Buddha Boy and Prahlad gyani- Modern Rishis and Yogi Live without intake

Ram Bahadur Bomjon, also called the Buddha Boy born in Nepal which is the birth place of Gautam Buddha. Coincidentally, the name of the mother of this Buddha Boy is also Mayadevi. There are a number of websites and blogs on the web which narrate stories of Ram Bahadur Bomjon. It is pretty good to inform you that Ratanpur, Nepal, birth place of Ram Bahadur Bomjon is just 10 km far from my hometown. This fact obliged me to write at least a single article about him. Even though, my words in this article about Ram Bahadur Bomjon are not my personal thoughts. What I believe and what I don't about Ram Bahadur Bomjon is part of another article. In this article I am writing those things about him which I knew from ave news, a news television of Nepal, Prateek Daily, a local Magazine from Birgunj  and some local people talking about him. So you can read and believe this article by using your own conscience.
This Buddha boy was highlighted by national and international Medias when he began his mysterious meditation without food and water on 16th May 2005. Discovery Channel also broadcasted a 45 minutes documentary about him. Then, thousand of devotees, researchers and common people began to flood in the Ratanpur Jungle to visit this new Buddha. People believed him as reincarnation of Gautam Buddha, the founder of Buddhism. His parents narrated his story of childhood as divine story. A specialized website was also created on him.

 WHERE IS HE NOW>>>
Ram Bahadur Bomjon has been forced to take sanctuary in his relative’s house after authorities banned him for meditating in Dharahara community forest in Bara District. He has been meditating in remote forest of Bara district since 2005.Since September 12, 2012 he has been staying at the home of Gopal yonjan in Syangwadanda in Pattharkot VDC-1 of Sarlahi district.

Ram Bahadur Bomjon aka Buddha Boy Pays Nrs 150,000 for Chartered Helicopter

Ram Bahadur Bomjon chartered a helicopter to reach Sindhupalchowk district on September 20th,2013.According to local news source, Ram Bahadur bomjon has been staying in cottage at Todke Bhir in Badegau VDC.

One of the disciple of Bomjon, Ram chandra Khadka who also happens to be the permanent resident of Badegau constructed the cottage and provided all the necessary arrangement for Bomjon's stay there.However, the land where cottage has been built is provided by the Local villagers there.In addition, around 16 people have been staying with Bomjon for his protection along with 8 dogs, 2 rabbits and 1 horse.

“The animals were brought for security purposes,” said Ram Chandra Khadka, adding, “Food for us will be managed for now by locals.” 

More than 500 Local villagers there came to see Ram bahadur Bomjon as soon as the news spread that he came here for meditation.

“He has come here to give continuity to his mediation. He will mediate for five to eight years,” said  Khadka

Ram Bahadur Bomjon who has been in media for all the wrong doing recently said to villagers of Badegau VDC that knowledge and salvation alone were the greatest religion in the world before going to Meditation.

astrology of Ram Bahadur Bomjon
 Ram Bahadur Bomjon (Sanskrit: राम बहादुर बामजान) (born c. 9 April 1990, sometimes spelled Bomjan, Banjan, or Bamjan), also known as Palden Dorje (his monastic name) and now Dharma Sangha, is from Ratanapuri, Bara district, Nepal. Some of his supporters have claimed that he is a reincarnation of Gautama Buddha, but Ram himself has denied this, and many practitioners of Buddhism agree that the Gautama Buddha has entered nirvana and cannot be reborn.^{[citation needed]}
He drew thousands of visitors and media attention by spending months in meditation. Nicknamed the Buddha Boy, he began his meditation on May 16, 2005. He reportedly disappeared from the hollow tree where he had been meditating for months on March 16, 2006, but was found by some followers a week later. He told them he had left his meditation place, where large crowds had been watching him, "because there is no peace". He then went his own way and reappeared elsewhere in Nepal on December 26, 2006, but left again on March 8, 2007. On March 26, 2007, inspectors from the Area Police Post Nijgadh in Ratanapuri found Bomjon meditating inside an underground chamber of about seven square feet.
On 10 November 2008, Bomjon reappeared in Ratanapuri and spoke to a group of devotees in the remote jungle.

Buddhist background

Bomjon is a member of the Tamang community.^{[citation needed]}
Bomjon's story gained popularity because it resembled a legend from the Jataka Nidanakatha about Gautama Budhha's enlightenment. This led some devotees to claim Ram was the reincarnation of Gautama Buddha. On 8 November 2005 Bomjon said, "Tell the people not to call me a Buddha. I don't have the Buddha's energy. I am at the level of a rinpoche." Rinpoche ("precious jewel") is an honorific used in Tibetan Buddhism for a teacher and adept. He said that he will need six more years of meditation before he can become a Buddha.^{[1][not in citation given]}
According to his followers Bomjon may have been or may be a bodhisattva, a person on the path to attaining full enlightenment or Buddhahood for the benefit of all sentient beings. According to the founder of Buddhism, Gautama Buddha, there were innumerable Buddhas before him and there are an infinite number of Buddhas to come. Others claim^{[citation needed]} that Bomjon may be Maitreya Bodhisattva, the predicted incarnation of the future Buddha. Scholars doubt the claims of his supporters.^[2] Mahiswor Raj Bajracharya, president of the Nepal Buddhist Council, has said, "We do not believe he is Buddha. He does not have Buddha's qualities".^[2]
His mother's name is Maya Devi Tamang, the same first name as Buddha's mother. It is reported that his mother fainted when she found out that her son intended to meditate for an indefinite period.^[3]

Wandering to Bara district

Bomjon went missing on 11 March 2006. His followers theorized that he went deeper into the woods to look for a quieter place to meditate.^[4]
On 19 March 2006, a group of Bomjon's followers met with him about 2 miles (3.2 km) southwest of his meditation site. They say they spoke to him for thirty minutes, during which Bomjon said, "There is no peace here," and that he would return in six years, which would be in 2011 or 2012. He left a message for his parents telling them not to worry.^{[5][not in citation given]}
On 25 December 2006, villagers in Bara district spotted Bomjon meditating. He was carrying a sword for protection in the jungle, reminding reporters that "Even Gautama Buddha had to protect himself," and claimed to have eaten nothing but herbs in the interim.^{[6][dead link]} He reiterated his six-year commitment to Buddhist devotion, and said he would allow people to come and observe him, as long as they remained at some distance and did not bother him. When a reporter pointed out that pilgrims to his meditation site would be making donations in his name, he asked for the donations to not be abused or used for commercial purposes. A new wave of visitors came to see him and pray at his new meditation spot.^[7]

Meditating in a pit

On 26 March 2007, news spread of Bomjon meditating underground. Inspector Rameshwor Yadav of the Area Police Post Nijgadh, found Bomjon inside an underground chamber, a bunker-like ditch seven feet square. "His face was clean and hair was combed well," Yadav said. According to him, the chamber had been cemented from all sides and fitted with a tiled roof. Indra Lama, a local deployed as Bomjon's caretaker since the beginning of his intensive meditation, said the chamber was prepared per Bomjon's request. "After granting audience a week ago, he expressed his desire to meditate inside the ground; so we built it," he said.^[8]

Preaching in Halkhoriya jungle

On 2 August 2007, Bomjon addressed a large crowd in Halkhoriya jungle in the Bara district of southern Nepal [see 2nd video down above - ed.]. The Namo Buddha Tapoban Committee, which is devoted to looking after Bomjon, assembled the meeting. A notice about the boy's first-ever preaching was broadcast by a local FM radio station, and the committee also invited people by telephone. Around three thousand people gathered to listen to Bomjon's address. A video was made of the event.^{[9][not in citation given]} According to Krishna Hari, a blogger who wrote an article and took pictures of the meeting, Bomjon's message was, "The only way we can save this nation is through spirituality".^[10]

