"Luca Paccioli biography" by J J O'Connor and E F Robertson

#581
Biography in English

Of interest with the following correction : the Bibliography is from the material available in 1999. Since in 2007was found a lost manuscript exceptional about Chess dedicated to Isabelle d' Este
Image

Paccioli and games :
The games he wrote about were :
A (lost) Treaty on Chess rediscovered 2007 by the bibliophile Duilio Contin
De ludo scacchorum
Times online : Renaissance chess master and the Da Vinci decode mystery
New York Times : Checkmate again for Leonardo? Chess book's diagrams are linked to artist
https://pt.wikipedia.org/wiki/De_Ludo_Scachorum

Magic trick cards
De viribus quantitatis, (1496-1508)
Mathknow: Mathematics, Applied Science and Real Life (Sous la direction de Michele Emmer et Alfio M. Quarteroni) page 193 (Editions Springer)

Maybe also the Treaty on Abacus was related to Boethius ?
Never published -Vatican Library codice Vaticano Urbinate 3129

Biography of Lucas Paccioli in English
http://www-history.mcs.st-andrews.ac.uk ... cioli.html

"Luca Pacioli's father was Bartolomeo Pacioli, but Pacioli does not appear to have been brought up in his parents house. He lived as a child with the Befolci family in Sansepolcro which was the town of his birth. This town is very much in the centre of Italy about 60 km north of the city of Perugia. As far as Pacioli was concerned, perhaps the most important feature of this small commercial town was the fact that Piero della Francesca had a studio and workshop in there and della Francesca spent quite some time there despite frequent commissions in other towns
Although we know little of Pacioli's early life, the conjecture that he may have received at least a part of his education in the studio of della Francesca in Sansepolcro must at least have a strong chance of being correct. One reason that this seems likely to be true is the extensive knowledge that Pacioli had of the work of Piero della Francesca and Pacioli's writings were very strongly influenced by those of Piero.

Pacioli moved away from Sansepolcro while he was still a young lad. He moved to Venice to enter the service of the wealthy merchant Antonio Rompiasi whose house was in the highly desirable Giudecca district of that city. One has to assume that Pacioli was already well educated in basic mathematics from studies in Sansepolcro and he certainly must have been well educated generally to have been chosen as a tutor to Rompiasi's three sons. However, Pacioli took the opportunity to continue his mathematical studies at a higher level while in Venice, studying mathematics under Domenico Bragadino. During this time Pacioli gained experience both in teaching, from his role as tutor, and also in business from his role helping with Rompiasi's affairs.

It was during his time in Venice that Pacioli wrote his first work, a book on arithmetic which he dedicated to his employer's three sons. This was completed in 1470 probably in the year that Rompiasi died. Pacioli certainly seemed to know all the right people for he left Venice and travelled to Rome where he spent several months living in the house of Leone Battista Alberti who was secretary in the Papal Chancery. As well as being an excellent scholar and mathematician, Alberti was able to provide Pacioli with good religious connections. At this time Pacioli then studied theology and, at some time during the next few years, he became a friar in the Franciscan Order.

In 1477 Pacioli began a life of travelling, spending time at various universities teaching mathematics, particularly arithmetic. He taught at the University of Perugia from 1477 to 1480 and while there he wrote a second work on arithmetic designed for the classes that he was teaching. He taught at Zara (now called Zadar or Jadera in Croatia but at that time in the Venetian Empire) and there wrote a third book on arithmetic. None of the three arithmetic texts were published, and only the one written for the students in Perugia has survived. After Zara, Pacioli taught again at the University of Perugia, then at the University of Naples, then at the University of Rome.
Certainly Pacioli become acquainted with the duke of Urbino at some time during this period. Pope Sixtus IV had made Federico da Montefeltro the duke of Urbino in 1474 and Pacioli seems to have spent some time as a tutor to Federico's son Guidobaldo who was to become the last ruling Montefeltro when his father died in 1482. The court at Urbino was a notable centre of culture and Pacioli must have had close contact with it over a number of years.

In 1489, after two years in Rome, Pacioli returned to his home town of Sansepolcro. Not all went smoothly for Pacioli in his home town, however. He had been granted some privileges by the Pope and there was a degree of jealousy among the men from the religious orders in Sansepolcro. In fact Pacioli was banned from teaching there in 1491 but the jealousy seemed to be mixed with a respect for his learning and scholarship for in 1493 he was invited to preach the Lent sermons.

