The planetary squares: the numerical tables of the seven planets

Among the most enigmatic pages of the second book are seven grids of numbers, one for each planet. Agrippa calls them tables of the planets and presents them as instruments endowed with great virtues, sacred and mysterious. From a mathematical point of view they are magic squares: arrangements of whole numbers in which the sum of every row, every column and both diagonals is always the same. In the De occulta philosophia this arithmetical curiosity becomes a key to celestial magic.

Seven squares for seven planets

Agrippa assigns each planet a square of a different size, according to the number that tradition attributed to it. To Saturn, the highest planet, falls the smallest square, three cells to a side, with the numbers from one to nine; to Jupiter the one of four to a side; to Mars that of five; to the Sun that of six; to Venus that of seven; to Mercury that of eight; to the Moon, the planet closest to the Earth, that of nine to a side, with eighty-one numbers.

The constant sums of each square have precise values: fifteen for Saturn, thirty-four for Jupiter, sixty-five for Mars, one hundred and eleven for the Sun, one hundred and seventy-five for Venus, two hundred and sixty for Mercury, three hundred and sixty-nine for the Moon. Agrippa also gives importance to the total of all the numbers in each table, and links these values to the names of the planetary intelligences and spirits through the numerical value of Hebrew letters, by kabbalistic procedures that he describes without explaining every step.

Distant origins

Agrippa did not invent magic squares. A Chinese legend tells of a three-by-three square, the Lo Shu, that appeared on the back of a turtle emerging from the river Luo, though its dating is uncertain. In the medieval Islamic world squares were studied both as mathematical problems and as objects endowed with virtues, and they were often associated with the planets. Through translations and Latin manuscripts the tradition reached Europe, where it was already circulating in the late Middle Ages.

Squares associated with the seven planets already appear in Latin manuscripts earlier than his work, and Agrippa's novelty lies above all in having placed it within a broad philosophical system and in having spread it through print. After 1533 these tables became one of the most frequently copied motifs of European magical literature. It was the printed book, more than manuscripts, that made them familiar to European readers.

Dürer's square

The most famous case of a magic square in Renaissance art is Albrecht Dürer's engraving Melencolia I, dated 1514. At the upper right, behind the winged figure lost in thought, appears a four-by-four square, with a sum of thirty-four, in whose bottom row the date of the work can be read. It is the same type of square that tradition assigned to Jupiter, the planet which according to Ficino could temper the effects of Saturn and of melancholy.

Some twentieth-century art historians, starting from the studies of Erwin Panofsky and Fritz Saxl, have suggested that Dürer knew the ideas of the first draft of the De occulta philosophia, which circulated in manuscript in the Germany of those years. The hypothesis remains debated, but it shows how close Agrippa's ideas were to the visual culture of the time.

The planetary squares are a perfect example of the meeting of mathematics and symbolism in Renaissance thought. For Agrippa number was not a mere quantity but a form endowed with power, and a perfectly balanced numerical arrangement appeared as a reflection of the harmony of the heavens. Today those same grids interest mathematicians for their combinatorial properties.