270 valid triples below, between and above the lines 1-15 of Plimpton 322

CDLN 2023:5

Cuneiform Digital Library Notes (ISSN: 1546-6566)

Published on 2023-02-22

© The Authors

Creative Commons License This work is licensed under a Creative Commons Attribution 4.0 International License except when noted otherwise and in the case of artifact images which follow the CDLI terms of use.

Jens Kleb ORCID logo

mail@artefacts.info

artefacts.info

Abstract

The data table provided below contains 270 valid triples that correspond to the triples given in the well-known tablet Plimpton 322, but are above, between, or below the values of rows 1-15 in their slope ratios. The form, the selection of the triangle/rectangle sides and the order of the columns obtained were adhered to. From the possible triples calculated by the author, only those have been included whose number of sexagesimal digits could also have been given in the original table.

§1 Introduction

Previous research on Plimpton 322 have, in addition to the 15 triples listed on the actual tablet, discussed a total of 23 possible triples, resulting in an overall 38 specific triples being available from the scholarly literature (Neugebauer and Sachs 1945; Price 1964; Britton et al. 2011; Abdulaziz 2010). However, further calculations demonstrate that there are up to 270 existing and valid triples within the definition area given by the heading above column I of Plimpton 322 (l. 1-2).

These could also have fit in terms of the number of sexagesimal places on this or a similarly sized cuneiform tablet with the same column width. While the lower end of the table supplied below (see Table 1, last row) is bounded by the slope of zero where no triangle would be established, the upper end of the table is bounded by the subtraction of "1" defined in the heading of column I. Above the triple of: 59.29.0.27.16.54.36.33.45; 40.47.11; 41.8.1, (see Table 1, first row) the value in the first column exceeds the maximum value of the first decimal place (which starts from 1.00... and goes up to 59.59...). This creates a new sexagesimal place in front of the other. From this point (or more accurately "value") on, the subtraction of the "1" specified in the heading of Column I no longer generates a valid result.

For example, the following triple upwards of the first row of Table 1 with the values 1.9.56.43.14.24; 41.31; 41.49 is a little bit steeper and the value in Column I exceeds the limit just described. Nevertheless, this triple also has a valid solution, but the "1" must be subtracted from the "second" sexagesimal place of the first value to make the triple valid. For this reason, this triple would not satisfy the given definition area of Plimpton 322. In addition, there are numerous other valid triples, which, however, are not included in Table 1, due to the actual widths of columns II and III of the original tablet. (see Figure 1).

§2 Examples of additional triples

Regardless of how the triples or lines were used in the Old Babylonian period, the table provided here is made available for the interested public and for the purpose of checking and excluding previous hypotheses. For instance, it seems very unusual for a "trigonometric table" meant to aid further calculations, as proposed by some scholars (Mansfield and Wildberger 2017), to leave out existing triples between the lines recorded on Plimpton 322. Why should the creators leave wider steps between each triple, if a solution to adding further triples was available?

As an example: 1.31.9.9.25.42.2.15; 3.12.9; 5.28.41 is a valid triple situated between lines 11 and 12 of Plimpton 322. Moreover, the number of sexagesimal digits is shorter than in line 10 of the original tablet. Another example is: 1.40.6.47.17.32.36.15; 3.55.29; 6.12.1, which would fit between lines 8 and 9 of Plimpton 322. Because of that, the triples given in Table 1 were set in the same form as on the Plimpton 322. As examples can be checked for their validity easily and uncomplicatedly, this data may form a basis for further research.

Following Neugebauer, the scientifically accepted reading of Plimpton 322 is the following (Neugebauer and Sachs 1945, 38-41; Neugebauer 1969):

Column I: squared ratio of to = (l)2 ; Column II: Column III: d

Where l is the base of the triangle, b the width or height of the triangle, and d the Babylonian diagonal or hypotenuse of the triangle. Ratios given in Column I are intended as additional, giving the squared value of the 'normalized' diagonal of the specific triangle. The value given in Column I is then =   (d’)², because d’ is the coefficient of the diagonal of the specific triangle (Kleb 2021).

