Extractions: International Study Group on the Relations Between HISTORY and PEDAGOGY of MATHEMATICS NEWSLETTER An Affiliate of the International Commission on Mathematical Instruction: No. 48, November 2001 A particular rule for finding the arc length of a bow-figure (i.e. segment of a circle) has been found on an ancient Babylonian tablet. fig.1 Let s, c, h be, respectively, the length of the arc PNQ (see fig.1), chord PQ, and the arrow or height MN of the circular segment (assumed to be not greater than a semicircle). Then the formula extracted from the procedure given in the old Babylonian text BM85194 (dated about 1600BC) is equivalent to s c h Actually, the scribe used (1) for finding h (without specifying it so) correctly equal to 10 from given s = 60 and c = 50. The true formula is c h d h But the exact formula (2) is not expected to be known in that remote pre-trigonometric antiquity and the empirical rule (1) can be regarded to be quite practical. Surprisingly the rule (1) is found preserved in some later traditions (see below). As an application of (1), consider the old common formula
Extractions: Note: there are also a chronological lists of mathematical works and mathematics for China , and chronological lists of mathematicians for the Arabic sphere Europe Greece India , and Japan 1700 B.C.E. 100 B.C.E. 1 C.E. To return to this table of contents from below, just click on the years that appear in the headers. Footnotes (*MT, *MT, *RB, *W, *SB) are explained below Ahmes (c. 1650 B.C.E.) *MT Baudhayana (c. 700) Thales of Miletus (c. 630-c 550) *MT Apastamba (c. 600) Anaximander of Miletus (c. 610-c. 547) *SB Pythagoras of Samos (c. 570-c. 490) *SB *MT Anaximenes of Miletus (fl. 546) *SB Cleostratus of Tenedos (c. 520) Katyayana (c. 500) Nabu-rimanni (c. 490) Kidinu (c. 480) Anaxagoras of Clazomenae (c. 500-c. 428) *SB *MT Zeno of Elea (c. 490-c. 430) *MT Antiphon of Rhamnos (the Sophist) (c. 480-411) *SB *MT Oenopides of Chios (c. 450?) *SB Leucippus (c. 450) *SB *MT Hippocrates of Chios (fl. c. 440) *SB Meton (c. 430) *SB
Aryabhata_II aryabhata ii. Born about 920 in India Died about 1000 in India. Essentiallynothing is known of the life of aryabhata ii. Historians http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Aryabhata_II.html
Extractions: Essentially nothing is known of the life of Aryabhata II. Historians have argued about his date and have come up with many different theories. In [1] Pingree gives the date for his main publications as being between 950 and 1100. This is deduced from the usual arguments such as which authors Aryabhata II refers to and which refer to him. G R Kaye argued in 1910 that Aryabhata II lived before al-Biruni but Datta [2] in 1926 showed that these dates were too early. The article [3] argues for a date of about 950 for Aryabhata II's main work, the Mahasiddhanta, but R Billiard has proposed a date for Aryabhata II in the sixteenth century. Most modern historians, however, consider the most likely dates for his main work as around 950 and we have given very approximate dates for his birth and death based on this hypothesis. See [7] for a fairly recent discussion of this topic. The most famous work by Aryabhata II is the Mahasiddhanta which consists of eighteen chapters. The treatise is written in Sanskrit verse and the first twelve chapters form a treatise on mathematical astronomy covering the usual topics that Indian mathematicians worked on during this period. The topics included in these twelve chapters are: the longitudes of the planets, eclipses of the sun and moon, the projection of eclipses, the lunar crescent, the rising and setting of the planets, conjunctions of the planets with each other and with the stars.
Aryabhata_II Biography of aryabhata ii. (9201000) Essentially nothing is known of the life of aryabhata ii. Historians have argued about his date and have come up with http://www-history.mcs.st-and.ac.uk/history/Mathematicians/Aryabhata_II.html
Extractions: Essentially nothing is known of the life of Aryabhata II. Historians have argued about his date and have come up with many different theories. In [1] Pingree gives the date for his main publications as being between 950 and 1100. This is deduced from the usual arguments such as which authors Aryabhata II refers to and which refer to him. G R Kaye argued in 1910 that Aryabhata II lived before al-Biruni but Datta [2] in 1926 showed that these dates were too early. The article [3] argues for a date of about 950 for Aryabhata II's main work, the Mahasiddhanta, but R Billiard has proposed a date for Aryabhata II in the sixteenth century. Most modern historians, however, consider the most likely dates for his main work as around 950 and we have given very approximate dates for his birth and death based on this hypothesis. See [7] for a fairly recent discussion of this topic. The most famous work by Aryabhata II is the Mahasiddhanta which consists of eighteen chapters. The treatise is written in Sanskrit verse and the first twelve chapters form a treatise on mathematical astronomy covering the usual topics that Indian mathematicians worked on during this period. The topics included in these twelve chapters are: the longitudes of the planets, eclipses of the sun and moon, the projection of eclipses, the lunar crescent, the rising and setting of the planets, conjunctions of the planets with each other and with the stars.
