Instruct: Given a question, retrieve relevant documents that can help answer the question.
Query: When was the  Brāhmasphuṭasiddhānta written?
Brahmagupta () (born , died ) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the "Brāhmasphuṭasiddhānta" (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical treatise, and the "Khaṇḍakhādyaka" ("edible bite", dated 665), a more practical text.
Tamil-Brahmi, or Tamili aka Tamizhi is a variant of the Brahmi script used to write the Tamil language. These are the earliest documents of a Dravidian language, and the script was well established in the Chera and Pandyan states, in what is now Tamil Nadu, Kerala and Sri Lanka. Inscriptions have been found on cave beds, pot sherds, Jar burials, coins, seals, and rings. The language is Archaic Tamil, and led to classical Sangam literature.
The Indian mathematician Brahmagupta (597–668 AD) explicitly described the quadratic formula in his treatise "Brāhmasphuṭasiddhānta" published in 628 AD, but written in words instead of symbols. His solution of the quadratic equation was as follows: "To the absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is the value."
This is equivalent to:
Taking a day to be 24 hours, the smallest time unit, "prāṇa", or one respiratory cycle, equals 4 seconds, a value consistent with the "normal breathing frequency" of 15 breaths/min used in modern medical research. The Surya Siddhanta also described a method of converting local time to the standard time of Ujjain. Despite these early advances, standard time was not widely used outside astronomy. For most of India's history, ruling kingdoms kept their own local time, typically using the Hindu calendar in both lunar and solar units. For example, the Jantar Mantar observatory built by Maharaja Sawai Jai Singh in Jaipur in 1733 contains large sundials, up to 90 ft (27 m) high, which were used to accurately determine the local time.
According to Dasharatha Sharma, the dramatists Kalidasa and Bhavabhuti utilized the "Arthashastra" of Kautilya while composing their famous works. Kalidasa is indebted to Kautilya's "Arthashastra" for material in the "Raghuvamsa". Similarly, Bhavabhuti utilizes words and ideas from the "Arthashastra" in the "Malatimadhava" and the "Mahaviracharita". There is indeed a striking resemblance between the methods advocated by "Ravana's" minister, "Malayavana" and the policies suggested by Kautilya in the "Arthashastra".
It was invented between the 1st and 4th centuries by Indian mathematicians. The system was adopted in Arabic mathematics by the 9th century. Influential were the books of Muḥammad ibn Mūsā al-Khwārizmī ("On the Calculation with Hindu Numerals", ) and Al-Kindi ("On the Use of the Hindu Numerals", ). The system later spread to medieval Europe by the High Middle Ages.
According to the Indologist Alexis Sanderson, before Islam arrived in India, the "Sanskrit sources differentiated Vaidika, Vaiṣṇava, Śaiva, Śākta, Saura, Buddhist, and Jaina traditions, but they had no name that denotes the first five of these as a collective entity over and against Buddhism and Jainism." This absence of a formal name, states Sanderson, does not mean that the corresponding concept of Hinduism did not exist. By late 1st-millennium CE, the concept of a belief and tradition distinct from Buddhism and Jainism had emerged. This complex tradition accepted in its identity almost all of what is currently Hinduism, except certain antinomian tantric movements. Some conservative thinkers of those times questioned whether certain Shaiva, Vaishnava and Shakta texts or practices were consistent with the Vedas, or were invalid in their entirety. Moderates then, and most orthoprax scholars later, agreed that though there are some variations, the foundation of their beliefs, the ritual grammar, the spiritual premises and the soteriologies were same. "This sense of greater unity", states Sanderson, "came to be called Hinduism".
The most commonly used system of numerals is the Hindu–Arabic numeral system. Two Indian mathematicians are credited with developing it. Aryabhata of Kusumapura developed the place-value notation in the 5th century and a century later Brahmagupta introduced the symbol for zero. The numeral system and the zero concept, developed by the Hindus in India, slowly spread to other surrounding countries due to their commercial and military activities with India. The Arabs adopted and modified it. Even today, the Arabs call the numerals which they use "Rakam Al-Hind" or the Hindu numeral system. The Arabs translated Hindu texts on numerology and spread them to the western world due to their trade links with them. The Western world modified them and called them the Arabic numerals, as they learned them from the Arabs. Hence the current western numeral system is the modified version of the Hindu numeral system developed in India. It also exhibits a great similarity to the Sanskrit–Devanagari notation, which is still used in India and neighbouring Nepal.
The decimal Hindu–Arabic numeral system with zero was developed in India by around AD 700. The development was gradual, spanning several centuries, but the decisive step was probably provided by Brahmagupta's formulation of zero as a number in AD 628. The system was revolutionary by including zero in positional notation, thereby limiting the number of individual digits to ten. It is considered an important milestone in the development of mathematics. One may distinguish between this positional "system", which is identical throughout the family, and the precise glyphs used to write the numerals, which varied regionally.
