calculating in floating sexagesimal place value notation...

Calculating in floating sexagesimal place value notation, 4000 years ago

ARITH 22 22nd Symposium on Computer Arithmetic

Lyon June 22-24, 2015

Christine Proust Laboratoire SPHère (CNRS & Université Paris-Diderot)

François Thureau-Dangin (1872-1944) 1938 Textes Mathématiques Babyloniens.

Otto Neugebauer (1899-1990) 1935-1937 Mathematische Keilschrifttexte I Neugebauer & Sachs, 1945 Mathematical Cuneiform Texts.

School tablet from Nippur , Old Babylonian period (HS 217a, University of Jena)

Multiplication table by 9

20x9 = 180 = 3x60 In base 60 : 3:0 But the scribes wrote: 3 This is puzzling.

20x9 is written 3 20x9 = 3 The square of 30 is written 15 30x30 = 15 The square root of 15 is written 30

1 9 2 18 3 27 4 36 5 45 6 54 7 1:3 8 1:12 9 1:21 10 1:30 11 1:39 12 1:48 13 1:57 14 2:6

15 2:15 16 2:24 17 2:33 18 2:42 20-1 2:51 20 3 30 4:30 40 6 50 7:30 8.20 a-ra2 1 8.20

HS 2

17a

Is the lack of graphical systems to determine the place of the unit in the number an imperfection of

the cuneiform script?

(Sachs 1947)

The algorithm for reciprocals according to Abraham Sachs (1947)

yes

Scribal schools in Old Babylonian period (ca. 2000-1600 BCE)

Scribal school in Nippur

The curriculum at Nippur

Level Content Elementary Metrological lists: capacities, weights, surfaces, lengths

Metrological tables: capacities, weights, surfaces, lengths, heights Numerical tables: reciprocals, multiplications, squares Tables of square roots and cube roots

Intermediate Exercises: multiplications, reciprocals, surface and volume calculations

School tablet from Nippur , Old Babylonian period (Ist Ni 5376, Istanbul)

A list of proverbs 1. Someone who cannot produce "a-a”,

from where will he achieve fluent speech?

2. A scribe who does not know Sumerian -- from where will he produce a translation?

3. The scribe skilled in counting is deficient in writing. The scribe skilled in writing is deficient in counting.

4. A chattering scribe. Its guilt is great. 5. A junior scribe is too concerned with

feeding his hunger; he does not pay attention to the scribal art.

6. A disgraced scribe becomes a priest. …

capacity

weight

surface

length

Metrological lists

School tablet from Nippur, Old Babylonian period (HS 249, University of Jena)

Metrological lists: measurements of capacity


1 sila ca. 1 liter 1 ban ca. 10 liters

1/3 sila 1/2 sila 2/3 sila 5/6 sila 1 sila 1 1/3 sila 1 1/2 sila 1 2/3 sila 1 5/6 sila 2 sila 3 sila 4 sila 5 sila 6 sila 7 sila 8 sila 9 sila 1 ban še 1 ban 1 sila 1 ban 2 sila 1 ban 3 sila 3

1 šusi 10 2 šusi 20 3 šusi 30 4 šusi 40 5 šusi 50 6 šusi 1 7 šusi 1:10 8 šusi 1:20 9 šusi 1:30 1/3 kuš 1:40 1/2kuš 2:30 2/3 kuš 3:20 5/6 kuš 4:10 1 kuš 5 1 1/3 kuš 6:40 1 1/2kuš 7:30 1 2/3 kuš 8:20 2 kuš 10


1 šusi = 1 finger, ca. 1.6 cm 1 kuš = 1 cubit, ca. 50 cm

Reciprocals Multiplication tables by 50 45 44:26:40 40 36 30 25 24 22:30 20 18 16:40 16 15 12:30 12 10 9 8:20 8

7:30 7:12 7 6:40 6 5 4:30 4 3:45 3:20 3 2:30 2:24 2 1:40 1:30 1:20 1:15

Table of squares

Numerical tables

School tablet from Nippur, Old Babylonian period (Ist Ni 2733, Istanbul Museum)

2 30 3 20 4 15 5 12 6 10 8 7:30 9 6:40 10 6 12 5 15 4 16 3:45 18 3:20 20 3 24 2:30 25 2:24 27 2:13:20 30 2 32 1:52:30 36 1:40 40 1:30 45 1:20

48 1:15 50 1:12 54 1:6:40 1 1 1:4 56:15 1:21 44:26:40

Table of reciprocals

School tablet from unknown provenance, Old Babylonian period (MS 3874, Schøyen collection, copy Friberg)

The division by a number was performed by mean of the multiplication by the reciprocal of this number.

