history of mathematics - National STEM Centre

5 Arab mathematics
solutions the cubic has. He therefore classified the cubics differently, into those
types with at least one positive solution and those types that may and may not have
a positive solution depending on the exact coefficients.
The following activity illustrates his approach.
Activity 5.10 Graphical deductions
The Renaissance is the
name given to the period
from the 14th to 16th
centuries in Europe when
there was a revival of
cultural thinking.
1 Al-Tusi notes that for an equation x-
3 + c = ax 2 expressed in the form
x (a — x) = c, whether the equation has a positive solution depends on whether the
expression of the left-hand side reaches c or not. He then states that, for any value of
x lying between 0 and a, x 2 (a — x) < (•} a) (i a) • Why is this true?
2 Show that a local maximum occurs for x in the expression x 2 (a - x) when
JCG = -| a. There is no indication of how al-Tusi found this value. Suggest how he
might have found it.
3 He then proceeds to make the following inferences:
• if ^-a 3 < c no positive solution exists
• if -If a 3 = c, there is only one positive solution, x — ja
• if -fa a 3 > c, two positive solutions x l and x2 exist where 0 < x ]