1. References For Bhaskara_II 1930), 727736. RC Gupta, bhaskara ii's derivation for the surfaceof a sphere, Math. Education 7 (1973), A49-A52. RC Gupta, The http://www-gap.dcs.st-and.ac.uk/~history/References/Bhaskara_II.html

References for Bhaskara Biography in Dictionary of Scientific Biography (New York 1970-1990). Biography in Encyclopaedia Britannica. Books: R Calinger (ed.), Classics of Mathematics (New Jersey, 1995). G Ifrah, A universal history of numbers : From prehistory to the invention of the computer (London, 1998). G G Joseph, The crest of the peacock (London, 1991). K S Patwardhan, S A Naimpally and S L Singh, Lilavati of Bhaskaracarya (Delhi 2001). Articles: S P Arya, On the Brahmagupta- Bhaskara equation, Math. Ed. B Chaudhary and P Jha, Studies of Bhaskara's works in Mithila, Ganita Bharati B Datta, The two Bhaskaras, Indian Historical Quarterly R C Gupta, Bhaskara II's derivation for the surface of a sphere, Math. Education R C Gupta, The last combinatorial problem in Bhaskara's Lilavati, Ganita Bharati M G Inamdar, A formula of Bhaskara for the chord of a circle leading to a formula for evaluating sin a°, Math. Student A A Krishnaswami Ayyangar, Remarks on Bhaskara's approximation to the sine of an angle, Math. Student

2. Encyclopædia Britannica Encyclopædia Britannica, bhaskara ii Encyclopædia Britannica Article. MLAstyle bhaskara ii. 2003 Encyclopædia Britannica Premium Service. http://www.britannica.com/eb/article?eu=81187

3. Bhaskara_II Bhaskara is also known as bhaskara ii or as Bhaskaracharya, this lattername meaning Bhaskara the Teacher . Since he is known in http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Bhaskara_II.html

Bhaskara Born: 1114 in Vijayapura, India Died: 1185 in Ujjain, India Show birthplace location Previous (Chronologically) Next Biographies Index Previous (Alphabetically) Next Main index Bhaskara is also known as Bhaskara II or as Bhaskaracharya, this latter name meaning "Bhaskara the Teacher". Since he is known in India as Bhaskaracharya we will refer to him throughout this article by that name. Bhaskaracharya's father was a Brahman named Mahesvara. Mahesvara himself was famed as an astrologer. This happened frequently in Indian society with generations of a family being excellent mathematicians and often acting as teachers to other family members. Bhaskaracharya became head of the astronomical observatory at Ujjain, the leading mathematical centre in India at that time. Outstanding mathematicians such as Varahamihira and Brahmagupta had worked there and built up a strong school of mathematical astronomy. In many ways Bhaskaracharya represents the peak of mathematical knowledge in the 12th century. He reached an understanding of the number systems and solving equations which was not to be achieved in Europe for several centuries. Six works by Bhaskaracharya are known but a seventh work, which is claimed to be by him, is thought by many historians to be a late forgery. The six works are:

4. Bhaskara Bhaskara is also known as bhaskara ii or as Bhaskaracharya, this latter name meaning "Bhaskara the Teacher". http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Bhaskara.html

Bhaskara This biography is now under You should be automatically forwarded to the correct page. If your browser leaves you on this page, then press HERE JOC/EFR November 2000

5. História Da Matemática Na Índia História da Matemática na Índia Bhaskara escreveu Siddhanta Siromani, em 1150. http://www.7mares.terravista.pt/mjl1/India/BhaskaraII.htm

A página que procura foi realojada, se dentro em 5 segundos esta página não for direccionada para o novo endereço, Clique http://www.malhatlantica.pt/mathis/India/BhaskaraII.htm

