Mathematics in Modern India

8/10/2019 Mathematics in Modern India

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MATHEMATICS IN INDIA

IN THE MODERN PERIOD (1750 ONWARDS)

M.D.SRINIVAS

CENTRE FOR POLICY STUDIES

[email protected]


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OUTLINE

Introduction: Continuing tradition of Indian Astronomy (1700-1850)

Surveys of Indigenous Education in India (1825-1835)

Survival of Indigenous education system till 1880

Modern Scholarship on Indian Mathematics and Astronomy (1700-1900)

Rediscovering the Tradition (1850-1950)

Development of Modern Education and Modern Mathematics in India

(1850-1950)

Srinivasa Ramanujan (1887-1920)

Modern Scholarship on Indian Mathematics (1900-2010)

Development of Modern Mathematics in India (1950-2010)

Development of Education in India (1950-2010)

Halting Growth of Higher Education and Scientific Research in India (1980-

2010)


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BACKGROUND: CONTINUED DEVELOPMENT OF

MATHEMATICS IN MEDIEVAL INDIA

Gaitasrakaumud(in Prkta)ofhakkura Pher(c.1300) and otherworks inregional languages such as Vyavahragaita (Kannaa) of

Rjditya, Pvulrigaitamuof Pvulri Mallaa in Telugu.

GaitakaumudandBjagaitvatasaof Nryaa Paita (c. 1350)

Mdhava (c.1350): Founder of the Kerala School. Infinite series for ,

Sine and cosine functions and fast convergent approximations to them

Works of Paramevara (c.1380-1460)

Works of Nlakaha Somayj(c.1444-1540): Revised planetary

model

Systematic exposition of Mathematics and Astronomy with proofs in

Yukti-bh(in Malayalam) of Jyehadeva (c.1530) and commentariesKriykrmakarand Yuktidpikof akara Vriyar (c.1540).

Works of Jnarja (c.1500), Gaea Daivaja (b.1507), Sryadsa

(c.1541) and Ka Daivaja (c.1600): Commentaries with upapatties

Works of Munvara (b.1603) and Kamalkara (b.1616)

Mathematics and Astronomy in the Court of Sawai Jayasiha (1700-1743). Translation from Persian of Euclid and Ptolemy.

Works of later Kerala astronomers Acyuta Pirai (c.1550-1621),

Putumana Somayaj(c.1700)


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A EUROPEAN ACCOUNT OF INDIAN ASTRONOMY (c.1769)

'While waiting in Pondicherry for the Transit of 1769, Le Gentil tried to gather

information about native astronomy...Le Gentil eventually contacted a Tamilwho was versed in the astronomical methods of his people. With the help of an

interpreter he succeeded in having computed for him the circumstances of the

lunar eclipse of 1765 August 30, which he himself had observed and checked

against the best tables of his times, the tables of Tobias Mayer [1752]. The

Tamil Method gave the duration of the Eclipse 41 second too short, the tables

of Mayer 1 minute 8 seconds too long; for the totality the Tamil was 7 minutes48 seconds too short, Mayer 25 seconds too long. These results of the Tamil

astronomer were even more amazing as they were obtained by computing with

shells on the basis of memorised tables and without any aid of theory. Le

Gentil says about these computations: They did their astronomical

calculations with swiftness and remarkable ease without pen and pencil; theironly accessories were cauries... This method of calculation appears to me to be

more advantageous in that it is faster and more expeditious than ours. '

[Neugebauer,A History of Ancient Mathematical Astronomy , Vol. III, Springer , 1975, p.20]

*Translated from French.

Note: What Neugebauer is referring to as Tamil method is nothing but the Vkya method

developed in south India, especially by Kerala Astronomers. Neugebauer also refers to the

report of John Warrren about the calculation of a lunar eclipse in 1825 June 1, where the Tamil

method predicted midpoint of the eclipse equally accurately with an error of about 23 minutes.


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CONTINUING TRADITION OF INDIAN ASTRONOMY (c.1850)

akaravarman (1784-1839):Raja of Kaattan in Malabar. Due to the wars

with Hyder and Tipu, he is supposed to have spent his early years withMahrja Svti Tirunl at Tiruvanantapuram. In 1819, He wrote Sadratnaml

(one of the four works mentioned by Whish in 1835), an Astronomical manual

following largely the Parahita system. He also wrote his own Malayalam

commentary, perhaps a few years later (published along with the text in

Kozhikode in 1899). Chapter I has interesting algorithms for calculation ofsquare and cue roots. Chapter IV deals with computation of sines.

Candraekhara Smanta (1835-1904): Popularly known as Paihni

Smanta, he had traditional Sanskrit education. Starting from around 1858, he

carried out extensive observations for over eleven years, with his own versatileinstruments, with a view to to improve the almanac of Puri Temple. He wrote

his Siddhntadarpaa with nearly 2500 verses in 1869 (published at Calcutta

1899). Based on his observations, Smanta improved the parameters of the

traditional works, he detected and included all the three major irregularities of

lunar motion, and improved the traditional estimates of the Sun-Earth distance.In Chapter V of his work, Smanta has presented his planetary model where all

the planets move around th Sun, which moves around the Earth.


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MAHATMA GANDHI ON INDIGENOUS EDUCATION

IN THE 19THCENTURY

We have the education of this future State. I say without fear of my figures

being challenged successfully, that today India is more illiterate than it

was fifty or a hundred years ago, and so is Burma, because the British

administrators, when they came to India, instead of taking hold of things

as they were, began to root them out. They scratched the soil and began tolook at the root, and left the root like that, and the beautiful tree perished.

The village schools were not good enough for the British administrator, so

he came out with his programme. I defy anybody to fulfil a programme

of compulsory primary education of these masses inside of a century. This

very poor country of mine is ill able to sustain such an expensive method

of education. Our State would revive the old village schoolmaster and dot

every village with a school both for boys and girls.

Mahatma Gandhi, Speech at Chatham House London, October 30, 1931


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REPORTS ON INDIGENOUS EDUCATION IN 19THCENTURY

If a good system of agriculture, unrivalled manufacturing skill, a capacity

to produce whatever can contribute to convenience or luxury; schoolsestablished in every village, for teaching reading, writing, and

arithmetic; the general practice of hospitality and charity among each

other; and above all a treatment of the female sex, full of confidence,

respect and delicacy, are among the signs which denote a civilised people,

then the Hindus are not inferior to the nations of Europe; and if civilisation

is to become an article of trade between the two countries, I am convinced

that this country [England] will gain by the import cargo.

[Thomas Munros Testimony before a Committee of House of Commons April 12, 1813]

We refer with particular satisfaction upon this occasion to that

distinguished feature of internal polity which prevails in some parts of

India, and by which the instruction of the people is provided for by a

certain charge upon the produce of the soil, and other endowments in

favour of the village teachers, who are thereby rendered public servants ofthe community.

[Public Despatch from London to Bengal, June 3, 1814]


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REPORTS ON INDIGENOUS EDUCATION IN 19THCENTURY

There are probably as great a proportion of persons in India who can read,

write and keep simple accounts as are to be found in European countries[Annual Report of Bombay Education Society 1819]

I need hardly mention what every member of the Board knows as well as I

do, that there is hardly a village, great or small, throughout our territories,

in which there is not at least one school, and in larger villages more; manyin every town, and in large cities in every division; where young natives

are taught reading, writing and arithmetic, upon a system so economical,

from a handful or two of grain, to perhaps a rupee per month to the school

master, according to the ability of the parents, and at the same time so

simple and effectual, that there is hardly a cultivator or petty dealer who is

not competent to keep his own accounts with a degree of accuracy, in my

opinion, beyond what we meet with amongst the lower orders in our own

country; whilst the more splendid dealers and bankers keep their books

with a degree of ease, conciseness, and clearness I rather think fully equalto those of any British merchants.

Minute of G. Prendargast, Member Bombay Governors Council, April 1821


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INDIGINOUS EDUCATION IN MADRAS PRESIDENCY (c.1825)

The British Government conducted a detailed survey of the indigenous system

of education covering all the Districts of the Madras Presidency during 1822-

25. The Survey found 11,575 schools and 1094 colleges in the Presidency.

Summarising the survey information the then Governor Thomas Munro wrote

in his Minute of March 10, 1826:

It is remarked by the Board of Revenue, that of a population of 12

millions, there are only 188,000, or 1 in 67 receiving education. This is

true of the whole population, but not as regards the male part of it, of

which the proportion educated is much greater than is here estimated if

we reckon the male population between the ages of five and ten years,

which is the period which boys in general remain at school, at one-ninth

the number actually attending the schools [and colleges] is only

184,110, or little more than one-fourth of that number. I have taken theinterval between five and ten years of age as the term of education,

because, though many boys continue at school till twelve or fourteen,

many leave it under ten. I am, however, inclined to estimate the portion

of the male population who receive school education to be nearer to

one-thirdthan one-fourth of the whole, because we have no returns from

the provinces of the numbers taught at home....The state of education here

exhibited, low as it is compared with that of our own country, is higher

than it was in most European countries at no very distant period.


