The Main Trends in the Foundations of Geometry in the 19th Century

Abstract

Publisher Summary The best-known chapter in the history of mathematics is the history of non-Euclidean geometry. The reception of non-Euclidean geometry started with three contemporary events: (1) Helmholtz's “Uber die Thatsachen, die der Geometrie zum Grunde liegen,” (2) Beltrami's interpretation of non-Euclidean geometry in differential geometry, and (3) J. Houel's translations and re-editions of texts on non-Euclidean geometry. After spreading awareness among mathematicians for non-Euclidean geometry, F. Klein discovered in Cayley's famous “Sixth Memoir upon Quantics,” a model of non-Euclidean geometry. Yet this discovery did not contribute to clarifying the foundations of geometry. The digression at the beginning of the 19th century from the Euclidean to the projective approach was one of the historical presuppositions of the axiomatization of geometry that was to be achieved at the end of the century. Euclidean geometry is a logically involved structure. As long as axiomatic thinking had not grown into a habit, it could not be easy to draw up a complete system of the axioms of geometry.