During this time in Sansepolcro, Pacioli worked on one of his most famous books the Summa de arithmetica, geometria, proportioni et proportionalita which he dedicated to Guidobaldo, the duke of Urbino.
Pacioli travelled to Venice in 1494 to publish the Summa. The work gives a summary of the mathematics known at that time although it shows little in the way of original ideas. The work studies arithmetic, algebra, geometry and trigonometry and, despite the lack of originality, was to provide a basis for the major progress in mathematics which took place in Europe shortly after this time. As stated in [1] the Summa was:-
... not addressed to a particular section of the community. An encyclopaedic work (600 pages of close print, in folio) written in Italian, it contains a general treatise on theoretical and practical arithmetic; the elements of algebra; a table of moneys, weights and measures used in the various Italian states; a treatise on double-entry bookkeeping; and a summary of Euclid's geometry. He admitted to having borrowed freely from Euclid, Boethius, Sacrobosco, Fibonacci, ...
The geometrical part of Pacioli's Summa is discussed in detail in [6]. The authors write:-
The geometrical part of L Pacioli's Summa [Venice, 1494] in Italian is one of the earliest printed mathematical books. Pacioli broadly used Euclid's Elements, retelling some parts of it. He referred also to Leonardo of Pisa (Fibonacci).

Another interesting aspect of the Summa was the fact that it studied games of chance. Pacioli studied the problem of points, see [9], although the solution he gave is incorrect.
Ludovico Sforza was the second son of Francesco Sforza, who had made himself duke of Milan. When Francesco died in 1466, Ludovico's elder brother Galeazzo Sforza became duke of Milan. However, Galeazzo was murdered in 1476 and his seven year old son became duke of Milan. Ludovico, after some political intrigue, became regent to the young man in 1480. With very generous patronage of artists and scholars, Ludovico Sforza set about making his court in Milan the finest in the whole of Europe. In 1482 Leonardo da Vinci entered Ludovico's service as a court painter and engineer. In 1494 Ludovico became the duke of Milan and, around 1496, Pacioli was invited by Ludovico to go to Milan to teach mathematics at Ludovico Sforza's court. This invitation may have been made at the prompting of Leonardo da Vinci who had an enthusiastic interest in mathematics.

At Milan Pacioli and Leonardo quickly became close friends. Mathematics and art were topics which they discussed at length, both gaining greatly from the other. At this time Pacioli began work on the second of his two famous works, Divina proportione and the figures for the text were drawn by Leonardo. Few mathematicians can have had a more talented illustrator for their book! The book which Pacioli worked on during 1497 would eventually form the first of three books which he published in 1509 under the title Divina proportione (see for example [3]).
This was the first of the three books which finally made up this treatise, and it studied the 'Divine Proportion' or 'golden ratio' which is the ratio a : b = b : (a + b). It contains the theorems of Euclid which relate to this ratio, and it also studies regular and semiregular polygons (see in particular [4] for a discussion of Pacioli's work on regular polygons). Clearly the interest of Leonardo in this aesthetically satisfying ratio both from a mathematical and artistic point of view was an important influence on the work. The golden ratio was also of importance in architectural design and this topic was to form the second part of the treatise which Pacioli wrote later. The third book in the treatise was a translation into Italian of one of della Francesca's works.

Louis XII became king of France in 1498 and, being a descendant of the first duke of Milan, he claimed the duchy. Venice supported Louis against Milan and in 1499 the French armies entered Milan In the following year Ludovico Sforza was captured when he attempted to retake the city. Pacioli and Leonardo fled together in December 1499, three months after the French captured Milan. They stopped first at Mantua, where they were the guests of Marchioness Isabella d'Este, and then in March 1500 they continued to Venice. From Venice they returned to Florence, where Pacioli and Leonardo shared a house[/u].

The University of Pisa had suffered a revolt in 1494 and had moved to Florence. Pacioli was appointed to teach geometry at the University of Pisa in Florence in 1500. He remained in Florence, teaching geometry at the university, until 1506. Leonardo, although spending ten months away working for Cesare Borgia, also remained in Florence until 1506. Pacioli, like Leonardo, had a spell away from Florence when he taught at the University of Bologna during 1501-02. During this time Pacioli worked with Scipione del Ferro and there has been much conjecture as to whether the two discussed the algebraic solution of cubic equations. Certainly Pacioli discussed this topic in the Summa and some time after Pacioli's visit to Bologna, del Ferro solved one of the two cases of this classic problem.

