"Sire, all you have to do is to stab lightly. The dagger has been soaked in poison. A slight p.r.i.c.k is enough to end my life."
King Zheng sat still and raised his hands to signal the guards rushing into the Great Hall to stop. Without changing his expression, he said, "I do not have to kill you to feel safe. Your words have convinced me that you do not have the heart of an a.s.sa.s.sin."
In a single, smooth motion, Jing Ke wrapped the fingers of his right hand around the handle of the dagger. The tip of the dagger was aimed at his own chest as though he was about to commit suicide.
"You"re a learned man." King Zheng"s voice was cold. "Dying now would be a waste. I"d like to have your skills and knowledge a.s.sisting my army. If you insist on dying, do so only after you"ve accomplished some things for me first." He waved at Jing Ke, dismissing him.
The a.s.sa.s.sin from Yan gently put the dagger down on the table, and still bowing, backed out of the Great Hall.
King Zheng stood up and walked out of the Great Hall. The sky was perfectly clear and he saw the pale white moon in the blue sky like a delicate dream left behind by the night.
"Jing Ke," he called after the a.s.sa.s.sin still descending the steps. "Does the moon appear during the day often?"
The a.s.sa.s.sin"s white robe reflected the sunlight like a bright flame. "It"s not unusual to have the sun and the moon appear in the sky simultaneously. On the lunar calendar, between the fourth and twelfth days of each month, it"s possible to see the moon at different times during the day as long as the weather is good."
King Zheng nodded. "Oh, not an uncommon sight," he muttered to himself.
Two years later, King Zheng summoned Jing Ke to an audience.
When Jing Ke arrived outside the palace in Xianyang, he saw three officials being marched out of the Great Hall by armed guards. Having been stripped of the insignia of their rank, their heads were bare. Two of them walked between the guards with faces drained of blood while the third was so frightened that he could no longer walk and had to be carried by two soldiers. This last man continued to mumble, begging King Zheng to spare his life. Jing Ke heard him muttering the word "medicine" a few times. He guessed that the three men had been sentenced to death.
King Zheng"s mood was jovial when he saw Jing Ke, as though nothing had happened. He pointed to the three departing officials and said, by way of explanation, "Xu Fu"s fleet has never returned from the East Sea. Someone has to be held responsible."
Jing Ke knew that Xu Fu was an occultist who claimed that he could go to three magical mountains on islands out in the East Sea to find the elixir of eternal life. King Zheng gave him a large fleet of ships loaded with three thousand youths and maidens and heaped with treasure, gifts for the immortals that held the secret of eternal life. But the fleet had set sail three years ago, and not a peep had been heard from him since.
King Zheng waved the sore topic away. "I hear that you"ve invented many wonders in the last couple years. The new bow you designed can shoot twice as far as the old models; the war chariots you devised are equipped with clever springs to ride smoothly over rough ground without having to slow down; the bridges whose construction you supervised use only half as much material, but are even stronger-I"m very pleased. How did you come up with these ideas?"
"When I follow the order of the Heavens, all things are possible."
"Xu Fu said the same thing."
"Sire, please permit me to be blunt. Xu Fu is nothing more than a fraud. Casting lots and empty meditation are not appropriate ways to understand the order of the universe. Men like him cannot understand the way the Heavens speak at all."
"What is the language of the Heavens then?"
"Mathematics. Numbers and shapes are the means by which the Heavens write to the world."
King Zheng nodded thoughtfully. "Interesting. So what are you working on now?"
"I"m always striving to understand more of the Heavens" messages for Your Majesty."
"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 compa.s.s or other tools to a.s.sist 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 that 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 circ.u.mference 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 ill.u.s.tration 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 circ.u.mscribing 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 ... 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 ox hide pa.s.sed 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 cutting the rope to drop it, you will have to sit below it so that it can be your tombstone."
Jing Ke lifted his eyes to gaze at the gigantic suspended stone that filled his field of vision. Against the moving clouds in the sky, the dark ma.s.s appeared oppressive.
Jing Ke turned to the king and said, "Your Majesty had spared my life once, and even if I could finish the calculations in time, it wouldn"t be enough to erase the crime of my attempt on your life. I"m not afraid to die. Please give me five more days to think about this. If I still can"t come up with a plan, I"ll sit under the monument willingly."
Four days later, Jing Ke requested an audience with the king, which was immediately granted. Calculating the circular ratio was the most important task on the king"s mind.
"From your expression, I gather that you have indeed come up with a plan." The king smiled.
Jing Ke did not answer directly. "Your Majesty, you once said that you would give me all the resources that I require. Is that still the case?"
"Of course."
"I need three million men from your army."
The number did not astonish the king. His eyebrows only lifted briefly. "What kind of soldiers?"
"The common soldiers under your command now are sufficient."
"I think you should be aware that most of the soldiers in my army are illiterate. In two years, you cannot possibly teach them complex mathematics, let alone finish the calculations."
