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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. 230
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### Transcript of Lecture 33. Non-Euclidean Geometryshanyuji/History/h-33.pdf · 2014-04-21 · Lecture 33....

• 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 ofassumptions (23 definitions, five common notions, and five postulates) and sought to proveall the other results (propositions) in the work. The most famous part of The Elements isthe 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 radiusand 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 innerangles on one side is less than two right angles, then the two lines inevitably mustintersect each other on that side if extended far enough (see above picture). Thispostulate 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 avoidusing it in The Elements. In fact, the first 28 theorems in The Elements are proved withoutusing the fifth postulate.

Many mathematicians attempted to find a simpler form of this property, or to prove thefifth postulate from the other four. Over a period of at least a thousand years, regardlessof the forms of the postulate that have been found, it consistently appears to be morecomplicated than Euclid’s other postulates. Many attempts have been accepted as proofsfor long periods of time until a mistake was found. Invariably the mistake was assumingsome ‘obvious’ property which turned out to be equivalent to the fifth postulate. Geometerstried and failed many times, but still believed that it could be proved as a theorem from theother four.

Gauss The first person to really come to understand the problem of the parallels wasGauss. He began work on the fifth postulate in 1792 while only 15 years old, at firstattempting to prove the parallels postulate from the other four. Soon he recognized theprofound difficulties involved. In 1804, he wrote to his friend Wolfgang Bolyai to mentionthe existence of a “group of rocks” and the hope that “ the rocks sometimes, before mydeath, will permit a passage.” By 1813 he had made little progress and wrote: “ In thetheory of parallels we are even now not further than Euclid. This is a shameful part ofmathematics. ”

By 1817 Gauss had become convinced that the fifth postulate was independent of theother four postulates. He began to work out the consequences of a new geometry whichexcludes the fifth postulate. Perhaps most surprisingly of all Gauss never published thiswork but kept it a secret. Like Newton, Gauss had an intense dislike of controversy andhe was sure that this discovery on the alternative geometry would shock the mathematicalcommunity. 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 impossi-ble...... In any case, consider this a private communication, of which no publicuse of us leading to publicity is to be made.

Gauss continued his investigation and was considering writing them up, possibly to bepublished after his death. He wrote to Freidrich Bessel in 1829:

It may take a very long time before I make publick my investigations on thisissue. 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 ofJános Bolyai.

Figure 33.2 János Bolyai

János Bolyai János Bolyai(1802-1860)’s father Wolfgang Bolyai(1775-1856) (or MarkasBolyai) was a friend of Gauss. Wolfgang learned the parallel problem from Gauss and hadmade several false proofs of the problem.

Wolfgang Bolyai taught his son, János Bolyai, mathematics. The father wrote to Gauusin 1816 to hope that his son would go to Götingen to study with Gauss. But Gauss did notanswer this letter and never write again for 16 years.

János then entered the Imperial Engineering Academy in Vienna in 1817. After gradua-tion in 1823, he entered on an army career. In the period of next ten years, János built upa reputation as a mathematician.

The young Bolyai was interested in the parallel postulate problem even though his fatheradvised him not to waste one hour’s time on that problem. In a letter of 1820, the fatherwrote:

“...... It could deprive you of all your leisure, your health, your rest, and thewhole happiness of your life. This abysmal darkness might perhaps devour athousand 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 astoundedout of nothing. I have created a strange new world.

Bolyai wrote down his discovery which was published in 1832 as an appendix to a bookby his father. The worrying father wrote to Gauss for advice.

Gauss’s reaction was typical —– sincere approval, bu lack of support in print. One monthlater, 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 inwhich I find all my own ideas and results developed with great elegance, althoughin a form so concise as to offer great difficulty to anyone not familiar with thesubject ..... 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 surprisedfor a moment; but I cannot do otherwise: To praise it would amount to praisingmyself. For the entire content of the work, the approach which your son hastaken, and the results to which he is led, coincide almost exactly with my ownmeditations which have occupied my mind for the past third years or thirty fiveyears...... I am overjoyed that it happened to be the son of my old friend whooutstrips me in such a remarkable way.

Even though he was regarded by Gauss as “a genius of the first order,” young Jánoswas deeply disappointed by the letter robbing him of his priority. His mental depressiondepended when the Appendix met with complete indifference from other mathematiciansand for a long period he did not do creative work. The final disappointment came in 1848when he saw a book by Lobachevsky on non-Euclidean geometry, which was published in1829 and its German translation is in 1840.

It was long after Bolyai’s death that recognition as one of the founders of non-Euclideangeometry finally came to him.

Figure 33.3. Lobachevsky

Lobachevsky Nicolai Lobachevsky (1792-1856), to parents of Polish origin, was the son ofa poor government clerk who died when the boy was seven. His mother moved the family toremote 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 panningto 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 atCaroline College in Burnswick. Under Bartels’s influence, Lobachevsky soon found himselfinterested in mathematics. Since Bertels was familiar with Gauss’s special interests in non-Euclidean geometry, Lobachevsky may first hear about it from him.

It is known that long after Bertels had returned to germany, Lobachevsky was stillworking on conventional lines, not searching for a new geometry. After many failed attemptsto prove the parallel postulate, Lobachevsky discovered the non-Euclidean geometry and hisresearch was first reported in 1826 and was published inKazan Messenger in 1829.

Lobachevsky submitted his work to the St. Petersburg Academy of Sciences but wasrejected. Without giving up, Lobachevsky continued to write a series of papers to convincethe mathematical world. The rejected manuscript was expanded into a new paper, New El-ements of Geometry, with a Complete Theory of Parallels, appeared in the recently foundedjournal of Kazan University.

In 1835, he published another paper Imaginary Geometry in Moscow University’s Mes-senger of Europe, which was the first time his work was printed outside of Kazan. The bestsummary 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 receiveda copy of this book. Gauss replied to him in congratulatory fashion. Gauss wrote toSchumacher 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 sameconviction. I have found in Lobachevsky’s work nothing that is new to me, butthe development is made in a way different from that which I have followed, andcertainly 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 fullprofessor, 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 wasgiven by Eugenio Beltrami (1835-1899) in 1868 as a so-called pseudosphere. Later there wasthe Calyley-Klein model in which points are represented by the points in the interior of theunit disk and lines are represented by the chords (straight line segments with endpoints onthe boundary circle)

Another model of Lobachevskian geometry in the interior of a circle was developed byHenri Poincaré in 1882. In this model, straight lines are represented by arcs of circlesthat are orthogonal to the boundary circle. Parallel lines are then represented by cir-cular arcs that intersect at the boundary. This model has the advantage that angles betweencircles are measured in the Euclidean way. It is this model that convinced mathematiciansby the end of the century that non-Euclidean geometry was as valid as Euclid’s. Now it isknown that it is impossible to prove that postulate as a theorem.

Figure 33.4 Poincaré disc hyperbolic parallel lines that are througha 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 hisdecision 2200 years earlier to take the parallel postulate as a postulate !

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