6. História Da Matemática Na Índia - Bhaskara II Translate this page textos História da Matemática na Índia. bhaskara ii. Bhaskaraescreveu Siddhanta Siromani, em 1150. O seu livro está dividido http://www.malhatlantica.pt/mathis/India/BhaskaraII.htm

textos: História da Matemática na Índia Bhaskara II Bhaskara escreveu Siddhanta Siromani Lilavati (A Bela) sobre aritmética; Bijaganita sobre a álgebra, Goladhyaya sobre a esfera, ou seja sobre o globo celeste e Grahaganita sobre a matemática dos planetas. Lilavati contém 278 versos. Trata de vários assuntos: Definições e tabelas O sistema de numeração Oito operações numéricas com números inteiros (adição, subtracção, multiplicação, divisão, quadrados, raízes quadradas, cubos, raízes cúbicas) As oito operações com fracções Oito regras relativas ao zero Descobrir quantidades desconhecidas Equações quadráticas Regra de três, proporção inversa, regra de cinco Juros ... Medições (teorema de Pitágoras) Volumes Problemas geométricos de sombras (trigonometria) Modificação da Kuttaka (a equação a x + c = b y ), da varga prakrit (a equação n x y , com n inteiro positivo, também conhecida como equação de Pell) Permutações e partições Definições e Tabelas Definições relativas à moeda Duas vezes dez varataka s [caurim ] são um kakini [concha]

7. 8 V. Bhaskaracharya II Bhaskaracharya, or bhaskara ii, is regarded almost without question as the greatest Hindu mathematician of all time and http://www-groups.dcs.st-and.ac.uk/~history/Miscellaneous/Pearce/Lectures/Ch8_5.

Indian Mathematics Previous page (8 IV. Mathematics over the next 400 years (700AD-1100AD)) Contents Next page (8 VI. Pell's equation) 8 V. Bhaskaracharya II Bhaskaracharya , or Bhaskara II, is regarded almost without question as the greatest Hindu mathematician of all time and his contribution to not just Indian, but world mathematics is undeniable. As L Gurjar states: ...Because of his work India gave a definite 'quota' to the forward world march of the science. [LG, P 104] Born in 1114 AD (in Vijayapura, he belonged to Bijjada Bida) he became head of the Ujjain school of mathematical astronomy ( Varahamihira and Brahmagupta had helped to found this school or at least 'build it up'). There is some confusion amongst the texts I have referred to as to the works that he wrote. C Srinivasiengar claims he wrote Siddhanta Siromani in 1150 AD, which contained four sections: Lilavati (arithmetic) Bijaganita (algebra) Goladhyaya (sphere/celestial globe) Grahaganita (mathematics of the planets) E Robertson and J O'Connor claim that he wrote 6 works, 1), 2) and SS (which contained two sections) and three further astronomical works, including two commentaries on the SS. G Joseph claims his mathematically significant works were 1), 2), and SS (which indeed he wrote in 1150 and is a highly influential astronomical work). S Sinha however agrees with C Srinivasiengar that

8. História Da Matemática Na Índia - Bhaskara II - Equações Translate this page História da Matemática na Índia. Lilavati de bhaskara ii. Descobrirquantidades desconhecidas. Verso 54 De um grupo de elefantes http://www.malhatlantica.pt/mathis/India/BhaskaraII-1.htm

textos: Problemas de Lilavati: Quantidades desconhecidas Equações quadráticas Regra de três Lucros Combinações e permutações ... Volumes História da Matemática na Índia Lilavati de Bha skara II Descobrir quantidades desconhecidas Verso 54 De um grupo de elefantes, metade e um terço da metade foram para uma gruta. Um sexto e 1/7 de um sexto foram beber água de um rio. Um oitavo e 1/9 de um oitavo estavam a brincar num pequeno lago cheio de lótus. O amado rei dos elefantes estava conduzindo três elefantes fêmeas. Nesta situação, quantos elefantes estavam na manada. (a partir da tradução de Patwardhan et al.) Verso 55 Um terço, um quinto e um sexto de uma quantidade de lótus foram oferecidos ao Lorde Siva, ao Lorde Visnu, e ao Sol, um quarto a Parvati. Os seis lótus que sobraram foram dados ao venerável preceptor [ guru Diz depressa o número total de lótus. (citado em http://www.brown.edu/Departments/History_Mathematics/lila/lilavati_stdtechs.html) Verso 56 ... o colar do pescoço da esposa partiu-se. Um terço das pérolas caíram no chão, um quinto foram para debaixo da cama. A esposa apanhou um sexto e o seu amado um décimo. Seis pérolas ficaram no fio original. Descobre o número total de pérolas no colar.

