Saccheri is primarily known today for his last publication, in 1733 shortly before his death. Now considered an early exploration of
non-Euclidean geometry,
Euclides ab omni naevo vindicatus (
Euclid Freed of Every Flaw) languished in obscurity until it was rediscovered by
Eugenio Beltrami, in the mid-19th century. The intent of Saccheri's work was ostensibly to establish the validity of Euclid by means of a
reductio ad absurdum proof of any alternative to
Euclid's
parallel postulate. To do so, he assumed that the parallel postulate was false and attempted to derive a contradiction. Since Euclid's postulate is equivalent to the statement that the sum of the internal angles of a triangle is 180°, he considered both the hypothesis that the angles add up to more or less than 180°. The first led to the conclusion that straight lines are finite, contradicting Euclid's second postulate. So Saccheri correctly rejected it. However, the principle is now accepted as the basis of
elliptic geometry, where both the second and fifth postulates are rejected. The second possibility turned out to be harder to refute. In fact he was unable to derive a logical contradiction and instead derived many non-intuitive results; for example that triangles have a maximum finite area and that there is an absolute unit of length. He finally concluded that: "the hypothesis of the acute angle is absolutely false; because it is repugnant to the nature of straight lines". Today, his results are theorems of
hyperbolic geometry. There is some minor argument on Saccheri's real intentions, since he came extremely close to discovering non-Euclidean geometry. Some believe Saccheri concluded as he did only to avoid the criticism that might come from seemingly illogical aspects of hyperbolic geometry. One tool that Saccheri developed in his work (now called a
Saccheri quadrilateral) has a precedent in the 11th-century Persian polymath
Omar Khayyám's
Discussion of Difficulties in Euclid (''Risâla fî sharh mâ ashkala min musâdarât Kitâb 'Uglîdis''). Khayyam, however, made no significant use of the quadrilateral, whereas Saccheri explored its consequences deeply. ==Works==