21. ISHM | Ganita Bharati | Contents 59. VN JHA aryabhata ii’s Methods for finding Cube Root of a Number6068. MD PANDIT Mathematicians in Ancient India 69-85. NOTES http://www.indianshm.com/ganitabharati/toc/show.php?id=000040

22. All Comments On MATHEMATICS IN ANCIENT INDIA By Bhattathiri On Sulekha Weblogs c. 598c. 670) Govindaswami (c. 800-850) Mahavira (Mahaviracharya) (850) Pruthudakaswami(850) Sridhara (900) Manjula (930) aryabhata ii (950) Prashastidhara http://www.sulekha.com/weblogs/commentsbyentries.asp?cid=9086

23. .::Vedic Mathematics::. Apastamba Aryabhata I aryabhata ii Baudhayana Bhaskara I Bhaskara II Bose BrahmadevaBrahmagupta De Morgan, Govindasvami HarishChandra Jagannatha Jyesthadeva http://www.sanalnair.org/articles/vedmath/india-1.htm

Home Thinking Page Home VEDIC MATHEMATICS Home Introduction Examples Links Mathematicians born in India Apastamba Aryabhata I Aryabhata II Baudhayana ... Yavanesvara The ORIGINAL URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/BirthplaceMaps/India.html Thinking Pages Home Feedback Contact Powered by thingal.com Webahn.com

24. Indian Mathematics The main mathematicians of the tenth century in India were aryabhata ii and Vijayanandi,both adding to the understanding of sine tables and trigonometry to http://202.38.126.65/mirror/www-history.mcs.st-and.ac.uk/history/HistTopics/Indi

An overview of Indian mathematics Ancient Indian Mathematics index History Topics Index 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

25. Changes Index Apastamba Baudhayana Brahmadeva Cannell Dowker Eilenberg Gnedenko GovindasvamiGuenther Hopkinson Bhaskara I aryabhata ii, Jagannatha Jyesthadeva Kamalakara http://202.38.126.65/mirror/www-history.mcs.st-and.ac.uk/history/Indexes/Changes

Changes in the archive up to January 2001 New history topics added in December and January: Topology and Scottish mathematical physics Greek number systems Arabic numerals An overview of Egyptian mathematics ... Babylonian Pythagoras's theorem New biographies for December and January Kaluznin Oskar Klein Sneddon Extended biographies for December and January Artin Clausius FitzGerald Tait New history topics added in November: An Overview of Indian mathematics Indian Numerals Indian Sulbasutras Jaina Mathematics ... Chrystal and the Royal Society of Edinburgh Major changes to the biographies are shown below. The following mathematicians had their biographies added to the archive from August to November 2000 Apastamba Baudhayana Brahmadeva Cannell ... Yavanesvara The following mathematicians had new and extended biographies from August to November 2000 Aryabhata Besicovitch Bhaskara Brahmagupta ... Time lines JOC/EFR January 2001 The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Indexes/Changes.html

26. This Article Appeared In The Indian Journal The medieval mathematicians like Aryabhata I (AD 476), Brahmagupta (AD 598), BhaskaraI (AD 600), Mahaviracarya (AD 850), aryabhata ii (AD 950), Sridharacarya http://www.vmacademy.com/kenneth/articles/kansara/kansara.htm

This article appeared in the Indian Journal ‘Sambodhi’ Vol. XXIII, 2000 and is reproduced here with permission. Diacritical marks have not been copied, nor have the few Sanskrit/Hindu lines of text. This article has been scanned into text and though we have done our best there may still be a few errors. For the full text please see the Journal version. VEDIC SOURCES OF THE 'VEDIC MATHEMATICS' Dr. N. M. Kansara Director, Akshardham Centre for Applied Research in Social Harmony (AARSH), Akshardham, Gandhinagar - (382 020) Jagadguru Shankaracharya Swami Shri Bharati Krishna Tirthaji Maharaja of Govardhan Peeth Matha, Puri, wrote or dictated a book entitled `Vedic Mathematics' based on 29 Sutras, of which 16 deal with the ‘general case’, while the rest 13 treat the special cases. The declaration of the Sutras as "Vedic" or as belonging to the Vedas, particularly to the Atharvaveda, and his claim that "the Sutras (aphorisms) apply to and cover each and every part of each and every chapter of each and every branch of mathematics including arithmetic, algebra, geometry ­plane and solid, trigonometry - plane and spherical, conics - geometrical and analytical, astronomy, calculus - differential and integral etc., etc.", and that "there is no part of mathematics, pure or applied, which is beyond their jurisdiction" has raised a controversy amongst the mathematicians of India, some of whom have questioned the Vedicity of the Sutras on the ground of their language, and the level of mathematics it deals with. It is endeavoured here to deal with the problem in all possible aspects, and examine the validity or otherwise of the claim.

