Science and technology of the Han dynasty

The Han dynasty (206 BCE – 220 CE) of early imperial China, divided between the eras of Western Han (206 BCE – 9 CE, when the capital was at Chang'an), the Xin dynasty of Wang Mang (r. 9–23 CE), and Eastern Han (25–220 CE, when the capital was at Luoyang, and after 196 CE at Xuchang), witnessed some of the most significant advancements in premodern Chinese science and technology.

There were great innovations in metallurgy. In addition to Zhou-era China's (c. 1046 – 256 BCE) previous inventions of the blast furnace and cupola furnace to make pig iron and cast iron, respectively, the Han period saw the development of steel and wrought iron by use of the finery forge and puddling process. With the drilling of deep boreholes into the earth, the Chinese used not only derricks to lift brine up to the surface to be boiled into salt, but also set up bamboo-crafted pipeline transport systems which brought natural gas as fuel to the furnaces. Smelting techniques were enhanced with inventions such as the waterwheel-powered bellows; the resulting widespread distribution of iron tools facilitated the growth of agriculture. For tilling the soil and planting straight rows of crops, the improved heavy-moldboard plough with three iron plowshares and sturdy multiple-tube iron seed drill were invented in the Han, which greatly enhanced production yields and thus sustained population growth. The method of supplying irrigation ditches with water was improved with the invention of the mechanical chain pump powered by the rotation of a waterwheel or draft animals, which could transport irrigation water up elevated terrains. The waterwheel was also used for operating trip hammers in pounding grain and in rotating the metal rings of the mechanical-driven astronomical armillary sphere representing the celestial sphere around the Earth.

The quality of life was improved with many Han inventions. The Han Chinese had hempen-bound bamboo scrolls to write on, yet by the 2nd century CE had invented the papermaking process which created a writing medium that was both cheap and easy to produce. The invention of the wheelbarrow aided in the hauling of heavy loads. The maritime ''junk'' ship and stern-mounted steering rudder enabled the Chinese to venture out of calmer waters of interior lakes and rivers and into the open sea. The invention of the grid reference for maps and raised-relief map allowed for better navigation of their terrain. In medicine, they used new herbal remedies to cure illnesses, calisthenics to keep physically fit, and regulated diets to avoid diseases. Authorities in the capital were warned ahead of time of the direction of sudden earthquakes with the invention of the seismometer that was tripped by a vibration-sensitive pendulum device. To mark the passing of the seasons and special occasions, the Han Chinese used two variations of the lunisolar calendar, which were established due to efforts in astronomy and mathematics. Han-era Chinese advancements in mathematics include the discovery of square roots, cube roots, the Pythagorean theorem, Gaussian elimination, the Horner scheme, improved calculations of '' pi'', and negative numbers. Hundreds of new roads and canals were built to facilitate transport, commerce, tax collection, communication, and movement of military troops. The Han-era Chinese also employed several types of bridges to cross waterways and deep gorges, such as beam bridges, arch bridges, simple suspension bridges, and pontoon bridges. Han ruins of defensive city walls made of brick or rammed earth still stand today.

Modern perspectives

Jin Guantao, a professor of the Institute of Chinese Studies at the Chinese University of Hong Kong, Fan Hongye, a research fellow with the Chinese Academy of Sciences' Institute of Science Policy and Managerial Science, and Liu Qingfeng, a professor of the Institute of Chinese Culture at the Chinese University of Hong Kong, assert that the latter part of the Han dynasty was a unique period in the history of premodern Chinese science and technology.Jin, Fan, & Liu (1996), 178–179. They compare it to the incredible pace of scientific and technological growth during the Song dynasty (960–1279). However, they also argue that without the influence of proto-scientific precepts in the ancient philosophy of Mohism, Chinese science continued to lack a definitive structure:

From the middle and late Eastern Han to the early Wei and Jin dynasties, the net growth of ancient Chinese science and technology experienced a peak (second only to that of the Northern Song dynasty) ...Han studies of the Confucian classics, which for a long time had hindered the socialization of science, were declining. If Mohism, rich in scientific thought, had rapidly grown and strengthened, the situation might have been very favorable to the development of a scientific structure. However, this did not happen because the seeds of the primitive structure of science were never formed. During the late Eastern Han, disastrous upheavals again occurred in the process of social transformation, leading to the greatest social disorder in Chinese history. One can imagine the effect of this calamity on science.

Joseph Needham (1900–1995), a late Professor from the University of Cambridge and author of the groundbreaking ''Science and Civilisation in China'' series, stated that the "Han time (especially the Later Han) was one of the relatively important periods as regards the history of science in China."Needham (1972), 111. He noted the advancements during Han of astronomy and calendrical sciences, the "beginnings of systematic botany and zoology", as well as the philosophical skepticism and rationalist thought embodied in Han works such as the '' Lunheng'' by the philosopher Wang Chong (27–100 CE).

Writing materials

The most common writing mediums found in archaeological digs from ancient sites predating the Han period are shells and bones as well as bronzewares.Loewe (1968), 89. In the beginning of the Han period, the chief writing mediums were bamboo () and clay tablets, silk cloth, strips of soft wood, and rolled scrolls made of strips of bamboo sewn together with hempen string passed through drilled holes (册) and secured with clay stamps. The written characters on these narrow flat strips of bamboo were arranged into vertical columns.

