Chapter 52: Drawing a Circle with the Left Hand

After all, Yang Xiong had decades of friendship with the State Preceptor — those near the tower catch the moonlight first. Diwu Lun had already made oblique inquiries and gotten this man's entire background crystal clear.

Old Yang Xiong also told him: "Liu Zijun's family has a tradition of changing names. His father was originally named Liu Gengsheng, and only later changed it to Liu Xiang."

Thus he learned that Liu Xiu, Liu Yingshu, was a name changed later — he was originally called "Liu Xin," courtesy name Zijun.

Liu Xin, together with his father, supervised the collation of the imperial library during the Heping era of Han Emperor Cheng, and it was then that he became acquainted with Yang Xiong. Though Yang Xiong was quite disparaging of Liu Xin, one could still hear his respect for the man's talent, praising him as one who "left no branch of numerology, technique, or skill uninvestigated."

His reputation was immense: standard-bearer of the Old Text classics, grand master of the Zuo Zhuan and Zhou Li schools, plus achievements like compiling the Santong Calendar and collating pre-Qin texts to produce the Seven Summaries. The uncultured Diwu Lun understood none of this — until Yang Xiong mentioned one matter.

"In the time of Emperor Ai of the former dynasty, Liu Xin also collated the ancient text Classic of Mountains and Seas from the era of Tang and Yu, eighteen chapters in all, and presented it to the Son of Heaven."

At that, Diwu Lun's eyes lit up. Who did they think they were dealing with? The Classic of Mountains and Seas — of course he knew it!

Having only heard of it and never read it was beside the point.

And so, the State Preceptor's identity became extraordinarily complex: great Confucian, classicist, literary scholar, calendrical expert — plus Wang Mang's closest comrade-in-arms and in-law. This made Diwu Lun secretly wonder, "This Liu Xiu... could he really be that Liu Xiu, the Child of the Plane? Surely it's not just a shared name."

First help Wang Mang supplant the Western Han, then turn against Mang and rebuild an Eastern Han? The script felt somewhat strange.

And now, finally seeing Liu Xin in the flesh, Diwu Lun realized — what Child of the Plane was this...

You could call him the Grandfather of the Plane!

According to Yang Xiong's description, Liu Xin's age should be about the same as his — sixty-seven or sixty-eight this year. Following Wei Ao and Liu Gong into the inner courtyard of the State Preceptor's residence, Diwu Lun spotted from afar an old man sitting beneath a peach tree with sparse branches and leaves. He wore a plain white robe, a fox-fur mantle draped over his shoulders, his hair grizzled.

"Uncle, Diwu Lun is here." Liu Gong called softly, making Liu Xin lift his head for a glance, then say nothing, lowering his head again to ponder in deep concentration.

Liu Gong quietly instructed Diwu Lun: "Unlucky timing — the State Preceptor is in the middle of mathematical calculations. Go sit on that rush mat over there and wait. If the State Preceptor does not call for you, you absolutely must not make a sound. Disturb the State Preceptor, and you'll be driven out with heavy staves."

That serious? Diwu Lun assented. Now that he was closer, he noticed that Liu Xin was far from showing the decrepitude his age should bring. Perhaps due to skill in self-cultivation, plus the practice of life-nourishing arts, he looked only in his early fifties. Every strand of hair and every wisp of beard was impeccably groomed, and in his every gesture there was a certain transcendent, Daoist air — visually, far superior to that old drunkard Yang Xiong.

On the ground was a pattern closely resembling the Eight Trigrams, the circular ring filled with sand. Its color differed from river sand — it might well have been top-grade fine sand transported from the seashore.

Liu Xin could be seen holding compass and square, ceaselessly drawing circles in the sand, then using ruler and leather cord to take measurements, or carefully drawing polygons outside the circles. Beside his hand lay a pile of counting rods, and with these ancient, backward tools, Liu Xin continuously assembled complex mathematical operations.

Diwu Lun watched this master of both letters and sciences with curiosity, then glimpsed an instrument tossed to one side. He tiptoed over, picked it up, and after examining it, drew a sharp breath.

"This is... a vernier caliper?"

The object was made of bronze, over a foot long, complete with fixed scale, fixed jaw, fish-shaped handle, guide slot, guide pin, assembly sleeve, sliding scale, sliding jaw, pull handle, and all other parts — strikingly similar to the vernier calipers one encounters in middle school or high school physics classes in later ages.

But after Diwu Lun tried it, he found that while it resembled one, it could not slide at all. One could only rely on indicator lines and visual estimation to gauge measurements below the "fen" unit of length — imprecise.

"Give it here."

Liu Xin, probably reaching a point in his calculations where he needed this tool, stretched out his hand to ask for it, yet did not raise his head, treating Diwu Lun as though he were a mere servant.

