Aside from many original inventions, the Chinese were also early original pioneers in the discovery of natural phenomena which can be found in the human body, the environment of the world, and the immediate solar system. They also discovered many concepts in mathematics. The list below contains discoveries which found their origins in China.

**Chinese remainder theorem**: The Chinese remainder theorem, including simultaneous congruences in number theory, was first created in the 3rd century AD in the mathematical book*Sunzi Suanjing*posed the problem: "There is an unknown number of things, when divided by 3 it leaves 2, when divided by 5 it leaves 3, and when divided by 7 it leaves a remainder of 2. Find the number."^{[1]}This method of calculation was used in calendrical mathematics by Tang Dynasty (618–907) mathematicians such as Li Chunfeng (602–670) and Yi Xing (683–727) in order to determine the length of the "Great Epoch", the lapse of time between the conjunctions of the moon, sun, and Five Planets (those discerned by the naked eye).^{[1]}Thus, it was strongly associated with the divination methods of the ancient*Yijing*.^{[1]}Its use was lost for centuries until Qin Jiushao (c. 1202–1261) revived it in his*Mathematical Treatise in Nine Sections*of 1247, providing constructive proof for it.^{[1]}**Circadian rhythm in humans**: The*Huangdi Neijing*, compiled by the 2nd century BC during the Han Dynasty (202 BC – 220 AD), noted the symptoms, behavior, and reactions of people with different diseases (i.e. of the liver, heart, spleen, lung, or kidneys) during different times of a 24-hour day.^{[2]}The idea of any organism following a daily circadian rhythm was not accepted in mainstream modern medical science even up until the 1960s, yet it is now well established that patients with Parkinson's disease lose much of their debilitating symptoms between 9 pm and midnight, while paroxysms of patients with asthma usually occur at night when secretion of hormones from the cortexes of the adrenal glands falls to a minimum.^{[3]}Although the ancient Chinese explained symptoms of diseased patients that followed the pattern of their circadian rhythms in terms of superstitious numerology and cyclic lore, they still documented such cases and expounded on them long before anyone else.^{[4]}The observation of a circadian or diurnal process in humans is mentioned in Chinese medical texts dated to around the 13th century, including the*Noon and Midnight Manual*and the*Mnemonic Rhyme to Aid in the Selection of Acu-points According to the Diurnal Cycle, the Day of the Month and the Season of the Year*.^{[5]}**Decimal fractions**: As proven by inscriptions from the 13th century BC, the decimal system existed in China since the Shang Dynasty (c. 1600–c. 1050 BC).^{[6]}The earliest evidence of a decimal fraction, where the fraction's denominator is a power of ten, appears on an inscription of a standard measure of volume used by the mathematician and astronomer Liu Xin (c. 46 BC–23 AD), dated precisely 5 AD.^{[7]}The first significant piece of Chinese literature to feature decimal fractions was*The Nine Chapters on the Mathematical Art*.^{[8]}This text was first mentioned in 179 AD,^{[9]}although Liu Hui (fl. 3rd century AD) asserts that some of its material predates the infamous Qin book burning in 213 BC (i.e. older than the oldest surviving Chinese mathematical treatise, the*Book on Numbers and Computation*, 202–186 BC).^{[10]}Liu Hui used decimal fractions with measurements and as solutions to equations.^{[8]}At first decimal fractions were written in word form, since it was Han Yan (fl. late 8th century) of the Tang Dynasty (607–907) who first used modern decimal notation to write out decimal fractions.^{[8]}Decimal fractions were vital to the work of Song (960–1279) mathematicians such as Yang Hui (1238–1298) and Qin Jiushao (c. 1201–1261).^{[8]}Jamshīd al-Kāshī (1380–1429), director of the astronomical observatory at Samarkand, adopted the use of decimal fractions; they were first mentioned in Europe by Christoff Rudolff of Augsburg in his*Exempel-Buechlin*of 1530, yet not given serious attention until the 1585 work of the Flemish mathematician Simon Stevin (1548–1620).^{[8]}

**Deficiency diseases, correction by proper diet**: As early as the 4th century BC, Warring States period (403–221 BC), records indicate that Imperial Dieticians were appointed at royal courts.^{[11]}The first explicit description of a regulated diet used to curb certain diseases is found in the*Systematic Treasury of Medicine*written by Zhang Zhongjing (c. 150 – c. 219) during the late Han Dynasty.^{[11]}Although Zhang did not understand the true nature of vitamins, he prescribed foods now known to be rich in certain vitamins, which were discovered to be useful after much trial and error.^{[11]}The Yuan Dynasty (1271–1368) physician and Imperial Dietician Hu Sihui (fl. 1314–1330) published his book*Principles of Correct Diet*which compiled a large amount of previous material written on the subject.^{[11]}**Diabetes, recognition and treatment of**: The*Huangdi Neijing*compiled by the 2nd century BC during the Han Dynasty identified diabetes as a disease suffered by those who had made an excessive habit of eating sweet and fatty foods, while the*Old and New Tried and Tested Prescriptions*written by the Tang Dynasty physician Zhen Quan (died 643) was the first known book to mention an excess of sugar in the urine of diabetic patients.^{[12]}^{[13]}While his book is now lost, quotations of it were preserved in the*Important Medical Formulae and Prescriptions Now Revealed by the Governor of a Distant Province*, written by Wang Tao in 752.^{[12]}The Tang physician Sun Simiao (581–682) wrote in his*Thousand Golden Remedies*of 655 that for diabetic patients "three things must be renounced, wine, sex, and eating salted, starchy cereal products; if this regimen can be observed, cure may follow without drugs."^{[14]}Robert Temple writes that this is similar to the modern method of avoiding alcohol and starchy foods.