While maps drawn in ink on flat silk cloths have been found in the tomb of the Marquess of Dai (interred in 168 BCE at Mawangdui, Hunan province), the earliest known paper map found in China, dated 179–141 BCE and located at Fangmatan (near Tianshui, Gansu province), is incidentally the oldest known piece of paper.Buisseret (1998), 12. Yet Chinese hempen paper of the Western Han and early Eastern Han eras was of a coarse quality and used primarily as wrapping paper.Needham (1986e), 1–2, 40–41, 122–123, 228. The papermaking process was not formally introduced until the Eastern Han court eunuch Cai Lun (50–121 CE) created a process in 105 where mulberry tree bark, hemp, old linens, and fish nets were boiled together to make a pulp that was pounded, stirred in water, and then dunked with a wooden sieve containing a reed mat that was shaken, dried, and bleached into sheets of paper. The oldest known piece of paper with writing on it comes from the ruins of a Chinese watchtower at Tsakhortei, Alxa League, Inner Mongolia, dated precisely to 110 CE when the Han garrison abandoned the area following a nomadic Xiongnu attack.Cotterell (2004), 11. By the 3rd century, paper became one of China's chief writing mediums.Needham (1986e), 1–2.

Ceramics

The Han ceramics industry was upheld by private businesses as well as local government agencies.Wang (1982), 146–147. Ceramics were used in domestic wares and utensils as well as construction materials for roof tiles and bricks.Wang (1982), 147–149.

Han dynasty grey pottery—its color derived from the clay that was used—was superior to earlier Chinese grey pottery due to the Han people's use of larger kiln chambers, longer firing tunnels, and improved chimney designs.Wang (1982), 142–143. Kilns of the Han dynasty making grey pottery were able to reach firing temperatures above . However, hard southern Chinese pottery made from a dense adhesive clay native only in the south (i.e. Guangdong, Guangxi, Hunan, Jiangxi, Fujian, Zhejiang, and southern Jiangsu) was fired at even higher temperatures than grey pottery during the Han. Glazed pottery of the Shang (c. 1600 – c. 1050 BCE) and Zhou (c. 1050 – 256 BCE) dynasties were fired at high temperatures, but by the mid Western Han (206 BCE – 9 CE), a brown-glazed ceramic was made which was fired at the low temperature of , followed by a green-glazed ceramic which became popular in the Eastern Han (25–220 CE).Wang (1982), 143–145.

Wang Zhongshu states that the light-green stoneware known as celadon was thought to exist only since the Three Kingdoms period (220–265 CE) onwards, but argues that ceramic shards found at Eastern Han (25–220 CE) sites of Zhejiang province can be classified as celadon.Wang (1982), 145. However, Richard Dewar argues that true celadon was not created in China until the early Song dynasty (960–1279) when Chinese kilns were able to reach a minimum furnace temperature of , with a preferred range of for celadon.Dewar (2002), 42.

Metallurgy

Furnaces and smelting techniques

A blast furnace converts raw iron oxide into iron, which can be remelted in a cupola furnace to produce cast iron. The earliest specimens of cast iron found in China date to the 5th century BCE during the late Spring and Autumn period, yet the oldest discovered blast furnaces date to the 3rd century BCE and the majority date to the period after Emperor Wu of Han (r. 141–87 BCE) established a government monopoly over the iron industry in 117 BCE (most of the discovered iron works sites built before this date were merely foundries which recast iron that had been smelted elsewhere). Iron ore smelted in blast furnaces during the Han was rarely if ever cast directly into permanent molds; instead, the pig iron scraps were remelted in the cupola furnace to make cast iron.Wagner (2001), 75–76. Cupola furnaces utilized a cold blast traveling through tuyere pipes from the bottom and over the top where the charge of charcoal and pig iron was introduced. The air traveling through the tuyere pipes thus became a hot blast once it reached the bottom of the furnace.

Although Chinese civilization lacked the bloomery, the Han Chinese were able to make wrought iron when they injected too much oxygen into the cupola furnace, causing decarburization. The Han-era Chinese were also able to convert cast iron and pig iron into wrought iron and steel by using the finery forge and puddling process, the earliest specimens of such dating to the 2nd century BCE and found at Tieshengguo near Mount Song of Henan province. The semisubterranean walls of these furnaces were lined with refractory bricks and had bottoms made of refractory clay. Besides charcoal made of wood, Wang Zhongshu states that another furnace fuel used during the Han were "coal cakes", a mixture of coal powder, clay, and quartz.Wang (1982), 126.

Use of steel, iron, and bronze

Donald B. Wagner writes that most domestic iron tools and implements produced during the Han were made of cheaper and more brittle cast iron, whereas the military preferred to use wrought iron and steel weaponry due to their more durable qualities. During the Han dynasty, the typical bronze sword found in the Warring States period was gradually replaced with an iron sword measuring roughly in length. The ancient dagger-axe (''ge'') made of bronze was still used by Han soldiers, although it was gradually phased out by iron spears and iron ''ji'' halberds.Wang (1982), 123. Even arrowheads, which were traditionally made of bronze, gradually only had a bronze tip and iron shaft, until the end of the Han when the entire arrowhead was made solely of iron. Farmers, carpenters, bamboo craftsmen, stonemasons, and rammed earth builders had at their disposal iron tools such as the plowshare, pickaxe, spade, shovel, hoe, sickle, axe, adze, hammer, chisel, knife, saw, scratch awl, and nails.Wang (1982), 122. Common iron commodities found in Han dynasty homes included tripods, stoves, cooking pots, belt buckles, tweezers, fire tongs, scissors, kitchen knives, fish hooks, and needles. Mirrors and oil lamps were often made of either bronze or iron. Coin money minted during the Han was made of either copper or copper and tin smelted together to make the bronze alloy.

Agriculture

Tools and methods

Modern archaeologists have unearthed Han iron farming tools throughout China, from Inner Mongolia in the north to Yunnan in the south.Wang (1982), 53. The spade, shovel, pick, and plow were used for tillage, the hoe for weeding, the rake for loosening the soil, and the sickle for harvesting crops. Depending on their size, Han plows were driven by either one ox or two oxen.Wang (1982), 54. Oxen were also used to pull the three-legged iron seed drill (invented in Han China by the 2nd century BCE), which enabled farmers to plant seeds in precise rows instead of casting them out by hand. While artwork of the Wei (220–266 CE) and Jin (266–420) periods show use of the harrow for breaking up chunks of soil after plowing, it perhaps first appeared in China during the Eastern Han (25–220 CE).Wang (1982), 55. Irrigation works for agriculture included the use of water wells, artificial ponds and embankments, dams, canals, and sluice gates.

