Dattaraya Ramchandra Kaprekar (1905-01-17 - 1986) was an Indian mathematician who discovered many interesting properties in number theory. He was born in the town Devlali, Maharashtra. Having never received any formal postgraduate training, for his entire career (1930-1962) he was a schoolteacher in the small town of Devlali in Maharashtra, India. Yet he became well known in recreational mathematics circles, and has a number, a constant, and a magic square named after him.
Kaprekar received his secondary school education in Thana and studied at Fergusson College in Pune. He attended the University of Bombay, receiving his bachelor"s degree in 1929. He published extensively, writing about such topics as recurring decimals, magic squares, and integers with special properties.
Working largely alone, Kaprekar discovered a number of results that have opened up important avenues of research in number theory. In addition to the Kaprekar constant and the Kaprekar number which were named after him, he also discovered the Self number or Devlali number, and also the important series called the Harshad number. He also constructed certain types of magic squares related to the Copernicus magic square.
Kaprekar"s famous works are below:
Kaprekar constant
The Kaprekar constant, named after him, is a fascinating general property of all number bases and may demonstrate some important but unknown theorem in number theory.The Kaprekar constant, or 6174 (1949). He showed that 6174 is reached in the limit as one repeatedly subtracts the highest and lowest numbers that can be constructed from four digits (not all the same). This series converges to 6174 in fewer than seven iterations.
Ex: lets take a non-zero 4 digit number 5, 5, 9, 2,
9552-2559=6993, 9963-3699=6264, 6642-2466=4176, 7641-1467=6174, 7641-1467=6174...this goes on.
Similarly the Kaprekar constant for 3 digits is 495.
Ex: lets take a non-zero 3 digit number 5, 5, 9,
955-559=396, 963-369=594, 954-459=495, 954-495=495...this goes on.
Kaprekar number
The Kaprekar number (also called Kaprekar series, based on the Kaprekar operation). This is a number with the interesting property that if it is squared, then two equal parts of this square also add up to the original number. This operation, of taking the last n digits of a square, and adding it to the number formed by the first (n-1) or n digits, is the Kaprekar operation.
Ex: an example 297
297^2 = 88,209 and 88 + 209 = 297!!!
So, just square the number, split it in half (leave the larger portion to the right) and add the two halves - if you get the original number back, then its a Kaprekar Number.
A few more examples of Kaprekar Numbers are - 9, 45, 297, 4879, 17344, 538461, ...
Devlali or Self number
In 1963, he also defined the property which has come to be known as self numbers, which are integers that cannot be generated by taking some other number and adding its own digits to it. For example, 21 is not a self number, since it can be generated from 15: 15 + 1 + 5 = 21. But 20 is a self number, since it cannot be generated from any other integer. He also gave a test for verifying this property in any number. These are sometimes referred to as Devlali numbers (after the town where he lived, Devlali,Maharashtra); though this appears to have been his preferred designation, the term self-number is more widespread. Sometimes these are also designated Colombian numbers after a later designation.
Harshad Number
He also discovered the Harshad numbers which he named harshad, meaning "giving joy" (Sanskrit harsha, joy +da taddhita pratyaya, causative); these have the property that they are divisible by the sum of their digits. Thus 12, which is divisible by 1 + 2 = 3, is a Harshad number. These were later also called Niven numbers after a 1997 lecture on these by the Canadian mathematician Ivan M. Niven. Numbers which are Harshad in all bases (only 1, 2, 4, and 6) are called all-Harshad numbers. Much work has been done on Harshad numbers, and their distribution, frequency, etc. are a matter of considerable interest in number theory today.
In 1927 he won the Wrangler R. P. Paranjpe Mathematical Prize for an original piece of work in mathematics.
Although he was largely unknown outside recreational mathematics circles in his lifetime, D. R. Kaprekar"s work and its impact on number theory has become widely recognized in recent years.
courtesy:internet
Monday, December 17, 2007
Subrahmanyan Chandrasekhar
Subrahmanyan Chandrasekhar (19 Oct 1910 - 21 Aug 1995) was known throughout his life as Chandra. Chandra wanted to be a scientist and his mother encouraged him to follow this route. He had a role model in his paternal uncle Sir Chandrasekhara Venkata Raman who went on to win the Nobel prize in 1930 for his 1928 discovery of Raman scattering and the Raman effect, which is a change in the wavelength of light occurring when a beam of light is deflected by molecules.
Chandra studied at Presidency College, University of Madras, and he wrote his first research paper while still an undergraduate there. The paper was published in the Proceeding of the Royal Society where it had been submitted by Ralph Fowler.
Chandra obtained a scholarship from the Indian government to finance his studies in England, and in 1930 he left India to study at Trinity College, Cambridge, England. From 1933 to 1937 he undertook research at Cambridge, but he returned to India in 1936 to marry Lalitha on 11 September.
At first he worked in Yerkes Observatory, part of the University of Chicago in Wisconsin. Later he moved to work on the university campus in the city of Chicago. During World War II he worked in the Ballistic Research Laboratories at the Aberdeen Proving Ground in Maryland.
Two reports, written in 1943, show the type of problems he was working on at this time: the first is On the decay of plane shock waves while the second is The normal reflection of a blast wave. He was honoured with being appointed Morton D Hull distinguished service professor of the University of Chicago in 1952.
Chandrasekhar published around 400 papers and many books.
His research interests were exceptionally broad but we can divide them into topics and rough periods when he was concentrating on these particular topics.
- stellar structure, including the theory of white dwarfs, from 1929 to 1939.
- stellar dynamics from 1939 to 1943.
- the theory of radiative transfer and the quantum theory of the negative ion of hydrogen from 1943 to 1950.
- Hydrodynamic and hydromagnetic stability from 1950 to 1961.
- During most of the 1960s he studied the equilibrium and the stability of ellipsoidal figures of equilibrium but during this period he also began work on topics from general relativity, the radiation reaction process, and the stability of relativistic stars.
- During the period from 1971 to 1983 he undertook research into the mathematical theory of black holes, then for the last period of his life he worked on the theory of colliding gravitational waves.
In 1930 Chandra showed that a star of a mass greater than 1.4 times that of the Sun (now known as the Chandrasekhar's limit) had to end its life by collapsing into an object of enormous density unlike any object known at that time.
Many years later Chandra was awarded the Nobel prize for Physics in 1983 for his theoretical studies of the physical processes of importance to the structure and evolution of the stars.
He found the Mathematical Theory of Black Holes (1983).
His other books include
- An Introduction to the Study of Stellar Structure (1939)
- Principles of Stellar Dynamics (1942)
- Radiative Transfer (1950)
- Plasma Physics (1960)
- Hydrodynamic and Hydromagnetic Stability (1961)
- Ellipsoidal Figures of Equilibrium (1969)
- Truth and Beauty: Aesthetics and Motivations in Science (1987)
- Newton's Principia for the Common Reader (1995).
These texts have played a major role in mathematical astronomy.
From 1952 until 1971 Chandrasekhar was editor of the Astrophysical Journal . This journal was originally a local University of Chicago publication, but it grew in stature to become national publication of the American Astronomical Society, then a leading international journal.
Chandrasekhar received many honours for his outstanding contributions some of which, such as
- the Nobel prize for Physics in 1983
- the Royal Society's Royal Medal of 1962
- the Royal Society's Copley Medal of 1984
- the Bruce medal of the Astronomical Society of the Pacific
- the Henry Draper medal of the National Academy of Sciences (United States)
- the Gold Medal of the Royal Astronomical Society.
Chandra retired in 1980 but continued to live in Chicago where he was made professor emeritus in 1985. He continued to give thought-provoking lectures such as Newton and Michelangelo which he delivered at the 1994 Meeting of Nobel Laureates held in Lindau. Other lectures in a similar vein include Shakespeare, Newton and Beethoven or patterns of creativity and The perception of beauty and the pursuit of science.
Courtesy:Internet
Chandra studied at Presidency College, University of Madras, and he wrote his first research paper while still an undergraduate there. The paper was published in the Proceeding of the Royal Society where it had been submitted by Ralph Fowler.
Chandra obtained a scholarship from the Indian government to finance his studies in England, and in 1930 he left India to study at Trinity College, Cambridge, England. From 1933 to 1937 he undertook research at Cambridge, but he returned to India in 1936 to marry Lalitha on 11 September.
At first he worked in Yerkes Observatory, part of the University of Chicago in Wisconsin. Later he moved to work on the university campus in the city of Chicago. During World War II he worked in the Ballistic Research Laboratories at the Aberdeen Proving Ground in Maryland.
Two reports, written in 1943, show the type of problems he was working on at this time: the first is On the decay of plane shock waves while the second is The normal reflection of a blast wave. He was honoured with being appointed Morton D Hull distinguished service professor of the University of Chicago in 1952.
Chandrasekhar published around 400 papers and many books.
His research interests were exceptionally broad but we can divide them into topics and rough periods when he was concentrating on these particular topics.
- stellar structure, including the theory of white dwarfs, from 1929 to 1939.
- stellar dynamics from 1939 to 1943.
- the theory of radiative transfer and the quantum theory of the negative ion of hydrogen from 1943 to 1950.
- Hydrodynamic and hydromagnetic stability from 1950 to 1961.
- During most of the 1960s he studied the equilibrium and the stability of ellipsoidal figures of equilibrium but during this period he also began work on topics from general relativity, the radiation reaction process, and the stability of relativistic stars.