5 ÷ 30 = 5 × 2 = 10 2 ÷ 44:26:40 = 2 × 1:21 = 2:42

2 30 3 20 4 15 5 12 6 10 8 7:30 9 6:40 10 6 12 5 15 4 16 3:45 18 3:20 20 3 24 2:30 25 2:24 27 2:13:20 30 2 32 1:52:30 36 1:40 40 1:30 45 1:20 48 1:15 50 1:12 54 1:6:40 1 1 1:4 56:15 1:21 44:26:40

Multiplying 4 50 4 50 ----------------------- 41 40 3 20 3 20 16 ----------------------- 23 21 40

17:46:40 Its reciprocal 3:22:30 ==================== 17:46:40 [9] 2:40 22:[30] 3:22:[30] A chattering scribe, his guilt is great. ----------------------------------- A chattering scribe, his guilt is great ====================== 17:46:40 9 1:30*

Reciprocal

Calculation of surface

2.10 2.10 4.26!.40

1/3 kuš3 3 šu-si its side --------------------- Its surface what ? --------------------- Its surface 13 še igi-4! gal2 še ==============

Ni 18 School tablet from Nippur

Istanbul Museum

SPVN

Metrological notations

Table of lengths length measures → SPVN

Table of surfaces SPVN → surface measure

Multiplication table (SPVN)

School tablets from Nippur

2 times 2 4 3 times 3 9 4 times 4 16 5 times 5 25 6 times 6 36 7 times 7 49 8 times 8 1.4 9 times 9 1.21 10 times 10 1.40 11 times 11 2.1 12 times 12 2.24

Extract of table of squares

6 šu-si 1 7 šu-si 1.10 8 šu-si 1.20 9 šu-si 1.30 1/3 kuš3 1.40 1/3 kuš3 1 šu-si 1.50 1/3 kuš3 2 šu-si 2 1/3 kuš3 3 šu-si 2.10 1/3 kuš3 4 šu-si 2.20

Extract of metrological table for lengths

1/3 sar 20 1/2 sar 30 2/3 sar 40 5/6 sar 50 1 sar 1 1 1/3 sar 1.20 1 1/2 sar 1.30 1 2/3 sar 1.40 1 5/6 sar 1.50

Extract of metrological table for surfaces

SPVN SPVN

SPVN Metrological notations Metrological notations

Obverse 4:26:40 Its reciprocal 13:30 ============== reverse 4:26:40 9 40* 1:30 13:30 *error of the scribe: he wrote 41 instead of 40

The algorithm for reciprocal

4:26:40 9 40 1:30 13:30

2 30 3 20 4 15 5 12 6 10 8 7:30 9 6:40 10 6 12 5 15 4 16 3:45 18 3:20 20 3 24 2:30 25 2:24 27 2:13:20 30 2 32 1:52:30 36 1:40 40 1:30 45 1:20 48 1:15 50 1:12 54 1:6:40 1 1 1:4 56:15 1:21 44:26:40

4:26:40 ends with the regular number 6:40, so 4:26:40 is "divisible" by 6:40 . To divide 4:26:40 by 6:40, we must multiply 4:26:40 by the reciprocal of 6:40. The reciprocal of 6:40 is 9 . The number 9 is placed in the right hand column. 4:26:40 multiplied by 9 is 40, thus, 40 is the quotient of 4:26:40 by 6:40. This quotient is placed in the left hand column. The reciprocal of 40 is 1:30. The number 1:30 is placed in the right hand olumn. To find the reciprocal of 4:26:40, one only has to multiply the reciprocals of the factors of 4:26:40, that is to say, the numbers 9 and 1:30 placed in the right hand column. This product is 13:30, the reciprocal sought.

4:26:40 9 40 1:30 13:30

2 30 3 20 4 15 5 12 6 10 8 7:30 9 6:40 10 6 12 5 15 4 16 3:45 18 3:20 20 3 24 2:30 25 2:24 27 2:13:20 30 2 32 1:52:30 36 1:40 40 1:30 45 1:20 48 1:15 50 1:12 54 1:6:40 1 1 1:4 56:15 1:21 44:26:40

Left hand column: 4:26:40 = 6:40 × 40

Right hand column

9 × 1:30 = 13:30

2.5 4.10 8.20 16.40 33.20 1.6.40 2.13.20 4.26.40 … 10.6.48.53.20

CBS 1215 Provenance: unknown Datation: OB period (ca. 1800 BCE) University of Pennsylvania, Philadelphia Publication: Sachs 1947, Babylonian Mathematical Texts 1 Copy Robson 2000: 14

