Zhang Heng

Mathematics

For centuries the Chinese approximated pi as 3; Liu Xin (d. CE 23) made the first known Chinese attempt at a more accurate calculation of 3.154, but there is no record detailing the method he used to obtain this figure.^[51][52] In his work around 130,^[53] Zhang Heng compared the celestial circle to the diameter of the earth, proportioning the former as 736 and the latter as 232, thus calculating pi as 3.1724.^[54] In Zhang's day, the ratio 4:3 was given for the area of a square to the area of its inscribed circle and the volume of a cube and volume of the inscribed sphere should also be 4²:3².^[54] In formula, with D as diameter and V as volume, D³:V = 16:9 or V=${\displaystyle {\tfrac {9}{16}}}$D³; Zhang realized that the value for diameter in this formula was inaccurate, noting the discrepancy as the value taken for the ratio.^[52][54] Zhang then attempted to remedy this by amending the formula with an additional ${\displaystyle {\tfrac {1}{16}}}$D³, hence V=${\displaystyle {\tfrac {9}{16}}}$D³ + ${\displaystyle {\tfrac {1}{16}}}$D³ = ${\displaystyle {\tfrac {5}{8}}}$D³.^[54] With the ratio of the volume of the cube to the inscribed sphere at 8:5, the implied ratio of the area of the square to the circle is √8:√5.^[54][55] From this formula, Zhang calculated pi as the square root of 10 (or approximately 3.162).^{[17][18][54][55][56]} In the 3rd century, Liu Hui made the calculation more accurate with his π algorithm, which allowed him to obtain the value 3.14159.^[57] Later, Zu Chongzhi (429–500) approximated pi as ${\displaystyle {\tfrac {355}{113}}}$ or 3.141592, the most accurate calculation for pi the ancient Chinese would achieve.^[58]

Astronomy

In his publication of AD 120 called The Spiritual Constitution of the Universe (靈憲, Ling Xian, lit. "Sublime Model"),^[18][59] Zhang Heng theorized that the universe was like an egg "as round as a crossbow pellet" with the stars on the shell and the Earth as the central yolk.^[4][60] This universe theory is congruent with the geocentric model as opposed to the heliocentric model. Although the ancient Warring States (403–221 BC) Chinese astronomers Shi Shen and Gan De had compiled China's first star catalogue in the 4th century BC, Zhang nonetheless catalogued 2,500 stars which he placed in a "brightly shining" category (the Chinese estimated the total to be 14,000), and he recognized 124 constellations.^[18][60] In comparison, this star catalogue featured many more stars than the 850 documented by the Greek astronomer Hipparchus (c. 190 – c. 120 BC) in his catalogue, and more than Ptolemy (AD 83–161), who catalogued over 1,000.^[61] Zhang supported the "radiating influence" theory to explain solar and lunar eclipses, a theory which was opposed by Wang Chong (AD 27–97).^[62] In the Ling Xian, Zhang wrote:

夫日譬猶火，月譬猶水，火則外光，水則含景。故月光生於日之所照，魄生於日之所蔽，當日則光盈，就日則光盡也。
The Sun is like fire and the Moon like water. The fire gives out light and the water reflects it. Thus the Moon's brightness is produced from the radiance of the Sun, and the Moon's darkness is due to (the light of) the Sun being obstructed. The side which faces the Sun is fully lit, and the side which is away from it is dark.

衆星被燿，因水轉光。當日之衝，光常不合者，蔽於地也。是謂闇虛。在星星微，月過則食。
The planets (as well as the Moon) have the nature of water and reflect light. The light pouring forth from the Sun does not always reach the Moon owing to the obstruction of the Earth itself—this is called "ànxū", a lunar eclipse. When (a similar effect) happens with a planet (we call it) an occultation; when the Moon passes across (the Sun's path) then there is a solar eclipse.

