Differential geometry arises from applying calculus and analytic geometry to curves and surfaces. This video begins with a discussion of planar curves and the work of C. Huygens on involutes and evolutes, and the related notions of curvature and osculating circle. We discuss involutes of the catenary (yielding the tractrix), cycloid and parabola. The evolute of the parabola is a semi-cubical parabola. For space curves we describe the tangent line, osculating plane, principle normal and binormal.Surfaces were studied by Euler, who investigated curvatures of planar sections and by Gauss, who realized that the product of Euler's two principal curvatures gave a new notion of curvature intrinsic to a surface. Curvature was ultimately extended by Riemann to higher dimensions, and plays today a major role in modern physics, due to the work of Einstein. If you like this topic, and want to learn more, make sure you don't miss Prof. Wildberger's exciting new course on Differential Geometry.

1. History of Pythagoras' theorem
2. History of Pythagoras' Theorem II
3. History of Greek Geometry I
4. History of Greek Geometry II
5. History of Greek Number Theory
6. History of Greek Number Theory II
7. Infinity in Greek Mathematics
8. History of Number Theory and Algebra in Asia
9. History of Number Theory and Algebra in Asia II
10. History of Polynomial Equations
11. History of Polynomial Equations II
12. History of Analytic Geometry and the Continuum
13. History of Analytic Geometry and the Continuum II
14. History of Projective Geometry
15. History of Calculus
16. History of Infinite series
17. Mechanics and the Solar System
18. History of Non-Euclidean Geometry
19. The Number Theory Revival
20. Mechanics and Curves
21. Complex Numbers and Algebra
22. History of Differential Geometry
23. History of Topology
24. Hypercomplex Numbers
25. History of Complex Numbers and Curves
26. History of Group Theory
27. History of Galois Theory I
28. History of Galois Theory II
29. History of Algebraic Number Theory and Rings I
30. History of Algebraic Number Theory and Rings II
31. Simple groups, Lie groups, and the Search for Symmetry I
32. Simple groups, Lie groups, and the Search for Symmetry II

In this course, Prof. N.J. Wildberger from UNSW provides a great overview of the history of the development of mathematics. The course roughly follows John Stillwell's book 'Mathematics and its History' (Springer, 3rd ed)Starting with the ancient Greeks, we discuss Arab, Chinese and Hindu developments, polynomial equations and algebra, analytic and projective geometry, calculus and infinite series, number theory, mechanics and curves, complex numbers and algebra, differential geometry, topology and hyperbolic geometry.  This course is meant for a broad audience, not necessarily mathematics majors. All backgrounds are welcome to take the course and enjoy learning about the origins of mathematical ideas. Generally the emphasis will be on mathematical ideas and results, but largely without proofs, with a main eye on the historical flow of ideas. At UNSW, this is MATH3560 and GENS2005. NJ Wildberger is also the developer of Rational Trigonometry: a new and better way of learning and using trigonometry.