Amazing Traces of a Babylonian Origin in Greek Mathematics

Contents......Page 16
Preface......Page 6
1. Elements II and Babylonian Metric Algebra......Page 22
1.1. Greek Lettered Diagrams vs. OB Metric Algebra Diagrams......Page 23
1.2. El. II.2-3 and the Three Basic Quadratic Equations......Page 28
1.3. El. II.4, II.7 and the Two Basic Additive Quadratic-Linear Systems of Equations......Page 31
1.4. El. II.5-6 and the Two Basic Rectangular-Linear Systems of Equations......Page 33
1.5. El. II.8 and the Two Basic Subtractive Quadratic-Linear Systems of Equations......Page 35
1.6. El. II.9-10, Constructive Counterparts to El. II.4 and II.7......Page 37
1.7. El. II.11* and II.14*, Constructive Counterparts to El. II.5-6......Page 39
1.8. El. II.12-13, Constructive Counterparts to El. II.8......Page 43
1.9. Summary. The Three Parts of Elements II......Page 45
1.10. An Old Babylonian Catalog Text with Metric Algebra Problems......Page 48
1.11. A Large Old Babylonian Catalog Text of a Similar Kind......Page 50
1.12 a. Old Babylonian problems for rectangles and squares......Page 56
1.12 b. Old Babylonian problems for circles and chords......Page 63
1.12 c. Old Babylonian problems for non-symmetric trapezoids......Page 69
1.13 a. Problems for rectangles and squares......Page 71
The seed measure of a hundred-cubit-square. Metric squaring......Page 72
A rectangle of given front and seed measure. Metric division......Page 74
A square of given seed measure. Metric square side computation......Page 75
A rectangle of given side-sum and seed measure. Basic problem of type B1a......Page 76
A rectangle of given side-difference and seed measure. Type B1b......Page 78
A square band of given width and seed measure. Type B3b......Page 79
A circle of given seed measure divided into five bands of equal width......Page 80
A circle of given circumference divided into five bands of equal width......Page 82
A Seleucid pole-against-a-wall problem......Page 85
Seleucid parallels to El. II.14* (systems of equations of type B1a)......Page 87
1.14. Old Akkadian Square Expansion and Square Contraction Rules......Page 89
1.15. The Long History of Metric Algebra in Mesopotamia......Page 90
2.1. Euclid’s Proof of El. I.47......Page 94
2.2. Pappus’ Proof of a Generalization of El. I.47......Page 95
2.3. The Original Discovery of the OB Diagonal Rule for Rectangles......Page 97
2.4. Chains of Triangles, Trapezoids, or Rectangles......Page 100
Euclid’s Generating Rule in the Lemma El. X.28/29 1a......Page 104
The Generating Rules Attributed to Pythagoras and Plato......Page 105
Metric Algebra Derivations of the Greek Generating Rules......Page 106
3.2. Old Babylonian igi-igi.bi Problems......Page 107
3.3. Plimpton 322: A Table of Parameters for igi-igi.bi Problems......Page 109
4.1. Division of a right triangle into a pair of right sub-triangles......Page 116
4.2. A Metric Algebra Proof of Lemma El. X.32/33......Page 117
4.3. An Old Babylonian Chain of Right Sub-Triangles......Page 118
5.1. The Pivotal Propositions and Lemmas in Elements X......Page 122
A Concise Outline of the Contents of Elements X......Page 123
5.2. Binomials and Apotomes, Majors and Minors......Page 124
5.3. Euclid’s Application of Areas and Babylonian Metric Division......Page 134
5.4. Quadratic-Rectangular Systems of Equations of Type B5......Page 137
An Outline of the Contents of Elements IV......Page 144
6.2. Figures Within Figures in Mesopotamian Mathematics......Page 146
7.1. El. VI.30: Cutting a Straight Line in Extreme and Mean Ratio......Page 162
An Outline of the Contents of El. XIII.1-12......Page 163
7.3. An Extension of the Result in El. XIII.11......Page 167
7.4. An Alternative Proof of the Crucial Proposition El. XIII.8......Page 170
7.5. Metric Analysis of the Regular Pentagon in Terms of its Side......Page 172
7.6. Metric Analysis of the Regular Octagon......Page 176
7.7. Equilateral Triangles in Babylonian Mathematics......Page 180
7.8. Regular Polygons in Babylonian Mathematics......Page 182
