Amazing traces of a babylonian origin in Greek mathematics:
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| Language: | English |
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2007
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| Physical Description: | XX, 476 S. Ill., graph. Darst. |
| ISBN: | 9789812704528 |
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| 100 | 1 | |a Friberg, Jöran |d 1934- |e Verfasser |0 (DE-588)1051628709 |4 aut | |
| 245 | 1 | 0 | |a Amazing traces of a babylonian origin in Greek mathematics |c Jöran Friberg |
| 264 | 1 | |a New Jersey [u.a.] |b World Scientific |c 2007 | |
| 300 | |a XX, 476 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
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| _version_ | 1804136654066155520 |
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| adam_text | Contents
Preface
v
1. Elements
II and Babylonian Metric Algebra
1
1.1.
Greek Lettered Diagrams vs. OB Metric Algebra Diagrams
................2
1.2. El. II.2-3
and the Three Basic Quadratic Equations
......................7
1.3.
El. II.4, II.7 and the Two Basic Additive Quadratic-Linear Systems of Equations
10
1.4. El. II.5-6
and the Two Basic Rectangular-Linear Systems of Equations
..... 12
1.5. El. II.8
and the Two Basic Subtractive Quadratic-Linear Systems of Equations
14
1.6.
El. II.9-
10,
Constructive Counterparts to
El. II.4
and
11.7 ................ 16
1.7.
El.
П.
11 *
and II.
14*,
Constructive Counterparts to El. II.5-6
............. 18
1.8.
El.
11.12-13,
Constructive Counterparts to
El. II.8......................22
1.9.
Summary. The Three Parts of Elements II
............................24
1.10.
An Old Babylonian Catalog Text with Metric Algebra Problems
.........27
1.11.
A Large Old Babylonian Catalog Text of a Similar Kind
...............29
1.12.
Old Babylonian Solutions to Metric Algebra Problems
.................35
1.12
a. Old Babylonian problems for rectangles and squares
...............35
1.12
b. Old Babylonian problems for circles and chords
...................42
1.12
с
Old Babylonian problems for non-symmetric trapezoids
............48
1.13.
Late Babylonian Solutions to Metric Algebra Problems
................50
1.13
a. Problems for rectangles and squares
............................50
The seed measure of a hundred-cubit-square. Metric squaring
...........51
A rectangle of given front and seed measure. Metric division
............53
A square of given seed measure. Metric square side computation
.........54
A rectangle of given side-sum and seed measure. Basic problem of type
В
1
a
55
A rectangle of given side-difference and seed measure. Type Bib
.........57
A square band of given width and seed measure. Type B3b
.............58
1.13 b. Problems for circles
.........................................59
A circle of given seed measure divided into five bands of equal width
.....59
A circle of given circumference divided into five bands of equal width
.... 61
A Seleucid pole-against-a-wall problem
.............................64
Seleucid parallels to El. II.
14*
(systems of equations of type
В
1
a)
........66
1.14.
Old Akkadian Square Expansion and Square Contraction Rules
..........68
1.15.
The Long History of Metric
Algebra
in Mesopotamia
..................69
xv
xvi
Amazing Traces of a Babylonian Origin in Greek Mathematics
2.
El.
1.47
and the Old Babylonian Diagonal Rule
73
2.1.
Euclid s Proof of El.
1.47..........................................73
2.2.
Pappus Proof of a Generalization of El.
1.47..........................74
2.3.
The Original Discovery of the OB Diagonal Rule for Rectangles
..........76
2.4.
Chains of Triangles, Trapezoids, or Rectangles
........................79
3.
Lemma El. X.28/29 la, Plimpton
322,
and Babylonian igi-igi.bi Problems
83
3.1.
Greek Generating Rules for Diagonal Triples of Numbers
................83
Euclid s Generating Rule in the Lemma El. X.28/29 la
.................83
The Generating Rules Attributed to Pythagoras and Plato
...............84
Metric Algebra Derivations of the Greek Generating Rules
.............85
3.2.
Old Babylonian igi-igi.bi Problems
.................................86
3.3.
Plimpton
322:
A Table of Parameters for igi-igi.bi Problems
.............88
4.
Lemma El. X.32/33 and an Old Babylonian Geometric Progression
95
4.1.
Division of a right triangle into a pair of right sub-triangles
...............95
4.2.
A Metric Algebra Proof of Lemma El. X.32/33
........................96
4.3.
An Old Babylonian Chain of Right Sub-Triangles
......................97
5.
Elements X and Babylonian Metric Algebra
101
5.1.
