Learning through failure in
mathematics
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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
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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 bestknown mathematicians ever
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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.
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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
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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
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A Saccheri Quadrilateral has
congruent sides that are
perpendicular to the base
Using congruent triangles, we see
that the “summit angles” must be
congruent
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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
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Consider the three cases for the summit angles
of the Saccheri quadrilateral
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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
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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!
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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.
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Not surprisingly, mathematicians were not
convinced by this argument
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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
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So, Euclid’s 5th Postulate was necessary for Euclidean
geometry, but Euclid’s geometry was not the only
possibility
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The most easily visualized
branch of non-Euclidean
geometry is spherical
geometry
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Earth, with its lines of
latitude and longitude, is an
excellent model
Other branches also exist,
including hyperbolic and
elliptic geometry
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The history of mathematics is littered with long
stretches of failure by bright minds to prove or
disprove various propositions
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Saccheri provides a great example of why we
should avoid allowing our own preconceptions to
distract us from deductive logic
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Despite these failings, much of the work that takes place
along the journey serves to further understanding in the
field
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
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“Journey through Genius: The Great Theorems of Mathematics”
by William Dunham (1991)
http://www.cut-the-knot.org/triangle/pythpar/Attempts.shtml
http://en.wikipedia.org/wiki/Giovanni_Girolamo_Saccheri
http://mathworld.wolfram.com/EuclidsPostulates.html
http://www.learner.org/courses/mathilluminated/units/8/text
book/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_geom
etry%29.jpg