Euclid Cleared of Every Flaw

In 1733, Girolamo Saccheri, an Italian Jesuit professor of mathematics, published Euclides ab Omni Naevo Vindicatus ("Euclid Cleared of Every Flaw"). His work aimed to prove that Euclid's fifth postulate (the parallel postulate) was a necessary truth. Saccheri explored alternative hypotheses to Euclid's postulate, specifically examining the angles formed by a quadrilateral with two parallel lines and two perpendicular sides.12

He considered three possibilities for the internal angles: right, obtuse, or acute. The first possibility aligns with Euclidean geometry. While Saccheri intended to demonstrate that the obtuse and acute alternatives led to contradictions, his detailed investigation inadvertently produced theorems of what would later be known as non-Euclidean geometry. For instance, he showed that the sum of angles in a triangle could be equal to, exceed, or fall short of 180 degrees, depending on which hypothesis was true. Although Saccheri believed he had "cleared Euclid of every defect," his work laid crucial groundwork for the development of non-Euclidean geometries in the 19th century.12