Commandery Princess Jianing and Commandery Prince Feng entered the capital under the banner of celebrating the Empress Dowager’s birthday.
On the twentieth of December, on the Empress Dowager’s birthday, titled ladies entered the palace to offer congratulations. The Emperor held a family banquet, bringing his consorts, princesses, and others to celebrate the Empress Dowager’s birthday.
Matters of the inner palace had nothing to do with Xie Xuanying, and previously he had no interest in inquiring about them.
But if he wanted to inquire, it would only take a moment.
He worked in the imperial guards and was excessively popular. As long as he spoke with someone, basically no one could hold out against him. Before they knew it, they would have blurted out a whole pile of things, and afterward they would not even feel it was inappropriate. Instead, they would hate themselves for being poor speakers, since the beauty had not even smiled.
Very quickly, he found out Commandery Princess Jianing’s recent movements.
Ever since the incident at the Wang family’s plum-viewing gathering, Commandery Princess Jianing had suddenly become very low-key. On ordinary days, she either accompanied the Empress Dowager in worshipping the Buddha or kept the Imperial Consort company in conversation. In her leisure time, she went to temples to burn incense, and she no longer had any dealings with officials. She was very well-behaved.
This, on the contrary, made Xie Xuanying more vigilant.
This woman was not simple; he must be careful.
By comparison, Commandery Prince Feng’s recent actions were somewhat impatient.
It was unclear whether it was because he had become a suspect in a scheme, but he frequently attended banquets, openly and implicitly clarifying that he had not yet met Commandery Princess Jianing, and was even less likely to have done anything to her beloved pet.
Whether others believed it or not was hard to say, but now his purpose had become known to everyone.
The assembled ministers all watched to see what the Emperor thought.
But the Emperor took no action at all, single-mindedly focusing on the New Year.
The New Year in the palace was very lively. These past two days, fireworks and firecrackers had been set off every day, and an Aoshan lantern had even been erected at the palace gate. This was the Hermès among lanterns. The name Aoshan was related to the immortal islands on the sea; it was the ancients’ imagination of a fairyland.
This year’s lantern was “ten meters tall and five hundred meters wide,” with all kinds of scenes: golden dragons coiling around pillars, bodhisattvas riding lions and white elephants, pavilions and terraces, flying waterfalls and flowing water.
The water was real flowing water, drawn to the top by a power mechanism and then allowed to flow down. Spray splashed and reflected the bright lantern light, luxurious and magnificent to the extreme.
Every day when Xie Xuanying passed by on his way to work, he could see the lantern become more complex and intricate. By the day it was lit, presumably the whole capital would be captivated by it.
In addition, the Emperor also had to distribute festival gift money to important ministers and their relatives.
Xie Xuanying received quite a few gold ingots, heavy in the hand, with the seal “All Under Heaven Is at Peace” carved on the bottom.
On the first day of January, there was a great banquet for the assembled ministers.
This was lunch, set out in the Fengtian Hall. Members of the Grand Secretariat and the ministers all had one table each; after that came tables for two, tables for four, tables for six, and tables for eight.
The food was very hard to eat.
Most of it was auspicious dishes, elaborately arranged and beautifully decorated, but actually inedible – just for looking at. What could be eaten was all steamed dishes and stewed dishes cooked to a mush, and also hot pots. But for the sake of appearances, the chicken, duck, fish, and meat were all whole, leaving nowhere to put one’s chopsticks; and if one really ate, it would not look good either.
The assembled ministers basically only ate a bit of shredded meat, drank a bowl of soup, ate a couple of bites of pastry, and watched the beauties eat.
Unfortunately, the beauties likewise had no appetite.
When the Emperor arrived, everyone began to toast. You drank a round, I drank a round, and everyone kept flattering him without stop.
After the Emperor drank two rounds, he named Commandery Prince Feng, Xie Xuanying, and the Imperial Consort’s nephew to drink on his behalf. They all belonged to the Emperor’s junior generation. Since he had no biological son, he could only have relatives step up.
Xie Xuanying drank several pots of wine, and when he got home he slept all afternoon. At night he ate a little hastily, and his head hurt as if it would split.
On the second day of January, his eldest sister, who had married out, and her husband returned to her natal home, so inevitably there was another banquet. In the evening was a family banquet, where children and grandchildren paid respects to their parents and elders and continued to toast.
After the third watch, he went back to his room, vomited once, and slept.
Lying in bed with his head splitting with pain, he thought with complete sincerity: If only he were already married. She would not need to do anything. Just like when she was in Jiaxiang, staying by his side and giving him a bowl of hot water would be enough.
Cheng Danruo, however, felt that this year’s Spring Festival was very meaningful.
She did not have to wait on a seriously ill elder, and could spend most of her time tucked away in her room reading, giving her a rather peaceful feeling like that of a haven beyond the world. Occasionally, when Lady Hong called her to meet guests, she only had to bring a greeting gift and accompany them in conversation for a while.
