Vātsyāyana

Vātsyāyana is the name of a Hindu philosopher in the Vedic tradition who is believed to have lived during time of the Gupta Empire (4th to 6th centuries CE) in India. His name appears as the author of the Kama Sutra and of Nyāya Sutra Bhāshya, the first commentary on Gotama's Nyāya Sutras.

His name is sometimes confused with Mallanaga, the prophet of the Asuras, to whom the origin of erotic science is attributed. This is an error; as Danielou says: 

The attribution of the first name Mallanaga to Vatsyayana is due to the confusion of his role as editor of the Kama Sutra with that of the mythical creator of erotic science.

Hardly anything is known about him, although it is believed that his disciples went on his instructions, on the request of the Hindu Kings in the Himalayan range to influence the hill tribals to give up the pagan cult of sacrifices. He is said to have created the legend of Tara among the hill tribes as a tantric goddess. Later as the worship spread to the east Garo hills,the goddess manifest of a 'yoni' goddess Kamakhya was created.His interest in human sexual behavior as a medium of attaining spirituality was recorded in his treatise Kama Sutra.

At the close of the Kama Sutra this is what he writes about himself:

“

After reading and considering the works of Babhravya and other ancient authors, and thinking over the meaning of the rules given by them, this treatise was composed, according to the precepts of the Holy Writ, for the benefit of the world, by Vatsyayana, while leading the life of a religious student at Benares, and wholly engaged in the contemplation of the Deity. This work is not to be used merely as an instrument for satisfying our desires. A person acquainted with the true principles of this science, who preserves his Dharma (virtue or religious merit), his Artha (worldly wealth) and his Kama (pleasure or sensual gratification), and who has regard to the customs of the people, is sure to obtain the mastery over his senses. In short, an intelligent and knowing person attending to Dharma and Artha and also to Kama, without becoming the slave of his passions, will obtain success in everything that he may do.'

”

It is impossible to fix the exact date either of the life of Vatsyayana or of his work. It is believed that he must have lived between the 1st and 6th century AD, on the following grounds: He mentions that Satakarni Satavahana, a king of Kuntal, killed Malayevati his wife with an instrument called Katamari by striking her in the passion of love. Vatsyayana quotes this case to warn people of the danger arising from some old customs of striking women when under the influence of sexual passion. This king of Kuntal is believed to have lived and reigned during the 1st century AD, and consequently Vatsyayana must have lived after him. On the other hand, another author, Varahamihira, in the eighteenth chapter of his "Brihatsanhita", discusses of the science of love, and appears to have borrowed largely from Vatsyayana on the subject. Varahamihira is believed to have lived during the 6th century, and therefore Vatsyayana must have written his works before the 6th century.

The Kama Sutra (Sanskrit: कामसूत्र  , Kāmasūtra) is an ancient Indian Hindu  text widely considered to be the standard work on human sexual behavior in Sanskrit literature written by Vātsyāyana. A portion of the work consists of practical advice on sexual intercourse.  It is largely in prose, with many inserted anustubh poetry verses. "Kāma" which is one of the three goals of Hindu life, means sensual or sexual pleasure, and "sūtra" literally means a thread or line that holds things together, and more metaphorically refers to an aphorism (or line, rule, formula), or a collection of such aphorisms in the form of a manual. Contrary to popular perception, especially in the western world, Kama sutra is not just an exclusive sex manual; it presents itself as a guide to a virtuous and gracious living that discusses the nature of love, family life and other aspects pertaining to pleasure oriented faculties of human life. 

The Kama Sutra is the oldest and most notable of a group of texts known generically as Kama Shastra (Sanskrit: Kāma Śāstra).  Traditionally, the first transmission of Kama Shastra or "Discipline of Kama" is attributed to Nandi the sacred bull, Shiva's doorkeeper, who was moved to sacred utterance by overhearing the lovemaking of the god and his wife Parvati and later recorded his utterances for the benefit of mankind. 

Historians attribute Kamasutra to be composed between 400 BCE and 200 CE.  John Keay says that the Kama Sutra is a compendium that was collected into its present form in the 2nd century CE. 

Content

Vatsyayana's Kama Sutra has 1250 verses, distributed in 36 chapters, which are further organized into 7 parts.  According to both the Burton and Doniger translations, the contents of the book are structured into 7 parts like the following:

1. General remarks

5 chapters on contents of the book, three aims and priorities of life, the acquisition of knowledge, conduct of the well-bred townsman, reflections on intermediaries who assist the lover in his enterprises.

