It Didn’t Belong to Pythagoras

Gabrielle Birchak/ March 8, 2022/ Uncategorized

PODCAST TRANSCRIPTS

In a few of my pre­vi­ous pod­casts, I briefly men­tion the Pythagore­an the­o­rem and how it was not the cre­ation of Pythagoras.

There is a leg­end that Pythago­ras was so proud of dis­cov­er­ing this the­o­ry that, accord­ing to Proclus’s com­men­tary on Euclid’s proof, he writes that “if we lis­ten to those who like to record antiq­ui­ties, we shall find them attribut­ing this the­o­rem to Pythago­ras and say­ing that he sac­ri­ficed an ox upon its dis­cov­ery.”[1]

How­ev­er, that is just a leg­end. Pythago­ras did not dis­cov­er this the­o­rem, and there is tan­gi­ble evi­dence to prove that this the­o­rem exist­ed 1,000 years before Pythago­ras was even born. This evi­dence includes four ancient cuneiform tablets.

The first one is called YBC-7289, and it dates to between 1800 and 1600 BCE. It was most like­ly found in south­ern Iraq.

The tablet YBC 7289 was deter­mined to have been cre­at­ed by a stu­dent between 1800 and 1600 BCE. The YBC 7289, which is now on dis­play at Yale University’s Insti­tute for the Preser­va­tion of Cul­tur­al Her­itage, indi­cates four adja­cent tri­an­gles and etch­ings that show numer­i­cal proof of the Pythagore­an theorem.

This tablet shows the Pythagore­an the­o­rem using an isosce­les right tri­an­gle to define 

\sqrt 2

uti­liz­ing the sex­a­ges­i­mal sys­tem. An isosce­les right tri­an­gle, specif­i­cal­ly a right tri­an­gle, has two sides of equal length, such that the units are mea­sured as 

1:1:\sqrt2

 as shown below:

The etch­ings and carv­ings on the YBC 7289 are a quar­tered square and a series of chips. It is one of our old­est intro­duc­tions into the infa­mous the­o­rem: a² +b² = c², which shows that the square of the two short­er sides of a right tri­an­gle, when added togeth­er, equals the square of the hypotenuse.

The math on the YBC 7289 uses base 60, which was com­mon­ly used in ancient Mesopotami­an math­e­mat­ics. We cur­rent­ly use base 10, as not­ed in our dec­i­mal sys­tem. So, imag­ine that instead of count­ing to 10 before car­ry­ing over to the next col­umn, ancient Mesopotami­ans would add up to 60 before car­ry­ing over to the next column.

Base 60 Conventions

In the ear­ly years of Mesopotamia, Sume­ri­ans used wedges to indi­cate a num­ber. Though the sym­bols evolved and changed over hun­dreds of years, the process of using base 60 did not. As a result, the fol­low­ing process remained the same, regard­less of numer­i­cal inscription.

In our cur­rent num­ber­ing con­ven­tion, we use base 10 to count 1, 2, 3,4,5,6,7,8,9, and 10. How­ev­er, unlike Baby­lon­ian math­e­mat­ics, we read our num­bers from left to right, apply­ing the pow­ers of 10 as we read the numbers.

Thus, for the num­ber 52,537, because it is in base ten, we under­stand it as:

The largest base 10 num­ber is 50,000.	5 x 104 =	50,000
The sec­ond-largest base 10 num­ber is 2,000.	2 x 103 =	2,000
The third-largest base 10 num­ber is 500.	5 x 102 =	500
The fourth-largest base 10 num­ber is 30.	30 x 101 =	30
Final­ly, the small­est num­ber is 7.	7 x 1 =	7
	TOTAL	52,537

If we were to use a dec­i­mal val­ue, we use the same process as fol­lows for the num­ber 37.25 as follows:

Base 60 is much like base 10 in that for each val­ue in its place, the num­ber is between 1 and 59. How­ev­er, unlike base 10, in base 60, we read the num­bers from left to right, not know­ing the pow­ers of 60 until we add up the numbers.

So, the num­ber 4,160,425 is writ­ten in base 60 as 19, 15, 40, and 25. We process it as follows:

The fol­low­ing expla­na­tion of the etch­ings will be con­vert­ed between base 10 and base 60.