- During the period from 1971 to 1983 he undertook research into the mathematical theory of black holes, then for the last period of his life he worked on the theory of colliding gravitational waves.
In 1930 Chandra showed that a star of a mass greater than 1.4 times that of the Sun (now known as the Chandrasekhar's limit) had to end its life by collapsing into an object of enormous density unlike any object known at that time.
Many years later Chandra was awarded the Nobel prize for Physics in 1983 for his theoretical studies of the physical processes of importance to the structure and evolution of the stars.
He found the Mathematical Theory of Black Holes (1983).
His other books include
- An Introduction to the Study of Stellar Structure (1939)
- Principles of Stellar Dynamics (1942)
- Radiative Transfer (1950)
- Plasma Physics (1960)
- Hydrodynamic and Hydromagnetic Stability (1961)
- Ellipsoidal Figures of Equilibrium (1969)
- Truth and Beauty: Aesthetics and Motivations in Science (1987)
- Newton's Principia for the Common Reader (1995).
These texts have played a major role in mathematical astronomy.
From 1952 until 1971 Chandrasekhar was editor of the Astrophysical Journal . This journal was originally a local University of Chicago publication, but it grew in stature to become national publication of the American Astronomical Society, then a leading international journal.
Chandrasekhar received many honours for his outstanding contributions some of which, such as
- the Nobel prize for Physics in 1983
- the Royal Society's Royal Medal of 1962
- the Royal Society's Copley Medal of 1984
- the Bruce medal of the Astronomical Society of the Pacific
- the Henry Draper medal of the National Academy of Sciences (United States)
- the Gold Medal of the Royal Astronomical Society.
Chandra retired in 1980 but continued to live in Chicago where he was made professor emeritus in 1985. He continued to give thought-provoking lectures such as Newton and Michelangelo which he delivered at the 1994 Meeting of Nobel Laureates held in Lindau. Other lectures in a similar vein include Shakespeare, Newton and Beethoven or patterns of creativity and The perception of beauty and the pursuit of science.
Courtesy:Internet
Sunday, December 16, 2007
Srinivasa Aiyangar Ramanujan
Srinivasa Aiyangar Ramanujan (22 Dec 1887 - 26 April 1920) was one of India's greatest mathematical geniuses. He was born in Erode, Tamil Nadu state, India. He made substantial contributions to the analytical theory of numbers and worked on elliptic functions, continued fractions, and infinite series.
In 1900 he began to work on his own on mathematics summing geometric and arithmetic series. Ramanujan was shown how to solve cubic equations in 1902 and he went on to find his own method to solve the quartic. The following year, not knowing that the quintic could not be solved by radicals, he tried (and of course failed) to solve the quintic.
By 1904 Ramanujan had begun to undertake deep research. He investigated the series (1/n) and calculated Euler's constant to 15 decimal places. He began to study the Bernoulli numbers, although this was entirely his own independent discovery. He continued his mathematical work, however, and at this time he worked on hypergeometric series and investigated relations between integrals and series. He was to discover later that he had been studying elliptic functions.
By 1908 Ramanujan studied fractions and divergent series.
In 1910 Ramanujan continued to develop his mathematical ideas and began to pose problems and solve problems in the Journal of the Indian Mathematical Society. He devoloped relations between elliptic modular equations.
In 1911 after publication of a brilliant research paper on Bernoulli numbers in the Journal of the Indian Mathematical Society he gained recognition for his work. Despite his lack of a university education, he was becoming well known in the Madras area as a mathematical genius.
On 16 March 1916 Ramanujan graduated from Cambridge with a Bachelor of Science by Research (the degree was called a Ph.D. from 1920). He had been allowed to enrol in June 1914 despite not having the proper qualifications. Ramanujan's dissertation was on Highly composite numbers and consisted of seven of his papers published in England.
On 18 February 1918 Ramanujan was elected a fellow of the Cambridge Philosophical Society and then three days later, the greatest honour that he would receive, his name appeared on the list for election as a fellow of the Royal Society of London.
Ramanujan worked out the Riemann series, the elliptic integrals, hypergeometric series and functional equations of the zeta function.
Ramanujan independently discovered results of Gauss, Kummer and others on hypergeometric series. Ramanujan's own work on partial sums and products of hypergeometric series have led to major development in the topic. Perhaps his most famous work was on the number p(n) of partitions of an integer n into summands. MacMahon had produced tables of the value of p(n) for small numbers n, and Ramanujan used this numerical data to conjecture some remarkable properties some of which he proved using elliptic functions. Other were only proved after Ramanujan's death.
Ramanujan left a number of unpublished notebooks filled with theorems that mathematicians have continued to study. G N Watson, Mason Professor of Pure Mathematics at Birmingham from 1918 to 1951 published 14 papers under the general title Theorems stated by Ramanujan and in all he published nearly 30 papers which were inspired by Ramanujan's work. Hardy passed on to Watson the large number of manuscripts of Ramanujan that he had, both written before 1914 and some written in Ramanujan's last year in India before his death.
Ramanujan's formulae have found applications in the field of crystallography and in string theory.
During the year 1987 (Ramanujan's centennial), the printed form of Ramanujan's Lost Notebook by the Narosa publishing house of Springer-Verlag was released by the late Indian prime minister, Rajiv Gandhi, who presented the first copy to S. Janaki Ammal Ramanujan (Ramanujan's late widow) and the second copy to George Andrews in recognition of his contributions in the field of number theory.
A stamp was issued by the Indian Post Office to celebrate the 75th anniversary of his birth.
A prize for young mathematicians from developing countries has been created in the name of Ramanujan by the International Centre for Theoretical Physics (ICTP), in cooperation with the International Mathematical Union, who nominate members of the prize committee.
Courtesy:Internet
In 1900 he began to work on his own on mathematics summing geometric and arithmetic series. Ramanujan was shown how to solve cubic equations in 1902 and he went on to find his own method to solve the quartic. The following year, not knowing that the quintic could not be solved by radicals, he tried (and of course failed) to solve the quintic.
By 1904 Ramanujan had begun to undertake deep research. He investigated the series (1/n) and calculated Euler's constant to 15 decimal places. He began to study the Bernoulli numbers, although this was entirely his own independent discovery. He continued his mathematical work, however, and at this time he worked on hypergeometric series and investigated relations between integrals and series. He was to discover later that he had been studying elliptic functions.
By 1908 Ramanujan studied fractions and divergent series.
In 1910 Ramanujan continued to develop his mathematical ideas and began to pose problems and solve problems in the Journal of the Indian Mathematical Society. He devoloped relations between elliptic modular equations.
In 1911 after publication of a brilliant research paper on Bernoulli numbers in the Journal of the Indian Mathematical Society he gained recognition for his work. Despite his lack of a university education, he was becoming well known in the Madras area as a mathematical genius.
On 16 March 1916 Ramanujan graduated from Cambridge with a Bachelor of Science by Research (the degree was called a Ph.D. from 1920). He had been allowed to enrol in June 1914 despite not having the proper qualifications. Ramanujan's dissertation was on Highly composite numbers and consisted of seven of his papers published in England.
On 18 February 1918 Ramanujan was elected a fellow of the Cambridge Philosophical Society and then three days later, the greatest honour that he would receive, his name appeared on the list for election as a fellow of the Royal Society of London.
Ramanujan worked out the Riemann series, the elliptic integrals, hypergeometric series and functional equations of the zeta function.
Ramanujan independently discovered results of Gauss, Kummer and others on hypergeometric series. Ramanujan's own work on partial sums and products of hypergeometric series have led to major development in the topic. Perhaps his most famous work was on the number p(n) of partitions of an integer n into summands. MacMahon had produced tables of the value of p(n) for small numbers n, and Ramanujan used this numerical data to conjecture some remarkable properties some of which he proved using elliptic functions. Other were only proved after Ramanujan's death.
Ramanujan left a number of unpublished notebooks filled with theorems that mathematicians have continued to study. G N Watson, Mason Professor of Pure Mathematics at Birmingham from 1918 to 1951 published 14 papers under the general title Theorems stated by Ramanujan and in all he published nearly 30 papers which were inspired by Ramanujan's work. Hardy passed on to Watson the large number of manuscripts of Ramanujan that he had, both written before 1914 and some written in Ramanujan's last year in India before his death.
Ramanujan's formulae have found applications in the field of crystallography and in string theory.
During the year 1987 (Ramanujan's centennial), the printed form of Ramanujan's Lost Notebook by the Narosa publishing house of Springer-Verlag was released by the late Indian prime minister, Rajiv Gandhi, who presented the first copy to S. Janaki Ammal Ramanujan (Ramanujan's late widow) and the second copy to George Andrews in recognition of his contributions in the field of number theory.
A stamp was issued by the Indian Post Office to celebrate the 75th anniversary of his birth.
A prize for young mathematicians from developing countries has been created in the name of Ramanujan by the International Centre for Theoretical Physics (ICTP), in cooperation with the International Mathematical Union, who nominate members of the prize committee.
Courtesy:Internet
Vijayanandi
Vijayanandi (940-1010) was an Indian mathematician and astronomer whose most famous work was the Karanatilaka. We should note that there was another astronomer named Vijayanandi who was mentioned by Varahamihira in one of his works. Since Varahamihira wrote around 550 and the Karanatilaka was written around 966, there must be two astronomers both named "Vijayanandi".
