The Forgotten Mathematician

Bhaskaracharya II was an Indian mathematician and astronomer in the 12th century. He was born in Bijapur in modern Karnataka.He was the greatest mathematician of medieval India.Bhaskaracharya wrote Siddhanta Shiromani in 1150 AD. This is a mammoth work containing about 1450 verses. It is divided into four parts, Lilawati, Beejaganit, Ganitadhyaya and Goladhyaya. In fact each part can be considered as separate book. The numbers of verses in each part are as follows, Lilawati has 278, Beejaganit has 213,Ganitadhyaya has 451 and Goladhyaya has 501 verses.These four sections deal with arithmetic, algebra, mathematics of the planets,and spheres respectively.

One of the most important characteristic of Siddhanta Shiromani is, it consists of simple methods of calculations from Arithmetic to Astronomy. Essential knowledge of ancient Indian Astronomy can be acquired by reading only this book. Siddhanta Shiromani has surpassed all the ancient books on astronomy in India. After Bhaskaracharya nobody could write excellent books on mathematics and astronomy anywhere in the world !

Bhaskaracharya's work on calculus predates Newton and Leibniz by over half a millennium. He is particularly known in the discovery of the principles of differential calculus and its application to astronomical problems and computations. Bhaskaracharya was a pioneer in some of the principles of differential calculus. He was the first to conceive the differential coefficient and differential calculus.

Bhaskaracharya gives his date of birth, and date of composition of his major work, in a verse in the Arya metre:

rasa-guṇa-pūrṇa-mahīsama

śaka-nṛpa samaye 'bhavat mamotpattiḥ /

rasa-guṇa-varṣeṇa mayā

siddhānta-śiromaṇī racitaḥ //

This reveals that he was born in 1036 of the Śaka era (1114 CE), and that he composed the Siddhānta Śiromanī when he was 36 years old.He wrote Karana-kutūhala when he was 69 (in 1183)

Bhaskaracharya's work in Algebra, Arithmetic and Geometry catapulted him to fame and immortality. His renowned mathematical works called Lilavati and Bijaganita are considered to be unparalleled and a memorial to his profound intelligence. Its translation in several languages of the world bear testimony to its eminence. In his treatise Siddhant Shiromani he writes on planetary positions, eclipses, cosmography, mathematical techniques and astronomical equipment. In the Surya Siddhant he makes a note on the force of gravity:

"Objects fall on earth due to a force of attraction by the earth. Therefore, the earth, planets, constellations, moon, and sun are held in orbit due to this attraction."

Bhaskaracharya was the first to discover gravity, 500 years before Isaac Newton. His contribution to mathematics is unmatched and incomparable. Here is a summary of a few of his contributions :

- A proof of the Pythagorean theorem by calculating the same area in two different ways and then canceling out terms to get a² + b² = c².

- In Lilavati, solutions of quadratic, cubic and quartic indeterminate equations.

- Solutions of indeterminate quadratic equations (of the type ax² + b = y²).

- Integer solutions of linear and quadratic indeterminate equations (Kuttaka). The rules he gives are (in effect) the same as those given by the Renaissance European mathematicians of the 17th century

- A cyclic Chakravala method for solving indeterminate equations of the form ax² + bx + c = y. The solution to this equation was traditionally attributed to William Brouncker in 1657, though his method was more difficult than the chakravala method.

- His method for finding the solutions of the problem x² − ny² = 1 (so-called "Pell's equation") is of considerable interest and importance.

- Solutions of Diophantine equations of the second order, such as 61x² + 1 = y².This very equation was posed as a problem in 1657 by the French mathematician Pierre de Fermat, but its solution was unknown in Europe until the time of Euler in the18th century.

- Solved quadratic equations with more than one unknown, and found negative and irrational solutions.

- Preliminary concept of mathematical analysis.

- Preliminary concept of infinitesimal calculus, along with notable contributions towards integral calculus.

- Conceived differential calculus, after discovering the derivative and differential coefficient.

