MOSCHOPOULOS, MANUEL

(b. c. 1265, Crete [?]; d. after 1430),

mathematics.

Moschopoulos, a Byzantine scholar, was a nephew of Nicephoros Moschopoulos, Metropolitan of Crete. He was a student of Maximus Planudes who wrote to the uncle an enthusiastic letter about the young man. Some letters of his dated between 1295 and 1316 have been preserved.

Manuel Moschopoulos was mainly a philologist and a grammarian. He left numerous work on classical authors: editions of classical texts, a dictionary of Attic words, a paraphrase of the first two books of the Iliad, a paraphrase of Works and Days of Hesiod, grammatical exercises, and a Discourse against the Latins.

Around 1305–1306, he was involved in a plot and spent some time in jail. According to the letters written in his prison, it seems that he took part in a conspiracy against the emperor Andronic II, but it is difficult to discover the precise reason of Moschopoulos’s misfortune.

His unique scientific work consists of a Treatise on Magic Squares written at the request of his friend Nicolas Rhabdas. The question is how to organize numbers from 1 to n² in a square, in such way that the sum of numbers in each row, column, or diagonal is always equal to

m = ½ n (n² + 1)

with n being the number of cells of the side of the square.

This kind of exercise occurs in Arabic writings from the ninth century and passed to the Latin world, but Moschopoulos’s treatise did not show any hint of Arabic or Latin influence. In the Arabic world and the Latin Middle Ages, those squares were commonly associated with magical practices. This is the reason why one usually speaks of magic squares. Moschopoulos ignored the magical side of the subject and simply referred to squared numbers or numbers in a square. It was a pure arithmetical exercise.

The author thus explained methods for constructing such squares. It is a construction that varies according to the parity of the number n, which can be odd (3, 5, 7 …), even-even (4, 8, 12 …) or odd-even (6, 10, 14 …). This last case was not developed in the treatise, in spite of the announcement made by Moschopoulos. The origin of Moschopoulos’s treatise is not clear. There existed no Greek writing on the matter before Moschopoulos’s work and no Arabic treatise can be identified as the source of Moschopoulos. He may have learned something on the subject with help of oral explanations given by a traveler or a scholar having been in Persia. At the end of the thirteenth century and in the beginning of the fourteenth century, Persian astronomical and astrological treatises were imported from Constantinople, and it would not be surprising if the magic squares had excited the curiosity of an unknown amateur of mathematical games who brought them to Byzantium.

BIBLIOGRAPHY

Descombes, René. Les carrés magiques. Histoire, théorie et technique des carrés magiques de l’Antiquité aux recherches actuelles. Paris: Vuibert, 2000.

Sesiano, Jacques. “Hestellungsverfahren magischer Quadrate aus islamicher Zeit.” Suddhoffs Archiv 64 (1980): 187–196; 65 (1981): 251–265; 71 (1987): 78–89; 79 (1995): 193–226.

_____. Un traité médiéval sur les carrés magiques. Lausanne, Switzerland: Presses polytechniques et universitaires romandes, 1996.

_____. “Les carrés magiques de Manuel Moschopoulos.”

Archive for History of Exact Sciences 53, no. 5 (1998): 377–397.

_____. “Le traité d’Abu$ l-Wafa$ sur les carrés magiques.”

Zeitschrift für Geschichte der arabisch-islamichen Wissenschaften 12 (1998): 121–244.

_____. Les carrés magiques dans les pays islamiques. Lausanne, Switzerland: Presses polytechniques et universitaires romandes, 2004.

_____. “Magic Squares for Daily Life.” In Studies in the History of the Exact Sciences in Honour of David Pingree, edited by Charles Burnett, Jan P. Hogendijk, Kim Plofker, et al., 715–734. Leiden, Netherlands: Brill, 2004.

Ševč, Ihor. “The Imprisonment of Manuel Moschopoulos in the Year 1305 or 1306.” Speculum 27 (1952): 133–157.

Tannery, Paul. “Le traité de Manuel Moschopoulos sur les carrés magiques.” In Mémoires scientifiques IV, 27–60. Toulouse, France, 1920. Greek text with French translation.

_____. “Manuel Moschopoulos et Nicolas Rhabdas.” In Mémoires scientifiques IV, 1–19. Toulouse, France, 1920.

Anne Tihon