Note added Aug. 23: this post is changed from its original version, to correspond to Alain's revised version
of August 18, 2015.
There are still a few places where I have included a "translator's comment". In one place he says, from his first version of the essay, that the sequence 1 + 4 + 7 + 10 of the pentagonal number 22 works for the B order, might work for the C order, and doesn't work for the A order; but he has omitted the rationale for these statements. One seems to me needed, and I put in a very short version of what is left of his argument in the first version. Alain no doubt otherwise engaged. Another omission is a reference for the 1585 work by Jacques Perrache. I have added a link to the relevant paper by Thierry Depaulis.
There is also the question, which we have discussed in this thread, about the meaning of "triomphe" or "atout", that it has two meanings, such that in one there are 22 of them, and in another there are 21. I have noted this double meaning in the appropriate place.
One major change from the preceding version (now revised) is that I have more clearly marked off the quotations from Alain's own words. I have arranged the long quotations the way they usually are done on this Forum. There are also additions on Alain's part: a few more explanations and pictures. (This note is of August 23, 2016.)
The arithmological sequence of the pentagonal number 22 = 1 + 4 + 7 + 10
At one point in history, the number of cards [in the tarot deck] finally settled at 78.
The game of tarot is based on 56 cards called Italian, but actually of Arab origin (coins, cups, staves and swords), which arrived in Italy in the fourteenth century, and 22 cards as Triumphs introduced during the fifteenth century in Italy. This game was probably inspired by Petrarch’s Triumphs...
The tarot sequence ... arrived, during the fifteenth century, at 22 pictorial allegories of a new game. 22 is a number that in Christian mysticism signifies the importance of an introduction to the wisdom and divine teachings inculcated and imbued in men. These cards are a development that proclaims the evolution of "playing cards" to religious requirements, perhaps to meet the wishes of the Church, which had regularly considered card games diabolical.
Medieval theology assigns to the universe a precise order, consisting of a symbolic staircase that goes from earth to heaven: the top of the stairs, God, the First Cause, governs the world, without intervening directly but operating ex gradibus, that is, through a continuous series of intermediaries. Thus His divine power is transmitted to the lower creatures – even to the humblest beggar. On the other hand, if we read this symbology from the Depths to the Heights, we are taught that man can gradually rise in spiritual climbing the peaks of the bonum, verum and nobile, and that science and the virtues bring him closer to God.
The first known list of tarot cards, that of the Sermones de ludo cum aliis, appears at the beginning of the sixteenth century, written by a preacher denouncing the game. However, the sequence conforms precisely to the ethical teachings of the Church.
The Magician [ Bateleur--trans.] represents a man who is a sinner. He has been given temporal guides, the Empress and Emperor, and spiritual guides, the Pope and Popess (the Faith). Human instincts must be tempered by the virtues: Love by Temperance, the desire for power (the Triumphal Chariot) by Strength. The Wheel of Fortune teaches that success is ephemeral and that even the mighty are destined to become dust.
Thus the Hermit, who comes after the Wheel, represents Time, to which every being must submit, while the Hanged Man warns of the danger of giving in to temptation and sin before the arrival of physical Death. Even life after death is represented according to its appropriate design in the Middle Ages: Hell, and thus the Devil, are placed at the center of the earth, while the celestial spheres are above the earth.
According to the Aristotelian vision of the cosmos, the terrestrial sphere is surrounded by "heavenly fires", represented in the Tarot by the Lightning falling on a Tower. The planetary spheres are "synthesized" into three main celestial bodies: Venus, the preeminent Star, the Moon and the Sun. The highest celestial body is the Empyrean, where the Angels sit that at the Judgment will be responsible for waking the Dead in their graves when divine Justice will triumph, weighing souls to separate the good from the bad. At the summit of all this arrangement is the World, namely God the Father, as wrote the anonymous monk who commented on the Tarot in the late fifteenth century. This same author places the Fool [Matto in Italian, Fol or Mat in French--trans.] after the World as if to indicate that he is foreign to all the rules and all the lessons because, given his lack of reason, he is not able to understand the revealed truth.
The thought of Scholasticism, which aimed to strengthen the truths of the faith through the use of reason, brings into the category of fools all those who do not believe in God. In the Tarot, the Fool's presence subsequently acquires a deeper meaning: the fool possessing reason but not being a believer, was to become, through the teachings expressed by the Mystical Staircase, the Fool of God, as Francis, the most popular saint, will become, named "Holy Minstrel of God and Holy Fool of God".
