DOCSLIB.ORG

Jtor Y of HINDU MATHEMATICS

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Jtor Y of HINDU MATHEMATICS HI.:jTOR Y OF HINDU MATHEMATICS A SOURCE BOOK PARTS I AND II BY BIBHUTIBHUSAN DATTA AND AVADHESH NARAYAN SINGH ASIA PUBLISHING HOUSE BOMBAY CALCUTTA NEW DELHI: MADRAS· LONDON NEW YORK @ 1935, 1938 AVADHESH NARAYAN SINGH Part I First Published: 1935 Part II First Published: 1938 Single Volume Edition: 1962 All Rights Reserved PART I: pp. 1-250 PART II: pp. 1-308 PRINTED IN INDrA BY SOMESHWAR DAYAL AT THE MUDRAN KALA MANDm. LVCKNOW AND PUBLISHED BY P. S. JAYASINGHE, ASU. PUBLISHING HOUSE, BOMBAY TRANSLITERATION VOWELS Short: ~ ~ ~ ~ ~ a I u r ! Long: om i 3: ~ ~ Q; ~ ~) ~( ... a i u .r 1 c ai 0 all Anusvara: ":"=m Visarga: := h , Non-aspirant: s= , CONSONANTS Classified : CJ) ~ iT 'q :g: " .. " ..... k kh g gh Ii :q --.. Et ~--.. ~--.. 01--.. c ch J jh Ii c 0- G G ~ ... "' --.. "' " ! !h (I (lh !1 tt I!.f ;;: \:l ~ '" '"' t th d dh n q 'I:fi ijJ ~ t{ '" '" -- -- P ph b bh m, f!1 = final q_ Un-classified: ~ ~ c:!i", q Q' ~ it, '"' '"' .... ~'" "' '" y r I v J ! s h Compound: ~-.. '31' ~.... "' k! tr jii Pall : i'li = I' LIST OF ABBREVIATIONS A Aryabhatia A] Ar~a-Jyoti~a ApSI Apastamba Sulba AV Atharvaveda BBi BijagaQ.ita of B~askara II BCMS Bulletin of the Calcutta Mathematical Society BMs Bakhsh3.li Manuscript BrSpSi Brahma-sphuta-siddhanta BSI Baudhlyana 5ulba DhC,. Dhy:1nagrahopadda EI Epigraphia Indica GK - GaQ.ita-kaumudi GL Geaha-Iaghava GSS GaQ.ita-sara-samgraha GT GaQ.ita-tilaka IA Indian Antiquary IHQ Indian Historical Quarterly ]A Journal Asiatique JASB J ouenal of the Asiatic Society of Bengal ]IMS Journal of the Indian Mathematical Society ]RAS Journal of the Royal Asiatic Society of Great Britain and Ireland KapS Kap,isthala SaIhhita KK KhaQc;la~khadyaka til Katyayana 5ulba KtS Ka~haka Sarilhita L Uiavati LBh Laghu-Bhaskariya L1ST OF ABBREVIATIONS LMa Laghu-minasa MaiS MaitrayaQi Samhita MaSJ Manava Sulba MBh Maha-Bhaskariya MSi Maha-siddMnta NBi BijagaQ.ita of NarayaQ.a PdSa Pitt-sara of Munisvara PLM Pradna-lipi-mala PSi Pafica-siddhantild RV ~gveda SEr Satapatha Brahmal,1a SmVr Si~Y!l-dhl-vrddhida SiSe Siddhanta-sekhara SiS; Siddhanta-siromani SiT Vi Siddhanta-tattVa-viveka SuSi Surya-siddhinta TB,. Taittiriya Bra.hmaQa Trif Trisatik:i TS Taittiriya Sarhhita YJ Yaju~a- ]yoti~a ZDMG Zeitschrift def deutschen morgenlandischen Gessels-, chaft HISTORY OF HINDU MATHEMATICS A SOURCE BOOK PART I NUMERAL NOTATION AND ARI'l'BMETlC HISTORY OF HINDU MATHEMATICS A SOURCE BOOK PART I NUMERAL NOTATION AND ARITHMETIC BY BIBHUTIBHUSAN DATTA AND AVADHESH NARAYAN SINGH COPYRIGHt', 1938, BY AVADHESH NARAYAN SINGH ALL RIGHTS RESERVED ~<t ;::n:r ~f~P::lf: ~"j::lf: ~;:f'l-'lf: qf~~"lf: (RV, x. 14. Z~) To the Seers, our Ancestors, the first Path-makers PREFACE Little is known at present to historians of mathe­ matics regarding the achievements of the early Hindu mathematicians and our indebtedness to them. Though it is now generally admitted that the decimal place­ value system of numeral notation. was invented and first used by the Hindus, it is not yet fully realized to what extent we are indebted to them for our elementary mathematics. This is due to the lack of a reliable and authentic history of Hindu mathematics. Our object in writing the present book has been to make up' for this deficiency by giving a comprehensive account of the growth and development of the science of mathematics in India from the earliest known times down to the seventeenth century of the Christian era. The subject is treated by topics. Under each topic are collected together and set forth in chronological order translations of relevant Sanskrit texts as found in the Hindu works. The texts have been elucidated, wherever necessary, by adding explanatory notes and comments, and also by illustrative examples