Beltrami's model and the independence of the parallel postulate

Abstract

E. Beltrami in 1868 did not intend to prove the consistency of non-euclidean plane geometry nor the independence of the euclidean parallel postulate. His approach would have been unsuccessful if so intended. J. Houel in 1870 described the relevance of Beltrami's work to the issue of the independence of the euclidean parallel postulate. Houel's method is different from the independence proofs using reinterpretation of terms deployed by Peano about 1890, chiefly in using a fixed interpretation for non-logical terms. Comparing the work of Beltrami and Houel with the treatment of non-euclidean geometry after the development of the axiomatic method in the 1890s indicates an important shift in mathematicians’ attitudes towards mathematical theories.