Alain Bougearel's essay  La séquence arithmologique du nombre pentagonal 22 = 1+4+7+10 now appears above this one as a "Host Essay" in both French and English. Here I would like to make some comments in appreciation, defense, and further development of Alain's argument. Here I would like to make some comments in appreciation, defense, and further development of Alain's argument.

1. The Pythagorean meanings of 1, 4, 7, and 10 as defining the content of the corresponding groups of tarot cards.

It has been notoriously difficult to put the cards of the tarot sequence into groups that actually have something in common, as opposed to a hodgepodge. In Game of Tarot (1980) and Il Mondo e l'Angelo (1993) Dummett did not even try to characterize the three groups he identified as being united by one theme; instead, he simply named the cards that were in each. In his FMR article “Tarot Triumphant” (1986, pp. 46-53) he did make an effort. After excluding the Fool as "strictly speaking, not part of the trionfi", he says that "the lowest is the bagatto, almost always called bagatella in the early lists" (p. 46). After that are "the papal and imperial cards" (here) 3rd column). That is three groups, conceptually! After him, others, such as Hurst, have said that, including the Fool, they have in common their being the extremes of the ranks of society, high and low. Yes, that works, after a fashion. If the Fool is added, there are two low-lifes. Many early lists actually did have the Fool first, as can be seen in M. M. Filesi's lists (here). Exactly why the Bagatto is on the low end, and not the Traitor, who was more reviled than either, is not said. This characterization does not explain much about why the first group should be the way it was.

Dummett then called the next group "conditions of human life" (p. 47). For Dummett in FMR, that includes Death as well as all three of the virtue cards (here). 

Others, including Dummett himself before and after the FMR article, have decided that Death belongs in the next group, and the virtues are not in any group. After all, the type B tarot put Justice in Dummett's third group, while the type C put Temperance in that group! Why "conditions of human life" should exclude the Bagatto, the Fool and the virtues but does include a Traitor (an early name for the Hanged Man) is not said. The classification explains some things but not enough. The final group is again hard to unite. What do death, the devil, the moon, and the Last Judgment have in common, so as to make them one group? Dummett in FMR says they are all "spiritual and celestial powers". That is two groups. Is Death really one or the other of these? Perhaps so, since it is shown as a walking skeleton. Dummett in the FMR article conveniently makes Death part of the second group, thus avoiding the question. The fluidity of Dummett's three groups, in which a card can be either the last card of one group or the first card of another, is not much of a help in deciding what characterizes each group.

It seems to me that Alain's division 1 + 4 + 7 + 10 gives a certain rationale--not the only one possible, to be sure--to the order and interpretation of the cards in each group, if the meanings of these numbers, in Pythagorean terms, is given with them.

The ordinal "First", with the cardinal "One", is the number, in monotheism, Pythagorean or otherwise, of God the Father, the Prima Causa and Creator. Also called the "Monad", it is the beginning of everything. The Pythagorean basis of this idea is expressed by Vincent Foster Hopper in his Medieval Number Theory, 1938 (reprinted 2000), p. 38:

"Hence it is very natural that the Pythagoreans should have considered the monad as the first principle from which the other numbers flow (20). Itself not a number, it is an essence rather than a being (21) and is sometimes, like the duad, designated as a potential number, since the point, though not a plane figure, can originate plane figures (22) As first originator, the monad is good and God (24). It is both even and odd, male and female (24), for when added to odd it produces even, and when added to even it produces odd (25). It is the basis and creator of number, but, although it is actually the great Even-Odd, its nature is considered to be more akin to masculine oddness than to feminine evenness. In short, it is always taken to represent all that is good and desirable and essential, indivisible and uncreated. (26)" 

If 1, the point, is the Father of number, it follows that the duad, the line, is the Mother of number (27)...
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20. Nichomachus, Introduction [to Arithmetic], II, vi, 3; Plotinus, Enneads, V, I, y; Photius, Biography of Pythagoras, 7; Proclus, Elements of Theology, A; C, 21.
21. Plotinus, Enneads, VI, 9, 3.
22. Nichomachus, op. cit,. II, vi, 3.
23. Enneads, VI, I; 9, 6; V, I, 7.
24. Macrobius, In Somn. Scip. I, 6.
25. De E apud Delphos, 8.
26. Capella, De nuptiis, VII.
27. Capella, ibid.; Plutarch, De animae procreatione in Timaeo, II.

I include the footnotes because it is important to know if this account would have been known during the Renaissance. Nichomachus, Macrobius, On the Dream of Scipio, and Capella, Marriage of Philology and Mercury, were in Latin and well known throughout the Middle Ages  These alone would have been enough to convey the basic principles. Plotinus was translated by Ficino, as was Proclus. Plutarch was known in Greek at least from the 1420s and in Latin after 1500. Photius, having been a major polemicist against the Latins at the time of the "great schism", would have been studied by those interested in that issue, which surely were many given the conclave of reconciliation of 1438-39 in Florence.