The Karanatilaka has not survived in its original form but we know of the text through an Arabic translation by al-Biruni. It is a work in fourteen chapters covering the standard topics of Indian astronomy.
It deals with the topics of:
- units of time measurement;
- mean and true longitudes of the sun and moon;
- the length of daylight;
- mean longitudes of the five planets;
- true longitudes of the five planets;
- the three problems of diurnal rotation;
- lunar eclipses, solar eclipses;
- the projection of eclipses;
- first visibility of the planets;
- conjunctions of the planets with each other and with fixed stars;
- the moon's crescent;
- the patas of the moon and sun.
Like other Indian astronomers, Vijayanandi made contributions to trigonometry and it appears that his calculation of the periods was computed by using tables of sines and versed sines. It is significant that accuracy was need in trigonometric tables to give accurate astronomical theories and this motivated many of the Indian mathematicians to produce more accurate methods of approximating entries in tables.
Courtesy:Internet
The Karanatilaka has not survived in its original form but we know of the text through an Arabic translation by al-Biruni. It is a work in fourteen chapters covering the standard topics of Indian astronomy.
It deals with the topics of:
- units of time measurement;
- mean and true longitudes of the sun and moon;
- the length of daylight;
- mean longitudes of the five planets;
- true longitudes of the five planets;
- the three problems of diurnal rotation;
- lunar eclipses, solar eclipses;
- the projection of eclipses;
- first visibility of the planets;
- conjunctions of the planets with each other and with fixed stars;
- the moon's crescent;
- the patas of the moon and sun.
Like other Indian astronomers, Vijayanandi made contributions to trigonometry and it appears that his calculation of the periods was computed by using tables of sines and versed sines. It is significant that accuracy was need in trigonometric tables to give accurate astronomical theories and this motivated many of the Indian mathematicians to produce more accurate methods of approximating entries in tables.
Courtesy:Internet
Varahamihira
Varahamihira (505-587) was born in Kapitthaka, India. The most famous work by Varahamihira is the Pancasiddhantika (The Five Astronomical Canons) dated 575 AD. This work is important in itself and also in giving us information about older Indian texts which are now lost. The work is a treatise on mathematical astronomy and it summarises five earlier astronomical treatises, namely the Surya, Romaka, Paulisa, Vasistha and Paitamaha siddhantas.
One treatise which Varahamihira summarises was the Romaka-Siddhanta which itself was based on the epicycle theory of the motions of the Sun and the Moon given by the Greeks in the 1st century AD.
Varahamihira made some important mathematical discoveries. Among these are certain trigonometric formulae which translated into our present day notation correspond to
sin x = cos(ð/2 - x),
sin2x + cos2x = 1, and
(1 - cos 2x)/2 = sin2x.
Another important contribution to trigonometry was his sine tables where he improved those of Aryabhata I giving more accurate values. This motivated much of the improved accuracy they achieved by developing new interpolation methods.
The Jaina school of mathematics investigated rules for computing the number of ways in which r objects can be selected from n objects over the course of many hundreds of years. They gave rules to compute the binomial coefficients nCr which amount to
nCr = n(n-1)(n-2)...(n-r+1)/r!
However, Varahamihira attacked the problem of computing nCr in a rather different way. He wrote the numbers n in a column with n = 1 at the bottom. He then put the numbers r in rows with r = 1 at the left-hand side. Starting at the bottom left side of the array which corresponds to the values n = 1, r = 1, the values of nCr are found by summing two entries, namely the one directly below the (n, r) position and the one immediately to the left of it. Of course this table is none other than Pascal's triangle for finding the binomial coefficients despite being viewed from a different angle from the way we build it up today.
Varahamihira's work on magic squares. In particular he examines a pandiagonal magic square of order four which occurs in Varahamihira's work.
Courtesy:Internet
One treatise which Varahamihira summarises was the Romaka-Siddhanta which itself was based on the epicycle theory of the motions of the Sun and the Moon given by the Greeks in the 1st century AD.
Varahamihira made some important mathematical discoveries. Among these are certain trigonometric formulae which translated into our present day notation correspond to
sin x = cos(ð/2 - x),
sin2x + cos2x = 1, and
(1 - cos 2x)/2 = sin2x.
Another important contribution to trigonometry was his sine tables where he improved those of Aryabhata I giving more accurate values. This motivated much of the improved accuracy they achieved by developing new interpolation methods.
The Jaina school of mathematics investigated rules for computing the number of ways in which r objects can be selected from n objects over the course of many hundreds of years. They gave rules to compute the binomial coefficients nCr which amount to
nCr = n(n-1)(n-2)...(n-r+1)/r!
However, Varahamihira attacked the problem of computing nCr in a rather different way. He wrote the numbers n in a column with n = 1 at the bottom. He then put the numbers r in rows with r = 1 at the left-hand side. Starting at the bottom left side of the array which corresponds to the values n = 1, r = 1, the values of nCr are found by summing two entries, namely the one directly below the (n, r) position and the one immediately to the left of it. Of course this table is none other than Pascal's triangle for finding the binomial coefficients despite being viewed from a different angle from the way we build it up today.
Varahamihira's work on magic squares. In particular he examines a pandiagonal magic square of order four which occurs in Varahamihira's work.
Courtesy:Internet
Vijay Kumar Patodi
Vijay Kumar Patodi (12 March 1945-21 Dec 1976) born in Guna, India. Mathematical fame for Patodi came early in his career with papers of great importance coming for the work of his Ph.D. His doctoral thesis, Heat equation and the index of elliptic operators, was supervised by M S Narasimhan and S Ramanan and the degree was awarded by the University of Bombay in 1971.
Patodi's first paper Curvature and the eigenforms of the Laplace operator was part of his thesis. The second paper which came from his thesis was An analytic proof of the Riemann- Roch- Hirzebruch theorem for Kaehler manifolds which extended the methods of his first paper to a much more complicated situation.
The years 1971 to 1973 were ones which Patodi spent on leave at the Institute for Advanced Study at Princeton. There he worked with M F Atiyah and made several visits to work with others in his field at various centres in the United States and England. During this time he also collaborated with R Bott and I M Singer.
On his return to Bombay and the Tata Institute in 1973 Patodi was promoted to associate professor. He was promoted to full professor in 1976.
Patodi's publications, in addition to the two mentioned above, include a number of joint ones with Atiyah and Singer. These papers introduce a spectral invariant of a compact Riemannian manifold. In another paper he studies the relationship between Riemannian structures and triangulations. Other work gives a combinatorial formula for Pontryagin classes.
Courtesy:Internet
Patodi's first paper Curvature and the eigenforms of the Laplace operator was part of his thesis. The second paper which came from his thesis was An analytic proof of the Riemann- Roch- Hirzebruch theorem for Kaehler manifolds which extended the methods of his first paper to a much more complicated situation.
The years 1971 to 1973 were ones which Patodi spent on leave at the Institute for Advanced Study at Princeton. There he worked with M F Atiyah and made several visits to work with others in his field at various centres in the United States and England. During this time he also collaborated with R Bott and I M Singer.
On his return to Bombay and the Tata Institute in 1973 Patodi was promoted to associate professor. He was promoted to full professor in 1976.
Patodi's publications, in addition to the two mentioned above, include a number of joint ones with Atiyah and Singer. These papers introduce a spectral invariant of a compact Riemannian manifold. In another paper he studies the relationship between Riemannian structures and triangulations. Other work gives a combinatorial formula for Pontryagin classes.
Courtesy:Internet
Sripati
Sripati (1019-1066) was born in Rohinikhanda, Maharashtra, India. Sripati was a follower of the teaching of Lalla writing on astrology, astronomy and mathematics. His mathematical work was undertaken with applications to astronomy in mind, for example a study of spheres. His work on astronomy was undertaken to provide a basis for his astrology. Sripati was the most prominent Indian mathematicians of the 11th Century.
Sripati's works are: Dhikotidakarana written in 1039, a work of twenty verses on solar and lunar eclipses; Dhruvamanasa written in 1056, a work of 105 verses on calculating planetary longitudes, eclipses and planetary transits; Siddhantasekhara a major work on astronomy in 19 chapters; and Ganitatilaka an incomplete arithmetical treatise in 125 verses based on a work by Sridhara.
Other mathematics included in Sripati's work includes, in particular, rules for the solution of simultaneous indeterminate equations of the first degree that are similar to those given by Brahmagupta.
He wrote the Jyotisaratnamala which was an astrology text in twenty chapters based on the Jyotisaratnakosa of Lalla. Sripati wrote a commentary on this work in Marathi and it is one of the oldest works to have survived that is written in that language. Another work on astrology written by Sripati is the Jatakapaddhati or Sripatipaddhati which is in eight chapters.
Genethlialogy was the science of casting nativities and it was the earliest branch of astrology which claimed to be able to predict the course of a person's life based on the positions of the planets and of the signs of the zodiac at the moment the person was born or conceived.
There is one other work on astrology the Daivajnavallabha which some historians claim was written by Sripati while other claim that it is the work of Varahamihira. As yet nobody has come up with a definite case to show which of these two is the author, or even whether the author is another astrologer.
Courtesy:Internet
Sripati's works are: Dhikotidakarana written in 1039, a work of twenty verses on solar and lunar eclipses; Dhruvamanasa written in 1056, a work of 105 verses on calculating planetary longitudes, eclipses and planetary transits; Siddhantasekhara a major work on astronomy in 19 chapters; and Ganitatilaka an incomplete arithmetical treatise in 125 verses based on a work by Sridhara.
