YBC 7289 is a

Babylonia Babylonia (; , ) was an Ancient history, ancient Akkadian language, Akkadian-speaking state and cultural area based in the city of Babylon in central-southern Mesopotamia (present-day Iraq and parts of Kuwait, Syria and Iran). It emerged as a ...

clay tablet In the Ancient Near East, clay tablets (Akkadian language, Akkadian ) were used as a writing medium, especially for writing in cuneiform, throughout the Bronze Age and well into the Iron Age. Cuneiform characters were imprinted on a wet clay t ...

notable for containing an accurate

sexagesimal Sexagesimal, also known as base 60, is a numeral system with 60 (number), sixty as its radix, base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified fo ...

approximation to the

square root of 2 The square root of 2 (approximately 1.4142) is the positive real number that, when multiplied by itself or squared, equals the number 2. It may be written as \sqrt or 2^. It is an algebraic number, and therefore not a transcendental number. Te ...

, the length of the diagonal of a

unit square In mathematics, a unit square is a square whose sides have length . Often, ''the'' unit square refers specifically to the square in the Cartesian plane with corners at the four points ), , , and . Cartesian coordinates In a Cartesian coordinat ...

. This number is given to the equivalent of six decimal digits, "the greatest known computational accuracy ... in the ancient world". The tablet is believed to be the work of a student in southern

Mesopotamia Mesopotamia is a historical region of West Asia situated within the Tigris–Euphrates river system, in the northern part of the Fertile Crescent. Today, Mesopotamia is known as present-day Iraq and forms the eastern geographic boundary of ...

from some time between 1800 and 1600 BC.

Content

The tablet depicts a square with its two diagonals. One side of the square is labeled with the sexagesimal number 30. The diagonal of the square is labeled with two sexagesimal numbers. The first of these two, 1;24,51,10 represents the fraction ≈ 1.414213, a numerical approximation of the square root of two that is off by less than one part in two million. The second of the two numbers is 42;25,35 = ≈ 42.426. This number is the result of multiplying 30 by the given approximation to the square root of two, and approximates the length of the diagonal of a square of side length 30. Because the Babylonian sexagesimal notation did not indicate which digit had which place value, one alternative interpretation is that the number on the side of the square is = . Under this alternative interpretation, the number on the diagonal is ≈ 0.70711, a close numerical approximation of

\frac

, the length of the diagonal of a square of side length , that is also off by less than one part in two million. David Fowler and Eleanor Robson write, "Thus we have a reciprocal pair of numbers with a geometric interpretation…". They point out that, while the importance of reciprocal pairs in Babylonian mathematics makes this interpretation attractive, there are reasons for skepticism. The reverse side is partly erased, but Robson believes it contains a similar problem concerning the diagonal of a rectangle whose two sides and diagonal are in the ratio 3:4:5.

Interpretation

Although YBC 7289 is frequently depicted (as in the photo) with the square oriented diagonally, the standard Babylonian conventions for drawing squares would have made the sides of the square vertical and horizontal, with the numbered side at the top. The small round shape of the tablet, and the large writing on it, suggests that it was a "hand tablet" of a type typically used for rough work by a student who would hold it in the palm of his hand. The student would likely have copied the sexagesimal value of the square root of 2 from a table of constants, but an iterative procedure for computing this value can be found in another Babylonian tablet, BM 96957 + VAT 6598. A table of constants that includes the same approximation of the square root of 2 as YBC 7289 is the tablet YBC 7243. The constant appears on line 10 of the table along with the inscription, "the diagonal of a square". The mathematical significance of this tablet was first recognized by Otto E. Neugebauer and Abraham Sachs in 1945. The tablet "demonstrates the greatest known computational accuracy obtained anywhere in the ancient world", the equivalent of six decimal digits of accuracy. Other Babylonian tablets include the computations of areas of hexagons and heptagons, which involve the approximation of more complicated algebraic numbers such as

\sqrt3

. The same number

\sqrt3

can also be used in the interpretation of certain ancient Egyptian calculations of the dimensions of pyramids. However, the much greater numerical precision of the numbers on YBC 7289 makes it more clear that they are the result of a general procedure for calculating them, rather than merely being an estimate. The same sexagesimal approximation to

\sqrt2

, 1;24,51,10, was used much later by Greek mathematician Claudius Ptolemy in his '' Almagest''. Ptolemy did not explain where this approximation came from and it may be assumed to have been well known by his time.