27. 8 II. Aryabhata And His Commentators Next page (8 iiI. Brahmagupta, and the influence on Arabia). 8 ii.aryabhata and his commentators. aryabhata, who is occasionally http://www-history.mcs.st-andrews.ac.uk/history/Projects/Pearce/Chapters/Ch8_2.h

Indian Mathematics MacTutor Index Previous page (8: The Classical period: I Introduction) Contents Next page (8 III. Brahmagupta, and the influence on Arabia) 8 II. Aryabhata and his commentators Aryabhata , who is occasionally known as Aryabhata I, or Aryabhata the elder to distinguish him from a tenth century astronomer of the same name, stands as a pioneer of the revival of Indian mathematics, and the so called 'classical period ', or 'Golden era' of Indian mathematics. Arguably the Classical period continued until the 12 th century, although in some respects it was over before Aryabhata 's death following a costly, if ultimately successful, war with invading Huns which resulted in the eroding of the Gupta culture (D Duncan P 171). As mentioned, the classical period arose following a 'dark period' of significant political instability 200-400 AD, which caused the widespread stagnation of mathematical development. We can accurately claim that Aryabhata was born in 476 AD, as he writes that he was 23 years old when he wrote his most significant mathematical work the Aryabhatiya (or Arya Bhateeya ) in 499 AD. He was a member of the Kusuma Pura School, but is thought to have been a native of Kerala (in the extreme south of India), although unsurprisingly there is some debate. Further debate surrounds how important the work of

28. Syndex II A Synergetic Revisioning Of Number Dynamics In Light Of Ancient Metrol 500 AD aryabhata argues for a moveable and rotating earth First brought to Europeby Moors; introduced by Gerbert of Aurillac (Pope Sylvester ii), about 1000 AD http://www.geocities.com/ResearchTriangle/Campus/7107/syn2intr.html

INTRODUCTION to the Philosophical and Scientific Background "Number makes its appearance in this context as the vinculum amoris , the bond of love which unites the two principles [psyche and matter] by jointly ordering them. In its profoundest sense number thus possesses the significance of an all-uniting Eros, although it connotes something different from the usual sense of the words love and Eros...Because there seems to exist such a clear spiritual "objective" order at the base of Eros, it is expressed in the seemingly abstract, feelingless, impersonal order of numbers, as a clear, immutable factor free from illusions...this cosmic ordering of the Self constitutes the ultimate mystery behind all human desire and behavior, an unfathomable and fearsome mystery." M.L. von Franz, Number and Time "The unexpectedness of a mathematical result gives us the feeling that it is not our own creation, that the world of number exists in its own right, while its necessity and symmetry are balm after the ragged edges of life, or pure joy to those who do not yet know them. The appeal of mathematical form reaches deep into human character." L. L. Whyte

29. Philosophengallery Translate this page PHILOSOPHENGALLERY ii de Gournay Die Mathematiker vlnr Gauss, Boole, Cantor frühereAhmes, Euklid, Eratosthenes, Diophantos, aryabhata, Brahmagupta, Al http://www.schepart.ch/mho/Philosophie/Philosophengallery/Philosophengallery2.ht

