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

Nicomaque : Livre I chapitre XI paragraphes 2 et 3

The prime and incomposite

P. 202 ... hmetic.pdf

Extrait :

Now the first species, the prime and inc1omposite, is found when an odd number admits of no other factor save the one woth the number itself as denominator which is always unity 1; for example,

Thus 31 is the Tenth prime and composite

Now this I had notived already. But what I did not find is the relation between this Tenth prime and composite (31) linked to the harmonical proportions :
1 + 2 +4 +8 +16=31
This os why I agreed with Michael saying :
The sum of these five numbers 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 for that reason
.) ... Biographie

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

In respect of Gosselin's analysis : yes it is a pythagorean analysis, but it hardly amounts to a demonstration of pythagorean origins for the playing card game structure, any more that his analysis of tennis does. It is interesting to me in historical terms as an early Pythagoric interpretation, but I tend to agree with Pratesi, and am not sure what has happened in 'the last 25 years' that demands a reconsideration

I didn't believe I 'll be followed on this revision... (smile)

rEgard to Gosselin, to mee, it appears clearly as a Pythagorean lecture applied to cards.
But i also underlined platonical influence (the order of the Elements).

. ... Biographie

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

BOUGEAREL Alain wrote:Nicomaque : Livre I chapitre XI paragraphes 2 et 3

...You find it i Nicomacus. ll give you the references in a little moment''...

The prime and incomposite

P. 202 ... hmetic.pdf

Extrait :

Now the first species, the prime and incomposite, is found when an odd number admits of no other factor save the one woth the number itself as denominator which is always unity 1; for example,

Thus 31 is the Tenth prime and composite

The relationship I mentioned, between perfect and prime numbers (1+2+4+8+16=31, a prime number, which multiplied by the last term results in a perfect number), is on p.210/211:

"You must set forth the even-times even numbers from unity, advancing
in order in one line, as far as you please: I, 2,4,8, 16,32,64,
128, 256, 5 I 2, 1,024, 2,048, 4,096. . .. Then you must add them together,
one at a time, and each time you make a summation observe
the result to see what it is. If you find that it is a prime, incomposite
number, multiply it by the quantity of the last number added, and
the result will always be a perfect number. If, however, the result is
secondary and composite, do not multiply, but add the next and observe
again what the resulting number is, if it is secondary and composite,
again pass it by and do not multiply; but add the next; but
if it is prime and incomposite, multiply it by the last term added, and
the result will be a perfect number; and so on to infinity. In similar
fashion you will produce all the perfect numbers in succession, overlooking

...When these have been discovered, 6 among the units and 28 in the 6
tens, you must do the same to fashion the next. Again add the next 7
number, 8, and the sum is 15. Observing this, I find that we no longer
have a prime and incomposite number, but in addition to the factor
with denominator like the number itself,' it has also a fifth and a
third, with unlike denominators. Hence I do not multiply it by 8,
but add the next number, 16, and 31 results. As this is a prime, incomposite
number, of necessity it will be multiplied, in accordance
with the general rule of the process, by the last number added, 16, and
the result is 496, in the hundreds; and then comes 8,128 in the thousands,
and so on, as far as it is convenient for one to follow."

The proposition for this is first found in Euclid, and obviously may subsequently found in books on arithmetic (neo-pythagorean or otherwise).
Immature poets imitate; mature poets steal; bad poets deface what they take, and good poets make it into something better, or at least something different.
T. S. Eliot

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

Alain wrote, about my breakdown of what is Pythagorean and what is not in Gosselin's account:
2.2. In general, Pythagoreans look for commonalities between different natural groups of four.

Needs to be detailled : ... 6/document

2.3. His argument for why each of the four suits corresponds to a particular one of the elements,
however, has nothing Pythagorean about it that I can see.

This is the most controversial point. From where did Gosselin take his correspondences?
Did he invent them? Did they seem to him as deriving from common sense? Or as I suggest :

Par contre, l ’explication qu’il offre pour justifier de ces correspondances des Eléments d’avec les Couleurs - la pesanteur ( du plus lourd “lourd”, la Terre , au moins “lourd”, le Feu) respective des Eléments hierarchisés depuis le plus stable (la Terre) jusqu’au plus léger (le Feu) pose problème.
On pressent là une ascension verticale liée à la densité des Eléments.
Ces équivalences peuvent provenir de différentes sources : l’une platonicienne [respectivement le Cube et le Tetraedre] pour la Terre et le Feu, l” autre moyenageuse sinon antique pour le Trèfle [Monde végétal] succédant au Monde minéral : Elément Terre] et étymologique pour le Coeur= Air [anima=souffle].