2. Amorous advances/Sexual union

10 chapters on stimulation of desire, types of embraces, caressing and kisses, marking with nails, biting and marking with teeth, on copulation (positions), slapping by hand and corresponding moaning, virile behavior in women, superior coition and oral sex, preludes and conclusions to the game of love. It describes 64 types of sexual acts.

3. Acquiring a wife

5 chapters on forms of marriage, relaxing the girl, obtaining the girl, managing alone, union by marriage.

4. Duties and privileges of the wife

2 chapters on conduct of the only wife and conduct of the chief wife and other wives.

5. Other men's wives

6 chapters on behavior of woman and man, how to get acquainted, examination of sentiments, the task of go-between, the king's pleasures, behavior in the women's quarters.

6. About courtesans

6 chapters on advice of the assistants on the choice of lovers, looking for a steady lover, ways of making money, renewing friendship with a former lover, occasional profits, profits and losses.

7. Occult practices

2 chapters on improving physical attractions, arousing a weakened sexual power.

Pleasure and spirituality

Dharma: Virtuous living.

Some Indian philosophies follow the "four main goals of life",  known as the purusharthas: 

1. Artha: Material prosperity.
2. Kama: Aesthetic and erotic pleasure. 
3. Moksha: Liberation.

Dharma, Artha and Kama are aims of everyday life, while Moksha is release from the cycle of death and rebirth. The Kama Sutra (Burton translation) says:

"Dharma is better than Artha, and Artha is better than Kama. But Artha should always be first practised by the king for the livelihood of men is to be obtained from it only. Again, Kama being the occupation of public women, they should prefer it to the other two, and these are exceptions to the general rule." (Kama Sutra 1.2.14) 

Of the first three, virtue is the highest goal, a secure life the second and pleasure the least important. When motives conflict, the higher ideal is to be followed. Thus, in making money virtue must not be compromised, but earning a living should take precedence over pleasure, but there are exceptions.

In childhood, Vātsyāyana says, a person should learn how to make a living; youth is the time for pleasure, and as years pass one should concentrate on living virtuously and hope to escape the cycle of rebirth.^[18]The Kama Sutra acknowledges that the senses can be dangerous: 'Just as a horse in full gallop, blinded by the energy of his own speed, pays no attention to any post or hole or ditch on the path, so two lovers, blinded by passion, in the friction of sexual battle, are caught up in their fierce energy and pay no attention to danger'(2.7.33).

Also the Buddha preached a Kama Sutra, which is located in the Atthakavagga (sutra number 1). This Kama Sutra, however, is of a very different nature as it warns against the dangers that come with the search for pleasures of the senses.

Many in the Western world wrongly consider the Kama Sutra to be a manual for tantric sex.  While sexual practices do exist within the very wide tradition of Hindu Tantra, the Kama Sutra is not a Tantric text, and does not touch upon any of the sexual rites associated with some forms of Tantric practice.

Translations

The most widely known English translation of the Kama Sutra was privately printed in 1883. It is usually attributed to renowned orientalist and author Sir Richard Francis Burton, but the chief work was done by the pioneering Indian archaeologist, Bhagwanlal Indraji, under the guidance of Burton's friend, the Indian civil servant Forster Fitzgerald Arbuthnot, and with the assistance of a student, Shivaram Parshuram Bhide. Burton acted as publisher, while also furnishing the edition with footnotes whose tone ranges from the jocular to the scholarly. Burton says the following in its introduction:

It may be interesting to some persons to learn how it came about that Vatsyayana was first brought to light and translated into the English language. It happened thus. While translating with the pundits the 'Anunga Runga, or the stage of love', reference was frequently found to be made to one Vatsya. The sage Vatsya was of this opinion, or of that opinion. The sage Vatsya said this, and so on. Naturally questions were asked who the sage was, and the pundits replied that Vatsya was the author of the standard work on love in Sanskrit literature, that no Sanscrit library was complete without his work, and that it was most difficult now to obtain in its entire state. The copy of the manuscript obtained in Bombay was defective, and so the pundits wrote to Benares, Calcutta and Jaipur for copies of the manuscript from Sanskrit libraries in those places. Copies having been obtained, they were then compared with each other, and with the aid of a Commentary called 'Jayamanglia' a revised copy of the entire manuscript was prepared, and from this copy the English translation was made. The following is the certificate of the chief pundit:

'The accompanying manuscript is corrected by me after comparing four different copies of the work. I had the assistance of a Commentary called "Jayamangla" for correcting the portion in the first five parts, but found great difficulty in correcting the remaining portion, because, with the exception of one copy thereof which was tolerably correct, all the other copies I had were far too incorrect. However, I took that portion as correct in which the majority of the copies agreed with each other.'