Obverse: 3 columns, # 1-16 Reverse: 3 columns, #16-21

Entries

colonne I colonne II colonne III #1

2.5 12 25 2.24 28.48 1.15 36 1.40 2.5

#2 4.10 6 25 2.24 14.24 2.30 36 1 .40 4.10

#3 8.20 3 25 2.24 7.12 5 36 1.40 8.20

#4 16.40 9 2.30 24 3.[36] [1.40] 6 10 15sic.40

#5 33.20 18 10 6 1.48 1.15 2.15 4 8sic 6.40 26.40 33.20

#6 1.6.40 9 10 6 54 1.6.40

#7 [2].13.20 18 [40] 1.30 [27] 2.13.20

#8 4.26.40 9 40 1.30 13.30 2 27 2.13.20

4.26.40

#9 8.53.20 18 2.40 22.30 6.45 1.20 9 6.40 8.53.20

#10 17.46.40 9 2.40 22.30 3.22.30 2 6.45 1.20 9 6.40 8.53.20

17.46.40 #11

36sic.2sic3.20 18 10.40 1.[30] [16] 3.4[5] 5.37.30 [1.41.1]5 4 [6.45] 1.20 [9] 6.40 [8.53].20

[35.33].20 #12

[1].11.6.[40] 9 10.40 1.[30] 16 3.4[5] 5.37.30 50.37.30 2 1.41.15 4 6.4[5] 1.20 9 6.40 [8.5]3.[20] 35.33.20

1.11.6.40 #13

2.22.13.20 [18] 42.40 22.30 16 3.45 1.24.22.30 25.18.45* [16]

6.45 1.20 9 [6.40] 8.53.20 [2.2]2.13.[20]

#14 4.44.26.40 [9] 42.40 2[2.30] 16 3.[45] 1.24.22.30 [12.3]9.22.30 [2] [25.18].45* [16] [6.45] [1.20] [9] [6.40] [8].53.20 [2.22.13. 20] [4.44.26.40]

#15 [9.28].53.[20] [18] 2.50.40 [1.30] [4.16] [3.45] [16] [3.45] 14.3.[45] [2]1.5.3[7.30] [6.19.4]1.15 [4] [25.18.45]* [16] [6.45] [1.20] [9] [6.40] [8.53.20] 2.[22.13.20] 9.[28.53.20]

#16 18.57.[46.40] [9] [2.50.40] [1.30] 4.[16] [3.45] 16 [3.45] [14].3. [45] [21.5.37.30] [3.9.50.37.30] [2] [6.19.41.15] [4]

(suite sur le revers)

colonne III colonne II colonne I #21

10.6.48.53.20 18 3.2.2.40 22.[30] 1.8.16 3.4[5] 4.16 3.[45] 16 3.[45] 1[4.3.4]5 52.44.[3.4]5 19.46.31.24.22.[30] 5.55.57.25.18.4[5] 16 1.34.55.18.45* 16 25.18.45* [16] 6.45 [1.20] 9 [6.40] 8.53.20 2.22.13. 20 37.55.33.20 10.6.48.53.20

#19 [2.31.42.13.20 18] [45.30.40 1.30] [1.8.16 3.45] [4.16 3.45] 16 [3.45] 14.[3.45] 5[2.44.3.45] 1.18sic.6.[5.37.30] 23.43.49.[41.15] [4] 1.[3]4.55.18.45* [16] [25].18.45* 1[6] [6].45 1.[20] [9] 6.40 8.53.20 2.22.13.20 37.55.3[3.20] 2.31.42.13.[20]

#20 5.3.24.26.40 [9] 45.30.40 1.30 1.8.16 3.45 4.16 3.45 16 3.45 14.3.45 5[2.44].3.45 1.19.6.5.37.30 11.51.54.50.37.30 2 23.43.49.41.15 4 1.34.55.18.45* 16 25.18.45* 16 6.45 1.20 9 6.[40] 8.53.20 2.22.13. 20 37.55.33.20 2.31.42.13.20 5.3.24.26.40

#16 (suite) [25.18.45* 16] [6.45 1.20] [9 6.40] [8.53.20] [2.22.13. 20] [9.28.53.20] [18.57.46.40]

#17 [37.55.33.20 18] [11.22.40 22.30] [4.16 3.45] [16 3.45] [14.3.45] [5.16.24.22.30] [1.34.55.18.45* 16] [25.18.45* 16] [6.45 1.20] 9 [6.40] [8.53.20] 2.22.13.[20] 37.55.33.[20]

