New viewpoint entirely given by Riemann i. On the sphere with folding into 4-space ii. A cross-ratio involving two arbitrary points P and Q inside the circle presents itself immediately if the segment PQ is extended to meet the circle in O and S . But this definition must be modified slightly to make it workable.
Lobachevskian geometry is consistent if and only if Euclidean geometry is consistent.
Some persons have contended that mathematics ought to be taught by making the illustrations obvious to the senses. Samuel Taylor Coleridge Klein s model also shows that Euclid s fifth postulate is independent of the others. It is impossible to prove the fifth postulate. Pasch went beyond Euclid in that he realized what he was doing, and did it deliberately, whereas Euclid seems to have been guided by visual imagery, and in consequence to have overlooked hidden assumptions. In any event, Pasch profoundly influenced the geometrical thought of his contemporaries and successors.