In the introduction to her own translation, Wendy Doniger, professor of the history of religions at the University of Chicago, writes that Burton "managed to get a rough approximation of the text published in English in 1883, nasty bits and all". The philologist and Sanskritist Professor Chlodwig Werba, of the Institute of Indology at the University of Vienna, regards the 1883 translation as being second only in accuracy to the academic German-Latin text published by Richard Schmidt in 1897.

A noteworthy translation by Indra Sinha was published in 1980. In the early 1990s its chapter on sexual positions began circulating on the internet as an independent text and today is often assumed to be the whole of the Kama Sutra.  

Alain Daniélou contributed a noteworthy translation called The Complete Kama Sutra in 1994. This translation, originally into French, and thence into English, featured the original text attributed to Vatsyayana, along with a medieval and a modern commentary. Unlike the 1883 version, Alain Daniélou's new translation preserves the numbered verse divisions of the original, and does not incorporate notes in the text. He includes English translations of two important commentaries:

• The Jayamangala commentary, written in Sanskrit by Yashodhara during the Middle Ages, as page footnotes.
• A modern commentary in Hindi by Devadatta Shastri, as endnotes.

Daniélou  translated all Sanskrit words into English (but uses the word "brahmin"). He leaves references to the sexual organs as in the original: persistent usage of the words "lingam" and "yoni" to refer to them in older translations of the Kama Sutra is not the usage in the original Sanskrit; he argues that "to a modern Hindu "lingam" and "yoni" mean specifically the sexual organs of the god Shiva and his wife, and using those words to refer to humans' sexual organs would seem irreligious." The view that lingam means only "sexual organs" is disputed by academics likeS.N.Balagangadhara. 

An English translation by Wendy Doniger and Sudhir Kakar, an Indian psychoanalyst and senior fellow at Center for Study of World Religions at Harvard University, was published by Oxford University Press in 2002. Doniger contributed the Sanskrit expertise while Kakar provided a psychoanalytic interpretation of the text. 

Varāhamihira &Vishnu Sharman

Varāhamihira   (Devanagari: वराहमिहिर) (505–587 CE), also called Varaha or Mihira, was an Indian astronomer,mathematician, and astrologer who lived in Ujjain.  He is considered to be one of the nine jewels (Navaratnas) of the court of legendary rulerVikramaditya (thought to be the Gupta emperor Chandragupta II Vikramaditya).

Works

He was the first one to mention in his work Pancha Siddhantika that the ayanamsa, or the shifting of the equinox is 50.32 seconds.

Pancha-Siddhantika

Varahamihira's main work is the book Pañcasiddhāntikā (or Pancha-Siddhantika, "[Treatise] on the Five [Astronomical] Canons) dated ca. 575 CE gives us information about older Indian texts which are now lost. The work is a treatise on mathematical astronomy and it summarises five earlier astronomical treatises, namely the Surya Siddhanta, Romaka Siddhanta, Paulisa Siddhanta,Vasishtha Siddhanta and Paitamaha Siddhantas. It is a compendium of Vedanga Jyotisha as well as Hellenistic astronomy (including Greek, Egyptian and Roman elements).  He was the first one to mention in his work Pancha Siddhantika that the ayanamsa, or the shifting of the equinox is 50.32 seconds.6655

The 11th century Arabian scholar Alberuni also described the details of "The Five Astronomical Canons":

"They [the Indians] have 5 Siddhāntas:

• Sūrya-Siddhānta, ie. the Siddhānta of the Sun, composed by Lāṭadeva,
• Vasishtha-siddhānta, so called from one of the stars of the Great Bear, composed by Vishnucandra,
• Pulisa-siddhānta, so called from Paulisa, the Greek, from the city of Saintra, which is supposed to be Alexandria, composed by Pulisa.
• Romaka-siddhānta, so called from the Rūm, ie. the subjects of the Roman Empire, composed by Śrīsheṇa.