- Stated Rolle's theorem, a special case of one of the most important theorems in analysis, the mean value theorem. Traces of the general mean value theorem are also found in his works.

- Calculated the derivatives of trigonometric functions and formulae.

- In Siddhanta Shiromani, Bhaskara developed spherical trigonometry along with a number of other trigonometric results.

Astronomical achievements of Bhaskaracharya :

- The Earth is not flat, has no support and has a power of attraction.

- The north and south poles of the Earth experience six months of day and six months of night.

- One day of Moon is equivalent to 15 earth-days and one night is also equivalent to 15 earth-days.

- Earth’s atmosphere extends to 96 kilometers and has seven parts.

- There is a vacuum beyond the Earth’s atmosphere.

- He had knowledge of precession of equinoxes. He took the value of its shift from the first point of Aries as 11 degrees. However, at that time it was about 12 degrees.

- Ancient Indian Astronomers used to define a reference point called ‘Lanka’. It was defined as the point of intersection of the longitude passing through Ujjaini and the equator of the Earth. Bhaskara has considered three cardinal places with reference to Lanka, the Yavakoti at 90 degrees east of Lanka, the Romak at 90 degrees west of Lanka and Siddhapoor at 180 degrees from Lanka. He then accurately suggested that, when there is a noon at Lanka, there should be sunset at Yavkoti and sunrise at Romak and midnight at Siddhapoor.

- Bhaskaracharya had accurately calculated apparent orbital periods of the Sun and orbital periods of Mercury, Venus, and Mars.

Bhaskaracharya is one of the greatest mathematician ever to have lived, and yet he is forgotten. Lets bring him back by rediscovering all the knowledge that is already there, hidden, in our ancient texts.

The Tree of Life

When President Obama visited Rajghat, to pay tribute to Gandhiji, he planted a Peepal tree there. Though most of us know that the Peepal tree is considered sacred in our country, very few know the reason behind it. 

The Peepal tree is also known as "Ashvattha" in Sanskrit. In the Bhagwat Geeta, Lord Krishna says, amongst the trees, I am Ashvattha.The word 'Ashvattha' means "One which does not remain the same tomorrow" like the whole universe itself ! A seal discovered at Mohenjodaro, one of the cities of the Indus Valley Civilisation depicts the Peepal being worshiped. And we have been worshipping it ever since.

So what makes the tree so special ?

According to the Brahma Purana and the Padma Purana, when the demons defeated the gods, Lord Vishnu hid himself in the Peepal

Tree and that is why it is believed that the Peepal Tree is a symbol of Vishnu and is worshiped since a long period of time.There is another belief that the tree represents the Trimurti-the roots being Brahma, the trunk Vishnu and the leaves Shiva.Some says that Lord Krishna is believed to have died under this tree, after which the present Kali Yuga started.

The Peepal tree is used in traditional medicine for about 50 types of disorders including asthma, diabetes, diarrhea,epilepsy, gastric problems, inflammatory disorders, infectious and several other disorders. According to Ayurveda, Peepal leaves, fruits and bark are killers of diseases. The Peepal leaf has both sweet and bitter taste and has a cooling property. Licking honey placed on peepal leaves is believed to cure speech irregularities.Its leaves, when heated in ghee, are applied to cure wounds. Ingesting the bark, fruit and buds with different combination of things cures diseases related with phlegm, bile, inflammation swelling and indisposition etc.The powdered form of the fruit of this tree increases appetite and cures numerous diseases

Scientific research has proved that the Peepal is a unique tree. It is dangerous to sit or to sleep under any other tree in the night, except the Peepal.Trees exhale oxygen during the day – it is beneficial to be near them – and during the night they exhale carbon dioxide. It is only the Peepal tree which exhales oxygen for all of the twenty-four hours. You can be near it any time; it is beneficial to life. This is perhaps the reason Gautam Buddha choose to meditate under the Peepal. He attained enlightenment (bodhi) while meditating underneath the Bodhi tree or the Peepal tree. The site is in present day Bodh Gaya, northern India. The original tree was destroyed, and has been replaced several times. A branch of the original tree was rooted in Anuradhapura, Sri Lanka in 288 BCE and is known as Jaya Sri Maha Bodhi; it is the oldest flowering plant (angiosperm) in the world.

