21. IANDANAE: (Greek) Papyri With Accounts, Math 6.30. Selenius, ClasOlof. “Rationale of the Chaklavala Process of Jayadevaand bhaskara ii”. Historia Mathematica 2 (1975), 167–184. 6.31. http://www.mathorigins.com/I.htm

Home Color Guide Abbreviation Guide Personal Library Master key ... Y-Z IANDANAE: (Greek) papyri with accounts, math (as per E. G. Turner) P. Iand. = Papyri Iandanae, cum discipulis edidit C. Kalbfleisch, Leipzig, 1912. Pt i, Voluminum Codicumque Fragmenta Graeca cum Amuleto Christiano, ed. E. Schaefer, 1912. Pt. ii, Epistulae Privatae Graecae, ed. L. Eisner, 1913. Pt iii, Instrumenta, Graeca Publica et Privata, pt. i, ed. L. Spohr, 1913. Pt iv, Instrumenta, Graeca Publica et Privata, pt. ii, ed. G. Speiss, 1914. Pt v, Literarische Stucke und Verwandtes, ed. J. Sprey, 1931. Pt. vi, Greichische Privatbriefe, ed. G. Rosenberger, 1934. Pt vii, Greichische Verwaltungsurkunden, ed. D, Curschmann, 1934. Pt. viii, Greichische Wirtschaftsrechnungen und Verwandtes. ed. J. Hummel, 1938. P. Iand . Inv. 653 = A Sixth Century Account of Hay, ed. T. Reekmans, Brussels, 1962. P. Iand . 2.8. Letter from Ischyriôn to Antôninos: (Greek; Arsinoite) http://perseus.csad.ox.ac.uk/cgi-bin/ptext?doc=Perseus%3Atext%3A1999.05.0138 P.Iand.inv. 653: A Sixth Century Account of Hay: (Greek) With dated accounts and inventory?

22. 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"

23. 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. A detailed version of the proof, for those who do not feel up to the challenge, appears in Alan M. Selby 's Appetizers and Lessons for Math and Reason (although he only uses the righthand picture with algebra, not geometry, to prove the required identity). 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.) 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

24. The Indian Road To Space APPLE, India's first experimental geosynchronous satellite, 1981. bhaskara iilaunched, 1981. INSAT IA launched, 1982. INSAT IB successfully launched, 1983. http://www.spaceengineer-india.org/indian_road_to_space.htm

The Indian Road to Space... Indian Space research programme is one of the most remarkable in its kind. With relatively small budget allocations compared to other space faring nations, India was able to achieve space self reliance. Some important milestones of Indian space programme are listed below. Only the major events are included.... Click here to have a look at the Indian Launchers... Indian National Space Committee for Space Research (INCOSPAR) formed by the Dept. of Atomic energy. First Sounding Rocket Launch from Thumba Equatorial Rocket Launching Station (TERLS) TERLS dedicated to the United Nations ISRO is formed under Dpt. of Atomic energy Formation of Space Commission and Dpt. of space First Indian satellite, Aryabhatta, launched (from USSR) Bhaskara I, experimental remote sensing satellite First Experimental launch of SLV-3, India's own launch vehicle Successful Launch of SLV-3 , with Rohini satellite APPLE, India's first experimental geosynchronous satellite

25. Amy Debner 8 18 381 5381 45381. A person who was very influential to arithmetic operationswas a mathematician named bhaskara ii. bhaskara ii. Encyclopedia Britannica. http://www.sienahts.edu/~adebner/mathstuff/arith.htm

