Amazing Traces of a Babylonian Origin in Greek Mathematics

The following sections are included:

• Preface

• Contents

The following sections are included:

• Greek Lettered Diagrams vs. OB Metric Algebra Diagrams

• El. II.2-3 and the Three Basic Quadratic Equations

• El. II.4, II.7 and the Two Basic Additive Quadratic-Linear Systems of Equations

• El. II.5-6 and the Two Basic Rectangular-Linear Systems of Equations

• El. II.8 and the Two Basic Subtractive Quadratic-Linear Systems of Equations

• El. II.9-10, Constructive Counterparts to El. II.4 and II.7

• El. II.11* and II.14*, Constructive Counterparts to El. II.5-6

• El. II.12-13, Constructive Counterparts to El. II.8

• Summary. The Three Parts of Elements II

• An Old Babylonian Catalog Text with Metric Algebra Problems

• A Large Old Babylonian Catalog Text of a Similar Kind

• Old Babylonian Solutions to Metric Algebra Problems
  • Old Babylonian problems for rectangles and squares
  • Old Babylonian problems for circles and chords
  • Old Babylonian problems for non-symmetric trapezoids

• Late Babylonian Solutions to Metric Algebra Problems
  • Problems for rectangles and squares
    • The seed measure of a hundred-cubit-square. Metric squaring
    • A rectangle of given front and seed measure. Metric division
    • A square of given seed measure. Metric square side computation
    • A rectangle of given side-sum and seed measure. Basic problem of B1a
    • A rectangle of given side-difference and seed measure. Type B1b
    • A square band of given width and seed measure. Type B3b
  • Problems for circles
    • A circle of given seed measure divided into five bands of equal width
    • A circle of given circumference divided into five bands of equal width
    • A Seleucid pole-against-a-wall problem
    • Seleucid parallels to El. II.14* (systems of equations of type B1a)

• Old Akkadian Square Expansion and Square Contraction Rules

• The Long History of Metric Algebra in Mesopotamia

The following sections are included:

• Euclid's Proof of El. I.47

• Pappus' Proof of a Generalization of El. I.47

• The Original Discovery of the OB Diagonal Rule for Rectangles

• Chains of Triangles, Trapezoids, or Rectangles

The following sections are included:

• Greek Generating Rules for Diagonal Triples of Numbers
  • Euclid's Generating Rule in the Lemma El. X.28/29 1a
  • The Generating Rules Attributed to Pythagoras and Plato
  • Metric Algebra Derivations of the Greek Generating Rules

• Old Babylonian igi-igi.bi Problems

• Plimpton 322: A Table of Parameters for igi-igi.bi Problems

The following sections are included:

• Division of a right triangle into a pair of right sub-triangles

• A Metric Algebra Proof of Lemma El. X.32/33

• An Old Babylonian Chain of Right Sub-Triangles

The following sections are included:

• The Pivotal Propositions and Lemmas in Elements X
  • A Concise Outline of the Contents of Elements X

• Binomials and Apotomes, Majors and Minors

• Euclid's Application of Areas and Babylonian Metric Division

• Quadratic-Rectangular Systems of Equations of Type B5

The following sections are included:

• Elements IV, a Well Organized Geometric Theme Text
  • An Outline of the Contents of Elements IV

• Figures Within Figures in Mesopotamian Mathematics

The following sections are included:

• El. VI.30: Cutting a Straight Line in Extreme and Mean Ratio

• Regular Pentagons and Equilateral Triangles in Elements XIII
  • An Outline of the Contents of El. XIII.1-12

• An Extension of the Result in El. XIII.11

• An Alternative Proof of the Crucial Proposition El. XIII.8

• Metric Analysis of the Regular Pentagon in Terms of its Side

• Metric Analysis of the Regular Octagon

• Equilateral Triangles in Babylonian Mathematics

• Regular Polygons in Babylonian Mathematics

• Geometric Constructions in Mesopotamian Decorative Art

The following sections are included:

