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

#491
https://en.wikipedia.org/wiki/Arithmetic
Arithmetic or arithmetics (from the Greek ἀριθμός arithmos, "number") is the oldest[1] and most elementary branch of mathematics. It consists of the study of numbers, especially the properties of the traditional operations between them—addition, subtraction, multiplication and division.
Arithmetic is NOT Pythagorean. Pythagoras might have done some arithmetic as a long row of persons before.

There were people before him, who already knew, that 1+1=2.

Old China also had a sort of musicology ...
https://en.wikipedia.org/wiki/Chinese_musicology
In my youth I'd a book of Marcel Granet, a French sinologue. He wrote in extensio about it. I wasn't very interested.
Huck
http://trionfi.com

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

#492
The argument is that this game is not based on simple Arithmetic but on Boethius treaty called Arithmetic or should I say Arithmology
http://www.history.upenn.edu/people/faculty/ann-e-moyer


NB
Boethius Arithmetic is a pythagorean arithmetic related to figurate numbers and harmonical proportions.
https://archive.org/stream/aniciimanlii ... 4/mode/2up
http://www.sgdl-auteurs.org/alain-bouge ... Biographie

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

#493
As far I got it, Boethius wrote about the Quadrivium ...

https://en.wikipedia.org/wiki/Quadrivium
The quadrivium consisted of arithmetic, geometry, music and astronomy. These followed the preparatory work of the trivium made up of grammar, logic and rhetoric. In turn, the quadrivium was considered preparatory work for the serious study of philosophy (sometimes called the "liberal art par excellence") and theology.

Origins

These four studies compose the secondary part of the curriculum outlined by Plato in The Republic, and are described in the seventh book of that work (in the order Arithmetic, Geometry, Astronomy, Music.) The quadrivium is implicit in early Pythagorean writings and in the De nuptiis of Martianus Capella, although the term "quadrivium" was not used until Boethius early in the sixth century. As Proclus wrote:

The Pythagoreans considered all mathematical science to be divided into four parts: one half they marked off as concerned with quantity, the other half with magnitude; and each of these they posited as twofold. A quantity can be considered in regard to its character by itself or in its relation to another quantity, magnitudes as either stationary or in motion. Arithmetic, then, studies quantities as such, music the relations between quantities, geometry magnitude at rest, spherics [astronomy] magnitude inherently moving.
Huck
http://trionfi.com

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

#494
Yes, Boethius was the "inventor" of the Quatrivium

" Boèce a forgé le terme de « quadrivium », ou quadruple voie vers la connaissance, comprenant l'arithmétique, la géométrie, la musique et l'astronomie. Ce terme routier d'origine romaine trouve son analogue dans le « trivium », comprenant la grammaire, la dialectique et la rhétorique"
https://fr.wikipedia.org/wiki/Bo%C3%A8ce

Boethius Arthmetic and Music is a kind of loose translation of Nicomacus of Gerasa Arithmetic and now lost Music "with addiionnal material from Ptolemy's Harmonics and other works".'
(Anne E MOYER, The Philosophers’ Game: Rithmomachia in Medieval and Renaissance Europe. Ann Arbor: The University of Michigan, Press, 2001.)

About the Quadrivium and the " philosopher game" in late Middle Ages and early Renaissance :

I. Introdution pp 3,4,5,6
II. From Cathedral Schools to Unversity pp.18 and 52, 53

Many foot-notes!

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

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

#495
Well, I am convinced that Alain is right. Pythagoreanism is a tradition of ancient thought that among other things saw numbers as the principles upon which the universe was based. Plato drew on Pythagoras, and his Timaeus utilizes Pythagorean principles. The book Theologumena Arithmeticae, Arithmetical Theology, is the major ancient compendium of such materials. It uses the results of works like Nichomachus's Introduction to Arithmetic The last Pythagorean in natural science was probably Kepler, but the idea of number as the organizing principle of the universe is basic to modern science. It is in contrast to Aristotelian science, which was biologically based and saw the universe as an organism. Boethian arithmetic is based on the Pythagorean school's work, particular Nichomachus. For the Pythagoreans, it was meant as a basis for philosophical speculation, natural science, or musical theory. Boethius has some of that, at least musical theory. Pythagorean arithmetic consisted of theorems, like plane geometry. These theorems (of which 2+2 is not one) are not arithmology in themselves, just the basis for arithmology. They are what min modern mathematics would be called number theory. The philosopher's game is seems Pythagorean, in that it teaches arithmetic in the context of harmony and I'm not sure what else. I haven't studied it in detail.

