Chapter 129: The Mathematical Mirror, The Comprehensive Methods of Calculation, and Western Mathematics

The standard punishment for false accusation is to impose the same penalty on the accuser.

In Han times, the Book of Han equated false accusation with murder or bodily harm, explicitly stating: "For those aged eighty or above, unless guilty of false accusation or causing injury or death, all other offenses shall be pardoned."

False accusation is equivalent to verbal defamation and weapon-based assault.

In Tang times, the Tang Code Commentary on Fighting and Litigation III stipulated: "Those who falsely accuse others shall suffer the same penalty they imposed." "Counter-punishment" means applying the punishment due to the falsely accused crime onto the accuser.

Ming inherited the Tang system; the Great Ming Code imposed even harsher penalties for false accusation, increasing the sentence by three grades: for false accusation punishable by caning, add two grades to the falsely accused crime; for false accusation punishable by exile, penal servitude, or beating, add three grades. All punishments were capped at one hundred strokes and exile three thousand li; if false accusation led to a capital crime and the accused was already executed, the accuser shall be put to death. If the accused had not yet been executed, the accuser shall receive one hundred strokes and exile three thousand li, plus three additional years of penal servitude.

The Gangxian Gangxian Shilei and the Great Ming Huidian's Bureau of Censorship section also required censors not to frequent brothels, not to escort courtesans on outings, not to sneak into theaters, and prohibited false accusation, with counter-punishment increased by three grades beyond the falsely accused crime.

The censorial officials were an essential component of the Great Ming's corrective mechanism; if the censorial corps became weaponized or instrumentalized as tools to eliminate dissent and aid tyranny, the Ming bureaucracy would immediately collapse entirely.

From the actual surveillance practices of the Ming dynasty, the role of censors showed a declining trend: in the early and middle periods, they functioned well, but by the late period, they transformed from purifiers of governance into catalysts of its corruption and decay.

Zhang Zhang Juzheng had no good solution for the censorial officials; he could not act against them, or he would confirm all accusations made by Fu Fu Yingzhen against him.

It did not matter that Zhang Zhang Juzheng could not act—Zhu Zhu Yijun could.

Commander Zhao Meng, sharpening his blade, had been waiting. After dragging the four men onto the bench, he raised the imperial cane and brought it down heavily; the censors cried and wailed under each successive blow.

Zhu Zhu Yijun watched coldly, only rising after the punishment ended. He did not even glance at them, left the Huangji Gate, and returned to the Zhao Wenhua Hall to resume imperial audience.

The punishment for these four men was not yet over: after the cane, they were stripped of their official ranks and ordered to return to their home regions in retirement, forbidden from signing any official documents—this eliminated any possibility of their reinstatement. They ended up in this fate for impeaching Lu Lu Guangzu's retention in office after his father's death; they had only themselves to blame.

Soon, an imperial edict was issued to the realm: if any censor still believed the emperor had harmed his ears and eyes or suppressed upright integrity, they could continue submitting memorials. But Zhu Zhu Yijun did not wait for further consecutive petitions; the young emperor's chain of maneuvers was too cunning—fabricating charges, administering the cane, then inviting further petitions would surely not end with just one beating.

Thus, Zhu Zhu Yijun used the conflict between retention in office and mourning rites to severely crush the arrogance of the base Confucians, partially corrected the censorial faction's habit of false accusation, and restored righteousness: rectifying the unjust, purifying the moral climate.

Recently, Zhu Zhu Zaiyu was busy with a major project: compiling an official textbook on mathematics. He thought it would be simple, but the Great Ming emperor repeatedly rejected his drafts; the emperor's sole evaluation was: unsatisfactory.

It was not that Zhu Zhu Zaiyu's mathematical ability was lacking—it was that the young emperor's demands were too high.

"Uncle, what is the principle of mathematics?" Zhu Zhu Yijun, in the side hall of the Zhao Wenhua Hall, looked at Zhu Zhu Zaiyu's latest draft and rejected it again, saying with slight exasperation: "As the early Ming scholar Yan Gong wrote in his Tongyuan Suanfa: 'The principle of one root producing myriad variations is realized through the method of Tongyuan.'"

"That is the principle of mathematics."

"During the Emperor Jingtai era, Wu Jing compiled the Nine Chapters Detailed Annotations and Classified Mathematical Methods, the culmination of Ming's mathematical tradition. But what is this 'Mathematical Enlightenment' you submitted? Methods for diagnosing illness, predicting the sex of unborn children—can mathematics determine whether one will fall ill or bear sons or daughters?"