9. História Da Matemática Na Índia - Bhaskara II - Lucros Lilavati de bhaskara ii. Lucros. Verso 98 http://www.malhatlantica.pt/mathis/India/BhaskaraII-4.htm

textos: Problemas de Lilavati: Quantidades desconhecidas Equações quadráticas Regra de três Lucros Combinações e permutações Progressões Medições Volumes História da Matemática na Índia Lilavati de Bha skara II Lucros Verso 98 Se o juro de 100 durante um mês é 5, a quantidade ao fim de um ano é 1000 niskas . Descobre a quantia e o lucro. (a partir da tradução de Patwardhan et al.) Verso 100 A soma de 94 niskas , foram emprestadas em três partes com um juro de 5%, 3% e 4%, obteve-se um lucro igual de cada uma das partes, em 7, 10 e 5 meses, respectivamente. Diz, matemático a quantidade de cada parte. Solução: 24, 28 e 42 niskas Verso 104 Diz depressa, amigo, em que porção de um dia [quatro] fontes, todas abertas, encherão uma cisterna, as quais sozinhas a encheriam num dia, em meio dia, num terço de um dia e na sexta parte de um dia, respectivamente? Solução: 1/12 de um dia. (citados por Shen Kangshen et al.) Verso 106 T rês unidades e meia de arroz e 8 unidades de feijão mung é comprado por um dramma . Ó merceeiro, eu tenho 13

10. Chakravala bhaskara ii perfects Bramagupta's method for solving the varga prakriti by providinga method for solving the equation Nx^2 + k = y^2, when k = 4, -2, -1, +1 http://www.math.sfu.ca/histmath/India/12thCenturyAD/Chakravala.html

Chakravala Bhaskara II perfects Bramagupta's method for solving the varga prakriti by providing a method for solving the equation Nx^2 + k = y^2, when k = -4, -2, -1, +1, +2, +4. Bhaskara calls this method the Chakravala or the cyclic process. I will explain the cyclic method by means of an example taken from Siddhanta Siroman of Bhaskara. Bhaskara demonstrates how to find the solution of the equation y^2 = 67x^2 + 1. He starts with the pairs (1,8) and (1,s). (1,8) satisfies y^2 = 67x^2 -3, with "additive" -3, while (1,s) satisfies the equation y^2 = 67x^2 + (s^2 - 67) with "additive" (s^2 - 67). Bhaskara then applies the principle of composition or Samasa process of Bramagupta modification to Kuttaka ) to Aryabhata 's solution of this equation ( Kuttaka Go back to the Indian History timeline

11. Did You Know? Aryabhata I, Varahamihira, Brahmagupta, Aryabhata II, Sripati, bhaskara ii (known popularly as Bhaskaracarya), Madhava, http://www.infinityfoundation.com/mandala/t_dy/t_dy_Q13_frameset.htm

12. Dream 2047-Article far from any educational activity, proved to be the most important work publishedin India after SiddhantaShiromani (written in AD 1150) by bhaskara ii. http://www.vigyanprasar.com/dream/august99/AUGUSTArticle2.htm

