 Direction is, nonetheless, a more plausible candidate than distance; Euclid did not start with the idea that the straight line joining two distinct points is the shortest curve joining them. The relevant primitive concept in the Elements is that of equality of segments, such as all the radii of a given circle. Euclid stated as Common Notion 4 that if two segments can be made to coincide then they are equal, and (in the troublesome I.4) he used the converse, that if two segments are equal then they can be made to coincide. Segments are such that either one is smaller than the other or they are equal, and in I.20 Euclid showed that “in any triangle two sides taken together in any manner are greater than the remaining one.” This result has become known as the triangle inequality, and it goes a long way to proving that the line segment joining any two distinct points is the shortest curve through those points. Once the parallel postulate is introduced Euclid showed that opposite sides of a parallelogram are equal, and so the distance between a pair of parallel lines is a constant.