Chapter 773: A Single Math Problem Sets Everything in Motion

He had barely managed to solve that problem using the Stacked Piles method when the next several problems all required the Dayan Seeking Unity and Stacked Piles algorithms.

These problems touched on practical matters of engineering, corvée labor, and military logistics — without these algorithms, they simply could not be answered.

By this point, the arithmetic problems were mostly drawn from the Mathematical Treatise in Nine Sections and Sunzi's Mathematical Manual, reaching a deeper level than The Nine Chapters on the Mathematical Art. Huang Bowen had even skipped two problems, waiting to solve them when he had time.

For instance, this problem read: "A mighty camp stands west of the garrison, its troop count unknown within. Three thousand six hundred forty bowls, used to the last with none to spare. Three men share one bowl of rice, four men share one bowl of soup. Pray, sir of clear reckoning, calculate how many soldiers are in the camp."

Though not terribly difficult, time was limited, and he could not linger long on any single problem. He solved the ones below first — after all, these surplus-and-deficit problems were tedious and time-consuming.

Huang Bowen was already past thirty, yet still unmarried. Though his parents had hoped since his childhood that he would study diligently and earn an official degree, he remained a mere Tongsheng after all these years.

It was precisely because he had been obsessed since childhood with what others dismissed as a petty craft — arithmetic — that his eight-legged essays were poorly written, and naturally he could not even pass the Licentiate examination.

Moreover, his father had also been a down-and-out Licentiate. The family lived in hardship, barely scraping by with enough for food and clothing. After both his parents passed away, his circumstances grew even more straitened, and he survived only by traveling the country peddling goods. The bitterness and suffering of life had numbed him — yet deep in his heart, how could he be without dreams?

The Xuanfu Garrison's wide recruitment of clerks, with its emphasis on practical arithmetic, had given him hope. The opportunity was right before his eyes — how could he not fight with everything he had?

He breathed on his hands, stiff and numb from the cold, warming them over the brazier beside the desk, then vigorously rubbed his face. He refocused his mind and continued solving the problems below.

Then he froze, staring at the next problem. "A geometry problem?"

Liu Dongyang, seated several rows ahead of him, also narrowed his eyes. "A geometry problem?"

Liu Dongyang's family background and life experience were somewhat happier than Huang Bowen's. After all, both his parents were still alive, he had taken a wife, and he had a son and a daughter — a life that could be called full and content. His ancestors had even once been prosperous, which had enabled him to attain the Licentiate degree and live for a time the life of a young master, with clothes brought to him and meals served at his call.

Liu Dongyang had received a good family education from childhood and likewise possessed a keen interest in arithmetic. Early on, he had discovered on his father's bookshelf the ancient mathematical classic The Nine Chapters on the Mathematical Art, and from then on he had been captivated by mathematics.

Later, he collected quite a number of arithmetic books, even acquiring a copy of Euclid's Elements, translated by Xu Guangqi under the title Jihe Yuanben.

The respective strengths and characteristics of Chinese and Western mathematics broadened Liu Dongyang's horizons immensely.

Western learning emphasized rigorous logic, clear reasoning, and proceeding layer by layer to reach a final result. Chinese learning leaned toward solution methods, stressing computational technique — whatever the process, the result was what mattered. It somewhat resembled the quality-oriented education of later ages, while Western learning was more akin to exam-oriented education.

Yet each had its own allure, and the utterly different mathematical ways of thinking from the two traditions left Liu Dongyang lost in the ocean of mathematics. While studying The Nine Chapters on the Mathematical Art, he simultaneously absorbed the new approaches of the Elements, and his mathematical attainment grew increasingly profound by the day.

But the good times did not last. Liu Dongyang's family fortunes declined, and he had no choice but to turn to trade, scraping together a meager living to support his family. After all, being good at mathematics did not necessarily mean one was good at business.

When the Xuanfu Garrison recruited clerks, Liu Dongyang's family was doing business in the capital. After only a brief discussion among themselves, they resolutely packed up the entire household and came to apply.

In the Elements, Xu Guangqi had designated "form studies" by the name jihe. Within it were one translated term after another — "parallel lines," "triangle," "opposite angles," "right angle," "acute angle," "obtuse angle," "similarity," and other such Chinese terminology, all of which would become thoroughly familiar in later ages.

So the moment he saw this problem, a thought surfaced in Liu Dongyang's mind: "A geometry problem..."

Like Huang Bowen, he too skipped the problem. In truth, it was not only Liu Dongyang and Huang Bowen — Li Tanran, Zhao Zhongju, and the others were all skipping problems just the same.