Mahamahopadhyaya Samanta Chandra Sekhara Harichandan Mohapatra 100 Years of Siddhanta-Darpana -Subodh Mahanti Leading astronomers of this period were Aryabhata I (born A.D. 476), Varahamihira (6th century A.D.), Bhaskara I (born c. A.D. 600), Brahmagupta (born c. A.D. 598), and Bhaskara II (born A.D. 1114). Besides the compilation work of Varahamihira, the immortal works of this period were Aryabhatia (by Aryabhata I), Brahmasphuta-siddhanta (by Brahmagupta) and Siddhanta-Shiromani (by Bhaskara II). with the help of commentaries. By the age of 15 he mastered the rules for calculating the ephemerides (tables showing the positions of heavenly bodies at regular intervals in time) of the planets. While calculating the positions of the planets he found that neither the stars appeared on the horizon at the right moment nor could the planets be seen in the right places. He began to observe and calculate the movement of heavenly bodies night after night. At the age of 23 he began to note down systematically the results of his observations. The journal Knowledge which reviewed the book in 1899 wrote: Pathani Samanta made contributions to the following four important aspects of astronomy: (1) Observations (2) Calculation (3) Method of measurement and instrumentation (4) Theory and models Siddhanta-Darpana wrote: The instruments used for his practical observation of the night sky were made by himself indigenously. His instruments which were mostly made up of wood and bamboo pieces can be broadly classified into three categories :

13. Bhaskara Translate this page inspiré. On parle parfois de Bhaskara I pour évoquer ce mathématicienhomonyme, bhaskara ii désignant celui qui nous intéresse ici. http://www.sciences-en-ligne.com/momo/chronomath/chrono1/Bhaskara.html

BHASKARA (Bhaskaracharya) indien, 1114-1185 un autre Brahmagupta Brahamagupta Trois oeuvres principales nous sont parvenues : Le Le Lilavati fausse position Brahamagupta Le Bija Ganita calcul des inconnues Pythagore Oresme : Dans son Lilavati diophantiennes ardues de la forme : x = ny x x x (x - 4) Les solutions sont donc : x = 9 et x = -1 Dis-moi, jeune fille au regard vif quel est le nombre qui : en multipliant par trois en ajoutant les trois quarts en divisant par 7 en enlevant le tiers en soustrayant ensuite 52 , Ed. Flammarion, Paris - 1997 fausse position Pour en savoir plus : S ur ChronoMath : Un livre : A HISTORY OF MATHEMATICS , an introduction, par Victor J. KATZ Addison-Wesley Educational Publishers -1998 Le Lilivati : http://www.brown.edu/Departments/History_Mathematics/lilivati.html Al Khayyam Fibonacci

14. COLEBROOK, Henry Thomas, Algebra With Arithmetic And Mensuration From The Sanskr of Brahmagupta, the greatest Indian mathematician of his period, as well as translationsof the works on arithmetic and algebra by his successor bhaskara ii. http://www.polybiblio.com/watbooks/2443.html

W. P. Watson Antiquarian Books COLEBROOK, Henry Thomas Algebra with Arithmetic and Mensuration from the Sanskrit of Brahmagupta and Bhascara. London: John Murray, 1817 4to (260 x 210 mm), pp [viii] lxxxiv 378; some occasional foxing, a very good copy in modern (but not recent) quarter-morocco and marbled boards, marbled endpapers and edges, old stamp of King's Inn Library, Dublin on verso of title and last leaf, with a little showthrough to recto. £1500 First edition of the most important early English work on Indian mathematics. It contains the first English translation (and the first translation into any western language) of any mathematical work of Brahmagupta, the greatest Indian mathematician of his period, as well as translations of the works on arithmetic and algebra by his successor Bhaskara II. Brahmagupta was born in 598 A.D. in northwestern India. His major work, the Brahmasphutasiddhanta (Correct Astronomical System of Brahma) was written when he was 30; the mathematical work translated here constitutes its 12th chapter. It treats the solution of linear and quadratic equations, and linear congruences, leading to the solution of the so-called Pell's equation ax2 + b = y2 (a and b being given integers and x and y integers to be found). This equation occurs for the first time in Brahmagupta's work, although it was misnamed by Euler after the 17th century English mathematician John Pell. Brahmagupta's work on Pell's equation was not really superseded until Fermat studied it a millennium later.

