Plane Geometry - Bruce E. Shapiro

Section 8Hilbert’s AxiomsDavid Hilbert (1862-1943) boiled geometry down to 20 axioms which heclassified into seven axioms of connection (we now use the term incidence);five axioms of order; one axiom of parallels; six axioms of congruence; andone axiom of continuity. He did this because it had been discovered overthe centuries that Euclid had left out parts of his arguments and Hilbertwas attempting to fill in all the blanks. The axioms below are taken fromthe lecture notes of his course in geometry given in 1898 and translated byE.J. Townsend in 1902.Hilbert’s system begins with the followingundefined terms: point, line, plane,lie on, between, congruent. His axiomsare divided up into different sub-areasof geometry: connection, order, parallels,congruence, continuity, and completeness.The axioms of connection definethings like how points form linesand planes. The axioms of order expressthe concept of betweenness of points.They are classified into four linear axiomsof order and one plane axiom of order.The axiom of parallels is a equivalentto Euclid’s parallel postulate. Theaxioms of congruence formalize our intuitivenotions of equivalences among linesegments, angles, and triangles. The ax-33