LIVE THE IDEAL
Swami Chinmayananda
You are not, unfortunately, what you think yourself to be. You think you are successful; others in your office are attributing your chair, perhaps to the greatness of your father-in-law. You think you are efficient and strict; others consider you as weak and bad tempered. You count yourself to be a kind father; others say that you are spoiling your son. You imagine that you are a great Romeo to your neighbor’s sister; she takes you to be a lunatic though of an interesting and entertaining type.
Thus, what you estimate yourself to be and what others judge you, from your actions, are poles apart from each other. Your so-called estimate of yourself is but your intellectual belief in some ideal picture of yourself; but in your dealing with the world outside you are acting just the opposite and the only one who is totally unconscious of the ugliness is yourself.
We have thus, in each one of us, a double personality playing in us at all times; one a personality which we believe in and another with which we act out. Roughly we can attribute the former to our Intellectual Entity and the latter to our Physical Person. Educated as we are, as a product of our study, appreciation and admiration of some of the perfect qualities, both moral and ethical, in other great men, we have a vague picture of the Ideal that we wish to be. This picture of the ideal, when entertained for long in ourselves becomes so clearly impressed in ourselves that we come to believe that we are the ideal that we cherish. But alas, Others suffer and groan under persecutions of what we actually are in our life.
Our Self-unfoldment, our growth in ourselves, our personality charm can burst forth in all its fragrance and joy into full bloom only when we plan for and bring about an actual translation of our idealism into our day – to – day life. We must live what we believe in. We must act upon our own cherished ideas.
We must strive to live our accepted philosophy of life. Or else, though a Bhakta we will run amuck to carry away our Sita, thereby inviting not only our own destruction but a host of sorrows to a million others around us.
As seekers and sincere aspirants to a higher cultural life you all must learn to translate your ideal picture of yourself in your own work-a-day-life. This is only possible through regular and sincere self-examination and long and careful introspection. Each day see the drama of your own day’s life, thrown upon the screen of your own mind and watch. Edit the play, take new decisions; discover the faults; avoid them in your next 24-hour’s life. Thus, one by one, weed out your weakness in your physical contacts and dealings, in your mental feelings and emotions, and in your intellectual estimations of things and beings.
An idler alone is satisfied by shop window gazing. Walk out into the world. Sweat and toil. Earn and save. And, rich with the fruits of your own honest labour, return to the shop. Push, open the door, walk in. Approach the counter. Order the thing of your fascination. And come to possess it as your own. So too, never allow to feel satisfied by recognizing glories in some body’s life; when you admire the meekness of Christ, the compassion of Buddha, the gorgeous joys of Krishna, the dignified suffering of Rama, the wisdom of Vasishta or the dynamic philosophy of Sankara, never sit back only to gaze at them and in futile wonderment, merely come to believe in them, but feel the urge to cultivate such a noble divine personality in yourself.
Be aggressively good. Strive hard and develop these traits and come to live and express it in the world. Be a Dhananjaya. What they did we too can do. There is nothing we cannot achieve. There is no thing impossible to a determined, sincere and industrious man of right judgement. Spare not yourself until you climb to the peak. Leave the valleys for others-the lesser ones-who are satisfied by the thrill of their crowded stink and mutual conflicts. Members of the Chinmaya Mission are not satisfied except by sunlit peak of Spiritual Perfection – therein awaits for you the realization of True Hinduism.