culled from original sources. We have tried to avoid repetition as far as has ~een consistent with our aim. However, on several occasions it has been considered desirable to repeat the same rule in the words of different authors in order to emphasize the continuity or rather the gradual evolution of mathematical thought and termino­ logy in India. Comparative study of this kind has helped us to throw light on certain obscure Sanskrit passages and technical terms whose full significance PREFACE had not. been understood before. In translating the texts we have tried to be as literal and faithful as possiQle without sacrificing the spirit of the original. Sometimes it has not been possible to find exact parallels to Sanskrit words and technical terms in English. In all such cases we have tried to maintain the spirit of the original in the English version. The above plan of the book has been adopted in pursuance of our intention to place before those who have no access to the Sanskrit sources all evidence, unfavourable as well as favourable, so that they can judge for rhemselves the claims of Hindu mathematics, without depending solely on our statements. In order to facilitate comparison with the development of mathematics in other countries the various topics have been arranged generally in accordance with the se­ quence in Professor D. E. Smith's History of Mathematics, Vol. II. This has sometimes necessitated divergence from the arrangement of topics as found in the Hindu works on mathematics. In search of material for the book we had to examine the literature of the Hindus, non-mathematical as well as mathematical, whether in Sanskrit or in Pdkrit (Pali and Ardha Magadhi). Very f~w of the Hindu treatises on mathematics have been printed so far, and even these are not generally known. The manuscript works that exist in the various Sanskrit libraries in India and Europe are still less known. We have not spared labour in collecting as many of these as we could. Sanskrit mathematical works mentioned in the bibliography given at the end of this volume have been specially consulted by us. We are thankful to the authorities of the libraries at Madras, Bangalore, Trivandrum, Trippunithura and Benares, and those of the India Office (London) and the Asiatic Society of P-REFACE Bengal (Calcutta) for supplying us tra~scripts of the manuscripts required or sending us manuscripts for consultation. We are indebted also to Dr. R. P. Paranjpye, Vice-Chancellor of the Lucknow University, for help in securing for our use several manuscripts or their transcripts from the state libraries iri India and the India Office, London. It would not have been possible to carry our study as far as has been done without the spade work of previous writers. Foremost among these must be men­ tioned the late Pandit Sudhakar Dvivedi of Benares, whose editions of the Lfldvatf, Brdhmasphttta-siddhanta, Triiatikd, .Mahasiddhanta, Siddhanta-tattva-viveka, etc., have been of immense help. Colebrooke's translations of the arithmetic and algebra of Brahmagupta and Bhas­ kara II. Kern's edition of the Aryabha![ya and Ranga­ carya's edition (with English translation) of the Gapita­ sdra-samgraha of Mahavira have also been of much use. The recent work of G. R. Kaye, however, has been found to be extremely unreliable. His translation of the Gapitapaaa of the At:yabha![ya and his edition of the Bakhshall Manuscript are full of mistakes and are misleading. I t has been decided to publish the book in three parts. The first part deals with the history of the numeral notation and of arithmetic. The second is devoted to algebra, a science in which the ancient Hindus made remarkable progress. The third part contains the history of geometry, trigonometry, calculus and various other topics' such as magic squares, theory of series and permutations and com~inations. Each part is complete in itself, so that one interested in any particular branch of mathematics need not consult all of them. Part I which is now being published contains two chapters. Chapter I gives an account of the various CONTENTS CHAPTER I NUMERAL NOTATION PAGE "1. A GLIMPSE OF ANCIENT INDIA 1 z. HINDUS AND MATHEMATICS 3 Appreciation of Mathematics-Mathematics In Hindu Education. 