In the upper left cor­ner are three etch­es, which rep­re­sent the num­ber 30, there­by indi­cat­ing that the edge of the square rep­re­sents 30 units.

The mid­dle set of num­bers 1, 24, 51, and 10 rep­re­sent the whole num­ber 1, along with the fol­low­ing dec­i­mal dig­its divid­ed by 60, as follows:

1 + \frac{24}{60} +\frac{51}{60^2} +\frac{10}{60^3} =1.41421 = \sqrt2

When we apply the length of the edge of the tri­an­gle to the Pythagore­an the­o­rem, we get:

a² + b² = c²

30² + 30² = c²

Since 30² + 30² = c² = 900 + 900 = 1800 = c²

To find the val­ue of c, we take the square root of 1800 as follows:

c^2 = 1800

c=\sqrt{1800}

c=\sqrt{1800} = 30\sqrt{2} = 42.4264

This val­ue can be seen on the cuneiform tablet in the bot­tom set of num­bers 42, 25, 35. When we take the whole num­ber 42 and divide the oth­er two num­bers by 60 since we are work­ing in base 60, we get the fol­low­ing value:

42+\frac{25}{60}+\frac{35}{60^2}=42.4264

Since

c=42.4264=30\sqrt2

 we find that

30^2+30^2=(30\sqrt2)^2

Thus, we see an Isosce­les tri­an­gle with a ratio of sides that equal 

30:30:30\sqrt2.

If we divide this ratio by 30, we get 

1:1:\sqrt2

which val­i­dates this theorem. 

Si.427

The next tablet is labeled Si.427. This lit­tle tablet was exca­vat­ed in 1894 by the French expe­di­tion at Tell Abu Hab­ba in Syr­ia. It was then pre­served and archived at the Instan­bul Arche­o­log­i­cal Muse­um. How­ev­er, this lit­tle tablet did not become famous until 2021, when David Mans­field, an Aus­tralian pro­fes­sor and math­e­mati­cian, deter­mined the pur­pose of this cuneiform tablet. This tablet shows a series of tri­an­gles and rec­tan­gles, which rep­re­sent a schemat­ic dia­gram of a field. This tablet also uti­lizes Pythagore­an triples to dis­tin­guish mea­sure­ments of the land boundaries.

The Plimp­ton 322

The Plimp­ton 322 dates to 1800 BCE. This tablet shows etch­ings that indi­cate a the­o­ret­i­cal knowl­edge of the the­o­rem because it includes a series of Pythagore­an triples. It is not a large tablet. It is about the size of a cell phone, with the spe­cif­ic dimen­sions being 12.7 cen­time­ters by 8.8 cen­time­ters. So, it is small.

In the late nine­teenth cen­tu­ry, an antiq­ui­ties deal­er and arche­ol­o­gist, Edgar Banks, pur­chased hun­dreds of cuneiform tablets in South­east­ern Europe as the Ottoman Empire was end­ing. He then sold these tablets to muse­ums and col­lec­tors around the world. New York pub­lish­er, George Arthur Plimp­ton, pur­chased the Plimp­ton 322 from Banks for only $10.[3] The tablet remained in Plimpton’s per­son­al pos­ses­sion until short­ly before he died in 1936, upon which it was donat­ed to Colum­bia University.

This tablet con­sists of fif­teen Pythagore­an triples. What is most impres­sive about this tablet is that it is small with many etch­ings. All these Pythagore­an triples were etched in base 60. A Pythagore­an triple is a set of three pos­i­tive inte­gers that serve to solve the fol­low­ing Pythagore­an equation

a² + b² = c²

For exam­ple, the most com­mon Pythagore­an triple includes the val­ues 3, 4, 5. When applied to the equa­tion, we get 3²+4²=5² which leads to 9+16=25.

There are a mul­ti­tude of oth­er Pythagore­an triples. Some of the most com­mon­ly rec­og­nized triples include the following:

• 5, 12, 13 (5² + 12² = 13²)
• 7, 24, 25 (7² + 24² = 25²)
• 8, 15, 17 (8² + 15² = 17²)
• 9, 40, 41 (9² + 40² = 41²)
• 11, 60, 61 (11² + 60² = 61²)

The fact that there is a list of Pythagore­an triples reveals that there had been exten­sive analy­sis on the Pythagore­an the­o­rem, which was not named the Pythagore­an the­o­rem at this time. These triples show that math­e­mati­cians had been study­ing this math­e­mat­i­cal con­cept and try­ing out a vari­ety of num­bers to obtain dif­fer­ent results.