^{[14]}The sweetness of urine in diabetic patients is also noted in an ancient text of India, but unlike the Chinese texts its date is ambiguous.^{[14]}**Endocrinology, isolation of sex and pituitary hormones from urine**: In 1110, a Chinese medical text specified the use of gypsum (containing calcium sulfate) as well as saponin from the beans of*Gleditschia sinensis*to extract hormones from urine, a process of using natural soaps which was not discovered elsewhere until the use of digitonin by Adolf Windaus (1876–1959) in 1909.^{[15]}In 1927, Selmar Ascheim (1878–1965) and Bernhard Zondek (1891–1966) discovered that urine of pregnant women had a high concentration of steroid sex hormones; a subsequent discovery was made that urine contained sex hormones of androgens and estrogens, as well as the pituitary hormone gonadotrophin.^{[16]}In modern medicine, the extraction of these hormones from urine is a standard practice, yet centuries before this the Chinese had used it to treat hypogonadism, impotence, spermatorrhea, dysmenorrhea, leukorrhea, and even stimulating the growth of beards (since they knew that castration resulted in the loss of ability to grow a beard).^{[17]}

**Equal temperament**: During the Han Dynasty (202 BC–220 AD), the music theorist and mathematician Jing Fang (78–37 BC) extended the 12 tones found in the 2nd century BC*Huainanzi*to 60.^{[20]}While generating his 60-divisional tuning, he discovered that 53 just fifths is approximate to 31 octaves, calculating the difference at ; this was exactly the same value for 53 equal temperament calculated by the German mathematician Nicholas Mercator (c. 1620–1687) as 3^{53}/2^{84}, a value known as Mercator's Comma.^{[21]}^{[22]}The Ming Dynasty (1368–1644) music theorist Zhu Zaiyu (1536–1611) elaborated in three separate works beginning in 1584 the tuning system of equal temperament; in an unusual event in music theory's history, the Flemish mathematician Simon Stevin (1548–1620) discovered the mathematical formula for equal temperament at roughly the same time (within 1 to 25 years of Zhu), yet he did not publish his work and it remained unknown until 1884; therefore, it is debatable who discovered equal temperament first, Zhu or Stevin.^{[23]}^{[24]}^{[25]}In order to obtain equal intervals, Zhu divided the octave (each octave with a ratio of 1:2, which can also be expressed as 1:2^{12/12}) into twelve equal semitones while each length was divided by the 12th root of 2.^{[26]}He did not simply divide the string into twelve equal parts (i.e. 11/12, 10/12, 9/12, etc.) since this would give unequal temperament; instead, he altered the ratio of each semitone by an equal amount (i.e. 1:2^{11/12}, 1:2^{10/12}, 1:2^{9/12}, etc.) and determined the exact length of the string by dividing it by^{12}√2 (same as 2^{1/12}).^{[26]}The*Harmonie Universelle*(1636) written by Marin Mersenne (1588–1648) was the first publication in Europe outlining equal temperament, a new system of tuning that was passionately defended by J.S. Bach (1685–1750) in his*Well-Tempered Clavier*of 1722.^{[25]}**First law of motion, partial description**: The Mohist philosophical canon of the*Mojing*, compiled by the followers of Mozi (c. 470 – c. 390 BC), provides the earliest known attempt to describe inertia: "The cessation of motion is due to the opposing force...If there is no opposing force...the motion will never stop. This is as true as that an ox is not a horse."^{[27]}However, like many of the Hundred Schools of Thought during the Warring States period (403–221 BC), the doctrine of the Mohist sect had little impact on the course of later Chinese thought, while this passage and others from the*Mojing*were only given serious attention by modern scholarship after the work of Joseph Needham in 1962.^{[27]}**Gaussian elimination**: First published in the West by Carl Friedrich Gauss (1777–1855) in 1826, the algorithm for solving linear equations known as Gaussian elimination is named after this Hanoverian mathematician, yet it was first expressed as the Array Rule in the Chinese*Nine Chapters on the Mathematical Art*, written at most by 179 AD during the Han Dynasty (202 BC–220 AD) and commented on by the 3rd century mathematician Liu Hui.^{[28]}^{[29]}^{[30]}

**Geobotanical prospecting**: Geobotanical prospecting can be defined as the connection made between the types of vegetation that grow in certain areas and the minerals that can be found underground in those same areas; this observation was first made in China.^{[31]}It is now established in modern geobotany that only certain plants can grow in soils which are rich in certain types of minerals, such as*Viola calaminaria*and*Thlaspi*which grow in soils rich in zinc.^{[31]}The Zhou Dynasty (c. 1050–256 BC) Chinese*Classic of Mountains and Rivers*, compiled from the 6th to 2nd centuries BC, states that a certain "huitang" plant only grows near ore deposits of gold.^{[31]}As seen in the 5th century BC text*Tribute of Yu*, geobotanical prospecting in ancient China was mainly concerned with describing the nature of soil in different regions for agricultural purposes.^{[31]}The*Book of Master Wen*, compiled by 380 AD and containing material from as far back as the 3rd century BC, states that the branches of trees tend to droop in soils where an abundance of jade is to be found.^{[32]}In about 290 AD, Zhang Hua (232–300) wrote that hematite was found in abundance in any soil where smartweed grew.^{[32]}In the*Illustrated Mirror of the Earth*, written in the early 6th century AD, there is a description of a plant with an elegant yellow stalk which was found to grow above copper, and another description of a plant with green leaves and a red stalk where lead is often found below.^{[32]}In his*Miscellaneous Morsels from Youyang*, the Tang Dynasty (618–907) author Duan Chengshi (d. 863) noted that silver could often be found in the soil where ciboule onion grew, gold where a certain kind of shallot grew, and copper where ginger grew.^{[32]}Su Song (1020–1101) of the Song Dynasty (960–1279) described how*Portulaca oleracea*could yield mercury if pounded, dried, and allowed to decay.