Cf: http://letarot.it/cgi-bin/pages/mostra_ ... r_sito.pdf
These 78 cards include 56 emblematic cards (40 numeral [10x4] + 16 courts [4x4]) and 22 allegorical subjects.
The Pythagorean arithmology of Nichomachus of Gerasa, like that of Iamblichus or of Theon of Smyrna, is one that mathematically defines triangular, square, and pentagonal numbers, teaching that:
10 is a Triangular number of base 4
16 is a Square number of base 4.
22 is a Pentagonal Number of base 4
It is the geometric representation of polygonal numbers: square, triangular, and pentagonal:
http://remacle.org/bloodwolf/erudits/ni ... etique.htm
The figures are lacking here in the manuscripts and editions of the Introduction to Arithmetic of Nicomachus; but they are found in the edition of the Commentary by Iamblichus and in the two editions of the Arithmetic of Theon of Smyrna.
The hexagonal, pentagonal, octagonal, etc., numbers give rise to similar diagrams. These tables are explained by the following observations:
1. Each number of the natural series is the side of the corresponding polygonal number..
2. Each series of polygonal numbers, begins with unity, which, according to an expression borrowed from Aristotle's Metaphysics, is a triangular, square, pentagonal, etc., in power (δυνάμει), that is to say, potentially, while the following numbers of each series are triangular, square, pentagonal, etc., in act (ένεργεία), that is to say, actually. Each polygonal number of the same series contains the number of the previous polygon, plus the difference, and therefore each of these numbers is the sum of all the previous differences.
3. Thus, the differences are at the same time the numbers composing the polygonal numbers. The composing numbers or differences form the series whose first term is always unity, and in which each number is equal to the previous + 1 for the formation of numbers triangles, the previous + 2 for the formation of square numbers, the previous + 3 for the formation of pentagonal numbers, proceeding + m - 2 for the formation of numbers that have a number of sides m.
4. The polygonal number next to each polygon has as many units as there are component numbers included in the corresponding polygonal number. The names triangular, square, pentagonal, etc., given to these numbers can be explained by geometrical figures, where the units, represented by points, are distributed regularly, so as to show how each polygonal number of a series includes the number of the previous polygon plus the difference, and is the sum of all the preceding differences or component numbers.
(Th. Henri MARTIN, Dean of the Faculty of Arts of Rennes, France, Corresponding Member of the Academy of Sciences of Berlin, Germany. French translation of 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
Series of polygonal numbers (which can be triangular, square or pentagonal)
"Natural Series" of numbers noted on the bottom side of the diagrams:
Triangular....................................1 2 3 4
Square.......................................1 2 3 4
Pentagonal..................................1 2 3 4
Series of component numbers or “differences” noted on the lower left and above each diagram
Triangular..........................................1 2 3 4
Square..............................................1 3 5 7
Pentagonal.........................................1 4 7 10
Series of polygonal numbers noted above each figure:
Triangular......................................1 3 6 10
Square.........................................1 2 9 16
Pentagonal...................................1 5 12 22
[The small lines connecting all the dots pertaining to each of the component numbers were superimposed by hand to figures that are arithmological in origin, i.e. diagrams of triangles, squares and pentagons - "and to highlight these numbers, which, in being added, form each polygonal number."]
Cf : http://remacle.org/bloodwolf/erudits/ni ... etique.htm
The arithmological arrangement of the 22 allegorical subjects of the Tarot is that of the Pythagorean generation of the Pentagonal Number 22: 1 + 4 + 7 + 10 = 22.
https://commons.wikimedia.org/wiki/File ... nomons.svg
Cf: https://www.loebclassics.com/view/pytha ... 335.99.xml
Pentagonal Number 22 of order 4
Natural succession of the Numbers:
1 2 3 4
Series of the component numbers or “differences”:
1 + 4 + 7 + 10
Series of Pentagonal Numbers:
1 5 12 22
The points in the arithmological diagram have been replaced by small diamonds.
The lines were superimposed by hand in order to connect:
- The component numbers, in order to highlight them
The successive Four Enclosures of the component numbers together so as to emphasize visually their order of succession: first 1, then 4, then 7 and finally 10.
It is inserted into the overall arithmological layout of the 78:
The 78 cards of the Tarot (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12) are represented by diamonds arranged in a triangle with a 90° angle at the top and two angles on the sides of 45° each:
- 40 numerals [10x4]: 10 is a Triangular Number of base 4
16 figures [Court Cards--trans.] [4x4]): 16 is a Square Number of base 4.