Other mathematics included in Sripati's work includes, in particular, rules for the solution of simultaneous indeterminate equations of the first degree that are similar to those given by Brahmagupta.
He wrote the Jyotisaratnamala which was an astrology text in twenty chapters based on the Jyotisaratnakosa of Lalla. Sripati wrote a commentary on this work in Marathi and it is one of the oldest works to have survived that is written in that language. Another work on astrology written by Sripati is the Jatakapaddhati or Sripatipaddhati which is in eight chapters.
Genethlialogy was the science of casting nativities and it was the earliest branch of astrology which claimed to be able to predict the course of a person's life based on the positions of the planets and of the signs of the zodiac at the moment the person was born or conceived.
There is one other work on astrology the Daivajnavallabha which some historians claim was written by Sripati while other claim that it is the work of Varahamihira. As yet nobody has come up with a definite case to show which of these two is the author, or even whether the author is another astrologer.
Courtesy:Internet
Satyendranath Bose
Satyendranath Bose (1 Jan 1894-4 Feb 1974)born in Calcutta, India. He did important work in quantum theory, in particular on Planck's black body radiation law. Bose sent his paper Planck's Law and the Hypothesis of Light Quanta (1924) to Einstein. This paper was only four pages long but it was highly significant. The derivation of Planck's formula had not been to Planck's satisfaction, and Einstein too was unhappy with it. Now Bose was able to derive the formula for radiation from Boltzmann's statistics. The paper, and his method of deriving Planck's radiation formula, was enthusiastically endorsed by Einstein who saw at once that Bose had removed a major objection against light quanta. The paper was translated into German by Einstein and submitted with a strong recommendation to the Zeitschrift für Physik. Einstein extended Bose's treatment to material particles whose number is conserved and published several papers on this extension.
An important consequence of Einstein's response to Bose's article was that his application to the University of Dacca for two years research leave beginning in 1924 was approved. He now had the chance of meeting European scientists and travelled first to Paris where he met Langevin and de Broglie. In October 1925 Bose travelled from Paris to Berlin where he met Einstein. Much progress had been made by Einstein following his receipt of Bose's paper for he was able to see how the ideas could be taken forward. While he was in Berlin Bose attended a course on quantum theory given by Born.
Bose published on statistical mechanics leading to the Einstein-Bose statistics. Dirac coined the term boson for particles obeying these statistics. Through these terms his name is rightly known and remembered, for indeed his contributions are remarkable, especially given the fact that he made his important discoveries working in isolation from the mainstream developments in Europe.
He gave leadership in many ways: as president of the physics section of the Indian Science Congress in 1939, as general president of the Indian Science Congress in Delhi in 1944, and as president of the National Institute of Science of India in 1949. His greatest honour was election to the Royal Society of London in 1958.
After Bose retired from Calcutta University in 1956 he was appointed as vice-chancellor of Viswa-Bharati University, Santiniketan. Two years later he was honoured with the post of national professor.
Courtesy:Internet
An important consequence of Einstein's response to Bose's article was that his application to the University of Dacca for two years research leave beginning in 1924 was approved. He now had the chance of meeting European scientists and travelled first to Paris where he met Langevin and de Broglie. In October 1925 Bose travelled from Paris to Berlin where he met Einstein. Much progress had been made by Einstein following his receipt of Bose's paper for he was able to see how the ideas could be taken forward. While he was in Berlin Bose attended a course on quantum theory given by Born.
Bose published on statistical mechanics leading to the Einstein-Bose statistics. Dirac coined the term boson for particles obeying these statistics. Through these terms his name is rightly known and remembered, for indeed his contributions are remarkable, especially given the fact that he made his important discoveries working in isolation from the mainstream developments in Europe.
He gave leadership in many ways: as president of the physics section of the Indian Science Congress in 1939, as general president of the Indian Science Congress in Delhi in 1944, and as president of the National Institute of Science of India in 1949. His greatest honour was election to the Royal Society of London in 1958.
After Bose retired from Calcutta University in 1956 he was appointed as vice-chancellor of Viswa-Bharati University, Santiniketan. Two years later he was honoured with the post of national professor.
Courtesy:Internet
Prasanta Chandra Mahalanobis
Prasanta Chandra Mahalanobis (June 29, 1893–June 28, 1972) was an Indian scientist and applied statistician. He is best known for the Mahalanobis distance, a statistical measure. He did pioneering work on anthropometric variation in India. He founded the Indian Statistical Institute, and contributed to large scale sample surveys.
Inspired by Biometrika and mentored by Acharya Brajendra Nath Seal he started his statistical work. Initially he worked on analyzing university exam results, anthropometric measurements on Anglo-Indians of Calcutta and some metrological problems. He also worked as a meteorologist for some time. In 1924, when he was working on the probable error of results of agricultural experiments, he met Ronald Fisher, with whom he established a life-long friendship. He also worked on schemes to prevent floods.
His most important contributions are related to large scale sample surveys. He introduced the concept of pilot surveys and advocated the usefulness of sampling methods. His name is also associated with the scale free multivariate distance measure, the Mahalanobis distance. He founded the Indian Statistical Institute on 17 December, 1931.
In later life, he contributed prominently to newly independent India's five-year plans starting from the second. His variant of Wassily Leontief's neo-Marxist Input-output model, the Mahalanobis model, was employed in the Second Five Year Plan, which worked towards the rapid industrialization of India and with other colleagues at his institute, he played a key role in the development of a statistical infrastructure.
He was awarded the Weldon Medal from Oxford University in1944, Fellow of the Royal Society, in 1945, Honorary President, International Statistical Institute in1957 and Padma Vibhushan in 1968.
The government of India has decided to celebrate his birthday, 29 June, as National Statistical Day.
Courtesy:Internet
Inspired by Biometrika and mentored by Acharya Brajendra Nath Seal he started his statistical work. Initially he worked on analyzing university exam results, anthropometric measurements on Anglo-Indians of Calcutta and some metrological problems. He also worked as a meteorologist for some time. In 1924, when he was working on the probable error of results of agricultural experiments, he met Ronald Fisher, with whom he established a life-long friendship. He also worked on schemes to prevent floods.
His most important contributions are related to large scale sample surveys. He introduced the concept of pilot surveys and advocated the usefulness of sampling methods. His name is also associated with the scale free multivariate distance measure, the Mahalanobis distance. He founded the Indian Statistical Institute on 17 December, 1931.
In later life, he contributed prominently to newly independent India's five-year plans starting from the second. His variant of Wassily Leontief's neo-Marxist Input-output model, the Mahalanobis model, was employed in the Second Five Year Plan, which worked towards the rapid industrialization of India and with other colleagues at his institute, he played a key role in the development of a statistical infrastructure.
He was awarded the Weldon Medal from Oxford University in1944, Fellow of the Royal Society, in 1945, Honorary President, International Statistical Institute in1957 and Padma Vibhushan in 1968.
The government of India has decided to celebrate his birthday, 29 June, as National Statistical Day.
Courtesy:Internet
Lalla
Lalla (720-790) was an Indian astronomer and mathematician who followed the tradition of Aryabhata I. Lalla's most famous work was entitled Shishyadhividdhidatantra.
This major treatise was in two volumes.
First volume: On the computation of the positions of the planets, was in thirteen chapters and covered topics such as: mean longitudes of the planets; true longitudes of the planets; the three problems of diurnal rotation; lunar eclipses; solar eclipses; syzygies; risings and settings; the shadow of the moon; the moon's crescent; conjunctions of the planets with each other; conjunctions of the planets with the fixed stars; the patas of the moon and sun, and a final chapter in the first volume which forms a conclusion.
Second volume: On the sphere. In this volume Lalla examined topics such as: graphical representation; the celestial sphere; the principle of mean motion; the terrestrial sphere; motions and stations of the planets; geography; erroneous knowledge; instruments; and finally selected problems.
Lalla also wrote a commentary on Khandakhadyaka, a work of Brahmagupta. Lalla's commentary has not survived but there is another work on astrology by Lalla which has survived, namely the Jyotisaratnakosa. This was a very popular work which was the main one on the subject in India for around 300 years.
Courtesy:Internet
This major treatise was in two volumes.
First volume: On the computation of the positions of the planets, was in thirteen chapters and covered topics such as: mean longitudes of the planets; true longitudes of the planets; the three problems of diurnal rotation; lunar eclipses; solar eclipses; syzygies; risings and settings; the shadow of the moon; the moon's crescent; conjunctions of the planets with each other; conjunctions of the planets with the fixed stars; the patas of the moon and sun, and a final chapter in the first volume which forms a conclusion.
Second volume: On the sphere. In this volume Lalla examined topics such as: graphical representation; the celestial sphere; the principle of mean motion; the terrestrial sphere; motions and stations of the planets; geography; erroneous knowledge; instruments; and finally selected problems.
Lalla also wrote a commentary on Khandakhadyaka, a work of Brahmagupta. Lalla's commentary has not survived but there is another work on astrology by Lalla which has survived, namely the Jyotisaratnakosa. This was a very popular work which was the main one on the subject in India for around 300 years.