^{[32]}The*Precious Secrets of the Realm of the King of Xin*, written in 1421 during the Ming Dynasty (1368–1644), described how mineral trace elements were observed and could be extracted from certain plants, such as copper from*Oxalis corniculata*, gold from rape turnip, silver from weeping willows, and lead and tin from mugwort, chestnut, barley, and wheat.^{[32]}Geobotanical prospecting was unknown in the rest of the world until about 1600 when Sir Thomas Challoner and his first cousin Thomas Challoner discovered alum mines on the former's property of Belman Bank, Guisborough, Yorkshire, England.^{[27]}Both Challoner relatives realized here (and later in Italy) that leaves of oak trees were a much darker, richer green and their branches stronger and more spread out where the alum was to be found.^{[27]}

**Geomorphology**: In his*Dream Pool Essays*of 1088, Shen Kuo (1031–1095) wrote about a landslide (near modern Yan'an) where petrified bamboos were discovered in a preserved state underground, in the dry northern climate zone of Shanbei, Shaanxi; Shen reasoned that since bamboo was known only to grow in damp and humid conditions, the climate of this northern region must have been different in the very distant past, postulating that climate change occurred over time.^{[33]}^{[34]}Shen also advocated a hypothesis in line with geomorphology after he observed a stratum of marine fossils running in a horizontal span across a cliff of the Taihang Mountains, leading him to believe that it was once the location of an ancient shoreline that had shifted hundreds of km (mi) east over time (due to deposition of silt and other factors).^{[35]}^{[36]}**Horner scheme**: Although named after English mathematician William George Horner (1786–1837), the Horner scheme, an algorithm used to estimate the root of an equation and evaluate polynomials in monomial form, was actually first invented in China to find the cube root of the number 1,860,867 (the answer given being 123).^{[37]}This is found in the Han Dynasty (202 BC–220 AD) work*The Nine Chapters on the Mathematical Art*, commented on by Liu Hui (fl. 3rd century) in 263 AD.^{[37]}The original*Nine Chapters*found the root of equations through continued fractions, just like the later Italian mathematician Joseph Louis Lagrange (1736–1813), while Liu Hui achieved this by increasing decimals, just like William George Horner in his work of 1819.^{[37]}**Jia Xian triangle**: This triangle was the same as Pascal's Triangle, discovered by Jia Xian in the first half of the 11th century, about six centuries before Pascal. Jia Xian used it as a tool for extracting square and cubic roots. The original book by Jia Xian titled*Shi Suo Suan Shu*was lost; however, Jia's method was expounded in detail by Yang Hui, who explicitly acknowledged his source: "My method of finding square and cubic roots was based on the Jia Xian method in*Shi Suo Suan Shu*."^{[38]}A page from the Yongle Encyclopedia preserved this historic fact.

**Leprosy, first description of its symptoms**: The*Feng zhen shi*封診式 (*Models for sealing and investigating*), written between 266 and 246 BC in the State of Qin during the Warring States period (403–221 BC), is the earliest known text which describes the symptoms of leprosy, termed under the generic word*li*癘 (for skin disorders).^{[39]}This text mentioned the destruction of the nasal septum in those suffering from leprosy (an observation that would not be made outside of China until the writings of Avicenna in the 11th century), and according to Katrina McLeod and Robin Yates it also stated lepers suffered from "swelling of the eyebrows, loss of hair, absorption of nasal cartilage, affliction of knees and elbows, difficult and hoarse respiration, as well as anaesthesia."^{[39]}Leprosy was not described in the West until the writings of the Roman authors Aulus Cornelius Celsus (25 BC – 37 AD) and Pliny the Elder (23–79 AD).^{[39]}Although it is alleged that the Indian*Sushruta Samhita*, which describes leprosy,^{[40]}is dated to the 6th century BC, India's earliest written script (besides the then long extinct Indus script)—the Brāhmī script—is thought to have been created no earlier than the 3rd century BC.^{[41]}

**Li Shanlan’s Summation Formulae**: discovered by the mathematician Li Shanlan in 1867.^{[42]}**Liu Hui's π algorithm**: Liu Hui's π algorithm was invented by Liu Hui (fl. 3rd century), a mathematician of Wei Kingdom.**Magic squares**: The earliest magic square is the Lo Shu square, dating to 4th century BCE China. The square was viewed as mystical, and according the Chinese mythology, and "was first seen by Emperor Yu."^{[43]}**Negative numbers, symbols for and use of**: In the*Nine Chapters on the Mathematical Art*compiled during the Han Dynasty (202 BC–220 AD) by 179 AD and commented on by Liu Hui (fl. 3rd century) in 263,^{[9]}negative numbers appear as black rods and positive numbers as red rods in the Chinese counting rods system.^{[44]}Liu Hui also used slanted counting rods to denote negative numbers.^{[44]}Negative numbers denoted by a "+" sign also appear in the ancient Bakhshali manuscript of India, yet scholars disagree as to when it was compiled, giving a collective range of 200 to 600 AD.^{[45]}Negative numbers were known in India certainly by about 630 AD, when the mathematician Brahmagupta (598–668) used them.^{[44]}Negative numbers were first used in Europe by the Greek mathematician Diophantus (fl. 3rd century) in about 275 AD, yet were considered absurd in the West until*The Great Art*written in 1545 by the Italian mathematician Girolamo Cardano (1501–1576).^{[44]}- : The ancient Egyptians, Babylonians, Indians, and Greeks had long made approximations for π by the time the Chinese mathematician and astronomer Liu Xin (c. 46 BC–23 AD) improved the old Chinese approximation of simply 3 as π to 3.1547 as π (with evidence on vessels dating to the Wang Mang reign period, 9–23 AD, of other approximations of 3.1590, 3.1497, and 3.1679).
