Girolamo Saccheri

In all the furor to prove the Parallel Postulate, one particular attempt is worth mentioning. In 1733, a Jesuit priest names Girolamo Saccheri published Euclides abi Omni Naevo Vindicatus (Euclid Vindicated of Every Blemish). This work contained Saccheri's proof of Parallel Postulate. Saccheri was a professor of mathematics, logic, philosophy at the University of Pavia, Italy. He is said to have had an incredible memory; commonly mentioned was his ability to play three games of chess simultaneously, without looking at chessboard.

Saccheri was the first to examine the consequences of assuming the Parallel Postulate to be false; that is, he attempted a proof by contradiction. Reductio ad absurdum, or proof by contradiction, is an indirect means of proving a statement. Simply put, in order to prove something, you assume it to be false. If you can show that the assumption lead to contradiction, then the assumed falsity of the statement is itself false, implying that the statement is true. This is acceptable line of reasoning that has been used since the days of classic Greek logic. Euclid himself use it in Elements, but not in reference to the Parallel Postulate.




In Saccheri's quest to prove the Parallel Postulate, he assumed it to be false. Thereby hoping to obtain some sort of logical contradiction. The denial of the Parallel Postulate consist of two alternatives.

1. Given a line and a point not on the line, there are no lines through the point parallel to the original line (parallel doesn't exist.
2. Given a line and a point not on the line, there are at least two lines through the point parallel to the original line

Surely each of these assumption would lead to a contradiction. To his surprise, Sacchari couldn't find one! How could this be? Were the foundation of Euclid's world built in upon rock or upon sand? Sacchari's work have a little impact that time. Later, Gauss, Bolyai, and Lobachevsky make it to the top that there is exist another shape, kind of geometry.

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