Alternating fields

During Emperor Wu's (r. 141–87 BCE) reign, the Grain Intendant Zhao Guo (趙過) invented the alternating fields system (''daitianfa'' 代田法).Nishijima (1986), 561. For every ''mou'' of land—i.e. a thin but elongated strip of land measuring wide and long, or an area of roughly —three low-lying furrows (''quan'' 甽) that were each wide were sowed in straight lines with crop seed. While weeding in the summer, the loose soil of the ridges (''long'' 壟) on either side of the furrows would gradually fall into the furrows, covering the sprouting crops and protecting them from wind and drought. Since the position of the furrows and ridges were reversed by the next year, this process was called the alternating fields system.

This system allowed crops to grow in straight lines from sowing to harvest, conserved moisture in the soil, and provided a stable annual yield for harvested crops.Nishijima (1986), 562. Zhao Guo first experimented with this system right outside the capital Chang'an, and once it proved successful, he sent out instructions for it to every commandery administrator, who were then responsible for disseminating these to the heads of every county, district, and hamlet in their commanderies. Sadao Nishijima speculates that the Imperial Counselor Sang Hongyang (d. 80 BCE) perhaps had a role in promoting this new system.

Rich families who owned oxen and large heavy moldboard iron plows greatly benefited from this new system. However, poorer farmers who did not own oxen resorted to using teams of men to move a single plow, which was exhausting work.Nishijima (1986), 563–564. The author Cui Shi (催寔) (d. 170 CE) wrote in his ''Simin yueling'' (四民月令) that by the Eastern Han Era (25–220 CE) an improved plow was invented which needed only one man to control it, two oxen to pull it, had three plowshares, a seed box for the drills, a tool which turned down the soil, and could sow roughly of land in a single day.

Pit fields

During the reign of Emperor Cheng of Han (r. 33–37 BCE), Fan Shengzhi wrote a manual (i.e. the ''Fan Shengzhi shu'') which described the pit field system (''aotian'' 凹田).Nishijima (1986), 564–565.Hinsch (2002), 67–68. In this system, every ''mou'' of farmland was divided into 3,840 grids which each had a small pit that was dug deep and wide and had good quality manure mixed into the soil. Twenty seeds were sowed into each pit, which allegedly produced 0.6 L (20 oz) of harvested grain per pit, or roughly 2,000 L (67,630 oz) per ''mou''. This system did not require oxen-driven plows or the most fertile land, since it could be employed even on sloping terrains where supplying water was difficult for other methods of farming. Although this farming method was favored by the poor, it did require intensive labor, thus only large families could maintain such a system.

Rice paddies

Han farmers in the Yangzi River region of southern China often maintained paddy fields for growing rice. Every year, they would burn the weeds in the paddy field, drench it in water, sow rice by hand, and around harvest time cut the surviving weeds and drown them a second time.Nishijima (1986), 568–569. In this system, the field lays fallow for much of the year and thus did not remain very fertile. However, Han rice farmers to the north around the Huai River practiced the more advanced system of transplantation.Nishijima (1986), 570–572. In this system, individual plants were given intensive care (perhaps in the same location as the paddy field), their offshoots separated so that more water could be conserved, and the field could be heavily fertilized since winter crops were grown while the rice seedlings were situated nearby in a plant nursery.

Mechanical and hydraulic engineering

Literary sources and archaeological evidence

Evidence of Han-era mechanical engineering comes largely from the choice observational writings of sometimes disinterested Confucian scholars. Professional artisan-engineers (''jiang'' 匠) did not leave behind detailed records of their work. Han scholars, who often had little or no expertise in mechanical engineering, sometimes provided insufficient information on the various technologies they described.

Nevertheless, some Han literary sources provide crucial information. As written by Yang Xiong in 15 BCE, the belt drive was first used for a quilling device which wound silk fibers onto the bobbins of weaver shuttles.Temple (1986), 54–55. The invention of the belt drive was a crucial first step in the development of later technologies during the Song dynasty, such as the chain drive and spinning wheel.

The inventions of the artisan and mechanical engineer Ding Huan (丁緩) are mentioned in the ''Miscellaneous Notes on the Western Capital''. The official and poet Sima Xiangru (179–117 BCE) once hinted in his writings that the Chinese used a censer in the form of a gimbal, a pivot support made of concentric rings which allow the central gimbal to rotate on an axis while remaining vertically positioned.Needham (1986c), 233–234. However, the first explicit mention of the gimbal used as an incense burner occurred around 180 CE when the artisan Ding Huan created his 'Perfume Burner for use among Cushions' which allowed burning incense placed within the central gimbal to remain constantly level even when moved. Ding had other inventions as well. For the purpose of indoor air conditioning, he set up a large manually operated rotary fan which had rotating wheels that were in diameter. He also invented a lamp which he called the 'nine-storied hill-censer', since it was shaped as a hillside.Temple (1986), 87; Needham (1986b), 123. When the cylindrical lamp was lit, the convection of rising hot air currents caused vanes placed on the top to spin, which in turn rotated painted paper figures of birds and other animals around the lamp.

When Emperor Gaozu of Han (r. 202–195 BCE) came upon the treasury of Qin Shi Huang (r. 221–210) at Xianyang following the downfall of the Qin dynasty (221–206), he found an entire miniature musical orchestra of puppets tall who played mouth organs if one pulled on ropes and blew into tubes to control them.Needham (1986c), 158. Zhang Heng wrote in the 2nd century CE that people could be entertained by theatrical plays of artificial fish and dragons. Later, the inventor Ma Jun ( fl. 220–265) invented a theater of moving mechanical puppets powered by the rotation of a hidden waterwheel.

