³⁷ at Jambūsaras, Gujarāt, in 1420 (Gaṇitāmṛtasāgarī); by Moṣadeva³⁸ before 1472; by the Lakṣmīdāsa who commented on the SŚB in 1501; by Sūryadāsa³⁹ at Pārthapura on the Godāvarī in 1541 (Gaṇitāmṛtakūpikā); by the well known Gaṇeśa at Nandigrāma in 1545 (Buddhivilāsinī); by Nārāyaṇa⁴⁰ in Kerala in about 1550 (Karmapradīpikā); by Śaṅkara, but completed by the above mentioned Nārāyaṇa, in Kerala in about 1556 (Kriyākramakari); by Mahīdhara at Vārāṇasī in 1587; by Munīśvara, the author of the SSBM, at Kāśī in the early seventeenth century (Nisṛṣṭārthadūtī); by his contemporary, Raṅganātha, also at Vārāṇasī

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³³ See M. M. BHAT, "Mathematics in Karnataka of the Middle Ages," Bhārata-Kaumudī, vol. 1, Allahabad 1945, pp. 127–136. His Vyavahāra-gaṇita in Kannaḍa was edited by M. M. BHAT, Madras 1955.

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³⁴ CESS A 1, 44b.

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³⁵ CESS A 4.

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³⁶ CESS A 4.

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³⁷ CESS A 2, 81a–82a; A 3, 26b; and A 4.

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³⁸ CESS A 4.

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³⁹ K. M. K. SARMA, "The Bhāskara Bhūṣaṇa of Sūrya Paṇḍita", PO 11 (1946), 54–66, and "Siddhānta-samhitāsāra-samuccaya of Sūrya Paṇḍita," Siddha-Bhāratī, pt. 2, VIS 2, Hoshiarpur 1950, pp. 222–225.

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⁴⁰ CESS A 3, 151a, and A 4.

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1893, 2nd ed. Calcutta 1927; edited with a Marāṭhī ṭīkā by V. P. KHĀNĀPŪRKAR, Poona 1897; edited with a Hindī ṭīkā by R. S. ŚARMĀ, Bombay 1907; edited by SUDHĀKARA DVIVEDIN as BSS 39, Benares 1912; edited by RĀDHĀVALLABHA, Calcutta 1913; edited with his own Sanskrit ṭīkā by MURALĪDHARA ṬHĀKURA as HNM 3, Benares 1928, 2nd ed. Benares 1938; edited with the ṭīkās of Gaṇeśa and Mahīdhara by D. ĀPAṬE as ASS 107, 2 vols., Poona 1937; edited with the ṭīkā of Dāmodara Miśra by D. JHĀ as MSVG 8, Darbhanga 1959; edited with the Sanskrit and Hindī ṭīkās of Laṣaṇa Lāla JHĀ by S. ŚARMAN as VSG 62, Benares 1961; edited with his own Sanskrit and Hindī ṭīkā by SITĀRĀMA JHĀ, 4th ed., Benares 1970; and edited with the ṭīkā of Śaṅkara and Nārāyaṇa by K. V. SARMA as VIS 66, Hoshiarpur 1975. An English translation by J. TAYLOR was published at Bombay in 1816, and another by H. T. COLEBROOKE in his Algebra, with Arithmetic and Mensuration, London 1817, pp. 1–127; the latter was reprinted by H. C. BANERJI, Colebrooke's Translation of the Līlāvatī, Calcutta 1893, 2nd ed. Calcutta 1927.

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(Mitabhāṣiṇī); by Paraśurāma⁴¹ before 1659; by Rāmakṛṣṇa⁴² at Jalapura in the Sahyādri in 1687 (Gaṇitāmṛtalaharī); by Śrīdhara Mahāpātra at Dalapura north of Nilagiri in Orissa in 1717 (Sarvabodhini); by Kṛpārāma⁴³ at Aḥmadābād in about 1790; and by Nīlāmbara Jhā,⁴⁴ probably at Alavara in Rājasthān where he was the court astrologer in the middle of the nineteenth century. To this impressive list of explicators of the Līlāvatī could be added many more whose dates are uncertain, or whose names are unknown.

