The Axioms of Non-Metrical Geometry

Extract

As Geometry was developed in recent centuries, it became realised that some of the more interesting results were not metrical in nature, and that even some of the metrical theorems were immediately derivable from more general non-metrical ones. For example, mid-point theorems were often derivable from harmonic theory, and confocal conics from general tangential pencils. It seemed inelegant as well as unnecessary to use the ladder of a metric to achieve non-metrical results and then to kick it away as if it had never existed; a far more satisfactory procedure would be to develop pure geometry from pure non-metrical axioms.