There are several rituals around the sacred Peepal tree . Usually seven pradakshinas are done around the tree, early in the morning while chanting "Vriksha Rajaya Namah", meaning "salutation to the king of trees." It is believed to bring good luck !

The juice of the tree has an enzyme that can increase child bearing capacity. In olden days the tree used to be in the center of the village so the atmosphere will be clean. The tree emits oxygen at night time too. Peepal tree leaves absorb moisture from the atmosphere and regulates the hydrological cycle.The tree releases large quantity of Isoprene (volatile organic compound) which is needed for preventing of decreasing the ozone layer. The Tree is important for preserving the ozone layer.

Peepal tree is antidote for pollution like water, air and noise. It is recommended to plant the tree around water bodies. Calcium is high in the tree so it helps in the purification of water. Perhaps, the reason why Peepal trees are almost always found in temple premises.

The Peepal tree has very high tolerance to pollution and reduces it most effectively.The ancient Indians understood the importance of the tree and called it sacred, in order to discourage people from cutting it down. Lets learn something from the ancient wisdom and vow to plant, at least one Peepal tree, not just for us, but as a gift for generations to come !

http://www.thehindu.com/seta/2004/12/30/stories/2004123000231700.htm

http://www.ijp-online.com/article.asp?issn=0253-7613%3Byear%3D2011%3Bvolume%3D43%3Bissue%3D6%3Bspage%3D662%3Bepage%3D666%3Baulast%3DGulecha

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3249921/

http://iasir.net/IJETCASpapers/IJETCAS13-541.pdf

The Land of Meluha

The Indus Valley civilization was entirely unknown until 1921, when excavations, revealed the cities of Harappa and Mohenjo Daro.This mysterious culture emerged nearly 9000 years ago and thrived for a thousands of years, on from the highly fertile lands of the Indus river.More than a thousand settlements are now known from the Indus valley civilization. The civilization is known to have had a unity of culture, art, script, and technology (even weights and measures). The largest city, Mohenjo Daro is thought to have reached a population of approximately 30,000, suggesting a total population of approximately 5 million people.It was the largest of the four ancient civilizations of Egypt, Mesopotamia, India and China. However, of all these civilizations the least is known about the Indus Valley people. This is because the Indus script has not yet been deciphered.

Mohenjo-daro built around 2600 B.C.E., located in the Sindh Province of Pakistan. The discovery of Mohenjo-daro led to tremendous excitement in the archaeological and historical worlds. Although the beginning of the Indus Valley Civilization predated the founding of Mohenjo-daro, and thousands of villages and towns had been discovered throughout the region occupied by the Indus Valley people, Mohenjo-daro represented the largest, most sophisticated city discovered. The city had a sophisticated sewage system with regular manhole covers. Public latrines for every block and sewers large enough to walk in.House drains, which were enclosed systems, were made of clay pipes and were connected to the sewers by open brick gutters.

One of the most spectacular structures at Mohenjo is the 'Great Bath', which is astonishingly well preserved and measures 180 feet north to south and 108 feet east to west. It is described as a vast hydropathic establishment and the most imposing of all the remains unearthed at Mohenjo-Daro.

Ancient Mesopotamian texts speak of trading with at least two seafaring civilizations - Makkan and Meluha - in the neighbourhood of India in the third millennium B.C. This trade was conducted with real financial sophistication in amounts that could involve tons of copper. The Mesopotamians speak of Meluha as an aquatic culture, where water and bathing played a central role.A number of Indus Valley objects have been found buried with Mesopotamians.

The ancient script of the Indus valley people has been found in another location, Easter Island (situated in the middle of the Pacific ocean) which is exactly diametrically opposed to Mohenjo daro. This curious coincidence has yet to be properly explained. The Indus Valley Culture was extant thousands of years before the Easter Island culture began, yet the similarity between the two scripts is uncanny.

