“Any progress?”
“Yes, some. At times, I even feel I’m standing right in front of the door to the treasury filled with the secrets of the universe.”
“How do the Heavens tell you these mysteries? Just now, you explained that the language of the Heavens consists of numbers and shapes.”
“The circle.”
Seeing that King Zheng was utterly confused, Jing Ke asked for and received permission to pick up a brush. He drew a circle on the silk cloth spread out on the low table. Though he didn’t use a compass or other tools to assist him, the circle appeared to be perfect.
“Sire, other than objects made by men, have you ever seen a perfect circle in nature?”
King Zheng pondered this for a moment. “Very rarely. Once, a falcon and I stared at each other, and I noticed that its eyes were very round.”
“Yes, that’s true. I can also suggest as examples eggs laid by certain aquatic creatures, the intersecting plane between a dewdrop and a leaf, and so on. But I’ve carefully measured all of these, and none of them are perfect circles. It’s the same with the circle I drew here: It may look round, but it contains errors and imperfections undetectable by the naked eye. In fact, it’s an oval, not a perfect circle. I’ve been searching for the perfect circle for a long time, and I finally realized it does not exist in the world below, but only in the Heavens above.”
“Oh?”
“Sire, please accompany me outside the palace.”
Jing Ke and King Zheng strode outside the palace. It was another beautiful day with the moon and the sun both visible in the clear sky.
“The sun and the full moon are both perfect circles,” said Jing Ke as he pointed at the sky. “The Heavens placed the perfect circle—impossible to find on earth—in the sky. Not just one, but two examples, and they’re the most notable features of the firmament. The meaning couldn’t be clearer: the secret of the Heavens resides inside the circle.”
“But the circle is the simplest of shapes. Other than a straight line, it’s the least complicated figure.” King Zheng turned around and returned inside the palace.
“That apparent simplicity disguises a profound mystery,” Jing Ke said as he followed the king back inside. When they returned to the low table, he drew a rectangle on the silk with the brush. “Observe this rectangle if you would. The longer dimension measures four inches, and the shorter dimension two. The Heavens speak also through this figure.”
“What does it say?”
“The Heavens tell me that the ratio between the longer side and the shorter side is two.”
“Are you mocking me?”
“I wouldn’t dare. This is just an example of a simple message. Please observe this other figure.” Jing Ke drew another rectangle. “This time, the long side is nine inches and the shorter seven. The ideas expressed by the Heavens in this figure are far richer.”
“From what I can see, it’s still extremely simple.”
“Not so. Sire, the ratio between the longer side and the shorter side in this rectangle is 1.285714285714285714… The sequence ‘285714’ repeats forever. Thus, you can calculate the ratio to be as precise as you like, but it will never be exact. Though the message is still simple, much more meaning can be extracted from it.”
“Interesting,” said King Zheng.
“Next, let me show you the most mysterious shape the Heavens gifted us: the circle.” Jing Ke drew a straight line through the center of the circle he had drawn earlier. “Observe that the ratio between the circumference and the diameter of a circle is an endless string of numbers beginning with 3.1415926. But it keeps on going after that, never repeating itself.”
“Never?”
“Yes. Imagine a silk cloth as large as all-under-heaven. The string of numbers in the circle’s ratio could be written in tiny script, each numeral no bigger than the head of a fly, all the way from here to the edge of the sky, and then coming back here start on a new line. Continued this way, the entire cloth could be filled, and there would still be no end to the numbers, and the sequence still wouldn’t repeat. Your Majesty, this endless string of numbers contains the mysteries of the universe.”
King Zheng’s expression didn’t change, but Jing Ke saw that his eyes had brightened. “Even if you obtained this number, how would you read from it the message the Heavens want to express?”
“There are many ways. For example, by treating the numbers as coordinates, it’s possible to turn the numbers into new shapes and pictures.”
“What will the pictures show?”
“I don’t know. Maybe it will be an illustration of the enigma of the universe. Or maybe it will be an essay, or perhaps even a whole book. But the key is that we must obtain enough digits of the circular ratio first. I estimate that we must compute tens of thousands, perhaps even hundreds of thousands, of digits before the meaning can be discerned. Right now, I’ve only computed about a hundred digits, inadequate to detect any hidden meaning.”
“A hundred? That’s all?”
“Sire, even so few digits have taken me more than ten years of effort. To compute the circular ratio, one must approach it by inscribing and circumscribing a circle with polygons. The mores sides in the polygons, the more precise the calculations and the more digits that can be obtained. But the complexity of the calculations increases rapidly, and progress is slow.”
King Zheng continued to stare at the circle crossed by a straight line. “Do you think you’ll find the secret of eternal life in it?”
“Yes, of course.” Jing Ke grew excited. “Life and death are the basic rules given to the world by the Heavens. Thus, the mystery of life and death must be contained in this message as well, including the secret of eternal life.”
“Then you must compute the circular ratio. I will give you two years to compute ten thousand digits. In five more years, I need you to get to a hundred thousand digits.”
“That’s… that’s impossible!”
King Zheng whipped his long sleeves across the table, scattering the silk cloth and ink and brush onto the floor. “You have but to name whatever resources you need.” He stared at Jing Ke coldly. “But you must complete the calculations in time.”
Five days later, King Zheng called for Jing Ke again. This time, Jing Ke didn’t come to the palace in Xianyang; instead, he met the royal entourage on the road as the king was touring around his domain. Right away, King Zheng asked Jing Ke for updates on the progress of the calculations.
Jing Ke bowed and spoke. “Sire, I gathered all the mathematicians who are capable of such calculations in the entire realm: they number only eight. Based on the amount of calculations necessary, even if all nine of us devote the rest of our lives to this task, we will only be able to obtain about three thousand digits of the circular ratio. In two years, the best that we can do is three hundred digits.”
King Zheng nodded and indicated that Jing Ke should walk with him. They came to a granite monument about twenty feet high. A hole was drilled through the top of the monument, and a thick rope made of twisted oxhide passed through the hole to suspend the monument from a wooden platform like the weight of a giant pendulum. The smooth bottom of the monument hovered about a man’s height above the ground. There were no inscriptions on the monument.
King Zheng pointed to the hanging monument and said, “Look, if you can finish the calculations in time, this will become a monument of your triumph. We will erect it on the ground and fill it with inscriptions of your many accomplishments. But if you can’t finish the calculations, this will become a memento of your shame. In that case, it will also be erected on the ground, but before the rope is cut to drop it, you will have to sit below it so that it can be your tombstone.”