They wanted to see whether there were problems further on that they could solve. After all, unlike yesterday's examination, today's arithmetic problems still had patterns one could follow.

Upon seeing the geometry problems, each man brought out his carpenter's square and compass. When they had entered the examination hall, the examiners had also issued pencils. The carpenter's square, also called a try square, was mostly used by woodworkers, but for solving arithmetic problems, the square was naturally needed as well — and the compass had been in use since as early as the Xia dynasty.

Huang Bowen studied this problem carefully: "Suppose there is a circular city, its circumference and diameter unknown, with gates opened on all four sides, and broad crossroads running vertically and horizontally outside each gate. The head of the northwest crossroads is designated the Qian point, the head of the northeast crossroads the Gen point, the head of the southeast crossroads the Xun point, and the head of the southwest crossroads the Kun point.... It is asked: Two men, A and B, both stand at the Qian point. B walks east three hundred twenty bu and stops. A walks south six hundred bu and sees B. What is the diameter in li?"

"The answer must include the solution method and calculation draft."

Huang Bowen exhaled deeply. "I shall answer it with the circle-cutting method!"

Using his carpenter's square, he drew a triangle on the draft paper, labeling the three vertices as Tian, Di, and Gan, then used the compass to draw an inscribed circle representing the circular city. He calculated: "It is said that all methods of number arise from the circle and the square; the circle arises from the square, the square arises from the try square, and the try square arises from nine nines, eighty-one. Therefore, bend the try square: take the base as three, the height as four, the hypotenuse as five..."

He constructed the diagram with base three and height four, designating the center of the inscribed circle as Xin. The vertical line passing through the center intersected the triangle and the inscribed circle at the Ri, Nan, and Bei points from top to bottom. The horizontal line passing through the center intersected the triangle and the inscribed circle at the Chuan, Dong, and Xi points from left to right, and so on. He separately calculated the base and the height, then sought the hypotenuse.

Liu Dongyang also drew a deep breath and began sketching his draft diagram. "I set a right triangle, designating the three points as A, B, and C..."

Huang Bowen clattered away on the abacus. "From the base and height, seek the hypotenuse: multiply the base by the height, double it to serve as the dividend..."

Looking at the result on the abacus, he picked up his brush with satisfaction and wrote it down. At that moment, Liu Dongyang had also used Euclidean geometric formulas to calculate the base and height, then applied the gougu theorem to obtain the result. He wrote: "Answer: the city's diameter is two hundred forty bu."

Although the geometry problems had by now reached content from the Zhoubi Suanjing, the first problem was relatively simple. Even Zhao Zhongju, after dawdling and drawing diagrams for ages, finally solved it as well.

The next few geometry problems were relatively easy, such as this one: "Now there is a bamboo one zhang tall. Its tip bends and touches the ground, three chi from the base. What is the height of the broken section?"

Huang Bowen and Liu Dongyang, using Chinese and Western methods respectively, both solved it at the same time.

But the problems below grew difficult. Continuing from the circular city above, it asked: "It is said: walking south four hundred eighty bu from the west gate, there is a tree; walking east two hundred bu from the north gate, one sees it. What is the diameter in li?"

Huang Bowen rubbed his face vigorously for a moment, then deployed the Celestial Element method. "I set the Celestial Element, one, as the radius. Place the southward-walking steps on the earth..."

He calculated: "Multiply the two walking steps to make the dividend; add the two walking steps to make the linear term; one step is the constant divisor. Obtain the radius."

Liu Dongyang was also drawing a diagram. "I set the radius as the unknown..."

Zhao Zhongju turned back to work on it again. Li Tanran finally solved that chicken problem and pressed forward with renewed vigor.

"Now there is an area, multiplied by the sum, subtract the area; the remainder, multiplied by the breadth, add the sum, yields one hundred seventy thousand one hundred sixty-two bu. It is only said that the sum is the augmented dividend, four is the augmented linear term, three is the upper adjacent term, two is the augmented lower adjacent term, one is the positive corner term. Extract the cube root, as one-fourth of the breadth. Question: what are the length and breadth?"

Huang Bowen answered: "The breadth is twelve bu, the length thirty bu."

"Now there is the product of the yellow square and the rectangular area, obtaining twenty-four bu. It is only said that the sum of the height and hypotenuse is nine bu. Question: what is the base?"

Huang Bowen answered: "Three bu."

"Now there is the square of the height minus the difference between the hypotenuse and the difference, equal to the product of the height and the base. It is only said that the square of the base plus the sum of the hypotenuse and the difference is the same as the product of the base and the hypotenuse. Question: what is the height?"

