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The timeline below shows the date of publication of possible major scientific breakthroughs, theories and discoveries, along with the discoverer. This article discounts mere speculation as discovery, although imperfect reasoned arguments, arguments based on elegance/simplicity, and numerically/experimentally verified conjectures qualify (as otherwise no scientific discovery before the late 19th century would count). The timeline begins at the Bronze Age, as it is difficult to give even estimates for the timing of events prior to this, such as of the discovery of counting, natural numbers and arithmetic. To avoid overlap with timeline of historic inventions, the timeline does not list examples of documentation for manufactured substances and devices unless they reveal a more fundamental leap in the theoretical ideas in a field.

Many early innovations of the Bronze Age were prompted by the increase in trade, and this also applies to the scientific advances of this period. For context, the major civilizations of this period are Egypt, Mesopotamia, and the Indus Valley, with Greece rising in importance towards the end of the third millennium BC. The Indus Valley script remains undeciphered and there are very little surviving fragments of its writing, thus any inference about scientific discoveries in that region must be made based only on archaeological digs. The following dates are approximations. 3000 BC: Units of measurement are developed in the Americas as well as the major Bronze Age civilizations: Egypt, Mesopotamia, Elam and the Indus Valley. 3000 BC: The first deciphered numeral system is that of the Egyptian numerals, a sign value system (as opposed to a place value system). 2650 BC: The oldest extant record of a unit of length, the cubit rod ruler, is from Nippur. 2600 BC: The oldest attested evidence for the existence of units of weight, and weighing scales date to the Fourth Dynasty of Egypt, with Deben (unit) balance weights, excavated from the reign of Sneferu, though earlier usage has been proposed. 2100 BC: The concept of area is first recognized in Babylonian clay tablets, and 3 dimensional volume is discussed in an Egyptian papyrus. This begins the study of geometry. 2100 BC: Quadratic equations, in the form of problems relating the areas and sides of rectangles, are solved by Babylonians. 2000 BC: Multiplication tables in a base 60, rather than base 10 (decimal), system from Babylon. 2000 BC: Primitive positional notation for numerals is seen in the Babylonian cuneiform numerals. However, the lack of clarity around the notion of zero made their system highly ambiguous (e. g. 13200 would be written the same as 132). Early 2nd millennium BC: Similar triangles and side ratios are studied in Egypt for the construction of pyramids, paving the way for the field of trigonometry. Early 2nd millennium BC: Ancient Egyptians study anatomy, as recorded in the Edwin Smith Papyrus. They identified the heart and its vessels, liver, spleen, kidneys, hypothalamus, uterus, and bladder, and correctly identified that blood vessels emanated from the heart (however, they also believed that tears, urine, and semen, but not saliva and sweat, originated in the heart, see Cardiocentric hypothesis). 1800 BC: The Middle Kingdom of Egypt develops Egyptian fraction notation. 1800 BC 1600 BC: A numerical approximation for the square root of two, accurate to 6 decimal places, is recorded on YBC 7289, a Babylonian clay tablet believed to belong to a student. 1800 BC 1600 BC: A Babylonian tablet uses 25/8 = 3.125 as an approximation for , which has an error of 0.5%. 1550 BC: The Rhind Mathematical Papyrus (a copy of an older Middle Kingdom text) contains the first documented instance of inscribing a polygon (in this case, an octagon) into a circle to estimate the value of .