Saturday, July 28, 2007
Gayatri Mantra
Gayatri Mantra
“OM bhuur bhuvaH svaH
tatsaviturvareNyaM
bhargo devasya dhiimahi
dhiyo yonaH pracodayaat “
The Rig Veda (10:16:3)
OM - the Infinite beyond Conception, Brahman, Supreme Lord and State of Beingbhuur - Protector of the earth, the material sheath, The Life breath of the Universe. From the feet to the navel center.bhuvaH - the Mental Worlds One of the meanings is the sky. The Lord who pervades and eliminates all miseries. From the navel to the throat center.svaH - I AM ONE WITH GOD - the Spiritual Worlds, One of the other meanings is the heavens. He is all Bliss and blesses His devotees with happiness. From above the throat center to the thousand petaled lotus.tat - That (That Thou Art)savitur - The SUN, Creator, Preserver and Self Luminous, the central mantra of the Solar Science.vareNyaM - Most fit to be worshipped. Most choice worthy – most adorable, enchanting.bhargo - The burning splendor of the Sun that dispels ignorance, and therefore sorrows and miseries.devasya - The 'Deva', The God, He who is All Bliss, The Joyful One, The Shining One, The Revealer of All Glorydhiimahi - We meditate upon Himdhiyo - Bhuddi, intelligence, Wisdomyo - whichnaH - ourpracodayaat - Inspires, enlightens, illumnsThis is one of the oldest and most powerful of all known mantras. You should chant this mantra and firmly establish it in your mind, as your spiritual ideal. If so, you will be able to do the work alotted to you, your life will be one of light and happiness.The word "Gayatri" itself explains the reason for the existence of this mantra. It has its origin in the Sanskrit phrase Gayantam Triyate iti, and refers to that mantra which rescues the chanter from all adverse situations that may lead to mortality.
Goddess Gayatri is also called "Veda-Mata" or the Mother of the Vedas — Rig, Yajur, Saam and Atharva — because it is the very basis of the Vedas. It is the basis, the reality behind the experienced and the cognized universe.The Gayatri mantra is composed of a metre consisting of 24 syllables — generally arranged in a triplet of eight syllables each. Therefore, this particular meter (tripadhi) is also known as the Gayatri Meter or "Gayatri Chhanda."Just like the Sun annihilates darkness, so does the Gayatri Mantra destroy ignorance. The Gayatri Mantra invokes the power and radiance of the Sun to energize all earthly life, to destroy our sins and to reveal the Supreme Self to us. Krishna in the Bhagvad Geeta says: "Amongst the Mantras, I am the Gayatri"In a loose translation that cannot possibly do justice to the original it implores:
OM, O Lord! You are the all pervading Source of Light, Sustainer, Protector and Bestower of Happiness, Kindle, Enlighten and inspire our Intelligence to possess Eternal Qualities.The Gayatri Mantra is the essence of the three Vedas. It is the Vedas simplified. The more one contemplates and meditates upon the Gayatri, the more complexities are revealed to one.The story goes that once the Devas, the Shining Ones wanted to learn the secrets of the Universe. Since learning about the Scientific aspect 'atoms' etc. did not quite work out, they went to the Lord. The Lord gave them the Rig Veda, The Yajur Veda and the Sama Veda to study. After a couple of eons of study, the Devas lost their patience and wanted to quit. The Lord urged them not to give up and decided to teach the essence of the 3 Vedas in 3 fragments:Essence of Rig Veda Tat Savitur VarenyamEssence of Yajur Veda Bhargo Devasya DheemahiEssence of Sama Veda Dheeyo Yonah PrachodayaatThe more the Devas practiced and contemplated upon the Gayatri, the more complexities were revealed to them. So they lost patience again and went to the Lord to make it simpler still. So the Lord gave them the essence of the 3 segments, in the 3 words:Bhur, Bhuvaha, SvahaThe Devas spent a a few cycles of Creation, probably a quadrillion human years pondering upon the 'maha vyahritis' ie 'Utterances of the Lord' but became impatient again and urged the Lord to make it simpler yet. So He Said "All right I'll make it simpler"And He taught them AUMThe great western thinker, Arthur Koestler, had once said that the Gayatri Mantra has the power of a thousand atomic bombs! and the famous scientists J.B.S. Haldana wrote: "The Gayatri Mantra should be carved on the doors of every laboratory of the world".The Gayatri with the Mahavyahritis is uttered as-
Om, bhur, bhuvah, svah tat Savitur varenyam bhargo devasya dhimahi dhiyo yo nah prachodayat.
This mantra is believed to have been composed by Sage Vishvamitra. According to others, however, it is so ancient that the four Vedas were born of it.Initially, this mantra was a simple invocation to the Sun to bless all on earth. Gradually it came to be regarded as a mystic formula of universal power. This was probably due to its simplicity and itspower to evolve the idealistic notion of a world that originated from an all-pervading Intelligence.
