Plimpton 322

Plimpton 322 is a famous stone tablet of the Babylonian era. It has been described as "one of the most remarkable documents of Old Babylonian mathematics".

Plimpton 322 was purchased by the American publisher George Arthur Plimpton from a Mr Banks, from Florida, in around 1923: it is uncertain from where Banks obtained it but it is speculated that it was dug up in the ancient city of Larsa, Mesopotamia, in modern-day Iraq. It holds catalogue number 322 in Plimpton's collection, giving rise to its - at first glance - unusual name.

Originally, until the studies and successful transcription offered by Otto E. Neugebauer, the tablet was catalogued as a "commercial account".

Description of the tablet and its contents

The left side of Plimpton 322 has been broken off since its excavation. The tablet has four columns of numbers, with words at the head of each. The heading of the first column is partially destroyed. The text headings for the second and third columns read 'Ib-sa of the front' and 'Ib-sa of the diagonal'. Ib-sa is a Sumerian word, the meaning of which is unknown.

Transcription

This transcription was offered by Otto E. Neugebauer in his 1969 book The Exact Sciences in Antiquity. Neugebauer made certain assumptions regarding the missing left-hand side which he has denoted with square brackets.

Source: The Open University It is worth remembering that the Babylonians used a sexagesimal (Base 60) system. Figure 1

Column A	Column B	Column C	Column D
[1; 59, 0,] 15	1, 59	2, 49	1
[1; 56, 56,] 58, 14, 5O, 6, 15	56, 7	3, 12, 1	2
[1; 55, 7,] 41, 15, 33, 45	1, 16, 41	1, 50, 49
[1;] 5 [3, 1]0, 29, 32, 52, 16	3, 31, 49	5.9, 1	4
[1;] 48, 54, 1, 40	1, 5	1, 37	5
[1;] 47, 6, 41, 40	5, 19	8, 1	6
[1 ;] 43, 11, 56, 28, 26.4O	38, 11	59, 1	7
[1;] 41, 33, 45, 14, 3, 45	13, 19	20, 49	8
[1;] 38, 33, 36, 36	9, 1	12, 49	9
1; 35, 10, 2, 28, 27, 24, 26, 40	1, 22, 41	2, 16, 1	10
1 ; 33, 45	45	1, 15	11
1;29, 21, 54, 2, 15	27, 59	48, 49	12
[1;]27, 0, 3, 45	7, 12, 1	4, 49	13
1; 25, 48, 51, 35, 6, 40	29, 31	53, 49	14
[1;]23.13, 46, 40	56	53	15

Figure 1's Column D's list incrementing by 1 each time indicates a pattern not dissimilar to a list. The numbers in Column A decrease in a semi-regular pattern, from just under 2 (if it was 1, 60, 0 it would be equivalent to 2 due to it being a sexagesimal system) to just over 1⅓. Due to their system of fractions, whether one place is worth 60, 1, 0.6 or 0.06 was up to them.

However, Neugebauer discovered the numbers in each line that the numbers in each line are related, such that $A=C^{2}\div (C^{2}-B^{2})$.

Line 11 offers a simple case:

1. B² = 45² = 2,025
2. C² = 75² = 5,625
3. C² ÷ (C² - B²) = 5,625 ÷ 3,600 = 1 ⁵⁶²⁵⁄₃₆₀₀ = 1 ²⁰²⁵⁄₃₆₀₀ = 1 ³³⁄₆₀ ⁴⁵⁄₃₆₀₀ = 1; 33; 45 = A

Further to this, Figure 2 explains:

Source: The Open University Figure 2

B	C	$D={\sqrt {(}}C^{2}-B^{2})$ (decimal)	$D={\sqrt {(}}C^{2}-B^{2})$ (sexagesimal)
119	169	120	2; 0
3,367	4,825	3,456	57; 36
4,601	6,649	4,800	1; 20; 0
12,709	18,541	13,500	3; 45; 0
65	97	72	1; 12
319	481	360	6; 0
2,291	3,541	2,700	45; 0
799	1,249	960	16; 0
481	769	600	10; 0
4,961	8,161	6,480	1; 48; 0
45	75	60	1; 0
1,679	2,929	2,400	40; 0
161	289	240	4; 0
1,771	3,229	2,700	45; 0
56	106	90	1; 30

Especially compared to Columns A and B, Column D's numbers are fairly nice, particularly when represented in sexagesimal form. That is because all numbers in Column D are regular numbers: these are numbers that evenly divide powers of 60. This explains why their squares divide into C².

