1. Cubic Equations

3

x2

-h px of the hatched region using a calculation corresponding to

the formula

P_

4

x

4

hH hH

p.

x

p.

4 4

Figure 1.1. Al-Khwarizmi's treatment of the quadratic equa-

tion

x2+px

= q. The hatched area corresponds to the constant

term q.

1.2 Following this digression on quadratic equations, we shall turn

our attention again to the question posed by Fior. When Fior posed

his problems, mathematics had made little progress in the seven hun-

dred years since al-Khwarizmi. European mathematics was marked

by its static adherence to the knowledge gained in earlier periods of

activity, notably from the Arabs and the ancient Greeks. Arabic nu-

merals had been introduced into Europe, which served the need of the

mathematical calculations required in trade. Such calculations were

generally carried out by professional masters of calculation.

Although mathematics was used in many commercial applica-

tions, negative numbers remained unknown. Moreover, mathemati-

cal notation was slow to develop. Thus, for example, in the fifteenth

century the notation R3 V31 m R16 was used for the expression

\j$\ - VT6.