Well known is the revolutionary idea, due to Ren\'e Descartes (1596--1650),of translating problems of geometry to algebra by means of the use of co-ordinates.Not so commonly known is another equally revolutionary instance of the introduction of algebra into geometry, due to Gauss, from about 1795,which lead to spectacular solutions of certain long standing problems of geometry:Can regular polygons be constructed? Can angles be trisected? Can the circle be squared?This talk is about this second instance of the application of algebra to geometry.