We find the value d’ in other cuneiform tablets also, where it is used for a normalised (and doubled) triangle with base length l of sexagesimal 1. Thus, it was named the “diagonal” also in the Old Babylonian period, as seen on YBC 7243, obv. 10 and TMS 25, obv. 31 and obv. 32. This diagonal value = d' is the same as the coefficient of the diagonal of its specific inscribed rectangle, (Square or Pythagorean triangle 3;4;5 ).

This again proves the efficient use of coefficients or their underlying ratios for the description and documentation of geometric shapes. Thus, a scale-independent, but fixed formal definition of a geometric object derived from the length ratios (i.e. side lengths at a gradient triangle) can be documented in a single value, if necessary. As YBC 7243 shows, 1.24.51.10 describes the diagonal of a general square, and as we can see from YBC 7289, this coefficient multiplied by the real side length can immediately give the real length of the diagonal, or vice versa. Might it be possible that elements of the data included in Table 1, including the square roots and minus 1 values of Column I, could also be found on cuneiform tablets that have not been transcribed yet?

A further refinement and gradation of the slope triangles is also conceivable through the use of approximations, so the coefficient of the diagonals of a square (see for example YBC 7243 and YBC 7289) can also be displayed in the same manner as seen in Plimpton 322 (see for example the highlighted triple 1.59.59.59.38.1.40 ( ∼ 2.0); 6; 8.29.7 in Table 1). It seems very plausible that, if one wanted to create a continuous system of gradient triangles, such approximate solutions for important ratios (1:1) could also be integrated if required.

Table 1: Triples above, between, and below l. 1-15 of Plimpton 322

Column I

Column II

Column III

Notes1

59.29.0.27.16.54.36.33.45

40.47.11

41.8.1

82,55°

56.45.4

7.28

7.32

 

55.22.14.57.53.21.40

44.14.31

44.38.49

 

52.20.28.20.25

1.25.59

1.26.49

 

51.4.8.53.38.3.59.3.45

22.38.35

22.52.05

 

49.56.23.4.0.15

13.59.29

14.08.01

 

48.43.35.46.36.9.36

8.38.08

8.43.32

 

46.3.49.56.17.46.40

1.00.25

1.01.05

 

44.56.45.3.45

26.31

26.49

 

41.27.41.29.35.23.26.15

3.23.33

3.26.3

 

39.33.50.45.36

31.03

31.27

 

37.24.26.20.47.52.25.40.44.26.40

16.17.29

16.30.49

 

36.30.6.15

11.55

12.05

 

35.37.6.9.37.7.36.15

28.14.31

28.38.49

 

33.40.46.8.30.3.45

22.51.59

23.12.49

 

32.8.33.21.40

33.29

34.01

 

31.21.58.24.04

2.45.19

2.48.01

 

29.39.43.42.57.46.40

48.11

49.01

 

28.56.47.54.36.33.45

1.24.35

1.26.5

 

27.37.44.32.39.50.24

4.18.01

4.22.49

 

26.7.52.50.29.56.17.46.40

9.01.25

9.12.05

 

25.30.09

4.57

5.03

 

23.32.33.45.56.15

37.59

38.49

 

22.28.31.48.9

1.32.41

1.34.49

 

21.56.10.59.29.8.16

17.9.31

17.33.49

 

20.45.11.6.40

13.20

13.40

 

20.15.22.30.6.15

1.45.19

1.48.01

 

19.9.56.41.46.46.40

12.47.11

13.08.01

 

18.42.28.21.34.37.44.3.45

4.29.19

4.36.49

 

18.18.4.54.27.21.40

4.59.29

5.08.01

 

17.51.52.57.36

5.08

5.17

 

16.54.22.43.2.42.57.46.40

1.47.41

1.51.01

 

16.30.14.3.45

15.45

16.15

 

15.45.46.23.10.24.36

12.48.27

13.14.3

 

15.14.59.25.7.38.26.15

10.3.59

10.24.49

 

14.55.14.8.33.7.9.56.17.46.40

16.07.05

16.41.13

 

14.34.1

3.41

3.49

 

14.13.19.7.6.46.40

10.54.31

11.18.49

 

