 The first four Books of Euclid’s Elements are about straight lines and circles, but it is well known that the concept of a straight line receives only a most unsatisfactory definition. A line is said to be “a breadthless length”, and a straight line to be a line “which lies evenly with the points on itself”. This may help convince readers that they share a common conception of the straight line, but it is no use if unexpected difficulties arise in the creation of a theory—as we shall see.