The History of Mathematics: A Brief Course

QUESTIONS AND PROBLEMS^407

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FIGURE 3. Another scenario to "fit" a text on cuneiform tablet AO 6670.

Questions and problems

13.1. What do the two problems of recovering two numbers from their sum and
product or from their difference and product have to do with quadratic equations
as we understand them today? Can we conclude that the Mesopotamians "did
algebra"?

13.2. You can verify that the solution of the problem from tablet AO 8862 (15
and 12) given by the author is not the only possible one. The numbers 14 and 13
will also satisfy the conditions of the problem. Why didn't the author give this
solution?

13.3. Of what practical value are the problems we have called "algebra"? Taking
just the quadratic equation as an example, the data can be coastrued as the area
and the semiperimeter of a rectangle and the solutions as the sides of the rectangle.
What need, if any, could there be for solving such a problem? Where are you ever
given the perimeter and area of a rectangle and asked to find its shape?

13.4. Figure 3 gives a scenario that can be fit to the data in AO 6670. Given
a square 1 unit on a side, in the right angle opposite one of its corners construct
a rectangle of prescribed area A that will be one-third of the completed gnomon.
Explain how the figure fits the statement of the problem. (As in Section 2, this
scenario is not being proposed as a serious explanation of the text.)

13.5. Given a cubic equation

ax^3 + bx^2 + cx = d,

where all coefficients are assumed positive, let A = d + be/(3a) - 2b^3 /(9a^2 ), Â =

b^2 /(3a) - c, and t = 3aA/(3aBx - bB), that is, ÷ = A/(Bt) + 6/(3a). Show that in

terms of these new parameters, this equation is

It could therefore be solved numerically by consulting a table of values of f^3 +t^2.

[Again a caution: The fact that such a table exists and could be used this way does

not imply that it was used this way, any more than the fact that a saucer can be

used to hold paper clips implies that it was designed for that purpose.]

13.6. Considering the origin of algebra in the mathematical traditions we have

studied, do you find a point in their development at which mathematics ceases to

be a disjointed collection of techniques and becomes systematic? What criteria

would you use for defining such a point, and where would you place it in the

mathematics of Egypt, Mesopotamia, Greece, China, and India?