However, Euclid's reasoning from assumptions to conclusions remains valid independently from the physical reality. 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. The platonic solids are constructed.Īxioms The parallel postulate (Postulate 5): If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.Įuclidean geometry is an axiomatic system, in which all theorems ('true statements') are derived from a small number of simple axioms. A typical result is the 1:3 ratio between the volume of a cone and a cylinder with the same height and base. It is proved that there are infinitely many prime numbers.īooks XI–XIII concern solid geometry. Notions such as prime numbers and rational and irrational numbers are introduced. 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 I 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)īooks V and VII–X deal with number theory, with numbers treated geometrically as lengths of line segments or areas of surface regions. 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.īooks I–IV and VI discuss plane geometry. The Elements is mainly a systematization of earlier knowledge of geometry.

This is in contrast to analytic geometry, introduced almost 2,000 years later by René Descartes, which uses coordinates to express geometric properties by means of algebraic formulas. Įuclidean 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. 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). Today, however, many other self-consistent non-Euclidean geometries are known, the first ones having been discovered in the early 19th century. įor more than two thousand years, the adjective 'Euclidean' was unnecessary becauseĮuclid's axioms seemed so intuitively obvious (with the possible exception of the parallel postulate) that theorems proved from them were deemed absolutely true, and thus no other sorts of geometry were possible. Much of the Elements states results of what are now called algebra and number theory, explained in geometrical language. It goes on to the solid geometry of three dimensions.

The Elements begins with plane geometry, still taught in secondary school (high school) as the first axiomatic system and the first examples of mathematical proofs. Although many of Euclid's results had been stated earlier, Euclid was the first to organize these propositions into a logical system in which each result is proved from axioms and previously proved theorems. Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions ( theorems) from these.

Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry, Elements.

Line geometry example