Claims of media

Some supporters believe that claims of media are less relevant than Bomjon's undisputed ability to remain nearly motionless in the same position day after day, with no regard for extremes of weather including cold winter and monsoon rains. American writer George Saunders visited Bomjon and observed him through a single night, and was impressed by Bomjon's perfectly still stature, even during an evening climate which seemed unbearably cold to the much better clothed journalist.^[11]
In December 2005, a nine-member government committee led by Gunjaman Lama watched Bomjon carefully for 48 hours and observed his not taking any food or water during that time. A video recording was also made of this test from a distance of 3 meters.^[12]

Reappearance in Ratanpuri jungle

On 10 November 2008, Bomjon reappeared and gave blessings to approximately 400,000 pilgrims over a 12-day period in the remote jungle of Ratanpuri, 150 km (93 mi) southeast of Kathmandu, near Nijgadh. His hair was shoulder-length and his body was wrapped in a white cloth. He made two speeches in which he urged people to recognize the compassion in their hearts, and their connection to one another through the all-encompassing soul.^{[13][dead link]}

Controversies

The BBC quoted a local Nepali newspaper which claimed that Bomjon had admitted to slapping some local villagers after having been physically assaulted by them on July 22, 2010. Bomjon said locals had been interrupting his meditation by climbing onto his platform, mimicking him, and attempting to manhandle him, and that he was "therefore forced to beat them". According to the newspaper, he claims he slapped them "two or three times", while the attackers alleged that they had been assaulted more seriously. Bomjon had been fasting before the altercation.^[14]

See also

• Buddhism
• Yogi

References

Jump up ^ Bell, Thomas (21 November 2005). "Pilgrims flock to see 'Buddha boy' said to have fasted six months". The Telegraph (Bara District, Nepal). Retrieved 5 February 2014.

^ Jump up to: ^{a b} "Nepal 'Buddha Boy' returns to jungle". Yahoo! News. 2008-11-22. Archived from the original on December 8, 2008. Retrieved February 11, 2014.

Jump up ^ Navin Singh Khadka (30 November 2005). "Scientists to check Nepal Buddha boy". BBC (Kathmandu). Retrieved 5 February 2014.

Jump up ^ Bhagirath Yogi (11 March 2006). "Nepal's 'Buddha' boy goes missing". BBC.

Jump up ^ "Nepalese Buddha Boy 'reappears'". BBC. 20 March 2006.

Jump up ^ Daily Telegraph, Buddha Boy found after retreating into jungle. 27 December 2006

Jump up ^ "Nepal Buddha Boy 'sighted again'". BBC. 26 December 2006.

Jump up ^ Buddha Boy Update: Ram Bahadur Bomjon Now Meditating in Pit. 28 March 2007

Jump up ^ "Video Clip Taken in Halkhoriya Jungle in August 2, 2007(Sharawan 17 th)". Official Site of Ram Bahadur Bomjan. Archived from the original on December 7, 2008.

Jump up ^ Ram Bahadur Bomjom, the Buddha Boy, Starts Preaching: Arrival of a Meditation Guru or a Religious Zealot?. 3 August 2007

Jump up ^ GQ. The Incredible Buddha Boy

Jump up ^ Indra Adhikari (12 March 2006). "The "Little Buddha" goes missing". Nepalnews.com. Archived from the original on 28 March 2006.

Jump up ^ "Om Namo Guru Buddha Gyani". Paldendorje.com. Retrieved 2012-02-17.

Jump up ^ Lang, Olivia (2010-07-27). "Nepal's 'Buddha boy' investigated for attacking group". BBC. Retr

 

Virgin Atlantic CEO Richard Branson goes Back to His Roots and Discovers He's of Indian Descent

Finding my roots

 

My father’s family left a paper trail that traced back to Madras, India in the 1700s. In 1793, my third great-grandfather, John Edward Branson set sail from Britain to India. After a gruelling six-month journey, in which his boat circled the Cape of Good Hope and crossed the Indian Ocean, he reached South East India – a trading hub of the fast-growing British Empire. He was eventually joined by his father, my fourth great-grandfather, Harry Wilkins Branson; and by 1808 three generations of my ancestors were living in Madras (or Chennai, as the city is known today).
When I heard this, I hoped that they had made the move for the love of adventure and in the spirit of entrepreneurship; and it turns out I was right. The paper trail showed that they moved in search of fortune, and within 10 years became successful businessmen – my great-great-great-grandfather, John, a shopkeeper and my great-great-great-great-grandfather, Harry, an auctioneer. I was terribly excited to discover that the entrepreneur gene runs deep in the Branson blood line.

What’s more exciting is that the Madras archives combined with analysis of my DNA uncovered a very surprising family secret. The baptismal record of my second great-grandmother Eliza Reddy strangely didn’t list her mother. Analysis of my DNA revealed that the reason for this was because my third great-grandmother was Indian. Yes, it turns out I’m part Indian. I couldn’t wipe the smile off my face when I found this out. I’m honored. I wish that my father had got to see these records; he would have been fascinated too.
Like my paternal ancestors, it appears that my maternal side also embraced the spirit of adventure. While probing into my mother’s family lineage something odd happened: my mother’s great-grandparents, Henry and Fanny Flindt, disappeared from English census records. They appeared to completely vanish after 1861, but luckily showed up in Australia!
Baptismal records from Prahran, Melbourne show that some time after 1861, Henry and Fanny moved their entire family to Australia. Again, I had no idea about this part of my family’s history. No wonder I have always loved Australia – it’s in the blood!

