sides by 8 to show that x^19 / 8. Algebra was unavailable to Ahmes of
Egypt, who contributed a section entitled “Directions for Knowing All
Dark Things” (many modern-day students would doubtless agree with
this definition of mathematics) to the Rhind papyrus, one of the first
mathematical manuscripts. Ahmes solved this problem using a method
that can only be described as tortuous.^3
The Hanging Gardens may have been the only physical contribution of
Babylonians to the wonders of the ancient world, but their mathematical
accomplishments were quite impressive for the times. They were capable
of solving certain quadratic equations (equations of the form ax^2 bxc0)
using the method of completing the square,^4 which is used in high-school
algebra to generate the full solution to this equation. The resulting for-
mula is known as the quadratic formula. It was described early in the
ninth century by the Arab mathematician Al-Khowarizmi, who is also re-
sponsible for giving the name algebra to algebra.
Del Ferro and the Depressed Cubic
Time passed—approximately seven centuries. There would be no signifi-
cant advance in equation solving until the middle of the fifteenth century,
when a collection of brilliant Italian mathematicians embarked upon a
quest to solve the equation ax^3 bx^2 cxd0. This equation, which is
known as the general cubic, was to prove a far tougher nut to crack.
As the degree of the polynomial increases, different types of numbers
are needed to solve it. Equations such as 2x 6 0 can be solved with
positive integers, but 2x 6 0 requires negative numbers, and 2x 5 0
requires fractions. Quadratic equations introduced square roots and com-
plex numbers into the mix, and it was clear that an equation such as
x^3 2 0 would require cube roots. Roots that are not whole numbers are
known as radicals, and the goal was to find a formula that could be con-
structed from integers, radicals, and complex numbers that would give all
solutions to the general cubic equation. Such a formula is referred to as
“solution by radicals.”
The first mathematician to make a dent in solving cubics by radicals
was Scipione del Ferro, who late in the fifteenth century managed to find
a formula that solved a restricted case of the general cubic, the case where
b0. These “depressed cubics” have the form ax^3 cxd0, and del
Ferro’s mathematical fame would undoubtedly have increased signifi-
cantly had the world learned of his advance. This, however, was an era in
which Machiavelli was writing of the importance of subterfuge—and
subterfuge, in Italian academe, was often how one survived.
The Hope Diamond of Mathematics 83