History of Mathematics Project

History of
Mathematics

Project

around 1790–1750 BCE

Plimpton 322

Pythagorean triples: teacher's aid or trigonometry table?

The text of this Babylonian tablet, dating from the reign of Hammurabi the Great between 1790 and 1750 BCE, includes an enumeration of Pythagorean triples. The meaning of these triples has been given various interpretations, including as a teacher's aid designed for generating problems involving right triangles or a trigonometric table.

Plimpton 322 is known throughout the world to those interested in the history of mathematics as a result of the interest that Otto Neugebauer, chair of Brown University's History of Mathematics Department. In the early 1940s, Neugebauer and Abraham Sachs interpreted the tablet as a listing of Pythagorean triples, integer solutions of the equation a² + b² = c², written a thousand years before the age of Pythagoras. Recently, Eleanor Robson, an authority on Mesopotamian mathematics, suggested that the tablet was created as a teacher's aid, designed for generating problems involving right triangles and reciprocal pairs. More recently still, it has been persuasively argued that Plimpton 322 is a very early trigonometry table.

Artifact dimensions

13 cm × 9 cm × 2 cm

Original artifact location

Larsa (historical name), An Nasiriyah, Dhi-Qar, Iraq (current name)

Current artifact location

Rare Book & Manuscript Library, Butler Library, Columbia University in the City of New York

Catalog number

CULC 460

Timeline

Interactive Content

Computational Explanation

Plimpton 322 is a 3700-year-old clay tablet from the Old Babylonian period. Its unusual sequence of numbers corresponding to Pythagorean triples proves that the Babylonians knew about the Pythagorean theorem a thousand years before Pythagoras was born. But while there is agreement on what the tablet contains, there has been no consensus on its purpose. A recent research paper by Mansfield and Wildberger (2017) gives a new interpretation of the tablet as a trigonometric table in which rows of sexagesimal numbers represent side ratios for a series of right-angled triangles ranging from almost a square to almost a flat line.

Each column of Plimpton 322 has a heading written in the Akkadian language. The following table summarizes what is generally agreed to be the correct content of Plimpton 322, including a number of corrections:

In[]:=

Plimpton322Sexagesimal=Transpose[{{{1,59,00,15},{1,56,56,58,14,50,06,15},{1,55,07,41,15,33,45},{1,53,10,29,32,52,16},{1,48,54,01,40},{1,47,06,41,40},{1,43,11,56,28,26,40},{1,41,33,45,14,03,45},{1,38,33,36,36},{1,35,10,02,28,27,24,26,40},{1,33,45},{1,29,21,54,02,15},{1,27,00,03,45},{1,25,48,51,35,06,40},{1,23,13,46,40}},{{1,59},{56,07},{1,16,41},{3,31,49},{1,05},{5,19},{38,11},{13,19},{8,01},{1,22,41},{45},{27,59},{2,41},{29,31},{28}},{{2,49},{1,20,25},{1,50,49},{5,09,01},{1,37},{8,01},{59,01},{20,49},{12,49},{2,16,01},{1,15},{48,49},{4,49},{53,49},{53}},Range[15]}];

The columns are traditionally referred to using Roman numerals as I', II', III' and IV', where the prime indicates that this column numbering may have been different in the original. Written out as sexagesimal digits (and including positional zeros, which cuneiform number representations lack but whose presence can be inferred based on considerations such as those to be discussed shortly):

Out[]=

I' : 2 δ n	II' : b n	III' : d n	IV' : n
The takiltum of the diagonal from which 1 is subtracted and that of the width comes up	ib-si of the width	ib-si of the diagonal	its place
1.59.0.15	1.59	2.49	ki 1
1.56.56.58.14.50.6.15	56.7	1.20.25	ki 2
1.55.7.41.15.33.45	1.16.41	1.50.49	ki 3
1.53.10.29.32.52.16	3.31.49	5.9.1	ki 4
1.48.54.1.40	1.5	1.37	ki 5
1.47.6.41.40	5.19	8.1	ki 6
1.43.11.56.28.26.40	38.11	59.1	ki 7
1.41.33.45.14.3.45	13.19	20.49	ki 8
1.38.33.36.36	8.1	12.49	ki 9
1.35.10.2.28.27.24.26.40	1.22.41	2.16.1	ki 10
1.33.45	45	1.15	ki 11
1.29.21.54.2.15	27.59	48.49	ki 12
1.27.0.3.45	2.41	4.49	ki 13
1.25.48.51.35.6.40	29.31	53.49	ki 14
1.23.13.46.40	28	53	ki 15

