[The following is the text of a note sent on 16 Feb 2007 to Prof. Peter Ullrich
at the University of Koblenz, with CC to Prof. Len Berggren at Simon Fraser U.]

Dear Peter,

Google introduced me to your nice paper about the history of latin squares
in Metrika (2002). So I thought you would be interested in something that
I stumbled on last night, as I was preparing part of The Art of Computer
Programming (vol 4): Orthogonal latin squares were implicitly present
already in amulets used in the Maghreb in the 14th century!

You can find the story by looking at an article by W. H. Ahrens in
Der Islam 7 (1917), especially pages 228--238, in conjunction with
a book by E. Doutt\'e that he cites: Magie et Religion dans l'Afrique
du Nord (Algiers: 1909). Doutt\'e discusses several amulets that appeared
in a well-known work of Muhammad Ibn al-Hajj called Kitab Shumus al-Anwar
(Cairo, 1322); al-Hajj died in 1336. (His work is reminiscent of
cabbalistic Hebrew traditions; I know little about him except from a
few Internet sources, which stress his somewhat rigid fundamentalism.)

Ahrens corrects errors of transcription and makes a convincing case for
his corrections. Indeed, such scribal errors were quite common when a
scribe of those days was copying mathematical data. Ahren's reconstruction
of several 4x4 squares (page 234 of his article) is
         d+0 c+1 b+2 a+3
         a+2 b+3 c+0 d+1
         c+3 d+2 a+1 b+0
         b+1 a+0 d+3 c+2
where a, b, c, d are constants chosen so that you get four words of
religious significance in the top row or in the corners.

This is of course a pair of orthogonal latin squares of order 4 ---
equivalent to the solution to Ozanam's problem (which you illustrated
so nicely in your paper)! (Incidentally, one must of course correct
four little errors made by Ozanam's artist, who confused some of
the suits on the cards.) This pattern occurs in the north African
amulet discussed on page 194 of Doutt\'e, and also in some 17th
century amulets found in Vienna that originated in Baghdad.
(Ibn al-Hajj spent time in Baghdad, visited magicians who apparently
showed him some astonished tricks. Genies out of the bottle!)

Ahrens should have realized this connection, because he had written
about those greco-latin squares on page 65 of his Mathematisches
Unterhaltungen und Spiele, volume 2. But he didn't.

He also corrected a 14th-century example of 5x5 orthogonal latin
squares, which are found on Doutt'\e's page 213. This pattern is
        b+2 e+4 c+1 a+3 d+0
        a+1 d+3 b+0 e+2 c+4
        e+0 c+2 a+4 d+1 b+3
        d+4 b+1 e+3 c+0 a+2
        c+3 a+0 d+2 b+4 c+1
and it also comes from Ibn al-Hajj.

Doutt\'e's book contains another example on page 247, not cited by
Ahrens---possibly because it has more than the usual amount of
scribal errors, and he didn't see how to fix them.
But really the restoration is not difficult, once one knows
a bit about latin squares. The diagram in Ibn al-Hajj's (corrupt)
text is
       30  70   1   3
        3   4  92  81
        5   3  68  38
       69  23   6   2
and I am quite certain that the original amulet really said
       30  70   1   3
        0   4  29  71
        5   3  68  38
       69  27   6   2.
(The two 3's are actually written differently --- in the top line
it's a letter j, but in the third line it's a Hindu-Arabic numeral.)

I have no time to pursue this, but someone clearly should take a
closer look, because there no doubt are further examples to
be found. In particular, Ibn al-Hajj has examples of pure latin
squares (4x4, 5x5, 6x6, 7x7, 8x8); Doutt\'e and Ahrens consider
some of them to be "poor-man's magic squares" by giving numeric
interpretations to the letters, but I think they are mistaken ---
the letters were just placeholders in a pattern, not numbers.
Some of these latin squares are "pandiagonal" ... they solve
the n queens problem for several sets of queens simultaneously.

Anyway I thank you for reading this long note, and I hope
you enjoy this unexpected blast from the past!

Cordially, Don Knuth