Huang Bowen set the Celestial Element, one, as the height, and the Earth Element, one, as the sum of the base and hypotenuse. Finally, he answered: "Four bu."

He felt waves of dizziness wash over his head, and at his temples, a throbbing, stabbing pain pulsed. Liu Dongyang also set down his brush and closed his eyes briefly to rest. But Huang Bowen shook his head and looked at the next problem: "Now there is the difference between the height and hypotenuse divided by the sum of the hypotenuse and the difference, equal to the rectangular area. It is only said that the difference between the base and height divided by the sum of the hypotenuse and the difference is the same as the base. Question: what is the hypotenuse?"

Huang Bowen struggled to sit steady and slowly exhaled. To have answered this many problems by this point in the examination filled him with deep pride. But the final problem set by the Marquis of Eternal Peace still lay ahead — he absolutely had to reach it.

He picked up his carpenter's square and pencil and drew a diagram on the draft paper. "I set the Celestial Element, one, as the base; the Earth Element, one, as the height; the Human Element, one, as the hypotenuse; the Material Element, one, as the root index!"

At last, he finished that problem, and at last, he saw the climactic final problem set by Wang Dou, the Marquis of Eternal Peace.

"In the Pacification Army, there is a general, Xiancai, skilled with firearms, and a general, Xuan, skilled with cannon. Xiancai daily shoots turtledoves and piles them up — the greater the stack, the higher the pile. Xuan laughs and says: With one cannon shot, I shall scatter your entire turtledove pile."

"It should be known that the turtledove pile is a pointed cone. It should be known that all pointed cones follow the principle of accumulation by stacking. The fundamental numbers begin from the infinitesimal and increase incrementally, and stacking them forms a flat pointed cone. Stacking a constant fundamental number forms a square; stacking fundamental numbers that are fewer above and more below forms a flat pointed cone. Square numbers begin from the infinitesimal, gradually increase, and stacking them forms a solid pointed cone."

"Stacking a constant square forms a cube; stacking squares that are fewer above and more below forms a solid pointed cone. Cube numbers begin from the infinitesimal, gradually increase and transform into surfaces, and stacking them forms a third-power pointed cone. It should be known that the bases of pointed cones of the third power and above are all square, but the four upper faces do not form flat surfaces — they become concave, and the higher the power, the more pronounced the concavity."

"It should be known that third-power numbers begin from the infinitesimal, gradually increase and transform into surfaces, and stacking them forms a fourth-power pointed cone. From this, proceeding to the infinite, lines, surfaces, and solids all follow a principle of cyclic recurrence."

"Pray, sir of clear reckoning: what is the algorithm for this pointed cone? What is the volume of the pointed cone? When Xuan strikes it with his cannon, what is the area within this circle?"

Huang Bowen stared at it, dumbfounded, his mind buzzing. Instinctively, he thought: "I shall set up the circle-cutting method, stacked piles..."

Then he froze again. Though the circle-cutting method could be used to approach the area of a circle infinitely — it is said that the finer the cutting, the less is lost; cut and cut again until it can be cut no more, and it merges with the circle, with nothing lost — yet it was insufficient to solve the problem before him.

This problem involved infinitesimal subdivision and infinite summation, yet it also set fixed conditions: "It should be known that all powers have pointed cones," "It should be known that all pointed cones follow the principle of accumulation by stacking." Then, within this framework of limit thinking, the pointed cone seemed to involve curves and also motion.

It further required calculating the area of Zhao Xuan's cannon-strike motion, and all of this seemed, in turn, to be mutually interactive.

Because the cone continuously expands and contracts, the area covered along the cannonball's trajectory is also different.

This... how does one calculate the area?

All manner of images spun through his mind. Huang Bowen sat in a daze, the buzzing in his head growing louder.

"My son, passing the civil examinations is the proper path!"

His mother's eyes, moist with tears.

"Wen'er, if you go on like this, what will become of you?"

His father, always stern, said nothing to him now — only shook his head, a flash of disappointment in his eyes, then turned and walked away.

"Huang Bowen, after all these decades you're still only a Tongsheng?"

The contemptuous, mocking stares cut to the bone — humiliation, blows, past events flashing swiftly before his eyes.

"No, I will solve this problem and win fame across the land in one stroke!"

Huang Bowen struggled. He reached for the abacus, but bursts of golden sparks swam before his eyes, making him unsteady. He braced himself against the table with all his strength, yet his body still swayed and tottered. Suddenly, he spat out a mouthful of fresh blood. The world spun before him, and he could hold on no longer — he collapsed to the ground just like that.