— Zhang Heng (AD 120), Sublime Model (Joseph Needham, trans.)^[63]

Zhang Heng viewed these astronomical phenomena in supernatural terms as well. The signs of comets, eclipses, and movements of heavenly bodies could all be interpreted by him as heavenly guides on how to conduct affairs of state.^[18] Contemporary writers also wrote about eclipses and the sphericity of heavenly bodies. The music theorist and mathematician Jing Fang (78–37 BC) wrote about the spherical shape of the Sun and Moon while discussing eclipses:

The Moon and the planets are Yin; they have shape but no light. This they receive only when the Sun illuminates them. The former masters regarded the Sun as round like a crossbow bullet, and they thought the Moon had the nature of a mirror. Some of them recognized the Moon as a ball too. Those parts of the Moon which the Sun illuminates look bright, those parts which it does not, remain dark.^[64]

The theory posited by Zhang and Jing was supported by later pre-modern scientists such as

The outflow clepsydra was a timekeeping device used in China as long ago as the Shang dynasty (c. 1600–c. 1050 BC), and certainly by the Zhou dynasty (1122–256 BC).^[67] The inflow clepsydra with an indicator rod on a float had been known in China since the beginning of the Han dynasty in 202 BC and had replaced the outflow type.^[67] The Han Chinese noted the problem with the falling pressure head in the reservoir, which slowed the timekeeping of the device as the inflow vessel was filled.^[67] Zhang Heng was the first to address this problem, indicated in his writings from 117, by adding an extra compensating tank between the reservoir and the inflow vessel.^[2][18] Zhang also mounted two statuettes of a Chinese immortal and a heavenly guard on the top of the inflow clepsydra, the two of which would guide the indicator rod with their left hand and point out the graduations with their right.^[68] Joseph Needham states that this was perhaps the ancestor of all clock jacks that would later sound the hours found in mechanical clocks by the 8th century, but he notes that these figures did not actually move like clock jack figurines or sound the hours.^[68] Many additional compensation tanks were added to later clepsydras in the tradition of Zhang Heng. In 610 the Sui dynasty (581–618) engineers Geng Xun and Yuwen Kai crafted an unequal-armed steelyard balance able to make seasonal adjustments in the pressure head of the compensating tank, so that it could control the rate of water flow for different lengths of day and night during the year.^[69] Zhang mentioned a "jade dragon's neck", which in later times meant a siphon.^[70] He wrote of the floats and indicator-rods of the inflow clepsydra as follows:

以銅為器，再疊差置，實以清水，下各開孔，以玉虯吐漏水入兩壺。右為夜，左為晝。
Bronze vessels are made and placed one above the other at different levels; they are filled with pure water. Each has at the bottom a small opening in the form of a 'jade dragon's neck'. The water dripping (from above) enters two inflow receivers (alternately), the left one being for the night and the right one for the day.

蓋上又鑄金銅仙人，居左壺；為胥徒，居右壺。皆以左手抱箭，右手指刻，以別天時早晚。
On the covers of each (inflow receiver) there are small cast statuettes in gilt bronze; the left (night) one is an immortal and the right (day) one is a policeman. These figures guide the indicator-rod (lit. arrow) with their left hands, and indicate the graduations on it with their right hands, thus giving the time.^[70]

Water-powered armillary sphere

What were the factors leading to the first escapement clock in China? The chief tradition leading to Yi Xing (AD 725 ) was of course the succession of 'pre-clocks' which had started with Zhang Heng about 125. Reason has been given for believing that these applied power to the slow turning movement of computational armillary spheres and celestial globes by means of a water-wheel using clepsydra drip, which intermittently exerted the force of a lug to act on the teeth of a wheel on a polar-axis shaft. Zhang Heng in his turn had composed this arrangement by uniting the armillary rings of his predecessors into the equatorial armillary sphere, and combining it with the principles of the water-mills and hydraulic trip-hammers which had become so widespread in Chinese culture in the previous century.[76]

Zhang did not initiate the Chinese tradition of hydraulic engineering, which began during the mid Zhou dynasty (c. 6th century BC), through the work of engineers such as Sunshu Ao and Ximen Bao.^[77] Zhang's contemporary, Du Shi, (d. AD 38) was the first to apply the motive power of waterwheels to operate the bellows of a blast furnace to make pig iron, and the cupola furnace to make cast iron.^[78][79] Zhang provided a valuable description of his water-powered armillary sphere in the treatise of 125, stating:

