 In fact, Legendre’s attempts to restore rigour to the treatment of elementary geometry was no better than Euclid’s, and in some ways worse, not only because his attempts to prove the parallel postulate inevitably failed, but because he smuggled more into his account than he realised. But its chief significance for present purposes is that it exemplifies the attempt to ground elementary geometry on a concept of distance, or more precisely, on the idea that a straight line is the curve of shortest distance between any of its points. Distance itself is not defined.