The Pivotal Propositions and Lemmas in Elements X
................... 101
A Concise Outline of the Contents of Elements X
.................... 102
5.2.
Binomials and Apotomes, Majors and Minors
........................ 103
5.3.
Euclid s Application of Areas and Babylonian Metric Division
.......... 113
5.4.
Quadratic-Rectangular Systems of Equations of Type B5
............... 116
6.
Elements IV and Old Babylonian Figures Within Figures
123
6.1.
Elements IV, a Well Organized Geometric Theme Text
.................. 123
An Outline of the Contents
oí
Elements IV
......................... 123
6.2.
Figures Within Figures in Mesopotamian Mathematics
................. 125
7.
El. VI.30,
XIII. 1-12,
and Regular Polygons in Babylonian Mathematics
141
7.1.
El. VI.30:
Cutting a Straight Line in Extreme and Mean Ratio
............ 141
7.2.
Regular Pentagons and Equilateral Triangles in Elements
XIII........... 142
An Outline of the Contents of El.
ХШ.1-12
......................... 142
7.3.
An Extension of the Result in El.
ХШ.11
............................ 146
7.4.
An Alternative Proof of the Crucial Proposition
El. XIII.8
.............. 149
7.5.
Metric Analysis of the Regular Pentagon in Terms of its Side
........... 151
7.6.
Metric Analysis of the Regular Octagon
............................ 155
7.7.
Equilateral Triangles in Babylonian Mathematics
..................... 159
Contents xvii
7.8.
Regular
Polygons in
Babylonian Mathematics
........................ 160
7.9. Geometrie
Constructions in Mesopotamian Decorative Art
.............. 164
8. El. XIII.13-18
and Regular Polyhedrons in Babylonian Mathematics
171
8.1.
Regular Polyhedrons in Elements
ΧΠΙ
.............................. 171
An Outline of the Contents of
И
XIII.13-18
........................ 171
Conclusion
................................................... 181
8.2.
MS
3049 § 5.
The Inner Diagonal of a Gate
.......................... 181
8.3.
The Weight of an Old Babylonian Colossal Copper Icosahedron
......... 184
9.
Elements
XII
and Pyramids and Cones in Babylonian Mathematics
189
9.1.
Circles, Pyramids, Cones, and Spheres in Elements
XII ................ 189
9.2.
Pre-literate Plain Number Tokens from the Middle East in the Form of
Circular Lenses, Pyramids, Cylinders, Cones, and Spheres
.............. 192
9.3.
Pyramids and Cones in OB Mathematical Cuneiform Texts
............. 195
9.3
a. The volume and grain measure of a ridge pyramid
................. 196
9.3
b. The grain measure of a ridge pyramid truncated at mid-height
........200
9.3
с
Problems for cones and truncated cones
.........................202
9.4.
Pyramids and Cones in Ancient Chinese Mathematical Texts
............202
9.4 a. The fifth chapter in
Jiu
Zhang Suan Shu
.........................202
9.4
b. Liu
Hui
s
commentary to
Jiu
Zhang Suan Shu, Chapter V
............206
9.5.
A Possible Babylonian Derivation of the Volume of a Pyramid
..........207
10.
El.
1.43-44,
El. VI.24-29, Data
57-59, 84-86,
and Metric Algebra
211
10.1.
El.
1ЛЪ-АА
&
Data
57:
Parabolic Applications of Parallelograms
........212
10.2.
El. VI.
28 &
Data
58.
Elliptic Applications of Parallelograms
..........217
10.3.
El. VI.
29 &
Data
59.
Hyperbolic Applications of Parallelograms
.......219
10.4.
El. VI.25
and£taa55
..........................................220
10.5.
Data
84-85.
Rectangular-Linear Systems of Equations
................225
10.6.
Data
86.
A Quadratic-Rectangular System of Equations of Type B6
......227
10.7.
Zeuthen s Conjecture: Intersecting Hyperbolas
......................232
10.8.
A Kassite Series Text with Modified Systems of Types B5 and B6
......233
11.
Euclid
s
Lost Book On Divisions and Babylonian Striped Figures
235
11.1.
Selected Division Problems in On Divisions
.........................236
OD
1-2, 30-31.
To divide a triangle by lines parallel to the base
.........236
OD
3.
To bisect a triangle by a line through a point on a side
...........237
OD
4-5.
To divide
a trapezoid
by lines
parallel
to the base
..............237
OD
8,12.
To bisect
a trapezoid
by a line through a point on a side
.......238
OD
19-20.
To divide a triangle by a line through an interior point
........239
xviii
Amazing Traces
afa
Babylonian Origin in Greek Mathematics
OD
32.
