Mathematicians From India

"Panini’s grammar is the earliest scientific grammar in the world, the earliest extant grammar of any language, and one of the greatest ever written. It was the discovery of Sanskrit by the West, at the end of the 18th century, and the study of Indian methods of analyzing language that revolutionized our study of language and grammar, and gave rise to our science of comparative philology … The study of language in India was much more objective and scientific than in Greece or Rome. The interest was in empirical investigation of language, rather than philosophical and syntactical. Indian study of language was as objective as the dissection of a body by an anatomist."

"The grammar of Panini stands supreme among the grammars of the world, alike for its precision of statement, and for its thorough analysis of the roots of the language and of the formative principles of words. By employing an algebraic terminology, it attains a sharp succinctness unrivalled in brevity, but at times enigmatical. It arranges, in logical harmony, the whole phenomena which the Sanskrit language presents, and stands forth as one of the most splendid achievements of human invention and industry. So elaborate is the structure, that doubts have arisen whether its complex rules of formation and phonetic change, its polysyllabic derivatives, its ten conjugations with their multiform aorists and long array of tenses, could ever have been the spoken language of a people."

"Sanskrit is constructed like geometry and follows a rigorous logic. It is theoretically possible to explain the meaning of the words according to the combined sense of the relative letters, syllables and roots. Sanskrit has no meanings by connotations and consequently does not age. Panini's language is in no way different from that of Hindu scholars conferring in Sanskrit today."

"The author of the oldest extant Sanskrit grammar was Panini, a native of extreme north-west India, ... His work consists of nearly 4000 aphorisms, each of which owing to the extreme conciseness of the style, generally consists of not more than two or three words. Hence, the whole grammar could be printed within the compass of about thirty-five octavo pages. Yet it describes the entire Sanskrit language with a completeness which has never been equalled elsewhere. Thus it is at once the shortest and the fullest grammar in the world."

"The most interesting non-Western grammatical tradition—and the most original and independent—is that of India, which dates back at least two and one-half millennia and which culminates with the grammar of Panini, of the 5th century BCE. There are three major ways in which the Sanskrit tradition has had an impact on modern linguistic scholarship."

"Roman alphabet: siṃho vyākaraṇasya kartur aharat prāṇān priyān pāṇineḥ"

"English translation: A lion took the dear life of Panini, author of the grammatical treatise."

"His teacher was Varsa, and he was a contemporary of Katyayana, Vyadi, and Indradatta. Panini is said to have secured the favour of Shiva and obtained from him the alphabet in the form of fourteen pratyadhara sutras."

"Pingala was the brother of Panini."

"There is in the rules or definitions (sutras) of Panini a remarkably subtle and penetrating account of Sanskrit grammar. The construction of sentences, compound nouns, and the like is explained through ordered rules operating on underlying structures in a manner strikingly similar in part to modes of modern theory. As might be imagined, this perceptive Indian grammatical work held great fascination for 20th-century theoretical linguists. A study of Indian logic in relation to Paninian grammar alongside AristotleAristotelian]] and Western logic in relation to Greek grammar and its successors could bring illuminating insights."

"The word `Sanskrit' means “prepared, pure, refined or prefect”. It was not for nothing that it was called the `devavani' (language of the Gods). It has an outstanding place in our culture and indeed was recognized as a language of rare sublimity by the whole world. Sanskrit was the language of our philosophers, our scientists, our mathematicians, our poets and playwrights, our grammarians, our jurists, etc. In grammar, Panini and Patanjali (authors of Ashtadhyayi and the Mahabhashya) have no equals in the world; in astronomy and mathematics the works of Aryabhata, Brahmagupta and Bhaskara opened up new frontiers for mankind, as did the works of Charaka and Sushruta in medicine."

"The issue of theism vis-à-vis atheism, in the ordinary senses of the English words, played an important role in Indian thought. The ancient Indian tradition, however, classified the classical systems (darshanas) into orthodox (astika) and unorthodox (nastika). Astika does not mean “theistic,” nor does nastika mean “atheistic.” Panini, a 5th-century-BCE grammarian, stated that the former is one who believes in a transcendent world (asti paralokah) and the latter is one who does not believe in it (nasti paralokah)."

"We pass at once into the magnificent edifice which bears the name of Panini as its architect and which justly commands the wonder and admiration of everyone who enters, and which, by the very fact of its sufficing for all the phenomenon which language presents, bespeaks at once the marvelous ingenuity of its inventor and his profound penetration of the entire material of the language."