There was also, for some, the Theologumena Arithemticae, in Greek only, brought to Italy by Bessarion, available in Rome in the 1460s, Venice thereafter (with a copy in Florence, according to Vittorio de Falco in the 1922 Leipzig edition of the text), and printed in 1543 Paris (per WorldCat). Attributed then to the Roman-era Neoplatonist Iamblicus, its chapter on the monad (Theology of Arithmetic, trans. Robin Waterfield, 1988, pp. 35-40) confirms Hopper's summary for example (Waterfield p. 37),

"Nichomachus says that God coincides with the monad, since he is seminally everything that exists, just as the monad is in the case of number;..."

Since in Christianity God the Father is also God the Son, the Christian God is also one of the physically weakest and lowest of his own creation, at least among humanity. So besides the Bagat's association with the number One, there is also his position in the hierarchy, as the "little one". Yet this very figure can be seen as the Creator, who in the Gospel of John (1:3) is the logos by whom “all things were made” (the Vulgate’s omnia per ipsum facta sunt). The four types of object on his table and in his hands then can symbolize the four elements out of which the Universe classically was thought to be made, as well as the regular four suits which even he, in the game, triumphs over. This is to be sure a different interpretation than Alain gives to the card, but it is the one suggested by Pythagorean philosophy.

We need not go into the significance of the number 2, that of the Popess or Empress), as we are concerned only with the numbers of Alain's sequence 1, 4, 7, 10.

The number 4 is in Pythagoreanism the number of the three-dimensional universe, the Creator’s creation. It takes a minimum of 4 points to define a solid, as opposed to 1 for a point, 2 for a line, and 3 for a plane figure). And the first four numbers added together make the sacred "tetraktys". Here is Hopper, pp. 42-43:

"The tetrad completes the list of the 'archetypal' numbers, representing the point, line, surface, and solid (55). But the particular glory of the archetypal numbers is that they produce the decad, either as a sum (1+2+3+4=10) or in the figured presentation of 10 as a triangular number. This figure was known as the tetraktys (56), the legendary oath of the Pythagoreans (57). ... Four is also the number of the square (60), and is represented in the elements, the seasons, the 4 elements of man, the 4 principles of a reasonable animal, the lunar phases, and the 4 virtues (61)".
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55. Philo, On the Ten Commandments, 7.
56. Photius, Biography, 4; Iamblichus, Biography, 28; Capella, op. cit., VII.
57. Iamblichus, Biography, 28.
      ... 
60. Plutarch, De animae procreatique, 1.
61. Diogenes Laertius, Biography of Pythagoras, 19, 7; Theolog. Arith., 22. Enneads, VI, 6, 16. Capella, op. cit. VII.

(For specific quotations from Diogenes Laertius and Capella, see my post at http://forum.tarothistory.com/viewtopic.php?f=11&t=1102&p=17476&hilit=Capella#p17476.  They are also listed in Nichomachus, Introduction to Arithmetic, Book II, Ch. 1, p. 230 of D'Ooge translation.)

Christianity extended the list of tetrads to include Adam, hence humanity, and  even the four gospels, which Irenaeus argued arithmologically could not be more or less, as pertaining to God's existence in the three-dimensional world. Here is Hopper, pp. 83-84:

"The principal Christian innovation in number science was the identification of this spiritual-temporal duality with the archetypal numbers, 3 and 4. Four, by the known analogues of the 4 winds, the 4 elements, the 4 seasons, and the 4 rivers, is specifically the number of the mundane sphere; and, as the first 3 days of creation foreshadow the Trinity, so the fourth is the 'type of man' (56). Mystically, the fact that man is a tetrad is evidenced in the name, Adam, whose letters are the 4 winds (57). For this reason, knowledge of divine things is disseminated throughout the world by the 4 gospels, evangelists or beasts, emblemized by the 4 extremities of the cross (58), the 4-fold division of Christ's clothing, and the 4 virtues, or forms of love, as Augustine names them (59). 'It is not possible', says Irenaeus, 'that the gospels can be either more or fewer than they are'. (60)"
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56. Theophilus of Antioch, To Autolycus; AN III, 82; also Ambrose, De fide II, Introduction; Augustine, On John, IX, 14.
57. Augustine, On John, IX, 14; see above, p. 31.
58. The cross was conceived to have 4 or 5 points--5 if the intersection was included. As the image of 4, it is encompassed man in the universe. As an emblem of 5, it coincided with the 4 wounds in providing the salvation of man, with his 5 senses, or of those living under the Old Dispensation of the Pentateuch.
59. Of the Morals of the Catholic Church, XV, 25.
60 Against Heresies, III, 11, 8.