"What is mathematics? Mathematics is the language of infinite things, the only way to make all things speak. If such a textbook were circulated nationwide, how would it differ from witchcraft? Would it not invite ridicule?"

Zhu Zhu Zaiyu fell silent for a moment, then bowed his head and said: "No, Your Majesty is wise."

Zhu Zhu Zaiyu himself was dissatisfied with his Mathematical Enlightenment—he had given his all.

As the emperor's mathematics tutor, he had nothing left to teach; the young emperor's mathematical skill was extraordinary.

Zhu Zhu Yijun continued with a serious expression: "Mathematics expresses the principles of infinite things. It must be as concise and clear as possible, with a rigorous, logically structured argumentative process. It must boldly hypothesize solutions to practical problems, then carefully verify, meticulously summarize, and repeatedly test them."

"Mathematics is the solid foundation of all development; mathematics is the master text governing heaven, earth, and all things."

"Your Mathematical Enlightenment is like a village shaman dancing before the sun, unaware of what he is doing, praying to clay idols, then swallowing a talisman—like eating a steamed bun stained with blood to cure tuberculosis."

"Uncle, this is not the mathematics I want, nor the mathematics the Great Ming should have."

"Uncle?"

Zhu Zhu Zaiyu hurriedly replied: "Your servant is here."

Zhu Zhu Yijun continued: "Your Mathematical Enlightenment does not enlighten—it misleads. It teaches people how to calculate mechanically, not how the scholarly world ought to calculate, nor why they should calculate that way. Like the corrupt Confucians who chant 'the ancient kings' ways,' you teach how to act, not how one ought to act, nor why."

"More precisely: know the what, know the why, know the necessity. Your Mathematical Enlightenment only teaches the what—it lacks the why."

"I am deeply disappointed."

"Your mathematics is so simple that even a twelve-year-old like me can master it—is this even a discipline?"

Zhu Zhu Zaiyu screamed silently inside: Can you, Your Majesty, be like ordinary people? Your mathematics has already reached transcendence—why would you need to be taught?

During the six-month mathematics course, Zhu Zhu Zaiyu was numb. It was not he teaching the young emperor—it was the emperor teaching him. The emperor needed no instruction; he could calculate the exact time of winter solstice to the minute using the gnomon's shadow, just like Zhu Zhu Zaiyu and Zhang Zhang Juzheng—both were child prodigies.

"Your Majesty, is it possible that the mathematics I compiled is precisely what our Ming students can accept?" Zhu Zhu Zaiyu proposed a hypothesis: perhaps the problem was not his compilation, but the emperor's expectations—overestimating the students' capacity.

Ming students read nothing but the Four Books and Five Classics; they did not even read the Great Ming Huidian, let alone mathematics. The reason Ming students avoided mathematics was that the imperial examinations did not test mathematical ability—not only in Ming, but since Song, and even in early Tang, the mathematical examination had been a fleeting novelty, disappearing entirely by the late Tang and Five Dynasties.

Thus, after mathematics was excluded from the imperial examinations, the development of mathematics in the Central Plains nearly halted.

"Your uncle speaks wisely," Zhu Zhu Yijun nodded helplessly. His demands were indeed too high—he sought mathematical principles, while Zhu Zhu Zaiyu offered only calculation. This gap between expectation and reality was the core conflict between emperor and uncle.

"Your Majesty, should we not first let students accept basic calculation? Being able to compute numbers is already excellent," Feng Bao spoke up for the uncle. Not everyone is a born prodigy; not everyone can grasp the idea that mathematics is the master text of heaven, earth, and all things.

Walk step by step. Eat bite by bite.

Zhu Zhu Yijun considered and said: "Your uncle's mathematics lacks principle—it needs revision."

"Your Majesty, Ge Ge Shouli requests an audience outside the hall," a eunuch rushed in.

"Why is he here?" Zhu Zhu Yijun asked curiously. "Admit him."

Ge Ge Shouli entered the Zhao Wenhua Hall carrying fifty-five scrolls, placed them on the ground, and kowtowed five times with three prostrations, loudly declaring: "Your servant bows before Your Majesty. May Your Majesty live ten thousand years, ten thousand years, ten thousand ten thousand years!"

Zhu Zhu Yijun looked at Ge Ge Shouli and said: "Rise, Grand Secretary. What brings you here?"