15. Indologie Tübingen: Vorlesungsverzeichnis Sommersemester 2001 Translate this page Zusammenhang mit astronomischen Untersuchungen ist auch das unendlich Kleine thematisiertworden, wobei die Ansätze zur Infinitesimalrechnung bei bhaskara ii. http://www.uni-tuebingen.de/indologie/ss01/8.html

Dr. Eberhard Guhe Seminar: Der Unendlichkeitsbegriff in der indischen Philosophie und Mathematik Literatur: Eine Literaturliste wird zu Semesterbeginn verteilt. Qualifikation: Home Vorlesungsverzeichnis Weiter Webmaster: aidinfo@uni-tuebingen.de und matthias.ahlborn@epost.de - Stand: 28.02.2001

16. Did You Know? of the Siddhanta period, in a chronological order were Aryabhata I, Varahamihira,Brahmagupta, Aryabhata II, Sripati, bhaskara ii (known popularly as http://www.infinityfoundation.com/mandala/t_dy/t_dy_Q13.htm

Did You Know? By D.P. Agrawal Question: Did you know Bhaskaracharya? What was he famous for and when did he live? Bhaskaracarya was a mathematician-astronomer of exceptional abilities. He was born in 1114 AD. Mathematics became the hand-maiden of astronomy and, from the time of Aryabhata I, it began to be incorporated in astronomical treatises. Thus all components of mathematics came to be developed: geometry, trigonometry, arithmetic and algebra. The great astronomers had to be great mathematicians too. The great astronomer-mathematicians of the Siddhanta period, in a chronological order were: Aryabhata I, Varahamihira, Brahmagupta, Aryabhata II, Sripati, Bhaskara II (known popularly as Bhaskaracarya), Madhava, Paramesvara and Nilakantha. These great scientists, except the last three, grew in different parts of this vast sub-continent. Perhaps such isolated growth may explain the apparent abruptness in astronomical and mathematical development in India. Even before Bhaskara made his mark on Indian Jyotisa, there were three distinct schools, the Saura, the Arya and Brahma. Bhaskara was respected and studied even in distant corners of India. Bhaskara was perhaps the last and the greatest astronomer that India ever produced. Brahmagupta was Bhaskara's role model and inspirer. To Brahmagupta he pays homage at the beginning of his

17. Sourcebook 199204. 8. CO Selenius, 1975. Rationale of the chakravala process of Jayadevaand bhaskara ii. Historia Mathematica, 2, pp. 167-184. 9. KV Sarma, 1972. http://www.infinityfoundation.com/sourcebook.htm

Home Grant Recipients Projects Announcements ... Feedback/Contact Us Sourcebook on Indic Contributions in Math and Science Subhash Kak, Editor This sourcebook will consist primarily of reprinted articles on Indic contributions in math and science, as well as several new essays to contextualize these works. It will bring together the works of top scholars which are currently scattered thoughout disparate journals, and will thus make them far more accessible to the average reader. There are two main reasons why this sourcebook is being assembled. First, it is our hope that by highlighting the work of ancient and medieval Indian scientists we might challenge the stereotype that Indian thought is "mystical" and "irrational". Secondly, by pointing out the numerous achievements of Indian scientists, we hope to show that India had a scientific "renaissance" that was at least as important as the European renaissance which followed it, and which, indeed, is deeply indebted to it. Currently, the following table of contents is proposed for this volume:

18. Mathsindiennes Translate this page équation ci-dessus. Les solutions ont été trouvées par la méthodeChakravala imaginée par bhaskara ii. Dans les temps modernes http://pages.intnet.mu/ramsurat/Bharatmata/maths.html

RESSUSCITER LES ANCIENNES MATHEMATIQUES INDIENNES Kamal Kanti Nandi "La vraie méthode de prévision du futur des mathématiques est d'étudier leur histoire et leur état actuel"

