Note on the Parallel-Postulate

Extract

A recent report on the teaching of geometry by a committee of the Mathematical Association (The Teaching of Geometry in Schools) suggests that a coherent geometrical system could be erected on a foundation other than that of Euclid’s axioms and postulates. Indeed, a cogent argument is advanced for the replacement of the principle of superposition and the parallel-postulate by the “principles of congruence and similarity” which are regarded as the expressions of more profound beliefs. The success of the recommendation, to substitute the principle of congruence for the principle of superposition, is undoubted, and the comparative reluctance to adopt the twin “principle of similarity” may be due, in part, to an obscurity in the treatment of parallels with this principle as basis.