Geometry

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• ObjectEuclid's Elements

  The Elements is a mathematical treatise in thirteen books, written in Greek by Euclid of Alexandria around 300 BC. Books I to VI treat plane geometry, and Books VII to IX treat the arithmetic of whole numbers. Book X treats incommensurable magnitudes, and Books XI to XIII treat solid geometry. Book I starts from definitions, five postulates and a list of common notions, and derives every proposition from these and from earlier propositions. The fifth postulate of Book I is the parallel postulate. Book I ends with the Pythagorean theorem and its converse. Book VII contains the Euclidean algorithm, Book IX contains Euclid's theorem that there are infinitely many primes, and Book XIII constructs the five Platonic solids.

• ConceptEuclidean geometry

  Euclidean geometry is the geometry of the Euclidean plane and of Euclidean space: the study of points, lines, distances and angles in a flat space. It can be built synthetically from axioms, as in Euclid's Elements and Hilbert's axioms, or analytically with coordinates in $\R^n$. Its characteristic axiom is the parallel postulate; dropping it leads to absolute geometry and to non-Euclidean geometry.

• FactParallel postulate

  Let two lines in a plane be cut by a third line, the transversal. Suppose that the two interior angles on one side of the transversal have a sum less than two right angles. Then the two lines, extended far enough, meet on that side of the transversal.

• FactPlayfair's axiom

  Let $\ell$ be a line in a plane and let $P$ be a point of the plane not on $\ell$. Then there is at most one line in the plane through $P$ that does not meet $\ell$.

• ConceptHilbert's axioms

  Hilbert's axioms are the axiom system for Euclidean geometry that David Hilbert gave in his book Grundlagen der Geometrie of 1899. They speak of points, lines and planes and of the relations of incidence, betweenness and congruence, and they fall into five groups. They fill the gaps of Euclid's Elements. They made it possible to prove that geometry is consistent if the real numbers are. They also made it possible to prove the independence of single axioms, such as the parallel postulate.

• ConceptTarski's axioms

  Tarski's axioms are an axiom system for plane Euclidean geometry in first-order logic, due to Alfred Tarski. They speak only of points and of two relations: betweenness of three points and congruence of two segments. Tarski proved that the resulting theory is complete and decidable: an algorithm decides, for each sentence, whether it is a theorem.