Re: Le Tarot arithmologique - la séquence 1+4+7+10 = 22

#472
There is more than one source for the relevant lot-book type ...

Conradus Bollstatter version ... c. 1450, 22 animals, with astrological context
Fränkisches Losbuch c. 1425-1450, 22 identical animals, with astrological context
Printed version of 1520, 22 animals, with astrological context
------------------
Each of these has minor variants in the row of the animals against the 2 others.
Each uses 22 questions, 22 Kings, 22 Prophets and 22 animals.

Then there is a Trier-version with differences in the figures, c. 1515

And a version with 32 animals, which includes all 22 animals of the upper three versions. It's suggested, that it was made in the time of Ludwig III von der Pfalz, who died 1437 and reigned since 1410.

The Splendor Solis (c. 1530, 22 pictures) has distant similarities in its astrology.

All these works appeared in Southern parts of Germany.

Parts of the astrological system look, as if they are very old, reaching back to astronomical ideas between 500 BC- 70 AD (a 13th zodiac sign looks, as if it has developed from a 13-months-calendar; two months in 59 days)
Or it might be just relatable to a religious Jewish calendar, which was very old and still was used. Lot book systems were often imported from Eastern versions.

Italy likely hadn't much lot books. When Lorenzo Spirito made one (maybe 1473, probably 1482), it didn't get a 22x22x22x22-scheme, but a 20x20x20x20-scheme. It became a big success .. many reprints.
Huck
http://trionfi.com

Re: Le Tarot arithmologique - la séquence 1+4+7+10 = 22

#473
I do not understand why your 10x22 = (1+2+3+4+5+6+7+8+9+10) x 4 is a necessary as opposed to contingent fact about the tarot deck (I mean, it is a necessary fact in and of itself, but contingent in relation to the tarot deck). P. 86 of the Theologumena (pp. 114-115 of Waterfield translation) is indeed impressive on the number 55 in relation to the Decad. However it never mentions either 220 or 22. That seems to be your idea. Please explain further.

I would not argue for any influence upon the tarot of Kabbalah, with its evident affinities to Pythagoreanism before Pico. He was its great popularizer. However it is not true that Pico was the first Christian humanist to learn about Kabbalah. Lazzarelli learned some aspects before him, probably from Alemanno in Padua (on Alemanno see my post at viewtopic.php?f=11&t=1049&p=15795&hilit ... ari#p15795). It is likely that others knew at least a little about it earlier. Ghiberti's Solomon and Sheba panel of the 1430s has a detail that could only have been derived from a Jewish midrash, of course known only to Jews, and mostly to rabbis. It is a detail probably suggested by Traversari, an important figure in Florence at the presumed time of the early development of the tarot (died 1439, and if him, then others), presumed to have had contact with his Jewish counterpart in Florence (for documentation see my post at viewtopic.php?f=11&t=1049&p=15795&hilit ... ari#p15800). Jews were an active new presence in Florence from 1429, and also very present in Cremona (again, viewtopic.php?f=11&t=1049&p=15795&hilit ... ari#p15795). The Bembo workshop even seems to have done an illumination in marriage contract for one of them (http://www.jpost.com/Arts-and-Culture/A ... istian-art) (so much for the idea that Jews didn't approve of images).

Re: Le Tarot arithmologique - la séquence 1+4+7+10 = 22

#474
ABOUT KABBALAH AND TAROT

Mikeh : "I do not understand why your 10x22 = (1+2+3+4+5+6+7+8+9+10) x 4 is a necessary as opposed to contingent fact about the tarot deck (I mean, it is a necessary fact in and of itself, but contingent in relation to the tarot deck). P. 86 of the Theologumena (pp. 114-115 of Waterfield translation) is indeed impressive on the number 55 in relation to the Decad. However it never mentions either 220 or 22. That seems to be your idea. Please explain further."

Alain :
My use of the qualificatifs "contingent" and "necessary" were specific of a possible influence of kabbalistic thoughts in the time of the standartisation of the classical Tarot.
" la référence à la Qabbale juive même christiannisée - Kabbale - relève encore de la contingence et non de la nécessité."