From literary sources it is known that the collapsible umbrella was invented during Wang Mang's reign, although the simple parasol existed beforehand. This employed sliding levers and bendable joints that could be protracted and retracted.

Modern archaeology has led to the discovery of Han artwork portraying inventions which were otherwise absent in Han literary sources. This includes the crank handle. Han pottery tomb models of farmyards and gristmills possess the first known depictions of crank handles, which were used to operate the fans of winnowing machines.Needham (1986c), 116–119, 153–154 & Plate CLVI; Temple (1986), 46; Wang (1982), 57. The machine was used to separate chaff from grain, but the Chinese of later dynasties also employed the crank handle for silk-reeling, hemp-spinning, flour-sifting, and drawing water from a well using the windlass. To measure distance traveled, the Han-era Chinese also created the odometer cart. This invention is depicted in Han artwork by the 2nd century CE, yet detailed written descriptions were not offered until the 3rd century. The wheels of this device rotated a set of gears which in turn forced mechanical figures to bang gongs and drums that alerted the travelers of the distance traveled (measured in '' li''). From existing specimens found at archaeological sites, it is known that Han-era craftsmen made use of the sliding metal caliper to make minute measurements. Although Han-era calipers bear incised inscriptions of the exact day of the year they were manufactured, they are not mentioned in any Han literary sources.

Uses of the waterwheel and water clock

By the Han dynasty, the Chinese developed various uses for the waterwheel. An improvement of the simple lever-and-fulcrum tilt hammer device operated by one's foot, the hydraulic-powered trip hammer used for pounding, decorticating, and polishing grain was first mentioned in the Han dictionary ''Jijiupian'' of 40 BCE.Needham (1986c), 183–184, 390–392. It was also mentioned in the '' Regional Speech'' (''Fangyan'') dictionary written by Yang Xiong (53 BCE – 18 CE) in 15 BCE, the philosophical ''Xinlun'' 新論 written by Huan Tan (43 BCE – 28 CE) in 20 CE, the poetry of Ma Rong (79–166 CE), and the writings of Kong Rong (153–208 CE).

In his '' Balanced Discourse'' (''Lunheng''), the philosopher Wang Chong (27–100 CE) was the first in China to describe the square-pallet chain pump used to lift water (and other substances).Needham (1986c), 89, 110, & 344. Although some models were operated manually by foot pedals, some chain pumps were powered by a horizontal waterwheel which rotated large toothed gears and a horizontal axis beam.Needham (1986c), 342–346. Their primary use was for lifting water into irrigation ditches, but chain pumps were also used in public works programs, such as when Zhang Rang (d. 189) had an engineer build several of them to lift water into pipes that provided the capital Luoyang and its palaces with clean water.

While acting as administrator of Nanyang in 31 CE, Du Shi (d. 38) invented a water-powered reciprocator which worked the bellows of the blast furnace and cupola furnace in smelting iron; before this invention, intensive manual labor was required to work the bellows.

Although the astronomical armillary sphere (representing the celestial sphere) had existed in China since the 1st century BCE, the mathematician and court astronomer Zhang Heng (78–139 CE) provided it with motive power by using the constant pressure head of an inflow water clock to rotate a waterwheel that acted on a set of gears. Zhang Heng was also the first to address the problem of the falling pressure head in the inflow water clock (which gradually slowed the timekeeping) by setting up an additional tank between the reservoir and inflow vessel.

Seismometer

The Han court was responsible for the major efforts of disaster relief when natural disasters such as earthquakes devastated the lives of commoners.Ebrey (1986), 621. To better prepare for calamities, Zhang Heng invented a seismometer in 132 CE, which provided instant alert to authorities in the capital Luoyang that an earthquake had occurred in a location indicated by a specific cardinal or ordinal direction. Although no tremors could be felt in the capital when Zhang told the court that an earthquake had just occurred in the northwest, a message came soon afterwards that an earthquake had indeed struck northwest of Luoyang (in what is now modern Gansu). Zhang called his device the 'instrument for measuring the seasonal winds and the movements of the Earth' (Houfeng didong yi 候风地动仪), so-named because he and others thought that earthquakes were most likely caused by the enormous compression of trapped air.Needham (1986a), 626.

As described in the '' Book of the Later Han'', the frame of the seismometer was a domed bronze vessel in the shape of a wine jar, although it was in diameter and decorated with scenes of mountains and animals. The trigger mechanism was an inverted pendulum (which the ''Book of the Later Han'' calls the "central column") that, if disturbed by the ground tremors of earthquakes located near or far away, would swing and strike one of eight mobile arms (representing the eight directions), each with a crank and catch mechanism. The crank and a right angle lever would raise one of eight metal dragon heads located on the exterior, dislodging a metal ball from its mouth that dropped into the mouth of one of eight metal toads below arranged like the points on a compass rose, thus indicating the direction of the earthquake.Needham (1986a), 627–631. The ''Book of the Later Han'' states that when the ball fell into any one of eight toad mouths, it produced a loud noise which gained the attention of those observing the device.Needham (1986a), 626–627. While Wang Zhenduo (王振铎) accepted the idea that Zhang's seismometer had cranks and levers which were disturbed by the inverted pendulum, his contemporary Akitsune Imamura (1870–1948) argued that the inverted pendulum could have had a pin at the top which, upon moving by force of the ground vibrations, would enter one of eight slots and expel the ball by pushing a slider.Needham (1986a), 631. Since the ''Book of the Later Han'' states that the other seven dragon heads would not subsequently release the balls lodged up into their jaws after the first one had dropped, Imamura asserted that the pin of the pendulum would have been locked into the slot it had entered and thus immobilized the instrument until it was reset.