Ge Ge Shouli smiled and said: "Your servant comes to relieve Your Majesty's worries. I have seen the Chief Minister revive ancient statutes to reform education and cultivate talent, and the Minister of Revenue apply mathematical principles to tax collection. Your Majesty also seeks mathematical talent for tax auditing. I possess a treasure of fifty-five scrolls: the New Collection of the Comprehensive Mirror of Ancient and Modern Mathematics."

"Oh? Bring it here." Zhu Zhu Yijun examined it for a long time before setting it down. "Where did this book come from?"

Ge Ge Shouli hurriedly replied: "Mathematics is indispensable in all public and private affairs. This book was written during the During Chenghua era by the Shanxi merchant Wang Wang Wensu, widely circulated among Shanxi merchants for commercial convenience. The Minister of Revenue Yang's household had a copied version. I found it novel; at the time I was Minister of Revenue overseeing state tax accounting, I copied one for personal use. Now, hearing Your Majesty and the Prince of Zheng are troubled by mathematical enlightenment, I have come to present it."

Zhu Zhu Yijun looked at the Mathematical Mirror beside him, then at Ge Ge Shouli, smiling: "So you obtained it from Minister Yang. Excellent. Grand Secretary, you have rendered great service. What reward do you request?"

"Your servant seeks no reward for relieving Your Majesty's worries. I hold the high office of Left Censor-in-Chief, overseeing remonstrance, yet I could not restrain these four reckless youths who provoked chaos before the Huangji Gate, forcing Your Majesty to harbor suspicion toward your ministers, and forcing me to bear the guilt of deceiving my sovereign."

"Mutual suspicion between sovereign and minister breeds distrust, stifles honest discourse, and undermines peace and tranquility—this is what I deeply lament."

Zhu Zhu Yijun understood: Ge Ge Shouli wanted to distance himself from the Huangji Gate petition incident.

Wu Wu Zhongxing, Zhao Zhao Yuxian, and others were censors; as head of the censorial corps, it was natural for people to suspect Ge Ge Shouli had orchestrated the incident. Ge Ge Shouli was truly innocent, so when he saw the emperor, the uncle, and the Chief Minister all troubled by mathematics, he came forward with his Mathematical Mirror.

Zhu Zhu Yijun deliberated carefully: "Grand Secretary, you misunderstand. I have always known your loyal heart. As leader of the Shanxi faction, you uphold the sovereign's authority. Today, for presenting this treasure, I appoint you Junior Tutor to the Crown Prince, and grant one of your sons the post of Secretary of the Secretariat—a special exception. Additionally, bestow upon you one embroidered python robe and one embroidered bull robe, one hundred taels of silver, four bolts of silk gauze, five thousand strings of paper money, and five bottles of wine—showing special favor, do not refuse."

The wine was Zhu Zhu Yijun's own sweet potato liquor—giving five bottles at once meant he was truly delighted.

Junior Tutor to the Crown Prince—henceforth, Ge Ge Shouli would be called "Junior Tutor Ge" in court, though the title was honorary.

"Your servant kowtows to thank Your Majesty's boundless grace," Ge Ge Shouli said, bowing again, forehead to the ground. "Your Majesty, I… am old. I can no longer relieve Your Majesty's worries."

"Huh? Are you retiring? If you fear being implicated by the censors' petition, do not worry—I am not unreasonable. Even you, the Chief Minister, the late emperor, or the Emperor Jiajing Emperor—could any of them control the censors? A group of men who risk their reputations to defy imperial edicts for private gain." Zhu Zhu Yijun thought Ge Ge Shouli feared association with the censors' protest and wished to retire for peace.

"Your Majesty, I am seventy this year—already beyond the age of rare longevity." Ge Ge Shouli said. "Truly, I am old."

"Grand Secretary, rise and speak! You are seventy? I thought you were fifty! Grand Secretary, you must have excellent health practices." Zhu Zhu Yijun stared, wide-eyed, at Ge Ge Shouli—he could not see a single sign of seventy.

"Your Majesty," Ge Ge Shouli smiled broadly, "at this age, clinging to office is to become a nuisance—greedy for position, despised by all."

"Do you suffer from chronic illness?" Zhu Zhu Yijun asked, puzzled.

Ge Ge Shouli shook his head: "No."

"Then what's the issue?" Zhu Zhu Yijun laughed. "I want to see who dares use your age against you. Do not speak of retirement again."

Ge Ge Shouli was still useful; under him, the Shanxi faction had been revitalized. If he retired now, Zhang Zhang Siwei and Wang Wang Chonggu would lose internal competition—how could Zhu Zhu Yijun let Zhang Zhang Siwei be happy?

