21. Science, Civilization And Society bhaskara ii (Bhaskaracarya). bhaskara ii did not recognize this in all its aspectsbut tried to analyze what happens if a number is divided by zero. http://www.es.flinders.edu.au/~mattom/science society/lectures/illustrations/lec

22. Sources In The History Of Algebra Supp. Arabe de la Bibliothèque Impériale, MS 952, Paris. bhaskara ii, (1150),Bijaganita. Edinburgh Univ. 500. bhaskara ii, (1150), Lilavati. Edinburgh Univ. http://logica.ugent.be/albrecht/math.php

Home Logic Math Publications Shogi Sites About Me Sources in the history of algebra: arithmetical and recreational problems A comprehensive database of problems is in progress. Some sources are listed below. The database currently contains 171 manuscripts, books, and reprints. [EDITIONS] gives an expanded list of (all) editions and translations. [CONSPECTUS] gives an overview of some selected problems in PDF format. [IMAGE] If this work in some edition is available in digital form.o If you find errors or additions, please send me a mail Brahmagupta, (628) Bráhma-sphuta-siddhânta Alcuin, (800) Propositiones alcuini doctoris caroli magni imperatoris ad acuendos juvenes CONSPECTUS Mahâvirâ, (850) Ganita-sâra-samgraha Kitab al-Jabr wal-Muqabala Alkarkhî, Aboû Beqr Mohammed (1010) Kitâb al-Fakhr . Supp. Arabe de la Bibliothèque Impériale, MS 952, Paris. Bhaskara II, (1150) Bijaganita . Edinburgh Univ. Library, Or MS. 500. Bhaskara II, (1150) Lilavati . Edinburgh Univ. Library, Or MS. 499. von Stade, Abbot Albert (1240) Annales Stadenses ben Ezra, Abraham (1325)

23. 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 :

24. 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

25. 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:

26. References For Bhaskara Ed. 8 (1) (1991), 2327. CO Selenius, Rationale of the chakravala processof Jayadeva and bhaskara ii, Historia Math. 2 (1975), 167-184. http://intranet.woodvillehs.sa.edu.au/pages/resources/maths/History/~DZAA2A.htm

References for Bhaskara Biography in Dictionary of Scientific Biography (New York 1970-1990). Biography in Encyclopaedia Britannica. Articles: S P Arya, On the Brahmagupta- Bhaskara equation, Math. Ed. C-O Selenius, Rationale of the chakravala process of Jayadeva and Bhaskara II, Historia Math. Acta Acad. Abo. Math. Phys. B Chaudhary and P Jha, Studies of Bhaskara's works in Mithila, Ganita Bharati 12 (1-2) (1990), 27-32. B Datta, The two Bhaskaras, Indian Historical Quarterly Gupta, R. C. Bhaskara II's derivation for the surface of a sphere, Math. Education V Madhukar Mallayya and K Jha, Bhaskara's concept of numeration in decuple proportions - earliest reference in Vedas with Yaska's 'Nirukta' throwing light on the notion of succession in enumeration : an anticipation of Peano's axioms, Ganita-Bharati S R Sinha, Bhaskara's Lilavati, Bull. Allahabad Univ. Math. Assoc. D A Somayaji, Bhaskara's calculations of the gnomon's shadow, Math. Student Close this window or click this link to go back to Bhaskara Welcome page Biographies Index History Topics Index Famous curves index ... Search Suggestions JOC/EFR December 1996 The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Bhaskara.html

27. Biography Of Sridhara The works of bhaskara ii (writing around 1150), Makkibhatta, and cite the booksthe Bijaganita, Navasati, and Brhatpati as being written by Sridhara. http://www.cabrillo.edu/~stappero/Sridhara.htm

Biography of Sridhara Sridhara was born around the year 870, although historians are unsure about this date. Estimates indicate that he wrote in about 900 AD, from examining other works of mathematics that he was familiar with and seeing which later mathematicians were familiar with his work. Historians place his birthplace in two very different location; one hypothesis is that he was born in Bengal, India, while other historians would assert that he was born in southern India. Sridhara penned his name to two mathematical treatises, called the Trisatika (sometimes called the Patiganitasara ) and the Patiganita However, three other mathematicians have attributed other works to his name. The works of Bhaskara II (writing around 1150), Makkibhatta, and cite the books the Bijaganita Navasati , and Brhatpati as being written by Sridhara. The Patiganita is written in the form of a poem; it details tables of monetary and metrological units. The algorithms in the novel are designed to carry out basic arithmetic operations such as: squaring, cubing, and square and cube root extraction. Sridhara gives rules to solve problems in a verse format, as was the trend in Indian texts at that time.

28. 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

29. 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.

