Math 4060 -- Foundations of Geometry

Spring 2023

Class Meets:   MWF 10-10:50 in LANG 223.

Instructor: William Cherry

Office Location: General Academic Building (GAB) 405.
Office Hours:  MW 4:30-6:30 and by appointment


All students should have successfully completed Math 3000 (Real Analysis I) before taking this course. The reason for this is that this course is a proof intensive course and students should already be familiar with basic proof writing techniques and strategies. Prior or concurrent enrollment in either Math 3610 (Real Analysis II) or Math 3510 (abstract algebra) is also strongly recommended. This is meant to ensure familiarity with proof and proof technique. Nothing from real analysis or abstract algbra will actually be needed. Students who have not yet taken Math 3610 or Math 3510, but who are confident in their proof-writing ability, are welcome in the course.

Course Description

Early interest in geometry was almost certainly motivated by a desire to improve building techniques and to be able to create interesting shapes for temples, altars, toys, and machines. Scholars in ancient Greece "abstracted" the study of shapes into an idealized form. The most widely read textbook in history is a geometry text by Euclid known as the Elements. Thousands of years later, much of what we learn in high school geometry is based on the ideas in Euclid's text. Euclid's goal was to create a logical foundation for idealized geometry. He wanted to provide proofs for geometric facts based on as few axioms as possible. The course will begin with a careful study of parts of Euclid and an exploration of what he was trying to do and why. The course will then move on to the contributions of 19th and 20th century scholars who built on Euclid's work. The second half of the course will look at alternative geometries where the so-called parallel postulate need not hold.

The course will emphasize the role of proof in geometry, its historical development, and the philosophical implications that proof had on the development of scientific thought. Students will begin by reading Euclid and continually develop their own proof writing skills and geometric intuition.


Robin Hartshorne, Geometry: Euclid and Beyond, Springer Verlag, 2000. ISBN 9780387986500
Euclid's Elements, The Thomas L. Heath translation edited by Dana Densmore, Green Lion Press, 2002. ISBN 9781888009194.
Euclid is also available online at
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