So also in the tarot, there are in the tarot the two pairs, spiritual and temporal, who hold sway over the human world, the combination of 3 (for God) and 4 (for Man). 

The number 7, in Christian Pythagoreanism, is that of the virtues, the vices, and many other things pertaining to the world of humans' and their choices for good or evil:

"From the triune principle of God and the quadruple principle of man are produced the universal symbols, 7 and 12. The addition of 3 and 4, spiritual and temporal, produces 7, which is therefore the first number which implies totality (61). It is the number of the universe and of man, signifying the creature as opposed to the Creator (62). Seven gifts of the Holy Spirit were derived from Isaiah XI: 1-3 (63) The Lord's Prayer was found to contain 7 petitions (64). Similarly, the Beatitudes were found to be 7, and by the principle of contraries these septenaries were balanced by the 7 deadly sins ((66). Later, the addition of the 3 theological virtues (Faith, Hope, Charity) to the 4 cardinal virtues produced one of the best known heptads of Catholicism. The habit of presenting these spiritual entities in precise numerical groupings indicates that a relationship was felt between them, but it remained for Augustine to show the precise connection of the 7 petitions of the Lord's Prayer to the 7 beatitudes, which in turn relate to the 7 gifts of the spirit or to the 7 steps to wisdom (67). Seven is the number of the Sabbath and Salvation, but it is also the number of sin (68). Necessarily the churches on earth are 7, forming a likeness of the universe (69)".
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61. Augustine, Civ. Dei, XX, 5.
62. Augustine, On the Sermon on the Mount, II, 10, 36; Letter LV, 15, 28.
63. Tertullian, Against Marcion, V, 8; Victorinus, On Creation.
64. Cyprian, On the Lord's Prayer; Tertullian, On Prayer, II. 8.
65. Augustine, On the Sermon on the Mount, II, 10-11.
66. Tertullian, Against Marcion, IV, 9; Augustine, Harmony of the Gospels, VI, 13.
67. On the Sermon on the Mount, II, 10-11. Contra Faustum, XII, 15; On Christian Doctrine, II, y, 9-11.
68. Augustine, Harmony of the Gospels, VI, 13.
69. Augustine, Letter LV, 5, 9.

Seven, it seems to me, is the number of humanity's being tested for worthiness. This group of tarot subjects, besides including at least two of the four traditional cardinal virtues, show the situations in which people are tested: love, triumph, fortune, shame, and the limits of time (you never know when the proctor will call "time"). The Bagatto, as the tester, is easily excluded, as is the Fool, who not being a creature who reasons, selfishly or not, cannot be tested.

The number 10 is that of the cycle of the basic numbers, after which one can go on forever. Hopper observes that for the Pythagoreans "all things are contained within the decad, since after 10 the numbers merely repeat themselves" (p. 34). Also (p. 44):

"Ten and 1 are mystically the same, as are also the 100 and 1,000, the 'boundaries' of number. In the decad, multiplicity again returns to unity". 

There is also Augustine (p. 84):

"Ten had long been recognized in the image of unity, but it was Augustinian Pythagoreanism that produced it by adding the Trinity of the Creator to the hebdomad of the created (75). In Christian usage, its great type is always the 10 Commandments, whose traditional division into 2 groups of 5 was soon to be altered to 3 and 7, in recognition of this doctrine".
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75. Augustine, Against the Epistle of Manicaeus, Called Fundamental, X, 11.

Besides Augustine, some in the Renaissance and after (see my comments on the number 1) would have known the Theolgumena Arithmeticae, then attributed to Iamblicus, which says of the Decad (The Theology of Arithmetic, trans. Robin Waterfield, 1988, pp. 109-110):

"Hence  the Pythagoreans in their theology called it sometimes 'universe', sometimes 'heaven', sometimes 'all', sometimes 'Fate' and 'eternity', 'power' and 'trust' and 'Necessity', 'Atlas' and 'unwearying', and simply 'God' and 'Phanes' and 'sun'."

It is called "Atlas", the text explains, "because he carries heaven on his shoulders" (p. 111). It adds, "The spheres of the universe are ten and fall under the decad."

In the Tarot, the group here begins in the sphere of earth, at death, and ends in heaven, the Empyrean sphere; it is the course of the soul from underground to eternity, through the medieval cosmograph of concentric circles, although not all are represented. Alain identifies the next card in the B order, the Devil, with earth's center, where Dante put him. That is one possibility. But Devils were associated with the sphere of Air. They were shown in frescoes grabbing, or trying to grab, souls on their way upward, e.g. the so-called "Triumph of Death" in Pisa of the 1330s. Like angels, they had wings. The Discorso on the tarot of Piscina put the demoni "fra l' Aria" (in the air: http://www.tarotpedia.com/wiki/Francesco_Piscina_Discorso_1565). After air comes fire, the realm from whence lightning comes. The earliest names for the Tower card were Saeta or Sagitta, Arrow, i.e. lightning,  and Fuoco, fire. The next card, the Star (representing either the Fixed Stars or Venus), is out of sequence. That is for the sake of the players. It is easier to remember a sequence of increasing light than it is the Ptolemaic universe.

