Greek Geometry

Geometry was the first branch of mathematics to become highly developed. The concepts of “theorem” and “proof” originated in geometry, and most mathematicians until recent times were introduced to their subject through the geometry in Euclid’s

Elements

. In the

Elements

one finds the first attempt to derive theorems from supposedly self-evident statements called

axioms

. Euclid’s axioms are incomplete and one of them, the so-called

parallel

axiom, is not as obvious as the others. Nevertheless, it took over 2000 years to produce a clearer foundation for geometry. The climax of the

Elements

is the investigation of the regular polyhedra, five symmetric figures in three-dimensional space. The five regular polyhedra make several appearances in mathematical history, most importantly in the theory of symmetry—

group theory

—discussed in Chapters 19 and 23. The

Elements

contains not only proofs but also many

constructions

, by ruler and compass. However, three constructions are conspicuous by their absence: duplication of the cube, trisection of the angle, and squaring the circle. These problems were not properly understood until the 19th century, when they were resolved (in the negative) by algebra and analysis. The only curves in the

Elements

are circles, but the Greeks studied many other curves, such as the conic sections. Again, many problems that the Greeks could not solve were later clarified by algebra. In particular, curves can be classified by

degree

, and the conic sections are the curves of degree 2, as we will see in Chapter 7.