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⁴¹ CESS A 4.

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⁴² P. K. GODE, “Date of Gaṇitāmṛtalaharī of Rāmakṛṣṇa,” ABORI 11 (1930), 94–95.

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⁴³ CESS A 2, 48b–49a.

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⁴⁴ CESS A 3, 193a–195a.

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Less popular than the Līlāvatī because it is more difficult was Bhāskara’s Bījagaṇita.⁴⁵ This, the standard Sanskrit textbook on algebra, describes arithmetical operations involving positive and negative numbers and zero; irrational numbers; the pulverizer (virtually identical with the section on this subject in the Līlāvatī); “Pell’s equation” (the vargaprakṛti); the cyclic solution (cakravāla) thereof; various kinds of linear and quadratic equations with one or more unknowns; and equations involving the product of different unknowns (bhāvita). The Bījagaṇita was translated into Persian by ‘Aṭa’ Allāh Rushdī⁴⁶ for the Mughal Shāh Jahān (1628–1659) in 1634/5. It was commented on by Sūryadāsa at Pārthapura in 1538 (Sūryaprakāśa); by Kṛṣṇa⁴⁷ at Vārāṇasī in about 1600 (Bījāṅkura = Navāṅkura = Bījapallava = Kalpalatāvatāra); by Bhāskara⁴⁸ at Rājagiri in 1652 (Śiśubodhana); by Rāmakṛṣṇa⁴⁹ at Jalapura in the Sahyādri in 1687 (Bījaprabodha); by Haridāsa before 1725 (Vāsanā- bhāṣya); by Kṛpārāma at Ahmadābād in 1792 (Bālabodhinī); and by an obscure Nijānanda⁵⁰ (Bījālavāla).

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⁴⁵ Published Calcutta 1834, rev. ed. Calcutta 1834; published Calcutta 1838 and Calcutta 1846; partial edition with German translation in H. BROCKHAUS, “Über die Algebra des Bhāskara,” BVKSGWL, Phil.-hist. Kl. 4 (1852), 1–46; published Calcutta 1853; edited by G. PĀṬHAKA, Benares 1864; edited by JĪVĀNANDA VIDYĀSĀGARA, Calcutta 1878; edited with his own Sanskrit ṭīkā by JĪVANĀTHA JHĀ, Benares 1885; edited with his own Sanskrit ṭīkā by SUDHĀKARA DVIVEDIN, Benares 1888; edited with a Marāṭhi translation and ṭīkā by V. P. KHĀNĀPŪRKAR, Poona 1913; edited with his own Sanskrit ṭīkā by RĀDHĀVALLABHA, Calcutta 1917; edited with S. DVIVEDIN’S and his own Sanskrit ṭīkās by MURALĪDHARA JHĀ as BSS 40, Benares 1927; edited with the ṭīkā of Kṛṣṇa by D. APAṬE as ASS 99, Poona 1930; edited with his own Sanskrit and Hindi ṭīkās by DURGĀPRASĀDA DVIVEDIN, 3rd ed., Lakṣmaṇapura 1941; edited with J. JHĀ’S Sanskrit and his own Sanskrit and Hindi ṭīkās by ACYUTĀNANDA JHĀ as KSS 148, Benares 1949; and edited with the ṭīkā of Kṛṣṇa by T. V. RĀDHĀKṚṢṆA ŚĀSTRIN as TSMS 78, Tanjore 1958. An English translation was published in H. T. COLEBROOKE, Algebra, with Arithmetic and Mensuration, London 1817, pp. 129–276; the section on the pulverizer was translated in D. M. MEHTA, Theory of Simple Continued Fractions, Bhavnagar [1931?], pp. 26–71.