Mathematics and astronomy

Mathematical treatises

One of the earliest surviving mathematical treatises of ancient China is the ''Book on Numbers and Computation'' (''Suan shu shu''), part of the Zhangjiashan Han bamboo texts dated 202 to 186 BCE and found in Jiangling County, Hubei. Another mathematical text compiled during the Han was ''The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven'' (''Zhoubi Suanjing''), dated no earlier than the 1st century BCE (from perhaps multiple authors) and contained materials similar to those described by Yang Xiong in 15 BCE, yet the ''zhoubi'' school of mathematics was not explicitly mentioned until Cai Yong's (132–192 CE) commentary of 180. A preface was added to the text by Zhao Shuang 趙爽 in the 3rd century.Dauben (2007), 214. There was also the ''Nine Chapters on the Mathematical Art'' (''Jiuzhang Suanshu''); its full title was found on two bronze standard measurers dated 179 CE (with speculation that its material existed in earlier books under different titles) and was provided with detailed commentary by Liu Hui (fl. 3rd century) in 263.Needham (1986a), 24–25. It is worth noting in this context that many of the documents excavated from Qin and Han sites contain evidence of the practical mathematics used by administrators for inventories and taxes, as well as for calculating labor needed for public works projects, just as described in the mathematical treatises.

Innovations in the treatises

The ''Suan shu shu'' presents basic mathematics problems and solutions. It was most likely a handbook for day-to-day business transactions or affairs of government administration.Dauben (2007), 213. It contains problems and solutions for field measurements of area, proportional exchange rates for agricultural millet and rice, distribution by proportion, short width division, and excess and deficiency.Dauben (2007), 212. Some of the problems found in the ''Suan shu shu'' appear in the later text ''Jiuzhang suanshu''; in five cases, the titles are exact matches. However, unlike the ''Jiuzhang suanshu'', the ''Suan shu shu'' does not deal with problems involving right-angle triangles, square roots, cube roots, and matrix methods, which demonstrates the significant advancements made in Chinese mathematics between the writings of these two texts.

The ''Zhoubi suanjing'', written in dialogue form and with regularly presented problems, is concerned with the application of mathematics to astronomy. In one problem which sought to determine the height of the Sun from the Earth and the diameter of the Sun, Chen Zi (陳子) instructs Rong Fang (榮方) to wait until the shadow cast by the 8 ''chi'' tall gnomon is 6 '' chi'' (one ''chi'' during the Han was 33 cm), so that a 3-4-5 right-angle triangle can be constructed where the base is 60,000 ''li'' (one ''li'' during the Han was the equivalent of 415 m or 1362 ft), the hypotenuse leading towards the sun is 100,000 ''li'', and the height of the sun is 80,000 ''li''.Dauben (2007), 219. Like the ''Jiuzhang suanshu'', the ''Zhoubi suanjing'' also gives mathematical proof for the "Gougu Theorem" (勾股定理; i.e. where ''c'' is the length of the hypotenuse and ''a'' and ''b'' are the lengths of the other two sides, respectively, a² + b² = c²), which is known as the Pythagorean theorem in the West after the Greek mathematician Pythagoras (fl. 6th century BCE).

The ''Jiuzhang suanshu'' was perhaps the most groundbreaking of the three surviving Han treatises. It is the first known book to feature negative numbers, along with the Bakhshali manuscript (200? – 600? CE) of India and the book of the Greek mathematician Diophantus (fl. 3rd century) written in about 275 CE. Negative numbers appeared as black counting rods, while positive numbers appeared as red counting rods. Although the decimal system existed in China since the Shang dynasty (c. 1600 – c. 1050 BCE), the earliest evidence of a decimal fraction (i.e. the denominator is a power of ten) is an inscription on a standard volume-measuring vessel dated 5 CE and used by the mathematician and astronomer Liu Xin (46 BCE – 23 CE).Temple (1986), 139 & 142–143. Yet the first book to feature decimal fractions was the ''Jiuzhang suanshu'', as a means to solve equations and represent measurements. Gaussian elimination, an algorithm used to solve linear equations, was known as the Array Rule in the ''Jiuzhang suanshu''. While the book used continued fractions to find the roots of equations, Liu Hui built on this idea in the 3rd century when he increased the decimals to find the cube root of 1,860,867 (yielding the answer 123), the same method used in the Horner scheme named after William George Horner (1786–1837).Temple (1986), 142.

Approximations of pi

For centuries, the Chinese had simply approximated the value of pi as 3, until Liu Xin approximated it at 3.154 sometime between 1–5 CE, although the method he used to reach this value is unknown to historians.Needham (1986a), 99–100. Standard measuring vessels dating to the reign of Wang Mang (9–23 CE) also showed approximations for pi at 3.1590, 3.1497, and 3.167.Berggren, Borwein & Borwein (2004), 27. Zhang Heng is the next known Han mathematician to have made an approximation for pi. Han mathematicians understood that the area of a square versus the area of its inscribed circle had an approximate ratio of 4:3, and also understood that the volume of a cube and the volume of its inscribed sphere would be 4²:3². With ''D'' as diameter and ''V'' as volume, D³:V = 16:9 or V=⁹⁄₁₆D³, a formula Zhang found fault with since he realized the value for diameter was inaccurate, the discrepancy being the value taken for the ratio. To fix this, Zhang added ¹⁄₁₆D³ to the formula, thus V = ⁹⁄₁₆D³ + ¹⁄₁₆D³ = ⁵⁄₈D³. Since he found the ratio of the volume of the cube to the inscribed sphere at 8:5, the ratio of the area of a square to the inscribed circle is :. With this formula, Zhang was able to approximate pi as the square root of 10, or 3.162. After the Han, Liu Hui approximated pi as 3.14159, while the mathematician Zu Chongzhi (429–500) approximated pi at 3.141592 (or ³⁵⁵⁄₁₁₃), the most accurate approximation the ancient Chinese would achieve.