The equatorial ring goes around the belly of the armillary sphere 91 and 5/19 (degrees) away from the pole. The circle of the ecliptic also goes round the belly of the instrument at an angle of 24 (degrees) with the equator. Thus at the summer solstice the ecliptic is 67 (degrees) and a fraction away from the pole, while at the winter solstice it is 115 (degrees) and a fraction away. Hence (the points) where the ecliptic and the equator intersect should give the north polar distances of the spring and autumn equinoxes. But now (it has been recorded that) the spring equinox is 90 and 1/4 (degrees) away from the pole, and the autumn equinox is 92 and 1/4 (degrees) away. The former figure is adopted only because it agrees with the (results obtained by the) method of measuring solstitial sun shadows as embodied in the Xia (dynasty) calendar.^[80]

Zhang Heng's water-powered armillary sphere had profound effects on Chinese astronomy and mechanical engineering in later generations. His model and its complex use of gears greatly influenced the water-powered instruments of later astronomers such as

During the Han dynasty, many learned scholars—including Zhang Heng—believed in the "oracles of the winds".^[84] These oracles of the occult observed the direction, force, and timing of the winds, to speculate about the operation of the cosmos and to predict events on Earth.^[85] These ideas influenced Zhang Heng's views on the cause of earthquakes.

In 132, Zhang Heng presented to the Han court what many historians consider to be his most impressive invention, the first seismoscope. A seismoscope records the motions of Earth's shaking, but unlike a seismometer, it does not retain a time record of those motions.

To indicate the direction of a distant earthquake, Zhang's device dropped a bronze ball from one of eight tubed projections shaped as dragon heads; the ball fell into the mouth of a corresponding metal object shaped as a toad, each representing a direction like the points on a compass rose.^[93] His device had eight mobile arms (for all eight directions) connected with cranks having catch mechanisms at the periphery.^[94] When tripped, a crank and right angle lever would raise a dragon head and release a ball which had been supported by the lower jaw of the dragon head.^[94] His device also included a vertical pin passing through a slot in the crank, a catch device, a pivot on a projection, a sling suspending the pendulum, an attachment for the sling, and a horizontal bar supporting the pendulum.^[94] Wang Zhenduo (王振鐸) argued that the technology of the Eastern Han era was sophisticated enough to produce such a device, as evidenced by contemporary levers and cranks used in other devices such as crossbow triggers.^[95]

Later Chinese of subsequent periods were able to reinvent Zhang's seismoscope. They included the 6th-century mathematician and surveyor Xindu Fang of the Northern Qi dynasty (550–577) and the astronomer and mathematician Lin Xiaogong of the Sui dynasty (581–618).^[96] Like Zhang, Xindu Fang and Lin Xiaogong were given imperial patronage for their services in craftsmanship of devices for the court.^[97] By the time of the Yuan dynasty (1271–1368), it was acknowledged that all devices previously made were preserved, except for that of the seismoscope.^[98] This was discussed by the scholar Zhou Mi around 1290, who remarked that the books of Xindu Fang and Lin Xiaogong detailing their seismological devices were no longer to be found.^[98] Horwitz, Kreitner, and Needham speculate if Tang dynasty (618–907) era seismographs found their way to contemporary Japan; according to Needham, "instruments of apparently traditional type there in which a pendulum carries pins projecting in many directions and able to pierce a surrounding paper cylinder, have been described."^[99]

Hong-sen Yan states that modern replicas of Zhang's device have failed to reach the level of accuracy and sensitivity described in Chinese historical records.

Anthony J. Barbieri-Low, a professor of early Chinese history at the University of California, Santa Barbara, names Zhang Heng as one of several high-ranking Eastern-Han officials who engaged in crafts that were traditionally reserved for artisans (gong 工), such as mechanical engineering.^[105] Barbieri-Low speculates that Zhang only designed his seismoscope, but did not actually craft the device himself. He asserts that this would most likely have been the job of artisans commissioned by Zhang.^[106] He writes: "Zhang Heng was an official of moderately high rank and could not be seen sweating in the foundries with the gong artisans and the government slaves. Most likely, he worked collaboratively with the professional casters and mold makers in the imperial workshops."^[106]