13.27.53.21.40

21.11

22.01

 

13.08.48.54.32.15.56.15

5.34.35

5.48.5

 

12.51.52.50.00.03.45

13.46.41

14.20.49

 

12.33.41.26.24.38.24

4.15.01

4.25.49

 

12.26.57.0.27.24.1

3.23.1.11

3.31.42.1

 

11.53.46.00.11.6.40

59.25

1.02.05

 

11.6.8.12.22.45.10.26.24

1.45.56.49

1.51.4.1

 

11.0.12.47.36.29.37.46.40

7.06.59

7.27.49

 

10.44.45.58.21.33.45

49.57

52.27

 

10.31.3.7.26.31.57.21.40

44.25.29

46.42.1

72,04°

10.16.19.17.24

10.09

10.41

71,82°

10.1.57.10.20.41.0.26.40

1.7.37.19

1.11.16.1

 

9.43.59.26.56.29.42.42.57.46.40

15.57.29

16.50.49

 

9.30.25

35

37

 

9.17.10.34.1.50.25

6.54.31

7.18.49

 

9.5.25.15.44.6.20.15

37.55.29

40.12.1

 

8.52.47.41.30.19.33.45.36

1.45.16.1

1.51.44.49

 

8.48.6.58.00.15

5.35.11

5.56.1

 

8.35.54.55.12.54.54.8.26.15

17.38.31

18.46.1

 

8.25.5.00.25

32.41

34.49

 

8.13.26.56.16

40.19

43.1

 

7.47.54.51.51.6.40

5.52

6.17

 

7.37.11.38.26.15

4.07

4.25

 

7.27.40.35.31.58.46.33.45

2.2.0.41

2.11.6.49

 

7.17.27.14.24.21.36

4.10.49

4.30.1

 

7.13.39.57.36.42.41.46.40

1.52.17.59

2.0.58.49

 

7.3.47.17.32.11.29.3.45

13.07.57

14.10.27

 

6.55.1.7.56.0.11.6.40

8.45.25

9.28.5

 

6.45.36

36

39

 

6.36.24.46.45.55.44.26.40

42.37.19

46.16.1

 

6.28.15.25.15.4.42.37.21

1.43.57.21

1.53.3.29

 

6.16.15.3.45

9.11

10.01

 

6.7.47.12.20.22.31.33.45

21.44.35

23.46.5

 

6.0.16.21.2.15

1.29.29

1.38.1

 

5.52.12.6.49.53.4

2.4.8

2.16.17

 

5.49.12.40.59.15.2.15

1.27.49.11

1.36.30.1

 

5.34.29.26.40

1.17

1.25

 

5.27.3.20.25

25.19

28.1

 

5.20.27.19.4.3.33.53.26.15

5.33.21.29

6.9.46.1

 

5.13.21.59.8.55.42.14.24

1.42.44.49

1.54.16.1

 

5.10.44.23.7.6.40

3.3.59

3.24.49

 

5.3.53.26.18.30.56.15

1.4.31

1.12.1

 

4.57.48.38.3.30.25

4.46.41

5.20.49

 

4.57.12.38.10.4.52.21.41.5.6.15

4.34.52.7

5.7.40.25

 

4.51.16.50.24

9.49

11.01

 

4.36.57.7.10.51.51.6.40

1.42.41

1.56.1

 

4.30.56.15

3.45

4.15

 

4.25.4.18.15.22.30.41.40

26.37.19

30.16.1

 

4.19.51.53.33.12.9

4.3.21

4.37.29

 

4.14.16.23.6.52.32.4.16

1.41.13.1

1.55.47.49

 

4.7.17.3.47.52.6.0.11.6.40

15.15.53

17.32.25

 

4.2.0.15

3.29

4.1

 

3.56.51.17.20.26.40

2.34.31

2.58.49

 

3.54.56.49.4.14.58.12.20.15

22.46.3.11

26.23.4.1

 

3.45.33.26.40

4.59

5.49

 

3.40.48.58.9.3.45

15.43

18.25

 