3. SCOPE AND DEVELOPMENT OF HINDU MATHEMATICS 7 4· NUMERAL TERMINOLOGY 9 Scale of Notation-Numerals in Spoken Language. 5. THE DEVELOPMENT OF NUMERICAL SYM- BOLISM 16 Writing in Ancient India-Earliest Numerals. ,6. KHARO~THt NUMERALS 2. I Early Occurrence-Forms and their Origin. 7. BRAHM! NUMERALS Early Occurrence and Forms-Difference from other Notations-Theories about their Origin-Relation with Letter Forms-Indraji's Theory-Period of In­ vention-Resume. 8. THE DECIMAL PLACE-VALUE SYSTEM Important Features-Forms-Nagad Forms-Epi­ graphic Instances-Their Supposed Unreliability­ Place of Invention of the New System-Inventor Unknown-Time of Invention. 9. PERSISTENCE OF THE OLD SYSTEM CONTENTS PAGE 10. WORD NUMERALS 53 Explanation of the System-List of Words-Word Numerals without Place-value-Word Numerals with Place-value-Word Numerals in Inscriptions­ Origin and Early History-Date of Invention. II. ALPHABETIC NOTATIONS Alphabetic System of Aryabha~a I-Explanation­ Ka!apCfYddi Systems-Ak!arpalli-Other Letter-Sys­ tems. IZ. THE ZERO SYMBOL 7; Earliest use-Form of the Symbol-Other uses of the Symbol. 13. THE PLACE-VALUE NOTATION IN HINDU LITERATURE 83- Jaina Canonical Works-Puralfas-Works on Philo­ sophy-Literary Works. 14. DATE OF INVENTION OF THE PLACE-VALUE NOTATION 86 15. HINDU NUMERALS IN ARABIA 88 16. HINDU NUMERALS IN EUROPE 9~ Boethius Question-Definite Evidence. J7. MISCELLANEOUS REFERENCES TO THE HINDU NUMERALS 95 Syrian Reference-Arabic References-The Terms Hindasi, etc.-European References. 18. TABLES 105 CHAPTER II ARITHMETIC 1. GENERAL SURVEY 12.} Terminology and Scope-Sources-Exposition and Teaching-Decay of Mathematics-The Fundamen­ tal Operations.
Load more

Recommended publications
  • Leprosy and Other Skin Disorders
    Copyright by Robert Joseph Gallagher 2014 The report committee for Robert Joseph Gallagher Certifies that this is the approved version of the following report: An Annotated Translation of Chapter 7 of the Carakasaṃhitā Cikitsāsthāna: Leprosy and Other Skin Disorders APPROVED BY SUPERVISING COMMITTEE: Supervisor: __________________________________ Donald R. Davis _________________________________ Joel Brereton An Annotated Translation of Chapter 7 of the Carakasaṃhitā Cikitsāsthāna: Leprosy and Other Skin Disorders by Robert Joseph Gallagher, B.A., M.A. Report Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Fulfillment for the degree of Master of Arts University of Texas at Austin May 2014 Dedication To my wife Virginia and our two daughters Michelle and Amy, who showed patience and understanding during my long hours of absence from their lives, while I worked on mastering the intricacies of the complex but very rewarding language of Sanskrit. In addition, extra kudos are in order for thirteen year-old Michelle for her technical support in preparing this report. Acknowledgements I wish to thank all the members of the South Asia team at UT Austin, including Prof. Joel Brereton, Merry Burlingham, Prof. Don Davis, Prof. Oliver Freiberger, Prof. Edeltraud Harzer, Prof. Patrick Olivelle, Mary Rader, Prof. Martha Selby and Jennifer Tipton. Each one has helped me along this path to completion of the M.A. degree. At the time of my last serious academic research, I used a typewriter to put my thoughts on paper. The transition from white-out to pdf has been challenging for me at times, and I appreciate all the help given to me by the members of the South Asia team.