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Community Profile of Boys Attending School

District Brahmin Kshatriya Vaisya SudraOtherCastes Muslims Total

TotalPopulation

Telugu Districts 13,893 121 7,676 10,076 4,755 1,639 38,160 4,029,408

% Total 36.41 0.32 20.12 26.40 12.46 4.30Boys of Sch-going ge 9,111 2,507 7,387 134,896 59,479 10,387 223,856

% of Community 152.49 4.83 103.91 7.47 7.99 15.78 17.05

Malabar 2,230 84 3,697 2,756 3,196 11,963 907,575

% of Total 18.64 0.70 30.90 23.04 26.72Boys of Sch-going age 953 15 620 25,447 9,893 13,286 50,421

% of Community 234.01 0.00 13.54 14.53 27.86 24.06 23.73

Tamil Districts 11,557 369 4,442 57,873 13,196 5,453 92,890 6,622,474

% of Total 12.44 0.40 4.78 62.30 14.21 5.87Boys of Sch-going age 10,191 1,619 7,910 255,260 77,373 14,901 367,915

% of Community 113.40 22.79 56.16 22.67 17.06 36.60 25.25

TOTAL 29,721 490 13,449 75,943 22,925 10,644 153,172 12,850,941% of Total 19.40 0.32 8.78 49.58 14.97 6.95

Boys of Sch-going age 23,203 4,212 16,778 457,279 169,275 42,051 713,941

% of the Community 128.09 11.63 80.16 16.61 13.54 25.31 21.45

Source data from Dharampal, The Beautiful Tree, Impex India, Delhi 1983, pp.21-22.

Boys of school going age in each community, estimated by using the community profile of the

total population as per 1871 Census, and by estimating the boys of school-going age

(5-10 years) as one-ninth of total population, following Munro.


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The languages of instruction in most of the 11,575 schools were the

regional languages. The average period of instruction was around 5-7

years. The subjects taught were reading writing and arithmetic. The

Collector of Rajahmandry gave a list of books which includes the titles

Gaitam, and Pvalri Gaitam.

The instruction in most of the 1,094 "colleges" or institutions of higher

learning was in Sanskrit. Details of the subjects taught are available for the618 colleges four districts - 418 taught Vedam, 198 Law, 34 Astronomy

and Gaita and 8 taught ndhra stram.

Further, in Malabar, 1594 scholars were receiving higher instruction

privately, of whom 808 studied Astronomy (of whom 96 were dvijas) and

154 Medicine (of whom 31 were dvijas).

As regards the financial support received by the indigenous schools and

colleges the situation was clearly stated by the Collector of Bellary:

Of the 533 institutions for education, now existing in this district, I

am ashamed to say not one now derives any support from the state...There is no doubt that in former times, especially under the Hindoo

Governments very large grants, both in money, and in land, were

issued for the support of learning.


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INDIGINOUS EDUCATION IN BENGAL PRESIDENCY (c.1835)

William Adams survey (1835) of indigenous education in selected

districts of Bengal and Bihar showed the following interesting subject-wise distribution of institutions of higher learning.


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INDIGINOUS EDUCATION IN BENGAL PRESIDENCY (c.1835)

Adams survey also showed that textbooks used in these institutions of

higher learning included, apart from the ancient canonical texts of thevarious disciplines, many of the important advanced treatises

commentaries and monographs composed in during the late medieval

period.

These included the works of Bhaoji Dkita (1625), Kauabhaa

(c.1650), Hari Dkita and Nagea Bhaa (c.1700) in Vyakaraa, the

works of Raghuntha (c.1500), Mathurantha (c.1570), Viwantha

(c.1650), Jagada (c.1650) and Gaddhara (c.1650) in Navya-nyya, the

works of Raghunandana (c.1550) in Dharmastra and the works

Vedntasra(c.1450) and Vedntaparibh(c.1650) in Vednta.

The period of study in these institutions of higher learning was between

ten and twenty-five years. In many of these centres of higher learning a

large part of the students came from outside, many from even different

regions of India. All the students were taught gratis and outside students

were provided in addition free food and lodging.


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THE UNIVERSITY OF NAVADVPA

On visiting Navadvpa or Nuddeah in 1787, William Jones wrote to Earl of

Spencer that 'This is the third University of which I am a member'.

An account of Navadvpa published in Calcutta Monthly Register in January

1791 noted:

'The grandeur of the foundation of the Nuddeah University is generally

acknowledged. It consists of three colleges Nuddeah, Santipore and

Gopulparra. Each is endowed with lands for maintaining masters in everyscience.in the college of Nuddeah alone, there are at present about

eleven hundred students and one hundred and fifty masters. Their

numbers, it is true, fall very short of those in former days. In Rajah

Roodres time (circa 1680) there were at Nuddaeah no less than four

thousand students and masters in proportion. The students that comefrom distant parts are generally of a maturity in years, and a proficiency in

learning to qualify them for beginning the study of philosophy

immediately on their admission, but yet they say, to become a real Pundit,

a man ought to spend twenty years at Nuddeah.'

According to Adam, in 1829 there were reported to be 25 schools of learning

in Navadvpa with 500 to 600 students. Some of these schools were supported

by a small allowance from British Government.


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ORIENTALIST-ANGLICIST DEBATE

While the vast indigenous system of school education received no aid or

support from the British Government, there were a few centres of higherlearning which received some grants, though it was on a much reduced

scale from what prevailed prior to British rule. For instance, the famous

Dakshina Fund of the Peshwas which amounted to several lakhs of Rupees

distributed each year in the period prior to 1818 was reduced to 35,000

Rupees annually, by 1824.

In the history of modern Indian education, the so called Orientalist-

Anglicist debate of the 1830s has often been misrepresented by portraying

the Orientalists as great admirers of indigenous learning. In fact, the

Orientalists held the same view as the Anglicists that the indigenouslearning was erroneous and outmoded, but they vehemently argued that

the study of English and true science is best engrafted upon the course

of education best esteemed by the Indian people.


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ORIENTALIST-ANGLICIST DEBATE

The Orientalist position is clearly set forth by James Prinsep in his note of

February 18, 1835 written in response to Macaulays Minute:


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MACAULAYs PRONOUNCEMENT

But it was Macaulays imperious dismissal all indigenous learning, clearly

formulated in his Minute of February 2, 1835, which carried the day in the

formulation of the British policy on Indian education:


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INDIGENOUS EDUCATION IN MADRAS PRESIDENCY (1855-1880)

At the time when the Department of Education waas established in 1855,

there were only 83 schools under it, while nearly 12,500 indigenous

schools were still functioning with a total of 1.6 lakh students.

The situation was similar even till 1870-1, except that about 3,000 schools

had been brought under the scheme of Aided Schools.

It was only duing the decade 1870-1880, that the Education Department

seems to have managed to bring nearly a lakh indigenous schools underthe aided scheme. In tishs way around 1875, for the first time, the number

of students studying under the aegis of the Department of Education

become comparable to the number who studied in the indigfenous schools

fifty years earlier in 1825.

Year Departmental

Extra

Departmental Aided

Unaided

Indigenous Total

Schools Students Schools Students Schools Students Schools Students Schools Students

18556 83 2093 1112 32843 12,498 161,687 13,693 196,623

18701 98 5463 3352 84,239 12,624 149,003 16,074 238,705

18812 1,263 46,975 13,223 313,668 2,828 54,064 17,314 414,707

Source: Report of the Education Commission 1882.


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EMERGENCE OF MODERN EUROPEAN SCHOLARSHIP ON

INDIAN ASTRONOMY AND MATHEMATICS (c.1700-1800)

In 1687, Simon de La Loubere (1642-1729) the French Ambassador to Siam,

brought to Paris a manuscript describing Indian methods of computation of

Solar and Lunar longitude. In his book, Du Royaume de Siam (1691), he

discussed these methods, along with comments by Cassini. La Loubre also

gave an account of the Indian methods of construction of magic squares.

The methods of calculation in the Siamese tables were commented upon in a

book by Bayer (1738) along with a note by Leonard Euler on Indian Solar

year.

Le Gentil (1725-1792) who visited India during 1761 and 1769, to observe ther

transit of Venus, gave a detailed account of Indian Astronomy in 1770s based

on Tables aand Texts obtained in Pondicherry. This led to the treatise Traite de

lAstronomie Indienne et Orientalle (1787)by Jean Sylvain Bailly (1736-

1793). This was reviewed in detail by John Playfair (1748-1819) in the

Transactions of Royal Society in 1790.

The Asiatic Society was founded in Calcutta under the leadership of William

Jones (1746-1794) in 1784 and started publishing the Journal Asiatic

Researches from 1788. Articles by William Jones, Samuel Davis and John

Bentley dealt with the Indian Zodiac and methods of computations in texts

such as Sryasiddhnta, Grahalghava andMakaranda-srii.


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TRANSLATIONS AND EDITIONS OF INDIAN TEXTS ON

ASTRONOMY AND MATHEMATICS IN 19THCENTURY

The Bjagaita of Bh

skara, was translated in to English from the PersianTranslation of Ata Allah Rushdi (1634) by Edward Strachey (1812-1901) with

Notes by Samuel Davis (London, 1813). This was closely followed by the

translation ofLlvatby John Taylor (Bombay, 1816).