 virgin.com

fox 

Ancient Indian Chariots

Chariots are the trademarks of the Aryans. So it's very logical that everything they do would have chariots in it. Rig Veda has several references to chariots. Chariot and its spoked wheel appear in double meanings to represent multiple things. In the discussions on Rig Vedic Gods we've seen that one hymn says Indra rules over the world like the rim, nemi, of the wheel containing the spokes, ara. Even intricate parts of a wheel like dhura, the peg with which the axle pole, aksha, is fastened to the center or navel, nabha, of the wheel, appears in hymns. Almost each and every part of a spoked wheel and chariot is used in RV.
Unlike the chariots and spoked wheels constellations are not the trademarks of the Aryans. In fact the Babylonians were the first in the world, at least in the recorded history, to have stared at the night sky with bewilderment and amazement. They were surely the earliest sky gazers, predating the Aryans by at least a thousand years, if we consider the dare of RV from 1700 BC onwards. It may not be unnatural for any one to stare at the night sky and feel a profoundness within. The unending darkness sprinkled with an unending number of sparkles of stars vanishing into nowhere has always aroused lot of questions in the minds of humans. The Aryans were not unique in this regard.
Staring at the sky for hours can very easily bring out some basic facts:
the sky in the shape of an inverted bowl along with millions of stars seem to revolve around you: this is nothing but the celestial sphere in astronomical terminology.
the sun, moon and the visible planets always appear in a line, rather, an arc: this arc or line is the ecliptic; in astronomical lingo it's the projection of the earth's orbital plane on the celestial sphere; simplistically it signifies the plane on which earth orbits round the sun; it's also roughly the plane on which all the planets orbit round the sun.
the line, on which appear the sun, moon and the planets, is marked by a number of bright fixed stars which can divide the line into several compartments - the sun, moon and the planets seem to be moving from one of these compartments to the other in a fixed cycle of time: the Babylonians were the first to study these stars, which are located close to the earth's orbital plane and which later constituted the twelve zodiac constellations; for the Rig Vedic Aryans these stars that compartmentalize the ecliptic constituted the twenty seven asterisms, nakshatras; the number twenty seven comes from the twenty seven lunar days that roughly make a lunar month - dividing the ecliptic into twenty seven compartments makes the moon appear everyday in a new compartment, which is eventually called lunar mansion.
there's only one star that doesn't seem to revolve - it doesn't move, doesn't rise, doesn't set, remains at the same place as long as the night sky is visible: this is the Polestar, the star that's located exactly to the north of or above the earth's north pole; each and every star appears to be rotating around the Polestar.
It can be assumed that it didn't take much intelligence for the Rig Vedic Aryans to observe these basic things about celestial sphere, ecliptic, lunar mansions and Polestars. Incidentally almost all the ancient civilizations made the same observations. What's different in the case of the Rig Vedic Aryans is that they used these basic astronomical observations liberally in their double meanings and poetic creations.
Before proceeding further let's see how the night sky looked like around 2000 BC in Arkaim - the site of an early Aryan settlement.
Below are a number of sky-maps, all with the same legend and in the same format - the red arc is the ecliptic; the constellation boundaries are marked in green; the names of the constellations and the bright stars visible very easily with naked eyes are marked in yellow and red.
The first sky map show how the night sky looked like on 10th April, 2000 BC in Arkaim. It was just a day before the full moon nearest to Vernal Equinox. Some of the lunar mansions with very bright stars like Spica, Arcturus, Antares and Shaula are marked on the ecliptic. In 2000 BC Thuban of the Draco constellation was very close to being the Polestar (it was the Polestar around 2800 BC). Due to the precession of equinoxes, discussed in details earlier, different stars, all arranged in a circle, become Polestars at various points of time. Thuban (2800 BC), Polaris (now) and Vega (12000 BC & 14000 AD) are marked in the sky-map. On this particular day, 10th April, the moon is in the nakshatra Anuradha.
Next are the sky-maps of 9th and 8th April, 2000 BC. 9th was a full moon coinciding with the Vernal Equinox, something that happens once in roughly nineteen years. On 9th and 8th the moon was in the adjoiningnakshatras Vishakha (full moon) and Swati. These sky maps show how the moon passes from one nakshatra to another with each passing day.
Next is the sky-map at the time of sun rising on the Vernal Equinox, 9th Apr, 2000 BC, in Arkaim. The sun is in Krittika and the full moon in Vishakha. Identifying the stars during the day time is not that trivial. But it may not be tough to interpolate the nakshatras from the previous knowledge of their locations in the night. It has been observed earlier that at a place like Arkaim, where the ecliptic comes quite close to the horizon, it's quite trivial to observe the stars and map them to their respective nakshatras on ground.
Next is the sky-map on 4th October, 2000 BC - the full moon close to Autumnal Equinox. It can be recalled that the location of full moon around Autumnal Equinox is same as that of sun on Vernal Equinox and vice versa. The full moon at Vernal Equinox was in Vishakha and the sun in Krittika. Hence the full moon around Autumnal Equinox should be in Krittika - that's what is seen in the sky-map too.
These observations, which can be noticed in the night sky without much difficulty, had left deep impact in the minds of the peoples of Rig Veda. Very poetically they have merged their favorite chariots with these and composed some wonderful hymns. Let's see some of these hymns.
sapta yuñjanti ratham ekacakram eko aśvo vahati saptanāmā |
trinābhi cakram ajaram anarvaṃ yatremā viśvā bhuvanādhi tasthuḥ || 1.164.02
Seven to the one-wheeled chariot yoke the Courser; bearing seven names the single Courser draws it.
