Synthetic geometry is the branch of geometry which makes use of theorems and synthetic observations to draw conclusions, as opposed to analytic geometry which uses algebra to perform geometric computations and solve problems. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position In Mathematics, a theorem is a statement proven on the basis of previously accepted or established statements Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of Geometry Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity.
The geometry of Euclid was indeed synthetic, though not all of the books covered topics of pure geometry. Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry The heyday of synthetic geometry can be considered to have been the 19th century; when methods based on coordinates and calculus were ignored by some geometers such as Jakob Steiner, in favour of a synthetic development of projective geometry. The 19th century of the Common Era began on January 1, 1801 and ended on December 31, 1900, according to the Gregorian calendar In Mathematics and its applications a coordinate system is a system for assigning an n - Tuple of Numbers or scalars to each point Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives A geometer is a Mathematician whose area of study is Geometry. Jakob Steiner ( 18 March, 1796 &ndash April 1, 1863) was a Swiss Mathematician. Projective geometry is a non- metrical form of Geometry, notable for its principle of duality.
For example, the treatment of the projective plane starting from axioms of incidence is actually a broader theory (with more models) than is found by starting with a vector space of dimension three. See Real projective plane and Complex projective plane, for the cases met as manifolds of respective dimension 2 and 4 In Mathematics In Mathematics, model theory is the study of (classes of mathematical structures such as groups, Fields graphs or even models In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added The close axiomatic study of Euclidean geometry led to the discovery of non-Euclidean geometry. Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. In mathematics non-Euclidean geometry describes how this all works--> hyperbolic and Elliptic geometry, which are contrasted with Euclidean geometry
If the axiom set is not categorical (so that there is more than one model) one has the geometry/geometries debate to settle. In Model theory, a branch of Mathematical logic, a theory is &kappa- categorical (or categorical in &kappa) if it has exactly one model of That's not a serious issue for a modern axiomatic mathematician, since the implication of axiom is now starting point for theory rather than self-evident plank in platform based on intuition. In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject And since the Erlangen program of Klein the geometrical nature of a geometry has been seen as the connection of symmetry and the content of propositions, rather than the style of development. An influential research program and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or
In relation with computational geometry, a computational synthetic geometry has been founded, having close connection, for example, with matroid theory. Computational geometry is a branch of Computer science devoted to the study of algorithms which can be stated in terms of Geometry. In Combinatorics, a branch of Mathematics, a matroid ( or independence structure is a structure that captures the essence of a notion of "independence" Synthetic differential geometry is an application of topos theory to the foundations of differentiable manifold theory. In Mathematics, synthetic differential geometry is a reformulation of Differential geometry in the language of Topos theory. In Mathematics, a topos (plural "topoi" or "toposes" is a type of category that behaves like the category of sheaves of sets A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus.