You are here

Magic Squares in the Tenth Century

Jacques Sesiano
Publication Date: 
Number of Pages: 
Sources and Studies in the History of Mathematics and Physical Sciences
[Reviewed by
John J. Watkins
, on

There has been much fanciful that has been written about the history of magic squares. Jacques Sesiano’s remarkably detailed and beautifully researched book should correct many of the common misconceptions about them. The earliest magic square is the famous \(3\times 3\) Luoshu magic square in China, but no higher order magic squares appeared there (or in India) until the 12th century, and these were clearly of Arabic or Persian origin. The game of chess arrived in Persia from India in the 7th century, and appears quickly to have become connected with the construction of magic squares of small orders utilizing moves of chess pieces such as the knight and the bishop. Many centuries later these same methods would be attributed to various European mathematicians, and unfortunately, frequently still are.

The author has wisely chosen two texts, by al-Anṭākī and by al-Būzjānī, that illustrate the state of the study of magic squares in the Islamic civilization in the 10th century. (For example, the well-known \(4\times 4\) magic square that appears in Albrecht Dürer’s etching Melancolia I of 1514 had been thoroughly studied by this time; clearly it did not originate with Dürer.) These two Arabic texts present not only ingenious methods for constructing small-order magic squares, but also powerful methods for constructing a variety of special magic squares of extraordinary beauty and of arbitrarily large size.

The book is in three parts: the Arabic texts themselves, a fully annotated translation of the texts, and a general overview of magic squares with emphasis on 10th century methods of construction (including proofs of general methods). I have listed the three parts of the book in reverse order to encourage readers to explore the book in this reverse order. First, look at the original Arabic texts and absorb the beauty — especially the diagrams of magic squares that almost need no translation to appreciate. Then read the translations that detail marvelous general constructions. For example, there are inductive constructions of “bordered” magic squares where the basic idea is that if you have an \(n\times n\) magic square, you can put a border around this square that creates an \((n+2)\times (n+2)\) magic square. If I were reviewing this today as a paper with the same idea, I would be amazed by its cleverness.

One thing I have always been amused by in the Luoshu magic square is that the four even numbers are at the corners and the five odd numbers are in the middle — that is they are contained within an inscribed square whose vertices lie at the midpoints of the sides of the original square. It had never occurred to me that there might be a general pattern here. But in the 10th century a general construction was developed with this same remarkable property: a magic square of any order so that the inner inscribed square contains only the odd numbers, and the outer triangles on the four corners contain the remaining even numbers. For me this was the single most beautiful 10th century construction, but there were other impressive constructions. One of these was constructing magic squares, say of order 9, such that each of the nine \(3\times 3\) subsquares are themselves magic squares. This same construction was also achieved for magic squares of order 12, both with \(3\times 3\) subsquares and with \(4\times 4\) subsquares.

Although this book is almost entirely concerned with magic squares, it contains a nice bonus: an English version of the Arabic translation by al-An\d{t}ākī of Nicomachos’s Introduction to Arithmetic — this influential 1st century Greek work had also been translated into Arabic by Thābit ibn Qurra in the 9th century.

John J. Watkins is Professor Emeritus of Mathematics at Colorado College.

See the table of contents in the publisher's webpage.