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⁴⁶ CESS A 1, 39b. An English translation of this Persian version by E. STRACHEY with notes by S. DAVIS was published at London in 1813.

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⁴⁷ CESS A 2, 53a–55b, and A 4.

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⁴⁸ CESS A 4.

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⁴⁹ P. K. GODE, “Date of Bījaprabodha,” ABORI 10 (1929), 160–161.

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⁵⁰ CESS A 3, 173a.

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After Bhāskara the next major authority on mathematics was Nārāyaṇa,⁵¹ who completed his Gaṇitakaumudī⁵² in 1356. The GK contains the traditional paribhāṣā, parikarmas, and vyavahāras, followed by the pulverizer, “Pell’s equation,” factors,⁵³ combinations, and magic squares. Before writing the GK Nārāyaṇa had written a work on algebra, the Bījagaṇitāvataṃsa.⁵⁴ This work covers arithmetical operations involving positive and negative numbers and zero; irrational numbers; the pulverizer and “Pell’s equation” (these two were drawn upon for the corresponding section in the GK); various kinds of linear and quadratic equations with one or more unknowns (the text is not extant beyond the beginning of the section on linear equations); and equations involving the product of different unknowns. Thus it is modeled directly on the BG.

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⁵¹ CESS A 3, 156b–157a.

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⁵² Edited by PADMĀKARA DVIVEDIN as PWSBT 57, 2 vols., Benares 1936–1942.

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⁵³ See R. C. GUPTA, “Nārāyaṇa’s Method for Evaluating Quadratic Surds,” ME 7 (1973), B 93–96.

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⁵⁴ The fragment in the Benares manuscript was edited by K. S. SHUKLA, Lucknow 1970.

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Sometime after Bhāskara one Devarāja⁵⁵ composed a Kuṭṭākāraśiromaṇi,⁵⁶ on which he wrote his own ṭīkā, entitled Mahālakṣmīmuktāvalī. His objective was to expand upon the pulverizer as propounded by Āryabhaṭa. Also subsequent to Bhāskara Rāmacandra wrote a Kautukalīlāvatī.⁵⁷ In this work, after the paribhāṣā, come problems of plane and solid geometry and estimates of the quantities of various substances in differently shaped containers; involved are, among other things, specific gravities. Traditional mathematics was continued in the sixteenth century by the Bījādhyāya of Jñānarāja, who wrote the SSJ in 1503 and was the father of Sūryadāsa, the commentator on Bhāskara’s L and BG; and by the Gaṇitamañjarī of Jñānarāja’s great nephew, Gaṇeśa.⁵⁸ Neither of these works has been published.

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⁵⁵ CESS A 3, 120b–121a.

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⁵⁶ Edited with the ṭīkā by B. D. ĀPAṬE as ASS 125, Poona 1944.

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⁵⁷ I have used Leipzig 986; Oxford, CS d. 796 (6); and University of Pennsylvania 1861.

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⁵⁸ CESS A 2, 107a–110a; A 3, 28b; and A 4.

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Except for commentaries, no significant traditional Sanskrit works on mathematics were written after the sixteenth century. At Jayasiṃha’s court at Jayapura, however, in the 1720’s and 1730’s, an effort was made to translate Arabic and Persian versions of Greek mathematical treatises into Sanskrit. The two leading translators were Nayanasukhopādhyāya⁵⁹ and Jagannātha; the former translated, at the dictation of Muḥammad Ābida, the Σφαιρικά of Theodosius under the title Ukāra (Arabic Ukarr) or Kaṭara (Arabic quṭr), the latter Euclid’s Στοιχεῖα under the title Rekhāgaṇita.⁶⁰

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⁵⁹ CESS A 3, 132a, and A 4.

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⁶⁰ Edited by H. DHRUVA and K. TRIVEDIN as Bombay SS 61–62, 2 vols., Bombay 1901–1902.

