By Barnabas Hughes (eds.)
Leonardo da Pisa, maybe larger referred to as Fibonacci (ca. 1170 - ca. 1240), chosen the main necessary components of Greco-Arabic geometry for the e-book often called De practica geometrie. starting with the definitions and structures chanced on early on in Euclid's parts, Fibonacci advised his reader the best way to compute with Pisan devices of degree, locate sq. and dice roots, make sure dimensions of either rectilinear and curved surfaces and solids, paintings with tables for oblique dimension, and maybe ultimately fireplace the mind's eye of developers with analyses of pentagons and decagons. His paintings passed what readers may count on for the subject.
Practical Geometry is the identify of the craft for medieval landmeasurers, differently referred to as surveyors nowa days. Fibonacci wrote De practica geometrie for those artisans, a becoming supplement to Liber abbaci. He were at paintings at the geometry venture for it slow whilst a chum inspired him to accomplish the duty, which he did, going past the purely useful, as he remarked, "Some components are provided in keeping with geometric demonstrations, different components in dimensions after a lay type, with which they need to interact based on the extra universal practice."
This translation deals a reconstruction of De practica geometrie because the writer judges Fibonacci wrote it. to be able to savour what Fibonacci created, the writer considers his command of Arabic, his education, and the assets to be had to him. to those are additional the authors personal perspectives on translation and feedback approximately early Renaissance Italian translations. A bibliography of fundamental and secondary assets follows the interpretation, accomplished via an index of names and exact words.
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This ebook by way of Jakob Nielsen (1890-1959) and Werner Fenchel (1905-1988) has had
a lengthy and complex heritage. In 1938-39, Nielsen gave a sequence of lectures on
discontinuous teams of motions within the non-euclidean airplane, and this led him - in the course of
World battle II - to put in writing the 1st chapters of the booklet (in German). whilst Fenchel,
who needed to get away from Denmark to Sweden end result of the German profession,
returned in 1945, Nielsen initiated a collaboration with him on what turned 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) was once complete in 1948 and it used to be deliberate to be released within the Princeton
Mathematical sequence. besides the fact that, as a result of speedy improvement of the topic, they felt
that sizeable 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 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 comprise a part of a corre-
spondence (first in German yet later in Danish) among Nielsen and Fenchel, the place
Nielsen makes designated 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 centred his examine within the thought of convexity, heading
the overseas Colloquium on Convexity in Copenhagen 1965. for nearly two decades
he additionally placed a lot attempt into his task as editor of the newly all started magazine Mathematica
Scandinavica. a lot to his dissatisfaction, this job left him little time to complete the
Fenchel-Nielsen venture 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 - stumbled on time to
finish the publication common Geometry in Hyperbolic area, which was once released through
Walter de Gruyter in 1989 almost immediately after his loss of life. concurrently, and with an analogous
collaborators, he supervised a typewritten model of the manuscript (manuscript 2) on
discontinuous teams, elimination a number of the imprecise issues that have been within the unique
manuscript. Fenchel advised me that he pondered elimination elements of the introductory
Chapter I within the manuscript, considering this may be coated via the e-book pointed out above;
but to make the Fenchel-Nielsen e-book self-contained he finally selected to not do
so. He did choose to miss
27, entitled Thefundamental team.
As editor, i began in 1990, with the consent of the felony 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 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 college) with the evidence analyzing of the TEX-manuscript (manuscript three)
against manuscript 2 in addition to with a common dialogue of the difference to the fashion
of TEX. In such a lot respects we determined to keep on with Fenchel's intentions. although, turning
the typewritten variation of the manuscript into TEX helped us to make sure that the notation,
and the spelling of convinced key-words, will be uniform during the ebook. additionally,
we have indicated the start and finish of an explanation within the traditional 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 put up the manuscript of their sequence
Studies in arithmetic. i'm so much thankful for this optimistic and speedy 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 carry 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. even if, the writer provided an answer
whereby I should still carry exact drawings of the figures (Fenchel didn't go away such
for Chapters IV and V), after which they might arrange the construction of the figures in
electronic shape. i'm very thankful to Marcin Adamski, Warsaw, Poland, for his superb
collaboration in regards to the genuine creation of the figures.
My colleague Bent Fuglede, who has personaHy recognized 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 e-book reproductions of pictures of the 2
authors that are within the ownership of the Academy.
Since the manuscript makes use of a few specific symbols, a listing of notation with brief
explanations and connection with the particular definition within the ebook has been incorporated. additionally,
a accomplished index has been additional. In either circumstances, all references are to sections,
not pages.
We thought of including an entire checklist 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 money owed appropriate 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 lifestyles.
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Additional resources for Fibonacci’s De Practica Geometrie
Example text
Similarly, take from line bd line bz equal to line ae. 4]. Now zd is equal to line eg, namely, to 8 rods. Again, from lines ez and gd take straight lines ei and gt, each of which has 21 16 rods Now join line it and it equals both lines eg and zd. Now when lines ei and gt are removed from lines ez and gd, each of which measures 21 16 rods, there remains for each of the straight lines iz and td 21 2 rods. And so when we shall have multiplied the 19 rods from above into the 33, then we have the square measure of quadrilateral az and there remains for us the quadrilateral ezdg out of the whole quadrilateral abgd and the area of quadrilateral eabz was 21 9 staria.
Although today the Arabic format can be adjusted to that of ascending continued fractions, Leonardo thought of his fractions and wrote of them as parts of parts. The calculation may appear fortuitous. However, Fibonacci would use the format to join fractional parts of a denier to larger units. Thus, a long verbal answer can be read from a short presentation in a kind of positional notation. For example, in problem [2], the abbreviated answer is 0 4 0 2 4. 6 6 11 6 The lone whole number indicates staria.
Then multiply the 3 rods [that remain from subtracting 110 from 113] by 54 rods, that is by 9 panes and 21 4 rods, to obtain 27 panes and 21 13 rods that are 29 panes and 21 7 soldi. Add this to 90 staria to obtain 92 staria, 5 panes, and 21 7 soldi. [11] Likewise if you wish to multiply 72 rods by 149 rods, first multiply 66 rods by 149 rods to obtain 149 staria. Then multiply 149 rods by 6 rods [the difference between 66 and 72]. Then first multiply by 132 rods or 2 staria and afterwards by 17 rods [the difference between 132 and 149] to obtain 12 staria and 102 rods; that is 13 staria, 6 panes, and 9 soldi.