To divide
a trapezoid
by a parallel in a given ratio
.............242
11.2.
Old Babylonian Problems for Striped Triangles
......................244
11.2
a. Str.
364 § 2.
A model problem for a 3-striped triangle
...........244
11.2
b. Str.
364 § 3.
A quadratic equation for a 2-striped triangle
........244
11.2
с
Str. 364 §§ 4-7.
Quadratic equations for 2-striped triangles
.......249
11.2
d. Str.
364 § 8.
Problems for 5-striped triangles
..................252
11.2
e. TMS
18.
A cleverly designed problem for a 2-striped triangle
.....255
11.2
f. MLC
1950.
An elegant solution procedure
....................258
11.2
g. VAT
8512.
Another cleverly designed problem
................259
11.2
h. YBC
4696.
A series of problems for a 2-striped triangle
.........261
11.2
і
.
MAH
16055.
A table of diagrams for 3-striped triangles
.........264
11.2
j.
IM 43996.
A 3-striped triangle divided in given ratios
...........267
11.3.
Old Babylonian Problems for 2-Striped Trapezoids
...................269
11.3
a. IM
58045,
an Old Akkadian problem for a bisected
trapezoid
.....269
11.3 b. VAT
8512,
interpreted as a problem for a bisected
trapezoid
.....271
11.3
с.
YBC
4675.
A problem for a bisected quadrilateral
.............272
11.3
d. YBC
4608.
A 2-striped
trapezoid
divided in the ratio
1: 3........274
11.3
e. Str.
367.
A 2-striped
trapezoid
divided in the ratio
29 : 51........277
11.3
f.
1st.
Si.
269.
Five 2-striped trapezoids divided in the ratio
60 : 1___279
11.3
g. The Bloom of Thymaridas and its relation to
Old Babylonian generating equations for transversal triples
....... 282
11.3
h. Relations between diagonal triples and transversal triples
........ 283
11.4.
Old Babylonian Problems for 3-and 5-Striped Trapezoids
............. 285
11.5.
Erm.
15189.
Diagrams for Ten Double Bisected Trapezoids
............ 287
11.6.
АО
17264.
A Problem for a Chain of
3
Bisected Quadrilaterals
......... 292
11.7.
VAT
7621
#1.A2 ^-striped
trapezoid
........................... 296
11.8.
VAT
7531.
Cross-wise striped trapezoids
........................... 297
11.9.
TMS
23.
Confluent Quadrilateral Bisections in Two Directions
......... 299
11.10.
Erm.
15073.
Divided Trapezoids in a Recombination Text
............304
12.
Hippocrates
Lunes
and Babylonian Figures with Curved Boundaries
309
12.1.
Hippocrates
Lunes
According to Alexander
.........................309
12.2.
Hippocrates
Lunes
According to Eudemus
.........................311
12.3.
Some Geometric Figures in the OB Table of Constants BR
............316
12.3
a. BR
10-12.
The bow field
...................................316
12.3
b. BR
13-15.
The boat field
...................................317
12.3
c. BR
16-18.
The barleycorn field
..............................318
12.3
d. BR
19-21.
The ox-eye
.....................................319
12.3
e. BR
22-24.
The lyre-window
................................319
12.3
f. BR
25.
The lyre-window of
3 ................................320
12.4.
W
23291-x
§ 1.
A Late Babylonian Double Segment and
Lune
.........321
Contents xix
12.5.
A Remark by Neugebauer Concerning BM
15285 #33................326
13.
Traces of Babylonian Metric Algebra in the Arithmetica of Diophantus
327
13.1.
Determinate Problems in Book I of Oiophaxuas1 Arithmetica
............328
13.1.
Four Basic Examples in Book II of Diophantus Arithmetica
............332
13.2
a.
Ar. II.8
(Sesiano, GA
(1990), 84)............................332
13.2
Ь.Аг. П.9
(Sesiano, GA
(1990), 85)............................334
13.2
с
Ar. 11.10
(Sesiano,
GA
(1990), 86)...........................336
13.2
d.
Ar. 11.19
(Sesiano, GA
(1990), 86)...........................337
13.2.
Ar. V.Ç. Diophantus
Method of Approximation to Limits
...........338
13.3. Ar. III.19.
A Square Number Equal to a Sum of Two Squares
in Four Different Ways
......................................... 341
Everywhere rational cyclic quadrilaterals
........................... 343
Diophantus
Ar.
HI.
19,
Birectangles, and the OB Composition Rule
..... 345
13.4.
Аг. У .ЗО.