Courtesy:Internet
Pandita Jagannatha Samrat
Pandita Jagannatha Samrat (1652-1744) was an Indian astronomer and mathematician in the court of Jai Singh II of Amber. He learned Arabic and Persian in order to study Islamic astronomy. His works include Rekhaganita, a translation of Euclid's Elements from Arabic; Siddhantasarakaustubha, a translation of the Almagest from Arabic; and two works on astronomical instruments such as the astrolabe, Siddhanta-samrat and Yantraprakara, which also record astronomical observations made by Jagannatha.
Courtesy:Internet
Courtesy:Internet
Saturday, December 15, 2007
Kamalakara
Kamalakara (1616-1700) born in Benaras, india, was an Indian astronomer and mathematician who came from a family of famous astronomers. Kamalakara's father was Nrsimha who was born in 1586. Two of Kamalakara's three brothers were also famous astronomer/ mathematicians, these being Divakara, who was the eldest of the brothers born in 1606, and Ranganatha who was younger than Kamalakara.
Kamalakara combined traditional Indian astronomy with Aristotelian physics and Ptolemaic astronomy as presented by Islamic scientists (especially Ulugh Beg). Following his family's tradition he wrote a commentary, Manorama, on Ganesa's Grahalaghava and, like his father, Nrsimha, another commentary on the Suryasiddhanta, called the Vasanabhasya.
Kamalakara's most famous work, the Siddhanta-tattva-viveka in 1658. The Siddhanta-tattva-viveka is a Sanskrit text and in it Kamalakara makes frequent use of the place-value number system with Sanskrit numerals.
Courtesy:Internet
Kamalakara combined traditional Indian astronomy with Aristotelian physics and Ptolemaic astronomy as presented by Islamic scientists (especially Ulugh Beg). Following his family's tradition he wrote a commentary, Manorama, on Ganesa's Grahalaghava and, like his father, Nrsimha, another commentary on the Suryasiddhanta, called the Vasanabhasya.
Kamalakara's most famous work, the Siddhanta-tattva-viveka in 1658. The Siddhanta-tattva-viveka is a Sanskrit text and in it Kamalakara makes frequent use of the place-value number system with Sanskrit numerals.
Courtesy:Internet
Munishvara
Munishvara (17th century) Indian mathematician, presented accurate sine tables.
Courtesy : Internet
Courtesy : Internet
Achyuta Pisharati
Achyuta Pisharati (1550–1621) was a renowned Sanskrit grammarian, astrologer and mathematician of his time. He was a student of Jyestadeva and a member of Madhava of Sangamagrama's Kerala school. He is remembered mainy for his part in the composition of his student Melpathur Narayana Bhattathiri's devotional poem, Narayaneeyam.
He discovered the technique of 'the reduction of the ecliptic'. He authored Sphuta-nirnaya , Raasi-gola-sphuta-neeti (raasi meaning zodiac, gola meaning sphere and neeti roughly meaning rule), Karanottama (1593) and a four- chapter treastise Uparagakriyakrama on lunar and solar eclipses.
Courtesy:Internet
He discovered the technique of 'the reduction of the ecliptic'. He authored Sphuta-nirnaya , Raasi-gola-sphuta-neeti (raasi meaning zodiac, gola meaning sphere and neeti roughly meaning rule), Karanottama (1593) and a four- chapter treastise Uparagakriyakrama on lunar and solar eclipses.
Courtesy:Internet
Jyesthadeva
Jyesthadeva (1500-1575) lived on the southwest coast of India in the district of Kerala. He belonged to the Kerala school of mathematics built on the work of Madhava, Nilakantha Somayaji, Paramesvara and others. Jyesthadeva wrote a famous text Yuktibhasa which he wrote in Malayalam, the regional language of Kerala. The work contains proofs of the theorems and gives derivations of the rules it contains. It is one of the main astronomical and mathematical texts produced by the Kerala school. The work was based mainly on the Tantrasamgraha of Nilakantha.
The Yuktibhasa is a major treatise, half on astronomy and half on mathematics, written in 1501. The Yuktibhasa is very important in terms of the mathematics Jyesthadeva presents. In particular he presents results discovered by Madhava and the treatise is an important source of the remarkable mathematical theorems which Madhava discovered. Written in about 1550, Jyesthadeva's commentary contained proofs of the earlier results by Madhava and Nilakantha which these earlier authors did not give.
Yuktibhasa describing Madhava's series, but remember that even this passage by Jyesthadeva was written more than 100 years before James Gregory rediscovered this series expansion. Other mathematical results presented by Jyesthadeva include topics studied by earlier Indian mathematicians such as integer solutions of systems of first degree equation solved by the kuttaka method, and rules of finding the sines and the cosines of the sum and difference of two angles.
Not only does the mathematics anticipate work by European mathematicians a century later, but the planetary theory presented by Jyesthadeva is similar to that adopted by Tycho Brahe.
Courtesy:Internet
The Yuktibhasa is a major treatise, half on astronomy and half on mathematics, written in 1501. The Yuktibhasa is very important in terms of the mathematics Jyesthadeva presents. In particular he presents results discovered by Madhava and the treatise is an important source of the remarkable mathematical theorems which Madhava discovered. Written in about 1550, Jyesthadeva's commentary contained proofs of the earlier results by Madhava and Nilakantha which these earlier authors did not give.
Yuktibhasa describing Madhava's series, but remember that even this passage by Jyesthadeva was written more than 100 years before James Gregory rediscovered this series expansion. Other mathematical results presented by Jyesthadeva include topics studied by earlier Indian mathematicians such as integer solutions of systems of first degree equation solved by the kuttaka method, and rules of finding the sines and the cosines of the sum and difference of two angles.
Not only does the mathematics anticipate work by European mathematicians a century later, but the planetary theory presented by Jyesthadeva is similar to that adopted by Tycho Brahe.
Courtesy:Internet
Shankara Variyar, Vasudeva Sarvabhauma, Raghunatha Shiromani, Mathuranatha Tarkavagisha, Jagadisha Tarkalankara, Gadadhara Bhattacharya
Shankara Variyar (1530) Shankara Variyar, wrote the celebrated Yuktidipika, a commentary on the Tantra-sangraha of Nilakantha Somayaji.
Vasudeva Sarvabhauma (1450-1525), Logician, Navadvipa school
Raghunatha Shiromani (1475-1550), Logician, Navadvipa school
Mathuranatha Tarkavagisha (1575)Logician, Navadvipa school
Jagadisha Tarkalankara (1625) Logician, Navadvipa school
Gadadhara Bhattacharya (1650) Logician, Navadvipa school
Courtesy: Internet
Vasudeva Sarvabhauma (1450-1525), Logician, Navadvipa school
Raghunatha Shiromani (1475-1550), Logician, Navadvipa school
Mathuranatha Tarkavagisha (1575)Logician, Navadvipa school
Jagadisha Tarkalankara (1625) Logician, Navadvipa school
Gadadhara Bhattacharya (1650) Logician, Navadvipa school
Courtesy: Internet
Mahendra Suri
Mahendra Suri (1340-1410) is the 14th century Jain astronomer who wrote the Yantraraja, the first Indian treatise on the astrolabe.
Astrolobe [Universe within one's palm] is a higly sophisticated astromical intrument of the pre-modern times. It is a versatile observational and computational instrument. As an observational instrument, it was employed for measuring the altitudes of heavenly bodies and for measuring the heights and distances in land survery. As an computational device, it can simulate the motion of the heavens at any given locality and time. It was also an analog computer for solving numerous problems in sphercial trignometry.
He was a pupil of Madana Suri. Mahendra Suri acted as a mediator between the Islamic and sanskritic tradition of learning.
The Yantraraja or "the king of astronomical instruments" is divided into five chapters
Chapter 1: Ganitadhyaya provides trigonometical parameters needed for the construction of astrolabe.
Chapter 2: Yantraghatanadhaya enumerates the different parts of astrolabe
Chapter 3: Yantraracanadhyaya construciton of common northern astrolabe and other variants
Chapter 4: Yantrasodhanadhyaya the method of verifying whether the astrolabe is properly constructed or not
Chapter 5: Yantravicaranadhyaya the use of astrolabe as an observational and computational instrument and dwells on the various problems in astronomy and spherical trinometry that can be solved using astrolabe
Courtesy:Internet
Astrolobe [Universe within one's palm] is a higly sophisticated astromical intrument of the pre-modern times. It is a versatile observational and computational instrument. As an observational instrument, it was employed for measuring the altitudes of heavenly bodies and for measuring the heights and distances in land survery. As an computational device, it can simulate the motion of the heavens at any given locality and time. It was also an analog computer for solving numerous problems in sphercial trignometry.
He was a pupil of Madana Suri. Mahendra Suri acted as a mediator between the Islamic and sanskritic tradition of learning.
The Yantraraja or "the king of astronomical instruments" is divided into five chapters
Chapter 1: Ganitadhyaya provides trigonometical parameters needed for the construction of astrolabe.
