When Han Qian returned to the eastern courtyard, he saw lamps lit in the hall. Zhao Ting’er was bent over the desk calculating something, while Madam Xi knelt on the opposite side of the desk, helping Zhao Ting’er organize calculation drafts. Seeing Han Qian approach, she retreated far to the side.
Han Qian glanced at Madam Xi, secretly feeling the timing still wasn’t quite right. He walked over to sit beside Zhao Ting’er. Seeing her troubled expression, he took the calculation sheets from her, only to discover that Zhao Ting’er was actually working on trigonometric functions.
After manufacturing the angle-measuring instrument, they would still need trigonometric function values to calculate elevation differences.
Although trigonometry in later eras was merely elementary mathematics, in this age, trigonometry based on the Pythagorean theorem mainly derived from astronomical observations and calendar calculations. All related knowledge was fragmentary and not yet systematized.
No matter how exceptionally intelligent Zhao Ting’er was, she had been studying with him for less than a year. Expecting her to independently calculate all the trigonometric function values using this unsystematized trigonometric knowledge was truly asking too much of her.
The simplest method for determining trigonometric function values was to draw diagrams for actual measurement and then calculate the numerical values. However, no matter how precise the drawings, the calculated values would still contain deviations.
Yet trying to derive all the trigonometric function values purely through mathematical calculation was actually an enormously vast and difficult undertaking.
“Silly girl, even I might not be able to calculate these,” Han Qian said, tapping Zhao Ting’er with the calculation sheets and telling her and Madam Xi to go rest first.
The person from his dreams, Zhai Xinping, wasn’t particularly strong in physics and chemistry, but financial analysis required extensive numerical calculations, so his foundation in mathematical computation wasn’t weak.
However, Han Qian didn’t remember the direct calculation methods for trigonometric function values. Trying to gradually derive them using the unsystematized trigonometric principles already summarized by predecessors would still be extremely difficult.
Han Qian sat at the desk through the entire night. When daylight broke brightly, he still had no solution.
That night, Zhao Ting’er and Madam Xi didn’t abandon Han Qian to return to their rooms to rest either—they kept him company in the hall.
Madam Xi held a fan to create a breeze or helped drive away mosquitoes. Zhao Ting’er forced herself to stay awake, lying by the desk watching Han Qian derive various formulas. Seeing Han Qian finally furiously crumple a large stack of calculation sheets into a ball and toss them into the waste paper box, she said with a laugh, “So there actually are things the young master doesn’t know…”
Han Qian didn’t want to waste precious time on boundless mathematical derivations. He thought perhaps this was work requiring several generations to complete, yet the sword hanging over their heads would fall within three or four years—this was time he truly couldn’t afford to waste.
Settling for second best, he could only draw diagrams as precisely as possible, then actually measure the angles and sides to calculate values. He could first create a rough trigonometric function value table.
Han Qian vaguely recalled that in later eras, the earliest trigonometric function value tables should have been created through this method as well.
Even if using this table to measure and calculate elevation differences resulted in errors of three to five meters, during relatively large-scale water conservancy projects, they could still make corrections through other auxiliary means.
Although this would waste more manpower and resources than originally envisioned, the technical standards of this era were simply at this level—not something Han Qian alone could shake heaven and change.
Han Qian had already taught Zhao Ting’er the drawing and measurement method, and this task also required sufficient patience to complete. So Han Qian delegated everything to Zhao Ting’er, having her organize it together with Madam Xi, while he called Qing Yun to find some food from the back kitchen. After eating his fill, he returned to his bedroom to catch up on sleep.
Han Qian slept until afternoon before getting up. Walking to the hall, he saw Zhao Ting’er had actually fallen asleep directly sprawled across the desk, while Madam Xi was curled up sitting before the desk, retrieving and examining the calculation sheets he had thrown into the waste paper box that morning.
Madam Xi was still wearing leg shackles. Her feet were tucked to the side, snow-white and delicate. Aside from the red marks the shackles had chafed on her ankles, there wasn’t the slightest blemish.
Madam Xi was quite alert. She suddenly turned her head to look back at him, then expressionlessly tossed the calculation sheets back into the waste paper box.
“You won’t understand them,” Han Qian said without much expression. “Go fetch washing water for me first.”
Hearing Han Qian’s contemptuous and dismissive tone, Xi Ren felt thoroughly displeased, yet had to admit that these purely mathematical derivation sheets looked like heavenly scriptures to her eyes—incomparably more profound than the mathematics she had studied in childhood.
Han Qian smiled, thinking that in this age, perhaps only someone like Marquis of Liyang Yang En might understand these calculation sheets. Even though Madam Xi was the daughter of Gao Long with deep family scholarship and had received a good education from childhood, the possibility of her reaching Marquis of Liyang Yang En’s level in mathematics was infinitesimally small.
However, if this could pique Madam Xi’s interest, that was something Han Qian was happy to see.
……
……
Although Han Qian had already confirmed that directly calculating trigonometric function values couldn’t be accomplished in a short time, if anyone later wished to research this deeply, these calculation sheets should still provide some help. They couldn’t truly be treated as waste paper thrown into the waste box to be sent to the privy for wiping purposes.
Han Qian sat down beside Zhao Ting’er, spreading out and organizing the calculation sheets neatly. He recalled Madam Xi’s confused yet alluring expression when examining the calculation sheets earlier, as well as her pair of snow-white jade feet. He secretly felt it was no wonder that even though Feng Changyu knew she might be a hidden danger, he ultimately couldn’t resist taking her into his household.
Zhao Ting’er had an extremely beautiful face, but her hands and feet were still somewhat rough—her jade feet weren’t as alluring as Madam Xi’s. Thinking of this, Han Qian again regretted that this era couldn’t manufacture amusing items like stockings.