    [Show full text]
  • Invest in Afghan Women: a Report on Education in Afghanistan a Checkered History
    INVEST IN AFGHAN WOMEN: — A REPORT ON — EDUCATION IN AFGHANISTAN Presented by the George W. Bush Institute’s Women’s Initiative OCTOBER 2O13 “I HOPE AMERICANS WILL JOIN OUR FAMILY IN WORKING TO INVEST IN AFGHAN GIRLS ENSURE THAT DIGNITY AND OPPORTUNITY WILL BE SECURED FOR ALL THE WOMEN AND CHILDREN OF AFGHANISTAN.” In October 2012, a Taliban operative shot 15-year-old Pakistani student Malala Yousafzai in the face and neck while she traveled — MRS. LAURA BUSH home on a school bus. The assassination attempt was punishment for her “crime” of advocating for girls’ education. After surgeons repaired her shattered skull, Malala made a full recovery. And on July 12, 2013, she gave a rousing speech at the United Nations, becoming a global voice for girls’ access to education. Malala’s story is inspiring, but unfortunately the evils she’s combating are all too common in her region of the world. Just next door, in Afghanistan, religious fanaticism and deeply entrenched cultural practices have led to the systematic oppression of women and young girls. The Afghan situation is particularly desperate. While her peers in the United States prepare for their freshman year of high school, a typical 14-year-old Afghan girl has already been forced to leave formal education and is at acute risk of mandated marriage and early motherhood. If she beats the odds and attends school, she has reason to fear an attack on her schoolhouse with grenades or poison. A full 76 percent of her countrywomen have never attended school. And only 12.6 percent can read.
    [Show full text]
  • Introduction
    INTRODUCTION ’ P”(Bha.t.tikāvya) is one of the boldest “B experiments in classical literature. In the formal genre of “great poem” (mahākāvya) it incorprates two of the most powerful Sanskrit traditions, the “Ramáyana” and Pánini’s grammar, and several other minor themes. In this one rich mix of science and art, Bhatti created both a po- etic retelling of the adventures of Rama and a compendium of examples of grammar, metrics, the Prakrit language and rhetoric. As literature, his composition stands comparison with the best of Sanskrit poetry, in particular cantos , and . “Bhatti’s Poem” provides a comprehensive exem- plification of Sanskrit grammar in use and a good introduc- tion to the science (śāstra) of poetics or rhetoric (alamk. āra, lit. ornament). It also gives a taster of the Prakrit language (a major component in every Sanskrit drama) in an easily ac- cessible form. Finally it tells the compelling story of Prince Rama in simple elegant Sanskrit: this is the “Ramáyana” faithfully retold. e learned Indian curriculum in late classical times had at its heart a system of grammatical study and linguistic analysis. e core text for this study was the notoriously difficult “Eight Books” (A.s.tādhyāyī) of Pánini, the sine qua non of learning composed in the fourth century , and arguably the most remarkable and indeed foundational text in the history of linguistics. Not only is the “Eight Books” a description of a language unmatched in totality for any language until the nineteenth century, but it is also pre- sented in the most compact form possible through the use xix of an elaborate and sophisticated metalanguage, again un- known anywhere else in linguistics before modern times.
    [Show full text]
  • Roman Numerals
    History of Numbers 1c. I can distinguish between an additive and positional system, and convert between Roman and Hindu-Arabic numbers. Roman Numerals The numeric system represented by Roman numerals originated in ancient Rome (753 BC–476 AD) and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. By the 11th century, the more efJicient Hindu–Arabic numerals had been introduced into Europe by way of Arab traders. Roman numerals, however, remained in commo use well into the 14th and 15th centuries, even in accounting and other business records (where the actual calculations would have been made using an abacus). Roman numerals are still used today, in certain contexts. See: Modern Uses of Roman Numerals Numbers in this system are represented by combinations of letters from the Latin alphabet. Roman numerals, as used today, are based on seven symbols: The numbers 1 to 10 are expressed in Roman numerals as: I, II, III, IV, V, VI, VII, VIII, IX, X. This an additive system. Numbers are formed by combining symbols and adding together their values. For example, III is three (three ones) and XIII is thirteen (a ten plus three ones). Because each symbol (I, V, X ...) has a Jixed value rather than representing multiples of ten, one hundred and so on (according to the numeral's position) there is no need for “place holding” zeros, as in numbers like 207 or 1066. Using Roman numerals, those numbers are written as CCVII (two hundreds, plus a ive and two ones) and MLXVI (a thousand plus a ifty plus a ten, a ive and a one).
