Download Reminder of Euclid`s five postulates Postulates

Reminder of Euclid’s five postulates
Postulates:
1. To draw a straight line from any point to any point.
2. To produce a finite straight line continuously in a straight line.
3. To describe a circle with any center and distance.
4. That all right angles are equal to one another.
5. That, if a straight line falling on two straight lines make the interior angles on that same side
less than two right angles, the two straight lines, if produced indefinitely, meet on that side on
which are the angles less than the two right angles.
Do you notice anything about the fifth postulate? How does it compare to the first four?
The Parallel Postulate
The fifth postulate is often called the Parallel Postulate. Even though it does not specifically
talk about parallel lines, it actually does deal with ideas of parallelism.
Euclid defines parallel lines as:
Two lines in the same plane which, if extended indefinitely in both directions, never meet
in either direction.
The first 28 theorems in Euclid’s book, the Elements, does not use the parallel postulate for their
proof.
Some people believe the Euclid originally started off with only 4 postulates and later only added
the fifth one, when he realized it would be necessary.
Many mathematicians (amongst the Greeks, the Arabs, Europeans) believed that the 5th postulate
actually was not necessary, that it could have been proven as a theorem, but that Euclid was just
not clever enough to figure out its proof.
Statements equivalent to the fifth postulate
Given a line l and a point P not on l, there exists a unique line m that contains P and is parallel to
l.
Parallel lines are equidistant.
There exists at least one triangle in which the sum of the measures of the interior angles is 180o.
The sum of the measures of the interior angles of a triangle is the same for all triangles.
(The first is known as Playfair’s Postulate, and is the most commonly used equivalent. It is
named for John Playfair, a Scottish scientist from the 18th century who made it popular.
Proculus had first stated this statement and explored it in the 5th century.)
The necessity of the fifth postulate
For more than 2000 years, mathematicians tried proving the 5th postulate using only the first four
postulates, but were unsuccessful.
During the early 18th century, Girolamo Saccheri, and Italian priest and scholar, came up with a
new idea:
• If we assume that the parallel postulate is false and then get a contradiction, that means
that it must be true. That means that the first four postulates must imply the fifth
postulate.
Consider Playfair’s version of the postulate:
Given a line l and a point P not on l, there exists a unique line m that contains P and is parallel to
l.
How can we make this statement be false?
The negation of Playfair’s Postulate has two parts:
1. Given a line l and a point P not on l, there exists no lines through P parallel to l.
2. Given a line l and a point P not on l, there exists more than one line through P parallel to
l.
Saccheri was able to show that the first of these statements was not possible given the other four
postulates.
He was unable to show a contradiction using the second part (although he thought he had come
up with a contradiction).
Nearly 100 years later, around 1810, Carl Gauss explored the idea of replacing the parallel
postulate with the second statement mentioned above and may have been the first person to
discover a non-Euclidean geometry. Because of the politics of that time, however, it was
considered dangerous to “rock the boat” and so Gauss did not publish his work.
In 1829, Nicolai Lobachevsky, a Russian mathematician, was the first to publish an investigation
of the non-Euclidean geometry. He devoted much of his professional life to studying it.
A short time later (1831) a young Hungarian mathematician named Janos Bolyai published, as an
appendix to his father’s work, a short exposition of a non-Euclidean geometry based on this
second statement. He sent his work to Gauss. Gauss responded that he had come up with the
same results previously. This devastated Bolyai and he withdrew from mathematical research
permanently.
All three of these mathematicians had independently discovered a non-Euclidean geometry and
by doing so had come to the same conclusion:
• The Parallel Postulate cannot be proven from the other four postulates of Euclid’s
geometry!
These three mathematicians had all discovered Hyperbolic Geometry!
Postulates of Euclidean Geometry:
1. To draw a straight line from any point to any point.
2. To produce a finite straight line continuously in a straight line.
3. To describe a circle with any center and distance.
4. That all right angles are equal to one another.
5. Given a line l and a point P not on l, there exists a unique line m that contains P and is
parallel to l.
Postulates of Hyperbolic Geometry:
1. To draw a straight line from any point to any point.
2. To produce a finite straight line continuously in a straight line.
3. To describe a circle with any center and distance.
4. That all right angles are equal to one another.
5. Given a line l and a point P not on l, there exists more than one line through P that is parallel
to l.
Postulates of Spherical Geometry:
1. To draw a straight line from any point to any point.
2. Straight lines are boundless.
3. To describe a circle with any center and distance.
4. That all right angles are equal to one another.
5. Given a line l and a point P not on l, there exists no lines through P that are parallel to l.