Brihat-Samhita

Varahamihira's other most important contribution is the encyclopedic Brihat-Samhita. It covers wide ranging subjects of human interest, including astrology, planetary movements, eclipses, rainfall, clouds, architecture, growth of crops, manufacture of perfume, matrimony, domestic relations, gems, pearls, and rituals. The volume expounds on gemstone evaluation criterion found in the Garuda Purana, and elaborates on the sacred Nine Pearls from the same text. It contains 106 chapters and is known as the "great compilation".

On Astrology

He was also an astrologer. He wrote on all the three main branches of Jyotisha astrology:

• Brihat Jataka - is considered as one the five main treatises on Hindu astrology on horoscopy.
• Daivaigya Vallabha
• Laghu Jataka
• Yoga Yatra
• Vivaha Patal

• His son Prithuyasas also contributed in the Hindu astrology; his book "Hora Saara" is a famous book on horoscopy.

Western influences

The Romaka Siddhanta ("Doctrine of the Romans") and the Paulisa Siddhanta ("Doctrine of Paul") were two works of Western origin which influenced Varahamihira's thought, though this view is controversial as there is much evidence to suggest that it was actually Vedic thought indigenous to India which first influenced Western astrologers and subsequently came back to India reformulated

A comment in the Brihat-Samhita by Varahamihira says: "The Greeks, though foreign, must be honored since they have shown tremendous interest in our science....." ("mleccha hi yavanah tesu samyak shastram kdamsthitam/ rsivat te 'p i pujyante kim punar daivavid dvijah" (Brihat-Samhita 2.15)).

Some important trigonometric results attributed to Varahamihira

He not only presented his own observations, but embellished them in attractive poetic and metrical styles. The usage of a large variety of meters is especially evident in his Brihat Jataka andBrihat-Samhita.

1. ^ "the Pañca-siddhāntikā ("Five Treatises"), a compendium of Greek, Egyptian, Roman and Indian astronomy. Varāhamihira's knowledge of Western astronomy was thorough. In 5 sections, his monumental work progresses through native Indian astronomy and culminates in 2 treatises on Western astronomy, showing calculations based on Greek and Alexandrian reckoning and even giving complete Ptolemaic mathematical charts and tables. Encyclopædia Britannica (2007) s.v.Varahamihira ^ 2. E. C. Sachau, Alberuni's India (1910), vol. I, p. 153

Vishnu Sharman (Sanskrit: विष्णु शर्मा) was an Indian scholar and author who is believed to have written the Panchatantra collection of fables.  The exact period of the composition of the Panchatantra is uncertain, and estimates vary from 1200 BCE to 300 CE. Some scholars place him in the 3rd century BCE. 

Vishnu Sharman is one of the most widely translated non-religious authors in history. The Panchatantra was translated into Pahlavi in 570 CE byBorzūya and into Arabic in 750 CE by Persian scholar Abdullah Ibn al-Muqaffa as Kalīlah wa Dimnah.   In Baghdad, the translation commissioned by Al-Mansur, the second Abbasid Caliph, is claimed to have become "second only to the Qu'ran in popularity. As early as the eleventh century this work reached Europe, and before 1600 it existed in Greek, Latin, Spanish, Italian, German, English, Old Slavonic,Czech, and perhaps other Slavonic languages. Its range has extended from Java to Iceland.  In France, "at least eleven Panchatantra tales are included in the work of La Fontaine. 

Legend

The prelude to the Panchatantra identifies Vishnu Sharma as the author of the work. Since there is no other independent external evidence about him, "it is impossible to say whether he was the historical author . . .or is himself a literary invention.  Based on analysis of various Indian recensions and the geographical features and animals described in the stories, Kashmir is suggested to be his birthplace by various scholars. 

The prelude narrates the story of how Vishnu Sharma supposedly created the Panchatantra. There was a king called Sudarshan who ruled a kingdom, most likely in patliputra. The capital of his kingdom was a city called Mahilaropya (महिलारोप्य), whose location on the current map of India is unknown.  King Sudarshan had three sons named Bahushakti, Ugrashakti and Anantshakti.  Though King Sudarshan himself was both a scholar and a powerful ruler, his sons were "all dullards.  The king despaired of his three princes' inability to learn, and approached his ministers for counsel. They presented him with conflicting advice, but the words of one, called Sumati, rang true to the king. He said that the sciences, politics and diplomacy were limitless disciplines that took a lifetime to master formally. Instead of teaching the princes scriptures and texts, they should somehow be taught the wisdom inherent in them, and the aged scholar Vishnu Sharma was the man to do it. 