    [Show full text]
  • Adits, Caves, Karizi-Qanats, and Tunnels in Afghanistan: an Annotated Bibliography by R
    Adits, Caves, Karizi-Qanats, and Tunnels in Afghanistan: An Annotated Bibliography by R. Lee Hadden Topographic Engineering Center November 2005 US Army Corps of Engineers 7701 Telegraph Road Alexandria, VA 22315-3864 Adits, Caves, Karizi-Qanats, and Tunnels In Afghanistan Form Approved REPORT DOCUMENTATION PAGE OMB No. 0704-0188 Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing this collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden to Department of Defense, Washington Headquarters Services, Directorate for Information Operations and Reports (0704-0188), 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to any penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS. 1. REPORT DATE 30-11- 2. REPORT TYPE Bibliography 3. DATES COVERED 1830-2005 2005 4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER “Adits, Caves, Karizi-Qanats and Tunnels 5b. GRANT NUMBER In Afghanistan: An Annotated Bibliography” 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) 5d. PROJECT NUMBER HADDEN, Robert Lee 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION REPORT US Army Corps of Engineers 7701 Telegraph Road Topographic Alexandria, VA 22315- Engineering Center 3864 9.ATTN SPONSORING CEERD / MONITORINGTO I AGENCY NAME(S) AND ADDRESS(ES) 10.
    [Show full text]
  • A Quartically Convergent Square Root Algorithm: an Exercise in Forensic Paleo-Mathematics
    A Quartically Convergent Square Root Algorithm: An Exercise in Forensic Paleo-Mathematics David H Bailey, Lawrence Berkeley National Lab, USA DHB’s website: http://crd.lbl.gov/~dhbailey! Collaborator: Jonathan M. Borwein, University of Newcastle, Australia 1 A quartically convergent algorithm for Pi: Jon and Peter Borwein’s first “big” result In 1985, Jonathan and Peter Borwein published a “quartically convergent” algorithm for π. “Quartically convergent” means that each iteration approximately quadruples the number of correct digits (provided all iterations are performed with full precision): Set a0 = 6 - sqrt[2], and y0 = sqrt[2] - 1. Then iterate: 1 (1 y4)1/4 y = − − k k+1 1+(1 y4)1/4 − k a = a (1 + y )4 22k+3y (1 + y + y2 ) k+1 k k+1 − k+1 k+1 k+1 Then ak, converge quartically to 1/π. This algorithm, together with the Salamin-Brent scheme, has been employed in numerous computations of π. Both this and the Salamin-Brent scheme are based on the arithmetic-geometric mean and some ideas due to Gauss, but evidently he (nor anyone else until 1976) ever saw the connection to computation. Perhaps no one in the pre-computer age was accustomed to an “iterative” algorithm? Ref: J. M. Borwein and P. B. Borwein, Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity}, John Wiley, New York, 1987. 2 A quartically convergent algorithm for square roots I have found a quartically convergent algorithm for square roots in a little-known manuscript: · To compute the square root of q, let x0 be the initial approximation.
    [Show full text]
  • Kharosthi Manuscripts: a Window on Gandharan Buddhism*
    KHAROSTHI MANUSCRIPTS: A WINDOW ON GANDHARAN BUDDHISM* Andrew GLASS INTRODUCTION In the present article I offer a sketch of Gandharan Buddhism in the centuries around the turn of the common era by looking at various kinds of evidence which speak to us across the centuries. In doing so I hope to shed a little light on an important stage in the transmission of Buddhism as it spread from India, through Gandhara and Central Asia to China, Korea, and ultimately Japan. In particular, I will focus on the several collections of Kharo~thi manuscripts most of which are quite new to scholarship, the vast majority of these having been discovered only in the past ten years. I will also take a detailed look at the contents of one of these manuscripts in order to illustrate connections with other text collections in Pali and Chinese. Gandharan Buddhism is itself a large topic, which cannot be adequately described within the scope of the present article. I will therefore confine my observations to the period in which the Kharo~thi script was used as a literary medium, that is, from the time of Asoka in the middle of the third century B.C. until about the third century A.D., which I refer to as the Kharo~thi Period. In addition to looking at the new manuscript materials, other forms of evidence such as inscriptions, art and architecture will be touched upon, as they provide many complementary insights into the Buddhist culture of Gandhara. The travel accounts of the Chinese pilgrims * This article is based on a paper presented at Nagoya University on April 22nd 2004.
