Dear David,
Joe Farrell has already given you a perfect answer by pointing you to
Brown's _Numeri Vergiliani_ (thank you, Joe, this was new to me!) If I may
add some brief explanations for those who wonder what the heck we are
talking about:
The Golden section or Golden means ('sectio aurea', division by 'media et
extrema ratio'), known to ancient and to some medieval mathematicians
though not half as prominent in their writings as some modern hunters of
Golden Sections woul lead us to believe, divides a given quantity "a" into
unequal parts "b" (major) and "c" (minor), in a way that the relation
between the whole (a) and the maor (b) is the same as the relation between
the major and the minor (a:b = b:c, with a=b+c). The value of the Golden
Section is an irrational number 1/2 (SQRT[5-1]) = 0,618... (SQRT means
'square root') You can find or rather approximate the major (B) by dividing
the whole (a) by 0,618..., or if you want to reckon the minor you divide
a/0,382... If you prefer a geometrical solution: draw a square with
sidelength a, draw a line XY from the midpoint X of one of the sides to one
of the facing corners (Y), then use point X as center of a circle with
radius=a so that it will divide the line XY at a point which I call Z: if
you now copy this line XY with its dividing point Z onto one of the sides
of your square, point C will divide the length a of this side by mean and
extreme ratio. Just trust me that doing it is far easier than describing it
in my German variant of English and with my self-made symbols!
Now to Fibonacci numbers, first discussed by Leonardo da Pisa ('figlio di
Bonacci') in his _Liber abbaci_, of which only the second redaction from
1228 is extant (this second version also mentiones a first redaction from
1202): Fibonacci numbers form a series [0, 1,] 1, 2, 3, 5, 8, 13,..., which
Leonardo illustrates by his famous rabbit problem, in the chapter _Quot
paria coniculorum in una anno ex uno pario germinentur_ (Liber abbaci, ed.
Boncompagni, p.283s.): "Qvidam posuit unum par cuniculorum in quodam loco,
qui erat undique pariete circundatus, ut sciret, quot ex eo paria
germinarentur in uno anno: cum natura eorum sit per singulum mensem aliud
par germinare; et in secundo mense ab eorum natiuitate germinant. Quia
suprascriptum par in primo mense germinat, dublicabis ipsum, erunt paria
duo in uno mense. Ex quibus unum, scilicet primum, in secundo mense
geminat; et sic sunt in secundo mense paria 3; ex quibus in uno mense duo
pregnantur; et geminantur in tercio mense paria 2 coniculorum; et sic sunt
paria 5 in ipso mense; ex quibus in ipso pregnantur paria 3; et sunt in
quarto mense paria 8; ex quibus paria 5 geminant alia paria 5: quibus
additis cum parijs 8, faciunt paria 13 in quinto mense; ex quibus paria 5,
que geminata fuerunt in ipso mense, non concipiunt in ipso mense, sed alia
8 paria pregnantur; et sic sunt in sexto mense paria 21 (...). Leonardo
goes on to count his incestous rabbits up to the 12th month, reaching a sum
of 377, then refers to a table in the margin where the sums are listed for
each month, and explains: "Potes enim uidere in hac margine, qualiter hoc
operati fuimus, scilicet quod iunximus primum numerum cum secundo,
uidelicet 1 cum 2; et secundum cum tercio; et tercium cum quarto; et
quartum cum quinto, et sic deinceps, donec iunximus decimum cum undecimo,
uidelicet 144 cum 233; et habuimus suprascriptorum cuniculorum summam,
uidelicet 377; et sic posses facere per ordinem de infinitis numeris
mensibus."
And finally to the relation between Fibonacci numbers and the Golden
Section, according to the series running under the name of the
mathematician Lame': if you take each Fibonacci number and *divide* it by
the subsequent number in the series, you get
1/1=1 1/2=0,5 2/3=0,666... 3/5=0,6 5/8=0,625 8/13=0,61538...
This means that, if you have enough rabbits at hand (and a knife to divide
them), you can use the Fibonacci series for approaching the value of the
Golden Section (the size of the 'major') by whole numbers.
Today it is fairly well known, though not easily accepted or reflected by
hunters of Golden Sections in Poetry, that these mathematical ratios occur
a bit everywhere in nature, even where nature's products are not taylored
by strange human breeding measures as applied by Leonardo to his rabbits.
>From the viewpoint of statistics, there are few things that are similarly
unexciting as the discovery that a portion of poem like the _Aeneid_ can be
divided by mean and extreme ratio. This does not mean that findings of this
kind are necessarily irrelevant. But it means that you need compelling
corroborating evidence in order to reach reasonable certainty that you are
dealing with a textual design purposefully created by the author.
Some time ago we have discussed the matter on AnSax-L with regard to David
Howlett's analyis of Hebrew, Greek and Latin texts of the Bible, where
David at times finds couples of verses where the number of letters as well
as the number of syllables (or words) divides by mean and extreme ratio
(very approximately, of course, because David has to use whole numbers, and
this adds to the likelyhood of obtaining results of this kind). I myself
had argued against his assessment of these results by adducing popular
verses and proverbs which yield similar or even better results but are
clearly not divided purposefully in order to reproduce the mathematical
ratios in question. This discussion (and other discussions of Golden
Sections in literature) is proably still available in the AnSax-L archives
on the WWW.
To David: I would tell your BBC man to be a bit careful with informations
about Fibonacci. Many or most historians of mathematics who write about him
have never daigned to read the _Liber abbaci_, the main reason being not so
much his somewhat dull Latin but the rarity of the only printed edition of
this absolutely outstanding book. If your man wants to be historically
correct, he might want to have his documentary checked by one of the very
few living experts, Heinz Lueneburg (more correctly Lu"neburg with
u-umlaut), who is professor of mathematics at the U of Kaiserslautern,
author of, among other things, the very readable book _Leonardi Pisani
Liber Abbaci, oder Lesevergnu"gen eines Mathematikers_, Mannheim [et al.]:
B.I. Wissenschaftsverlag, 3nd ed. 1993, and can be reached by email at
<[EMAIL PROTECTED]>. He also knows and loves to tell all the
anecdotes about Fibonacci and his reception that I myself would want to put
into a broadcasted science documentary.
Otfried
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Otfried Lieberknecht, Schoeneberger Str. 11, D-12163 Berlin
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