Amy Debner Toni Carroll Arithmetic Operations There are several ideas and people that played a major role in developing arithmetic operations through the early history. Treviso Arithmetic was the earliest known printed arithmetic book. It appeared, in 1478, even before the printed edition of Euclid’s Elements The original book was written in Venetian dialect, had no formal title, and it has an unknown author. The reason it is named Treviso Arithmetic is because Treviso is where the book originated. The great thing about this mathematics book was that it was aimed at a broad audience and it was intended for individual study. Treviso Arithmetic was a very practical book because Venice and Treviso were major trading centers during the 15 th Century. “The book’s language, examples, and problems reflected a wide range of commercial concerns” (Peterson, 1996). And it “contained all that was known in that day of arithmetic, algebra, and trigonometry” (Struik, 1967). Treviso Arithmetic “introduced a ‘new math’, promoting the use of Hindu-Arabic numeral system and the pen-and-ink computational algorithms that accompanied this notion” (Peterson, 1996). This is a problem that appeared in the Treviso Arithmetic textbook: “Three merchants have invested their money in a partnership.

26. 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.

27. Sulbasutra Geometry is somewhat unclear. Also important are Bhaskara I (c. 522 CE), Bramhagupta(628 CE) and bhaskara ii (b. 1114 CE). In this period http://www.math.ubc.ca/~cass/courses/m309-01a/kong/sulbasutra_geometry.htm

Sulbasutra Geometry For Prof. W. Casselman By Susanna Kong Math 309 April 2001 Introduction The basis and inspiration for the whole of Indian mathematics is geometry. The beginnings of algebra can be traced to the constructional geometry of the Vedic priests preserved in the Sulbasutras , a manual of geometrical constructions from the 5 th to the 8 th centuries. Earlier remnants of geometrical knowledge of the Indus Valley Civilisation can be found in excavations at Harappa and Mohenjo-daro where there is evidence of circle-drawing instruments from as early as 2500 B.C.E. (Amma 1) Early geometry of the Sulbasutras was based on religious needs with regards to the construction of altars such as agnicitis, vedis, mandapas etc. that are required for sacrificial ritual (Kulkarni 19, Amma 3). The history of Indian geometry can be divided into three distinct periods: pre-Aryan, such as excavated in Indus Valley. Vedic or Sulbasutra post-Christian. Evidence of the pre-Aryan period includes well-planned towns and geometric designs including circles, squares and triangles. A link between this and the Vedic period can be found in the motif of a rectangle with the four sides curved inwards resembling a stretched hide; in the former period it can be seen as a decorative pattern while in the later period it is seen in the shape of the sacrificial altars or vedi . The vedi as well as the fireplaces or agni had such exact measurements and geometric shapes that they were codified and became the Sulbasutras . However, it is not known from how far back such knowledge originated as the sacrificial act is as old as the Vedas or older (Amma 5).

28. QUERIES ON ORIENTAL SOURCES IN RECREATIONAL MATHEMATICS By David Singmaster Both versions are common throughout medieval European mathematics and some occurin the Chiu Chang Suan Ching (c150), in Sridhara (c900) and bhaskara ii (1150 http://anduin.eldar.org/~problemi/singmast/mideastr.html

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 QUERIES ON MIDDLE-EASTERN SOURCES IN RECREATIONAL MATHEMATICS by David Singmaster 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 Receive email when this page changes Click Here Powered by Netmind Resources provided by Brad Spencer

29. Encyclopædia Britannica bhaskara ii the leading mathematician of the 12th century, who wrote thefirst work with full and systematic use of the decimal number system. http://search.britannica.com/search?query=x2&ct=eb&fuzzy=N&show=10&start=18

30. îÏ×ÁÑ áÓÔÒÏÌÏÇÉÞÅÓËÁÑ üÎÃÉËÌÏÐÅÄÉÑ etogo uch¸nogo Bhaskara I, poskol'ku neskol'kimi stoletiyami pozzhe nego zhil irabotal drugoi uch¸nyi, nosivshii takoe zhe imya (bhaskara ii; 1115 ?) i http://encyclopedia.astrologer.ru:8005/cgi-bin/guard/B/bhaskara.html

31. Winners This type of equations was first solved by Brahmagupta and BhaskaraII. Later, bhaskara ii (AD 12th century) simplified this method. http://www.angelfire.com/ak/ashoksandhya/winners2.html

