Cheng Danruo had come to make her rounds, but she had not expected to stumble upon an ancient lesson in geometry and arithmetic.
As luck would have it, Yan Hongzhi had gone to change his clothes, while Abbot Mengjue was reciting scriptures. She happened to wander over for a look – and immediately found herself captivated.
She had never expected ancient people to make geometry problems so interesting. They expanded an irregular geometric figure to six times its original size before calculating it. When she mentally worked through the problem herself using equations, however, she found that the formula they arrived at in the end was indeed correct.
Impressive.
“Doctor Cheng, have you studied arithmetic as well?” Xie Xuanying asked in surprise, momentarily forgetting the proprieties between men and women.
Ever since the School of Mind had risen to prominence, it was no longer particularly unusual for women to receive an education. Families of status would have their daughters study the Four Books for Women. More enlightened households might even teach them a few lines of poetry and classical texts, both to display their learning and because, should they one day achieve marital harmony with their husbands, it would make for a fine story.
But given Cheng Danruo’s background, being able to recognize even a few characters was already quite remarkable. Even in merchant families, girls would generally only be taught practical calculations involving land measurements and grain. Few would ever learn anything involving hydraulic engineering.
And it was not only Xie Xuanying. Even Yan Hongzhi could not help showing a look of curiosity.
Cheng Danruo hesitated for a moment.
She had never studied mathematics in ancient times, so she had no idea what level mathematics had reached in this era. She could not tell whether they were astonished simply because a woman understood mathematics, or because they thought her knowledge was unusually advanced. So she answered cautiously, “I know a little.”
Xie Xuanying pressed his lips together and looked away.
“That could not be better.” Abbot Mengjue remained composed as he explained the matter of repairing the embankment. “Would you be willing to lend our temple a hand, Miss?”
Regardless of the flooding of the Yangtze, everyone would suffer from the disaster, herself included. Even if she were far away, if there was something she could do to help, she had no reason to refuse.
Cheng Danruo nodded. “If you do not think me inadequate, I am willing to give it a try.”
There was only a single counting rod on the table. After hesitating for a moment, Xie Xuanying pretended to be casually tidying the inkstone and quietly pushed it toward her.
But Cheng Danruo did not know how to use one.
She flipped through General Treatise on River Works and discovered that the ancients had already accumulated an astonishing amount of experience in hydraulic engineering. Exactly what kind of timber was required for constructing a river embankment, how many pieces were needed, how many bundles of grass were required for binding the joints, how many line boards, how many bamboo ropes – the details were all recorded clearly.
The ancients really were remarkable.
As she marveled at this, formulas and equations she had thought she had forgotten gradually began to surface in her mind.
Writing numbers with a calligraphy brush was awkward, though, and the ink kept bleeding into little blobs.
Abbot Mengjue said, “What you are using is the Indian method of writing numbers.”
Cheng Danruo froze for a moment, then immediately remembered that Arabic numerals had originated in India. Since Abbot Mengjue had devoted himself to studying Buddhist scriptures, it was hardly surprising that he would recognize them. She smiled and said, “Yes. What I learned isn’t the usual method of arithmetic.”
“It seems to have originated in the Western countries,” Yan Hongzhi remarked. “In recent years, works from the West have often been brought into China. I hear there is much worth learning from them, but the languages are difficult to understand, which makes reading them rather laborious.”
A flicker of interest crossed Cheng Danruo’s face.
It was clear that this old gentleman was a man of no ordinary standing. He was acquainted with the abbot, and even had a disciple who was related to the Gu family. He must have quite an impressive background. For someone like him, a single word could carry the weight of a hundred words spoken by an ordinary person.
“Sir, you really are remarkable.” She suppressed her thoughts and tried to keep the conversation as natural as possible. “What I learned was indeed Western mathematics. If you would like to know more, once you have recovered, I can tell you about it. Just don’t think too poorly of my limited knowledge.”
If she remembered correctly, mathematics reached its peak in ancient China during the Song and Yuan dynasties, but gradually fell behind after the Yuan dynasty. And this handsome, refined gentleman seemed to know quite a lot about hydraulic engineering. If he could help solve the flooding problem, who knew how many lives might be saved?
An opportunity like this was hard to come by. It was worth taking a risk.
Yan Hongzhi was a free-spirited sort. Although he was so weak that he needed someone to support him even when walking, once he became interested in something, he agreed immediately.
“That would be wonderful. I wonder how many days you can stay, Miss Cheng?”
Cheng Danruo paused, and a sudden ache rose in her heart.
“I’ll do my best.” She avoided the question and said solemnly, “Please rest assured.”
Her earnestness actually made Yan Hongzhi feel a little guilty.
