The Magic Squares of Manuel Moschopoulos - Magic Squares before Moschopoulos

While I have no intention of giving a detailed history of the development of magic squares, a summary of what is known prior to the writing of Moschopoulos would be pertinent. (See [3], [9], [10], and [4] ad loc. for more detailed accounts of the history of magic squares.) \[\begin{array} {| c | c | c |} \hline 4 & 9 & 2 \\ \hline 3 & 5 & 7 \\ \hline 8 & 1 & 6 \\ \hline \end{array}\]

Figure 1

The Lo Shu of the Chinese is a pictorial representation of the magic square shown in Figure 1, which makes its first appearance in the first century A.D. It is undoubtedly much older but there is no direct evidence as to how far back it goes, and claims of 2000 B.C. are probably far too extreme (see [6] and [18]). No earlier magic square is known and Tannery's amazing claim that such squares were known to, or even anticipated by the ancient Greeks is to be rejected outright (see [16] and [5]). Andrewes' ([1], pp. 148ff.) equally absurd comments regarding magic squares in relation to Plato's Republic Book 9, and Timaeus 35 should also be ignored.

The idea of the magic square was transmitted to the Arabs from the Chinese, probably through India, in the eighth century and is discussed by Thabit ibn Qurra (known for his formula for amicable numbers) in the early ninth century. A list of squares of all orders from 3 to 9 are displayed in the Encyclopaedia (the Rasa`il) compiled about 990 by a group of Arabic scholars known as the "brethren of purity" (the Ikhwan al-safa) (see [11], Vol. 1, pp. 660f.). Despite all this, no general constructive methods appeared until slightly later. In 1225, Ahmed al-Buni showed how to construct magic squares using a simple bordering technique, but he may not have discovered the method himself. Biggs ([3], p. 120), referring to a paper by Camman ([5]), suggests that the methods explained by Moschopoulos may have been of Persian origin and be linked to those expounded by al-Buni. Camman indeed claims that the two methods given by Moschopoulos for constructing odd magic squares were known to the Persians, citing an anonymous Persian manuscript (Garrett Collection no. 1057, Princeton University). Even so, this document contains examples and not explicit methods.

It appears that magic squares were introduced to Europe through Spain. Indeed, Abraham ben Meir ibn Ezra (c. 1090-1167), an Hispano-Jewish philosopher and astrologer, translated many Arabic works into Hebrew and had a deep interest in magic squares and numerology in general. He travelled widely throughout Italy and beyond, and may have been the one of the people responsible for the introduction of magic squares into Europe.

Moschopoulos' mathematical writings on the other hand, seem to have had little influence at the time. They were in fact "lost" until the geometer Philippe de la Hire (1640-1718) found them in the Royal Library in Paris and produced a translation.