# Old Babylonian 'Quadratic' Problems

Old Babylonian mathematicians were much taken with problems involving two unknowns and square roots, what we would term 'quadratic' problems. These problems usually involved finding lengths, widths or diagonals of rectangles. The simplest example would be a problem giving the sum of the length and width of a rectangle (or field) and its area. The problem is to find the length and the width. So we might read, 'Length plus width is 50. Area is 600. What are the length and the width?' In all of Old Babylonian mathematics, it is understood that the length (uš) is at least as large as the width (sag).

A modern student would probably write down the formulas and , solve the first as, say, , substitute for w in the second equation to get , and then solve the quadratic equation for l using the quadratic formula to obtain , from which it follows that . The Old Babylonian procedure was rather different.

First, it is important to note that Mesopotamian was largely algorithmic in style. That is, instead of writing down a formula and substituting particular values for the variables, Old Babylonian mathematicians concentrated on following a particular procedure. The procedure for the type of problem given above was as follows:

Step 1: Take half the sum of the length and width (we'll call this the half-sum): 25

Step 2: Square the half-sum: 625.

Step 3: Subtract the area: 25

Step 4: Take the square root: 5

Length is half-sum + square root: 30

Width is half-sum - square root: 20.

Another popular type of problem is where the student is given the difference of the length and the width as well as the area. So we would read, 'The length exceeds the width by 10. The area is 600. What are the length and the width?' The procedure for this type of problem is very similar.

Step 1: Take half the difference of the length and width (the half-difference): 5

Step 2: Square the half-difference: 25

Step 3: Add the area: 625

Step 4: Take the square root: 25

Length is square root + half-difference: 30

Width is square root - half-difference: 20.

Hundreds of these 'rectangular' problems are known. In many cases, just the problem is stated, but in others the procedure is given so that we can see how they solved the particular types of problems. Interestingly, we have no examples of the two basic types given above. They must have been considered too easy to bother writing them down. However, in the case of more complicated types of problems (such as being given the difference of the length and width and the area minus the square of the difference) the first step is to reduce the problem to one of the two standard types above and then solve it with the standard procedure.

The first major analysis of Babylonian rectangular problems was by Solomon Gandz1 in 1937, in a massive paper in Osiris. He divided Mesopotamian quadratic problems into nine types, of which the simple ones we gave above are Types I and II. In that paper, Gandz also speculated as to how the Babylonians derived their procedures and noted the similarities between their approaches and the procedures of Diophantus, contrasting these with both the Arabic and modern approaches.

1. Gandz, S. 'The origin and development of the quadratic equations in Babylonian, Greek and early Arabic algebra,' Osiris 3 (1937) 405-557.

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Last modified: 14 March 2005

Duncan J. Melville
dmelville@stlawu.edu