Hilbert's Statement of Pasch's Axiom
David Hilbert uses Pasch's axiom in his book Foundations of Geometry which provides an axiomatic basis for Euclidean geometry. Depending upon the edition, it is numbered either II.4 or II.5 and is stated as:
- Let A, B, C be three points that do not lie on a line and let a be a line in the plane ABC which does not meet any of the points A, B,C. If the line a passes through a point of the segment AB, it also passes through a point of the segment AC, or through a point of segment BC.
The fact that both segments AC and BC are not intersected by the line a is proved in Supplement I,1, which was written by P. Bernays.
In Hilbert's treatment, this axiom appears in the section concerning axioms of order and is referred to as a plane axiom of order. Since he does not phrase the axiom in terms of the sides of a triangle there is no need to talk about internal and external intersections of the line a with the sides of the triangle ABC. However, if one applies the triangle terminology to Hilbert's version of the axiom, the fact that it is equivalent to Pasch's version is apparent.
Read more about this topic: Pasch's Axiom
Famous quotes containing the words statement and/or axiom:
“Most personal correspondence of today consists of letters the first half of which are given over to an indexed statement of why the writer hasn’t written before, followed by one paragraph of small talk, with the remainder devoted to reasons why it is imperative that the letter be brought to a close.”
—Robert Benchley (1889–1945)
“It is an axiom in political science that unless a people are educated and enlightened it is idle to expect the continuance of civil liberty or the capacity for self-government.”
—Texas Declaration of Independence (March 2, 1836)