I have elaborated on this cosmographic progression in the essay to which Alain, at http://forum.tarothistory.com/viewtopic.php?f=11&t=1102#p16911, graciously gave a link, "The Astral Journey of the Soul", http://letarot.it/page.aspx?id=454&lng=ENG. For an exposition of the entire sequence in the interpretation which I am overlaying onto Alain's structure, see my "Platonism and the Tarot", at http://platonismandtarot.blogspot.com/.

It is important also to understand the Christian style of incorporating Pythagorean number theory into a text. It is not necessary to cite Pythagoras by name; it is merely enough to use the principles, which were known as Pythagorean by all already. For example, Hopper quotes Augustine (p. 81):

"That the flood came 7 days after Noah entered the ark, as we are baptised in the hope of future rest, which was denoted by the 7th day..."

This is is an essay (Contra Faustem) in which Augustine discusses one number after another, for its Christian significance. So also D’Oncieu, discussing the meaning of 3, 4, and 5 in the game of tarot, does not have to mention Pythagoras specifically to be seeing the game in a Pythagorean way. 

In Alain's diagram, the 1 is in the center, with the 4, 7, and 10, as concentric enclosures, U-shaped, around it. If the tops are joined, they can be seen as concentric squares, each enclosed in its own circle passing through the four vertices of each. In the center is the Creator, a kind of "big bang" point. The authorities in this world get their legitimacy from that center; they are the divinely appointed earthly representatives of that Creator. Beyond that are humans in general, all confronted with the situations of the middle section. Beyond that is the cosmograph, now represented as dots on a U, or square, or circle. Enclosing all is the "infinite circle whose center is everywhere and whose circumference is nowhere", a definition of God of uncertain origin that Nicholas of Cusa (https://en.wikiquote.org/wiki/Nicholas_of_Cusa) revived in the mid-15th century. The Fool is then the soul entering the realm of God, who is beyond concepts. St. Paul characterized Christians as "fools for Christ's sake" (I Cor. 4:10)  The Fool is even God himself from that perspective: Paul called Christ's message "the foolishness of God" (I Cor 1:25). In the same vein, one poem in 16th century Ferrara characterized the Fool as “divine brain” (cervel divino; see Andrea Vitali at http://www.letarot.it/page.aspx?id=199&lng=eng).

Alain speaks of the Bateleur as the “Alpha” of the sequence and the Fool as the “Omega”. This way of speaking was well known at the time from the words of Jesus in the Bible’s vision of the Apocalypse: “I am Alpha and Omega, the beginning and the end, the first and the last” (Rev. 1:8 and 22:13, echoing Isaiah 44:6. "I am the first and I am the last"). This utterance when applied to the tarot is entirely in accord with Pythagorean symbolism. The Bateleur is Jesus as the creator of John 1:3; the Fool is Jesus as Lord of the New Jerusalem, in a place beyond all places and concepts. 

2. The position of the Fool at the high end of the sequence.

For Alain’s interpretation to work, the Fool has to be at the end of the sequence. Yet the Sermones de Ludo give it the number 0 and calls it “nulla”. Moreover, most of the numerous lists of Triumphs in Italy put it at the Bagato’s end of the sequence rather than the other end (although there is one, of Type A, which does put the Fool above the World, a villanelle at the end of Andrea Vitali's http://www.associazioneletarot.it/page.aspx?id=261&lng=ENG).

That it gets the number “0” in the Sermones and is listed first in numerous poems incorporating the titles of the cards does not preclude its being 22nd, however. I would call attention to the comment on the Fool card by Court de Gébelin (http://www.tarock.info/gebelin.htm):

"Quant à cet Atout, nous l'appellons Zero, quoiqu'on le place dans le jeu après le XXI, parce qu'il ne compte point quand il est seul, & qu'il n'a de valeur que celle qu'il donne aux autres, précisément comme notre zero: montrant ainsi que rien n'existe sans sa folie".

(As for this Triumph, we call it Zero, which one places in the game after the XXI, because it counts for nothing when it is alone, and has no value other than what it gives to others, precisely like our zero: showing thus that nothing exists without folly.)

I think he means that it only has value in relation to other cards, just as the value of zero is as a place-holder, for example, in 1000, in which case it is of unending value. This is something to be borne in mind: “0” is ambiguous. Besides denoting "nothing" it is a place-holder in the decimal system, an Arabic innovation (although with Greco-Egyptian precedents) that the Renaissance seems to have been quite impressed with. So a card numbered "0" can be either before the beginning or--standing for the idea of the place-holder, after any given end whatever.

