• Scott Wolpert to lead NSF-funded project on DEI in mathematics and statistics

    Congratulations to Scott Wolpert, professor emeritus of mathematics, who was named principal investigator of a new project to improve diversity, equity and inclusion in mathematics departments. The project is funded by a $600,000 grant from the NSF, and it will provide DEI training to six representatives of math and statistics departments Read More
  • Abba Gumel Featured in Scientific American Article

    Congratulations to Abba Gumel being featured in a new Scientific American Article. The title is “How Mathematics Can Predict and Help Prevent the Next Pandemic” (link to the article). What a great advertisement to Maryland. It is a great honor to have Abba as our colleagues.   Congratulations Abba! Read More
  • Congratulations to Perrin Ruth and Elliot Kienzle

    Eighteen current students and recent alums of the University of Maryland’s College of Computer, Mathematical, and Natural Sciences (CMNS) received prestigious National Science Foundation (NSF) Graduate Research Fellowships, which recognize outstanding graduate students in science, technology, engineering, and mathematics. Across the university, 34 current students and recent alums were among the Read More
  • Congrats to CMNS' Deven Bowman, 2023 Goldwater Scholar

    Congratulations to Deven Bowman, a junior physics and mathematics dual-degree student, who was named a 2023 Goldwater Scholar! Deven has done research with Eun-Suk Seo and Steve Rolston and studied abroad in Florence with Luis Orozco. This summer he’ll be at Caltech working on LIGO. And, fun fact, both of his parents Read More
  • Maryland finishes fourth in the 2022 Putnam competition

     We are very excited to share the news that the University of Maryland Putnam team ranked fourth among 456 institutions in the 2022 Putnam Competition. This is the best result for our team in more than four decades. The first three teams are MIT, Harvard and Stanford. Our team members, Read More
  • 1
  • 2
  • 3
  • 4
  • 5


Hilbert's axioms for Euclidean Geometry. Neutral Geometry: The consistency of the hyperbolic parallel postulate and the inconsistency of the elliptic parallel postulate with neutral geometry. Models of hyperbolic geometry. Existence and properties of isometries.


1 course with a minimum grade of C- from (MATH240, MATH341, MATH461).

Level of Rigor


Sample Textbooks

The Four Pillars of Geometry, by J. Stillwell

Foundations of Geometry, by G.A. Venema.


If you like this course, you might also consider the following courses

Additional Notes


Logical deficiencies in Euclidean Geometry

Flawed proofs

Correction of some flawed proofs by clearly stating certain postulates and giving rigorous deductions

Overview of the structure of Euclidean Geometry

Importance of Euclid's Parallel Postulate as opposed to the other postulates

Equivalence of certain postulates (Play fair, etc) to Euclid's Parallel Postulate

Absolute or Neutral geometry. Work of Legendre and Saccheri

Negation of Euclid's Parallel Postulate; non-Euclidean geometry.

Discussion of models for non-Euclidean geometry

Axiom systems

Incidence axiom and ruler postulate


Segments, Rays and Convex sets

Angles and Triangles

Pasch's Postulate and Plane Separation Postulate

Perpendiculars and inequalities

SAS postulate

Parallel postulates

Models for non-Euclidean geometry

Proof of the consistency of non-Euclidean geometry by means of models

  • 1
  • 2