Mathematics in Astronomy

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The mathematics utilized in astronomical texts is most developed in the areas of analemmata and of trigonometry—as well as, of course, the application of the pulverizer to astronomical problems that we have already considered. Some of the most significant mathematical work undertaken in India was in connection with trigonometry.⁶¹ Briefly, after the initial success in constructing tables of Sines and Versines and in inventing a rule for finding the approximate Sine of any angle in the fifth through seventh centuries, the next advance was the discovery by the second Bhāskara of the formula for the addition of Sines. The more advanced trigonometry of the seventeenth and eighteenth century astronomers of northern India seems to be due to the influence of al-Kāshī and Ulugh Beg.

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⁶¹ See, for example, A. N. SINGH, “Hindu Trigonometry,” PBMS, NS 1 (1939), 77–92; and the following articles, among others, by R. C. GUPTA: “Early Indians on Second Order Sine Differences,” IJHS 7 (1972), 81–86; “Bhāskara I’s Approximation to Sine,” IJHS 2 (1967), 121–136; “Addition and Subtraction Theorems for the Sine and the Cosine in Medieval India,” IJHS 9 (1974), 164–177; “Sine of Eighteen Degrees in India up to the Eighteenth Century,” IJHS 11 (1976), 1–10; and “Sines of Submultiple Arcs as found in the Siddhānta-tattvaviveka,” RUMJ 5 (1974), 21–27.

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But in Kerala truly extraordinary advances were made by a school of astronomer-mathematicians building on the pioneering work of Mādhava of Saṅgamagrāma in the late fourteenth and early fifteenth century. He evidently discovered what is called “Gregory’s Series” for π and the power series for sines and cosines associated with Newton and Leibniz.⁶² Mādhava’s work was continued by, and is largely known from the works of, Nīlakaṇṭha, Jyeṣṭhadeva, and Śaṅkara in the sixteenth century, and Putumana in the eighteenth.

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⁶² After the work of C. M. WHISH, “On the Hindú Quadrature of the Circle and the Infinite Series of the Proportion of the Circumference to the Diameter in Four Śāstras, the Tantrasangraham, Yucti Bhāshá, Carana Paddhati, and Sadratnamálá,” TRAS 3 (1830), 509–523, the investigation of this topic was revived by C. T. RAJAGOPAL; see K. M. MARAR and C. T. RAJAGOPAL, “On the Hindu Quadrature of the Circle,” JBBRAS, NS 20 (1944), 65–82, and “Gregory’s Series in the Mathematical Literature of Kerala,” MS 13 (1945), 92–98; C. T. RAJAGOPAL and A. VENKATARAMAN, “The Sine and Cosine Power-series in Hindu Mathematics,” JRASB/S 15 (1949), 1–13; C. T. RAJAGOPAL, “A Neglected Chapter of Hindu Mathematics,” SM 15 (1949), 201–209; C. T. RAJAGOPAL and T. V. V. AIYAR, “In the Hindu Proof of Gregory’s Series,” SM 17 (1951), 65–74, and “A Hindu Approximation to Pi,” SM 18 (1952), 25–30; and C. T. RAJAGOPAL and M. S. RANGACHARI, “On an Untapped Source of Medieval Keralese Mathematics,” AHES 18 (1978), 89–102. His investigations have been continued by T. A. SARASWATHI, “The Development of Mathematical Series in India after Bhāskara II,” BNISI 21 (1963), 320–343; by

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A. K. BAG, “Trigonometrical Series in the Karaṇapaddhati and the Probable Date of the Text,” IJHS 1 (1966), 98–106, and “Mādhava’s Sine and Cosine Series,” IJHS 11 (1976), 54–57; and by R. C. GUPTA, “The Mādhava-Gregory Series,” ME 7 (1973), B 67–70, “An Indian Form of Third Order Taylor Series Approximation of the Sine,” HM 1 (1974), 287–289, and “Mādhava’s and Other Medieval Indian Values of Pi,” ME 9 (1975), B 45–48.

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