In the early winter of 1787, in the east wing of the Palace of Versailles.
Joseph sat in a room adorned with rococo gold trimmings and large oil paintings, shaking his head with a wry smile at the exam paper in front of him.
The light from a luxurious two-meter diameter crystal chandelier flickered, casting reflections on his fair skin, as well as his youthful and delicate features, making him look like the handsome Paris from a painting.
At his side, an elderly man with a white curly wig and wearing a lace cravat sighed and with disappointment flashing in his brown eyes, he bowed to Joseph and said, "Crown Prince, if you find this difficult, perhaps you can start with the basic courses..."
Joseph was startled and quickly returned to his senses from his reverie, nodding politely to the old man:
"Mr. Lagrange, I think you may be mistaken, I meant to take your course's final examination, not the entrance exam."
Indeed, this unassuming old man was none other than the founder of analytical mechanics, a pioneer of group theory, and the celebrated French mathematician and physicist known as the prince of mathematics—Lagrange.
"Final examination?" Lagrange frowned at the boy of only thirteen, "Your Highness, the courses I teach are university level, you might..."
The other extravagantly dressed noble youths who were taking the exam immediately turned their heads and looked over, their eyes filled with curiosity.
At that moment, a sixteen-year-old boy wearing an embroidered silk coat with laced collar and with slightly droopy eyes showed a disdainful expression and loudly said,
"Crown Prince, I remember you still have two years to complete the basic courses." He nodded to the old man, "Mr. Lagrange often says that one must climb the ladder of mathematics step by step, aiming too high will surely lead to a nasty fall. Perhaps Your Highness should heed this advice."
Joseph ignored him and earnestly said to Lagrange:
"Sir, I have studied university mathematics courses on my own. I really do need to take the final examination."
The elderly mathematician sighed with resignation and turned to his assistant, "Andrei, please bring me the exam paper from the very bottom of my folder."
"Certainly, Professor."
Soon, several exam papers were laid out in front of Joseph.
He quickly scanned them and found that the difficulty had increased several times compared to the previous one, but most of it was still high school level from later generations, involving a limited amount of calculus content—it really wasn't difficult for him.
Yes, just over half a month ago, he was still in the 21st century, studying for his master's degree. That day, he went to France with his tutor to work on a wind turbine project, accidentally fell from the top of a tower, and when he woke up, he found himself reincarnated as Louis XVI's eldest son—Louis Joseph. Perhaps due to the effects of time travel, Joseph was born a few years earlier than in history, and was already thirteen.
As Joseph quickly wrote down the answer to the first question under Lagrange's scrutinizing gaze, his mind wandered to France's historical trajectory: the French Revolution would erupt next year, the royal family would be executed, and as the Crown Prince, he wouldn't be able to escape... King Louis XVI was inept at everything except locksmithing, France was in debt for more than two billion, but had an annual income of only 500 million.
Due to financial collapse, civil servants were backlogged with wages, government operations were struggling, foreign trade stalled, and the colonies were deteriorating. In order to bolster the finances, the Cabinet had no choice but to levy heavy taxes, sucking the common people dry to the bone, while the tax-exempt nobility lived a life of extravagance.
Moreover, next summer France would face a severe hailstorm, and with the impact of previous years' droughts, a national famine was imminent. This would be followed by a mob of hungry people storming the Bastille, triggering the immense upheaval of the Revolution, with hundreds of thousands meeting their end at the guillotine...
So, to keep his own head, he counted his options on his fingers: one, solve France's fiscal deficit; two, secure enough food to prevent starvation; three, deal with treacherous nobles; and lastly, fend off the covetous British and Prussians.
With a famine starting in July, he had barely more than half a year left, and he rubbed his forehead in annoyance, for being too young to participate in state affairs, having no outlet for his efforts.
It was an absolute hellish start, with scant hope...
```
Not far away, the young man with the droopy eyes saw his movements and assumed he was fretting because he couldn't solve the problem. He immediately sneered with disdain: "What an idiot, actually coming here claiming he can handle university courses, how embarrassing! Why is this strawman the Crown Prince and not me?!"
While thinking about how to save his life, Joseph quickly wrote down the answers, soon completing the first page of the exam.
He flipped through the pages with some impatience, for once he had passed Lagrange's subject, he could consider his studies at the University of Paris complete!
Half a month earlier, he had proposed to Queen Mary, his stepmother, that he wanted to participate in politics in order to reverse the dire situation he was in, but she promptly refused him, telling him to focus on his studies and wait until he had achieved academic success.
So he had to make an agreement with the queen that after finishing his courses at the University of Paris, he would formally join the political scene.
Of course, with his level of knowledge during that era, he was a study prodigy among the fighter jets. In the past half-month, he had already passed most of the subjects, and that was because he had to remember incorrect knowledge which delayed his time—many truths widely accepted in that era were actually errors.
Lagrange, watching the Crown Prince's pen fly across the paper, had stopped paying attention to the other students, his eyes growing wider and wider.
This was work meant to be completed with five years of study at the University of Paris, yet the Crown Prince was answering effortlessly, with clear thoughts and not a single mistake!
He was only 13 years old, and he was self-taught! Lagrange's heart shook violently—could it be that another Leibniz had been born?
Suddenly, Lagrange glanced at his assistant, his eyes narrowing slightly, wondering if Andrei had leaked the exam questions to the Crown Prince. After all, the Crown Prince's performance was too unusual, considering that a super prodigy like Leibniz had only started attending university at the age of 14.
He immediately took a piece of paper and quickly wrote down a few lines and handed them to Joseph, saying,
"Your Highness, there is no need to finish the rest; just complete these few questions and I'll consider you to have passed."
Seeing this, the young man with droopy eyes sneered to himself: Hah, is Lagrange trying to make it easy because he sees that he doesn't know how? What a fool clinging to the royal family! I must find a way to let everyone see the Crown Prince's exam paper later, to let him properly embarrass himself.
Joseph looked at the paper in surprise—there were only 5 questions. The difficulty remained the same, but the quantity was less. Good news.
He quickly finished the first two questions, and then he saw the third question, "Please write out the proof of Rolle's Theorem." This he knew very well, without a second thought, he wrote on the blank space:
Rolle's Theorem states that for a continuous function f on a closed interval [a, b], and f is differentiable on the open interval
(a, b), if f(a) = f(b), then there exists at least one point c in the open interval (a, b) such that the derivative of f at that point is zero.
Proof: Because the function f(x) is continuous within [a, b], it attains its maximum value (M) and minimum value (m)...
Joseph completed the proof quickly but then felt Lagrange's breath quicken next to him. He looked up and saw the old mathematician looking at the exam paper with the excitement of seeing a first love.
Joseph immediately looked down at the questions again, and hesitated, "I shouldn't have written it wrong, should I?"
Lagrange grabbed the exam paper and after examining the proof several times, muttered to himself, "So it holds true for differentiable functions as well! Why didn't I think of it?"
He then turned his fervent gaze back to Joseph: "Your Highness, how did you come to think of this?"
"Ah? Isn't it just..." Joseph suddenly recalled that Rolle had simply proven that in a polynomial equation with two adjacent roots, the equation has at least one root in between, and it was not until the nineteenth century that it was extended to differentiable functions.
He had been careless and not careful enough...
"Cough!" He hurriedly took back the paper, changing the subject, "Lagrange, sir, I'm going to do the last two questions now."