Lecture 33. Non-Euclidean Geometry
Figure 33.1. Euclid’s fifth postulate
Euclid’s fifth postulate
In the Elements, Euclid began with a limited number of
assumptions (23 definitions, five common notions, and five postulates) and sought to prove
all the other results (propositions) in the work. The most famous part of The Elements is
the following five 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. That all right angles are equal to each other.
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 (see above picture). This
postulate is equivalent to what is known as the parallel postulate.
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It is apparent that the fifth postulate, which can also be called “Euclid’s Fifth Postulate”,
is radically different from the first four. Euclid was not satisfied with it, so he tried to avoid
using it in The Elements. In fact, the first 28 theorems in The Elements are proved without
using the fifth postulate.
Many mathematicians attempted to find a simpler form of this property, or to prove the
fifth postulate from the other four. Over a period of at least a thousand years, regardless
of the forms of the postulate that have been found, it consistently appears to be more
complicated than Euclid’s other postulates. Many attempts have been accepted as proofs
for long periods of time until a mistake was found. Invariably the mistake was assuming
some ‘obvious’ property which turned out to be equivalent to the fifth postulate. Geometers
tried and failed many times, but still believed that it could be proved as a theorem from the
other four.
Gauss
The first person to really come to understand the problem of the parallels was
Gauss. He began work on the fifth postulate in 1792 while only 15 years old, at first
attempting to prove the parallels postulate from the other four. Soon he recognized the
profound difficulties involved. In 1804, he wrote to his friend Wolfgang Bolyai to mention
the existence of a “group of rocks” and the hope that “ the rocks sometimes, before my
death, will permit a passage.” By 1813 he had made little progress and wrote: “ In the
theory of parallels we are even now not further than Euclid. This is a shameful part of
mathematics. ”
By 1817 Gauss had become convinced that the fifth postulate was independent of the
other four postulates. He began to work out the consequences of a new geometry which
excludes the fifth postulate. Perhaps most surprisingly of all Gauss never published this
work but kept it a secret. Like Newton, Gauss had an intense dislike of controversy and
he was sure that this discovery on the alternative geometry would shock the mathematical
community. In a letter of 1824, Gauss wrote to F. Taurinus:
...The theorems of this geometry appear to be paradoxical and, to the unlimited,
absurd, but calm, steady reflection reveals that they contain nothing impossible...... In any case, consider this a private communication, of which no public
use of us leading to publicity is to be made.
Gauss continued his investigation and was considering writing them up, possibly to be
published after his death. He wrote to Freidrich Bessel in 1829:
It may take a very long time before I make publick my investigations on this
issue. In fact, it may not happen during my lifetime.
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Gauss stopped writing up his study when he received a copy of the famous Appendix of
János Bolyai.
Figure 33.2 János Bolyai
János Bolyai János Bolyai(1802-1860)’s father Wolfgang Bolyai(1775-1856) (or Markas
Bolyai) was a friend of Gauss. Wolfgang learned the parallel problem from Gauss and had
made several false proofs of the problem.
Wolfgang Bolyai taught his son, János Bolyai, mathematics. The father wrote to Gauus
in 1816 to hope that his son would go to Götingen to study with Gauss. But Gauss did not
answer this letter and never write again for 16 years.
János then entered the Imperial Engineering Academy in Vienna in 1817. After graduation in 1823, he entered on an army career. In the period of next ten years, János built up
a reputation as a mathematician.
The young Bolyai was interested in the parallel postulate problem even though his father
advised him not to waste one hour’s time on that problem. In a letter of 1820, the father
wrote:
“...... It could deprive you of all your leisure, your health, your rest, and the
whole happiness of your life. This abysmal darkness might perhaps devour a
thousand towering Newtons, it will never be light on earth......”
Nevertheless János Bolyai inherited his father’s passion and worked on the problem. In 1823,
Bolyai wrote to his father saying I have discovered things so wonderful that I was astounded
out of nothing. I have created a strange new world.
Bolyai wrote down his discovery which was published in 1832 as an appendix to a book
by his father. The worrying father wrote to Gauss for advice.
Gauss’s reaction was typical —– sincere approval, bu lack of support in print. One month
later, Gauss wrote to his former student C.L. Gerling:
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I have this day received from Hungary a little work on non-Euclidean geometry in
which I find all my own ideas and results developed with great elegance, although
in a form so concise as to offer great difficulty to anyone not familiar with the
subject ..... I regarded this young geometer Bolyai as a genius of first order.