After that day's clerk examinations, Xuanfu Garrison quickly boiled over with excitement — especially after this "Two Generals' Cannon Problem," also called the "Wang's Arithmetic Problem," was released. Xuanfu Garrison plunged into a garrison-wide problem-solving frenzy.

This final, high-stakes problem personally set by the Marquis of Eternal Peace drew the interest of countless people. From what was learned afterward, of the several thousand scholars who took the exam, not one could produce a solution — they could not even propose a viable approach. Some even coughed blood from the strain, drawing widespread concern.

Fortunately, he was not seriously hurt, which set everyone's minds at ease.

Difficulty is what makes a challenge. Who would not be stirred by the prospect of founding a new school and becoming a grandmaster? Not only the primary, secondary, and tertiary academy students of Xuan Garrison, but even the garrison's officials, commoners, gentry, townsfolk, and soldiers all worked calculations on paper in their spare time. Books on arithmetic of every kind sold like wildfire throughout Xuanfu Garrison.

Even Li Banghua gathered Zhu Zhifeng, Ma Guoxi, Wu Zhi, and others to discuss this problem together.

Zhao Xuan was now famous too.

In truth, Zhong Xiancai was just as famous as he was, but Zhong Xiancai was currently at Guihua City, outside the storm's center of public attention. Zhao Xuan, however, was in the garrison city, and many officers and officials who saw him would tease him: "Brother Zhao... with one cannon shot from me, your entire flock of doves is routed."

Zhao Xuan paid no mind to everyone's teasing. His whole heart was absorbed in this arithmetic problem designed by Wang Dou. He keenly sensed the Generalissimo's many painstaking intentions behind this problem, and even more keenly foresaw that this arithmetic problem was the key to giving his own cannon battalion wings.

As the commanding officer of the cannon battalion, Zhao Xuan had some modest study of arithmetic, and among his subordinate officers, clerks, and others, those proficient or partially versed in mathematics were not few.

He gathered his finest troops and strongest officers and spent days studying this "Wang's Arithmetic Problem." Yet looking at the dense equations and symbols covering the blackboard — his subordinates had used chalk to fill several blackboards with calculations, rows upon rows of four-variable formulas laid out — they still could not even find the door to solving it.

Zhao Xuan's head throbbed as if about to split. There is a saying: cut the Gordian knot with one swift stroke. He keenly felt he must first find a single key point — but where was that key point?

Staring blankly at that math problem, he felt the urge to vomit blood just the same.

Very soon, all the questions from this clerk examination appeared in that issue of the Xuan Garrison Times. The "Two Generals' Cannon Problem" was prominently included. Paired with the shocking content of a scholar coughing blood from the exam, the storm it stirred swept outward beyond the garrison like a hurricane.

Wherever the newspaper reached, every place seemed to boil over. When it reached the capital, the whole city was likewise thrown into an uproar. Teahouses and taverns buzzed with discussion, and even illiterate commoners would mention a word or two: "Did you hear? At the Xuanfu Garrison clerk exam, a scholar calculated until he coughed blood..."

"Really? What kind of problem is so hard?"

"I hear the Marquis of Eternal Peace personally set it. They're all calling it Wang's Arithmetic Problem now..."

Countless people grew interested in the Xuanfu Garrison exam questions. For the first time, many felt that arithmetic could be so fascinating.

The Xuanfu Garrison clerk examinations naturally drew the attention of the capital's hundred officials. They all read the newspaper, analyzing the questions one by one — though they unanimously ignored the earlier classical literature questions. How many of the questions set by Xuanfu Garrison could they themselves answer correctly?

Could it be that after decades of bitter study, they were not even qualified to serve as clerks in Xuanfu Garrison?

This was too devastating — and too terrifying.

Moreover, the policy essay questions set by Xuanfu Garrison were each more pointed than the last. Take the very first one: "A certain prefecture at the dynasty's founding had a population of over three hundred forty thousand, with summer tax and autumn grain totaling over one hundred forty thousand shi per annum. Now it has a population of over six hundred thousand, yet summer tax and autumn grain total fifty thousand shi per annum. What is the reason? Please explain."

Any clear-sighted person could see at a glance that this touched upon the gentry problem. Without the gentry's tax evasion and resistance, without their vast annexation of land, how could the population increase while tax grain decreased?

People speak of the mandate being exhausted — but is it not, in truth, the problems of land, population, and finances erupting to their breaking point?