Three-naved the wheel is, sound and undecaying, whereon are resting all these worlds of being. 1.164.02
imaṃ ratham adhi ye sapta tasthuḥ saptacakraṃ sapta vahanti aśvāḥ |
sapta svasāro abhi saṃ navante yatra ghavāṃ nihitā sapta nāma || 1.164.03
The seven who on the seven-wheeled car are mounted have horses, seven in tale, who draw them onward.
Seven Sisters utter songs of praise together, in whom the names of the seven Cows are treasured. 1.162.03
The first verse talks about a single wheeled chariot, ekachakra ratha, yoked to seven horses and driven by a single horse. In the next line it stresses on the fact that the wheel is three naved, trinabhi, undecaying and strong, ajaram anarvam, and on it rests the whole world, vishva bhuvana. It's really tempting to identify the single wheel with seven horses with the ecliptic, which is also a sort of wheel that carries the seven horses - the sun, moon and the five visible planets, Mercury, Venus, Mars, Jupiter and Saturn. The stress on the point that the wheel is three naved, or with three centers, again point very simplistically to the ecliptic. The part of ecliptic that's visible to anyone during night does appear like an ellipse - with the three foci an ellipse really has three centers or naves. Another interesting thing in this verse is the term 'driven by a single horse'. Perhaps this is the first usage of an expression that means horse power - the gravitational power that drives the wheel, the ecliptic, is wonderfully and poetically referred to as the power of a single horse.
The next verse talks about seven horses, sapta ashva, driving a seven wheeled chariot, sapta chakra ratham, on which rests the Seven. This seems to be a reference to the Big Dipper asterism, the seven stars of the Ursa Major constellation representing the seven sages, Saptarshi.
In another verse there's a reference to the One that's beyond the seven sages, sapta rishi; the One that has fixed firmly, tastambha, the six regions of the sky, rajamsi; the One on whom rests the whole world, vishvani bhuvanani tasthu; the One that supports the sky as if it's the peg, dhura, with which the axle pole of the entire wheel of the sky, rajas, is affixed. The outer and simplistic identity of this One is surely the Polestar, the One that's at the center of the celestial sphere, the entire sky that rotates round it. Knowing that many natural things are used as the outer layers for something more profound and philosophic it's very likely that the Polestar, that doesn't move, doesn't rise or set, remains unchanged, unaltered and firm since ages, would be used in Rig Veda very effectively.
Indeed it's used in several verses to refer to someone who's regarded even by Vushwakarma, who's strong in mind,vimana vihaya, and who's the Maker and Disposer, dhata vidhata; someone who's sought after even by He who has made us and who knows the whole world, bhuvanani vishva; someone who has created all things that have existence, bhutani; someone who's older than the Gods and the Asuras and earlier than the earth and heaven, prithivi and diva; someone who's like the germ primeval, garbham prathamam.
The following few verses talk about this One in the typical Rig Vedic style of double meaning.
acikitvāñ cikituṣaś cid atra kavīn pṛchāmi vidmane na vidvān |
vi yas tastambha ṣaḷ imā rajāṃsi ajasya rūpe kimapi svid ekam || 1.164.06
I ask, unknowing, those who know, the sages, as one all ignorant for sake of knowledge,
What was that ONE who in the Unborn's image hath stablished and fixed firm these worlds' six regions. 1.164.06
indraś ca yā cakrathuḥ soma tāni dhurā na yuktā rajaso vahanti || 1.164.19
And what so ye have made, Indra and Soma, steeds bear as ’twere yoked to the region's car-pole. 1.164.19
viśvakarmā vimanā ād vihāyā dhātā vidhātā paramota sandṛk |
teṣām iṣṭāni sam iṣā madanti yatrā saptaṛṣīn para ekam āhuḥ || 10.82.02
Mighty in mind and power is Visvakarman, Maker, Disposer, and most lofty Presence.
Their offerings joy in rich juice where they value One, only One, beyond the Seven Ṛṣis. 10.82.02
yo naḥ pitā janitā yo vidhātā dhāmāni veda bhuvanāni viśvā |
yo devānāṃ nāmadhā eka eva taṃ sampraśnam bhuvanā yanti anyā || 10.82.03
Father who made us, he who, as Disposer, knoweth all races and all things existing,
Even he alone, the Deities' narne-giver,him other beings seek for information. 10.82.03
ta āyajanta draviṇaṃ sam asmā ṛṣayaḥ pūrve jaritāro nabhūnā |
asūrte sūrte rajasi niṣatte ye bhūtāni samakṛṇvan imāni || 10.82.04
To him in sacrifice they offered treasures,—Ṛṣis of old, in numerous troops, as singers,
Who, in the distant, near, and lower region, made ready all these things that have existence. 10.82.04
paro divā para enā pṛthivyā paro devebhir asurair yad asti |
kaṃ svid garbhaṃ prathamaṃ dadhra āpo yatra devāḥ samapaśyanta viśve || 10.82.05
That which is earlier than this earth and heaven, before the Asuras and Gods had being,—
What was the germ primeval which the waters received where all the Gods were seen together? 10.82.05
tam id garbhaṃ prathamaṃ dadhra āpo yatra devāḥ samaghachanta viśve |
ajasya nābhāv adhi ekamarpitaṃ yasmin viśvāni bhuvanāni tasthuḥ || 10.82.06
The waters, they received that germ primeval wherein the Gods were gathered all together.
It rested set upon the Unborn's navel, that One wherein abide all things existing. 10.82.06
Here it's worth mentioning that the Seven Sages of Saptarshi, the Big Dipper asterism, made a perfect circle around the Polestar Thuban during the timeline of Rig Veda (~ 1700 BC). But now the stars of Big Dipper don't make a tight circle around the current Polestar Polaris. The term 'beyond the seven sages' surely made much more sense when the Polestar was really at the center.
The following sky-map shows how the Big Dipper stars revolved round the Polestar Thuban forming a perfect circle in 2000 BC. The dotted lines show the locations of the asterism at different times of the same night.
The next sky-map shows the locus of the Big Dipper asterism as seen now. Had Rig Veda been written now the Saptarshi stars might not have been so important - it no longer makes a proper circle around the current polestar Polaris.
Finally let's see how the ideas about these constellations impacted the calendar system in India. It's worth mentioning that a very precise and scientific calendar system has been in place in India since the Rig Vedic times.

