This document provides a history of non-Euclidean geometry, beginning with Euclid's fifth postulate and early attempts to prove it from the other four postulates. In the early 19th century, Bolyai, Lobachevsky, and Gauss independently developed hyperbolic geometry by replacing the fifth postulate. However, their work was initially rejected by the mathematical community. Later, Riemann generalized the concept of geometry and Beltrami provided a model showing the consistency of non-Euclidean geometry. Klein classified the three types of geometry as hyperbolic, elliptic and Euclidean. Non-Euclidean geometry has since found applications in Einstein's theory of relativity and GPS systems.

1. History of Non-Euclidean
Geometry
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Non-
Euclidean_geometry.html
http://en.wikipedia.org/wiki/Non-Euclidean_geometry

2. Euclid’s Postulates
from Elements, 300BC
To draw a straight line from any point to any
1.
other.
To produce a finite straight line continuously in a
2.
straight line.
To describe a circle with any centre and distance.
3.
That all right angles are equal to each other.
4.
That, if a straight line falling on two straight lines
5.
make the interior angles on the same side less
than two right angles, if produced indefinitely,
meet on that side on which are the angles less
than the two right angles.

3. What is up with #5?
 5. That, if a straight line falling on two straight lines make the
interior angles on the same side less than two right angles, if
produced indefinitely, meet on that side on which are the
angles less than the two right angles.
 Equivalently,
 Playfair’s Axiom: Given a line and a point not on the line, it is
possible to draw exactly one line through the given point parallel
to the line.
 To each triangle, there exists a similar triangle of arbitrary
magnitude.
 The sum of the angles of a triangle is equal to two right angles.
 Through any point in the interior of an angle it is always possible
to draw a line which meets both sides of the angle.

4. Can the 5th Postulate be proven
from the other 4?
 Ptolemy tried (~150 BC)
 Proclus tried (~450BC)
 Wallis tried (1663)
 Saccheri tried (1697)
 This attempt was important, he tried proof by contradiction
 Legendre tried… for 40 years (1800s)
 Others tried, making the 5th postulate the hot problem in
elementary geometry
D’Ambert called it
“the scandal of elementary geometry”

5. Gauss and his breakthrough
 Started working on it at age 15 (1792)
 Still nothing by age 36
 Decided the 5th postulate was independent of the other 4.
 Wondered, what if we allowed 2 lines
through a single point to BOTH be parallel to
a given line
The Birth of
non-Euclidean Geometry!!!
 Never published his work, he wanted to avoid controversy.

6. Bolyai’s Strange New World
 Gauss talked with Farkas Bolyai about the 5th
postulate.
 Farkas told his son Janos, but said don’t “waste one
hour's time on that problem”.
 Janos wrote daddy in 1823 saying
“I have discovered things so wonderful
that I was astounded ... out of nothing
I have created a strange new world.”

7. Bolyai’s Strange New World
 Bolyai took 2 years to write a 24 page appendix
about it.
 After reading it, Gauss told a friend,
“I regard this young geometer Bolyai as a
genius of the first order”
 Then wrecked Bolyai by telling him that he
discovered this all earlier.

8. Lobachevsky
 Lobachevsky also published a work about replacing the 5th
postulate in 1829.
 Published in Russian in a local university publication, no one
knew about it.
 Wrote a book, Geometrical investigations on the theory of
parallels in 1840.
Lobachevsky's Parallel Postulate. There exist two lines
parallel to a given line through a given point not on
the line.

9. 5th postulate controversy
 Bolyai’s appendix
 Lobachevsky’s book
 the endorsement of Gauss…
but the mathematical community
wasn’t accepting it.
WHY?

10. 5th postulate controversy
 Many had spent years trying to prove the 5th
postulate from the other 4. They still clung to the
belief that they could do it.
 Euclid was a god. To replace one of his postulates
was blasphemy.
 It still wasn’t clear that this new system was
consistent.

11. Riemann
 Riemann wrote his doctoral dissertation
under Gauss (1851)
 he reformulated the whole concept of geometry,
now called Riemannian geometry.
 Instead of axioms involving just points and lines, he
looked at differentiable manifolds (spaces which are
locally similar enough to Euclidean space so that one
can do calculus) whose tangent spaces are inner
product spaces, where the inner products vary smoothly
from point to point.
 This allows us to define a metric (from the inner
product), curves, volumes, curvature…

12. Consistent by Beltrami
 Beltrami wrote Essay on the interpretation
of non-Euclidean geometry
 In it, he created a model of 2D
non-Euclidean geometry within Consistent
by Beltrami
3D Euclidean geometry.
 This provided a model for showing the
consistency on non-Euclidean geometry.

