The Garden of Archimedes   A Museum for Mathematics

Pythagoras and his theorem

The Pythagorean theorem in very ancient times.

	The first known evidence about the Pythagorean theorem is contained in an early Babylonian table, dated to between 1800 and 1600 b. C., where a square with two diagonals is drawn. The side of the square carries the number 30, and along the diagonal we find the numbers (in sexagesimal numbering) 1, 24, 51, and 10, or rather 1+24/60+51/602+10/603, and 42,25,35, or rather 42+25/60+35/602 . In decimal form these emerge as 1,414213 and 42,42639. The former is an exellent approximation of the root of 2, while the latter is the diagonal of the square with a side that measures 30, and equal to the product of 30 by the former number. The fact that the diagonal of the square can be found by multiplying its side by the root of 2 reveals the knowledge of the Pythagorean theorem, at least in the case of the triangle with equal cathets.
	More uncertain are the other attributions. One very often quoted example claimes that Egyptian surveyors, to find a right angle, used a string on which they marked the lengths 3, 4 and 5, forming the sides of a right angle triangle. This can hardly be considered as an understanding of the Pythagorean theorem - it is more likely to prove the converse of the theorem (see specifications).
	Also the Chinese figure "hsuan-thu", dating back to 1200 B.C. (with no great certainty as to the date) was seen by some as proof of the knowledge of the Pythagorean theorem. This claim is controversial. In fact, the figure shows a triangle with sides measuring 3, 4 and 5, with the square with side 7=3+4 containing the one with side 5. That in turn is formed by four triangles and a little square with side 1=4-3. There is no evidence of the squares on the cathets 3 and 4. In general, if we indicate with the cathets with a and b and the hypotenuse with c , the square with side a + b can be considered to be formed by 8 triangles, and by the little square with side b - a, or also by the square on the hypotenuse c and four triangles, from the relation 4ab+ (b - a) 2 = c2 +2ab is derived. Developing (b - a)2 = b2 + a2 –2ab, the result is b2 + a2 = c2 and thus the Pythagorean theorem, as long as the formula of the square of the binomial is known (b - a)2 =b2 + a2 –2ab. Needless to say this last formula - especially in the geometric version that should be used here - isn't any more simple than the Pythagorean theorem that we want to demonstrate. In any case, we have neither a precise statement of the theorem nor a proof.