^{[46]}^{[47]}Next, Zhang Heng (78–139 AD) made two approximations for π, by proportioning the celestial circle to the diameter of the earth as = 3.1724 and using (after a long algorithm) the square root of 10, or 3.162.^{[47]}^{[48]}^{[49]}In his commentary on the Han Dynasty mathematical work*The Nine Chapters on the Mathematical Art*, Liu Hui (fl. 3rd century) used various algorithms to render multiple approximations for pi at 3.142704, 3.1428, and 3.14159.^{[50]}Finally, the mathematician and astronomer Zu Chongzhi (429–500) approximated pi to an even greater degree of accuracy, rendering it , a value known in Chinese as Milü ("detailed ratio").^{[51]}This was the best rational approximation for pi with a denominator of up to four digits; the next rational number is , which is the best rational approximation. Zu ultimately determined the value for π to be between 3.1415926 and 3.1415927.^{[52]}Zu's approximation was the most accurate in the world, and would not be achieved elsewhere for another millennium,^{[53]}until Madhava of Sangamagrama^{[54]}and Jamshīd al-Kāshī^{[55]}in the early 15th century.

**Snowflake, observation of its hexagonal structure**: In his*Moral Discourses Illustrating the Han Text of the Book of Songs*of 135 BC, the Han Dynasty (202 BC– 220 AD) author Han Ying wrote: "Flowers of plants and trees are generally five-pointed, but those of snow, which are called*ying*, are always six pointed."^{[56]}This was the first explicit reference in world history to the hexagonal structure of snowflakes.^{[56]}From then on, Chinese writers throughout the centuries mentioned the hexagonal structure of snowflakes, including the crown prince and poet Xiao Tong (501–531) and the Neo-Confucian philosopher Zhu Xi (1130–1200).^{[56]}In contrast to Western ideas of snowflakes, Olaus Magnus (1490–1557) wrote in his*A Description of the Northern Peoples*in 1555 that snowflakes could take on many shapes, including crescents, arrows, nails, bells, and even the shape of the human hand.^{[27]}It was not until 1591 that Thomas Hariot (1560–1621) recognized the snowflake's hexagonal structure, but he did not publish his jotted private notes on the subject.^{[27]}Finally, the astronomer Johannes Kepler (1571–1630) wrote the first known European publication on the subject in 1611, the fifteen-page*A New Year's Gift, or On the Six-Cornered Snowflake*.^{[56]}**Solar wind, observation of via comet tails**: In the*Book of Jin*compiled during the Tang Dynasty (618–907), a passage written in 635 AD states: "In general, when a comet appears in the morning, its tail points towards the west, and when it appears in the evening, its tail points towards the east. This is a constant rule. If the comet is north or south of the Sun, its tail always points following the same direction as the light radiating from the Sun."^{[57]}In other words, as Robert Temple states, "the Chinese observations of comet tails had been refined enough to establish the principle that comet tails always point away from the sun."^{[57]}Furthermore, the text reveals that astronomers by at least the Tang Dynasty understood that, like the Moon, the light shining from a comet was merely reflected sunlight;^{[57]}from the writings of Jing Fang (78–37 BC), Wang Chong (27–100), Zhang Heng (78–139), and others it is apparent that the Chinese already by the Han Dynasty (202 BC – 220 AD) understood that the Moon was illuminated solely by the Sun's rays of light.^{[58]}Although the Chinese explained this constant rule about comets in terms of supernatural*qi*, it is now understood in modern astronomy as the concept of 'solar wind', where the powerful force of radiation from the Sun causes comets to turn away from it.^{[57]}**Spontaneous combustion, recognition of**: In his*Record of Strange Things*written sometime before 290 AD, the Jin Dynasty official and poet Zhang Hua (232–300) wrote the earliest known account acknowledging spontaneous combustion: "If ten thousand piculs of oil are accumulated in store, the oil will ignite itself spontaneously. The calamitous fire which occurred in the arsenal of the time of the Emperor Wu [of the Jin Dynasty] in the Taishi reign-period [265–74 AD] was caused by the stored oil."^{[59]}There were other mentionings of spontaneous combustion in early Chinese literary works, while more often than not fires were blamed on arsonists.^{[60]}The 13th-century work*Parallel Cases Solved by Eminent Judges*recounts an event in 1050 where imperial guards were charged in a court of law with the crime of allowing a fire to spread in the palace at Kaifeng; their sentence was commuted from the death penalty to a light punishment when artisans confessed that the chemical-enhanced (perhaps quicklime) oily curtains they made had the propensity to catch fire spontaneously when left out in the open, a statement which convinced Emperor Renzong (r. 1022–1063) since a random fire had recently started in oiled garments of Emperor Zhenzong's (r. 997–1022) mausoluem.