Vishnu Sharma was invited to the court, where the king offered him a hundred land grants if he could teach the princes.^[15] Vishnu Sharma declined the promised award, saying he did not sell knowledge for money, but accepted the task of making the princes wise to the ways of politics and leadership within six months. Vishnu Sharma knew that he could never instruct these three students through conventional means. He had to employ a less orthodox way, and that was to tell a succession of animal fables - one weaving into another - that imparted to them the wisdom they required to succeed their father.The collection of his work was done by pandit Narayana and is called hitopdesh. Adapting stories that had been told for thousands of years in India, panchatantra was composed into an entertaining five part work to communicate the essence of diplomacy, relationships, politics and administration to the princes. These five discourses, became the Panchatantra, meaning the five (pancha) treatises (tantra).

After Vishnu Sharma taught king's son, he tried to give land and other gifts to him but he declined then he gave Vishnu Sharma, title of "Pathak" and after him his family started using "Pathak" as surname. Now also in villages there is belief in Pathak's that they don't have to charge anything for knowledge transfer.

Aryabhata

Aryabhata    or Aryabhata I (476–550 CE)  was the first in the line of greatmathematician-astronomers from the classical age of Indian mathematics and Indian astronomy. His most famous works are the Āryabhaṭīya (499 CE, when he was 23 years old)  and the Arya-siddhanta.

The works of Aryabhata dealt with mainly mathematics and astronomy. He also worked on the approximation for pi.

Name

While there is a tendency to misspell his name as "Aryabhatta" by analogy with other names having the "bhatta" suffix, his name is properly spelled Aryabhata: every astronomical text spells his name thus, including Brahmagupta's references to him "in more than a hundred places by name".  Furthermore, in most instances "Aryabhatta" does not fit the metre either. 

Time and place of birth

Aryabhata mentions in the Aryabhatiya that it was composed 3,630 years into the Kali Yuga, when he was 23 years old. This corresponds to 499 CE, and implies that he was born in 476. 

Aryabhata was born in Bihar, India Ansari, S.M.R. (March 1977). "Aryabhata I, His Life and His Contributions". Bulletin of the Astronomical Society of India 5 However, early Buddhist texts describe Ashmaka as being further south, in dakshinapath or the Deccan, while other texts describe the Ashmakas as having fought Alexander.

Education

It is fairly certain that, at some point, he went to Kusumapura for advanced studies and lived there for some time.  Both Hindu and Buddhist tradition, as well as Bhāskara I (CE 629), identify Kusumapura as Pāṭaliputra, modern Patna.  A verse mentions that Aryabhata was the head of an institution (kulapati) at Kusumapura, and, because the university of Nalanda was in Pataliputra at the time and had an astronomical observatory, it is speculated that Aryabhata might have been the head of the Nalanda university as well. Aryabhata is also reputed to have set up an observatory at the Sun temple in Taregana, Bihar. 

Other hypotheses

Some archeological evidence suggests that Aryabhata could have originated from the present day Kodungallur which was the historical capital city of Thiruvanchikkulam of ancient Kerala.  For instance, one hypothesis was that aśmaka (Sanskrit for "stone") may be the region in Kerala that is now known as Koṭuṅṅallūr, based on the belief that it was earlier known as Koṭum-Kal-l-ūr ("city of hard stones"); however, old records show that the city was actually Koṭum-kol-ūr ("city of strict governance"). Similarly, the fact that several commentaries on the Aryabhatiya have come from Kerala were used to suggest that it was Aryabhata's main place of life and activity; however, many commentaries have come from outside Kerala.

Aryabhata mentions "Lanka" on several occasions in the Aryabhatiya, but his "Lanka" is an abstraction, standing for a point on the equator at the same longitude as his Ujjayini. 

Works

Aryabhata is the author of several treatises on mathematics and astronomy, some of which are lost. His major work, Aryabhatiya, a compendium of mathematics and astronomy, was extensively referred to in the Indian mathematical literature and has survived to modern times. The mathematical part of the Aryabhatiya covers arithmetic, algebra, plane trigonometry, and spherical trigonometry. It also contains continued fractions, quadratic equations, sums-of-power series, and a table of sines.

The Arya-siddhanta, a lot work on astronomical computations, is known through the writings of Aryabhata's contemporary, Varahamihira, and later mathematicians and commentators, includingBrahmagupta and Bhaskara I. This work appears to be based on the older Surya Siddhanta and uses the midnight-day reckoning, as opposed to sunrise in Aryabhatiya. It also contained a description of several astronomical instruments: the gnomon (shanku-yantra), a shadow instrument (chhAyA-yantra), possibly angle-measuring devices, semicircular and circular (dhanur-yantra /chakra-yantra), a cylindrical stick yasti-yantra, an umbrella-shaped device called the chhatra-yantra, and water clocks of at least two types, bow-shaped and cylindrical. 

