Mathematical achievements from Mesopotamia had some influence on the development of mathematics in India, and there were confirmed transmissions of mathematical ideas between India and China, which were bidirectional. Nevertheless, the mathematical and scientific achievements in India and particularly in China occurred largely independently from those of Europe and the confirmed early influences that these two civilizations had on the development of science in Europe in the pre-modern era were indirect, with Mesopotamia and later the Islamic World acting as intermediaries.
India Mathematics The earliest traces of mathematical knowledge in the Indian subcontinent appear with the
Indus Valley Civilisation (). The people of this civilization made bricks whose dimensions were in the proportion 4:2:1, which is favorable for the stability of a brick structure. They also tried to standardize measurement of length to a high degree of accuracy. They designed a ruler—the
Mohenjo-daro ruler—whose length of approximately was divided into ten equal parts. Bricks manufactured in ancient Mohenjo-daro often had dimensions that were integral multiples of this unit of length. The
Bakhshali manuscript contains problems involving
arithmetic,
algebra and
geometry, including
mensuration. The topics covered include fractions, square roots,
arithmetic and
geometric progressions, solutions of simple equations,
simultaneous linear equations,
quadratic equations and
indeterminate equations of the second degree. In the 3rd century BCE,
Pingala presents the
Pingala-sutras, the earliest known treatise on
Sanskrit prosody. He also presents a numerical system by adding one to the sum of
place values. Pingala's work also includes material related to the
Fibonacci numbers, called ''''. Indian astronomer and mathematician
Aryabhata (476–550), in his
Aryabhatiya (499) introduced the
sine function in
trigonometry and the number 0. In 628,
Brahmagupta suggested that
gravity was a force of attraction. He also lucidly explained the use of
zero as both a placeholder and a
decimal digit, along with the
Hindu–Arabic numeral system now used universally throughout the world.
Arabic translations of the two astronomers' texts were soon available in the
Islamic world, introducing what would become
Arabic numerals to the Islamic world by the 9th century.
Narayana Pandita (1340–1400) was an Indian
mathematician.
Plofker writes that his texts were the most significant Sanskrit mathematics treatises after those of
Bhaskara II, other than the
Kerala school. He wrote the
Ganita Kaumudi (lit. "Moonlight of mathematics") in 1356 about mathematical operations. The work anticipated many developments in
combinatorics. Between the 14th and 16th centuries, the
Kerala school of astronomy and mathematics made significant advances in astronomy and especially mathematics, including fields such as trigonometry and analysis. In particular,
Madhava of Sangamagrama led advancement in
analysis by providing the infinite and taylor series expansion of some trigonometric functions and pi approximation.
Parameshvara (1380–1460), presents a case of the Mean Value theorem in his commentaries on
Govindasvāmi and
Bhāskara II. The
Yuktibhāṣā was written by
Jyeshtadeva in 1530.
Astronomy c. 1650 The first textual mention of astronomical concepts comes from the
Vedas, religious literature of India. According to Sarma (2008): "One finds in the
Rigveda intelligent speculations about the genesis of the universe from nonexistence, the configuration of the universe, the
spherical self-supporting earth, and the year of 360 days divided into 12 equal parts of 30 days each with a periodical intercalary month.".
Jai Singh II of
Jaipur constructed five
observatories called
Jantar Mantars in total, in
New Delhi,
Jaipur,
Ujjain,
Mathura and
Varanasi; they were completed between 1724 and 1735.
Grammar Some of the earliest linguistic activities can be found in
Iron Age India (1st millennium BCE) with the analysis of
Sanskrit for the purpose of the correct recitation and interpretation of
Vedic texts. The most notable grammarian of Sanskrit was (c. 520–460 BCE), whose grammar formulates close to 4,000 rules for Sanskrit. Inherent in his analytic approach are the concepts of the
phoneme, the
morpheme and the
root. The
Tolkāppiyam text, composed in the early centuries of the common era, is a comprehensive text on Tamil grammar, which includes sutras on orthography, phonology, etymology, morphology, semantics, prosody, sentence structure and the significance of context in language.
Medicine or Sahottara-Tantra'' from
Nepal, Findings from
Neolithic graveyards in what is now Pakistan show evidence of proto-dentistry among an early farming culture. The ancient text
Suśrutasamhitā of
Suśruta describes procedures on various forms of surgery, including
rhinoplasty, the repair of torn ear lobes, perineal
lithotomy, cataract surgery, and several other excisions and other surgical procedures. The
Charaka Samhita of
Charaka describes ancient theories on human body,
etiology,
symptomology and
therapeutics for a wide range of diseases. It also includes sections on the importance of diet, hygiene, prevention, medical education, and the teamwork of a physician, nurse and patient necessary for recovery to health.
Politics and state An ancient Indian treatise on
statecraft,
economic policy and
military strategy by Kautilya and , who are traditionally identified with Chanakya| (c. 350–283 BCE). In this treatise, the behaviors and relationships of the people, the King, the State, the Government Superintendents, Courtiers, Enemies, Invaders, and Corporations are analyzed and documented.
Roger Boesche describes the
Arthaśāstra as "a book of political realism, a book analyzing how the political world does work and not very often stating how it ought to work, a book that frequently discloses to a king what calculating and sometimes brutal measures he must carry out to preserve the state and the common good."
Logic The development of Indian logic dates back to the Chandahsutra of Pingala and
anviksiki of Medhatithi Gautama (c. 6th century BCE); the
Sanskrit grammar rules of
Pāṇini (c. 5th century BCE); the
Vaisheshika school's analysis of
atomism (c. 6th century BCE to 2nd century BCE); the analysis of
inference by
Gotama (c. 6th century BCE to 2nd century CE), founder of the
Nyaya school of
Hindu philosophy; and the
tetralemma of
Nagarjuna (c. 2nd century CE).