They Answered Them... Solution to Puzzle 3 The following solution is sent by Umesh PN Each one contributed , amounting to $360. Ashok paid $9 extra, so total is $169; 3 frustrated people left. the remaining people ate for $41 each. Basically, you want me to solve the equation in integers . Look at that equation. 12n is divisible by 3, 9 is divisible 3, so 41m also must be divisible by 3, which means m is divisible by 3. Putting m = 3k, we get a simpler equation Such equations are called indeterminate or Diophantine equations and there are many methods to solve them. I give two of them. Correct Answer was sent by : Kishore M., CA, USA Umesh PN, IL, USA Balaji R, CA, USA Solution 1: I am using the symbol == to denote "is congruent to" as there is no symbol for congruence in standard ASCII set. So, 41k == 3 (mod 4) 4n == 3 (mod 41) Since 41 and 4 are mutually prime, this is soluble by considering the complete residue system. Thus, by putting k = 0, 1, 2, 3, we get 41k is congruent to 0, 1, 2, 3 respectively (mod 4). . So k = 3 is a solution to the congruence 41k == 3 (mod 4). So k = 3 , which means n and m Solution 2: To solve Express (41/4) as a continued fraction in such a way that there are an odd number of terms in the expansion. Now, leave the last term and evaluate the fraction. Let it be (a/b). Now 41b - 4a = 1. Multiplying all the terms of the equation by 3, we get the solution to 41k - 4n = 3. The continued fraction expansion of (41/4) is 10 + (1/4), giving the values (10/1) and (41/4). The penultimate fraction is (10/1). So 41.1 - 4.10 = 1. Multiplying it by 3, we get 41.3 - 4.30 = 3. Thus

32. SCIAMVS: Volume 3 Takao Hayashi. Notes on the Differences between the Two Recensions of the Lilavatiof bhaskara ii .. http://plaza.harmonix.ne.jp/~sciamvs/vol_03.html

Last update August 12, 2002 HOME SCIAMVS Volume 3 is now Available Contents of Volume 3 Contents in PDF File (24KB) Contents + Editorial + Title page of each article in PDF File (257KB) Editorial ..............................................................................................1 in PDF File (28KB) Lis Brack-Bernsen and Hermann Hunger. TU 11: A Collection of Rules for the Prediction of Lunar Phases and of Month Lengths........................................3 (with photos of the tablet TU 11) Title Page in PDF File (22KB) Charles Burnett. The Abacus at Echternach in ca. 1000 A.D..................................91 (with photos of the abacus) Title Page in PDF File (46KB) Reviel Netz, Ken Saito and Natalie Tchernetska. A New Reading of Method Proposition 14: Preliminary Evidence from the Archimedes Palimpsest (Part 2)....109 (with photos of a part of the palimpsest) Title Page in PDF File (49KB) Ken'ichi Takahashi, Takako Mori and Youhei Kikuchihara. A Paraphrased Latin Version of Euclid's Optica A Text of De visu in MS Add.17368

33. Timelinescience - 1101 To 1200 bhaskara ii, an Indian mathematician, modifies a 5th century idea from Sanskritwritings to describe a wheel which he claims will run indefinitely an early http://www.timelinescience.org/years/1200.htm

1101 to 1200 Setting the scene Islamic culture is the most advanced in the western world. Many scientific and mathematical terms (eg "algebra" and "algorithm") are of Arabic origin, reflecting their roots in these early days of recorded science. The Islamic empire is vast, and much of its success is down to trade and commerce. Many countries become part of the Islamic empire and many others trade with it, so there is an input to scientific ideas from many different cultures including Iran, Turkey, India and China. The Arabic language becomes a unifying factor allowing ideas to be exchanged freely, and centres of learning and wisdom arise in a number of places, including Baghdad, Al-Ma'mum and Cordoba in Spain. Many areas of science and mathematics move forward during these years. Increasingly accurate astronomical observations are made, and mathematics benefits enormously from the introduction of Indian numerals - referred to today as Arabic numerals. With these numerals great strides are made in solving equations (algebra), trigonometry and numerical calculations. Chemistry becomes an experimental subject at last, as does physics. And health care is comprehensive, with doctors, hospitals and even special care for the mentally ill.

