- EUROPE 427
He then turned to the five types that could have no (positive) solutions for some
values of the data. As an example, one of these forms is
x^3 + ax^2 + c = bx.For each of these cases, al-Tusi considered a particular value of x, which for this
example is the value m satisfying
3m^2 + 2am = b.Let us denote the positive root of this equation (the larger root, if there are two)
by rn. The reader will undoubtedly have noticed that the equation can be obtained
by differentiating the original equation and setting ÷ equal to m. The point m is
thus in all cases a relative minimum of the difference of the left- and right-hand
sides of the equation. That, of course, is precisely the property that al-Tusi wanted.
Hogendijk comments that it is unlikely that al-Tusi had any concept of a derivative.
In fact, the equation for m can be derived without calculus, by taking m as the
value at which the minimum occurs, subtracting the values at ÷ from the value at
m, and dividing by ÷ - TO. The result is the inequality m^2 + mx + x^2 + a(m + x) > b
for ÷ > m and the opposite inequality for ÷ < m. Therefore equality must hold
when ÷ = TO, that is, 3m^2 + 2am = b, which is the condition given by al-Tusi.
After finding the point m, al-Tusi concluded that there will be no solutions if the
left-hand side of the equation is larger than the right-hand side when ÷ = m. There
will be one unique solution, namely ÷ = m if equality holds there. That left only
the case when the left-hand side was smaller than the right-hand side when ÷ = m.
For that case he considered the auxiliary cubic equation
y^3 +py^2 = d,where ñ and d were determined by the type of equation. The quantity d was simply
the difference between the right- and left-hand sides of the equation at ÷ = TO, that
is, bm — TO^3 — am^2 — c in the present case, with ñ equal to 3m + a. Al-Tusi was
replacing ÷ with y = x — m here. The procedure was precisely the method we know
as Horner's method, and the linear term drops out because the condition by which
TO was chosen ordains that it be so (see Problem 14.9.) The equation in y was
known to have a root because it was one of the other 13 types, which always have
solutions. Thus, it followed that the original equation must also have a solution,
÷ = m+y, where y was the root of the new equation. The added bonus was that
a lower bound of m was obtained for the solution.
6. Europe
As soon as translations from Arabic into Latin became generally available in the
twelfth and thirteenth centuries, Western Europeans began to learn about algebra.
The first of these was a Latin translation of al-Khwarizmi's Algebra, made in 1145
by Robert of Chester (dates unknown). Several talented mathematicians appeared
early on who were able to make original contributions to its development. In
some cases the books that they wrote were not destined to be published for many
centuries, but at least one of them formed part of an Italian tradition of algebra
that continued for several centuries.