Note 104 — Introductory material

But unfortunately, the task of discovering the axioms of projective Geometry is far from easy. They have, as yet, found no Riemann or Helmholtz to formulate them philosophically. Many geometers have constructed systems, which they intended to be, and which, with sufficient care in interpretation, really are, free from metrical presuppositions. But these presuppositions are so rooted in all the very elements of Geometry, that the task of eliminating them demands a reconstruction of the whole geometrical edifice. Thus Euclid, for example, deals, from the start, with spatial equality—he employs the circle, which is necessarily defined by means of equality, and he bases all his later propositions on the congruence of triangles as discussed in Book I.[118] Before we can use any elementary proposition of Euclid, therefore, even if this expresses a projective property, we have to prove that the property in question can be deduced by projective methods. This has not, in general, been done by projective geometers, who have too often assumed, for example, that the quadrilateral construction—by which, as we saw in Chap. I., they introduce projective coordinates—or anharmonic ratio, which is primâ facie metrical, could be satisfactorily established on their principles. Both these assumptions, however, can be justified, and we may admit, therefore, that the claims of projective Geometry to logical independence of measurement or congruence are valid. Let us see, then, how it proceeds.