Science in the ancient world encompasses the earliest history of science from the protoscience of prehistory and ancient history through to late antiquity. In ancient times, culture and knowledge were passed on generation to generation by means of oral tradition. The development of writing further enabled the ability to preserve knowledge and culture, allowing communication to travel across generations with greater fidelity. The earliest scientific traditions of the ancient world developed in the Ancient Near East with Ancient Egypt and Babylonia in Mesopotamia. Later traditions of science during classical antiquity were advanced in Ancient Persia, Ancient Greece, Ancient Rome, Ancient India, Ancient China, and ancient Pre-Columbian Mesoamerica. Aside from alchemy and astrology that waned in importance during the Age of Enlightenment, civilizations of the ancient world laid the roots of various modern sciences. These include astronomy, calendrical science, mathematics, horology and timekeeping, cartography, botany and zoology, medicine and pharmacology, hydraulic and structural engineering, metallurgy, archaeology, and many other fields.

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Scientific thought in Classical Antiquity becomes tangible from the 6th century BC in pre-Socratic philosophy (Thales, Pythagoras). In c. 385 BC, Plato founded the Academy. With Plato's student Aristotle begins the "scientific revolution" of the Hellenistic period culminating in the 3rd to 2nd centuries with scholars such as Eratosthenes, Euclid, Aristarchus of Samos, Hipparchus and Archimedes.

In Classical Antiquity, the inquiry into the workings of the universe took place both in investigations aimed at such practical goals as establishing a reliable calendar or determining how to cure a variety of illnesses and in those abstract investigations known as natural philosophy. The ancient people who are considered the first scientists may have thought of themselves as natural philosophers, as practitioners of a skilled profession (for example, physicians), or as followers of a religious tradition (for example, temple healers).

The earliest Greek philosophers, known as the pre-Socratics, provided competing answers to the question found in the myths of their neighbours: "How did the ordered cosmos in which we live come to be?"^[15] The pre-Socratic philosopher Thales dubbed the "father of science", was the first to postulate non-supernatural explanations for natural phenomena such as lightning and earthquakes. Pythagoras of Samos founded the Pythagorean school, which investigated mathematics for its own sake and was the first to postulate that the Earth is spherical. Subsequently, Plato and Aristotle produced the first systematic discussions of natural philosophy, which did much to shape later investigations of nature. Their development of deductive reasoning was particularly useful to later scientific inquiry.

The important legacy of this period included substantial advances in factual knowledge, especially in anatomy, zoology, botany, mineralogy, geography, mathematics and astronomy; an awareness of the importance of certain scientific problems, especially those related to the problem of change and its causes; and a recognition of the methodological importance of applying mathematics to natural phenomena and of undertaking empirical research.^[16] In the Hellenistic age scholars frequently employed the principles developed in earlier Greek thought: the application of mathematics and deliberate empirical research, in their scientific investigations.^[17] Thus, clear unbroken lines of influence lead from ancient Greek and Hellenistic philosophers, to medieval Muslim philosophers and scientists, to the European Renaissance and Enlightenment, to the secular sciences of the modern day. Neither reason nor inquiry began with the Ancient Greeks, but the Socratic method did, along with the idea of Forms, great advances in geometry, logic, and the natural sciences. Benjamin Farrington, former Professor of Classics at Swansea University wrote:

"Men were weighing for thousands of years before Archimedes worked out the laws of equilibrium; they must have had practical and intuitional knowledge of the principles involved. What Archimedes did was to sort out the theoretical implications of this practical knowledge and present the resulting body of knowledge as a logically coherent system."

and again:

"With astonishment we find ourselves on the threshold of modern science. Nor should it be supposed that by some trick of translation, the extracts have been given an air of modernity. Far from it. The vocabulary of these writings and their style are the source from which our own vocabulary and style have been derived."^[18]

The level of achievement in Hellenistic astronomy and engineering is impressively shown by the Antikythera mechanism (150-100 BC). The astronomer Aristarchus of Samos was the first known person to propose a heliocentric model of the solar system, while the geographer Eratosthenes accurately calculated the circumference of the Earth.^[19] Hipparchus (c. 190 – c. 120 BC) produced the first systematic star catalog. In medicine, Herophilos (335 - 280 BC) was the first to base his conclusions on the dissection of the human body and to describe the nervous system. Hippocrates (c. 460 BC – c. 370 BC) and his followers were the first to describe many diseases and medical conditions. Galen (129 – c. 200 AD) performed many audacious operations—including brain and eye surgeries— that were not tried again for almost two millennia. The mathematician Euclid laid down the foundations of mathematical rigour and introduced the concepts of definition, axiom, theorem and proof still in use today in his Elements, considered the most influential textbook ever written.^[20] Archimedes, considered one of the greatest mathematicians of all time,^[21] is credited with using the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, and gave a remarkably accurate approximation of pi.^[22] He is also known in physics for laying the foundations of hydrostatics and the explanation of the principle of the lever.