"The grammar of Panini is one of the most remarkable literary works that the world has ever seen, and no other country can produce any grammatical system at all comparable to it, either for originality of plan or analytical subtlety."

"For example, the great linguist Panini gave the concept for meta-language-and constructed one-thousands of years before computer scientists began exploring the same idea. No one has been able to match him to this day."

"Classical Sanskrit theatre flourished during the first nine centuries CE. Aphorisms on acting appear in the writings of Panini, the Sanskrit grammarian of the 5th century BCE, and references to actors, dancers, mummers, theatrical companies, and academies are found in Kautilya’s book on statesmanship, the Artha-shastra (4th century BCE)."

"Pāṇini had before him a list of irregularly formed words, which survives, in a somewhat modified form, as the Uṇādi Sūtra. There are also two appendixes to which Pāṇini refers: one is the Dhātupāṭha, "List of Verbal Roots," containing some 2000 roots, of which only about 800 have been found in Sanskrit literature, and from which about fifty Vedic verbs are omitted; the second is the Gaṇapāṭha, or "List of Word-Groups," to which certain rules apply. These gaṇas were metrically arranged in the Gaṇaratna-mahodadhi, composed by Vardhamāna in 1140 A.D."

"Among the earliest attempts to explain Pāṇini was the formulation of rules of interpretation or paribhāshās; a collection of these was made in the last century by Nāgojibhaṭṭa in his Paribhāshenduçekhara."

"Next we have the Vārttikas or "Notes" of Kātyāyana (probably third century B.C.) on 1245 of Pāṇini's rules, and, somewhat later, numerous grammatical Kārikās or comments in metrical form: all this critical work was collected by Patanjali in his Mahābhāshya or "Great Commentary," with supplementary comments of his own. He deals with 1713 rules of Pāṇini."

"The Mahābhāshya was commented on in the seventh century by Bhartṛihari in his Vākyapadīya which is concerned with the philosophy of grammar, and by Kaiyaṭa (probably thirteenth century). About 650 A.D. was composed the first complete commentary on Pāṇini, the Kāçikā Vṛitti or "Benares Commentary," by Jayāditya and Vāmana."

"In the fifteenth century Rāmachandra, in his Prakriyā-kaumudī, or "Moonlight of Method," endeavoured to make Pāṇini's grammar easier by a more practical arrangement of its matter. Bhaṭṭoji's Siddhānta-kaumudī (seventeenth century) has a similar aim; an abridgment of this work, the Laghu-kaumudī, by Varadarāja is commonly used as an introduction to the native system of grammar. Among non-Pāṇinean grammarians may be mentioned Chandra (about 600 A.D.), the pseudo-Çākaṭāyana (later than the Kāçikā), and, the most important, Hemachandra (12th century), author of a Prākrit grammar."

"Panini was a Sanskrit grammarian who gave a comprehensive and scientific theory of phonetics, phonology, and morphology. Sanskrit was the classical literary language of the Indian Hindus and Panini is considered the founder of the language and literature. It is interesting to note that the word "Sanskrit" means "complete" or "perfect" and it was thought of as the divine language, or language of the gods."

"A treatise called Astadhyayi (or Astaka) is Panini's major work. It consists of eight chapters, each subdivided into quarter chapters. In this work Panini distinguishes between the language of sacred texts and the usual language of communication. Panini gives formal production rules and definitions to describe Sanskrit grammar. Starting with about 1700 basic elements like nouns, verbs, vowels, consonants he put them into classes. The construction of sentences, compound nouns etc., is explained as ordered rules operating on underlying structures in a manner similar to modern theory. In many ways Panini's constructions are similar to the way that a mathematical function is defined today."

"Sanskrit is a scientific and systematic language. Its grammar is perfect and has attracted scholars worldwide. Sanskrit has a perfect grammar which has been explained to us by the world's greatest grammarian Panini."

"The most successful, hence most prominent amongst these grammarians was Panini. His grammar, surpassing all others in tightness and precision, became the standard and remained so undisputedly until today. Panini was able to joint the original devanagari language into an exact framework of rules, thus preserving it for the posterity. Since his time, this language is called Sanskrit, “joined together, refined”."

"Pāṇini's grammar defines Classical Sanskrit, so Pāṇini by definition lived at the end of the Vedic period. He notes a few special rules, marked chandasi ("in the hymns") to account for forms in the Vedic scriptures that had fallen out of use in the spoken language of his time. These indicate that Vedic Sanskrit was already archaic, but still a comprehensible dialect."