Provenance and curation

It is unknown where in Mesopotamia YBC 7289 comes from, but its shape and writing style make it likely that it was created in southern Mesopotamia, sometime between 1800BC and 1600BC. At Yale, the Institute for the Preservation of Cultural Heritage has produced a digital model of the tablet, suitable for 3D printing. The original tablet is currently kept in the Yale Babylonian Collection at Yale University.

References

{{reflist, 30em, refs= {{citation , last = Robson , first = Eleanor , author-link = Eleanor Robson , editor-last = Katz , editor-first = Victor J. , page = 143 , publisher = Princeton University Press , contribution = Mesopotamian Mathematics , title = The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook , url = https://books.google.com/books?id=3ullzl036UEC , year = 2007, isbn = 978-0-691-11485-9 {{citation , last = Friberg , first = Jöran , editor1-first = Jöran , editor1-last = Friberg , doi = 10.1007/978-0-387-48977-3 , isbn = 978-0-387-34543-7 , mr = 2333050 , page = 211 , publisher = Springer, New York , series = Sources and Studies in the History of Mathematics and Physical Sciences , title = A remarkable collection of Babylonian mathematical texts , year = 2007 {{citation , last1 = Fowler , first1 = David , author1-link = David Fowler (mathematician) , last2 = Robson , first2 = Eleanor , author2-link = Eleanor Robson , doi = 10.1006/hmat.1998.2209 , issue = 4 , journal = Historia Mathematica , mr = 1662496 , pages = 366–378 , title = Square root approximations in old Babylonian mathematics: YBC 7289 in context , volume = 25 , year = 1998, doi-access = free {{citation , last1 = Neugebauer , first1 = O. , author1-link = Otto E. Neugebauer , last2 = Sachs , first2 = A. J. , author2-link = Abraham Sachs , mr = 0016320 , page = 43 , publisher = American Oriental Society and the American Schools of Oriental Research, New Haven, Conn. , series = American Oriental Series , title = Mathematical Cuneiform Texts , year = 1945 {{citation , last = Neugebauer , first = O. , author-link = Otto E. Neugebauer , mr = 0465672 , pages = 22–23 , publisher = Springer-Verlag, New York-Heidelberg , title = A History of Ancient Mathematical Astronomy, Part One , url = https://books.google.com/books?id=6tkqBAAAQBAJ&pg=PA22 , year = 1975, isbn = 978-3-642-61910-6 {{citation, url=https://books.google.com/books?id=8eaHxE9jUrwC&pg=PA57, page=57, title=A Survey of the Almagest, series=Sources and Studies in the History of Mathematics and Physical Sciences, first=Olaf, last=Pedersen, editor-first=Alexander, editor-last=Jones, publisher=Springer, year=2011, isbn=978-0-387-84826-6 {{citation , last = Rudman , first = Peter S. , isbn = 978-1-59102-477-4 , mr = 2329364 , page = 241 , publisher = Prometheus Books, Amherst, NY , title = How mathematics happened: the first 50,000 years , url = https://books.google.com/books?id=BtcQq4RUfkUC&pg=PA241 , year = 2007 {{citation , last1 = Beery , first1 = Janet L. , author1-link = Janet Beery , last2 = Swetz , first2 = Frank J. , date = July 2012 , doi = 10.4169/loci003889 , journal = Convergence , publisher = Mathematical Association of America , title = The best known old Babylonian tablet?, doi-broken-date = 10 February 2025 , doi-access = free {{citation, title=A 3,800-year journey from classroom to classroom, first=Patrick, last=Lynch, magazine=Yale News, date=April 11, 2016, url=https://news.yale.edu/2016/04/11/3800-year-journey-classroom-classroom, access-date=2017-10-25 {{citation, title=A 3D-print of ancient history: one of the most famous mathematical texts from Mesopotamia, date=January 16, 2016, url=http://ipch.yale.edu/news/3d-print-ancient-history-one-most-famous-mathematical-texts-mesopotamia, publisher=Yale Institute for the Preservation of Cultural Heritage, access-date=2017-10-25 {{citation , title=Mesopotamian tablet YBC 7289 , last=Kwan , first=Alistair , date=April 20, 2019 , publisher=University of Auckland , doi = 10.17608/k6.auckland.6114425.v1 Babylonian mathematics Mathematics manuscripts Clay tablets 18th-century BC works

Content

Interpretation

Provenance and curation

See also

External links

References