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
Syllabus
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
Syllabus PDF
DMAT 288 - Learning Outcomes
- To understand the historical development of the discovery of Non-Euclidean Geometry
- To understand the foundations of the axiomatic method
- To understand rigorous geometric proofs
- To understand the history and flaws of Euclid
- To understand the statements of the Parallel Postulate
- To understand the concepts of consistent geometry
- To understand the foundations of Neutral Geometry
- To understand the Non-Euclidean Parallel Postulate
- 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:
Agnes Scott College • Aiken Technical College • Albany College of Pharmacy and Health Science • Alma College • American Graduate University • American Public University • American University • Andrews University • Arizona State University • Armstrong Atlantic State Univeristy • Athens State University • Auburn University • Auburn University MBA Program • Augusta State University • Austin Peay State University • Azusa Pacific University • Babson College • Baruch College • Baylor University • Belmont University • Beloit College • Bentley University • Berklee College of Music • Berry College • Bethany College • Binghamton University • Bloomsburg University • Bluefield State College • Bluegrass Community and Technical College • Borough of Manhattan Community College • Boston Conservatory • Boston University • Bryant University • Buena Vista University • California Lutheran University • California Polytechnic State University, San Luis Obispo • California state University • California State 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Distance Calculus - Student Reviews
Posted: Feb 28, 2020
Courses Completed: Precalculus, Calculus I
Pros: once you get going, you can go really fast. The visual textbook is pretty cool. The instructors were very responsive.
Cons: the movies are great, but the software crashes more than it should. Sometimes it is just a hassle doing things in the software instead of on paper, but once I got used to the software, it was ok.
Cons: the movies are great, but the software crashes more than it should. Sometimes it is just a hassle doing things in the software instead of on paper, but once I got used to the software, it was ok.
Transferred Credits To: Texas Christian University
Posted: Dec 8, 2020
Courses Completed: Differential Equations
This course may be more difficult than your average differential equations course, which better prepares you to use these skills in your degree. The self-learning does make learning some of the concepts challenging, but you get the help you need to understand these concepts.
Transferred Credits To: Johns Hopkins University
Posted: May 5, 2025
Courses Completed: Calculus II
The course material was well explained through the videos and notebooks, and it was nice to receive feedback on each assignment and correct any mistakes. However, the software required for the course was ridiculously outdated, to the point where I had to reinstall a very old operating system on my computer to even be able to run it. It was annoying and tedious to use, and it sometimes crashed. The graders were also sometimes overly picky about the way that concepts had to be explained to be considered correct (specific wording, etc), but it wasn’t too bad. Having multiple final exams seemed a bit excessive but again was not too bad. Overall, the main strength of this course is that it is completely asynchronous and self-paced, so I do recommend it if that is the most important factor to you. You will learn all of the material thoroughly, but you must be willing to deal with the more annoying aspects of the course (primarily the outdated technology).
Posted: Jun 25, 2026
Courses Completed: Multivariable Calculus
The professor as well as the TAs give great feedback when you need help with problems and the videos are great at explaining concepts. Return time on work is good and the work is not too much to handle.
Transferred Credits To: University of Wisconsin
Posted: Jun 24, 2026
Courses Completed: Linear Algebra
This course gives a perspective on Linear algebra that no traditional course does. I’d say i gained much more intuition for this subject from the DC course than my friends who took traditional courses elsewhere. As a cs major, this version of learning with visualization has helped me a lot in understand ML models. However the course doesn’t have videos for the last 2 chapers so i had to self learn with the mathematica notebooks. Response times are a little slow but since it’s a remote class, i guess it’s justified. Overall amazing course and definitely take this over traditional lin alg classes.
Posted: Jun 24, 2026
Courses Completed: Calculus I
As a full-time business owner completing an Executive MBA, I needed to satisfy a calculus prerequisite without putting my work on hold. Distance Calculus made that possible. The fully self-paced structure let me work early mornings and weekends around an unpredictable schedule, which a fixed-semester classroom course never would have allowed.
The course covered the core business calculus material thoroughly — derivatives, optimization, integration techniques including u-substitution, the Fundamental Theorem of Calculus, improper integrals, and numerical methods. The LiveMath computer algebra environment was central to the experience: it forced me to build each step explicitly rather than just arriving at an answer, which actually deepened my understanding of the mechanics.
Communication through the student portal was responsive when I had questions. For working professionals who need a rigorous, accredited calculus course on a flexible timeline, I'd recommend it.
The course covered the core business calculus material thoroughly — derivatives, optimization, integration techniques including u-substitution, the Fundamental Theorem of Calculus, improper integrals, and numerical methods. The LiveMath computer algebra environment was central to the experience: it forced me to build each step explicitly rather than just arriving at an answer, which actually deepened my understanding of the mechanics.
Communication through the student portal was responsive when I had questions. For working professionals who need a rigorous, accredited calculus course on a flexible timeline, I'd recommend it.
Transferred Credits To: MIT Ebma
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