"It is not certain whether Pāṇini used writing for the composition of his work, though it is generally agreed that he knew of a form of writing, based on references to words such as "script" and "scribe" in his Ashtadhyayi. These must have referred to Aramaic or early Kharosthi writing. It is believed by some that a work of such complexity would have been difficult to compile without written notes, though others have argued that he might have composed it with the help of a group of students whose memories served him as 'notepads' (as is typical in Vedic learning). Writing first reappears in India in the form of the Brāhmī script from the 3rd century BC in the Ashokan inscriptions."

"While Pāṇini's work is purely grammatical and lexicographic, cultural and geographical inferences can be drawn from the vocabulary he uses in examples, and from his references to fellow grammarians, which show he was a northwestern person. New deities referred to in his work include Vasudeva (4.3.98). The concept of dharma is attested in his example sentence dharmam carati "he observes the law" (cf. Taittiriya Upanishad 1.11)."

"Pingala and Panini (fifth century BCE) along with the likes of Aryabhata, Bhaskara and Brahmagupta are the pillars of ancient Indian mathematics. Astonishingly, Panini’s immortal fame is not even as a mathematician but as the definitive Sanskrit grammarian. But he also “introduced abstract symbols to denote various subsets of letters and words that would be treated in some common way in some rules; and he produced rewrite rules that were to be applied recursively in a precise order”, notes Mumford, “one could say without exaggeration that he (Panini) anticipated the basic ideas of modern computer science”."

"At a very early date India began to trace the roots, history, relations and combinations of words. By the fourth century B.C. she had created for herself the science of grammar, and produced probably the greatest of all known grammarians, Panini. The studies of Panini, Patanjali (ca. 150 A.D.) and Bhartrihari (ca. 650) laid the foundations of philology; and that fascinating science of verbal genetics owed almost its life in modern times to the rediscovery of Sanskrit."

"Linguistics is an Indian science par excellence, and the entire modern discipline of modern linguistics in indebted to the Indian grammarians of the 1st millennium BCE. In India, it has briefly missed out on the recent innovation of comparison between seriously different languages (as opposed to dialectal differences, well-known among Indian scholars). But here too, Indians should take some pride in the official birth of IE linguistics in Kolkata 1786... Linguistics started in Takṣaśĭla university, where Pāṇini taught 26 (or so) centuries ago. Indo-European linguistics started at the feet of Brahmin informers in Kolkata (as shown on a freeze in Oxford showing William Jones learning from Hindu Pandits)."

"His formalism even served in the twentieth century as the basis for the first high-level programming languages, as ALGOL60, which also work on the basis of a fully specified system of rules. Virtually all programming languages are written in formalism that uses Pāṇini’s linguistic notion of grammar... Compared with Pāṇini, the other linguistics from antiquity appears to be from a different world. No other work from Chinese, Greek or Roman literature comes close to Pāṇini’s grammar in terms of complexity of precision.... Western linguistics continued to be dominated by the taxonomic study of words after Priscian, and this situation did not change until the later Middle ages."

"He hailed from the Asmaka country, which comprised the present South Gujarat and North Maharashtra, through which the rivers Godavari and Narmada flowed. He flourished at Pataliputra (modern Patna) in the ancient Magadha country (now Bihar) where he composed his works, the 'Aryabhatiya and arya-siddhanta'."

"...spherical astronomy in Gola, where he applied plane trigonometry to spherical geometry by projecting points and lines on the surface of a sphere onto appropriate planes. Topics include prediction of solar and lunar eclipses and an explicit statement that the apparent westward motion of the stars is due to the spherical Earth’s rotation about its axis. He also correctly ascribed the luminosity of the Moon and planets to reflected sunlight."

"With Kala-kriya he turned to astronomy — in particular, treating planetary motion along the ecliptic. The topics include definitions of various units of time, eccentric and epicyclic models of planetary motion (see Hipparchus for earlier Greek models), planetary longitude corrections for different terrestrial locations, and a theory of “lords of the hours and days” (an astrological concept used for determining propitious times for action)."

"In Ganita, he names the first 10 decimal places and gives algorithms for obtaining square and cubic roots, utilizing the decimal number system. Then he treats geometric measurements — employing 62,832/20,000 (= 3.1416) for π—and develops properties of similar right-angled triangles and of two intersecting circles."