Ancient Mathmatics Part 3

Ancient Indian Mathematics. Part Three.
Numerals and the decimal number system[edit]
It is well known that the decimal place-value system in use today was first recorded in India, then transmitted to the Islamic world, and eventually to Europe. The Syrian bishop Severus Sebokht wrote in the mid-7th century CE about the "nine signs" of the Indians for expressing numbers. However, how, when, and where the first decimal place value system was invented is not so clear.
The earliest extant script used in India was the Kharoṣṭhī script used in the Gandhara culture of the north-west. It is thought to be of Aramaic origin and it was in use from the 4th century BCE to the 4th century CE. Almost contemporaneously, another script, the Brāhmī script, appeared on much of the sub-continent, and would later become the foundation of many scripts of South Asia and South-east Asia. Both scripts had numeral symbols and numeral systems, which were initially not based on a place-value system.
The earliest surviving evidence of decimal place value numerals in India and southeast Asia is from the middle of the first millennium CE. A copper plate from Gujarat, India mentions the date 595 CE, written in a decimal place value notation, although there is some doubt as to the authenticity of the plate. Decimal numerals recording the years 683 CE have also been found in stone inscriptions in Indonesia and Cambodia, where Indian cultural influence was substantial.
There are older textual sources, although the extant manuscript copies of these texts are from much later dates. Probably the earliest such source is the work of the Buddhist philosopher Vasumitra dated likely to the 1st century CE. Discussing the counting pits of merchants, Vasumitra remarks, "When [the same] clay counting-piece is in the place of units, it is denoted as one, when in hundreds, one hundred." Although such references seem to imply that his readers had knowledge of a decimal place value representation, the "brevity of their allusions and the ambiguity of their dates, however, do not solidly establish the chronology of the development of this concept."
A third decimal representation was employed in a verse composition technique, later labelled Bhuta-sankhya (literally, "object numbers") used by early Sanskrit authors of technical books. Since many early technical works were composed in verse, numbers were often represented by objects in the natural or religious world that correspondence to them; this allowed a many-to-one correspondence for each number and made verse composition easier. According to Plofker 2009, the number 4, for example, could be represented by the word "Veda" (since there were four of these religious texts), the number 32 by the word "teeth" (since a full set consists of 32), and the number 1 by "moon" (since there is only one moon). So, Veda/teeth/moon would correspond to the decimal numeral 1324, as the convention for numbers was to enumerate their digits from right to left. The earliest reference employing object numbers is a ca. 269 CE Sanskrit text, Yavanajātaka (literally "Greek horoscopy") of Sphujidhvaja, a versification of an earlier (ca. 150 CE) Indian prose adaptation of a lost work of Hellenistic astrology.[53] Such use seems to make the case that by the mid-3rd century CE, the decimal place value system was familiar, at least to readers of astronomical and astrological texts in India.
It has been hypothesized that the Indian decimal place value system was based on the symbols used on Chinese counting boards from as early as the middle of the first millennium BCE. According to Plofker 2009,
These counting boards, like the Indian counting pits, ..., had a decimal place value structure ... Indians may well have learned of these decimal place value "rod numerals" from Chinese Buddhist pilgrims or other travelers, or they may have developed the concept independently from their earlier non-place-value system; no documentary evidence survives to confirm either conclusion."
Bakhshali Manuscript
The oldest extant mathematical manuscript in South Asia is the Bakhshali Manuscript, a birch bark manuscript written in "Buddhist hybrid Sanskrit" in the Śāradā script, which was used in the northwestern region of the Indian subcontinent between the 8th and 12th centuries CE. The manuscript was discovered in 1881 by a farmer while digging in a stone enclosure in the village of Bakhshali, near Peshawar (then in British India and now in Pakistan). Of unknown authorship and now preserved in the Bodleian Library in Oxford University, the manuscript has been variously dated—as early as the "early centuries of the Christian era" and as late as between the 9th and 12th century CE.The 7th century CE is now considered a plausible date, albeit with the likelihood that the "manuscript in its present-day form constitutes a commentary or a copy of an anterior mathematical work."
The surviving manuscript has seventy leaves, some of which are in fragments. Its mathematical content consists of rules and examples, written in verse, together with prose commentaries, which include solutions to the examples. The topics treated include arithmetic (fractions, square roots, profit and loss, simple interest, the rule of three, and regula falsi) and algebra (simultaneous linear equations and quadratic equations), and arithmetic progressions. In addition, there is a handful of geometric problems (including problems about volumes of irregular solids). The Bakhshali manuscript also "employs a decimal place value system with a dot for zero."[55] Many of its problems are of a category known as 'equalisation problems' that lead to systems of linear equations. One example from Fragment III-5-3v is the following:
One merchant has seven asava horses, a second has nine haya horses, and a third has ten camels. They are equally well off in the value of their animals if each gives two animals, one to each of the others. Find the price of each animal and the total value for the animals possessed by each merchant.
The prose commentary accompanying the example solves the problem by converting it to three (under-determined) equations in four unknowns and assuming that the prices are all integers.
Classical Period (400–1600)
This period is often known as the golden age of Indian Mathematics. This period saw mathematicians such as Aryabhata, Varahamihira, Brahmagupta, Bhaskara I, Mahavira, Bhaskara II, Madhava of Sangamagrama and Nilakantha Somayaji give broader and clearer shape to many branches of mathematics. Their contributions would spread to Asia, the Middle East, and eventually to Europe. Unlike Vedic mathematics, their works included both astronomical and mathematical contributions. In fact, mathematics of that period was included in the 'astral science' (jyotiḥśāstra) and consisted of three sub-disciplines: mathematical sciences (gaṇita or tantra), horoscope astrology (horā or jātaka) and divination (saṃhitā). This tripartite division is seen in Varāhamihira's 6th century compilation—Pancasiddhantika[61] (literally panca, "five," siddhānta, "conclusion of deliberation", dated 575 CE)—of five earlier works, Surya Siddhanta, Romaka Siddhanta, Paulisa Siddhanta, Vasishtha Siddhanta and Paitamaha Siddhanta, which were adaptations of still earlier works of Mesopotamian, Greek, Egyptian, Roman and Indian astronomy. As explained earlier, the main texts were composed in Sanskrit verse, and were followed by prose commentaries.
Fifth and sixth centuries
Surya Siddhanta
Though its authorship is unknown, the Surya Siddhanta (c. 400) contains the roots of modern trigonometry.[citation needed] Because it contains many words of foreign origin, some authors consider that it was written under the influence of Mesopotamia and Greece.
This ancient text uses the following as trigonometric functions for the first time: Sine (Jya). Cosine (Kojya). Inverse sine (Otkram jya). It also contains the early uses of tangent,.secant.
.
Later Indian mathematicians such as Aryabhata made references to this text, while later Arabic and Latin translations were very influential in Europe and the Middle East.
