Saccheri discovered hyperbolic geometry in 1733 while trying to disprove it
Giovanni Girolamo Saccheri set out to rescue Euclid from every flaw. Assuming the parallel postulate false, he derived result after result about a strange geometry, then declared it impossible, though no real contradiction existed. He had stumbled on a valid new geometry, hyperbolic geometry, without realising what he held.
The trouble centred on Euclid's fifth postulate. His Elements starts from 23 definitions, five common notions and five postulates, and the first four are short: join two points, extend a segment, draw a circle, all right angles are equal. The fifth, about when two lines crossed by a third must eventually meet, is far wordier, and it is equivalent to Playfair's version: through a point off a given line, exactly one line never meets it. For at least a millennium geometers suspected it could be proved from the others.
Ibn al-Haytham in the 11th century, Omar Khayyám in the 12th and Nasīr al-Dīn al-Tūsī in the 13th all tried. Khayyám studied a quadrilateral whose summit angles could be right, obtuse or acute, ruled out the last two, and recovered Euclid's postulate without noticing that his starting assumption was equivalent to it. Their quadrilateral theorems have been called the first results of hyperbolic and elliptic geometry. A 1298 book by al-Tūsī's son, printed in Rome in 1594, reached European readers including Saccheri, who criticised it along with John Wallis's efforts.
Saccheri's 1733 book, Euclid Freed from All Flaws, quickly dismissed the elliptic option and pursued the acute case at length. In 1766 Johann Lambert wrote a treatise on parallels he never published, using a quadrilateral with three right angles. Unlike Saccheri, he never believed he had hit a contradiction, and he proved that under the acute assumption a triangle's angle sum grows as its area shrinks. That led him to wonder about a sphere of imaginary radius, an idea he dropped.
The payoff is three distinct worlds. Take two lines that share a perpendicular. In Euclid's plane they stay a fixed distance apart forever. In hyperbolic geometry they spread apart, and through an outside point infinitely many lines avoid a given line. In elliptic geometry they close in and cross, and every line through the point meets the original. Such geometries won wide acceptance only in the 19th century. A broader usage also applies the name to kinematic geometries built from planar algebras.
Source: Non-Euclidean geometry