An Applied Problem and Quadratic Inequalities
........... 349
An indeterminate combined price problem
......................... 349
13.5. Ar.
VI . A
Theme Text with Equations for Right Triangles
............ 352
Ar.
VI .
16.
A right triangle with a rational bisector
.................. 357
13.6. Ar. V.7-12.
A Section of a Theme Text with Cubic Problems
........... 358
13.7. Ar. IV.17.
Another Appearance of the Term Representable
........... 360
14.
Heron s, Ptolemy s, and Brahmagupta s Area and Diagonal Rules
361
14.1.
Metrica
1.8
1 Dioptra
31.
Heron s Triangle Area Rule
.................361
14.2.
Two Simple Metric Algebra Proofs of the Triangle Area Rule
..........363
14.3.
Simple Proofs of Special Cases of Brahmagupta s Area Rule
...........365
14.4.
Simple Proofs of Special Cases of Ptolemy s Diagonal Rule
...........368
14.5.
Simple Proofs of Special Cases of Brahmagupta s Diagonal Rule
.......370
14.6.
A Proof of Brahmagupta s Diagonal Rule in the General Case
..........370
15.
Theon of Smyrna s Side and Diagonal Numbers and Ascending Infinite
Chains of Birectangles
373
15.1.
The Greek Side and Diagonal Numbers Algorithm
....................375
15.2.
MLC
2078.
The Old Babylonian Spiral Chain Algorithm
..............377
15.3.
Side and Diagonal Numbers When Sq.
ρ
= Sq.
q
■
D
- 1...............381
15.4.
Side
and
Diagonal
Numbers When
Sq.
í> =
Sq. q-D+ 1...............382
16.
Greek and Babylonian Square Side Approximations
385
16.1.
Metrica
1.8
b.
Heron s Square Side Rule
............................385
16.2.
Heronic Square Side Approximations
.............................386
16.3.
A New Explanation of Heron s Accurate Square Side Rule
............387
xx
Amazing Traces of a Babylonian Origin in Greek Mathematics
16.4.
Third Approximations in Ptolemy s
Syntaxis
1.10.................... 390
16.5.
The General Case of Formal Multiplications
........................ 391
16.6.
A New Explanation of the Archimedian Estimates for Sqs.
3........... 392
16.7.
Examples of Babylonian Square Side Approximations
................ 394
The additive and subtractive square side rales
....................... 394
Late and Old Babylonian approximations to sqs.
2 ................... 395
Late and Old Babylonian approximations to sqs.
3 ................... 397
A Late Babylonian approximation to sqs.
5......................... 399
Late and Old Babylonian exact computations of square sides
........... 399
17. Theodoms
of Cyrene s Irrationality Proof and Descending Infinite Chains
of Birectangles
405
17.1.
Theaetetus
147
С
-D.
Theodorus Metric Algebra Lesson
............... 405
17.2.
A Number-Theoretical Explanation of Theodoras Method
............ 406
17.3.
An Anthyphairetic Explanation of Theodorus Method
................ 407
17.4.
A Metric Algebra Explanation of Theodorus Method
................ 409
18.
The Pseudo-Heronic
Geometrica
415
18.1.
Geometrica
as a Compilation of Various Sources
.................... 415
18.2.
Geometrica mss
AC
........................................... 417
18.3.
Geometrica
ms
S
24........................................... 420
18.4.
Metrica
3.4.
A
Division
of
Figures
Problem
......................... 429
Appendix
1.
A Chain of
Trapezoide
with Fixed Diagonals
431
A.I.I. VAT
8393.
A New Old Babylonian Single Problem Text
............. 431
A.
1.2.
VAT
8393.
About the Clay Tablet
............................... 440
Appendix
2.
A Catalog of Babylonian Geometric Figures
443
Index of Texts, Propositions, and Lemmas
447
Index of Subjects
, 453
Bibliography
463
Comparative Mesopotamian, Egyptian, and Babylonian Timelines
476
|
| adam_txt |
Contents
Preface
v
1. Elements
II and Babylonian Metric Algebra
1
1.1.
Greek Lettered Diagrams vs. OB Metric Algebra Diagrams
.2
1.2. El. II.2-3
and the Three Basic Quadratic Equations
.7
1.3.
El. II.4, II.7 and the Two Basic Additive Quadratic-Linear Systems of Equations
10
1.4. El. II.5-6
and the Two Basic Rectangular-Linear Systems of Equations
. 12
1.5. El. II.8
and the Two Basic Subtractive Quadratic-Linear Systems of Equations
14
1.6.