Mikeh
"I would not argue for any influence upon the tarot of Kabbalah, with its evident affinities to Pythagoreanism before Pico. He was its great popularizer. However it is not true that Pico was the first Christian humanist to learn about Kabbalah. Lazzarelli learned some aspects before him, probably from Alemanno in Padua (on Alemanno see my post at viewtopic.php?f=11&t=1049&p=15795&hilit=Traversari#p15795). It is likely that others knew at least a little about it earlier. Ghiberti's Solomon and Sheba panel of the 1430s has a detail that could only have been derived from a Jewish midrash, of course known only to Jews, and mostly to rabbis. It is a detail probably suggested by Traversari, an important figure in Florence at the presumed time of the early development of the tarot (died 1439, and if him, then others), presumed to have had contact with his Jewish counterpart in Florence (for documentation see my post at viewtopic.php?f=11&t=1049&p=15795&hilit=Traversari#p15800). Jews were an active new presence in Florence from 1429, and also very present in Cremona (again, viewtopic.php?f=11&t=1049&p=15795&hilit=Traversari#p15795). The Bembo workshop even seems to have done an illumination in marriage contract for one of them (http://www.jpost.com/Arts-and-Culture/A ... istian-art) (so much for the idea that Jews didn't approve of images)."

Alain : Agreement.
My focus was mainly on Dr FORTUNA Pic de la Mirandola because of his belonging to the Platonician Medicean AcademIe...
http://www.sgdl-auteurs.org/alain-bouge ... Biographie

Re: Le Tarot arithmologique - la séquence 1+4+7+10 = 22

#475
Mikeh : "I do not understand why your 10x22 = (1+2+3+4+5+6+7+8+9+10) x 4 is a necessary as opposed to contingent fact about the tarot deck (I mean, it is a necessary fact in and of itself, but contingent in relation to the tarot deck). P. 86 of the Theologumena (pp. 114-115 of Waterfield translation) is indeed impressive on the number 55 in relation to the Decad. However it never mentions either 220 or 22. That seems to be your idea. Please explain further."

Alain : Well, I failed to explain myself.

Let's go back to the beginning on this issue.
Decker's take about the succession of decades ad infinitum
I wrote it was orthodox to present the series of numbers this way.
But that this manner did not (enough?) take in account the pythagorean figurative geometry of Number 22 as an expansion 1+4+7+10.

Could there be a way of conciliating the two lectures?

Nicolas de Cuse shows very well the succession of the Decades ad infinitum (though he stops his example at one thousand)
"Il est important de diriger votre attention sur la progression du nombre et vous réaliserez que cette progression s’accomplit par le nombre 4. Car 1, 2, 3 et 4 additionnés ensemble font 10, qui déploie la puissance de la simple unité.

...
En effet, à partir du nombre dix, qui est une unité de second ordre, le déploiement au carré de la racine dix est atteint par une progression similaire en quatre étapes : 10, 20, 30, et 40, qui additionnés font 100, carré de la racine 10. De manière semblable, par le même mouvement, la centaine au carré donne origine au millier
".

In Tarot the Numericals are 4 Decades :
40 =10 +10+10+10
Counting the numbers of the Emblems on each card of each Decade :
220 = 55+55+55+55

(In modern Maths, we'd say that 10 is the 'Triangular Root' (?) = Racine triangulaire of 55:
Nombres 1 2 3 4 5 6 7 8 9 10
Racine triangulaire 1 3 6 10 15 21 28 36 45 55)

The Decade is repeated 4 times.
So this is the relation :
- between the Number 40 and 220
- between the Number 10 and 55

Though the Number 220 is not cited in the Thelogumena, one finds, in the Thelogumena, it's fourth part or the Decade (55x4 = 220) as Number 55 = sum of the Decade ten numbers

1+2+3+4+5+6+7+8+9+10 = 55
This specific pythagorean summation of the ten first numbers of the Decade as 55 is repeated 4 times in the Four Decades of the Tarot's numericals.
55x4

From my point of view, it is not random that the result of the summation is 220 = 55x4
This is , in my mind, an indication of a relation of the Decade (Suits Numericals) to the 22 Trumps and vice-versa.
22x10 or 10x22

Nota : there is also the Tetrade of the 4 Honours repeated 4 times in the 4 Emblems
Giving value 1+2+3+4 to them, we'd have the 4 Tetractys specific of the 4 Emblems.