According to Ge Ge Shouli, the Mathematical Mirror was published in Emperor Jiajing 2, widely circulated among Shanxi merchants. Did Zhang Zhang Siwei not know? If he had even a shred of deference, why did he not present it to the throne to solve the empire's urgent need? Zhang Zhang Siwei did not—Ge Ge Shouli did.

"Then I shall continue relieving Your Majesty's worries?" Ge Ge Shouli asked.

"Excellent, excellent. Sweet potato liquor—not… Guojiao—this wine is strong. Grand Secretary, drink sparingly." Zhu Zhu Yijun beamed.

This was Ge Ge Shouli's test: through presenting the book, he probed whether the emperor still saw him as a sycophant, whether he should retire to preserve himself.

Clearly, the young emperor was very satisfied with Ge Ge Shouli. He had the scholar's petty acidity, but he was no base Confucian or corrupt scholar—he had become an amused observer, spending his days watching the spectacle.

The censorial officials were too vicious for Ge Ge Shouli to control—not because he was weak, but because their roots were rotten.

"Your Majesty, the Chief Minister requests an audience outside," another eunuch rushed in.

"Admit him."

Zhang Zhang Juzheng also entered carrying a stack of books, placed them on the ground, and brushed his sleeves aside: "Your servant bows before Your Majesty. How is Your Majesty's health?"

"Rise. Are you here to present a book too? What books are these?" Zhu Zhu Yijun eyed the dozen scrolls and asked curiously.

Zhang Zhang Juzheng bowed: "Your Majesty, the book is the Direct Guide to the Comprehensive Mathematical Methods, written by Cheng Cheng Dawei, a merchant from Huizhou Prefecture, Nanzhili, totaling seventeen volumes. Since Gui E promoted the Single Whip Law, I have been attentive to mathematical talent. When I proposed reviving ancient statutes to reform education and heard Your Majesty and the Prince of Zheng were troubled by this, I present this treasure."

Zhu Zhu Yijun carefully flipped through the book. Zhang Zhang Juzheng presenting a book was unsurprising—he had proposed reviving ancient statutes and added mathematics to the curriculum. To promote mathematics without a textbook would make his policy a mirage; thus, he presented the book.

"The Great Ming truly has abundant talent," Zhu Zhu Yijun said, deeply moved after a quick glance.

Zhu Zhu Yijun smiled: "What reward do you request?"

"I proposed the policy—I must implement it. I humbly accept your grace with utmost gratitude," Zhang Zhang Juzheng declined reward, for the emperor had already rewarded him: summoning his parents to the capital to fulfill filial piety, allowing Liang Liang Menglong to retain office despite mourning, and Zhao Meng's fabricated case against Lu Lu Guangzu—all these moves, orchestrated by the young emperor, resolved Zhang Zhang Juzheng's greatest political vulnerability: his parents were still alive.

What reward could Zhang Zhang Juzheng possibly need? To serve as Chief Minister and implement reform was his greatest gratitude to heaven's grace.

Zhu Zhu Yijun shook his head: "You decline, but I must give. If I reward no merit, the world will say I am cold and ungrateful. You presented a book—you have earned merit. I raise your salary to first-rank grade. Do not refuse."

"Your servant kowtows to thank heaven's grace," Zhang Zhang Juzheng finally accepted after careful thought—he had indeed managed reform well.

Zhu Zhu Yijun looked at the two piles of books: "Where is Cheng Cheng Dawei, author of the Comprehensive Mathematical Methods?"

"He is a merchant in Nan Prefecture, Huizhou," Zhang Zhang Juzheng answered honestly. The book had been sent to the capital by the Nanjing Governor Song Yangshan to support Zhang Zhang Juzheng's educational reforms.

Zhu Zhu Yijun listened, then said seriously: "Talents remain in obscurity because I have not cultivated virtue. Master, such a great mind—should he not be recommended to return to court and serve the state?"

Talents in obscurity, not in court, meant the emperor lacked virtue—this was the base Confucians' favorite argument. Since such a mathematician still lived, he must be brought to court to serve the state.

"Your servant obeys the decree," Zhang Zhang Juzheng bowed.

Zhang Zhang Juzheng looked at the other pile: "Is Grand Secretary Ge also here to present a book?"

"What a coincidence—you are here to present a book too?" Ge Ge Shouli smiled. On upholding the sovereign's authority, Ge Ge Shouli had acted faster—he presented the Mathematical Mirror before Zhang Zhang Juzheng.