During his time in Florence Pacioli was involved with Church affairs as well as with mathematics. He was elected the superior of his Order in Romagna and then, in 1506, he entered the monastery of Santa Croce in Florence. After leaving Florence, Pacioli went to Venice where he was given the sole rights to publish his works there for the following fifteen years. In 1509 he published the three volume work Divina proportione and also a Latin translation of Euclid's Elements. The first printed edition of Euclid's Elements was the thirteenth century translation by Campanus which had been published in printed form in Venice in 1482. Pacioli's edition was based on that of Campanus but it contained much in the way of annotation by Pacioli himself.

In 1510 Pacioli returned to Perugia to lecture there again. He also lectured again in Rome in 1514 but by this time Pacioli was 70 years of age and nearing the end of his active life of scholarship and teaching. He returned to Sansepolcro where he died in 1517 leaving unpublished a major work De Viribus Quantitatis on recreational problems, geometrical problems and proverbs. This work makes frequent reference to Leonardo da Vinci who worked with him on the project, and many of the problems in this treatise are also in Leonardo's notebooks. Again it is a work for which Pacioli claimed no originality, describing it as a compendium.

Despite the lack of originality in Pacioli's work, his contributions to mathematics are important, particularly because of the influence which his book were to have over a long period. In [10] the importance of Pacioli's work is discussed, in particular his computation of approximate values of a square root (using a special case of Newton's method), his incorrect analysis of certain games of chance (similar to those studied by Pascal which gave rise to the theory of probability), his problems involving number theory (similar problems appeared in Bachet's compilation), and his collection of many magic squares.

In 1550 there appeared a biography of Piero della Francesca written by Giorgio Vasari. This biography accused Pacioli of plagiarism and claimed that he stole della Francesca's work on perspective, on arithmetic and on geometry. This is an unfair accusation, for although there is truth that Pacioli relied heavily on the work of others, and certainly on that of della Francesca in particular, he never attempted to claim the work as his own but acknowledged the sources which he used."

Article by: J J O'Connor and E F Robertson
JOC/EFR © July 1999
Copyright information School of Mathematics and Statistics
University of St Andrews, Scotland
The URL of this page is:
http://www-history.mcs.st-andrews.ac.uk ... cioli.html


Some Notes
1. S A Jayawardene, Biography in Dictionary of Scientific Biography (New York 1970-1990).
3. G M Biggiogero, Luca Pacioli e la sua ’Divina proportione’, Rendiconti dell’Istituto lombardo di scienze e lettere 94 (1960), 3-30.
4. J F Field, Rediscovering the Archimedean polyhedra : Piero della Francesca, Luca Pacioli, Leonardo da Vinci, Albrecht Dürer, Daniele Barbaro, and Johannes Kepler, Arch. Hist. Exact Sci. 50 (3-4) (1997), 241-289.
6. F R Glushkova and S S Glushkov, The geometrical part of Pacioli’s ’Summa’ (Russian), in History and methodology of the natural sciences XXIX (Moscow, 1982), 57-63.

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

#582
About the Tetractys Game of 56 : reminder
https://drive.google.com/file/d/0B5Hg6j ... FmdTg/view

Other curiosities

56 cards in 4 sets of 14 as 10+4

4 Tetractys of 10 numerals :

1............ 1
2............ 3
3............ 6
4.............10
5.............15
6.............21
7.............28
8.............36
9.............45
10............55
Image




4 Tetrads of 4 Honours

V.............1
C.............2
D.............3
R.............4
Image



The 16 Court cards in 4 emblems are 16 as 4x4
https://drive.google.com/file/d/0B5Hg6j ... FmdTg/view
They may be also understood as a Magic Square of 4.

Example of MAGIC SQUARE with 1.2.3.4 (4 sums of 10)

1/4 // 4/1
3/2 // 2/3
------------
2/3 // 3/2
4/1 // 1/4

This Magic Square of 2x2 listing 4 times the series of numbers from 1 to 4 has it's magic constant equal to 10 and the addition of all the numbers of the square is the sum 40.


Example of MAGIC SQUARE with
1.2.3.4.5.6.7.8.9.10.11.12.13.14. 15. 16 = 136 (4 sums of 34)
Image


Image

Durer 1514
https://fr.wikipedia.org/wiki/Melencolia_de_D%C3%BCrer


16/03 // 02/13
05/10 // 11/08
09/06 // 07/12
04/15 // 14/01

This Magic Square of order 4 listing the series of numbers from 1 to 16 has it's magic constant equal to :
34 and the addition of all the numbers of the square is the sum 136.