Numerous objects found in excavation include seated and standing figures, copper and stone tools, carved seals, balance-scales and weights,gold and jasper jewelry, and children's toys. Many of which, are on display in the Harappan gallery in National Museum in New Delhi. The whole collection of this gallery represents the advanced technology and sophisticated lifestyle of the Harappan people.Most prominent among the objects are the Priest Head, the Dancing Girl made in Bronze and belongs to the early Harappan period, skeleton excavated from Rakhigarhi in Haryana, terracotta images of Mother Goddess and clay pottery. Apart from these, the gallery has sculptures in bronzes & terracotta, Bone objects, Ivory, Steatite, semi-precious Stones, painted pottery and jewelry items.Many seals have been discovered during numerous excavations. These seals were probably used for trading purposes. These seals depict bulls, elephants, unicorns, tigers, crocodiles,and other unknown symbols.

What a fine and well developed civilization it was, is evident from the following video.

http://vimeo.com/65393454

So, how did this ancient civilization vanish, some believe, that it was an ancient nuclear explosion that ended the civilization ! But it is yet to validated scientifically. The mystery is waiting to be solved.

http://www.globalpost.com/dispatch/news/regions/asia-pacific/india/121116/indus-civilization-2000-years-old-archaeologists

KAMDHENU

' dhenunam asmi kamadhuk '

Among cows, I am the wish fulfilling cow
- Lord Krishna ( Bhagvat geeta, Chapter 10, verse 28 )

Kamdhenu, also called as Surabhi is described as the mother of all cows. She is a divine creature, who provides her owner everything he desires. She is depicted as a white cow with several deities in her body.

One of the most common questions from the West to us is - “Why do you worship cows?” Most of the times,there is a mocking undertone attached to the question. It is understandable, because predominant western religions or ideologies have conditioned people not to respect the so-called lower forms of life and to think that those who do so, are somehow primitive or superstitious.

In India, we not only worship cows, but also elephants, monkeys and lots of other animals too. Infact we have more than 33 million gods !

We believe, that each and everything, in the natural world, is a manifestation of the supreme god and hence it should be respected, revered and treated as god himself.Sanatan Dharma advocates a respect for all creatures and for our environment, because God’s presence is everywhere.In the history of our civilization, we have taken a lot from cows. The cow’s milk is taken to nourish ourselves.The milk was used to make 'Ghee', which is an essential part of almost all our rituals. Even the cows’ dung was extensively used, as fuel and a building material.It is often said, in the West, “a dog is a man’s best friend”. But if we look at the history of the human race, it is more true to say that cows are man’s best friend.They helped man settle down from thier nomadic lifestyles. Ancient Indians saw it as a mark of ingratitude to take so much from an animal, and then to kill and eat it. Such ungratefulness and lack of respect creates a negative and exploitative tendency in us as a species.

Today, some of the major crises facing our planet are due to unrestrained exploitation of nature, including other creatures. In our quest for more, humans have not hesitated to take so much, without any sense of balance. This has left many species on the verge of extinction and has caused seemingly irreversible environmental changes, such as global warming, ozone layer disruption and huge deforestation.We are constantly exploiting the same mother nature who is our provider.

What does reverence of the cow have to do with this? Not much you may think at first. But if the issue is examined with a deeper vision, we will find that cow protection stems from an outlook towards life and nature that could help prevent and overcome the current problems. Aspects of Vedic culture such as respect for the cow and other forms of life create and reinforce a mentality and outlook of gratefulness, appreciation and humility. Humanity has shamelessly exploited nature. The whole problem arises from a mistaken feeling that nature and other creatures are simply here at our disposal, and we owe nothing to the nature. If we strive to appreciate how much we receive, we would not have abused our planet and its inhabitants.