    [Show full text]
  • OSU WPL # 27 (1983) 140- 164. the Elimination of Ergative Patterns Of
    OSU WPL # 27 (1983) 140- 164. The Elimination of Ergative Patterns of Case-Marking and Verbal Agreement in Modern Indic Languages Gregory T. Stump Introduction. As is well known, many of the modern Indic languages are partially ergative, showing accusative patterns of case- marking and verbal agreement in nonpast tenses, but ergative patterns in some or all past tenses. This partial ergativity is not at all stable in these languages, however; what I wish to show in the present paper, in fact, is that a large array of factors is contributing to the elimination of partial ergativity in the modern Indic languages. The forces leading to the decay of ergativity are diverse in nature; and any one of these may exert a profound influence on the syntactic development of one language but remain ineffectual in another. Before discussing this erosion of partial ergativity in Modern lndic, 1 would like to review the history of what the I ndian grammar- ians call the prayogas ('constructions') of a past tense verb with its subject and direct object arguments; the decay of Indic ergativity is, I believe, best envisioned as the effect of analogical develop- ments on or within the system of prayogas. There are three prayogas in early Modern lndic. The first of these is the kartariprayoga, or ' active construction' of intransitive verbs. In the kartariprayoga, the verb agrees (in number and p,ender) with its subject, which is in the nominative case--thus, in Vernacular HindOstani: (1) kartariprayoga: 'aurat chali. mard chala. woman (nom.) went (fern. sg.) man (nom.) went (masc.
    [Show full text]
  • Recognition of Online Handwritten Gurmukhi Strokes Using Support Vector Machine a Thesis
    Recognition of Online Handwritten Gurmukhi Strokes using Support Vector Machine A Thesis Submitted in partial fulfillment of the requirements for the award of the degree of Master of Technology Submitted by Rahul Agrawal (Roll No. 601003022) Under the supervision of Dr. R. K. Sharma Professor School of Mathematics and Computer Applications Thapar University Patiala School of Mathematics and Computer Applications Thapar University Patiala – 147004 (Punjab), INDIA June 2012 (i) ABSTRACT Pen-based interfaces are becoming more and more popular and play an important role in human-computer interaction. This popularity of such interfaces has created interest of lot of researchers in online handwriting recognition. Online handwriting recognition contains both temporal stroke information and spatial shape information. Online handwriting recognition systems are expected to exhibit better performance than offline handwriting recognition systems. Our research work presented in this thesis is to recognize strokes written in Gurmukhi script using Support Vector Machine (SVM). The system developed here is a writer independent system. First chapter of this thesis report consist of a brief introduction to handwriting recognition system and some basic differences between offline and online handwriting systems. It also includes various issues that one can face during development during online handwriting recognition systems. A brief introduction about Gurmukhi script has also been given in this chapter In the last section detailed literature survey starting from the 1979 has also been given. Second chapter gives detailed information about stroke capturing, preprocessing of stroke and feature extraction. These phases are considered to be backbone of any online handwriting recognition system. Recognition techniques that have been used in this study are discussed in chapter three.
    [Show full text]
  • Brahmagupta, Mathematician Par Excellence
    GENERAL ARTICLE Brahmagupta, Mathematician Par Excellence C R Pranesachar Brahmagupta holds a unique position in the his- tory of Ancient Indian Mathematics. He con- tributed such elegant results to Geometry and Number Theory that today's mathematicians still marvel at their originality. His theorems leading to the calculation of the circumradius of a trian- gle and the lengths of the diagonals of a cyclic quadrilateral, construction of a rational cyclic C R Pranesachar is involved in training Indian quadrilateral and integer solutions to a single sec- teams for the International ond degree equation are certainly the hallmarks Mathematical Olympiads. of a genius. He also takes interest in solving problems for the After the Greeks' ascendancy to supremacy in mathe- American Mathematical matics (especially geometry) during the period 7th cen- Monthly and Crux tury BC to 2nd century AD, there was a sudden lull in Mathematicorum. mathematical and scienti¯c activity for the next millen- nium until the Renaissance in Europe. But mathematics and astronomy °ourished in the Asian continent partic- ularly in India and the Arab world. There was a contin- uous exchange of information between the two regions and later between Europe and the Arab world. The dec- imal representation of positive integers along with zero, a unique contribution of the Indian mind, travelled even- tually to the West, although there was some resistance and reluctance to accept it at the beginning. Brahmagupta, a most accomplished mathematician, liv- ed during this medieval period and was responsible for creating good mathematics in the form of geometrical theorems and number-theoretic results.
