Is this a fair assessment?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.) Correspondingly, he goes on, 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; Gosselin's order of presentation, a somewhat unusual one, is that of Plato's Timaeus, 31b and 32b: fire, earth, water, air. In general, Pythagoreans look for commonalities between different natural groups of four. His argument for why each of the four suits should correspond to a particular one of the elements, however, has nothing Pythagorean about it that I can see.). Regarding the "most excellent harmony", 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.) 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 in virtue of being such a sum.) 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.)
I have been trying to formulate what is and is not Pythagorean about Gosselin's analysis of the game of Trente et Un. Here is what I have come up with, a summary of his discussion of the game, with my parenthetical comments about what is Pythagorean or not about it.