Thanks for explaining the mystery of 4x55 = 10x22, Alain. Seeing it as basic to the tarot is a Pythagorean way of thinking, It is the kind of thing D'Oncieu might have said. Whether anyone actually ever thought in such terms about the tarot is unknown.

Often Pythagorean thinking goes too far, at least as an explanation for something, as opposed to a way of looking at something better explained in other terms. This is one of those times, I think.

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

#496
An Appendix to viewtopic.php?f=11&t=1102&start=460#p18457 and then viewtopic.php?f=11&t=1102&start=460#p18464

I want to pursue further the lists of the various orders that I constructed, to see what is suggested regarding "early" vs. "late" cards. This is off-topic, but I am putting it here because it follows a train of thought I began here. It has nothing to do with the Pythagoreans, other than keeping the 1 + 4 + 7 + 10 format. Even then, it doesn't really matter here where the Fool goes.

Here again are the various orders, by section as defined by 1 + 4 + 7 + 10. For Alciati, I put translations of his Latin titles:

A, B, C (1 card): Bagat

A (4 cards):
Rosenwald: Popess, Empress, Emperor, Pope
Bologna: 4 papi.

B (4 cards):
Sermones: Empress, Emperor, Popess, Pope;
Metropolitan: Popess, Empress, Emperor, Pope

C order (4 cards)
Geoffroy, Noblet, etc.: Popess, Empress, Emperor, Pope
Alciati (same order, different titles): Priestess (or Priest's Wife), Queen, King, Priest
Maison Academique: Popess, Emperor, Empress, Pope

A order (7 cards). I put the confirmed ones in bold:
Bologna: Love, Chariot, Temperance, Fortitude, Justice, Wheel, Old Man
Charles VI, Rosenwald (for which last 2 unclear): Love, Temperance, Fortitude, Justice, Chariot, Wheel, Old Man
Orfeo, Catania: Love, Temperance, Fortitude, Justice, Wheel, Chariot, Old Man
Colonna: Love, [Temperance, Fortitude, Justice, Old Man, ?] Chariot, Wheel

B order (7 cards):
Sermones: Temperance, Love, Chariot, Fortitude, Wheel, Old Man, Hanged Man
Garzoni/Bertoni, Met., Pomeran, Rouen, Anonymous (appropriati), Imperiali, Citolini: Temperance, Chariot, Love, Fortitude, Wheel, Old Man, Traitor/Hanged Man.

C order (7 cards):
Susio, Piscina, Vieville: Love, Justice, Chariot, Fortitude, Wheel, Old Man, Traitor/Hanged Man
Alciati: Justice, Fortitude, Love, Chariot, Wheel, Old Man, Cross (Crux)
Geoffroy, Noblet, etc.: Love, Chariot, Justice, Old Man, Wheel, Fortitude, Hanged Man

A order (10 cards) Traitor, Death, Devil, Fire, Star, Moon, Sun, World, Angel, Fool

B order (10 cards): Death, Devil, Fire, Star, Moon, Sun, World, Justice, Angel, Fool

C order (10 cards): Death, Temperance (Fama for Alciati), Devil (Daemon in Alciati), Lightning (Maison-Dieu in Noblet etc), Star, Moon, Sun (Phoebus for Alciati), World, Angel, Fool

It seems to me that the degree of variability in the three sections tells us something. The middle section has by far the most variability. The virtues are in no particular order; in B and C one is even outside the section. The other triumphs generally follow a recognizable order: Love, Chariot, Wheel, Old Man, Death. This fits a Petrarch-like progression.

Cards 2-5 basically have only 1 variation, 1 card, if we ignore the Maison Academique, which would make 2.

Cards 15-19 are always in the same order, exactly, with no variation. Finally, the Angel and World are always at the end, if not always in the same order, usually the last 2 but interrupted by Justice in B. And we are stipulating the Fool at the end, even if not confirmed and in fact in literary sources mostly disconfirmed. That is not important for the present post.

Dummett hypothesized that differences in the order were due to local variations that developed because players in one locality were isolated from players in other localities. They are like random mutations, apparently, but influenced by individuals' preferences and perceptions of what should follow what. If so, the more variation from region to region, the earlier the cards would have spread to all those regions. So 15-19 would be late cards and the virtues early ones. Probably also Love/Chariot/Wheel/Old Man/Death/ is early, too, even if mostly in the same order, because of the variations that happen around them. That quasi-Petrarchan sequence makes easy sense, unlike the order of the virtues. I would expect the other two, Angel and World, to be early, too, because of their variability relative to each other.

