Euclid's Postulate Research Articles

Ontological Convictions and Epistemological Obstacles in Bolzano's Elementary Geometry.

Bernard Bolzano (1781-1848) was a contemporary of the founders of non-Euclidean geometry and of the renovation of projective geometry. However, he did not participate in the movement transforming concepts and methods which crystallized in a new order of geometry at the beginning of the nineteenth century. On the contrary, throughout his life Bolzano tried to demonstrate Euclid's postulate of parallel lines.

Les bases de la géométrie et de la physique: l'in-variance de l'espace euclidien

THE author claims to have proved Euclid's postulate of parallels and to have pointed out the errors and contradictions of Lobatschewsky. The book is an attack on non-Euclidean geometry and the theory of relativity, which M. Laures describes respectively as “une des plus stupides inventions du XIXe siecle,” and “une autre stupidite, fille ainee de la premiere.”

On the substitution of Wallis's postulate of similarity for Euclid's postulate of parallels

The following is an alternative to Proposition II of the former paper. It does not assume the second of the initial assumptions in Art. 2, nor does it assume that the angles supplementary to equal angles are equal, which Hilbert regards as a proposition (Foundations of Geometry, p. 18 of the English translation).

On the substitution of Wallis's Postulate of Similarity for Euclid's Postulate of Parallels

In the year 1663 John Wallis, Savilian Professor of Geometry in the University of Oxford, delivered a lecture in which he claimed that he had proved Euclid's Postulate of Parallels according to the strictest laws of demonstration after Euclid's manner.