Ancient Greece

Rise of Geometric Algebra

When the Greeks solved problems involving unknown quantities, they considered the quantities as shapes, lines, and other physical/geometric objects. As a result, they would only work with geometric problems, and their solution/final result would be a geometric object, not a number or finite quantity. The greeks defined magnitude as the ranking of a class (numbers, lines, areas, solids, ... e.t.c.). For example, the Pythagorean theorem (A translated proof is shown on the left) was proved with geometry and was documented verbally/rhetorically – no algebraic symbols were used at all.  

Euclid

Euclid, one of the most famous greek mathematicians, created a textbook full of the mathematical knowledge of geometry and geometric algebra, titled “Euclid’s Elements”. In the book, he showcased what would now be considered algebraic identities and processes to solve algebraic equations. Not only was this extremely influential among the greeks, but it also inspired a new generation of mathematicians from various countries, including India and, most notably, Europe. 

Diophantus

Geometric algebra remained relatively unchanged until Diophantus started publishing books about his methods for solving algebraic problems. Diophantus provided solutions to specific algebraic equations but did not detail a general process/procedure in his work. Most importantly, Diophantus's work is the first known literature where symbolism was involved in solving algebraic equations –syncopated algebra. However, syncopated algebra was only widely adopted several centuries later. It's also worth noting that the term "Diophantine equations" refer to Diophantus. Those equations involved two or more variables and were introduced symbolically with Diophantus's work.