Geometry and Algebra in Ancient Civilizations

Bartel L. van der Waerden 

Springer | 2011 - reprint of the original 1st ed. 1983 edition |235 páginas | rar - pdf | 5,3 Mb

link (password: matav)

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 ProblemsTwo Babylonian Problems.- Egyptian Problems.- The “Bloom of Thymaridas”.C. Greek Arithmetical EpigramsD. 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 SquareAn 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 RootG. 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.