Musical tuning and theory

Mathematics were also used in musical tuning and music theory. The 2nd-century-BCE ''Huainanzi'', compiled by eight scholars under the patronage of King Liu An (179–122 BCE), outlined the use of twelve tones on a musical scale.McClain & Ming (1979), 207–208. Jing Fang (78–37 BCE), a mathematician and music theorist, expanded these to create a scale of 60 tones. While doing so, Jing Fang realized that 53 just fifths is approximate to 31 octaves.McClain & Ming (1979), 212; Needham (1986b), 218–219. By calculating the difference at ¹⁷⁷¹⁴⁷⁄₁₇₆₇₇₆, Jing reached the same value of 53 equal temperament duly discovered by the German mathematician Nicholas Mercator (1620–1687) (i.e. 3⁵³/2⁸⁴, known as Mercator's comma). Later, the prince Zhu Zaiyu (1536–1611) in Ming China and Simon Stevin (1548–1620) of the Flemish Region in Europe would simultaneously (but separately) discover the mathematical formula for equal temperament.

Astronomical observations

The ancient Chinese made careful observations of heavenly bodies and phenomena since observations of the cosmos were used for astrology and prognostication. The astronomer Gan De (fl. 4th century BCE) from the State of Qi was the first in history to acknowledge sunspots as genuine solar phenomena (and not obstructing natural satellites as thought in the West after Einhard's observation in 807), while the first precisely dated sunspot observation in China occurred on May 10, 28 BCE, during the reign of Emperor Cheng of Han (r. 33–7 BCE). Among the Mawangdui Silk Texts dated no later than 168 BCE (when they were sealed in a tomb at Mawangdui Han tombs site, Changsha, Hunan province), the ''Miscellaneous Readings of Cosmic Patterns and Pneuma Images'' (''Tianwen qixiang zazhan'' 天文氣象雜占) manuscript illustrates in writings and ink drawings roughly 300 different climatic and astronomical features, including clouds, mirages, rainbows, stars, constellations, and comets. Another silk text from the same site reports the times and locations of the rising and setting of planets in the night sky from the years 246–177 BCE.

The Han-era Chinese noted the passage of the same comet seen in Persia for the birth of Mithridates II of Parthia in 135 BCE, the same comet the Romans observed close to the assassination of Julius Caesar in 44 BCE, Halley's comet in 12 BCE, the same comet noted by Roman historian Cassius Dio (c. 155 – c. 229 CE) for 13 CE, and (what is now known to have been) a supernova in AD 185.Loewe (1994), 69. For various comets discussed in the Han-era history books ''Records of the Grand Historian'' and ''Book of Han'', details are given for their position in the sky and direction they were moving, the length of time they were visible, their color, and their size.

The Han-era Chinese also made star catalogues, such as historian Sima Qian's (145–86 BCE) ''A Monograph on Celestial Officials'' (''Tianguanshu'' 天官書) and Zhang Heng's 2nd-century-CE star catalogue which featured roughly 2,500 stars and 124 constellations. To create a three-dimensional representation of such observations, Astronomer Geng Shouchang (耿壽昌) provided his armillary sphere with an equatorial ring in 52 BCE. By 84 CE, the elliptical ring was added to the armillary sphere, while Zhang Heng's model of 125 added the celestial horizon ring and meridian ring.

Han calendars

The Han Chinese used astronomical studies mainly to construct and revise their calendar. In contrast to the Julian calendar (46 BCE) and Gregorian calendar (1582) of the West (but like the Hellenic calendars of Classical Greece), the Chinese calendar is a lunisolar calendar, meaning that it uses the precise movements of the Sun and Moon as time-markers throughout the year. During the spring and autumn periods of the 5th century BCE, the Chinese established the Sifen calendar (古四分历), which measured the tropical year at 365¹⁄₄ days (like the Julian calendar of Rome).Deng (2005), 67. Emperor Wu replaced this with the new Taichu calendar (太初历) in 104 BCE, which measured the tropical year at 365³⁸⁵⁄₁₅₃₉ days and the lunar month at 29⁴³⁄₈₁ days. Since the Taichu calendar had become inaccurate over two centuries, Emperor Zhang of Han (r. 75–88 CE) halted its use and revived use of the Sifen calendar. Later, astronomer Guo Shoujing (1233–1316) would set the tropical year at 365.2425 days for his Shoushi calendar (授時曆), the same value used in the Gregorian calendar. Besides the use of the calendar for regulating agricultural practices throughout the seasons, it was also used to mark important dates in the Sexagenary cycle—constructed by celestial stems (''gan'' 干) and Earthly Branches (''zhi'' 支), each of the latter associated with an animal of the Chinese zodiac.

Astronomical theory

Zhao Shaung's 3rd-century commentary in the ''Zhoubi suanjing'' describes two astronomical theories: in one, the heavens are shaped as a hemi-spherical dome extending over the Earth, while the other compares the Earth to the central yolk of an egg, where the heavens are shaped as a celestial sphere around the earth. The latter astronomical theory was mentioned by Yang Xiong in his ''Model Sayings'' (''Fayan'' 法言) and expounded on by Zhang Heng in his ''Spiritual Constitution of the Universe'' (''Lingxian'' 靈憲) of 120 CE. Thus, the Han-era Chinese believed in a geocentric model for the immediate Solar System and greater universe, as opposed to a heliocentric model.

The Han-era Chinese discussed the illumination and shapes of heavenly bodies: were they flat and circular, or were they rounded and spherical? Jing Fang wrote in the 1st century BCE that Han astronomers believed the Sun, Moon, and planets were spherical like balls or crossbow bullets.Needham (1986a), 227. He also wrote that the Moon and planets produce no light of their own, are viewable to people on Earth only because they are illuminated by the Sun, and those parts not illuminated by the Sun would be dark on the other side. For this, Jing compared the Moon to a mirror illuminating light. In the 2nd century CE, Zhang Heng drew a similar comparison to Jing's by stating that the Sun is like fire and the Moon and planets are like water, since fire produces light and water reflects it.Needham (1986a), 414. He also repeated Jing's comment that the side of the Moon not illuminated by the Sun was left in darkness. However, Zhang noted that sunlight did not always reach the Moon since the Earth obstructs the rays during a lunar eclipse. He also noted that a solar eclipse occurred when the Moon and Sun crossed paths to block sunlight from reaching earth.