"Aryabhatiya...written in verse couplets ...contains astronomical tables and Aryabhata’s system of phonemic number notation, the work is characteristically divided into three sections: Ganita (“Mathematics”), Kala-kriya (“Time Calculations”), and Gola (“Sphere”)."

"...he flourished in Kusumapura—near Patalipurta (Patna), then the capital of the Gupta dynasty — where he composed at least two works, Aryabhatiya (c. 499) and the now lost Aryabhatasiddhanta. Aryabhatasiddhanta circulated mainly in the northwest of India and, through the Sāsānian dynasty (224–651) of Iran, had a profound influence on the development of Islamic astronomy. Its contents are preserved to some extent in the works of Varahamihira (flourished c. 550), Bhaskara I (flourished c. 629), Brahmagupta (598–c. 665), and others. It is one of the earliest astronomical works to assign the start of each day to midnight."

"In geometry his greatest achievement was an accurate value of π. His rule is stated as: dn^2+(2a-d)n=2s, which implies the approximation 3.1416 which is correct to the last decimal place."

"The development of Indian trigonometry, based on sine as against chord of the Greeks, a necessity for astronomical calculations with his own concise notation which expresses the full sine table in just one couplet for easy remembrance. One of the two methods suggested by him for the sine table is based on the property that the second order sine differences were proportional to sines themselves."

"The peculiar system of alphabetic numerals evolved by him with 33 consonants of the Sanskrit alphabet (Nagari script) denoted various numbers in conjunction with vowels which themselves did not represent any numerical value. For example khyughr (=khu+yu+ghr) is denoted by 2x100^2 +30x100^2+4x10^3 =4,300,000 which is the number of revolutions of the Sun in a yuga (epoch)"

"He was the father of the Indian epicyclic astronomy which resulted in the planetary theory that determines more accurately the true positions and distances of the planets (including the Sun and the Moon)...was also the first to produce celestial latitudes...proposed the scientific cause of eclipses as against the mythological demon Rahu [Moon's node]. His ideas resulted in the new school of Indian Astronomy – the Āryabhata School Āryapakșa based on the text of Āryabhatīțya."

"Use of better planetary parameters, the innovations in astronomical methods, and the concise style of exposition of Aryabhatiya makes it an excellent text book on Astronomy. As opposed to the geostationary theory, Aryahabata held the view that the earth rotates on its axis. His estimate of the period of the sidereal rotation of earth was 23 hours 56 min, and 4.1 s is close to the actual value."

"An Arabic translation of the Aryabhatiya entitled Zij-al-Arjabar (800 AD) is attributed to Ahwazi."

"The Aryabhatiya consists of four sections:1.Dasagitika (10+3 couplets in Giti meter); 2.Ganitapada (33 verses on mathematics); 3.Kala-kriyapada 25 verses on time-reckoning), and 4. Golapada (50 verses on spherical astronomy)"

"Aryabhatiya, an improved work, is product of mature intellect, which he wrote when he was 23 years old. Unlike in the Aryabhata siddhanta, the civil days are reckoned from one sunrise to the next, a practice which is still prevalent among the followers of Hindu calendar."

"His fame rests mainly on his Aryabhatiya, but from the writings of Varahamihira (Sixth century AD), Bhaskara I, and Brahmagupta (seventh century) it is clear that earlier he composed the Aryabhata Siddhantha (voluminous) is not extant. It is also called Ardharatrika Siddhanta, because in it the civil days were reckoned from one midnight to the next; 34 verses on astronomical instruments from this have been quoted by Ramakrishna Aradya."

"He was a celebrated astronomer and mathematician of the classical period of the Gupta Dynasty...He played an important role in shaping scientific astronomy in India. He is designated as Arayabhata I to differentiate him from Arya Bhata II who flourished much later (ca. AD 950-1100) and who wrote the Mahasiddhanta."

"The second reason adduced, viz., that Aryabhata should have hailed from Kerala is fragile. Besides the Aryabhatan system being prevalent in this land,-“all” commentaries on Aryabhaflya have been produced by Kerala astronomers really does not stand scrutiny."

"He gives a clue to his date of birth in his Aryabhatiya... The date works out to the end of the Kali year 3600, corresponding to the Saka year 421, the date being 21 March 499 ...and that he composed the Aryabhatiya when he was 25 years old, i.e. in Saka 444 or AD 522. Page 4"