Plane Geometry - Bruce E. Shapiro

Section 9Birkhoff/MacLaneAxiomsGeroge Birkhoff (1884-1944) is bestknown for his works on differential equations,taught at Univ. of Wisconsin,Princeton, and Harvard. In 1932 he proposeda very compact set of axioms whichallow you to use a ruler (a straight edgewith marks on it) and a protractor. Hispurpose was to make geometry more understandableto high school students. 1Birkhoff (left); MacLane (right)Saunders MacLane (1909-2005) was afriend (and professional collaborator) of George Birkhoff’s son Garrett, andworked primarily at Harvard (where he met the Birkhoffs) and the Univ.of Chicago. 2 In 1959 [MacLane, 1959] proposed an extension of Birkhoff’sAxioms that included a distance measure thereby making the system somewhatmore intuitive than Hilbert’s. MacLane introduced the concept ofdistance metrics into the axioms, and added an axiom of continuity.1 The system was published in the paper ”A Set of Postulate for Plane Geometry basedon a Scale and Protractor,” in Annals of Mathematics, 33:329-345 (1932).2 he photo is by Konrad Jacobs (CCASA license) and from Wikimedia).37