HG 20
Neutral (or absolute) geometry is essentially a list of definitions, postulates, and theorems that are independent of a "parallel postulate". Euclid's (about 300 B.C.) famous parallel postulate (E on page HG 18) appeared in his Elements in about the same position thai it appears on page HG 18. By making this assumption, Euclid was able to prove a great many facts in Euclidean geometry, among which are the following:
rectangles and squares exist an angle inscribed in a semicircle is a right angle opposite sides of a parallelogram are congruent the diagonals of a parallelogram bisect each otherthe sum of the angles of any triangle must equal 180°
Over the centuries, geometers criticized Euclid's parallel postulate because it was not brief, simple, and self-evident, as postulates were suppose to be. Some wished to rectify this matter by simply substituting briefer and apparently simpler statements than the postulate, notably Playfair (about 1780). Playfair's postulate appears as P on page HG 18. Others included Posidonius (first century B.C.), Prolemy (second century B.C.), and Vitale (1633-1711). Many others sought to prove the postulate on the basis of Euclid's other assumptions, believing that these other assumptions were adequate for the complete development of Euclidean geometry. It was their efforts that eventually led to the discovery of non-Euclidean geometry.
Beginning with Proclus (fifth century A.D.), continued attempts to solve the problem were made, first by the Arabs, and then, by Italians, Frenchmen, and Englishmen. All of the "solutions" found were marred by the use of some unproved statement, or unjustified procedure.
Historians believe that the boldest and most creative attempt to solve the problem was first taken by the Italian priest Girolamo Saccheri (1667-1733). Saccheri was first to use the revolutionary procedure of trying to prove the postulate by assuming a proposition contrary to it and reaching a contradiction.
Saccheri attempted to prove Euclid's parallel postulate by assuming it to be false, and arriving at a contradiction by logical reasoning. This would validate the postulate by the principle of indirect method. Saccheri's point of departure was the study of quadrilaterals, which have two sides, which are equal and perpendicular to a third side. Without assuming any parallel postulate he made an extensive study of such quadrilaterals, now called Saccheri quadrilaterals.