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A Concise History of Science in India c7.s6: Interrelationship between Indian and Greek, Arabic and Chinese Mathematics and the Part Played by Indian Mathematics in European Renaissance

A Concise History of Science in India c7.s6

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English

Interrelationship between Indian and Greek, Arabic and Chinese Mathematics and the Part Played by Indian Mathematics in European Renaissance

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Situated geographically in the centre of the old world, India was from the beginning destined to play an important role in the transmission and diffusion of scientific ideas. In this diffusion mathematics formed no exception. Commercial and cultural contacts of India with Western Asia and Egypt extend to prehistoric times. The Persian Empire under Darius served as a bridge between India and Western Asia with its numerous settlements of the Ionian Greeks. Alexander’s invasion and the founding of the Bactrian Greek kingdoms on the borders of, and within, India provided a further opportunity of intensification of East-West relationships. Ptolemy’s Alexandria evinced great interest in establishing commercial contacts with India, an interest which was transformed into a practical reality through Rome’s eastern policies. The Sino-Indian intercourse depended partly on the thriving trade along the silk roads, but largely on the spread of Buddhism into China. The rise of Islam and its bid for political and intellectual supremacy provided yet another opportunity for the appreciation of Indian sciences, particularly mathematics and astronomy, in West Asia, North Africa and in Spain.

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The Indian emphasis on arithmetic and algebra is often contrasted with the Greek preoccupation with geometry. Nevertheless, the close similarity between the philosophical and mathematical writings of Pythagoras and his school on the one hand and the Vedic Hindus on the other has never failed to attract the attention of scholars. Leopold Von Schroeder, in one of his exhaustive studies, tried to show Pythagoras’ indebtedness to India in respect of (1) his doctrine of transmigration and metempsychosis, (2) irrational numbers, (3) the theorem known after his name and (4) the doctrine of five elements.a Regarding Pythagorean geometry and irrational number, Schroeder held the view, shared by Garbe, Hopkins and Macdonell, that the square relation among the sides of a right-angled triangle and the concept of irrational numbers ultimately had their origin in the *śulba-sūtras* of Baudhāyana and Āpastamba. Others, notably Keith, have disputed such views.

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As to Sino-Indian intercourse in mathematics, the catalogue of the Sui dynasty listed a number of Brāhmaṇical works, such as *Po-lo-mên Suan-fa* (Brāmaṇical methods of calculations, in 3 books), *Po-lo-mên-wen-ching* (Brāhmaṇical astronomy, in 21 books) and a number of similar Hindu astronomical tracts already referred to. The establishment, in the seventh century A.D., of an Astronomical School or Board in Chang-Nan for the propagation of Brāhmaṇical astronomy and the reform of the calendar after the Hindu model, played an important part in transmitting Hindu mathematics in China. In the eighteenth century, during the Thang times, Ch’u-t’an Hsi-ta (Chinese version of Gautama Siddha or Siddhārtha), translated a Sanskrit calendar under the Chinese title *Chiu-Chi-li* in which were discussed, among others, the Indian decimal notation and arithmetical rules. It is further on record that the Tantric-Buddhist astronomer I-hsing was ordered by the emperor to investigate the chronological and arithmetical ideas introduced to China from India by Ch'u-t'an Hsi-ta. Sarton observes: 'The Chinese treatises of Ch'u-t'an Hsi-ta and I-hsing are of special value as witnesses of the penetration of Hindu mathematics into China. It is possible that the Hindu numerals were introduced into China at this time, though we have no positive evidence of it.'a Writing about the arithmetical computations employed by Hsi-ta in his calendrical work, Mikami informs us that the technical terms used therein 'were so peculiar that the Chinese scholars who are ever fond of following conservative tendencies were little able to understand its contents'.b He further observes: 'The employment of the symbol 0 for zero or for the digit that is wanting was probably derived from this translation of an Indian calendrical treatise, or from some other Indian source. This employment does not, however, occur until the middle of the thirteenth century, so far as we are informed.' Studies in other areas, such as the problem of the circle-measurement and accurate determination of the value of π, appear for the first time in China after her contact with Indian civilization.

