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Non-Euclidean geometry | Math History | NJ Wildberger

The development of non-Euclidean geometry is often presented as a high point of 19th century mathematics. The real story is more complicated, tinged with sadness, confusion and orthodoxy, that is reflected even the geometry studied today. The important insights of Gauss, Lobachevsky and Bolyai, along with later work of Beltrami, were the end result of a long and circuitous study of Euclid's parallel postulate. But an honest assessment must reveal that in fact non-Euclidean geometry had been well studied from two thousand years ago, since the geometry of the sphere had been a main concern for all astronomers. This lecture gives a somewhat radical and new interpretation of the history, suggesting that there is in fact a much better way of thinking about this subject, as perceived already by Beltrami and Klein, but largely abandoned in the 20th century. This involves a three dimensional linear algebra with an unusual inner product, looked at in a projective fashion. This predates and anticipates the great work of Einstein on relativity and its space-time interpretation by Minkowski. ************************ Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of the lectures for various Playlists: great for review, study and summary. My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/... My blog is at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things. Online courses will be developed at openlearning.com. The first one, already underway is Algebraic Calculus One at https://www.openlearning.com/courses/... Please join us for an exciting new approach to one of mathematics' most important subjects! If you would like to support these new initiatives for mathematics education and research, please consider becoming a Patron of this Channel at   / njwildberger   Your support would be much appreciated. *********************** Here are all the Insights into Mathematics Playlists: Elementary Mathematics (K-6) Explained:    / playlist   list=PL8403C2F0C89B1333 Year 9 Maths:    • Year9Maths   Ancient Mathematics:    • Ancient Mathematics   Wild West Banking:    • Wild West Banking   Sociology and Pure Mathematics:    • Sociology and Pure Mathematics   Old Babylonian Mathematics (with Daniel Mansfield):    / playlist   list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov Math History:    • MathHistory: A course in the History ...   Wild Trig: Intro to Rational Trigonometry:    • WildTrig: Intro to Rational Trigonometry   MathFoundations:    • Math Foundations   Wild Linear Algebra:    • Wild Linear Algebra   Famous Math Problems:    • Famous Math Problems   Probability and Statistics: An Introduction:    • Probability and Statistics: an introd...   Boole's Logic and Circuit Analysis:    • Boole's Logic and Circuit Analysis   Universal Hyperbolic Geometry:    • Universal Hyperbolic Geometry   Differential Geometry:    • Differential Geometry   Algebraic Topology:    • Algebraic Topology   Math Seminars:    • MathSeminars   ************************ And here are the Wild Egg Maths Playlists: Triangle Centres:    • ENCYCLOPEDIA OF TRIANGLE CENTERS   Six: An elementary course in pure mathematics:    • Six: An elementary course in Pure Mat...   Algebraic Calculus One:    • Algebraic Calculus One   Algebraic Calculus Two:    • Algebraic Calculus Two   м