The following dates are approximations. 700 BC: Pythagoras's theorem is discovered by Baudhayana in the Hindu Shulba Sutras in Upanishadic India. However, Indian mathematics, especially North Indian mathematics, generally did not have a tradition of communicating proofs, and it is not fully certain that Baudhayana or Apastamba knew of a proof. 700 BC: Pell's equations are first studied by Baudhayana in India, the first diophantine equations known to be studied. 700 BC: Grammar is first studied in India (note that Sanskrit Vyākaraṇa predates Pāṇini). 600 BC: Thales of Miletus discovers Thales's theorem. 600 BC: Maharshi Kanada gives the ideal of the smallest units of matter. According to him, matter consisted of indestructible minutes particles called paramanus, which are now called as atoms. 600 BC 200 BC: The Sushruta Samhita shows an understanding of musculoskeletal structure (including joints, ligaments and muscles and their functions) (3. V). It refers to the cardiovascular system as a closed circuit. In (3. IX) it identifies the existence of nerves. 500 BC: Anaxagoras identifies moonlight as reflected sunlight. 5th century BC: The Greeks start experimenting with straightedge and compass constructions. 5th century BC: The earliest documented mention of a spherical Earth comes from the Greeks in the 5th century BC. It is known that the Indians modeled the Earth as spherical by 300 BC
460 BC: Empedocles describes thermal expansion. Late 5th century BC: Antiphon discovers the method of exhaustion, foreshadowing the concept of a limit. 4th century BC: Greek philosophers study the properties of logical negation. 4th century BC: The first true formal system is constructed by Pāṇini in his Sanskrit grammar. 4th century BC: Eudoxus of Cnidus states the Archimedean property. 4th century BC: Thaetetus shows that square roots are either integer or irrational. 4th century BC: Thaetetus enumerates the Platonic solids, an early work in graph theory. 4th century BC: Menaechmus discovers conic sections. 4th century BC: Menaechmus develops co ordinate geometry. 4th century BC: Mozi in China gives a description of the camera obscura phenomenon. 4th century BC: Around the time of Aristotle, a more empirically founded system of anatomy is established, based on animal dissection. In particular, Praxagoras makes the distinction between arteries and veins. 4th century BC: Aristotle differentiates between near sighted and far sightedness. Graeco Roman physician Galen would later use the term "myopia" for near sightedness. 4th century BC: Pāṇini develops a full fledged formal grammar (for Sanskrit). Late 4th century BC: Chanakya (also known as Kautilya) establishes the field of economics with the Arthashastra (literally "Science of wealth"), a prescriptive treatise on economics and statecraft for Mauryan India. 4th 3rd century BC: In Mauryan India, The Jain mathematical text Surya Prajnapati draws a distinction between countable and uncountable infinities. 350 BC 50 BC: Clay tablets from (possibly Hellenistic era) Babylon describe the mean speed theorem. 300 BC: Greek mathematician Euclid in the Elements describes a primitive form of formal proof and axiomatic systems. However, modern mathematicians generally believe that his axioms were highly incomplete, and that his definitions were not really used in his proofs. 300 BC: Finite geometric progressions are studied by Euclid in Ptolemaic Egypt. 300 BC: Euclid proves the infinitude of primes. 300 BC: Euclid proves the Fundamental Theorem of Arithmetic. 300 BC: Euclid discovers the Euclidean algorithm. 300 BC: Euclid publishes the Elements, a compendium on classical Euclidean geometry, including: elementary theorems on circles, definitions of the centers of a triangle, the tangent secant theorem, the law of sines and the law of cosines. 300 BC: Euclid's Optics introduces the field of geometric optics, making basic considerations on the sizes of images. 3rd century BC: Archimedes relates problems in geometric series to those in arithmetic series, foreshadowing the logarithm. 3rd century BC: Pingala in Mauryan India studies binary numbers, making him the first to study the radix (numerical base) in history. 3rd century BC: Pingala in Mauryan India describes the Fibonacci sequence. 3rd century BC: Pingala in Mauryan India discovers the binomial coefficients in a combinatorial context and the additive formula for generating them , i. e. a prose description of Pascal's triangle, and derived formulae relating to the sums and alternating sums of binomial coefficients. It has been suggested that he may have also discovered the binomial theorem in this context. 3rd century BC: Eratosthenes discovers the Sieve of Eratosthenes. 3rd century BC: Archimedes derives a formula for the volume of a sphere in The Method of Mechanical Theorems. 3rd century BC: Archimedes calculates areas and volumes relating to conic sections, such as the area bounded between a parabola and a chord, and various volumes of revolution. 3rd century BC: Archimedes discovers the sum/difference identity for trigonometric functions in the form of the "Theorem of Broken Chords". 3rd century BC: Archimedes further develops the method of exhaustion into an early description of integration. 3rd century BC: Archimedes calculates tangents to non trigonometric curves. 3rd century BC: Archimedes uses the method of exhaustion to construct a strict inequality bounding the value of within an interval of 0.002. 3rd century BC: Archimedes develops the field of statics, introducing notions such as the center of gravity, mechanical equilibrium, the study of levers, and hydrostatics. 3rd century BC: Eratosthenes measures the circumference of the Earth. 260 BC: Aristarchus of Samos proposes a basic heliocentric model of the universe. 200 BC: Apollonius of Perga discovers Apollonius's theorem. 200 BC: Apollonius of Perga assigns equations to curves. 200 BC: Apollonius of Perga develops epicycles. While an incorrect model, it was a precursor to the development of Fourier series. 2nd century BC: Hipparchos discovers the apsidal precession of the Moon's orbit. 2nd century BC: Hipparchos discovers Axial precession. 2nd century BC: Hipparchos measures the sizes of and distances to the Moon and Sun. 190 BC: Magic squares appear in China. The theory of magic squares can be considered the first example of a vector space. 165 BC 142 BC: Zhang Cang in Northern China is credited with the development of Gaussian elimination.