#18 1.15.51.6.40 9 11.22.40 22.30 4.16 3.45 16 [3.45] 14.[3.45] 5.16.[24.22.30] 47.27.[39.22.30 2] [1.34.55.18.45* 16] [25.18.45* 16] [6.45 1.20] [9 6.40] 8.[53.20] 2.2[2.13. 20] 37.55.[33.20] 1.15.51.[6.40]

Obverse (columns from left to right) Reverse (columns from right to left)

CBS 1215: transliteration

2:5 12 25 2:24 28:48 1:15 36 1:40

2:5

CBS 1215 #1 2 30 3 20 4 15 5 12 6 10 8 7:30 9 6:40 10 6 12 5 15 4 16 3:45 18 3:20 20 3 24 2:30 25 2:24 27 2:13:20 30 2 32 1:52:30 36 1:40 40 1:30 45 1:20 48 1:15 50 1:12 54 1:6:40 1 1 1:4 56:15 1:21 44:26:40

2:5 12 25 2:24 28:48 1:15 36 1:40

2:5

The factorization of 2:5 appears in the left column: 2:5 × 5 = 25 The factorization of the reciprocal of 2:5 appears in the right hand column: 12 × 2:24 = 28:48 The reciprocal of 2:5 is thus 28:48.

CBS 1215 #8 2 30 3 20 4 15 5 12 6 10 8 7:30 9 6:40 10 6 12 5 15 4 16 3:45 18 3:20 20 3 24 2:30 25 2:24 27 2:13:20 30 2 32 1:52:30 36 1:40 40 1:30 45 1:20 48 1:15 50 1:12 54 1:6:40 1 1 1:4 56:15 1:21 44:26:40

4.26.40 9 40 1.30 13.30 2 27 2.13.20 4.26.40

Transcription Copy [Robson 2000, p. 23] 5.3.24.26.40 [9] 45.30.40 1.30 1.8.16 3.45 4.16 3.45 16 3.45 14.3.45 5[2.44].3.45 1.19.6.5.37.30 11.51.54.50.37.30 2 23.43.49.41.15 4 1.34.55.18.45* 16 25.18.45* 16 6.45 1.20 9 6.[40] 8.53.20 2.22.13. 20 37.55.33.20 2.31.42.13.20 5.3.24.26.40

CBS 1215 #20: iteration 2 30 3 20 4 15 5 12 6 10 8 7:30 9 6:40 10 6 12 5 15 4 16 3:45 18 3:20 20 3 24 2:30 25 2:24 27 2:13:20 30 2 32 1:52:30 36 1:40 40 1:30 45 1:20 48 1:15 50 1:12 54 1:6:40 1 1 1:4 56:15 1:21 44:26:40

Left hand column: 5:3:24:26:40 = 6:40 × 40 × 16 × 16 × 16 Right hand column: 9 × 1:30 × 3:45 × 3:45 × 3:45 = 11:51:54:50:37:30

2 30 3 20 4 15 5 12 6 10 8 7:30 9 6:40 10 6 12 5 15 4 16 3:45 18 3:20 20 3 24 2:30 25 2:24 27 2:13:20 30 2 32 1:52:30 36 1:40 40 1:30 45 1:20 48 1:15 50 1:12 54 1:6:40 1 1 1:4 56:15 1:21 44:26:40

Transcription 5:3:24:26:40 [9] 45:30:40 1:30 1:8:16 3:45 4:16 3:45 16 3:45 14:3:45 5[2:44]:3:45 1:19:6:5:37:30 11:51:54:50:37:30

2 23:43:49:41:15 4 1:34:55:18:45 16 25:18:45 16 6:45 1:20 9 6:[40] 8:53:20 2:2:22:2:13: 20 37:55:33:20 2:31:42:13:20 5:3:24:26:40

Explanation: n → 5:3:24:26:40 Factors of n factors of inv(n) 6:40 9 40 1:30 16 3:45 16 3:45 16 3:45 Products 14:3:45 5[2:44]:3:45 1:19:6:5:37:30 11:51:54:50:37:30 n→ 11:51:54:50:37:30 30 2 15 4 3:45 16 3:45 16 45 1:20 9 6:[40] Products 8:53:20 2:22:13: 20 37:55:33:20 2:31:42:13:20 5:3:24:26:40

Aspects of CBS 1215: Paradigmatic examples: Calculation performed on common values (2.5 and doubles) that provide the ability to control the outputs, known in advance. The execution of the algorithm is guided by a codified layout: the layout of the numbers indicates the nature of the operation carried out and the meaning of the calculations. The implementation of the reverse algorithm illustrates the property "the reciprocal of a reciprocal of a number is this number itself."