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a Sen (S. N.) (3), pp. 8–30.

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Kaye has cited a number of parallel examples occurring simultaneously in the Chinese and the Indian mathematical texts. The Chinese arithmetical classic, *Chiu Chang Suan Shu* (Arithmetic in Nine Sections) (second century B.C.), gives the area of the segment of a circle in a form which is also found in Mahāvīra's work (*c.* A.D. 830). The following problem on right-angled triangle given in the same work appears in Hindu works after the sixth century A.D.: 'There is a bamboo 10 feet high, the upper end of which being broken reaches to the ground 3 feet from the ground. What is the break?' The remainder problem, without any rule for solution, appears in the *Sun-Tzu Suan-Ching*, another arithmetical classic (fourth or fifth century A.D.), a little earlier than such examples are met with in the Indian texts. The rules for solving such remainder problems, or what is the same thing as the solution of indeterminate equations of the first degree, appear in Indian texts from the time of Āryabhaṭa much earlier than they do in the Chinese works. Accordingly, questions of borrowings on the ground of chronological lag, although tempting, cannot be decided for certain. From the scanty materials that have survived so far it is safe to say that it was by no means a one-way traffic and the mathematical advancement of the one stimulated that of the other.

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The part played by India in Arabic literary renaissance is well known. When the Arabs, after their successful military conquests of the Middle East and North Africa, set themselves upon the intellectual conquest, they first turned their attention to the Persian literature of the Sassanian Empire.

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a Sarton, I, pp. 504, 513, 514.

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b Mikami, p. 59.

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Their interest in astronomy was roused by a Persian work on astronomy, *Zīj-ashshahriyar*, which was largely a book of tables based on astronomical observations carried out in the Sassanian period. This *Zīj* itself was based on Indian elements, and it was felt necessary to study Indian astronomy in detail for a better understanding of the work. An opportunity presented itself when, during the reign of the second Abbāsid Caliph al-Mansūr (753–774), a direct contact with India was established through the Arab territory of Sind. Al-Adamī reports that an Indian astronomer visited the court of al-Mansūr and discoursed on Hindu astronomy. Abu-Māshar, an astrologer from Balkh, acknowledged his debt to India in obtaining his knowledge of the Hindu great cycles. Obviously, these early contacts were very useful, for shortly thereafter a number of Arab astronomers were commissioned to translate into Arabic Brahmagupta’s *Brāhmasphuṭasiddhānta* and *Khaṇḍakhādyaka*. Muhammad ibn Ibrāhīm al-Fazārī (d. 796 or 806) and Ya’qūb ibn Ṭāriq (d. 796) produced these translations with the assistance of Hindu pandits. Sarton suggests that these translations were possibly the vehicle by which Hindu numerals were transmitted to the Arabs.a Most probably, the knowledge of Hindu numerals and number-writing had spread to Western Asia much earlier, as in the seventh century Severus Sebokht, a Christian monk, already mentioned in appreciative terms the Hindu numerals.

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The next important figure in the transmission of Hindu astronomy and mathematics among the Arabs was al-Khwārizmī, one of the greatest mathematicians of his time. Al-Khwārizmī developed a great liking for Indian mathematics and astronomical system, learnt Sanskrit, prepared an abridged version of *Sindhind* (Arabic title of *Brāhmasphuṭa-siddhānta*) and wrote an arithmetic explaining the Hindu system of numeration. His algebra was also inspired by his Indian studies. Indian studies were so fashionable at his time that a number of his notable contemporaries also participated actively in this work of transmission. Thus al-Kindī wrote four books on the use of the Hindu numerals and computation *Hiṣābu’l hindi*; Ḥabash al-Ḥāsib, al-Nairizī, al-Ḥasan ibn Miṣbāḥ and ibn al-Adamī constructed astronomical tables on Hindu astronomical model. In the eleventh century al-Bīrūnī revived Indian studies with fresh energy and thoroughness, himself checking corrupt translations of the previous generations, producing new translations and critical studies of various aspects of Hindu mathematics and astronomy. On Hindu numerals, his was the best medieval account.