El. II.9-
10,
Constructive Counterparts to
El. II.4
and
11.7 . 16
1.7.
El.
П.
11 *
and II.
14*,
Constructive Counterparts to El. II.5-6
. 18
1.8.
El.
11.12-13,
Constructive Counterparts to
El. II.8.22
1.9.
Summary. The Three Parts of Elements II
.24
1.10.
An Old Babylonian Catalog Text with Metric Algebra Problems
.27
1.11.
A Large Old Babylonian Catalog Text of a Similar Kind
.29
1.12.
Old Babylonian Solutions to Metric Algebra Problems
.35
1.12
a. Old Babylonian problems for rectangles and squares
.35
1.12
b. Old Babylonian problems for circles and chords
.42
1.12
с
Old Babylonian problems for non-symmetric trapezoids
.48
1.13.
Late Babylonian Solutions to Metric Algebra Problems
.50
1.13
a. Problems for rectangles and squares
.50
The seed measure of a hundred-cubit-square. Metric squaring
.51
A rectangle of given front and seed measure. Metric division
.53
A square of given seed measure. Metric square side computation
.54
A rectangle of given side-sum and seed measure. Basic problem of type
В
1
a
55
A rectangle of given side-difference and seed measure. Type Bib
.57
A square band of given width and seed measure. Type B3b
.58
1.13 b. Problems for circles
.59
A circle of given seed measure divided into five bands of equal width
.59
A circle of given circumference divided into five bands of equal width
. 61
A Seleucid pole-against-a-wall problem
.64
Seleucid parallels to El. II.
14*
(systems of equations of type
В
1
a)
.66
1.14.
Old Akkadian Square Expansion and Square Contraction Rules
.68
1.15.
The Long History of Metric
Algebra
in Mesopotamia
.69
xv
xvi
Amazing Traces of a Babylonian Origin in Greek Mathematics
2.
El.
1.47
and the Old Babylonian Diagonal Rule
73
2.1.
Euclid's Proof of El.
1.47.73
2.2.
Pappus' Proof of a Generalization of El.
1.47.74
2.3.
The Original Discovery of the OB Diagonal Rule for Rectangles
.76
2.4.
Chains of Triangles, Trapezoids, or Rectangles
.79
3.
Lemma El. X.28/29 la, Plimpton
322,
and Babylonian igi-igi.bi Problems
83
3.1.
Greek Generating Rules for Diagonal Triples of Numbers
.83
Euclid's Generating Rule in the Lemma El. X.28/29 la
.83
The Generating Rules Attributed to Pythagoras and Plato
.84
Metric Algebra Derivations of the Greek Generating Rules
.85
3.2.
Old Babylonian igi-igi.bi Problems
.86
3.3.
Plimpton
322:
A Table of Parameters for igi-igi.bi Problems
.88
4.
Lemma El. X.32/33 and an Old Babylonian Geometric Progression
95
4.1.
Division of a right triangle into a pair of right sub-triangles
.95
4.2.
A Metric Algebra Proof of Lemma El. X.32/33
.96
4.3.
An Old Babylonian Chain of Right Sub-Triangles
.97
5.
Elements X and Babylonian Metric Algebra
101
5.1.
The Pivotal Propositions and Lemmas in Elements X
. 101
A Concise Outline of the Contents of Elements X
. 102
5.2.
Binomials and Apotomes, Majors and Minors
. 103
5.3.
Euclid's Application of Areas and Babylonian Metric Division
. 113
5.4.
Quadratic-Rectangular Systems of Equations of Type B5
. 116
6.
Elements IV and Old Babylonian Figures Within Figures
123
6.1.
Elements IV, a Well Organized Geometric Theme Text
. 123
An Outline of the Contents
oí
Elements IV
. 123
6.2.
Figures Within Figures in Mesopotamian Mathematics
. 125
7.
El. VI.30,
XIII. 1-12,
and Regular Polygons in Babylonian Mathematics
141
7.1.
El. VI.30:
Cutting a Straight Line in Extreme and Mean Ratio
. 141
7.2.
Regular Pentagons and Equilateral Triangles in Elements
XIII. 142
An Outline of the Contents of El.
ХШ.1-12
. 142
7.3.
An Extension of the Result in El.
ХШ.11
. 146
7.4.
An Alternative Proof of the Crucial Proposition
El. XIII.8
. 149
7.5.
Metric Analysis of the Regular Pentagon in Terms of its Side
. 151
7.6.
Metric Analysis of the Regular Octagon
. 155
7.7.
Equilateral Triangles in Babylonian Mathematics
. 159
Contents xvii
7.8.