L’algorithme d’inversion selon l’interprétation d’Abraham Sachs (1947)

Abstract numbers are not quantities, but calculation tools

Plimpton 322 Columbia University, New York Old Babylonian period Provenience unknown (probably Southern Mesopotamia)

Sexagesimal number = number with finite sexagesimal development Sexagesimal rectangle = length, width and diagonal are represented by sexagesimals numbers Diagonal rule = Pythagorean rule

1

59,30 1,24,30

obv I' II' III' IV' 1 ta-k ]I- il- ti şi - li - ip - tim íb.si8 sag íb.si8 şi-li-ip-tim mu.bi.im2 ša 1 in ]-na-as-sà-hu-ú-ma sag i- [il ]-lu-ú3 1 59 15 1 59 2 49 ki 14 1 56 56 58 14 56 15 56 7 3 12 1 ki 25 1 55 7 41 15 33 45 1 16 41 1 50 49 ki 36 1 53 10 29 32 52 16 3 31 49 5 9 1 ki 47 1 48 54 1 40 1 5 1 37 ki 58 1 47 6 41 40 5 19 8 1 ki 69 1 43 11 56 28 26 40 38 11 59 1 ki 7

10 1 41 33 59 3 45 13 19 20 49 ki 811 1 38 33 36 36 9 1 12 49 ki 912 1 35 10 2 28 27 24 26 40 1 22 41 2 16 1 ki 1013 1 33 45 45 1 15 ki 1114 1 29 21 54 2 15 27 59 48 49 ki 1215 1 27 3 45 7 12 1 4 49 ki 1316 1 25 48 51 35 6 40 29 31 53 49 ki 1417 1 23 13 46 40 56 53 ki 15

δ β

1

d b

l

× l

The square (takiltum) of the diagonal (from) which 1 is torn out (i.e. subtracted) and (that of) the width comes up.

width diagonal Line n°

δ2 b d

δ² − 1 = β²

1 1² + 59,0,15 59,30² = 1,59,0,15 1,24,30²

1

2,49 1,59 59,30 1,24,30

I’ II’ III’ IV’

The square of the diagonal (from) which 1 is torn out (i.e. subtracted) and (that of) the width comes up.

width diagonal line

1,59,0,15 1,59 2,49 n°1

Reduced rectangle Unit rectangle

1,59,0,15 1,24,30² − 1 1² = 59,0,15 59,30²

txt r /s = in descending order txt r /s = in descending orderNo s r r/s = α s/r = 1/ α No s r r/s = α s/r = 1/ α

1 5 12 2 24 25 20 5 8 1 36 37 302 27 64 2 22 13 20 25 18 45 21 16 25 1 33 45 38 243 32 75 2 20 37 30 25 36 22 2 3 1 30 404 54 125 2 18 53 20 25 55 12 23 27 40 1 28 53 20 40 305 4 9 2 15 26 40 24 25 36 1 26 24 41 406 9 20 2 13 20 27 0 25 45 64 1 25 20 42 11 157 25 54 2 9 36 27 46 40 26 32 45 1 24 22 30 42 408 15 32 2 8 28 7 30 27 18 25 1 23 20 43 129 12 25 2 5 28 48 28 20 27 1 21 44 26 40

10 40 81 2 1 30 29 37 46 40 29 3 4 1 20 4511 1 2 2 30 30 25 32 1 16 48 46 52 3012 25 48 1 55 12 31 15 31 4 5 1 15 4813 8 15 1 52 30 32 0 32 5 6 1 12 5014 27 50 1 51 6 40 32 24 33 27 32 1 11 6 40 50 37 3015 5 9 1 48 33 20 34 8 9 1 7 30 53 2016 9 16 1 46 40 33 45 35 9 10 1 6 40 5417 16 27 1 41 15 35 33 20 36 25 27 1 4 48 55 33 2018 3 5 1 40 36 37 15 16 1 4 56 1519 50 81 1 37 12 37 2 13 20 38 24 25 1 2 30 57 36

1<r/s < (<2;25) r is a regular number s is a 1-place regular number r/s is irreducible

List of all the values of r/s (John Britton according to Price 1964)

21+δ = ½(r/s + s/r) β = ½(r/s - s/r) We have: δ2 - β² = 1

β

β δ δ2 = 1+β2 b d No

[-I'] [0'] I' II' III' IV' sag şi-li-ip-tum ta]-ki-il-ti şi-li-ip - tim íb.si8 sag íb.si8 şi-li-ip-tim mu.bi.im