Theophrastus wrote some of the earliest descriptions of plants and animals, establishing the first taxonomy and looking at minerals in terms of their properties such as hardness. The ancient Roman military officer, philosopher, and historian Pliny the Elder produced what is one of the largest encyclopedias of the natural world in 77 AD, and must be regarded as the rightful successor to Theophrastus.

For example, he accurately describes the octahedral shape of the diamond. He proceeds to mention that diamond dust is used by engravers to cut and polish other gems owing to its great hardness. His recognition of the importance of crystal shape is a precursor to modern crystallography, while mentioning numerous other minerals presages mineralogy. He also recognises that other minerals have characteristic crystal shapes, but in one example, confuses the crystal habit with the work of lapidaries. He was also the first to recognise that amber was a fossilized resin from pine trees because he had seen samples with trapped insects within them.

Excavations at Harappa, Mohenjo-daro and other sites of the Indus Valley civilization (IVC) have uncovered evidence of the use of "practical mathematics". The people of the IVC manufactured bricks whose dimensions were in the proportion 4:2:1, considered favourable for the stability of a brick structure. They used a standardised system of weights based on the ratios 1⁄20, 1⁄10, 1⁄5, 1⁄2, 1, 2, 5, 10, 20, 50, 100, 200, and 500, with the unit weight equaling approximately 28 grams (and approximately equal to the English ounce or Greek uncia). They mass-produced weights in regular geometrical shapes, which included hexahedra, barrels, cones, and cylinders, thereby demonstrating knowledge of basic geometry.^[23]

The inhabitants of the Indus civilisation also tried to standardise the measurement of length to a high degree of accuracy. They designed a ruler—the Mohenjo-Daro ruler—whose unit of length (approximately 1.32 inches or 3.4 centimetres) was divided into ten equal parts. Bricks manufactured in ancient Mohenjo-Daro often had dimensions that were integral multiples of this unit of length.^[24][25]

Mehrgarh, a Neolithic IVC site, provides the earliest known evidence for in vivo drilling of human teeth, with recovered samples dated to 7000–5500 BCE.^[26]

Early astronomy in India, as in other cultures, was intertwined with religion.^[27] The first textual mention of astronomical concepts comes from the Vedas—religious literature of India.^[27] 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."^[27]

Classical Indian astronomy documented in literature spans the Maurya (Vedanga Jyotisha, c. 5th century BCE) to the Vijayanagara (South India) (such as the 16th century Kerala school) periods. The first named authors writing treatises on astronomy emerged from the 5th century, the date when the classical period of Indian astronomy can be said to begin. Besides the theories of Aryabhata in the Aryabhatiya and the lost Arya-siddhānta, we find the Pancha-Siddhāntika of Varahamihira. The astronomy and the astrology of ancient India (Jyotisha) are based upon sidereal calculations. However, a tropical system was also used in a few cases.

Alchemy (Rasaśāstra in Sanskrit) was popular in India. It was the Indian alchemist and philosopher Kanada who introduced the concept of 'anu' which he defined as the matter which cannot be subdivided.^[28] This is analogous to the concept of the atom in modern science.

Linguistics (along with phonology, morphology, etc.) first arose among Indian grammarians studying the Sanskrit language. Aacharya Hemachandrasuri wrote grammars of Sanskrit and Prakrit, poetry, prosody, lexicons, texts on science and logic and many branches of Indian philosophy. The Siddha-Hema-Śabdanuśāśana includes six Prakrit languages: the "standard" Prakrit (virtually^{[clarification needed]} Maharashtri Prakrit), Shauraseni, Magahi, Paiśācī, the otherwise-unattested Cūlikāpaiśācī and Apabhraṃśa (virtually Gurjar Apabhraṃśa, prevalent in the area of Gujarat and Rajasthan at that time and the precursor of Gujarati language). He gave a detailed grammar of Apabhraṃśa and illustrated it with the folk literature of the time for better understanding. It is the only known Apabhraṃśa grammar.^[29] The Sanskrit grammar of Pāṇini (c. 520 – 460 BCE) contains a particularly detailed description of Sanskrit morphology, phonology and roots, evincing a high level of linguistic insight and analysis.^{[citation needed]}