A third text, which may have survived in the Arabic translation, is Al ntf or Al-nanf. It claims that it is a translation by Aryabhata, but the Sanskrit name of this work is not known. Probably dating from the 9th century, it is mentioned by the Persian scholar and chronicler of India, Abū Rayhān al-Bīrūnī. 

Aryabhatiya

Direct details of Aryabhata's work are known only from the Aryabhatiya. The name "Aryabhatiya" is due to later commentators. Aryabhata himself may not have given it a name. His discipleBhaskara I calls it Ashmakatantra (or the treatise from the Ashmaka). It is also occasionally referred to as Arya-shatas-aShTa (literally, Aryabhata's 108), because there are 108 verses in the text. It is written in the very terse style typical of sutra literature, in which each line is an aid to memory for a complex system. Thus, the explication of meaning is due to commentators. The text consists of the 108 verses and 13 introductory verses, and is divided into four pādas or chapters:

1. Gitikapada: (13 verses): large units of time—kalpa, manvantra, and yuga—which present a cosmology different from earlier texts such as Lagadha's Vedanga Jyotisha (c. 1st century BCE). There is also a table of sines (jya), given in a single verse. The duration of the planetary revolutions during a mahayuga is given as 4.32 million years.
2. Ganitapada (33 verses): covering mensuration (kṣetra vyāvahāra), arithmetic and geometric progressions, gnomon / shadows (shanku-chhAyA), simple, quadratic, simultaneous, andindeterminate equations
3. Kalakriyapada (25 verses): different units of time and a method for determining the positions of planets for a given day, calculations concerning the intercalary month (adhikamAsa),kShaya-tithis, and a seven-day week with names for the days of week.
4. Golapada (50 verses): Geometric/trigonometric aspects of the celestial sphere, features of the ecliptic, celestial equator, node, shape of the earth, cause of day and night, rising ofzodiacal signs on horizon, etc. In addition, some versions cite a few colophons added at the end, extolling the virtues of the work, etc.

The Aryabhatiya presented a number of innovations in mathematics and astronomy in verse form, which were influential for many centuries. The extreme brevity of the text was elaborated in commentaries by his disciple Bhaskara I (Bhashya, c. 600 CE) and by Nilakantha Somayaji in his Aryabhatiya Bhasya, (1465 CE). He was not only the first to find the radius of the earth but was the only one in ancient time including the Greeks and the Romans to find the volume of the earth.

Mathematics

Place value system and zero

The place-value system, first seen in the 3rd century Bakhshali Manuscript, was clearly in place in his work. While he did not use a symbol for zero, the French mathematician Georges Ifrahexplains that knowledge of zero was implicit in Aryabhata's place-value system as a place holder for the powers of ten with null coefficients^[13]

However, Aryabhata did not use the Brahmi numerals. Continuing the Sanskritic tradition from Vedic times, he used letters of the alphabet to denote numbers, expressing quantities, such as the table of sines in a mnemonic form. 

Approximation of π

Aryabhata worked on the approximation for pi (), and may have come to the conclusion that  is irrational. In the second part of the Aryabhatiyam (gaṇitapāda 10), he writes:

caturadhikam śatamaṣṭaguṇam dvāṣaṣṭistathā sahasrāṇām
ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ.
"Add four to 100, multiply by eight, and then add 62,000. By this rule the circumference of a circle with a diameter of 20,000 can be approached."  

This implies that the ratio of the circumference to the diameter is ((4 + 100) × 8 + 62000)/20000 = 62832/20000 = 3.1416, which is accurate to five significant figures.

It is speculated that Aryabhata used the word āsanna (approaching), to mean that not only is this an approximation but that the value is incommensurable (or irrational). If this is correct, it is quite a sophisticated insight, because the irrationality of pi was proved in Europe only in 1761 by Lambert. 

After Aryabhatiya was translated into Arabic (c. 820 CE) this approximation was mentioned in Al-Khwarizmi's book on algebra. 

Trigonometry

In Ganitapada 6, Aryabhata gives the area of a triangle as

tribhujasya phalashariram samadalakoti bhujardhasamvargah

that translates to: "for a triangle, the result of a perpendicular with the half-side is the area." 

