Ancient Algebra
Algebra. What is it?
Our word `Algebra' is derived from the Arabic expression
{em
al-jabr wa'l muqabala}
which occurs in the title of the first Arabic text on algebra written by
Al-Khwarizmi in the 
century.
We have the words:
So, to Al-Khwarizmi, `Algebra' is the art of reducing and solving equations. In modern algebra, the emphasis has shifted to structure. Its roots started with the work of Galois on the possibility of solving equations by means of radicals (1830).
In ancient times there were: Three Kinds of Algebra
A. Mixed Algebra. This is Babylonnian type in which line segments and areas, etc. are added together and set equal to numbers.
B. Numerical Algebra. Herein only rational numbers
are admitted as coefficients and solutions of equations.
(cf. Diophantus)
C. Geometric Algebra. Here line segments, areas, and volumes are kept strictly apart, e.g. when lengths are multiplied, area results. This is found in Greek mathematics, we know, but also in Chinese and Indian mathematics. Solutions to quadratics and linear equations are accomplished geometrically.