Henry Thomas Colbrooke (1756-1837) published several articles on Indian

astronomy and many other disciplines such as Law, Linguistics Philosophy

etc. His most important work is Algebra withArithmetic and Mensuration

from the Sanskrit of Brahmagupta and Bhascara (London, 1817), which

included a tranlsation of Gaitdhyyaand KuakdhyyaofBrhmasphua-

siddhnta as well as the Llvat and Bjagaita of Bhskara II, along with

notes some times drawn from the ancient commentaries.

John Warren (1769-1830) wrote on Indian calendrical computations based on

both the siddhnta and vkya methods in Klasakalita(1825). The book also

includes notes of exchanges between Warren, B.Heyne and C.M.Whish on the

various infinite series with which the contemporary south Indian astronomers

of seemed to be acquainted with.

Charles Matthew Whish (1794-1833) collected several important manuscriptsof the Kerala School and made extensive notes on them. His seminal article on

the Kerala School was published in 1835.


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TRANSLATIONS AND EDITIONS OF INDIAN TEXTS

Lancelot Wilkinson, political agent at Bhopal, edited the Siddhntairomai

of Bhskarcrya with Vsan(Calcutta 1842) and Grahalghavaof Gaea

with commentary of Mallri (Calcutta 1843). His translation of Goldhyyaof

Siddhntairomaiwas edited by Bapudeva Sastri (Calcutta 1861).

Fitz-Edward Hall (1825-1901), in collaboration with Bapudeva Sastri, edited

Sryasiddhnta with the commentary of Ragantha (Calcutta 1854). Rev.

Ebenzer Burgess published an English translation of Sryasiddhnta with

detailed notes with the help of William Dwight Whitney (New Haven 1860).

Albrecht Weber edited the Vedga Jyotiawith the commentary Somkara

(Berlin 1862)

Johann Hendrick Caspar Kern (1833-1917) edited the Bhatsahit of

Varhamihira (Calcutta, 1865) and partially translated it (JRAS, 1873). He also

edited the ryabhaya with the commentary of Paramevara (Leiden 1875)

George Frederick William Thibaut (1848-1914) edited the Baudhyana-

ulvastra with the commentary of Dvrakntha (1874). He also edited

Vedga-Jyotia(1877) and Ktyyana-ulvastra(in part) with commentary

(1882). In collaboration with Sudhakara Dvivedi, he edited and translated the

Pacasiddhntikof Varhamihira (1884). Thibauts essay, The ulva Stras,

was reprinted as a book (Calcutta, 1875). He also wrote an overview

Astronomie Astrologie und Mathematik(Strassburg, 1899).


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REDISCOVERING THE TRADITION (1850-1950)

Some editions and translations into Bengali, Telugu, Marathi etc, of Indian

source-works such as Llvat, Bjagaita, Grahalghava, were published in

the first half of 19th century. Around the same time, several Indian scholars,who were often from traditional learned families, but had also studied in the

English education system, embarked on a process of rediscovery of Indian

tradition. We mention some of the seminal figures in this movement.

Bapudeva Sastri (1821-1900) studied with Pandit Dhundiraja Misra and later

Pandit Sevarama and Wilkinson at Sehore Sanskrit College. He became aProfessor at Benares Sanskrit College where he is said to have taught

Euclidean Geometry. He published editions of Siddhntairomaiwith Vsan

of Bhskarcrya (1866) and Llvat with his own commentary (1883). He

collaborated with Lancelot Wilkinson and edited his translation of Goldhyya

of Siddh

ntairomai(1861).

Bhau Daji Laud (1821-1874), trained in medicine at Grants College, Mumbai,

was also a Sanskrit and an expert in numismatics. He was the first to locate a

manuscript of ryabhayain 1864.

Shankar Balakrishna Dikshit (1853-1898),a mathematics teacher and Principal

of Teachers Training College, Pune, wrote a voluminous history of IndianAstronomy, Bhratya Jyotia strch Prchna i Arvchn Itihs in

Marathi (Pune, 1896). Along with Robert Sewell, he also wrote on theIndian

Calendar(London, 1896)


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Sudhakara Dvivedi (1855-1910) studied with Pandits Devakrishna and

Bapudeva Sastri at Benares Sanskrit College and later Professor there. He

edited a very large number of ancient texts which included Llvat (1878),

Karaakuthala of Bhskarcrya II (1881), Yantrarja with Malayendus

commentary (1882), Siddhnta-tattvaviveka with eavsan of Kamalkara

(1885), iyadhvddhid of Lalla (1886), Bjagaita with his own

commentary (1888), Bhatsahit with Utpalas commentary (1895-7),

Triatik of rdhara (1899), Karaapraka of Brahmadeva (1899),Brhmasphuasiddhnta with his own Sanskrit commentary (1902),

Grahalghavawith commentaries of Mallri and Vivantha (1904), Yjua-

Jyautiamwith Somkara commentary (1908),Mahsiddhntaof ryabhaa II

with his own commentary (1910), ryabhaya with his own commentary

(1911), Sryasiddhnta with his own commentary (1911). With Thibaut, he

edited Pacasiddhntikwith his own Sanskrit commentary (1889).

Dvivedi wrote many original works such as Drghavtta-lakaam (1878),

Vstavacandra-gonnati-sdhana (1880), Bhbhramarekh-nirpaam

(1882), Calanakalana on differential calculus in Hindi (1886) and

Gaakataragi (1890) on the lives of Indian mathematicians andastronomers. In 1910 he wrote A History of Mathematics, Part I (Arithmetic)

in Hindi. His son, Padmakara Dvivedi, edited Gaitakaumud of Nryana

Paita in two volumes (1936, 1942).


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Several important texts of Indian mathematics and astronomy were

published in the period 1900-1950.

Harilal Dhruva published the Rekhgaita, translation of Euclid from

Tusis Persian Version (Bombay Sanskrit Series 1901).

Vidhyesvari Prasad Dvivedi published the available ancient siddhntas in

Jyotiasiddhnta-sagraha(Benares 1912).

Babuaji Misra edited the Khaakhdyaka of Brahmagupta with

marjas commentary (Calcutta 1925) andSiddhntaekhara of rpati

with Makkibhaas commentary (Calcutta 1932, 47).

Gopinatha Kaviraja edited the Siddhntasrvabhauma of Munvara, 2

Vols. (Benares 1932, 3); 3rdVol. Ed. Mithalal Ojha (Benres 1978)

Kapadia edited the Gaitatilakaof rdhara with commentary (Gaekwad

Oriental Series 1935)


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Several important works were published from the Anandasrama Pune:

Karaakaustubha ofKa Daivaj

a (1927),L

lvat

with commentaries

of Gaea Daivaja and Mahdhara (1937), Bjagaita with Ka

Daivajas commentary (1930), Siddhntairomai Gaitdhyaya with

commentary of Gaea (1939, 41), Kukrairomaiof Devarja (1944),

Mahbhskarya with Paramevara's commentary (1945),

Laghubhskarya with Paramevara's commentary (1946), Laghumnasa

with Paramevara's commentary (1952). Siddhntairomai Goldhyaya

with Munivaracommentary (1943, 52)

Several important works of Kerala School were published: Goladpikof

Paramevara, ed. by T. Ganapati Sastri (1916), ryabhayabhya of

Nlakaha in 3 Volumes, ed. by K. Sambasiva Sastry and S.Kunjan Pillai

(1930. 31, 1957), Karaapaddhati of Putumana Somayaji, ed. By

K.Sambasiva Sastri (1937), Tantrasagraha of Nlakaha with

Laghuvivrti of ankara ed. by S.Kunjan Pillai (1957).


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Malur Rangacharya (1861-1916), a science graduate from Madras Christian

College, later became Professor of Sanskrit at Presidency College. He edited

Gaitasrasagraha of Mahvra along with English translation and notes(Madras 1912). This was the first detailed exposition of an Indian

mathematical work after Colebrookes translations of Brahmagupta and

Bhskara in 1817.

Prabodh Chandra Sengupta (1876-1962), a Professor of Mathematics at

Bethune College, Calcutta, published several technical articles on Indianmathematics and astronomy highlighting the distinct nature of Indian methods

from the Greek. He published a translation of ryabhaya (1927) and a

translation of Khaakhdyaka with detailed notes and examples (Calcutta,

1934). He later edited Khaakhdyaka with Pthdakas Commentary

(Calcutta, 1941). He also wrote on Ancient Indian Chronology(1947)

Attippattu A.Krishnaswamy Ayyangar (1892-1953) wase ducated at

Pachaiyyappa college, Madras, and later worked as a Professor of Mathematics

at Maharajas College, Mysore. He published several articles bringing out

many technical aspects of Indian Mathematics. In a series of articles (1929-

1942), he showed that the cakravla process always leads to a solution of the

Vargaprakti equation and that it corresponds to a semi-regular continued

fraction expansion which is a contraction of the simple continued fraction

associated with the Euler-Lagrange method of solution.


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Bibhutibhusan Datta (1888-1958) passed MSc in Mixed Mathematics in

1914, and was awarded a research scholarship at Calcutta University.

However, right form an young age Datta, had an inclination for Sannyasa,

the life of renunciation. He was initiated by Swami Vishnu Tirtha

Maharaja in 1920. After receiving the DSC degree of Calcutta University

for his thesis on hydrodynamics in 1921, Datta became interested in

History of Mathematics under the influence of Prof. Ganesh Prasad.