Yet such questions have been evaded through the ages. Everyone understood the truth in their hearts; no one would raise it openly. Xuanfu Garrison, however, presented it as an exam question this time. Could it be that the Marquis of Eternal Peace intended to challenge the privileged exemptions that officials and gentry had enjoyed for millennia? What game was he playing?

The several questions that followed were all of the same nature, which stirred discontent in many hearts.

But Wang Dou was now at the height of his power, his troops strong and his horses vigorous — who dared openly challenge his authority? Did they not know that Li Banghua, Left Censor-in-Chief of the Chief Surveillance Bureau, had been banished to a frontier military garrison by His Majesty the Emperor precisely because he targeted Marquis of Eternal Peace Wang Dou?

And so the officials ignored these questions, shifting their attention entirely to the arithmetic problems at the end — and deliberately guided public opinion in that direction.

Openly, they still dismissed these arithmetic problems with disdain, stressing that a scholar's energy should be devoted to the classics. As Senior Grand Secretary Zhou Yanru declared on behalf of the assembled officials: "Mathematics and computation are but minor arts. The Four Books and Five Classics, the subtle words and profound meanings of the sages — that is the grand and proper path."

His words appeared in the newly founded Imperial Ming Times, yet privately, Zhou Yanru and his advisors were studying these arithmetic problems with keen interest — especially that "Wang's Arithmetic Problem." The more they studied it, the more unfathomably profound it seemed.

To solve it, he even dug out from his piles of old papers the Nine Chapters on the Mathematical Art and other books he had long since forgotten, and began to study them meticulously.

If Zhou Yanru was like this, the hundred officials of the capital followed the trend all the more, tossing out a few mathematical terms in their leisure time, as though without doing so they would fall behind the fashion.

"Click-clack, click-clack..."

The sound of abacuses was like a sudden rainstorm. In the Eastern Warm Pavilion, the large gilded plaque reading "Rising Before Dawn, Eating After Dark" hung high. Inside the pavilion, the Chongzhen Emperor stood with his hands behind his back, holding the Xuan Garrison Times, merely staring blankly outside in a trance.

Within the pavilion, numerous eunuchs were gathered, led by Wang Dehua, the Seal-holding Director of the Directorate of Ceremonial, busily calculating under great tension.

Before them, long narrow tables were arranged in rows, on which abacuses were placed — all the large, long, black abacuses used by the Ministry of Revenue. These great abacuses were extremely long, some reaching five or six meters, and their counting units were correspondingly vast.

One could see from the digits marked on them: from units, tens, hundreds, thousands, ten-thousands, and continuing all the way back — hundred-millions, trillions, jing, gai, zi were all included, and some were even marked with the extreme units rang, gou, jian, zheng, zai.

Of course, in actual calculation, reaching hundred-millions or trillions was already quite good.

At this moment, the many eunuchs and abacuses were arrayed in a single line, beads clicking and clacking as they worked them — a magnificent sight.

These large abacuses were also shared by several people working one together, to boost computing power.

Some eunuchs were also writing and calculating on blackboards. The blackboards and chalk from Xuanfu Garrison had made their way into the capital as well, and were indeed more convenient for drafting calculations.

How could the Chongzhen Emperor not care about the Xuanfu Garrison clerk examinations? After the newspaper arrived, he studied the questions one by one. Wang Dou's earlier policy essay topics actually pleased him greatly, for they reminded him of recent events in the capital.

Chen Xinjia had proposed large-scale training of new armies in the capital, which the Chongzhen Emperor strongly approved of — but training troops required provisions and pay, and the ministers discussed this. Senior Grand Secretary Zhou Yanru proposed having wealthy households and the hundred officials contribute aid, and even prepared a yellow-silk register for the purpose.

Zhou Yanru's plan seemed perfect: at this critical juncture of peril for the court, surely the capital's gentry, officials, and wealthy households would all generously open their purses to meet the nation's urgent need. Yet — what happened in reality?

The officials all passed the buck to one another. No one was willing to donate. In the end, under the Emperor's implicit prompting, Senior Grand Secretary Zhou Yanru took the lead by donating ten thousand taels of silver. Then the Grand Secretaries donated — one ten thousand, another five thousand. Down to the hundred officials, it became one a thousand, another five hundred. The final sum obtained was a mere drop in the bucket.

The Emperor was greatly displeased. The officials then said the ennobled relatives were wealthy and could be ordered to assist with provisions — especially the imperial relative Jiading Bo Zhou Kui, who had just been promoted to Marquis. It was said Zhou Kui was richly endowed, and as an imperial relative, he should take the lead. The Chongzhen Emperor thought this reasonable. Zhou Kui and the others were, after all, relatives, and Zhou Kui himself was the father of his own consort — surely he would help him in this matter.

