Seki, Founder of Modern Mathematics in Japan : A Commemoration on His Tercentenary.

By: Knobloch, EberhardContributor(s): Komatsu, Hikosaburo | Liu, DunMaterial type: TextTextSeries: eBooks on DemandSpringer Proceedings in Mathematics and Statistics Ser: Publisher: Tokyo : Springer, 2013Copyright date: ©2013Edition: 1st edDescription: 1 online resource (604 pages)Content type: text Media type: computer Carrier type: online resourceISBN: 9784431542735Subject(s): Mathematicians -- Japan -- Biography.;Mathematics, JapaneseGenre/Form: Electronic books.Additional physical formats: Print version:: Seki, Founder of Modern Mathematics in Japan : A Commemoration on His TercentenaryDDC classification: 510.92 LOC classification: QA1-939Online resources: Click here to view this ebook.
Contents:
Intro -- Foreword -- Preface -- Contents -- Program -- August 25, 2008 (Monday) -- Introductory Lectures in Japanese -- Lecture in Japanese -- August 26 -- August 27 -- August 28 -- August 29 -- August 30 -- August 31 -- Lectures in Japanese -- List of Contributors -- Part I Contributed papers -- Seki Takakazu, His Life and Bibliography -- Introduction -- 1 Personal Particulars -- 1.1 Family Name, Given Name and Popular Name: -- 1.2 Date of Birth: -- 1.3 Place of Birth: -- 1.4 Marital Status: -- 1.5 Home: -- 1.6 Date of Death: -- 1.7 Family Crest: -- 2 Professional Experience -- 2.1 On the Seki Family and His First Career -- 2.2 Land Survey -- 2.3 Chief of the Division of Provisions of the Kōfu Fief -- 2.4 Drawing Maps -- 2.5 Examiner of the Division of Accounts -- 2.6 Chief of a Team of Ceremonies in the Household of the West Castle -- 2.7 Retirement on a Pension -- 3 Education and Supervisors -- 4 Teaching Experience: -- 5 Published Research Works -- 5.1 Mathematical Methods for Exploring Subtle Points -- 5.2 Compendium of Mathematics -- 5.3 Complete Book of Mathematics -- Source Books: -- References -- Some Reflections on Main Lines of Mathematical Development -- 1 Algorithms in Chinese Mathematics -- 1.1 Nine Chapters on the Mathematical Art -- 1.2 Qin Jiushao's Book of Mathematics in Nine Chapters -- 1.3 Zhu Shijie's Four Elements Method -- 2 Algorithms in Wasan -- 2.1 Theory of Circles -- 2.2 Higher Interpolation Formulas -- 2.3 Elimination theory for higher equations of multi-unknowns -- 3 René Descartes' Geometry -- 4 Deductive vs. Algorithmic Mathematics -- References -- Babylonian Number Theory and Trigonometric Functions: Trigonometric Table and Pythagorean Triples in the Mathematical Tablet Plimpton 322 -- 1 Introduction -- 2 The technical terms in the headings -- 3 How to construct the numbers in Column I.
4 How to construct the numbers in Columns II and III -- 5 Conclusion -- References -- Archimedes in China: Archimedes and His Works in Chinese Literature of the Ming and Qing Dynasties -- 1 Mathematics -- 1.1 The Circle -- 1.2 The Sphere -- 1.3 Conics -- 1.4 Spirals -- 1.5 Spheroids and Conoids -- 2 Mechanics -- 2.1 The Lever Principle -- 2.2 Specific Gravity -- 2.3 Barycenter -- 2.4 Machine Designed -- 2.5 Flotage -- 3 Legends -- 3.1 The Gold Crown of King Hiero -- 3.2 Defending Syracuse -- 3.3 Lifting the Earth -- 3.4 Tombstone -- Conclusion -- References -- The Nine Chapters on the Mathematical Procedures and Liu Hui's Mathematical Theory -- 1 Procedures in Nine Chapters and its Style in which Questions Associated with Procedures as Examples -- 2 Liu Hui's Mathematical Definitions and Deductive Reasoning -- 2.1 Liu Hui's Definitions -- 2.2 Liu Hui's Deductive Reasoning -- 3 Liu Hui's Mathematical Proofs -- Acknowledgement -- References -- On the Alternative Algorithm of the 7th Problem in the Sea Island Mathematical Canon -- 1 Introduction -- 2 Original Text -- 3 Restoration of the Alternative Algorithm -- References -- A Comparative Study on Traditional Mathematics of Korea and Japan -- 1 Ancient Mathematics of Korea and Japan -- 2 Mathematical System in the Legal System -- 3 Japan -- 4 Mathematics of Chosǒn Dynasty -- 5 Paradigm of Chosǒn Dynasty -- 5.1 Mathematics of Nobles -- 5.2 Mathematics of Practical Scholars -- 5.3 Intermediary Class -- 5.4 Mathematicians from Intermediary Class (Technical Officials) -- 5.5 Collaboration with Intermediary Class and Noble Mathematicians -- 6 The Comparison with Japanese Mathematics -- 6.1 Abacus -- 6.2 Chonon-Sul -- 7 Paradigm of Japanese Mathematics -- 8 The Idea of Simplification and Generalization -- 9 Conclusion -- References.
