Magic squares

The teaching

As the lore of the grid from the Lo Shu to Agrippa teaches it.

The rule of the square

A magic square of order n holds the numbers from 1 to n squared, each once, so that every row, every column and both long diagonals add to the same sum. Since all the numbers together make n squared times n squared plus one, halved, each of the n rows must take an equal share, and the sum of a line is n times n squared plus one, halved: 15 for a square of three, 34 for four, 65 for five, 111 for six, 175 for seven, 260 for eight and 369 for nine.

In the square of three the middle cell must hold five, and every pair of cells on opposite sides of the middle adds to ten.

The Lo Shu

The oldest is the Lo Shu, the writing from the river Luo. Its rows read 4, 9, 2 at the top, 3, 5, 7 across the middle and 8, 1, 6 at the bottom. Legend says it was seen on the shell of a turtle that rose from the river in the days of Yu the Great, who tamed the floods. The odd numbers, which are yang, stand on the cross through the centre; the even numbers, which are yin, stand at the corners; and five, at the centre, is the pivot of all.

Chinese teaching lays it out as the nine palaces, with south at the top as on old Chinese maps. Nine is the south and fire, one the north and water, three the east and thunder, seven the west and the lake; four is the south-east and wind, two the south-west and earth, eight the north-east and the mountain, six the north-west and heaven. So the square joins the numbers to the eight trigrams of the Book of Changes and the eight directions, and the art of feng shui moves the numbers through the palaces year by year as the flying stars.

Building the even squares

Odd squares are built by a stepping rule; even ones need others. For a square of four, write the numbers 1 to 16 in order, row by row, then replace each number on the two long diagonals by its partner, the number that adds to it to make 17. The rows then read 16, 2, 3, 13; 5, 11, 10, 8; 9, 7, 6, 12; 4, 14, 15, 1, and every line adds to 34. For a square of eight the same is done on the diagonals of each of its four smaller squares, with partners that make 65.

Squares of six, ten and the other even orders not divisible by four are the hardest: they are made by joining four odd squares and swapping certain cells between them.

Dürer's square

Dürer set a square of four in his engraving Melencolia I: 16, 3, 2, 13; 5, 10, 11, 8; 9, 6, 7, 12; 4, 15, 14, 1. Every row, column and long diagonal adds to 34, and so do the four corners, the four middle cells and each of its four quarters, while every pair of cells set opposite each other through the centre adds to 17. The two middle cells of the bottom row give the year, 1514.

Agrippa's tables of the planets

Agrippa gives a square to each of the seven planets, in the old order from the slowest to the swiftest, and the size of the square rises with it. Saturn has the square of three, with lines of 15 and a total of 45; Jupiter four, 34 and 136; Mars five, 65 and 325; the Sun six, 111 and 666; Venus seven, 175 and 1225; Mercury eight, 260 and 2080; the Moon nine, 369 and 3321. Each was to be engraved on a plate of metal at a fortunate hour, lead for Saturn, gold for the Sun, silver for the Moon.

With each square come names whose letters, counted as numbers in Hebrew, give the square's own numbers: for Saturn the intelligence Agiel and the spirit Zazel both count 45, and for the Sun the spirit Sorath counts 666. The square of the Sun is thus the square of the number that Revelation gives the beast: Here is wisdom. Let him that hath understanding count the number of the beast: for it is the number of a man; and his number is Six hundred threescore and six.

Seals traced on the squares

Agrippa draws for each planet a seal made from the lines of its square. Later magicians, the Golden Dawn among them, made a sigil for any name the same way: turn each letter of the name into its number, find the numbers in turn on the planet's square, and join them with one line, beginning with a small circle and ending with a short bar. The line, lifted off the square, is the sigil of that name on that planet's table.