Chhedi calendar
This Chhedi calendar (594) contains an early use of the modern place-value Hindu-Arabic numeral system now used universally (see also Hindu-Arabic numerals).
Aryabhata I
Aryabhata (476–550) wrote the Aryabhatiya. He described the important fundamental principles of mathematics in 332 shlokas. The treatise contained:
Quadratic equations
Trigonometry
The value of π, correct to 4 decimal places.
Aryabhata also wrote the Arya Siddhanta, which is now lost. Aryabhata's contributions include:
Trigonometry: (See also : Aryabhata's sine table)
Introduced the trigonometric functions.
Defined the sine (jya) as the modern relationship between half an angle and half a chord.
Defined the cosine (kojya).
Defined the versine (utkrama-jya).
Defined the inverse sine (otkram jya).
Gave methods of calculating their approximate numerical values.
Contains the earliest tables of sine, cosine and versine values, in 3.75° intervals from 0° to 90°, to 4 decimal places of accuracy.
Contains the trigonometric formula sin(n + 1)x − sin nx = sin nx − sin(n − 1)x − (1/225)sin nx.
Spherical trigonometry.
Arithmetic: Continued fractions. Algebra:
Solutions of simultaneous quadratic equations.
Whole number solutions of linear equations by a method equivalent to the modern method.
General solution of the indeterminate linear equation .
Mathematical astronomy:
Accurate calculations for astronomical constants, such as the:
Solar eclipse.
Lunar eclipse.
The formula for the sum of the cubes, which was an important step in the development of integral calculus.
Varahamihira
Varahamihira (505–587) produced the Pancha Siddhanta (The Five Astronomical Canons). He made important contributions to trigonometry, including sine and cosine tables to 4 decimal places of accuracy and the following formulas relating sine and cosine functions:
\sin^2(x) + \cos^2(x) = 1
\sin(x)=\cos\left(\frac{\pi}{2}-x\right)
\frac{1-\cos(2x)}{2}=\sin^2(x)
Seventh and eighth centuries[edit]
Brahmagupta's theorem states that AF = FD.
In the 7th century, two separate fields, arithmetic (which included measurement) and algebra, began to emerge in Indian mathematics. The two fields would later be called pāṭī-gaṇita (literally "mathematics of algorithms") and bīja-gaṇita (lit. "mathematics of seeds," with "seeds"—like the seeds of plants—representing unknowns with the potential to generate, in this case, the solutions of equations). Brahmagupta, in his astronomical work Brāhma Sphuṭa Siddhānta (628 CE), included two chapters (12 and 18) devoted to these fields. Chapter 12, containing 66 Sanskrit verses, was divided into two sections: "basic operations" (including cube roots, fractions, ratio and proportion, and barter) and "practical mathematics" (including mixture, mathematical series, plane figures, stacking bricks, sawing of timber, and piling of grain). In the latter section, he stated his famous theorem on the diagonals of a cyclic quadrilateral:
Brahmagupta's theorem: If a cyclic quadrilateral has diagonals that are perpendicular to each other, then the perpendicular line drawn from the point of intersection of the diagonals to any side of the quadrilateral always bisects the opposite side.
Chapter 12 also included a formula for the area of a cyclic quadrilateral (a generalisation of Heron's formula), as well as a complete description of rational triangles (i.e. triangles with rational sides and rational areas).
Brahmagupta's formula: The area, A, of a cyclic quadrilateral with sides of lengths a, b, c, d, respectively, is given by
A = \sqrt{(s-a)(s-b)(s-c)(s-d)} \,
where s, the semiperimeter, given by s=\frac{a+b+c+d}{2}.
Brahmagupta's Theorem on rational triangles: A triangle with rational sides a, b, c and rational area is of the form:
a = \frac{u^2}{v}+v, \ \ b=\frac{u^2}{w}+w, \ \ c=\frac{u^2}{v}+\frac{u^2}{w} - (v+w)
for some rational numbers u, v, and w .
Chapter 18 contained 103 Sanskrit verses which began with rules for arithmetical operations involving zero and negative numbers and is considered the first systematic treatment of the subject. The rules (which included a + 0 = \ a and a \times 0 = 0 ) were all correct, with one exception: \frac{0}{0} = 0 . Later in the chapter, he gave the first explicit (although still not completely general) solution of the quadratic equation:
\ ax^2+bx=c
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: x = \frac{\sqrt{4ac+b^2}-b}{2a}
Also in chapter 18, Brahmagupta was able to make progress in finding (integral) solutions of Pell's equation,
\ x^2-Ny^2=1,
where N is a nonsquare integer. He did this by discovering the following identity:
Brahmagupta's Identity: \ (x^2-Ny^2)(x'^2-Ny'^2) = (xx'+Nyy')^2 - N(xy'+x'y)^2 which was a generalisation of an earlier identity of Diophantus: Brahmagupta used his identity to prove the following lemma:
Lemma (Brahmagupta): If x=x_1,\ \ y=y_1 \ \ is a solution of \ \ x^2 - Ny^2 = k_1, and, x=x_2, \ \ y=y_2 \ \ is a solution of \ \ x^2 - Ny^2 = k_2, , then:
x=x_1x_2+Ny_1y_2,\ \ y=x_1y_2+x_2y_1 \ \ is a solution of \ x^2-Ny^2=k_1k_2
He then used this lemma to both generate infinitely many (integral) solutions of Pell's equation, given one solution, and state the following theorem:
Theorem (Brahmagupta): If the equation \ x^2 - Ny^2 =k has an integer solution for any one of \ k=\pm 4, \pm 2, -1 then Pell's equation:
\ x^2 -Ny^2 = 1
also has an integer solution. Brahmagupta did not actually prove the theorem, but rather worked out examples using his method. The first example he presented was: Example (Brahmagupta): Find integers \ x,\ y\ such that: \ x^2 - 92y^2=1
In his commentary, Brahmagupta added, "a person solving this problem within a year is a mathematician." The solution he provided was:
\ x=1151, \ y=120
Bhaskara I
Bhaskara I (c. 600–680) expanded the work of Aryabhata in his books titled Mahabhaskariya, Aryabhatiya-bhashya and Laghu-bhaskariya. He produced:
Solutions of indeterminate equations.
A rational approximation of the sine function.
A formula for calculating the sine of an acute angle without the use of a table, correct to two decimal places.
Ninth to twelfth centuries.
Virasena
Virasena (8th century) was a Jain mathematician in the court of Rashtrakuta King Amoghavarsha of Manyakheta, Karnataka. He wrote the Dhavala, a commentary on Jain mathematics, which:
Deals with the concept of ardhaccheda, the number of times a number could be halved, and lists various rules involving this operation. This coincides with the binary logarithm when applied to powers of two, but differs on other numbers, more closely resembling the 2-adic order.
The same concept for base 3 (trakacheda) and base 4 (caturthacheda).
Virasena also gave:
The derivation of the volume of a frustum by a sort of infinite procedure.
It is thought that much of the mathematical material in the Dhavala can attributed to previous writers, especially Kundakunda, Shamakunda, Tumbulura, Samantabhadra and Bappadeva and date who wrote between 200 and 600 CE.
Mahavira
Mahavira Acharya (c. 800–870) from Karnataka, the last of the notable Jain mathematicians, lived in the 9th century and was patronised by the Rashtrakuta king Amoghavarsha. He wrote a book titled Ganit Saar Sangraha on numerical mathematics, and also wrote treatises about a wide range of mathematical topics. These include the mathematics of:
Zero
Squares
Cubes
square roots, cube roots, and the series extending beyond these
Plane geometry
Solid geometry
Problems relating to the casting of shadows
Formulae derived to calculate the area of an ellipse and quadrilateral inside a circle.
Mahavira also:
Asserted that the square root of a negative number did not exist
Gave the sum of a series whose terms are squares of an arithmetical progression, and gave empirical rules for area and perimeter of an ellipse.
Solved cubic equations.
Solved quartic equations.
Solved some quintic equations and higher-order polynomials.
Gave the general solutions of the higher order polynomial equations:
\ ax^n = q
a \frac{x^n - 1}{x - 1} = p
Solved indeterminate quadratic equations.
Solved indeterminate cubic equations.
Solved indeterminate higher order equations

