Chapter 1 Introduction

Early winter, 1787, the East Wing of the Palace of Versailles.

Joseph sat in a room adorned with Rococo-style golden patterns and large oil paintings, shaking his head with a bitter smile at the test in front of him.

The light of a luxurious crystal chandelier, spanning two meters in diameter, danced across his fair complexion and delicate features, making him look as picturesque as Paris himself in an oil painting.

Beside him, an elder man with white curled powdered wig and a lace cravat let out a sigh, his brown eyes filled with disappointment. He bowed slightly and said, "Your Highness the Crown Prince, if you find this challenging, perhaps it would be better to start with basic courses..."

Joseph froze for a moment, snapping back from his daze, and nodded politely to the elder man, replying,

"Mr. Lagrange, I think there's been a misunderstanding. I requested your graduation examination, not an entrance test."

Indeed, this seemingly unremarkable elderly man was none other than the founder of analytical mechanics, pioneer of group theory, and the man hailed as the "Prince of Mathematics," the renowned French mathematician and physicist—Lagrange.

"Graduation examination?" Lagrange frowned as he looked at the thirteen-year-old boy sitting before him. "Your Highness, the courses I teach are university-level. You may find them..."

The other noble boys in opulent attire who were answering their own tests turned to look over immediately, their eyes filled with curiosity.

At this moment, a sixteen-year-old boy in a silk jacket adorned with lace trim and slightly slanted eyes sneered in disdain and spoke loudly,

"Your Highness the Crown Prince, I recall you still have two years left to finish your basic studies." He nodded to the elder man. "Mr. Lagrange often says, 'Mathematics must be climbed step by step; aiming too high will only result in a catastrophic failure.' Your Highness the Crown Prince should keep this wisdom in mind."

Joseph ignored him and seriously said to Lagrange, "Sir, I've self-taught university-level mathematics. I really require a graduation examination."

The elderly mathematician sighed helplessly and turned to his assistant. "Andrei, fetch the test paper at the bottom of my bookcase."

"Yes, Professor."

Soon, several test papers were placed before Joseph.

He quickly skimmed them and discovered the difficulty level was several times higher than the previous test, but most of it still resembled modern high school material, with some calculus included. It posed little challenge to him.

Yes, just over half a month ago, he had been a graduate student in the 21st century. That day, while accompanying his advisor on a wind turbine project in France, he had accidentally fallen from the top of a tower. Upon waking up, he found himself transformed into Louis XVI's eldest son—Louis Joseph of France. Perhaps due to the effects of time travel, Joseph was born a few years earlier than his historical counterpart, and he was now thirteen years old.

Under Lagrange's scrutinizing gaze, Joseph quickly wrote down the answer to the first question, his mind simultaneously occupied with thoughts of France's historical trajectory: next year, the French Revolution would erupt; the royal family would all be guillotined, and as Crown Prince, there was no escaping his fate. King Louis XVI, adept only at locksmithing, had left France drowning in over 20 billion livres of debt while its annual revenue was only 5 billion.

Due to financial collapse, public servants hadn't been paid, basic government operations had ground to a halt, foreign trade was stagnant, and the colonies were in increasing decay. To ease the financial strain, the Cabinet had to impose excessive taxes, leaving the lower classes drained to the bone, while tax-exempt nobles indulged in extravagance and debauchery.

What's more, next summer, France would face severe hailstorms compounded by prior years of drought, resulting in a nationwide famine. This would be followed by peasant uprisings, the storming of the Bastille, and the kickoff of the French Revolution expansion pack. Chaos would reign, tens of thousands would end up on the guillotine...

To save his own neck, Joseph began counting on his fingers: First, he'd have to resolve France's financial deficit; second, secure enough grain to prevent starvation; third, deal with the scheming nobles. Lastly, he'd have to fend off the watchful British and Prussians.

With famine expected in July, he had barely half a year left. Frustrated, he rubbed his temples; his young age prevented him from engaging in politics. He had energy but no outlet. *It felt like the ultimate "hell start," with no hope in sight...*

Not far away, the boy with slanted eyes watched Joseph's gesture and assumed he was struggling over the exam questions. He scoffed disdainfully, *What an idiot. Coming here to claim he knows advanced math is absolutely humiliating! Why is this fool the Crown Prince, and not me?!*

Joseph, alternating between crisis management strategies and rapidly answering questions, soon completed the first page of the exam.

Impatient, he flipped to the next page. Completing this course under Lagrange meant he'd have finished his studies at the University of Paris!

Half a month ago, he had begged Queen Mary, his so-called "conveniently assigned mother," to let him participate in governance, hoping to alter his doomed fate. She, however, promptly refused, telling him to focus on his studies and discuss politics once he excelled academically.

Thus, Joseph made a pact with the Queen: once he completed the University of Paris courses, he'd formally engage in governance.

Of course, given his capabilities, he was practically a "battle-hardened genius" in this era. Over the past couple of weeks, he'd completed most of the subjects—not without delay due to memorizing erroneous concepts. *Many so-called "truths" of this time were outright falsehoods.*

Lagrange watched the Crown Prince writing at lightning speed, entirely ignoring the other students by now. His eyes widened by the second.

The questions required five years of coursework at the University of Paris, yet the Crown Prince answered them effortlessly, as if the solutions poured directly from his mind—clear, accurate, without a single mistake!

He was only thirteen, and entirely self-taught. *Lagrange's heart pounded. Could another Leibniz have been born?!*

Suddenly, Lagrange cast a side glance at his assistant, narrowing his eyes. *Had Andrei leaked the exam questions to the Crown Prince? After all, Joseph's performance was unnaturally brilliant.* Even Leibniz, the legendary prodigy, had started university at fourteen.

Immediately, Lagrange grabbed a pen and paper, scribbled down a few problems, and placed them in front of Joseph. "Your Highness, you don't need to complete the rest of the exam. Finish these five questions, and I will consider you passed."

The boy with slanted eyes smirked internally. *Hmph, Lagrange must be coaxing him along, preparing to show favoritism toward the royal family. What a fool! I have to find a way to expose the Crown Prince's answers later so he can be humiliated.*

Surprised, Joseph looked down at the paper. There were only five questions, with the difficulty unchanged but the workload reduced. That was good news.

Quickly, he tackled the first two problems, then arrived at the third: "Write the proof for Rolle's Theorem." He knew this well and immediately began to write:

Rolle's Theorem: Let f be continuous on a closed interval [a, b] and differentiable on an open interval (a, b). If f(a) = f(b), then there exists at least one point in (a, b) where the derivative is zero.

Proof: Since the function f(x) is continuous on [a, b], it attains a maximum value (M) and minimum value (m) within the closed interval...

Joseph finished in no time, but suddenly noticed Lagrange's breathing had grown rapid. Startled, he glanced up, only to see the elderly mathematician staring at the paper with an expression akin to witnessing his first love.

Nervously, Joseph scanned the question again and hesitated, saying, "I... shouldn't have made a mistake, right?"

Lagrange seized the test paper, examined it thoroughly several times, and muttered under his breath, "So it also applies to differentiable functions! Why didn't I think of that?"

He looked at Joseph again, the intensity in his gaze almost fiery. "Your Highness, how did you come up with this?"

"Ah? I mean..." Joseph suddenly remembered that Rolle had only proved the theorem for polynomial equations between adjacent real roots, and the generalization to differentiable functions hadn't occurred until the 19th century.

*Careless. Didn't hold back...*

"Ahem!" Quickly, he snatched the paper back and shifted the topic. "Mr. Lagrange, I need to finish the remaining questions now."

(Panoramic oil painting of the Palace of Versailles)