22 allegorical subjects [1 = 4 = 7 = 10]: 22 is a Pentagonal Number of base 4.
The methodology requires that the Triumphs be arranged as if one ignored their iconography or ludic value, taking into account only their ordinal value i.e. from the first to the twenty-second position in the order of the generation of the Pentagonal number 22.
https://commons.wikimedia.org/wiki/File ... 53-0.jpg?u
The problem of the Order of the Trumps [Fr. Atouts, "trumps" in a sense that applies only to tarot, meaning the allegorical subjects--trans.] renders complex and delicate the replacement of the ordinal numbers of the Pentagonal Number 22 by the corresponding allegorical subject of the Triumphs.
I am indebted to Michael Hurst, in his "critical examination" of my theory of the sequence of Trumps + 4 + 7 + 10 = 22, for drawing my attention to the work of Michael Dummett [The Game of Tarot
and the 1985 FMR article "Tarot Triumphant"] and to the analysis of the "Problem of the Order of the Trumps” by Thierry Depaulis in Tarot Jeu et Magie
"His analyses [those of Mr. Dummett], particularly that published in "The Game of Tarot "are not so different from the analysis you present here. (The differences are just your odd placement of the Fool; and your segregation of the Bateleur). Both of you divide the Triumphs at the Pope and Death, which gives a quite intelligible grouping of the series in three types of subject. The only "apparent" difference between your two analyses is that you focus on what is incidental, the number of images in each group, while Dummett fixed on what is essential, the type of subject in each group.
(Michael Hurst, Ltarot, "Alain's 1 + 4 + 7 + 10 theory".)
What Mr. Hurst considers incidental, the Number, and as essential, the Image, is not important here - even if Hurst's belief in the primacy of Signified (Image) over Signifying (the Number) raises subjectivity over objectivity: how else to account for the sequence 22 when it comes to the poems of Boiardo or the 22 Figures of the Sola Busca?
Nothing prevents us from thinking that the final predominance of the structure of 78 cards, broken down into 22 Triumphs and 56 cards (4 × 10 + 4 × 4), belongs to so random a register of contingency - these numbers would then be purely incidental and random. Nonetheless, it will be understood, I do not share this opinion. Is it not important that the arithmology of the pentagonal Number 22 (quantitative aspect) coincides, not incidentally, with the symbolic and historical analysis of the allegorical series of Triumphs (qualitative)?
In fact, the only difference between the sequence of pentagonal Number 22 as 1 + 4 + 7 + 10 and Dummett’s analysis into 3 groups is that one member of the whole constitutes a group of one by itself (The Bateleur [Magician] at Alpha) and the Fool is added to the end of the fourth group, in the last position, as twenty second in the sequence.
T. Depaulis for his part examines 12 different orders of "decks existing now", "old decks"
and "literary sources". The analysis of the 12 orders leads to his putting forward for the sequence of Triumphs 3 “blocs” – Typologies called A, B, or C.
Type A corresponds to the “existing Tarocchino Bolognese, the Sicilian Tarot (with some modifications), the Minchiate (likewise), the so-called Charles VI and some other ancient decks
” (T. Depaulis, op cit.
Type B is (“the oldest known
” (T. Depaulis, op. cit.
Type C, “That of the Tarot de Marseille and its descendants, such as the Tarot de Besançon and the Tarocco Piemontese
” (T. Depaulis, op. cit.
Type A would not correspond to the ordinal arrangement of the generation of the pentagonal Number 22 according to the sequence 1+4+7+10. [In that sequence, the beginning of the last group, i.e. the 12th from the lowest, would be the Hanged Man, which is not part of the medieval cosmograph that shapes those 10, but belongs to the second section—trans.]
Type B does correspond. [The 12th triumph in this case is Death, which does correspond to a position on the cosmograph, namely, on the circumference of the sphere of earth, of which the Devil is the center--trans.]
Type C could correspond. [The 12th triumph is again Death; but there is no extant account until de Gébelin in 1781 that puts the Fool in that last position--trans.]
Iconographically, the Pythagorean arrangement of the 22 allegories in four enclosures raises two questions: the places of the Alpha
and the Omega
of the sequence of triumphs - namely the Bateleur [Magician] at Alpha and the Fool at Omega in the sequence.
It should be noted that the Sermones,
which is of Order B, positions the Bagatella
[Magician--trans.] in 1st position and places the Matto
[Fool], while joining to him the epithet "nulla", at the twenty-second line, after the World in 21st position (Sermones de ludo
, c. 1500).