Regular
Polygons in
Babylonian Mathematics
. 160
7.9. Geometrie
Constructions in Mesopotamian Decorative Art
. 164
8. El. XIII.13-18
and Regular Polyhedrons in Babylonian Mathematics
171
8.1.
Regular Polyhedrons in Elements
ΧΠΙ
. 171
An Outline of the Contents of
И
XIII.13-18
. 171
Conclusion
. 181
8.2.
MS
3049 § 5.
The Inner Diagonal of a Gate
. 181
8.3.
The Weight of an Old Babylonian Colossal Copper Icosahedron
. 184
9.
Elements
XII
and Pyramids and Cones in Babylonian Mathematics
189
9.1.
Circles, Pyramids, Cones, and Spheres in Elements
XII . 189
9.2.
Pre-literate Plain Number Tokens from the Middle East in the Form of
Circular Lenses, Pyramids, Cylinders, Cones, and Spheres
. 192
9.3.
Pyramids and Cones in OB Mathematical Cuneiform Texts
. 195
9.3
a. The volume and grain measure of a ridge pyramid
. 196
9.3
b. The grain measure of a ridge pyramid truncated at mid-height
.200
9.3
с
Problems for cones and truncated cones
.202
9.4.
Pyramids and Cones in Ancient Chinese Mathematical Texts
.202
9.4 a. The fifth chapter in
Jiu
Zhang Suan Shu
.202
9.4
b. Liu
Hui'
s
commentary to
Jiu
Zhang Suan Shu, Chapter V
.206
9.5.
A Possible Babylonian Derivation of the Volume of a Pyramid
.207
10.
El.
1.43-44,
El. VI.24-29, Data
57-59, 84-86,
and Metric Algebra
211
10.1.
El.
1ЛЪ-АА
&
Data
57:
Parabolic Applications of Parallelograms
.212
10.2.
El. VI.
28 &
Data
58.
Elliptic Applications of Parallelograms
.217
10.3.
El. VI.
29 &
Data
59.
Hyperbolic Applications of Parallelograms
.219
10.4.
El. VI.25
and£taa55
.220
10.5.
Data
84-85.
Rectangular-Linear Systems of Equations
.225
10.6.
Data
86.
A Quadratic-Rectangular System of Equations of Type B6
.227
10.7.
Zeuthen's Conjecture: Intersecting Hyperbolas
.232
10.8.
A Kassite Series Text with Modified Systems of Types B5 and B6
.233
11.
Euclid'
s
Lost Book On Divisions and Babylonian Striped Figures
235
11.1.
Selected Division Problems in On Divisions
.236
OD
1-2, 30-31.
To divide a triangle by lines parallel to the base
.236
OD
3.
To bisect a triangle by a line through a point on a side
.237
OD
4-5.
To divide
a trapezoid
by lines
parallel
to the base
.237
OD
8,12.
To bisect
a trapezoid
by a line through a point on a side
.238
OD
19-20.
To divide a triangle by a line through an interior point
.239
xviii
Amazing Traces
afa
Babylonian Origin in Greek Mathematics
OD
32.
To divide
a trapezoid
by a parallel in a given ratio
.242
11.2.
Old Babylonian Problems for Striped Triangles
.244
11.2
a. Str.
364 § 2.
A model problem for a 3-striped triangle
.244
11.2
b. Str.
364 § 3.
A quadratic equation for a 2-striped triangle
.244
11.2
с
Str. 364 §§ 4-7.
Quadratic equations for 2-striped triangles
.249
11.2
d. Str.
364 § 8.
Problems for 5-striped triangles
.252
11.2
e. TMS
18.
A cleverly designed problem for a 2-striped triangle
.255
11.2
f. MLC
1950.
An elegant solution procedure
.258
11.2
g. VAT
8512.
Another cleverly designed problem
.259
11.2
h. YBC
4696.
A series of problems for a 2-striped triangle
.261
11.2
і
.
MAH
16055.
A table of diagrams for 3-striped triangles
.264
11.2
j.
IM 43996.
A 3-striped triangle divided in given ratios
.267
11.3.
Old Babylonian Problems for 2-Striped Trapezoids
.269
11.3
a. IM
58045,
an Old Akkadian problem for a bisected
trapezoid
.269
11.3 b. VAT
8512,
interpreted as a problem for a bisected
trapezoid
.271
11.3
с.
YBC
4675.
A problem for a bisected quadrilateral
.272
11.3
d. YBC
4608.
A 2-striped
trapezoid
divided in the ratio
1: 3.274
11.3
e. Str.
367.