30. 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"

31. Il Bambu' Spezzato bhaskara ii, 1150. http://digilander.libero.it/basecinque/pitagora/bambu.htm

BASE Cinque Appunti di Matematica ricreativa BASE Cinque Collezione Il bambù spezzato Una ricreazione pitagorica nata in Cina? Il bambù spezzato Una canna di bambù di altezza h si spezza ad una distanza x dal suolo. La parte spezzata rimane attaccata al bambù per un'esile fibra e cade al suolo ruotando attorno al punto di frattura. L'estremo superiore della parte spezzata tocca il suolo ad una distanza d dalla base del bambù. Calcolare x sapendo che h = 10 e d = 3. Nota storica La prima versione di questo problema si trova nel libro cinese Chiu Chang Suan Shu, Nove capitoli sulle arti matematiche di autore ignoto, risalente al 300-200 a. C. Qualcuno sa risolverlo senza usare equazioni? Il bambù spezzato: la versione originale C'è un bambù alto 1 zhang la cui estremità superiore, essendo spezzata, tocca il terreno ad una distanza di 3 chih dalla base del fusto. A quale altezza si trova la frattura? Nota: 1 zhang = 10 chih Illustrazione del Chiu Chang Suan Shu tratta da G. Ghevergeese J., C'era una volta un numero, Il Saggiatore, 2000

32. Scale Incrociate Translate this page La figura qui sotto illustra la situazione. Nota storica. Le prime versioni diquesti problemi risalgono a Mahavira 850, bhaskara ii 1150, Fibonacci 1202. http://digilander.libero.it/basecinque/pitagora/scale.htm

BASE Cinque Appunti di Matematica ricreativa BASE Cinque Collezione Scale incrociate 1. Scale incrociate facile Due case si trovano l'una di fronte all'altra lungo una via piuttosto stretta. Ci sono inoltre due scale incrociate che partono dalla base di ciascuna casa e si appoggiano al muro della casa di fronte rispettivamente alle altezze a , b. A quale altezza c si incrociano le due scale? La figura qui sotto illustra la situazione. 2. Scale incrociate difficile Due case si trovano l'una di fronte all'altra lungo una via piuttosto stretta. Ci sono inoltre due scale incrociate lunghe rispettivamente x = 30, y = 20 che partono dalla base di ciascuna casa e si appoggiano al muro della casa di fronte. Il punto d'incrocio delle scale si trova ad un'altezza c = 10. Quanto è larga la strada? La figura qui sotto illustra la situazione. Nota storica. Le prime versioni di questi problemi risalgono a Mahavira 850, Bhaskara II 1150, Fibonacci 1202. 3. La scala e la scatola facile

33. Search from 5th century AD, Aryabhatta I, Prabhakara, Bhaskara I, Brahma Gupta, Vateswara,Aryabhatha II Someswara, Sutananda, bhaskara ii, Amaraja, Parameswara http://www.vichar.nic.in/Astronomy/astronomy_chapter6.asp

Search Home All in the Game About NCSTC NCSTC Comm. ... CONTENTS Astronomy in India According to this calender, the year was divided into five seasons viz. Grashma, Vasanta, Sisira, Sarada and Varsha. The year was divided into 12 months (as shown in the figure) and the lunar belt of 27 nakshtras. Yantras of Jantar Mantar Jantar Mantar or the Yantra mandir - the temple of instruments. Talking of Yantras, what comes to one's mind are the observatories of Jantar Mantar built by Maharaja Sawai Jai Singh II. Year 1719 A.D., Place : Delhi, Location : Red Fort - A noisy session about the auspicious time for the emperor Mohammed Shah to embark upon a big expedition. The maulvis and pandits did not have astronomical laboratories to verify the calculations and hence the confusion and debate. The spectator was Maharaj Sawai Jai Singh II. He decided to construct huge stone astronomical observatories to educate people. By 1724 A.D., the first observatory at Delhi was completed. The maharaja himself carried out experiments and observations for nearly seven years at this observatory. The next one was constructed at Jaipur in 1728 A.D. and the others at Ujjain, Varanasi and Mathura. All the yantras were made of red sand stone, marble and iron. Sawai Jai Singh wanted to promote the scientific approach to astonomy and astrology. To acquaint people with the scientific aspect of the Sun, moon, stars and the various astronomical phenomena, he designed the various yantras. Let us learn how to use some of these yantras. You can also make your own DhoopGari (Sundial) and star clock.

34. Applications Of Integration For example, bhaskara ii, a well known Indian mathematician of middle ageswrote a math book Lilavati in 1150 AD in the memory of his daughter. http://www.mathwright.com/book_pgs/book680.html

Microworld: Applications of Integration Click the Hyperlink above to visit the Microworld. Author Ravinder Kumar This 9-page microworld explores arc length of a curve, area under a curve, and surface area and volume of revolution. For simplicity we explore only those surfaces of revolution that can be obtained by revolving a curve about x-axis. Arc length, area, surface area, and volume can be found by dividing the arc, region, or solid into tiny portions in Riemannian spirit. You will be living in Riemannian spirit as you conduct explorations on the following interactive pages. The theory will be briefly explained on the help pages that can be viewed by pressing the button “math for this page”. Often an example or two may be used to explain the theory. When a page of the microworld contains a button named “instructions”, you can press it to view instructions for using the interactivity of the page in order to make explorations. Seeds for the ideas of integration that lead up to finding area and volume were sown much earlier than the advent of calculus.