3.36.36.28.30.0.56.15

12.55.29

15.12.01

58,24°

3.32.5.20.38.33.36

3.58.49

4.42.01

57,87°

3.30.24.53.40.0.4

47.29.59

56.10.49

 

3.26.3.0.27.25.50.23.26.15

16.38.31

19.46.1

 

3.22.10.34.29.26.40

11.5

13.13

 

3.18.1

1.31

1.49

 

3.10.21.40.40.9.8.0.36

1.38.16.1

1.58.44.49

 

3.8.53.34.25.58.31.6.40

49.28

59.53

 

3.5.3.53.26.15

3.51

4.41

 

3.1.40.3.39.2.26.0.25

41.00.41

50.06.49

 

3.1.19.57.2.38.5.27.36.15

16.22.55

20.1.37

 

2.58.1.13.21

9.21

11.29

 

2.54.27.51.59.46.15.6.40

15.32.19

19.11.01

 

2.53.8.49.58.54.50.3.45

36.37.11

45.18.01

 

2.50.1.18.42.13.50.51.51.6.40

14.37.29

18.10.49

 

2.46.40

4

5

53,13°

2.43.23.45.0.41.40

1.34.31

1.58.49

 

2.40.29.37.10.25.35.3.45

34.30.41

43.36.49

 

2.37.22.41.59.38.15.2.24

1.35.32.49

2.1.28.1

 

2.36.13.28.1

1.15.59

1.36.49

 

2.33.13.0.51.39.36.33.45

3.59.19

5.6.49

 

2.30.32.55.6.15

29.29

38.01

 

2.27.41.5.4

4.32

5.53

 

2.26.37.26.40.56.28.48.36

12.0.55.59

15.37.56.49

 

2.21.24.27.24.26.40

10.29

13.49

 

2.18.46.33.45

55

1.13

 

2.16.26.30.39.39.41.38.26.15

1.48.21.29

2.24.46.1

 

2.13.56.13.33.41.24

3.42.1

4.58.49

 

2.13.0.34.36.26.47.6.40

24.49.11

33.30.1

 

2.10.35.34.8.45.56.15

57.51

1.18.41

 

2.8.26.58.13.4.29.26.40

1.32.17

2.6.25

 

2.8.14.17.13.58.41.1.52.53.26.15

4.5.42.31

5.36.50.1

 

2.6.9

21

29

 

2.3.54.35.27.25.11.6.40

9.17.19

12.56.1

 

2.1.55.24.46.42.20.15.20.15

1.30.18.9

2.6.42.41

45,45°

! Square of √2 approx.: 1.24.51.10

1.59.59.59.38.1.40  (∼ 2.0)

6

8.29.7

YBC 7243+YBC 7289; 45,00°

1.59.47.34.27.27.58.38.7.21.36

37.26.6.49

52.59.14.1

 

1.59.00.15

1.59

2.49

1

1.57.8.2.9.6.31.41.48.30.25

5.37.14.41

8.2.52.49

 

1.56.56.58.14.50.06.15

56.07

1.20.25

2

1.55.07.41.15.33.45

1.16.41

1.50.49

3

1.54.56.54.43.23.8.58.34.45.50.4.37.46.40

1.39.13.10.31

2.23.30.22.49

 

1.54.23.8.31.6.8.30.52.2.57.46.40

8.1.58.59

11.38.59.49

 

1.53.10.29.32.52.16

3.31.49

5.9.1

4

1.52.27.6.59.24.9

18.41.59

27.22.49

 

1.50.44.17.0.28.3.20.30.51.51.6.40

1.59.10.41

2.56.4.1

 

1.50.34.8.41.43.8.22.44.3.45

58.45.19

1.26.52.49

 

1.48.54.01.40

1.05

1.37

5

1.47.06.41.40

5.19

8.01

6

1.45.31.35.31.36.53.28.21.33.45

4.38.44.41

7.4.22.49

 

1.45.22.13.1.56.23.32.51.36.17.46.40

9.23.29.19

14.18.44.1

 

1.44.52.50.23.4.20.24.50.15.38.16.17.46.40

17.30.49.29

26.46.22.49

 

1.43.49.39.19.46.33.57.36

1.25.28.1

2.11.32.49

 