He had merely asked out of curiosity, tossing out the question without much thought, yet she had taken it so seriously and agreed to help. Then he remembered that she had to provide free medical consultations during the day and would inevitably be exhausted. He was about to tactfully explain himself, but before he could say anything, a sharp pain suddenly shot through his wound.
Since he was particularly afraid of pain, he immediately forgot what he had been about to say and began sucking in breaths through his teeth.
“Doctor Cheng, let’s put the mathematics aside for now.” Yan Hongzhi leaned back against the couch and gave a bitter smile. “When will this wound of mine finally start getting better?”
Cheng Danruo’s answer was exceedingly professional. “Take your medicine properly and get plenty of rest. Then you’ll recover faster.”
Yan Hongzhi was left speechless.
Still, although that was what Cheng Danruo said, she continued to do her duty as a physician. She examined his wound carefully and took his pulse before finally saying, “Sir, please don’t worry. The wound is healing quite well. It probably wasn’t a particularly poisonous snake. A couple more days of rest should do it.”
Yan Hongzhi shook his head and dejectedly sank against the soft pillow.
Cheng Danruo still had the matter of mathematics on her mind, but she did not want to make her interest too obvious. So she said, “Since you’ve kept your promise and taken your medicine on time every day, I’ll teach you the ‘eye-reading technique’ now.”
This time, the others in the room became interested as well.
“Once you see through the trick, it’s actually quite simple.” Cheng Danruo picked up the two Buddhist scriptures from earlier and revealed the secret. “When I first got this book, I simply flipped through it and memorized the first character on the first line of page thirty-six.”
Yan Hongzhi was skeptical. “But the page number was under my control. How could you possibly know which page I would stop on?”
Cheng Danruo smiled. “No matter which page I happened to have open when you told me to stop, I would simply say it was page thirty-six.”
“Really?” Xie Xuanying could not help asking. “But at the time, you clearly showed me…”
“Did you see it clearly?” Cheng Danruo asked him.
He fell silent.
“For the human eye to make out something that small, it needs a little time,” she explained. “As long as I move quickly and speak with enough confidence, who would doubt what I say?”
“I see.” Yan Hongzhi suddenly understood, then laughed. “You certainly have quite some nerve, young lady.”
“Not quite,” Cheng Danruo replied. “The key to a trick is to distract the audience. They think the trick is being performed while I’m supposedly ‘reading’ with my eyes, but in fact, it’s the opposite. Everything is arranged casually and inconspicuously beforehand.”
Everyone nodded in agreement and could not help praising her cleverness once again.
Seeing that it was still early and nowhere near dinnertime, Cheng Danruo was in no hurry to leave.
According to the timeline of the Ming dynasty, Matteo Ricci would not arrive in China until the end of the sixteenth century, and Xu Guangqi’s translation of The Elements of Geometry was still some time away.
She did not have much time. Perhaps she would have to return to the Chen household in a couple of days. If she missed this opportunity, who knew whether she would ever again find someone capable of recognizing the value of what she knew and helping spread mathematics further?
So she pretended not to notice anything amiss and picked up her brush again to work through the calculations.
Calculating the required materials was not difficult, merely tedious. The troublesome part was calculating the surface area of the section of embankment that needed repairing.
In other words, solid geometry.
Cheng Danruo instinctively drew the figure and added auxiliary lines.
To be honest, she was a little slower at calculating than Xie Xuanying. The General Treatise on River Works already contained established formulas for many common calculations, so he could simply plug the numbers into them.
Cheng Danruo did not understand those methods particularly well. She had to work things out using her own approach, which required a little more thought.
But when the two of them compared their answers, they arrived at the same result.
She felt a quiet sense of satisfaction. Even after all these years since crossing into this world, I haven’t actually forgotten my mathematics. I must not have slacked off too much when I was studying back then. Knowledge really does repay those who learn it.
But then she thought about her current circumstances, and the satisfaction turned to sadness.
Across from her, Xie Xuanying glanced over, and a ripple stirred in his heart.
For scholars of this era, the Four Books and Five Classics were naturally considered the most important subjects. But apart from inheriting Li Wu’s School of Pure Sincerity, Yan Hongzhi also had ideas of his own. He opposed empty theorizing and advocated putting learning to practical use. In his view, scholarship was the foundation of establishing oneself, while practical knowledge was what could be used to govern the country; the two complemented each other, each serving as the outward expression of the other.
Xie Xuanying studied under him and was therefore not restricted to the orthodox subjects. He was free to study a wide range of miscellaneous disciplines.
And given his status and position, the education he had received was among the very best available in the Xia dynasty. Even if he only knew “a little,” he would still be far more knowledgeable than most people.