^{[60]}The author of*Parallel Cases Solved by Eminent Judges*noted that Zhang Hua had once believed oil stored in an arsenal spontaneously combusted, yet he concludes that what happened in that ancient arsenal was most likely the result of oiled garments, not just oil by itself.^{[60]}The first acknowledgement of spontaneous combustion anywhere else in the world was made by J. P. F. Duhamel in a French scientific paper published in 1757, in which he described oiled canvas sails catching fire after being left out in the summer sun for only a few hours.^{[60]}**Sunspots, recognition of as solar phenomena**: The astronomer Gan De (fl. 4th century BC) from the State of Qi during the Warring States period (403–221 BC) was the first known writer to attribute sunspots as characteristics of the sun and true solar phenomena.^{[61]}The next known recording of a sunspot in China was in 165 BC, yet the first precisely dated sunspot observed from China occurred on May 10, 28 BC, during the Han Dynasty (202 BC – 220 AD).^{[61]}From 28 BC to 1368 AD, a total of 112 other instances of sunspots were recorded by the Chinese.^{[62]}In the West, from the time of Aristotle (384–322 BC) of ancient Greece to the time of Galileo Galilei (1564–1642), it was commonly believed that the heavens were perfect, including the sun.^{[61]}After the first written observation in the West of sunpots by Einhard (d. 840) in his*Life of Charlemagne*in 807 AD, the sun's periodic blemishes were explained by Western thinkers as being small invisible satellites or transits of Mercury and Venus; it was only in the 17th century that these beliefs were overturned.^{[63]}**True north, concept of**: The Song Dynasty (960–1279) official Shen Kuo (1031–1095), alongside his colleague Wei Pu, improved the orifice width of the sighting tube to make nightly accurate records of the paths of the moon, stars, and planets in the night sky, for a continuum of five years.^{[64]}By doing so, Shen fixed the outdated position of the pole star, which had shifted over the centuries since the time Zu Geng (fl. 5th century) had plotted it; this was due to the precession of the Earth's rotational axis.^{[65]}^{[66]}When making the first known experiments with a magnetic compass, Shen Kuo wrote that the needle always pointed slightly east rather than due south, an angle he measured which is now known as magnetic declination, and wrote that the compass needle in fact pointed towards the magnetic north pole instead of true north (indicated by the current pole star); this was a critical step in the history of accurate navigation with a compass.^{[67]}^{[68]}^{[69]}

**Arteminisinin, anti-malarial treatment**: The antimalarial drug of compound artemisinin found in*Artemisia annua*, the latter being a plant long used in traditional Chinese medicine, was discovered in 1972 by Chinese scientists in the People's Republic led by Tu Youyou and has been used to treat multi-drug resistant strains of*Plasmodium falciparum*malaria.^{[70]}^{[71]}^{[72]}Artemisinin remains the most effective treatment for malaria today and has saved millions of lives and is yielded one of the greatest drug discoveries in modern medicine.^{[73]}**Chen's theorem**: Chen's theorem states that every sufficiently large even number can be written as the sum of either two primes, or a prime and a semiprime, and was first proven by Chen Jingrun in 1966,^{[74]}with further details of the proof in 1973.^{[75]}**Chen prime**: A prime number*p*is called a**Chen prime**if*p*+ 2 is either a prime or a product of two primes (also called a semiprime). The even number 2*p*+ 2 therefore satisfies Chen's theorem.The Chen primes are named after Chen Jingrun, who proved in 1966 that there are infinitely many such primes. This result would also follow from the truth of the twin prime conjecture.^{[76]}**Cheng's eigenvalue comparison theorem**: Cheng's theorem was introduced in 1975 by Hong Kong mathematician Shiu-Yuen Cheng.^{[77]}It states in general terms that when a domain is large, the first Dirichlet eigenvalue of its Laplace–Beltrami operator is small. This general characterization is not precise, in part because the notion of "size" of the domain must also account for its curvature.^{[78]}**Chern class**: Chern classes are characteristic classes in mathematics first introduced by Shiing-Shen Chern in 1946.^{[79]}^{[a]}**Chow's moving lemma**: In algebraic geometry,**Chow's moving lemma**, named after Wei-Liang Chow, states: given algebraic cycles*Y*,*Z*on a nonsingular quasi-projective variety*X*, there is another algebraic cycle*Z'*on*X*such that*Z'*is rationally equivalent to*Z*and*Y*and*Z'*intersect properly. The lemma is one of key ingredients in developing the intersection theory, as it is used to show the uniqueness of the theory.**Culturing Chlamydia trachomatis bacteria**: Chlamydia trachomatis agent was first cultured in the yolk sacs of eggs by Chinese scientists in 1957^{[80]}**Feathered theropods**: The first feathered dinosaur outside of Avialae,*Sinosauropteryx*, meaning "Chinese reptilian wing," was discovered in the Yixian Formation by Chinese paleontologists in 1996.