Of the Imperial and Spiritual powers, Popess is always before Pope, and Pope always 5th, whereas Empress and Emperor occasionally switch the order relative to each other and the other cards. So Empress and Emperor, being more variable in relation to each other, are probably earlier than Pope and Popess. Hanged Man is late because it is invariably before Death.

It is difficult to say anything about the Bagat and the Fool, because their positions are unique. Because they are always where they are, it is tempting to say they are late; on the other hand, it is easy to remember which card is first and which card is not in the order at all, the first by its name "bagatella", meaning "thing of little value" as well as "illusionist"; the second by its role in the game, if it had that role from the first.

A surprising result is that the Old Man is almost always just in front of the Hanged Man. Perhaps it is late, too--although its position might be the result of an "late" decision to make Time refer to time before death, rather than cosmic time, as it would have early on. Or else Time early on was not the Old Man, but instead the Sun (the opening image in Petrarch's poem, complete with chariot), stuck in between World and Angel, later refigured as a Celestial at the same time the Old Man was put in. That would require a less drastic change in the sequence.

So when would this change have occurred, from either an Old Man or the Sun as cosmic time, in the sequence between Fame and Eternity, to human time, put before Death? Perhaps the illuminations of Petrarch's poem, in different places and times, can give us some clues.

From Lombardy, there is an illumination of what I think is the Triumph of Time. I think it is Apollo in his chariot, just as in the opening lines of Petrarch's poem, making the horses go fast. It is by an illuminator active from 1418 to 1459. Trapp dates this one to c. 1440. For more see viewtopic.php?f=11&t=858&p=13655&hilit= ... rch#p13655:
Image


For Florence, Susan Cohen in "Early Renaissance personifications of time and changing concepts of temporality" (Renaissance studies 14:3 (2000), pp. 301-328) makes the point that the pre-1450 illustrations there of Petrarch's Trionfi showed him holding a sphere rather than an hourglass: in one, it is an armillary sphere, in another it is divided into three parts labeled "Europe - Africa - Asia", in another it is just a sphere.

Armillary sphere (Florence, Biblioteca Lauriziana, Strozzi 461, fol. 44r):
Image


Tripartite sphere (1442, Biblioteca Lauriziana, Pal. 72, fol. 86r):


Both are attributed to Apollonio di Giovanni. Since hourglass versions started appearing after 1450, that is a reasonable time for the change to have occurred.

Even then, the sphere persisted in a few cases. There is this one, probably later than the other two, attributed to Zanobi Strozzi, about which I have no further information: (https://commons.wikimedia.org/wiki/Cate ... _tempo.JPG). Wikipedia says he was engaged in illumination production, with Pesellino and Domenico di Michelino, "between 1446 and 1454", and another stint, with Cheriaco. These periods were for choir books, but his collaborators are known Trionfi illustrators..

When Time started being represented by an old man with an hourglass, the hourglass was a natural symbol of the brevity of life (as Cohen says), hence no longer (I think) cosmic time, but human time. The image would then fit better before Death in the tarot sequence rather than after (I cannot speculate on which came first, the tarot card or the Trionfi illumination with the hourglass). Then the celestials would serve to represent cosmic time.

In this vein, there is a late 15th century Florentine engraving where Time holds an armillary sphere and there are a sun and a moon overhead
.

It remains true that there is no extant tarot card for Time as I have imagined him, as the Sun/Apollo or holding a sphere. It is only a reasonable hypothesis. What I have, from the extant versions, is 11 "early" cards (Empress, Emperor, Love, Temperance, Fortitude, Justice, Chariot, Wheel, Death, World, Angel; 7 "late" ones (Popess, Pope, Hanged Man, Devil, Tower, Star, Moon, Sun), and 4 where it is hard to say (Fool, Bagat, Old Man, Sun), but probably the first two are late and at least one of the others. Early Hope, Faith, Charity seems almost required, making 14 "early" cards.

If Prudence is a separate card (as opposed to being put in place of, or amalgamated with, something else, as happens in a few literary sources, e.g. in place of Temperance by Piscina, or Hanged Man by Imperiali), and either Old Man or Sun is there as Time, that would be 16.

