Geometry on the Surface of the Sphere
This paper was originally intended to serve as one section of a larger project designed to assist Dr. Juliet Vogel in her study of mathematically gifted children. Under the notion that mathematically gifted students may approach geometry problems from a non-Euclidean standpoint, I began to study several non-Euclidean geometries and the existing research on the approaches most students take in solving geometry problems. I started my study of non-Euclidean geometries with Euclid's Fifth Postulate (about parallelism), which led to a study of the hyperbolic, elliptic, and double-elliptic axioms of parallelism. The hyperbolic axiom led me to study asymptotic triangles, Saccheri Quadrilaterals, and the Poincare plane, while the double-elliptic axiom led me to study the information contained in this paper. Having found no evidence of the validity of my original hypothesis, my attention on the existing research on students' approaches to geometry problems shifted to a model of the development of geometric thought. Since my work on non-Euclidean geometries didn't fit into the new framework of my study, I have treated it in this paper (separately) and have narrowed it to include only double-elliptic (spherical) geometry.