^{[81]}The discovery is seen as evidence that dinosaurs originated from birds, a theory proposed and supported decades earlier by paleontologists like Gerhard Heilmann and John Ostrom, but "no true dinosaur had been found exhibiting down or feathers until the Chinese specimen came to light."^{[82]}The dinosaur was covered in what are dubbed 'protofeathers' and considered to be homologous with the more advanced feathers of birds,^{[83]}although some scientists disagree with this assessment.^{[84]}**Finite element method**: In numerical analysis, the finite element method is a technique for finding approximate solutions to systems of partial differential equations. The FEM was developed in the West by Alexander Hrennikoff and Richard Courant, and independently in China by Feng Kang.**Grunwald–Wang theorem**: In algebraic number theory, the**Grunwald–Wang theorem**states that—except in some precisely defined cases—an element*x*in a number field*K*is an*n*th power in*K*if it is an*n*th power in the completion for almost all (i.e. all but finitely many) primes of*K*. For example, a rational number is a square of a rational number if it is a square of a*p*-adic number for almost all primes*p*. The Grunwald–Wang theorem is an example of a local-global principle.It was introduced by Wilhelm Grunwald (1933), but there was a mistake in this original version that was found and corrected by Shianghao Wang (1948).**Hua's identity**: In algebra,**Hua's identity**^{[85]}states that for any elements*a*,*b*in a division ring, : whenever . Replacing with gives another equivalent form of the identity: :**Hua's lemma**: In mathematics,**Hua's lemma**,^{[86]}named for Hua Loo-keng, is an estimate for exponential sums.**Heterosis in rice, three-line hybrid rice system**: A team of agricultural scientists headed by Yuan Longping applied heterosis to rice, developing the three-line hybrid rice system in 1973.^{[87]}The innovation allowed for roughly 12,000 kg (26,450 lbs) of rice to be grown per hectare (10,000 m^{2}). Hybrid rice has proven to be greatly beneficial in areas where there is little arable land, and has been adopted by several Asian and African countries. Yuan won the 2004 Wolf Prize in agriculture for his work.^{[88]}**Huang-Minglon modification**: The Huang-Minglon modification, introduced by Chinese chemist Huang Minglon,^{[89]}^{[90]}is a modification of the Wolff–Kishner reduction and involves heating the carbonyl compound, potassium hydroxide, and hydrazine hydrate together in ethylene glycol in a one-pot reaction.^{[91]}**Ky Fan norms**: The sum of the*k*largest singular values of*M*is a matrix norm, the Ky Fan*k*-norm of*M*.The first of the Ky Fan norms, the Ky Fan 1-norm is the same as the operator norm of*M*as a linear operator with respect to the Euclidean norms of*K*^{m}and*K*^{n}. In other words, the Ky Fan 1-norm is the operator norm induced by the standard*l*^{2}Euclidean inner product.**Lee–Yang theorem**: The Lee-Yang theorem in statistical mechanics was first proved for the Ising model by future Nobel laureates Tsung-Dao Lee and Chen Ning Yang in 1952. The theorem states that if partition functions of certain models in statistical field theory with ferromagnetic interactions are considered as functions of an external field, then all zeros are purely imaginary, or on the unit circle after a change of variable.^{[92]}^{[b]}**Pu's inequality**: In differential geometry,**Pu's inequality**is an inequality proved by Pao Ming Pu for the systole of an arbitrary Riemannian metric on the real projective plane RP^{2}.**Siu's semicontinuity theorem**: In complex analysis, the**Siu semicontinuity theorem**implies that the Lelong number of a closed positive current on a complex manifold is semicontinuous. More precisely, the points where the Lelong number is at least some constant form a complex subvariety. This was conjectured by Harvey & King (1972) and proved by Siu (1973, 1974).**Sun's curious identity**: In combinatorics,**Sun's curious identity**is the following identity involving binomial coefficients, first established by Zhi-Wei Sun in 2002:**Tsen rank**: A Tsen rank of a field describes conditions under which a system of polynomial equations must have a solution in the field. It was introduced by mathematician Chiungtze C. Tsen in 1936.^{[93]}**Wu's method**: Wu's method was discovered in 1978 by Chinese mathematician Wen-Tsun Wu.^{[94]}The method is an algorithm for solving multivariate polynomial equations, based on the mathematical concept of characteristic set introduced in the late 1940s by J.F. Ritt.^{[95]}**Yunnan Baiyao**^{[96]}

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- List of China-related topics

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^{a}^{b}^{c}^{d}Ho (1991), 516. **^**Temple (1986), 125.**^**Temple (1986), 124–125.**^**Temple (1986), 126.**^**Lu, Gwei-Djen (25 October 2002).*Celestial Lancets*. Psychology Press. pp. 137–140. ISBN 978-0-7007-1458-2.**^**Temple (1986), 139.**^**Temple (1986), 142–143.- ^