Chapter 2: Yantraghatanadhaya enumerates the different parts of astrolabe
Chapter 3: Yantraracanadhyaya construciton of common northern astrolabe and other variants
Chapter 4: Yantrasodhanadhyaya the method of verifying whether the astrolabe is properly constructed or not
Chapter 5: Yantravicaranadhyaya the use of astrolabe as an observational and computational instrument and dwells on the various problems in astronomy and spherical trinometry that can be solved using astrolabe
Courtesy:Internet
Nilakantha Somayaji
Nilakantha Somayaji (1444-1544), from Kerala, was a major Indian mathematician and astronomer of the Kerala school of astronomy and mathematics and was a student of Damodara. Later, he lived in Tryambakeshwar. Among his many influential books, he wrote the comprehensive astronomical treatise Tantrasamgraha in 1501. He also wrote the Aryabhatiya Bhasya, which contains work on infinite series expansions, problems of algebra, spherical geometry, and many results of calculus. Grahapareeksakrama is a manual on making observations in astronomy based on instruments of the time.
The Tantrasamgraha 432 Sanskrit verses divided into 8 chapters, and it covers various aspects of Indian astronomy. It is based on the epicyclic and eccentric models of planetary motion.
Description of chapters below
Chapter 1: deal with the motions and longitudes of the planets.
Chapter 2: deal with the motions and longitudes of the planets.
Chapter 3: Treatise on shadow deal;
Various problems related with the sun's position on the celestial sphere;
Chapter 5: treat various aspects of the eclipses of the sun and the moon
Chapter 6: vyatipata and deals with the complete deviation of the longitudes of the sun and the moon
Chapter 7: visibility computation discusses the rising and setting of the moon and planets
Chapter 8: elevation of the lunar cusps examines the size of the part of the moon which is illuminated by the sun and gives a graphical representation of it.
Courtesy:Internet
The Tantrasamgraha 432 Sanskrit verses divided into 8 chapters, and it covers various aspects of Indian astronomy. It is based on the epicyclic and eccentric models of planetary motion.
Description of chapters below
Chapter 1: deal with the motions and longitudes of the planets.
Chapter 2: deal with the motions and longitudes of the planets.
Chapter 3: Treatise on shadow deal;
Various problems related with the sun's position on the celestial sphere;
The relationships of its ex-pressions in the three systems of coordinates namely ecliptic, equatorial and horizontal coordinates.
Chapter 4: on the lunar eclipse and On the solar eclipseChapter 5: treat various aspects of the eclipses of the sun and the moon
Chapter 6: vyatipata and deals with the complete deviation of the longitudes of the sun and the moon
Chapter 7: visibility computation discusses the rising and setting of the moon and planets
Chapter 8: elevation of the lunar cusps examines the size of the part of the moon which is illuminated by the sun and gives a graphical representation of it.
Courtesy:Internet
Parameshvara Vatasseri
Parameshvara Vatasseri (1360 - 1425) was a major Indian mathematician of Madhava of Sangamagrama 's Kerala school, as well as an astronomer and astrologer . He presented a series form of the sine function that is equivalent to its Taylor series expansion. The family home was Vatasseri (also called Vatasreni) in the village of Alattur, Kerala.
Parameshvara's teachers included Rudra, Madhava and Narayana Pundit.Parameshvara wrote many commentaries on many mathematical and astronomical works, such as those by Bhaskara I and Aryabhatta. He made eclipse observations over a 55 year period, and constantly attempted to compare these with the theoretically computed positions of the planets. He revised planetary parameters based on his observations.
Paramesvara most significant contributions is his mean value type formula for inverse interpolation of sine. He is the first mathematician to give the radius of circle with inscribed cyclic quadrilateral , an ex-pression that is normally attributed to Lhuilier (1782), 350 years later.
Courtesy:Internet
Parameshvara's teachers included Rudra, Madhava and Narayana Pundit.Parameshvara wrote many commentaries on many mathematical and astronomical works, such as those by Bhaskara I and Aryabhatta. He made eclipse observations over a 55 year period, and constantly attempted to compare these with the theoretically computed positions of the planets. He revised planetary parameters based on his observations.
Paramesvara most significant contributions is his mean value type formula for inverse interpolation of sine. He is the first mathematician to give the radius of circle with inscribed cyclic quadrilateral , an ex-pression that is normally attributed to Lhuilier (1782), 350 years later.
Courtesy:Internet
Narayana Pandit
Narayana Pandit (1340 – 1400) was a major mathematician of the Kerala school. He wrote the Ganita Kaumudi in 1356 about mathematical operations. The work anticipated many developments in combinatorics.He considered an algorithm for multiplying numbers and he then looked at the special case of squaring numbers. One of the unusual features of Narayana's work Karmapradipika is that he gave seven methods of squaring numbers which are not found in the work of other Indian mathematicians.
He discussed another standard topic for Indian mathematicians namely that of finding triangles whose sides had integral values. In particular he gave a rule of finding integral triangles whose sides differ by one unit of length and which contain a pair of right-angled triangles having integral sides with a common integral height. In terms of geometry Narayana gave a rule for a segment of a circle.
Courtesy : Internet
He discussed another standard topic for Indian mathematicians namely that of finding triangles whose sides had integral values. In particular he gave a rule of finding integral triangles whose sides differ by one unit of length and which contain a pair of right-angled triangles having integral sides with a common integral height. In terms of geometry Narayana gave a rule for a segment of a circle.
Courtesy : Internet
Lagadha
Lagadha (around 1300 BC) is the author of Vedanga Jyotisha, the text on Vedic astronomy that has been dated to 1350 BC. This text describes rules for tracking the motions of the sun and the moon. Lagadha praises astronomy as the crowning subject in the ancillary Vedic sciences.
Courtesy:Internet
Courtesy:Internet
Yajnavalkya
Yajnavalkya (perhaps 1800 BC) advanced a 95-year cycle to synchronize the motions of the sun and the moon. He is also credited with the authorship of the Shatapatha Brahmana, in which the references to the motions of the sun and the moon are found. He is also a major figure in the Upanishads. His deep philosophical teachings of the Brhadaranyaka Upanishad.
Courtesy : Internet
Courtesy : Internet
Madhava
Madhava (1350-1425) was born in town of Irinjalakkuda, near Cochin, Kerala was at the time known as Sangamagrama has major mathematician of the Kerala school who is considered the father of mathematical analysis for having taken the decisive step from the finite procedures of ancient mathematics to treat to their limit-passage to infinity, which is the kernel of modern classical analysis.He is the first to have developed infinite series approximations for a range of trigonometric functions, which has been called the "decisive step onward from the finite procedures of ancient mathematics to treat their limit-passage to infinity". His discoveries opened the doors to what has today come to be known as mathematical analysis.
One of the greatest mathematician-astronomers of the Middle Ages, Madhava contributed to infinite series, calculus, trigonometry, geometry and algebra.
The following presents a summary of results that have been attributed to Madhava by various scholars.
Infinite series
He discovered the infinite series for the trigonometric functions of sine, cosine, tangent and arctangent, and many methods for calculating the circumference of a circle.
Trigonometry
Madhava also gave a most accurate table of sines, defined in terms of the values of the half-sine chords for twenty-four arcs drawn at equal intervals in a quarter of a given circle.
Algebra
Madhava also carried out investigations into other series for arclengths and the associated approximations to rational fractions of ð, found methods of polynomial expansion, discovered tests of convergence of infinite series, and the analysis of infinite continued fractions.He also discovered the solutions of transcendental equations by iteration, and found the approximation of transcendental numbers by continued fractions.
Calculus, Geometry
Madhava has been called "the greatest mathematician-astronomer of medieval India".
Courtesy : Internet
Gangesha Upadhyaya
Gangesha Upadhyaya (13th century) Indian mathematician and philosopher from the kingdom of Mithila. He established the Navya-Nyâya ("New Logic") school. His Tattvacintamani ("The Jewel of Thought on the Nature of Things") is the basic text for all later developments. The logicians of this school were primarily interested in defining their terms and concepts.
Tattvacintamani which composed by Gangesopadhyaya of Mithila, deals with all the important aspects of Indian philosophy, logic, and epistemology and sets a standard for the scholarly discussions over philo-sophical topics in Modern India. The Tattvacintamani, which recognized only the first of the sixteen categories of the Nyaya-sutra, viz., means of valid knowledge (pramana), strictly adhered to the theory of four pramanas, and brought all the other categories under the means of knowledge.
The Sabdakhanda (Word Section), which is divided into sixteen chapters, provides a comprehensive view of word as a means of knowledge with all the relevant topics, viz., the authoritativeness of word, associate causes of word such as speaker's intention, eternality of word, injunctive and other statements, signification (expressive power) and parts of speech (i.e., verbal root, conjugational ending and prefix).
Courtesy : Internet
Tattvacintamani which composed by Gangesopadhyaya of Mithila, deals with all the important aspects of Indian philosophy, logic, and epistemology and sets a standard for the scholarly discussions over philo-sophical topics in Modern India. The Tattvacintamani, which recognized only the first of the sixteen categories of the Nyaya-sutra, viz., means of valid knowledge (pramana), strictly adhered to the theory of four pramanas, and brought all the other categories under the means of knowledge.
The Sabdakhanda (Word Section), which is divided into sixteen chapters, provides a comprehensive view of word as a means of knowledge with all the relevant topics, viz., the authoritativeness of word, associate causes of word such as speaker's intention, eternality of word, injunctive and other statements, signification (expressive power) and parts of speech (i.e., verbal root, conjugational ending and prefix).
Courtesy : Internet
Friday, December 14, 2007
Sridhara
Sridhara (870–930) was an Indian mathematician known for two treatises: Trisatika (sometimes called the Patiganitasara) and the Patiganita. At least three other works have been attributed to him, namely the Bijaganita, Navasati, and Brhatpati. Sridhara gives rules for operating with rational fractions. He gives a wide variety of applications including problems involving ratios, barter, simple interest, mixtures, purchase and sale, rates of travel, wages, and filling of cisterns.