šá 1 in]-na-as-šà-hu-ú-ma sag i-[il-lu]-ú59 30 1 24 30 1 59 0 15 1 59 2 49 ki 158 27 17 30 1 23 46 2 30 1 56 56 58 14 50 6 15 56 7 1 20 25 ki 257 30 45 1 23 6 45 1 55 7 41 15 33 45 1 16 41 1 50 49 ki 356 29 4 1 22 24 16 1 53 10 29 32 52 16 3 31 49 5 9 1 ki 454 10 1 20 50 1 48 54 1 40 1 5 1 37 ki 553 10 1 20 10 1 47 6 41 40 5 19 8 1 ki 6 obv50 54 40 1 18 41 20 1 43 11 56 28 26 40 38 11 59 1 ki 749 56 15 1 18 3 45 1 41 33 45 14 3 45 13 19 20 49 ki 848 6 1 16 54 1 38 33 36 36 8 1 12 49 ki 945 56 6 40 1 15 33 53 20 1 35 10 2 28 27 24 26 40 1 22 2 16 1 ki 1045 1 15 1 33 45 45 1 15 ki 1141 58 30 1 13 13 30 1 29 21 54 2 15 27 59 48 49 ki 1240 15 1 12 15 1 27 0 3 45 2 41 4 49 ki 1339 21 20 1 11 45 20 1 25 48 51 35 6 40 29 31 53 49 ki 1437 20 1 10 40 1 23 13 46 40 28 53 ki 15

36 27 30 1 10 12 30 1 22 9 12 36 15 2 55 5 37 ki 1632 50 50 1 8 24 10 1 17 58 56 24 1 40 7 53 16 25 ki 17 lo.e32 1 8 1 17 4 8 17 ki 1830 4 53 20 1 7 7 6 40 1 15 4 53 43 54 4 26 40 1 7 41 2 31 1 ki 1929 15 1 6 45 1 14 15 33 45 39 1 29 ki 2027 40 30 1 6 4 30 1 12 45 54 20 15 6 9 14 41 ki 2125 1 5 1 10 25 5 13 ki 2224 11 40 1 4 41 40 1 9 45 22 16 6 40 14 31 38 49 ki 2322 22 1 4 2 1 8 20 16 4 11 11 32 1 ki 2421 34 22 30 1 3 45 37 30 1 7 45 23 26 38 26 15 34 31 1 42 1 ki 2520 51 15 1 3 31 15 1 7 14 53 46 33 45 16 41 50 49 ki 26 rev20 4 1 3 16 1 6 42 40 16 5 1 15 49 ki 2718 16 40 1 2 43 20 1 5 34 4 37 46 40 5 29 18 49 ki 2817 30 1 2 30 1 5 6 15 7 25 ki 2914 57 45 1 1 50 15 1 3 43 52 35 3 45 6 39 27 29 ki 3013 30 1 1 30 1 3 2 15 9 41 ki 3111 1 1 1 2 1 11 1 1 ki 3210 14 35 1 0 52 5 1 1 44 55 12 40 25 4 55 29 13 ki 337 5 1 0 25 1 0 50 10 25 17 2 25 ki 346 20 1 0 20 1 0 40 6 40 19 3 1 ki 354 37 20 1 0 10 40 1 0 21 21 53 46 40 1 44 22 34 ki 363 52 30 1 0 7 30 1 0 15 0 56 15 31 8 1 ki 37 u.e.2 27 1 0 3 1 0 6 0 9 49 20 1 ki 38

