Fall 2015

Instructor: Roberto Hasfura



SCHEDULE:  TR 9:55-11:10 room MMH 130

PREREQUISITES: MATH 3326 or equivalent 

TEXT: Euclidean and Non-Euclidean Geometries, M. J. Greenberg, Third or Fourth Edition, (Fourth Edition is about $160; Third Edition can still be found online for far less money)

OFFICE HOURS: MW 4-5 p.m., TR 3:30-4:30 p.m.; and by appointment. In Room MMS 115K

INSTRUCTOR’S COORDINATES---Office: MMS 115K; Phone Extension: X8240; E-mail:





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.




The homework will consist of eight to ten take-home problem sets throughout the semester. Your final grade in the course will depend mostly 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.)


Your final grade in this course will be computed in the following manner:


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


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.




“All students are covered by a policy that prohibits dishonesty in academic work. Under Trinity’s Honor Code, a faculty member will (or a student may) report an alleged violation to the Academic Honor Council. Students who are under the Honor Code are required to pledge all written work that is submitted for a grade:  `On my honor, I have neither given nor received any unauthorized assistance on this work’ and their signature.  The pledge may be abbreviated `pledged’ with a signature. As your instructor, I support and adhere to the principles of the honor code. You will be assumed trustworthy in all your dealings with me and with your fellow classmates. However, should a violation of this trust be discovered, it will be reported to the Council which will investigate and adjudicate. The goal is not vengeance against those who violate the honor code, but fairness for those who adhere to it.”



1. Please do not use cell phones in the classroom. If you need do it during class, please discreetly leave the classroom and return when you are finished.





94<#<100 A  

90<#<94 A-

87<#<90 B+

83<#<87 B

80<#<83 B-

77<#<80 C+

73<#<77 C

70<#<73 C-

65<#<70 D+

60<#<65 D

#<60 F





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. Due in my office (please use the accordion envelope labeled Geometry) before 5 p.m. on Friday 9/11.