Diwu Lun carefully skirted the circles in the sand. He knew all too well the fury of a science-and-engineering type when a calculation in progress was disrupted. He simply handed the caliper to Liu Xin.

He had just been about to say something — perhaps offer Liu Xin a few pointers to build rapport — when the old man impatiently waved him off, dismissing his guest.

"I have now seen the man. Erudite, take him away."

So it really was just "seeing" him! Diwu Lun found it utterly baffling.

After they exited the inner courtyard, the Lower Grand Master Liu Gong said with a smile: "Uncle's temperament has always been like this. Never mind you, Boyu — once the Crown Prince came to the residence and happened upon him in the middle of calculations; he was equally ignored. The Crown Prince could only leave in frustration."

The Xin Dynasty Crown Prince, Wang Lin, was precisely Liu Xin's son-in-law, currently locked in covert and overt struggle with the Duke of Chonggong, Wang Zong. Liu Xin naturally stood on the Crown Prince's side. Diwu Lun knew that had he accepted Wang Zong's offer today, this door to the State Preceptor's residence would likely have been closed to him forever.

But having entered, it was of little use — he had not exchanged a single word with Liu Xin.

Fortunately, Liu Gong treated him with great courtesy, inviting Diwu Lun to drink some warmed wine to chase off the chill before leaving.

As the wine was being set out, Diwu Lun seized the chance to ask: "Dare I ask, Grand Master Liu, was the State Preceptor just now calculating the ratio of the circumference to the diameter?"

Diwu Lun had heard Yang Xiong mention it — something about "of the ancient Nine Numbers, the circumference ratio is three, the diameter ratio one," meaning the ancients believed a circle's circumference was three times its diameter, and dividing the two gave the circular constant.

"Indeed so," Liu Gong said. "Has Boyu ever seen the 'Jia Liang'? That was made by my uncle under His Majesty's command."

The Jia Liang was a standard-measure instrument of the Xin Dynasty. Diwu Lun had been astonished the first time he saw it: a bronze cylinder, square inside, with small ears on the left and right. It looked ordinary, but in truth it encompassed all five measures: the hu measure formed the main body, the ring-foot was the dou measure, the left ear the sheng, the right ear's upper part the ge, and its lower part the yue — weighing two jun in total.

On its back was the inscription: "The standard Jia Liang hu: a square foot circumscribed by a circle, the excess at the sides nine li five hao, the area one hundred sixty-two square cun, depth one chi, volume one thousand six hundred twenty cubic cun, capacity ten dou."

Five standards of measurement combined in a single small vessel — truly an ingenious design, and a glimpse of Liu Xin's talent.

Liu Gong said: "When my uncle was making the Jia Liang, he discovered that the ancients' belief in 'circumference three, diameter one' was far too erroneous, so much so that a circle was no circle, harming the sacred dynasty's method of standardizing measures. He then devised a new method of calculation, breaking the angular to form the circular, and derived a new circular constant — and thus the Jia Liang was completed."

"But recently, my uncle has found that the circular constant is still not precise enough, and so he has been repeatedly recalculating."

"What was the previously calculated figure?" Diwu Lun pressed. Seeing that Liu Gong would not continue, he deliberately said: "To be honest, Grand Master Liu, my master, the Ziyun Elder, has also been at home cutting circles and calculating with rods, and has made some gains — perhaps they could benefit the State Preceptor."

"The Ziyun Elder is calculating it too?"

Liu Gong did not realize this was Diwu Lun's fabrication. After a moment's surprise, he understood and could not help but laugh heartily.

Fine — the two old rivals were at it again! Over the past few years, because Yang Xiong would not let Liu Xin see his Fangyan, Liu Xin had been nursing a grudge and wanted to compile a similar work himself. Having seen it so often, Liu Gong had grown accustomed to it.

Liu Gong then gestured: "Taking the circle's diameter as one zhang, the circumference's excess value is three zhang, one chi, five cun, eight fen, six li."

In this era, there was no convenient way to express digits after the decimal point, so the eight units — zhang, chi, cun, fen, li, hao, miao, hu — were used as integers to express it. Diwu Lun understood.

"3.1586, eh. Not bad already, but still quite far off..."

Diwu Lun had come today to see Liu Xin, yet had seen nothing but emptiness. He recalled how earlier Yang Xiong, on his behalf, had gone to the State Preceptor's residence to plead his case, only to be mocked by Liu Xin and suffer great indignity, remaining sullen for days.

So, with a touch of sly, vengeful mischief, he rose to take his leave and said: "My master and I, after days of arduous calculation, have already obtained the most precise circular constant — even if Zhang Cang were reborn, he could not surpass it."

"Therefore, the State Preceptor need not waste any more time in fruitless, solitary calculation."

Liu Gong's face stiffened: "You whelp, what grand talk. A few days' work, and you think you can surpass my uncle's decade and more of calculation?"