In his '' Balanced Discourse'' (''Lunheng''), Wang Chong (27–100 CE) wrote that some Han thinkers believed that rain fell from the Heavens (i.e. where the stars were located).Needham (1986a), 468. Wang argued that, although rain fell from above, this common theory was false. He agreed with another theory that stated clouds were formed by the evaporation of water on Earth, and that since clouds disperse rain, clouds and rain are in fact one and the same; in essence, he accurately described the water cycle.

Structural engineering and public works

Materials and construction

Timber was the chief building material in Han architecture.Ebrey (1999), 76. It was used for grand palace halls, multi-story towers, multi-story residential halls, and humble abodes. However, due to wood's rapid decay over time and susceptibility to fire, the oldest wooden buildings found in China (i.e. several temple halls of Mount Wutai) date no earlier than the Tang dynasty (618–907). Architectural historian Robert L. Thorp describes the scarcity of Han-era archaeological remains, as well as the often unreliable Han-era literary and artistic sources used by historians for clues about non-existent Han architecture. What remains of Han-dynasty architecture are ruins of brick and rammed earth walls (including aboveground city walls and underground tomb walls), rammed earth platforms for terraced altars and halls, funerary stone or brick pillar-gates, and scattered ceramic roof tiles that once adorned timber halls. Sections of the Han-era rammed earth Great Wall still exist in Gansu province, along with the Han frontier ruins of thirty beacon towers and two fortified castles with crenellations. Han walls of frontier towns and forts in Inner Mongolia were typically constructed with stamped clay bricks instead of rammed earth.

Thatched or tiled roofs were supported by wooden pillars, since the addition of brick, rammed earth, or mud walls of these halls did not actually support the roof. Stone and plaster were also used for domestic architecture. Tiled eaves projecting outward were built to distance falling rainwater from the walls; they were supported by ''dougong'' brackets that were sometimes elaborately decorated. Molded designs usually decorated the ends of roof tiles, as seen in artistic models of buildings and in surviving tile pieces.

Courtyard homes

Valuable clues about Han architecture can be found in Han artwork of ceramic models, paintings, and carved or stamped bricks discovered in tombs and other sites. The layout of Han tombs were also built like underground houses, comparable to the scenes of courtyard houses found on tomb bricks and in three-dimensional models. Han homes had a courtyard area (and some had multiple courtyards) with halls that were slightly elevated above it and connected by stairways. Multi-story buildings included the main colonnaded residence halls built around the courtyards as well as watchtowers. The halls were built with intersecting crossbeams and rafters that were usually carved with decorations; stairways and walls were usually plastered over to produce a smooth surface and then painted.

Chang'an and Luoyang, the Han capitals

The ruins of the walls of Han's first capital Chang'an still stand today at in height with a base width of .Wang (1982), 1–2. Modern archaeological surveys have proven that the eastern wall was long, the southern wall was long, the western wall was long, and the northern wall was 7,200 (23,622 ft) long. Overall the total length of walls equalled , and formed a roughly square layout (although the southern and northern walls had sections which zigzagged due to topographical concerns: rough terrain existed along the southern wall and the course of the Wei River obstructed the straight path of the northern wall). The city's moat was wide and deep; the remains of what were wooden bridges have been discovered along the moat.Wang (1982), 2. Chang'an had twelve gatehouses leading into the city, three for each side of the wall, and acted as terminus points for the main avenues.Wang (1982), 2–3. Every gatehouse had three gateway entrances that were each wide; Han-era writers claimed that each gateway could accommodate the traffic of four horse-drawn carriages at once. The drainage system included many drainholes that were dug under these gates and lined with bricks that form arches, where ceramic water pipes have been found that once connected to the ditches built alongside the major streets.Wang (1982), 4. Only some wall sections and platform foundations of the city's once lavish imperial palaces remain.Wang (1982), 4–6. Likewise, the stone foundations of the armory were also discovered, but its wooden architecture had long since disappeared.

Some sections of the wall ruins of Han's second capital Luoyang still stand at in height and in width at the base. The eastern wall was long, the western wall was long, and the northern wall was long, yet the southern wall was washed away when the Luo River changed its course centuries ago; by using the terminus points of the eastern and western walls, historians estimate that the southern wall was long. The overall walled enclosure formed a rectangular shape, yet with some disruptive curves due to topographical obstructions.Wang (1982), 30. Like Chang'an, Luoyang had twelve gatehouses, three for each side of the wall, while each gatehouse had three gateway entrances which led to major avenues within the city.Wang (1982), 30–31. The rammed earth foundational platforms of religious altars and terraces still stand today outside of the walled perimeter of Luoyang, dedicated to the worship of deities and where state sacrifices were conducted.Wang (1982), 39. They were approached by long ramps and once had timber halls built on top with verandas on the lower levels.

Underground tombs

By the 1980s, over ten thousand brick-and-stone underground Han tombs had been discovered throughout China.Wang (1982), 175. Earlier Chinese tombs dating to the Warring States were often vertically dug pits lined with wooden walls.Wang (1982), 176. In digging the tomb sites, Han workers would first build vertical pits and then dig laterally, hence the name "horizontal pits" for Han tombs; this method was also used for tomb sites dug into the sides of mountains. The walls of most Western Han tombs were built of large hollow bricks while the smaller, non-hollow brick type that dominated Eastern Han tomb architecture (with some made out of stone) appeared in the late Western Han.Wang (1982), 175, 177–178. The smaller brick type was better-suited for Han tomb archways at entrances, vaulted chambers, and domed roofs. Underground vaults and domes did not require buttress supports since they were held in place by earthen pits.Watson (2000), 108. The use of brick vaults and domes in aboveground Han structures is unknown.