Ancient Indian Mathematics. Part Two.

Ancient Indian Mathematics. Part Two.

Pingala
Among the scholars of the post-Vedic period who contributed to mathematics, the most notable is Pingala (piṅgalá) (fl. 300–200 BCE),a musical theorist who authored the Chhandas Shastra (chandaḥ-śāstra, also Chhandas Sutra chhandaḥ-sūtra), a Sanskrit treatise on prosody. There is evidence that in his work on the enumeration of syllabic combinations, Pingala stumbled upon both the Pascal triangle and Binomial coefficients, although he did not have knowledge of the Binomial theorem itself. Pingala's work also contains the basic ideas of Fibonacci numbers (called maatraameru). Although the Chandah sutra hasn't survived in its entirety, a 10th-century commentary on it by Halāyudha has. Halāyudha, who refers to the Pascal triangle as Meru-prastāra (literally "the staircase to Mount Meru"), has this to say:
Draw a square. Beginning at half the square, draw two other similar squares below it; below these two, three other squares, and so on. The marking should be started by putting 1 in the first square. Put 1 in each of the two squares of the second line. In the third line put 1 in the two squares at the ends and, in the middle square, the sum of the digits in the two squares lying above it. In the fourth line put 1 in the two squares at the ends. In the middle ones put the sum of the digits in the two squares above each. Proceed in this way. Of these lines, the second gives the combinations with one syllable, the third the combinations with two syllables, ...
The text also indicates that Pingala was aware of the combinatorial identity:
{n \choose 0} + {n \choose 1} + {n \choose 2} + \cdots + {n \choose n-1} + {n \choose n} = 2^n
Katyayana
Katyayana (c. 3rd century BCE) is notable for being the last of the Vedic mathematicians. He wrote the Katyayana Sulba Sutra, which presented much geometry, including the general Pythagorean theorem and a computation of the square root of 2 correct to five decimal places.
Jain Mathematics (400 BCE – 200 CE)
Although Jainism as a religion and philosophy predates its most famous exponent, the great Mahavira (6th century BCE), most Jain texts on mathematical topics were composed after the 6th century BCE. Jain mathematicians are important historically as crucial links between the mathematics of the Vedic period and that of the "Classical period."
A significant historical contribution of Jain mathematicians lay in their freeing Indian mathematics from its religious and ritualistic constraints. In particular, their fascination with the enumeration of very large numbers and infinities led them to classify numbers into three classes: enumerable, innumerable and infinite. Not content with a simple notion of infinity, they went on to define five different types of infinity: the infinite in one direction, the infinite in two directions, the infinite in area, the infinite everywhere, and the infinite perpetually. In addition, Jain mathematicians devised notations for simple powers (and exponents) of numbers like squares and cubes, which enabled them to define simple algebraic equations (beejganita samikaran). Jain mathematicians were apparently also the first to use the word shunya (literally void in Sanskrit) to refer to zero. More than a millennium later, their appellation became the English word "zero" after a tortuous journey of translations and transliterations from India to Europe.
In addition to Surya Prajnapti, important Jain works on mathematics included the Vaishali Ganit (c. 3rd century BCE); the Sthananga Sutra (fl. 300 BCE – 200 CE); the Anoyogdwar Sutra (fl. 200 BCE – 100 CE); and the Satkhandagama (c. 2nd century CE). Important Jain mathematicians included Bhadrabahu (d. 298 BCE), the author of two astronomical works, the Bhadrabahavi-Samhita and a commentary on the Surya Prajinapti; Yativrisham Acharya (c. 176 BCE), who authored a mathematical text called Tiloyapannati; and Umasvati (c. 150 BCE), who, although better known for his influential writings on Jain philosophy and metaphysics, composed a mathematical work called Tattwarthadhigama-Sutra Bhashya.
Oral Tradition
Mathematicians of ancient and early medieval India were almost all Sanskrit pandits (paṇḍita "learned man"), who were trained in Sanskrit language and literature, and possessed "a common stock of knowledge in grammar (vyākaraṇa), exegesis (mīmāṃsā) and logic (nyāya)." Memorisation of "what is heard" (śruti in Sanskrit) through recitation played a major role in the transmission of sacred texts in ancient India. Memorisation and recitation was also used to transmit philosophical and literary works, as well as treatises on ritual and grammar. Modern scholars of ancient India have noted the "truly remarkable achievements of the Indian pandits who have preserved enormously bulky texts orally for millennia."
Styles of memorization
Prodigous energy was expended by ancient Indian culture in ensuring that these texts were transmitted from generation to generation with inordinate fidelity. For example, memorisation of the sacred Vedas included up to eleven forms of recitation of the same text. The texts were subsequently "proof-read" by comparing the different recited versions. Forms of recitation included the jaṭā-pāṭha (literally "mesh recitation") in which every two adjacent words in the text were first recited in their original order, then repeated in the reverse order, and finally repeated again in the original order. The recitation thus proceeded as:
word1word2, word2word1, word1word2; word2word3, word3word2, word2word3; ...
In another form of recitation, dhvaja-pāṭha (literally "flag recitation") a sequence of N words were recited (and memorised) by pairing the first two and last two words and then proceeding as:
word1word2, wordN − 1wordN; word2word3, wordN − 3wordN − 2; ..; wordN − 1wordN, word1word2;
The most complex form of recitation, ghana-pāṭha (literally "dense recitation"), according to (Filliozat 2004, p. 139), took the form:
word1word2, word2word1, word1word2word3, word3word2word1, word1word2word3; word2word3, word3word2, word2word3word4, word4word3word2, word2word3word4; ...
That these methods have been effective, is testified to by the preservation of the most ancient Indian religious text, the Ṛgveda (ca. 1500 BCE), as a single text, without any variant readings. Similar methods were used for memorising mathematical texts, whose transmission remained exclusively oral until the end of the Vedic period (ca. 500 BCE).
The Sutra genre
Mathematical activity in ancient India began as a part of a "methodological reflexion" on the sacred Vedas, which took the form of works called Vedāṇgas, or, "Ancillaries of the Veda" (7th–4th century BCE). The need to conserve the sound of sacred text by use of śikṣā (phonetics) and chhandas (metrics); to conserve its meaning by use of vyākaraṇa (grammar) and nirukta (etymology); and to correctly perform the rites at the correct time by the use of kalpa (ritual) and jyotiṣa (astrology), gave rise to the six disciplines of the Vedāṇgas. Mathematics arose as a part of the last two disciplines, ritual and astronomy (which also included astrology). Since the Vedāṇgas immediately preceded the use of writing in ancient India, they formed the last of the exclusively oral literature. They were expressed in a highly compressed mnemonic form, the sūtra (literally, "thread"):
The knowers of the sūtra know it as having few phonemes, being devoid of ambiguity, containing the essence, facing everything, being without pause and unobjectionable.
Extreme brevity was achieved through multiple means, which included using ellipsis "beyond the tolerance of natural language," using technical names instead of longer descriptive names, abridging lists by only mentioning the first and last entries, and using markers and variables. The sūtras create the impression that communication through the text was "only a part of the whole instruction. The rest of the instruction must have been transmitted by the so-called Guru-shishya paramparai, 'uninterrupted succession from teacher (guru) to the student (śisya),' and it was not open to the general public" and perhaps even kept secret. The brevity achieved in a sūtra is demonstrated in the following example from the Baudhāyana Śulba Sūtra (700 BCE).
The domestic fire-altar in the Vedic period was required by ritual to have a square base and be constituted of five layers of bricks with 21 bricks in each layer. One method of constructing the altar was to divide one side of the square into three equal parts using a cord or rope, to next divide the transverse (or perpendicular) side into seven equal parts, and thereby sub-divide the square into 21 congruent rectangles. The bricks were then designed to be of the shape of the constituent rectangle and the layer was created. To form the next layer, the same formula was used, but the bricks were arranged transversely. The process was then repeated three more times (with alternating directions) in order to complete the construction. In the Baudhāyana Śulba Sūtra, this procedure is described in the following words:
II.64. After dividing the quadri-lateral in seven, one divides the transverse [cord] in three.
II.65. In another layer one places the [bricks] North-pointing.
According to (Filliozat 2004, p. 144), the officiant constructing the altar has only a few tools and materials at his disposal: a cord (Sanskrit, rajju, f.), two pegs (Sanskrit, śanku, m.), and clay to make the bricks (Sanskrit, iṣṭakā, f.). Concision is achieved in the sūtra, by not explicitly mentioning what the adjective "transverse" qualifies; however, from the feminine form of the (Sanskrit) adjective used, it is easily inferred to qualify "cord." Similarly, in the second stanza, "bricks" are not explicitly mentioned, but inferred again by the feminine plural form of "North-pointing." Finally, the first stanza, never explicitly says that the first layer of bricks are oriented in the East-West direction, but that too is implied by the explicit mention of "North-pointing" in the second stanza; for, if the orientation was meant to be the same in the two layers, it would either not be mentioned at all or be only mentioned in the first stanza. All these inferences are made by the officiant as he recalls the formula from his memory.
The written tradition: prose commentary.
With the increasing complexity of mathematics and other exact sciences, both writing and computation were required. Consequently, many mathematical works began to be written down in manuscripts that were then copied and re-copied from generation to generation.
India today is estimated to have about thirty million manuscripts, the largest body of handwritten reading material anywhere in the world. The literate culture of Indian science goes back to at least the fifth century B.C. ... as is shown by the elements of Mesopotamian omen literature and astronomy that entered India at that time and (were) definitely not ... preserved orally.
The earliest mathematical prose commentary was that on the work, Āryabhaṭīya (written 499 CE), a work on astronomy and mathematics. The mathematical portion of the Āryabhaṭīya was composed of 33 sūtras (in verse form) consisting of mathematical statements or rules, but without any proofs. However, according to (Hayashi 2003, p. 123), "this does not necessarily mean that their authors did not prove them. It was probably a matter of style of exposition." From the time of Bhaskara I (600 CE onwards), prose commentaries increasingly began to include some derivations (upapatti). Bhaskara I's commentary on the Āryabhaṭīya, had the following structure:
Rule ('sūtra') in verse by Āryabhaṭa
Commentary by Bhāskara I, consisting of:
Elucidation of rule (derivations were still rare then, but became more common later)
Example (uddeśaka) usually in verse.
Setting (nyāsa/sthāpanā) of the numerical data.
Working (karana) of the solution.
Verification (pratyayakaraṇa, literally "to make conviction") of the answer. These became rare by the 13th century, derivations or proofs being favoured by then.
Typically, for any mathematical topic, students in ancient India first memorized the sūtras, which, as explained earlier, were "deliberately inadequate" in explanatory details (in order to pithily convey the bare-bone mathematical rules). The students then worked through the topics of the prose commentary by writing (and drawing diagrams) on chalk- and dust-boards (i.e. boards covered with dust). The latter activity, a staple of mathematical work, was to later prompt mathematician-astronomer, Brahmagupta (fl. 7th century CE), to characterize astronomical computations as "dust work" (Sanskrit: dhulikarman).