P H I L O S O P H E N G A L L E R Y I I TIMELINE: 17.-19. JAHRHUNDERT Die Empiristen und Rationalisten * v.l.n.r. F. Bacon, Descartes, Spinoza, Locke, Leibniz, Wolff, Hume * Die Tradition der Philosophiegeschichte sieht die Empiristen und Rationalisten im strengen Gegensatz, Als Empiristen gelten F. Bacon, Locke und Hume, als Rationalisten Descartes, Spinoza, Leibniz und Wolff. [Kritischer Rationalismus (20.Jh.): Popper, Albert, Lakatos] Die Naturwissenschaftler Comenius Die Okkasionalisten v.l.n.r. Geulincx, de Malebranche v.l.n.r. Bayle, Montesquieu, Voltaire, Reid, Rousseau, Diderot, Kant, Lessing [weitere: Thomasius, Shaftesbury, Reimarus, Gottsched, Crusius, d'Alembert, Jacobi, Knigge; Neukantianer: Liebmann, Cohen, Natorp, Cassirer, Lask] Der Historist Vico Die Idealisten und Romantiker v.l.n.r. Berkeley, Fichte, Hegel, Schlegel, Schelling Der Theosoph * Swedenborg Die Materialisten v.l.n.r. Smith, Bentham, Malthus, Ricardo, Mill Die Feministin Wollstonecraft Die Mathematiker v.l.n.r. Gauss, Boole, Cantor Die Transzendentalisten v.l.n.r. Channing, Emerson

30. Bhaskara II --  Britannica Student Encyclopedia of aryabhata I (born 476). More results . 100 Student Encyclopedia Britannica articles,specially written for elementary and high school students. , Bhaskara ii http://www.britannica.com/ebi/article?eu=342278&query=decimal number system&ct=e

31. SATELLITE TECHNOLOGY DEVELOPMENT aryabhata carried three payloads one each for X-ray, astronomy, solar physics and Bhaskara-I ii Bhaskara - I and ii were the first two experimental remote http://ceos.cnes.fr:8100/cdrom-00b2/ceos1/isro/abisro/old_sat.htm

S ATELLITE TECHNOLOGY DEVELOPMENT Aryabhata: Aryabhata was the first Indian satellite launched into a near earth orbit on April 19, 1975 by an Intercosmos rocket of the erstwhile USSR. Aryabhata carried three payloads - one each for X-ray, astronomy, solar physics and aeronomy. The mission was a success and provided the early experience to ISRO for planning, developing and managing a satellite mission. Bhaskara - I and II were the first two experimental remote sensing satellites launched on June 7, 1979 and November 20, 1981 respectively by Intercosmos rockets. Both the satellites carried two TV cameras, one in visible and the other in near-infrared band and having a 1 Km resolution and a 3-frequency Passive Microwave Radiometer. The Bhaskara mission saw the birth of a systematic ground segment and applications programme. Together with the capabilities developed in satellite and sensor development, a truly indigenous space-capability was the result of the Bhaskara mission. Rohini: A series of technological/scientific satellites, launched by India's own launch vehicle, SLV-3. First one was used to measure the performance of SLV-3, second and third carried a SMARTsensor - the first Charge Coupled Device (CCD) camera developed in-house. Rohini saw the entry of ISRO to the use of CCD technology and system design for sensors.

32. Ariel Foo Project iiOwen Gingerich, Islamic Astronomy, April 1986, http//home.naxs.com/hiro ofÀryabhata, http//www.geocities.com/Athens/Olympus/4844/aryabhata.html . http://courses.nus.edu.sg/course/phibalas/dialogue2001/Contribute/Ariel Foo Proj

33. Bineesha Project CCSP04 Scientific Method and Strategies of Critical and Creative Thinking (Project Semester ii, 2000-2001). The Indian astronomer aryabhata is regarded as http://courses.nus.edu.sg/course/phibalas/dialogue2001/Contribute/Bineesha proje

34. Old Satellites aryabhata First Indian satellite, launched into a near earth orbit on April 19 BhaskaraI ii Experimental remote sensing satellites launched on June 7, 1979 http://www.isro.org/old_sat.htm

About ISRO ISRO Centres Programmes Decade Plan ... Press Release Satellites Aryabhata First Indian satellite, launched into a near earth orbit on April 19, 1975, by an Intercosmos rocket of erstwhile USSR. Carried three payloads, one each for X-ray astronomy, solar physics and aeronomy. Experimental remote sensing satellites launched on June 7, 1979 and November 20, 1981 'respectively' by Intercosmos rockets. Carried two TV cameras, one in visible and the other in near-infrared band, and a 3-frequency passive microwave radiometer. APPLE (Ariane Passenger Payload Experiment) An experimental communication satellite with a C-band transponder, launched on June 19, 1981 by an Ariane launch vehicle of European Space Agency (ESA). Used to conduct several communication experiments. Rohini A series of technological / scientific satellites, launched by India's own launch vehicle, SLV-3. First one was used to measure the performance of SLV-3, second and third carried land-mark sensor payloads. Stretched Rohini Series Satellites (SROSS-C and SROSS-C2) Launched by India's Augmented Satellite Launch Vehicle, ASLV, on May 20, 1992 and May 4, 1994 respectively. Carried a Retarding Potential Analyser and a Gamma Ray Burst (GRB) detector. SROSS-C2 is still providing valuable scientific data.