1.43.11.56.28.26.40

38.11

59.1

7

1.42.10.27.11.59.45.34.18.40.28.26.40

35.22.11.1

55.3.9.49

 

1.41.42.33.47.51.56.32.36.20.26.0.11.6.40

1.26.26.39.25

2.14.59.22.5

 

1.41.33.45.14.03.45

13.19

20.49

8

1.40.6.47.17.32.36.15

3.55.29

6.12.1

 

1.39.58.13.1.43.26.28.57.40.25

56.24.55

1.29.13.13

 

1.39.31.21.38.45.33.27.10.11.46.35.23.27.24.26.40

1.17.2.22.28

2.2.15.2.53

 

1.38.33.36.36

8.01

12.49

9

1.37.59.8.47.23.52.33.4

5.58.3.11

9.35.4.1

 

1.37.2.58.16.9.53.22.57.46.40

53.2.31

1.25.50.49

 

1.36.29.27.1.9.22.38.26.15

13.51.51

22.32.41

 

1.35.42.43.22.51.50.58.19.13.21

34.17.18.49

56.8.2.1

 

1.35.10.02.28.27.24.26.40

1.22.41

2.16.1

10

1.33.45

45

1.15

11 ; 36,87°

1.32.29.44.5.18.12.27.2.15.56.15

14.7.47.29

23.50.20.1

 

1.32.22.19.11.52.16.6.40

5.17.19

8.56.1

 

1.31.9.9.25.42.2.15

3.12.9

5.28.41

 

1.30.39.22.19.27.37.5.11.6.40

6.1.52

10.22.17

 

1.29.50.50.40.5.41.37.4

1.19.20.49

2.17.40.1

 

1.29.21.54.02.15

27.59

48.49

12

1.28.13.22.35.19.15.34.29.26.40

59.15.25

1.44.46.5

 

1.28.6.37.40.40.47.15.56.15

13.8.31

23.16.1

 

1.27.00.03.45

2.41

4.49

13

1.26.53.30.27.28.14.23.1.56.31.26.24.1.40

55.31.44.7

1.39.48.56.25

 

1.25.48.51.35.06.40

29.31

53.49

14

1.25.22.33.25.44.2.25.21

4.20.7.59

7.57.8.49

 

1.24.45.55.14.9.6.53.37.38.26.15

11.25.17.29

21.7.50.1

 

1.24.39.43.27.25.11.37.24.10.12.20.44.26.40

51.55.50.31

1.36.13.2.49

 

1.24.20.19.1.18.27.57.21.58.31.6.40

1.43.10.41

3.12.4.1

 

1.24.14.11.23.25.33.20.31.20.51.33.45

13.33.31.19

25.16.38.49

 

1.23.38.37.39.7.15.33.8.38.24

31.23.14.1

59.2.6.49

 

1.23.13.46.40

28

53

15

1.22.14.59.43.6.35.40.20.36.11.0.25

42.5.28.41

1.20.55.38.49

 

1.22.9.12.36.15

2.55

5.37

16

1.21.12.11.7.30.14.3.45

9.30.41

18.36.49

 

1.21.6.34.26.34.6.21.40.4.37.46.40

5.7.29.19

10.2.44.1

 

1.20.49.0.7.30.23.8.12.31.56.2.57.46.40

14.54.34.29

29.22.37.49

 

1.20.11.16.19.14.24

2.54.1

5.46.49

 

1.19.48.47.37.36.16

8.37.11

17.18.1

 

1.19.12.13.2.56.59.25.52.47.24.26.40

4.46.24.19

9.41.39.1

 

1.18.55.39.25.56.48.37.24.27.10.51.51.6.40

14.33.29.29

29.43.42.49

 

1.18.50.25.49.43.21.33.45

2.59.19

6.6.49

 

1.17.58.56.24.01.40

7.53

16.25

17

1.17.53.52.34.9.35.24.25.6.54.3.45

12.35.1.19

26.15.8.49

 

1.17.04

32

1.08

18

1.16.43.44.36.8.51.4.57.45.36

1.19.12.17.11

2.49.37.38.1

 

