Explore the fascinating world of spherical geometry in this 32-minute video lecture. Delve into Archimedes' groundbreaking formulas for the surface area and volume of a sphere, and discover his remarkable findings relating spheres to circumscribing cylinders. Learn about rational turn angles, a revolutionary approach to angle measurement that simplifies trigonometric calculations. Examine the Proportionality Principle and its applications in spherical geometry. Investigate Harriot's theorem, which relates the area of a spherical triangle to the total surface area of a sphere. Gain insights into lunes, spherical triangles, and other geometric concepts that bridge classical and modern mathematics.