According to the Skanda Purana, nothing in the Vedas is superior to the Gayatri. No invocation is equal to it just as no city is equal to Kashi (see Tirtha). The Gayatri is the mother of the Vedas, itcontains the essence of all the Vedas and of the Brahmanas, for the Gayatri is believed to embody Brahma, Vishnu, Shiva and the Vedas.According to the Aitareya Brahmana (see Brahmana), those who desire to go to heaven should recite this mantra a thousand times. The Gayatri Mantra must be recited by all 'twice-born' (see Upanayanam) Hindus, Gayatri Mantra especially the Brahmins, who are expected to recite it every morning and evening. This mantra was however not allowed be reciting or even hearing by a Shudra or a woman. In the early Vedic age however, the status of women was considerably higher than in the later Vedic age. Rishinis or women rishis like Gargi and Lopamudra are said to have undergone the Upanayanam and the former indeed, engaged in debate none less than the law giver, Yajnavalkya. Later, when this mantra became exclusive to 'twice-born' males, carewas taken not to recite it loudly.The Gayatri Mantra is also recited at various rituals pertaining to divine worship and the ritual of the manes (see Shradha). It is said that the recitation of the Gayatri five times a day is as effective as performing the panchamahayagya.This mantra is taught for the first time during the Upanayanam ceremony when the guru whispers it into the ears of the newly Recitation of the Pupil (brahmopadesham) . Thereafter the pupil isexpected to recite it every morning and evening throughout his lifeIt should ideally be recited 16 times a day. To keep track, the right hand is used. When the mantra is recited the first time, the thumb is placed on the third joint of the ring finger and is held there till the mantra is complete. With each completion, the thumb moves one more joints, down the ring finger, up the little finger, over the tips, down the index finger, up the middle finger and the sixteenth recitation is completed on the third joint of the middle finger. When reciting the Gayatri Mantra, the sacred thread (see Upanayanam) is held across the thumb of the right hand.
“OM bhuur bhuvaH svaH
tatsaviturvareNyaM
bhargo devasya dhiimahi
dhiyo yonaH pracodayaat “
The Rig Veda (10:16:3)
OM - the Infinite beyond Conception, Brahman, Supreme Lord and State of Beingbhuur - Protector of the earth, the material sheath, The Life breath of the Universe. From the feet to the navel center.bhuvaH - the Mental Worlds One of the meanings is the sky. The Lord who pervades and eliminates all miseries. From the navel to the throat center.svaH - I AM ONE WITH GOD - the Spiritual Worlds, One of the other meanings is the heavens. He is all Bliss and blesses His devotees with happiness. From above the throat center to the thousand petaled lotus.tat - That (That Thou Art)savitur - The SUN, Creator, Preserver and Self Luminous, the central mantra of the Solar Science.vareNyaM - Most fit to be worshipped. Most choice worthy – most adorable, enchanting.bhargo - The burning splendor of the Sun that dispels ignorance, and therefore sorrows and miseries.devasya - The 'Deva', The God, He who is All Bliss, The Joyful One, The Shining One, The Revealer of All Glorydhiimahi - We meditate upon Himdhiyo - Bhuddi, intelligence, Wisdomyo - whichnaH - ourpracodayaat - Inspires, enlightens, illumnsThis is one of the oldest and most powerful of all known mantras. You should chant this mantra and firmly establish it in your mind, as your spiritual ideal. If so, you will be able to do the work alotted to you, your life will be one of light and happiness.The word "Gayatri" itself explains the reason for the existence of this mantra. It has its origin in the Sanskrit phrase Gayantam Triyate iti, and refers to that mantra which rescues the chanter from all adverse situations that may lead to mortality.