Yet now, a young orphan girl who had lost her parents at an early age was also proficient in mathematics. And not merely in the common practical mathematics used for managing households and conducting business – plane geometry and land measurement, proportion and conversion, or problems involving surpluses and deficits – but in the mathematics of engineering and volume calculations.
It reminded him of another of his half-teachers: his teacher’s wife’s cousin, Hong Yuexia, a Palace Matron.
She was highly skilled in astronomy and calendrical calculations, a field in which few women had earned any reputation for learning. After her husband died, she entered the palace as a female official and became well regarded. A few years earlier, she had been ordered to join the Directorate of Astronomy, where she helped compile the annual calendars and had even drawn astronomical charts.
Xie Xuanying had studied astronomy and calendrical calculations under her. He knew what an esoteric and difficult field it was, and therefore held her in considerable admiration.
But no matter what he was thinking, he would never let it show on his face. He still did not look at Cheng Danruo at all, merely lowering his head and turning the pages of his book.
The atmosphere was somewhat subtle.
Cheng Danruo came back to herself and realized that it had already grown too late. She therefore took the initiative to bid them farewell.
When she returned to the side room, Bai Zhi had already brought back dinner. There were four dishes and a soup: vegetarian Dongpo-style meat, eggplant with preserved mustard greens, wood ear mushrooms with tofu skin, wheat gluten stir-fried with seasonal vegetables, and an asparagus and lily bulb soup.
Cheng Danruo was surprised. “Why is there so much?”
Bai Zhi replied, “The old gentleman’s people instructed them to prepare it. They said that you should have the same meals as they do.”
So it was another gift of thanks from the patient’s family.
After a brief moment of thought, she did not refuse.
A family like theirs probably cared about owing others favors. Since they had already incurred a debt of gratitude to her, it was better to let them repay a little of it rather than have them feel they owed her. And besides, it was a good way to improve her meals.
So Cheng Danruo accepted it without the slightest guilt and enjoyed a hearty dinner.
After Cheng Danruo left, Xie Xuanying’s gaze remained fixed on the sheets of paper where she had been working out her calculations.
He very much wanted to take them and have a look. But secretly reading a woman’s writing was not something a gentleman ought to do. So he could only steal one glance, then another.
No one in the room was paying any attention to him.
Abbot Mengjue had already left, while Yan Hongzhi had taken his medicine and fallen asleep. The only other person in the meditation room was a young servant, who held a fly whisk and lazily waved it now and then to drive away the bothersome insects.
Slowly, Xie Xuanying reached out and picked up the paper from the table.
To be fair, Cheng Danruo’s handwriting was nothing remarkable. The person who had written it seemed to have tried her best to make the characters neat and orderly, but that was about all there was to it. They lacked any real strength or structure.
People often said that one’s handwriting reflected one’s character. If he had seen handwriting like this before meeting Cheng Danruo, he would certainly have assumed that its owner was an ordinary, unremarkable woman.
But after witnessing Cheng Danruo’s medical skills and mathematical ability, such an impression was obviously no longer possible. Xie Xuanying thought for a moment and guessed that perhaps she simply had no time – or perhaps not enough paper and brushes – to practice.
She was an orphan living under someone else’s roof, and she also had to care for her seriously ill great-aunt by marriage. Her life must be extremely difficult.
He remembered that day during the Shangsi Festival, when all the other women had been dressed in gold and silver and wrapped in fine silks. Only she had worn a simple cloth dress, plain and impoverished.
Xie Xuanying suddenly felt a pang of sympathy for her.
It was a very subtle feeling.
He had been born into a marquis’s household and was a young gentleman of an eminent family. He had long since grown accustomed to his own wealth and privilege, and to the poverty and lowly circumstances of others. It had never occurred to him that there was anything wrong with it. It was simply the way the rules of the world worked – just as officials and scholars were exempt from punishment, while commoners were excluded from certain rites and privileges.
Yet he truly had felt a tiny bit of sympathy for Cheng Danruo, if only for a brief moment.
Still, the thoughts of youth came and went quickly.
Once he had the paper in his hands, his attention immediately shifted to the diagrams drawn on it.
Cheng Danruo drew a diagram of the embankment cross-section on the paper, marking it with some strange symbols and adding lines.
Was she dividing the figure into sections before calculating it?
Xie Xuanying could not understand the symbols, but the numerical data was the same. By comparing his own calculations with hers, he quickly deciphered the meanings of ten of the symbols.
He then rewrote them using Chinese characters and retraced her method of calculation.
It was an entirely new way of calculating. He became so absorbed in working it out that he failed to notice when Yan Hongzhi woke up.
It was only when his teacher spoke that he realized.
“San-lang, are you still calculating?”
Xie Xuanying started and instinctively hid the paper.