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There is no doubt that Indian mathematical elements, particularly arithmetic based on decimal place-value numeration, passed into Latin Europe in the course of transmission of Arabic mathematical knowledge there. The leaders of this transmission were, among others, Adelard of Bath (c. 1142), John of Seville, Robert of Chester, Villedieu, Sacrobosco and Leonardo Pisano.b At this time and up to a much later date, the abacus of Gerbert was in general use for all calculations and counting purposes. Adelard (*c.* 1142), English philosopher, mathematician and scientist, was probably the earliest Latin exponent of Hindu arithmetic, trigonometry and astronomy through his translations of al-Khwārizmī's mathematical and astronomical works. He probably translated an arithmetical work attributed to al-Khwārizmī under the title *Liber ysagogarum Alchorismi*. But the earliest Latin version of al-Khwārizmī's arithmetic *Algoritmi de numero Indorum*, of which the Arabic version is lost, is due to an unknown translator. Hispano-Jewish John of Seville (*c.* 1135) wrote *Liber algorismi*, an arithmetical work which, although based on al-Khwārizmī, drew upon other Arabic sources. Robert of Chester (*c.* 1141) introduced the study of algebra in Latin Europe by his translation of al-Khwārizmī's *Ḥiṣāb al-jabr wal-muqābala* which bears the impress of Hindu algebraic thought. Villedieu's (Alexandre de Villedieu, d. 1240) *Carmen de algorismo*, composed in hexameter, closely followed John's *Liber algorismi*. Translations of *Carmen* into English, French and Icelandic, several commentaries on it and a large number of manuscript copies found in the libraries of Europe bear testimony to the influence it exerted and the wide circulation it enjoyed in the thirteenth century. The work possibly played an important role in the diffusion of Hindu numerals in Latin Europe. Villedieu's contemporary John Sacrobosco's *Algorismus Vulgaris*, another arithmetical tract, also enjoyed great popularity. Another date often taken to be the starting point of European mathematical renaissance is 1202 in which year appeared Leonardo Pisano's arithmetical classic *Liber abaci*, containing probably the first complete exposition in Latin of Hindu and Arabic arithmetic, including decimal place-value numeration.

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a Sarton I, p. 530.

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b Sen (S. N.) (6), pp. 55–59.

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This is the background of the introduction into Europe of the new arithmetic, called ‘algorism’, based on decimal place-value numeration. In time algorism became one of the active promoters of the Renaissance itself. This is corroborated by the sudden appearance in quantity of printed arithmetical works from the sixteenth century in several European countries. To mention a few, in Italy, Cardano's *Practica arithmetice et mensurandi singularis* appeared in 1501 and Tartaglia's *La Prima Parte del general trattato di numeri e misure* in 1556. In England, Robert Recorde's *The grounde of artes, teachyng the worke and practise of arithmetike* was reprinted seventeen times before 1601 and Digg's *Stratiotios* appeared in 1579. In Germany, the arithmetical movement was led by Jacob Köbel, author of *Rechenbiechlin* (1514), Stifel, author of *Arithmetica integra* (1514), and Christopher Clavius, author of *Epitome arithmeticae practice* (1583). In France, Boissière's *L'art d'arythmétique* (1554) and Forcadel's *L'Arithmétique* (1556–57) helped popularize arithmetic. Elementary as these arithmetical tracts may now appear, their potentialities were soon felt in the rising trade and commerce of the Renaissance period, in the teaching programmes of universities and schools and in the higher pursuits of mathematics in general.

Cites (7)