Indian logic stands as one of the three original traditions of
logic, alongside the
Greek and the
Chinese logic. The Indian tradition continued to develop through early to modern times, in the form of the
Navya-Nyāya school of logic. In the 2nd century, the
Buddhist philosopher
Nagarjuna refined the
Catuskoti form of logic. The Catuskoti is also often glossed
Tetralemma (Greek) which is the name for a largely comparable, but not equatable, 'four corner argument' within the tradition of
Classical logic. Navya-Nyāya developed a sophisticated language and conceptual scheme that allowed it to raise, analyse, and solve problems in logic and epistemology. It systematised all the Nyāya concepts into four main categories: sense or perception (pratyakşa), inference (anumāna), comparison or similarity (
upamāna), and testimony (sound or word; śabda).
China 's survey of a sea island from the
Haidao Suanjing, 3rd century AD
Chinese mathematics From the earliest the Chinese used a positional decimal system on counting boards in order to calculate. To express 10, a single rod is placed in the second box from the right. The spoken language uses a similar system to English: e.g. four thousand two hundred and seven. No symbol was used for zero. By the 1st century BCE, negative numbers and decimal fractions were in use and
The Nine Chapters on the Mathematical Art included methods for extracting higher order roots by
Horner's method and solving linear equations and by
Pythagoras' theorem. Cubic equations were solved in the
Tang dynasty and solutions of equations of order higher than 3 appeared in print in 1245 CE by
Ch'in Chiu-shao.
Pascal's triangle for binomial coefficients was described around 1100 by
Jia Xian. Although the first attempts at an axiomatization of geometry appear in the
Mohist canon in 330 BCE,
Liu Hui developed algebraic methods in geometry in the 3rd century CE and also calculated
pi to 5 significant figures. In 480,
Zu Chongzhi improved this by discovering the ratio \tfrac{355}{113} which remained the most accurate value for 1200 years.
Astronomical observations s from
Su Song's
Xin Yi Xiang Fa Yao published in 1092, featuring a cylindrical projection similar to
Mercator, and the corrected position of the
pole star thanks to
Shen Kuo's astronomical observations. Astronomical observations from China constitute the longest continuous sequence from any civilization and include records of sunspots (112 records from 364 BCE), supernovas (1054), lunar and solar eclipses. By the 12th century, they could reasonably accurately make predictions of eclipses, but the knowledge of this was lost during the Ming dynasty, so that the Jesuit
Matteo Ricci gained much favor in 1601 by his predictions. By 635 Chinese astronomers had observed that the tails of comets always point away from the sun. From antiquity, the Chinese used an equatorial system for describing the skies and a star map from 940 was drawn using a cylindrical (
Mercator) projection. The use of an
armillary sphere is recorded from the 4th century BCE and a sphere permanently mounted in equatorial axis from 52 BCE. In 125 CE
Zhang Heng used water power to rotate the sphere in real time. This included rings for the meridian and ecliptic. By 1270 they had incorporated the principles of the Arab
torquetum. In the
Song Empire (960–1279) of
Imperial China, Chinese
scholar-officials unearthed, studied, and cataloged ancient artifacts.
Inventions 's
seismometer of 132 CE 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. There are many notable contributors to early Chinese disciplines, inventions, and practices throughout the ages. One of the best examples would be the medieval Song Chinese
Shen Kuo (1031–1095), a
polymath and statesman who was the first to describe the
magnetic-needle
compass used for
navigation, discovered the concept of
true north, improved the design of the astronomical
gnomon,
armillary sphere, sight tube, and
clepsydra, and described the use of
drydocks to repair boats. After observing the natural process of the inundation of
silt and the find of
marine fossils in the
Taihang Mountains (hundreds of miles from the Pacific Ocean), Shen Kuo devised a theory of land formation, or
geomorphology. He also adopted a theory of gradual
climate change in regions over time, after observing
petrified bamboo found underground at
Yan'an, Shaanxi. If not for Shen Kuo's writing, the architectural works of
Yu Hao would be little known, along with the inventor of
movable type printing,
Bi Sheng (990–1051). Shen's contemporary
Su Song (1020–1101) was also a brilliant polymath, an astronomer who created a celestial atlas of star maps, wrote a treatise related to
botany,
zoology,
mineralogy, and
metallurgy, and had erected a large
astronomical clocktower in
Kaifeng city in 1088. To operate the crowning
armillary sphere, his clocktower featured an
escapement mechanism and the world's oldest known use of an endless power-transmitting
chain drive. The
Jesuit China missions of the 16th and 17th centuries "learned to appreciate the scientific achievements of this ancient culture and made them known in Europe. Through their correspondence European scientists first learned about the Chinese science and culture." Western academic thought on the history of Chinese technology and science was galvanized by the work of
Joseph Needham and the Needham Research Institute. Among the technological accomplishments of China were, according to the British scholar Needham, the
water-powered celestial globe (Zhang Heng),
dry docks, sliding
calipers, the double-action
piston pump, the multi-tube
seed drill, the
wheelbarrow, the
suspension bridge, the
winnowing machine,
gunpowder, the
raised-relief map, toilet paper, the efficient harness, along with contributions in
logic,
astronomy,
medicine, and other fields. However, cultural factors prevented these Chinese achievements from developing into "modern science". According to Needham, it may have been the religious and philosophical framework of Chinese intellectuals which made them unable to accept the ideas of laws of nature: ==Pre-Columbian Mesoamerica==