How to Use History to Clarify Common Confusions in Geometry

Abstract

We have found that students and even mathematicians are often confused about the history of geometry. Many expository descriptions of geometry (especially non-Euclidean geometry) contain confusing and sometimes-incorrect statements. Therefore, we found it very important to give some historical perspective of the development of geometry, clearing up many common misconceptions. In this paper we use history to clarify the following questions, which often have confusing or misleading (or incorrect) answers: 1. What is the first non-Euclidean geometry? 2. Does Euclid's parallel postulate distinguish the non-Euclidean geometries from Euclidean geometry? 3. Is there a potentially infinite surface in 3-space whose intrinsic geometry is hyperbolic? 4. In what sense are the Models of Hyperbolic Geometry 'models'? 5. What does 'straight' mean in geometry? How can we draw a straight line? We noticed that most confusions related to the above questions come from not recognizing certain strands in the history of geometry. The main aspects of geometry today emerged from four strands of early human activity that seem to have occurred in most cultures: art/patterns, building structures, motion in machines, and navigation/stargazing. These strands developed more or less independently into varying studies and practices that eventually from the 19th century on were woven into what we now call geometry. In this paper we describe how these strands can be used to clarify issues surrounding these questions.