I was extremely happy to read in the papers recently, that Mr Manohar Lal Khattar, chief minister of Haryana, has promised to ban cow slaughter in his state. Finally, someone makes a beginning! We as a nation, have to not just respect, but more importantly, protect our divine mother cow.

The Legend of Agastya

Maharishi Agastya, also known as Agathiyar, was one of the great sages of ancient India. He was the son of MitraVaruna Rishi and the apsara Urvashi. He is also referred to as Kumbhasambhava or Kumbha Muni - meaning born from a pot. About a hundred years ago, the whole idea of someone being born from a pot or a vessel must have sounded extremely ridicules,unbelievable or even super natural romanticism. However, with so many in vitro fertilization methods and test tube babies, amongst us now, it doesn't sound all that unbelievable, does it ? There was a time, in 1600s, when Galileo an Italian physicist, mathematician claimed that the earth is not flat, but round and it revolves around the sun. He was dramatically persecuted by the Church and ridiculed upon. However, in Ancient India,Yajnavalkya (c. 9th–8th century BCE) recognized that the Earth is spherical in his astronomical text Shatapatha Brahmana. This is also recognized in another Vedic Sanskrit text Aitareya Brahmana composed around the same time, and in a later Sanskrit commentary Vishnu Purana.

So you see, science is a very relative thing !

Coming back to Rishi Agastya, the marriage of Lord Shiva to Uma, the daughter of the king of mountains Himavan was going to be conducted at Mount Kailas. All the people in the world and all the creatures began to flock to the north. The earth was in danger of toppling over, as the weights of North and South were not in balance. Upon beholding this the Lord said, "This cannot be allowed to happen. We need a suitable weight to counterbalance this northern tilt. Only Agastya can do this job." He then summoned Agastya and bade him to go south and use his yogic powers to balance the earth. Agastya was pleased with this honor, but was also saddened, because then he won't be able to witness this divine marriage. Perceiving the thoughts in the Rishi's mind, Shiva granted him a boon that himself and Uma will appear before Agastya, whenever the latter happened to think of them. Pleased, Agastya began his southward journey." He is one of the most revered sages of south India.

Siddhar were spiritual adepts who possessed the ashta siddhis, or the eight supernatural powers. Sage Agathiyar is considered the guru of all Siddhars, and the Siddha medicine system is believed to have been handed over to him by Lord Muruga, son of the Lord Shiva and Goddess Parvathi. Siddhars are the followers of Lord Shiva. Agathiyar is the first Siddhar. His disciples and other siddhars contributed thousands of texts on Siddhar literature, including medicine and form the propounders of the system in this world.He is also considered as the Father of Tamil literature and compiled the first Tamil grammar called Agathiyam. He is regarded to have lived in the 6th or 7th century B.C and specialized in language, alchemy, medicine and spirituality (yogam and gnanam).There are 96 books in the name of Agathiyar. In Tamil language, the term 'Agam' means inside and 'iyar' means belong. One who belongs inside (soul) is Agathiyar.

There are several stories around Agastya Rishi's life, but I find this one very interesting.

IIvala and Vatapi were two demon brothers. Vatapi could changed himself into any form he desired, like a shape shifter. IIvala knew the Sanjivini Mantra.Hence, the demons used to trick people into eating Vatapi who would assume the form of a goat and then, IIlava would say the mantra, enabling Vatapi to tear the stomach and come out. Agastya wanted to put an end to their ways. Hence we met them and as expected Ilava offered Vatapi as food to the Sage. The Sage after eating simply said “Vatapi Jeerno Bhava " ( may Vatapi be digested ). By his yogic powers, he completely dissolved Vatapi. Enraged by his brother's death, Ilvala attacked the Rishi, who promptly reduced him to ashes !

A lot of people strongly object to mixing of mythology with science and spirituality. But I think there are no boundary walls separating the three. Where one begins and the other ends is for us to find out!