Thus de Gébelin says we put it after XXI. While saying this, however, he nonetheless lists this card first, before the Bateleur. Putting it first in a list does not exclude its also being last! This double meaning of “zero” is something the Renaissance would have been well aware of, since it was an innovation not present in classical Greek and Roman literature and was an elegant solution to the problem of how to write large numbers unambiguously and in a way easy to use in arithmetical calculations. And at the same time it is the lowest card of all, incapable of winning any trick whatsoever!

The Fool is also the highest trump in another sense, that of having the potential for getting the most points, at least in the some versions of the game, from filling in blank spaces in sequences in the scoring of game-points. That is another way it is like “zero”: adding 0 after some other number increases its value tenfold. The Fool’s value is “what it gives to others”, as de Gébelin put it.

There is also another point, made by Alain in his note 5: that is the Fool’s role in the “slam”, in French "chelem", giving the Fool the power to take the last trick, provided that player has taken all the tricks before then. The "chelem" seems not to have been alluded to in print before around 1765, as "vole" and only in relation to the game of Whist, 1765 (here) then Hombre, 1770 (here), and then others. (I owe these references to Lothar Teikemeier, here.) But if the rules in 1585 and earlier (in Ferrara and Mantua) were similar to those of 1637, then the situation would have surely occurred sometime, with experienced players, when one player, or pair of players, could win all the points except in the trick playing the Fool card. In such a case, as an added incentive to achieve such a goal, it would have been just as natural then as later to grant the Fool the last trick and so attain the supreme triumph, even if such a rule was not "written in stone", so to speak.

There is also the fact, described on p. 6 of Dummett and McLeod's Games Played with the Tarot Pack, that in the 18th century some versions of the game in German-speaking regions did in fact make the Fool the 22nd triumph, dispensing with its role as Excuse. We might ask, why 22 instead of 0, i.e. below the Bateleur? Perhaps it was already considered allegorically the highest trump.

I conclude that is a matter of perspective and interpretation whether the Fool is highest or lowest. or not in the sequence at all. Alain’s thesis uses one interpretation out of three, which brings out certain aspects of the cards over others.

3. Application of Alain’s division to the order of the triumphs.

Alain  concludes that his schema of four groups  applies well to the B order, is possible for the C order  and the A order not at all. These proposition needs to be examined in relation to the Pythagorean meanings of the four groups explicated in section 1.

The B order has Justice in the 4th group. If so, on my account of what the 4th group is about, it is Justice  not as a human virtue to be practiced in this world, but as something in the celestial world or even higher, the world of the angels. Given that it occurs between "Angel"--i.e. the Angel of Judgment--and the World, i.e. Heaven, what seems to be implied is God's Justice. Dummett, for example, observed (Game of Tarot, p. 400):

"In orders of type B, ... the World is the highest trump, and Justice is promoted to the second highest position in the sequence, coming immediately below the World and above the Angel, the third highest card. There is clearly here an association of ideas: the Angel proclaims the last Judgment, at which justice will be dispensed".

However this does not mean that the card loses its value as a human virtue; it is still one of the four cardinal virtues of the Church. It is just that the primary meaning in the sequence is now divine justice.

The C order puts Temperance immediately after Death. If we see Death as the sphere of earth and Temperance, with its lady pouring water from one vessel to another, as water, it fits Alain’s cosmographic interpretation of the last 10 trumps perfectly. The card could also be seen as part of the ritual of Holy Communion, a necessary prelude to the soul's rising to heaven. Or perhaps it could be taken as illustrating the soul's move from a physical body to something less material in the air, as Dante had described (Purgatorio XXV, 97-102, translated at http://www.italianstudies.org/comedy/Purgatorio25.htm ):

"...e simigliante poi a la fiammella

che segue il foco là ‘vunque si muta,
segue lo spirto sua forma novella.                                 99

Però che quindi ha poscia sua paruta,
è chiamata ombra; e quindi organa poi
ciascun sentire infino a la veduta..."    

 

(And then, in the same way a flame will follow

After the fire whichever way it moves,

So the new form is following the spirit.

-

"Since it has its visibility from air,

It’s called a shade, and out of air it forms

Organs for all the senses, even sight...")

Again, this meaning of the two jugs, with the soul in between, does not exclude Temperance as a cardinal virtue; it is just that the position in the order brings other meanings to the fore.

All these interpretations fit the theme of ascent to heaven after death. In that case, the Judgment card represents the level before the Empyrian, that of the primum mobile, the “first moved”, which is where angels were historically put.