1.16.10.48.11.46.11.36.17.46.40

28.2.31

1.0.50.49

 

1.15.51.11.42.49.51.57.57.30.56.15

9.8.17.33

19.59.20.3

 

1.15.23.55.27.53.38.36.9

1.7.32.49

2.29.28.1

 

1.15.04.53.43.54.04.26.40

1.07.41

2.31.01

19

1.14.15.33.45

39

1.29

20

1.13.32.8.4.23.23.10.6.15

27.21.29

1.3.46.1

 

1.13.27.52.7.34.35.10.25

2.43.43

6.22.25

 

1.12.45.54.20.15

6.09

14.41

21

1.12.28.53.28.18.58.45.27.6.40

2.33.55.59

6.10.56.49

 

1.12.1.15.38.20.33.46.40

1.15.32

3.4.53

 

1.11.44.50.10.3.20.15

5.53.59

14.34.49

 

1.11.6.7.32.59.8.27.24.26.40

9.17.29

23.30.49

 

1.11.2.19.38.18.43.51.9.8.26.15

2.44.42.31

6.57.50.1

 

1.10.25

25

65

22

1.9.45.22.16.06.40

14.31

38.49

23

1.9.10.37.15.12.23.45.56.15

20.51.29

57.16.1

 

1.9.7.12.53.15.5.21.41.16.7.54.4.26.40

25.15.50.31

1.9.33.2.49

 

1.8.56.33.53.15.45.55.53.5.11.6.40

1.18.10.41

3.37.4.1

 

1.8.33.46.58.8.0.9.36

56.40.1

2.40.20.49

 

1.8.20.16.04

11.11

32.01

24

1.7.58.22.22.59.53.12.12.33.59.18.31.6.40

11.32.2.8

33.40.38.17

 

1.7.48.30.3.45.9.55.18.4.29.26.40

1.14.48.17

3.40.26.25

 

1.7.45.23.26.38.26.15

34.31

1.42.1

25

1.7.14.53.46.33.45

16.41

50.49

26

1.7.11.54.43.40.37.33.45.48.54.1.40

11.58.14.35

36.34.28.5

 

1.7.2.35.14.27.38.2.41.32.29.49.37.46.40

43.21.44.11

2.13.47.5.1

 

1.6.42.40.16

5.1

15.49

27

1.6.30.52.37.25.45.38.24

1.38.51.11

5.15.52.1

 

1.6.11.48.20.27.3.43.32.20.44.26.40

2.10.9.19

7.5.24.1

 

1.6.3.13.18.7.55.56.59.4.1.58.31.6.40

10.17.29.29

33.59.42.49

 

1.6.0.31.9.12.5.56.29.3.45

33.45.19

1.51.52.49

 

1.5.34.04.37.46.40

5.29

18.49

28

1.5.06.15

7

25

29

1.4.42.4.46.34.55.24.36.33.45

53.44.41

3.19.22.49

 

1.4.39.43.22.57.30.4.37.46.40

12.2.31

44.50.49

 

1.4.16.43.13.49.21

1.46.49

6.54.1

 

1.4.7.29.28.11.8.26.40

2.56.59

11.37.49

 

1.3.52.37.51.43.45.25.48.26.40

42.53.49

2.54.7.1

 

1.3.43.52.35.3.45

6.39

27.29

30

1.3.23.29.29.33.54.1.40

41.5

2.57.37

 

1.3.21.30.39.36.57.27.31.33.45

27.15.19

1.58.22.49

 

1.3.2.15

9

41

31

1.2.42.13.55.36.17.46.40

57.19

4.36.1

 

1.2.35.00.3.23.51.3.2.15

55.19.59

4.32.20.49

 

1.2.25.5.3.25.13.27.50.3.45

35.41.21

3.1.19.29

 

1.2.18.15.56.22.47.54.4.26.40

5.17.29

27.30.49

 

1.2.7.22.7.54.32.52.29.26.24

14.6.26.1

1.16.18.54.49

 

1.2.1

11

1.01

32

1.1.46.19.56.3.47.33.13.47.36.15

1.58.47.29

11.41.20.1

 