See my hypothesis : ... xNRk0/view
Yes, my "In general, Pythagoreans look for commonalities between different natural groups of four" needed more detail. I should have lingered a bit longer. Thanks for catching this. What could I have meant? Let me focus on some examples. What is in common between the four directions and the four winds? Well, the four winds come from the four directions. But that is better called a relationship, not something they have in common. Maybe I should have said "Pythagoreans look for relationships between natural groups of four". Another example: the four elements and the four seasons. What they have in common is that both can be constructed from the four qualities. Fire is hot and dry, and so is summer. And so on. In this case, we do find a relationship to something they have in common. But relationship is more fundamental.

Another example, more like Gosselin's in being modern, is Durer's painting of the four apostles as representatives of the four temperaments. John is youthful and red-headed, like spring and the sanguine temperament. Peter is imperturbable as a rock, phlegmatic. Mark, always associated with his solar lion, has fiery eyes, choleric. Paul is the Hellenistically trained philosopher contemptuous of this world, melancholic. This is a Pythagorean way of connecting groups of four

But then it is clear that I was wrong in saying that there is nothing Pythagorean about Gosselin's statements about the relationship of the four suits to the four elements. He clearly finds relationships and commonalities between the two groups of four things:
The tiles (diamonds) that are painted on the cards signify Earth because just as Earth supports all heavy things, so tiles support the heavy things that one puts on them.

The clovers (clubs) which are painted on the cards represent Water, because the clover is a grass that grows in wet environments and is nourished by Water.

The hearts which are painted on the cards represent Air, for our hearts cannot live without air.

The pikes (spades) which are painted on the cards represent Fire, because Fire is the most penetrating of the four elements just as pikes are the most penetrating instruments of war.
What tiles and Earth have in common is that they both support heavy things. He must mean floor tiles.

What clovers and Water have in common is that they both depend on wetness. Or: clover is related to water in needing it for nourishment.

How hearts are related to air is that hearts need air. He has perhaps observed, or heard from others, that without air the heart stops beating.

How pikes are related to fire is that both are the most penetrating of their respective group. They have that in common.

It may well be true that this set of relationships and commonalities has no precedent among Pythagoreans. But that makes no difference. When Kepler, for example, investigated commonalities between a certain arithmetical series (I forget what, maybe 1, 2, 4, 8...) and the distances of the orbits of the planets from the sun, he was doing something not done by previous Pythagoreans (Kepler did think of himself as a Pythagorean and he was investigating a Pythagorean commonality). They had not considered the Sun as at the center of a series of concentric circles representing planetary orbits around it. When Durer likened apostles to temperaments, this was not something done by the ancient Pythagoreans. Likewise the Pythagoreans of ancient and medieval times had not considered playing cards in Pythagorean terms, probably because they didn't use them. But it is still a Pythagorean way of proceeding, in that it looks for relationships between two groups of four.

So you are right, Alain, and I am wrong on these two points. I will have to rewrite what I wrote.

However I do not think that Gosselin is thinking in terms of the Platonic solids. It is possible, as you say, that he is thinking of a tetrahedron when he says that both fire and Pikes are the most penetrating. He could be imagining pushing the flat side of a tetrahedron away from himself so as to pierce the skin of his opponent, like a "poignard" (blade).

But how does an octohedron suggest a heart that needs air? You say in addition that air = breath = soul = heart. I understand how breath was soul for the Stoics. And if the soul is between spirit and body, so is the heart between the head and the lower organs of appetite. Perhaps that is what you have in mind. It is possible such relationships in philosophy were in his mind. But there is no necessity, and no icosohedrons enter in.

With water, all I can find you saying is that Poseidon reigns over the water and has a trident. The relationship of trident to clover is rather distant. Clover suggests water without any such reference.

With tiles, you point to a relationship between carreau and the Latin for quadrilateral, quadribis or something.. It seems to me that there is an even more direct relationship between carreau and the similar-sounding carré, meaning square. A square might possibly be suggested by a cube, but not necessarily. You also say that the cube is the most stable solid, like the earth. Yes, but Gosselin's relationship is that of heaviness: not that they are both heavy, but both support heavy things.

Your appeal to the Platonic solids is more mathematical and to that extent more Pythagorean. But Gosselin's account of the relationship of suits to elements is much more simple-minded and just what he says, no more and no less. What I didn't realize is that that it is a Pythagorean way of thinking, even though it doesn't re-use a typical Pythogorean set of comparisons.

About Pratesi: It is not a question of agreeing with Alain and me or with him, or something in between. He was interested in the questionw of (a) what Gosselin had to say about the origin of the four suits and the card game of Trente et Un, and (b) is any of it useful? We are interested in a different question, namely, to what extent does Gosselin give a Pythagorean explanation of the four suits and the game of Trente et Un? Is it just using the word "Pythagorean" or is there more to it, related to Pythagoreanism in other contexts? I can't see that Pratesi was interested in that question.