Ayurveda medicine traces its origins to the Vedas, Atharvaveda in particular, and is connected to Hindu religion.^[30] The Sushruta Samhita of Sushruta appeared during the 1st millennium BCE.^[31] Ayurvedic practice was flourishing during the time of Buddha (around 520 BCE), and in this period the Ayurvedic practitioners were commonly using Mercuric-sulphur combination based medicines. An important Ayurvedic practitioner of this period was Nagarjuna, accompanied by Surananda, Nagbodhi, Yashodhana, Nityanatha, Govinda, Anantdev, Vagbhatta and others. During the regime of Chandragupta Maurya (375-415 CE), Ayurveda was part of mainstream Indian medical techniques, and continued to be so until the Colonial period.^{[citation needed]}

The main authors of classical Indian mathematics (400 CE to 1200 CE) were scholars like Mahaviracharya, Aryabhata, Brahmagupta, and Bhaskara II. Indian mathematicians made early contributions to the study of the decimal number system, zero, negative numbers, arithmetic, and algebra. In addition, trigonometry, having evolved in the Hellenistic world and having been introduced into ancient India through the translation of Greek works, was further advanced in India, and, in particular, the modern definitions of sine and cosine were developed there. The Hindu-Arabic numeral system was developed in ancient India and spread to the later Islamic world all the way to Al-Andalus in the Iberian peninsula, where it was adopted (without the zero) by the French monk Gerbert of Aurillac, who would become Pope Sylvester II (r. 999–1003 CE) and spread its usage throughout medieval Europe in the 11th century with the reintroduction of the Greco-Roman abacus calculating tool.^[32]

Joseph Needham (1900–1995), who outlined China's "Four Great Inventions" (papermaking, compass, printing, and gunpowder) in his Science and Civilisation in China, highlights the Han dynasty (202 BC – 220 AD) in particular as one of the most pivotal eras for Chinese sciences, noting the period's significant advancements in Chinese astronomy and calendar making, the systematic documentation of living organisms in early forms of botany and zoology, and the philosophical skepticism and rationalism of the age embodied in works such as the Lunheng by Wang Chong (27–100 AD).^[33] Concurring with Needham, professors Jin Guantao (Chinese University of Hong Kong), Fan Hongye (Chinese Academy of Sciences), and Liu Qingfeng (Chinese University of Hong Kong) emphasize the Han dynasty as a unique period for Chinese scientific advancements comparable to the medieval Song dynasty (960–1279 AD), but stress that the protoscientific ideas of philosophical Mohism (one of the Hundred Schools of Thought) developed during the Warring States period (475–221 BC) that could have provided a definitive structure for Chinese science was hindered by Chinese theology and dynastic royal promotion of Confucianism and its literary classics.^[34] Needham and other sinologists indicate that cultural factors prevented these Chinese achievements from developing into what might be considered modern science, as the religious and philosophical framework of Chinese intellectuals hampered their efforts to rationalize the laws of nature:

It was not that there was no order in nature for the Chinese, but rather that it was not an order ordained by a rational personal being, and hence there was no conviction that rational personal beings would be able to spell out in their lesser earthly languages the divine code of laws which he had decreed aforetime. The Taoists, indeed, would have scorned such an idea as being too naïve for the subtlety and complexity of the universe as they intuited it.

— Joseph Needham, Science and Civilization in China, vol. 2, p. 581.

Nevertheless, nascent scientific ideas were established during the late Zhou dynasty (1046–256 BC) and proliferated in the Han dynasty. Much like the earlier Aristotle in Greece, the aforementioned Wang Chong accurately described the water cycle of Earth dismissed by his contemporaries.^[35] However, Wang (similar to the contemporary Roman Lucretius) inaccurately criticized the by then mainstream Han Chinese hypotheses of lunar and solar eclipses that the Sun and Moon are spherical and that the Moon is illuminated by the reflection of sunlight, the correct hypotheses being advocated by astronomer and music theorist Jing Fang (78–37 BC) and expanded upon by the polymath scientist and inventor Zhang Heng (78–139 AD).^[36] Zhang theorized that the celestial sphere was as round as a crossbow pellet and structured like an egg with the Earth as its yoke, a geocentric model that was largely accepted in the contemporary Greco-Roman world.^[37] Analytical approaches were even applied to writing itself. Though the Erya of the Warring States period provides a basic dictionary, the first analytical Chinese dictionary to explain and dissect the logographic Chinese written characters, with 9,353 characters listed and categorized by radicals, was the Shuowen Jiezi composed by the Eastern Han philologist and politician Xu Shen (c. 55–149 AD).^[38]

