Geometry and algebra in ancient civilizations
Originally, my intention was to write a "History of Algebra", in two or three volumes. In preparing the first volume I saw that in ancient civilizaƯ tions geometry and algebra cannot well be separated: more and more secƯ tions on ancient geometry were added. Hence the new title of the book: "Geometry and Algebra in Ancient Civilizations". A subsequent volume on the history of modem algebra is in preparation. It will deal mainly with field theory, Galois theory and theory of groups. I want to express my deeply felt gratitude to all those who helped me in shaping this volume. In particular, I want to thank Donald Blackmore Wagner (Berkeley) who put at my disposal his English translation of the most interesting parts of the Chinese "Nine Chapters of the Art of ArithƯ metic" and of Liu Hui's commentary to this classic, and also Jacques SeƯ siano (Geneva), who kindly allowed me to use his translation of the reƯ cently discovered Arabic text of four books of Diophantos not extant in Greek. Warm thanks are also due to Wyllis Bandler (Colchester, England) who read my English text very carefully and suggested several improveƯ ments, and to Annemarie Fellmann (Frankfurt) and Erwin NeuenschwanƯ der (Zurich) who helped me in correcting the proof sheets. Miss Fellmann also typed the manuscript and drew the figures. I also want to thank the editorial staff and production department of Springer-Verlag for their nice cooperation
History
1 online resource (XII, 223 pages)
9783642617799, 3642617794
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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
Electronic reproduction, [Place of publication not identified], HathiTrust Digital Library, 2010
English