Cartography

Ancient Chinese texts describe the mechanical carriage's functions; after one li was traversed, a mechanically driven wooden figure struck a drum, and after ten li had been covered, another wooden figure struck a gong or a bell with its mechanically operated arm.^[116] However, there is evidence to suggest that the invention of the odometer was a gradual process in Han dynasty China that centered on the "huang men"—court people (i.e. eunuchs, palace officials, attendants and familiars, actors, acrobats, etc.) who followed the musical procession of the royal "drum-chariot".^[117] There is speculation that at some time during the 1st century BC the beating of drums and gongs was mechanically driven by the rotation of the road wheels.^[117] This might have actually been the design of Luoxia Hong (c. 110 BC), yet by at least 125 the mechanical odometer carriage was already known, as it was depicted in a mural of the Xiao Tang Shan Tomb.^[117]

The south-pointing chariot was another mechanical device credited to Zhang Heng.^[17] It was a non-magnetic compass vehicle in the form of a two-wheeled chariot. Differential gears driven by the chariot's wheels allowed a wooden figurine (in the shape of a Chinese state minister) to constantly point to the south, hence its name. The Song Shu (c. AD 500 ) records that Zhang Heng re-invented it from a model used in the Zhou dynasty era, but the violent collapse of the Han dynasty unfortunately did not allow it to be preserved. Whether Zhang Heng invented it or not, Ma Jun (200–265) succeeded in creating the chariot in the following century.^[118]

Legacy

Science and technology

Zhang Heng's mechanical inventions influenced later Chinese inventors such as Yi Xing, Zhang Sixun, Su Song, and Guo Shoujing. Su Song directly named Zhang's water-powered armillary sphere as the inspiration for his 11th-century

Based on the theories of his predecessors, Zhang Heng systematically developed the celestial sphere theory. An armillary constructed on the basis of his hypotheses bears a remarkable similarity to Ptolemy's earth-centered theory. However, Zhang Heng did not definitely propose a theoretical model like Ptolemy's earth-centered one. It is astonishing that the celestial model Zhang Heng constructed was almost a physical model of Ptolemy's earth-centered theory. Only a single step separates the celestial globe from the earth-centered theory, but Chinese astronomers never took that step.

Here we can see how important the exemplary function of the primitive scientific structure is. In order to use the Euclidean system of geometry as a model for the development of astronomical theory, Ptolemy first had to select hypotheses which could serve as axioms. He naturally regarded circular motion as fundamental and then used the circular motion of deferents and epicycles in his earth-centered theory. Although Zhang Heng understood that the sun, moon and planets move in circles, he lacked a model for a logically structured theory and so could not establish a corresponding astronomical theory. Chinese astronomy was most interested in extracting the algebraic features of planetary motion (that is, the length of the cyclic periods) to establish astronomical theories. Thus astronomy was reduced to arithmetic operations, extracting common multiples and divisors from the observed cyclic motions of the heavenly bodies.^[119]

Poetic literature

Zhang's poetry was widely read during his life and after his death. In addition to the compilation of Xiao Tong mentioned above, the Eastern Wu official Xue Zong (d. 237) wrote commentary on Zhang's poems "Dongjing fu" and "Xijing fu".^[125] The influential poet Tao Qian wrote that he admired the poetry of Zhang Heng for its "curbing extravagant diction and aiming at simplicity", in regards to perceived tranquility and rectitude correlating with the simple but effective language of the poet.^[126] Tao wrote that both Zhang Heng and Cai Yong "avoided inflated language, aiming chiefly at simplicity", and adding that their "compositions begin by giving free expression to their fancies but end on a note of quiet, serving admirably to restrain undisciplined and passionate nature".^[127]

Posthumous honors

Zhang was given great honors in life and in death. The philosopher and poet

In honor of Zhang's achievements in science and technology, his friend Cui Ziyu (Cui Yuan) wrote a memorial inscription on his burial stele, which has been preserved in the Guwen yuan.[8] Cui stated, "[Zhang Heng's] mathematical computations exhausted (the riddles of) the heavens and the earth. His inventions were comparable even to those of the Author of Change. The excellence of his talent and the splendour of his art were one with those of the gods."^[129] The minor official Xiahou Zhan (243–291) of the Wei dynasty made an inscription for his own commemorative stele to be placed at Zhang Heng's tomb. It read: "Ever since gentlemen have composed literary texts, none has been as skillful as the Master [Zhang Heng] in choosing his words well ... if only the dead could rise, oh I could then turn to him for a teacher!"^[130]

Several things have been named after Zhang in modern times, including the lunar crater

• Han poetry
• Fu (poetry)
• Return to the Field
• Yu Xi

References

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