When written in cuneiform, the preceding digits indeed reproduce the tablet contents:

GraphicsGridMap

[◼]	AncientNumberRepresentation

[FromDigits[Flatten[{#}],60],"Babylonian"]&,Plimpton322Sexagesimal,{2},

opts



The Greek astronomer Hipparchus, who lived around 120 BCE, has long been regarded as the father of trigonometry, and his table of chords on a circle is considered the oldest known trigonometric table. A modern trigonometric table consists of a list of right triangles with hypotenuse

and approximations to the side lengths sin

and

cosθ

, along with the ratio tan θ  sin θ/cos θ, for example:

Out[]=

angle α	x  cos α	y  sin α	y/x  tan α	triangle
0	1	0	0
18°	5 8 + 5 8	1 4 (-1+ 5 )	1- 2 5
36°	1 4 (1+ 5 )	5 8 - 5 8	5-2 5
54°	5 8 - 5 8	1 4 (1+ 5 )	1+ 2 5
72°	1 4 (-1+ 5 )	5 8 + 5 8	5+2 5
90°	0	1	ComplexInfinity

Mansfield and Wildberger suggest that the 15 rows and four columns of Plimpton 322 actually comprise a trigonometric table in which the top row contains relatively equal ratios corresponding to a near-equilateral triangle, while the rows descending down the tablet give ratios that decrease the triangle's inclination, creating skinnier triangles. This is consistent with interpreting the number in the second column as the shorter side

of a right triangle, the number in the third column as the hypotenuse

and noting that these lengths always give a longer side



that is also an integer, thus making

(

)

a Pythagorean triple:

SideData=MapIndexed[Flatten[{#,Sqrt[

#[[2]]

#[[1]]

],#2[[1]]}]&,Map[FromDigits[#,60]&,Plimpton322Sexagesimal[[All,2;;]],{2}]]

Out[]=

{{119,169,120,1},{3367,4825,3456,2},{4601,6649,4800,3},{12709,18541,13500,4},{65,97,72,5},{319,481,360,6},{2291,3541,2700,7},{799,1249,960,8},{481,769,600,9},{4961,8161,6480,10},{45,75,60,11},{1679,2929,2400,12},{161,289,240,13},{1771,3229,2700,14},{28,53,45,15}}

Out[]=

b n	d n	a n	n
119	169	120	1
3367	4825	3456	2
4601	6649	4800	3
12709	18541	13500	4
65	97	72	5
319	481	360	6
2291	3541	2700	7
799	1249	960	8
481	769	600	9
4961	8161	6480	10
45	75	60	11
1679	2929	2400	12
161	289	240	13
1771	3229	2700	14
28	53	45	15

Visualizing these right triangles including the implied longer side length

Out[]=

Mansfield and Wildberger also interpret the numbers in the first column

as rational numbers defined by either



if a leading 1 is included, or



1+



if it is not. Since numbers expressed in cuneiform do not use place values, there is ambiguity in interpreting the power of 60 by which the original base-60 digits in the first column must be multiplied. However, given the just-stated interpretation of

, this ambiguity is removed by expressing

in terms of

and

In[]:=

Plimpton322Decoded=MapIndexedAppend[#,#2[[1]]]&,With{b=FromDigits[#2,60],d=FromDigits[#3,60]},

,N

,Log60,

FromDigits[#1,60]

,b,

,d&@@@Plimpton322Sexagesimal;

The following table gives the values of

and n computed from the original (possibly corrected) cuneiform digits, together with the power of 60 by which the digits in the first column must be divided:

Out[]=

2 δ n (exact)	2 δ n (approx.)	power of 60	b n	d n	a n	n
28561 14400	1.9834	3	119	120	169	1
23280625 11943936	1.94916	7	3367	3456	4825	2
44209201 23040000	1.9188	6	4601	4800	6649	3
343768681 182250000	1.88625	6	12709	13500	18541	4
9409 5184	1.81501	4	65	72	97	5
231361 129600	1.78519	4	319	360	481	6
12538681 7290000	1.71998	6	2291	2700	3541	7
1560001 921600	1.69271	6	799	960	1249	8
591361 360000	1.64267	4	481	600	769	9
66601921 41990400	1.58612	8	4961	6480	8161	10
25 16	1.5625	2	45	60	75	11
8579041 5760000	1.48942	5	1679	2400	2929	12
83521 57600	1.45002	4	161	240	289	13
10426441 7290000	1.43024	6	1771	2700	3229	14
2809 2025	1.38716	4	28	45	53	15

For example, the first row consists of:

In[]:=

Row

[◼]	AncientNumberRepresentation

[#,"Babylonian"]&/@Plimpton322[[1]]

Out[]=





… corresponding to the base-60 digits:

In[]:=

Plimpton322[[1]]

Out[]=

{{1,59,0,15},{1,59},{2,49}}

… and the integers:

In[]:=

FromDigits[#,60]&/@Plimpton322[[1]]

Out[]=

{428415,119,169}

But from the values 119 and 169, it can be determined that the first number must be divided by

since:

In[]:=

Log60,

428415

169

119



Out[]=

… giving:

In[]:=

428415

Out[]=

28561

14400

As expected, this agrees with:

In[]:=

169

119

Out[]=

28561

14400

In every case, the long side

is a regular number—that is, an integer divisor of a power of 60 or, equivalently, a product of powers of 2, 3 and 5. It is for this reason that the numbers

in the first column are also exact in Babylonian sexagesimal notation, because dividing an integer by a regular number produces a terminating sexagesimal number.

Students these days know about the (3, 4, 5) triangle, which is equivalent to the (45, 75, 60) triangle encoded in row

In[]:=

Plimpton322[[11,2;;]]

Out[]=

{{45},{1,15}}

In[]:=

45,60+15,

(60+15)



Out[]=

{45,75,60}

… corresponding to:

In[]:=

{45,75,60}/15

Out[]=

{3,5,4}

The fact that this tablet contains Pythagorean triples makes it a scientific marvel of the ancient world. And if it is actually a trigonometric table, it is the most ancient one known. Using the Plimpton 322 values augmented by hypothetical rows introduced by Mansfield and Wildberger, it can easily be transformed into a modern table by making use of some simple trigonometry relations:

In[]:=

MansfieldWildbergerData={{119,169},{3367,4825},{4601,6649},{12709,18541},{65,97},{319,481},{2291,3541},{799,1249},{481,769},{4961,8161},{45,75},{1679,2929},{161,289},{1771,3229},{56,106},{175,337},{473,985},{8,17},{4061,9061},{39,89},{369,881},{5,13},{871,2329},{671,1921},{2071,6121},{1001,3049},{301,949},{329,1129},{7,25},{399,1649},{9,41},{11,61},{295,1753},{17,145},{19,181},{52,677},{31,481},{49,1201}};

Out[]//TraditionalForm=

n	α (degrees)	α (radians)	α (exact)	cos α	sin α	tan α
1	44.76	0.781214	-1 tan 119 120	120 169	119 169	119 120
2	44.25	0.772355	-1 tan 3367 3456	3456 4825	3367 4825	3367 3456
3	43.79	0.764233	-1 tan 4601 4800	4800 6649	4601 6649	4601 4800
4	43.27	0.755227	-1 tan 12709 13500	13500 18541	12709 18541	12709 13500
5	42.08	0.734348	-1 tan 65 72	72 97	65 97	65 72
6	41.54	0.725088	-1 tan 319 360	360 481	319 481	319 360
7	40.32	0.703633	-1 tan 2291 2700	2700 3541	2291 3541	2291 2700
8	39.77	0.694123	-1 tan 799 960	960 1249	799 1249	799 960
9	38.72	0.675756	-1 tan 481 600	600 769	481 769	481 600
10	37.44	0.653402	-1 tan 4961 6480	6480 8161	4961 8161	4961 6480
11	36.87	0.643501	-1 tan 3 4	4 5	3 5	3 4
12	34.98	0.610446	-1 tan 1679 2400	2400 2929	1679 2929	1679 2400
13	33.86	0.590882	-1 tan 161 240	240 289	161 289	161 240
14	33.26	0.58053	-1 tan 1771 2700	2700 3229	1771 3229	1771 2700
15	31.89	0.556599	-1 tan 28 45	45 53	28 53	28 45
16	31.28	0.546017	-1 tan 175 288	288 337	175 337	175 288
17	28.70	0.500886	-1 tan 473 864	864 985	473 985	473 864
18	28.07	0.489957	-1 tan 8 15	15 17	8 17	8 15
19	26.63	0.464733	-1 tan 4061 8100	8100 9061	4061 9061	4061 8100
20	25.99	0.453598	-1 tan 39 80	80 89	39 89	39 80
21	24.76	0.43217	-1 tan 369 800	800 881	369 881	369 800
22	22.62	0.394791	-1 tan 5 12	12 13	5 13	5 12
23	21.96	0.383297	-1 tan 871 2160	2160 2329	871 2329	871 2160
24	20.44	0.356821	-1 tan 671 1800	1800 1921	671 1921	671 1800
25	19.78	0.345156	-1 tan 2071 5760	5760 6121	2071 6121	2071 5760
26	19.17	0.334508	-1 tan 1001 2880	2880 3049	1001 3049	1001 2880
27	18.49	0.32275	-1 tan 301 900	900 949	301 949	301 900
28	16.94	0.295699	-1 tan 329 1080	1080 1129	329 1129	329 1080
29	16.26	0.283794	-1 tan 7 24	24 25	7 25	7 24
30	14.00	0.24439	-1 tan 399 1600	1600 1649	399 1649	399 1600
31	12.68	0.221314	-1 tan 9 40	40 41	9 41	9 40
32	10.39	0.18132	-1 tan 11 60	60 61	11 61	11 60
33	9.69	0.169088	-1 tan 295 1728	1728 1753	295 1753	295 1728
34	6.73	0.117512	-1 tan 17 144	144 145	17 145	17 144
35	6.03	0.105166	-1 tan 19 180	180 181	19 181	19 180
36	4.41	0.0768852	-1 tan 52 675	675 677	52 677	52 675
37	3.70	0.0644938	-1 tan 31 480	480 481	31 481	31 480
38	2.34	0.0408107	-1 tan 49 1200	1200 1201	49 1201	49 1200

