SACCHERI’S GREAT MISTAKE

Learning through failure in mathematics

First, a bit of history…

Euclid’s Elements Around 300 B.C., Euclid began work in Alexandria

on Elements, a veritable “bible of mathematics” Euclid set up an axiomatic system upon which

propositions are proven using deductive reasoning The foundation was just 23 definitions, 5 postulates,

and 5 general axioms The end result was 13 books of 465 propositions

Though little of the work in Elements was unchartered mathematics, the beauty and simplicity in logic of the axiomatic system that Euclid created has made Euclid one of the best-known mathematicians ever

Euclid’s 5 Postulates 1. A straight line segment can be drawn joining any two

points. 2. Any straight line segment can be extended indefinitely

in a straight line. 3. Given any straight line segment, a circle can be drawn

having the segment as radius and one endpoint as center.

4. All right angles are congruent. 5. If two lines are drawn which intersect a third in such a

way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate.

Euclid’s 5th Postulate Also known as the Parallel Postulate Clearly a far longer and more complex

postulate than any of the other four postulates

Though “it was universally agreed that the postulate was a logical necessity,” many felt it could be derived from the first four postulates and thus should have been a proposition

The debate raged on for centuries as mathematicians tried and failed to prove it

Fast forward 2,000 years!

Giovanni Girolamo Saccheri(1677-1733) Saccheri was one such mathematician convinced in

his gut that Euclid’s Parallel Postulate was not independent of the other four

Though about 2,000 years had passed since Euclid authored Elements, Saccheri pressed on

While many mathematicians had attempted to directly prove the Parallel Postulate using the other four postulates, Saccheri had a different approach He built upon the work of Nasir Eddin (one of Genghis

Khan’s many grandchildren) nearly 500 years earlier Eddin and Saccheri studied quadrilaterals (now known as

Saccheri quadrilaterals) and looked to prove the Parallel Postulate using a proof by contradiction

Saccheri Quadrilaterals A Saccheri Quadrilateral has

congruent sides that are perpendicular to the base

Using congruent triangles, we see that the “summit angles” must be congruent Critically, this could be proven

without the use of Euclid’s Parallel Postulate

These summit angles must then both be either obtuse, right, or acute

Saccheri’s Plan of Attack Consider the three cases for the summit

angles of the Saccheri quadrilateral If summit angles are obtuse, that would imply zero

parallel lines If summit angles are acute, that would imply more

than one parallel line If summit angles are right, there is exactly one

parallel line Saccheri’s aim was to show that the first two

scenarios were impossible, and thus prove that Euclid’s Parallel Postulate is unnecessary

Case #1:Obtuse summit angles

The Saccheri-Legendre Theorem states that “The sum of the measures of the angles of any triangle is less than or equal to 180 degrees.”

A corollary of this result in neutral geometry (and thus without the use of Euclid’s Parallel Postulate) is that the angle sum of any convex quadrilateral is less than or equal to 360 degrees.

A Saccheri quadrilateral with obtuse summit angles clearly contradicts this corollary, and thus we can rule out the case.

Saccheri was just one step from his goal of proving that Euclid’s Parallel Postulate was unnecessary!

Case #2:Acute summit angles

Saccheri now set out to find a contradiction assuming the summit angles were acute

Unfortunately (or was it?), he found that he was able to derive many of Euclid’s propositions while using this assumption

Unable to find a contradiction, Saccheri nonetheless published a book called “Euclid Freed of Every Flaw,” and in it stated under Proposition XXXIII: The hypothesis of acute angle is absolutely false;

because repugnant to the nature of the straight line. Not surprisingly, mathematicians were not

convinced by this argument

Saccheri’s Legacy After publishing “Euclid Freed of Every Flaw,”

Saccheri went on to publish another work to convince mathematicians of his claim

Regardless, to his great dismay, Saccheri died having neither proven nor disproven the need for Euclid’s Parallel Postulate

Despite Saccheri’s failure, he had actually laid much of the groundwork for an entire world of unexplored geometries, non-Euclidean geometry So, Euclid’s 5th Postulate was necessary for Euclidean

geometry, but Euclid’s geometry was not the only possibility

Non-Euclidean Geometries The most easily

visualized branch of non-Euclidean geometry is spherical geometry Earth, with its lines of

latitude and longitude, is an excellent model

Other branches also exist, including hyperbolic and elliptic geometry

The bottom line The history of mathematics is littered with long

stretches of failure by bright minds to prove or disprove various propositions Despite these failings, much of the work that takes place

along the journey serves to further understanding in the field

Saccheri provides a great example of why we should avoid allowing our own preconceptions to distract us from deductive logic Had Saccheri recognized the implications of his work, he

could have become the “father of non-Euclidean geometry” Finally, students will be relieved to see that even the

brightest minds struggled immensely

Sources “Journey through Genius: The Great Theorems of

Mathematics” by William Dunham (1991) http://www.cut-the-knot.org/triangle/pythpar/Attempts.sht

ml http://en.wikipedia.org/wiki/Giovanni_Girolamo_Saccheri http://mathworld.wolfram.com/EuclidsPostulates.html http://www.learner.org/courses/mathilluminated/units/8/te

xtbook/03.php

http://en.wikipedia.org/wiki/File:Saccheri_quads.svg http://www.wcfchess.org/wp/hall-of-fame/ http://en.wikipedia.org/wiki/File:Noneuclid.svg http://en.wikipedia.org/wiki/File:Triangles_%28spherical_ge

ometry%29.jpg