"Sire, the skills they would need can be taught to the least intelligent solider in an hour. Please give me three soldiers so that I can demonstrate."
"Three? Only three? I can easily give you three thousand."
"I only need three."
King Zheng waved his hand and summoned three soldiers. They were all very young. Like other Qin soldiers, they moved like order-obeying machines.
"I don"t know your names," Jing Ke said, tapping the shoulders of two of the soldiers. "The two of you will be responsible for number input, so I"ll call you "Input One" and "Input Two."" He pointed to the last soldier. "You will be responsible for number output, so I"ll call you "Output."" He shoved the soldiers to where he wanted them to stand. "Like this. Form a triangle. Output is the apex. Input One and Input Two form the base."
"You could have just told them to stand in the Wedge Attack Formation," King Zheng said, glancing at Jing Ke with a smile.
Jing Ke took out six small flags-three white, three black-and handed them out to the three soldiers so that each held a black flag and a white flag. "White represents zero; black represents one. Good. Now, listen to me. Output, you turn around and look at Input One and Input Two. If they both raise the black flags, you raise the black flag as well. Under all other circ.u.mstances, you raise the white flag. Specifically, there are three such conditions: Input One is white and Input Two is black; Input One is black and Input Two is white; Input One and Input Two are both white."
Jing Ke repeated the instructions one more time to be sure that the three soldiers understood. Then he began to shout orders. "Let"s begin! Input One and Input Two, you can raise whichever flag you want. Good. Raise! Good. Raise again! Raise!"
Input One and Input Two raised their flags three times. The first time they were black-black, the second time white-black, and the third time black-white. Output reacted correctly each time, raising the black flag once and the white one twice.
"Very good. Sire, your soldiers are very smart."
"Even an idiot would be capable of this. Tell me, what are they really doing?" King Zheng looked baffled.
"The three soldiers form a component for a calculation system, which I call an AND gate. If both numbers input into the gate are one, the result output is also one; otherwise, if one of the numbers input is zero, such as zero-one, one-zero, or zero-zero, the result output is zero." Jing Ke paused to let King Zheng digest the information.
The king said impa.s.sively, "All right. Continue."
Jing Ke turned to the three soldiers again. "Let"s form another component. You, Output, if you see either Input One or Input Two raise a black flag, you raise the black flag. There are three situations where that will be true: black-black, white-black, black-white. When it"s white-white, you raise the white flag. Understand? Good lad, you"re really clever. You"re the key to the correct functioning of the gate. Work hard, and you will be rewarded! Let"s begin operation. Raise! Good, raise again! Raise again! Perfect. Your Majesty, this component is called an OR gate. Whenever one of the two inputs is one, the output is also one."
Then, Jing Ke used the three soldiers to form what he called a NAND gate, a NOR gate, an XOR-gate, an XNOR-gate, and a tristate gate. Finally, using only two soldiers, he made the simplest gate, a NOT gate: Output always raised the flag that was opposite in color from the one raised by Input.
Jing Ke bowed to the Emperor. "All calculating components have been demonstrated. This is the scope of the skills that the three million soldiers must learn."
"How can you perform complex calculations using such simple, childish tricks?" King Zheng"s face was full of distrust.
"Great King, the complexity in everything in the universe is built up from the simplest components. Similarly, an immense number of simple components, when structured together appropriately, can generate extremely complex capabilities. Three million soldiers can form a million of these gates I"ve just demonstrated, and these gates can be put together into a whole formation capable of any complex calculation. I call my invention a calculating formation."
"I still don"t understand how the calculations will be carried out."
"The precise process is complicated. If Your Majesty continues to be interested, I can explain in detail later. For now, it"s enough to know that the operation of the calculating formation is based on a novel method of thinking about and writing down numbers. In this method, only two numerals, zero and one, corresponding to the white and black flags, are needed. But this new method can use zero and one to represent any number, and this allows the calculating formation to use a large number of simple components to collectively carry out high-speed calculations."
"Three million is almost the entirety of my army, but I will give them to you." King Zheng sighed meaningfully. "Hurry. I"m feeling old."
A year pa.s.sed.
It was another beautiful day with the sun and the moon both out. King Zheng and Jing Ke stood together on a high stone dais, with the king"s numerous ministers in rows behind them. Below them, a magnificent phalanx of three million Qin soldiers stood arrayed on the ground, the entire formation a square three miles* on each side. Lit by the freshly risen sun, the phalanx remained still like a giant carpet made of three million terra-cotta warriors. But when a flock of birds wandered above the phalanx, the birds immediately felt the potential for death from below and scattered anxiously.
Jing Ke said, "Your Majesty, your army is truly matchless. In an extremely short time, we have completed such complex training."
King Zheng held on to the hilt of his long sword. "Even though the whole is complex, what each soldier must do is very simple. Compared to the military training they went through, this is nothing."
"Then, Your Majesty, please give the great order!" Jing Ke"s voice trembled with excitement.