The History of Mathematics: A Brief Course

402 13. PROBLEMS LEADING TO ALGEBRA

area

width length - width^1

+ length 1

FIGURE 1. Reduction of a problem to standard form.

difference between the square of the average and the product will be the square
of the semidifference of the two numbers whose sum is 29 and whose product is
3,30, that is, 210). What is not clear is the following: Why add 27 to the number
3,3 in the first place, and why add 2 to 27? Possibly the answer is contained in
Fig. 1, which shows that adding the difference between length and width to the
area amounts to gluing a smaller rectangle of unit width onto a larger rectangle.
Then adding the sum of length and width amounts to gluing a gnomon onto the
resulting figure in order to complete a rectangle two units wider than the original.
Finding the dimensions of that rectangle from its perimeter and area is the standard
technique of solving a quadratic equation, and that is what the author does.
The tablet AO 6670, discussed by van der Waerden (1963, pp. 73-74) con-
tains a rare explanation of the procedure for solving a problem that involves two
unknowns and two conditions, given in abstract terms without specific numbers.
Unfortunately, the explanation is very difficult to understand. The statement of the
problem is taken directly from Neugebauer's translation: Length and width as much
as area; let them be equal. Thereafter, the translation given by van der Waerden,
due to Frangois Thureau-Dangin (1872-1944), goes as follows:

The product you take twice. From this you subtract 1. You form

the reciprocal. With the product that you have taken you multiply,

and the width it gives you.

Van der Waerden asserts that the formula y = (l/(x — 1)) • ÷ is "stated in the

text" of Thureau-Dangin's translation. If so, it must have been stated in a place

not quoted by van der Waerden, since ÷ is not a "product" here, nor is it taken

twice. Van der Waerden also notes that according to Evert Marie Bruins (1909-

1990), the phrase "length and width" does not mean the sum of length and width.

Van der Waerden says that "the meaning of the words has to be determined in

relation to the mathematical content." The last two sentences in the description

tell how to determine the width once the length has been found. That is, you take