Math 410: Modern Geometry

Course Website for Math 410 (Spring 2010). Last updated 10 May 2010.


These cover my version of Hilbert's rigorous approach to Euclidean and hyperbolic geometry.

1. The Incidence-Betweenness Geometry handout.

2. The IBC Geometry handout.

3. The Neutral Geometry handout.

4. The Euclidean Geometry handout.

5. The Quadrilateral handout.

6. The Legendre's Defect Zero Theorem handout.

7. The Euclidean and Hyperbolic Conditions handout.

8. The Topics in Hyperbolic Geometry handout.

Course Description

This is a course on Euclidean and non-Euclidean geometries with emphasis on (i) the contrast between the traditional and modern approaches to geometry, and (ii) the history and role of the parallel postulate. This course will be useful to students who want to teach and use Euclidean geometry, to students who want to learn more about the history of geometry, and to students who want an introduction to non-Euclidean geometry.

Euclid wrote the first preserved Geometry book which has traditionally been held up as a role model for logical reasoning inside and outside mathematics for thousands of years. However, Euclid has several subtle logical omissions, and in the late 1800s it was necessary to revise the foundations of Euclidean geometry. The need for such a revision was partly due to advances in mathematical logic and changes in the conception of an axiom system. In this course we will review the traditional approach, and then a modern approach based on Hilbert's axioms developed around 1900. The famous mathematician David Hilbert, building on work of several other mathematicians, was able to develop axioms that allow one to develop geometry without any overt or covert appeals to intuition. His idea was that, although intuitions are important in discovering, motivating, communicating and appreciating the theorems, rigorous proofs should not appeal to them. With the more modern approach to the axiomatic method that is not logically dependent on intuition, mathematicians are free to develop more types of geometries than the traditional Euclidean geometry. We will discuss different types and models of geometry that are used today. These include finite geometries with applications in discrete mathematics and number theory, spaces of more than three dimensions, geometries whose coordinates are not real numbers, and geometries where a line can pass through a circle without actually intersecting the circle. Many of these geometries are useful, and not just curious examples.

A second major theme of the course will be the history and role of the parallel postulate. The parallel postulate makes the assumption that anytime you have a point P and a line l not going through P, there is one and only one line m going through P that is parallel to l (this is closely related to Euclid's original fifth postulate). Modern geometry began in the 1800s with the realization that there are interesting consistent geometries for which the parallel postulate is false. For example, hyperbolic and elliptic geometry do not satisfy the parallel postulate.

Since this postulate is less intuitively obvious than the other axioms of geometry, many mathematicians, especially medieval Arab mathematicians and later several European mathematicians of the 1700s, tried to make the parallel postulate a theorem and not an axiom. This goes along with the traditional idea that axioms should be restricted to a few simple, self-evident propositions, and the rest of the subject should be built upon these using proof. However, no mathematician was able to show that the parallel postulate followed as a theorem from the other axioms. Several prominent mathematicians thought that they had a proof of the parallel postulate, but subtle flaws were later discovered in their proofs. Finally mathematicians such as Lobachevsky and Bolyai started to believe that it is possible for there to be geometries where the parallel postulate fails, and they proved theorems about such non-Euclidean geometries. After the discovery of (Euclidean) models of non-Euclidean geometries in the late 1800s, no one was able to doubt the existence and consistency of non-Euclidean geometry. Also, these models show that the parallel postulate is independent of the other axioms of geometry: you cannot prove the parallel postulate from the other axioms.

Required Texts (2)

There are two required texts.

1. Euclidean and Non-Euclidean Geometries : Development and History
by Marvin Jay Greenberg
ISBN: 0716799480
Publisher: W. H. Freeman and Company (4th edition 2008)
See Amazon or Barnes and Noble for descriptions of this textbook.

2. Book I of the Elements of Euclid. You can use the recent translation by Richard Fitzpatrick or the classic translation by Thomas Heath.

You have at least several options for obtaining Book 1 of Euclid.

A. I recommend downloading Fitzpatrick's translation which is free, contains all of Euclid, and as a bonus has the original Greek. Instead of printing this out, you might want to buy the Hard bound edition.

B. Alternatively, there is an inexpensive complete edition published by Green Lion Press.
Click here for the first few pages of the Elements from this edition.

C. If instead you want an edition with extensive commentary, you can get the Dover edition of Euclid instead (same translation, but with Heath's commentary): Volume 1 has all the text we need, so Volume 2 and 3 are optional.
See Amazon or Barnes and Noble for descriptions. For the optional volumes: Amazon (vol 2), Barnes and Noble (vol 2), Amazon (Volume 3), Barnes and Noble. (Volume 3)

D. There is an online version designed by David E. Joyce. The translation is basically Heath's but slightly less literal in order to make it more readable.

Optional Texts (3)

There are three optional texts. These are books that I use when preparing for class, and they are suitable for students who want to dig deeper into the subject.

1. Geometry: Euclid and Beyond
by Robin Hartshorne
ISBN: 0387986502
Publisher: Springer (2000)
See Amazon or Barnes and Noble for descriptions of this textbook.

2. Foundations of Geometry
by David Hilbert
ISBN: 0875481647
Publisher: Open Court (1971 translation of an early 20th century classic)
See Amazon or Barnes and Noble for descriptions of this book.
Get an older (1902) edition for free from Project Gutenberg.

3. Non-Eulcidean Geometry
by Roberto Bonola
ISBN: 0486600270
Publisher: Dover (reprint of the 1912 edition).
This book includes translations of articles by Lobachevski and Bolyai, two originators of non-Euclidean geometry.
See Amazon or Barnes and Noble for descriptions of this book.


Experience with logic, mathematical proofs, and basic set theory.
This experience can be obtained by taking Math 350 or Math 370 (with a grade of C or higher).

Instructor Information

Dr. Wayne Aitken
Office: SCI2 327
Telephone: (760) 750-4155 (but I prefer e-mail)