The Saccheri Quadrilaterals

 

"The hypothesis of the acute angle is absolutely false, because it is repugnant to the nature of the straight line!"

-Girolamo Saccheri

Girolamo Saccheri was a man who was troubled by a math problem he could not accomplish. This problem plagued him for many years and in the end he threw up his arms and said "The hypothesis of the acute angle is absolutely false, because it is repugnant to the nature of the straight line!". Little did he know he was a pioneer for the discovery of the non-Euclidean geometries, including Hyperbolic Geometry. He had found the forest but was blinded by the trees.

Giovanni Saccheri was born in Italy and decided to join the Jesuit priesthood around 1685 in Genoa. In Milan he studied philosophy and theology at the Jesuit College, where he later took on mathematics as a study. Saccheri became a full-fledged priest in 1694 at Como. He taught philosophy and theology at Pavia from 1697 until his death and also held the chair of mathematics at Pavia.

The problem that plagued Saccheri for so many years was Euclid's Fifth Postulate (The Parallel Postulate) Saccheri tried to use a popular mathematical technique called reductio ad adsurdum (proof by contradiction) to prove the 5th postulate. Specifically he studied certain quadrilaterals whose base angles were right angles and the adjacent sides were congruent to each other. These quadrilaterals were studied centuries earlier by Omar Khayyam (600 years earlier) and Nasir Deddin, but they never made the conclusions that Saccheri came to. Saccheri's method broke the proof into three cases:

Case 1: The summit angles are right angles.

Case 2: The summit angles are obtuse.

Case 3: The summit angles are acute.

Saccheri's idea was to prove that case 2 and case 3 would lead to a contradiction, therefore case 1 must be true. He was successful in showing that case 2 lead to a contradiction which states that the angles were obtuse. If the angles' sums were obtuse then the angle sums for a convex quadrilateral would be more than 360 degrees. However, Case 3 was not as easy to disprove; no matter how hard he tried, Saccheri could never get the contradiction he needed. Girolamo Saccheri was the first mathematician to suggest the possibilities of non-Euclidean geometry. Although he did not follow through with his speculation because he could not believe what he discovered, Saccheri is still considered to be a pioneer for non-Euclidean geometry.