So he dispatched the eunuch Xu Gao to proclaim an edict seeking aid: "In weal and woe we are bound together; you must coordinate your efforts and make arrangements, to prepare for emergencies."

Zhou Kui, however, replied: "How could this old minister have much money?"

Xu Gao tearfully exhorted him again and again. Zhou Kui had no choice and in the end donated ten thousand taels. Then each ennobled relative donated ten thousand or five thousand, each high eunuch ten thousand or five thousand. The wealthy households of the capital, seething with helpless frustration, likewise dug into their purses, throwing the entire capital into noisy commotion. In the end, only three hundred thousand taels of silver were raised — far short of even the initial investment for the new army. How could training begin?

And because of this, Zhou Yanru was constantly impeached.

Then there was Hong Chengchou, who had been ordered to reorganize the capital garrison. Though his methods were gentle yet seasoned, advancing step by step, there were always those whose interests were harmed and who were dissatisfied. A storm of impeachment was about to rise, and whether Hong Chengchou would follow in Li Banghua's footsteps was extremely hard to say.

Furthermore, Cao Bianjiao and Wang Tingchen returned to their garrisons at the start of the intercalary month and nearly incited a mutiny. It turned out that during their long absence from their towns, and with heavy casualties among their subordinates, local officials and gentry had seized the opportunity to swallow up the New Army's mature farmland. Cao Bianjiao flew into a rage and executed a large number of the local gentry who had seized the land, causing public sentiment to boil over.

Memorials attacking Cao and Wang came again like clouds, and there were even cold, whispered words spreading: "The New Army is truly a scourge upon the Great Ming, only knowing how to contend with the people for profit..."

Although the Chongzhen Emperor kept all the attacking memorials in the palace without issuing them, he was still utterly exhausted in mind and body. He had intended to let the two men continue organizing and training the New Army in places like Yutian, but for lack of funds and provisions, he had to abandon the idea. In the end, he decided to grant Cao Bianjiao and Wang Tingchen's request and transfer them to Liaodong, to guard places like Yizhou.

All these matters gave the Chongzhen Emperor a splitting headache. The money in his hands, in particular, vexed him beyond measure. But was the Great Ming truly without money? Holding the Xuanfu Times in his hand, the Emperor disagreed, and felt even more deeply that Xuanfu Garrison's essay topic this time had been excellently chosen.

However, this was something he could only ponder in his heart. The Great Ming's civil examination system was extremely mature; for several hundred years, scholars had all followed these exams. A sudden change would only throw the situation into chaos upon chaos. The matter of the New Army was already headache-inducing enough; the Chongzhen Emperor did not want to add more vexations.

Still, the Emperor was very curious about the "Wang Family Arithmetic Problem" within. He had heard people say this problem was extremely difficult, and that officials and scholars throughout the capital had all calculated it privately, yet not a single person could solve it.

The Chongzhen Emperor simply could not believe that, with his hundreds of millions of subjects and countless talents, he could not even solve a mere arithmetic problem posed by a frontier warlord. He had Wang Dehua find eunuchs in the palace skilled in arithmetic and resolved to solve this problem.

Yet after half a day, the crowd still showed no sign of stopping their calculations; the storm-like clatter of abacus beads continued without pause.

The Chongzhen Emperor furrowed his brow, paced back to his desk, and flipped through the Imperial Ming Times lying on it. Compared to the Xuanfu Times, the Imperial Ming Times was essentially a reprint of the official gazette, rather dry and flavorless. Few people bought it voluntarily, so after its launch, it was apportioned to every ministry and household, with a monthly newspaper fee deducted from every official's salary.

He reviewed memorials for a while longer, but the sound of abacus beads in the hall still did not cease. The Chongzhen Emperor grew impatient. A mere arithmetic problem, and so many people still could not solve it?

"Well? Is there still no result?"

The Emperor's brow furrowed deeper. His impatience was matched by an even greater curiosity. Was this "Wang Family Arithmetic Problem" truly so difficult?

Wang Dehua smiled bitterly and begged forgiveness: "Your servants are incompetent, please punish us, Your Majesty..."

Wiping his forehead, he said: "It is truly that the Marquis of Yongning's problem, this problem..."

In the bitter cold, fine beads of sweat had appeared on his head, making his sleek, shiny face glisten — clearly the result of excessive mental exertion. Finally, he offered the Emperor a suggestion: "Why not send this problem to the Imperial Academy? After all, they are proper scholars, and within it there are even those who specialize in the arithmetic discipline."