Goddess Gayatri is also called "Veda-Mata" or the Mother of the Vedas — Rig, Yajur, Saam and Atharva — because it is the very basis of the Vedas. It is the basis, the reality behind the experienced and the cognized universe.The Gayatri mantra is composed of a metre consisting of 24 syllables — generally arranged in a triplet of eight syllables each. Therefore, this particular meter (tripadhi) is also known as the Gayatri Meter or "Gayatri Chhanda."Just like the Sun annihilates darkness, so does the Gayatri Mantra destroy ignorance. The Gayatri Mantra invokes the power and radiance of the Sun to energize all earthly life, to destroy our sins and to reveal the Supreme Self to us. Krishna in the Bhagvad Geeta says: "Amongst the Mantras, I am the Gayatri"In a loose translation that cannot possibly do justice to the original it implores:
OM, O Lord! You are the all pervading Source of Light, Sustainer, Protector and Bestower of Happiness, Kindle, Enlighten and inspire our Intelligence to possess Eternal Qualities.The Gayatri Mantra is the essence of the three Vedas. It is the Vedas simplified. The more one contemplates and meditates upon the Gayatri, the more complexities are revealed to one.The story goes that once the Devas, the Shining Ones wanted to learn the secrets of the Universe. Since learning about the Scientific aspect 'atoms' etc. did not quite work out, they went to the Lord. The Lord gave them the Rig Veda, The Yajur Veda and the Sama Veda to study. After a couple of eons of study, the Devas lost their patience and wanted to quit. The Lord urged them not to give up and decided to teach the essence of the 3 Vedas in 3 fragments:Essence of Rig Veda Tat Savitur VarenyamEssence of Yajur Veda Bhargo Devasya DheemahiEssence of Sama Veda Dheeyo Yonah PrachodayaatThe more the Devas practiced and contemplated upon the Gayatri, the more complexities were revealed to them. So they lost patience again and went to the Lord to make it simpler still. So the Lord gave them the essence of the 3 segments, in the 3 words:Bhur, Bhuvaha, SvahaThe Devas spent a a few cycles of Creation, probably a quadrillion human years pondering upon the 'maha vyahritis' ie 'Utterances of the Lord' but became impatient again and urged the Lord to make it simpler yet. So He Said "All right I'll make it simpler"And He taught them AUMThe great western thinker, Arthur Koestler, had once said that the Gayatri Mantra has the power of a thousand atomic bombs! and the famous scientists J.B.S. Haldana wrote: "The Gayatri Mantra should be carved on the doors of every laboratory of the world".The Gayatri with the Mahavyahritis is uttered as-
Om, bhur, bhuvah, svah tat Savitur varenyam bhargo devasya dhimahi dhiyo yo nah prachodayat.
This mantra is believed to have been composed by Sage Vishvamitra. According to others, however, it is so ancient that the four Vedas were born of it.Initially, this mantra was a simple invocation to the Sun to bless all on earth. Gradually it came to be regarded as a mystic formula of universal power. This was probably due to its simplicity and itspower to evolve the idealistic notion of a world that originated from an all-pervading Intelligence.
According to the Skanda Purana, nothing in the Vedas is superior to the Gayatri. No invocation is equal to it just as no city is equal to Kashi (see Tirtha). The Gayatri is the mother of the Vedas, itcontains the essence of all the Vedas and of the Brahmanas, for the Gayatri is believed to embody Brahma, Vishnu, Shiva and the Vedas.According to the Aitareya Brahmana (see Brahmana), those who desire to go to heaven should recite this mantra a thousand times. The Gayatri Mantra must be recited by all 'twice-born' (see Upanayanam) Hindus, Gayatri Mantra especially the Brahmins, who are expected to recite it every morning and evening. This mantra was however not allowed be reciting or even hearing by a Shudra or a woman. In the early Vedic age however, the status of women was considerably higher than in the later Vedic age. Rishinis or women rishis like Gargi and Lopamudra are said to have undergone the Upanayanam and the former indeed, engaged in debate none less than the law giver, Yajnavalkya. Later, when this mantra became exclusive to 'twice-born' males, carewas taken not to recite it loudly.The Gayatri Mantra is also recited at various rituals pertaining to divine worship and the ritual of the manes (see Shradha). It is said that the recitation of the Gayatri five times a day is as effective as performing the panchamahayagya.This mantra is taught for the first time during the Upanayanam ceremony when the guru whispers it into the ears of the newly Recitation of the Pupil (brahmopadesham) . Thereafter the pupil isexpected to recite it every morning and evening throughout his lifeIt should ideally be recited 16 times a day. To keep track, the right hand is used. When the mantra is recited the first time, the thumb is placed on the third joint of the ring finger and is held there till the mantra is complete. With each completion, the thumb moves one more joints, down the ring finger, up the little finger, over the tips, down the index finger, up the middle finger and the sixteenth recitation is completed on the third joint of the middle finger. When reciting the Gayatri Mantra, the sacred thread (see Upanayanam) is held across the thumb of the right hand.