^{a}^{b}^{c}^{d}^{e}Temple (1986), 143. - ^
^{a}^{b}Needham (1986), Volume 3, 24–25. **^**Straffin (1998), 165.- ^
^{a}^{b}^{c}^{d}^{e}Temple (1986), 131. - ^
^{a}^{b}Temple (1986), 132. **^**Medvei (1993), 49.- ^
^{a}^{b}^{c}Temple (1986), 133. **^**Temple (1986), 128–129.**^**Temple (1986), 127.**^**Temple (1986), 130.**^**Temple (1986), 199.**^**McClain and Ming (1979), 206.**^**McClain and Ming (1979), 207–208.**^**McClain and Ming (1979), 212.**^**Needham (1986), Volume 4, Part 1, 218–219.**^**Kuttner (1975), 166–168.**^**Needham (1986), Volume 4, Part 1, 227–228.- ^
^{a}^{b}Temple (1986), 209. - ^
^{a}^{b}Needham (1986), Volume 4, Part 1, 223. - ^
^{a}^{b}^{c}^{d}^{e}^{f}Temple (1986), 161. **^**Needham (1986), Volume 3, 24–25, 121.**^**Shen, Crossley, and Lun (1999), 388.**^**Straffin (1998), 166.- ^
^{a}^{b}^{c}^{d}Temple (1986), 159. - ^
^{a}^{b}^{c}^{d}^{e}^{f}Temple (1986), 160. **^**Chan, Clancey, Loy (2002), 15.**^**Needham (1986), Volume 3, 614.**^**Sivin (1995), III, 23.**^**Needham (1986), Volume 3, 603–604, 618.- ^
^{a}^{b}^{c}Temple (1986), 142. **^**Wu Wenjun chief ed,*The Grand Series of History of Chinese Mathematics*Vol 5 Part 2, chapter 1, Jia Xian- ^
^{a}^{b}^{c}McLeod & Yates (1981), 152–153 & footnote 147. **^**Aufderheide et al., (1998), 148.**^**Salomon (1998), 12–13.**^**https://link.springer.com/chapter/10.1007/978-3-540-33783-6_18#page-1**^**C. J. Colbourn; Jeffrey H. Dinitz (2 November 2006).*Handbook of Combinatorial Designs*. CRC Press. p. 525. ISBN 978-1-58488-506-1.- ^
^{a}^{b}^{c}^{d}Temple (1986), 141. **^**Teresi (2002), 65–66.**^**Neehdam (1986), Volume 3, 99–100.- ^
^{a}^{b}Berggren, Borwein & Borwein (2004), 27 **^**Arndt and Haenel (2001), 177**^**Wilson (2001), 16.**^**Needham (1986), Volume 3, 100–101.**^**Berggren, Borwein & Borwein (2004), 24–26.**^**Berggren, Borwein & Borwein (2004), 26.**^**Berggren, Borwein & Borwein (2004), 20.**^**Gupta (1975), B45–B48**^**Berggren, Borwein, & Borwein (2004), 24.- ^
^{a}^{b}^{c}^{d}Temple (1986), 162. - ^
^{a}^{b}^{c}^{d}Temple (1986), 34. **^**Needham (1986), Volume 3, 227 & 411–414.**^**Temple (1986), 166–167.- ^
^{a}^{b}^{c}^{d}Temple (1986), 167. - ^
^{a}^{b}^{c}Temple (1986), 29. **^**Temple (1986), 30.**^**Temple (1986), 29–30.**^**Sivin (1995), III, 17–18.**^**Sivin (1995), III, 22.**^**Needham (1986), Volume 3, 278.**^**Sivin (1995), III, 21–22.**^**Elisseeff (2000), 296.**^**Hsu (1988), 102.**^**Croft, S.L. (1997). "The current status of antiparasite chemotherapy". In G.H. Coombs; S.L. Croft; L.H. Chappell.*Molecular Basis of Drug Design and Resistance*. Cambridge: Cambridge University Press. pp. 5007–5008. ISBN 0-521-62669-2.**^**O'Connor, Anahad (12 September 2011). "Lasker Honors for a Lifesaver". The New York Times.**^**Tu, Youyou (11 October 2011). "The discovery of artemisinin (qinghaosu) and gifts from Chinese medicine". Nature Medicine.**^**McKenna, Phil (15 November 2011). "The modest woman who beat malaria for China". New Scientist.**^**Chen, J.R. (1966). "On the representation of a large even integer as the sum of a prime and the product of at most two primes".*Kexue Tongbao*.**17**: 385–386.**^**Chen, J.R. (1973). "On the representation of a larger even integer as the sum of a prime and the product of at most two primes".*Sci. Sinica*.**16**: 157–176.**^**Chen, J. R. (1966). "On the representation of a large even integer as the sum of a prime and the product of at most two primes". Kexue Tongbao 17: 385–386.**^**Cheng, Shiu Yuen (1975a). "Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Stanford Univ., Stanford, Calif., 1973), Part 2". Providence, R.I.: American Mathematical Society: 185–193. MR 0378003.`|chapter=`

ignored (help)**^**Chavel, Isaac (1984). "Eigenvalues in Riemannian geometry". Pure Appl. Math.**115**. Academic Press.**^**Chern, S. S. (1946). "Characteristic classes of Hermitian Manifolds".*Annals of Mathematics. Second Series*. The Annals of Mathematics, Vol. 47, No. 1.**47**(1): 85–121. ISSN 0003-486X. JSTOR 1969037. doi:10.2307/1969037.**^**S Darougar, B R Jones, J R Kimptin, J D Vaughan-Jackson, and E M Dunlop. Chlamydial infection. Advances in the diagnostic isolation of Chlamydia, including TRIC agent, from the eye, genital tract, and rectum. Br J Vener Dis. 1972 December; 48(6): 416–420; TANG FF, HUANG YT, CHANG HL, WONG KC. Further studies on the isolation of the trachoma virus. Acta Virol. 1958 Jul-Sep;2(3):164-70; TANG FF, CHANG HL, HUANG YT, WANG KC. Studies on the etiology of trachoma with special reference to isolation of the virus in chick embryo. Chin Med J. 1957 Jun;75(6):429-47; TANG FF, HUANG YT, CHANG HL, WONG KC. Isolation of trachoma virus in chick embryo. J Hyg Epidemiol Microbiol Immunol. 1957;1(2):109-20**^**Ji Qiang; Ji Shu-an (1996). "On the discovery of the earliest bird fossil in China and the origin of birds" (PDF).*Chinese Geology*.**233**: 30–33.**^**Browne, M.W. (19 October 1996). "Feathery Fossil Hints Dinosaur-Bird Link".