The Imperial Academy in the capital was located inside Anding Gate in the eastern part of the city. Though not as vast as the Imperial Academy in Nanjing, it still had a multitude of students. Besides studying the Four Books and Five Classics, they also had to study concurrently, as well as courses in laws and decrees, calligraphy, and mathematics. It even had dedicated Erudites for the disciplines of law, calligraphy, and mathematics.

But truth be told, these people held low and humble status. Even the highest Erudite was no more than a junior ninth rank, how could they compare to the Erudites of the Five Classics, each of whom was at least a senior fifth rank? From this, one could also see the lowly status of the law, calligraphy, and mathematics disciplines in the Great Ming.

Yet today, the Erudite of the mathematics discipline suddenly received an imperial edict, ordering them to calculate that "Wang Family Arithmetic Problem." It must be solved, to demonstrate the imperial court's strength and authority.

This caused a sensation throughout the entire Imperial Academy. The mathematics Erudite dared not be negligent and immediately gathered all the assistant instructors, lecturers, and elite students of the mathematics discipline to calculate it by imperial order.

But despite their utmost efforts, they encountered the same problem as Zhao Xuan: where was the entry point? With dynamic solid geometry, where should one begin?

Watching the mathematics Erudite at a complete loss, his face pale and swaying unsteadily as he calculated, the Chancellor of the Imperial Academy and the Vice-Chancellors waiting nearby were greatly alarmed. Could this problem be so profound? That even their own academy's mathematics Erudite was utterly helpless?

Since it concerned an imperial edict, they dared not slack off, continuously throwing academy students proficient in arithmetic into the calculation. In the end, the entire Imperial Academy was mobilized.

"Fathers, our opportunity has come..."

The speaker was a Westerner in the Imperial Observatory, with a high nose and deep-set eyes, wearing Great Ming official robes. He was about fifty years old, with a thick, full beard, and deep eyes that occasionally flickered with the light of wisdom. This was the Western missionary Johann Adam Schall von Bell, currently serving in the Imperial Observatory.

This German, born in the twentieth year of the Wanli reign, was originally named Johann, with the surname Adam. He had studied at the Three Kings' Crown Secondary School founded by the Jesuits, and after further studies at the Roman College and the Academy of the Lynx-Eyed in Rome, he had crossed east to the Great Ming at the end of the Wanli era, together with Johann Schreck, Giacomo Rho, and several other missionaries, under the nominal dispatch of the Portuguese government.

At this time, the missionaries were continuing Matteo Ricci's evangelistic methods of "expelling Buddhism to supplement Confucianism" and "uniting with Confucianism to surpass Confucianism." After setting foot on Chinese soil, everyone had to study the Chinese language and culture, and research China's classics, histories, and ethics, in order to find a breakthrough point for their mission.

Ricci's approach had achieved great success for a time. Toward these Western missionaries, each of whom adopted a Chinese name, wore Chinese robes, and were versed in astronomy above and geography below, thoroughly familiar with Chinese classical texts, the literati and scholar-officials of the time held them in great esteem and trust, even producing high-ranking Jesuit members like Xu Guangqi.

However, after Ricci's death, some zealous missionaries felt that Ricci's thinking had accommodated the Chinese too much, affecting the "purity" of Catholicism and making the conversion of followers too slow. So they began to alter Ricci's missionary route, adopting more radical methods of evangelism.

They firmly rejected Confucian thought and strictly forbade Chinese converts from offering sacrifices to Heaven, to ancestors, or to Confucius, arousing immense antipathy and suspicion among the people of the time. Some then said: "There are men like Li Madou and Ai Rulüe, who claim to come from the Great West, borrow the name of Confucian learning, attack Buddhist teachings as false, and call themselves the Heavenly Lord's Teaching, also called the Heavenly Learning."

They also said: "These people substitute one thing for another, stealthily replacing our sacred teachings' concepts of the Lord on High and the Holy Scriptures. This is a presumptuous intent to use barbarians to change the Xia. Furthermore, the Heavenly Lord's Teaching forbids the keeping of spirit tablets for sovereign and parents, and forbids sacrifices to ancestors, fathers, and mothers. This is truly leading the world to become a place without sovereigns and fathers."

After the Nanjing Incident, all missionaries were expelled and could only reside in Macau. Johann was determined to change this situation. He likewise changed into Chinese robes, chose a name from among them, taking the Chinese name Tang Ruowang. Seizing the opportunity of the war with the Later Jin, he and the other missionaries arrived in the capital under the title of military advisors.

