Geometry and algebra in ancient civilization (Book, 1983) []
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Geometry and algebra in ancient civilization

Author: Bartel L van der Waerden
Publisher: Berlin Heidelberg New York Tokyo Springer 1983
Edition/Format:   Print book : GermanView all editions and formats

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Waerden, Bartel L. van der , 1903-1996
Geometry and Algebra in Ancient Civilizations
Berlin, Heidelberg : Springer Berlin Heidelberg, 1983
Document Type: Book
All Authors / Contributors: Bartel L van der Waerden
ISBN: 9783540121596 3540121595 9780387121598 0387121595
OCLC Number: 239761263
Notes: In Deutsche Bibliographie, Reihe N, angezeigt u.d.T.: Waerden, Bartel Leendert van der: A history of algebra. Vol. 1
Description: XII, 223 S. 98 graph. Darst. 25 cm
Contents: 1. Pythagorean Triangles.- A. Written Sources.- Fundamental Notions.- The Text Plimpton 322.- A Chinese Method.- Methods Ascribed to Pythagoras and Plato.- Pythagorean Triples in India.- The Hypothesis of a Common Origin.- Geometry and Ritual in Greece and India.- Pythagoras and the Ox.- B. Archaeological Evidence.- Prehistoric Ages.- Radiocarbon Dating.- Megalithic Monuments in Western Europe.- Pythagorean Triples in Megalithic Monuments.- Megalith Architecture in Egypt.- The Ritual Use of Pythagorean Triangles in India.- C. On Proofs, and on the Origin of Mathematics.- Geometrical Proofs.- Euclid's Proof.- Naber's Proof.- Astronomical Applications of the Theorem of Pythagoras?.- Why Pythagorean Triangles?.- The Origin of Mathematics.- 2. Chinese and Babylonian Mathematics.- A. Chinese Mathematics.- The Chinese "Nine Chapters".- The Euclidean Algorithm.- Areas of Plane Figures.- Volumes of Solids.- The Moscow Papyrus.- Similarities Between Ancient Civilizations.- Square Roots and Cube Roots.- Sets of Linear Equations.- Problems on Right-Angled Triangles.- The Broken Bamboo.- Two Geometrical Problems.- Parallel Lines in Triangles.- B. Babylonian Mathematics.- A Babylonian Problem Text.- Quadratic Equations in Babylonian Texts.- The Method of Elimination.- The "Sum and Difference" Method.- C. General Conclusions.- Chinese and Babylonian Algebra Compared.- The Historical Development.- 3. Greek Algebra.- What is Algebra?.- The Role of Geometry in Elementary Algebra.- Three Kinds of Algebra.- On Units of Length, Area, and Volume.- Greek "Geometric Algebra".- Euclid's Second Book.- The Application of Areas.- Three Types of Quadratic Equations.- Another Concordance Between the Babylonians and Euclid.- An Application of II, 10 to Sides and Diagonals.- Thales and Pythagoras.- The Geometrization of Algebra.- The Theory of Proportions.- Geometric Algebra in the "Konika" of Apollonios.- The Sum of a Geometrical Progression.- Sums of Squares and Cubes.- 4. Diophantos and his Predecessors.- A. The Work of Diophantos.- Diophantos' Algebraic Symbolism.- Determinate and Indeterminate Problems.- From Book A.- From Book B.- The Method of Double Equality.- From Book ?.- From Book 4.- From Book 5.- From Book 7.- From Book ?.- From Book E.- B. The Michigan Papyrus 620.- C. Indeterminate Equations in the Heronic Collections.- 5. Diophantine Equations.- A. Linear Diophantine Equations.- Aryabhata's Method.- Linear Diophantine Equations in Chinese Mathematics.- The Chinese Remainder Problem.- Astronomical Applications of the Pulverizer.- Aryabhata's Two Systems.- Brahmagupta's System.- The Motion of the Apogees and Nodes.- The Motion of the Planets.- The Influence of Hellenistic Ideas.- B. PelVs Equation.- The Equation x2= 2y2+-l.- Periodicity in the Euclidean Algorithm.- Reciprocal Subtraction.- The Equations x2= 3y2 +1 and x2= 3y2- 2.- Archimedes' Upper and Lower Limits for w3.- Continued Fractions.- The Equation x2= Dy2+- 1 for Non-squareD.- Brahmagupta's Method.- The Cyclic Method.- Comparison Between Greek and Hindu Methods.- C. Pythagorean Triples.- 6. Popular Mathematics.- A. General Character of Popular Mathematics.- B. Babylonian, Egyptian and Early Greek Problems.- Two Babylonian Problems.- Egyptian Problems.- The "Bloom of Thymaridas".- C. Greek Arithmetical Epigrams.- D. Mathematical Papyri from Hellenistic Egypt.- Calculations with Fractions.- Problems on Pieces of Cloth.- Problems on Right-Angled Triangles.- Approximation of Square Roots.- Two More Problems of Babylonian Type.- E. Squaring the Circle and Circling the Square.- An Ancient Egyptian Rule for Squaring the Circle.- Circling the Square as a Ritual Problem.- An Egyptian Problem.- Area of the Circumscribed Circle of a Triangle.- Area of the Circumscribed Circle of a Square.- Three Problems Concerning the Circle Segment in a Babylonian Text.- Shen Kua on the Arc of a Circle Segment.- F. Heron of Alexandria.- The Date of Heron.- Heron's Commentary to Euclid.- Heron's Metrika.- Circles and Circle Segments.- Apollonios' Rapid Method.- Volumes of Solids.- Approximating a Cube Root.- G. The Mishnat ha-Middot.- 7. Liu Hui and Aryabhata.- A. The Geometry of Liu Hui.- The "Classic of the Island in the Sea".- First Problem: The Island in the Sea.- Second Problem: Height of a Tree.- Third Problem: Square Town.- The Evaluation of ?.- The Volume of a Pyramid.- Liu Hui and Euclid.- Liu Hui on the Volume of a Sphere.- B. The Mathematics of Aryabhata.- Area and Circumference of a Circle.- Aryabhata's Table of Sines.- On the Origin of Aryabhata's Trigonometry.- Apollonios and Aryabhata as Astronomers.- On Gnomons and Shadows.- Square Roots and Cube Roots.- Arithmetical Progressions and Quadratic Equations.
Responsibility: B. L. van der Waerden
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