*New York Times*. p. Section 1 page 1 of the New York edition.**^**Chen Pei-ji, Pei-ji; Dong Zhiming; Zhen Shuo-nan (1998). "An exceptionally preserved theropod dinosaur from the Yixian Formation of China".*Nature*.**391**(6663): 147–152. Bibcode:1998Natur.391..147C. doi:10.1038/34356.**^**Sanderson, K. (23 May 2007). "Bald dino casts doubt on feather theory". doi:10.1038/news070521-6. Retrieved 14 January 2011.**^**Cohn 2003, §9.1**^**Hua Loo-keng (1938). "On Waring's problem".*Quarterly Journal of Mathematics*.**9**(1): 199–202. doi:10.1093/qmath/os-9.1.199.**^**Sant S. Virmani, C. X. Mao, B. Hardy, (2003).*Hybrid Rice for Food Security, Poverty Alleviation, and Environmental Protection*. International Rice Research Institute. ISBN 971-22-0188-0, p. 248**^**Wolf Foundation Agricultural Prizes**^**Huang-Minlon (1946). "A Simple Modification of the Wolff-Kishner Reduction".*Journal of the American Chemical Society*.**68**: 2487–2488. doi:10.1021/ja01216a013.**^**Huang-Minlon (1949). "Reduction of Steroid Ketones and other Carbonyl Compounds by Modified Wolff--Kishner Method".*Journal of the American Chemical Society*.**71**: 3301–3303. doi:10.1021/ja01178a008.**^**Organic Syntheses, Coll. Vol. 4, p. 510 (1963); Vol. 38, p. 34 (1958). (Article)**^**Yang, C. N.; Lee, T. D. (1952). "Statistical Theory of Equations of State and Phase Transitions. I. Theory of Condensation".*Physical Review*.**87**: 404–409. Bibcode:1952PhRv...87..404Y. ISSN 0031-9007. doi:10.1103/PhysRev.87.404.**^**Tsen, C. (1936). "Zur Stufentheorie der Quasi-algebraisch-Abgeschlossenheit kommutativer Körper".*J. Chinese Math. Soc*.**171**: 81–92. Zbl 0015.38803.**^**Wu, Wen-Tsun (1978). "On the decision problem and the mechanization of theorem proving in elementary geometry".*Scientia Sinica*.**21**.**^**P. Aubry, D. Lazard, M. Moreno Maza (1999). On the theories of triangular sets. Journal of Symbolic Computation, 28(1–2):105–124**^**Exum, Roy (December 27, 2015). "Roy Exum: Ellen Does It Again".*The Chattanoogan*.

- Arndt, Jörg, and Christoph Haenel. (2001).
*Pi Unleashed*. Translated by Catriona and David Lischka. Berlin: Springer. ISBN 3-540-66572-2. - Aufderheide, A. C.; Rodriguez-Martin, C. & Langsjoen, O. (1998).
*The Cambridge Encyclopedia of Human Paleopathology*. Cambridge University Press. ISBN 0-521-55203-6. - Berggren, Lennart, Jonathan M. Borwein, and Peter B. Borwein. (2004).
*Pi: A Source Book*. New York: Springer. ISBN 0-387-20571-3. - Chan, Alan Kam-leung and Gregory K. Clancey, Hui-Chieh Loy (2002).
*Historical Perspectives on East Asian Science, Technology and Medicine*. Singapore: Singapore University Press. ISBN 9971-69-259-7 - Elisseeff, Vadime. (2000).
*The Silk Roads: Highways of Culture and Commerce*. New York: Berghahn Books. ISBN 1-57181-222-9. - Gupta, R C. "Madhava's and other medieval Indian values of pi," in
*Math*, Education, 1975, Vol. 9 (3): B45–B48. - Ho, Peng Yoke. "Chinese Science: The Traditional Chinese View,"
*Bulletin of the School of Oriental and African Studies*, University of London, Vol. 54, No. 3 (1991): 506-519. - Hsu, Mei-ling (1988). "Chinese Marine Cartography: Sea Charts of Pre-Modern China".
*Imago Mundi*.**40**: 96–112. doi:10.1080/03085698808592642. - McLeod, Katrina C. D.; Yates, Robin D. S. (1981). "Forms of Ch'in Law: An Annotated Translation of The Feng-chen shih".
*Harvard Journal of Asiatic Studies*.**41**(1): 111–163. doi:10.2307/2719003. - McClain, Ernest G.; Shui Hung, Ming (1979). "Chinese Cyclic Tunings in Late Antiquity".
*Ethnomusicology*.**23**(2): 205–224. doi:10.2307/851462. - Medvei, Victor Cornelius. (1993).
*The History of Clinical Endocrinology: A Comprehensive Account of Endocrinology from Earliest Times to the Present Day*. New York: Pantheon Publishing Group Inc. ISBN 1-85070-427-9. - Needham, Joseph. (1986).
*Science and Civilization in China: Volume 3, Mathematics and the Sciences of the Heavens and the Earth*. Taipei: Caves Books, Ltd. - Needham, Joseph (1986).
*Science and Civilization in China: Volume 4, Physics and Physical Technology; Part 1, Physics*. Taipei: Caves Books Ltd. - Salomon, Richard (1998),
*Indian Epigraphy: A Guide to the Study of Inscriptions in Sanskrit, Prakrit, and the Other Indo-Aryan Languages*. Oxford: Oxford University Press. ISBN 0-19-509984-2. - Sivin, Nathan (1995).
*Science in Ancient China: Researches and Reflections*. Brookfield, Vermont: VARIORUM, Ashgate Publishing. - Straffin Jr, Philip D. (1998). "Liu Hui and the First Golden Age of Chinese Mathematics".
*Mathematics Magazine*.**71**(3): 163–181. - Temple, Robert. (1986).
*The Genius of China: 3,000 Years of Science, Discovery, and Invention*. With a forward by Joseph Needham. New York: Simon and Schuster, Inc. ISBN 0-671-62028-2. - Teresi, Dick. (2002).
*Lost Discoveries: The Ancient Roots of Modern Science–from the Babylonians to the Mayas*. New York: Simon and Schuster. ISBN 0-684-83718-8. - Wilson, Robin J. (2001).
*Stamping Through Mathematics*. New York: Springer-Verlag New York, Inc.

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