GOTTFRIED Leibniz was on top of the world when he saw Shao Yong’s two diagrams brought back from China by Joachim Bouvet. These diagrams explained the hexagrams of 64 gua of I Ching in a structured manner.

He believed that the order of gua happened to coincide with the binary math (2-base of 0 or 1) he had invented and had been exploring for nearly 20 years. Merely based on these hexagrams, Leibniz calculated the ID number for each gua. For instance, Qian Gua ­— 63, Kun Gua ­— 0, Pi Gua ­— 7, Tai Gua­ — 56, Tun Gua ­— 34, Meng Gua ­— 17. In addition, Leibniz determined the sequence of Bagua, or the eight trigrams.

As we know, each of the 64 6-yao gua (hexagram) consists of two 3-yao gua (trigrams). For example, Tai Gua is Sky (☰) underneath Earth (☷), while Pi Gua is to the contrary — Sky being on top of Earth. The sequence of Bagua is critical for us to understand the underlying order of all the 64 hexagrams, which is the key to reveal the core meaning of “I Ching.”

Leibniz came up with the ID number of each 3-yao gua’s based on the position of yao from top down to base. He defined yang yao (whole line) as 1 and yin yao (broken line) as 0. Here are some examples according to Leibniz-Shao Yong approach.

Mountain (☶) can be converted to 100. Its sequential number is 1*20 + 0*21+ 0*22 = 1. Similarly, Water (☵ , 010) obtains its sequential number 2 (0*20 + 1*21 + 0*22 = 2). So Wind (☴, 110) acquires its ID 3 (1*20 + 1*21 + 0*22 = 3), etc. The results turn out to be:

If Leibniz had defined yang yao as 0, and yin yao as 1, then the ID numbers should have been ☷ 7 (Earth), ☶ 6 (Mountain), ☵ 5(Water), ☳ 4(Thunder), ☴ 3 (Wind), ☲ 2 (Fire), ☱ 1 (Valley), and ☰ 0 (Sky). As a result, the sequence is almost identical to what Shao Yong had reached: ☷ 8, ☶ 7, ☵ 6, ☳ 5, ☴ 4, ☲ 3, ☱ 2 and ☰ 1. As for a sequence, it is the relative rankings, rather than the specific ID numbers, that matters.

It is worth mentioning that in his calculation of the Bagua sequence, Leibniz had made a big compromise.

According to the authentic rules of binary method, the value conversion is bottom-up. The ID numbers of Mountain (☶, 100) and Thunder (☳, 001) should be reversed, since ☶ 100 = 1*22 + 0*21+ 0*20 = 4, while ☳ 001 = 0*22 + 0*21+ 1*20 = 1. The same reversion should apply to Wind and Valley, too. It must be easy to anyone with some basic knowledge of the binary coding, the ID numbers for Valley’s ID (☱,011) should be 3, while Wind’s ID (☴, 110) should be 6.

We have reasons to believe that Leibniz might have been confused when he attempted to apply the binary method to figure out Bagua’s sequence in line with Shao Yong’s arrangement, or known as the Fu Xi’s Arrangement.

Leibniz communicated with J. Bouvet for years over the confusion. At least seven of their correspondences were well preserved, and can be referenced to in Leibniz Library in Germany.

As a telling fact, the full title of Leibniz’s landmark paper on the binary method, published in 1703, is “Explanation of the Binary Arithmetic, which uses only the characters 1 and 0, with some remarks on its usefulness, and on the light it throws on the ancient Chinese figures of Fu Xi.”

Fortunately, we may offer an easier way to gain an insight into the sequence of Bagua. You may have seen the diagram above last week. To look at the eight 3-yao gua from the center of the circle outward, their sequential numbers are 0-Earth, 1-Mountain, 2-Water, 3-Wind, 4-Thunder, 5-Fire, 6-Valley, and 7-Sky. Linking them together from 0 through 7 forms an S-shaped line.