During the period 1926-35 he published over fifty papers on variousaspects of Indian Mathematics. He also collected and studied a large

number of manuscripts. All of this led to the preparation of the

monumentalHistory of Hindu Mathematics.

Datta resigned from Professorship of Calcutta University in 1929 'in

aspiration of the life of a Vedantist residing in the Brahman, the Infinite

Self' as he mentioned in a letter to Prof. Karpinsky in 1934. In 1931, he

briefly returned to the University and, in deference to the wishes of Prof.

Ganesh Prasad, delivered a series of lectures which were published as The

Science of ulba (1932). In 1933, he retired from the University and in

1938 took up Sanyasa and became Swami Vidyaranya. He spent most of

his later life at Pushkara. In this period he wrote several scholarly volumes

in Bengali on Indian Philosophy.


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It is said that, prior to leaving the University in 1929, Datta handed over

the manuscript of the bookHistory of Hindu Mathematics: A Source Book,

to his junior colleague Avadhesh Narayan Singh (1901-1954). The lattersaw through the publication of two volumes (Arithmetic and Algebra) in

1935 and 1938. Singh went on to initiate study of Indian mathematics at

Lucknow University. He died in 1954 without completing the publication

of the third volume - which was to contain the history of 'geometry,

trigonometry, calculus and various other topics such as magic squares,theory of series and permutations and combinations' as noted in the

preface of the first volume.

The third volume was not included even when the book was reprinted in

1961. It seems that, prior to his death in 1958, Swami Vidyaranya gave the

manuscript of the third volume to Kripa Shankar Shukla. A copy was alsoobtained by R.C.Gupta in 1979 from S.N.Singh, the son of A.N.Singh.

Shukla published a revised version of the third volume as a series of seven

articles in the Indian Journal of History of Science during 1980-1994.

Ramavarma Maru Thampuran published, in collaboration with

Akhileswarayyar, the Mathematics section of Yuktibh (in Malayalam)of Jyeha Deva with detailed mathematical notes in Malayalam (Trichur

1948). This formed the basis of all later work on Yuktibh.


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DEVELOPMENT OF HIGHER EDUCATION IN INDIA (1850-1900)

The Universities of Calcutta, Bombay and Madras were set up in 1857. It

has often been remarked that these (and the later Universities in India)

were established as examining bodies with affiliated colleges on the modelof the then London University and not on the model of the renowned

Oxford and Cambridge Universities with extensive research and teaching

activities.

The Indian Association of Cultivation of Science was established by

Mahendra Lal Sircar (1833-1904) in 1876 with a view the object of

enabling Indians to cultivate science in all its departments with a view to

its advancement by original research. However, during the first thirty

years, the main efforts of the institution were directed towards the

development of science teaching at the collegiate level.

In 1855, there were 15 Arts Colleges with 3246 students, and 13

Professional Colleges with 912 students. By 1901, there were 5

Universities, 145 Arts Colleges with 17,651 students and 46 Professional

Colleges with 5,358 students in the whole of India (including Burma).

The medium of instruction in all the High Schools and Colleges wasEnglish. English was also the most important subject of study in them.


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DEVELOPMENT OF EDUCATION IN INDIA (1850-1900)

DEVELOPMENT OF EDUCATION IN INDIA (1900 1950)


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DEVELOPMENT OF EDUCATION IN INDIA (1900 1950)


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DEVELOPMENT OF MODERN MATHEMATICS IN INDIA (1850 1950)


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DEVELOPMENT OF MODERN MATHEMATICS IN INDIA (1850-1950)

Yesudas Ramchundra (1821-1880), a teacher of science in Delhi College,

wrote a Treatise on Problems of Maxima and Minima (1850), which

approached these problems purely algebraically. Augustus de Morgan got itrepublished, with his own introduction, from London in 1859.

The Indian Mathematical Society began as the Analytical Club in 1907 at

the initiative of V. Ramaswamy Aiyar, a civil servant (then a Deputy Collector

at Gutti). It was renamed Indian Mathematical Society in 1910. It started a

journal in 1909 which was edited by M.T.Narayaniyengar and S.NarayanaAiyar. The Society also started the journalMathematics Studentin 1932.

The Calcutta Mathematical Society was formed in 1908 at the initiative of

Prof. Asutosh Mukherjee (1864-1924) the then Vice Chancellor of Calcutta

University. It also began publishing theBulletinof the Society in 1909.

Syamdas Mukhopadhyaya (1866-1937) was A student of Presidency College,he proved the famous four-vertex theorem in global differential geometry in

the Bulletin of the Calcutta Mathematical Society in 1909. He continued to be

a major contributor to the Journal for several decades.

Ganesh Prasad (1876-1935) studied in Cambridge and Gottingen and taught at

Allahabad, Benares (1905-1923), and Calcutta Universities. He worked on

potential theory and summability. His two volume work Some Great

mathematicians of the Nineteenth Century(1933) was a landmark publication

in the history of mathematics.



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K.Ananda Rau (1893-1966) was a student of Hardy at the time Ramanujan

was in Cambridge and served as a Professor at Presidency College,

Madras, from 1919. He was an outstanding analyst and teacher of manyprominent Indian mathematicians.

Prasanta Chandra Mahalonobis (1893-1972), was a contemporary of

Ramanujan at Cambridge. He founded the Indian Statistical Institute in

1931, the first institute devoted to Mathematics research in modern India.

The Journal Sankhya was started in 1933. Mahalanobis was elected anFRS in 1945.

Ramaswamy S. Vaidyanathaswamy (1894-1960) was a student of

E.T.Whittaker and H.F.Baker. He served as a Professor at the Presidency

College, Madras, from 1927. He worked on Lattice theory and Topology.

He served as the editor of the Journal of Indian Mathematical Society

during 1927-1950. He wrote a pioneering Treatise on Point Set Topology

(1947).

Raj Chandra Bose (1901-1987) was also associated with the Indian

Statistical Institute. He did important work on design theory and errorcorrecting codes. He migrated to the United States in 1947.



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Subbayya Sivasankaranarayana Pillay (1901-1950) was a student of

Ananda Rau and worked in the Annamalai University during 1929-41. He

is well known for his work on the Waring problem in Number Theory. Healso worked on Diophantine approximation.

Tirukkannapuram Vijayaraghavan (1902-1955)was a student of Hardy at

Oxford during 1925-28 when he did notable work on summability. He

later joined Andre Weil at Aligharh University during 1930-32 and moved

on to Dhakka. He did important work on nonlinear differential equationsand Diophantine approximation. He was the first Director (1950-55) of the

Ramanujan Institute of Mathematics set up by the munificence of

Alagappa Chettiar in Madras.

Subbaramaiah Minakshisundaram (1913-1968), a student of Ananda Rau

at Madras, he later specialised in partial differential equations. He worked

at Andhra University at Vishakhapatnam. His work with Pleijel on the

eigen-functions of Laplace operator in Riemannian Manifolds is highly

acclaimed.

Hansraj Gupta (1902-1988) and Sarvadaman Chowla (1907-1995) wereimportant number theorists from Punjab.

SRINIVASA RAMANUJAN (1887 1920)


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SRINIVASA RAMANUJAN (1887-1920)

Srinivasa Ramanuja Iyengar, the greatest mathematician India has produced in

recent times, was born on December 22, 1887 at Erode. In 1892, he enrolled in

a primary school in Kumbakonam. In 1898, having passed his primary

examinations with distinction, he joined the Town High School of

Kumbakonam. He passed out of the school as an outstanding student in 1904

and received a scholarship to study at the Government College Kumbakonam.

While at school, he got a copy of Loneys Plane Trigonometry which he soonmastered and also derived several advanced results by himself. Around 1903,

Ramanujan went through Carrs Synopsis of Pure and Applied Mathematics (a

compendium of about 5000 results) which, by its concise style, is said to have

influenced his writing.

Ramanujan seems to have started discovering new results and recording them

his Notebook by 1904. However, at the college, owing to his weakness in

English as Hardy notes, Ramanujan failed and lost his scholarship. He joined

the Pachaiyappas College at Madras where also he faced the same fate.

Ramanujan got married in 1909 and with great effort managed to get a job inMadras Port Trust as a clerk in 1912, by the good will of various personalities

who were impressed by his mathematical results.

EARLY PUBLICATIONS OF RAMANUJAN


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During 1911-1913, Ramanujan published 5 papers in the Journal of Indian

Mathematical Society. On the first paper, On certain properties of Bernoulli

Numbers, it is said that the principal results are well-known and proofs are

incomplete. The paper Modular equations and approximations to ,

published later from London (QJM 1914, 350-372), is also said to embody

Ramanujans early Indian work. Here is a sample of his results:

Ramanujan also notes that the last series "is extremely rapidly convergent"Indeed in late 1980s, it blazed a new trail in the saga of computation of



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During 1911-13, Ramanujan also posed about 30 problems in the Journal

of Indian Mathematical Society for nearly twenty of which he also

provided the solution (as it was not solved by others in six months). Here

is a sample question published in 1911.

APPROACHING BRITISH MATHEMATICIANS (1912-13)


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APPROACHING BRITISH MATHEMATICIANS (1912 13)

In 1912, Ramanujan sent a sample of his results to Prof. M.J.M. Hill of

University College London through Prof. C.L.T. Griffith of the Madras

College of Engineering. Prof. Hill wrote back that Ramanujan had fallen into

the pit falls of ...divergent series and advised that he consult Bromwichs

book on infinite series. The issue had to do with Ramanujans Theory of

Divergent Series elucidated in Chapter 6 of his Second Notebook.

