Euclidean Geometry

euclidean geometry

[yü′klid·ē·ən jē′äm·ə·trē] (mathematics) The study of the properties preserved by isometries of two- and three-dimensional euclidean space.

Euclidean Geometry

a geometry, the systematic construction of which was first provided in the third century B.C. by Euclid. The system of axioms of Euclidean geometry is based on the following basic concepts: point, line, plane, motion, and the relations “a point lies on a line in a plane” and “a point lies between two other points.” In modern presentations of Euclidean geometry the axioms fall into five groups:

(I) Axioms of incidence. (1) One and only one line passes through any two points. (2) On every line there are at least two points. There exist at least three points that do not lie on one line. (3) One and only one plane passes through any three points that do not lie on one line. (4) There are at least three points in every plane, and there are at least four points not lying in the same plane. (5) If two points of a given line lie in a given plane, then the line itself lies in that plane. (6) If two planes have a point in common, then they have an additional point (and consequently, a line) in common.

(II) Axioms of order. (1) If point B lies between points A and C, then all three points lie on one line. (2) For any points A and B, there exists a point C such that B lies between A and C. (3) Of any three points on a line, just one lies between the other two. (4) If a line intersects one side of a triangle, then it also intersects another side or passes through a vertex. (A segment AB is defined as the set of points lying between A and B the sides of a triangle are defined in a similar manner.)

(III) Axioms of motion. (1) A motion carries points into points, lines into lines, and planes into planes while retaining incidence of points on lines and in planes. (2) Two successive motions are equivalent to a certain single motion, and for each motion there is an inverse motion. (3) Let A, A’ and a, a’ be two points and halflines going out from them and α, α, halfplanes bounded by the lines a and a’ produced; then there exists one and only one motion that carries A, a, and α to A’, a’, and α. (The halflines and halfplanes are easily defined using the concepts of order and incidence.)

(IV) Axioms of continuity. (1) Axiom of Archimedes: Let AB and CD be given segments. Then, using motions, it is possible to lay off enough copies of AB, say, end to end on CD to cover CD. (2) Cantor’s axiom: The intersection of a nested sequence of segments contains at least one point.

(V) Parallel axiom. Given a line a and point A not on a, there is just one line through A that lies in the plane containing both a and A and does not intersect a.

The emergence of Euclidean geometry is closely related to man’s intuitive concepts about the world around him (lines—taut strings, rays of light, and so forth). The prolonged process of ever deeper understanding of these concepts has led to a more abstract view of geometry. The discovery by N. I. Lobachevskii of a geometry that differs from Euclidean geometry demonstrated that our ideas of space are not a priori. In other words, Euclidean geometry cannot claim to be the only geometry describing the properties of the space surrounding us. The development of the natural sciences, primarily physics and astronomy, has shown that Euclidean geometry describes the structure of the surrounding space with only a certain degree of accuracy and does not adequately describe the properties of space connected with the displacements of bodies at velocities approaching the velocity of light. Thus Euclidean geometry may be viewed as a first approximation to a description of the structure of the real physical world.

E. G. POZNIAK

Euclidean geometry