The square of Jupiter and Dürer's Melencolia I

In 1514, in Nuremberg, Albrecht Dürer engraved one of the most enigmatic images in the history of art. A winged figure sits brooding among abandoned instruments: a sphere, a polyhedron, a pair of compasses, an hourglass, a set of scales. On the wall behind her, beneath a bell, is a square of sixteen numbers. It is often regarded as the first magic square to appear in a European work of art, and it is tied to the tradition of the square of Jupiter.

A square that carries a date

Dürer's square contains the numbers from one to sixteen, and every row, column and diagonal adds up to thirty-four. The four corner numbers and the four central ones also total thirty-four. In the two middle cells of the bottom row we read fifteen and fourteen, which together form 1514, the year of the engraving. On either side, the numbers four and one correspond, according to a widespread reading, to the letters D and A, the artist's initials.

In the astrological tradition the square of four was assigned to Jupiter. The version Agrippa published in 1533, however, has a different arrangement from Dürer's: the numbers are the same but ordered in another way. How the artist came to know this tradition is unclear. Agrippa had written a first draft of his treatise in 1510, but there is no evidence that Dürer read it.

Jupiter against melancholy

To understand the meaning of the square we must go back to Florence. In 1489 Marsilio Ficino published De vita libri tres, devoted to the life of men of learning. According to Ficino, scholars were naturally exposed to the influence of Saturn and therefore to melancholy, the humour of black bile. To temper its excesses he advised turning to Jupiter, a benevolent and moderate planet, through music, scents, colours and images.

From this perspective the table of Jupiter on the wall of the engraving acts as a symbolic counterweight to the gloom of the winged figure. Erwin Panofsky and Fritz Saxl, in studies of melancholy begun in the 1920s, proposed precisely this reading, which remains the most widely accepted, although it does not exhaust the many interpretations put forward over the century.

The mystery of the number one

The title too, inscribed on the wings of a bat in flight, has been much debated. One of the best-known hypotheses links the Roman numeral to Agrippa's doctrine, which distinguished three degrees of inspired melancholy: the first concerned the imagination, proper to artists and craftsmen, the second reason, the third the mind contemplating the divine. The engraving would thus depict the first degree. Other scholars think instead of a planned series of engravings that was never completed, but no contemporary source confirms this.

The square of Melencolia I has become a small icon of recreational mathematics and of the history of esotericism. In sixteen cells it sums up a whole system of Renaissance ideas: the link between planets and temperaments, faith in the power of images, the dignity of intellectual work. Anyone who looks at it closely can recognise in that grid of numbers a dialogue between a great artist and the culture of his time.