Durer Magic Square of 16 does not apply to the Courts cards of Tarot. It would suggest that one of the Courts card has a value of 1, and another of 16 - it is not the case. Their game value are : 1,2,3,4. In the case of the 16 Honours of Tarot,it is the Magic Square of 4 that is showed.

Nevertheless ...
220 = 55x 4
55 = 1+2+3+4+5+6+7+8+9+10
(Cf Theleogoumena, On the Decad From Anatolius [86], pp.114-115 and Note 25)

And, as J.M. DAVID of Aeclectic noted :
"Adding the 136 to the 220 does indeed give us 356, which, if divided by the 22 Atouts, makes (approx.) 16.18, and if divided by a tenfold aspect, and by a number also used in the generation of the 356 (ie, 220), then an approximation to Phi is given: 1.618.

It should be pointed out that any sequence which uses an additive method of generation (such as the famous Lucas or Fibonacci series) will result in adjacent numbers having close approximations to Phi when one is divided by the other. Given that 356 divided by 220 is an approximation of Phi, it will also be the case that 220 divided by (356-220) will also approximate it - though not as closely.
Of greater interest, for me at any rate, is that the 220 is a very close approximation of the Golden Angle (360 / Phi = approx. 220 degrees)."
"


1;Golden ratio and Number 16
1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16 = 136

"Sectio divina" : "In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The figure on the right illustrates the geometric relationship. Expressed algebraically, for quantities a and b with a > b > 0,"
https://en.wikipedia.org/wiki/Golden_ratio
Image

356:220= circa 220:136 = circa Phi
a = 220
b = 136



2. Square root of PHI in Numbers 28 and 22 :
(56/2 ) = 28 divided by 22 = Square Root of PHI
The value of [square root of ]PHI being here :
356:280= circa 280:220= racine of Phi

Square root of PHI (also called Golden Number) = 1,272 with PHI as 1.16179 circa real value as 1.618
Reminder : PHi also called Golden number : (1 + square root of 5) divided by 2 = circa 1.618

A "Golden" curiosity ?
BtW the 78 cards can also be divided in two series :
56 the Tetractys Game with 2 series of 2 "suits" (?) that is 56 = 2x (14x2=28)
22 the Pentagonal pogression of the iconographical figures.
56 and 22 are not in a Golden Ratio.
But are these numbers really random?
No. These Numbers (22) and (56 as 28x2) are nevertheless related to the square root of PHI .

56/2 ) = 28 divided by 22 = Square Root of PHI

Re: "Luca Paccioli biography" in French

#583
BOUGEAREL Alain wrote:Biography
BIOGRAPHY IN FRENCH /
https://litalieparsestimbres.wordpress. ... a-pacioli/

"Luca Bartolomeo de Pacioli naquit vers 1445 à Borgo San Sepolcro dans une famille modeste. Nous ne savons pas grand-chose ni de son enfance, ni de son adolescence, mais il n’est pas impossible qu’il ait reçu une partie de son éducation dans l’atelier de son compatriote, le peintre et mathématicien Piero della Francesca.

Jeune homme, il partit à Venise et entra au service d’Antonio Rompiansi, un marchand vénitien. Il vivait chez lui et s’occupait, entre autres, de l’éducation de ses trois fils. Pendant cette période, il étudia les mathématiques sous la direction de Domenico Bragadino, maître de mathématiques à l’école de San Giovanni di Rialto. L’expérience qu’il acquit en s’occupant des affaires de Rompiansi et les connaissances qu’il reçut grâce à Bragadino l’incitèrent à écrire, vers 1470, un traité sur les mathématiques qu’il dédicaça aux fils Rompiansi.