35. SIT-Dept. Of CS&E-Facilities aryabhata COMPUTER CENTRE. HARDWARE. DESKTOP SYSTEMS. (i) Pentium iiI @500 Mhz,Asus Board, 64 MB RAM, 8.4 GB HDD, 15” Color Monitor. (ii) Pentium ii @333 Mhz http://www.sit.ac.in/Departments/cse/facilities.html

Programmes Faculty Profile Supporting Staff Profile Facilities Computer/Internet access E-Books SiGMA Laboratory Resources BHASKARA COMPUTER CENTRE HARDWARE DESKTOP SYSTEMS (i) Pentium – IV 1.5 Ghz, 128 MB RDRAM, 40GB HDD, 15” Color Monitor (ii) Intel Celeron @1.2 Ghz, 128 MB SD RAM, 40 GB HDD, 15” Color Monitor WORKSTATION SUN BLADE 150 @650 Mhz, 733 Mhz PCI II Co-processor card, 256 MB RAM, 40 GB HDD, 21” Color Monitor Epson Dot matrix Ethernet Hub – 24 Port, HCL Switch – 24 Port, Hub Rack – 22U - 1 No. each SOFTWARE Operating System (OS) / Platform Solaris 8.0 Red Hat Linux 7.2 Compilers / Languages / Application GNU C, G++, Star Office Electric Power Resources UPS, Electric Panel Fire Extinguisher Air Conditioners, Fans, Precautionary Equipments ARYABHATA COMPUTER CENTRE HARDWARE DESKTOP SYSTEMS (i) Pentium III @500 Mhz, Asus Board, 64 MB RAM, 8.4 GB HDD, 15” Color Monitor

36. Mth1124 Engineering Calculus II c. The mathematician Qin Jiushao. 2. Use the algorithms of aryabhata and Brahmaguptato find all solutions to the. indeterminate linear equation 31x + 24y = 7. http://www.math.neu.edu/~gilmore/U201sp04files/201q3'.htm

History of Mathematics Quiz #3 Professor Gilmore 40 Minutes Mar. 17, 2004 Name Show All Your Work Write a paragraph on each of the following topics. Each paragraph must contain at least four distinct facts about the topic indicated: Continue your paragraphs on the back of this sheet if necessary a. The ancient Yangshao-Longshan culture. b. The Jiuzhang Suanshu. c. The mathematician Qin Jiushao. . Use the algorithms of Aryabhata and Brahmagupta to find all solutions to the indeterminate linear equation 31x + 24y = 7

37. Kuttaka 23 + 137m, for m an integer. Bhaskara ii later made a modificationto aryabhata s Kuttaka. Go back to the Indian History timeline. http://www.math.sfu.ca/histmath/India/5thCenturyAD/Kuttaka.html

Aryabhata's Kuttaka Aryabhata 's solution of the equation ax-by=c was known as the Kuttaka (pulverizer) in Hindu mathematics. The equation ax-by=c, arose as a result of the following: It is required to determine an integer N which when divided by a leaves a remainder r' and when divided by b leaves a remainder r''. From this we get: N = xa + r', N = yb + r''. By equating the first equation with the second we get xa + r' = yb + r'' or ax-by = c for c = r''-r'. Aryabhata noted that any factor common to a and b should be a factor of c, otherwise the equation has no solution. Dividing a, b and c by the greatest common factor of (a,b) we can reduce the equation to the form where a and b are relatively prime. In the discussion that follows we can assume that (a,b)=1 and without loss of generality c>0. Aryabhata's method of solution to the equation ax-by=c is given in stanza 32 and 33 of his Aryabhatiya. The following translation of his method was by Bhaskara I, who was a pupil of Aryabhata's teachings. Bhaskara I added some steps in the translation if it was missing in the original.