Although he is known by many attributes, Ganesha's elephant head makes him easy to identify. Ganesha is widely revered as the remover of obstacles, the patron of arts and sciences and the deva of intellect and wisdom. As the god of beginnings, he is honoured at the start of rites and ceremonies. Ganesha is also invoked as patron of letters and learning during writing sessions. Several texts relate mythological anecdotes associated with his birth and exploits and explain his distinct Ganesha likely emerged as a deity as early as the 2nd century CE, but most certainly by the 4th and 5th centuries CE, during the Gupta period, although he inherited traits from Vedic and pre-Vedic precursors. Hindu mythology identifies him as the restored son of Parvati and Shiva of the Shaivism tradition, but he is a pan-Hindu god found in its various traditions. In the "Ganapatya" tradition of Hinduism, Ganesha is the supreme deity. The principal texts on Ganesha include the "Ganesha Purana", the "Mudgala Purana", and the "Ganapati Atharvashirsa". "Brahma Purana" and "Brahmanda Purana" are other two Puranic genre encyclopedic texts that deal with Ganesha.
After Tamil Brahmi fell out of use, Tamil was written using a script called the ' amongst others such as Grantha and Pallava script. The current Tamil script consists of 12 vowels, 18 consonants and one special character, the "āytam". The vowels and consonants combine to form 216 compound characters, giving a total of 247 characters (12 + 18 + 1 + (12 x 18)). All consonants have an inherent vowel "a", as with other Indic scripts. This inherent vowel is removed by adding a tittle called a ', to the consonantal sign. For example, is "ṉa" (with the inherent "a") and is "ṉ" (without a vowel). Many Indic scripts have a similar sign, generically called virama, but the Tamil script is somewhat different in that it nearly always uses a visible "puḷḷi" to indicate a 'dead consonant' (a consonant without a vowel). In other Indic scripts, it is generally preferred to use a ligature or a half form to write a syllable or a cluster containing a dead consonant, although writing it with a visible virama is also possible. The Tamil script does not differentiate voiced and unvoiced plosives. Instead, plosives are articulated with voice depending on their position in a word, in accordance with the rules of Tamil phonology.
Hypatia (born 350–370; died 415 AD) was a Hellenistic Neoplatonist philosopher, astronomer, and mathematician, who lived in Alexandria, Egypt, then part of the Eastern Roman Empire. She was a prominent thinker of the Neoplatonic school in Alexandria, where she taught philosophy and astronomy. She is the first female mathematician whose life is reasonably well recorded. Hypatia was renowned in her own lifetime as a great teacher and a wise counselor. She is known to have written a commentary on Diophantus's thirteen-volume "Arithmetica", which may survive in part, having been interpolated into Diophantus's original text, and another commentary on Apollonius of Perga's treatise on conic sections, which has not survived. Many modern scholars also believe that Hypatia may have edited the surviving text of Ptolemy's "Almagest", based on the title of her father Theon's commentary on Book III of the "Almagest".
The numeral system employed, known as algorism, is positional decimal notation. Various symbol sets are used to represent numbers in the Hindu–Arabic numeral system, potentially including both symbols that evolved from the Brahmi numerals, and symbols that developed independently. The symbols used to represent the system have split into various typographical variants since the Middle Ages:The evolution of the numerals in early Europe is shown here in a table created by the French scholar Jean-Étienne Montucla in his "Histoire de la Mathematique", which was published in 1757:
Stefan of Pise translated into Latin around 1127 an Arab manual of medical theory. The method of algorism for performing arithmetic with Indian-Arabic numerals was developed by the Persian al-Khwarizmi in the 9th century, and introduced in Europe by Leonardo Fibonacci (1170–1250). A translation by Robert of Chester of the "Algebra" by al-Kharizmi is known as early as 1145. Ibn al-Haytham (Alhazen, 980–1037) compiled treatises on optical sciences, which were used as references by Newton and Descartes. Medical sciences were also highly developed in Islam as testified by the Crusaders, who relied on Arab doctors on numerous occasions. Joinville reports he was saved in 1250 by a “Saracen” doctor.
Malwa was the centre of Sanskrit literature during and after the Gupta period. The region's most famous playwright, Kalidasa, is considered to be the greatest Indian writer ever. His first surviving play is "Malavikagnimitra" (Malavika and Agnimitra). Kalidasa's second play, his masterpiece, is the Abhijñānaśākuntalam, which tells the story of king Dushyanta, who falls in love with a girl of lowly birth, the lovely Shakuntala. The last of Kalidasa's surviving plays is "Vikramuurvashiiya" ("Urvashi conquered by valour"). Kalidasa also wrote the epic poems "Raghuvamsha" ("Dynasty of Raghu"), "Ritusamhāra" and "Kumarasambhava" ("Birth of the war god"), as well as the lyric "Meghaduuta" ("The cloud messenger").
