Giovanni Girolamo Saccheri
Giovanni Girolamo Saccheri (; 5 September 1667 – 25 October 1733) was an Italian Jesuit priest, scholastic philosopher, and mathematician. Saccheri was born in Sanremo. He entered the Jesuit order in 1685 and was ordained as a priest in 1694. He taught philosophy at the University of Turin from 1694 to 1697 and philosophy, theology and mathematics at the University of Pavia from 1697 until his death. He was a protégé of the mathematician Tommaso Ceva and published several works including "Quaesita geometrica" (1693), "Logica demonstrativa" (1697), and "Neo-statica" (1708). Geometrical work. He 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 whether Saccheri really meant that, as he published his work in the final year of his life, came extremely close to discovering non-Euclidean geometry and was a logician. 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.