Diwu Lun smiled: "If you doubt it, Grand Master, just pass a number on to the State Preceptor and have him reverse-calculate it — then you'll know which is better!"

Wasn't this just stoking the fire between the two old rivals? Yet Liu Gong was also keen to watch the excitement.

Diwu Lun left behind a number and departed. For someone like Liu Xin, stuck on a mathematical problem, being told only the answer without the process of solving it was all the more maddening.

Never mind whether he was the Child of the Plane or not — first, vent some spleen on Yang Xiong's behalf.

A short while later, Liu Xin, who had been racking his brains beneath the peach tree, yet misled by his algorithm and the precision of his tools, unable to obtain a more accurate result, heard from Liu Gong's mouth a string of digits.

"Uncle, the circumference excess value calculated by Yang Xiong and Diwu Lun is..."

"Three zhang, one chi, four cun, one fen, five li, nine hao, two miao, six hu!"

……

Meanwhile, Diba Jiao also returned to the Imperial Academy dormitory beaming with delight and began packing his bags.

Liu Xiu, who happened to live next door, came over and saw this, and couldn't help being surprised.

"Where is Jizheng headed? A long journey, or back home?"

After the banner-raising rescue of Diwu Lun, Diba Jiao trusted Liu Xiu deeply, and being straightforward and honest by nature, he considered Liu Xiu a kindred spirit, so he cheerfully gave a rough account of the day's events.

"The day after tomorrow I'm going to the residence of the Duke of Gongchong to take up my post as Mounted Escort of the duchy. I can no longer remain a fellow student with Wenshu."

Hearing this, Liu Xiu was instantly alarmed, but his concern was for someone else: "Has Diwu Boyu also accepted an appointment from the Duke of Gongchong?"

Diba Jiao shook his head: "That he hasn't. Boyu intends to withdraw from the world and has no interest in an official career. He refused."

The moment these words were out, Liu Xiu couldn't help laughing, though Diba Jiao had no idea what he found so amusing.

Liu Xiu merely said he had come to thank the Diwu Lun brothers for gifting coal — as a token of thanks from Diwu Lun, every Imperial Academy student who had taken part in raising the banner had received a share.

And Liu Xiu, to his surprise, had received double the amount, matching Liu Long's share; it seemed Diwu Lun already knew who the chief contributors that day had been.

Though Liu Xiu's face betrayed nothing, inwardly he felt a small thrill of delight: "Diwu Lun has taken note of Liu Wenshu."

After this episode, Diwu Lun's name had already spread throughout Chang'an. And looking around, so many Imperial Academy students had taken part in the affair and received gifts from Diwu Lun — the gifts were modest, but the sentiment was weighty. They came from all corners of the realm, and when they finished their studies and returned to their hometowns, they would spread the word by mouth, scattering Diwu Lun's reputation across the land.

"If only my elder brother could gain the support of such a renowned scholar."

And now, learning that Diwu Lun had no intention of serving the Wang Mang dynasty, Liu Xiu was even more overjoyed.

Liu Xiu had heard the story of Gong Sheng, the great Confucian scholar of Chu. In the time of Emperor Ai of Han, this old scholar had already denounced harsh punishments and oppressive taxes, and was renowned as a man of unblemished integrity. Later, displeased that Emperor Ai favored Dong Xian, he had offered biting criticism. When Wang Mang took control of the government, Gong Sheng discerned Wang Mang's intentions and retired to his home village.

In the third year of Shijianguo of the Xin Dynasty (AD 11), Wang Mang wished to appoint Gong Sheng as tutor to the Crown Prince. Gong Sheng refused the command, resolutely would not board the carriage, and ultimately starved himself to death.

There was also Guo Dan, from the same commandery as Liu Xiu. Guo Dan was a native of Rang County, orphaned at seven, and famed for his filial devotion to his stepmother. He later entered the Imperial Academy at Chang'an, often served as head lecturer, and was revered by the scholars. After Wang Mang usurped the throne, he summoned Guo Dan twice, offering high office, but Guo Dan declined on grounds of illness and refused, eventually leading a group of students to the Northern Lands Commandery, where government authority was weak.

In Liu Xiu's eyes, refusing office and withdrawing from the world — these two things combined were practically synonymous with discontent toward Wang Mang!

Back in his room, the corners of Liu Xiu's mouth curved up into a perfect checkmark.

"These men who withdraw from the world and refuse to serve Wang Mang."

"Every last one of them, without exception, yearns for our Great Han!"

……

PS: Palace Attendant, Commandant of the Equipage, and Imperial Household Grandee, your servant Xiu, collated; the Secretariat's collation-officer and the Secretariat Grand Master of Ceremonies' subordinate, your servant Wang, collated the Classic of Mountains and Seas, in total thirty-two sections, now fixed at eighteen sections — finalized. — Memorial Presenting the Classic of Mountains and Seas.

Additionally, Liu Xin was the first person in China to attempt calculating pi.