References For Aryabhata_II References for aryabhata ii. SK Jha and VN Jha, Computation of sinetable basedon the Mahasiddhanta of aryabhata ii, J. Bihar Math. Soc. 14 (1991), 9-17. http://www-gap.dcs.st-and.ac.uk/~history/References/Aryabhata_II.html
Extractions: B Datta, Two Aryabhatas of al-Biruni, Bull. Calcutta Math. Soc. T Hayashi, T Kusuba and M Yano, Indian values for derived from Aryabhata's value, Historia Sci. No. S K Jha and V N Jha, Computation of sine-table based on the Mahasiddhanta of Aryabhata II, J. Bihar Math. Soc. V N Jha, Aryabhata II's method for finding cube root of a number, Ganita Bharati V N Jha, Indeterminate analysis in the context of the Mahasiddhanta of Aryabhata II, Indian J. Hist. Sci. D Pingree, On the date of the Mahasiddhanta of the second Aryabhata, Ganita Bharati Main index Birthplace Maps Biographies Index
References For Aryabhata_II References for the biography of aryabhata ii. S K Jha and V N Jha, Computation of sinetable based on the Mahasiddhanta of aryabhata ii, J. Bihar Math. http://www-history.mcs.st-and.ac.uk/References/Aryabhata_II.html
Extractions: B Datta, Two Aryabhatas of al-Biruni, Bull. Calcutta Math. Soc. T Hayashi, T Kusuba and M Yano, Indian values for derived from Aryabhata's value, Historia Sci. No. S K Jha and V N Jha, Computation of sine-table based on the Mahasiddhanta of Aryabhata II, J. Bihar Math. Soc. V N Jha, Aryabhata II's method for finding cube root of a number, Ganita Bharati V N Jha, Indeterminate analysis in the context of the Mahasiddhanta of Aryabhata II, Indian J. Hist. Sci. D Pingree, On the date of the Mahasiddhanta of the second Aryabhata, Ganita Bharati Main index Birthplace Maps Biographies Index
Indian Mathematics order were Aryabhata I, Varahamihira, Brahmagupta, aryabhata ii, Sripati, Bhaskara II (known popularly as Bhaskaracarya), http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Indian_mathematics.html
Extractions: It is without doubt that mathematics today owes a huge debt to the outstanding contributions made by Indian mathematicians over many hundreds of years. What is quite surprising is that there has been a reluctance to recognise this and one has to conclude that many famous historians of mathematics found what they expected to find, or perhaps even what they hoped to find, rather than to realise what was so clear in front of them. We shall examine the contributions of Indian mathematics in this article, but before looking at this contribution in more detail we should say clearly that the "huge debt" is the beautiful number system invented by the Indians on which much of mathematical development has rested. Laplace put this with great clarity:- The ingenious method of expressing every possible number using a set of ten symbols each symbol having a place value and an absolute value emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. the importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of Antiquity, Archimedes and Apollonius We shall look briefly at the Indian development of the place-value decimal system of numbers later in this article and in somewhat more detail in the separate article
CHRONOLOGY OF RECREATIONAL MATHEMATICS By David Singmaster 943 elMasudi Meadows of Gold - first Chessboard Problem. 950 aryabhata ii.10C Europeans learn chess from north Africa, probably via Moorish Spain. http://www.geocities.com/SiliconValley/9174/recchron.html
Extractions: WWW page processed by Mario VELUCCHI (velucchi@cli.di.unipi.it) with the consent of David Singmaster Computing, Information Systems and Mathematics 87 Rodenhurst Road South Bank University London, SW4 8AF, England London, SE1 0AA, England Tel/fax: 0181-674 3676 Tel: 0171-815 7411 Fax: 0171-815 7499 E-mail: ZINGMAST@VAX.SBU.AC.UK WWW page processed by Mario VELUCCHI (velucchi@cli.di.unipi.it) with the consent of David Singmaster
Indian Mathematics Index 500. Yativrsabha. 920. aryabhata ii. 1500. Jyesthadeva http://www-groups.dcs.st-and.ac.uk/~history/Indexes/Indians.html
Madhurima's Page - Scientific Literature Of Ancient India as al sind al Arkhand respectively; Vateshwara's (880AD) Vateshwara Siddhanta;Manjulacharya's (932 AD) Laghumanasa; aryabhata ii's (950AD) Mahasddhanta http://www.geocities.com/fisik_99/sci_liter.htm
Extractions: About me Food Arts Places ... Miscellaneous SCIENTIFIC LITERATURE OF ANCIENT INDIA Science is essentially the systematic study of anything. It is a well known fact that science was well developed in ancient India. Science was cultivated by the brahminical schools. It was preserved and written in the form of Sutras - formulae. Later commentaries were written to explain these sutras. A vast collection of scientific literature is available in India of which a few are mentioned here. GRAMMAR The earliest known work on grammar is the Asthadhyi of Panini (circa 4-5 C BC)which refers to previous works. The Vartikas of Katyayana (3 C BC) are the critical, explanatory and commentary works of some rules of Panini. Patanjali's Mahabhasha (2 C BC) is a commentary on the Vartikas. Vakyapadiya of Bhartrihari (7 C AD) is more a work on the philosophy of language. LEXICOGRAPHY (KOSHAS) Koshas were a collection of rare and important works and their meanings. Unlike the modern dictionaries the Koshas were in the form of verses. They were of two types: those of synonyms and those of homonyms. The best known works are Amarasimha's Namalinganushasana (or AMARAKOSHA) - a three section dictionary of synonyms and Shasvata's Anekarthasamuchchaya - a dictionary of homonyms.