One more month later, Gauss wrote to Wolfgang Bolyai:
If I begin by saying that I dare not praise this work, you will of course be surprised
for a moment; but I cannot do otherwise: To praise it would amount to praising
myself. For the entire content of the work, the approach which your son has
taken, and the results to which he is led, coincide almost exactly with my own
meditations which have occupied my mind for the past third years or thirty five
years...... I am overjoyed that it happened to be the son of my old friend who
outstrips me in such a remarkable way.
Even though he was regarded by Gauss as “a genius of the first order,” young János
was deeply disappointed by the letter robbing him of his priority. His mental depression
depended when the Appendix met with complete indifference from other mathematicians
and for a long period he did not do creative work. The final disappointment came in 1848
when he saw a book by Lobachevsky on non-Euclidean geometry, which was published in
1829 and its German translation is in 1840.
It was long after Bolyai’s death that recognition as one of the founders of non-Euclidean
geometry finally came to him.
Figure 33.3. Lobachevsky
Lobachevsky Nicolai Lobachevsky (1792-1856), to parents of Polish origin, was the son of
a poor government clerk who died when the boy was seven. His mother moved the family to
remote Kazan and succeeded in getting her three sons admitted into the secondary schools
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on public scholarship. In 1807, the young Lobachevsky entered Kazan University panning
to study for a medical career.
At the time, the Kazan university had acquired four distinguished German professors.
Among them was Johann Bartels (1769-1836) who was one of Gauss’s early teacher at
Caroline College in Burnswick. Under Bartels’s influence, Lobachevsky soon found himself
interested in mathematics. Since Bertels was familiar with Gauss’s special interests in nonEuclidean geometry, Lobachevsky may first hear about it from him.
It is known that long after Bertels had returned to germany, Lobachevsky was still
working on conventional lines, not searching for a new geometry. After many failed attempts
to prove the parallel postulate, Lobachevsky discovered the non-Euclidean geometry and his
research was first reported in 1826 and was published inKazan Messenger in 1829.
Lobachevsky submitted his work to the St. Petersburg Academy of Sciences but was
rejected. Without giving up, Lobachevsky continued to write a series of papers to convince
the mathematical world. The rejected manuscript was expanded into a new paper, New Elements of Geometry, with a Complete Theory of Parallels, appeared in the recently founded
journal of Kazan University.
In 1835, he published another paper Imaginary Geometry in Moscow University’s Messenger of Europe, which was the first time his work was printed outside of Kazan. The best
summary of his new geometry was a little book of 61 pages, published in Berlin in 1840.
Gauss first learned of Lobachevski’s work on non-Euclidean geometry when he received
a copy of this book. Gauss replied to him in congratulatory fashion. Gauss wrote to
Schumacher in 1846:
I have recently had occasion to look through again that little volume by Lobachevsky
... You know that for fifty-four years now (even since 1792) I have held the same
conviction. I have found in Lobachevsky’s work nothing that is new to me, but
the development is made in a way different from that which I have followed, and
certainly by Lobachevsky in a skillful way and a truly geometric spirit.
In 1814, Lobachevsky became a lecturer at Kazan University, and, in 1822, became a full
professor, teaching mathematics, physics, and astronomy. Nicolai retired (or was dismissed)
from the university in 1846, ostensibly due to his deteriorating health: by the early 1850s,
he was nearly blind and unable to walk. He died in poverty in 1856.
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Models of non-Euclidean geometry The first model of non-Euclidean geometry was
given by Eugenio Beltrami (1835-1899) in 1868 as a so-called pseudosphere. Later there was
the Calyley-Klein model in which points are represented by the points in the interior of the
unit disk and lines are represented by the chords (straight line segments with endpoints on
the boundary circle)
Another model of Lobachevskian geometry in the interior of a circle was developed by
Henri Poincaré in 1882. In this model, straight lines are represented by arcs of circles
that are orthogonal to the boundary circle. Parallel lines are then represented by circular arcs that intersect at the boundary. This model has the advantage that angles between
circles are measured in the Euclidean way. It is this model that convinced mathematicians
by the end of the century that non-Euclidean geometry was as valid as Euclid’s. Now it is
known that it is impossible to prove that postulate as a theorem.
Figure 33.4 Poincaré disc hyperbolic parallel lines that are through
a given point and are parall to a given line.
With the work of many mathematicians including Gauss, Labachevsky, Bolyai, Beltrami,
Klein and Poincaré, Euclid was truly vindicated. Euclid has been completely correct in his
decision 2200 years earlier to take the parallel postulate as a postulate !
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