13. Eternity by Klein
 Klein finished the work started by
Beltrami
 Showed there were 3 types of
(non-)Euclidean geometry:
Hyperbolic Geometry (Bolyai-Lobachevsky-Gauss).
1.
Elliptic Geometry (Riemann type of
2.
spherical geometry)
Euclidean geometry.
3.

14. The Geometries
Comparison of Major Two-Dimensional Geometries. Smith, The Nature of Mathematics, p. 501

15. Hyperbolic geometry
There are infinitely many lines through
a single point which are parallel to a
given line
The Klein Model The Poincare Model

16. Hyperbolic geometry
 Used in Einstein's theory of general relativity
 If a triangle is constructed out of three rays of light, then in general
the interior angles do not add up to 180 degrees due to gravity. A
relatively weak gravitational field, such as the Earth's or the sun's, is
represented by a metric that is approximately, but not exactly,
Euclidean.

17. Hyperbolic geometry
 Used in Einstein's theory of general relativity
 If a triangle is constructed out of three rays of light, then in general
the interior angles do not add up to 180 degrees due to gravity. A
relatively weak gravitational field, such as the Earth's or the sun's, is
represented by a metric that is approximately, but not exactly,
Euclidean.

18. Theory of Relativity
General relativity is a theory of gravitation
Some of the consequences of general relativity are:

Time speeds up at higher gravitational potentials.

Even rays of light (which are weightless) bend in the presence of a gravitational field.

Orbits change in the direction of the axis of a rotating object in a way unexpected in Newton's

theory of gravity. (This has been observed in the orbit of Mercury and in binary pulsars).
The Universe is expanding, and the far parts of it are moving away from us faster than the speed of

light. This does not contradict the theory of special relativity, since it is space itself that is
expanding.
Frame-dragging, in which a rotating mass quot;drags alongquot; the space time around it.

http://en.wikipedia.org/wiki/Theory_of_relativity

19. Theory of Relativity
Special relativity is a theory of the structure of spacetime.
Special relativity is based on two postulates which are contradictory in classical

mechanics:
1. The laws of physics are the same for all observers in uniform motion relative to one another

(Galileo's principle of relativity),
2. The speed of light in a vacuum is the same for all observers, regardless of their relative motion

or of the motion of the source of the light.
The resultant theory has many surprising consequences. Some of these are:

Time dilation: Moving clocks are measured to tick more slowly than an observer's quot;stationaryquot;

clock.
Length contraction: Objects are measured to be shortened in the direction that they are moving

with respect to the observer.
Relativity of simultaneity: two events that appear simultaneous to an observer A will not be

simultaneous to an observer B if B is moving with respect to A.
Mass-energy equivalence: E = mc², energy and mass are equivalent and transmutable.

http://en.wikipedia.org/wiki/Theory_of_relativity

20. Einstein and GPS
 GPS can give position, speed, and
heading in real-time, accurate to
without 5-10 meters.
 To be this accurate, the atomic
clocks must be accurate to within
20-30 nanoseconds.
Special Relativity predicts that the on-

board atomic clocks on the satellites
should fall behind clocks on the
ground by about 7 microseconds
per day because of the slower ticking
rate due to the time dilation effect of their
http://www.ctre.iastate.edu/educweb/ce352/lec24/gps.htm
relative motion.

21. Einstein and GPS
 Further, the satellites are in orbits high above the Earth, where the
curvature of spacetime due to the Earth's mass is less than it is at the
Earth's surface.
 A prediction of General Relativity is that clocks closer to a massive
object will seem to tick more slowly than those located further away.
As such, when viewed from the surface of the Earth, the clocks on the
satellites appear to be ticking faster than identical clocks on the
ground.
 A calculation using General Relativity predicts that the clocks
in each GPS satellite should get ahead of ground-based clocks
by 45 microseconds per day.

22. Einstein and GPS
If these effects were not properly taken into account, a
navigational fix based on the GPS constellation
would be false after only 2 minutes, and errors in
global positions would continue to accumulate at a
rate of about 10 kilometers each day!

23. Elliptic geometry
There are no parallel lines
http://www.joelduffin.com/opensource/globe/
http://gc.kls2.com/

24. Elliptic geometry
 Captain Cook was a mathematician.
‘An Observation of an Eclipse of the Sun at the Island of
Newfoundland, August 5, 1766, with the Longitude of the place
of Observation deduced from it.’
 Cook made an observation of the eclipse in latitude 47° 36’ 19”, in Newfoundland.
He compared it with an observation at Oxford on the same eclipse, then computed
the difference of longitude of the places of observation, taking into account the
effect of parallax, and the the shape of the earth.
 Parallax: the apparent shift of an object against the background that is caused by a
change in the observer's position.

25. Projective geometry
Projective Geometry developed
independent of non-Euclidean
geometry.
In the beginning, mathematicians used
Euclidean geometry for their calculations.
Riemann showed it was consistent without
the 5th postulate.