Post scriptum
Though in a different take, it is also of interest that the structure of 22 repeated 4 times was also present in these times
viewtopic.php?f=11&t=1102&p=18484#p18484
http://www.sgdl-auteurs.org/alain-bouge ... Biographie

Re: Le Tarot arithmologique - la séquence 1+4+7+10 = 22

#476
Huck wrote:There is more than one source for the relevant lot-book type ...

Conradus Bollstatter version ... c. 1450, 22 animals, with astrological context
Fränkisches Losbuch c. 1425-1450, 22 identical animals, with astrological context
Printed version of 1520, 22 animals, with astrological context
------------------
Each of these has minor variants in the row of the animals against the 2 others.
Each uses 22 questions, 22 Kings, 22 Prophets and 22 animals.

Then there is a Trier-version with differences in the figures, c. 1515

And a version with 32 animals, which includes all 22 animals of the upper three versions. It's suggested, that it was made in the time of Ludwig III von der Pfalz, who died 1437 and reigned since 1410.

The Splendor Solis (c. 1530, 22 pictures) has distant similarities in its astrology.

All these works appeared in Southern parts of Germany.

Parts of the astrological system look, as if they are very old, reaching back to astronomical ideas between 500 BC- 70 AD (a 13th zodiac sign looks, as if it has developed from a 13-months-calendar; two months in 59 days)
Or it might be just relatable to a religious Jewish calendar, which was very old and still was used. Lot book systems were often imported from Eastern versions.

Italy likely hadn't much lot books. When Lorenzo Spirito made one (maybe 1473, probably 1482), it didn't get a 22x22x22x22-scheme, but a 20x20x20x20-scheme. It became a big success .. many reprints.
Huck
It is quite interesting to see that the structure of 22 pictures repeated 4 times was already present at this time ...
http://www.sgdl-auteurs.org/alain-bouge ... Biographie

Re: Le Tarot arithmologique - la séquence 1+4+7+10 = 22

#477
Aparte
For a better understanding about fIgurate numbers, these two links may be of some help for beginners...
" Figurate numbers or Polygonal numbers (Old name: figural numbers).
Such numbers represent an ancient link between geometry and number theory.
Their origins can be traced back to the Greeks,
where properties of oblong, triangular, and square numbers
were investigated and discussed
by the sixth century BC, pre-Socratic philosopher Pythagoras of Samos and his followers"


Wikipedia : Nombre Figuré
https://fr.wikipedia.org/wiki/Nombre_figur%C3%A9

DicoNombre Math : curiosités Pythagore
http://villemin.gerard.free.fr/Wwwgvmm/ ... thagor.htm
http://www.sgdl-auteurs.org/alain-bouge ... Biographie

Re: Le Tarot arithmologique - la séquence 1+4+7+10 = 22

#478
La disposition arithmo-géométrique du Nombre Pentagone est
1, 5, 12, 22
Image

1=1
5 =1+4
12= 5=7
22=12+10

Pythagorean literary references to the arithmological sequence of Pentagone Number 22 as a figurate number :

1. Iamblicus and Theon de Smyrna : 1+4+7+10
2. Boethius : 1,5,12,22


1.Commentary of Iamblicus and Arithmetic of Theon de Smyrna : 1+4+7+10

Geometrical configuration of triangular, square, and pentagonal numbers:
"The manuscripts and editions of the Introduction to Arithmetic of Nicomachus lack these figures here; but they are found in the edition of the Commentary of Iamblichus and in both editions of the Arithmetic of Theon of Smyrna."

Image


(Th. Henri MARTIN, Dean of the Faculty of Arts of Rennes, France, Corresponding Member of the Academy of Sciences of Berlin, Germany. Translation from the Greek of the Introduction to arithmetic of Nicomachus of Gerasa, Chapters IX and XX, Book II, with Notes by the translator, 1856, Rennes, France, )
http://remacle.org/bloodwolf/erudits/ni ... etique.htm



2. Boethius : 1,5, 12, 22

Anicius Manlius Severinus Boethius, De institutione arithmetica libri duo, Book 2, sections 13-14.
Link : https://archive.org/stream/aniciimanlii ... 4/mode/2up

Sorry for the images - I don't know how to make them smaller...
Serie of Numbers : 22 = 1,5,12,22

I was only able t access to section 13.
Sections 14 is blank!