19. TIMELINE 12th CENTURY Page Of ULTIMATE SCIENCE FICTION WEB GUIDE See 11701180 1114 Birth of Indian Astronomer Bhaskara (aka bhaskara ii) 1117P'ingchow Table Talk by Chu Yu, has the first known mention in China of a http://www.magicdragon.com/UltimateSF/timeline12.html

TIMELINE 12th CENTURY Return to Timeline Table of Contents Return to Ultimate SF Table of Contents TIMELINE 12th CENTURY May be posted electronically provided that it is transmitted unaltered, in its entirety, and without charge. We examine both works of fiction and important contemporaneous works on non-fiction which set the context for early Science Fiction and Fantasy. There are hotlinks here to authors, magazines, films, or television items elsewhere in the Ultimate Science Fiction Web Guide or beyond. Most recently updated: 14-15 June 2000 [37 kilobytes]. Facts were also checked against "The 1979 Hammond Almanac" [ed. Martin A. Bacheller et al., Maplewood, New Jersey, 1978], p.795. It also utilizes facts from Volume I of D.E. Smith's "History of Mathematics" [(c) 1921 by David Eugene Smith; (c) 1951 by May Luse Smith; New York: Dover, 1958]. Jump Straight to the Chronology , or else first read: 12th Century: Executive Summary The 12th Century, according to D.E. Smith, "was to Christian Europe what the 9th Century was to the eastern Mohammedan world, a period of translations . In the case of Baghdad, these translations were from the Greek into Arabic; in the case of Christian Europe, from the Arabic into Latin. The reasons for this desire to know the science of the East are not difficult to find.... the advancement of Moorish Spain in the arts and sciences was already causing intellectual unrest in the higher classes of Church schools in France, Italy, and England. The result of this unrest was an influx of students into Spain, an acquiring of some knowledge of Arabic on the part of various scholars, and a strong desire to know and make known the science of the East. Just as Baghdad never translated the Greek literature, but sought to known Greek science, so Europe gave little attention to Arab letters, but devoted great care to those works on

20. A História De Lilavati Translate this page O livro Lilavati, na verdade, é a quarta parte do livro Siddhanta Siromani, escritopor bhaskara ii (1114-1185), possivelmente o mais famoso matemático http://www.reniza.com/matematica/novidades/0011.htm

Malba Tahan Newsletter nº2 - A História de Lilavati Novembro de 2000 O Sérgio Ratto escreveu-me este mês: "Desde de que auxiliei ao filho de um amigo, procurando na internet sobre Baskara, fiquei intrigado do porquê de Lilavati nunca ter se casado. Inclusive esta era uma das questões do trabalho em questão." Sobre Lilavati, conta Malba Tahan, em seu livro O Homem que Calculava "Baskara tinha uma filha chamada Lilavati . Quando essa menina nasceu, consultou ele as estrelas e verificou, pela disposição dos astros, que sua filha, condenada a permanecer solteira toda a vida, ficaria esquecida pelo amor dos jovens patrícios. Não se conformou Baskara com essa determinação do Destino e recorreu aos ensinamentos dos astrólogos mais famosos do tempo. Como fazer para que a graciosa Lilavati pudesse obter marido, sendo feliz no casamento? Um astrólogo, consultado por Baskara, aconselhou-a a casar Lilavati com o primeiro pretendente que aparecesse, mas demonstrou que a única hora propícia para a cerimónia do enlace seria marcada, em certo dia, pelo cilindro do Tempo. Os hindus mediam, calculavam e determinavam as horas do dia com o auxílio de um cilindro colocado num vaso cheio d'água. Esse cilindro, aberto apenas em cima, apresentava um pequeno orifício no centro da superfície da base. À proporção que a água, entrando pelo orifício da base, invadia lentamente o cilindro, este afundava no vaso e de tal modo que chegava a desaparecer por completo em hora previamente determinada.

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