Shao Yong

The Huangji jingshi is an attempt to give history a mathematical shape. Shao Yong divides time by twelves and thirties in the way a calendar divides a day, and continues the division upward past the year until he arrives at a complete world cycle: one yuan of 129,600 years, made of twelve hui, each of thirty yun, each of twelve shi. Within that cycle a world comes into being, flourishes, declines and ends, and the sequence of dynasties can be located on it.

The machinery of the system is a repeated fourfold division taken from the Yijing. Heaven has four images, sun, moon, stars and zodiacal spaces; earth has water, fire, soil and stone; a person has ear, eye, nose and mouth; there are four kinds of ruler, four kinds of government, four ways a thing can be affected. Everything is generated by doubling: two into four, four into eight, and so on to sixty-four and beyond. Shao Yong claimed to be recovering the method by which the sages made the hexagrams rather than inventing a scheme of his own.

What he wanted from it was a way of seeing things as they are rather than as the observer wishes them to be, which he called viewing things from the standpoint of things. The numbers are a discipline of detachment as much as a description of the cosmos, which is why contemporaries who thought the calculations eccentric could still regard him as an exemplary man.

Two arrangements of the trigrams and hexagrams circulate in the Yijing tradition. The received or after-heaven order is the one carried by the transmitted text. The before-heaven order, associated with Shao Yong and attributed by him to Fuxi, arranges the sixty-four hexagrams in a sequence generated by successive division, and displays them in a square and a surrounding circle.

Zhu Xi reproduced these diagrams at the front of his own work on the Yijing, and in doing so made them standard furniture of the tradition, which is the main reason Shao Yong's influence outran the number of people who read him. The diagrams travelled further than that. In 1701 the Jesuit Joachim Bouvet, working in Beijing, sent the before-heaven arrangement to Leibniz, who had been developing binary arithmetic and who recognised the sequence as a binary count. He took it as evidence that the ancient Chinese had possessed a knowledge since lost.

Modern historians treat that reading with care. The order does correspond to a binary enumeration, because both are generated by the same operation of repeated division, but nothing in the Chinese tradition treats the broken and solid lines as digits or performs arithmetic with them. The correspondence is real and the inference from it was Leibniz's. That episode, in which a Song cosmologist's diagram becomes an exhibit in a European argument about the antiquity of mathematics, is one of the odder passages in the history of the Yijing.