Reconstruction of complete tablet, by John Britton

A B 2

3 4 5 6 8 9 10 12 15 16 18 20 24 25 27 30 32 36 40 45 48 50 54 1

1:4 1:12 1:15 1:20 1:21 1:30 1:36 1:40 1:48 2:5

2 3 4 5 6 8 9 10 12 15 16 18 20 24 25 27 30 32 36 40 45 48 50 54 1

A = 1-place value regular numbers + 2-place value regular numbers until 2:5 B = 1-place value regular numbers C = {a/b, a є A and b є B} = {{1},{2},{3},{4},{5},{6},{8},{9},{10},{12},{15},{16},{18},{20},{24},{25},{27},{30},{32},{36},{40},{45},{48},{50},{54},{1,4},{1,12},{1,15},{1,20},{1,21},{1,30},{1,36},{1,40},{1,48},{2,5},{2,8},{2,15},{2,24},{2,30},{2,40},{2,42},{3,12},{3,20},{3,36},{3,45},{4,3},{4,10},{4,16},{4,30},{4,48},{5,20},{5,24},{6,15},{6,24},{6,40},{6,45},{7,12},{7,30},{8,6},{8,20},{9,36},{10,25},{10,40},{10,48},{11,15},{12,30},{12,48},{13,20},{13,30},{14,24},{16,12},{16,40},{18,45},{19,12},{20,15},{20,50},{21,20},{21,36},{22,30},{26,40},{28,48},{31,15},{32,24},{33,20},{33,45},{37,30},{38,24},{40,30},{41,40},{42,40},{43,12},{53,20},{56,15},{57,36},{1,2,30},{1,4,48},{1,6,40},{1,7,30},{1,16,48},{1,23,20},{1,25,20},{1,26,24},{1,33,45},{1,37,12},{1,41,15},{1,46,40},{1,52,30},{1,55,12},{2,1,30},{2,9,36},{2,13,20},{2,33,36},{2,36,15},{2,46,40},{2,48,45},{2,52,48},{3,7,30},{3,14,24},{3,22,30},{3,28,20},{3,33,20},{3,50,24},{4,19,12},{4,26,40},{4,41,15},{5,3,45},{5,12,30},{5,33,20},{5,37,30},{5,45,36},{6,56,40},{7,6,40},{7,40,48},{7,48,45},{8,26,15},{8,38,24},{8,53,20},{9,22,30},{10,7,30},{11,6,40},{11,31,12},{12,57,36},{13,53,20},{14,3,45},{15,21,36},{15,37,30},{16,52,30},{17,16,48},{17,46,40},{22,13,20},{23,2,24},{23,26,15},{25,18,45},{25,55,12},{27,46,40},{28,7,30},{30,43,12},{35,33,20},{38,52,48},{42,11,15},{44,26,40},{46,52,30},{50,37,30},{55,33,20},{1,11,6,40},{1,15,56,15},{1,24,22,30},{1,28,53,20},{1,51,6,40},{1,57,11,15},{2,18,53,20},{2,22,13,20},{2,31,52,30},{2,57,46,40},{3,42,13,20},{3,54,22,30},{4,37,46,40},{5,55,33,20},{7,24,26,40},{11,51,6,40},{14,48,53,20},{18,31,6,40},{23,42,13,20},{29,37,46,40},{37,2,13,20},{47,24,26,40},{1,32,35,33,20}}

Select numbers of C between 1 and 1+ √2 (<2,25), in the lexicographic order. This list is the same as the Price’s one. This list generates the first 15 entries of Plimpton 322, as well as the 23 additional entries reconstructed by Price, Britton, and others. These entries are obtained from the values n of C as follows: the diagonal is half the sum of n and its reciprocal: Column I’ contain the squares of these diagonals.

MS 3971 #3 (Friberg 2007, 252-3)

3 1-2

3 5-8

In order for you to see five diagonals: 1,4 (is) the igi, and the igibi 56.15 […]

3b 1 2 3 4

5 6 7

The 2nd (example). 1,40 the igi, 36 the igibi. 1,40 and 36 heap, 2,16 it gives. ½ of 2,16 break, 1,8 it gives. 1,8 square, 1,17,4 it gives. 1 from 1,17,4 tear off, 17,4 it gives. 17,4 makes 32 equalsided. 32, the width, it gives.

3c 1

2 3 4

5

The 3rd. 1,30 the igi, 40 the igibi. 1,30 and 40 heap, 2,10 it gives. ½ of 2,10 break, 1,5 it gives. 1,5 square, 1,10,25. 1 from 1,10,25 tear off, 10,25 it gives. 10,25 makes <25 equalsided>. 25, the 3rd width.

3d 1 2 3 4

5 6

The 4th. 1,20 the igi, 45 the igibi. 1,30 and 45 heap, 2,5 it gives. ½ of 2,5 break, 1,2,30 it gives. 1,2,30 square, 1,5,6,15. 1 from the length tear off, 5,6,15 it gives. 5,6,15 makes 17,30 equalsided. 17,30, the width of the 4th diagonal.

3e 1

2 3 4

5

The 5th. 1,12 the igi, 50 the igibi. 1,12 and 50 heap, 2,2 it gives. ½ of 2,2 break, 1,1. 1,1 square, 1,2,1. 1 from 1,2,1 tear off, 2,1 it gives. 2,1 makes 11equalsided. 11, the 5th width.

5 diagonals.