Mathematics and astronomy flourish during the Golden Age of India (4th to 6th centuries AD) under the Gupta Empire. Meanwhile, Greece and its colonies have entered the Roman period in the last few decades of the preceding millennium, and Greek science is negatively impacted by the Fall of the Western Roman Empire and the economic decline that follows. 1st to 4th century: A precursor to long division, known as "galley division" is developed at some point. Its discovery is generally believed to have originated in India around the 4th century AD, although Singaporean mathematician Lam Lay Yong claims that the method is found in the Chinese text The Nine Chapters on the Mathematical Art, from the 1st century AD. 60 AD: Heron's formula is discovered by Hero of Alexandria. 2nd century: Ptolemy formalises the epicycles of Apollonius. 2nd century: Ptolemy publishes his Optics, discussing colour, reflection, and refraction of light, and including the first known table of refractive angles. 2nd century: Galen studies the anatomy of pigs. 100: Menelaus of Alexandria describes spherical triangles, a precursor to non Euclidean geometry. 150: The Almagest of Ptolemy contains evidence of the Hellenistic zero. Unlike the earlier Babylonian zero, the Hellenistic zero could be used alone, or at the end of a number. However, it was usually used in the fractional part of a numeral, and was not regarded as a true arithmetical number itself. 150: Ptolemy's Almagest contains practical formulae to calculate latitudes and day lengths. 3rd century: Diophantus discusses linear diophantine equations. 3rd century: Diophantus uses a primitive form of algebraic symbolism, which is quickly forgotten. 210: Negative numbers are accepted as numeric by the late Han era Chinese text The Nine Chapters on the Mathematical Art. Later, Liu Hui of Cao Wei (during the Three Kingdoms period) writes down laws regarding the arithmetic of negative numbers. By the 4th century: A square root finding algorithm with quartic convergence, known as the Bakhshali method (after the Bakhshali manuscript which records it), is discovered in India. By the 4th century: The present Hindu–Arabic numeral system with place value numerals develops in Gupta era India, and is attested in the Bakhshali Manuscript of Gandhara. The superiority of the system over existing place value and sign value systems arises from its treatment of zero as an ordinary numeral. 4th to 5th centuries: The modern fundamental trigonometric functions, sine and cosine, are described in the Siddhantas of India. This formulation of trigonometry is an improvement over the earlier Greek functions, in that it lends itself more seamlessly to polar co ordinates and the later complex interpretation of the trigonometric functions. By the 5th century: The decimal separator is developed in India, as recorded in al Uqlidisi's later commentary on Indian mathematics. By the 5th century: The elliptical orbits of planets are discovered in India by at least the time of Aryabhata, and are used for the calculations of orbital periods and eclipse timings. By 499: Aryabhata's work shows the use of the modern fraction notation, known as bhinnarasi. 499: Aryabhata gives a new symbol for zero and uses it for the decimal system. 499: Aryabhata discovers the formula for the square pyramidal numbers (the sums of consecutive square numbers). 499: Aryabhata discovers the formula for the simplicial numbers (the sums of consecutive cube numbers). 499: Aryabhata develops Kuṭṭaka, an algorithm very similar to the Extended Euclidean algorithm. 499: Aryabhata develops an algorithm to solve the Chinese remainder theorem. 499: Historians speculate that Aryabhata may have used an underlying heliocentric model for his astronomical calculations, which would make it the first computational heliocentric model in history (as opposed to Aristarchus's model in form). This claim is based on his description of the planetary period about the Sun (śīghrocca), but has been met with criticism. 499: Aryabhata creates a particularly accurate eclipse chart. As an example of its accuracy, 18th century scientist Guillaume Le Gentil, during a visit to Pondicherry, India, found the Indian computations (based on Aryabhata's computational paradigm) of the duration of the lunar eclipse of 30 August 1765 to be short by 41 seconds, whereas his charts (by Tobias Mayer, 1752) were long by 68 seconds.

