Math and magic: a block-printed wafq amulet from the beinecke library at Yale.

• Author: Muehlhaeusler, Mark
• Position: Report

The Beinecke Rare Books Reading Room at Yale University houses an important collection of Arabic papyri and paper documents. Among them are at least two printed texts: P.CtYBR inv. 2016 and P.CtYBR inv. 2367. The latter is a small fragment containing but a few words, and is therefore of little value, while the former is a large fragment of an amulet, and is of considerable interest because it contains two magic squares, that is, squares in which numbers are arranged in such a way as to produce a constant sum in all rows and columns (Ar. wafq, pl. awfaq). Although the use of magic squares in amulets is well attested through theoretical works such as al-Buni's Shams al-maarif al-kubra, and through manuscript specimens, only one other example of a block-printed square is known. (1) What is more, the present piece contains a rare occurrence of a magic square of the order 13 -- perhaps the only such occurrence in all amulets published to date.

P. CTYBR INV. 2016

Provenance: purchased from Alan Edouard Samuel (University of Toronto) in New York, 24 February 1992, who in turn bought the item in Cairo in 1965.

Description: block-printed text on light cream paper; in two fragments: one large fragment (ca. 12x12 cm), representing the lower right-hand corner of a large sheet (approx. 24 x 24 cm or larger), and another small piece of that same sheet.

The verso is blank and smooth; it is devoid of strong creases that would offer clues as to how the sheet was once folded. However, the print of the recto is clearly visible on this side, and in some places -- he recto, a small gap runs though the print from the lower margin upwards, through the small magic square and its containing circle. This appears to stem from a fold in the paper rather than from a cleft in the print matrix.

The print was framed by a band of text enclosed within double borders. Set within this frame, there were a number of separate elements, of which the following remain (counterclockwise from bottom left): a large magic square, a small magic square enclosed by a ring of text, and parts of at least two circles containing text (see the image on p. 618). The large magic square was apparently positioned at the center of the lower margin, while the smaller circular elements were arranged along the right-hand side of the sheet. Assuming that the elements were laid out symmetrically, one can conjecture that at least three more circles are now lost on the left-hand side of the talisman. In addition, there may have been another large magic square at the center of the upper margin of the sheet. Here follows a description of each of the extant elements in turn.

A. Border

The text of the border is in naskhi script, fully pointed and with some vocalization. It is framed on each side by a double line, and its base line is oriented toward the center of the sheet. Beginning on the lower left, the text runs towards the right, then wraps around the corner to continue towards the top. I read

[TEXT NOT REPRODUCIBLE IN ASCII]

= Qur'an 48:6:

[TEXT NOT REPRODUCIBLE IN ASCII]

And that He may chastise the hypocrites, men and women alike, and the idolaters men and women alike, and those who think evil thoughts of God; against them shall be the evil turn of fortune. God is wroth with them, and has cursed them, and has prepared for them Gehenna -- an evil homecoming! (2) Note the omission of three words in this verse. The text continues in the border on the right-hand side:

[TEXT NOT REPRODUCIBLE IN ASCII]

= Qur'an 2:7-8:

TEXT NOT REPRODUCIBLE IN ASCII]

To God belong the hosts of the heavens and the earth; God is All-mighty, All-wise. Surely We have sent thee as a witness, good tidings to bear, and warning. B. Large Magic Square

On the lower left-hand side of the fragment are the remains of a magic square. A block of 6 x 8 cells has been preserved in the bottom right-hand corner of that square, as well as an irregular section of some 23 cells. Altogether, this leaves us with just over one third of the original square of 169 cells, as we shall see below. It is fortunate that this particular corner of the square is still extant, because it includes the number one as well as the beginning of the numerical sequence that follows.

Two peculiarities should be noted: the number four is of the Indio-Iranian type (*), and the number five is represented in the shape of an angular and mirrored capital B, similar to the letter m in Epigraphic South Arabian.

It appears that the square was constructed according to the method for odd-order squares described in detail by Jacques Sesiano, (3) and one can therefore reconstruct the original square in its entirety.

Since this method works only with odd-order squares (i.e., squares with n = 2k + l cells on each side), since the number one is always placed beneath the central cell, and since there are six cells to the right of the central column, one can conclude that the square was once thirteen cells in length on either side, and that it contained [13.sup.2] = 169 cells. Indeed, the number 169 appears in its proper place, that is, in the cell just above the central one. The square would originally have appeared as follows (extant cells are in bold italics):

a b c d e f g h i j k 1 m A 79 164 67 152 55 140 43 128 31 116 19 104 7 B 8 80 165 68 153 56 141 44 129 32 117 20 92 C 93 9 81 166 69 154 57 142 45 130 33 105 21 D 22 94 10 82 167 70 155 58 143 46 118 34 106 E 107 23 95 11 83 168 71 156 59 131 47 119 35 F 36 108 24 96 12 84 169 72 144 60 132 48 120 G 121 37 109 25 97 13 85 157 73 145 61 133 49 H 50 122 38 110 26 98 1 86 158 74 146 62 134 I 135 51 123 39 111 14 99 2 87 159 75 147 63 J 64 136 52 124 27 112 15 100 3 88 160 76 148 K 149 65 137 40 125 28 113 16 101 4 89 161 77 L 78 150 53 138 41 126 29 114 17 102 5 90 162 M 163 66 151 54 139 42 127 30 115 18 103 6 91 The initial cell containing the number one is just below the center at position Hg, and all subsequent numbers proceed to the right and diagonally downwards. When the edge of the square is reached, the sequence continues into an imaginary adjacent square, and is then transferred to the corresponding cell in the main square. For example, we see that the sequence leaves the square after 6 at position Ml; it would enter an adjacent square below at position Am, and 7 is therefore entered at that position in the main square. When the sequence meets a cell that is already occupied -- as with 13 at Gf -- the next number is entered two cells below (If, in this example), and the sequence then continues as before. The sequence ends as the last number ([n.sup.2], where n = number of cells on one side, or "order" of the square) is reached and placed in the cell just above the center -- 169 at position Fg. The sum of each row, column, and central diagonal is determined by the order of the square (when filled in with a sequence of positive...