About Euclidean Geometry
The word geometry comes from the Greek geometrein (geo meaning earth, and metrein meaning to measure); geometry was originally the science of measuring the land. Euclid was a disciple of the Platonic school and around 300 B.C. he produced the definitive treatment of geometry in his 13-volume Elements: Books I-IV, VII came from the Pythagoreans, Book VIII came from Archytas, Books V, VI, and XII came from Eudoxus, and Books X and XIII came from Theaetetus, and so in compiling his masterpiece Euclid built on the experiences and achievements of his predecessors. Some of Euclid's assumptions used in his proofs were not stated explicitly; for example, there is nothing in Euclid's postulates from which we can deduce that an angle bisector of a triangle will intersect the opposite side, and thus Euclid simply assumed the existence of the needed point. Even so, Euclid's geometry reigned supreme until the 19th century when the discovery of non-Euclidean geometry brought about a re-examination of the foundations of Euclidean geometry. The concepts of incidence (i.e. a point lies on a line), betweeness (i.e. a point is between two other points), and congruence (i.e. line segments are congruent) have been the main improvements in Euclidean geometry since Euclid's time. Several great mathematicians including Pasch, Peano, Pieri, Veblen, Forder, Robinson, Levi, Hilbert, Birkhoff, and MacLane, and several groups including the School Mathematics Study Group (SMSG) and the University of Chicago School Mathematics Project (UCSMP) have made stunning improvements in Euclidean geometry as a mathematical (axiomatic) system. Eventually there evolved a consensus that the validity of a geometric axiomatic system was dependent on the consistency, independence, and completeness of the axiom set on which it is built and not on a physical or biased rationale. What is astonishing is the number and variety of propositions that can be deduced from so few assumptions.
Cite this as:About Euclidean Geometry
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/about-euclidean-geometry.html