On Little New Year’s Eve, Second Master Yan returned.
He had not become an official, and had only passed the provincial examination. Most of the time he followed his teacher surveying the hydrology of the Yellow River, searching for ways to solve the Yellow River’s flooding.
At Tianxin Temple, Yan Hongzhi had said that he had a good friend who was good at arithmetic – none other than Second Master Yan’s teacher, He Shuo. This man was extremely skilled in mathematics and water conservancy, and served as a yuanwailang in the Waterways Bureau of the Ministry of Works, specifically tasked with managing the Yellow River.
Because he was away traveling year-round, exposed to wind and sun, Second Master Yan was unlike the refined and courteous First Master Yan. His skin was dark, and he was straightforward and bold in manner. When he saw her, he grinned, showing white teeth. “I did not even know the family had gained a little sister. I did not prepare a greeting gift.”
He thought for a moment, then had a young servant bring an abacus made of ivory. “Sister, take it to play with.”
Cheng Danruo thanked him and accepted it.
Yan Hongzhi said, “You brought back another pile of things to calculate again this time?”
Second Master Yan sighed and said, “This year’s autumn floods brought disaster to quite a few places again. My teacher is already preparing to memorialize the throne, requesting that the dikes be rebuilt in early spring. If it succeeds, I will go with him.”
First Master Yan said, “The Ministry of Revenue does not have that much money.”
Second Master Yan sneered, “What is more important than governing the river? The Empress Dowager’s birthday?”
Yan Hongzhi clutched his forehead, his head aching, and shot a look at Lady Hong.
Lady Hong said expressionlessly, “In that case, before spring begins, you should get married. After that, wherever you want to go, none of us will interfere with you.”
Second Master Yan suddenly rose and clasped his hands in salute. “Mother, I said it before, if I cannot marry the Han family’s young lady, I would rather remain unmarried for life!”
The veins at Lady Hong’s temples throbbed. “The Han family’s young lady? That is the Zhao family’s daughter-in-law!”
Cheng Danruo: “…” Good heavens.
“That fellow is already dead.” Second Master Yan was utterly self-assured. “A widow remarrying – what is wrong with that?”
Lady Hong could endure it no longer. She slapped the table and rose. “Is it I who will not let her remarry? It is she who says that if she is to remarry, she must take the Zhao family’s son with her. That is not even her son. Is there any such reasoning?”
Second Master Yan said, “Can our family not even afford to feed two extra mouths?”
Lady Hong said, “To forcibly take another’s child – does that sound good if word gets out?”
“It is mutually willing. Where is the force?”
The mother and son quarreled fiercely, and it took Cheng Danruo quite a while to understand.
Second Master Yan liked the Han family’s daughter, but the Han and Zhao families had arranged a betrothal when the two were young. Although the Zhao family’s young master was frail and sickly, the Han family still insisted on marrying their daughter to him.
The Han family’s young lady was also a clever one. After marrying over and seeing that her husband was about to die, she wasted no time and immediately took a concubine, hurrying and rushing to produce a concubine-born son before he kicked the bucket.
Holding the concubine-born son in her arms and relying on her natal family, it became somewhat difficult for the clan members to devour the family’s property. But last year her parents-in-law passed away one after the other, and the clan members again began coming to the door.
Second Master Yan wanted to marry her. She agreed to remarry, with the sole condition that she bring the concubine and the concubine-born son with her, and not leave them in the Zhao family to be picked clean.
But Lady Hong was concerned that taking away another family’s son might lead the Zhao family to bear a grudge and slander them everywhere, which would make the Yan family’s reputation very ugly. Therefore, she did not agree to take in the concubine and the concubine-born son.
The Han family’s young lady then refused to remarry.
Second Master Yan felt that she was full of love and righteousness, and became even more determined not to marry anyone but her.
First Master Yan tried to smooth things over. “It is the New Year. This matter can be brought up again after the New Year.”
This was a universal reason. Mother and son beat a retreat and first celebrated the New Year.
Thus, all the way until New Year’s Eve, the mother and son never mentioned the matter again.
Cheng Danruo and Second Master Yan, however, gradually became familiar with each other.
Because Yan Hongzhi said that she was good at arithmetic, Second Master Yan, on a whim, came to ask for her help.
And then, quite naturally, it turned into one-sided teaching.
Cheng Danruo knew that once she entered the palace, it would be difficult to pass on mathematical knowledge anymore. It was rare for Second Master Yan to be obsessed with arithmetic, and the teacher he had taken as his master was also a mathematician, making him an absolutely ideal person to transmit it to.
There was no time to proceed step by step. Every day, in addition to reading, she stuffed modern mathematics into Second Master Yan’s head.
Although ancient mathematics was likewise brilliant, it had to be admitted that its methods of calculation were too complicated, a profound learning that only a minority could master.