Ramanujan is said to have also contacted Profs H.F.Baker and E.W.Hobson at

Cambridge and received no response. On January 16, 1913 Ramanujan wrote

to Prof Godfrey Harold Hardy (1877-1947) at Cambridge, enclosing an eleven

page list of over one hundred formulas and theorems.

The impact of this letter can be gauged from the fact that on February 2, 1913,

Bertrand Russell wrote to Lady Morell that he found Hardy and Littlewood ina state of wild excitement, because they have discovered a second Newton, a

Hindu clerk on 20 a year. There is also a note of Littlewood to Hardy in

March 1913 with the comment I can believe that he is at least a Jacobi

On February 8, Hardy wrote expressing his interest in the work of Ramanujan,

while adding, But I want particularly to see your proofs of your assertions

here. You will understand that in this theory everything depends on rigorous

exactitude of proof. The same point is repeated twice again in the same letter.


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RAMANUJANS WORK IN ENGLAND


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Ramanujan arrived in London on April 14, 1914 and left for India on February

27, 1919. Of the nearly five years he spent there, he was very ill for more than

two years. From around the spring of 1917, he was in hospitals most of thetime. On his work during 1914, Ramanujan wrote to his friend B.Krishna Rao

on November 14, 1914:

I am attending only some of the University lectures...I am very slowly

publishing my results owing to the present war....I changed my plan ofpublishing my results. I am not going to publish any of the old results in my

notebook till the war is over. After coming here I have learned some of their

methods. I am trying to get new results by their methods by their methods so

that I can easily publish these results without delay. In a week or so I am going

to send a along paper to the London Mathematical Society. The results in thispaper [on highly composite numbers] have nothing to do with those of my old

results. I have published only three short papers....

Ramanujan reiterated this in a letter to S.M.Subramanian in January 1915: I

am doing my work very slowly. My note-book is sleeping in a corner for these

four or five months. I am publishing only my present researches as I have not

yet proved the results in my notebooks rigorously. I am at present working in

arithmetical functions...

RAMANUJANS WORK IN ENGLAND


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During 1914-1919, Ramanujan wrote about 30 papers, 7 of them in

collaboration with Hardy which were mostly concerning properties of

various arithmetical functions.

His work was highly acclaimed. On March 16, 1916 he was awarded

Bachelor of Science degree by Research from the Cambridge University

He was elected a Fellow of Royal Society on February 28, 1918, the

second Indian to be so honoured. On October 23, 1918 he was elected a

Fellow of the Trinity College (It appears that the College failed to elect

Ramanujan as a fellow in 1917 for various non-academic reasons).

In late 1918, the Madras University also offered a matching grant of 250

a year. On receipt of this communication Ramanujan wrote to the

Registrar of the University on January 11, 1919 that after meeting his

basic expenses the surplus should be used for some educational purpose,

such in particular as the reduction of school-fees for poor boys and

orphans and provision of books in schools.

RAMANUJANS LOST NOTEBOOK


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On March 27, 1919 Ramanujan returned to India. He was in very poor health.

He stayed for while in Madras and then moved to Kodumudi, then to his home

town of Kumbakonam, and finally returned to Madras by January 1920.

Though seriously ill, he was continuing his work all the while. On January 12,

1920, Ramanujan wrote to Hardy (for the first time after returning to India): I

discovered very interesting functions recently which I call Mock -functions.

Unlike the False -functions (studied partially by Prof. Rogers in hisinteresting paper) they enter into mathematics as beautifully as the ordinary -

functions. I am sending you with this letter some examples.... This was

followed by a few pages containing definitions and some properties of the

mock -functions.

The so called Lost Notebook of Ramanujan is a sheaf of over hundred sheets

containing about 600 results that Ramanujan had found during the last year of

his life. This seems to have been sent to Hardy along with all other papers of

Ramanujan in 1923. It was finally discovered by George Andrews in Trinity

College Library in 1976.

Ramanujan passed away in Madras on April 26, 1920.

HARDYS ASSESSMENT OF RAMANUJAN


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Soon after Ramanujans death, Hardy wrote an obituary (Proc. Lond.

Math. Soc. 19, 1921, pp. 40-58), which was later reproduced in the

Collected Papers of Ramanujan (Cambridge 1927). There, Hardy wasfrank enough to present his assessment of Ramanujan in detail. Hardy first

presents his assessment of Ramanujan he arrived in England:



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Hardy then notes that, after their interaction, In a few years time, he

[Ramanujan] had a very tolerable knowledge of the theory of functions,

and the analytic theory of numbers. He was never a mathematician of the

modern school...but he knew when he had proved a theorem and when he

had not. And his flow of original ideas shewed no symptom of abetment.

Hardy also states that Ramanujan adhered with a severity most unusual in

an Indian resident in England to the religious observance of his caste; but

his religion was a matter of observance and not of intellectual conviction,

and I remember well his telling me (much to my surprise) that all religions

seem to him more or less to be equally true.

Hardy then raises the issue: I have often been asked whether Ramanujan

had any special secret; whether his method differed in any kind from those

of other mathematicians; whether there was anything really abnormal in

his mode of thought. I cannot answer these questions with any confidence

or conviction; but I do not believe it. My belief is that all mathematiciansthink, at bottom in the same kind of way, and that Ramanujan was no

exception....



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Hardy then goes onto declare: It was his insight in to algebraic formulae,

transformation of infinite series, and so forth, that was most amazing. On

this side most certainly I have never met his equal, and I can compare him

only with Euler or Jacobi. He worked, far more than the majority of

modern mathematicians, by induction from numerical examples; all of his

congruence properties of partitions, for example, were discovered this

way. But with his memory, his patience, and his power of calculation, hecombined a power of generalisation, a feeling for form, and a capacity for

rapid modification of his hypothesis that were often really startling...

Hardy concludes by the observation: Opinions may differ as to the

importance of Ramanujans work, the kind of standard by which it should

be judged, and the influence which it is likely to have on the mathematics

of the future. It has not the simplicity and the inevitability of the very

greatest work; it would be greater if it were less strange. One gift it has

which no one can deny, profound and invincible originality. He would

probably have been a greater mathematician if he had been caught andtamed a little in his youth; he would have discovered more that was new,

and that, no doubt of greater importance.

HOW IS RAMNUJANS WORK ASSESSED TODAY


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Till the latter half of 20thcentury, the corpus of work of Ramanujan that was

generally available comprised of the 37 papers that he had published in various

Journals during 1911-1920. These, together with, the 57 questions and

solutions published by him in the Journal of Indian Mathematical Society, and

extracts from his two letters to Hardy which contained statements of around

120 results, were edited by G.H.Hardy, P.V.Seshu Aiyar and B.M.Wilson and

published by the Cambridge University in 1927 as the Collected Papers of

Srinivasa Ramanujan.

This of course excluded most of Ramanujans work done both before he left

for England and after his return to India.

The corpus of work done before leaving to England is available in the form of

three notebooks which are said to contain around 3250 results. The corpus of

work done after the return from England is contained in the Lost Note Book

which is said to have about 600 results.

Detailed analysis of this large corpus began only in the last quarter of the 20 th

century. Though much of the work is still in progress, it has already so to say

revolutionised our understanding and appreciation of Ramanujans work.

THE SAGA OF RAMANUJANS NOTEBOOKS


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It seems Ramanujan started recording his results in a Notebook around the

time he entered the Government College Kumbakonam in 1904 (though there

are some entries in the first book such as those on magic squares which arebelieved to have been made during his school days). Sometime during 1911-

13, Ramanujan copied these results in to a second notebook. As Ramanujan

noted in his letters to Krishna Rao and Subramaniam, he perhaps did not add

any further results to these notebooks, nor did he try to publish the results

contained in there, during his stay in England.

The following is a brief description of the notebooks due to Bruce Berndt:

Ramanujan left three notebooks. The first notebook totalling 351 pages

contains 16 chapters of loosely organised material with the remainder

unorganised. ...The second notebook is a revised enlargement of thefirstThis notebook contains 21 chapters comprising 256 pages followed by

100 pages of miscellaneous material. The third short notebook contains 33

pages of unorganised entries. ...in preparingRamanujans Notebooks Parts I

V, we counted 3254 results, although we emphasize that different people will

tally such a count in different ways. Each Chapter contains approximately

50-150 entries.

[B.Berndt, An Overview of Ramanujans Notebooks 1998]



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When Ramanujan left England he left his first notebook with Hardy. The

second and third notebooks were given to Madras University after his

death. Later, Hardy sent the first notebook back to Madras with Prof. S.R.

Ranganathan.

In January 1921, Prof. K Ananda Rau of Presidency College wrote to

Hardy: Mr. R. Ramachandra Rao told me that you had written to him

some months ago that Ramanujan was working on a certain topic in hislast days and possibly there may be some record of this work left. If you

will please tell us the nature of this investigation, we may find it easier to

sift the papers. The whole of the manuscripts will of course be sent to you

in accordance with the resolution of the syndicate. You will have noticed

also in the Minutes that the syndicate has asked Mr. Seshu Iyer and me to

arrange for the preparation of a transcript of Ramanujans note book, with

a view to having it incorporated as an Appendix to the Memorial volume. I

do not know if this will serve any useful purpose. I fear it may look a little

incongruous by the side of his mature work. But there are some here, whothink that the Note Book may contain valuable algorithms providing

starting points for future investigations.



