The beautiful formulas for the surface area and volume of a sphere go back to Archimedes, who also discovered some other remarkable facts relating spheres to circumscribing cylinders. We describe these results.Then we introduce rational turn angles---a renormalization of the notion of angle so that perpendicular lines are represented not by 90 degrees, or by pi/2 radians, but rather by 1/4 turn. This is mathematically the most natural parametrization of an angle, and we restate the sum of angles in a triangle and quadrilateral in terms of turn angles. We state a useful Proportionality Principle. A famous theorem of Harriot (or Girard) gives the ratio of the area of a spherical triangle to the area of the sphere in terms of the sum of turn angles.

1. Introduction to Universal Hyperbolic Geometry
2. Apollonius and Polarity
3. Apollonius and Harmonic Conjugates
4. Pappus' Theorem and the Cross Ratio
5. First Steps in Hyperbolic Geometry
6. The Circle and Cartesian Coordinates
7. Duality, Quadrance and Spread in Cartesian Coordinates
8. The Circle and Projective Homogeneous Coordinates
9. The Circle and Projective Homogeneous Coordinates II
10. Computations with Homogeneous Coordinates
11. Duality and Perpendicularity
12. Existence of Orthocenters
13. Theorems using Perpendicularity
14. Null Points and Null Lines
15. Apollonius and Polarity Revisited
16. Reflections in Hyperbolic Geometry
17. Reflections and Projective Linear Algebra
18. Midpoints and Bisectors
19. Medians, Midlines, Centroids and Circumcenters
20. Parallels and the Double Triangle
21. The J function, sl(2) and the Jacobi identity
22. Pure and Applied Geometry: understanding the continuum
23. Quadrance and Spread
24. Pythagoras' Theorem in Universal Hyperbolic Geometry
25. The Triple Quad Formula in Universal Hyperbolic Geometry
26. Visualizing Quadrance with Circles
27. Geometer's Sketchpad and Circles in Universal Hyperbolic Geometry
28. Trigonometric Laws in Hyperbolic Geometry using Geometer's Sketchpad
29. The Spread Law in Universal Hyperbolic Geometry
30. The Cross Law in Universal Hyperbolic Geometry
31. Thales' Theorem, Right Triangles and Napier's Rules
32. Isosceles Triangles in Hyperbolic Geometry
33. Menelaus, Ceva and the Laws of Proportion
34. Trigonometric Dual Laws and the Parallax Formula
35. Introduction to Spherical and Elliptic Geometries
36. Introduction to Spherical and Elliptic Geometries II
37. Areas and Volumes for a Sphere
38. Classical Spherical Trigonometry
39. Perpendicularity, Polarity and Duality on a Sphere
40. Parametrizing and Projecting a Sphere
41. Rational Trigonometry: An Overview
42. Rational Trigonometry in Three Dimensions

This is a complete and relatively elementary course explaining a new, simpler and more elegant theory of non-Euclidean geometry; in particular hyperbolic geometry. It is a purely algebraic approach which avoids transcendental functions like log, sin, tanh etc, relying instead on high school algebra and quadratic equations. The theory is more general, extending beyond the null circle, and connects naturally to Einstein's special theory of relativity. This course is meant for mathematics majors, bright high school students, high school teachers, engineers, scientists, and others with an interest in mathematics and some basic algebraic skills. NJ Wildberger is also the developer of Rational Trigonometry: a new and better way of learning and using trigonometry. Look up for his course in order to familiarize with this new development. He also has recorded very organized and detailed lecture series on Algebraic Topology, History of Mathematic, Linear Algebra, and more.