Aryabhata discussed the concept of sine in his work by the name of ardha-jya, which literally means "half-chord". For simplicity, people started calling it jya. When Arabic writers translated his works from Sanskrit into Arabic, they referred it as jiba. However, in Arabic writings, vowels are omitted, and it was abbreviated as jb. Later writers substituted it with jaib, meaning "pocket" or "fold (in a garment)". (In Arabic, jiba is a meaningless word.) Later in the 12th century, when Gherardo of Cremona translated these writings from Arabic into Latin, he replaced the Arabic jaib with its Latin counterpart, sinus, which means "cove" or "bay"; thence comes the English since. Alphabetic code has been used by him to define a set of increments. If we use Aryabhata's table and calculate the value of sin(30) (corresponding to hasjha) which is 1719/3438 = 0.5; the value is correct. His alphabetic code is commonly known as the Aryabhata cipher. 

Indeterminate equations

A problem of great interest to Indian mathematicians since ancient times has been to find integer solutions to equations that have the form ax + by = c, a topic that has come to be known asdiophantine equations. This is an example from Bhāskara's commentary on Aryabhatiya:

Find the number which gives 5 as the remainder when divided by 8, 4 as the remainder when divided by 9, and 1 as the remainder when divided by 7

That is, find N = 8x+5 = 9y+4 = 7z+1. It turns out that the smallest value for N is 85. In general, diophantine equations, such as this, can be notoriously difficult. They were discussed extensively in ancient Vedic text Sulba Sutras, whose more ancient parts might date to 800 BCE. Aryabhata's method of solving such problems is called the kuṭṭaka (कुट्टक) method. Kuttaka means "pulverizing" or "breaking into small pieces", and the method involves a recursive algorithm for writing the original factors in smaller numbers. Today this algorithm, elaborated by Bhaskara in 621 CE, is the standard method for solving first-order diophantine equations and is often referred to as the Aryabhata algorithm. The diophantine equations are of interest in cryptology, and theRSA Conference, 2006, focused on the kuttaka method and earlier work in the Sulbasutras.

Algebra

In Aryabhatiya Aryabhata provided elegant results for the summation of series of squares and cubes: 

and

Astronomy

Aryabhata's system of astronomy was called the audAyaka system, in which days are reckoned from uday, dawn at lanka or "equator". Some of his later writings on astronomy, which apparently proposed a second model (or ardha-rAtrikA, midnight) are lost but can be partly reconstructed from the discussion in Brahmagupta's khanDakhAdyaka. In some texts, he seems to ascribe the apparent motions of the heavens to the Earth's rotation. He may have believed that the planet's orbits as elliptical rather than circular. 

Motions of the solar system

Aryabhata correctly insisted that the earth rotates about its axis daily, and that the apparent movement of the stars is a relative motion caused by the rotation of the earth, contrary to the then-prevailing view in other parts of the world, that the sky rotated. This is indicated in the first chapter of the Aryabhatiya, where he gives the number of rotations of the earth in a yuga,  and made more explicit in his gola chapter: 

In the same way that someone in a boat going forward sees an unmoving [object] going backward, so [someone] on the equator sees the unmoving stars going uniformly westward. The cause of rising and setting [is that] the sphere of the stars together with the planets [apparently?] turns due west at the equator, constantly pushed by the cosmic wind.

Aryabhata described a geocentric model of the solar system, in which the Sun and Moon are each carried by epicycles. They in turn revolve around the Earth. In this model, which is also found in the Paitāmahasiddhānta (c. CE 425), the motions of the planets are each governed by two epicycles, a smaller manda (slow) and a larger śīghra (fast).   The order of the planets in terms of distance from earth is taken as: the Moon, Mercury, Venus, the Sun, Mars, Jupiter, Saturn, and the asterisms." 

The positions and periods of the planets was calculated relative to uniformly moving points. In the case of Mercury and Venus, they move around the Earth at the same mean speed as the Sun. In the case of Mars, Jupiter, and Saturn, they move around the Earth at specific speeds, representing each planet's motion through the zodiac. Most historians of astronomy consider that this two-epicycle model reflects elements of pre-Ptolemaic Greek astronomy.  Another element in Aryabhata's model, the śīghrocca, the basic planetary period in relation to the Sun, is seen by some historians as a sign of an underlying heliocentric model. 

