ABSTRACT

IT has often been maintained that the twentieth proposition of the first book of Euclid-that two sides of a triangle are together greater than the third side-is evident even to asses. This does not, however, seem to me generally true. I once asked a coastguardsman the distance from A to B ; he replied : " Eight miles." On further inquiry I elicited the fact that the distance from A to C was two miles and the distance from C to B was twenty-two miles. Now the paths from A to B and from C to B were by sea; while the path from A to C was by land. Hence if the path by land was rugged and the distance along the road was two miles, it would appear that the coastguardsman believed that not only could one side of a triangle be greater than the other two, but that one straight side of a triangle might be greater than one straight side and any curvilinear side of the same triangle. The only escape from part of this astonishing creed would be by assuming that the distance of two miles from A to C was measured " as the crow flies," while the road A to C was so hilly that a pedestrian would traverse more than fourteen miles when proceeding from A to C. Then indeed the coastguardsman could maintain the true proposition that there is at least one triangle ABC, with the side AC curvilinear, such that the sum of the lengths of AB and AC is greater than the length of BC, and only deny the twentieth proposition of the first book of Euclid.