A la mort de Rompiansi, l’engagement de Pacioli à Venise prit fin et il partit plusieurs mois à Rome, comme invité de l’architecte Leon Battista Alberti. Peu de temps après, il entra chez les Franciscains et devint moine au couvent San Francisco della Vigna de Venise. A la fin de ses études de théologie, il fut envoyé dans diverses villes enseigner les mathématiques. De 1477 à 1489, il résida ainsi à Perugia (où il écrivit un autre traité de mathématiques, Tractatus mathematicus ad discipulos perusinos), à Zara, à Naples, à Rome et à Urbino, où il fut l’ami du duc Federico da Montefeltro.
En1490, il retourna à San Sepolcro où il écrivit son oeuvre majeure, Summa de arithmetica, geometria, proportioni et proportionalità. Cet ouvrage représente la première version imprimée et écrite en langue vernaculaire sur l’algèbre et il résume l’ensemble des connaissances mathématiques de l’époque. Il est particulièrement impostant car il explique pour la première fois de manière claire la méthode vénitienne de tenue des comptes, maintenant connue sous le nom de comptabilité en partie double. Cette méthode était couramment utilisée depuis le XIIIe siècle par les marchants et les banques italiennes et aurait été découverte par les Egyptiens il y a 3 700 ans. Même si Pacioli n’est pas l’inventeur de ce système comptable et ne fit que l’expliquer, il est considéré comme le « père de la comptabilité », car son traité permit aux étudiants de le comprendre et de l’utiliser. Pacioli dédicaça son livre au duc Guidobaldo da Montefeltro qui fut son élève lorsqu’il enseignait à Urbino.

En 1497, Pacioli fut invité à la cour de Ludovic Sforza, duc de Milan, pour enseigner les mathématiques. Il y rencontra Léonard de Vinci avec lequel il noua une étroite amitié et qui le consulta pour des questions relatives aux mathématiques. Entre 1496 et 1498, Pacioli écrivit De divina proportione, un traité sur les proportions mathématiques et artistiques illustré par de Vinci. L’œuvre traite aussi de l’usage de la perspective par les peintres della Francesca, Melozzo de Forlì et Marco Palmezzano. La troisième partie de l’ouvrage, Libellus in tres partiales tractatus divisus, est une traduction en italien de l’ouvrage en latin de della Francesca sur les cinq solides de Platon, De Corporibus regalaribus, mais elle n’inclut aucune référence à l’auteur originel. Giorgio Vasari traita Pacioli d’« usurpateur », pour avoir publié sous son nom les écrits de della Francesca qui étaient en sa possession depuis la mort du peintre. Il existait trois exemplaires du manuscrits. Deux nous sont restés: un à la Bibliothèque publique et universitaire de Genève; le second, dédié à Galeazzo Sanseverino, est conservé à la bibliothèque Ambrosienne de Milan. Le troisième, dédié à Pier Soderini, a disparu.

En 1499, lorsque les troupes de Louis XII de France prirent le duché de Milan et destituèrent Ludovic Sforza, les deux amis s’enfuirent à Mantoue, à Venise, puis à Florence où ils partagèrent le même appartement. Ils se séparèrent finalement en 1506. Pacioli rentra dans son village natal où il traduisit les Eléments d’Euclide. Il mourut en 1517."

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

#584
About 22 remembering that the 22 expand in Tarot in the Four Emblems as 220.

1. Pentagonal Number 22 as the arithmetic serie 1,5,12,22 [gnomons : 1+4+7+10]
Boece « De institutione arithmetica » edition 1867
viewtopic.php?f=11&t=1102&start=550#p18861

2. In the scholastic thought of the Church, the number 22 makes sense.
(Cf: Steve Mangan on THF : viewtopic.php?f=11&t=1102&start=340#p17660)
Also :
"In the order of the numbers, each individual number contains some strength and power over things. The Creator of the Universe has made use of this power and strength, either for the constitution of the universe itself or to express the nature of each thing as it appears. It follows, according to the Scriptures, that we must observe and calculate these aspects which belong to the numbers themselves. And in truth, the books of the Bible itself, such as the Jews have transmitted them, are [in number] twenty-two, equal to the number of Hebrew letters, and this is not without reason. Indeed, twenty-two letters [that] seem to be the introduction to God's wisdom and the knowledge of the world."
(Select in Ps I - PG 12, 1084)
(Cf : A. Quacquarelli, s.v. Numeri, in DPAC,
pages 2447-2448)

These notions were known at the time and place of the standartisation of Tarot


Considering that the iconography of the 22 allégorical figures also refered to a mystical stair then their numbers also would be a mystical arithmetic.
Considering that "the Books of the Bible are [in number] twenty-two, equal to the number of Hebrew letters" - I finally for the first time accept to consider the possibility of an implicit indirect Christian Pythagorean "Kabbalah" influence in the early 1500- 1505 upon the tarocchi or taraux fifth suit finally settled as 22 ... not strictly as Jew Qabbalah but as an indirect reference to the 22 Letters Numbers of the mosaic alphabet in correlation to the 22 Books of the Bible.