Early Chinese astronomy provides an example of the exhaustive documentation of the natural world and observable universe that often preoccupied Chinese scholars. Chinese star names are mentioned in oracle bone inscriptions of the Shang dynasty (c. 1600–1046 BC).^[40] Lists of stars along the ecliptic in the Chinese twenty-eight mansions were provided on lacquerware of the 433 BC Tomb of Marquis Yi of Zeng and in the Lüshi Chunqiu encyclopedia of Qin statesman Lü Buwei (291–235 BC), but it wasn't until the Han dynasty that full star catalogues were published that listed all stars in the observable celestial sphere.^[39] The Mawangdui Silk Texts interred within a Western Han tomb in 168 BC provide writings and ink illustrations of Chinese star maps showing Chinese constellations as well as comets.^[41] The Warring States era astronomers Shi Shen and Gan De are traditionally thought to have published star catalogues in the 4th century BC,^[42] but it was the star catalogue of Sima Qian (145–86 BC) in his "Book of Celestial Offices" (天官書 Tianguan shu) chapter of the Shiji (Records of the Grand Historian, the groundbreaking professional work of Chinese historiography as the first of the Twenty-Four Histories) that provided the model for all later Chinese star catalogues.^[43] Chinese constellations were later adopted in medieval Korean astronomy and Japanese astronomy.^[44] Building upon the star catalogue of Sima Qian that featured 90 constellations,^[45] the star catalogue of Zhang Heng published in 120 AD featured 124 constellations.^[46]

Works by Zhang Heng were highly influential throughout later Chinese history. As a horologist, Zhang demonstrated the movement of recorded stars and planets by being the first to apply the hydraulic power of waterwheels and water clock timer for automatically rotating the assembled rings of his armillary sphere,^[47] a model that would directly inspire the liquid escapement in astronomical clockworks pioneered in the Tang dynasty by Yi Xing (683–727 AD) and used by Song dynasty scientist Su Song (1020–1101 AD) in building his chain drive and water-driven astronomical clocktower.^[48] Greek astronomer Eratosthenes is the first known inventor of the armillary sphere in 255 BC and it is uncertain when it first appeared in China, though the Western Han astronomer Geng Shouchang was the first in China to add an equatorial ring to its design in 52 BC, with Jia Kui (30–101 AD) adding an ecliptic ring in 84 AD, followed by Zhang Heng adding the horizon and meridian rings.^[49] Zhang was not the first in China to utilize the motive power of waterwheels, since they were used in ferrous metallurgy by Du Shi, (d. AD 38 AD) to operate the bellows of a blast furnace to make pig iron, and the cupola furnace to make cast iron.^[50] Zhang invented a seismometer device with an inverted pendulum that detected the cardinal direction of distant earthquakes.^[51] It is unclear if Zhang invented or simply improved the designs of the odometer cart for measuring traveled distances and the non-magnetic south-pointing chariot that used differential gears to constantly point southward for navigation,^[52] though Three Kingdoms era engineer Ma Jun (200–265 AD) created a successful model of the chariot.^[53] The odometer cart, depicted in Eastern Han art, was most likely invented in Western Han China by Luoxia Hong around 110 BC and separately by the Greeks (either Archimedes in the 3rd century BC or Heron of Alexandria in the 1st century AD).^[54] Zhang, also a mathematician, approximated pi as 3.162 using the square root of 10 (with an 8:5 ratio of the volume of a cube to an inscribed sphere),^[55] though this was less accurate than the earlier Liu Xin (d. 23 AD) who calculated it as 3.154 using an unknown method.^[56] Zhang's calculation was improved upon by Three Kingdoms era mathematician Liu Heng in his 263 AD commentary on the Jiuzhang Suanshu (Nine Chapters on the Mathematical Art), providing a π algorithm with a value of 3.14159,^[57] while Liu Song and Southern Qi era mathematician Zu Chongzhi (429–500 AD) reached a value of 3.141592, the most accurate figure Chinese would achieve before exposure to Western mathematics.^[58] In cartography, Qin maps dating to the 4th century BC have been discovered and the Western Jin dynasty official Pei Xiu (224–271 AD) is the first known Chinese cartographer to have used a geometric grid reference that allowed for measurements on a graduated scale and for topographical elevation,^[59] though this might have been based on a rectangular grid system in maps made by Zhang Heng that are now lost.^[60]