The Axes of Mathematical Methodology in the Song and Yuan Dynasties: The Construction of Mathematical Models -- 1 Introduction -- 2 Li Ye's Algebra and his Illustration of A Circle Town -- 2.1 The Academic Demand for Construction of Mathematical Models -- 2.2 Illustration of a Circle Town-a Useful Mathematical Model -- 2.3 The Meaning of the Illustration of a Circle Town -- 3 Zhu Shijie's Algebra and the Five Sums and Five Differences of Right-angled Triangle -- 3.1 From the Heavenly Element to the Four Elements -- 3.2 The Model of the five Sums and five Differences of a Right-angled Triangle -- 4 Re-evaluation of Qin Jiushao's Algebra -- 5 Concluding Remarks -- References -- The Suanxue Qimeng and Its Influence on Japanese Mathematics -- 1 Chinese sources of Japanese mathematics -- 1.1 Early years of the wasan -- 1.2 Zhu Shijie and his two books -- 1.3 Acceptance of the Introduction to Mathematics in Japan -- 1.4 The Complete Colloquial Commentary on the Introduction to Mathematics -- 2 Counting board algebra -- 2.1 Counting-rods -- 2.2 Counting Board -- 2.3 Counting Board Algebra -- 3 Method of side writing -- 3.1 Seki Takakazu and Takebe Katahiro -- 3.2 Method of side writing -- Note -- References -- Power Series Expansions in India Around A. D. 1400 -- 1 Introduction -- 2 Mādhava and his school -- 3 Mādhava's series for calculating circumferences -- 3.1 Source of the Text -- 3.2 Original Expression of the Series -- 3.3 Derivation of the Series -- 3.4 Other Formulas Based on the Same Principle -- 3.5 Corrective Term -- References -- An Early Japanese Work on Chinese Mathematics in Vietnam: Yoshio Mikami's Study of the Vietnamese Mathematical Treatise Chi Minh Toan Phap -- 1 Introduction -- 2 The book Mikami studied -- 3 The Chi Minh Lap Thanh Toan Phap -- 4 The Preface by Phan Huy Khuong -- 5 The Appendix of Mikami's book.
6 The Appendix of A.1240 -- 7 The counting rods in Vietnam: Mikami's evidence -- 8 Discussion and Conclusions -- References -- The Jinkōki of Yoshida Mitsuyoshi -- 1 Introduction -- 2 Contents of the Jinkōki -- 3 Revised Edition of the Jinkōki -- 4 The Influence of the Jinkōki on Japanese Mathematicians -- References -- Résumé of Works on Mathematics of Seki Takakazu -- Introduction -- Book 1 -- Book 2 -- Book 3 -- Book 4 -- References -- Seki Takakazu's Measuring Process of the Volume of Solids Derived from Spheres -- 1 Volume of a sphere -- 2 Area of spheric surfaces -- 3 Solids made by revolution -- 4 Sphere segments and finger rings -- 5 Thinking patterns hereditary in Wasan -- 6 Barycenters of arc-figures -- 7 Onglets and the barycenters of arc-figures -- 8 Awonderful relation -- 9 An umbrella-shaped solid -- References -- Seki Takakazu's Method on the Remainder Problem -- 1 The Remainder Problem -- 2 Traditional Chinese remainder problem and its solution -- 3 The spread of remainder problems in Japan -- 4 The Compendium of Mathematics -- 5 The Complete Book of Mathematics -- 6 The Mathematical Treatise in Nine Chapters -- 7 Seki's method on the remainder problem -- References -- Seki Takakazu's Method of Calculating the Volume of Solids of Revolution and His Mathematical Object -- 1 Introduction -- 2 Method of calculating the volume of solids of revolution -- 2.1 Torus -- 2.2 Arc Ring -- 3 Volume 4 of the Complete Book of Mathematics -- 4 length of arc -- 5 Summary -- References -- Leibniz's Theory of Elimination and Determinants -- 1 The Fundamental Ideas -- 2 Inhomogeneous Systems of Linear Equations: motivation, rules of signs, general theorems -- 2.1 Motivation -- 2.2 Rules of sign -- 2.3 General theorems -- 3 Elimination of a Common Variable -- 3.1 Sylvester's dialytic method -- 3.2 B´ezout's method and Euler's second method.