34. A/AC.105/INF.390 Remarks Information on bhaskara ii. Avaliable Format Adobe PDF. AvaliableLanguages Arabic; Chinese; English; French; Russian; Spanish. http://www.oosa.unvienna.org/Reports/inf390.html

Online Index of Objects Launched into Outer Space: Reports Document Symbol: A/AC.105/INF.390 Notifying State/Organization: India Main Title: Information Furnished in Conformity with General Assembly Resolution 1721 B (XVI) by States Launching Objects into Orbit or Beyond Sub Title: Letter dated 5 January 1982 from the Permanent Representative of India to the United Nations addressed to the Secretary-General Remarks: Information on Bhaskara II Avaliable Format: Adobe PDF Avaliable Languages: Arabic Chinese English French Russian Spanish Online Index Space Law COPUOS SAP ... Home

35. Pell'sche Gleichung Translate this page bhaskara ii (1114-1191) beschrieb als erster eine allgemeine Methode ur Lösungder Pell'schen Gleichung (ohne Beweis - den Beweis führte Lagrange). http://www.mathematik.uni-bielefeld.de/~ringel/vortrag/pell.htm

Differenzen von Quadraten Wie konstruiert man das geometrische Mittel zweier Zahlen a,b? Gesucht ist also x mit a : x = x : b, also x mit x ab = ((a+b)/2) - ((a-b)/2) und verwendet nun Pythagoras, um aus der Differenz zweier Quadratzahlen die Wurzel zu ziehen. Statt von einer Quadratzahl x eine andere Quadratzahl y nur einmal abzuziehen, betrachten wir nun allgemeiner Differenzen der Form x - dy Die Pell'sche Gleichung Dies ist die Pell'sche Gleichung: x - dy Einige Namen John Pell (1610-1685) nach ihm ist die Gleichung benannt - er hat aber damit nichts zu tun! Archimedes (287-212 v.u.Z.): auf ihn soll noch eingegangen werden! Brahmagupta (598-670) schrieb 628 das Buch Brahmasphutasiddhanta , in dem er viele Pell'sche Gleichungen analysierte. Zur Theorie: Satz: - dy n = x n +y n n ,y n n x y Von Brahmagupta stammt der Satz: = 1 findet, ist ein Mathematiker. Das Rinder-Problem des Archimedes Farbe schwarz gescheckt braun Stiere x y z t x' y' z' t' Die ersten Bedingungen: x = (1/2+1/3)y+t y = (1/4+1/5)z+t z = (1/6+1/7)x+t Die zweiten Bedingungen: x' = (1/3+1/4)(y+y') y' = (1/4+1/5)(z+z') z' = (1/5+1/6)(t+t') t' = (1/6+1/7)(x+x') (x,y,z,t) = m(2226,1602,1580,891)

36. CHRONOLOGY OF RECREATIONAL MATHEMATICS By David Singmaster 1141 Abu Ishaq first recorded Arabic Knight's tour, possibly due to alAdlior as-Suli. 1150 bhaskara ii Lilivati Bijaganita. 1150 ibn Ezra. http://www.geocities.com/SiliconValley/9174/recchron.html

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 CHRONOLOGY OF RECREATIONAL MATHEMATICS by David Singmaster WWW page processed by Mario VELUCCHI (velucchi@cli.di.unipi.it) with the consent of David Singmaster

37. Newsletter 46, March 2001 century) in his Ganitatilaka (rule 45) gives a ÷ 0 = 0 4. bhaskara ii in his famousLilavati (12th century) gives the wrong rule (a x 0) = a. His commentator http://www.hpm-americas.org/nl46/nl46art2.html

