We discuss aspects of Euclidean geometry including isometries of the plane, affine tranformations in the plane and symmetry groups. We then explore similar concepts in the sphere and projective space, and explore elliptic and hyperbolic geometry.

In two articles published in 1868 and 1869, Eugenio Beltrami provided three models in Euclidean plane (or space) for non-Euclidean geometry. Our main aim here is giving an extensive account of the two articles’ content. We will also try to understand how the way Beltrami, especially in the first article, develops his theory depends on a changing attitude with regards to the definition of surfac...

We use Herbrand’s theorem to give a new proof that Euclid’s parallel axiom is not derivable from the other axioms of first-order Euclidean geometry. Previous proofs involve constructing models of nonEuclidean geometry. This proof uses a very old and basic theorem of logic together with some simple properties of ruler-and-compass constructions to give a short, simple, and intuitively appealing p...

Projective geometry is called geometry of joining and intersecting in old German textbooks. This suggested that we might base projective geometry on assumptions concerning two operations of joining and intersecting as the algebra of numbers may be based on assumptions concerning two operations of adding and multiplying. The classical treatments of geometry did not mention assumptions as simple ...