A 2-striped
trapezoid
divided in the ratio
29 : 51.277
11.3
f.
1st.
Si.
269.
Five 2-striped trapezoids divided in the ratio
60 : 1_279
11.3
g. The Bloom of Thymaridas and its relation to
Old Babylonian generating equations for transversal triples
. 282
11.3
h. Relations between diagonal triples and transversal triples
. 283
11.4.
Old Babylonian Problems for 3-and 5-Striped Trapezoids
. 285
11.5.
Erm.
15189.
Diagrams for Ten Double Bisected Trapezoids
. 287
11.6.
АО
17264.
A Problem for a Chain of
3
Bisected Quadrilaterals
. 292
11.7.
VAT
7621
#1.A2 ^-striped
trapezoid
. 296
11.8.
VAT
7531.
Cross-wise striped trapezoids
. 297
11.9.
TMS
23.
Confluent Quadrilateral Bisections in Two Directions
. 299
11.10.
Erm.
15073.
Divided Trapezoids in a Recombination Text
.304
12.
Hippocrates'
Lunes
and Babylonian Figures with Curved Boundaries
309
12.1.
Hippocrates'
Lunes
According to Alexander
.309
12.2.
Hippocrates'
Lunes
According to Eudemus
.311
12.3.
Some Geometric Figures in the OB Table of Constants BR
.316
12.3
a. BR
10-12.
The 'bow field'
.316
12.3
b. BR
13-15.
The 'boat field'
.317
12.3
c. BR
16-18.
The 'barleycorn field'
.318
12.3
d. BR
19-21.
The 'ox-eye'
.319
12.3
e. BR
22-24.
The 'lyre-window'
.319
12.3
f. BR
25.
The 'lyre-window of
3'.320
12.4.
W
23291-x
§ 1.
A Late Babylonian Double Segment and
Lune
.321
Contents xix
12.5.
A Remark by Neugebauer Concerning BM
15285 #33.326
13.
Traces of Babylonian Metric Algebra in the Arithmetica of Diophantus
327
13.1.
Determinate Problems in Book I of Oiophaxuas1 Arithmetica
.328
13.1.
Four Basic Examples in Book II of Diophantus' Arithmetica
.332
13.2
a.
Ar. II.8
(Sesiano, GA
(1990), 84).332
13.2
Ь.Аг. П.9
(Sesiano, GA
(1990), 85).334
13.2
с
Ar. 11.10
(Sesiano,
GA
(1990), 86).336
13.2
d.
Ar. 11.19
(Sesiano, GA
(1990), 86).337
13.2.
Ar. "V.Ç. Diophantus'
Method of Approximation to Limits
.338
13.3. Ar. III.19.
A Square Number Equal to a Sum of Two Squares
in Four Different Ways
. 341
Everywhere rational cyclic quadrilaterals
. 343
Diophantus'
Ar.
HI.
19,
Birectangles, and the OB Composition Rule
. 345
13.4.
Аг. "У'.ЗО.
An Applied Problem and Quadratic Inequalities
. 349
An indeterminate combined price problem
. 349
13.5. Ar.
"VI". A
Theme Text with Equations for Right Triangles
. 352
Ar.
"VI".
16.
A right triangle with a rational bisector
. 357
13.6. Ar. V.7-12.
A Section of a Theme Text with Cubic Problems
. 358
13.7. Ar. IV.17.
Another Appearance of the Term 'Representable'
. 360
14.
Heron's, Ptolemy's, and Brahmagupta's Area and Diagonal Rules
361
14.1.
Metrica
1.8
1 Dioptra
31.
Heron's Triangle Area Rule
.361
14.2.
Two Simple Metric Algebra Proofs of the Triangle Area Rule
.363
14.3.
Simple Proofs of Special Cases of Brahmagupta's Area Rule
.365
14.4.
Simple Proofs of Special Cases of Ptolemy's Diagonal Rule
.368
14.5.
Simple Proofs of Special Cases of Brahmagupta's Diagonal Rule
.370
14.6.
A Proof of Brahmagupta's Diagonal Rule in the General Case
.370
15.
Theon of Smyrna's Side and Diagonal Numbers and Ascending Infinite
Chains of Birectangles
373
15.1.
The Greek Side and Diagonal Numbers Algorithm
.375
15.2.
MLC
2078.
The Old Babylonian Spiral Chain Algorithm
.377
15.3.
Side and Diagonal Numbers When Sq.
ρ
= Sq.
q
■
D
- 1.381
15.4.
Side
and
Diagonal
Numbers When
Sq.