Cf : http://www.tarotpedia.com/wiki/Sermones ... _Cum_Aliis
of La Maison académique
, dated by T. Depaulis at 1637*, lists 22 Triumphs:
Ce jeu qui est composé de soixante & dix-huict Cartes, se peut distribuer en cinq bandes, la première & la plus noble de toutes appellée triomphes qui sont au nombre de vingt-deux : & les quatre autres couleurs sont nommées d'espées, bastons, couppes & deniers, chacune desquelles a quatorze cartes : Sçauoir le Roy, la Royne, le Cheualier, & le Faon, qui s'appellent aussi les quatre honneurs & le reste depuis le dix iusques à laz,...
(This pack, which consists of seventy-eight cards, can be divided into five sets. The first and noblest of them all is called Trumps [in the special sense applying only to the tarot deck--trans.], of which there are twenty-two. The other four plain suits (1) are known as swords, batons, cups and coins: each of them consists of fourteen cards : namely the King, Queen, Cavalier and Jack (2)who are also called the four honors & the rest from ten to ace.)
*1637: Dieser nunmehr datierte und Michel de Marolles zugeordnete Text erscheint in der Zeitschrift "Le Vieux Papier" unter dem Titel "Quand l'abbé de Marolles jouait au tarot" auf 1637.
(1637: This, now dated, with associated text by Michel de Moralles, appears in the journal "Le Vieux Papier" under the title "When the abbot of Marolles plays tarot" in 1637.)
(French text and German footnote from http://www.tarock.info/depaulis.htm
. English translation of text from http://trionfi.com/0/p/1637.html
This dating is included in Lois et règles de jeux pour l’esprit et le bonheur des hommes
by Manfred Zollinger: http://expositions.bnf.fr/jeux/arret/03_3.htm
However it is noted that from 1585, in the Triomphe du Berlan
by PERRACHE, already specific expressions are present in the text that are literally identical to ones later reproduced in the Rule
of the Maison Académique
, said of 1637.
[Cf. http://www.academia.edu/15317283/_%C3%8 ... p._386-392
The Position of the Fool at the Omega of the sequence after the World does not mean the triumph of the Matto over the World, but suggests the idea of a Madman falling perhaps into divine madness, also the playful and subversive theme of the medieval Feasts of Fools.
The other side of the Mass of Fools in the Middle Ages is the wonderful innocence personified by the FOU, eliminating the chaos. The 'crazy one', the village idiot, the most deprived, the youngest of the community - embodied and fixed in the TRUE CENTER, and it is through him that we hunt the scapegoat...
The Mass of Fools takes place in the cathedral, ideal because its architecture meets all the medieval theological grandeur and life of the masses who have congregated to be elevated. Nevertheless we know from the work of Julio Caro Banoja that the Feasts of Fools quickly made the street: the people participating in these festivals were not the sole prerogative of cleric-insiders. Mass religious movements of in the Middle Age were mostly mobilized by disease, starvation or despair; they left little trace on institutions or medieval thought. Yet the shifting crowd in search of the supernatural and miraculous 'were immensely sure of themselves and contemptuous of traditional disciplines and restraints. In their zeal they swamped the ordinary organization of the church, broke through the barrier between the illiterate populace and the learned rulers of society, and spoke with certainty about things to come' (R. W. Southern, Western Society and the Church in the Middle Ages, p. 308).
The Mass of Fools will also gather the creative energy of this people without writing.
Berry Hayward, La Messe des Fous
Placed in a carnival perspective, the figure of the jester would affix itself to the festive triumphal processions of the second half of the fifteenth century.
"Is the jester not supposed to operate outside of the carnival procession, following it and moving about along the parade" like the Fool of the so-called Charles VI Tarot?
Copyright Alain Bougearel
For fuller information, consult the discussion of the thesis at: viewtopic.php?f=11&t=1102
Nicomachus of Gerasa, Introduction to Arithmetic
, English translation by Martin Luther D’Ooge,
with Studies in Greek Arithmetic
, by Fank Egleston Robbins and Louis Charles Karpinski,
London: Macmillan and Company, Ltd., 1916.
https://ia600709.us.archive.org/27/item ... hmetic.pdf
Translator's note added Aug. 23: I have now incorporated most of the changes that Alain has introduced since his first draft. There are still a few problems, such as the size of some of the images and one that I could not find on the Internet to reproduce.