Image


InsT. Arith. II, 13
p.97 paragraphe XIII
Image


Also on :
II, 17 p.101 paragraphe XVII
Image



PS Maybe something of interest looking through these references in English..

REFERENCES
T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, pages 2 and 311.
R. Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963; p. 129.
A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189.
C. K. Cook and M. R. Bacon, Some polygonal number summation formulas, Fib. Q., 52 (2014), 336-343.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 1.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 284.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 64.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A. Weil, Number theory: an approach through history; from Hammurapi to Legendre, Birkhaeuser, Boston, 1984; see p. 186.
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, pp. 98-100 Penguin Books 1987.
D. B. West, Introduction to Graph Theory, 2nd ed., Prentice-Hall, NJ, 2001.
LINKS
T. D. Noe, Table of n, a(n) for n = 0..1000
Jordan Bell, Euler and the pentagonal number theorem, arXiv:math/0510054 [math.HO], 2005-2006.
George E. Andrews, Euler's "De Partitio Numerorum", Bull. Amer. Math. Soc., 44 (No. 4, 2007), 561-573.
Anicius Manlius Severinus Boethius, De institutione arithmetica libri duo, Book 2, sections 13-14.
Stephan Eberhart, Letter to N. J. A. Sloane, Jan 06 1978, also scanned copy of Mathematical-Physical Correspondence, No. 22, Christmas 1977.
Leonhard Euler, De mirabilibus proprietatibus numerorum pentagonalium, par. 1
Leonhard Euler, Observatio de summis divisorum p. 8.
Leonhard Euler, An observation on the sums of divisors, arXiv:math/0411587 [math.HO], 2004, 2009. See p. 8.
Leonhard Euler, On the remarkable properties of the pentagonal numbers, arXiv:math/0505373 [math.HO], 2005.
Rodney T. Hansen, Arithmetic of pentagonal numbers, Fib. Quart., 8 (1970), 83-87.
Alfred Hoehn, Illustration of initial terms of A000326, A005449, A045943, A115067
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 339 [Broken link]
Milan Janjic and Boris Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From N. J. A. Sloane, Feb 13 2013
Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2002), 65-75.
Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.
Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.
R. P. Loh, A. G. Shannon, A. F. Horadam, Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients, Preprint, 1980.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.
Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.
Omar E. Pol, Illustration of initial terms of A000217, A000290, A000326, A000384, A000566, A000567


Alison Schuetz and Gwyneth Whieldon, Polygonal Dissections and Reversions of Series, arXiv:1401.7194 [math.CO], 2014.
W. Sierpiński, Sur les nombres pentagonaux, Bull. Soc. Royale Sciences Liège, 33 (No. 9-10, 1964), 513-517. [Annotated scanned copy]
N. J. A. Sloane, Illustration of initial terms of A000217, A000290, A000326
Michel Waldschmidt, Continued fractions, Ecole de recherche CIMPA-Oujda, Théorie des Nombres et ses Applications, 18 - 29 mai 2015: Oujda (Maroc).
Eric Weisstein's World of Mathematics, Pentagonal Number
Wikipedia, Mycielskian
Wikipedia, Pentagonal number
Index entries for "core" sequences
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1)


Credits : https://oeis.org/search?q=1%2C5%3B12%2C ... o=Chercher
http://www.sgdl-auteurs.org/alain-bouge ... Biographie

Re: Le Tarot arithmologique - la séquence 1+4+7+10 = 22

#479
Question : may in the late Middle Ages and Renaissance Boethius Arithmetic be clearly related to a "pythagorean" Game?
It seems so.

Anne E. MOYER, The Philosopher's game, Rithmomachia in Renaissance and Medieval Europe, University of Michogan Press, 2001 :
I. Introduction
II. From Cathedral Schools to University
III. Games of Philosphers and Astrologers from the late Middle Ages to the Renaissance : Ludus Philosophorum

Available on Google Books , section I and II... section III is circa blank...I guess Cusanus and Bessarion ..

https://books.google.fr/books?id=SNM2tj ... ce&f=false
http://www.sgdl-auteurs.org/alain-bouge ... Biographie

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