• Regular Polyhedrons in Elements XIII
  • An Outline of the Contents of El. XIII.13-18
  • Conclusion

• MS 3049 § 5. The Inner Diagonal of a Gate

• The Weight of an Old Babylonian Colossal Copper Icosahedron

The following sections are included:

• Circles, Pyramids, Cones, and Spheres in Elements XII

• Pre-literate Plain Number Tokens from the Middle East in the Form of Circular Lenses, Pyramids, Cylinders, Cones, and Spheres

• Pyramids and Cones in OB Mathematical Cuneiform Texts
  • The volume and grain measure of a ridge pyramid
  • The grain measure of a ridge pyramid truncated at mid-height
  • Problems for cones and truncated cones

• Pyramids and Cones in Ancient Chinese Mathematical Texts
  • The fifth chapter in Jiu Zhang Suan Shu
  • Liu Hui's commentary to Jiu Zhang Suan Shu, Chapter V

• A Possible Babylonian Derivation of the Volume of a Pyramid

The following sections are included:

• El. I.43-44 & Data 57: Parabolic Applications of Parallelograms

• El. VI. 28 & Data 58. Elliptic Applications of Parallelograms

• El. VI. 29 & Data 59. Hyperbolic Applications of Parallelograms

• El. VI.25 and Data 55

• Data 84-85. Rectangular-Linear Systems of Equations

• Data 86. A Quadratic-Rectangular System of Equations

• Zeuthen's Conjecture: Intersecting Hyperbolas

• A Kassite Series Text with Modified Systems of Types B5 and B6

The following sections are included:

• Selected Division Problems in On Divisions
  • OD 1-2, 30-31. To divide a triangle by lines parallel to the base
  • OD 3. To bisect a triangle by a line through a point on a side
  • OD 4-5. To divide a trapezoid by lines parallel to the base
  • OD 8, 12. To bisect a trapezoid by a line through a point on a side
  • OD 19-20. To divide a triangle by a line through an interior point
  • OD 32. To divide a trapezoid by a parallel in a given ratio

• Old Babylonian Problems for Striped Triangles
  • Str. 364 § 2. A model problem for a 3-striped triangle
  • Str. 364 § 3. A quadratic equation for a 2-striped triangle
  • Str. 364 §§ 4-7. Quadratic equations for 2-striped triangles
  • Str. 364 § 8. Problems for 5-striped triangles
  • TMS 18. A cleverly designed problem for a 2-striped triangle
  • MLC 1950. An elegant solution procedure
  • VAT 8512. Another cleverly designed problem
  • YBC 4696. A series of problems for a 2-striped triangle
  • MAH 16055. A table of diagrams for 3-striped triangles
  • IM 43996. A 3-striped triangle divided in given ratios

• Old Babylonian Problems for 2-Striped Trapezoids
  • IM 58045, an Old Akkadian problem for a bisected trapezoid
  • VAT 8512, interpreted as a problem for a bisected trapezoid
  • YBC 4675. A problem for a bisected quadrilateral
  • YBC 4608. A 2-striped trapezoid divided in the ratio 1:3
  • Str. 367. A 2-striped trapezoid divided in the ratio 29 : 51
  • Ist. Si. 269. Five 2-striped trapezoids divided in the ratio 60 : 1
  • The Bloom of Thymaridas and its relation to Old Babylonian generating equations for transversal triples
  • Relations between diagonal triples and transversal triples

• Old Babylonian Problems for 3-and 5-Striped Trapezoids

• Erm. 15189. Diagrams for Ten Double Bisected Trapezoids

• AO 17264. A Problem for a Chain of 3 Bisected Quadrilaterals

• VAT7621 # 1. A2 • 9-stripedtrapezoid

• VAT 7531. Cross-wise striped trapezoids

• TMS 23. Confluent Quadrilateral Bisections in Two Directions

• Erm. 15073. Divided Trapezoids in a Recombination Text

The following sections are included:

• Hippocrates' Lunes According to Alexander

• Hippocrates' Lunes According to Eudemus

• Some Geometric Figures in the OB Table of Constants BR
  • BR 10-12. The 'bow field'
  • BR 13-15. The 'boat field'
  • BR 16-18. The 'barleycorn field'
  • BR 19-21. The 'ox-eye'
  • BR 22-24. The 'lyre-window'
  • BR 25. The 'lyre-window of 3'

• W 23291-x § 1. A Late Babylonian Double Segment and Lune

• A Remark by Neugebauer Concerning BM 15285 # 33

The following sections are included:

• Introduction

• Determinate Problems in Book I of Diophantus' Arithmetica

• Four Basic Examples in Book II of Diophantus' Arithmetica
  • Ar. II.8 (Sesiano, GA (1990), 84; Thomas, op. cit., 551)
  • Ar. II.9 (Sesiano, GA (1990), 85; Heath, DA (1964), 145)
  • Ar. II.10 (Sesiano, GA (1990), 86; Heath, DA (1964), 146)
  • Ar. II.19 (Sesiano, GA (1990), 86; Heath, DA (1964), 146)

• Ar. "V".9. Diophantus' Method of Approximation to Limits

• Ar. III.19. A Square Number Equal to a Sum of Two Squares in Four Different Ways
  • Everywhere rational cyclic quadrilaterals
  • Diophantus' Ar. III.19, Birectangles, and the OB Composition Rule

• Ar. "V".30. An Applied Problem and Quadratic Inequalities
  • An indeterminate combined price problem

• Ar. "VI". A Theme Text with Equations for Right Triangles
  • Ar. "VI".16. A right triangle with a rational bisector

• Ar. V.7-12. A Section of a Theme Text with Cubic Problems

• Ar. IV.17. Another Appearance of the Term 'Representable'

The following sections are included:

• Metrica I.8/Dioptra 31. Heron's Triangle Area Rule

• Two Simple Metric Algebra Proofs of the Triangle Area Rule

• Simple Proofs of Special Cases of Brahmagupta's Area Rule

• Simple Proofs of Special Cases of Ptolemy's Diagonal Rule

• Simple Proofs of Special Cases of Brahmagupta's Diagonal Rule

• A Proof of Brahmagupta's Diagonal Rule in the General Case

The following sections are included:

• The Greek Side and Diagonal Numbers Algorithm

• MLC 2078. The Old Babylonian Spiral Chain Algorithm

• Side and Diagonal Numbers When Sq. p = Sq. q · D - 1

• Side and Diagonal Numbers When Sq. p = Sq. q · D + 1

The following sections are included:

• Metrica I.8 b. Heron's Square Side Rule

• Heronic Square Side Approximations

• A New Explanation of Heron's Accurate Square Side Rule

• Third Approximations in Ptolemy's Syntaxis I.10

• The General Case of Formal Multiplications

• A New Explanation of the Archimedian Estimates for Sqs. 3

• Examples of Babylonian Square Side Approximations
  • The additive and subtractive square side rules
  • Late and Old Babylonian approximations to sqs. 2
  • Late and Old Babylonian approximations to sqs. 3
  • A Late Babylonian approximation to sqs. 5
  • Late and Old Babylonian exact computations of square sides

The following sections are included:

• Theaetetus 147 C-D. Theodoras' Metric Algebra Lesson

• A Number-Theoretical Explanation of Theodoras' Method

• An Anthyphairetic Explanation of Theodoras' Method

• A Metric Algebra Explanation of Theodoras' Method

The following sections are included:

• Geometrica as a Compilation of Various Sources

• Geometrica mss AC

• Geometrica ms S 24

• Metrica III.4. A Division of Figures Problem

The following sections are included:

• Appendix 1. A Chain of Trapezoids with Fixed Diagonals
  • VAT 8393. A New Old Babylonian Single Problem Text
  • VAT 8393. About the Clay Tablet

• Appendix 2. A Catalog of Babylonian Geometric Figures

• Index of Texts, Propositions, and Lemmas

• Index of Subjects

• Bibliography

• Comparative Mesopotamian, Egyptian, and Babylonian Timelines