The square of Saturn: from the Chinese Lo Shu to the planetary tables
The magic square of order three, summing to fifteen: from the Chinese legend of the Luo river turtle to the square of Saturn in the European tradition.
The smallest possible magic square has three rows and three columns and contains the numbers from one to nine. Arranged in the right order, they add up to fifteen in every row, every column and along both diagonals. In the European tradition this figure is the square of Saturn, but its story begins far away, in China, where it was known as the Lo Shu, the “writing of the Luo river”.
The legend of the turtle
According to Chinese myth, during a great flood the ruler Yu, legendary founder of the Xia dynasty, saw a turtle emerge from the Luo river bearing a pattern of dots on its shell. That figure is said to have revealed the order of the universe and the way to govern the waters. The legend is ancient, but the story grew over time and its versions do not agree in every detail.
The numerical pattern itself is clearly attested in texts of the Han period, around the beginning of the Christian era. In China the Lo Shu was associated with the Hetu, the “Yellow River diagram”, and with the cosmology of the five phases and the eight directions. It thus became a tool of geomancy and cosmological thought, far more than an arithmetical game.
The square in the Islamic world and in Europe
The square of order three was known to Arabic-speaking authors at least from the tenth century and circulated in treatises on numerical squares, called wafq. In the Islamic world it was credited with a protective function, especially during childbirth. Through Spain and Latin translations, the tradition of planetary squares reached Europe, where the figures were linked to the seven planets in order of increasing size.
In the system that Agrippa made famous in 1533, the square of three belongs to Saturn, the slowest and most distant of the planets visible to the naked eye. The traditional arrangement is four, nine, two in the first row; three, five, seven in the second; eight, one, six in the third. Five, the central number, sits at the heart of the figure, and the sum of all the cells is forty-five.
The meanings attributed to Saturn
In ancient and medieval astrology Saturn was the planet of cold, old age, melancholy and deep meditation. Its metal was lead, its day Saturday. For this reason Renaissance tables named lead as the traditional material for the Saturnine square, and tradition attributed to it qualities connected with stability, endurance and the protection of long undertakings.
Renaissance authors regarded Saturn as ambivalent. Marsilio Ficino, who saw Saturn in a dominant position in his own horoscope, described it as the planet of contemplatives and scholars, but also as a source of sadness. This double nature explains why the square of Saturn was thought to be powerful and, at the same time, to call for caution.
Few mathematical objects have had such a long and varied life. The same pattern of nine numbers has served Chinese cosmology, the protection of women in childbirth in the Islamic world and the astrological magic of the Renaissance. It is still studied today as the first example of a magic square, and it is the only one of order three: every other solution is simply a rotation or reflection of the same figure.