It is true that there is no early C listing that has the Fool on the 22nd line. As Alain observes, the 1637 Rules indicate 22 triumphs. If so, the Fool must be one of them, but it is not said where in the order it would go, if anywhere. There are only hints, which Alain highlights, from the way in which the order "Monde, Math, Bagat" keeps repeating, as though to say that the Fool was both after the World and before the Math; perhaps in a kind of circular movement. That the person who inspired the printing of the Rules was Louise-Marie de Gonzague-Nevers is also suggestive, because she was not only the daughter of a Duke of Mantua but also the great-great-granddaughter of Isabella d'Este (I owe this point to Lothar Teikemier, on Tarot History Forum); she surely played something like the same tarot as that condemned by the c. 1500 Sermones. These games pass down through the families devoted to them.

There is also La Maison Academique contenant les Jeux, in its editions from 1659 to 1702. The 1702 says on p. 168 that there are 22 triumphs; but on p. 172, there are 21 (for both pages, go here). When listing the 21 by name and number, it lists the Fool as well, but on its own line, unnumbered, after the others (p. 173). The editions of 1659 (http://www.tarock.info/maison_academique.htm)  and 1697 (here, p. 176), both have "21" in the place where the 1702 edition has "22", and no other changes. So the "22" in 1702 is most likely a misprint. But it seems significant that such a misprint could happen. These people seem to have been as confused about 21 vs. 22 as we are! The 1659 and 1697 editions still list the Fool after the others. If it is thought of as after the others, perhaps it doesn't matter, for our purposes, whether it is called a triumph or not.

We must recall also Court de Gebelin’s 1781 account of the Fool—zero, but also specifically after the World card. This description matches the Sermones' account of the card in both regards, suggesting to me that the same tradition regarding the Fool expressed by the Sermones carried over to the places where the C order later obtained. "Zero" has two meanings, one as the number below 1 and the other as a place-holder that can add value forever. It is the same for the Fool.

Finally, is it clear that the type-A tarot is an exception to the division 1 + 4 + 7 + 10? The A order in Bologna had the "four papi", all with the same trick-taking power; that sounds very much like Alain's group of four. In that order the three moral virtues are all in the third group, associated with the number 7; that was the number traditionally associated with the virtues, the 4 cardinal and 3 theological. But putting all three before the Hanged Man makes him 13th; the third group then has too many cards and the fourth group too few, even if the Fool is included. That is why Alain says that the "arithmological" schema does not fit the A order.

But if the Hanged Man is made the beginning of the 4th group, then everything comes out OK. Is that wrong, conceptually?  The theme of the last section is ascent to heaven. In the early type A cards (the “Charles VI”, the Rosenwald Sheet, the Rothschild Sheet), the man is shown clutching bags of coins. That suggests Judas and his “30 pieces of silver” for which he betrayed Christ. But Judas's betrayal is what leads to the crucifixion, and without the crucifixion, there is no admittance to heaven. In that way the card is not only the moment of failed trial of types B and C, but also involved in the ascent to heaven, which is the theme of the last section. The division 1 + 4 + 7 + 10 in that case still holds for the A order.

In fact Dummett's rationale for his three groups cannot specify by itself which group the Hanged Man is in, any more than it can specify which group Death is in. Dummett says (1980 p. 394:

"Ignoring the Virtues, we can say that the sequence of the remaining trumps falls into three distinct segments, an initial one, a middle one and a final one, all variation in order occurring only within these different segments".

But where one group ends and the next begins cannot be determined by this method, if a card is invariably in the same place in the sequence. So Dummett puts Death in the last group in Game of Tarot, 1980 (p. 399), and Il Mondo e L'Angelo, 1993 (p. 174), and in the next-to-last group in "Tarot Triumphant", 1986 (p. 47). The same can be said for the Hanged Man, which immediately precedes Death in all the orders except one, the Sicilian. The tarot was most likely introduced there in 1663-4 by a particular 17th century governor (Dummett 1980, p. 376); it has many innovations not seen in other tarot decks, although perhaps reflecting in some ways, such as the suits, 17th century decks elsewhere in Italy (1980, p. 378). There is no way of knowing why or when the Hermit there came to be immediately before Death, with the Hanged Man coming before the Hermit (1980, p. 375). But this fact can be ignored as most likely irrelevant to the orders of the pre-17th century tarot. Unlike all the early A-order Hanged Men, this one holds no money bags: nor is he hanging from his ankles, the characteristic pose of the traitor. The allegory has been lost.

However it is a rather radical idea, to make Judas part of the road to salvation, perhaps too radical for the time. In which case Alain is right that his way of dividing the sequence does not hold for type A.