Re: Le Tarot arithmologique

#586
The Arithmological Tarot : 22 and 56
Image

- The Pentagonal Number 22 = 1 + 4 + 7 + 10
http://letarot.it/page.aspx?id=603&lng=ITA
- The Tetractys Game of 56 =16 [4x4] + 40 {4x10]
https://drive.google.com/file/d/0B5Hg6j ... FmdTg/view

Recapitulation
Image


My first book edited in 1997 was a kind of "Unicorn research"
http://data.bnf.fr/13165391/alain-jacques_bougearel/
[A propos de mon livre publié en 1997. ce fut une première "mouture" dirais-je aujourd'hui avec le recul - un ouvrage de jeunesse qui mêlait intuitions , recherches et et convictions - d'où le pluriel "Origines ..." du titre. L'ouvrage qui désormais date, pourtant, a eu le mérite de mettre en relief la possibilité d'une matrice mathématique ayant présidé à la standartisation finale du "Tarot" en un ensemble de 78 cartes]

A few years later, Andrea Vitali kindly published a short presentation on the central thesis on line : Tarot and Neo-Pythagoreanism
The numerological structure of the Tarots http://letarot.it/page.aspx?id=83&lng=ENG
In 2010, the Editions Lo Scarabeo published a collective study Il Castello dei Tarocchi with an Italian version of the article : Tarocchi e Neopitagorismo di Alain Bougearel
La struttura numerologica dei Tarocchi e la sua derivazione dall’aritmologia pitagorica.
http://letarot.it/page.aspx?id=219&lng=ENG
Finally , twenty years after the intuition published in 1997 i.e the main thesis is now presented as a plausible hypothesis in historical research.
My hypothesis is precisely that the final state of the 78 cards of the Tarocchi were influenced by Pythagorean Number theory such the arithmological structure of the 78.
Discussions :
viewtopic.php?f=11&t=1168&start=100#p19190
And :
viewtopic.php?f=11&t=1168&start=100#p19193

Reminder .
Link to the hypothesis
The Pentagonal Number 22
http://letarot.it/page.aspx?id=603&lng=ITA
The Tetractys Game of 56 :
https://drive.google.com/file/d/0B5Hg6j ... FmdTg/view

M. Howard translated the essay in English and wrote a review :
The arithmological sequence of the pentagonal number 22 = 1 + 4 + 7 + 10
Translation from French by Michael S. Howard
http://letarot.it/page.aspx?id=603&lng=ENG
In Appreciation of Alain Bougearel's "1+4+7+10=22"
An essay by Michael S. Howard
http://letarot.it/page.aspx?id=608

Actual considerations and updates :

1.The 22 expand in Tarot in the Four Emblems as 220
viewtopic.php?f=11&t=1102&start=570#p18947
viewtopic.php?f=11&t=1102&start=570#p18949
Steve Mangan :
viewtopic.php?f=11&t=1102&start=570#p18948


2. Pentagonal Number 22 as the arithmetic serie 1,5,12,22 [gnomons or differences : 1+4+7+10]
Boece « De institutione arithmetica »
viewtopic.php?f=11&t=1102&start=550#p18861

3. In the scholastic thought of the Church, the number 22 makes sense.
Cf: : Steve Mangan on THF : viewtopic.php?f=11&t=1102&start=340#p17660

Also ORIGENE :
"In the order of the numbers, each individual number contains some strength and power over things. The Creator of the Universe has made use of this power and strength, either for the constitution of the universe itself or to express the nature of each thing as it appears. It follows, according to the Scriptures, that we must observe and calculate these aspects which belong to the numbers themselves. And in truth, the books of the Bible itself, such as the Jews have transmitted them, are [in number] twenty-two, equal to the number of Hebrew letters, and this is not without reason. Indeed, twenty-two letters [that] seem to be the introduction to God's wisdom and the knowledge of the world."
(Select in Ps I - PG 12, 1084)
(Cf : A. Quacquarelli, s.v. Numeri, in DPAC,
pages 2447-2448)

These notions were known at the time and place of the standartisation of "Tarot"

4. Considering that the iconography of the 22 allégorical figures also refered to a mystical stair then their numbers also would be a mystical arithmetic.
Considering that "the Books of the Bible are [in number] twenty-two, equal to the number of Hebrew letters" - I finally accept to consider the possibility of an implicit indirect Christian Pythagorean "Kabbalah" influence in the early 1500- 1505 upon the tarocchi or taraux fifth suit finally settled as 22 ... not strictly as Jew Qabbalah but as an indirect reference to the 22 Letters Numbers of the mosaic alphabet in correlation to the 22 Books of the Bible.

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