'Vedic practices provided the inspiration for advances in astronomy and mathematics'
'Vedic practices provided the inspiration for advances in astronomy and mathematics'
(Excerpted from an article by B.V.Subbarayyappa in the book India 1000 to 2000, Editor : T.J.S.George, published in December 1999 by Express Publications (Madurai) Ltd, Express Estates, Anna Salai, Chennai - 600 002. The excerpt was also published in The New Indian Express on Sunday in the FYI column on April 8, 2001.)
Jyothisha (astronomy) was one of the six auxiliaries of the Vedas and the earliest Indian astronomical text goes by the name of Vedanga Jyotisha. Year-long sacrifices commenced from the day following the winter solstice and Vedic knowledge of both winter and summer solstices was fairly accurate. The Vedanga Jyotisha had developed a concept of a cycle of 5 years (one Yuga) for luni-solar and other time adjustments with intercalation at regular intervals.
Indian mathematics too owes its primary inspiration to Vedic practices. The Shulba sutras, part of another Vedic auxiliary called the Kalpa sutras, deal with the construction of several types of brick altars and in that context with certain geometrical problems including the Pythagorean theorem, squaring a circle, irrational numbers and the like. Yet another Vedic auxiliary, Metrics (chandah), postulated a triangular array for determining the type of combinations of 'n' syllables of long and short sounds for metrical chanting. This was mathematically developed by Halayudha who lived in Karnataka (10th Century) into a pyramidal expansion of numbers. Such an exercise appeared six centuries later in Europe, known as Pascal's triangle. Vedic mathematics and astronomy were pragmatic and integrated with Vedic religio-philosophical life.
But such an approach was not to last long. During the three centuries before and after the Christian era, there were new impulses. Astronomy became mathematics-based. In the succeeding centuries, while astronomy assimilated Hellenic ideas to some extent mathematics was really innovative. Indian astronomers were able mathematicians too. The doyen among them, Aryabhatta I (b.476 A.D.) gave the value of pi (3.1416 approx., a value used even today) worked out trigonometrical tables, areas of triangles and other plane figures, arithmetical progression, summation of series, indeterminate equations of the first order and the like. He expounded that the earth rotates about its own axis and the period of one sidereal rotation given by him is equivalent to 23h 56m 4s.1, while the modern value is 23h 56m 4s.091. He discarded the mythical Rahu-Ketu postulate concerning eclipses in favour of a scientific explanation.
Aryabhatta's junior contemporary Varahamihira, was well known for his compendium, the Panchasiddhantika, a compilation of the then extant five astronomical works called the Siddhantha- Surya, Paulisha, Romaka, Vasishta, and Paitamaha. Of them, the Suryasiddhanta, which he regarded as the most accurate, underwent revisions from time to time and continues to be an important text for computing pancangas.
Brahmagupta was a noted astronomer mathematician of the 7th Century. His remarkable contribution was his equation for solving indeterminate equations of the second order - an equation that appeared in Europe a thousand years later known as Pell's equation. His lemmas in this connection were rediscovered by Euler (1764) and Lagrange (1768). Brahmagupta was also the first to enunciate a formula for the area of a rational cyclic quadrilateral. In the latter half of the first millenium A.D. there were other noted astronomers and mathematicians like Bhaskara I, Lalla, Pruthudakasvamin, Vateshvara, Munjala, Mahavira (Jaina mathematician), Shripati, Shridhara, Aryabhatta II , and Vijayanandin. The tradition of astronomy and mathematics continued unabated - determination of procession of equinoxes, parallax, mean and true motions of planet, permutations and combinations, solving quadratic equations, square root of a negative number and the like.
