Euclidean geometry

Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system. The Elements begins with plane geometry, still taught in secondary school (high school) as the first axiomatic system and the first examples of formal proof. It goes on to the solid geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, explained in geometrical language.The Pons Asinorum or Bridge of Asses theorem states that in an isosceles triangle, α = β and γ = δ.The Triangle Angle Sum theorem states that the sum of the three angles of any triangle, in this case angles α, β, and γ, will always equal 180 degrees.The Pythagorean theorem states that the sum of the areas of the two squares on the legs (a and b) of a right triangle equals the area of the square on the hypotenuse (c).Thales' theorem states that if AC is a diameter, then the angle at B is a right angle.A surveyor uses a levelSphere packing applies to a stack of oranges.A parabolic mirror brings parallel rays of light to a focus.Geometry is used in art and architecture.The water tower consists of a cone, a cylinder, and a hemisphere. Its volume can be calculated using solid geometry.Geometry can be used to design origami....when we begin to formulate the theory, we can imagine that the undefined symbols are completely devoid of meaning and that the unproved propositions are simply conditions imposed upon the undefined symbols.If our hypothesis is about anything, and not about some one or more particular things, then our deductions constitute mathematics. Thus, mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.Geometry is the science of correct reasoning on incorrect figures.If proof simply follows conviction of truth rather than contributing to its construction and is only experienced as a demonstration of something already known to be true, it is likely to remain meaningless and purposeless in the eyes of students. Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system. The Elements begins with plane geometry, still taught in secondary school (high school) as the first axiomatic system and the first examples of formal proof. It goes on to the solid geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, explained in geometrical language. For more than two thousand years, the adjective 'Euclidean' was unnecessary because no other sort of geometry had been conceived. Euclid's axioms seemed so intuitively obvious (with the possible exception of the parallel postulate) that any theorem proved from them was deemed true in an absolute, often metaphysical, sense. Today, however, many other self-consistent non-Euclidean geometries are known, the first ones having been discovered in the early 19th century. An implication of Albert Einstein's theory of general relativity is that physical space itself is not Euclidean, and Euclidean space is a good approximation for it only over short distances (relative to the strength of the gravitational field). Euclidean geometry is an example of synthetic geometry, in that it proceeds logically from axioms describing basic properties of geometric objects such as points and lines, to propositions about those objects, all without the use of coordinates to specify those objects. This is in contrast to analytic geometry, which uses coordinates to translate geometric propositions into algebraic formulas. The Elements is mainly a systematization of earlier knowledge of geometry. Its improvement over earlier treatments was rapidly recognized, with the result that there was little interest in preserving the earlier ones, and they are now nearly all lost. There are 13 books in the Elements: Books I–IV and VI discuss plane geometry. Many results about plane figures are proved, for example 'In any triangle two angles taken together in any manner are less than two right angles.' (Book 1 proposition 17) and the Pythagorean theorem 'In right angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle.' (Book I, proposition 47) Books V and VII–X deal with number theory, with numbers treated geometrically as lengths of line segments or areas of regions. Notions such as prime numbers and rational and irrational numbers are introduced. It is proved that there are infinitely many prime numbers. Books XI–XIII concern solid geometry. A typical result is the 1:3 ratio between the volume of a cone and a cylinder with the same height and base. The platonic solids are constructed. Euclidean geometry is an axiomatic system, in which all theorems ('true statements') are derived from a small number of simple axioms. Until the advent of non-Euclidean geometry, these axioms were considered to be obviously true in the physical world, so that all the theorems would be equally true. However, Euclid's reasoning from assumptions to conclusions remains valid independent of their physical reality.