7.9. Geometric Constructions in Mesopotamian Decorative Art......Page 185
An Outline of the Contents of El. XIII.13-18......Page 192
8.2. MS 3049 § 5. The Inner Diagonal of a Gate......Page 202
8.3. The Weight of an Old Babylonian Colossal Copper Icosahedron......Page 18
9.1. Circles, Pyramids, Cones, and Spheres in Elements XII......Page 210
9.2. Pre-literate Plain Number Tokens from the Middle East in the Form of Circular Lenses, Pyramids, Cylinders, Cones, and Spheres......Page 213
9.3. Pyramids and Cones in OB Mathematical Cuneiform Texts......Page 216
9.3 a. The volume and grain measure of a ridge pyramid......Page 217
9.3 b. The grain measure of a ridge pyramid truncated at mid-height......Page 221
9.4 a. The fifth chapter in Jiu Zhang Suan Shu......Page 223
9.4 b. Liu Hui’s commentary to Jiu Zhang Suan Shu, Chapter V.......Page 227
9.5. A Possible Babylonian Derivation of the Volume of a Pyramid......Page 228
10. El. I.43-44, El. VI.24-29, Data 57-59, 84-86, and Metric Algebra......Page 232
10.1. El. I.43-44 & Data 57: Parabolic Applications of Parallelograms......Page 233
10.2. El. VI. 28 & Data 58. Elliptic Applications of Parallelograms......Page 238
10.3. El. VI. 29 & Data 59. Hyperbolic Applications of Parallelograms......Page 240
10.4. El. VI.25 and Data 55......Page 241
10.5. Data 84-85. Rectangular-Linear Systems of Equations......Page 246
10.6. Data 86. A Quadratic-Rectangular System of Equations of Type B6......Page 248
10.7. Zeuthen’s Conjecture: Intersecting Hyperbolas......Page 253
10.8. A Kassite Series Text with Modified Systems of Types B5 and B6......Page 254
11. Euclid’s Lost Book On Divisions and Babylonian Striped Figures......Page 256
OD 1-2, 30-31. To divide a triangle by lines parallel to the base......Page 257
OD 4-5. To divide a trapezoid by lines parallel to the base......Page 258
OD 8, 12. To bisect a trapezoid by a line through a point on a side......Page 259
OD 19-20. To divide a triangle by a line through an interior point......Page 260
OD 32. To divide a trapezoid by a parallel in a given ratio......Page 263
11.2 a. Str. 364 § 2. A model problem for a 3-striped triangle......Page 265
11.2 b. Str. 364 § 3. A quadratic equation for a 2-striped triangle......Page 268
11.2 c. Str. 364 §§ 4-7. Quadratic equations for 2-striped triangles......Page 270
11.2 d. Str. 364 § 8. Problems for 5-striped triangles......Page 273
11.2 e. TMS 18. A cleverly designed problem for a 2-striped triangle......Page 276
11.2 f. MLC 1950. An elegant solution procedure......Page 279
11.2 g. VAT 8512. Another cleverly designed problem......Page 280
11.2 h. YBC 4696. A series of problems for a 2-striped triangle......Page 282
11.2 i. MAH 16055. A table of diagrams for 3-striped triangles......Page 285
11.2 j. IM 43996. A 3-striped triangle divided in given ratios......Page 288
11.3 a. IM 58045, an Old Akkadian problem for a bisected trapezoid......Page 290
11.3 b. VAT 8512, interpreted as a problem for a bisected trapezoid......Page 292
11.3 c. YBC 4675. A problem for a bisected quadrilateral......Page 293
11.3 d. YBC 4608. A 2-striped trapezoid divided in the ratio 1: 3......Page 295
11.3 e. Str. 367. A 2-striped trapezoid divided in the ratio 29 : 51......Page 298
11.3 f. Ist. Si. 269. Five 2-striped trapezoids divided in the ratio 60 : 1......Page 300
11.3 g. The Bloom of Thymaridas and its relation to Old Babylonian generating equations for transversal triples......Page 303
11.3 h. Relations between diagonal triples and transversal triples......Page 304
11.4. Old Babylonian Problems for 3-and 5-Striped Trapezoids......Page 306
11.5. Erm. 15189. Diagrams for Ten Double Bisected Trapezoids......Page 308
11.6. AO 17264. A Problem for a Chain of 3 Bisected Quadrilaterals......Page 313
11.7. VAT 7621 # 1. A 2 · 9-striped trapezoid......Page 317
11.8. VAT 7531. Cross-wise striped trapezoids. .......Page 318
11.9. TMS 23. Confluent Quadrilateral Bisections in Two Directions......Page 320