The Power of One

A little more than 30 years ago, a teenager named Jadav "Molai" Payeng began burying seeds along a barren sandbar near his birthplace in northern India's Assam region to grow a refuge for wildlife. Not long after, he decided to dedicate his life to this endeavor, so he moved to the site so he could work full-time creating a lush new forest ecosystem. Incredibly, the spot today hosts a sprawling 1,360 acres of jungle that Payeng planted — single-handedly.

It all started way back in 1979, when floods washed a large number of snakes ashore on the sandbar. One day, after the waters had receded, Payeng, only 16 then, found the place dotted with the dead reptiles. That was the turning point of his life.

"The snakes died in the heat, without any tree cover. I sat down and wept over their lifeless forms. It was carnage. I alerted the forest department and asked them if they could grow trees there. They said nothing would grow there. Instead, they asked me to try growing bamboo. It was painful, but I did it. There was nobody to help me. Nobody was interested," says Payeng, now 47.

While it's taken years for Payeng's remarkable dedication to planting to receive some well-deserved recognition internationally, it didn't take long for wildlife in the region to benefit from the manufactured forest. Demonstrating a keen understanding of ecological balance, Payeng even transplanted ants to his burgeoning ecosystem to bolster its natural harmony. Soon the shadeless sandbar was transformed into a self-functioning environment where a menagerie of creatures could dwell. The forest, called the Molai woods, now serves as a safe haven for numerous birds, deer, rhinos, tigers and elephants — species increasingly at risk from habitat loss.

Despite the conspicuousness of Payeng's project, forestry officials in the region first learned of this new forest in 2008 — and since then they've come to recognize his efforts as truly remarkable, but perhaps not enough.

"We're amazed at Payeng," says Gunin Saikia, assistant conservator of Forests. "He has been at it for 30 years. Had he been in any other country, he would have been made a hero."

THE GURU

गुरुर्ब्रह्मा गुरुर्विष्णुः गुरुर्देवो महेश्वरः ।
गुरुरेव परंब्रह्म तस्मै श्रीगुरवे नमः ॥

Salutation to the noble Guru, who is Brahma, Vishnu and Maheswara, the direct Parabrahma, the Supreme Reality.

The teacher is like Lord Brahma as he generates the knowledge within us,like Lord Vishnu as he Operates the ideas/knowledge in our mind unto the right path, and like Lord Mahesh (Shiva) as he destroys the ignorance in our minds and leads us towards enlightenment.

Most of us can remember at least one teacher who had a profound effect on our early life, someone who had a positive impact on us and made us the person we are today.

A teacher's role in our lives and the society at large is very significant and invaluable. The ancient Indian understood this and hence had the great Guru-Shishya parampara. "Gu" means dark and "Ru" means light. So, "Guru" may be translated as darkness to light, or possibly one who leads from darkness to light. The Guru is a teacher who guides the Shishya. He is a mentor who leads the Shishya (student) from blindness or ignorance to bliss,wisdom, and enlightenment.

Guru Shishya Parampara has been the most ancient and is also known as the best system of eduction.The student while living in a Gurukul would help to the Guru with the household work and at the same time, living under a stringent discipline,spending moderate lifestyle and perpetually practicing whatever education has been given to the student by the Guru.Knowledge, whether it be Vedic studies, maths,science, architecture, music or spiritual, was imparted through the developing relationship between the guru and the disciple. It is considered that this relationship,based on the genuineness of the guru,and the respect, commitment, devotion and obedience of the student, is the best way for subtle or advanced knowledge to be conveyed. The student eventually masters the knowledge that the guru embodies.

The Britishers took away a lot from us. But amongst the worst things they did to our society was the destruction of the Gurukul form of education which in turn, eliminated the entire Guru-Shishya Parampara.It was one of the most wonderful system of education from the ancient India. In this day and age, there is rapid deterioration of education system, quality and morality of the teachers. At the same time, there is little respect for the teachers and the teaching profession at large. It is very important for us to revive the ancient system of education. It is the

need of the hour.

From early childhood itself, young minds should be driven towards seeking knowledge and blessed are those who get the guidance to move away from the darkness of ignorance towards illuminating wisdom !