35. Math History - Middle Ages 1140, bhaskara ii (sometimes known as Bhaskaracharya) writes Lilavati (The Beautiful)on arithmetic and geometry, and Bijaganita (Seed Arithmetic), on algebra. http://lahabra.seniorhigh.net/pages/teachers/pages/math/timeline/MmiddleAges.htm

36. Formulas Of Euclid And Archimedes Activity triangle.). Exercise 6 is from the Lilavati, written by the Indianmathematician and astronomer, bhaskara ii, in about 1150 CE. 6 http://newton.uor.edu/facultyfolder/beery/math115/day4.htm

[Today’s class: Maya arithmetic, especially subtraction (use toothpicks and small candies-or pencil and paper), review of Pythagorean Theorem and its converse, Puzzle Proofs of Pythagorean Theorem activity, Proofs of Pythagorean Theorem via area and algebra (see Pythagorean Theorem activity), Historical applications of Pythagorean Theorem Mathematics 115 Homework Assignment #4 Due Monday, January 14, 2002 Prof. Beery's office hours this week Thursday 1/ 10   10:30 a.m.-12:30 p.m. 4-5 p.m.                                                                      Friday 1/11   1:30 - 3:30 p.m.                                                                   Monday 1/14   10:30 a.m.-12:30 p.m. 4-5 p.m. and by appointment, Hentschke 203D, x3118 Tutorial session :  Sunday, Jan. 13, 4 - 5 p.m. , Hentschke 204 (Jody Cochrane) Read :   "No Stone Unturned (Early Southern California math artifacts?)"             "Kernel revealing history of humans in the New World "              "Mayan Arithmetic" (you may skip Section 4, Division)             "Mayan Head Variant Numerals"

37. Pythagoras' Theorem I have been told that this proof, with the exclamation `Behold! , isdue to the Indian mathematician bhaskara ii (approx. 11141185). http://www.math.ntnu.no/~hanche/pythagoras/

Behold! The above picture is my favourite proof of Pythagoras' theorem. Filling in the details is left as an exercise to the reader. Is this the oldest proof? This proof is sometimes referred to as the Chinese square proof , or just the Chinese proof . It is supposed to have appeared in the Chou pei suan ching (ca. 1100 B.C.E.), according to Ralph H. Abraham [see ``Dead links'' below,] who attributes this information to the book by Frank J. Swetz and T. I. Kao, Was Pythagoras Chinese? . See also Development of Mathematics in Ancient China According to David E. Joyce 's A brief outline of the history of Chinese mathematics , however, the earliest known proof of Pythagoras is given by Zhoubi suanjing (The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven) (c. 100 B.C.E.-c. 100 C.E.) In the The MacTutor History of Mathematics archive there is a section devoted to Chinese mathematics . The overview section at that section also mentions the Zhoubi suanjing is a proof as well. I have been told that this proof, with the exclamation `Behold!', is due to the Indian mathematician Bhaskara II (approx. 1114-1185). A web page at the

38. Profes.net Translate this page a la historia de las matemáticas. bhaskara ii tuvo una hija a la quepuso por nombre Lilavati. Al nacer, su padre consultó a los http://www.matematicas.profes.net/apieaula2.asp?id_contenido=41565

39. Pergunta Agora Translate this page Biruni, Avicenna, al-Baghdadi, Al-Jayyani, Al-Nasawi, Hermann of R., Sripati, Shen,Khayyam, Brahmadeva, Abraham, Adelard, Ezra, Aflah, bhaskara ii, Gherard, al http://www.apm.pt/pa/index.asp?accao=showtext&id=3407

40. Transmission Of Mathematical Ideas Author 20 bhaskara ii even declares that the Rule of Three pervades the whole field of arithmeticwith its many variations, just as Visnu pervades the entire universe http://www.iwr.uni-heidelberg.de/transmath/author20.html

2000 Years Transmission of Mathematical Ideas: Exchange and Influence from Late Babylonian Mathematics to Early Renaissance Science S. R. Sarma (Aligarh, India) "Rule of Three in Sanskrit Variations" In the history of transmission of mathematical ideas, the Rule of Three forms an interesting case. It was known in China as early as the first century AD. Indian texts dwell on it from the fifth century onwards. It was introduced into the Islamic world in about the eighth century. Renaissance Europe hailed it as the Golden Rule. The importance of the rule lies not so much in the subtlety of its theory as in the simple process of solving problems. This process consists of writing down the three given terms in a linear sequence (A -> B -> C) and then, proceeding in the reverse direction, multiplying the last term with the middle form and dividing their product by the first term (C x B : A). With this rule one can easily solve several types of problems even without a knowledge of the general theory of proportion. The writers in Sanskrit, however, were well aware of the theory.

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