This table can also be visualized by allowing angles α to correspond to the acute angles of right triangles, drawing all triangles at once and labeling each upper-right vertex according to the row number in which it is found:

In[]:=

ExtendedTable=Flatten/@Transpose[{{{119,169},{3367,4825},{4601,6649},{12709,18541},{65,97},{319,481},{2291,3541},{799,1249},{481,769},{4961,8161},{45,75},{1679,2929},{161,289},{1771,3229},{56,106},{175,337},{473,985},{8,17},{4061,9061},{39,89},{369,881},{5,13},{871,2329},{671,1921},{2071,6121},{1001,3049},{301,949},{329,1129},{7,25},{399,1649},{9,41},{11,61},{295,1753},{17,145},{19,181},{52,677},{31,481},{49,1201}},Range[38]}];

In[]:=

Graphics

,Withy=

,{Triangle[{{0,0},{1,y},{1,0}}],Point[{1,y}],Text[#3,{1,y},{-2.5,0}]}&@@@ExtendedTable,



Out[]=

Other Resources

Wolfram Demonstration
Babylonian Numerals

Wolfram Demonstration
Pythagorean Theorem

Wolfram Demonstration
Ptolemy's Table of Chords

Additional Reading

Bruins, E. M. "On Plimpton 322, Pythagorean Numbers in Babylonian Mathematics." Koninklijke Nederlandse Akademie van Wetenschappen Proceedings, Vol. 52, 629–632, 1949.
Buck, R. C. "Sherlock Holmes in Babylon." Amer. Math. Monthly, Vol. 87, pp. 335–345, 1980.
Friberg, J. "Methods and traditions of Babylonian Mathematics I: Plimpton 322, Pythagorean triples, and the Babylonian triangle parameter equations." Historia Math., Vol. 8, 277–318, 1981.
Mansfield, D. F. and Wildberger, N. J. "Plimpton 322 is Babylonian Exact Sexagesimal Trigonometry." Historia Mathematica, Vol. 44, pp. 395–419, 2017.
Maor, E. The Pythagorean Theorem: A 4000-Year History. Princeton, NJ: Princeton University Press, pp. 7–11, 2007.
Neugebauer, O. and Sachs, A. (Ed.). Mathematical Cuneiform Texts. New Haven, CT: American Oriental Society and the American Schools of Oriental Research, pp. 38–39 and pl. 25, 1945.
Pickover, C. The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics. New York: Sterling, pp. 34–35, 2012.
Quiñonez, F. "The Algorithm of Factor 12: Generating the Information Carved on the Plimpton 322 Tablet." arXiv preprint. 30 Jan 2020.
Robson, E. "Neither Sherlock Holmes nor Babylon: A Reassessment of Plimpton 322." Historia Mathematica, Vol. 28, pp. 167–206 , 2001.
Robson, E. "Words and Pictures: New Light on Plimpton 322." Amer. Math. Monthly, Vol. 109, pp. 105–120, 2002.
Robson, E. Fig 1.5 in "Tables and tabular formatting in Sumer, Babylonia, and Assyria, 2500 BCE–50 CE." Ch. 1 in The History of Mathematical Tables: From Sumer to Spreadsheets (Ed. M Campbell-Kelly, M. Croarken, R. Flood and E. Robson). Oxford, England: Oxford University Press, p. 34, 2007.