The Golden Age of Indian mathematics and astronomy continues after the end of the Gupta empire, especially in Southern India during the era of the Rashtrakuta, Western Chalukya and Vijayanagara empires of Karnataka, which variously patronised Hindu and Jain mathematicians. In addition, the Middle East enters the Islamic Golden Age through contact with other civilisations, and China enters a golden period during the Tang and Song dynasties. 6th century: Varahamira in the Gupta empire is the first to describe comets as astronomical phenomena, and as periodic in nature. 525: John Philoponus in Byzantine Egypt describes the notion of inertia, and states that the motion of a falling object does not depend on its weight. His radical rejection of Aristotlean orthodoxy lead him to be ignored in his time
628: Brahmagupta states the arithmetic rules for addition, subtraction, and multiplication with zero, as well as the multiplication of negative numbers, extending the basic rules for the latter found in the earlier The Nine Chapters on the Mathematical Art. 628: Brahmagupta writes down Brahmagupta's identity, an important lemma in the theory of Pell's equation. 628: Brahmagupta produces an infinite (but not exhaustive) number of solutions to Pell's equation. 628: Brahmagupta provides an explicit solution to the quadratic equation. 628: Brahmagupta discovers Brahmagupta's formula, a generalization of Heron's formula to cyclic quadrilaterals. 628: Brahmagupta discovers second order interpolation, in the form of Brahmagupta's interpolation formula. 628: Brahmagupta invents a symbolic mathematical notation, which is then adopted by mathematicians through India and the Near East, and eventually Europe. 629: Bhāskara I produces the first approximation of a transcendental function with a rational function, in the sine approximation formula that bears his name. 9th century: Jain mathematician Mahāvīra writes down a factorisation for the difference of cubes. 9th century: Algorisms (arithmetical algorithms on numbers written in place value system) are described by al Khwarizmi in his kitāb al ḥisāb al hindī (Book of Indian computation) and kitab al jam' wa'l tafriq al ḥisāb al hindī (Addition and subtraction in Indian arithmetic). 9th century: Mahāvīra discovers the first algorithm for writing fractions as Egyptian fractions,which is in fact a slightly more general form of the Greedy algorithm for Egyptian fractions. 816: Jain mathematician Virasena describes the integer logarithm. 850: Mahāvīra derives the expression for the binomial coefficient in terms of factorials, 10th century AD: Kashmiri astronomer Bhaṭṭotpala lists names and estimates periods of certain comets.

