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Saccheri

© Copyright 2000, Jim Loy

In 1733, Girolamo Saccheri wrote Euclides Vindicatus. It's entire title has been translated, Euclid Freed of All Blemish or "A geometric endeavor in which are established the foundation principles of universal geometry." In the book, Saccheri invented what are now called the two Non-Euclidean Geometries, also called Hyperbolic Geometry (or Lobachevskian Geometry) and Elliptical Geometry (or Riemannian Geometry), and he compared them to Euclidean Geometry. These three geometries are all based upon different versions of the Parallel Postulate. Saccheri used the quadrilateral that you see here (now called the Saccheri Quadrilateral) as the starting point of his exploration of these geometries. He called his versions of the parallel postulate, the Hypothesis of the Right Angle (Euclidean Geometry, where angle C is a right angle), the Hypothesis of the Acute Angle (Hyperbolic Geometry, where angle C is an acute angle), and the Hypothesis of the Obtuse Angle (Elliptical Geometry, where angle C is an obtuse angle).

Saccheri's aim was to disprove the Hypothesis of the Acute Angle, and the Hypothesis of the Obtuse Angle, and thereby prove the Euclid's Parallel Postulate. For this reason, Saccheri is not normally given credit for inventing the Non-Euclidean Geometries. I think that that is not fair.

Near the end of his book, Saccheri disproved the Hypothesis of the Obtuse Angle, as it violates the postulate that a straight line is of infinite length. He did not have to write a whole book to do that. It can be done in a page or two. Nowadays, Riemann's Elliptical Geometry contains the further postulate that lines are of finite length. From there, Saccheri thought that he had also disproved the Hypothesis of the Acute Angle. In this case, his reasoning was sloppy and incorrect.

Saccheri could have just disproved the Hypothesis of the Obtuse Angle at the beginning of the book, and not examined it in detail. But, because he went to great lengths to examine all three hypotheses, we are presented with a growing comparison of the three Geometries. For example, he showed that the sum of the angles of a triangle are less than 180 degrees (Hypothesis of the Acute Angle), equal to 180 degrees (Hypothesis of the Right Angle), or greater than 180 degrees (Hypothesis of the Obtuse Angle). But first of all, he showed that the three hypotheses are really incompatible, that if one of the three hypotheses is true in any one case it is true in all cases.

Nowadays, geometry that does not depend on one of the versions of the Parallel Postulate is called Universal Geometry. This is an echo of the subtitle of Saccheri's book. Perhaps he was the first person to use that expression.

See Non-Euclidean Geometries.

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