3.3 Euler's resultants -- 3.4 Rules of formation and sign rule of the resultant -- 3.5 Leibniz's explication theory-Euler's first method -- Epilogue -- References -- Algebra, Elimination and the Complete Book of Mathematics -- 1 Historical Background -- 1.1 Chinese mathematics -- 1.2 Mathematicians in Kyoto and Osaka -- 1.3 Methods of Mathematics, Old and New by Sawaguchi -- 2 Mathematical Methods without Secrets by Seki Takakazu -- 3 Quadrilateral Method or Six Obliques Procedure -- 4 Power Procedures or Diminishing and Stretching Method -- 5 Resolution of Entanglements in Mathematics by Tanaka -- 6 Descartes' too Optimistic View -- 6.1 Problems that can be constructed only by means of circles and straight lines -- 7 Applications to Geometry -- References -- Some Questions and Observations Around the Mathematics of Seki Takakazu -- 1 The mathematics of Seki and his predecessors seen from a modern standpoint -- 1.1 A problem on volumes -- 1.2 The formula for arc length in the Jugairoku -- 1.3 The value of π in the Sanso -- 1.4 Seki's solution of a problem of the Kokon Sanpōki -- 1.5 On Problems 4 and 14 of the Kokon Sanpōki -- 2 Did Seki or Takebe learn any mathematics from the Dutch? -- Appendix: Takebe's questions to the Dutch in Dejima -- References -- Ming Antu and His Power Series Expansions -- 1 Ming Antu's Life and Works -- 1.1 Ming Antu was a top astronomer in China -- 1.2 Ming's work on the area measurement in China -- 1.3 Ming Antu's main contributions to mathematics -- 2 The Power Series Expansions by Ming Antu -- 2.1 Ming Antu obtained six formulae of infinite series -- 2.2 Catalan numbers in the infinite power series expansions -- 2.3 Ming Antu's method of calculating infinite series -- 3 Our Commemoration -- 3.1 The researches on Ming Antu by historians and mathematicians -- 3.2 The nomination of Ming Antu Star -- References.
Standing on the Shoulders of the Giant Influence of Seki Takakazu on Takebe Katahiro's Mathematical Achievements.
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Intro -- Foreword -- Preface -- Contents -- Program -- August 25, 2008 (Monday) -- Introductory Lectures in Japanese -- Lecture in Japanese -- August 26 -- August 27 -- August 28 -- August 29 -- August 30 -- August 31 -- Lectures in Japanese -- List of Contributors -- Part I Contributed papers -- Seki Takakazu, His Life and Bibliography -- Introduction -- 1 Personal Particulars -- 1.1 Family Name, Given Name and Popular Name: -- 1.2 Date of Birth: -- 1.3 Place of Birth: -- 1.4 Marital Status: -- 1.5 Home: -- 1.6 Date of Death: -- 1.7 Family Crest: -- 2 Professional Experience -- 2.1 On the Seki Family and His First Career -- 2.2 Land Survey -- 2.3 Chief of the Division of Provisions of the Kōfu Fief -- 2.4 Drawing Maps -- 2.5 Examiner of the Division of Accounts -- 2.6 Chief of a Team of Ceremonies in the Household of the West Castle -- 2.7 Retirement on a Pension -- 3 Education and Supervisors -- 4 Teaching Experience: -- 5 Published Research Works -- 5.1 Mathematical Methods for Exploring Subtle Points -- 5.2 Compendium of Mathematics -- 5.3 Complete Book of Mathematics -- Source Books: -- References -- Some Reflections on Main Lines of Mathematical Development -- 1 Algorithms in Chinese Mathematics -- 1.1 Nine Chapters on the Mathematical Art -- 1.2 Qin Jiushao's Book of Mathematics in Nine Chapters -- 1.3 Zhu Shijie's Four Elements Method -- 2 Algorithms in Wasan -- 2.1 Theory of Circles -- 2.2 Higher Interpolation Formulas -- 2.3 Elimination theory for higher equations of multi-unknowns -- 3 René Descartes' Geometry -- 4 Deductive vs. Algorithmic Mathematics -- References -- Babylonian Number Theory and Trigonometric Functions: Trigonometric Table and Pythagorean Triples in the Mathematical Tablet Plimpton 322 -- 1 Introduction -- 2 The technical terms in the headings -- 3 How to construct the numbers in Column I.