After Han Qian finished organizing the calculation sheets, he took out the diagrams Zhao Ting’er had drawn to measure them—only if the drawings were sufficiently precise could they minimize deviations when measuring trigonometric function values.
This was a clumsy method resorted to out of desperation.
On the desk sat a plumb line used for alignment. Zhao Ting’er was sleeping soundly when she unconsciously flicked her small hand outward, causing the small conical bronze weight to roll toward the edge of the desk. Han Qian reached out to catch the line end of the plumb. Watching the conical bronze weight swing as it hung down, he suddenly realized there was actually a simpler way to construct the angle-measuring instrument.
The land-based angle-measuring instrument needed to ensure the base was as level as possible, but besides the level, didn’t the plumb line’s direction always maintain absolute perpendicularity to the horizontal plane?
Then wouldn’t using a plumb line to calibrate the base’s levelness be much simpler and more convenient than a level?
He hadn’t thought of this point before. Han Qian couldn’t help lightly slapping his own face, cursing himself for being a fool.
Xi Ren brought in washing water and saw Han Qian slapping his own face while cursing himself as a fool. She didn’t know what fit he was having.
After understanding this principle, when Han Qian redesigned the angle-measuring instrument, he discovered it could be designed extremely simply—there was no need for several craftsmen to make such a big fuss about it.
Zhao Ting’er woke up bleary-eyed and saw Han Qian disgustedly rubbing his own face. “What’s wrong?” she asked.
“Look, the angle-measuring instrument can actually be made like this!” Han Qian showed Zhao Ting’er the newly designed instrument draft.
The newly designed angle-measuring instrument only required an upright rod, a semicircular plate marked with angle graduations, and a plumb line. During actual measurement, the upright rod would be inserted into the ground, allowing the plumb to hang freely. Then one only needed to align the semicircular plate’s center line—that is, the zero-degree angle—with the plumb line.
This way, the straight edge of the semicircular plate would be guaranteed to be in a horizontal position.
Then by rotating the semicircular plate so that two points on its straight edge aligned in a straight line with the measurement target, one could directly read the angle between the semicircular plate and the plumb line—this was actually the target elevation angle they wanted to measure.
“Once the principle is explained, it’s truly this simple!” Zhao Ting’er exclaimed joyfully. “Young master, you didn’t think of this before and actually had the craftsmen use such a complicated method to make the angle-measuring instrument—how foolish!”
“Say I’m foolish one more time and see what happens!” Han Qian reached out to knock Zhao Ting’er’s head.
“I’ll go call someone right now to notify those two master craftsmen to build the angle-measuring instrument using the new method…” Zhao Ting’er jumped up excitedly to call someone.
Han Qian said, “Let them build the angle-measuring instrument the clumsy way. We can’t teach everyone all the good methods, otherwise we’ll have no way to earn our living later.”
Han Qian pulled Zhao Ting’er back, telling her that the more complex the angle-measuring instrument used in Xu Prefecture was made, the more it would appear grand and mysterious to ordinary people when taken out for actual measurements, making them know to feel reverence.
Under Xu Prefecture’s current circumstances, this had only benefits and no drawbacks.
As for the angle-measuring instrument they themselves would build using the new method, it could be used for measuring and calculating trigonometric function values.
Using the measurement method to calculate trigonometric function values essentially meant drawing right triangles at corresponding angles, measuring the specific data of the three sides, and then performing simple proportional calculations.
Under this age’s limited precision of measuring rulers, this empirical approach required triangles to be drawn large enough for the final trigonometric function deviations to be small enough.
If Han Qian and the others built a simplified version of the angle-measuring instrument, they could use a straight wall as one side of the triangle, the perpendicular distance between the angle-measuring instrument and the right angle as another side, and the hypotenuse could be simply calculated using the Pythagorean theorem.
This way, the triangles they could use to calculate trigonometric function values could actually be over ten meters in size. The trigonometric function values measured this way could naturally be controlled within extremely small deviations.
Furthermore, if each angle-measuring instrument could correspond to its own set of trigonometric function values, they could also eliminate the instrumental errors of different angle-measuring instruments.
Han Qian more deeply understood the principle behind the three characters “doing scholarship”—scholarship truly had to be “done.” If he hadn’t actually started doing this over the past two days, he wouldn’t have been able to consider things this carefully and thoroughly.
Watching the master-servant pair of Han Qian and Zhao Ting’er jump for joy, Xi Ren felt utterly confused. How could a person with such ruthless methods possess such profound and unfathomable scholarship in his breast?
When her mother died in the ninth year of Tianyou’s reign, Xi Ren was already fourteen years old. Although she had studied mathematics, it was quite rudimentary. This morning, helping Zhao Ting’er draw diagrams, she could barely understand the principles of making the angle-measuring instrument and using trigonometric function values to measure elevation, but there were still many doubtful points she hadn’t figured out.
However, Han Qian was her brother’s killer. No matter how much curiosity she felt inside, she wouldn’t actively seek his instruction.
At this moment, Gao Shao walked over carrying a stack of booklets. Seeing Madam Xi present, he stopped in the corridor without rushing inside.
Han Qian beckoned for Gao Shao to bring the booklets in.
These were all copies of memorials and official correspondence that Wang Yu had formally submitted in his name as prefectural governor to Emperor Tianyou and various court departments during his over four years as governor of Xu Prefecture. Chief Clerk Xue Ruogu had kept copies there.
After returning yesterday, Han Qian had specifically asked Gao Shao to retrieve these memorial copies from Xue Ruogu. He wanted to see exactly what Wang Yu had done in Xu Prefecture that ultimately made him intolerable to both the Four Surnames and Tan Prefecture.