Eclipses

Solar and lunar eclipses were scientifically explained by Aryabhata. He states that the Moon and planets shine by reflected sunlight. Instead of the prevailing cosmogony in which eclipses were caused by pseudo-planetary nodes Rahu and Ketu, he explains eclipses in terms of shadows cast by and falling on Earth. Thus, the lunar eclipse occurs when the moon enters into the Earth's shadow (verse gola.37). He discusses at length the size and extent of the Earth's shadow (verses gola.38–48) and then provides the computation and the size of the eclipsed part during an eclipse. Later Indian astronomers improved on the calculations, but Aryabhata's methods provided the core. His computational paradigm was so accurate that 18th century scientist Guillaume Le Gentil, during a visit to Pondicherry, India, found the Indian computations of the duration of the lunar eclipse of 30 August 1765 to be short by 41 seconds, whereas his charts (by Tobias Mayer, 1752) were long by 68 seconds. 

Sidereal periods

Considered in modern English units of time, Aryabhata calculated the sidereal rotation (the rotation of the earth referencing the fixed stars) as 23 hours, 56 minutes, and 4.1 seconds;  the modern value is 23:56:4.091. Similarly, his value for the length of the sidereal year at 365 days, 6 hours, 12 minutes, and 30 seconds (365.25858 days)  is an error of 3 minutes and 20 seconds over the length of a year (365.25636 days). 

Heliocentrism

As mentioned, Aryabhata advocated an astronomical model in which the Earth turns on its own axis. His model also gave corrections (the śīgra anomaly) for the speeds of the planets in the sky in terms of the mean speed of the sun. Thus, it has been suggested that Aryabhata's calculations were based on an underlying heliocentric model, in which the planets orbit the Sun, though this has been rebutted.  It has also been suggested that aspects of Aryabhata's system may have been derived from an earlier, likely pre-Ptolemaic Greek, heliocentric model of which Indian astronomers were unaware,  though the evidence is scant.  The general consensus is that a synodic anomaly (depending on the position of the sun) does not imply a physically heliocentric orbit (such corrections being also present in late Babylonian astronomical texts), and that Aryabhata's system was not explicitly heliocentric. 

Legacy

His definitions of sine (jya), cosine (kojya), versine (utkrama-jya), and inverse sine (otkram jya) influenced the birth of trigonometry. He was also the first to specify sine and versine (1 − cos x) tables, in 3.75° intervals from 0° to 90°, to an accuracy of 4 decimal places.

Aryabhata's work was of great influence in the Indian astronomical tradition and influenced several neighbouring cultures through translations. The Arabic translation during the Islamic Golden Age (c. 820 CE), was particularly influenced. Some of his results are cited by Al-Khwarizmi and in the 10th century Al-Biruni stated that Aryabhata's followers believed that the Earth rotated on its axis.

In fact, modern names "sine" and "cosine" are mistranscriptions of the words jya and kojya as introduced by Aryabhata. As mentioned, they were translated as jiba and kojiba in Arabic and then misunderstood by Gerard of Cremona while translating an Arabic geometry text to Latin. He assumed that jiba was the Arabic word jaib, which means "fold in a garment", L. sinus (c. 1150). 

Aryabhata's astronomical calculation methods were also very influential. Along with the trigonometric tables, they came to be widely used in the Islamic world and used to compute many Arabic astronomical tables (zijes). In particular, the astronomical tables in the work of theArabic Spain scientist Al-Zarqali (11th century) were translated into Latin as the Tables of Toledo (12th c.) and remained the most accurate ephemeris used in Europe for centuries.

Calendric calculations devised by Aryabhata and his followers have been in continuous use in India for the practical purposes of fixing thePanchangam (the Hindu calendar). In the Islamic world, they formed the basis of the Jalali calendar introduced in 1073 CE by a group of astronomers including Omar Khayyam,  versions of which (modified in 1925) are the national calendars in use in Iran and Afghanistan today. The dates of the Jalali calendar are based on actual solar transit, as in Aryabhata and earlier Siddhanta calendars. This type of calendar requires an ephemeris for calculating dates. Although dates were difficult to compute, seasonal errors were less in the Jalali calendar than in the Gregorian calendar.

India's first satellite Aryabhata and the lunar crater Aryabhata are named in his honour. An Institute for conducting research in astronomy, astrophysics and atmospheric sciences is theAryabhatta Research Institute of Observational Sciences (ARIOS) near Nainital, India. The inter-school Aryabhata Maths Competition is also named after him,  as is Bacillus aryabhata, a species of bacteria discovered by ISRO scientists in 2009.