International Study Group on the Relations Between HISTORY and PEDAGOGY of MATHEMATICS NEWSLETTER No. 46, March 2001 An Affiliate of the International Commission on Mathematical Instruction The Dangerous Hole of Zero History makes a man wise is a common saying. By studying history we can know the errors and mistakes committed in the past and save ourselves from repeating them. According to P. S. Jones "One use of the history of mathematics is to reveal to students come of the conceptual difficulties and errors which have impeded progress". G. A. Miller even says "The teachers of mathematics may frequently gain more from a clear exposition of failures than from such an exposition of successes on the part of the eminent mathematicians of the past". In this brief note we mention the mistakes, gathered from a few earlier works, in connection with some arithmetical operations involving the number zero (now denoted by the hole "0"). 1. The great Indian mathematician Brahmagupta (7th century AD) was the first to give explicitly in his Brahmasphuta-Siddhanta (chapter XVIII), the various rules involving zero (in arithmetical operations) but they also include his statement that "zero divided by zero is zero". That is, ÷ = which is not true in general. 2. The Ganitasara-sangraha (I, 49) of the Jaina mathematician Mahavira (9th century) contains a ÷ = a

38. Newsletter 44, November 2000: History And Culture In Mathematics Education 4. bhaskara ii in his famous Lilavati (12 th century) gives the wrong rule (ax 0)/0 = a His commentator Ganesa (1545) remarks that the rule comes by http://www.hpm-americas.org/nl47/nl47apologies.html

International Study Group on the Relations Between HISTORY and PEDAGOGY of MATHEMATICS NEWSLETTER An Affiliate of the International Commission on Mathematical Instruction: No. 47, July 2001 Apologies My apologies, but in the last HPM Newsletter there were two mistakes that I failed to notice before sending it to distributors. These were both in the article The Dangerous Hole of Zero Firstly I forgot to acknowledge the author. It was Professor R. C. Gupta from India who submitted the article, and my sincere apologies to him and all the HPM Newsletter readers for this oversight. Secondly in example 4 the “wrong rule” was misprinted. The correct paragraph is shown here. 4. Bhaskara II in his famous Lilavati th century) gives the wrong rule ( a x a His commentator Ganesa (1545) remarks that the rule comes by cancelling zero from the numerator and denominator! Return home. Web design by Dr. Katie Ambruso and maintained by Andrew K

39. HISTORIA MATHEMATICA VOLUME 2, PAGES 127252, MAY 1975 161166 Rationale of the Chakravala process of Jayadeva and bhaskara iiClasOlof Selenius .. http://www.chass.utoronto.ca/hm/table/02127252.html

Volume Index Previous VOLUME 2, PAGES 127252, MAY 1975 SOURCES ............................................................... 200202 The mathematical papers and library of Sir Edward Collingwood in the University of Durham I. Grattan-Guinness ............................................. 200202 Archives of mathematical journals J. D. Gray ........................................................... 202 BOOK REVIEWS Lazare Carnot Savant Africa Counts by Claudia Zaslavsky (R. W. Wilder) ................................................... 207-210 Statistical Papers in Honor of George W. Snedecor by T. A. Bancroft (Churchill Eisenhart) ........................................... 211218 Ming Kan No Syuzan Syo Euclid and his modern rivals by Lewis Carroll (Daniel Pedoe) .................................................. 219222 Joseph Fourier 17681830 by I. Grattan-Guinness (J. W. Herivel) ................................................. 222223 ABSTRACTS Next Back to Historia Mathematica homepage.

40. About State Observatory Nainital The works of Aryabhatta I ( born 476 AD) Varahmihir ( died 587 AD), Brahmagupta(born 598 AD) and bhaskara ii ( born 1114 AD) are still looked upon with http://upso.ernet.in/intro.html

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