Ancient Indian Mathematics. Part 1

Ancient Indian Mathematics.
Important facts the world should know about Indian Mathematics. This is a long article, it clearly explains the great achievements of Indian mathematicians ,and therefore of the accomplishments on the advancement of science and technology. Part One.
It has been suggested that Indian contributions to mathematics have not been given due acknowledgement in modern history and that many discoveries and inventions by Indian mathematicians are presently culturally attributed to their Western counterparts, as a result of Eurocentrism. According to G. G. Joseph's take on "Ethnomathematics":
[Their work] takes on board some of the objections raised about the classical Eurocentric trajectory. The awareness [of Indian and Arabic mathematics] is all too likely to be tempered with dismissive rejections of their importance compared to Greek mathematics. The contributions from other civilisations – most notably China and India, are perceived either as borrowers from Greek sources or having made only minor contributions to mainstream mathematical development. An openness to more recent research findings, especially in the case of Indian and Chinese mathematics, is sadly missing"
The historian of mathematics, Florian Cajori, suggested that he and others "suspect that Diophantus got his first glimpse of algebraic knowledge from India." However, he also wrote that "it is certain that portions of Hindu mathematics are of Greek origin".
More recently, as discussed in the above section, the infinite series of calculus for trigonometric functions (rediscovered by Gregory, Taylor, and Maclaurin in the late 17th century) were described (with proofs) in India, by mathematicians of the Kerala school, remarkably some two centuries earlier. Some scholars have recently suggested that knowledge of these results might have been transmitted to Europe through the trade route from Kerala by traders and Jesuit missionaries. Kerala was in continuous contact with China and Arabia, and, from around 1500, with Europe. The existence of communication routes and a suitable chronology certainly make such a transmission a possibility. However, there is no direct evidence by way of relevant manuscripts that such a transmission actually took place. According to David Bressoud, "there is no evidence that the Indian work of series was known beyond India, or even outside of Kerala, until the nineteenth century."
Both Arab and Indian scholars made discoveries before the 17th century that are now considered a part of calculus. However, they were not able, as Newton and Leibniz were, to "combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between the two, and turn calculus into the great problem-solving tool we have today." The intellectual careers of both Newton and Leibniz are well-documented and there is no indication of their work not being their own; however, it is not known with certainty whether the immediate predecessors of Newton and Leibniz, "including, in particular, Fermat and Roberval, learned of some of the ideas of the Islamic and Indian mathematicians through sources we are not now aware." This is an active area of current research, especially in the manuscripts collections of Spain and Maghreb, research that is now being pursued, among other places, at the Centre National de Recherche Scientifique in Paris.
Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics (400 CE to 1600 CE), important contributions were made by scholars like Aryabhata, Brahmagupta, Mahāvīra, Bhaskara II, Madhava of Sangamagrama and Nilakantha Somayaji. The decimal number system in use today was first recorded in Indian mathematics. Indian mathematicians made early contributions to the study of the concept of zero as a number, negative numbers, arithmetic, and algebra. In addition, trigonometry was further advanced in India, and, in particular, the modern definitions of sine and cosine were developed there. These mathematical concepts were transmitted to the Middle East, China, and Europe and led to further developments that now form the foundations of many areas of mathematics.
Ancient and medieval Indian mathematical works, all composed in Sanskrit, usually consisted of a section of sutras in which a set of rules or problems were stated with great economy in verse in order to aid memorization by a student. This was followed by a second section consisting of a prose commentary (sometimes multiple commentaries by different scholars) that explained the problem in more detail and provided justification for the solution. In the prose section, the form (and therefore its memorization) was not considered so important as the ideas involved. All mathematical works were orally transmitted until approximately 500 BCE; thereafter, they were transmitted both orally and in manuscript form. The oldest extant mathematical document produced on the Indian subcontinent is the birch bark Bakhshali Manuscript, discovered in 1881 in the village of Bakhshali, near Peshawar (modern day Pakistan) and is likely from the 7th century CE.
A later landmark in Indian mathematics was the development of the series expansions for trigonometric functions (sine, cosine, and arc tangent) by mathematicians of the Kerala school in the 15th century CE. Their remarkable work, completed two centuries before the invention of calculus in Europe, provided what is now considered the first example of a power series (apart from geometric series). However, they did not formulate a systematic theory of differentiation and integration, nor is there any direct evidence of their results being transmitted outside Kerala.
Excavations at Harappa, Mohenjo-daro and other sites of the Indus Valley Civilisation have uncovered evidence of the use of "practical mathematics". The people of the IVC manufactured bricks whose dimensions were in the proportion 4:2:1, considered favourable for the stability of a brick structure. They used a standardised system of weights based on the ratios: 1/20, 1/10, 1/5, 1/2, 1, 2, 5, 10, 20, 50, 100, 200, and 500, with the unit weight equaling approximately 28 grams (and approximately equal to the English ounce or Greek uncia). They mass-produced weights in regular geometrical shapes, which included hexahedra, barrels, cones, and cylinders, thereby demonstrating knowledge of basic geometry.
The inhabitants of Indus civilisation also tried to standardise measurement of length to a high degree of accuracy. They designed a ruler—the Mohenjo-daro ruler—whose unit of length (approximately 1.32 inches or 3.4 centimetres) was divided into ten equal parts. Bricks manufactured in ancient Mohenjo-daro often had dimensions that were integral multiples of this unit of length
Samhitas and Brahmanas[edit]
The religious texts of the Vedic Period provide evidence for the use of large numbers. By the time of the Yajurvedasaṃhitā- (1200–900 BCE), numbers as high as 1012 were being included in the texts.[2] For example, the mantra (sacrificial formula) at the end of the annahoma ("food-oblation rite") performed during the aśvamedha, and uttered just before-, during-, and just after sunrise, invokes powers of ten from a hundred to a trillion:
Hail to śata ("hundred," 102), hail to sahasra ("thousand," 103), hail to ayuta ("ten thousand," 104), hail to niyuta ("hundred thousand," 105), hail to prayuta ("million," 106), hail to arbuda ("ten million," 107), hail to nyarbuda ("hundred million," 108), hail to samudra ("billion," 109, literally "ocean"), hail to madhya ("ten billion," 1010, literally "middle"), hail to anta ("hundred billion," 1011,lit., "end"), hail to parārdha ("one trillion," 1012 lit., "beyond parts"), hail to the dawn (uṣas), hail to the twilight (vyuṣṭi), hail to the one which is going to rise (udeṣyat), hail to the one which is rising (udyat), hail to the one which has just risen (udita), hail to svarga (the heaven), hail to martya (the world), hail to all.
The solution to partial fraction was known to the Rigvedic People as states in the purush Sukta (RV 10.90.4):
With three-fourths Puruṣa went up: one-fourth of him again was here.
The Satapatha Brahmana (ca. 7th century BCE) contains rules for ritual geometric constructions that are similar to the Sulba Sutras.
Śulba Sūtras[edit]
Main article: Śulba Sūtras
The Śulba Sūtras (literally, "Aphorisms of the Chords" in Vedic Sanskrit) (c. 700–400 BCE) list rules for the construction of sacrificial fire altars.[22] Most mathematical problems considered in the Śulba Sūtras spring from "a single theological requirement,"[23] that of constructing fire altars which have different shapes but occupy the same area. The altars were required to be constructed of five layers of burnt brick, with the further condition that each layer consist of 200 bricks and that no two adjacent layers have congruent arrangements of bricks.
According to (Hayashi 2005, p. 363), the Śulba Sūtras contain "the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians."
The diagonal rope (akṣṇayā-rajju) of an oblong (rectangle) produces both which the flank (pārśvamāni) and the horizontal (tiryaṇmānī) <ropes> produce separately."
Since the statement is a sūtra, it is necessarily compressed and what the ropes produce is not elaborated on, but the context clearly implies the square areas constructed on their lengths, and would have been explained so by the teacher to the student.
They contain lists of Pythagorean triples, which are particular cases of Diophantine equations. They also contain statements (that with hindsight we know to be approximate) about squaring the circle and "circling the square."
Baudhayana (c. 8th century BCE) composed the Baudhayana Sulba Sutra, the best-known Sulba Sutra, which contains examples of simple Pythagorean triples, such as: (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25), and (12, 35, 37),[28] as well as a statement of the Pythagorean theorem for the sides of a square: "The rope which is stretched across the diagonal of a square produces an area double the size of the original square."] It also contains the general statement of the Pythagorean theorem (for the sides of a rectangle): "The rope stretched along the length of the diagonal of a rectangle makes an area which the vertical and horizontal sides make together." Baudhayana gives a formula for the square root of two:
\sqrt{2} \approx 1 + \frac{1}{3} + \frac{1}{3\cdot4} - \frac{1}{3\cdot 4\cdot 34} = 1.4142156 \ldots
The formula is accurate up to five decimal places, the true value being 1.41421356... This formula is similar in structure to the formula found on a Mesopotamian tablet from the Old Babylonian period (1900–1600 BCE):
\sqrt{2} \approx 1 + \frac{24}{60} + \frac{51}{60^2} + \frac{10}{60^3} = 1.41421297 \ldots
which expresses √2 in the sexagesimal system, and which is also accurate up to 5 decimal places (after rounding).
According to mathematician S. G. Dani, the Babylonian cuneiform tablet Plimpton 322 written ca. 1850 BCE[32] "contains fifteen Pythagorean triples with quite large entries, including (13500, 12709, 18541) which is a primitive triple,[33] indicating, in particular, that there was sophisticated understanding on the topic" in Mesopotamia in 1850 BCE. "Since these tablets predate the Sulbasutras period by several centuries, taking into account the contextual appearance of some of the triples, it is reasonable to expect that similar understanding would have been there in India." Dani goes on to say:
As the main objective of the Sulvasutras was to describe the constructions of altars and the geometric principles involved in them, the subject of Pythagorean triples, even if it had been well understood may still not have featured in the Sulvasutras. The occurrence of the triples in the Sulvasutras is comparable to mathematics that one may encounter in an introductory book on architecture or another similar applied area, and would not correspond directly to the overall knowledge on the topic at that time. Since, unfortunately, no other contemporaneous sources have been found it may never be possible to settle this issue satisfactorily.
In all, three Sulba Sutras were composed. The remaining two, the Manava Sulba Sutra composed by Manava (fl. 750–650 BCE) and the Apastamba Sulba Sutra, composed by Apastamba (c. 600 BCE), contained results similar to the Baudhayana Sulba Sutra.
Vyakarana
An important landmark of the Vedic period was the work of Sanskrit grammarian, Pāṇini (c. 520–460 BCE). His grammar includes early use of Boolean logic, of the null operator, and of context free grammars, and includes a precursor of the Backus–Naur form (used in the description programming languages).
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