Magic squares – the seal of Saturn and the ten
In the planetary magic of the Renaissance, the square of the numbers 1 to 9 belongs to Saturn. Every line gives 15, and every two numbers through the centre add up to ten.
A magic square is a field of numbers in which every row, every column and both diagonals give the same sum. For the scholars of the Renaissance this was no puzzle but an image of the order by which the powers of the heavens act.
The seven tables of the planets
In 1533, in De occulta philosophia, Heinrich Cornelius Agrippa von Nettesheim compiled the seven tables of the planets. Each of the seven classical planets receives a magic square, from the smallest to the largest: Saturn three by three fields, Jupiter four by four, Mars five, the Sun six, Venus seven, Mercury eight and the Moon nine by nine. Each table comes with divine names whose number values arise from the square, and with seals that were engraved on talismans. Agrippa drew on Arabic sources, in which magic squares were studied from the 10th century and later linked with the planets.
The smallest table belongs to Saturn, the outermost and slowest of the planets then known. He was held to rule over time, measure and limit, and he was the planet of melancholy and deep thought.
4 9 2 · 3 5 7 · 8 1 6
Fifteen in every direction
In the square of Saturn each of the eight lines gives 15. The numbers 1 to 9 add up to 45, and because the three rows share the 45, every line must give 15. No other square of three by three fields has this property, all others arise from this one by turning and mirroring. 15 is itself a triangular number, 1 + 2 + 3 + 4 + 5, and 45 is the triangular number of nine.
The path of numbers and the seal
Join the numbers in order, 1, 2, 3 up to 9, and a line runs across the square. The seals of the planets and their spirits were derived from such paths of numbers. Later, in the 19th century, the Hermetic Order of the Golden Dawn drew names into the tables in the same way: each letter has a number value, and the line from number to number gives the sign of the name.
The square of Saturn. “Path of numbers” joins the numbers 1 to 9 in order, the circle marks the start, the bar the end. “Equal sums” lights up the eight lines one after another. “Pairs” shows the four pairs through the centre, tapping a number highlights its pair.
Ten through the centre
Even more striking is what faces each other through the centre: 4 and 6, 9 and 1, 2 and 8, 3 and 7. Each of these four pairs gives 10. In the centre stands 5, half of ten and the middle of the numbers 1 to 9. The square thus consists of four tens around a five, 4 × 10 + 5 = 45.
Four pairs leading to ten: this is the same link between four and ten that the tetractys shows in a single figure (see The triangle of dots). The Pythagoreans laid out ten as a triangle of four rows, the square of Saturn lays it out as four pairs around a centre.
Even and odd numbers are ordered too. The odd ones stand in the cross through the centre, the even ones in the corners. Start at 1 at the bottom and keep multiplying by 3, and the last digits are 3, 9, 7 and 1 again. In exactly this order the odd numbers lie clockwise around the centre. Start at 2 at the top right and keep doubling, and you get 4, 8, 6 and 2 again, and this is how the even numbers lie, anticlockwise.
Dürer’s square
The most famous magic square is on Albrecht Dürer’s engraving Melencolia I of 1514. The figure of Melancholy sits brooding among tools of geometry, a polyhedron and a pair of scales, and on the wall hangs a square of four by four fields. This is the size Agrippa assigned to Jupiter. According to a widespread reading, based on the teachings of the Florentine Marsilio Ficino, the table of Jupiter is meant to soften Saturnine melancholy.
Every line gives 34, and the bottom row contains the year 1514, framed by 4 and 1, the positions of the letters D and A in the alphabet. Here too the numbers facing each other through the centre add up to the same sum, 17. The principle of the square of Saturn, pairs around a centre, also carries the larger square.
The square from Dürer's Melencolia I. All ten lines give 34, and every two numbers through the centre give 17. The bottom row contains the year 1514.
The oldest trace
The square of the numbers 1 to 9 is much older than Agrippa. The earliest known trace leads to China, where it is attested in texts from about the 1st century and was later called Lo Shu. Arab mathematicians developed methods to build squares of any size, and through their writings the knowledge reached Europe.
What the squares show
Magic squares show an order in which each element is related to a counterpart and everything to a centre. That is why they were assigned to the planets: as the image of a power acting equally in every direction. That the smallest of these tables consists of four pairs leading to ten links the planetary magic of the Renaissance with the Pythagorean tetractys. Agrippa himself describes it in his chapter on the number four, as the number by which the Pythagoreans swore.