3b 1 2 3 4

5 6 7


u 1 40 36 2 16 1 8 1 17 4 1 17 4 32

1

Configuration of gnomon Configuration of diagonal rule

36

1,40

β δ δ2 = 1+β2 b d No

[-I'] [0'] I' II' III' IV' sag şi-li-ip-tum ta]-ki-il-ti şi-li-ip - tim íb.si8 sag íb.si8 şi-li-ip-tim mu.bi.im

šá 1 in]-na-as-šà-hu-ú-ma sag i-[il-lu]-ú59 30 1 24 30 1 59 0 15 1 59 2 49 ki 158 27 17 30 1 23 46 2 30 1 56 56 58 14 50 6 15 56 7 1 20 25 ki 257 30 45 1 23 6 45 1 55 7 41 15 33 45 1 16 41 1 50 49 ki 356 29 4 1 22 24 16 1 53 10 29 32 52 16 3 31 49 5 9 1 ki 454 10 1 20 50 1 48 54 1 40 1 5 1 37 ki 553 10 1 20 10 1 47 6 41 40 5 19 8 1 ki 6 obv50 54 40 1 18 41 20 1 43 11 56 28 26 40 38 11 59 1 ki 749 56 15 1 18 3 45 1 41 33 45 14 3 45 13 19 20 49 ki 848 6 1 16 54 1 38 33 36 36 8 1 12 49 ki 945 56 6 40 1 15 33 53 20 1 35 10 2 28 27 24 26 40 1 22 2 16 1 ki 1045 1 15 1 33 45 45 1 15 ki 1141 58 30 1 13 13 30 1 29 21 54 2 15 27 59 48 49 ki 1240 15 1 12 15 1 27 0 3 45 2 41 4 49 ki 1339 21 20 1 11 45 20 1 25 48 51 35 6 40 29 31 53 49 ki 1437 20 1 10 40 1 23 13 46 40 28 53 ki 15

36 27 30 1 10 12 30 1 22 9 12 36 15 2 55 5 37 ki 1632 50 50 1 8 24 10 1 17 58 56 24 1 40 7 53 16 25 ki 17 lo.e32 1 8 1 17 4 8 17 ki 1830 4 53 20 1 7 7 6 40 1 15 4 53 43 54 4 26 40 1 7 41 2 31 1 ki 1929 15 1 6 45 1 14 15 33 45 39 1 29 ki 2027 40 30 1 6 4 30 1 12 45 54 20 15 6 9 14 41 ki 2125 1 5 1 10 25 5 13 ki 2224 11 40 1 4 41 40 1 9 45 22 16 6 40 14 31 38 49 ki 2322 22 1 4 2 1 8 20 16 4 11 11 32 1 ki 2421 34 22 30 1 3 45 37 30 1 7 45 23 26 38 26 15 34 31 1 42 1 ki 2520 51 15 1 3 31 15 1 7 14 53 46 33 45 16 41 50 49 ki 26 rev20 4 1 3 16 1 6 42 40 16 5 1 15 49 ki 2718 16 40 1 2 43 20 1 5 34 4 37 46 40 5 29 18 49 ki 2817 30 1 2 30 1 5 6 15 7 25 ki 2914 57 45 1 1 50 15 1 3 43 52 35 3 45 6 39 27 29 ki 3013 30 1 1 30 1 3 2 15 9 41 ki 3111 1 1 1 2 1 11 1 1 ki 3210 14 35 1 0 52 5 1 1 44 55 12 40 25 4 55 29 13 ki 337 5 1 0 25 1 0 50 10 25 17 2 25 ki 346 20 1 0 20 1 0 40 6 40 19 3 1 ki 354 37 20 1 0 10 40 1 0 21 21 53 46 40 1 44 22 34 ki 363 52 30 1 0 7 30 1 0 15 0 56 15 31 8 1 ki 37 u.e.2 27 1 0 3 1 0 6 0 9 49 20 1 ki 38

Plimpton 322

3 1-2 3

5-8

In order for you to see five diagonals: 1,4 (is) the igi, and the igibi 56.15 […]

3b 1 2 3 4

5 6 7


3c 1 2 3 4

5

The 3rd. 1,30 the igi, 40 the igibi. 1,30 and 40 heap, 2,10 it gives. ½ of 2,10 break, 1,5 it gives. 1,5 square, 1,10,25. 1 from 1,10,25 tear off, 10,25 it gives. 10,25 makes <25 equalsided>. 25, the 3rd width.

3d 1 2 3 4

5 6

The 4th. 1,20 the igi, 45 the igibi. 1,30 and 45 heap, 2,5 it gives. ½ of 2,5 br eak, 1,2,30 it gives. 1,2,30 square, 1,5,6,15. 1 from the length tear off, 5,6,15 it gives. 5,6,15 makes 17,30 equalsided. 17,30, the width of the 4th diagonal.

3e 1 2 3 4

5

The 5th. 1,12 the igi, 50 the igibi. 1,12 and 50 heap, 2,2 it gives. ½ of 2,2 break, 1,1. 1,1 square, 1,2,1. 1 from 1,2,1 tear off, 2,1 it gives. 2,1 makes 11equalsided. 11, the 5th width.

5 diagonals.

calculating in floating sexagesimal place value notation...

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