Using nine digits and zero, the decimal place value system had established itself by about the 4th century A.D. Says historian of science, George Sarton, "Our numbers and the use of zero were invented by the Hindus and transmitted by Arabs, hence the name Arabic numerals which we often give them.' Brahmagupta's Brahmasphuta Siddhanta and Khandakhadyaka were also rendered into Arabic in the 9th-10th century. The Brahmi numerical forms with some modifications along with the decimal place-value system developed in India have since become universal.
The beginning of the second millenium A.D. witnessed the emergence of the notable astronomer-mathematician, Bhaskaracharya II (b.1114). His cyclic (cakravala) method for solving indeterminate equations of the second order has been hailed by the German mathematician 'Henkel', as the finest thing achieved in the theory of numbers before Lagrange. Bhaskaracharya II had also developed basic Calculus. Between the 14th and 18th Centuries, there were schools of astronomers-mathematicians in Kerala and Maharashtra, Ganesha Daivajna simplified methods of computation for almanac makers. The Kerala school was well known for its keen observations of eclipses over 55 years. Parameshwara (1360-1455) was the first in the history of mathematics to have given the exact formula for the circumradius of a cyclic quadrilateral; this was rediscovered in Europe by L'Huiler nearly 300 years later.
Adhering to the Aryabhatta tradition, other Kerala savants like Govinda Bhatta, Damodara, Nilakantha Somayaji, Jyesthadeva, Acyuta Pisharati and Putumana Somayaji added to both astronomy and mathematics. The leader of this intellectual lineage was Madhava (14th century), who formulated the approximations for pi, trigonometrical sine, cosine, arc tan power series (now known as Gregory series) that were rediscovered in Europe three centuries later. Nilakantha Somayaji provided a convergent infinite geometric progression. In north India, Narayana Pandita worked out a rule for finding out factors of divisors of a number, much before such an attempt was made in Europe. He was well known for his analysis of sets of numbers, magic squares and the like. As for astronomy, the astrolabe began to be used during the Muslim rule. Maharaja Sawai Jai Singh erected five observatories in Benares, Mathura, Ujjain, Delhi and Jaipur (early 18th Century). Though Jai Singh knew the use of telescope and European heliocentric astronomy, he clung to the traditional geo-centric calculations of Indian astronomy, but raised the level of observational mathematics.
- India 1000 to 2000
(Excerpted from an article by B.V.Subbarayyappa in the book India 1000 to 2000, Editor : T.J.S.George, published in December 1999 by Express Publications (Madurai) Ltd, Express Estates, Anna Salai, Chennai - 600 002. The excerpt was also published in The New Indian Express on Sunday in the FYI column on April 8, 2001.)
Jyothisha (astronomy) was one of the six auxiliaries of the Vedas and the earliest Indian astronomical text goes by the name of Vedanga Jyotisha. Year-long sacrifices commenced from the day following the winter solstice and Vedic knowledge of both winter and summer solstices was fairly accurate. The Vedanga Jyotisha had developed a concept of a cycle of 5 years (one Yuga) for luni-solar and other time adjustments with intercalation at regular intervals.
Indian mathematics too owes its primary inspiration to Vedic practices. The Shulba sutras, part of another Vedic auxiliary called the Kalpa sutras, deal with the construction of several types of brick altars and in that context with certain geometrical problems including the Pythagorean theorem, squaring a circle, irrational numbers and the like. Yet another Vedic auxiliary, Metrics (chandah), postulated a triangular array for determining the type of combinations of 'n' syllables of long and short sounds for metrical chanting. This was mathematically developed by Halayudha who lived in Karnataka (10th Century) into a pyramidal expansion of numbers. Such an exercise appeared six centuries later in Europe, known as Pascal's triangle. Vedic mathematics and astronomy were pragmatic and integrated with Vedic religio-philosophical life.
