From the Babylonian Mathematics website:
In this article we examine four Babylonian tablets which all have some connection with Pythagoras’s theorem. Certainly the Babylonians were familiar with Pythagoras’s theorem. A translation of a Babylonian tablet which is preserved in the British museum goes as follows:From the Counting in Babylon website:4 is the length and 5 the diagonal. What is the breadth ? Its size is not known.All the tablets we wish to consider in detail come from roughly the same period, namely that of the Old Babylonian Empire which flourished in Mesopotamia between 1900 BC and 1600 BC.
4 times 4 is 16.
5 times 5 is 25.
You take 16 from 25 and there remains 9.
What times what shall I take in order to get 9 ?
3 times 3 is 9.
3 is the breadth.
Some of the clay tablets discovered contain lists of triplets of numbers, starting with (3, 4, 5) and (5, 12, 13) which are the lengths of sides of right angled triangles, obeying Pythagoras’ “sums of squares” formula. In particular, one tablet, now in a collection at Yale, shows a picture of a square with the diagonals marked, and the lengths of the lines are marked on the figure: the side is marked <<< meaning thirty (fingers?) long, the diagonal is marked:For further interesting mathematical history, see the website of the University of St Andrews School of Mathematics and Statistics’ History of Mathematics Archive, which also has a Famous Curves Index, pages about Egyptian, Indian, Arabic, and Mayan mathematics, and histories of things like Zero, Indian Numerals (which are weird), Arabic Numerals (which rotated), and Pi. From this last I learn that the value of Pi was calculated as 3.162 by the Mesopotamians, 3.16 by the Egyptians, 3.0 by the builders of King Solomon’s Temple (must try harder), 3.1418 by Archimedes of Syracuse, 355/113 by Tsu Ch’ung Chi, and (sigh of relief) 3.1416 by Abu Ja’far Muhammad ibn Musa Al-Khwarizmi; also that Al-Khwarizmi is where we get the word “algorithm”, a word I’ve often wondered about at moments when I wasn’t able to look it up, and forgotten to look up when I could.
<<<<11 <<11111 <<<11111.
This translates to 42, 25, 35, meaning 42 + 25/60 + 35/3600. Using these figures, the ratio of the length of the diagonal to the length of the side of the square works out to be 1.414213… Now, if we use Pythagoras’ theorem, the diagonal of a square forms with two of the sides a right angled triangle, and if we take the sides to have length one, the length of the diagonal squared equals 1 + 1, so the length of the diagonal is the square root of 2. The figure on the clay tablet is incredibly accurate—the true value is 1.414214… Of course, this Babylonian value is far too accurate to have been found by measurement from an accurate drawing—it was clearly checked by arithmetic multiplication by itself, giving a number very close to two.