When the function $D={\sqrt {(}}C^{2}-B^{2})$ is rearranged, it becomes $B^{2}+D^{2}=C^{2}$. This follows Pythagoras' theorem, that is the famous $A^2+B^2=C^2$.

It therefore appears that Columns B and C of Plimpton 322 relate to the two shortest sides of a right-angled triangle, indicating the Babylonians understood that adding the squares of the lengths of the two shorter sides of a right-angle triangle gives the same number as squaring the length of the diagonal.

However, the tablet begs the question of why those certain numbers were listed and why they were ordered in this way. Mathematical historians have reached varying interpretations as to why. Neugebauer and his colleague Abraham Sachs in 1945 predicted that rather than to do with geometry, Plimpton 322 was more about number theory.

Consider the previous table, but with two further columns added, as per Figure 3.

Figure 3

B	C	$D={\sqrt {(}}C^{2}-B^{2})$ (decimal)	$D={\sqrt {(}}C^{2}-B^{2})$ (sexagesimal)	p	q
119	169	120	2; 0	12	5
3,367	4,825	3,456	57; 36	64	27
4,601	6,649	4,800	1; 20; 0	75	32
12,709	18,541	13,500	3; 45; 0	125	54
65	97	72	1; 12	9	4
319	481	360	6; 0	20	9
2,291	3,541	2,700	45; 0	54	25
799	1,249	960	16; 0	32	15
481	769	600	10; 0	25	12
4,961	8,161	6,480	1; 48; 0	81	40
45	75	60	1; 0
1,679	2,929	2,400	40; 0	48	25
161	289	240	4; 0	15	8
1,771	3,229	2,700	45; 0	50	27
56	106	90	1; 30	9	5

From this table, each line can come from a pair of smaller regular numbers, such that $B=p^{2}-q^{2}$; $C=p^{2}+q^{2}$ and $D=2pq$. It is therefore potentially possible that the author of Plimpton 322 was aware of the aforementioned formulae, which themselves generate Pythagorean triples.

Plimpton 322 does contain errors: the last entry in Figure 1's Column C is 53, where the writer should have said 1, 46 (106). However it could be said Column B contains the error (but our p and q pairs do not add up if so).

The Swedish mathematician Jöran Friberg puts forward a hypothesis contrasting with Neugebauer and Sach's view. He instead suggested it may have been a teaching aid for problems concerning right-angled triangles. Due to the formula $B^{2}+C^{2}=D^{2}$, D would come out nicely.

Modern hypothesis

In 2002, Eleanor Robson put forward a new hypothesis, published by the Mathematical Association of America, contrasting with both the Neugebauer/Sach and Friberg hypotheses. She described the tablet as a method of describing quadratics in the form $x-(1\div x)=c$. The solver calculates the sequence of intermediate values $v_{1}=c\div 2$; $v_{2}=v_{1}^{2}$; $v_{3}=1+v_{2}$; and$v_{4}=v_{3}^{0.5}$. One can therefore calculate that $x=v_{4}+v_{1}$ and that $1\div x=v_{4}-v_{1}$.

Ms. Robson noted that Plimpton 322 can be interpreted as such that for regular number values of x and (1/x) in numerical order:

• v₃ in the first column
• v₁ (x - 1 / x) in the second column
• v₄ (x + 1 / x) / 2 in the third column.

Robson's interpretation states that x and 1/x would have appeared in the broken-off portion. Therefore, where x = 2, Line 11 can be generated this way. This interpretation reveals methods such as reciprocal pairs, completing the square and dividing by common factors. Robson's work won her the 2003 Lester R. Ford Award.