4. Is there any historical validity to seeing the sequence in such "arithmological" terms?

Alain in his footnote one focuses on the revival of interest in Plato during the time of the early tarot. That is certainly part of the story, since Plato had a positive attitude toward Pythagoreanism and used it in his dialogues, notably the Timaeus. The interest in Plato might also have stimulated an increased interest in Pythagorean works themselves, many of which were already available throughout in the Middle Ages. In the medieval trivium the subject of arithmetic was seen in the terms of Nichomachus's treatise, as an extension of geometry, including theorems involving series of so-called triangular, square, and pentagonal numbers. Below is an excerpt from Joannes Martinus, Arithmetica, 1526, on "plane" and "solid" numbers (reproduced in S. K. Heninger, Touches of Sweet Harmonies: Pythagorean Cosmology and Renaissance Poetics, 1974, p. 73):

In relation to the tarot, a Pythagorean analysis of the deck was given by Guillaume D’Oncieu in 1584 Savoy, part of a lengthy presentation on each of the ten primary numbers and the groups of things falling under them (see Andrea Vitali at http://www.associazioneletarot.it/page.aspx?id=293). When D’Oncieu speaks of a “quaternary” of 56 cards being made “ternary’ by the addition of the 21 trick-taking cards and the Fool, he has in mind the Pythagorean meanings of these terms, the mundane world including the divine. Also, his reference to the number 27 as a quaternary number is a Pythagorean notion; it is the fourth in the series 1, 3, 9, 27, as enumerated by the Pythagorean teacher Timaeus in Plato’s dialogue of that name (I thank Steve Mangan for this last, at http://forum.tarothistory.com/viewtopic.php?f=11&t=1102&p=17204&hilit=Lambda#p17204).

An explicitly Pythagorean analysis of the ordinary deck was given by Jean Gosselin in 1582, in a book whose title begins La signification de l 'ancien jeu des chartes pythagoriques...(https://play.google.com/books/reader?id=65s9VoMc6Q4C&printsec=frontcover&output=reader&hl=fr&pg=GBS.PT36, pp. 30-31; thanks to Alain for the transcription, http://forum.tarothistory.com/viewtopic.php?f=11&t=1102&start=160#p17344, and Steve Mangan for translating, http://forum.tarothistory.com/viewtopic.php?f=11&t=1102&start=220#p17453). Having presented the rudiments of Pythagorean musical theory, Gosselin introduces the section by saying:

"Après avoir expliqué le plus familièrement qu'il nous a été possible, les proportions des nombres, les consonances et Harmonies qui en proviennent: il convient declarer les secrets qui sont cachés en ce jeu des cartes - lequel a été inventé et mis en usage par quelques hommes savants en Philosophie Pythagorique: Attendu que les Pythagoriques affirmaient qu'il y a de très grands secrets de nature cachés sous les nombres; Et aussi, que la plus grande victoire du jeu des cartes consiste au nombre de trente et un, lequel selon ses parties contient une très excellente Harmonie comme nous le démontrons présentement".

(After having explained, as clearly as has been possible for us, the proportions of numbers and the consonances and Harmonies that arise from them, it is appropriate to declare the secrets hidden in this game of cards - which was invented and put into use by a few men learned in Pythagorean Philosophy: Considering that the Pythagoreans say that there are very great secrets of nature hidden in numbers; And also that the greatest victory in the game of cards consists in the number thirty-one, which by its parts contains a most excellent Harmony, as we demonstrate presently.)

 

What follows is his application of Pythagorean to the cards, and in particular to a game called Trente et Un, i.e. Thirty-One. Here I will summarize his presentation and add in parentheses what I think are the Pythagorean principles involved.

1. Gosselin observes that no card, including the court cards, exceeds in points the number 10, which is 1+2+3+4, in other words, he says, made of four parts that do not exceed four. (This relationship between 4 and 10 is the Pythagorean Tetratkys.)

2. Correspondingly, there are four suits, which themselves correspond to the four elements. (That there are four elements is an assumption of Pythagoreanism and most other ancient philosophies. In general, Pythagoreans looked for relationships and commonalities between the members of different natural groups of the same number, grounded in observation. For example the four seasons and the four elements could each be understood as a different combination of 2 of the 4 qualities dry, wet, warm, and cold. The four winds depend on the four directions and are related to the same qualities. In Christian times, some in the Church had associated the four evangelists with the four "animals" of Ezekiel and Revelation and the four elements, in different ways. Each is Pythagorean but in a different narrative.)

3. Between the French suit of Tiles (floor tiles?) and Earth there is the commonality of supporting heavy things. Between Pikes and Fire there is the commonality of penetrating, and being the most penetrating of its group. Hearts (in our bodies) are in a relationship of dependence on Air. Clover is in a relationship of dependence on much Water. (This follows the Pythagorean way of finding relationships. It is clearly not the only way the correspondences could be drawn.)

4. Regarding the "most excellent harmony" in the game, Gosselin observes that in music the series of diapasons (which we would call octaves) are of notes in perfect "consonance" with one another. A diapason exists when two vibrating strings are in a ratio of 2:1. Thus a series of four diapasons, starting from unity, is 1+2+4+8+16. (This is an application of Pythagorean musical theory, which Gosselin expounded in his previous section, to the "4" of the suits.)