The layout of tombs dug into the sides of mountains typically had a front chamber, side chambers, and a rear chambers designed to imitate a complex of aboveground halls. The tomb of King Liu Sheng (d. 113 BCE) in Hebei province not only had a front hall with window drapes and grave goods, carriages and horses in the southern separate side chamber, and storage goods in the northern side chamber, but also the remains of real timber houses with tiled roofs erected within (along with a house made of stone slabs and two stone doors in the rear chamber). Doors made completely out of stone were found in many Han tombs as well as tombs in later dynasties.

A total of twenty-nine monumental brick or stone-carved pillar-gates ('' que'') from the Han dynasty have survived and can be found in the aboveground areas around Han tomb and shrine sites. They often formed part of outer walls, usually flanking an entry but sometimes at the corners of walled enclosures. Although they lack wooden and ceramic components, they feature imitation roof tiles, eaves, porches, and balustrades.

Boreholes and mining shafts

On Han tomb brick reliefs of Sichuan province, scenes of borehole drilling for mining projects are shown. They show towering derricks lifting liquid brine through bamboo pipes to the surface so that the brine could be distilled in evaporation pans over the heat of furnaces and produce salt.Loewe (1968), 191–194. The furnaces were heated by natural gas brought by bamboo pipes, with gas brought up from below the surface. The drill bit for digging boreholes was operated by a team of men jumping on and off a beam while the boring tool was rotated by a draft animal, usually oxen or water buffaloes. Han boreholes dug for collecting brine could reach hundreds of meters (feet) beneath the Earth's surface. Mining shafts dating to the Han dynasty have been found which reach depths of hundreds of meters (feet) beneath the earth, complete with spacious underground rooms structured by timber frames along with ladders and iron tools left behind.

Ceramic model buildings

There are Han-era literary references to tall towers found in the capital cities; they often served as watchtowers, astronomical observatories, and religious establishments meant to attract the favor of immortals. The court eunuchs Zhao Zhong and Zhang Rang discouraged the aloof Emperor Ling of Han (r. 168–189 CE) from ascending to the top floors of tall towers (claiming it would cause bad luck), in order to conceal from him the enormous palatial mansions the eunuchs built for themselves in Luoyang. It is not known for certain whether or not miniature ceramic models of residential towers and watchtowers found in Han-dynasty tombs are completely faithful representations of such timber towers, yet they reveal vital clues about lost timber architecture.

There are only a handful of existing ceramic models of multi-story towers from pre-Han and Western Han eras; the bulk of the hundreds of towers found so far were made during the Eastern Han period.Steinhardt (2005), "Pleasure Tower Model," 275–278. Model towers could be fired as one piece in the kiln or assembled from several different ceramic pieces to create the whole.Steinhardt (2005), "Pleasure Tower Model," 275–277. No one tower is a duplicate of the other, yet they share common features. They often had a walled courtyard at the bottom, a balcony with balustrades and windows for every floor, roof tiles capping and concealing the ceiling rafters, human figures peering out the windows or standing on the balconies, door knockers, and pets such as dogs in the bottom courtyard. Perhaps the most direct pieces of evidence to suggest that miniature ceramic tower models are faithful representations of real-life Han timber towers are tile patterns.Steinhardt (2005), "Tower Model" 283–284. Artistic patterns found on the circular tiles that cap the eave-ends on the miniature models are exact matches of patterns found on real-life Han roof tiles excavated at sites such as the royal palaces in Chang'an and Luoyang, and even the tiles of the original White Horse Temple.

Besides towers, other ceramic models from the Han reveal a variety of building types. This includes multi-story storehouses such as granaries, courtyard houses with multi-story halls, kiosks, walled gate towers, mills, manufactories and workshops, animal pens, outhouses, and water wells.Steinhardt (2005), "Pleasure Tower Model," 278. Even models of single-story farmhouses show a great amount of detail, including tiled roofs, courtyards, steps leading to walkways, farmyards with troughs and basins, parapets, and privies. Models of granaries and storehouses had tiled rooftops, ''dougong'' brackets, windows, and stilt supports raising them above ground level. Han models of water wells sometimes feature tiny tiled roofs supported by beams that house the rope pulley used for lifting the bucket.

Roads, bridges, and canals

In order to facilitate commerce and communication as well as speed the process of tax collection and movement of military troops, the Han government sponsored the building of new roads, bridges, and canal waterways. These include repairs and renovation work on the Dujiangyan Irrigation System of Sichuan and Zhengguo Canal of Shaanxi, both of which were built by the previous State of Qin.Wang (1982), 59. Accepting the proposal of Ni Kuan, in 111 BCE Emperor Wu commissioned Er to lead the project of creating extensions to the Zhengguo Canal that could irrigate nearby terrain elevated above the main canal.Needham (1986d), 286. Since a large amount of silt had built up over time at the bottom of the Zhengguo Canal (causing flooding), in 95 BCE another project was initiated to tap irrigation waters from further up the Jing River, requiring the dredging of a new long canal following a contour line above the Zhengguo. The Han state also maintained a system of dikes to protect farmland from seasonal floods.

Roadways, wooden bridges, postal stations, and relay stations were occasionally repaired, while many new facilities such as these were established. As written by Han authors, roads built during the Han were tamped down with metal rammers, yet there is uncertainty over the materials used; Joseph Needham speculates that they were rubble and gravel. The widths of roads ranged from narrow footpaths where only a single horse or oxen could pass at once to large highways that could accommodate the simultaneous passage of nine horse-drawn chariots abreast. Fortified Han roadways were built as far west as Shanshan (Loulan

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