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In 1923, Hardy published the paper A Chapter from Ramanujans Notebook.

As Berndt notes, This chapter pertains almost entirely to hyper-geometric

series, and Hardy pointed out that Ramanujan discovered most of theimportant classical results in the theory as well as many new theorems. In

the introduction to his paper, Hardy remarks that a systematic verification

of the results (in the notebooks) would be a very heavy undertaking. In

his unpublished notes, B. M. Wilson reports a conversation with Hardy in

which Hardy told him that the writing of this paper was a very difficult

task to which he devoted three to four full months of hard work.[ B.Berndt, Ramanujans Notebooks Vol I (1985), p.5]

On August 30, 1923, transcripts of Ramanujans notebooks and a packet

of miscellaneous papers were despatched by the Registrar of University ofMadras, to Hardy, with a note that You may decide whether any or all of

them should find a place in the proposed Memorial Volume.

But the final volume of Collected Papersedited by Hardy et al in 1927,

did not include any of this material. In a letter to B.M.Wilson in June1925, Hardy had expressed his opinion that the notebooks may be

published later.



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In a recent article on the occasion of Ramanujans 125th birthday, Bruce

Berndt has recounted the saga of publication of Ramanujans notebooks:

As it transpired, ...published with the Collected Papers [of Ramanujan],

were the first two letters that Ramanujan had written to Hardy, which

contained approximately one hundred twenty mathematical claims. Upon

their publication, these letters generated considerable interest, with the

further publication of several papers establishing proofs of these claims.

Consequently, either in 1928 or 1929, at the strong suggestion of Hardy,

Watson and B. M. Wilson,... agreed to edit the notebooks

In an address to the London Mathematical Society on February 5, 1931,

Watson cautioned (in retrospect, far too optimistically), We anticipate

that it, together with the kindred task of investigating the work of other

writers to ascertain which of his results had been discovered previously,

may take us five years. Wilson died prematurely in 1935, and although

Watson wrote approximately thirty papers on Ramanujans work, hisinterest evidently flagged in the late 1930s, and so the editing was not

completed.


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RAMANUJANS NOTEBOOKS: THE VERDICT OF HISTORY

I h i l B B d h l d h f ll i ll


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In the same article, Bruce Berndt has also presented the following overall

assessment of Ramanujans notebooks:

Altogether, the notebooks contain over three thousand claims, almost all

without proof. Hardy surmised that over two-thirds of these results were

rediscoveries. This estimate is much too high; on the contrary, at least

two-thirds of Ramanujans claims were new at the time that he wrote

them, and two-thirds more likely should be replaced by a larger fraction.

Almost all the results are correct; perhaps no more than five to ten are

incorrect.

The topics examined by Ramanujan in his notebooks fall primarily under the

purview of analysis, number theory and elliptic functions, with much of his

work in analysis being associated with number theory and with some of hisdiscoveries also having connections with enumerative combinatorics and

modular forms. Chapter 16 in the second notebook represents a turning point,

since in this chapter he begins to examine the q-series for the first time in these

notebooks and also to begin an enormous devotion to theta functions.

Especially in the years 1912-14 and 1917-1920, Ramanujans concentration on

number theory was through q-series and elliptic functions.

[B.Berndt, Notices of AMS (2012), p.1533]

ONGOING WORK ON RAMANUJANS LOST NOTEBOOK

The manuscript of Ramanujan discovered in the Trinity College Library


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The manuscript of Ramanujan discovered in the Trinity College Library

(amongst Watson papers) by G.E.Andrews in 1976, is generally referred as

Ramanujans Lost Notebook. This seems to pertain to work done byRamanujan during 1919-20 in India. This manuscript of about 100 pages with

138 sides of writing has around 600 results. This notebook along with some

other unpblished manuscripts of Ramanujan were published during Ramanujan

Centenary in 1987. Profs G.E.Andrews and B.Berndt have embarked on a four

volume edition of all this material in four volumes; of which the first two have

appeared in 2005 and 2009. They note in the preface of the first volume that:

...only a fraction (perhaps 5%) of the notebook is devoted to the mock theta

functions themselves. What is in Ramanujans lost notebook besides mock

theta functions? A majority of the results fall under the purview of q-series.

These include mock theta functions, theta functions, partial theta function

expansions, false theta functions, identities connected with the RogersFineidentity, several results in the theory of partitions, Eisenstein series, modular

equations, the RogersRamanujan continued fraction, other q-continued

fractions, asymptotic expansions of q-series and q-continued fractions,

integrals of theta functions, integrals of q-products, and incomplete elliptic

integrals. Other continued fractions, other integrals, infinite series identities,

Dirichlet series, approximations, arithmetic functions, numerical calculations,

Diophantine equations, and elementary mathematics are some of the further

topics examined by Ramanujan in his lost notebook.

THE ENIGMA OF RAMANUJANS MATHEMATICS

For the past hundred years the problem in comprehending and assessing


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For the past hundred years, the problem in comprehending and assessing

Ramanujans mathematics and his genius has centred around the issue of

proof. In 1913, Hardy wrote to Ramanujan asking for proofs of his results.Ramanujan responded by asserting that he had a systematic method for

deriving all his results, but that could not be communicated in letters.

Ramanujans published work in India, and a few of the results contained in the

note books have proofs, but they were often found to be sketchy, not rigorous,

incomplete and sometimes even faulty. Ramanujan had never any doubts about

the validity of his results, but still he was often willing to wait and supply

proofs in the necessary format so that his results could be published. But, all

the time, he was furiously discovering more and more interesting results.

The Greco-western tradition of mathematics does almost equate mathematicswith proof, so that the process of discovery of mathematical results can only be

characterised vaguely as intuition, natural genius etc. Since mathematical

truths are believed to be non-empirical, there are no systematic ways of

arriving at them except by pure logical reason. There are some philosophers

who have argued that this philosophy of mathematics is indeed barren: itseems to have little validity when viewed in terms of mathematical

practice either in history or in our times.

RAMANUJAN: NOT A NEWTON BUT A MDHAVA

In the Indian mathematical tradition as is known from the texts of the last two


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In the Indian mathematical tradition, as is known from the texts of the last two

to three millennia, mathematics was not equated with proof. Mathematical

results were not perceived as being non-empirical and they could be validatedin diverse ways. Proof or logical argumentation to demonstrate the results was

important. But proofs were mainly for the purpose of obtaining assent for

ones results in the community of mathematicians.

In 1913, Bertrand Russell had jocularly remarked about Hardy and Littlewood

having discovered a second Newton in a Hindu clerk. If parallels are to be

drawn, Ramanujan may indeed be compared to the legendary Mdhava.

It is not merely in terms of his philosophy of mathematics that Ramanujan is

clearly in continuity with the Indian tradition of mathematics. Even in his

extraordinary felicity in handling iterations, infinites series, continued fractions

and transformations of them, Ramanujan is indeed a successor, a very worthy

one at that, of Mdhava, the founder of the Kerala School and a pioneer in the

development of calculus. It is perhaps an irony of history that Mdhavas

results are also known in the form of versified statements of results and the

proofs are only found in the Malayalam treatise Yuktibhcomposed a couple

of generations later.

MODERN SCHOLARSHIP ON INDIAN MATHEMATICS 1900-2000

George Rusby Kaye (1866-1929) was the Principal of Government Training


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George Rusby Kaye (1866-1929) was the Principal of Government Training

College, Allahabad and later a member of Bureau of Education at Simla. He

wrote monographs on Indian mathematics, tracing most of it to foreigninfluences. Some of his books are: Indian Mathematics (1915), The

Astronomical Observatories of Jaising (1918), Hindu Astronomy (1924) and

The Bakshali Manuscript:A Study in Medieval mathematics (1927).

A.Burk edited the pastamba ulvastrawith a German translation (1901-2).

Walter Eugene Clark, a Professor of Sanskrit at Harvard, translated the

ryabhaya(Chicago 1930). J.M.van Gelder edited and translated theMnava

ulvastra(1961-3).

Otto Edward Neugebauer (1899-1990) was born in Austria, studied

mathematics in Gottingen, and later shifted to the study of Egyptian,

Babylonian and Greek exact sciences. In 1939 he moved to the US and

founded the history of Mathematics Department at Brown University in 1947.During 1952-67 he published several papers on Indian astronomy highlighting

issues of transmission. He has translated the Astronomical Tables of al

Khwarizmi(1962) and Pacasiddhntikwith Pingree (1970).

Bertel Leendert Van der Waerden (1903-1996), a Dutch mathematician, and

author of one of the most influential volumes on Modern Algebra, he laterworked extensively on history of ancient science. Since 1950 he published

several articles on the history of Indian mathematics and astronomy.