Modern mathematics, by contrast, even if it was likewise equations, was simply more convenient to calculate and easier to understand.
Cheng Danruo changed the names of X and Y: X was called “cha” and Y was called “shu,” abbreviated as the “branch equation.” Because both symbols were not hard to write with a brush, they were very easily accepted.
Beyond equations, what she taught most was geometry.
On the night of New Year’s Eve, while the Yan family kept vigil in the main hall, Cheng Danruo was still explaining spatial geometry to him.
In order to explain three-dimensional structures clearly, she could not avoid drawing diagrams for him.
By the same principle, all letters had to be given a Chinese reading. She was too lazy to think much about it, and directly made z “root,” a “flower,” b “leaf,” c “bud,” t “stem,” and O “heart.”
Afterward, it was called flower-leaf geometry.
This was a very important stroke in the history of mathematics in the Xia Dynasty, only the person involved did not know it.
Like a mathematics teacher, Cheng Danruo nearly collapsed during the teaching process.
“Second Brother, take a rest.” It was the New Year, so she could not scold anyone. She chose to give up.
Second Master Yan looked embarrassed, and picked up Yin-niang, who was playing cat’s cradle. “Come, Uncle will take you to set off firecrackers.”
First Madam opened her mouth as if to say something. First Master Yan shot her a look and took the initiative to say, “Do not frighten the child.”
Second Master Yan asked, “Yin-niang, are you afraid of firecrackers?”
Yin-niang pressed her lips together and said adorably, “With Uncle here, I am not afraid.”
“Good child.” Second Master Yan lifted her onto his shoulder. “Third Sister, are you coming too?”
Cheng Danruo had been made dizzy and furious by him, and hurriedly said, “All right, I will get some fresh air too.”
The three of them went outside to set off firecrackers.
Fire trees and silver flowers, radiance brilliant. Outside the gate, the sound of firecrackers continued without end, and from time to time the laughter of neighbors came over, along with some family’s dog, frightened into barking wildly.
A rich smell of sulfur drifted in the air.
Cheng Danruo leaned against a pillar, slightly lost in thought: Another year had passed. This time next year, would I still be alive?
“Third Sister.” Second Master Yan was not stupid; it was impossible for him to really let his niece set off firecrackers. He covered her ears and had a young servant go light the fuse. “Tomorrow, can you explain that diagram to me again?”
He had little experience dealing with sisters, and did not know whether, if he spoke of the Yellow River’s floods, she could understand their horror. He could only describe it as plainly as possible. “This is truly important. It can save countless people.”
Cheng Danruo came back to herself and looked at this cheap elder brother of hers. The lantern light had limited reach; he was a dark figure, and had he not been wearing silk, he would have looked like a farmer’s son.
Born into the Yan family, he should have had no worries about food and clothing, yet he was willing to brave wind and dew to seek the welfare of the common people.
What reason did she have to refuse to help him?
“Of course.” She drew a deep breath, and her irritation immediately receded. “I am very willing.”
Even if she could not live to this day next year, at least she should strive to leave behind traces of having been in this world.
Public account: Three Thousand Years of History
Author: Math Class Representative
“Counting Ancient Mathematicians, Eighth Installment: Yan Guang”
Character profile: A famous mathematician and hydraulic engineer of the Xia Dynasty, second son of Yan Hongzhi, author of Collected Calculations, who perfected and promoted the branch equation and flower-leaf geometry, and presided over the construction of…
By the Yuan Dynasty, the heavenly element method capable of solving quaternary equations had already appeared, but its calculations were complex and difficult to promote. Although the branch equation did not increase the depth of ancient mathematics, it simplified a great deal of complex calculation, making the spread of higher mathematics possible… The appearance of flower-leaf geometry proved that our ancients already had a systematic understanding of spatial geometry and were capable of a great deal of abstract calculation, which was of enormous significance for river defense and water conservancy…
In Collected Calculations, Yan Guang clearly stated that he was not the founder, and that he had learned it from his sworn younger sister, Cheng Danruo. Considering that the symbols used closely match English letters, the current guess in historical circles is that even before the formal translation of Elements, someone had already come into contact with Euclidean geometry, made localized improvements to it, and spread it.
But whether that person was Cheng Danruo cannot currently be definitively confirmed by any historical sources. Especially since in her later years she made no achievements in mathematics, the conservative estimate is that she may have been a transmitter, not the founder.
…
The true founder of the branch equation and flower-leaf geometry is likely already difficult to verify.
But I believe that if he or she, beneath the springs, knew that their achievements had been passed down to later generations, they would certainly be very comforted.
Gratitude to all mathematicians, named and unnamed. It is precisely because of your efforts and explorations that humanity has gained a deeper understanding of the world.
Life does not cease, and inquiry does not stop!
Today’s mathematics still awaits our generation!