Sridhara was one of the first mathematicians to give a rule to solve a quadratic equation.
"Multiply both sides of the equation by a known quantity equal to four times the coefficient of the square of the unknown; add to both sides a known quantity equal to the square of the coefficient of the unknown; then take the square root. "
Courtesy : Internet
Sridhara was one of the first mathematicians to give a rule to solve a quadratic equation.
"Multiply both sides of the equation by a known quantity equal to four times the coefficient of the square of the unknown; add to both sides a known quantity equal to the square of the coefficient of the unknown; then take the square root. "
Courtesy : Internet
Brahmagupta
Brahmagupta (589–668) was an Indian mathematician and astronomer.Brahmagupta was born in 598 CE in Bhinmal city in the state of Rajasthan of northwest India. He likely lived most of his life in Bhillamala (modern Bhinmal in Rajasthan) in the empire of Harsha during the reign (and possibly under the patronage) of King Vyaghramukha. As a result, Brahmagupta is often referred to as Bhillamalacarya, that is, the teacher from Bhillamala Bhinmal. He was the head of the astronomical observatory at Ujjain, and during his tenure there wrote four texts on mathematics and astronomy: the Cadamekela in 624, the Brahmasphutasiddhanta in 628, the Khandakhadyaka in 665, and the Durkeamynarda in 672.
Brahmagupta's most famous work is his Brahmasphutasiddhanta. It is composed in elliptic verse, as was common practice in Indian mathematics, and consequently has a poetic ring to it. As no proofs are given, it is not known how Brahmagupta's mathematics was derived.
Famous extracts from Brahmasphutasiddhanta
Algebra
The solution of the general linear equation
Arithmetic Series
The sum of the squares and cubes of the first n integers.He gives the sum of the squares of the first n natural numbers as n(n+1)(2n+1)/6 and the sum of the cubes of the first n natural numbers as (n(n+1)/2)².
Zero
Brahmagupta made use of an important concept in mathematics, the number zero. The Brahmasphutasiddhanta is the earliest known text to treat zero as a number in its own right. He first describes addition and subtraction, multiplication and division.
Diophantine analysis
Pythagorean triples
Pell's equation
Nx2 + 1 = y2 (called Pell's equation) by using the Euclidean algorithm. The Euclidean algorithm was known to him as the "pulverizer" since it breaks numbers down into ever smaller pieces.
Geometry
Brahmagupta's formula
Brahmagupta's most famous result in geometry is his formula for cyclic quadrilaterals.
Pi
He gives values of [pi]
Measurements and constructions
Trigonometry
Sines
Astronomy
Courtesy:Internet
Brahmagupta's most famous work is his Brahmasphutasiddhanta. It is composed in elliptic verse, as was common practice in Indian mathematics, and consequently has a poetic ring to it. As no proofs are given, it is not known how Brahmagupta's mathematics was derived.
Famous extracts from Brahmasphutasiddhanta
Algebra
The solution of the general linear equation
Arithmetic Series
The sum of the squares and cubes of the first n integers.He gives the sum of the squares of the first n natural numbers as n(n+1)(2n+1)/6 and the sum of the cubes of the first n natural numbers as (n(n+1)/2)².
Zero
Brahmagupta made use of an important concept in mathematics, the number zero. The Brahmasphutasiddhanta is the earliest known text to treat zero as a number in its own right. He first describes addition and subtraction, multiplication and division.
Diophantine analysis
Pythagorean triples
Pell's equation
Nx2 + 1 = y2 (called Pell's equation) by using the Euclidean algorithm. The Euclidean algorithm was known to him as the "pulverizer" since it breaks numbers down into ever smaller pieces.
Geometry
Brahmagupta's formula
Brahmagupta's most famous result in geometry is his formula for cyclic quadrilaterals.
Pi
He gives values of [pi]
Measurements and constructions
Trigonometry
Sines
Astronomy
Courtesy:Internet
Panini
Panini (520-460 BC) was born in Shalatula, a town beside the Indus River, in Gandhara, which is in the modern day the Attock District of Pakistan's Punjab province, located between Rawalpindi and Peshawar.
He is known for his Sanskrit grammar, particularly for his formulation of the 3,959 rules of Sanskrit morphology in the grammar known as Ashtadhyayi (meaning "eight chapters"), the foundational text of the grammatical branch of the Vedanga, the auxiliary scholarly disciplines of Vedic religion.
The Ashtadhyayi is the earliest known grammar of Sanskrit (though scholars agree it likely built on earlier works), and the earliest known work on descriptive linguistics, generative linguistics, and together with the work of his immediate predecessors (Nirukta, Nighantu, Pratishakyas) stands at the beginning of the history of linguistics itself.
Panini's comprehensive and scientific theory of grammar is conventionally taken to mark the end of the period of Vedic Sanskrit, by definition introducing Classical Sanskrit.
Courtesy: Internet
He is known for his Sanskrit grammar, particularly for his formulation of the 3,959 rules of Sanskrit morphology in the grammar known as Ashtadhyayi (meaning "eight chapters"), the foundational text of the grammatical branch of the Vedanga, the auxiliary scholarly disciplines of Vedic religion.
The Ashtadhyayi is the earliest known grammar of Sanskrit (though scholars agree it likely built on earlier works), and the earliest known work on descriptive linguistics, generative linguistics, and together with the work of his immediate predecessors (Nirukta, Nighantu, Pratishakyas) stands at the beginning of the history of linguistics itself.
Panini's comprehensive and scientific theory of grammar is conventionally taken to mark the end of the period of Vedic Sanskrit, by definition introducing Classical Sanskrit.
Courtesy: Internet
Bhaskara-II
Bhaskara-II (1114-1185) is a well-known mathematician of ancient India. He was born in 1114 AD in Vijayapura, India. Bhaskara II is also known as Bhaskaracharya, which means "Bhaskara the Teacher". His father Mahesvara was as an astrologer, who taught him mathematics, which he later passed on to his son Loksamudra.
Bhaskara II was the head of the astronomical observatory at Ujjain, the chief mathematical center of ancient India. It goes to the credit of Varahamihira and Brahmagupta, the leading mathematicians who worked there and built up this school of mathematical astronomy. He wrote six books and the seventh book, which is attributed to him, is considered to be a forgery. The subjects of his six works are arithmetic, algebra, trigonometry, calculus, geometry, and astronomy. The six works are: Lilavati on mathematics; Bijaganita on algebra; the Siddhantasiromani which is divided into two parts: mathematical astronomy and sphere; the Vasanabhasya of Mitaksara which is Bhaskaracharya's views on the Siddhantasiromani ; the Karanakutuhala or Brahmatulya in which he simplified the concepts of Siddhantasiromani ; and the Vivarana which comments on the Shishyadhividdhidatantra of Lalla. From the mathematical point of view the first three of these works are the most interesting.
Bhaskara II wrote Siddhanta Shiromani at the age of 36 in 1150 AD. This colossal work is divided into four parts Lilawati, Beejaganit, Ganitadhyaya and Goladhyaya and consists of about 1450 verses. Each part of the book consists of huge number of verses and can be considered as a separate book: Lilawati has 278, Beejaganit has 213, Ganitadhyaya has 451 and Goladhyaya has 501 verses. He formulated simple ways of calculations from Arithmetic to Astronomy in this book. He wrote Lilawati is an excellent lucid and poetic language. It has been translated in various languages throughout the world.
Few important contributions of BhaskarII to mathematics are as follows:
Terms for numbers
Bhaskaracharya gave the terms for numbers in multiples of ten which are as follows:
eka(1), dasha(10), shata(100), sahastra(1000), ayuta(10,000), laksha(100,000), prayuta (1,000,000=million), koti(107), arbuda(108), abja(109=billion), kharva (1010), nikharva (1011), mahapadma (1012=trillion), shanku(1013), jaladhi(1014), antya(1015=quadrillion), Madhya (1016) and parardha(1017). (In English, the multiples of 1000 are termed as thousand, million, billion, trillion, quadrillion etc)
Kuttak
Chakrawaal
Simple mathematical methods
He died in 1185 in Ujjain, India
Courtesy:Internet
Bhaskara II was the head of the astronomical observatory at Ujjain, the chief mathematical center of ancient India. It goes to the credit of Varahamihira and Brahmagupta, the leading mathematicians who worked there and built up this school of mathematical astronomy. He wrote six books and the seventh book, which is attributed to him, is considered to be a forgery. The subjects of his six works are arithmetic, algebra, trigonometry, calculus, geometry, and astronomy. The six works are: Lilavati on mathematics; Bijaganita on algebra; the Siddhantasiromani which is divided into two parts: mathematical astronomy and sphere; the Vasanabhasya of Mitaksara which is Bhaskaracharya's views on the Siddhantasiromani ; the Karanakutuhala or Brahmatulya in which he simplified the concepts of Siddhantasiromani ; and the Vivarana which comments on the Shishyadhividdhidatantra of Lalla. From the mathematical point of view the first three of these works are the most interesting.
