The Saccheri Quadrilateral

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The Saccheri Quadrilateral

Saccheri was a Jesuit priest who lived from 1667 to 1733. Before he passed away he published a book entitled Euclides ab omni nævo vidicatus. (Euclid Freed of Every Flaw). Saccheri suggested that for a line and a point, there are three possibilities. The first is that there is exactly one line through the point parallel to the given line. The second, there is no line through the point parallel to the given line. The third is more than one line through the point parallel to the given line. Saccheri investigated the summit angles of a quadrilateral; his result depended on the postulate chosen. He tried to find a contradiction to prove that Euclids fifth postulate was true. He assumed the negation of the parallel Postulate and tried to arrive at a contradiction. He studied a family of quadrilateral that have come to be called Saccheri quadrilateral.

Saccheri tried to find a contradiction; he started with postulates other than Euclids. He didn’t find one, so he still claimed that Euclids was freed from all defects. Although Saccheri’s work did have on flaw. Many historians praised Saccheri’s book of 39 theorems. The first 70 pages is an ensemble of logic and geometric keenness, which can be called perfect. But all of a sudden Saccheri abruptly turns away from his carefully plotted course. In his 33rd theorem is where his flaw can be found. This is where h he breaks away from his strict logic and carefully crafted perfect work. He remarks “… but this contrary to our intuitive knowledge of a straight line”. Although Saccheri ends his book by admitting that he had not completely proven the acute case and for the reason is said to have with held publication of the book during
his lifetime.

Even though his book came to an abrupt conclusion, Saccheri’s investigation was a crucial step in the evolving of non- Eluclidian geometries. His major achievement was to break ground for the later geome...