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

About Plato and Pythagoreans

Speusippus, the son of Plato's sister Potone, and head of the Academy before Xenocrates, compiled a polished little book from the Pythagorean writings which were particularly valued at any time, and especially from the writings of Philolaus; he entitled the book On Pythagorean Numbers. In the first half of the book, he
elegantly expounds linear numbers, polygonal numbers and all sorts of plane numbers, solid numbers and the five figures which are assigned to the elements of the universe, discussing both their individual attributes and their shared features, and their proportionality and reciprocity.

(Cited as so in Theologumena)

About the Four Elements an their relation one with each other , the analogy given in the THeoologumena is :

And there are evidently also four elements (fire, air, water and earth) and their powers (heat, cold, wetness and dryness), which are disposed in things according to the nature of the tetrad.

See update on ; ... sp=sharing

About the Corps platoniciens (4 out of 5) associated with the 4 Elements :
Moreover, the sequence of the first five triangular numbers generates 55 [3, 6, 10, 15 and 21 make 55) and again, the sequence of the first five squares generates 55 [ 1,4,9,16 and 25 make 55); and according to Plato the universe is generated out of triangle and square,
For he constructs three figures out of equilateral triangles—pyramid, octahedron and icosahedron, which are the figures respectively of fire, air and water—and the cube, the figure of earth, out of squares.

Personnal comment out of context in this specific study :

Interesting to note that :
55 is also the sommation of the numbers in the Decade
1+2+3+4+5+6+7+8+9+10 = 55 ... Biographie

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

About the terminology ....

Perfect numbers:
6, 28, 496, 8128

Mersenne numbers:
The first four Mersenne primes M2 = 3, M3 = 7, M5 = 31 and M7 = 127 were known in antiquity. The fifth, M13 = 8191, was discovered anonymously before 1461; the next two (M17 and M19) were found by Pietro Cataldi in 1588. After nearly two centuries, M31 was verified to be prime by Leonhard Euler in 1772.

Well, if Gosselin had a little bit of a playing card researcher, he would have looked, what other suit signs were used in other countries ...

cups ... naturally filled with something liquid. Element water looks probable, usually translated as hearts in French suits. But Gosselin has no water for hearts.
swords ... metal, worked by fire. Fire? Sometimes translated as pique, but also as trefle.

Gosselin's interest in cards is very "national", limited to French conditions.


Tarot was known at the French court 1582. Gosselin doesn't mention it. Tarot (we don't know the Tarot rules of 1582) has a point system usually, either ...

4 points for Kings
3 points for Queens
2 points for Knights
1 point for Pages
... or ...
5 points for Kings
4 points for Queens
3 points for Knights
2 point for Pages

... the difference likely depending on the counting system, that tricks also count points. As already stated, we don't know, how Tarot was played in 1582 (especially we don't know, how it was played at the French court).

Gosselin would have had a splendid argument for his Pythagorean suspicions, if he would have connected ...

4 points for Kings
3 points for Queens
2 points for Knights
1 point for Pages

... to the Tetraktys. What shall we conclude from the condition, that he didn't take this in his calculations?

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

I have rewritten what I wrote about Gosselin, as follows. I added a part on Kepler's use of the Pythagoreans' idea that musical "consonances" are mathematical ratios, to describe the "music of the spheres". His formula--a little more complicated than Gosselin's, but not much more--became the basis of Newton's Law of Gravitation. I have highlighted my additions in bold.
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. This point value is particular to Trente et Un, i.e. Thirty-One.)

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, but Gosselin is using a Pythagorean style of reasoning. In 1618, Kepler would use the Pythagorean idea of musical "consonances" as arithmetical ratios to describe what he called, in the title of his most famous book, Harmonice Mundi, the Harmony of the Cosmos. See Charles H. Kahn, Pythagoras and the Pythagoreans, a Brief History, p. 168, in Google Books, ... an&f=false.)

6. For these reasons, in his view, the game is designed to illustrate Pythagorean philosophy. (This has not been proved. However the number 31 can indeed be arrived at from an application of Pythagorean principles, and in that sense the game can be used to illustrate Pythagoreanism as he has done.)
Alain might find Kepler worth studying for another reason: he tried to use the five Platonic solids nested one in another, as described by Euclid but in a Pythagorean application, to explain the orbits of the planets. That theory did not survive scrutiny, but the idea, inspired by Pythagorean musical and cosmological theory, that the planets' motions stand in relation to one another in a fixed ratio, did, as his Third Law. According to Kepler on the page linked to above, "the heavenly motions are nothing but a perpetual song for several voices, perceived by the intellect, not by the ear." The Third Law then became the basis for Newton's Law of Universal Gravitation.

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