Euclid’s Elements and the Axiomatic Method

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“There is no royal road to geometry.” – Euclid

Euclid’s Elements are predominantly the most fundamental concepts of mathematics, but his perspective on geometry was the model for over two millennia. He is believed by many to be the leading mathematics teacher of all time. However, little is known about his life outside of mathematics, or even when he was born or when he died. According to a passage written by Proclus, Euclid probably lived after Ptolemy and the pupils of Plato, but came before Archimedes and Eratosthenes. This places his existence sometime around 300 B.C. Euclid is most famous for having set the guidelines for geometry and arithmetic written in Euclid’s Elements, a series of thirteen books in which Euclid states definitions, postulates, and theorems for mathematical concepts that are still used today. What is most remarkable about the Elements is the simple, rational, and very logical structure in which Euclid presents the accumulated geometrical knowledge from the past several centuries of Greek mathematicians. The manner in which the propositions have been derived is considered to be the prime model of the axiomatic method. (Hartshorne 296).

Euclid’s axiomatic method works by “starting from a small number of definitions and assumptions at the beginning, [so that] all the succeeding results are proved by logical deduction from what has gone before.” In essence it is no more than “a method of proving that results are correct.” Many of Euclid’s proofs are constructions, all of which can be done using no more than a ruler and a compass and rely only on the theorems and rules of the system. Despite having developed this rigorous system of proofs, Euclid did not actually demonstrate everyt...

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... of Nebraska Press, 1991.

Blumenthal, Leonard M. A Modern View of Geometry. San Fransisco: W.H. Freeman and Company, 1961.

Greenberg, Marvin Jay. Euclidean and Non-Euclidean Geometries. New York: W.H. Freeman and Company, 1993.

Hartshorne, Robin. Geometry, Euclid and Beyond. New York: Springer, 2000.

Hofstadter, Douglas R.. Gödel, Escher, Bach: An Eternal Golden Braid. New York: Basic Books, 1975

Narkiewicz, Wladyslaw. The Development of Prime Number Theory: From Euclid to Hardy and Littlewood. Brlin: Springer-Verlag, 2000.

Singer, David A. Geometry: Plane and Fancy. New York: Springer, 1998.

Internet Sources:

Joyce, D. E. “Euclid’s Elements.” 1997, [online]. http://aleph0.clarku.edu/~djoyce/java/elements/toc.html (September 18, 2002)

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