Conceptual stage

Static equation-solving stage

Dynamic function stage

Abstarct stage

Geometric stage

Greek geometric algebraIt is sometimes alleged that the Greeks had no

algebra, but this is inaccurate By the time of Plato, Greek mathematics had undergone a drastic

change. The Greeks created a geometric algebra where terms were represented by sides of

geometric objects,usually lines, that had letters associated with them,and with this new form of

algebra they were able to find solutions to equations by using a process that they invented,

known as "the application of areas"."The application of areas" is only a part of geometric algebra and it is thoroughly covered in Euclid's

Elements.

Bloom of ThymaridasIamblichus in Introductio arithmatica tells us that Thymaridas worked with simultaneous linear equations.In particular, he created the then famous rule that was known as the "bloom

of Thymaridas" or as the "flower of Thymaridas", which states that:

:) If the sum of n quantities be given, and also the sum of every pair containing a particular quantity, then this particular

quantity is equal to 1/ (n - 2) of the difference between the sums of these pairs and the first given sum.

A proof from Euclid's Elements that, given a line segment, an equilateral triangle exists that includes the segment as one of

its sides.or using modern notion, the solution of the following system

of n linear equations in n unknowns.