SDDS Volume 1 Issue 16 VEDAS) 11. JYOTISH SASTRA (ASTRONOMY AND ASTROLOGY) i. aryabhata ii.VARAHAMIHIRA iii. PARASARA iv. GARGYA SAMHITA 12. GRIHYA SASTRA http://www.srivaishnava.org/sgati/sddsv1/v01016.htm
Extractions: (Limitations under which Bhakti and Prapatti operate) Certain people exaggerate the efficacy of Prapatti to absurd extents. This chapter seeks to disabuse the views so expressed on certain aspects. ( 1 ) That even though one is born in a lower caste, he becomes one of a higher caste on performing Prapatti. The answer is that so long as the body exists, the caste does not change. EVEN A TEMPLE COW, HOWEVER HOLY IT MIGHT BE, REMAINS ONLY A COW. Even though Sri Krishna eulogized VIDURA a person belonging to the fourth caste, he did not say that he changed his caste. Similarly, even Viswamitra never became a Brahmin. The story regarding his birth shows that since his mother partook of the potion meant for a Brahmin foetus, he was already a Brahmin indeed by birth but his brahminic traits remained eclipsed. This does not mean that one can afford to despise another on the basis of caste. EVEN A PERSON OF THE SO CALLED LOWER CASTE, IF THEY ARE DEVOTEES - DUE
Did You Know? great astronomermathematicians of the Siddhanta period, in a chronological orderwere Aryabhata I, Varahamihira, Brahmagupta, aryabhata ii, Sripati, Bhaskara http://www.infinityfoundation.com/mandala/t_dy/t_dy_Q13.htm
Extractions: By D.P. Agrawal 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
Extractions: Computing, Information Systems and Mathematics 87 Rodenhurst Road South Bank University London, SW4 8AF, England London, SE1 0AA, England Tel/fax: 0181-674 3676 Tel: 0171-815 7411 Fax: 0171-815 7499 E-mail: ZINGMAST@VAX.SBU.AC.UK last Web revision:December 22, 1998 Mario Velucchi's Web Index visitors since Dec. 22, 1998 Web page processed by Web Master - Mario Velucchi velucchi@bigfoot.com Mario Velucchi / Via Emilia, 106 / I-56121 Pisa - Italy
References For Govindasvami S K Jha and V N Jha, Computation of sinetable based on the Mahasiddhanta of aryabhata ii, J. Bihar Math. http://www-gap.dcs.st-and.ac.uk/~history/References/Govindasvami.html
Great Indian Mathematicians Mahavira (Mahaviracharya), (850). Pruthudakaswami, (850). Sridhara, (900). Manjula,(930). aryabhata ii. (950). Prashastidhara, (958). Halayudha, (975). Jayadeva,(1000). http://hinduism.about.com/library/weekly/extra/bl-indianmathematicians.htm
Sanscrito Translate this page Estan basados en un sistema decimal. Se describen 4 sistemas de codigos principales, a saber KATAPAYA aryabhata ii KATAPAYA II KATAPAYA III ( Pali). http://www.angelfire.com/fl/ugf/sanscrito.html
Extractions: Sanscrito El termino "Sanscrito" en su uso actual significa " bienhecho" o " rectificado". Los Vedas, unas de las Escrituras Sagradas mas antiguas conocidas fueron escritas en esta lengua. Asi otros textos como los Upanishads,Mahabharata, Bagavadgita, los Yoga Sastras,etc. De alli la importancia de su estudio y de conocer algunos de sus elementos mas basicos. El Idioma Sanscrito fue escrito y hablado en India Antigua y hoy es de uso de estudiosos e intelectuales pues otros dialectos lo han reemplazado en el uso comun. El Sanskrito ha sido llamado tambien Devanagari ( Ciudad Divina) pues se considera hablado por los Dioses ( devas) en su Morada ( nagari). Este Idioma pertenece al Grupo de Lenguas Indo-Europeas, que un grupo bastante amplio al cual pertenecen el Ingles,Español, Latin entre otros. Se puede ver cierta similitud: Ingles Mother Father Frances Mere Pere Español Madre Padre Sanscrito Mata Putra Latin Mater Pater Una gran mayoria de los terminos utilizados en la Filosofia India, de la cual Yoga es una de ellas ( existen 6 Dharsanas o filosofias tradicionales), provienen del Sanskrito o del Pali (otro dialecto antiguo). La Filosofia budista y sus textos utilizan el sanskrito, pero mas comunmente el Pali , que era un dialecto hablado mayormente por gente del pueblo.