5. The sum of 1+2+4+8+16 is 31, the highest number of points achievable in the game of Trente et Un, Thirty-One. (I cannot find where Pythagoreans attach any significance to the number 31 in virtue of being such a sum. All the same, Gosselin is using a Pythagorean mode of reasoning. In 1618, Kepler would use the Pythagorean idea of musical "consonances" as arithmetical ratios to describe, in his 3rd Law  of planetary motion and elsewhere, what he called, in the title of his most famous book, Harmonice Mundi, the Harmony of the Cosmos. (See online Charles H. Kahn, Pythagoras and the Pythagoreans, a Brief History, p. 168.) 

6. For these reasons, in his view, the game is designed to illustrate Pythagorean philosophy.

My conclusion is that Gosselin has in fact put into a Pythagorean framework, however convincingly, the four suits in the French deck and some important features of the game of Trente et Un, namely the maximum point value of each card at 10 and the maximum number of points possible to be earned at 31. He has not proved, however, that the deck and the game were designed with Pythagorean principles in mind, however, especially not in relation to French suits, which didn't exist until around 1470, long after 4 suits had become standard. Other numbers of suits were in fact tried; but except for the special tarot suit later, they didn't catch on. It might be that 4 makes for a better game than 3 and 5, if there are 2-4 players. Also, the number 4 has significance outside of Pythagoreanism, for example in the four corners of a game board, or indeed the four seasons etc. apart from Pythagorean arithmology. As for the number 31, it was not part of any standard Pythagorean list of significant numbers, in musical theory or otherwise. Any number can be given significance in relation to some series or other. And 10 is an easy number for adding up points at the end of a hand. There may be other considerations as well, depending on the rules of that particular game. 

Pythagorean number theory, found in the biblical commentaries of Augustine and other Church Fathers, is also in the background of the symbolic interpretations of other games, starting with Alfonso X’s Libro de Juegos in 1286 (for which see Los Libros de Acedrex Dados e Tablas: Historical, Artistic and Metaphysical Dimensions of Alfonso X's Book of Games, by Sonja Musser Golladay, 2007, at http://jnsilva.ludicum.org/HJT2k9/AlfonsoX.pdf). It also contributed significantly to medieval and Renaissance architecture (on medieval architecture, see Golladay, just cited; for the Renaissance, see George L. Hersey, Pythagorean Palaces: Magic and Architecture in the Italian Renaissance, Cornell University Press, Ithaca NY, 1976).

Finally, Dummett’s celebrated argument for his “three groups” has no force against Alain’s proposal. Dummett uses as his criterion for grouping that of looking at all the orders once the virtue cards have been removed, and seeing what groups result if one supposes that while there are variations within groups, there are no cards that go from one group to another (Game of Tarot, p. 394). Here  is Dummett again, a little fuller than my previous quote:

"Now the cards which wander most unrestrainedly within the sequence, from one ordering to another, are the three Virtues. If we remove these three cards, and consider the sequence formed by the remaining eighteen trump cards, it becomes very easy to state those features of their arrangement which remain constant in .all the orderings. Ignoring the Virtues, we can say that the sequence of the remaining trumps falls into three distinct segments, an initial one, a middle one and a final one, all variation in order occurring only within these different segments".

-

This criterion of “all variation in order occurring within the segment” does not allow for groups of one. So it begs the question at issue. In every order, excluding the Sicilian, the Bagat is the lowest in the hierarchy of trick-taking cards. That could be reason enough for considering it as a group in itself, since it is invariant in that position, as well as the fact that it isn’t a ruling power of society like the four after it.

Groups of one are not unheard of in the historical literature about the tarot; D’Oncieu, in dividing the 78 cards of the tarot into three groups (one of several ways he divided them), had one group of 56 cards, one of 21, and one of just one card, the Fool.  If the Fool can be a group of one in one way of dividing the cards (even the last mentioned), the Bagatto/Bateleur can surely be a group to itself in another. By itself it is not easily part of the same group as the other four. As for Hurst’s way of incorporating the Fool, making it part of the first group, that is, as we have seen, a matter of interpretation. By the same token, cards at the beginning and end of groups, if they are always at those ends, can, by Dummett's criterion, belong to either group. Thus Death can be in either the last or second- to-last group, and so can the Hanged Man, if the Sicilian order is ignored.

The theory remains speculative, since it is not found explicitly in the historical literature. But neither is Dummett’s or Hurst’s, and as a Pythagorean approach it has as much historical basis as theirs for putting into groups the 22 special cards of the tarot, if all 22 are desired. As far as having explanatory value, which Hurst says it lacks, it seems to me to explain well what the cards have in common with other cards that are near one another, in multiple meanings that vary in emphasis from one order to another. Dummett's account in "Tarot Triumphant", which Hurst endorses with one modification (putting the Fool first), has the difficulties mentioned at the start of this essay.