    [Show full text]
  • Indian Mathematics
    Indian Mathemtics V. S. Varadarajan University of California, Los Angeles, CA, USA UCLA, March 3-5, 2008 Abstract In these two lectures I shall talk about some Indian mathe- maticians and their work. I have chosen two examples: one from the 7th century, Brahmagupta, and the other, Ra- manujan, from the 20th century. Both of these are very fascinating figures, and their histories illustrate various as- pects of mathematics in ancient and modern times. In a very real sense their works are still relevant to the mathe- matics of today. Some great ancient Indian figures of Science Varahamihira (505–587) Brahmagupta (598-670) Bhaskara II (1114–1185) The modern era Ramanujan, S (1887–1920) Raman, C. V (1888–1970) Mahalanobis, P. C (1893–1972) Harish-Chandra (1923–1983) Bhaskara represents the peak of mathematical and astro- nomical knowledge in the 12th century. He reached an un- derstanding of calculus, astronomy, the number systems, and solving equations, which were not to be achieved any- where else in the world for several centuries...(Wikipedia). Indian science languished after that, the British colonial occupation did not help, but in the 19th century there was a renaissance of arts and sciences, and Indian Science even- tually reached a level comparable to western science. BRAHMAGUPTA (598–670c) Some quotations of Brahmagupta As the sun eclipses the stars by its brilliancy, so the man of knowledge will eclipse the fame of others in assemblies of the people if he proposes algebraic problems, and still more, if he solves them. Quoted in F Cajori, A History of Mathematics A person who can, within a year, solve x2 92y2 =1, is a mathematician.
    [Show full text]
  • The Kingdom of Afghanistan: a Historical Sketch George Passman Tate
    University of Nebraska Omaha DigitalCommons@UNO Books in English Digitized Books 1-1-1911 The kingdom of Afghanistan: a historical sketch George Passman Tate Follow this and additional works at: http://digitalcommons.unomaha.edu/afghanuno Part of the History Commons, and the International and Area Studies Commons Recommended Citation Tate, George Passman The kingdom of Afghanistan: a historical sketch, with an introductory note by Sir Henry Mortimer Durand. Bombay: "Times of India" Offices, 1911. 224 p., maps This Monograph is brought to you for free and open access by the Digitized Books at DigitalCommons@UNO. It has been accepted for inclusion in Books in English by an authorized administrator of DigitalCommons@UNO. For more information, please contact [email protected]. Tate, G,P. The kfn&ean sf Af&mistan, DATE DUE I Mil 7 (7'8 DEDICATED, BY PERMISSION, HIS EXCELLENCY BARON HARDINGE OF PENSHURST. VICEROY AND GOVERNOR-GENERAL OF INDIA, .a- . (/. BY m HIS OBEDIENT, SERVANT THE AUTHOR. il.IEmtev 01 the Asiniic Society, Be?zg-nl, S?~rueyof I~din. dafhor of 'I Seisinqz : A Menzoir on the FJisio~y,Topo~rcrphj~, A7zliquiiies, (112d Peo$Ie of the Cozi?zt~y''; The F/.o?zlic7,.~ of Baluchisia'nn : Travels on ihe Border.? of Pe~szk n?zd Akhnnistnn " ; " ICalnf : A lMe??zoir on t7ze Cozl7~try and Fnrrzily of the Ahntadsai Khn7zs of Iinlnt" ; 4 ec. \ViTkI AN INrPR<dl>kJCTOl2Y NO'FE PRINTED BY BENNETT COLEMAN & Co., Xc. PUBLISHED AT THE " TIMES OF INDIA" OFFTCES, BOMBAY & C.1LCUTT-4, LONDON AGENCY : gg, SI-IOE LANE, E.C.
    [Show full text]