4 How to construct the numbers in Columns II and III -- 5 Conclusion -- References -- Archimedes in China: Archimedes and His Works in Chinese Literature of the Ming and Qing Dynasties -- 1 Mathematics -- 1.1 The Circle -- 1.2 The Sphere -- 1.3 Conics -- 1.4 Spirals -- 1.5 Spheroids and Conoids -- 2 Mechanics -- 2.1 The Lever Principle -- 2.2 Specific Gravity -- 2.3 Barycenter -- 2.4 Machine Designed -- 2.5 Flotage -- 3 Legends -- 3.1 The Gold Crown of King Hiero -- 3.2 Defending Syracuse -- 3.3 Lifting the Earth -- 3.4 Tombstone -- Conclusion -- References -- The Nine Chapters on the Mathematical Procedures and Liu Hui's Mathematical Theory -- 1 Procedures in Nine Chapters and its Style in which Questions Associated with Procedures as Examples -- 2 Liu Hui's Mathematical Definitions and Deductive Reasoning -- 2.1 Liu Hui's Definitions -- 2.2 Liu Hui's Deductive Reasoning -- 3 Liu Hui's Mathematical Proofs -- Acknowledgement -- References -- On the Alternative Algorithm of the 7th Problem in the Sea Island Mathematical Canon -- 1 Introduction -- 2 Original Text -- 3 Restoration of the Alternative Algorithm -- References -- A Comparative Study on Traditional Mathematics of Korea and Japan -- 1 Ancient Mathematics of Korea and Japan -- 2 Mathematical System in the Legal System -- 3 Japan -- 4 Mathematics of Chosǒn Dynasty -- 5 Paradigm of Chosǒn Dynasty -- 5.1 Mathematics of Nobles -- 5.2 Mathematics of Practical Scholars -- 5.3 Intermediary Class -- 5.4 Mathematicians from Intermediary Class (Technical Officials) -- 5.5 Collaboration with Intermediary Class and Noble Mathematicians -- 6 The Comparison with Japanese Mathematics -- 6.1 Abacus -- 6.2 Chonon-Sul -- 7 Paradigm of Japanese Mathematics -- 8 The Idea of Simplification and Generalization -- 9 Conclusion -- References.

The Axes of Mathematical Methodology in the Song and Yuan Dynasties: The Construction of Mathematical Models -- 1 Introduction -- 2 Li Ye's Algebra and his Illustration of A Circle Town -- 2.1 The Academic Demand for Construction of Mathematical Models -- 2.2 Illustration of a Circle Town-a Useful Mathematical Model -- 2.3 The Meaning of the Illustration of a Circle Town -- 3 Zhu Shijie's Algebra and the Five Sums and Five Differences of Right-angled Triangle -- 3.1 From the Heavenly Element to the Four Elements -- 3.2 The Model of the five Sums and five Differences of a Right-angled Triangle -- 4 Re-evaluation of Qin Jiushao's Algebra -- 5 Concluding Remarks -- References -- The Suanxue Qimeng and Its Influence on Japanese Mathematics -- 1 Chinese sources of Japanese mathematics -- 1.1 Early years of the wasan -- 1.2 Zhu Shijie and his two books -- 1.3 Acceptance of the Introduction to Mathematics in Japan -- 1.4 The Complete Colloquial Commentary on the Introduction to Mathematics -- 2 Counting board algebra -- 2.1 Counting-rods -- 2.2 Counting Board -- 2.3 Counting Board Algebra -- 3 Method of side writing -- 3.1 Seki Takakazu and Takebe Katahiro -- 3.2 Method of side writing -- Note -- References -- Power Series Expansions in India Around A. D. 1400 -- 1 Introduction -- 2 Mādhava and his school -- 3 Mādhava's series for calculating circumferences -- 3.1 Source of the Text -- 3.2 Original Expression of the Series -- 3.3 Derivation of the Series -- 3.4 Other Formulas Based on the Same Principle -- 3.5 Corrective Term -- References -- An Early Japanese Work on Chinese Mathematics in Vietnam: Yoshio Mikami's Study of the Vietnamese Mathematical Treatise Chi Minh Toan Phap -- 1 Introduction -- 2 The book Mikami studied -- 3 The Chi Minh Lap Thanh Toan Phap -- 4 The Preface by Phan Huy Khuong -- 5 The Appendix of Mikami's book.

