It is believed that Euclid lived about 300 B.C. in Alexandria – at that time a center of science, with its fantastic library. Euclid's most famous work, Elements, contains virtually all the mathematics Greek scholars had arrived at up to that time. A major share of Elements is about geometry.
Euclidean geometry is geometry on a plane. Here, the sum of angles in a triangle is 180° and a line can be extended to any length whatsoever without meeting itself.
Non-Euclidean geometry is geometry on other surfaces than a plane. For example, a triangle drawn on a sphere has a sum of angles greater than 180°. A line extended far enough will meet itself. On a sphere, a line is thus a circle. Theories for non-Euclidean geometry were developed in the 19thcentury by Lobachevsky and Bolyai, among others.
No other book on mathematics has been so influential as Euclid's Elements.
Elements is very logical in structure, with definitions, axioms, theorems, and proofs.
Definitions, axioms: The cornerstones of a theory are definitions and axioms. New concepts are introduced in a theory by definitions. The first three in Elements are definitions of point, line, and plane. Axioms provide concepts with properties that cannot be proven, but since we regard them as true, they support the theory. One of the axioms in the first book says: ”It is possible to draw a straight line between any two points whatsoever.” Another says: ”A whole is greater than any of its parts.” Self-evident? An axiom should be self-evident.
Theorems and proofs: By proving relations and new properties for concepts, the theory is developed. For instance, Pythagoras' proposition is proven in theorem 47 in Book 1.