Old Man shows up as "late" in my lists, because he is mostly in one place in the order; but that "late" status may be due to being moved from the penultimate position originally, as Petrarchan "Time". Alternatively, the Sun might have had that function, and then been moved, so that it shows up as "late".

This result is much the same as what I have arrived at from others means. It is also like Phaeded's, if my view of Time and Prudence is rejected. We only differ by 2 cards (i.e. Fame and Prudence as separate cards early on, and Time, whatever it is, which he puts late). We have of course discussed these differences at great length with no resolution.

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

#497
Mikeh


"Here again are the various orders, by section as defined by 1 + 4 + 7 + 10. For Alciati, I put translations of his Latin titles:

A, B, C (1 card): Bagat

A (4 cards):
Rosenwald: Popess, Empress, Emperor, Pope
Bologna: 4 papi.

B (4 cards):
Sermones: Empress, Emperor, Popess, Pope;
Metropolitan: Popess, Empress, Emperor, Pope

C order (4 cards)
Geoffroy, Noblet, etc.: Popess, Empress, Emperor, Pope
Alciati (same order, different titles): Priestess (or Priest's Wife), Queen, King, Priest
Maison Academique: Popess, Emperor, Empress, Pope

A order (7 cards). I put the confirmed ones in bold:
Bologna: Love, Chariot, Temperance, Fortitude, Justice, Wheel, Old Man
Charles VI, Rosenwald (for which last 2 unclear): Love, Temperance, Fortitude, Justice, Chariot, Wheel, Old Man
Orfeo, Catania: Love, Temperance, Fortitude, Justice, Wheel, Chariot, Old Man
Colonna: Love, [Temperance, Fortitude, Justice, Old Man, ?] Chariot, Wheel

B order (7 cards):
Sermones: Temperance, Love, Chariot, Fortitude, Wheel, Old Man, Hanged Man
Garzoni/Bertoni, Met., Pomeran, Rouen, Anonymous (appropriati), Imperiali, Citolini: Temperance, Chariot, Love, Fortitude, Wheel, Old Man, Traitor/Hanged Man.

C order (7 cards):
Susio, Piscina, Vieville: Love, Justice, Chariot, Fortitude, Wheel, Old Man, Traitor/Hanged Man
Alciati: Justice, Fortitude, Love, Chariot, Wheel, Old Man, Cross (Crux)
Geoffroy, Noblet, etc.: Love, Chariot, Justice, Old Man, Wheel, Fortitude, Hanged Man

A order (10 cards) Traitor, Death, Devil, Fire, Star, Moon, Sun, World, Angel, Fool

B order (10 cards): Death, Devil, Fire, Star, Moon, Sun, World, Justice, Angel, Fool

C order (10 cards): Death, Temperance (Fama for Alciati), Devil (Daemon in Alciati), Lightning (Maison-Dieu in Noblet etc), Star, Moon, Sun (Phoebus for Alciati), World, Angel, Fool"

Alain

Thank you for this very clear presentation of the 3 Orders disposed in the Pentagonal expansion in the 4 successives sections of Number 22 : 1+4+7+10 (Nicomacus, Iamblicus, Theon de Smyrna)- that is in the sequence 1,5,12,22 (Boethius).
You also wrote :
"It has nothing to do with the Pythagoreans, other than keeping the 1 + 4 + 7 + 10 format"
I would not say it is not Pythagorean... (smile).

Dummett's hypothesis about the early or late dating from the degree of variability is of interest for sure.

Impressive post.
http://www.sgdl-auteurs.org/alain-bouge ... Biographie

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

#498
Image

http://www.hermes-press.com/philosophy_emboldening0.htm
Gregor Reisch's Arithmetica, 1503. A symbolic image depicting Boethius and Pythagoras in a mathematical competition. Pythagoras uses an abacus, while Boethius uses numerals from India. Boethius looks very proud, he is ready while the poor Pythagoras still tries to find the solution.

Philosophy Presenting the Seven Liberal Arts to Boethius; Coëtivy Master (Henri de Vulcop?), French, active about 1450 - 1485

I read from a report, that on a picture Boethius was paired with Arithmetic, Pythagoras with Music, and the 5 other Artes Liberalis with other persons.
It might be this one, but the name Boethius is not readable:

Image

Large picture: https://de.wikipedia.org/wiki/Gregor_Re ... 500ppi.jpg
Huck
http://trionfi.com

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