HG 27
In attempting to prove Euclid's parallel postulate, Saccheri stated three hypotheses:
1) hypothesis of the obtuse angle: the summit angles of a SQ are obtuse;
2) hypothesis of the acute angle: the summit angles of a SQ are acute; and
3) hypothesis of the right angle: the summit angles of a SQ are right.
His strategy was to show that hypotheses 1 and 2 reached contradictions in neutral geometry and thus hypothesis 3 had to be accepted. The validity of hypothesis 3 implies the existence of rectangle and this existence is equivalent to the Euclidean parallel postulate.
He was able to show that hypothesis 1 reached a contradiction. In attempting to eliminate hypothesis 2, he reached numerous unfamiliar conclusions, all of which are correct, including those listed on pages HG 21-26.
However, he reached the point of frustration and began to argue intuitively by involving his vague concept of infinitely distant points. The fact is that he could not eliminate hypothesis 2, and had he recognized this, he probably would be considered the founder (discoverer, inventor, ...) of non-Euclidean geometry.
Johann Lambert (1728-1777) continued the work of Saccheri by critically examining the question of the probability of the Euclidean parallel postulate. In attempting to eliminate the hypotheses of the acute and obtuse angles, Lambert employed a quadrilateral ABCD (called a Lambert quadrilateral) with three right angles, say at A, B, and C. In the process, Lambert made an extensive study of area and utilized the concept of defect of a triangle. Lambert is also known for developing hyperbolic trigonometry.
D20. Quadrilateral ABCD is called a Lambert quadrilateral (LQ) if angles A,B,
and C are right angles.
T41. The fourth angle of a LQ is less than or equal to a right angle.