In regards to mathematics, the Nine Chapters on the Mathematical Art, compiled in its entirety by 179 AD during the Eastern Han and well before Liu Hui's commentary, is perhaps also the first text to utilize negative numbers. These were symbolized by counting rods in a slanted position, while red rods symbolizing negative numbers versus black rods that symbolize positive numbers may date all the way back to the Western Han period.^[61] The Bakhshali manuscript of India also features negative numbers, but it was compiled at an uncertain date as early as 200 AD and as late as 600 AD,^[62] after which they were used with certainty by Indian mathematician Brahmagupta (598–668 AD).^[63] In his Arithmetica, Greek mathematician Diophantus described negative numbers around 275 AD but they were considered an absurd concept in the Western world until the Ars Magna of Italian mathematician Girolamo Cardano (1501–1576).^[63]

A seminal work of traditional Chinese medicine was the Huangdi neijing (Yellow Emperor's Inner Canon) compiled between the 3rd and 2nd centuries BC, which viewed the human body, its organs and tissues through the lens of the metaphysical five phases and yin and yang, and stated a belief in two circulatory channels of qi vital energy.^[64] Physicians of the Han dynasty believed that pulse diagnosis could be used to determine which organs in the body emitted qi energy and thus the type of ailments suffered by patients.^[65] The Huangdi neijing is the first known Chinese text to describe the use of acupuncture, while golden acupuncture needles have been discovered in the tomb of Liu Sheng, Prince of Zhongshan (d. 113 BC) and stone-carved artworks of the Eastern Han period depict the practice.^[66] The Huangdi neijing is also the first known text to describe diabetes and link it to the excessive consumption of sweet and fatty foods.^[67] The Mawangdui silk texts of the 2nd century BC provide illustrated diagrams with textual captions for exercises in calisthenics.^[68] In surgery, Han texts offered practical advice for certain procedures such as clinical lancing of abscesses.^[69] The first known physician in China to describe the use anesthesia for patients undergoing surgery was the Eastern Han physician Hua Tuo (d. 208 AD), who utilized his knowledge of Chinese herbology based in the Huangdi neijing to create an ointment that healed surgical wounds within a month.^[70] One of his surgical procedures was the removal of a dead fetus from the womb of a woman whom he diagnosed and cured of her ailments.^[70] Hua's contemporary physician and pharmacologist Zhang Zhongjing (150–219 AD) preserved much of the medical knowledge known in China by the Eastern Han period in his major work Shanghan Lun (Treatise on Cold Injury and Miscellaneous Disorders) as well as the Jinkui Yaolue (Essential Medical Treasures of the Golden Chamber ).^[71] Outside the major canon of Chinese medicine established during the Han period, modern archaeology has revealed previous Chinese discoveries in medicine, such as the Shuihudi Qin bamboo texts dated to the 3rd century BC that provide perhaps the earliest known descriptions of the symptoms of leprosy (predating 1st-century AD Roman author Aulus Cornelius Celsus and perhaps also the Indian Sushruta Samhita, the oldest version of which is indeterminable).^[72]

During the Middle Formative Period (c. 900 BC – c. 300 BC) of Pre-Columbian Mesoamerica, either the script of the Zapotec civilization or the script of the Olmec civilization (with the Cascajal Block being perhaps the earliest evidence) represent the earliest full writing systems of the Americas.^[73] The Zapotecs also created the first known astronomical calendar in Mesoamerica, though this was possibly under heavy influence by the Olmecs.^[74][75] The Maya script developed by the Maya civilization between 400–200 BC during its Preclassic period was rooted in the Olmec and Zapotec writing systems, and became widespread in use by 100 BC.^[76] The Classic Maya (c. 250 AD – c. 900 AD) built on the shared heritage of the Olmecs by developing the most sophisticated systems of writing, astronomy, calendrical science, and mathematics among urbanized Mesoamerican peoples.^[74] The Maya developed a positional numeral system with a base of 20 that included the use of zero for constructing their calendars, with individual symbolic characters for numbers 1 through 19.^[77][78] Maya writing contains easily discernible calendar dates in the form of logographs representing numbers, coefficients, and calendar periods amounting to 20 days (within their 360-day years) and even 20 years for tracking social, religious, political, and economic events such as the gathering of political tribute, exchange of gifts, the days that markets opened, and state rituals dedicated to gods and goddesses.^[78]