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Thirteen cuneiform clay tablets of ancient Mesopotamia, dating from 1900 to 1700 B.C., are on display until Dec. 17 at the Institute for the Study of the Ancient World, part of New York University. Many are exercises of students learning to be scribes, who were mastering mathematics based on texts in Sumerian, a language that even at the time was long since dead. The items are drawn from archaeological collections of Columbia, Yale and the University of Pennsylvania and include two celebrated tablets, known as YBC 7289 and Plimpton 322, that have played central roles in the reconstruction of Babylonian math.

Credit: Yale Babylonian Collection

YBC 7289 is a small clay disc containing a rough sketch of a square and its diagonals. Across one of the diagonals is scrawled 1,24,51,10 — a sexagesimal number that corresponds to the decimal number 1.4142129, an approximation of the square root of 2. Below is the answer to the problem of calculating the diagonal of a square whose sides are 0.5 units. This bears on the issue of whether the Babylonians had discovered Pythagoras’s theorem some 1,300 years before Pythagoras did.

Credit: Yale Babylonian Collection

No tablet bears the well-known algebraic equation, that the squares of the two smaller sides of a right-angled triangle equal the square of the hypotenuse. But Plimpton 322 contains columns of numbers that seem to have been used in calculating Pythagorean triples, sets of numbers that correspond to the sides and hypotenuse of a right triangle, like 3, 4 and 5.

Credit: Columbia University

A school tablet with an incomplete calculation. The early training of scribes consisted of copying lists of units of measure and arithmetic tables. Later, they practiced calculations and simple problem solving.

Credit: University of Pennsylvania Museum

A 59 x 59 multiplication table is far too large to memorize, so tablets were needed to provide essential look-up tables. But cuneiform numbers are simple to write because each is a combination of only two symbols, those for 1 and 10.

Credit: University of Pennsylvania Museum

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