Bhaskara II wrote Siddhanta Shiromani at the age of 36 in 1150 AD. This colossal work is divided into four parts Lilawati, Beejaganit, Ganitadhyaya and Goladhyaya and consists of about 1450 verses. Each part of the book consists of huge number of verses and can be considered as a separate book: Lilawati has 278, Beejaganit has 213, Ganitadhyaya has 451 and Goladhyaya has 501 verses. He formulated simple ways of calculations from Arithmetic to Astronomy in this book. He wrote Lilawati is an excellent lucid and poetic language. It has been translated in various languages throughout the world.
Few important contributions of BhaskarII to mathematics are as follows:
Terms for numbers
Bhaskaracharya gave the terms for numbers in multiples of ten which are as follows:
eka(1), dasha(10), shata(100), sahastra(1000), ayuta(10,000), laksha(100,000), prayuta (1,000,000=million), koti(107), arbuda(108), abja(109=billion), kharva (1010), nikharva (1011), mahapadma (1012=trillion), shanku(1013), jaladhi(1014), antya(1015=quadrillion), Madhya (1016) and parardha(1017). (In English, the multiples of 1000 are termed as thousand, million, billion, trillion, quadrillion etc)
Kuttak
Chakrawaal
Simple mathematical methods
He died in 1185 in Ujjain, India
Courtesy:Internet
Bhaskara-I
Bhaskara-I (600-680) was an Indian mathematician of the 7th century, who probably lived between c.600- c.680. He was most likely the first to use a circle for the zero in the Hindu-Arabic decimal system, and while commenting on Aryabhata's work, he evaluated an extraordinary rational approximation of the sine function. There is very little information about Bhaskara's life. He is said to be born near Saurashtra in Gujarat and died in Ashmaka. He was educated by his father in astronomy. He is considered to be a follower of Aryabhata I and one of the most renowned scholars of Aryabhata's astronomical school. Bhaskara I wrote two treatises, the Mahabhaskariya and the Laghubhaskariya. He also wrote commentaries on the work of Aryabhata I entitled Aryabhatiyabhasya. The Mahabhaskariya comprises of eight chapters dealing with mathematical astronomy. The book deals with topics such as: the longitudes of the planets; association of the planets with each other and also with the bright stars; the lunar crescent; solar and lunar eclipses; and rising and setting of the planets. Bhaskara I suggested a formula which was astonishingly accurate value of Sine. The formula is: sin x = 16x (p - x)/[5p2 - 4x (p - x)]
Bhaskara I wrote the Aryabhatiyabhasya in 629,, which is a commentary on the Aryabhatiya written by Aryabhata I. Bhaskara I commented only on the 33 verses of Aryabhatiya which is about mathematical astronomy and discusses the problems of the first degree of indeterminate equations and trigonometric formula. While discussing about Aryabhatiya he discussed about cyclic quadrilateral. He was the first mathematician to discuss about quadrilaterals whose four sides are not equal with none of the opposite sides parallel.
For many centuries, the approximate value of p was considered v10. But Bhaskara I did not accept this value and believed that p had an irrational value which later proved to be true. Some of the contributions of Bhaskara I to mathematics are: numbers and symbolism, the categorization of mathematics, the names and solution of the first degree equations, quadratic equations, cubic equations and equations which have more than one unknown value, symbolic algebra, the algorithm method to solve linear indeterminate equations which was later suggested by Euclid, and formulated certain tables for solving equations that occurred in astronomy.
Courtesy:Internet
Bhaskara I wrote the Aryabhatiyabhasya in 629,, which is a commentary on the Aryabhatiya written by Aryabhata I. Bhaskara I commented only on the 33 verses of Aryabhatiya which is about mathematical astronomy and discusses the problems of the first degree of indeterminate equations and trigonometric formula. While discussing about Aryabhatiya he discussed about cyclic quadrilateral. He was the first mathematician to discuss about quadrilaterals whose four sides are not equal with none of the opposite sides parallel.
For many centuries, the approximate value of p was considered v10. But Bhaskara I did not accept this value and believed that p had an irrational value which later proved to be true. Some of the contributions of Bhaskara I to mathematics are: numbers and symbolism, the categorization of mathematics, the names and solution of the first degree equations, quadratic equations, cubic equations and equations which have more than one unknown value, symbolic algebra, the algorithm method to solve linear indeterminate equations which was later suggested by Euclid, and formulated certain tables for solving equations that occurred in astronomy.
Courtesy:Internet
Aryabhatta-II
Aryabhata-II (920-1000) was an mathematician and astronomer of India, and wrote the well-known book the Mahasiddhanta. He is believed to be born in India during c.920 and died at c.1000. These dates are approximately suggested by the modern historians, however there are historians like G.R.Kaye believed that Aryabhata II lived before al-Biruni, whereas Datta in 1926 proved that these dates were too early. Though Pingree considers that Aryabhatta's main publications was published between 950 and 1100, but R.Billiard has proposed a date in the sixteenth century.
Aryabhatta II's most eminent work was Mahasiddhanta. The treatise consists of eighteen chapters and was written in the form of verse in Sanskrit. The initial twelve chapters deals with topics related to mathematical astronomy and covers the topics that Indian mathematicians of that period had already worked on. The various topics that have been included in these twelve chapters are: the longitudes of the planets, lunar and solar eclipses, the estimation of eclipses, the lunar crescent, the rising and setting of the planets, association of the planets with each other and with the stars.
The next six chapters of the book includes topics such as geometry, geography and algebra, which were applied to calculate the longitudes of the planets. In about twenty verses in the treatise, he gives elaborate rules to solve the indeterminate equation: by = ax + c. These rules have been applied to a number of different cases such as when c has a positive value, when c has a negative value, when the number of the quotients is a even number, when this number of quotients is an odd number, etc.
Aryabhata II also deduced a method to calculate the cube root of a number, but his method was already given by Aryabhata I, many years earlier. Indian mathematicians were very keen to give the correct sine tables since they played a vital role to calculate the planetary positions as accurately as possible. Aryabhatta II played a vital role in it by constructing a sine table, which was accurate up to five decimal places.
Courtesy : Internet
Aryabhatta II's most eminent work was Mahasiddhanta. The treatise consists of eighteen chapters and was written in the form of verse in Sanskrit. The initial twelve chapters deals with topics related to mathematical astronomy and covers the topics that Indian mathematicians of that period had already worked on. The various topics that have been included in these twelve chapters are: the longitudes of the planets, lunar and solar eclipses, the estimation of eclipses, the lunar crescent, the rising and setting of the planets, association of the planets with each other and with the stars.
The next six chapters of the book includes topics such as geometry, geography and algebra, which were applied to calculate the longitudes of the planets. In about twenty verses in the treatise, he gives elaborate rules to solve the indeterminate equation: by = ax + c. These rules have been applied to a number of different cases such as when c has a positive value, when c has a negative value, when the number of the quotients is a even number, when this number of quotients is an odd number, etc.
Aryabhata II also deduced a method to calculate the cube root of a number, but his method was already given by Aryabhata I, many years earlier. Indian mathematicians were very keen to give the correct sine tables since they played a vital role to calculate the planetary positions as accurately as possible. Aryabhatta II played a vital role in it by constructing a sine table, which was accurate up to five decimal places.
Courtesy : Internet
Aryabhatta-I
Âryabhatta (476–550 AD) is the first in the line of great mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy. His most famous works are the Aryabhatiya (499) and Arya-Siddhanta. He was born in 476 AD in Kerala. He studied at the University of Nalanda. One of his major work was Aryabhatiya written in 499 AD. The book dealt with many topics like astronomy, spherical trigonometry, arithmetic, algebra and plane trigonometry. He jotted his inventions in mathematics and astronomy in verse form. The book was translated into Latin in the 13th century. Through the translated Latin version of the Aryabhattiya, the European mathematicians learned how to calculate the areas of triangles, volumes of spheres as well as how to find out the square and cube root.
In the field of astronomy, Aryabhatta was the pioneer to infer that the Earth is spherical and it rotates on its own axis which results in day and night. He even concluded that the moon is dark and shines because of the light of sun. He gave a logical explanation to the theory of solar and lunar eclipses. He declared that eclipses are caused due to the shadows casted by the Earth and the moon. Aryabhatta proposed the geocentric model of the solar system which states that the Earth is in the center of the universe and also laid the foundation for the concept of Gravitation. His propounded methods of astronomical calculations in his Aryabhatta-Siddhatha which was used to make the the Panchanga (Hindu calendar). What Copernicus and Galileo propounded was suggested by Aryabhatta nearly 1500 years ago.
Aryabhatta's contribution in mathematics is unparalleled. He suggested formula to calculate the areas of a triangle and a circle, which were correct. The Gupta ruler, Buddhagupta, appointed him the Head of the University for his exceptional work. Aryabhatta gave the irrational value of pi. He deduced ? = 62832/20000 = 3.1416 claiming, that it was an approximation. He was the first mathematician to give the 'table of the sines', which is in the form of a single rhyming stanza, where each syllable stands for increments at intervals of 225 minutes of arc or 3 degrees 45'. Alphabetic code has been used by him to define a set of increments. If we use Aryabhatta's table and calculate the value of sin(30) (corresponding to hasjha) which is 1719/3438 = 0.5; the value is correct. His alphabetic code is commonly known as the Aryabhata cipher.
Mathematics
Place Value system and zero
Pi as Irrational
Mensuration and trigonometry
Indeterminate Equations
Astronomy
Motions of the Solar System
Eclipses
Sidereal periods
Heliocentrism
Courtesy : Internet
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