If we look from the outside toward the center of the circle, it becomes 0-Earth, 1-Thunder, 2-Water, 3-Valley, 4-Mountain, 5-Fire, 6-Wind, and 7-Sky, which also forms an S-shaped line. Thunder and Mountain exchange their positions; so do Wind and Valley. 0-Earth, 7-Sky, 2-Water, and 5-Fire remain the same.

Why is that? It’s because the last four triagrams are horizontally symmetrical. They look exactly the same from top to base or from base to top.

This is a critical clue in explaining the I Ching system of trigrams that four of Bagua change while the other half remain unchanged.

It is of fundamental importance in interpreting the transformation of 64 hexagrams (6-yao) and their paring. Before we start digging deeper, let’s take a look at how Bagua forms its four pairs.

Embrace in mind that the basic idea I Ching is “contrary and complimentary,” as grasped by N. Bohr into his family arms of coat (see the first story of this series http://dwz.cn/2hbnDk). The Table 1 makes clear that Earth-Sky, Mountain-Valley, Water-Fire, and Thunder-Wind make up four pairs. Each pair’s sequential numbers add up to 7. The yao at the same position of paired gua are opposite in yin and yang: if the yao is (—) in one gua then it must be (--) in its paired gua, and vice versa. We will explain this point with some details next time.

For now, let’s imagine the members of a Bagua family. Sky is viewed as Father while Earth as Mother. Mother (☷) conceives from Father (☰), when the base yao of Mother converts to yang yao, there comes the eldest son ☳ or Thunder; when Mother’s middle yao converts to yang yao, the middle son ☵ or Water results; when the top yao converts to yang yao, the youngest son ☶ or Mountain delivers. So Father ☰ and three sons ☳, ☵, and ☶ belong to the Yang Sphere.

Similarly, in the Yin Sphere, Father (☰) conceives from Mother (☷). When the base yao of Father converts to yin yao, there comes the eldest daughter ☴ or Wind; when Father’s middle yao converts to yin yao, it results in the middle daughter ☲ or Fire; when the top yao converts to yin yao, it delivers the youngest daughter ☱ or Valley. Therefore, Mother and three daughters ☷, ☴, ☲, and ☱ belong to the Yin Sphere. As such, Father (☰ 7, Sky) and Mother (☷ 0, Earth) are in pair; Eldest Son (☳ 1, Thunder) and Eldest Daughter (☴ 6,Wind), Middle Son (☵ 2, Water) and Middle Daughter (☲ 5, Fire), Youngest Son (☶ 4, Mountain) and Youngest Daughter (☱ 3, Valley) form the remaining three pairs of Bagua.

Gottfried Wilhelm Leibniz
(1646-1716)

He was a German philosopher and mathematician, hailed as a rare polymath in history, and commended as Aristotle of the 17th century. Leibniz developed calculus independently of Isaac Newton. Leibniz was perhaps the first major European intellect to take a close interest in Chinese civilization, which he knew by corresponding with Joachim Bouvet, a French Jesuit missionary in China. He noted with fascination how the I Ching hexagrams correspond to the binary numbers from 000000 to 111111, and concluded that this mapping was evidence of major Chinese accomplishments in the sort of philosophical mathematics he admired. Leibniz then tried to apply the binary coding in calculating the sequence of Bagua as well as the 64 hexagrams.

Joachim Bouvet (1656-1730)

He is also known as Bai Jin (白晋) in Chinese. He joined the Jesuits in 1678 and showed great interest in mathematics and physics. The Frenchman was sent to China by the French King Louis XIV and arrived in Beijing in 1688. They presented many scientific instruments and atlas to the Court of Qing Dynasty, and introduced geometry and arithmetic to the Kangxi Emperor.

Bouvet introduced “I Ching” to the West and tried to find a connection between the Chinese classics and the Bible. It was he who introduced I Ching to Leibniz. As a respected Sinologist, Bouvet delivered a speech titled “I Ching” in Paris in 1698. During his speech, Bouvet regarded “I Ching” as a reasonable and perfect philosophical system similar to those by Plato and Aristotle.