Euclid's Elements

An ancient Greek mathematical treatise that organizes geometry and number theory into definitions, postulates, propositions, and deductive proofs. Compiled around 300 BCE, it became a model of axiomatic mathematics.

Euclidean geometry: The geometry of flat space, characterized by Euclid's postulates and their consequences. The treatise's geometric books became the classical foundation of this subject.
Euclid's parallel postulate: Euclid's fifth postulate, which implies that through a point outside a line exactly one parallel can be drawn. It is the treatise's most distinctive assumption and underlies many results about parallel lines.
Euclid: An ancient Greek mathematician associated with teaching and mathematical writing in Alexandria around 300 BCE. The treatise bears his name, though its compilation draws on earlier mathematical work.
Euclidean algorithm: A procedure for finding the greatest common divisor of two integers through repeated division. Book VII presents the method as part of its arithmetic theory.
David Hilbert: A German mathematician whose work reshaped geometry, logic, and the foundations of mathematics. His 1899 Foundations of Geometry recast Euclidean geometry with explicit axioms and modern rigor.
Axiom: A statement accepted as a starting point in a system of reasoning. Euclid's initial assumptions let later propositions be proved rather than asserted.
Playfair's axiom: The assertion that exactly one parallel to a given line passes through a point outside it. This simpler equivalent expresses the geometric content of Euclid's fifth postulate.
Plato's Academy: A philosophical school founded by Plato in Athens in the fourth century BCE. Its mathematical culture formed part of the intellectual background to Greek deductive geometry.
Number theory: The branch of mathematics that studies integers and their properties. Books VII through IX develop divisibility, primes, and properties of numbers.
Foundations of Geometry: The study of the axioms and logical structure underlying geometric theories. Modern foundational work examined assumptions that Euclid left implicit.

Linked from 56 pages