By David Gans

Publication through Gans, David

**Read or Download An Introduction to Non-Euclidean Geometry PDF**

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**Discontinuous Groups of Isometries in the Hyperbolic Plane**

This ebook via Jakob Nielsen (1890-1959) and Werner Fenchel (1905-1988) has had

a lengthy and complex historical past. In 1938-39, Nielsen gave a chain of lectures on

discontinuous teams of motions within the non-euclidean airplane, and this led him - in the course of

World struggle II - to jot down the 1st chapters of the publication (in German). whilst Fenchel,

who needed to get away from Denmark to Sweden as a result German career,

returned in 1945, Nielsen initiated a collaboration with him on what turned recognized

as the Fenchel-Nielsen manuscript. at the moment they have been either on the Technical

University in Copenhagen. the 1st draft of the Fenchel-Nielsen manuscript (now

in English) was once complete in 1948 and it was once deliberate to be released within the Princeton

Mathematical sequence. even though, because of the swift improvement of the topic, they felt

that massive adjustments needed to be made ahead of book.

When Nielsen moved to Copenhagen college in 1951 (where he stayed till

1955), he was once a lot concerned with the foreign association UNESCO, and the

further writing of the manuscript used to be left to Fenchel. The documents of Fenchel now

deposited and catalogued on the division of arithmetic at Copenhagen Univer-

sity comprise unique manuscripts: a partial manuscript (manuscript zero) in Ger-

man containing Chapters I-II (

I -15), and an entire manuscript (manuscript I) in

English containing Chapters I-V (

1-27). The data additionally comprise a part of a corre-

spondence (first in German yet later in Danish) among Nielsen and Fenchel, the place

Nielsen makes precise reviews to Fenchel's writings of Chapters III-V. Fenchel,

who succeeded N. E. Nf/Jrlund at Copenhagen collage in 1956 (and stayed there

until 1974), was once a great deal concerned with an intensive revision of the curriculum in al-

gebra and geometry, and centred his examine within the concept of convexity, heading

the foreign Colloquium on Convexity in Copenhagen 1965. for nearly twenty years

he additionally positioned a lot attempt into his activity as editor of the newly begun magazine Mathematica

Scandinavica. a lot to his dissatisfaction, this job left him little time to complete the

Fenchel-Nielsen undertaking the best way he desired to.

After his retirement from the college, Fenchel - assisted through Christian Sieben-

eicher from Bielefeld and Mrs. Obershelp who typed the manuscript - came upon time to

finish the booklet effortless Geometry in Hyperbolic area, which used to be released through

Walter de Gruyter in 1989 almost immediately after his demise. at the same time, and with a similar

collaborators, he supervised a typewritten model of the manuscript (manuscript 2) on

discontinuous teams, removal a number of the imprecise issues that have been within the unique

manuscript. Fenchel advised me that he meditated elimination elements of the introductory

Chapter I within the manuscript, considering that this might be lined by way of the e-book pointed out above;

but to make the Fenchel-Nielsen booklet self-contained he eventually selected to not do

so. He did choose to omit

27, entitled Thefundamental staff.

As editor, i began in 1990, with the consent of the felony heirs of Fenchel and

Nielsen, to supply a TEX-version from the newly typewritten model (manuscript 2).

I am thankful to Dita Andersen and Lise Fuldby-Olsen in my division for hav-

ing performed an excellent task of typing this manuscript in AMS- TEX. i've got additionally had

much aid from my colleague J0rn B0rling Olsson (himself a scholar of Kate Fenchel

at Aarhus collage) with the facts studying of the TEX-manuscript (manuscript three)

against manuscript 2 in addition to with a basic dialogue of the difference to the fashion

of TEX. In such a lot respects we determined to persist with Fenchel's intentions. although, turning

the typewritten version of the manuscript into TEX helped us to make sure that the notation,

and the spelling of sure key-words, will be uniform through the booklet. additionally,

we have indicated the start and finish of an evidence within the traditional type of TEX.

With this TEX -manuscript I approached Walter de Gruyter in Berlin in 1992, and

to my nice reduction and delight they agreed to submit the manuscript of their sequence

Studies in arithmetic. i'm so much thankful for this optimistic and fast response. One

particular challenge with the ebook grew to become out to be the copy of the various

figures that are a vital part of the presentation. Christian Siebeneicher had at

first agreed to bring those in ultimate digital shape, yet through 1997 it turned transparent that he

would no longer have the capacity to locate the time to take action. although, the writer provided an answer

whereby I may still convey exact drawings of the figures (Fenchel didn't go away such

for Chapters IV and V), after which they'd arrange the construction of the figures in

electronic shape. i'm very thankful to Marcin Adamski, Warsaw, Poland, for his fantastic

collaboration about the genuine creation of the figures.

My colleague Bent Fuglede, who has personaHy recognized either authors, has kindly

written a quick biography of the 2 of them and their mathematical achievements,

and which additionally areas the Fenchel-Nielsen manuscript in its right point of view. In

this connection i want to thank The Royal Danish Academy of Sciences and

Letters for permitting us to incorporate during this ebook reproductions of images of the 2

authors that are within the ownership of the Academy.

Since the manuscript makes use of a few particular symbols, an inventory of notation with brief

explanations and connection with the particular definition within the ebook has been integrated. additionally,

a accomplished index has been extra. In either circumstances, all references are to sections,

not pages.

We thought of including an entire record of references, yet determined opposed to it because of

the overwhelming variety of examine papers during this sector. as an alternative, a miles shorter

list of monographs and different accomplished bills correct to the topic has been

collected.

My ultimate and so much honest thank you visit Dr. Manfred Karbe from Walter de Gruyter

for his commitment and perseverance in bringing this booklet into life.

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**Additional resources for An Introduction to Non-Euclidean Geometry**

**Sample text**

LEGENDRE There is nothing logically wrong with this proof, but it rests on more than the basis E. In taking the liberty of drawing a line through D that meets lines AB and AC, Legendre was, in effect, making the assumption that through any point within an angle a line can be drawn which meets both sides of the angle. This assumption is equivalent to Postulate 5, as was noted in Chapter I, Section 6, where it is listed as Substitute (j). Since we never proved this equivalence, let us do so now. First, from Legendre's work we see that Postulate 5 can be proved if we use statement ( j) and the basis E.

Prove Theorem 38. 2. The distance between two parallels with a common perpendicular increases in the same way on one side of the perpendicular as on the other. Prove this by taking M on A to the left of F in Fig. Ill, 10 so that FM = FB and showing that MN = BC, where N is the projection of M on g. 3. If two Saccheri quadrilaterals have equal arms and unequal bases, prove that the one with the greater base has smaller summit angles. (Use Ex. 2 and Theo. ) 4. In Fig. Ill, 12 prove that *EFP2 < *EFPU and hence that line FP2 is closer to line EF than is line FPl.

The men who conceived and developed this new geometry did their initial work on it approximately in the period 1800 to 1830. Since they worked for the most part independently of one another and lived in different countries, there is considerable variety in their approach to the new system and in the names they gave it. Among these names we find anti-Euclidean geometry, astral geometry, non-Euclidean geometry, logarithmic-spherical geometry, imaginary geometry, and pangeometry. Of these, only the name non-Euclidean geometry is still used today.