Geometry Versus Euclid

Euclid ' S Geometry

Teaching Geometry According to Euclid, Volume 47, Number 4

460 NOTICES OF THE AMS VOLUME 47, NUMBER 4 I n the fall semester of 1988, I taught an undergraduate course on Euclidean and nonEuclidean geometry. I had previously taught courses in projective geometry and algebraic geometry, but this was my first time teaching Euclidean geometry and my first exposure to non-Euclidean geometry. I used the delightful book by Greenberg [8], which I believe my stu...

Translating Euclid: Liberating the Cognitive Potential of Collaborative Dynamic Geometry

How should one translate the classic-education approach of Euclid’s geometry into the contemporary vernacular of computer visualization, social networking and discourse-centered pedagogy? The birth of geometry in ancient Greece and its systematization by Euclid played an important role in the development of deductive reasoning and science. As it was translated and refined over the centuries, ho...

Computational Euclid

We analyse the axioms of Euclidean geometry according to standard object-oriented software development methodology. We find a perfect match: the main undefined concepts of the axioms translate to object classes. The result is a suite of C++ classes that efficiently supports the construction of complex geometric configurations. Although all computations are performed in floating-point arithmetic...

Proofs, pictures, and Euclid

Though pictures are often used to present mathematical arguments, they are not typically thought to be an acceptable means for presenting mathematical arguments rigorously. With respect to the proofs in the Elements in particular, the received view is that Euclid’s reliance on geometric diagrams undermines his efforts to develop a gap-free deductive theory. The central difficulty concerns the g...