"Tamil Astronomy": In 1952 Neugebauer wrote a paper on 'Tamil


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Tamil Astronomy : In 1952, Neugebauer wrote a paper on Tamil

Astronomy' where he suggested that the vkya methods of computation

outlined in Warrens Klasakalita (1825) are indicative of an, earlier 'Tamiltradition' in astronomy. Van der Waerden followed suit and wrote a paper on

'Tamil Astronomy' in 1956. Neugebauer's work was cited by S. Chandrasekhar

in his Nehru Memorial Lecture on 'Astronomy in Science and Human Culture'

(1968). Now, it is well established that the vkya method goes back to

Vararuci and major improvements were made in it by the Kerala School. As,

Pingree was to later write on the vkya methods: 'misnamed "Tamil", they

generated some interest among non-Indian astronomers in 1950s and 60s'.

David Edwin Pingree (1933-2005): After completing his PhD in Harvard in

1960 on "Transmission of Greek Astrology to India" under the guidance of

D.H.H.Ingaals, Pingree worked in University of Chicago. He joined the Brown

University Department of History of Mathematics in 1971. He became one of

the foremost experts on Exact Sciences in ancient and medieval world, with a

focus on issues of transmission between cultures. He published around 20

books and over 60 articles on all aspects of Jyotistra, which include editions

of Pacasiddhntik (with Neugebauer) Vddha-Yavanajtaka and

Yavanajtaka, and a history of Jyotistra, apart from the volumes of theseminal Census of Exact Sciences. Many of his students have become leading

scholars of Indian mathematics and astronomy.


Census of Exact Sciences in Sanskrit: During the period 1970 95 David


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Census of Exact Sciences in Sanskrit: During the period 1970-95, David

Pingree brought out five volumes of this landmark publication, which lists the

works of each author (names listed alphabetically) together with detailedreferences to the available manuscripts as listed in various catalogues. Detailed

notices are also given regarding publications of these works (if any) and

secondary studies on them. It is an invaluable resource for all researchers. The

five published volumes run into over 1600 pages and cover all the authors till

the Sanskrit letter 'Va'. Unfortunately Pingree passed away in 2005 before the

last volume in this series could be brought out.

French Scholars of Indian Mathematics: Leon Rodet, studied the Algebra of

Al Khwarizimi in relation to Greek and Indian methods (1878). He also

translated the Gaitapda of ryabhaya (1879). Louis Renou (1896-1966)

and Jean Filliozat (1906-1982) were major scholars of Sanskrit, Indian

philosophy and sciences. Roger Billard (1922-2000) wrote LAstronomie

Indiene(1971) where he tried to use computational and statistical methods to

date the Indian texts. Recently, Karine Chemla (b.1957) has worked on the

inter relation between Chinese and Indian mathematical methods. Agathe

Keller has published a translation of theGaitapdaof ryabhayabhyaof

Bhskara I with detailed notes. Francois Patte has worked on BhskarasLlvatandBjagaitaand its commentaries.


Japanese Scholars of Indian Mathematics: Kiyosi Yabuuti (1906-2000), the


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Japanese Scholars of Indian Mathematics: Kiyosi Yabuuti (1906 2000), the

doyen of Japanese scholars on history of science, had written on the relation

between the Chinese text Chiu-Chih Liand Indian astronomy in 1963. TakaoHayashi (b.1949), Takanori Kusuba and Micho Yano have written several

papers on Indian mathematics and astronomy. They have written a book

Studies in Indian Mathematics, Series Pi and Trigonometryin Japanese (1997)

and, along with S.R.Sarma, they have translated Gaitasrakaumud of

hakkura Pheru (2011). Hayashi has edited and translated the Bakli

Manscript (2005) which was his thesis work with Pingree. He has also editedand translated theBjagaitaof Bhskara with Vsanand Kuttkrasiromani

(2012). Kusuba and Setsuro Ikeyama were also Pingrees students and have

worked on Gaitakaumud and Brhmasphuasiddhnta. Yukio Ohashi has

worked on Indian astronomical instruments under the guidance of K.S.Shukla.

Recent Publicatins: There have been several important works, which have

been published recently, which include significant material on Indian

mathematics and astronomy. These include: Helaine Selin (ed), Encyclopedia

of Science Technology and Medicine in Non-western Cultures (2008); van

Brummelen, The Mathematics of the Heavens and the Earth (2009); Kim

Plofker,History of Mathematics in India (2009); Clemency Montelle, ChasingShadows, Mathematics Astronomy and Early History of Eclipse Calculation

(2010).

MODERN INDIAN SCHOLARSHIP ON INDIAN MATHEMATICS

Cadambattur Tiruvenkatachrlu Rajagopal (1903-1978) taught in Madras


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j g p ( ) g

Christian College, and later joined the Ramanujan Institute in 1951 and

became its Director in 1955. He worked on Series and Summability. In series

of seminal papers, published from around 1944 onwards, Rajagopal and his

students, K.Mukunda Marar, A.Venkataraman, M.S.Rangachari, brought to

light for the first time, the full technical details of the proofs contained in

Yuktibhand other seminal works of the Kerala School.

Chcikamagalur Narayana Iyengar Srinivasa Iyengar (1901-1972) obtained

MSc at Calcutta and taught at Mysore, Bangalore and Dharwar.His research

ondifferential equations earned him a DSc in 1932 from Calcutta University.

Keenly interested in the history of mathematics, he published the

Gaitastrada-caritrein Kannada (1958), and an extremely lucid account of

theHistory of Ancient Indian Mathematics(1967). He supervised the thesis of

Dhulipala Arka Somayaji on A critical Study of Ancient Indian Astronomyin1971.

Tekkath Amayankoth Kalam Saraswati Amma (1918-2000) worked with V.

Raghavan in Madras University and then taught in Ranchi and later at

Dhanbad. Her thesis work led to the publication of the first authentic and

comprehensive study of Geometry in Ancient and Medieval India(1979). Shesupervised the thesis of R.C.Gupta on Trigonometry in Ancient and Medieval

India.


Indian National Commission for History of Science was constituted by the


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Indian National Science Academy in 1965. The Commission actively promotes

research on History of Science in India. In 1971 the Academy brought AConcise History of Science in India(1971), a landmark publication in the field,

edited by D.M.Bose, S.N.Sen and B.V.Subbarayappa. In 2009, Subbarayappa

brought out a revised edition with substantial additions. The Academy also

started the Indian Journal of History of Sciencewhich has been successfully

running from 1966 and is counted amongst the premier Journals of India.

Samarendra Nath Sen (1918-1992) was a graduate in Physics and served as

Registrar of Indian Association for Cultivation of Science. During 1947-49 he

was with UNESCO and came under the influence of Joseph Needham. In

1950s He wrote a two volume Vijner Itihsain Bengali. Sen was an active

member of the INSA Commission on history of Science. In 1966 he brought

out A Bibliography of Sanskrit Works in Astronomy and Mathematics with

assistance from A.K.Bag and S.R.Sarma). Sen and Bag published an edition

and translation of the ulva strasin 1983. Sen also edited, with K.S.Shukla, a

volume on the History of Astronomy in India (1985). In 1986 he edited thevolume on Science in the series on Cultural Heritage brought out by the

Ramakrishna Math.


Kripa Shankar Shukla (1918-2009) obtained his MSc from Allahabad


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University, and joined the Department of Mathematics Lucknow University. In

1955, he obtained his PhD degree under the supervision of A.N.Singh, for histhesis on Bhskara I and his works. This led to his landmark editions and

translations of Mahbhskarya (1960), Laghu-bhskarya(1963) and edition

of ryabhayabhsya (1976) of Bhskara I. Shukla continued to bring out

editions (and translations) of several other seminal texts such as

Sryasiddhnta with Paramevaras commentary (1957), Pgaita (1959),Dhkoidkaraa (1969), Bjagaitvatasa (1970), ryabhaya with

K.V.Sarma (1976), Karaaratna(1979), Vaewara-siddhntain two volumes

(1985,86), andLaghumnasa(1990).

Shukla has written over 30 papers on various aspects of Indian astronomy andmathematics. He also edited and published, as a series of articles, the various

chapters of the unpublished third volume of Datta and Singhs History of

Hindu Mathematics. He also edited the History of Astronomy in India with

S.N.Sen (1985). In 1992, the Japanese scholar Yukio Ohashi did his thesis on

A History of Astronmical Instruments in Indiaunder the supervision of Shukla.


Krishna Venkatesvara Sarma (1919-2005) did his MA in Sanskrit in

i d d l j i d h C l C l j i h


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Treivandrum, and later joined the Catalogus Catalagorum Project with

V.Raghavan at Madras University. Soon he embarked on his life-long pre-

occupation with Kerala Astronomy and edited the important works

Grahacranibandhana of Haridatta, Siddhntadarpaa of Nlakaha,

Vevroha of Mdhava and Goladpik of Paramewara. During this

period, Sarma worked with the renowned scholar T. S. Kuppanna Sastri,

in the edition of Vkyakaraa (1962). Sarma shifted to the

Visvesvaranand Institute at Hoshirapur, where he served as director during1975-80.During this period, he published more than 50 books, including

seminal works such as Dggaita of Paramevara, Golasra of

Nlakaha, A History of the Kerala School of Hindu Astronomy, Llvat

of Bhskarcrya with Kriykramakar of akara, Tantrasagraha of

Nlakaha with Yuktidpik of akara, Jyotirmms of Nlakaha,and Gaitayuktaya. In 1983 he returned to Madras, and was relentlessly

active till his very death in 2005.His important publications during this

period include:Indian Astronomy:A Source Bookwith B.V.Subbarayappa

(1985); Pacasiddhntikof Varhamihira, based on his work with T. S.

