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By William J. M'Clelland

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man containing Chapters I-II (

I -15), and an entire manuscript (manuscript I) in
English containing Chapters I-V (

1-27). The files additionally include a part of a corre-
spondence (first in German yet later in Danish) among Nielsen and Fenchel, the place
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who succeeded N. E. Nf/Jrlund at Copenhagen college in 1956 (and stayed there
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collaborators, he supervised a typewritten model of the manuscript (manuscript 2) on
discontinuous teams, removal some of the vague issues that have been within the unique
manuscript. Fenchel informed me that he pondered elimination elements of the introductory
Chapter I within the manuscript, given that this could be coated via the publication pointed out above;
but to make the Fenchel-Nielsen e-book self-contained he finally selected to not do
so. He did choose to omit
27, entitled Thefundamental workforce.

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 a superb 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 interpreting 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 so much respects we determined to keep on 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 during the e-book. additionally,
we have indicated the start and finish of an evidence within the ordinary sort of TEX.
With this TEX -manuscript I approached Walter de Gruyter in Berlin in 1992, and
to my nice aid and delight they agreed to submit the manuscript of their sequence
Studies in arithmetic. i'm so much thankful for this confident and quickly response. One
particular challenge with the ebook grew to become out to be the copy of the numerous
figures that are an essential component of the presentation. Christian Siebeneicher had at
first agreed to convey those in ultimate digital shape, yet by way of 1997 it turned transparent that he
would now not be capable to locate the time to take action. besides the fact that, the writer provided an answer
whereby I should still convey designated 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 superb
collaboration about 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 standpoint. In
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Letters for permitting us to incorporate during this publication reproductions of photos of the 2
authors that are within the ownership of the Academy.
Since the manuscript makes use of a couple of distinct symbols, a listing of notation with brief
explanations and connection with the particular definition within the booklet has been integrated. additionally,
a finished index has been additional. In either situations, all references are to sections,
not pages.
We thought of including an entire record of references, yet made up our minds opposed to it as a result of
the overwhelming variety of examine papers during this quarter. in its place, a far shorter
list of monographs and different accomplished debts suitable to the topic has been
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 ebook into lifestyles.

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Additional info for A treatise on the geometry of the circle and some extensions to conic sections by the method of reciprocation, with numerous examples.

Example text

Points invariably connected with the describe triangle, right lines which can at once be constructed by the above method. all other Moreover, circle, we know that and the variable if is line or fixed and P describes a Radius Vector OP be divided in Q, in a given ratio, the locus of Q is a circle. Now if Q be turned around through any given angle the locus is the same circle displaced through the same angle. Therefore if one vertex of a triangle of given and another vertex describe a circle, the vertex and all other points invariably conremaining species is fixed, nected with 1.

OXBL is I. 26) a cyclic quadrilateral, therefore Euc. (III. PX=1r\ PL PB is given hence the segment PB is known, and since it ; equal to 0(7, C is determined. In the general case the line A B does not meet the ment is stated : the seg- AB rectangle, having two remaining two on the 3. ] XY of a circle in a given direction such that the area of the quadrilateral A BX Y, where AB is a given diameter, is a maximum. [Draw a diameter YX\ and A YBX' is a rectangle, hence AX' is BY. Join JJJfand XX', and draw BC parallel to XX'.

COR. 1. If the line difference of the sides L cuts the base internally the (ACBC) is a maximum when it bisects internally the angle G. EXAMPLES. The 1. triangle of minimum perimeter inscribed in a given one is Z of the perpendiculars * let fall formed by joining the feet A", T, from the vertices on the opposite sides. [For the joining lines are equally inclined to sides on which they intersect (Euc. III. ] The polygon 2. one is Ex. ) of least perimeter that can be inscribed in a given that whose angles are bisected externally by its sides.

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A treatise on the geometry of the circle and some extensions to conic sections by the method of reciprocation, with numerous examples. by William J. M'Clelland

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