page_title: Magic square
section_title: Latin Europe
hierarchical_section_title: Magic square / History / Latin Europe
caption_attribution_description: English: Page 47 of the second part of Vol. II of Athanasius Kircher's "Oedipus Aegyptiacus". The chapter about "Hieroglyphic Mathematics" also deals with magic squares. This page shows "Jupiter's square" which is closely related to Dürer's magic square. Deutsch: Seite 47 des zweiten Teils des zweiten Bandes des „Oedipus Aegyptiacus“ von Athanasius Kircher. Das Kapitel „Hieroglyphische Mathematik“ behandelt unter anderem Magische Quadrate; auf dieser Seite ist das „Jupiter-Quadrat“ abgebildet, das eine Variante des Magischen Quadrats von Albrecht Dürer ist.
context_page_description: In recreational mathematics and combinatorial design, a magic square is a square grid filled with distinct positive integers in the range such that each cell contains a different integer and the sum of the integers in each row, column and diagonal is equal. The sum is called the magic constant or magic sum of the magic square. A square grid with n cells on each side is said to have order n.
  The mathematical study of magic squares typically deals with its construction, classification, and enumeration. Although completely general methods for producing all the magic squares of all orders do not exist, historically three general techniques have been discovered: by bordering method, by making composite magic squares, and by adding two preliminary squares. There are also more specific strategies like the continuous enumeration method that reproduces specific patterns. The magic squares are generally classified according to their order n as: odd if n is odd, evenly even if n = 4k, oddly even if n = 4k + 2. This classification is based on different techniques required to construct odd, evenly even, and oddly even squares.
context_section_description: Unlike in Persia and Arabia, we have better documentation of how the magic squares were transmitted to Europe. Around 1315, influenced by Arab sources, the Greek Byzantine scholar Manuel Moschopoulos wrote a mathematical treatise on the subject of magic squares, leaving out the mysticism of his Middle Eastern predecessors, where he gave two methods for odd squares and two methods for evenly even squares. Moschopoulos was essentially unknown to the Latin Europe until the late 17th century, when Philippe de la Hire rediscovered his treatise in the Royal Library of Paris.  However, he was not the first European to have written on magic squares; and the magic squares were disseminated to rest of Europe through Spain and Italy as occult objects. The early occult treaties that displayed the squares did not describe how they were constructed. Thus the entire theory had to be rediscovered.
  Magic squares had first appeared in Europe in Kitāb tadbīrāt al-kawākib (Book on the Influences of the Planets) written by Ibn Zarkali of Toledo, Al-Andalus, as planetary squares by 11th century. The magic square of three was discussed in numerological manner in early 12th century by Jewish scholar Abraham ibn Ezra of Toledo, which influenced later Kabbalists. Ibn Zarkali's work was translated as Libro de Astromagia in the 1280s, due to Alfonso X of Castille. In the Alfonsine text, magic squares of different orders are assigned to the respective planets, as in the Islamic literature; unfortunately, of all the squares discussed, the Mars magic square of order five is the only square exhibited in the manuscript.
  Magic squares surface again in Florence, Italy in the 14th century. A 6×6 and a 9×9 square are exhibited in a manuscript of the Trattato d'Abbaco (Treatise of the Abacus) by Paolo Dagomari. It is interesting to observe that Paolo Dagomari, like Pacioli after him, refers to the squares as a useful basis for inventing mathematical questions and games, and does not mention any magical use. Incidentally, though, he also refers to them as being respectively the Sun's and the Moon's squares, and mentions that they enter astrological calculations that are not better specified. As said, the same point of view seems to motivate the fellow Florentine Luca Pacioli, who describes 3×3 to 9×9 squares in his work De Viribus Quantitatis by the end of 15th century.
caption: This page from Athanasius Kircher's Oedipus Aegyptiacus (1653) belongs to a treatise on magic squares and shows the Sigillum Iovis associated with Jupiter
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