The Pseudo-Definition of the Straight Line

Extract

We recognise as decadence the substitution for Euclid’s theorem I. 20, “Any two sides of a triangle are together greater than the third side,” of the objectionable pseudo-axiom, “A straight line is the shortest line from one point to another.”

This phrase, as Hilbert has well said, is of necessity meaningless when the concept “length of a curve” has not been defined. Every book on elementary geometry which has introduced as axiom or definition of a straight line any phrase equivalent to “ A straight line is the shortest distance between two points,” every book which has at its beginning as axiom or definition of a straight line any phrase equivalent to “A straight line is the shortest distance between two points,” has been in error. We believe the advances in regard to the foundations of elementary geometry make timely the rejection of this piece of decay from the body of the elements, and that the time is ripe for the return at this point to the pristine purity of Euclid.