Instructor: Roberto Hasfura

SCHEDULE: MWF 10:30-11:20 AM in Room MMS 110

PREREQUISITES: MATH 1321 or equivalent

TEXT: Euclidean and Non-Euclidean Geometries, Development and History, by M. J. Greenberg

OFFICE HOURS: MW 1:30-4:00 PM, M 7-8 PM; TR 10:00 11:30; F 11:30-12:00 AM; and by appointment.

In Room MMS 115I; Tel. 736-8240; email:; Course Web page at


The discovery of non-euclidean geometries in the early part of the nineteenth century is an event not only of great importance in mathematics and physics, but also one with significant epistemological import.

In this course, after a brief discussion of the essentials of axiomatic systems we will examine the merits and deficiencies of the geometry in Euclid's Elements (c. 300 B.C.) and then we will present the system postulated by Hilbert, which redresses those shortcomings. We will also study the role of the parallel postulate in that geometry as well as the history of the controversy surrounding it. Finally, the geometry that arises by postulating the negation of the parallel postulate (a non-euclidean geometry) will be considered along with some of its models.

Hopefully, by the end of the semester, the first paragraph in this course description will have been justified.


Your final grade in this course will be a function of the quality of your work on the assignments I will give and of your attendance record, as follows:

Final Grade = .90(Average Grade on Assignments)+.10(Grade on Attendance).

The assignments will consist of eight to ten take-home problem sets throughout the semester. Notice that most of your final grade depends on the grades you get on these assignments so you shouldn't treat them lightly (as you might treat a regular, less weighty homework assignment.) The papers you will turn in must be thoughtful, clear, and as complete as possible. (I will not give credit to illegible or disorganized or otherwise messy work. I suggest that you work out the problems in their entirety and carefully review them before you attempt to produce the final draft. The solutions that you will hand in should be neatly presented with all work shown.) With regard to attendance, each unjustified absence will cost 20 (out of 100) points in your Attendance Grade so that if, for example, you have five unjustified absences your Attendance Grade will be 0 and you will not be able to get an A in the course.


1.- We will abide by the University Policy on Academic Integrity.

2. University regulations prohibit eating, drinking and smoking in the classroom.


01/13-02/10 - Chapters 1,2 & 3

02/12-03/01 - Chapter 4

03/03-03/19 - Chapter 5

03/22-04/12 - Chapter 6

04/14-04/26 - Chapter 7


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77<#<80 C+ 73<#<77 C 70<#<73 C- 65<#<70 D+ 69<#<65 D

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HOMEWORK. (You can find solution keys to the homework sets by following the links below. Those files are in pdf format, so you will need Adobe's Acrobat Reader, which can be downloaded from Adobe's web site or, faster, from the Trinity network. Once installed in your machine, the browser opens the reader when needed.)

Hwk. 1 Due 1/27 Read Chapter 1. Hand in solutions to all the Review Exercises and Exercises 1, 2, 4, 5, 10, 11, 13 in Chapter 1.

Hwk. 2 Due 2/3. Read Chapter 2. Hand in solutions to all the Review Exercises and Exercises 2, 3, 4, 5, 6, 7, 9, 11 in Chapter 2.

Hwk. 3 Due 2/10 Read Betweenness on Chapter 3. Hand in solutions to Exercises 1, 2, 3, 4, 5, 6, 9, 13, 16 in Chapter 3.

Hwk 4. Due 2/17 Read Congruence on Chapter 3. Hand in solutions to Exercises 22, 23, 25, 26, 27, 29, 30 and Major Ex. 3 in Chapter 3.

Hwk 5. Due 2/22 Read Chapter 3. Hand in solutions to the Review Exercises, Exercises 7ab, 34, 35, and Major Ex. 1 in Chapter 3.

Hwk 6. Due 3/15. Read Chapter 4. Hand in solutions to the Review Exercises and Exercises 3, 9, 16, 17ab, 18, 19, 23, 27, 28 and 29 in Ch. 4.

Hwk 7. Due 3/24. Read Chapter 5. Hand in solutions to the Review Exercises and Exercises 2, 3, 4, 10, 11 and 12.

Hwk. 8. Due 4/14. Read Chapter 6. Hand in solutions to the Review Exercises, Exercises 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15 and 16, and Major Exercises 10 and 11.

Hwk 9. Due 5/4. Read Chapter 7. Hand in solutions to exercises K-11, K-12, P-6, P-7, P-8, P-9, P-10 and P-11 in Chapter 7.