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• Print publication year: 2005
• Online publication date: January 2012

# 6 - How to Use History to Clarify Common Confusions in Geometry

from II - Geometry

## Summary

Most people judge the size of cities simply from their circumference. So that when one says that Megalopolis is fifty stades in contour and Sparta forty-eight, but that Sparta is twice as large as Megalopolis, what is said seems unbelievable to them. And when in order to puzzle them even more, one tells them that a city or camp with the circumference of forty stades may be twice as large as one of the circumference of which is one hundred stades, what is said seems to them absolutely astounding. The reason of this is that we have forgotten the lessons in geometry we learnt as children.

—Polybius, 2nd century B.C. [56]

Introduction

We have found that students who take our senior/graduate level geometry course usually have very little background in geometry. We have lead many week-long UFE and PREP workshops (funded by the National Science Foundation) for professors on teaching geometry and we found that even mathematicians are often confused about the history of geometry. In addition, many expository descriptions of geometry (especially non-Euclidean geometry) contain confusing and sometimes-incorrect statements — this is true even in expositions written by well-known research mathematicians. Therefore, we found it very important to give some historical perspective of the development of geometry, clearing up many common misconceptions and increasing people's interest both in geometry and in the history of mathematics.