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|Title: ||How to Use History to Clarify Common Confusions in Geometry|
|Authors: ||Taimina, Daina|
Henderson, David W.
|Keywords: ||history of geometry|
|Issue Date: ||15-May-2003|
|Publisher: ||Cornell Library Technical Reports and Papers|
|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.|
|Appears in Collections:||History and Theory of Machines and Mechanisms|
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