"Your Majesty, the Minister of Rites, Wan Shihé, requests an audience," another eunuch rushed in.

"What day is this? Everyone's come at once?" Zhu Zhu Yijun chuckled. "Admit him."

Wan Shihé entered carrying several books, kowtowed five times with three prostrations, and said: "Your servant bows before Your Majesty. How is Your Majesty's health?"

"Well. Rise. What book does Minister Wan bring?" Zhu Zhu Yijun asked, eyeing the books.

"The Director of the Court of Colonial Affairs, Chen Chen Xuehui, has compiled and edited the Western Mathematics, six volumes," Wan Shihé handed the books to Zhang Hong with utmost reverence.

In Macau and Luzon, the Ming had acquired many books. Chen Chen Xuehui sifted through them, discarded all the Confucian classics, and selected, organized, translated, and compiled the entire mathematical system.

Zhu Zhu Yijun flipped through them for a long time, nodding repeatedly: "Excellent. Minister Wan, you have rendered great service. Grant one of your sons the post of Secretary of the Secretariat. Promote Chen Chen Xuehui one rank. As a special exception, bestow each of you one hundred taels of silver, four bolts of silk gauze, five thousand strings of paper money, and five bottles of wine—to honor your translation and compilation."

"Excellent," Zhu Zhu Yijun said, flipping through the many works. All of this began with the telescope. As the language of all things, mathematics had become ever more vital. When the emperor studied mathematics, imperial officials would strive to present mathematical texts for his inspection.

Zhu Zhu Yijun smiled and said, "Your Highness, copy one copy and begin compiling the *Enlightenment of Arithmetic*."

Zhu Zhu Yijun held in his hands the *Suanxue Baojian of Calculation*, a monumental mathematical work compiled by Wang Wang Wensu through decades of tireless study, yet it circulated only among Shanxi merchants as a tool for trade—truly a pity.

In the *Suanxue Baojian of Calculation*, there is a method: Tongzheng Xinji.

Tongzheng means eliminating falsehood, preserving truth, filling gaps, restoring continuity, and clarifying origins—a synthesis and validation of past mathematics, explaining why calculations proceed as they do; Xinji, meanwhile, proposes original conjectures on certain problems, carefully verifying them through Tongzheng and summarizing conclusions.

This aligns with Zhu Zhu Yijun's requirements for mathematics: bold hypotheses, careful verification, and systematic summation.

Zhu Zhu Yijun saw that the *Suanxue Baojian of Calculation* studied numerical solutions to higher-order univariate equations; in this book, mathematical reasoning was as obscure as celestial script—concepts like Jia Zong, Wang Wang Yushi, Yi Jin Lian Zengcheng, Yi Zong, and Yi Fang were indeed difficult to comprehend.

By his side lay the *Western Arithmetic*, which introduced *Algebra*, published in Milan in the twenty-ninth year of Emperor Jiajing, summarizing symbols for addition, subtraction, multiplication, and division; using letters to represent unknowns made understanding far easier.

After reading all six volumes of *Algebra*, Zhu Zhu Yijun learned that Western arithmetic at the time still lacked the concept of decimal notation—decimal fractions, decimal decimals, computational methods, and notations were absent. (These would not be systematically introduced until 1585 by the Dutch mathematician Stevin.)

Yet in Zhu Zhu Yijun's mathematics textbook, the *Nine Chapters on the Mathematical Art* from the Wei-Jin-Northern and Southern Dynasties stated: "Micro-numbers without names are treated as numerators; the first descent uses ten as denominator, the second descent uses a hundred as denominator; the further it descends, the finer the fraction becomes." In the Southern Song's *Nine Chapters on Mathematical Methods*, the great mathematician Qin Jiushao calculated compound interest as: seventy-nine wen, three fen, four li, eight hao, four si, six hu, seven wei, seven sha, three mang, one qing, two qing, five yan.

In reality, no actual units existed below wen; fen, li, hao, si, hu, wei, sha, mang, qing, and yan were merely computational terms for interest.

Wang Wang Wensu's method for solving numerical solutions to higher-order univariate equations was particularly interesting.

For example, to approximate the root of x³ - 3x + 1 = 0, Wang Wang Wensu's approach was simple and blunt: he simply discarded the x³ term, yielding -3x + 1 = 0, so x = 1/3. Substituting this approximate root back, the left side became 1/27 ≈ 0.03; clearly, 0.03 ≠ 0, and error remained.

