By Robin Hartshorne
This ebook bargains a special chance to appreciate the essence of 1 of the good thinkers of western civilization. A guided studying of Euclid's components ends up in a serious dialogue and rigorous sleek remedy of Euclid's geometry and its newer descendants, with whole proofs. subject matters contain the advent of coordinates, the speculation of sector, background of the parallel postulate, a few of the non-Euclidean geometries, and the commonplace and semi-regular polyhedra.
Read or Download Geometry: Euclid and Beyond (Undergraduate Texts in Mathematics) PDF
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This ebook by means of Jakob Nielsen (1890-1959) and Werner Fenchel (1905-1988) has had
a lengthy and intricate heritage. In 1938-39, Nielsen gave a sequence of lectures on
discontinuous teams of motions within the non-euclidean aircraft, and this led him - in the course of
World struggle II - to write down the 1st chapters of the booklet (in German). while Fenchel,
who needed to get away from Denmark to Sweden due to the German career,
returned in 1945, Nielsen initiated a collaboration with him on what grew to become identified
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 accomplished in 1948 and it was once deliberate to be released within the Princeton
Mathematical sequence. even if, a result of quick improvement of the topic, they felt
that massive adjustments needed to be made ahead of ebook.
When Nielsen moved to Copenhagen college in 1951 (where he stayed till
1955), he was once a lot concerned with the overseas association UNESCO, and the
further writing of the manuscript used to be left to Fenchel. The data 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 files additionally comprise a part of a corre-
spondence (first in German yet later in Danish) among Nielsen and Fenchel, the place
Nielsen makes unique 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 a great deal concerned with a radical revision of the curriculum in al-
gebra and geometry, and targeted his study within the idea of convexity, heading
the overseas Colloquium on Convexity in Copenhagen 1965. for nearly twenty years
he additionally placed a lot attempt into his task as editor of the newly begun magazine Mathematica
Scandinavica. a lot to his dissatisfaction, this task left him little time to complete the
Fenchel-Nielsen venture the best way he desired to.
After his retirement from the collage, Fenchel - assisted by way of Christian Sieben-
eicher from Bielefeld and Mrs. Obershelp who typed the manuscript - came across time to
finish the booklet effortless Geometry in Hyperbolic house, which used to be released through
Walter de Gruyter in 1989 almost immediately after his dying. concurrently, and with an identical
collaborators, he supervised a typewritten model of the manuscript (manuscript 2) on
discontinuous teams, elimination some of the vague issues that have been within the unique
manuscript. Fenchel advised me that he reflected elimination components of the introductory
Chapter I within the manuscript, due to the fact this is able to be lined via the booklet pointed out above;
but to make the Fenchel-Nielsen ebook self-contained he eventually selected to not do
so. He did choose to omit
27, entitled Thefundamental team.
As editor, i began in 1990, with the consent of the criminal 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 activity of typing this manuscript in AMS- TEX. i've got additionally had
much support from my colleague J0rn B0rling Olsson (himself a pupil of Kate Fenchel
at Aarhus collage) with the facts examining 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. even though, turning
the typewritten variation of the manuscript into TEX helped us to make sure that the notation,
and the spelling of yes key-words, will be uniform during the booklet. additionally,
we have indicated the start and finish of an explanation within the ordinary variety 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 quickly response. One
particular challenge with the booklet grew to become out to be the copy of the various
figures that are an essential component of the presentation. Christian Siebeneicher had at
first agreed to convey those in ultimate digital shape, yet through 1997 it turned 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 carry specific drawings of the figures (Fenchel didn't go away such
for Chapters IV and V), after which they might set up the construction of the figures in
electronic shape. i'm very thankful to Marcin Adamski, Warsaw, Poland, for his nice
collaboration about the genuine creation of the figures.
My colleague Bent Fuglede, who has personaHy identified either authors, has kindly
written a brief biography of the 2 of them and their mathematical achievements,
and which additionally locations the Fenchel-Nielsen manuscript in its right standpoint. In
this connection i need to thank The Royal Danish Academy of Sciences and
Letters for permitting us to incorporate during this publication reproductions of images of the 2
authors that are within the ownership of the Academy.
Since the manuscript makes use of a few specified symbols, a listing of notation with brief
explanations and connection with the particular definition within the e-book has been incorporated. additionally,
a complete index has been additional. In either situations, all references are to sections,
not pages.
We thought of including a whole record of references, yet determined opposed to it because of
the overwhelming variety of study papers during this zone. as a substitute, a miles shorter
list of monographs and different accomplished bills correct 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 e-book into lifestyles.
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Extra info for Geometry: Euclid and Beyond (Undergraduate Texts in Mathematics)
Sample text
So the modern approach is to say these notions are undefined, that is, they can be anything at all, provided that they satisfy whatever postulates or axioms may be imposed on them later. In the algebraic definition of an abstract group, for example, you never say what 28 1. Euclid's Geometry the elements of the group are, nor what the group operation is. Those are undefined. However, they must satisfy the group axioms that the operation is associative, there exists an identity, and that there exist inverses.
We are used to thinking of a, b, c as the lengths of the sides of the triangle, in which case the theorem becomes an equation among real numbers. How can we reconcile these two points of view? The modern answer to this question, which we will discuss in more detail later (Section 23), is that after introducing coordinates in our geometry we can prove the existence of an area function. The area of a square of side a will be a 2 • Furthermore, we will show that having equal content in the sense of Euclid is equivalent to having equal area in the sense of the area function.
Line AL. 12. Line EL. Then MEL is the required triangle. The angles at E and at L will be equal to twice the angle at A. Proof From a modern point of view, it would seem that some theory of quadratic equations is essential for the proof. Euclid did not have any algebra available to him, but he was able to deal with quantities essentially equivalent to quadratic expressions via the theory of area. We can think of a rectangle as representing the product of its sides, or a square as the square of its side.