In the first part of this article, we encountered mathematician Zhao Da’s magical ability to use arithmetic to deduce and derive “impossible” hidden factors and their quantity down to the tiniest number. Looking into more accounts and records, we can feel that the ancients, particularly before the Song Dynasty, regarded mathematics as we know it today to be just a small part of the numbers world.
Cao Yuanli precisely calculates warehouse contents
Xijing Miscellaneous Notes is an ancient book that records many amazing anecdotes and secret history. It recounts that in the Western Han Dynasty, a man named Cao Yuanli once visited his friend Chen Guanghan in his hometown. Chen Guanghan had known Cao Yuanli for a long time. As soon as they met, Chen Guanghan asked him a mathematical problem: “My family has two big warehouses for storing rice, and I have forgotten how much we have stored in them. Please help me calculate the quantity.”
Cao Yuanli laughed aloud saying that this was a small case. He used a chopstick, measured more than 10 times around the warehouse, and then said: “The warehouse on the eastern side holds seven hundred and forty-nine stones worth, plus two buckets and seven ounces. The warehouse on the western side holds six hundred and ninety-seven stones and eight buckets.” So Chen Guanghan closed the warehouse doors and put a seal on each. He also wrote the numbers calculated by Cao Yuanli on the seals.
Later on, when Chen Guanghan went to sell the rice from the warehouse on the western side, there was a mouse that weighed the equivalent of 1 liter of rice. The weight of the rice taken out from the eastern warehouse was exactly the same number calculated by Cao Yuanli.
In the following year, Cao Yuanli met Chen Guanghan again. Chen Guanghan told Cao Yuanli the quantity of rice he had measured when he took it out from the warehouse and that the number for the western warehouse was off by one liter. Cao Yuanli hit the table with his hands and said: “Why didn’t I realize that the mouse would eat the rice? This should have been figured out as well.”
Accurately calculating everything even if it was out of sight
Another time, Chen Guanghan brought out wine and a few pieces of dried venison to share with his friend. While eating and drinking, Chen Guanghan asked Cao Yuanli to help calculate the situation of the Chen family’s property. Cao Yuanli calculated with chips and then said: “Twenty-five sugarcane fields should produce 1,536 stalks of sugarcane. Big taro has 37 acres and should result in 673 stones of produce. There are one thousand cows; two hundred calves will be born. There are ten thousand chickens; fifty thousand chicks will be hatched.”
What about the other animals such as sheep, pigs, geese, and ducks? Cao Yuanli could tell the number of them as well, not to mention melons, fruits, and vegetables, he could tell how many there would be. Cao Yuanli then jokingly said: “You have such great wealth, how can you treat me to a meal with such a small amount of food?”
Chen Guanghan shyly replied: “It is possible to have guests who come suddenly; it is also possible for the host to be in a rush. I’m afraid I won’t be able to get you something delicious in time.” Cao Yuanli then said: “It doesn’t matter. There is a steamed piglet in the kitchen, and there is a plate of lychees in the cupboard, just bring them out and we can enjoy them together.”
Chen Guanghan was quite surprised. A person who had never entered his house could calculate his family’s wealth in detail, down to how much food was in the kitchen and what was in the cupboard. It was really extraordinary. Chen Guanghan bowed to apologize and went into the kitchen to get the dishes out. The two ate and drank happily till evening.
When Cao Yuanli calculated the quantity of rice, livestock, and crops belonging to the Chen family, he did not go to the warehouse or to the farm to count everything one by one. Instead, he just used chopsticks and chips to calculate and yet the results were so accurate. Even with modern computers, there is really no method that can compare.
Yuan Hongyu calculates the leaves on a tree
Taiping Guangji, a collection of stories compiled in the early Song Dynasty, records that there lived an imperial official named Yuan Hongyu in the later Tang Dynasty who was an especially proficient mathematician. “It’s not good for me to say I’m good, but my colleagues must see it for themselves and can only be convinced by their own verification.” In one instance, Yuan Hongyu was asked to count how many leaves were on the tree in the yard. This was a tricky problem, as it was like counting the stars in the sky or counting the fine hairs on a sheep.
Yuan Hongyu immediately measured the tree, drew a circle around the tree seven feet away from the tree, and calculated the diameter of the circle. One would wonder what kind of algorithm this is, to count the number of leaves by measuring the waistline of the tree. After a long time, Yuan Hongyu came up with the number of leaves. Yuan Hongyu’s colleagues were unable to verify the number and they were thinking that Yuan Hongyu may have just given them a number at random to fool everyone.
So they came up with a solution: They took Yuan Hongyu off to another place first, then they returned to the tree and shook it. Some leaves fell off, but Yuan Hongyu could not see what his colleagues had done. The colleagues then told Yuan Hongyu that some leaves fell to the ground, and asked if he could calculate how many leaves had fallen.
Without looking at the leaves on the ground, Yuan Hongyu calculated, muttered to himself for a while, and then said: “The tree has twenty-one leaves less than the number that I gave initially.” Everyone went forward to check and found twenty-two leaves, as one leaf had split into two when it fell. Yuan Hongyu’s calculation was accurate.
Yuan Hongyu had even greater skills
Zhang Jingda, a military general and politician of the later Tang Dynasty, had two jade bowls. The Zhang family asked Yuan Hongyu to have a look at them, which was also a way to show how wealthy their family was. Yuan Hongyu looked at them, pondered for a while, measured the depth and width of the bowls, calculated, and then said to Zhang Jingda: “These two bowls will definitely be broken by the sixteenth day of the fifth lunar month next year between 9 and 11 a.m.”
On hearing this, Zhang Jingda thought to himself: “I’ll hide them carefully, and see if they will still be broken.” Then he ordered his servants to wrap the two jade bowls with cloth, pack them in a big bamboo cage, and place them in the warehouse.
At 9 a.m. on the 16th day of the fifth lunar month the following year, the roof beam of the warehouse suddenly collapsed and fell onto the bamboo cage where the two bowls were kept, thus smashing the bowls. At that time, a servant saw the incident with his own eyes. The news of Zhang Jinda’s broken bowls spread. Someone verified Yuan Hongyu’s prediction.
Yuan Hongyu not only counted the number of leaves on a tree, but also the number of leaves that just fell to the ground. What’s more, he was able to calculate the time, date, month, and year of when the two jade bowls would be broken. It seems his abilities entered an almost mystical realm.
These calculations are even more accurate than modern-day Probability Mathematics. From the preserved records that we are privy to, we can see the magic of ancient Chinese mathematicians! There seems to have been a supernatural dimension to these legendary abilities: The mathematicians may have had a very advanced level of intuition, esoteric divination abilities, or may have been blessed with a rare God-given gift. In any case, they all seem to have been professional, honorable, and ethical men.
See Part 1 here
Translated by Chua BC
