an aritmetrichal game

DoctorArcanus

This post has very little to do with historical research, yet here is a brief “historical” introduction for those who are not familiar with Aritmetricha:

The so-called Mantegna Tarot is a XV Century set of 50 engravings. The engravings were not intended to be used as “cards”, since a few of the original copies have survived with the original binding as booklets, each image was printed on a separate sheet, and none of the many surviving examples was mounted on thick card-board as playing-cards used to be. The engravings have been widely used as models and copied by many different artists. It is likely that their intended use was to provide such graphical models.
The so-called Mantegna Tarot is known in two versions: an older and better quality E-Series and a slightly later and artistically poorer S-Series.

Reading about the so-called Mantegna Tarot S-Series in “A Descriptive Catalogue of Playing and Other Cards In the British Museum”, William Hughes Willshire, 1876, I found this passage (pag 73):



On the tablet held by the emblematic figure C Aritmetricha, xxv. — 25, are these figures in the following arrangement : —

1.2.34.5.6*
.7.8.9.10
l4o8.5
Duchesne, Passavant, Merlin, and other writers, have considered the lowest row of numerals to imply the date 1485, and the period when the engraving was executed. Kolloff ridicules this opinion as “an highly wonderful conjecture.",
"The counting-slate is intended to represent a so-called magic quadrature, i.e. table with numbers so placed, that in whatever direction they may be added the resulting amount or sum shall be the same. Albert Durer also has represented arithmetic by such a table in his famous piece of the 'Melancholy.' I fare the numbers run from one to sixteen, and the sum is forty-four [sic]. The Italian engraver was not aware evidently of what he had to represent.” (Meyer's "Kunstler-Lexicon" p.594, vol. ii.)



Apparently, Kolloff was not aware of the theory according to which Duerer included the date of the engraving (1514) in the bottom row of his magic square. Anyway, I have decided to try and see if the sequence of symbols in Artitmetricha's table could be interpreted as a magic square. First thing, one must read all the symbols in the table as digits: for the symbols in the last row there is a certain ambiguity. I will not discuss here the other possible interpretations. The digits (?) on the table are 16:
1 2 3 4 5 6 7 8 9 1 0 1 4 0 8 5
Since 16 is a square number, the digits can be arranged in a square:

1 2 3 4
5 6 7 8
9 1 0 1
4 0 8 5

The total sum of the 16 digits also is a square number: 64 (not 44 as stated by Kolloff). 64 is a multiple of 4, which is necessary for the sum of the numbers in a 4x4 magic square.

The 16 numbers can indeed be arranged as a magic square, in which the 4 numbers in all rows, all columns, both diagonals and the inner 2x2 square sum to 16. Actually, many such arrangements are possible. Here is one:

1 3 5 7
5 9 1 1
2 0 6 8
8 4 4 0

A similar arrangement can be found for many (possibly most) sequences of 16 digits whose sum is a multiple of 4. I exclude that this little game I have played has anything to do with the intended meaning of Aritmetricha's table. Anyway, it was fun :)
 

kwaw

I thought a normal magic square includes the numbers from n (say 1) to the number of cells in the square (1 to 9 in a 3x3; 1 to 16 in a 4x4 - they do not always start at one, for instance the kubera kolam is 3x3 square with numbers from 20 to 28) - arithmetic here clearly simply shows the arabic numbers 1 -10 in sequence in the first two lines of the slate - I think the final line being a date is much more likely than any speculation by Kollof that it is a magic square...

While it would not be standard magic square, I suppose there may be non-standard ones in which it is permissible to repeat a digit in different squares (as you have here, using only the single digits 0-9 in a 16 cell square) ? Have you any other examples of such squares being made?

(The Sagrada family square repeats two numbers, 10 and 14, leaving out 12 and 16).
 

DoctorArcanus

I thought a normal magic square includes the numbers from n (say 1) to the number of cells in the square (1 to 9 in a 3x3; 1 to 16 in a 4x4 - they do not always start at one, for instance the kubera kolam is 3x3 square with numbers from 20 to 28) - arithmetic here clearly simply shows the arabic numbers 1 -10 in sequence in the first two lines of the slate - I think the final line being a date is much more likely than any speculation by Kollof that it is a magic square...

I agree.

While it would not be standard magic square, I suppose there may be non-standard ones in which it is permissible to repeat a digit in different squares (as you have here, using only the single digits 0-9 in a 16 cell square) ? Have you any other examples of such squares being made?

(The Sagrada family square repeats two numbers, 10 and 14, leaving out 12 and 16).

No, I do not have other examples.