IM 67118

Final­ly, there is the IM 67118, which is also referred to as Db2-146. It is a Baby­lon­ian clay tablet that dates to 1770 BCE. The IM 67118 con­tains text, a dia­gram, and aids to deter­mine the area of a rec­tan­gle and the length of its diag­o­nal. The text describes the last sec­tion of the solu­tion using the Pythagore­an the­o­rem.[4] Oth­er tablets also show us how the Pythagore­an the­o­rem was applied over 1,000 years before Pythago­ras became the acclaimed aca­d­e­m­ic to have “dis­cov­ered it.”

So now that we have estab­lished that Pythago­ras is not the orig­i­nal source of this equa­tion, where did this equa­tion come from? Well, let’s first look at the works that hail Pythago­ras as the genius behind this equa­tion. He was born around 570 BCE, over 1,000 years after the the­o­rem had been etched into cuneiform tablets. His teach­ings come to us from extracts of works copied from Eude­mus of Rhodes, Philo­laus of Cro­tona, and archivists of Tar­en­tum, Italy.[5]

Ancient his­to­ri­ans state that Pythago­ras was born in Italy and migrat­ed with his father to Samos.[6] Some ancient writ­ers such as Herodotus and Isocrates note in their writ­ings that Pythago­ras trav­eled to Egypt. Addi­tion­al­ly, the ancient writer Aris­tox­enus wrote that Pythago­ras left Samos around 535 BCE when the tyran­ny of Poly­crates sieged Samos. Thus, we can gath­er that he trav­eled to the east and even spent some time in Asia, includ­ing India. After his years of trav­el, he set­tled on the South­east­ern coast of Italy in Cro­tona, where he began teach­ing phi­los­o­phy and math­e­mat­ics.[7] Like many oth­er philoso­phers of his time, his works were writ­ten by his students.

Addi­tion­al­ly, oth­er ancient aca­d­e­mics wrote about Pythago­ras, includ­ing Aris­tox­enus, Dicaearchus, and Iamblichus, whom I have men­tioned in my pod­cast about Hypa­tia. Inter­est­ing­ly, the one philoso­pher who lived near the clos­est time of Pythago­ras is Pla­to. More­over, Pla­to only men­tioned him once in his writ­ings. Thus, Pythago­ras trav­eled and was a promi­nent philosopher.

What is notable is that this the­o­rem was also present in India around the eighth cen­tu­ry BCE, in a Vedic San­skrit text called the Sul­ba­su­tra writ­ten by Baud­hāyana. Baud­hāyana was most like­ly a Vedic Hin­du schol­ar who­lived in India and wrote sev­er­al San­skrit texts encom­pass­ing Dhar­ma, dai­ly activ­i­ty and rit­u­als, and math­e­mat­ics. The text that includes a vari­a­tion of the Pythagore­an the­o­rem is the Sul­ba­su­tra. Though Baud­hāyana was one of the first in India to write about this equa­tion 300 years before Pythago­ras was born, oth­er Vedic priests also wrote about this equa­tion after Baud­hāyana.[8]

Fur­ther­more, this the­o­rem is known in Chi­na as GouGu. There is a fable that states when the Myth­i­cal Deity, the Yel­low Emper­or, known as Huang Di, reigned around 2600 BCE; he charged his min­is­ter Li Shou with the respon­si­bil­i­ty of com­pil­ing a body of work called the Jiu Jang Suan Shu, 九章算术, which trans­lates to The Nine Chap­ters on Math­e­mat­i­cal Art.[9] In this work, the GouGu is explained such that “in the right tri­an­gle, the sum of the squares of two right sides is equal to the square of the hypotenuse.”  In this text, the labels for a, b and c are such that

a = 勾

b = 股

c = 弦

Where a (勾) is called Gou, b (股) is called Gu, and c (弦) is called Xian.

Though the sto­ry is based on a deity and steeped in a fable, there is some truth in that this equa­tion, this the­o­rem, had been around long before Pythago­ras and had been in exis­tence in oth­er areas oth­er than in Mesopotamia. India’s Baud­hāyana received the knowl­edge of this math­e­mat­i­cal appli­ca­tion through oral tra­di­tion. Some schol­ars pro­pose that this the­o­rem was passed down through oral tra­di­tion as ear­ly as 2000 BCE.[10] So now we are look­ing at the pos­si­bil­i­ty that this equa­tion and this the­o­rem is 1,500 years old­er than Pythago­ras. So, it could be much old­er than we ini­tial­ly thought.