Thereafter, Tang Ruowang continued Ricci's route, evangelizing gently and permitting converts to perform ancestral rites and venerate Confucius, and his work made great progress. He alone had personally baptized as many as fifty people, among them members of the imperial clan, the imperial family, and many eunuchs.

Because of his achievements, in the thirteenth year of the Chongzhen reign, he was appointed by the Church as the head of the Beijing mission district. And because Tang Ruowang was proficient in mathematics and astronomy, he was appointed by the Chongzhen Emperor as an official of the Imperial Observatory, translating and compiling calendrical texts, observing celestial phenomena, and crafting instruments.

Using this opportunity, he developed converts such as the eunuch Pang Tianshou of the Imperial Horse Administration, and also brought many missionaries into the Observatory to serve. At this moment, gathered around him were several golden-haired, blue-eyed Westerners, each wearing Great Ming official robes.

Tang Ruowang and the other missionaries had been secretly paying attention to Wang Dou for a long time. Especially after Wang Dou was enfeoffed as Count and then Marquis, Tang Ruowang and the others focused even more closely on him. When the Jingbian Army went to war in Liaodong, and after they returned in triumph to the court, they had also made a point of secretly observing them.

Afterwards, Tang Ruowang sighed to those around him: "What fine, elite young men, not in the least inferior to our soldiers in Europa. The future of the Great Ming lies with them. Fathers, we must bring this Great Ming warlord into the embrace of the Lord."

But to the disappointment of Tang Ruowang and the others, Wang Dou had always been indifferent toward them, and even forbade missionaries from entering his territory.

Moreover, what alarmed Tang Ruowang and the others was that Wang Dou seemed to understand the world very well, understood these missionaries very well, and even understood Europa very well. This left Tang Ruowang and the others utterly baffled. How could a warlord who rose from a common soldier possibly know so much?

Tang Ruowang even wondered: "The Easterners have a saying: a sage sent by Heaven knows by Heaven and perceives by Heaven. Could it be that this Marquis of the Great Ming is a sage sent by Heaven?"

Beyond this, he could not explain the "mystical strangeness" surrounding Wang Dou, which also made them strive ever more tirelessly, hoping to gain Wang Dou's favor, and ultimately, ideally, his conversion.

At this moment, Tang Ruowang smiled at those beside him: "Fathers, the opportunity has come. Display your talents, the accumulated knowledge of our Europa, the crystallization of the wisdom of Archimedes, Euclid, and others. Let the people of this Eastern land be utterly astonished."

A missionary beside him laughed: "Yes, Father. Word came from the Imperial Academy yesterday that they gathered the strength of the entire school and still could not solve it. It seems the Marquis's problem still needs us to solve it. After all, mathematics is a rigorous discipline."

Another Father also laughed: "Easterners are full of romantic fantasy in their logic, while we Europeans are like Jews counting gold coins... But it is also because of this trait that we are suited to studying rigorous and logical subjects. It is time to open the eyes of these Cathayans."

Tang Ruowang said: "Father, I must correct you on one point. After my careful research, the people of the Ming nation have no relation to the Cathayans. If one must speak of a relation, the Cathayans are like the barbarian tribes of Europe's past, a complex situation much like that between the Han people and the northeastern savages now. Even the old city of Khanbaliq was merely the Tatar name for Beijingcheng. We should not utter the term 'Cathayan' to them, or it will provoke their fury. After all, that was a branch of barbarians, and now they consider themselves civilized people."

He said: "We may call them the people of Serica, or the Seres. Of course, their own terms are very complex; now they are called the Ming people, and also the people of the Central Kingdom. The more closely I study this country, the more bewildered I become."

Another Father said: "The Central Kingdom — that is what the central empire calls itself. I think they are too arrogant."

Tang Ruowang said: "This is a wealthy and powerful nation; they have the right to be arrogant... Fathers, the name is unimportant, this is a matter of great significance. We should wield the weapon of our learning, properly display our brilliance, and ultimately make these heathens look up with admiration and convert to the embrace of the Lord."

All the Fathers made the sign of the cross together, each brimming with passion. The Emperor had issued an edict, yet the students of the Imperial Academy could not solve it. If they could calculate that mathematical problem, what a wave of sensation it would cause, how many people would look up to them?

Then they could even send one of their own to Xuanfu Garrison to serve as Grand Master, seizing the chance to draw closer to the Marquis.

Tang Ruowang composed his expression, took out the Xuanfu Times in his hand, and said: "Very well, Fathers, let us begin calculating. Solve these mathematical problems in the newspaper one by one."