11.10. Erm. 15073. Divided Trapezoids in a Recombination Text......Page 325
12.1. Hippocrates’ Lunes According to Alexander......Page 330
12.2. Hippocrates’ Lunes According to Eudemus......Page 332
12.3 a. BR 10-12. The ‘bow field’......Page 337
12.3 b. BR 13-15. The ‘boat field’......Page 338
12.3 c. BR 16-18. The ‘barleycorn field’......Page 339
12.3 e. BR 22-24. The ‘lyre-window’......Page 340
12.3 f. BR 25. The ‘lyre-window of 3’......Page 341
12.4. W 23291-x § 1. A Late Babylonian Double Segment and Lune......Page 342
12.5. A Remark by Neugebauer Concerning BM 15285 # 33......Page 347
Introduction......Page 348
13.1. Determinate Problems in Book I of Diophantus’ Arithmetica......Page 349
13.2 a. Ar. II.8 (Sesiano, GA (1990), 84)......Page 353
13.2 b. Ar. II.9 (Sesiano, GA (1990), 85)......Page 355
13.2 c. Ar. II.10 (Sesiano, GA (1990), 86)......Page 357
13.2 d. Ar. II.19 (Sesiano, GA (1990), 86)......Page 358
13.3. Ar. “V”.9. Diophantus’ Method of Approximation to Limits......Page 359
13.4. Ar. III.19. A Square Number Equal to a Sum of Two Squares in Four Different Ways......Page 362
Everywhere rational cyclic quadrilaterals......Page 364
Diophantus’ Ar. III.19, Birectangles, and the OB Composition Rule......Page 366
13.5. Ar. “V”.30. An Applied Problem and Quadratic Inequalities An indeterminate combined price problem......Page 370
13.6. Ar. “VI”. A Theme Text with Equations for Right Triangles......Page 373
Ar. “VI”.16. A right triangle with a rational bisector......Page 378
13.7. Ar. V.7-12. A Section of a Theme Text with Cubic Problems......Page 379
13.8. Ar. IV.17. Another Appearance of the Term ‘Representable’......Page 381
14.1. Metrica I.8 / Dioptra 31. Heron’s Triangle Area Rule......Page 382
14.2. Two Simple Metric Algebra Proofs of the Triangle Area Rule......Page 384
14.3. Simple Proofs of Special Cases of Brahmagupta’s Area Rule......Page 386
14.4. Simple Proofs of Special Cases of Ptolemy’s Diagonal Rule......Page 389
14.6. A Proof of Brahmagupta’s Diagonal Rule in the General Case......Page 391
15. Theon of Smyrna’s Side and Diagonal Numbers and Ascending Infinite Chains of Birectangles......Page 394
15.1. The Greek Side and Diagonal Numbers Algorithm......Page 396
15.2. MLC 2078. The Old Babylonian Spiral Chain Algorithm . .......Page 398
15.3. Side and Diagonal Numbers When Sq. p = Sq. q · D – 1......Page 402
15.4. Side and Diagonal Numbers When Sq. p = Sq. q · D + 1......Page 403
16.1. Metrica I.8 b. Heron’s Square Side Rule......Page 406
16.2. Heronic Square Side Approximations......Page 407
16.3. A New Explanation of Heron’s Accurate Square Side Rule......Page 408
16.4. Third Approximations in Ptolemy’s Syntaxis I.10......Page 411
16.5. The General Case of Formal Multiplications......Page 412
16.6. A New Explanation of the Archimedian Estimates for Sqs. 3......Page 413
The additive and subtractive square side rules......Page 415
Late and Old Babylonian approximations to sqs. 2......Page 417
Late and Old Babylonian approximations to sqs. 3......Page 418
Late and Old Babylonian exact computations of square sides......Page 420
17.1. Theaetetus 147 C-D. Theodorus’ Metric Algebra Lesson......Page 426
17.2. A Number-Theoretical Explanation of Theodorus’ Method......Page 427
17.3. An Anthyphairetic Explanation of Theodorus’ Method......Page 428
17.4. A Metric Algebra Explanation of Theodorus’ Method......Page 430
18.1. Geometrica as a Compilation of Various Sources......Page 436
18.2. Geometrica mss AC......Page 438
18.3. Geometrica ms S 24......Page 441
18.4. Metrica 3.4. A Division of Figures Problem......Page 450
A.1.1. VAT 8393. A New Old Babylonian Single Problem Text......Page 452
A.1.2. VAT 8393. About the Clay Tablet......Page 461
Appendix 2. A Catalog of Babylonian Geometric Figures......Page 464
Index of Texts, Propositions, and Lemmas......Page 468
Index of Subjects......Page 474
Bibliography......Page 484
Comparative Mesopotamian, Egyptian, and Babylonian Timelines......Page 497