This insight allows us to see a pos­si­ble evo­lu­tion­ary con­nec­tion between India, Chi­na, and Mesopotamia. Fur­ther­more, this per­cep­tion shows us the pow­er of shar­ing knowl­edge and how some of our most insight­ful ances­tors shared these bril­liant ideas as they migrat­ed around the world.

This rev­e­la­tion always brings me back to what I have said since my first pod­cast: that we are all math­e­mati­cians and that math has been in our insights for thou­sands and thou­sands of years. Our ancient ances­tors were capa­ble of deep, intel­lec­tu­al math­e­mat­i­cal thought found­ed on shar­ing knowl­edge. These cuneiform tablets and our rev­e­la­tions about them are evi­dence that our ances­tors were capa­ble of so much advanced knowl­edge. Like­wise, we are also capa­ble of much advance­ment. And I am not just talk­ing about progress in math­e­mat­ics. I am also talk­ing about evo­lu­tions toward a peace­ful world. Because I think we are capa­ble of that too.

Thanks for watch­ing. Until next time, carpe diem!

[1] Pro­clus, ca 410–485. A Com­men­tary on the First Book of Euclid’s Ele­ments. Prince­ton, N.J., Prince­ton Uni­ver­si­ty Press, 1970, 426.6. http://archive.org/details/commentaryonfirs0000proc.

[2] Wales, Uni­ver­si­ty of New South. “Math­e­mati­cian Reveals World’s Old­est Exam­ple of Applied Geom­e­try.” Accessed Feb­ru­ary 23, 2022. https://phys.org/news/2021–08-mathematician-reveals-world-oldest-geometry.html.

[3] Brit­ton, John P., Chris­tine Proust, and Steve Shnider. “Plimp­ton 322: A Review and a Dif­fer­ent Per­spec­tive.” Archive for His­to­ry of Exact Sci­ences 65, no. 5 (August 3, 2011): 245. https://doi.org/10.1007/s00407-011‑0083‑4.

[4] Brit­ton, John P., Chris­tine Proust, and Steve Shnider. “Plimp­ton 322: A Review and a Dif­fer­ent Per­spec­tive.” Archive for His­to­ry of Exact Sci­ences 65, no. 5 (August 3, 2011): 550–51. https://doi.org/10.1007/s00407-011‑0083‑4.

[5] Huff­man, Carl. “Pythago­ras (Stan­ford Ency­clo­pe­dia of Phi­los­o­phy.” Stan­ford Ency­clo­pe­dia of Phi­los­o­phy. Last mod­i­fied, Octo­ber 17, 2018. https://plato.stanford.edu/entries/pythagoras/.

[6] Smith, David E. His­to­ry of Math­e­mat­ics. Vol. I. North Chelms­ford: Couri­er Cor­po­ra­tion, 1958.

[7] Huff­man, Carl. “Pythago­ras (Stan­ford Ency­clo­pe­dia of Phi­los­o­phy.” Stan­ford Ency­clo­pe­dia of Phi­los­o­phy. Last mod­i­fied, Octo­ber 17, 2018. https://plato.stanford.edu/entries/pythagoras/.

[8] Katz, Vic­tor, ed. The Math­e­mat­ics of Egypt, Mesopotamia, Chi­na, India, and Islam. Prince­ton, NJ: Prince­ton Uni­ver­si­ty Press, 2007.

[9] Dauben, Joseph. “Ancient Chi­nese Math­e­mat­ics: The Jiu Zhyang Suan Shue vs Euclid’s Ele­ments. Aspects of Proof and the Lin­guis­tic Lim­its of Knowl­edge.” Inter­na­tion­al Jour­nal of Engi­neer­ing Sci­ence 36 (1998): 1339–59.

[10] Math­ews, Jerold. “A Neolith­ic Oral Tra­di­tion for the van Der Waerden/Seidenberg Ori­gin of Math­e­mat­ics.” Archive for His­to­ry of Exact Sci­ences 34, no. 3 (1985): 193–220.