Remarks on the 20th century history of axiom systems for Euclidean plane geometry

The axiom system we're using for Euclidean plane geometry is based on an axiom system introduced by George Birkhoff in the early 1920's. He published a paper on his axioms, "A set of postulates for plane geometry, based on scale and protractor" (Annals of Mathematics 33 (1932), 239-245). Later he wrote a high school geometry book with Ralph Beatley, Basic Geometry (Scott, Foresman, and Company, Boston 1940). The third edition (1959) has been reprinted by Chelsea Publishing Company, and it is available from the American Mathematical Society. Also available are the Answer Book and the Manual for Teachers. (A history of some of the ideas in Birkhoff's axioms can be found in the introduction to the Manual for Teachers.)

A modification of Birkhoff's axiom system was formulated by Sanders MacLane, who published his version as "Metric postulates for plane geometry" (American Mathematical Monthly 66 (1959), 543-555). We've used MacLane's version of the betweenness axiom, which he calls the continuity axiom.

The difference between Birkhoff's axiom systems and earlier axiom systems for plane geometry is that his axioms use the real numbers and their properties. I go even further in this direction: I take the properties of distance, angle measure, and area as three of my axioms. This is actually a very common viewpoint; for example, in modern geometry the concept of a "metric space" is fundamental. A metric space is just a set of points with a distance function satifying our distance axiom.

In older axiom systems, distance, angle measure, and area were not mentioned! Instead, the traditional approach used the concepts of "congruence" of segments, angles, and polygons. This traditional viewpoint was inherited from the ancient Greeks, whose system of geometry was preserved in Euclid's Elements (c. 300 BC). The Elements were used as a geometry textbook until well into the 20th century. A scholarly annotated version of the Elements is available from Dover Books. There's also a wonderful online version of the Elements (by David Joyce) with java animations.

Early in the 20th century mathematicians recognized that Euclid's axiom system was incomplete--concepts such as "between," "inside," "outside" were not made precise, but instead reasoning using pictures was used by Euclid. David Hilbert developed a modern axiom system which removed these defects, in his book Grundlagen der Geometrie (Foundations of Geometry), Leipzig 1902. (An English translation of the seventh edition is available in paperback from Open Court Press.) In Birkhoff's axiom system, questions of betweenness, et cetera, are handled through the properties of the real number line. (The properties of the real numbers were put on a solid logical foundation near the end of the 19th century.)

In the early 1960's Birkhoff's axioms were modified by the School Mathematics Study Group (SMSG) to provide a new standard for teaching high school geometry. (See Geometry, parts I and II, Student's Text and Teacher's Commentary, SMSG, Yale University Press, New Haven 1960.) Influential books written with this viewpoint are the college text Elementary Geometry from an Advanced Standpoint, by Edwin Moise (Addison-Wesley 1963, third edition 1990), and the high school text Geometry, by Moise and Floyd Downs (Addison Wesley 1964).

I prefer the simplicity of Birkhoff's original axiom system to the more elaborate system developed by Moise and SMSG. I believe that Birkhoff's axioms can be taught at either the college or the high school level, depending on whether the properties of the real numbers are used explicitly or taken for granted. My axiom system is appropriate for college students who have studied sets and functions; Birkhoff's is appropriate for high school students who have no experience with abstract mathematics.

A recent book which studies Euclidean geometry from an advanced viewpoint is Geometry: Euclid and Beyond, by Robin Hartshorne (Springer-Verlag 2000). (Actually Hartshorne follows Hilbert more than he follows Euclid.) Hartshorne emphasizes that Euclid's axioms do not need the properties of real numbers; instead Euclid's development of geometry (made precise by Hilbert) actually contains the construction of the real number system!

C. McCrory 9/17/04