The greatest writer of the time was Kalidasa. Poetry in the Gupta age tended towards a few genres: religious and meditative poetry, lyric poetry, narrative histories (the most popular of the secular literatures), and drama. Kalidasa excelled at lyric poetry, but he is best known for his dramas. We have three of his plays; all of them are suffused with epic heroism, with comedy, and with erotics. The plays all involve misunderstanding and conflict, but they all end with unity, order, and resolution.
The "Hansa-Sandesha" owes a great deal to its two poetic predecessors, Kalidasa's "Meghaduta" and Valmiki's "Ramayana". Vedanta Desika's use of the "Meghaduta" is extensive and transparently deliberate; his poem is a response to one of India's most famous poems by its most celebrated poet. Vedanta Desika's debt to Valmiki is perhaps more pervasive but less obvious, and possibly less deliberate too. Where the poet consciously plays with Kalidasa's verse, he treats the "Ramayana" more as a much-cherished story. Nevertheless, he is clearly as familiar with the details of Valmiki's poem as with Kalidasa's, and he echoes very specific images and details from the epic.
Subsequently, splits occurred at various instances in the mainstream Bohra community regarding the spiritual appointment in the succession of the representative of the Imam us Satr or Da’i in Ahmedabad between 1422-1640 AD. Two major splits during this period resulted in the formation of three major groups of Bohras: Alavis, Dawoodis and Sulaymanis.
Hinduism and Buddhism were the prominent religions of southeast Asia in the 1st millennium CE, prior to the Islamic conquest that started in the 14th century. The Hindus prevailed in Bali, Indonesia and they have two types of Hindu calendar. One is a 210-day based Pawukon calendar which likely is a pre-Hindu system, and another is similar to lunisolar calendar system found in South India and it is called the Balinese saka calendar which uses Hindu methodology. The names of month and festivals of Balinese Hindus for the most part are different, though the significance and legends have some overlap.
In India, the "Baudhayana Sulba Sutra", the dates of which are given variously as between the 8th and 5th century BC, contains a list of Pythagorean triples discovered algebraically, a statement of the Pythagorean theorem, and a geometrical proof of the Pythagorean theorem for an isosceles right triangle.
The "Apastamba Sulba Sutra" (c. 600 BC) contains a numerical proof of the general Pythagorean theorem, using an area computation. Van der Waerden believed that "it was certainly based on earlier traditions". Carl Boyer states that the Pythagorean theorem in "Śulba-sũtram" may have been influenced by ancient Mesopotamian math, but there is no conclusive evidence in favor or opposition of this possibility.
With contents known much earlier, but in surviving texts dating from roughly the 1st century BC, the Chinese text "Zhoubi Suanjing" (周髀算经), ("The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven") gives a reasoning for the Pythagorean theorem for the (3, 4, 5) triangle—in China it is called the "Gougu theorem" (勾股定理). During the Han Dynasty (202 BC to 220 AD), Pythagorean triples appear in "The Nine Chapters on the Mathematical Art", together with a mention of right triangles. Some believe the theorem arose first in China, where it is alternatively known as the "Shang Gao theorem" (商高定理), named after the Duke of Zhou's astronomer and mathematician, whose reasoning composed most of what was in the "Zhoubi Suanjing".
The Mahāsāṃghika, translated into Chinese as the "Móhēsēngzhī Lǜ" (Taishō Tripiṭaka 1425) describes several units of time, including "shùn" or "shùnqǐng" () and "niàn". According to this text, "niàn" is the smallest unit of time at 18 milliseconds and a "shùn" is 360 milliseconds. It also describes larger units of time, including a "tánzhǐ" () which is 7.2 seconds long, a "luóyù" () which is 2 minutes 24 seconds long, and a "xūyú" (), which is of a day at 48 minutes long.
Along with Yaʿqūb ibn Ṭāriq and his father he helped translate the Indian astronomical text by Brahmagupta (fl. 7th century), the "Brāhmasphuṭasiddhānta", into Arabic as "Az-Zīj ‛alā Sinī al-‛Arab"., or the "Sindhind". This translation was possibly the vehicle by means of which the Hindu numerals were transmitted from India to Islam.
The "Zīj", written around 770, was based on a Sanskrit work, thought to be similar to the "Brāhmasphuṭasiddhānta". This work was brought to the court of al-Mansūr from Sindh, reportedly by a Sindhi astronomer named Kankah.
Brahmagupta's Brāhmasphuṭasiddhānta is the first book that mentions zero as a number, hence Brahmagupta is considered the first to formulate the concept of zero. The current system of the four fundamental operations (addition, subtraction, multiplication and division) based on the Hindu-Arabic number system also first appeared in Brahmasphutasiddhanta. It was also one of the first texts to provide concrete ideas on positive and negative numbers.
The book was written completely in verse and does not contain any kind of mathematical notation. Nevertheless, it contained the first clear description of the quadratic formula (the solution of the quadratic equation).