Background reading
Magic squares
A wafq is a numerical grid in which each row, each column and both diagonals produce the same sum. That is the complete definition. The rest concerns the uses to which such a grid was put.
The simplest one
The grid above is the three-by-three arrangement, the smallest possible. Each row yields a total of fifteen. Each column does the same. Both diagonals also come to fifteen. If rotations and reflections are disregarded, only one such layout exists, and that singularity helps explain why it was regarded as meaningful rather than random.
In the Islamic world it carries a name: budūḥ. The label derives from the grid's own corners. When the four corner entries are read as abjad letters, two corresponds to bāʾ, four to dāl, six to wāw and eight to ḥāʾ. Those letters, b-d-w-ḥ, combine to form a term that is not a real word: budūḥ. For a thousand years it appeared on entrances, on amulets for childbirth, and on correspondence to ensure its swift arrival, across the breadth of the Islamic world. Most of those who wrote it down were unaware of its origin in a number square.
How a square becomes a talisman
The transition from arithmetic to talismanic science rests on a single principle. If a divine Name carries a numerical value, and a square can be devised whose constant matches that value, then the square is understood as a written rendering of the Name, organised so that it is evenly balanced in all directions.
The rationale is worth noting. The assertion is not that the numbers themselves possess power. Rather, it is that the Name holds the power, that the square offers a two-dimensional means of writing the Name as opposed to a single line of text, and that the symmetry of the grid reflects a symmetry in the thing it stands for. Whether one accepts that logic or not, it is a reasoned position rather than blind superstition, and it explains why so much care went into the method of construction. A grid assembled incorrectly was not merely a less effective talisman. It simply was not the Name.
The larger squares
Moving beyond the three-by-three makes construction more demanding and the range of categories grows accordingly. Squares with an odd order, squares with a singly even order, and squares with a doubly even order all call for a distinct approach. Al-Būnī and those who followed him devised methods for each category, together with squares built to a chosen constant rather than from the standard sequence of integers, a task that is genuinely challenging.
The four-by-four square in our article header is an example. Every row, column and diagonal totals thirty-four. The same happens, incidentally, with the four corners, the four central cells, and each of the four two-by-two subsections. That level of internal structure is not accidental, and identifying it required real mathematical skill.
This is why the diagrams matter. A translation of al-Būnī that refers to the squares in words alone, without showing them, has removed the very object the book addresses. Our complete edition presents them as diagrams, and the digital version carries them at print quality so that they can be printed correctly.
Where they came from
They did not originate with al-Būnī, nor in the Islamic world at all. The three-by-three square turns up in Chinese texts more than a millennium earlier as the Lo Shu, and it entered the Arabic-speaking world through translation, almost certainly by way of India and Persia. In the tenth century, Arab mathematicians examined the general problem of construction as pure mathematics, with no talismanic dimension attached.
The talismanic use developed later, building on a body of mathematics that was already established. This point matters, because it shows that the squares in Shams al-Maʿārif stand at a meeting point between two distinct fields: rigorous combinatorial mathematics on the one hand, and practical religious observance on the other. As one reads the book, the two can be seen coming together.