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Non-Euclidean Geometry Online Course for Academic Credit

Non-Euclidean Geometry explores hyperbolic and elliptic geometries, contrasting with classical Euclidean geometry, with computational visualization using Mathematica.

Course
DMAT 288 - Introduction to Non-Euclidean Geometry
Credits
4 Semester Credit Hours
Delivery
Fully Online, Asynchronous, Self-Paced

RWU Course Catalog - DMAT 288 • Introduction to Non-Euclidean Geometry
Course
DMAT 288
Course Title
Introduction to Non-Euclidean Geometry
Transcript Title
Non-Euclidean Geometry
Credits
4 Semester Credit Hours
Description
A historical and rigorous introduction to the subject of Non-Euclidean Geometry, spanning from Euclid's Elements through the attempts to prove the Parallel Postulate, ultimately resulting in the intellectual blossoming of multiple geometric realities. Topics include Euclid's postulates, formulating axiomatic systems, rigorous geometric proofs, Neutral Geometry, multiple Parallel Postulates, Hyperbolic and Projective Geometry.
Prerequisite
Successful completion with grade C or higher in Precalculus or equivalent, or consent of instructor.
E-Textbook
Euclidean and Non-Euclidean Geometries, Development and History, 2nd Edition, by Greenberg
Software
None

DMAT 288 - Learning Outcomes

  1. To understand the historical development of the discovery of Non-Euclidean Geometry
  2. To understand the foundations of the axiomatic method
  3. To understand rigorous geometric proofs
  4. To understand the history and flaws of Euclid
  5. To understand the statements of the Parallel Postulate
  6. To understand the concepts of consistent geometry
  7. To understand the foundations of Neutral Geometry
  8. To understand the Non-Euclidean Parallel Postulate
  9. To understand the historical relevance of the discovery of Hyperbolic Geometry

DMAT 288 - Syllabus of Topics

1. Getting Started
1.1 Email and Chat
1.2 Learning About the Course
1.3 Required Hardware
1.4 Software Fundamentals
2. Euclid Geometry
2.1 Origins of Geometry
2.2 Axiomatic Method
2.3 Undefined Terms
2.4 Euclid first four postulates
2.5 The Parallel Postulate
2.6 Attempts to Prove The Parallel Postulate
3. Logic
3.1 Informal Logic
3.2 Theorems and Proofs
3.3 RAA Proofs
3.4 Negation
3.5 Quantifiers
3.6 Implication
3.7 Law of Excluded Middle
3.8 Incidence Geometry
3.9 Models
3.10 Isomorphism of Models
3.11 The Danger in Diagrams
4. Hilbert Axioms
4.1 Flaws in Euclid
4.2 Axioms of Betweenness
4.3 Axioms of Congruence
4.4 Axioms of Continuity
4.5 Axiom of Parallelism
5. Neutral Geometry
5.1 Geometry Without The Parallel Axiom
5.2 Alternate interior angle theorem
5.3 Exterior angle theorem
5.4 Measure of angles and segments
5.5 Saccheri-Legendre theorem
5.6 Equivalence of parallel postulates
5.7 Angle sum of a triangle
6. History of the Parallel Postulate
6.1 Proclus
6.2 Wallis
6.3 Saccheri and Lambert
6.4 Wolfgang Bolyai
7. The Discovery of Non-Euclidean Geometry
7.1 Janos Bolyai
7.2 Gauss
7.3 Lobachevsky
7.4 Hyperbolic geometry
7.5 Angle sums (again)
7.6 Similar triangles
7.7 Parallels that admit a common perpendicular
7.8 Limiting parallel rays
7.9 Classification of parallels
7.10 Strange new universe?
8. Independence of the Parallel Postulate
8.1 Consistency of hyperbolic geometry
8.2 The Beltrami-Klein model
8.3 The Poincare models
8.4 Perpendicularity in the Beltrami-Klein model
8.5 Inversion in circles
8.6 The projective nature of the Beltrami-Klein model
9. Philosophical Implications
9.1 What is the geometry of physical space?
9.2 What is mathematics about?
9.3 The controversy about the foundations of mathematics
9.4 Perpendicularity in the Beltrami-Klein model
9.5 The mess

Distance Calculus Referenced Colleges & Universities (29 Years - 393+ Institutions)

Distance Calculus students have transferred course credits to these colleges and universities:

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Distance Calculus - Student Reviews

Joe★★★★★
Posted: Jan 13, 2020
Courses Completed: Calculus II
This is the most interactive and productive online course I have ever taken. I had taken calculus before but never understood some of the underlying concepts until I took this course.
If you want to really learn calculus in a way that will stay with you for the rest of your life, take this course.
Transferred Credits To: The college of New Jersey
Lauren B.★★★★★
Posted: May 5, 2025
Courses Completed: Linear Algebra, Probability Theory
You don’t just learn in this course you understand. As someone who has always struggled with math taking advanced courses was intimidating. Dr. Curtis and his TAs made math manageable and framed the questions in a way they were applicable to real world. Thanks again!
Catherine M.★★★★★
Posted: Apr 5, 2020
Courses Completed: Calculus I
Calculus I from Distance Calculus was wonderful! I took AB Calculus in high school, but I didn't take the AP Calc exam. Instead I took Calculus I with Distance Calculus, and it was so much better! It was a little review of topics, but not really. I really understood calculus when I finished!
Transferred Credits To: University of Chicago
M M.★★★★★
Posted: Feb 8, 2026
Courses Completed: Precalculus, Calculus I
The courses were excellent. Very flexible and engaging and the platform offers a lot of upper-level courses. Dr. Curtis is an outstanding professor and very responsive. I would take again.
Transferred Credits To: None yet
Tanja B.★★★★★
Posted: Jan 28, 2026
Courses Completed: Calculus I
After two failed attempts at my university, this course helped me understand Calculus. The live maths tool along with Dr. Curtis were especially helpful, allowing me to visualize concepts and expand my understanding. The explanations were clear, the examples practical, and I could learn at my own pace, which built my confidence. Thank you.
Transferred Credits To: University of Namibia
Henry F.★★★★★
Posted: Dec 18, 2025
Courses Completed: Differential Equations
Transferred Credits To: Saint Joseph High School
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