í> =
Sq. q-D+ 1.382
16.
Greek and Babylonian Square Side Approximations
385
16.1.
Metrica
1.8
b.
Heron's Square Side Rule
.385
16.2.
Heronic Square Side Approximations
.386
16.3.
A New Explanation of Heron's Accurate Square Side Rule
.387
xx
Amazing Traces of a Babylonian Origin in Greek Mathematics
16.4.
Third Approximations in Ptolemy's
Syntaxis
1.10. 390
16.5.
The General Case of Formal Multiplications
. 391
16.6.
A New Explanation of the Archimedian Estimates for Sqs.
3. 392
16.7.
Examples of Babylonian Square Side Approximations
. 394
The additive and subtractive square side rales
. 394
Late and Old Babylonian approximations to sqs.
2 . 395
Late and Old Babylonian approximations to sqs.
3 . 397
A Late Babylonian approximation to sqs.
5. 399
Late and Old Babylonian exact computations of square sides
. 399
17. Theodoms
of Cyrene's Irrationality Proof and Descending Infinite Chains
of Birectangles
405
17.1.
Theaetetus
147
С
-D.
Theodorus' Metric Algebra Lesson
. 405
17.2.
A Number-Theoretical Explanation of Theodoras' Method
. 406
17.3.
An Anthyphairetic Explanation of Theodorus' Method
. 407
17.4.
A Metric Algebra Explanation of Theodorus' Method
. 409
18.
The Pseudo-Heronic
Geometrica
415
18.1.
Geometrica
as a Compilation of Various Sources
. 415
18.2.
Geometrica mss
AC
. 417
18.3.
Geometrica
ms
S
24. 420
18.4.
Metrica
3.4.
A
Division
of
Figures
Problem
. 429
Appendix
1.
A Chain of
Trapezoide
with Fixed Diagonals
431
A.I.I. VAT
8393.
A New Old Babylonian Single Problem Text
. 431
A.
1.2.
VAT
8393.
About the Clay Tablet
. 440
Appendix
2.
A Catalog of Babylonian Geometric Figures
443
Index of Texts, Propositions, and Lemmas
447
Index of Subjects
, 453
Bibliography
463
Comparative Mesopotamian, Egyptian, and Babylonian Timelines
476 |
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| author | Friberg, Jöran 1934- |
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| discipline | Mathematik Geschichte |
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| geographic_facet | Babylonien Griechenland Altertum |
| id | DE-604.BV022540680 |
| illustrated | Illustrated |
| index_date | 2024-07-02T18:10:09Z |
| indexdate | 2024-07-09T20:59:50Z |
| institution | BVB |
| isbn | 9789812704528 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015747128 |
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| physical | XX, 476 S. Ill., graph. Darst. |
| publishDate | 2007 |
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| publisher | World Scientific |
| record_format | marc |
| spelling | Friberg, Jöran 1934- Verfasser (DE-588)1051628709 aut Amazing traces of a babylonian origin in Greek mathematics Jöran Friberg New Jersey [u.a.] World Scientific 2007 XX, 476 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Mathematik (DE-588)4037944-9 gnd rswk-swf Babylonien (DE-588)4004102-5 gnd rswk-swf Griechenland Altertum (DE-588)4093976-5 gnd rswk-swf Griechenland Altertum (DE-588)4093976-5 g Mathematik (DE-588)4037944-9 s Babylonien (DE-588)4004102-5 g DE-604 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015747128&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Friberg, Jöran 1934- Amazing traces of a babylonian origin in Greek mathematics Mathematik (DE-588)4037944-9 gnd |
| subject_GND | (DE-588)4037944-9 (DE-588)4004102-5 (DE-588)4093976-5 |
| title | Amazing traces of a babylonian origin in Greek mathematics |
| title_auth | Amazing traces of a babylonian origin in Greek mathematics |
| title_exact_search | Amazing traces of a babylonian origin in Greek mathematics |
| title_exact_search_txtP | Amazing traces of a babylonian origin in Greek mathematics |
| title_full | Amazing traces of a babylonian origin in Greek mathematics Jöran Friberg |
| title_fullStr | Amazing traces of a babylonian origin in Greek mathematics Jöran Friberg |
| title_full_unstemmed | Amazing traces of a babylonian origin in Greek mathematics Jöran Friberg |
| title_short | Amazing traces of a babylonian origin in Greek mathematics |
| title_sort | amazing traces of a babylonian origin in greek mathematics |
| topic | Mathematik (DE-588)4037944-9 gnd |
| topic_facet | Mathematik Babylonien Griechenland Altertum |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015747128&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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