Clearly, this approximate root was still insufficiently precise. To refine it further, let the error be u, meaning x = 1/3 + u. Substituting into the original equation yields: (1/3 + u)³ - 3(1/3 + u) + 1 = 0. This remains a higher-order equation—how to solve it? Again, discard the higher-order term, obtaining: 1/27 + 1/3u - 1 - 3u + 1 = 0. Solving gives u = 1/72, so x = 1/3 + 1/72 = 25/72.

Substituting x = 25/72 back, the left side ≈ 0.00025; clearly, 0.00025 ≠ 0, error still persists.

To refine further, let the error be i, so x = 25/72 + i. Substituting again and repeating the same process yields an even closer approximation.

On this foundation, Wang Wang Wensu employed an estimation method: first roughly determine an approximate root a, then assume an error b, and gradually refine it step by step.

To find an approximate solution to f(x) = 0, set x = a + b, substitute to obtain: f(a + b) = f(a) + kb + o(b). f(a) is a solvable constant term; o(b) is a higher-order term too difficult to compute—discard it directly, reducing it to a linear univariate equation. Once the coefficient k of the linear term is found, iteration yields the approximate solution, no matter how high the equation's degree—allowing infinite approximation.

This k was later called the derivative; this iterative method for solving higher-order equations was primarily an applied technique for approximating solutions, yet it was indeed an infinite subdivision of the derivative.

What came after? Nothing came after.

Even Wang Wang Wensu's decades of solitary, painstaking labor amounted to little more than a merchant's ledger tool—never widely disseminated. When Ge Ge Shouli presented these fifty-five volumes, it was merely to address an urgent shortage of textbooks.

Compared to the Song and Yuan dynasties, Ming mathematics had progressed, but this progress was fragmented and unstructured.

Zhu Zhu Yijun looked at his pile of mathematical masterpieces and knew he had a long road ahead.

Zhu Zhu Zaiyu removed sections on divination for illness and predicting fetal gender, recompiled the *Enlightenment of Arithmetic*—enlightenment meant enlightenment: addition, subtraction, multiplication, division, solving equations, roughly equivalent to elementary to middle school textbooks in later eras. He simplified mathematics; the six-volume *Western Arithmetic* was easily understood by Zhu Zhu Zaiyu, as its mathematical symbols and algebraic thinking made the subject more concise.

Higher-level mathematics textbooks could only be compiled after Zhu Zhu Zaiyu fully mastered the three monumental works in his possession.

Zhu Zhu Yijun was only twelve—he could wait.

After watching the young emperor rebuke his ministers in the capital, Chen Lin boarded his three-hulled hydrofoil sailboat once more and sailed south toward Songjiang Prefecture.

Back in Songjiang Prefecture, Chen Lin needed to carry out another oceanic testing mission: to Yuegang, to the Penghu Inspection Office, and to Lu Song. This time, seven hydrofoil sailboats would sail together to the Great Ming's Lu Song Viceroyalty—Yin Yin Zhengmao had been officially appointed Lu Song Viceroy.

This was not the first time the Great Ming appointed a Lu Song Viceroy; the first appointment occurred in the third year of Yongle, when Xu Xu Chailao received the Great Ming's seals and credentials as Lu Song Viceroy, overseeing military, civil, financial, and cultural authority.

In Yu Yu Dayou's coastal defense plan, a patrol office would also be established in Manila, Lu Song, specifically to combat smuggling.

Chen Lin's reasons for going to Lu Song were twofold: first, to continue testing the hydrofoil sailboats; second, to confirm whether the Red-Haired Barbarians' large ships would arrive on schedule this year.

The Great Ming needed silver—vast quantities of silver inflow to activate its commodity economy, enhance commodity circulation, and complete state tax reform. Yet the Great Ming had extended a claw, bringing the Spanish Lu Song Viceroyalty back under its dominion.

Spain, meanwhile, needed the Great Ming's vast array of goods to alleviate its domestic inflation crisis.

War or peace—that was the question.

Whether the concept of "Yi Fang" in the *Suanxue Baojian of Calculation* was truly a derivative remains debated. Yi Fang does equal the result of multiplying Jia by Jia to obtain the first derivative; from this alone, we can indeed discern the "derivative" in Wang Wang Wensu's method—but only as a shadow. The *Suanxue Baojian of Calculation*'s method for solving higher-order univariate equations remains a finite-difference approximation of differentiation. To write this still brings sighs and regret. Requesting monthly votes, woohoo!!!!!!

(End of Chapter)