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By Werner Fenchel, Jakob Nielsen, Asmus L. Schmidt

This publication by means of Jakob Nielsen (1890-1959) and Werner Fenchel (1905-1988) has had
a lengthy and intricate background. 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 conflict II - to jot down the 1st chapters of the booklet (in German). while Fenchel,
who needed to break out from Denmark to Sweden as a result of the German profession,
returned in 1945, Nielsen initiated a collaboration with him on what grew to become 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) used to be complete in 1948 and it used to be deliberate to be released within the Princeton
Mathematical sequence. even though, end result of the quick improvement of the topic, they felt
that colossal adjustments needed to be made ahead of book.
When Nielsen moved to Copenhagen collage in 1951 (where he stayed until eventually
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 information of Fenchel now
deposited and catalogued on the division of arithmetic at Copenhagen Univer-
sity include 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 documents additionally include a part of a corre-
spondence (first in German yet later in Danish) among Nielsen and Fenchel, the place
Nielsen makes certain 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), used to be greatly concerned with a radical revision of the curriculum in al-
gebra and geometry, and focused his learn 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 task 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 collage, Fenchel - assisted via Christian Sieben-
eicher from Bielefeld and Mrs. Obershelp who typed the manuscript - discovered time to
finish the booklet basic Geometry in Hyperbolic house, which used to be released by means of
Walter de Gruyter in 1989 almost immediately after his demise. concurrently, and with a similar
collaborators, he supervised a typewritten model of the manuscript (manuscript 2) on
discontinuous teams, elimination the various vague issues that have been within the unique
manuscript. Fenchel informed me that he meditated removal elements of the introductory
Chapter I within the manuscript, due to the fact that this could be coated by means of the booklet pointed out above;
but to make the Fenchel-Nielsen booklet self-contained he eventually selected to not do
so. He did choose to pass over
27, entitled Thefundamental staff.

As editor, i began in 1990, with the consent of the criminal heirs of Fenchel and
Nielsen, to provide 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 activity of typing this manuscript in AMS- TEX. i've got additionally had
much aid from my colleague J0rn B0rling Olsson (himself a pupil of Kate Fenchel
at Aarhus collage) with the evidence examining of the TEX-manuscript (manuscript three)
against manuscript 2 in addition to with a common dialogue of the variation to the fashion
of TEX. In so much respects we made up our minds to keep on with Fenchel's intentions. even though, turning
the typewritten variation of the manuscript into TEX helped us to make sure that the notation,
and the spelling of sure key-words, will be uniform during the ebook. 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 post the manuscript of their sequence
Studies in arithmetic. i'm so much thankful for this confident and quickly response. One
particular challenge with the e-book became out to be the replica 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 via 1997 it grew to become transparent that he
would no longer have the ability to locate the time to take action. besides the fact that, the writer provided an answer
whereby I may still convey specified drawings of the figures (Fenchel didn't go away such
for Chapters IV and V), after which they might arrange the creation of the figures in
electronic shape. i'm very thankful to Marcin Adamski, Warsaw, Poland, for his high-quality
collaboration in regards to the real construction 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 locations 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 pictures of the 2
authors that are within the ownership of the Academy.
Since the manuscript makes use of a couple of precise symbols, an inventory of notation with brief
explanations and connection with the particular definition within the booklet has been integrated. additionally,
a complete index has been further. In either circumstances, all references are to sections,
not pages.
We thought of including an entire record of references, yet made up our minds opposed to it because of
the overwhelming variety of examine papers during this zone. in its place, a far shorter
list of monographs and different accomplished bills suitable to the topic has been
collected.
My ultimate and such a lot honest thank you visit Dr. Manfred Karbe from Walter de Gruyter
for his commitment and perseverance in bringing this booklet into lifestyles.

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Extra info for Discontinuous Groups of Isometries in the Hyperbolic Plane

Sample text

How many points are determined if exactly two of the lines are parallel? 12. How many lines may be drawn between four points lying in a plane if no three of the points are on the same straight line? 13. What is the total number of angles formed by two intersecting transversals cutting two parallel lines? 6. Three Parallel Lines Cut by Two Transversals. Theorem 15 may be applied to prove an important property regarding the segments of two transversals intercepted by three parallel lines; namely, THEOREM 16.

Find the hypotenuse c of a right triangle ,whose legs are given as indicated. Use the formula c = Va 2 + b2• (a) a =6",b =8". (d) a = 21",b =45". (b) a = 10", b = 24". , b = v'ls in. (c) a = 9"', b = 12". /2 in. 7. One leg and the aypotenuse c of a right triangle are given as indicated. Use the formula b = V c2 - a2 to find the length of the other leg. (a) c = 17", a = 15". (d) c = 34", a = 16". (e) c = 100", a = 60". (b) c = 25", a = 24". (c) c = 35", a = 21". , a = 2Vs in. 8. Find the area of each of the right triangles whose sides are given as follows.

3. Parallels Perpendicular to the SaDle Line. THEOREM 8. Two lines in the saDle plane perpendiculn to the S8Dle line are parallel. GIVEN: The two lines II and 12 lying in the same plane and perpendicular to the line l3 at the points A and B, respectively. To PROVE: ll"~. · * This method of proof makes use of the fact that only one of two conclusions is possible. Arguments arising from the use of one conclusion show a violation of an established fact - a definition, proposition, axiom, or postulate - and hence prove that this conclusion is invalid.

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