The Discovery of Non-Euclidean Geometry
Professor Susan Goldstine

Wednesday, November 7 at 5:00 in MW 724

Euclid's Fifth Postulate (paraphrased): Given a line and a point not on that line, there is exactly one line through the given point parallel to the given line.
Euclid's Parallel Postulate spawned the longest-standing controversy in the history of mathematics. Could it really be that the fifth postulate does not follow from the other four? Why does Euclid prove the converse of his fifth postulate but not the fifth postulate itself? Only after two thousand years of failed attempts to demonstrate the Parallel Postulate was it finally established that the statement cannot be proven or disproven from Euclid's other postulates, and that assuming its opposite yields a new and radically different form of plane geometry.
The talk will give an overview of the events leading to this discovery, which heralded a revolution in modern mathematics, and a description of the properties of the non-Euclidean plane.
Technical prerequisites: basic Euclidean geometry (e.g., the sum of the angles in a triangle is 180 degrees) and a little bit of trigonometry, although the trigonometry is not essential.