Geometry in Modern Dress

Extract

When it is said of the Axioms of Projective Geometry that, for the “points”, “lines” and “planes” of which the Axioms treat, we may take what elements we please (provided only that the Axioms be satisfied), it is, I believe, tacitly assumed that the elements so to be taken are classes of algebraic symbols. That in point of fact the “elements” most commonly used, at least in the introductory stages of learning geometry, are not “points”, “lines” and “planes”, but dots, dashes and blackboards, is rarely acknowledged. Only too often the so-called “figure” of a theorem is apologised for and treated as a mere concession to human weakness. Whereas in fact the “dot”, “dash” and “blackboard”, as elements satisfying the Axioms, provide, in many cases, a most excellent symbolism for the expression of geometrical theorems.