The first great mathematicians were all Greek. As
Leonard Mlodinow said of the Egyptians and Babylonians that came before
the Greek, "like our political leaders, they sometimes accomplished
astonishing feats with surprisingly little comprehension of what they
were doing. Nor did they care." The Greeks, on the other hand, did care.
And so while the Egyptians built pyramids with their knowledge, the
Greek mathematicians-- Euclid, Pythagoras, Thales, Eratosthenes, Archimedes, Hypatia, and many more-- are immortalized through their work in the creation of the system of mathematics and logic called geometry.
The Greeks:
Euclid, while the eponym of Euclidean geometry, was by no means the only, or even the most revolutionary of the ancient Greek geometers. Why, then, is he the one remembered in modern day as the "father of geometry"? It was he who gathered and organized the mathematical discoveries of his peers into a single complete treatise or geometry conducted entirely through theory, completely unrelated to the material world-- the first of its kind.
Other eponymous mathematical concepts are also not as well-named as you might think. Even Pythagoras, with his notorious Pythagorean theorem, may not have been the first to realize that a^2 + b^2 = c^2. That distinction probably belongs to the Babylonians, who cataloged Pythagorean triples as large as 3367-3456-4825. Pythagoras was, however, the first to record and prove his theorem, allowing the Babylonians to escape the endless curses and hatred from modern geometry students directed at Pythagoras for his "invention" of such a troublesome theorem. Heron's formula may have been known by Archimedes, a fellow Greek mathematician, two hundred years before Heron proved it.
...And Their Words:
Not only do the people that created geometry hail from Greece, so do that words they use. Geometry translates to the measurement of land, a pseudosphere is a fake ball, and the hyperbola is "excessive," ellipses "falling short." Many, many more mathematical and scientific terms derive from Greek, a testament to their countless intellectual discoveries.
After the Greeks:
With the fall of Greek civilization came the demise of most mathematical advancement in Europe. While they languished in the Dark Ages, mathematicians in the Middle East and Asia continued the advancement of geometry and other intellectual pursuits. However, it was once more the Europeans that pioneered non-Euclidean geometries through their attempts at indirect proofs of Euclid's 5th postulate.
Don't miss the upcoming installment of NEG History, another adventure in geometry and geometers (and their eponyms), coming soon to a computer near you.