Author | Waerden, B. L. van der. author |
---|---|
Title | Geometry and Algebra in Ancient Civilizations [electronic resource] / by B. L. van der Waerden |
Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg, 1983 |
Connect to | http://dx.doi.org/10.1007/978-3-642-61779-9 |
Descript | XII, 226 p. online resource |
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 -- Euclidโs Proof -- Naberโs 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 โNine Chaptersโ -- 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 โSum and Differenceโ 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 โGeometric Algebraโ -- Euclidโs 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 โKonikaโ of Apollonios -- The Sum of a Geometrical Progression -- Sums of Squares and Cubes -- 4. Diophantos and his Predecessors -- A. The Work of Diophantos -- Diophantosโ 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 -- Aryabhataโs Method -- Linear Diophantine Equations in Chinese Mathematics -- The Chinese Remainder Problem -- Astronomical Applications of the Pulverizer -- Aryabhataโs Two Systems -- Brahmaguptaโs System -- The Motion of the Apogees and Nodes -- The Motion of the Planets -- The Influence of Hellenistic Ideas -- B. PelVs Equation -- The Equation x2= 2y2ยฑl -- Periodicity in the Euclidean Algorithm -- Reciprocal Subtraction -- The Equations x2= 3y2 +1 and x2= 3y2โ 2 -- Archimedesโ Upper and Lower Limits for w3 -- Continued Fractions -- The Equation x2= Dy2ยฑ 1 for Non-squareD -- Brahmaguptaโs 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 โBloom of Thymaridasโ -- 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 -- Heronโs Commentary to Euclid -- Heronโs Metrika -- Circles and Circle Segments -- Apolloniosโ 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 โClassic of the Island in the Seaโ -- 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 -- Aryabhataโs Table of Sines -- On the Origin of Aryabhataโs Trigonometry -- Apollonios and Aryabhata as Astronomers -- On Gnomons and Shadows -- Square Roots and Cube Roots -- Arithmetical Progressions and Quadratic Equations