Having formulated the axioms that re‡ect our intuitive knowledge of Euclidean and spherical geometry, we undertook in the preceding chapters a modest development of what has been called unied geometry from those axioms. In that development, it was tacitly assumed that if α (the least upper bound of distances) is Šnite, spherical geometry is obtained, and if α is inŠnite, Euclidean geometry results. It may come as a surprise to the reader to learn that along with Euclidean and spherical geometry, you were also studying a third geometry just as logical and just as important. This “hidden” geometry was the one which early nineteenth century mathematicians were trying to disprove and believed was inconsistent. Some were great giants of the past, notably, Gauss and Legendre. If you have trouble believing the validity of that third geometry, then you are in good company.