But such an approach was not to last long. During the three centuries before and after the Christian era, there were new impulses. Astronomy became mathematics-based. In the succeeding centuries, while astronomy assimilated Hellenic ideas to some extent mathematics was really innovative. Indian astronomers were able mathematicians too. The doyen among them, Aryabhatta I (b.476 A.D.) gave the value of pi (3.1416 approx., a value used even today) worked out trigonometrical tables, areas of triangles and other plane figures, arithmetical progression, summation of series, indeterminate equations of the first order and the like. He expounded that the earth rotates about its own axis and the period of one sidereal rotation given by him is equivalent to 23h 56m 4s.1, while the modern value is 23h 56m 4s.091. He discarded the mythical Rahu-Ketu postulate concerning eclipses in favour of a scientific explanation.
Aryabhatta's junior contemporary Varahamihira, was well known for his compendium, the Panchasiddhantika, a compilation of the then extant five astronomical works called the Siddhantha- Surya, Paulisha, Romaka, Vasishta, and Paitamaha. Of them, the Suryasiddhanta, which he regarded as the most accurate, underwent revisions from time to time and continues to be an important text for computing pancangas.
Brahmagupta was a noted astronomer mathematician of the 7th Century. His remarkable contribution was his equation for solving indeterminate equations of the second order - an equation that appeared in Europe a thousand years later known as Pell's equation. His lemmas in this connection were rediscovered by Euler (1764) and Lagrange (1768). Brahmagupta was also the first to enunciate a formula for the area of a rational cyclic quadrilateral. In the latter half of the first millenium A.D. there were other noted astronomers and mathematicians like Bhaskara I, Lalla, Pruthudakasvamin, Vateshvara, Munjala, Mahavira (Jaina mathematician), Shripati, Shridhara, Aryabhatta II , and Vijayanandin. The tradition of astronomy and mathematics continued unabated - determination of procession of equinoxes, parallax, mean and true motions of planet, permutations and combinations, solving quadratic equations, square root of a negative number and the like.
Using nine digits and zero, the decimal place value system had established itself by about the 4th century A.D. Says historian of science, George Sarton, "Our numbers and the use of zero were invented by the Hindus and transmitted by Arabs, hence the name Arabic numerals which we often give them.' Brahmagupta's Brahmasphuta Siddhanta and Khandakhadyaka were also rendered into Arabic in the 9th-10th century. The Brahmi numerical forms with some modifications along with the decimal place-value system developed in India have since become universal.
The beginning of the second millenium A.D. witnessed the emergence of the notable astronomer-mathematician, Bhaskaracharya II (b.1114). His cyclic (cakravala) method for solving indeterminate equations of the second order has been hailed by the German mathematician 'Henkel', as the finest thing achieved in the theory of numbers before Lagrange. Bhaskaracharya II had also developed basic Calculus. Between the 14th and 18th Centuries, there were schools of astronomers-mathematicians in Kerala and Maharashtra, Ganesha Daivajna simplified methods of computation for almanac makers. The Kerala school was well known for its keen observations of eclipses over 55 years. Parameshwara (1360-1455) was the first in the history of mathematics to have given the exact formula for the circumradius of a cyclic quadrilateral; this was rediscovered in Europe by L'Huiler nearly 300 years later.
Adhering to the Aryabhatta tradition, other Kerala savants like Govinda Bhatta, Damodara, Nilakantha Somayaji, Jyesthadeva, Acyuta Pisharati and Putumana Somayaji added to both astronomy and mathematics. The leader of this intellectual lineage was Madhava (14th century), who formulated the approximations for pi, trigonometrical sine, cosine, arc tan power series (now known as Gregory series) that were rediscovered in Europe three centuries later. Nilakantha Somayaji provided a convergent infinite geometric progression. In north India, Narayana Pandita worked out a rule for finding out factors of divisors of a number, much before such an attempt was made in Europe. He was well known for his analysis of sets of numbers, magic squares and the like. As for astronomy, the astrolabe began to be used during the Muslim rule. Maharaja Sawai Jai Singh erected five observatories in Benares, Mathura, Ujjain, Delhi and Jaipur (early 18th Century). Though Jai Singh knew the use of telescope and European heliocentric astronomy, he clung to the traditional geo-centric calculations of Indian astronomy, but raised the level of observational mathematics.
- India 1000 to 2000
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