38. Science After Aryabhatta aryabhata (born 476) is the author of the first of the later siddhantas chapters(i) the astronomical constants and the sine table, (ii) mathematics required http://www.hindunet.org/science_after_aryabhatta/

Sciences Yoga Ayurveda Indian Medicine ... Sciences Science After Aryabhatta Explore this topic in detail Recommend this page Comment on this page Print this page Create / Join Club Hindu Universe Links Articles Online Books Hindu Web Discussion Book- store Related sections In the earliest period of Indian science, it is exceptional when we know the authorship of a text or an idea. For example, although Yajnavalkya and Lagadha describe considerable astronomy, we do not know if this was developed by them or they merely summarized what was then well known. Likewise we are not sure of the individual contributions in the Shulba Sutras- of Baudhayana, Apastamba, and other authors- which describe geometry, or in Pingala's Chhandahsutra which shows how to count in a binary manner. The major exception to the anonymous nature of early Indian science is the grammatical tradition starting with Panini. This tradition is an application of the scientific method where the infinite variety of linguistic data is generated by means of a limited number of rules. With Aryabhata, we enter a new phase in which it becomes easier to trace the authorship of specific ideas. But even here there remain other aspects which are not so well understood. For example, the evolution of Indian medicine is not as well documented as that of Indian mathematics. Neither do we understand well the manner in which the philosophical basis underlying Indian science evolved.

39. ISRO Success Stories BhaskaraI and ii, which were evolved from aryabhata and carrying remote sensorsin the visible, infrared and microwave region of the electromagnetic spectrum http://www.flonnet.com/fl1809/18090180.htm

Volume 18 - Issue 09, Apr. 28 - May 11, 2001 India's National Magazine from the publishers of THE HINDU Table of Contents COVER STORY ISRO success stories The launch of the GSLV marks the beginning of a significant phase in launch vehicle technology development for the Indian Space Research Organisation, which already has an impressive record in the field of sophisticated satellite technology. R. RAMACHANDRAN FROM building the first experimental satellite Aryabhata in 1975 to the world class operational Indian Remote Sensing (IRS) satellite series on the one hand and the third generation communication satellite INSAT-3 on the other, is indeed an impressive track record by any standards for the 30-year-old Indian Space Research Organisation (ISRO). It could well be argued that the development of launch vehicle technology, which began in the mid-1960s, has not achieved the same degree of success. Indeed, many, both within the organisation and without, believe that the launch vehicle front did not receive the same kind of focus as satellite development did, particularly after Satish Dhawan retired from ISRO in 1984. At a time when ISRO's launch vehicle development has reached an important phase with the launch of the Geosynchronous Satellite Launch Vehicle, it would be appropriate to put satellite technology development also in perspective. At the launch of the first Rohini-75 rocket at Thumba, 1969.

40. A Timeline Of Ancient India in the rock at Bamiya, Bactria (Afghanistan) 390 Chandra Gupta ii extends the Gupta workat the Ajanta caves 499 the Hindu mathematician aryabhata writes the http://www.scaruffi.com/politics/indians.html

A time-line of ancient India World News Politics History Editor See a timeline of the Near East 5000 BC : the Kurgan culture in the steppes west of the Ural Mountains (Indo-Aryans) 3120 BC : mythical Indian war of the Mahabarata 3000 BC : the proto-indo-european language develops in Central Asia 3000 BC : Dravidian speaking people develop the civilization of the Indus Valley 2500 BC : the cities of Harappa and Mohenjo-Daro in the Indus Valley 2000 BC : the civilization of the Indus Valley declines 2000 BC : the Kurgan culture spreads to eastern Europe and northern Iran 1700 BC : Indo-Iranians separate from the other Indo-European tribes and migrate eastward to settle in Iran 1600 BC : Indo-Aryans invade India from the west and expel the Dravidians 1500 BC : religious texts are written in Vedic, an Indo-European language 1100 BC : the Indo-Aryans use iron tools 1000 BC : the Rig-Veda are composed 900 BC : Indo-Aryans discover iron and invade the Ganges Valley 750 BC : Indo-Aryans rule over 16 mahajanapadas ("great states") in northern India, from the Indus to the Ganges 700 BC : the caste system emerges, with the Brahman priests at the top

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