The Scientific Revolution occurs in Europe around this period, greatly accelerating the progress of science and contributing to the rationalization of the natural sciences. 16th century: Gerolamo Cardano solves the general cubic equation (by reducing them to the case with zero quadratic term). 16th century: Lodovico Ferrari solves the general quartic equation (by reducing it to the case with zero quartic term). 16th century: François Viète discovers Vieta's formulas. 16th century: François Viète discovers Viète's formula for .1500: Scipione del Ferro solves the special cubic equation Late 16th century: Tycho Brahe proves that comets are astronomical (and not atmospheric) phenomena. 1517: Nicolaus Copernicus develops the quantity theory of money and states the earliest known form of Gresham's law: ("Bad money drowns out good"). 1543: Nicolaus Copernicus develops a heliocentric model, rejecting Aristotle's Earth centric view, would be the first quantitative heliocentric model in history. 1543: Vesalius: pioneering research into human anatomy. 1545: Gerolamo Cardano discovers complex numbers. 1556: Niccolò Tartaglia introduces parenthesis. 1557: Robert Recorde introduces the equal sign. 1564: Gerolamo Cardano is the first to produce a systematic treatment of probability. 1572: Rafael Bombelli provides rules for complex arithmetic. 1591: François Viète's New algebra shows the modern notational algebraic manipulation.

1735: Carl Linnaeus described a new system for classifying plants in Systema Naturae. 1745: Ewald Georg von Kleist first capacitor, the Leyden jar. 1749 – 1789: Buffon wrote Histoire naturelle. 1750: Joseph Black: describes latent heat. 1751: Benjamin Franklin: lightning is electrical. 1755: Immanuel Kant: Gaseous Hypothesis in Universal Natural History and Theory of Heaven. 1761: Mikhail Lomonosov: discovery of the atmosphere of Venus. 1763: Thomas Bayes: publishes the first version of Bayes' theorem, paving the way for Bayesian probability. 1771: Charles Messier: publishes catalogue of astronomical objects (Messier Objects) now known to include galaxies, star clusters, and nebulae. 1778: Antoine Lavoisier (and Joseph Priestley): discovery of oxygen leading to end of Phlogiston theory. 1781: William Herschel announces discovery of Uranus, expanding the known boundaries of the Solar System for the first time in modern history. 1785: William Withering: publishes the first definitive account of the use of foxglove (digitalis) for treating dropsy. 1787: Jacques Charles: Charles's law of ideal gases. 1789: Antoine Lavoisier: law of conservation of mass, basis for chemistry, and the beginning of modern chemistry. 1796: Georges Cuvier: Establishes extinction as a fact. 1796: Edward Jenner: smallpox historical accounting. 1796: Hanaoka Seishū: develops general anaesthesia. 1800: Alessandro Volta: discovers electrochemical series and invents the battery.

1802: Jean Baptiste Lamarck: teleological evolution. 1805: John Dalton: Atomic Theory in (chemistry). 1820: Hans Christian Ørsted discovers that a current passed through a wire will deflect the needle of a compass, establishing the deep relationship between electricity and magnetism (electromagnetism). 1820: Michael Faraday and James Stoddart discover alloying iron with chromium produces a stainless steel resistant to oxidising elements (rust). 1821: Thomas Johann Seebeck is the first to observe a property of semiconductors. 1824: Carnot: described the Carnot cycle, the idealized heat engine. 1824: Joseph Aspdin develops Portland cement (concrete), by heating ground limestone, clay and gypsum, in a kiln. 1827: Evariste Galois development of group theory. 1827: Georg Ohm: Ohm's law (Electricity). 1827: Amedeo Avogadro: Avogadro's law (Gas law). 1828: Friedrich Wöhler synthesized urea, refuting vitalism. 1830: Nikolai Lobachevsky created Non Euclidean geometry. 1831: Michael Faraday discovers electromagnetic induction. 1833: Anselme Payen isolates first enzyme, diastase. 1837: Charles Babbage proposes a design for the construction of a Turing complete, general purpose Computer, to be called the Analytical Engine. 1838: Matthias Schleiden: all plants are made of cells. 1838: Friedrich Bessel: first successful measure of stellar parallax (to star 61 Cygni). 1842: Christian Doppler: Doppler effect. 1843: James Prescott Joule: Law of Conservation of energy (First law of thermodynamics), also 1847 – Helmholtz, Conservation of energy. 1846: Johann Gottfried Galle and Heinrich Louis d'Arrest: discovery of Neptune. 1847: George Boole: publishes The Mathematical Analysis of Logic, defining Boolean algebra; refined in his 1854 The Laws of Thought. 1848: Lord Kelvin: absolute zero.