 In all these editions Legendre took a firmly metrical point of view, taking as given algebra and the theory of proportions. His opening definition of the first edition proclaimed that “Geometry is a science that has as its object the measure of extent”. Extent, he explained, has three dimensions, length, breadth, and height; a line is a length without breadth, its extremities are called points and a point therefore has no extent. A straight line is the shortest path from one point to another; surfaces have length and breadth but no height or depth; and a plane is a surface in which if two arbitrary points are joined by a straight line this line lies entirely in the surface. Legendre then set out to prove even some truths that Euclid had preferred to assume, such as (Legendre’s first result): any two right angles are equal. The notion of congruence was central to his approach, and familiar congruence theorems follow in each edition until the parallel postulate could no longer be ignored. Once the existence of parallel lines was assured Legendre argued that they were equidistant.