6 The Appendix of A.1240 -- 7 The counting rods in Vietnam: Mikami's evidence -- 8 Discussion and Conclusions -- References -- The Jinkōki of Yoshida Mitsuyoshi -- 1 Introduction -- 2 Contents of the Jinkōki -- 3 Revised Edition of the Jinkōki -- 4 The Influence of the Jinkōki on Japanese Mathematicians -- References -- Résumé of Works on Mathematics of Seki Takakazu -- Introduction -- Book 1 -- Book 2 -- Book 3 -- Book 4 -- References -- Seki Takakazu's Measuring Process of the Volume of Solids Derived from Spheres -- 1 Volume of a sphere -- 2 Area of spheric surfaces -- 3 Solids made by revolution -- 4 Sphere segments and finger rings -- 5 Thinking patterns hereditary in Wasan -- 6 Barycenters of arc-figures -- 7 Onglets and the barycenters of arc-figures -- 8 Awonderful relation -- 9 An umbrella-shaped solid -- References -- Seki Takakazu's Method on the Remainder Problem -- 1 The Remainder Problem -- 2 Traditional Chinese remainder problem and its solution -- 3 The spread of remainder problems in Japan -- 4 The Compendium of Mathematics -- 5 The Complete Book of Mathematics -- 6 The Mathematical Treatise in Nine Chapters -- 7 Seki's method on the remainder problem -- References -- Seki Takakazu's Method of Calculating the Volume of Solids of Revolution and His Mathematical Object -- 1 Introduction -- 2 Method of calculating the volume of solids of revolution -- 2.1 Torus -- 2.2 Arc Ring -- 3 Volume 4 of the Complete Book of Mathematics -- 4 length of arc -- 5 Summary -- References -- Leibniz's Theory of Elimination and Determinants -- 1 The Fundamental Ideas -- 2 Inhomogeneous Systems of Linear Equations: motivation, rules of signs, general theorems -- 2.1 Motivation -- 2.2 Rules of sign -- 2.3 General theorems -- 3 Elimination of a Common Variable -- 3.1 Sylvester's dialytic method -- 3.2 B´ezout's method and Euler's second method.

3.3 Euler's resultants -- 3.4 Rules of formation and sign rule of the resultant -- 3.5 Leibniz's explication theory-Euler's first method -- Epilogue -- References -- Algebra, Elimination and the Complete Book of Mathematics -- 1 Historical Background -- 1.1 Chinese mathematics -- 1.2 Mathematicians in Kyoto and Osaka -- 1.3 Methods of Mathematics, Old and New by Sawaguchi -- 2 Mathematical Methods without Secrets by Seki Takakazu -- 3 Quadrilateral Method or Six Obliques Procedure -- 4 Power Procedures or Diminishing and Stretching Method -- 5 Resolution of Entanglements in Mathematics by Tanaka -- 6 Descartes' too Optimistic View -- 6.1 Problems that can be constructed only by means of circles and straight lines -- 7 Applications to Geometry -- References -- Some Questions and Observations Around the Mathematics of Seki Takakazu -- 1 The mathematics of Seki and his predecessors seen from a modern standpoint -- 1.1 A problem on volumes -- 1.2 The formula for arc length in the Jugairoku -- 1.3 The value of π in the Sanso -- 1.4 Seki's solution of a problem of the Kokon Sanpōki -- 1.5 On Problems 4 and 14 of the Kokon Sanpōki -- 2 Did Seki or Takebe learn any mathematics from the Dutch? -- Appendix: Takebe's questions to the Dutch in Dejima -- References -- Ming Antu and His Power Series Expansions -- 1 Ming Antu's Life and Works -- 1.1 Ming Antu was a top astronomer in China -- 1.2 Ming's work on the area measurement in China -- 1.3 Ming Antu's main contributions to mathematics -- 2 The Power Series Expansions by Ming Antu -- 2.1 Ming Antu obtained six formulae of infinite series -- 2.2 Catalan numbers in the infinite power series expansions -- 2.3 Ming Antu's method of calculating infinite series -- 3 Our Commemoration -- 3.1 The researches on Ming Antu by historians and mathematicians -- 3.2 The nomination of Ming Antu Star -- References.

Standing on the Shoulders of the Giant Influence of Seki Takakazu on Takebe Katahiro's Mathematical Achievements.

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