Kuppanna Sastri (1993); and his magnum opus, the edition and translationof Gaita-Yuktibh, which appeared in 2008, along with notes by

K.Ramasubramanian, M.S.Sriram and M.D.Srinivs.


Radha Charan Gupta (b.1935) studied in Lucknow University and did his PhD

i h S hi A R hi H d P f f M h i


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with Saraswathi Amma at Ranchi. He served as a Professor of Mathematics at

the Birla Institute of Technology, Ranchi. With his extraordinary passion for

history of mathematics, Gupta has investigated almost all aspects of Indian

mathematics. He has published nearly 500 papers on history of mathematics.

He founded the Journal Ganita Bharatiin 1979, which has played a major role

in promoting research on history of mathematics in India. Gupta was awarded

the prestigious K.O.May Prize in the History of Mathematics in 2009.

Amulya Kumar Bag (b.1937) worked with S.N.Sen from the time of inception

of the INSA Commission. He has also been associated with the Indian Journal

of History of Science all through and has contributed significantly to the

Journals success. Bag has published an overview of Mathematics in Ancient

and Medieval India(Benares 1981).

George Gheverghese Joseph was born in Kerala,but spent much of childhood

in Africa and later studied in England. He has been in the forefront of the

debate against Euro-centricism in the history mathematics. His book The Crest

of the Peacock: Non-European Roots of Mathematics, first published in 1991,

drew wide acclaim and was translated to several languages. The third edition

has appeared in 2011. Recently, Joseph has also writtenA Passage to InfinityMedieval Indian Mathematics from Kerala and its Impact (2009).


Among the prominent mathematicians associated with the Tata Institute of

F d t l R h P f K bil B l dh (1922 2012)


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Fundamental Research, Prof. Kopparambil Balagangadharan (1922-2012)

Ramiengar Sridharan (b.1935) and Shrikrishna Gopalrao Dani (b.1947) have

done significant work also on Indian mathematical tradition. Balagangadharan

has analysed the work of Kerala School on infinite series; Sridharan has

studied Indian work on algebra and the combinatorial techniques developed in

Indian works on prosody and music. Dani has investigated the geometrical and

algebraic techniques in ulvastras. P.P.Divakaran (b.1936) of the School of

Physics TIFR has been working on the conceptual issues surrounding thedevelopment of calculus by Kerala mathematicians.

Recently, there have been many important books published on Indian

Mathematics and astronomy. They include: The translations of Grahalghava

(2006) and Karaakuthalam (2008) with notes by S.Balachandra Rao and

S.K.Uma; C.K.Raju,The Cultural Foundations of Mathematics (2007); h.Selinand R.Narasimha (ed),Encyclopaedia of Classical Indian Sciences (2007);The

Tradition of Astronomy in India (2008) by B.V.Subbarayappa; S.R.Sarma,

The Archaic and the Exotic: Studies in the History of Indian Astronomical

Instruments (2008); J.V.Narlikar (ed), Science in India (2009); C.S.Seshadri

(ed), Studies in the History of Mathematics (2010); and Tantrasagraha,

Translated with notes by K.Ramasubramanian and M.S.Sriram (2011).


Harish Chandra (1923-1983):One of the most distinguished mathematicians


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from India in modern times, Harish Chandra did his Masters in Physics at the

University of Allahabad. During 1943-45 he worked with Bhabha at the IndianInstitute of Science. In 1945, he left for Cambridge University to work with the

renowned theoretical physicist P.A.M. Dirac and was awarded PhD for his

work on the Lorentz group in 1947.

During his stay at the Princeton Institute of Advanced Study in 1947-49,

Harish Chandra shifted to mathematics. He taught at Columbia Universityduring 1950-1963 and returned to the Institute of Advanced study in 1963,

where he served as IBM von Neumann research Professor.

Harish Chandra is considered to be a pioneer in the area of Harmonic analysis

of Lie groups. Robert Langlands wrote in an article on Harish Chandra that

He was considered for the Fields Medal in 1958, but a forceful member of theselection committee in whose eyes Thom was a Bourbakist was determined not

to have two. So Harish-Chandra, whom he also placed on the Bourbaki camp,

was set aside.

Harish Chandra was elected an FRS in 1973, the second Indian to be so

honoured in the field of mathematics after Ramanujan. Harish Chandra tookup US citizenship in 1980. Soon thereafter, in 1981, he was elected a Fellow of

the US National Academy of Sciences.


The Indian Statistical Institute (ISI) and the Tata Institute of Fundamental

h h b h i h i i i i I di i h i


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research have been the premier research institutions in India in mathematics.

Indian Statistical Institute, which was founded in 1933 by P.C.Mahalanobis,produced many distinguished statisticians and mathematicians.

Calyampudi Radhakrishna Rao (b.1920), the renowned statistician, was a

student of the Andhra and Calcutta Universities. Rao acquired his PhD degree

working with R.A. Fisher at Kings College, London, in 1948. He joined the

ISI and later served as its Director. Rao is well known for many outstanding

contributions such as Cramer-Rao bound and Rao-Blackwell theorem. There

have been a number of well known mathematicians and statisticians who have

been his students. Since 1978, Rao has been a distinguished professor at many

US universities.Srinivasa Ranga Iyengar Varadhan (b.1940) was a student of Rao at ISI. After

obtaining his PhD degree in 1963, Varadhan shifted to the Courant Institute in

New York where he is a distinguished professor. He is known for his highly

acclaimed work on diffusion processes and large deviations. Varadhan was

elected a Fellow of the US Academy of Sciences in 1995 and he became FRS

in 1998. He won the coveted Abel Prize, the first Indian to do so, in 2007.


The Tata Institute of Fundamental researchwas founded in 1945 at the


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initiative of Homi Bhabha (1909-1966) with the help of Sir Dorabji

Jamshedji Tata Trust. Komaravelu S. Chandrasekharan (b. 1920), astudent of Ananda Rau at Madras, was with the Princeton Institute of

Advanced Study when he was invited by Bhabha to organise the School of

Mathematics at TIFR. Chandrasekharan is known for his work on the

complex variables and multiple Fourier series. Chandrasekharan left TIFR

in 1965 and is currently professor emeritus at ETH Zurich.TIFR School of mathematics had many eminent mathematicians.

Kollagunta Gopalaier Ramanathan (1920-1992) was a student of Emil

Artin, who built up the number theory group and also worked on

Ramanujans unpublished works. C.P.Ramanujam (1938-1974) was a

student of Ramanathan, who worked on number theory and algebraicgeometry. Vijay Kumar Patodi (1945-1976), a student of Narasimhan and

Ramanan at TIFR, worked on differential geometry and topology.


Three members of the School of Mathematics at TIFR have distinguished

h l b h d f FRS


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themselves by the award of FRS

Conjeevaram Srirangachari Seshadri (b.1932) was a student ofChandrsekharan at TIFR. He has done highly acclaimed work in algebraic

geometry, especially on unitary vector bundles (in collaboration with

Narasimhan) and Schubert varieties. He founded the Chennai

Mathematical Institute in 1989. He was awarded FRS in 1992.

Mudumbai Seshachalu Narasimhan (b.1932) was a student of

Chandrasekharan at TIFR. He has done highly acclaimed work in algebraic

geometry. He worked as Head of the Mathematics group at the ICTP during

1992-1997. He was awarded FRS in 1996.

Madabusi Santanam Raghunathan (b.1941) was a student of Narasimhan in

TIFR. He has done highly acclaimed work on discrete subgroups of Lie

groups. He has been associated with the National Board of Higher

Mathematics with its inception and has been its chairman. He was awarded

FRS in 2000.

DEVELOPMENT OF SCHOOL EDUCATION IN INDIA (1950-2005)

Number of Institutions

Years Primary Upper Sec /Sr Colleges Colleges for Universities/


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Years Primary Upper

Primary

Sec./Sr.

Sec/

Inter

Colleges

of General

Education

Colleges for

Professional

Education

Universities/

Deemed

Univ../

1950-51 209671 13596 7416 370 208 27

1960-61 330399 49663 17329 967 852 45

1980-81 494503 118555 51573 3421 3542** 110

1990-91 560935 151456 79796 4862 886 184

2000-01 638738 206269 126047 7929 2223 254

2005-06 772568 288493 159667 11698 5284 350

**IncludesinstitutionsforPostMatriccourses

Enrolment in Millions

YEARPrimary(I V) Middle/ (VI-VIII) Sec./Sr.S (IX-XII)

Boys Girls Total Boys Girls Total Boys Girls Total

1950-51 13.8 5.4 19.2 2.6 0.5 3.1 1.3 0.2 1.5

1960-61 23.6 11.4 35.0 5.1 1.6 6.7 2.7 0.7 3.4

1980-81 45.3 28.5 73.8 13.9 6.8 20.7 7.6 3.4 11.0

1990-91 57.0 40.4 97.4 21.5 12.5 34.0 12.8 6.3 19.12000-01 64.0 49.8 113.8 25.3 17.5 42.8 16.9 10.7 27.6

2005-06 70.5 61.6 132.1 28.9 23.3 52.2 22.3 16.1 38.4

LOW

Mathematics in Modern India

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