The Date Of Mahabharata Based On The Indian Astronomical Works This is repeated in Brahmasputha-Siddhanta (i.4), Maha-Siddhanta of aryabhata ii(i.5), Siddhanta-sekhara (i.10), Siddhanta-siromani of Bhaskara II (Ii.15). http://www.hindunet.org/saraswati/colloquium/astronomy01.htm
Extractions: Mahabharata as the sheet-anchor of bharatiya itihasa International Colloquium The Date of Mahabharata Based on the Indian Astronomical Works K.V. Ramakrishna Rao, B.Sc., M.A., A.M.I.E., C.Eng.(I)., B.L., Introduction The date of Mahabharat is analyzed for determination only based on the Indian astronomical works. The following facts are taken into consideration for such critical study: The Indian astronomers of Siddhantic works and followers have recorded the date of Bharata implying Mahabharat war in particular and starting of Kaliyuga or Era, that is used to reckon the dates of themselves at many places and in conjunction with Saka era in some places later. Aryabhata makes a specific mention about Bharata in his Aryabhatiyam. Most of the scholars including westerners have taken the connotation of it as referring to Mahabharat and in particular Mahabharat war, because, that is considered as the staring point of Kaliyuga / era in Indian astronomy and history too. Therefore, taking the astronomical works - Siddhantas, Tantras and Karanas like - Aryabhatiyam, Mahabhaskariyam, Vatesvara - Siddhanta
Aryabhata Mahasiddhanta ( Aryasiddhanta ). Przez historyków nauki uczony ten zazwyczajokreslany jest jako aryabhata ii . Polecana literatura ?.?. http://www.damar.home.pl/Encyklopedia/A/aryabhata.htm
Extractions: Kliknij w powy¿szy banner - Pomo¿esz utrzymaæ serwis Aryabhata (ur. 476 r. Kusumapura k. Patna, Indie - zm. ok. 550 r.) Staroindyjski astronom, astrolog i matematyk. Jako jeden z pierwszych stosowa³ algebrê, dosyæ dok³adnie okre¶li³ warto¶æ liczby jako 3,1416. W swojej wierszowanej pracy "Aryabhatia" (499 r.) Aryabhata wykazywa³, ¿e Ziemia i inne planety systemu S³onecznego poruszaj± siê wokó³ s³oñca. Ponadto, jak wynika z jego dzie³a, wierzy³, ¿e planety poruszaj± siê po orbitach eliptycznych (na ponad tysi±c lat przed narodzinami Keplera !). Obja¶ni³ tam równie¿ takie zjawiska jak nastêpstwo dnia i nocy i naukowo okre¶li³ przyczyny ksiê¿ycowych i s³onecznych zaæmieñ . Istnieje takie mnóstwo komentarzy, napisanych przez staroindyjskich uczonych do tej pracy Aryabhata, ¿e wskazuje to na wielk± wa¿no¶æ tego dzie³a dla dalszego rozwoju indyjskiej nauki. Do Aryabhata nale¿a³ jeszcze jeden traktat, który jednak nie zachowa³ siê do naszych czasów. Miêdzy 950, a 1100 rokiem ¿y³ i pracowa³ jeszcze jeden astronom indyjski, nosz±cy imiê Aryabhata. Jest on autorem obszernego traktatu "Mahasiddhanta" ("Aryasiddhanta"). Przez historyków nauki uczony ten zazwyczaj okre¶lany jest jako "Aryabhata II". Polecana literatura: А.И. Володарский "Ариабхата" - М.: Наука, 1977.