1.1.44.55.12.40.25

4.55

29.13

33

1.1.31.19.18.53.26.15

25.29

2.42.1

 

1.1.30.0.58.39.59.26.6.52.28.27.24.26.40

8.11.50.31

52.29.2.49

 

1.1.25.58.00.57.12.27.28.11.51.6.40

19.33.28

2.8.3.53

 

1.1.17.28.44.49.4

1.6.1

7.34.49

 

1.1.12.33.43.12.36

1.25.11

10.6.1

 

1.1.1.22.41.39.19.55.48.27.24.26.40

33.50.41

4.21.24.1

 

1.1.00.18.46.52.33.30.56.15

4.8.31

32.16.1

 

1.00.50.10.25

17

2.25

34

1.00.40.6.40

19

3.01

35

1.00.31.56.55.18.13.22.5.23.26.15

1.00.17.29

10.42.50.1

 

1.00.31.11.4.17.36.49.27.54.4.26.40

30.9.19

5.25.24.1

 

1.00.28.50.18.56.21.39.21.2.33.5.11.6.40

3.37.29.29

40.39.42.49

 

1.00.28.6.47.11.38.19.24.24.36.33.45

3.8.31.19

35.41.38.49

 

1.00.24.3.00.7.12.5.24

16.20.49

3.20.40.1

 

1.00.21.21.53.46.40

52

11.17

36

1.00.15.32.45.57.41.23.40.46.20.34.29.26.40

2.43.31.5

41.33.41.13

 

1.00.15.00.56.15

31

8.1

37

1.00.10.12.56.3.9.3.45

30.41

9.36.49

 

1.00.9.47.12.14.45.11.24.1.40

1.48.7

34.36.25

 

1.00.6.00.9

49

20.1

38

1.00.4.42.13.31.39.48.16

8.7.59

3.45.8.49

 

1.00.2.54.15.26.28.20.44.26.40

57.31

33.45.49

 

1.00.2.1.31.1.30.33.45

12.39

8.53.29

 

1.00.00.33.20.4.37.46.40

2.41

3.36.1

 

1.00.00.27.33.10.39.46.45.33.59.3.45

26.1.19

38.24.8.49

 

1.00.00

0

60

392 ; 0,00°

Figure 1: Calculated triples between 0 and 90 degrees, according to the original Babylonian procedure

Notes

1 The table extends on valid triples (numbered) given in Price 1964; Abdulaziz 2010; Britton et al. 2011; Hajossy 2016; Mansfield and Wildberger 2017. Expressions in degrees are used only to illustrate the steadily decreasing slope of the gradient triangles.

2 See Hajossy 2016

Cite this Article
Kleb, Jens. 2023. “270 Valid Triples below, between and above the Lines 1-15 of Plimpton 322.” Cuneiform Digital Library Notes 2023 (5). https://cdli.mpiwg-berlin.mpg.de/articles/cdln/2023-5.
Kleb, J. (2023). 270 valid triples below, between and above the lines 1-15 of Plimpton 322. Cuneiform Digital Library Notes, 2023(5). https://cdli.mpiwg-berlin.mpg.de/articles/cdln/2023-5
Kleb, J. (2023) “270 valid triples below, between and above the lines 1-15 of Plimpton 322,” Cuneiform Digital Library Notes, 2023(5). Available at: https://cdli.mpiwg-berlin.mpg.de/articles/cdln/2023-5 (Accessed: April 22, 2024).
@article{Kleb2023270,
	note = {[Online; accessed 2024-04-22]},
	address = {Oxford; Berlin; Los Angeles},
	author = {Kleb, Jens},
	journal = {Cuneiform Digital Library Notes},
	issn = {1546-6566},
	number = {5},
	year = {2023},
	publisher = {Cuneiform Digital Library Initiative},
	title = {270 valid triples below, between and above the lines 1-15 of {Plimpton} 322},
	url = {https://cdli.mpiwg-berlin.mpg.de/articles/cdln/2023-5},
	volume = {2023},
}

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T2  - Cuneiform Digital Library Notes
TI  - 270 valid triples below, between and above the lines 1-15 of Plimpton 
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Y2  - 2024/4/22/
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