Section 5.5 A Hyperbolic Model
¶Although we have only verified, but not proved, that the SAS postulate holds in the Poincaré Disk, our constructions certainly make it plausible that it does so. Similarly, our modeling strongly supports the assertion that the other neutral postulates are satisfied. Which they are; a formal proof can be given either in terms of Euclidean geometric constructions, Cartesian algebraic computations, or both.
Although this demonstration of the existence of hyperbolic geometries is well known today, it was unexpected when first shown. Many of those whose work was essential to the demonstration, such as Saccheri, were in fact trying to disprove the assertion, that is, to show that the Euclidean parallel postulate follows from the neutral postulates, so that Euclidean geometry would be the only neutral geometry.
In this sense, hyperbolic geometry was the first geometry to earn the right to be called non-Euclidean.