The Arabic world and the
cubic
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The
Arabs made a great contribution to the development of mathematics, is said to
be their role as intermediary agent between Indian and Greek mathematics and
the modern world. They had access to the great Indian and Greek
mathematicians works in the original language and translated them to Arabic. Al-Khwarizmi's
(c. 780-c. 850) use of geometrical justifications of algebraic manipulations
together with the fact that the Elements existed in two distinct
translations from Greek into Arabic by his contemporary at the House of
Wisdom, al-Hajjaj ibn Yusuf ibn Matar, suggest a line of descent from
Euclid. On the
other hand, because his treatment of practical geometry so closely followed that
of the Hebrew text, Mishnat ha Middot, which dated from around A.D.
150, the evidence of Semitic ancestry exists. Al-Khwarizmi's concern with
practical algebra and his treatment of equations through the second degree
betray a vestige of the Babylonian line, while his totally rhetorical style
points to a remote Hindu ancestor and a lack of contact with later Greek
texts, particularly the Arithmetica of Diophantus. In fact,
since the first known Arabic translation of the Arithmetica was not
completed by Qusta ibn Luqa until the middle of the ninth century or later,
we can be fairly certain that the more theoretical ideas of Diophantus had
not yet entered the environment of, and so had not come into competition
with, Arabic mathematics. The
Arabs did not make full use of the advances of the Hindus so they had neither
negative quantities nor abbreviations for their unknowns. However
Muhammad ibn Musa Al-Khwarizmi gave a classification of different types of
quadratics (although only numerical examples of each). The different types
arise since al'Khwarizmi had no zero or negatives. He has six chapters each
devoted to a different type of equation, the equations being made up of three
types of quantities namely: roots, squares of roots and numbers i.e. x, x and
numbers. Squares
equal to roots. al'Khwarizmi gives the rule
for solving each type of equation, essentially the familiar quadratic formula
given for a numerical example in each case, and then a geometrical proof for
each example utilizing the method of completing the square. The
Arabs solved some cubics algebraically, and gave a geometrical explanation. This
was done, for example, by Tabit ibn Qorra (836-901) and by al-hasan ibn
al-haitham (965-1039). `Umar al-Khayyami (Omar
Khayyam) (c. 1048-c. 1125) went further and used conical sections to to
solve general equations of third degree. Khayyami
used pure algebra to solve quadraatics, but he verified the results using
constructions. He solved particular cubics also using constructions. He did
admit that he couldnīt find an algebraic formula for the general cubic, but
he invited other mathematicians to look for one. He also classified cubic
equations, and he described those which he can solve and those which he
couldnīt . Khayyam
believed that cubic equations could only be solved geometrically, by
using conic sections. His method, used in Algebra (c 1079) is based on
a geometrical construction where the solution is found at the intersection
between a parabola and a semicircle. He
solves the equation: x3 + Bx = C
A
positive solution of the equation is found in the intersection between the
curves. The following example shows the solution of x3
+ 4x = 16
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